UNITED STATES DEPARTMENT OF COMMERCE • John T. Connor, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

CRITICAL PHENOMENA

Proceedings of a Conference

Held in Washington, D.C., April 1965

Edited by

M. S. Green and J. V. Sengers

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

National Bureau of Standards

National Science Foundation

Office of Naval Research

Sponsored by

National Bureau of Standards Miscellaneous Publication 273

Issued December 1, 1966

For sale bv the Suuerintendent of Documents. I S. Government Printing Office Washington. D.C.. 20402 • Price S2.50
Library of Congress Catalog (lard Number: 66-60040
Conference on

PHENOMENA IN THE NEIGHBORHOOD OF

CRITICAL POINTS

Held at

The National Bureau of Standards

Washington, D.C.

April 5 to 8, 1965

Organ izing Committee:

M. S. GREEN. Chairman

G. B. Benedek

M. E. Fisher

E. W. Montroll

National Bureau of Standards

Massachusetts Institute of Technology

University of London King s College

Institute for Defense Analysis and

University of Maryland

C. J. Pings California Institute of Technology

iii

FOREWORD

One of the important responsibilities of the National Bureau of Standards is to help the Nation

meet its needs for accurately measured, critically evaluated, and well-understood properties of

materials. To aid in fulfilling this responsibility, the Bureau joined with the National Science

Foundation and the Office of Naval Research in sponsoring a Conference on Phenomena in the

Neighborhood of Critical Points, which was held at NBS in Washington. D.C.. April 5 to 8. 1965. The Conference was attended by approximately 200 scientists from this country and abroad,

representing many laboratories and disciplines. Most of the papers presented at the Conference,

as well as the subsequent discussions, are included in this volume. Primary responsibility for

their technical content must rest, of course, with the individual authors and their organizations.

A. V. Astin, Director.

PREFACE

In that system for denning, determining, communicating, and recording numerical descriptors

of physical systems which has recently been called the National Measurement System, the National Bureau of Standards has had from its earliest beginnings a concern with the part having to do with properties of substances. Indeed, accurately determined values of key data on properties

of substances are an essential means for the dissemination of measurement capability. The experimental determination, the critical evaluation and the theoretical understanding of well-defined

properties of well-characterized substances is one of the major modes through which the scientific enterprise contributes to the technological enterprise and to the well-being of the nation. To

facilitate and to contribute to this process and to bring it into close coupling with the national

standards is one of the primary responsibilities of \the Bureau and its component Institute for Basic

Standards.

The program of the Conference on Phenomena in the Neighborhood of Critical Points, whose

proceedings are recorded in the present volume, serves excellently to illustrate some of the ways

in which this process takes place. An important part of the program of the conference was con- cerned with measurement techniques. New techniques were introduced. Advantages and limi- tations of traditional techniques were discussed. In many instances it became apparent that new

techniques were necessary not only to provide more accurate data in new experimental ranges

but to answer fundamental theoretical questions. The Institute for Basic Standards has a deep

and perennial interest in techniques for measuring properties of substances.

The technological enterprise needs accurate data on the substances and materials from which

the designs of our engineers are constructed, which constitute the ambience which they must

take into account,! or which enter into the various stages of the productive process. In meeting

this need the task of the scientific enterprise does not end with the complete measurement of a property. Competent scientists must critically review and evaluate various measurements which

inevitably conflict and contradict each other. Many of the papers at the conference were just

such critical reviews and evaluation of experimental data. The literature was searched for rele- vant data. Factors affecting the accuracy and validity of the experimental approach were investi- gated. Patterns, meanings, and conclusions were drawn insofar as the data warranted. These

reviews represent an essential step in the process by which primary experimental data become

the reliable and easily accessible reference data which the technological enterprise requires.

Anyone who attempts even a rough estimate the dimensions of the need for data will realize

that not even all the laboratories and data centers of the whole world are sufficient to provide an

accurately measured and critically evaluated datum for every required property in every required

condition. The scientific enterprise has, of course, another resource available to satisfy the data

requirements of technology, the web of well-established theoretical relationships which relate properties in various substances and conditions among themselves and to the fundamental constants of nature. One of the purposes which has motivated the present conference was the expectation or hope that a common thread of explanation would be found among such diverse phenomena

as liquid-vapor phase transitions, magnetism, superconductivity, order-disorder transitions. If these expectations are realized, a few theoretical concepts will provide knowledge of properties

which is significant for a large number of very disparate fields of technology.

The task of providing adequate data on properties of substances is, of course, one which

must be carried out by the scientists of the nation and indeed the world. The Bureau and Institute contribute to this task through services they provide to the scientific community as well as by the

scientific work of its own laboratories. As I remarked in my address of welcome to the conference,

the Bureau and Institute were happy and honored to serve as a place of congregation for the dis-
tinguished group of speakers and participants for the purpose of discussing a subject so germane

to our function. We were happy also for the stimulus to our technical program which the con- ference provided through the activities of our staff as participants, speakers, and organizers.

More than these, however, we were happy to have this important group of scientists in our midst

so that we might learn from them about opportunities and needs to improve our services and they

in turn might perhaps learn from us about those of our services and activities which might be of help to them.

R. D. HUNTOON, Director

Institute for Basic Standards.
INTRODUCTION

As their temperature is raised, the properties of a liquid and of the vapor with which it is in equilibrium become more and more similar until, at a particular temperature, the difference dis- appears. The state of the fluid at this temperature is the critical point. At room temperature

iron has the power to acquire a magnetic moment which remains even after the magnetic field is removed. As its temperature is raised this remanent magnetism diminishes until, at a temperature

of some several hundred of degrees, it suddenly disappears. This state of iron is called the Curie

point. Below the temperature of its critical point He4 may exist in an ordinary liquid form which

is qualitatively the same as liquids existing at room temperature. At a temperature several degrees

below its critical point the ordinary liquid suddenly acquires new properties, especially those

relating to flow, which are so extraordinary that the state has been called superfluid. The state

at which this transition occurs is called the Appoint.

All these phenomena have in common that at a definite transition point a substance gains or

loses all at once what another age would have called a virtue. In this they differ from first order

phase transitions, melting, evaporation, sublimation, in which the physical change is not sudden,

but takes place by a piecemeal transformation of small portions of the substance from one state

to another. They differ also from first order phase transitions in that they seem to be the focus

or source of anomalous behavior in a wide region of the surrounding thermodynamic phase plane.

Critical points of liquid vapor transitions, consolute points in binary liquid mixtures, Curie and

Neel points in magnetism, second order phase transition points in solids, A-points in liquid helium

together with the anomalous phenomena which occur in their neighborhood were the subject of

the conference whose proceedings are reported in this volume.

That critical points in this general sense have much in common with each other is attested

by the fact that they have been treated from the beginning by very similar models and theories.

These models and theories are based on the idea that phase transitions and critical points are brought about through the mutual cooperative interaction of many particles. For this reason these

phenomena are often called cooperative phenomena. The essential similarity of these early

theories, the Van der Waals theory of the liquid-vapor critical point, Curie-Weiss theory of ferro- magnetism, the Bragg-Williams and Bethe theory of order-disorder phenomena is emphasized

in the introductory historical lecture of Uhlenbeck. They give, as Uhlenbeck pointed out, an

excellent qualitative picture of critical phenomena, but fail more and more seriously in their

quantitative predictions as the critical point is approached. Their ultimate inadequacy is demonstrated by their disagreement with Onsager's exact solution of the two-dimensional Ising model

of ferromagnetism.

Several recent scientific developments, both experimental and theoretical, contributed to the

feeling on the part of a number of scientists, and in particular, on the part of those who formed

themselves into an ad hoc committee to organize it, that April 1965 was an appropriate moment

for a conference on critical phenomena. Firstly, there was the theoretical prediction of nonclassical critical behavior for three dimensional lattice models of gases and magnets based on the

numerical generation and summation of a large number of terms of exact low and high temperature

series expansions. A second recent development, in this case experimental, was the discovery

that the specific heat at constant volume of a simple fluid near the critical point is apparently

singular and very similar in its behavior to the A-singularity in the specific heat of liquid helium.

A third development was the confirmation through measurement of the internal magnetic field by

nuclear magnetic resonance that the shape of the magnetization versus temperature curve of a

ferro- or antiferromagnet is similar if not identical to the shape of the coexistence curve of the

liquid-vapor phase transition and similar to the shape predicted by the series summation method.
A fourth development was the critique of the Ornstein-Zernike' theory of critical opalescence and

the experimental confirmation of deviations from this theory.

A common feature of these developments and of others not listed is that they point up a relationship between phenomena which are normally investigated by different institutional groupings of scientists using different experimental techniques who perhaps normally do not meet each

other at scientific conferences. It was the purpose of the conference to present these recent developments, to provide an opportunity to assess their significance both individually and as a

total picture, to bring together for mutual enlightenment scientists working in different specialities

and disciplines, to provide a forum for theoretical speculations.

The program of the conference consisted primarily of papers reviewing the experimental

situation for various important critical phenomena. In addition to these, each session contained

a selection of shorter papers presenting new data, a novel experimental method, or new theoretical

ideas. These proceedings contain the text of the papers substantially as presented except for the paper of Passell which is represented only as an abstract. Also included is a "'dictionary"

of corresponding parameters in various critical systems together with a tabulation of results on

critical exponents prepared by Fisher.

An attempt was made by the session chairmen to elicit as much discussion as time permitted

and this is included almost verbatim in these proceedings. Certain discussion remarks were in effect short papers. In some of these cases a manuscript was available and was included in the

text. In other bases it was possible to make references to a recent or shortly to appear publication.

It will be noted that the bulk of the papers at the conference were experimental. This does

not represent a bias towards empiricism on the part of the conference organizers but rather their feeling that the most appropriate way to further an eventual theoretical understanding of critical phenomena was by a complete and critical presentation of the experimental situation. However

in addition to the theoretical papers of Domb, Fisher, Buckingham, and Marshall which were pre sented in the main program, the last session of the conference was almost entirely devoted to short

theoretical presentations. Not all of these are published here. This supplementary session als included the experimental paper of Lorentzen. Although not actually presented at the confer ence, a review article by I. M. Firth, a participant, of certain techniques of low temperature calo rimetry was felt to be relevant and is included in this volume.

The contribution of a conference such as this one is to be judged not only on the questions it answers but also for the issues which it brings to clear statement. I close this introduction wit

my own opinion of what some of these outstanding issues are.

It may perhaps be regarded as established that the thermodynamic properties of substances

near critical points are singular functions in the mathematical sense. A primary issue then is what is the nature of these singularities?

One of the ways of expressing these singularities is through critical exponents as defined in the paper by Fisher. The fact, coming out of the papers by Rowlinson, Domb, Benedek. and

Fisher, that values of critical exponents predicted by the series summation method are in very good agreement with experiments both in fluids and in magnets is an important success of theory

Nevertheless, several questions remain.

The summation theory predicts different results for different physical models such as the

Heisenberg and Ising models of ferromagnetism. Can we exhibit these differences experimentally?

The paper of Wolf describes an experimental realization of an Ising ferromagnet. What is the significance of the quantum effects on critical exponents pointed out in the papers of Sherman and

M. H. Edwards?

Another aspect of the nature of the critical singularity is the behavior of the specific heat. Is the logarithmic specific heat singularity, so convincingly demonstrated for the K-transition in liquid

helium in the paper of Fairbank, a paradigm for the behavior ofspecific heat on all critical systems?

The paper of Moldover and Little reports experiments on this question for the liquid vapor critical point of helium while Yamamoto reviews the situation for some solid-state second order transitions. Teaney discusses the specific heat of some ferromagnetic and antiferromagnetic transitions from
this point of view, while Kierstead presents results on the singular behavior of the pressure coefficient near the \-point.

A question which was not represented in the program but which was touched upon in the

discussion is what is the relation of the superconducting transition point to the k-point and to the liquid vapor critical point?

The paper of Chu and the earlier work of Mclntyre referred to in the discussion indicate

deviations from the classical, Ornstein-Zernike theory of critical fluctuations within a few hundredths of a degree Kelvin from the critical consolute point of a liquid mixture. An obvious

experimental problem is to confirm these deviations, to correlate the results of light and x-ray scat- tering and to make precise the nature of the critical singularity in these phenomena. The present

status of results on elastic light and x-ray scattering was reviewed by Brumberger while Dietrich

and Als-Nielsen presented their results obtained by elastic neutron scattering on /3-brass. A

related question brought up in the discussion is why do deviations from the classical theory of

critical opalescence occur only within hundredths of a degree of the critical point while deviations

in thermodynamic quantities show nonclassical behavior much farther away?

Phenomenological theory predicts that magnetic fluctuations become slower and slower

near the Curie point, and therefore that critical magnetic scattering should be elastic, while the

experiments of Passell and Jacrot show a definite and unexpected amount of inelastic scattering.

Although Marshall's paper suggests a possible and even likely solution, the question still remains

why is critical magnetic neutron scattering inelastic?

The review of transport phenomena by Sengers indicates that the thermal conductivity of

simple fluids exhibits a large anomaly near the critical point while the viscosity exhibits no such

anomaly or a very minor one. The opposite seems to be the case for binary liquid mixtures. The

question arises, which transport coefficients are anomalous near critical points and why?

Several new techniques or techniques newly applied to critical phenomena were presented

at the conference. One of these was the inelastic scattering of laser light, which was represented

in the papers of Alpert and of Ford and Benedek. Another was nuclear magnetic resonance whose

application to fluids was given in the paper by Bloom and in a discussion remark of Trappeniers.

The application of this method to antiferromagnetism was presented in the talk of Heller. Not a new method but one which has not perhaps been fully exploited is the diagnosis, using

a light probe, of the density gradients produced by gravity in a fluid near its critical point. The

papers of Schmidt and L,orentzen present results obtained by this method. A somewhat related

method was brought up by Webb in the discussion in which the meniscus, which becomes more

and more diffuse as the critical point is approached, is probed by its reflectivity to light.

Results on ultrasonic propagation in fluids near critical points were reviewed by Sette while

the application of this method to a solid state second order transition is given in the paper of

Garland and Renard and to helium near its critical point by Chase and Williamson.

Perhaps the foregoing questions can be subsumed under two fundamental questions: the

first question is are all the phenomena considered in the conference under the rubric of critical phenomena truly analogous? This question can be perhaps answered quickly and affirmatively

when, say, the liquid-vapor transition is compared to the phase separation of a binary liquid mixture

or even to order-disorder phenomena in solids. The analogies become less clear when these are compared to magnetic phenomena, superfluidity, and superconductivity.

When available in any given case, the best experimental and theoretical evidence seems to show that the singular behavior of analogous properties in different systems is very similar. Of

course, the series summation method indicates small but definite differences between different systems. Without prejudging the significance of these differences, the similarity of behavior

already suggests a common explanation.

A second question is then, what is the common reason for the singular behavior of analogous

properties ? Melville S. Green.
ACKNOWLEDGEMENTS

The organizing committee wishes to express appreciation for the generous support of the

National Science Foundation and the Office of Naval Research which enabled the conference to benefit from the participation of many scientists from western Europe, Japan, and Australia whose

contribution otherwise would have been sorely missed. The committee also wishes to express

its thanks to the many members of the NBS staff who contributed to making the conference ar- rangements pleasant and convenient. It wishes especially to thank Mr. Robert Cooke of the

Office of Technical Information and Publications and Mrs. May J. Lum and Mrs. Hattie N. Brown

of the Heat Division. It also wishes to thank Mrs. Vivian R. Green for her help in organizing

the social program of the conference.
CONTENTS

Page FOREWORD V

PREFACE VII INTRODUCTION , IX ACKNOWLEDGEMENTS XII

EQUILIBRIUM CRITICAL PHENOMENA IN FLUIDS

Chairman: A. M. J. F. Michels

G. E. Uhlenbeck — The Classical Theories of the Critical Phenomena 3

R. H. Sherman — The Coexistence Curve of He3 7

J. S. Rowlinson — Critical States of Simple Fluids and Fluid Mixtures: a Review of the Experimental Position 9

E. H. W. Schmidt — Optical Measurements of the Density Gradients Produced by Gravity in C02 , N2 0, and CCIF3 Near the Critical Point 13 M. E. Fisher — Notes. Definitions, and Formulas for Critical Point Singularities 21 Discussion 25

CRITICAL PHENOMENA IN FERRO- AND ANTIFERROMAGNETS

Chairman: H. B. Callen

C. Domb — Critical Properties of Lattice Models 29

G. B. Benedek — Equilibrium Properties of Ferromagnets and Antiferromagnets in the Vicinity of the Critical Point 42 W. P. Wolf—Experimental Studies of Magnetic Ising Systems Near the Critical Point 49

D. T. Teaney — Specific Heats of Ferro- and Antiferromagnets in the Critical Region 50

P. Heller — Nuclear Resonance Studies of Magnetic Critical Fluctuations in MnF> 58

Discussion 64

LOGARITHMIC SINGULARITIES

Chairman: C. N. Yang

W. M. Fairbank — The Lambda Transition in Liquid Helium 71 M. R. Moldover and W. A. Little-The Specific Heat of He3 and He4 in the Neighborhood of Their Critical Points 79 M. H. Edwards — The Coexistence Curve of He4 82

T. Yamamoto. O. Tanimoto, Y. Yasuda, and K. Okada — Anomalous Specific Heats Associated with Phase Transitions

of the Second Kind 86

H. A. Kierstead — The Logarithmic Anomaly in the Pressure Coefficient of Helium Close to the Lambda Line 92 M. J. Buckingham — The Nature of the Cooperative Transition 95

Discussion 101

SCATTERING: ELASTIC

Chairman: P. Debye

P. Debye — Introductory Remarks 107 M. E. Fisher — Theory of Critical Fluctuations and Singularities 108

H. Brumberger— Scattering of Light and X Rays from Critically Opalescent Systems 116

B. Chu — Experiments on the Critical Opalescence of Binary Liquid Mixtures:

Elastic Scattering 123

Discussion 129

SCATTERING: INELASTIC

Chairman: K. S. Singwi W. Marshall — Critical Scattering of Neutrons by Ferromagnets 135

L. Passell — Critical Magnetic Scattering of Neutrons in Iron ^2

0. W. Dietrich and J. Als-Nielsen — Critical Neutron Scattering from Beta-Brass 144

N. C. Ford, Jr., and G. B. Benedek — The Spectrum of Light Inelastically Scattered by a Fluid Near Its Critical Point... 150

S. S. Alpert — Time-Dependent Concentration Fluctuations Near the Critical Temperature 157 Discussion IgO

xi
CONTENTS— Continued

TRANSPORT AND RELAXATION PHENOMENA

Chairman: R. W. Zwanzig

J. V. Sengers — Behavior of Viscosity and Thermal Conductivity of Fluids Near the Critical Point 165 M. Bloom — Nuclear Magnetic Resonance Measurements Near the Critical Point of Ethane 178

D. Sette — Ultrasonic Investigation of Fluid System in the Neighborhood of Critical Points 183

C. E. Chase and R. C. Williamson — Ultrasonic Investigation of Helium Near Its Critical Point 197

C. W. Garland and R. Renard — Ultrasonic Investigation of the Order-Disorder Transition in Ammonium Chloride 202

Discussion 209

SUPPLEMENTARY SESSION.

H. L. Lorentzen and B. B. Hansen — Effect of Gravity on the Equilibrium Mass Distribution of Two-Component Liquid

Systems. The Observation of Reversed Diffusion Flow 213

S. Katsura and B. Tsujiyama — Ferro- and Antiferromagnetism of Dilute Ising Model 219

S. F. Edwards — The Statistical Mechanics of a Single Polymer Chain 225

I. M. Firth — Some Comments on Techniques of Modern Low-Temperature Calorimetry 233 LIST OF PARTICIPANTS 239

xiv
EQUILIBRIUM CRITICAL PHENOMENA IN FLUIDS

Chairman: A. M. J. F. Michels

1

The Classical Theories of the Critical Phenomena

G. E. Uhlenbeck

The Rockefeller University, New York, N.Y.

1. When the organizers of this conference on

critical phenomena asked me to start the dis- cussions with a review of the classical theories

in order to provide the proper historical perspective,

I had hoped that I would find the time to do this in

a responsible way. However I soon found out

that is not easy! The literature is large and very

few (if any!) good historical studies have been made

about the development of thermodynamics and

statistical mechanics in the second half of the last century. I recommend it to the historian of science!

There are many themes and questions which seem

to me very worth while to study as for instance the

penetration of the ideas of Gibbs and his methods

(especially his graphical methods) into the Dutch

school of thermodynamics of van der Waals, Bakhuis

Roozeboom. Kamerlingh Onnes. and others; the

relation of the Strassbourg school of Weiss on fer- romagnetism with the ideas of van der Waals: the

early controversies about the critical region versus

the critical point which is now again so much in the

center of attention: and so on!

I hope that you will take therefore the short

historical remarks I will present with a grain of

salt, and that you will consider my talk more as a

kind of keynote address! I will try to mention some

of the names and some of the notions which you will hear over and over again, and I will try to explain

why I think the recent revival of the study of the

critical phenomena is of such deep interest for the

student of statistical mechanics.

2. Let me begin with some names and dates. 1869 Discovery of the critical temperature of CO2 by Andrews. The general form of the set of isotherms for the liquid-vapor equilibrium can be

1873

1880

found already in Andrews' paper

plus the suggestion that this was a general phenomenon.

Dissertation of J. D. van der Waals

on the continuity of the gas and

liquid state. You all know that the

miraculous van der Waals equation:

kT

v — b

together with the Maxwell equal area

rule (added by Maxwell in 1874 in a long review of the work of van der

Waals) explains the vapor-liquid

equilibrium and the existence of the

critical point:

0 , 1 a . „ 8 a

vc = 3b; pc =--; kTc=--

b

Of course this confirmed the idea

that the critical point was a general

phenomena and that there were no "permanent" gases. Made explicit

by the

law of corresponding states (van der

Waals)

Pc^= tt= F(a>, m w = -

vc # = ^T,

with F a universal function. The

role which this law played in the

liquefaction 1 of gases and especially

of Helium (Kamerlingh Onnes, 1908)

209-493 O-66—
was spectacular and I hope well

known.

The discovery of the magnetic transi- tion point came much later. So far as I know, it was only in: 1890 Hopkinson found that iron above a certain temperature ceased to be

ferromagnetic. Confirmed and thor- oughly investigated by Pierre Curie

in a classical paper of 1895, and

therefore now always called the

Curie temperature.

1907 P. Weiss introduced the notion of the

inner field and explained the exist- ence of the Curie temperature Tc and of the spontaneous magnetization for T < Tc . The third critical transition phenomena is that of the

order-disorder transformation in bi- nary alloys or better substitutional

solid solutions. Suspected in 1919

by Tammann, proved in 1925 by the

x-ray work of Johanson and Linde,

the theoretical explanation given in: 1934 by Bragg and Williams, was clearly

patterned after the Weiss theory.

One could still mention the discov- ery of the critical mixing temperature of two fluid phases for a binary

mixture. Found by Alexejew around

1800. Explanation by van der Waals

and a series of coworkers

1890-1905 (Kuenen, Kohnstamm, van Laar,

Korteweg) 1890-1905, by generalizing the van der Waals equation for mixtures and applying for the ther- modynamic discussion the graphical

methods of Gibbs.

3. Leaving for a moment this sort of schoolbook

history let me emphasize the great similarity of the

three classical theories of van der Waals, Weiss,

and Bragg-Williams although the derivations are

often quite different. It is now clear and generally

accepted that they are all based on the assumption

that the attractive forces between the molecules

which produce the cooperative effects have a very- long range. As a result the type of singularity

which the critical point represents is in all the the- ories exactly the same. To put this in evidence it

is good to direct attention (following M. Fisher)

to the following three theoretical predictions:

a. At the critical point the specific heat makes u

finite jump and then decreases again. For a van der

Waals gas at the critical density, one gets:

Ac,

0 for T > Tc 2STr -T

A 1 25 Tc

For a Weiss ferromagnet:

0 for T>Tr Acp = - 13 T(.-T

5 Tc

For a Bragg-Williams binary alloy:

Ac

0 for T > Tr 3

2 A

8 Tc - T

5 Tc

b. At the critical point the coexistence curve is parabolic. For a van der Waals gas (with reduced

quantities): 0)g — m = 4(1 — d) u -. For a Weiss

ferromagnet:

[also with reduced quantities; W= constant of

Weiss inner field, M= magnetization; Tc =

3k = Curie temperature. Putting H*=j-j, H. then

the ^magnetic equation of state becomes:

I* — —L(H* + I*) where L(x) is the Langevin u

Function].

c. At the critical point the compressibility or the

susceptibility become infinite as \j\T— Tc \. For ai van der Waals gas:

dir\

r'»aJ„ =i, r>rc = -6(#-l)

coex, T<TC

12(1 - &).

For a Weiss ferromagnet: (x is the susceptibility)

1 1

1 W 1 T> Tr

X =71W l-#

T< Tc

4
For a Bragg-Williams alloy, properties b and c can

be denned but tbey have less direct observational

meaning.

4. At this point one may well already raise the

question, why the classical theories represent so

well, at least qualitatively, the observed critical phenomena. Of course, as long as little was known

about the intermolecular forces, one could think

that the attractive forces had a very long range.

In fact van der Waals never doubted that, and probably Weiss also had always the long range dipoledipole forces in mind. However the later development showed more and more clearly that the cooperative forces were not of a very long

range, that each atom was influenced by at best a few shells of neighboring atoms. Of course, it took a long time before this was realized. It re- quired the elucidation, of the nature of the van der

Waals attractive forces by London and of the Weiss

inner field by Heisenberg which came in the late twenties as a byproduct of the quantum mechanical

revolution.

Partially because of this fact, the classical theories became more and more discredited in the

thirties. This was also because it turned out that

the classical theories although qualitatively correct could not be fixed up so as to give a precise

quantitative description of the phenomena. For

the van der Waals theory this became accepted

already around 1900, and as a result KamerlinghOnnes started then to represent the experimental

data by series expansions in the density, the so- called virial expansion, for which Ursell and Mayer

then gave the theoretical analysis in the thirties. For the Weiss and the Bragg-Williams theory the

attempts for improvement lasted a bit longer, but

also here one soon began to turn more and more to

series expansions. The reason is clear. The

terms in the expansions can be calculated exactly

(at least in principle) and related to the inter- molecular forces. And since one began to learn more and more about these forces, this was of course of the greatest interest. However, one lost the general point of view which the classical theories provided, and perhaps because of this one

also lost interest in these theories. They were

considered to give "convenient interpolation for- mula" or "semiphenomenological descriptions."

Remarkable for instance is the fact that in the great

international congress in 1938 to celebrate van der

Waals' 100th birthday, the van der Waals equation was mentioned only once, and all attention was

centered on the so-called rigorous theories. I think one is now coming back from these theories

because one has become dissillusioned about the

series expansions since no general point of view

emerges from them.

5. The first and unfortunately till now

1944 practically the only step forward beyond the

classical theories was made in the famous

paper by Onsager on the two-dimensional

Ising model. I will not say much about it, because you will hear much more about it later. It has dominated the subject for the

last 20 years. Let me remind you only that the Ising model (proposed by Ising in 1925) can be considered as a simplified version of

the ferromagnetic, and of the order-disorder

and even of the vapor-liquid problem (through

the lattice gas). Each atom on a lattice site can be in two states (spin up or down) and

the interaction is only between neighboring

pairs and is ±J depending on whether the

spins are parallel or antiparallel. It is therefore a model with short range forces. The

most striking result of Onsager was that there

is a critical point and that the specific heat

is logarithmically infinite there in contrast to the classical result.

6. So far I have only talked about the criti- cal point and its properties for the equation of

state. A deeper question is the question

what happens with the correlation function near a critical point. That the density fluc- tuations will get large near the critical point was pointed out by Smoluchowski, but that in addition the correlation of the density fluc- tuation at different points will become of

very long range and that this is the cause of

the critical opalescence was first clearly

1914 stated by Ornstein-Zernike in 1914. and then

worked out in the Groningen dissertation of

1916 Zernike of 1916. Angular distribution of the scattered radi- ation is related to the Fourier transform of

the correlation function:

g(k, v, T) dreKk - n>{r: v.T) — 1

, 2tt . e A= T

sm 2"

Ornstein and Zernike give specifically for

5
T > Tc v = vc : c»

g(k; iv. T)=— j

with

cfrv-Vattr(r) * 3

—77- versus k

2 (0-Z plot) should be a straight

line whose intercept with the ordinate axis

must go to zero if T —> Tc as (T— Tc ). There are many ways of justifying the O-Z

theory. I think though that always one again

has to assume that the forces are of long

range, so that it can be considered a natural

generalization of the van der Waals theory.

The corresponding theory for neutron

scattering near the Curie point is so recent

that it does not belong in a historical survey!

7. Outlook. What are the problems, and what

can one hope from the present revival of interest?

As always in statistical mechanics one must dis- tinguish the equilibrium from the nonequilibrium

properties. A. I think it is wonderful that so much new ex- perimental work is being done. For a long time

experiment was too much dominated by the theory,

and except in a few places (like in the van der Waals

laboratorium in Amsterdam) experimental work on

the properties of matter near the critical point nearly

ceased. I am glad that it is different now. What

can one hope to find? I hope that it will turn out

that again there is a deep analogy between the

various critical points, that there are similarities

like in the classical theories. You will hear a lot about the new experimental work, so I will only

mention a few points which seem to me very suggestive.

1. Near the critical point the coexistence curve seems much better represented by the empirical

7 Guggenheim law: &>,,— coj = -(l — 0) 1/3

. Also the

magnetization curve near Tc seems to show such a cubic behavior.

2. For Argon and O2, and now also for He, it seems that c r does not show a jump but becomes

logarithmically infinite. For T,

with B+ = B~ and C~-C+= 10k.

This surprising result by the Russians, is also found

for He" — He1 transition and I believe also for some ferromagnetic transitions. You will hear more about it tomorrow.

If there is such an universal, but nonclassical

behavior, then there must be an universal explanation which means that it should be largely independent of the nature of the forces. The only corner where this can come from is I think the fact that the forces are not long range. The Onsager

solution gives I think a strong hint. It may well be

so that away from the critical points the classical theories give a good enough description but that they fail close to the critical point where the sub- stance remembers so to say Onsager. I think that

to show something like this is the central theoretical

problem. One can call it the reconciliation of Onsager with van der Waals.

With regard to the correlation function and the

Ornstein-Zernike theory I hope that the new ex- perimental work will show that there is a critical region, where already deviations from the classical van der Waals-Ornstein-Zernike theories begin to show up. I think there are indications (as the

specific heat anomaly) that such a region exists. Of course there is also a singularity or a critical point. B. Finally a remark about nonequilibrium phe- nomena. There are very surprising experimental

results found about various transport coefficients,

which clearly are worth pursuing. Till now there

is no theory, not even a van der Waals like theory.

The development of the proper transport theory with

long range forces, van der Waals-like but probably

with one more constant, seems to me not impossible and very worth while.

6
The Coexistence Curve of He 3 *

Robert H. Sherman

Los Alamos Scientific Laboratory, Los Alamos, N. Mex.

In this report we wish to present the results of some recent P-V-T measurements on He3 in the

critical region. A significant departure was found

in the shape of the coexistence curve from the

behavior expected for classical systems such as Xe and CO2.

The P-V-T surface of liquid He 3 was investi- gated by Sherman and Edeskuty [1]

1 between 1.0 and 3.3 °K and for pressures from the vapor pressure

to solidification. The measurements reported by

these authors lacked detail in the region from 3.0

to 3.324 °K, the critical point, [2] and for pressures

less than ~ 3 atm. In addition, there were no measurements reported on the fluid for T > Tc . Recently 23 isochores of He3 were measured with

densities ranging from 0.015 to 0.066 g/cm3 using

the apparatus of reference [1] with only slight modification. Along each isochore data were taken with

temperature decreasing from 4 °K, a final observation being made along the liquid-vapor equilibrium curve (vapor pressure). Temperatures were re- duced to the 1958 He4 Scale of Temperatures [3]. Pressures were measured with a fused quartz

bourdon tube gage having an accuracy of 0.05 mm

Hg. While the cell has a vertical dimension of ~4.6 cm, measurements were not made very close

to the critical point and as a result, density gra- dients within the cell were not of any significance.

P-V-T data taken by the isochoric method have

the advantage of yielding P-T relations which are almost linear. Thus each isochore was extrapolated

to the vapor pressure [4] to provide a p-T point on

the coexistence curve. Values of pi and pt] were

read from a smooth p-T graph and p, —

p

(/ was

plotted as a function of T( — T using logarithmic

coordinates as shown in figure 1. It is easily seen

that the critical point exponent /3 [5, 6] in the

relationship

p,-p^ (Tc -T)i

Work performed under the auspices of the U.S. Atomic Energy Commission.

1 Figures in brackets indicate the literature references at the end of each paper.

I0" 3

I0" 2 10"' Tc-T, °K

Figure 1. A plot of px — p% versus Tc —T along the co- existence curve for He3

.

deviates from the "classical" value of ~ V3, and

(3^ 0.48 + 0.02 as T^ Tc. This slope appears to *be constant up to Tc — 7'= 0.07°.

Table 1. Densities of He3 along the coexistence curve.

T Pq Pi

°K gm cm~3 gm cm~3 2.4 0.00801 0.07404

2.5 .00915 .07288

2.6 .01044 .07159

2.7 .01192 .07015

2.8 .01371 .06852

2.9 .01573 .06668

3.0 .01808 .06458

3.05 .01951 .06338

3.1 .02113 .06200

3.15 .02317 .06033

3.2 .02557 .05815

3.25 .02867 .05498

3.3 .03373 .04970

3.324 .04178 .04178

7
Table 2. Properties of He3 at the critical point

Tc = 3.324 ±0.0018 °K a

pc = 873.0 ±1,5 mm a

pc = 0.0418 ±0.001 g/cm-3

a Reference 2.

Kerr [7] has reported results of P-V-T measure- ments for He'5 along the coexistence curve. Above

3.0 °K, however, his results show increasing deviations from the present measurements probably

due to the increasingly larger obnoxious volume

corrections which must be made as the critical point is approached. Below 2.9 °K, Kerr's gas

densities are in excellent agreement with those

calculated using virial coefficients [8]. In table 1 values are given for the densities of He:i liquid and

vapor along the coexistence curve from the present

series of experiments and in table 2 some of the

properties of He3 at the critical point are summarized.

References

[1] R. H. Sherman and F. J. Edeskuty, Annals of Physics 9, 522

(1960). An error of +0.30 percent was made in the cell volume calibration. As a result all volumes and entropy changes reported should be reduced in magnitude by 0.30 percent. Expansion and compressibility coefficients are unaffected.

[2] S. G. Sydoriak and R. H. Sherman. J. Res. NBS 68A (Phys. and Chem.) No. 6, 547 (1964).

[3] F. G. Brickwedde. H. Van Dijk. M. Durieux. J. R. Clement

and J. K. Logan, NBS Mono. 10. (U.S. Govt. Printing Office. 1960).

[41 R. H. Sherman, S. G. Sydoriak. and T. R. Roberts. J. Res. NBS 68A (Phys. and Chem.) No. 6, 579 (1964).

[5] For the critical point exponents the notation of Fisher [6] is used.

[6] M. E. Fisher, J. Math. Phys. 5, 944 (1964).

[7] E. C. Kerr, Phys. Rev. 96, 551 (1954).

[8] T. R. Roberts. R. H. Sherman. S. G. Sydoriak. J. Res. NBS

68A (Phys. and Chem.) No. 6, 567 (1964).

8
Critical States of Simple Fluids and Fluid Mixtures;

a Review of the Experimental Position

J. S. Rowlinson

Imperial College, London, Great Britain

An elementary and ultimately erroneous analysis

of the behavior of a system of one component may

be based on the assumption that the Helmholtz

free-energy is an analytic function of V and T at and near the critical point [1]. Let the first non- vanishing derivative of A with respect to volume be

of the order 2n. Thermodynamic stability requires

that n be integral, and van der Waals's equation

and all similar equations, require that n — 2. This

assumption leads to the following values of the

indices above

a

+ = c*7 = 0 oq =— (2 — n)l(n— 1) (2)

Property Index Conditions

Cv — a ct +

, limit along the critical isochore V= Vc . af, limit in the one-phase fluid along the outside of the co- existing curve.

cii, limit along the critical isochore in the two ^hase fluid.

(v-vx The subscript <r denotes the orthobaric phases, hence /3 + does not exist and the superscript may be omitted.

(dpldV)T y y

+ and yr are defined exactly as a+ and af-

(p—pc

) as a 8 8

+ is apparently equal to 8~ and no distinction is made here.

function of

(V-Vc) at r=Tc . (d*pldT% -e The derivative is the curvature of the vapor pressure line

and so 6

+ does not exist.

9

At a critical point the intensive properties of two

coexisting phases become identical. This review

is a summary of our experimental knowledge of the

rate of the approach to identity as T— Tc approaches

zero. Consider some property of the fluid X(T, V)

and define an index x by

X

± = \im[\nX(T,V)l\n±(T-Tc )]. (1)

The limit is usually taken along the path V— Vc , and, in general, \

+ and X~ wih be different. The following indices are discussed below:
)8 = l/2(n-l)

y

+ = yr = l

8 = 2/i-l

0 = 0.

(3)

(4)

(5)

(6)

Thermodynamic stability requires that certain

inequalities hold between the indices. Thus Rushbrooke [2] showed that for a magnetic critical point (and Fisher [3] extended the proof to the fluid of one component)

Griffiths [4] has obtained the inequalities

cx.7 + /3( 1 + 5)^2 Gl

a; + f3 G2.

(7)

(8)

(9)

Widom [5] has shown that R becomes an equation if the conditions attached to a are that it represents

the difference (CV)* — (Cr)i across the orthobaric

boundary. He has also made two conjectures

[6, 7] for which there are no rigorous proofs; namely

y

+ = 2(1-/3)

yr = 0(8-1)

Wl

W2

(10)

(11)

It is seen that W2, if correct, implies that R and Gl

are the same inequality.

The classical equations (2) to (6) satisfy R, Gl,

G2, and W2 for all values of n. The best experimental values do not fit (2) to (6) with any integral value of n, and certainly not with

ra = 2. One concludes that A is nonanalytic.

The most probable values of the indices will now

be reviewed in turn. a.— The best measurements of Cv appear to be

those of Voronel' and his colleagues for argon and

oxygen [8]. Little experimental detail is given but

the results are more consistent and extend to lower

values of (T— Tc ) than any previous sets. However,

they are restricted to the critical isochore. Fisher

[9] has analyzed them in detail, and his analysis need

not be repeated. The results of Michels and

Strijland [10] for carbon dioxide are at densities

above and below the critical, but no reliable value of ar could be obtained from them.

Fisher [9] concludes that the singularity is a little sharper than logarithmic, both above and, probably,

below. Hence

a

+ -0.1-0.2 a2 ~ 0.0-0.1

Oil ?

fi. — Here there is substantial agreement of a large

number of measurements. The most useful of these

appear to be those of Michels, Blaisse, and Michels

[11] (C02 ), those analyzed by Guggenheim [12]

(inert gases), Cook [13] (N2 0), Weinberger and

Schneider [14] (Xe) as analyzed by Fisher [3], Lorentzen [15] (CO2), and Edwards and Woodbury

[16] (He) as analyzed by Miller [17]. The conclusions are, as follows. MBM (3 = 0.36

G, C =0.33

WS(F) =0.345 ±0.015

L =0.33 ±0.02

EW(M) =0.41

Here, Lorentzen's value is that obtained with a

freshly filled cell and includes points to within

10"3 °C of Tc . All figures except the last suggest

that is close to V3 but probably above it. y. — An analysis of the results of Michels, Blaisse,

and Michels [11] (CO2) and of Habgood and Schneider [18] (Xe) is shown in figure 1.

y+ = 1.2

y~ is uncertain but probably exceeds unity.

cr. — Widom and Rice [19] have analyzed the re- sults of Michels, Blaisse, and Michels [11] (C02 ), Habgood and Schneider [18] (Xe) and Johnston,

Keller, and Friedman [20] (H2). The results are

Xe, H2 C02

4.2

= 4.0.

However more recent discussion [21, 22] of the

optical determinations of density gradients near

the critical point suggest that measurements of

8 from P-V-T may not be as reliable as the analysis

of Widom and Rice suggests. Because of the high

value of 8 the determination from P-V-T rests essentially on results at densities well-removed from the

10
Figure 1

critical and hence possibly they are not a true guide

to the limiting behavior.

6. —Careful measurements of vapor pressure in the

critical region [11, 23] suggest that 9 may not be

zero but might have a small positive value, since

the fitting of a smooth curve through experimental

points usually gives a deviation graph that shows

a sharp rising upwards as Tc is reached.

In summary,

0.1-0.2

0.0-0.1

ft 0.34 ±0.02

y

+ 1.2±0.1

rr 1.0-1.3

8 apparently 4.2 ± 0.2 e >0 ?

ft is the most certain of these, although 8 is certainly

of high apparent precision. If and 8 are taken

as known within the limits shown then the inequalities above lead to the following results

Gl 0.06-0.40

R oq ^ (0.08 — 0. 16) ± uncertainty in yf

W\ y+= 1.28-1.36

W2 y~ = 0.96 -1.22.

Hence,

Gl, the most useful inequality, probably requires

a-7 > 0 and suggests that both (3 and 8 are near to their upper limits, that is /3 = 0.36 and

8 = 4.4. G2 is trivially satisfied. Possibly [5] 6=a^.

R is too uncertain to be of practical value for

fluids. W\ is probably wrong.

W2 is probably correct but is subject to the same

uncertainty as R. It also requires /3 and 8

to be at their upper limits. A positive value

of 6 implies a "infinite asymmetry" in (Cv)2 as T approaches (Tc )~.

All these results can be extended to the critical point of a two-component system. Here there is no real distinction between a gas-liquid, a gas-gas, and a liquid-liquid point, but all precise work has

been done on systems at which P is essentially zero and which are therefore called conventionally

liquid-liquid critical points. The behavior of a two-component system can be obtained by analogy

from that of a one-component system by replacing

A(V, T) by G(x, T), where x is the mole fraction of one component and where the pressure of the sys- tem is fixed. Hence the indices above now refer

to the following properties

a Cp

ft x<r y (d2Glx2

) P , T 8 (/Xi — (x)k along T— Tc 6 {dCpldx)T in the two-phase region.

Thus Fisher's proof [3] of Rushbrooke's inequality

follows at once for a two-component system from

an equation [24] for (Gp

)-2. The experimental results are now more fragmentary, although this probably reflects lack of

effort rather than inherent difficulty. Precise

calorimetry and measurement of vapor pressures

should be easier than at the critical pressure of a one-component system since these critical points can be studied at negligible pressures. a.— The best measurements of C,, are those of

Schmidt, Jura, and Hildebrand [25] (CCL4 + CKFiiCF.i): Jura, Fraga, Maki, and Hildebrand

[26] (/ — ChHik + n — CtFk;): Amirkhanow. Gurvich,
and Matizen [27] (PhOH + rL>0). These suggest

either a weak infinity in C,, or possibly no infinity

at all.

/3. — The recent and, precise results of Thompson

and Rice [28] make it unnecessary to discuss earlier work, including their own. They studied CCI4

+ 7? — C7F16 to within 10~4 °C of Tc and obtain

£ = 0.33 + 0.02.

y, 8. — The results of Croll and Scott [29] for CH4 + CF4 suggest that y

+

is about unity. The

authors may have unpublished points on these

systems to justify some attempt to determine y

and 8; no earlier work suffices.

If, by analogy with the more extensive work on

systems with one component, it turns out that y

+ > 0

then there are interesting consequences. Thus

both {d2H/dx2 )p, T and (d2S/dx2

) P , rmust be zero at the

critical point. That is curves of the heat of mixing

as a function of composition must have a point of

inflexion. This is not inconsistent with, for ex- ample, the measurements of Copp and Everett

[30] and of Matizen and Kushova [31] on the lower

critical solution point of Et 3N + H20. There

appear to be no other accurate measurements close

to Tc. A further consequence of y

+ > 1 for which there

is some experimental evidence is that coexistent curves calculated from excess Gibbs free-energies

that have been extrapolated to Tc on the implicit

assumption that y

+ = 1 must always be too high

at an upper critical solution temperature and too low at a lower critical solution temperature. It is common experience that this is so [32, 33]. A

recent paper by Scatchard and Wilson [33] shows

a closed solubility loop calculated by such extrapolations which shows simultaneously an upper critical point that is too high and a lower one that is too low.

However more direct evidence of the magnitude

of (y+ — 1) would be welcome.

References

[1] J. S. Rowlinson. Liquids and Liquid Mixtures, ch. 3 (Butter- worths. London. 1959).

[2] G. S. Rushbrooke.J. Chem. Phys. 39,842(1963).

[3] M. E. Fisher, J. Math. Phys. 5, 944 (1964).

[4] R. B. Griffiths, Phys. Rev. Letters 14,623 (1965).

[5] B. Widom. private communication.

[6] B. Widom, J. Chem. Phys. 37,2703(1962).

[71 B. Widon, J. Chem. Phys. 41, 1633 (1964).

[8] M. I. Bagatskii, A. V. Voronel. and B. G. Gusak. Soviet Phys.

JETP 16, 517 (1963); A. V. Voronel' Y. R. Chashkin, V. A. Popov, and V. G. Simkin, ibid. 18, 568 (1964).

[9] M. E. Fisher. Phys. Rev. 136, A1599 (1964).

[10] A. Michels and J. Strijland. Physica 18, 613 (1952).

[11] A. Michels. B. Blaisse. and C. Michels, Proc. Roy. Soc. A160, 358, (1937).

[12] E. A. Guggenheim. J. Chem. Phys. 13, 253 (1945).

[13] D. Cook. Trans. Faraday Soc. 49, 716 (1953).

[14] A. M. Weinberger and W. G. Schneider, Canad. J. Chem.

30, 422 (1952).

[15] H. L. Lorentzen. Paper read at conference on Statistical Mechanics at Aachen. 1964.

[16] M. M. Edwards and W. C. Woodbury, Phys. Rev. 129,

1911 (1963).

[17] D. Miller, private communication.

[18] H. W. Habgood and W. G. Schneider, Canad. J. Chem. 32,

98. 164(1954).

[19] B. Widom and O. K. Rice. J. Chem. Phys. 23, 1250 (1955).

[20] H. L. Johnston. K. E. Keller, and A. S. Friedman. J. Amer. Chem Soc. 76 (1954) 1482.

[21] S. A. Ulybin and S. P. Malyshenko. Third Symposium on Thermophysical Properties. Amer. Soc. Mech. Eng., Purdue. 1965. p. 68.

[22] S. Y. Larsen and J. M. H. Levelt Sengers. Third Symposium

on Thermophysical Properties. Amer. Soc. Mech. Eng.. Purdue. 1965. p. 74.

[23] A. Michels. J. M. H. Levelt and W. De Graaff. Physica 24,

659 (1958): A. Michels. J. M. H. Levelt, and G. J. Wolkers.

ibid. 24, 769 (1958).

[24] Reference 1, p. 168.

[25] H. Schmidt. G. Jura and J. H. Hildebrand, J. Phys. Chem.

63, 297 (1959).

[26] G. Jura. D. Fraga. G. Maki. and J. H. Hildebrand. Proc. Mat. Acad. Sci. 39, 19 (1953).

[27] Kh. Amirkhanow, 1. G. Gurvich and E. M. Matizen. Dokl. Akad. Nauk. SSSR 100, 735 (1955).

[28] D. R. Thompson and O. K. Rice. J. Amer. Chem. Soc. 86,

3547 (1964).

[29] I. M. Croll and R. L. Scott, J. Phys. Chem. 68, 3853 (1964): see also N. Thorp and R. L. Scott, ibid. 60, 670. 1441

(1956).

[30] J. L. Copp and D. H. Everett. Discuss. Faraday Soc. 15,

174 (1953).

[31] E. M. Matizen and N. V. Kushova. Russ. J. Phys. Chem. 34,

1056 (1960).

[32] G. E. Williamson and R. L. Scott. J. Phys. Chem. 65,

275 (1961).

[33] G. Scatchard and G. Wilson, J. Amer. Chem. Soc. 86, 133

(1964).

12
Optical Measurements of the Density Gradients Produced by Gravity in

C02 , N2 0, and CC1F3 Near the Critical Point

E. H. W. Schmidt

Technische Hochschule, Miinohen, Germany

1. Introduction

When Dr. Green asked me to read a paper dealing

with my work on density gradients produced by

gravity in fluids near their critical points, I hesitated

to follow this invitation because I already presented some of such experiments with CO > at the Second

Symposium on Thermophysical Properties held in Princeton University January 1962 [1]. However,

meanwhile together with my collaborator J. Straub, we repeated these measurements with greater precision and extended them to N >0 and CC1F :! (Freon

13).First I may remind you of the original purpose and

the start of these experiments. The old conventional method for measuring the critical data of a substance by observing the disappearance of the

meniscus i.e., the interface between the liquid and

the vapor phase is not very exact because it needs a slow change of temperature or pressure and therefore the fluid is not in a state of equilibrium. Moreover it is difficult and depends upon the ability of

the observer to state exactly when the meniscus has

disappeared. This is the reason why the critical point of most substances is not very well known, and some observers even had the conception, that the

critical point had to be extended to a critical region.

In order to avoid the difficulties of the conventional method I came to the idea of using the refractive index for the determination of the critical state data, which index depends in a sjmple way

upon the density of the substance irrespective of

its liquid or gaseous state. As long as a meniscus is visible, the densities

and the refractive indices are different in the liquid

and in the vapor. However, when the critical state

is attained, refractive index and density should be

equal below and above the level where the meniscus

disappeared.

2. The Pressure Chamber and Its

Accessories

For our experiments we use now the installation shown in figure 1. The fluid is in a strong cylindrical chamber of stainless steel with horizontal

axis closed on both ends by parallel windows of optical glass with 28 mm free diameter.

For equalizing the temperature of the pressure

chamber it is placed in a big copper cylinder, which

on its outer side is supplied with screw-shaped

groves through which water from a thermostatic

apparatus flows bifilarly. In order to minimize the

fluctuations of the temperature produced by the

switching in and out of the heating coil of the ther- mostat with the help of a mercury contact ther- mometer, three thermostats are used in series so

that the first with its contact thermometer delivers warm water for heating the next one, and this again for the third, as shown in figure 2. In this way the temperature of the copper cylinder with

the pressure chamber in it could be held constant

for many hours and even for several days within

deviations of less than ±0.0015 °C. The copper cylinder has on both ends conical

holes for the optical observation. In these holes

two more plates of thick optical glass are placed,

which together with two outer thinner windows and

electrically heated coils inside these windows

prevent the cooling of the pressure chamber by

the surrounding air. The temperature of the room, in which the whole

experimental arrangement has its place is also regulated by way of contact thermometers together

with electric heaters and the fluctuations of the room temperature do not exceed ±0.1 °C. Nevertheless, the copper cylinder is covered with a thick

layer of polystyrol as heat insulation.

13
Figure I. The pressure chamber used for the experiments:

(a) Axial cross section

(b) Cross section A-B

(c) Cross section perpendicular to axis

Dimensions in mm; a. Copper cylinder, b. Pressure chamber, c. Strong glass windows, d. Sealing foils, e. Glass prism, f. Tube,

g. Filling valve, h. Screw shaped channels for water heating, i. Protective heating of the valve, k. Heating tube, 1. Glass plate with protective heating coil behind it. m. Glass plates, n. Holes for thermocouples, o. Welded joints.

I 11 ^

d C 0

FIGURE 2. Scheme of experimental arrangement:

a. First thermostat, b. Water container, c. Second thermostat, d. Third thermostat, e. Screw pump

for mercury, f. Pressure balance, g. Manometer, h. Pressure chamber, i. Dehydrator for gas, k. Container

for pressurized gas, I. Glass tube for testing the vacuum, m. Valve, n. Joint to vacuum pump.
Figure 3. Picture of the pressure chamber {heat insulation removed).

A picture of the pressure chamber and the copper

cylinder before surrounding it with the heat insulation is shown in figure 3. All tubes leaving the pressure chamber are provided with protective

electrical heating in order to prevent any loss of

heat which would cause local cooling. The purpose

of all these arrangements is to improve the constancy of temperature in the pressure chamber and

to avoid natural convection in the fluid, which dis- turbs the optical measurements.

The temperature of the fluid is measured with the

help of manganin-constantan thermocouples placed

in vertical holes drilled into the copper cylinder.

The pressure of the fluid is measured with a piston

manometer f shown in figure 2. This manometer

is filled with oil which is separated from the fluid

in the pressure chamber by mercury. With the

help of a screw pump more or less mercury can be

pumped into the pressure chamber changing the

volume and the density of the fluid contained in it.

3. Optical Installation

The optical installation for measuring the refractive index and from it the density of the fluid is shown in figure 4. The pressure chamber a with the

glass prism in it is placed in the middle of the

disk c, around which two arms can be swung of

which one supports the sodium lamp d. the con- densor e, the vertical slit f, and the lens g; whilst

the other arm bears the telescope h, the horizontal

slit i, and a small concave mirror k. The sodium

lamp with its monochromatic light illuminates with

the help of the condensor the vertical slit which is placed in the focus of the lens g. In this way a parallel beam of light enters the

pressure chamber and is refracted by the glass prism

and the prism-shaped fluid in the chamber. If the

telescope is turned in the right direction, a vertical picture of the slit is to be seen in it, and from the

angle of the position of the telescope the refractive

index of the fluid in the pressure chamber can be

calculated. This angle can be determined very

exactly with the help of a small concave mirror

placed above the middle of the pressure chamber

and moving with the telescope, to which it is fixed. A light pointer reflected by the mirror throws a mark on a circular scale with 2.05 m radius. In this way the angle of the telescope can be determined

with precision of better than 0.01°.

c b a

FIGURE 4. Optical installation for measuring the refractive index.

a. Pressure chamber, b. Glass prism, c. Circular disk, d. Sodium lamp. e. Condensor. f. Vertical slit, g. Lens. h. Telescope,

i. Horizontal slit, movable up and down, k. Concave mirror. 1. Lighting of the observation mark, rn. Circular scale, n. Picture of the observation mark.

15
FIGURE 5 Samples of photographs taken with the telescope.

If the density of the fluid in the pressure chamber

does not depend on height, the telescope shows a

vertical picture of the slit. If there is a meniscus

in the pressure chamber separating liquid and vapor, we get two different pictures in the telescope, one

produced by the light passing the liquid phase, the

other produced by the light passing the vapor phase,

the distance of these lines being proportional to the difference in density of liquid and vapor. One of the first photographs which I got in this way with CO> at its critical pressure is shown in figure 5. Here two horizontal slits are placed before

the telescope so that the upper slit gets light having passed the vapor phase, the lower light having

passed the liquid phase. On the right-hand side we see what the telescope shows. The distance

of the two lines decreases if we approach the critical temperature, but they do not coincide when the

meniscus has completely disappeared.

The last two pictures of figure 5 are taken without

the horizontal slits. Here both lines are indistinct,

showing that in the fluid there must be a change of

density with height. A better picture of the same

kind, also taken with two horizontal slits, is shown

in figure 6; here even at the critical temperature of 31.01 °C the two lines have a substantial distance,

as the third frame shows. The last frame was taken

after stirring the fluid thoroughly with the help of

a magnetic stirrer. However, if the stirrer is stopped the two lines of frame three reappear after some hours and we learn that in the field of gravity

the stable state has gradients of density.

FIGURE 6 Photographs taken below and at the critical temperature of CO ..

These experiences gave rise to use a horizontal

slit before the objective of the telescope, which can I be moved up and down, whereas the position of

the slit can be measured with the help of a cathe- tometer. In this way the density of the fluid can

be measured as a function of height.

4. Some Results of the Point by Point

Measurements of the Density of CO2

Near Its Critical Point as a Function

of Height

As an example figure 7 shows the results of density measurements depending on height for CO2 at its critical pressure and at a temperature

-x

0

b

\

0.90 0.95 1.00 1.05 1.10 REDUCED DENSITY pip

FIGURE 7. Reduced density p\p c of C02 at the critical pressure and § = 0.007 °C above the critical temperature.

a. Curve with measured points, b. Corrected curve.

16
= 0.007 °C above the critical temperature. We

see how well the measured values lie on the curve a

the scattering being less than the thickness of the

line. All curves which we shall see later are deter- mined by measurements with the same precision

and thus it may be allowed to omit the measured

points. The curve b is a slight correction of curve a

due to the fact that in a field with a gradient of density the ray of light is slightly curved. In the figure

the abscissa is the reduced density p/pc where pc is the density at the critical point. We see that for a change in height of only 10 mm (from + 5 until — 5) the change in density is about 10 percent.

Now already in 1892 Gouy [2] has emphasized that the critical pressure can only exist at a certain level of the fluid, because the pressure and thus the density decreases with height due to the weight of the

fluid. However, the measured gradients of densities are 30 to 50 times larger than those produced by

hydrostatic pressure if these are derived from a van

der Waals equation.

Figure 7 is measured at a temperature &=

+ 0.007 °C above the critical temperature and at a mean density very near the critical value. If we

change the temperature at constant mean density of the fluid we get a set of curves as shown in figure 8, taken at a mean density of plpc = 0.985. The curve

a corresponding to a temperature # = — 0.005° below

the critical shows a jump in density corresponding

to a visible meniscus, but nevertheless there is still a substantial gradient of density in the liquid and

0.95 1.00 1.05 REDUCED DENSITY^

Figure 8. Density plp c against height for C02 with a mean

relative density 0.985 p c and for the following temperatures # relative to the critical vilue: a. -0.005 °C; b. 0.000 C:

c. 0.003 °C: d. 0.012 °C; e. 0.035 °C;f. 0.095 °C.

in the gaseous phase. The curve b is observed at a temperature very near the critical value, the curves

c, d, e, and / are taken at temperatures from 0.003

until 0.095 °C above the critical. Here the jump

in density has changed to an inflection point with

a tangent growing steeper the more we surpass the

critical temperature.

\

(A)

?

1 e d c bo

0.90 0.95 1.00 1.10 0.90 REDUCED DENSITY pip

FIGURE 9. Density plpc against height for CO>.

A. Same as figure 8. B. Mean relative density of 1.033 pc and the following temperatures # relative to the critical value: a. -0.008 °C; b.- 0.003 °C; c. 0.002 °C; d. 0.005 °C;

e. 0.015 °C;f. 0.035 °C; g. 0.065 °C; h. 0.125 °C; i. 0.232 °C. 17
The immediate neighborhood of the critical point

is difficult to observe because here the gradient

of density has its highest value, and the light is deflected down. Therefore, in what follows we will calculate the temperature difference # using as the

starting point the highest temperature at which the

density versus height curve still shows a discontinuity.

If the mean density is increased by pumping more

mercury into the pressure chamber the level where

the meniscus disappears ascends as to be seen in figure 9, where the left-hand picture repeats figure

8 with the mean density p/p<= 0.985, and the righthand picture corresponds to a mean density p/p (

. = 1.033. The different curves are isotherms from — 0.008 up to 0.252 °C above critical temperature.

In figure 10 all curves are taken at the criti- cal pressure and at the same temperature # =

— 0.006 °C below the critical value, only the mean

densities differ from curve to curve and are denoted

by the small circle in each curve. Figure 11 shows the same very near the critical temperature. Figure 12 for # = + 0.007 °C; figurj

13 for # = + 0.03 °C; figure 14 for # = + 0.06 °cJ

These figures show how difficult it is to determine

the critical density without optical observation be- cause the inflection point of the density curve (the

level where the meniscus disappears) may have a higher or lower position. If it is high, more than I half of the chamber is filled with fluid of larger density, if it is low, only a smaller part of volume has I the higher density.

US 100 105 REDUCED DENSITY/?//? REDUCED DENSITY pip

FIGURE 10. Density against height for C02 at a relative tem- perature i? = — 0.006 °C and at different mean densities in- dicated by the small circle at each curve.

FIGURE 12. The same as figure 10. but for a relative temperature $ = 0,007 °C.

REDUCED DENSITY pip

FIGURE 11. The same as figure 10 but for a relative temperature § = 0.000 °C. that is at the critical temperature.

B.60 0.85 0 50 0.9S 1J>0 I.OS 1.10 1,15 UO | REDUCED DENSITY p/p^

Figure 13. The same as figure 10, but for a relative temperatures $ = 0,030 °C.

18
0.91 100 1.01 110 REDUCED DENSITY pip

FIGURE 14. The same as figure 10, but for a relative temperature $=0,060 °C.

5. Measurements With N2 0, CC1F :! , and H2 0

All measurements up to here are made with COiIn order to show that not only this substance undergoes gradients of density in the field of gravity,

experiments were also made with N^O and with

CC1F:! . Figure 15 shows isotherms of N2O for temperatures # = — 0.027 until +0.021 °C at a mean density

plpr — 0.998. In figure 16 two sets of isotherms are placed side by side, the left with a mean density

0.95 1,00 1.05 REDUCED DENSITY pip

1.10

Figure 15. Density against height for N2O at a mean relative density 0.998 pc and for temperatures ?? relative to the critical temperature tc = 36.585 °C:

a. -0.027 °C: b. -0.012 °C; c. -0.006 °C; d. 0.000 °C;

e. 0.012 °C: f. 0.021 °C.

plp< = 0.955 and temperatures # from 0.000 until + 0.056 °C, the right with p/pc = 1.003 and temperatures # from -0.002 until +0.218 °C. The figure

is very similar to figure 9. Figure 17 also relates to N2 0 at a temperature & = — 0.004 °C with different mean densities given by the small circles in the curves.

REDUCED DENSITY/0//3, FIGURE 16. Density against height for N2 0.

A. Mean density 0.995 pc and for the following relative temperatures: a. 0.000 °C,

b. 0.004 °C, c. 0.015 °C; d. 0.020 °C; e. 0.044 °C; f. 0.056 °C. B. Mean density 1.005 pc and for the following relative temperatures: a. -0.002 X; b. 0.003 °C; c. 0.010 X; d. 0.020

°C; e. 0.032 °C;f. 0.038 °C; g. 0.063 °C; h. 0.119 °C: i. 0.167 °C; k. 0.218 °C. 209-493 O-66—

3

19
0.66 0.90 1.00 REDUCED DENSITY pip

FIGURE 17. Density against height for N2 0 at a relative temperature i? = — 0.004 °C and at different mean densities given by the small circle at each curve.

REDUCED DENSITY pip

FIGURE 18. Density against height for CC1F: i with the mean:

density 0.989 pc and the following temperatures relative to the '

critical temperature tc = 28.715 °C: a. -0.065 °C; b.- 0.025 °C;

c. 0.000 °C; d. 0.021 °C; e. 0.040 °C; f. 0.093 °C; g. 0.183 °GM

h. 0.265 °C.

Finally figure 18 is measured with CClF:i at a mean density pip, — 0.989. and the different curves correspond to temperature differences with the

critical value from # = -0.065 until 4- 0.265 °C.

In the COj-molecule the two O-atoms are placed

symmetrically on opposite sides of the C-atom.

corresponding to the scheme 0=C=0. The N^Omolecule is asymmetric and therefore polarized

following the scheme N=N=0. The CClF:i molecule has the shape of a tetraeder with the C-atom

in the middle and the halogen atoms at the corners

of the tetraeder. As these three molecules, differ- ing in shape and chemical constitution show the same behavior in the neighborhood of the critical state, it is to be expected that this behavior is a genera] quality of all substances.

The critical data which we found are in good

agreement with the results of other authors.

Measurements with FFO are now being done.

This molecule has the shape of a triangle and its behavior near the critical state seems to be the same as for the other substances. However, the

measurements are much more difficult due to the

higher temperature ?,=374.15 °C and the higher

pressure p, =221.3 bar. Windows of glass and

even of quartz are impossible because these sub- stances are corroded and lose their optical quality.

Finally we found that sapphire windows withstand

corrosion and these are used now.

6. Summary

The above experiments show undoubtedly that near the critical state of substances considerable

gradients of density are produced by gravity. For

the three substances encountered, the density at the critical pressure and near the critical temperature changes by about 10 percent over a height of

about 10 mm. These distributions of density are

stable and reproducible. After any changes of

state of the fluid, they are restored by diffusion and some kind of sedimentation within times of about 2 hr for heights of some centimeters.

7. References

[1] Ernst H. W. Schmidt anc? K. Traube. Gradients of Density j of Fluids Near their Critical States in the Field of Gravity, pp. | 193-205. Progress in International Research on Thermodynamic | and Transport Properties A.S.M.E. (Academic Press. New York

and London 1962).

[2] R. Gouy. Effets de la pesanteur sur les fluides au point

critique, Compt. Rend. Acad. Sci. Paris 115, 720-22 (1892). I 20
Notes, Definitions, and Formulas for Critical Point Singularities

M. E. Fisher

University of London, King's College, London, Great Britain*

1. Introduction

For the convenience of participants a set of

definitions and notes concerning the various criti- cal point singularities and a table of formulas

relating the lattice gas and Ising ferromagnet

models were distributed at the Conference. These

are reproduced here with a few corrections and

extensions.

Most of the results quoted are described and

discussed in detail in references 1. 2. and 3 where

further references to the literature are given. A

few more recent results are contained in references

4 to 8 and, of course, in the work reported at this Conference particularly in connection with critical scattering by Chu, Passel, Nathans, Dietrich, and

Als-Nielsen.

2. Critical Point Exponents

For theoretical discussions and in the analysis

of experimental data it is useful to have a precise

definition of a critical point exponent. This may

be given as follows:

if lim log f{x)l logx= k.

jr-»0 + we say fix) ~ x

K as x —> 0 +

.

Note that with this definition the statement f(x) ~ x

A does not exclude the possibility of a logarithmically

divergent factor, i.e., AxK

\ log x\ ~ x

x

. In particular if A =0 the function fix) might diverge as | log | for example, or f(0) might be finite, the function f(x)

*Now at Cornell University, Ithaca. New York.

then being either continuous or discontinuous as x passes through zero. The definitions of the most important critical point exponents for gases and ferromagnets are

set out in the following table together with certain

relations between them. Analogous formulas

hold for binary alloys and binary fluid systems.

Note that the exponent tj enters directly into the

scattering of waves at the critical point through the

formula

l//,(k) ~ k-~\

where 1(h) is the intensity of scattering at wave

vector k with A = (47r/A.) sin h 0 (for d=3>).

The meanings of the symbols used in table 1 and

below are

Cr, Cp , Ch- Cm Specific heats at constant volume,

pressure, field, and magnetization,

respectively.

T. Tc Temperature, critical temperature,

p, p,-; p/., pa Density, critical density; densities

of coexisting liquid and gaseous

phases.

M, Mo Magnetization, spontaneous magnetization.

Kt, Ks Isothermal and adiabatic compressibilities. Xt, Xs Isothermal and adiabatic susceptibilities. p. pc po- Pressure, at the critical point and at coexistence (vapor pressure),

respectively.

H Magnetic field. gi(r) Pair correlation function (in a fluid).

21
TABLE 1. Critical point exponents

Exponent Fluid Magnet

Below Tca'P

y'

a*

At coexistence as T—> Tc Cv~(Tc -T)-a

'

In, — n, \ ~ IT — TV3 VPL pc) \1 c I r KT ~(Tc -T)-y'

Idtpv/d-Pl-iTc-T)-"*

At // = 0 as T^TCC„~(Tc -D-a

' MjT\ — IT — T\0

XT~(Tc -T)-y'

At T= Tc As p

-» pc As H^O

8 \P-Pc\~\p-Pc\ 6 Note the exponents j8, y', and 8 might differ on the gas and liquid

sides.

\H\~\M\ 8

At p = pc as i * 00 \g2(r)-l\~ l/r"- 2+ " At # = 0 as °° <S&Sf> ~

Above Tca

y

V

At p =

p

c , as 7

1 —» rc +

^-(r-r,.)-0 KT ~{T-Tcyy

Inverse Range of Correlation K

At // = 0, as T-* Tc +

CH ~ (T- TcYa xr ~ (TT

c)-y

T) ~ (T-TcY

Rigorous

inequalities

a' + (2)8 + y')mi„^2

«'+i8max + (/3S) min ^2

a*^a'+/3max

a' + 2p + y' ^2

a' + /3(l + 8)^2

Conjectured

relations

a' + 2i3 + y' = 2, 7' = £(8-1), y=(2 — t/)i/

o? Dimensionality of the system.

Si Spin variable at site r. ffi Hamiltonian.

z Activity. p, p„ Chemical potential, at coexistence.

In table 2 the exact, estimated and observed

values of the critical exponents defined above are presented for various theoretical models and for real fluids and ferromagnets [1-6]. The queries

indicate significant uncertainties or total ignorance!

Notice that it is not clear that the existing exper- imental results for a', /3, and 8 satisfy the rig22

orous inequality recently derived by Griffiths [7] ij (see table 1). For the standard three-dimensional (d=3) spherical model one may add the results: a' = a = 0

(continuous), fB—i, y' not defined, 8 = 5, 77 = 0, | y = 2, v = l. For a spherical model with long

range ferromagnetic forces decaying as l/rlt+ "~ {0

< cr < 2) one has more generally 1: a' — a = 0, j /3=f as before, but (a) for cr < |e/: y' — 7=1,1

8 = 3, r) = 2 — cr, v undefined; while (b) for cr 3= \d\\

y' undefined, 8 = (d+ ar)/(d— cr), 7) — 2 — cr, l

! y = <r/(d— cr), v undefined.

1 These results have been obtained by G. S. Joyce.
Table 2. Values of critical point exponents

Exponent

"Classical"

theory

Ising

d=2 d=3

Heisenberg

d = 3

Observation

Fluids Magnets

a' 0

(discon.)

0

(log)

> 0? ? > 0 (log) > 0 ?

P

i2 Vs 0.312 - 5/ie 0.303 to 0.308

9 0.33 to 0.36 0.33

±0.015

y' 1 l 3

/4 1.23 to 1.32 9 =*1.2 ? ?

a*

8

0

3

0

(log) 15

>0

5.20

±0.15

?

5=0 ?

4.2 ±0.14.22

±0.05

V 0 V4 0.059 - Vis

±0.006

0.08

± 0.04

>0 ? >0 ?

a 0

(discon.)

0

(log)

2=0

^0.2

-0 =3 0? >0 ?

7 1 l 3

/4 1V4 - 1V3 1.32 to 1.38 > 1.1 ? 1.35 ±0.02

V v% 1 0.644

±0.002

0.69

±0.02

> 0.55 ? >0.66

3. Relation Between a Lattice Gas

and an Ising Magnet

The mathematical equivalence of the lattice gas model of a fluid and the Ising model of a magnet,

[stressed originally by Yang and Lee, is well known

1 but it was felt that a more complete account of

I the relation between the two models than normally

given would be useful. Indeed some of the experimental results included in table 2 suggest that the

fluid-magnet analogy might be deeper than the

simplicity of the Ising model would indicate.

We consider a general (/-dimensional lattice of

./V sites, each occupying a cell, of volume vo- The

pair interactions are not necessarily limited to nearest neighboring sites. Positive values of the

exchange integral J(r) correspond to ferromagnetic

coupling but the formulas are valid for arbitrary

coupling.

23
Table 3. Correspondence between a lattice gas and an Ising magnet

Lattice of N sites; J-dimensional volume v() per site. (/3= l//.«r)

Total Potential

Energy of n atoms

&s = -yj(riJ)siss-Hyj s i

Occupation

Variables

ti = 1 or 0

Potentials </>(0) = £ <t>(ru)

Density

Pmax = 1/W0 = 2pf

Chemical Potential p

P/jL = \n(zA") A = (h 2 l27rmkH T) il - at condensation

p = p,r = ^(O)-AT ln,i>»/A")

ti= Hi- si)

pjpc =\-M

= -2H

5 , = 2Sf = ±l Spin

Variables

Exchange Integral

Magnetization per Spin

^raax 1

Pressure p(p, T)

Configurational

Entropy Density

Magnetic Field, H

units chosen so that magnetic

moment per spin is unity.

PVu F*-H-Vs(t>(0)= [l-M(H')]dH'

Free Energy per Spin

F*(//, T)

cunfiu — C *

Isothermal Compressibility

Configurational

Specific Heats

^Vi)p2 Kr = Xi

pvoCf«t* = CM =CH -Tc?Jxr

Entropy per Spin

Isothermal Susceptibility

Specific Heats

p^qonfm = T ( JlL^ Xr + CH + 2T f-^n | 1-M l-M

Note

Thermal Pressure

Coefficient

Correlation

Functions

Cp/Ci' — Kt/K.s ChICu = XtIxs

Note 1/xt = 0 in the

"two-phase" region.

(dM\

aH = Q for H=0, T > Tc

f„y. = vt) (dpldT) y = S* + (1 - M)anlxi

pVi)Qr) = pVi) [g.j{r) - 1 ] = ( < W- (s0y-)/((sl) -<s„> 2

) = Hi

at p = p„ T>,TC W(r) = <W = r(r) at H = 0, T>TC.

24
From'the correlation functions G(r) and T(r) one may derive the critical scattering intensities through

the formulas

/(k) =x(k)/o(k)

where /o(k) is the form factor for scattering from

the noninteracting system and where

X(k)=l+p 2 vl)e

ik *G(r)=l+ e' k T(r).

r ft 0 r * 0 The fluctuation relation may then be written

KtIK%=xW) = XtI7&

where the superscript zero refers to an ideal noninteracting system.

4. References

[1] M. E. Fisher. J. Math. Phys. 4, 124 (1963).

[2] M. E. Fisher. J. Math. Phys. 5 , 944 (1964).

[3] M. E. Fisher, Proc. Int. Conf. on Magnetism. Nottingham

1964, 79 (The Physical Society, London, 1965).

[4] J. S. Kouvel and M. E. Fisher. Phys. Rev. 136, A. 1626 (1964).

[5] M. E. Fisher. Phys. Rev. 136 A. 1599 (1964).

[6] D. S. Gaunt. M. E. Fisher. M. F. Sykes. and J. W. Essam.

Phys. Rev. Letters 13,713(1964).

[7] R. B. Griffiths. Phys. Rev. Letters. 14,623(1965).

[8] R. J. Burford. and M. E. Fisher, to be published.

Discussion

R. L. Scott: Dr. Rowlinson made some reference to our work on

mixtures of CF4 with methane. Recently Croll and I published some rather old work on mixtures of CF4 with ethane [1]. I did

not actually consider the work good enough to apply this kind

of consideration, but at the request of Dr. Widom, I looked at the results again. The data give the total pressure of CF4 with ethane at about 150.6 °K, which is about 0.5° above the critical solution temperature. The curves are, in fact, nearly flat. If one ignores the fact that the only relevant data are considerably removed from the critical composition (just the point that Dr. Rowlinson made and that really vitiates the whole result), then

the results are consistent with an exponent 8 = 4, but it would

not be very difficult to make the data consistent with 8 = 5.

C. J. Pings: The one enigma in the data showed by Dr. Rowlinson

for fi was in the results obtained by Edwards and Woodbury for He4

. I believe those were deduced from refractive index measurements. We have some unpublished data on the Lo- rentz-Lorenz function for argon and methane which suggest a possible anomaly in the Lorentz-Lorenz function in the critical region [2]. Until this is resolved, we must be dubious about any

conclusions of this type deduced from refractive index measurements.

J. S. Rowlinson: I mentioned the results of Edwards and Woodbury only in passing. I did not use them in any way in the analysis. They are clearly out of line with the results on gases at high temperatures. Whether this is a particular quantum effect or not, I do not know.

S. Y. Larsen: I would like to make a remark concerning the com- ment of Dr. Pings. Dr. Mountain, Dr. Zwanzig, and I have made

a theoretical studv of the behavior of the index of refraction [3]. We find no anomaly in the behavior of the real part of the index

of refraction in the supercritical region when the critical point

is approached.

O. K. Rice: I may mention that some years ago Rowden and I made vapor pressure measurements on the cyclohexane-aniline

system [4]. We approached the critical point to within 0.1 or 0.2 °C and it looked like y= 1.

J. S. Rowlinson: Did it?

O. K. Rice: We could not say definitely whether y was either greater or smaller then 1, but it did look very much like y= 1, as

far as we could tell from the measurements.

J. S. Rowlinson: I am sorry that I omitted your results in my

review. I did not try to deduce a value for y from your results on the cyclohexane-aniline system.

M. E. Fisher: I would just like to emphasize one point, which

Dr. Rowlinson made very well. There is an inconsistency with

the rigorous inequalities if one takes the values of the exponents

at their face value. Although one cannot be too sure that any

of them are as good as they seem to be, the fact is that for 8 to be consistent with the relationship (Gl) a rather big change in its value is needed. It means that 8 should be near 5 rather than

4.2, is not that right?

J. S. Rowlinson: It depends on what lower limit one puts on

aj. If one allows = 0.06, one can get away with /3 = 0.36 and

8 = 4.4.

M. E. Fisher: That is correct. However, since you multiply

8 with 1/3 in relation (Gl), the effect of a change in 8 in order

to satisfy the relation gets damped down. If you take the central values of 8 and 8. there really is quite a big discrepancy.

J. S. Rowlinson: I agree that the central values lead to a dis- crepancy, and if you prefer 8 = 5, I cannot say no.

E. Helfand: In a mixture there are many new thermodynamic

variables because one has one extra variable and consequently

many more derivatives. One of the interesting derivatives may be dp/dp. This property depends on the liquid correlation functions, that is density correlation function, whereas the second derivative of the free energy with respect to the concentration depends on the concentration correlation function. One

might well expect these correlation functions to have the same

range and the same analytic behavior. Thus perhaps dpldp would reflect the same y behavior.

J. S. Rowlinson: It would be very difficult to measure the dpldp in a dense fluid with the precision that is needed to get these indices.

W. W. Webb presented measurements of the reflectivity of the

interface between two phases in equilibrium just below the

critical consolute temperature [5]. These measurements indi- cate that the effective equilibrium thickness of the transition region increases roughly as [Tr — 7V"- 7 as the critical temperature

is approached, reaching ~ 1.000 A at {T,. — T)ITV ~ 10"3

.

S. Y. Larsen: Dr. J. M. H. Levelt-Sengers and I have looked at the disagreement between the optical data and the isotherms

derived from PVT measurements [6]. This disagreement was

first noticed by Dr. Schmidt and noted in the account of his optical work dven at the 1962 Symposium of the A.S.M.E. on

25
Thermophysical Properties and also has been taken up by Ulybin and Malyshenko in a recent paper delivered at the Third Symposium on Thermophysical Properties [71. Our conclusion is that the optical work may be in agreement

with the PVT data. The disagreement lies between the optical

results and the analytical forms for the isotherms which have

been fitted to the PVT data. These include both the Taylor

type expansions and the fit of Widom and Rice. We believe

that one of the main reasons for the disagreement is that these expressions are really fitted to data somewhat outside the critical region. We find that the conventional way of effecting PVT

measurements provides an exceedingly insensitive method of determining the pressure versus density relationship very close

to the critical point and that optical methods are certainly to be

preferred. Accordingly, we believe that the isotherms are very much flatter near the critical point, and the compressibilities very much larger, than has been inferred from PVT data.

G. B. Benedek: I wonder if those remarks could be translated

into the following terms: What do you think is the proper value

of 6, and what do you think is the proper value of y?

S. Y. Larsen: At the present time we have not attempted to reeval- uate the old data. The result that we have does not imply that the S of Widom and Rice, for example, is not the correct one

because one could still change the leading coefficient. I think

that one would like to start with the beautiful new data of Dr. Schmidt and try again to look at these questions.

References

[1] I. M. Croll and R. L. Scott, J. Phys. Chem. 68, 3853 (1964).

[2] C. P. Abbiss, C. M. Knobler, R. K. Teague, and C. J. Pings.

J. Chem. Phys. 42, 4145 (1965).

[3] S. Y. Larsen, R. D. Mountain, and R. Zwanzig, J. Chem.

Phys. 42, 2187 (1965).

[4] R. W. Rowden and O. K. Rice, J. Chem. Phys. 19, 1423

(1951); Changements de Phases, Comptes rendus de la 2e Reunion Annuelle tenue en commun avec la Commission de Thermodynamique de 1' Union Internationale de Physique, Paris, 1952, p. 78.

[5] G. H. Gilmer, W. Gilmore, J. Huang, and W. W. Webb,

Phys. Rev. Letters 14, 491 (1965).

[6] S. Y. Larsen and J. M. H. Levelt Sengers, Third Symposiun

on Thermophysical Properties, A.S.M.E., Purdue, 1965,

p. 74.

[7] S. A. Ulybin and S. P. Malyshenko, Third Symposium on Thermophysical Properties, A.S.M.E., Purdue, 1965,

p. 68.

26
CRITICAL PHENOMENA IN FERRO- AND

ANTIFERROMAGNETS

Chairman: H. B. Callen

27

Critical Properties of Lattice Models

C. Domb

University of London King's College, London, Great Britain

Introductory Remarks

I shall assume that you all know what is meant

by the Ising and Heisenberg models and are rea- sonably familiar with Onsager's exact solution

of the two-dimensional Ising model [l] 1. My

task is to discuss the three-dimensional model,

and the effect of changing the spin and range of

interaction, and for these problems exact methods

have so far proved to be of no avail. Some people

have no enthusiasm for methods which are not

exact and consider that detailed investigation of

higher order approximation terms is not the task

of a theoretical physicist. I am reminded of a conversation which I had with Yvon about his own

important contribution to this field (which I shall

refer to later). He told me that he developed the

method, tried it in lower order, and found that

it gave the same result as other approximations.

I asked if he had not been interested to go to higher

orders to see whether it gave any improvement.

He replied: "my task as a theoretical physicist is to develop methods, and see that they work and

are correct: I then hand over to the engineers."

By these standards, what I have to say to you this afternoon, is engineering, but at least I will spare

you the technical details of the engineering methods

and discuss only the results. Let us also remember

that engineering methods led to the discovery of

Fourier series, and I hope to terminate my talk with some remarks about the possibility of an engineering approach to an exact solution. The methods I shall discuss depend on developing large numbers of terms of series expansions

at high and low temperatures which are used to estimate critical behavior. Now any second-year

mathematics student will tell you that nothing can

be learned about the singularities of a function

from a finite number of terms of a series expansion:

for example, even if the first 15 terms of an expansion are positive and well-behaved, there is nothing

to ensure that an effect which is small and unnoticed

in the first few terms will not diminate asymptotically (and I shall later quote an example of such

behavior for the Heisenberg model), but the fol- lowing physical considerations enable one, nevertheless, in many cases, to make confident predictions about the singularities of functions from a

finite number of terms of series expansions.

(a) The two-dimensional solution can act as a guide since we expect the three-dimensional solu- tion to be similar to it in essential features.

(b) We have a few exact theorems which apply

to three-dimensional solutions.

(c) Physical considerations of what is expected*

of the model can lead us to transformations of the

series which reveal peculiarities of behavior.

We must be sure at the outset that we have

enough terms of our series to get over small number effects. No general rule can be given since the

number of terms needed varies from problem to problem. However, careful analysis of the data

for each particular problem usually shows if the

number of terms derived is adequate.

Before going into details of the behavior of the

Ising and Heisenberg models, I should make the

connection with this morning's lectures on liquids

and gases. Fortunately, Professor Fisher has made

my task very easy by circulating a sheet giving the

detailed connection between the Ising model and

the lattice gas. I should like, first, to point out

that two quite different lattice models are used in connection with fluids. The first, with which we

are concerned here, is a model of liquid vapor

equilibrium, and it corresponds to the Ising model

of a ferromagnet [2]. From the relations given by

Professor Fisher, I shall draw attention to three

which are of relevance to my talk. In critical be- havior C\ for the fluid corresponds to Ch for the

magnet in zero field at all temperatures: the density

of one component of the fluid corresponds to minus

the spontaneous magnetization: and the isothermal

compressibility of the fluid corresponds to the iso- thermal susceptibility of the magnet. It is impor29
tant, also, to note that the chemical potential for the fluid is proportional to minus the magnetic

field. The second model applies to solid-fluid equilibrium and corresponds to the Ising model of an anti- ferromagnet [3]. This model has been extensively

used as an analog of the condensation of a fluid of

hard spheres [4], and will not be discussed further

in the present talk.

Short range forces; Series expansions

May I start with a few elementary observations

about power series expansions. If all the terms

of a power series are positive, the dominant singularity is on the positive real axis [5]. If they alter- nate regularly the dominant singularity is on the

negative real axis. If they manifest more complicated behavior, the dominant singularities are elsewhere in the complex plane. Physical interest centers largely on singularities on the real axis. For the Ising and Heisenberg models in zero field, we expect to find a positive singularity on the real

axis corresponding to the Curie point, and possibly a singularity on the negative real axis cor- responding to the Neel point.

It is convenient to divide the series expansions

to be considered into three groups. Group I have all terms positive and from table 1 we select as an example the coefficients in the high temperaTable 1. Coefficients a n in the susceptibility expansion for a number of lattices

Type of lattice

Simple Face-centered n /Til Onl'oi'lP tjudurciiic cubic

Triangular Simple

cubic

0 i 1 1 1

1 4 6 6 12

2 12 30 30 132

3 36 138 150 1,404

4 100 606 726 14,652

5 276 2,586 3,510 151,116

6 740 10,818 16,710 1,546,332

7 1,972 44,574 79,494 15,734,460

8 5,172 181,542 375,174 139,425,580

9 13,492 732,678 1,769,686

10 34,876 2,935,218 8,306,862

11 89,764 11,687,202 38,975,286

12 229,628 46,296,210

13 585,508

14 1,486,308

15 3,763,460

16 9,497,380

30
ture series for the susceptibility of the Ising model

for triangular and F.C.C. lattices. These lattices are chosen because the absence of antiferromagnetic ordering leads one to expect a single clear and dominant singularity. The general behavior

of the terms suggests comparison with a binomial

expansion of the form

(1)

and we therefore investigate the ratio of successive

terms a„/fl„-i as a function of —. For a binomial n

expansion (1) we should have

a„lan -\ H,

1 + (2)

Figure 1 shows that the terms converge rapidly and

smoothly; for the triangular lattice the estimates

of wc and the slope are very close to the correct

values [6] 2- V3 and 3/4. For the F.C.C. lattice a good estimate can be obtained of wc and one can

u

5 conjecture that y ——.

Passing now to "even" lattices like the S.C.,

B.C.C. and Diamond, we again find the same value

for y. However, now because of the symmetry

between ferromagnetism and antiferromagnetism, we expect a corresponding singularity on the negative real axis which is much weaker and therefore

masked by the positive singularity. We should

therefore try to look for a function of the form

where the second term represents the antiferro- magnetic singularity. In order to focus attention on this second factor, Fisher and Sykes [7] took

In Xo, subtracted — - In 4 w

Wc and obtained a

well-behaved series with terms alternating in sign.

After some ingenious manipulation they concluded

that the second factor in (3) could be represented by

C

i+D[l +

H In 11 +-^

(4)

The form of the susceptibility at the Neel point is shown in figure 2; it is characterized by a maximum

above the Neel temperature and a logarithmically

infinite slope at the Neel temperature.

The second example in this group is the high

temperature expansions for the specific heat of

the Ising model for the triangular and F.C.C.

lattices [8]. The terms are again all positive but

converge more slowly (eq. (5)).

qa„-i face-centred cubic

triangular

l/n

Figure 1. Ising model. [33]

Successive ratios in the susceptibility expansions of the triangular and face-centered

cubic lattices as functions of l/n. tq = Coordination number of lattice = number of

nearest neighbors ot each atom.)

kT/q J

FIGURE 2. Comparison of the antiferromagnetic susceptibility of (a) the simple cubic lattice (q = 6) and (b) the plane honeycomb lattice (q = 3). [7] The critical point is indicated by a small circle.

31
ChIR = 0.06257*2(1 + 0.8169? + 0.6778;2 + 0.5680i3 + 0.4818?4 + 0.4190r5 + 0.3725r6 + 0.3364r7 +. . . .) (F.C.C.)

(5)

ChIR = 0.2263f2 (l + 1.0986* + 0.8298;2 + 0.5525i3 + 0.402k4 + 0.3384?5 + 0.297k6 + 0.2614r7 +. . . .) (Triangular).

It is clear from the analysis shown in table 2 that not enough terms are yet available to draw definite conclusions; the slope for the three-dimensional

lattice seems to be sharper than a logarithm cor- responding to the two-dimensional lattice, and a

figure of (l

YJ

could be regarded as reasonably consistent with the data. Table 2a shows an

extension of the triangular lattice (calculated by

D. L. Hunter) and it will be seen that for values of n greater than 11 the series settles down to steady

behavior; a linear extrapolation would yield a value

of a close to zero. The same steady behavior seems to set in earlier (ra = 5) in the three dimensional case, and this increases confidence in the

above estimate of critical behavior.

It is interesting to ask why the convergence to the asymptotic value of the specific heat coefficients

is so much slower than for the susceptibility. I think that this can best be understood by comparison with an analogous problem in the lattice model of polymer configurations. If c„ represents

Table 2a. Triangular lattice

n h

O8 0.117270

a

9 .122611

1 A10 1 OOA/1 o .122048

"1 "1 11 1 1 O AAA .113000

1 O12 1 A/1 TAC .104705

13 AAOCC7 .09858

/

1 A 14 AAOOOO .09ooo8

15

AAA A A/; .088486

16 AO A AC

C

.084055

17 .080042

18 .076592

1 a

19 .073056

20 .069993

on21 .067174

22 .064570

23 A/oi CO .062158

O/I 24 ACAAl

O

.059918

oc25 AC7O0O .05 /832

OA26 AC COOA .055886

(17 2/ AC/I AAC .054065

28 .052358

29 .050755

30 .049246

Table 2

n h (triangular) h (F.C.C.)

1 1.0986 0.8169

2 0.5106 .6594

3 -.0025 .5140

4 -.0887 .3930

5 .2079 .3483

6 .2676 .3341

7 .1599 .3215

the number of simple chains of n links on a lattice and u» the number of simple closed polygons, the' series Xc„:r" and %unx

n

, both have the same radius

of convergence; but the cn reach their asymptotic

value much more rapidly than the «„, as can be

seen from figure 3. The most important configurations contributing to the susceptibility are "chain-like," and those contributing to specific!

heat are "polygon-like."

Group II refers to series not consistent in sign; in which the singularity of physical interest is masked by a spurious singularity. Examples of

this group are given in table 3 of the coefficients'

in the low temperature expansion for the spon-|

taneous magnetization. Only a real positive!

singularity is of physical significance here and it! can be assumed that this singularity has beenj

32
tion of the type

FIGURE 3. Excluded volume problem. [33]

Asymptotic limits of c„ and u„ for self-avoiding walks.

located by the high temperature susceptibility

series. The triangular and S.C. lattices give rise

to a regular alternation in sign, and the dominant

singularity is therefore on the negative real axis:

for the F.C.C. lattice a more complicated distri- bution in the complex plane is indicated. We know

the exact solution for the triangular lattice [6], of

the form

(1 - 3uyl*(l + u?1 * a + 3 U yi«a-uf*' (6)

? We might therefore reasonably try for the spons taneous magnetization of three-dimensional latc tices a function of the form

V(u) = E(u- ucf{u -mfiu- u»f* (7)

It is for this type of series that the Pade approxi- mant. introduced into this field by G. A. Baker

[9], is of prime importance. The [N, M] approximant is defined by

flu)

_ P(u) =

J.p r u' (degree M)

Q(u)~2q,W (degree AO' (8)

The [N, N] approximant is the most useful since

it is invariant under any homographic transformau = A<oll + B<a), (9)

and we should expect it to be at least as useful in improving the convergence as the best such homographic transformation. Clearly, for a function of

the type iMu) it is best to approximate to -7- In Mu),

clu and the zeros of the denominator then locate the

singularities uc , u.\, u > . . . . Our interest is focused on B. and it is best to assume the value of uc from

the high temperature susceptibility expansion and

approximate to fi by

[(« — u c ) *M")]- (10)

There are a number of devices which can be used

to improve the estimate of /3. After careful examination it has been conjectured that B =— for three

16 dimensional lattices. For series of Group II we also find examples of

slow convergence, and these are illustrated in table

4, which gives the low temperature susceptibility

expansion for three-dimensional lattices [10]. The

Pade approximant technique has again been applied

assuming a function of the form (7), but even though

a substantial number of terms are available, it will be seen from the analysis in table 5 that the limits of the index y'(xo ~ A'(u — u,)y

) are still between

1.27 and 1.37. Similarly, the specific heat at low temperatures

[11] converges slowly and present data are consist- ent with logarithmic behavior

Ch ~ BYn \TC — T), (11)

but are not sufficiently convergent to draw reliable conclusions.

These last two quantities are of particular interest

because of thermodynamic inequalities derived

by Rushbrooke and Griffiths [12] which estimates

of the indices fail to satisfy (this aspect will be

discussed in more detail by Professor Fisher).

The best hope of obtaining more reliable infor- mation would seem to lie with the development of more terms for the diamond lattice [13] since

they belong to Group I. Even with the powerful

help of the Pade approximant. it is still true to say

33
Table 3. Coefficients bn in expansion of {d<j>(t. T)ldr}ofor large t

ft

Type of lattice

Facecciiicx cu Simple cubic

quadratic

Triangular Simple cubic

0 8 12 12 24

1 34 -2 -14 -26

2 152 78 90 0

3 714 -24 -192 0

4 3,472 548 792 48

5 17,318 -228 -2,148 252

6 88,048 4,050 7,716 -720

7 454,378 - 2,030 -23,262 /inn 438

8 2,373,048 30,960 79,512 192

9 1 9 51 5 634 — 1 7 670 — 95? 054 984

10 66 01 ft 242,402 Oti U,UZ.O 1,008

1

1

L 1 — 1 ^9 ^90 — 9 7^3 ^90 — 1 9 994

12 1 Q39 000 0 90^ 800 19,536

13 — 1 312 844 — 30 371 124 - 3,062

14 15,612,150 8,280

15

16

17

-11,297,052 — 26,694 - 153,536

507,948

that series of Group I provide the most reliable information. (Of course, the Pade approximants can be used to good effect on these series as well.)

This program is being currently pursued.

Group III consists of series in which there is unexpected asymptotic behavior of the coefficients

which does not manifest itself for many terms. Table 6 shows the coefficients in the high temperature susceptibility series for the Heisenberg model

of the B.C.C. lattice [14]. If they are analyzed

in the manner of Group I, they lead to a good

1 Kor subsequenl work (spin Vj) see Baker. Gilbert. Eve. and Rushbrooke. Phvs

Letters 20. 146 (1%6I.

34

estimate of Curie temperature and a susceptibility

index of 4/3, 1

[Xo~C'(7V-D-4/3

] (12)!

(and this index is given more clearly by the F.C.C.

!

lattice [15]). However, physical considerations lead us to | think that there is a corresponding antiferromag- ! netic singularity which is no longer symmetric j with the ferromagnetic singularity. The position of

this Neel point was located very strikingly by

Rushbrooke and Wood recently [16], who used the

staggered susceptibility, (i.e., susceptibility in a
Table 4

n

F.C.C. B.C.C. S.C.

q=\2 q = S q = 6

1 0 0 0

2 0 0 12

3 0 16 — 14

4 0 -18 135

5 24 0 -276

6 -26 252 1 520

7 0 -576 -4 056

o8 U

rirv

1 i (to

9 72 3 264 -54 392

10 378 - 12 468 213 522

11 -1 080 20 568 - 700 362

12 665 26 662 2 601 674

13 DO/1 384 — 215 568 — 8 836 812

14 1 968 528 576 31 925 046

15 2 016 - 164 616 -110 323 056

16 - 25 698 - 3 014 889

17 39 552 10 894 920

1 o18 — 3 872

19 20 880

20 - 65 727

21 -379 856

22 1 277 646

field which alternates in sign from one lattice point

to the next). They found that the Neel temperature

is 10 percent higher than the corresponding ferro- magnetic Curie temperature. If we take off the

ferromagnetic singularity (table 6) as for the Ising

model, we find clear evidence of this dominant

factor although insufficient terms are available to analyze the detailed critical behavior. Fisher

has advanced general theoretical arguments [17]

which lead us to expect a similar critical behavior

209-493 0-66—

4

to the Ising model, and if so. we can anticipate an

asymptotic form for the coefficients of the type

4 (9 7Q1 )" |(2.537)" + (-D" • (13) o n

The terms would then ultimately alternate in sign

but the first negative coefficient would not occur

until after about 60 terms.

35
Table 5. Pade approximants for y'

N F.C.C. B.C.C.

7=1 /=0 /=-l j=l 7 = 0

6 1.207 1.254 1.241 1.246 1.264

7 1.259 1.251 1.254 1.280 1.402 1.374

8 1.287 1 C% A A 1.244 1 C% A T 1.247 1.375 1.405 1.285

9 1.249 1.262 1.236 1.295 1.273 1.268

10 1.240 1.360 1.332 1.269 1.272 1.276

n 1.332 1.353 1.350 1.277 1.274 1.273

12 1.350 1.353 1.270 1.274

13 1.273 1.273

Table 6

n

Heisenberg

model

B.C.C. lattice Magnetic

susceptibility

X

d log xo (4/3) Delta dK a-KiKc)

1 4.00000 4 3.383236 + 0.616764

2 12.00000 8 8.584714 - .584714

3 34.66666 24 21.783088 + 2.216912

4 95.83333 52.6 55.272995 -2.606328

5 262.7000\) 154.16 140.251202 + 13.915465

6 708.04166 350.11663 355.8773 -5.760670

7 1893.28968 943.01682 903.0126 + 40.004

8 5012.10863 2189.44354 2291.329 -101.885

9 13235.59898 6111.4476 5814.018 + 297.43

Kc taken as 0.3941, inverse 2.5374.
Summary of Critical Behavior of

Nearest Neighbor Lattice Systems

We have already referred to the critical be- havior of the magnetic susceptibility of ferro- and antiferromagnets in the previous section.

Change in spin value from V2 to general s does not change this critical behavior either for the Ising or Heisenberg model [15]. Regarding the critical behavior of the spontaneous

magnetization, accurate information is available

only for the Ising model of spin V2 (see previous

section). A Green's function interpolation method

used by Tahir-Kheli [18] has suggested an index of

1 V3 for /3; this differs only slightly from the Ising

value of 5/i6. Critical values of thermodynamic functions,

particularly entropy, provide insight into the magnitude of the "tail" of the specific heat curve. For

a model of spin V2 the total entropy change is k\n2 per system. With the Ising interaction in two- dimensions the tail is large and corresponds to more than 50 percent of the entropy; in three- dimensions the tail is very much reduced and

corresponds to about 15 percent. For the three- dimensional Heisenberg model, the tail is increased

by a factor of more than two and corresponds to about 35 percent of the entropy [19]. When we change from spin V2 to general 5, the

total entropy changes from Aln2 per system to £ln(2s+l). However, most of the change takes

place below the Curie temperature and even in going from 5 = 1/2 to s = °°, the increase in entropy

above the Curie temperature is less than 30 percent.

This can readily be understood since the high temperature portion is derived from averages which

are not likely to be very sensitive to change in spin

value. However, the behavior at low temperatures

is determined by the excitation spectrum which is extremely sensitive to the form of interaction.

We have referred to the critical behavior of the

specific heat in the previous section. It is worth

pointing out that this critical behavior is more

dependent on the form of interaction than that of

the magnetic susceptibility. Although only limited

numbers of terms are available, they indicate an

appreciable sharpening on the high temperature

side as s changes from V2 to infinity; similarly,

there are marked differences between the Ising and

Heisenberg interactions [15]. Attention has recently been focused on the behavior of the magnetization as a function of field

at the Curie point. For the Ising model the index

S {H = Mh

) is about 15 in two dimensions and about

5.2 in three dimensions [20]. Data for the Heisenberg model are available but have not yet been

put into a convenient form for detailed analysis.

Critical properties of the correlations are being

discussed in detail by Professor Fisher.

Longer Range Forces

The precision of our knowledge of the effect of longer range forces has increased appreciably during the past few years. The idea had often been

expressed intuitively that if the number of neighbors

participating equally in the interaction tended to

infinity the mean field or Bragg-Williams approximation would result exactly. This has now been

established rigorously [21] by taking an interaction

of the form eJe~(R and allowing e to tend to zero; the

result is valid even for a one-dimensional model.

A solution of Bethe type including a small residual

specific heat above Tc results if the force is long

range in one direction and short range in the other

[22]. It is significant to observe this physical inter- pretation of the Bethe result which had previously

been regarded as a first approximation to the solu- tion for a nearest neighbor model, or the exact

solution for an unphysical "pseudo-lattice" containing no closed circuits [23].

It is interesting to see how the mean field solution

is approached as the range of force increases. For

this purpose it has been found convenient to intro- duce an "equivalent neighbor (r shell)" model in which all interactions are equal up to the rth neighbor shell, and zero outside. The series expansion

methods described in the previous section can be

applied to this model, and Dalton and I have been

able to take account of the second and third neighbor shells for a number of two and three dimensional lattices [24]. We find that the critical behavior (e.g., susceptibility index) is identical with

that of a nearest neighbor model.

However, we noticed empirically that for large q simple asymptotic formulae fit the Curie temperature and thermodynamic functions. For example,

and (14) —k~= Llq -

37
Here H and L are constants which depend on the

dimension and the type of interaction. Although

this is the kind of asymptotic formula given by

closed form approximations, the numerical values

of H and L which we obtained were quite different from the closed form values. We then tried to find a justification for this type

of limiting form. Series expansions for an equivalent neighbor model can be expressed in terms of

"lattice constants" which are the number of in- dependent multiply connected graphs of various

kinds which can be formed from the interacting

bonds on the lattice. We realized that for a range

of interaction which included many lattice points,

these lattice constants could be regarded as lattice sum approximations to hard sphere cluster in- tegrals, many of which have already been evalu-1 ated. The efficacy of this approximation can be

judged from table 7 which gives typical estimates,

and these are compared with the correct valuesi based on exact enumerations.

We can thus readily see why the constants H, L differ for different dimensions since the cor

TABLE 7. Asymptotic formula from cluster integrals

Three Dimensions

p5n = 0.05903(g + D

3 - 0.8360 q + l

2 + 3q + 1.25

Q p5 n(calc.) p5 „(exact) p5 r/(calc.)

p5rt(exact)

6

12

18

26

42

-0.46

25.7

158.3

637.8

3274.8

0

36

174,186

630,690

3,318

0.71

.88

.97

.99

p5 = 0.02381 <?+l4 - 0.3963 g+l3 + 3.1719 9 +l2 -8g-3.6

/ t \ , ^ p5(calc.)

q p5 (calc.) p5 (exact)

p5 (exact)

6

12

18

26

42

25.1

245.8

1382.2

6954.0

55418.3

0

168

1212

6852, 6276

54 564

1.5

1.14

1.06 (mean)

1.02

38
responding cluster integrals will then have different

; values. The Bethe approximation gives the correct

limiting value as the dimension becomes large, and

| not as the coordination number becomes large in

' a given dimension.

Using the above approximation based on cluster

integrals, we find that the simple polygons dominate

I for the early terms, and these can be calculated

I to any order using the methods of Montroll and

Mayer. The approximation must break down for higher order terms since it does not give correct

critical behavior for magnetic susceptibility, spej cific heat, etc. However, it can give valid approximations for the Curie point and critical values of thermodynamic functions since higher order terms

make only a small contribution to these. The

I approach to the mean field limit (linear inverse

susceptibility, finite and discontinuous specific

heat, etc.) occurs in a singular manner, and this is illustrated diagramatically for the specific heat

in figure 4. In the light of the above discussion, it is constructive to consider the effect of a 1/r'' force for varying p and this is being currently investigated

at King's College by one of my research students,

G. S. Joyce. It is much more difficult to derive

series expansions for such a law of force since all neighbors must be taken into account. A substantial simplification occurs for the Ising model

of spin V2 if the method of Yvon (see next section)

is used to eliminate all but multiply connected configurations. Joyce finds that a sufficiently small

value of p can change critical behavior, and produce

critical effects, even in one dimension.

LONG RANGE FORCE FINITE CUTOFF

Figure 4. Diagrammatic illustration of the difference in specific heat curve between a finite range force and a force extending to infinity sufficiently slowly.

For a finite cutoff the specific heat is always infinite although the residual entropy

tends to zero as the range becomes large.

For example, taking the interaction between i-j pairs as J/rjj in one dimension (assuming unit lat- tice spacing), the series expansion of magnetic

suceptibility is

kTXo. m

1 + 3.290 £/ + 8.659 (pjf + 20.686 (#/

)

46.411 (/3J)

4 + 99.699 (/3J ) 5 + 207.338 ((3J)6

. (15)

This leads us to a Curie temperature

^0.5ix£

J «j (16)

and a susceptibility index of about three. With an

interaction in two dimensions the corresponding susceptibility expansion is

rn1 + 9.033622 (#/) + 76.94741 (/3J)

2

+ 642.168 (/8J)

3 + 5302.1 (£/)4 + 43508 (/3J

)

5 + . (17)

and the susceptibility index is reduced from V4 for a short range force to about 1

• 13. This change in susceptibility index is demonstrated in figure 5. Joyce has also investigated the properties of the

spherical model in one dimension for long range

forces. Whilst one must be cautious in drawing

detailed conclusions about critical behavior from

Figure 5. Susceptibility of two-dimensional Ising model with

a long range 1/R3 law of force: {a) long range interaction,

{b) short range nearest and second neighbor interaction. [14]

(The final slope represents the susceptibility index.)

39
the spherical model, I think it can act as a general

guide to the influence of long range forces on

critical behavior. Joyce finds that there is a crit- ical point in one dimension when 1 < p < 2. When

3 3

1 < p < — the susceptibility index is 1 and when —

<p<2 it is ^—-, so that it is equal to 3 when

z p

p=1.75. Corresponding results can be derived

in two and three dimensions.

For the general spin s and for the Heisenberg

model, the calculations are more complicated since

all connected diagrams must be taken into account. However, if the spin is allowed to become infinite in the Heisenberg model (classical vector model),

the conditions of applicability of the second Mayer

theorem are satisfied, and only multiply connected

clusters enter. The expansions can then be pushed

much further, and this simplification is being cur- rently exploited.

Diagram Expansions; Approach to the

Exact Solution

Early work using closed form approximations

provided no reliable information on detailed be- havior near the critical point. One of the major

defects of this work was the inability to develop

a series of successive approximations which would

tend in the limit to the exact solution. Even when

the methods introduced could in principle produce

such a series of approximations, the practical cal- culations were prohibitive.

The cluster integral development of Mayer can

be adapted to the Ising model [25], and can be made

to furnish a steady series of approximations based

on all multiply connected graphs which can be

constructed from bonds of the lattice. The Ath

stage of the approximation could correspond for example, to including all graphs of A lines where k

is allowed to increase steadily. It was Yvon [26]

who showed how to take all graphs into account,

and not merely regular graphs in which vertices

and bonds are equivalent. His method involved

using the Mayer theory of multicomponent mixtures.

Hiley and I [27] investigated the critical behavior

of such a series of approximations to see how they

approached the exact solution as estimated from

series expansions. To focus attention on the susceptibility index, for example, we calculated

T

1-8-

1 0 1-5 20 T* 2-5

Figure 6. Plot of y(T*) against T*(=T/TC ) for the triangular | lattice. [27]

For each successive approximation y passes through a maximum and then falls

j to unity. The locus of the maxima extrapolates to a value 1.75 for T*=l. Thefl

limiting curve is derived from series expansions.

which should tend to 7/4 for a two dimensional and

5/4 for a three dimensional model as T—» Tc . Figure

6 shows the results for a two dimensional model.

For each stage of the closed form approximation

y(T) rises to a maximum and then falls to the mear

field value of 1. For temperatures nearer to T, than this maximum, the approximation cannot be

regarded as reliable. The locus of maxima tends

to 7/4, and we see that the approach of the closed

form approximations to the exact solution is nonuniform.

We may follow an alternative approach classi- fying graphs into groups of increasing complexity

and summing each group to infinity. Let us

look at the high temperature partition function in zero field which can be written in the simple

form [28] aN

In Z.v = /V In 2 +

1— In (cosh K) + ^ pHXf>M

(19

K = JlkT w = tanhK

40
where the p„x are the "lattice constants" repre- senting numbers of multiply connected graphs and

the fnxiw) can be determined quite simply. Confining attention first to simple polygons which

provide the dominant contribution, we obtain the

function S/J„ln(l + w"). Sykes [29] has suggested

that for three dimensional lattices

p„=^(A~ 1.75) (20)

arises for a single self-avoiding walk; for more

complex graphs the mutual interference of two

or more self-avoiding walks must be considered.

Even if this difficult empirical investigation is completely satisfactory the theory could only be con- sidered respectable if the asymptotic estimates

are established rigorously. But this at least is a

far less formidable problem than the Ising problem itself, and Professor Edwards has taken an

important step recently towards its solution [32].

where /jl is the self-avoiding walk limit. Hence, if we ignore all configurations other than simple

polygons, we will obtain a specific heat singularity of the form (T—Tc )~ l/4

, and an estimate of

the Curie point which differs from the correct value

by about 2V2 percent (the close connection between

I [i and the Ising critical value was pointed out several years ago by Fisher and Sykes [30]). To make further progress we must introduce

a suitable topological classification of multiply

connected graphs, and here Sykes [31] has sug- gested using the cyclomatic number c (number of

bonds — number of vertices + 1); the types of graph

corresponding to c= 1, 2. 3. and 4 are shown in figure 7. Even here the number of types increases

quite rapidly, but can be substantially reduced by

ignoring in the first instance multiple connections

I between two vertices (thus p. 0, 8, and Q can also be considered basically as one type).

1 One must then consider the asymptotic numbers

1 of each type of graph to understand how the as- ymptotic value of the coefficients in the high tem1 perature series is constituted. The constant /jl

G H l J K L

M N O P Q

Figure 7.

Helpful discussion with colleagues and research

students mentioned in the text is gratefully ac- knowledged. The remarks at the end of the final section were stimulated by a series of lectures given

at King's College recently by Dr. R. Brout.

References

[1] L. Onsager, Phys. Rev. 65, 117 (1944).

[2] C. N. Yang and T. D. Lee. Phys. Rev. 87, 404. 410 (1952).

[3] C. Domb, Phil. Mag. 42, 1316 (1951).

[4] C. Domb. II Nuovo Cimento Supplement to Vol. 9, 9 (1958); D. M. Burley, Proc. Phys. Soc. 77, 262 (1960); D. M.

Gaunt and M. E. Fisher, J. Chem. Phys. 43,2840(1965).

[5] P. Dienes, The Taylor Series Oxford 1931. ch. 14.

[6] C. Domb, Advan. in Physics 9, 149, 245 (1960).

7] M. F. Sykes and M. E. Fisher, Physica 28, 919, 939 (1962).

[8] C. Domb and M. F. Sykes, Phys. Rev. 108, 1415 (1957).

[9] G. A. Baker, Phys. Rev. 124, 768 (1961).

[10] J. W. Essam and M. E. Fisher, J. Chem. Phys. 38, 802 ( 1963).

[11] D. S. Gaunt and J. W. Essam, Proc. International Conference

on Magnetism, Nottingham 1964, p. 88.

[12] G. S. Rushbrooke, J. Chem. Phys. 39, 842 (1963); R. B.

Griffiths, Phys. Rev. Letters 14, 623 (1965).

[13] J. W. Essam and M. F. Sykes, Physica 29, 378 (1963).

[14] C. Domb and D. W. Wood, Proc. Phys. Soc. 86, 1 (1965). C. Domb, N. W. Dalton, G. S. Joyce and D. W. Wood,

Proc. International Conference on Magnetism, Nottingham 1964, p. 85.

[15] C. Domb and M. F. Sykes, Phys. Rev. 128, 168 (1962);

J. Gammel, W. Marshall and L. Morgan, Proc. Roy. Soc. A275, 257 (1963).

[16] G. S. Rushbrooke and P. J. Wood, Mol. Phys. 6, 409 (1963).

[17] M. E. Fisher, Phil. Mag. 7, 1731 (1962).

[18] R. A. Tahir Kheli, Phys. Rev. 132, 689 (1963).

[19] C. Domb and A. R. Miedema, Progress in Low Temperature

Physics, IV (1963), p. 296.

[20] D. S. Gaunt, M. E. Fisher, M. F. Sykes and J. W. Essam,

Phys. Rev. Letters 13, 713 (1964).

[21] G. A. Baker, Phys. Rev. 126, 2071 (1962); M. Kac and E. Helfand, J. Math. Phys. 4, 1078 (1963).

[22] G. A. Baker, Phys. Rev. 130, 1406 (1963).

[23] M. Kurata, R. Kikuchi and T. Watari. J. Chem. Phys. 21,

434 (1953); see also ref. 6, p. 284.

[24] N. W. Dalton, Thesis, London (1965); see also ref. 14. [25] G. S. Rushbrooke and I. Scoins, Proc. Roy. Soc. 230, 74

(1955).

[26] J. Yvon, Cah. Phys. No. 28 (1945), Nos. 31, 32 (1948).

[27] C. Domb and B. J. Hiley, Proc. Roy. Soc. 268, 506 (1962).

[28] Reference 6, p. 322, eq (59). [29] B. J. Hilev and M. F. Sykes, J. Chem. Phys. 34, 1531 (1961).

[30] M. E. Fisher and M. F. Sykes, Phys. Rev. 114, 45 (1959).

[31] M. F. Sykes, private communication. To be submitted

to J. Math. Phys.

[32] S. F. Edwards, This Conference.

[33] C. Domb. J. Royal Stat. Soc. London B26, 373 (1964).

41
Equilibrium Properties of Ferromagnets and Antiferromagnets

in the Vicinity of the Critical Point*

G. B. Benedek

Massachusetts Institute of Technology, Cambridge, Mass.

Within the past few years there has been a considerable advance both theoretically and experimentally in our knowledge of the behavior of magnetic systems in the vicinity of the critical temperature. The present paper consists of a presentation of some of the new experimental in- formation and its connection with the new theoretical predictions.

Some of the equilibrium properties of magnetic

systems which are important in the description of a ferromagnet in the critical region are the following:

1. The spontaneous magnetization versus temperature in zero external field,

2. the magnetic susceptibility (dA//d//)w=0 = x.

3. the magnetization versus field at the critical temperature Tc , 4. the specific heat at constant volume Cv , and

5. the thermal expansion coefficient. In the discussion which follows, we shall present

the experimental data and theoretical predictions now available on each of the properties listed above.

In view of the fact that each of these properties

has an analog in the liquid-gas system, as has been

emphasized by Fisher [1] and his coworkers [2], we shall make the connection between each magnetic property and its fluid analog.

1. The spontaneous magnetization. In figure

1 we present a schematic drawing of the temperature dependence of the spontaneous magnetization

M(T) of a ferromagnet normalized to its value at r=0, (M(0)), in zero external field (H = 0). Along

side this we present the fluid analog, namely the

temperature dependence of the liquid-gas density

along the coexistence curve. In the critical region

the magnetization and orthobaric density data can

be fit to equations of the form:

FERROMAGNET: SPONTANEOUS MAGNETIZATION FLUID: COEXISTENCE CURVE

P

Mill

M(O) , H = 0

(a)

FIGURE 1. (a) A schematic plot of the temperature dependence

of the magnetization of a ferromagnet. (b) The fluid analog,

the liquid and gas densities along the coexistence curve.

*D (l \M(0)) H=0 U V T

2Pc D' 1 —

(1)

(2)

This research was supported bv the Advanced Research Projects Agency under

Contract SD 90.

The first experimental measurements of D and (3 were reported in 1962 by P. Heller and G. Benedek'

[3] through a study of the nuclear resonance of' the F

19 nucleus in the antiferromagnet MnF2 . In table 1 we show the temperature range AT/T( over which (1) was found to fit the data, and the

corresponding values of D and /3. Equation (l)lj holds in MnF> over a temperature range 7 X 10~5

=S (AT/TC) < 0.1. The closest point measured! was about 3 mdeg from the critical point. This \ data shows that 0 = 0.333 ± 0.003. This result is

1

in disagreement with molecular field theories which

give j8 = V2 in the limit T—> Tc . In view of the

marked disparity between this result and that of

the classical theories of magnetism, Heller and

Benedek [4] later carried out a careful investigation

of the insulating ferromagnet EuS. This crystal

42
Table 1. Experimental and theoretical values for the parameters describing the temperature dependence of the spontaneous

magnetization of a ferromagnet or antiferromagnet

(f) \

1 c / D

MnF2 0.10 <-> 7 x 10"5 1.200 ±0.004 0.333 ±0.003

EuS 0.08 ^lx 10"2 1.145 ±0.02 0.33 ±0.015

CrBr3 0.04-^ 0.7 x 10~ 2 1.32 ±0.07 0.365 ±0.015

Molecular field

theory

1.48 (S = 5/2)

1.44 (S = 7/2)

0.500

Ising model 1.49 0.3125

is cubic, the magnetic europium spins are localized on the lattice sites and each magnetic ion has zero

orbital angular momentum and S = 7/2. This

crystal represents the closest known physical

approximation to the ideal "Heisenberg'' ferromagi net. The results of this investigation are summarized in table 1. The one-third power law was

found to apply also to this system.

It is worth noting that the 1/3 power law starts

to fit the data on EuS and MnF2 when M/Tc ~ 9

percent. The magnetization at this point is about

1/2 the saturation value. Thus the 1/3 power law

applies to the magnetization data over a rather

wide range in M, namely 1/2 > (M(T)/M(0)) 2: 0. In row 3 of table 1 we also present new measurements of D, f3 and AT/TC made very recently by

S. D. Senturia at MIT on the ferromagnetic insulator CrBr.3. This material is particularly interest- ing from the theoretical point of view because the

ferromagnetic chromium spins are arranged in planes, and each plane is separated from the next

plane by a plane of nonmagnetic Br ions. As a

result, the coupling between the magnetic spins

is very anisotropic. The exchange constant for coupling in the plane is 16 times stronger than the constant for coupling between the planes. Thus,

this ferromagnet is almost two-dimensional, with

only a small three-dimensional admixture. Nevertheless, this system behaves somewhat like the

ideal three-dimensional Heisenberg ferromagnet

EuS in that /3 = 0.365. The critical region for CrBr3 sets in a A777V ~ 5 X 10"- at which point M(T)/M(0) ~ 0.41.

In table 1 we also present the theoretical predictions of the Ising [5] and molecular field models.

No theoretical prediction has yet appeared for the

Heisenberg model. It should also be mentioned

that H. Callen and E. Callen [6] have calculated

M(T) theoretically using a sophisticated cluster

approximation, and report that M(T) follows the 1/3

power law provided that T is not too near Tc. In the limit T—> Tc , of course, their cluster theory

goes over to the /3= 1/2 value.

As an example of the analogous behavior of a

fluid we point out the work of Weinberger and

Schneider [7, 1]. They found that eq (2) fits the

data on xenon for AT/TC as small as 3 X 10~5

, and

that /3' = 0.345 ±0.015 and D' — 1.61. The van

der Waals equation gives /3 = 1/2. In fact, Landau's

theory [8] of the critical region shows that /3 = 1/2

follows from the very general assumption that the

free energy can be expanded as a power series in density and temperature around the critical point.

43
Thus the breakdown of this law reflects a rather

unusual mathematical and physical character of the critical point.

From an historical standpoint it should be noted

that the departure of from the classical value for magnetic systems was discovered at about the same time that important theoretical advances

were being made. In 1961 and 1962 Domb and

Sykes [9, 10] were able to show from a study of

their high temperature expansions that the sus- ceptability of a ferromagnet in the Ising and Heisenberg models did not diverge as slowly as was

expected by the molecular field theories. Also,

Baker [11, 12] introduced at this time the method

of Pade approximants as an accurate and reliable means of extrapolating the high temperature ex- pansions into the critical region. This extrapolation showed [13] that the susceptibility as calculated on the Heisenberg and Ising Hamiltonians diverged more rapidly than that predicted by the molecular

field theories. These developments and the observation, which was stressed particularly by M. E. Fisher, of the close similarity in the behavior of

systems as disparate as the ferromagnet and the

fluid greatly contributed to the present expansion

of work in this field.

2. The susceptibility. In figure 2 we show the

temperature dependence of the isothermal susceptibility of a ferromagnet as T~* Tc . Along side

it is shown the temperature dependence of the

analogous property of the fluid, the isothermal bulk

modulus along the critical isochore. The susFERROMAGNET: SUSCEPTIBILTY FLUID: COMPRESSIBILTY

ceptibility of the ferromagnet is defined as

x = (dM(T,H)ldH)H=o. (3)

As one approaches Tc from the high temperature

or paramagnetic side, x diverges like

x = CI(\-TITc

)y. (4)

As one approaches Tc from the ferromagnetic side,

X can be fit to an equation of the same form as (4) but one must leave the possibility that the coefficient D and exponent y have different values

for T < Tc . Thus, on this side of Tc we write

x = C'ia-(TITc ))y\ (5)

In the fluid the analog is the isothermal compres

sibility (3t-

(6)

This can also be written in terms of the chemical

potential p, using the thermodynamic identity that along an isotherm

Thus

(7):

(8)

As T—>TC from above, the compressibility at p — pc diverges as follows:

(3r, Pc = dl{l-(TITc)f.

(T+ ->TC) (9)

Below Tc the i ^levant compressibility is that evaluated right at the coexistence line for either the

liquid or the gas

(a) (b)

FIGURE 2. (a) The temperature dependence of the isothermal

susceptibility of a ferromagnet as T-*TC. (b) The temperature dependence of the isothermal compressibility of a fluid evaluated at p = pc for T > Tc and at p = p, or p = pQ forT < Tc

. 44

fa. p e or P,, (T =

Tc) d'Kl- (T/TcW. ( 10)

We see that in the liquid case, the pressure, or the

chemical potential plays the same role as does the

magnetic field in the ferromagnetic case.
The first comparison between the new theoretical

predictions [9, 10. 13. 14] and experiments was

provided by the work of Jacrot [15] who determined

X through a study of the scattering of neutrons from iron as T—> T, from above. His measure- ments indicated that y=1.3. Stimulated by the

departure of y from the classical value y = I, Noakes

and Arrott [16] reexamined their older measure- ments of the susceptibility of iron for T> 7V, and

they also carried out new measurements of this quantity. They found y— 1.37 ±0.04. This result

is tabulated in table 2 where we give y for each

ferromagnet which has been studied up till now. We also give in column 2 the maximum range over which eq (4) actually fits the experimental data. The range is 0 < A777V < &Tu/Tc, and &T.u/Tc rep- resents the outer limit of the critical region insofar

as the susceptibility is concerned. In their paper

Noakes and Arrott pointed out that earlier data

by S. Arajs [17], when analyzed in terms of eq (4) also gave a nonclassical value for y. This observation was later confirmed by Arajs [18] who has

measured y from his studies on Fe, Co, and Ni. His important results on y and AT

7,,,/ 7V are given in table 2. Kouvel and Fisher also examined the

divergence in the susceptibility of nickel by going

back to the very early careful work of P. Weiss

and R. Forrer [20]. This gave y=? 1.35 ±0.02.

Graham [21] has studied the susceptibility of Gd

above the Curie point and concludes 7 = 1.3 but

does not give error limits for this estimate. Fur- thermore, his data indicates that y depends on whether the magnetic field is applied along the

c-axis or perpendicular to it. In table 2 we also list in the last three rows the

values of y as obtained theoretically using the

molecular field, Ising and Heisenberg models along

with literature references to these calculation.

Table 2. Experimental and theoretical values for the parameters describing the temperature dependence of the susceptibility of a ferromagnet as T—*TC from above

Ferromagnet (ArM/rc) y References

Fe 3 x 10"2 1.3

1.37 ±0.04

1.33 ±0.04

15, 13

16

18

Co 1 x 10-2 1.21 ±0.04 18

Ni 2 x 10-2 1.35 ±0.02

1.29 ±0.03

19

18

Gd 4 x 10-2 1.3 21

Theory

Molecular field 1.00

Ising 1.25 9, 11

Heisenberg 1.33 ±0.01 13, 14

45
It is worth stressing that we do not know of data on y' the parameter which describes the divergence

of x as T^> Tc from the ferromagnetic side of the

Curie point. In the case of a fluid, the classical theories give

r=1.0. Because of the experimental difficulties,

data on the isothermal compressibility appears to be scanty and insufficiently precise in the critical region. The measurements of Habgood and

Schneider [22] indicate only that [1] T > 1.1. The

Ising theory for the liquid predicts r=1.25.

3. The critical isotherm. The next equilibrium

property of a ferromagnet which is of particular

interest in the critical region is the dependence of

M on H at T— Tc . The divergence of the susceptibility as T^-Tc shows that M is an extremely

strong function of H at Tc . In figure 3 we show a schematic plot of M versus H at Tc . Along side

it we present the fluid analog, namely the dependence of (p — pc ) on the pressure for T=TC , i.e., the

shape of the critical p—p isotherm. The data on

both these systems may be fit to equations of the

following form:

M(TC , H) = AH1 ' S (11)

and

(p-pc )= A'{p-pc ) l i 6 ' = A'\tJL-^yi6' (12) r=7v

where p. is the chemical potential. The index 8

or 8' is a measure of the verticality of the critical isotherms as sketched schematically in figure 3, as these indices are measures of the lowest nonvanishFERROMAGNET

M(TC)

FLUID

<>P-Pc)

COEXISTENCE

CURVE f

(p-pc

)

(a) (b)

FIGURE 3. (a) The dependence of the spontaneous magnetization M on the magnetic field H at the Curie point (T =TC). (b) The shape of the critical isotherm of a fluid.

ing derivative of H (or pS) relative to M (or p). In magnetic systems 6 has been found for two materials, nickel and gadolinium. The information on

nickel was obtained from Weiss and Forrer's 1926

measurements [20] by Kouvel and Fisher [19]. They

found 8 = 4.22 ±0.04. Graham [21] reports that

his studies of gadolinium give 8 = 4, without placing error limits on this figure. The molecular field theory predicts 8 = 3 while the best present analysis of the Ising model in three dimensions strongly

indicates that the value 8 = 5.2 is the correct the- oretical value for this Hamiltonian. There are as

yet- no calculations of 8 using the Heisenberg Hamiltonian. These results are collected in table 3. In the fluid there is considerably more data [23] on the shape of the critical isotherm. The data on Xe, C02, and H2 indicates [2, 24] that for the

fluid 8' = 4.2 ±0.1. Once again we see the close

similarity in behavior between the ferromagnet

and the fluid. Theoretically the van der Waals

theory, which is the analog of the molecular field theory of ferromagnetism, predicts a cubic equation for the p — p critical isotherm, i.e., 8' = 3. The

Ising model as analyzed by Gaunt, Fisher, Sykes,

and Essam [2] gives 8' = 5.2. We tabulate these

results in table 4. 4. The specific heat. Of all the physical quantities that vary rapidly near the critical point, the I specific heat appears to diverge most delicately.

Onsager's theoretical treatment [25] of the two- dimensional Ising system indicates that the specific

heat diverges logarithmically as T~^TC . Such a slow divergence requires very high precision in the measurement of the specific heat, and accurate temperature control very near to the critical point.

A very careful study of the specific heat of a magnetic system has recently been carried out by Skalyo

and Friedberg [26] on the antiferromagnet CoCl • 6H2O. These workers find that as one approaches

Tc from above (+) or from below (—) that the specific

heat appears to diverge like cP

(T) R y/h=o a In L

Tc (13)

This behavior begins when (T—TC)ITC is of the

order of a few percent. However, in the immediate

vicinity of Tc {(T-TC)ITC ~-5X 10~3 and + 1 X lO""5 ) this divergence does not continue. Presumably

this is due to the presence of crystal strains and the

resulting coexistence of the antiferromagnetic and

paramagnetic phases.

46
Table 3. Experimental and theoretical values of parameters describing the shape of the critical isotherm for a ferromagnet

Ferromagnet 8 Reference

TVT * Ni 4.22 ±0.04 19

Gd 4 21

Theory

Molecular field 3.0

Ising 5.2 ±0.15 2

Table 4. Experimental and theoretical values ofparameters describing the shape

of the critical isotherm for a fluid

Fluid 8 References

C02 , H2 , Xe 4.2±0.1 2, 23, 24

Theory

van der Waals 3.0

Ising 5.2±0.15 2

In addition, Yamomoto, et al. [27] have recently

surveyed the literature for specific heat measure- ments on magnetic systems. They fit the experimental data to a logarithmic divergence of the form

of eq (13), and present a table listing the resulting

values of a and (A+ — A_) for 28 ferromagnetic and

antiferromagnetic elements and compounds.

It may be well to stress that while the data does

show that Cp does get large very slowly as T^TC

the range over which the logarithmic dependence

applies is - 5 X lO"3 < (TT

c)/Tc < 5 X 10"2

. Measurements examining the question of continu- ance of the logarithmic divergence closer to the

critical point are needed. Along these same lines

it should be pointed out that what is measured ex- perimentally is Cp and what is calculated theoretically is Cv. The difference is proportional to

cx

2//3r where a is the thermal expansion and /3t

47
the isothermal compressibility. It is known that the thermal expansion increases markedly as T—> Tc , but accurate data is not now available giving the precise behavior of a{T) and fi(T). In view of the slow divergence of Cp

it is particularly

necessary to determine the importance of the

Cp — Cv correction term before reaching conclusions about the precise mathematical nature of the divergence of Cv

. In the case of the fluid, while the van der Waals

theory predicts no divergence in Cv as Tc , the

experimental data has indicated [23] that Cv along

the critical isochore grows very large as T—* Tc . Recently the work of Voronel and his coworkers

[28, 29] has shown that the specific heat at constant volume in argon and oxygen apparently diverges

logarithmically as T-+ Tc . The data of these workers has recently been very carefully examined and

discussed in the light of the best present theoretical

information [30, 31].Conclusions

Since 1961-1962 a considerable amount of ex- perimental data on the equilibrium properties of

magnetic systems has been obtained. The behavior of each of these properties in the critical region is in striking and clear-cut disagreement

with the predictions of the "classical" molecular

field theories of ferromagnetism. The development of the method of Pade approximants, in combination with the high temperature expansions of

the partition functions have enabled calculations

of the equilibrium properties of the ferromagnet

using the Ising Hamiltonian. The results of such

calculations are closer to the data than the "classi- cal" theories, however, by and large the experimental results indicate that the Ising model is inadequate to describe a real ferromagnet. The

Heisenberg Hamiltonian has been used successfully to calculate the divergence of the susceptibility and the results are in very good agreement

with the experiment.

At this point in the development of the subject

certain directions for the future work seem clear. On the experimental side there is a need for much

more information particularly on the shape of the

critical isotherm, the specific heat, the thermal

expansion, and the susceptibility below Tc. Also, we have to have more data on the localized spin

systems rather than the ferromagnetic metals.

In the insulators we may expect that the Heisenberg Hamiltonian will apply, while the appropriate

Hamiltonian for the metals is not known. Also,

it would be most desirable to measure several

equilibrium properties on a single ferromagnet.

On the theoretical side, the clear need is a cal- culation of the magnetization, the critical isotherm,

the susceptibility below Tc and the specific heat

using the Heisenberg Hamiltonian. And finally,

the close analogy between the fluid and the ferro- magnet, combined with the clear breakdown of

the conventional expansions of the free energy

in a power series around Tc suggest that the be- havior of three-dimensional ferromagnets and fluids near the critical point may be a result of some very

general property connected with the ordering and

is relatively independent of the details of the

Hamiltonian of the system.

References

[1] M. E. Fisher, J. Math. Phys. 5, 944 (1964).

[2] D. Gaunt, M. E. Fisher, M. Sykes, and J. W. Essam, Phys.

Rev. Letters 13, 713 (1964).

[3] P. Heller and G. B. Benedek, Phys. Rev. Letters 8, 428

(1962).

[4] P. Heller and G. B. Benedek, Phys. Rev. Letters 14, 71

(1965).

[5] J. W. Essam and M. E. Fisher, J. Chem. Phys. 38, 802 (1963).

[6] H. Callen and E. Callen, Conference on Magnetism, J. Appl.

Phys. (1965), in press.

[7] M. A. Weinberger and Schneider, Can. J. Chem. 30, 4222

(1952).

[8] L. D. Landau and Lifshitz, Statistical Physics (AddisonWesley, 1958).

[9] C. Domb and M. F. Sykes, J. Math. Phys. 2, 63 (1961).

[10] C. Domb and M. F. Sykes, Phys. Rev. 128, 168 (1961).

[11] G. A. Baker. Jr., Phys. Rev. 124, 168 (1961).

[12] G. A. Baker, Jr., Phys. Rev. 129, 99 (1962).

[13] J. Gammel, W. Marshall and L. Morgan, Proc. Roy. Soc. (London), A275 (1963).

[14] G. A. Baker, Jr., Phys. Rev. 136, A1376 (1964).

[15] B. Jacrot, J. Konstantinovic, G. Parette, and D. Cribier, Symposium on Inelastic Scattering of Neutrons in Solids and Liquids, Chalk River (1962). (See also reference 13.) [16] J. E. Noakes and A. Arrott, J. Appl. Phys. 35, 931 (1964).

[17] S. Arajs and P. S. Miller, J. Appl. Phys. 31, 986 (1960).

[18] S. Arajs, Conference on Magnetism and Magnetic Materials,

J. Appl. Phys. (1965), in press.

[19] J. Kouvel and M. E. Fisher, Phys. Rev. 136, A1626 (1964).

[20] P. Weiss and R. Forrer, Ann. Phys. (Paris), 5, 153 (1926).

[21] G. D. Graham, Jr., Conference on Magnetism and Magnetic

Materials, J. Appl. Phys. (1964), in press.

[22] H. Habgood and W. Schneider, Can. J. Chem. 32, 98(1954).

[23] J. S. Rowlinson, Liquids and Liquid Mixtures, ch. 3, pp.

93ff. (Butterworth's (London), 1959).

[24] B. Widom and O. K. Rice, J. Chem. Phys 23, 1250 (1955).

[25] L. Onsager, Phys. Rev. 65, 117 (1944).

[26] J. Skalyo, Jr. and S. A. Friedberg, Phys. Rev. Letters 13,

133 (1964).

[27] T. Yamamoto, H. Matsuda, O. Tanimoto, and Y. Yasuda,

Department of Chemistry, Kyoto University, Kyoto, Japan.

(See Proceedings of Conference on Critical Phenomena, Washington, 1965.)

[28] M. Bagatskii, A. Voronel, and V. Gusak, Soviet Phys.

JETP 43, 728 (1962).

[29] A. Voronel, Y. Chashkin, V. Popov, and V. Sinkin, Soviet Phys. JETP 45, 828 (1963).

[30] M. E. Fisher, Phys. Rev. 136, A1599 (1964).

[31] C. N. Yang and C. P. Yang, Phys. Rev. Letters 13, 303

(1964). 48
Experimental Studies of Magnetic Ising Systems

Near the Critical Point*

W. P. Wolf

Yale University, New Haven, Conn.

The Ising model has long been used as a con- venient approximation for the investigation of magnetic order-disorder transistions, but until recently no real magnetic substances were known in which

the assumptions of the model coincided with reality.

The advent of many new rare earth compounds

has changed this position. There are now compounds in which it is an excellent approximation

to treat the magnetic ions as effective spins s' = 1/2,

whose magnetic moments /x = g-s' are constrained

to one direction by an extremely anisotropic g

tensor [1]. A simple material of this kind is dysprosium aluminum garnet (DAG) which orders

antiferromagnetically at about 2.5 °K. The properties of DAG in the critical region have been

investigated by a number of workers over the past

three years [2-9].

The principal difference between such a material

and the Ising model lies in the importance of long

range magnetic dipole forces, which account for a major fraction of the total interactions. Dipolar

forces lead to a shape dependence of all the magnetic properties (even the "zero field" susceptibility) and at present there is no theory better

than the molecular field to allow for these effects. With a molecular field calculated from suitable

lattice sums, it is possible to remove most of the

long range effects, and to reduce measured magnetic

properties to those corresponding approximately

to a material with nearest neighbor forces only [7]. This has been done for the susceptibility of DAG in the region of the critical point. It was found that the results could be fitted to law of the form pre- dicted by the Ising model theory [10]:

XT= A He - (1 - TITC ) log 1 1 - TITcW

,

with k equal to the known Curie constant, 10.35/mol

and £c , 5+, and B- respectively 0.565, 0.30, and 0.80.

'This work was supported in part by the U.S. Atomic Energy Commission. A full account will be published at a later time.

Tc — 2.493 °K is critical (Neel) temperature in zero

field. The values for £c , B+, and B- are not strictly

comparable with those given by any of the theoretical estimates since none of these have considered

the garnet structure explicitly, but it is interesting

to note that the ratio B--.B+ is close to the value 3

found for other three dimensional structures. The "reduced" susceptibility measurements

have also been used to compare Fisher's predicted

relation for the specific heat [11], C — 4 —(X^'

dl

with the observed values [7]. (A is a slowly varying

function of temperature whose value at Tc can be

estimated from other measurements.) Excellent

agreement was found over a range of temperatures

0.5 TC <T< 1.5 Tc . Critical effects at temperatures below Tc have also been investigated as a function of magnetic field

[4]. The most striking result is that the transition

from the antiferromagnetic state to the paramagnetic

state seems to be first order for T < ~ 0.6 Tc and higher order for —0.6 Tc <T<TC [12]. A

first order transition is not predicted by any of the

simple Ising models at any finite temperature, and

it is probably a result of the long range interactions.

Accompanying the change from antiferromagnetic

to paramagnetic order in an applied field there are magneto-caloric effects and these have been studied

by measuring adiabatic changes of temperature with

field [13]. In contrast to the isothermal transitions,

which are completely reversible, the adiabatic

changes showed a large and as yet unexplained

hysteresis.

Although most of the critical point measurements

on Ising-like systems have so far been concentrated on dysprosium aluminum garnet, there is reason to believe that other rare earth compounds will be

found which also approximate to an Ising model [14].

It seems rather certain that almost all materials

of this kind will only order at low temperatures,

where long-range magnetic dipole interactions are' important, and it would therefore be very helpful

49
if critical point theories could be extended to in- clude these effects. Magnetic dipole forces are

of course present also in all other materials, and as a matter of principle their effect should be considered even for materials in which exchange interactions are dominant.

It is a pleasure to acknowledge a number of

stimulating discussions with Professor Michael E. Fisher. I should also like to thank Dr. A. F. G. Wyatt, Dr. B. Schneider, Dr. B. E. Keen and Mr.

D. Landau for making available to me the results of their experiments prior to publication.

References

[1] See for example W. P. Wolf, M. Ball, M. T. Hatchings, M. J. M. Leask, and A. F. G. Wyatt, J. Phys. Soc. Japan

17, Suppl. B-l, 443 (1962).

[2] M. Ball, M. T. Hutchings, M. J. M. Leask, and W. P. Wolf,

Proc. 8th Int. Conf. on Low Temperature Physics, p. 248

(Butterworth, London, 1963).

[3] M. Ball, M. J M. Leask, W. P. Wolf, and A. F. G. Wyatt.

J. Appl. Phys. 34, 1104 (1963).

[4] M. Ball, W. P. Wolf, and A. F. G. Wyatt, Physics Letters 10, 7 (1964).

[5] A. F. G. Wyatt, Thesis, Oxford University, 1963.

[6] A. Herpin and P. Meriel, C. R. Acad. Sc. Paris 2 5 9, 2416

(1964).

[7] W. P. Wolf, and A. F. G. Wyatt, Phys. Rev. Letters 13, 368

(1964).

[8] J. M. Hastings, L. M. Corliss, and C. G. Windsor, PhysJ

Rev. 138, A176 (1965).

[9] A. H. Cooke, K. A. Gehring, M. J. M. Leask, D. Smith, and

H. M. Thornley. Phys. Rev. Letters 14, 685 (1965).

[10] See for example M. E. Fisher, and M. Sykes, Physica 28,

939(1962).

[11] M. E. Fisher, Phil. Mag. 7, 1731 (1962).

[12] The temperature at which the transition changes from being

first order has recently been determined with greater accuracy at (0.65±0.02)rr by B. Schneider and W. P. Wolf (to be published). Above 0.657V it is apparently

higher than second order.

[13] B. E. Keen, D. Landau and W. P. Wolf (to be published).

i

[14] Substances already known to have Ising-like magneticproperties include several rare earth ethyl sulphates

(A. H. Cooke, D. T. Edmonds, C. B. P. Finn, and W. P. Wolf, J. Phys. Soc. Japan 17, Suppl. B-l, 481 (1962)) and a number of the anhydrous rare earth chlorides

(J. C. Eisenstein, R. P. Hudson, and B. W. Mangum, Phys. Rev. 137, A1886 (1965). However all these materials order at such low temperatures that it has not yet been

possible to investigate their critical properties in any]

detail.

Specific Heats of Ferro- and Antiferromagnets in the Critical Region

D. T. Teaney

IBM Watson Research Center, Yorktown Heights, N.Y.

1. Introduction

In this paper we shall describe for the first time

the results of high resolution specific heat measure- ments of two classical magnetic systems: a ferro- magnetic insulator, EuO, and an antiferromagnetic

insulator, MnF2 . For completeness we shall also review the earlier results of Friedberg and Skalyo

[1] on CoCl 2

• 6H20, a rather different kind of

antiferromagnet. A logarithmic form of the specific

heat is found in all cases, the experimental results being the most clear in the case of MnF2 . The

problem of separating lattice and magnetic specific

*W,,rk supported in part by the USAFOSR of the Office of Aerospace Research

under i:.>ntract AK 4916381-1379 and Contract AF 49(6381-1230. 50

heats will be discussed, and a number of other

hazards will be described. The prospects of even more refined experiments are examined and found

to be encouraging.

2. Choice of Material and the Lattice

Subtraction Problem

The heat capacity of a typical magnetic system

is shown in figure 1. The anomaly at the Curie

point of EuO [2] (69 °K) is clearly displayed on top I of the lattice specific heat, which is of about the! same magnitude. From the figure it seems that I EuS [3] would be a better case to study, but suffi- ciently large single crystals are not yet available. But the choice of material goes beyond the avail- ability of crystals.
TCK)

Figure 1. The specific heat o/EuO and EuS.

At least for a start, it is desirable to study critical point phenomena in simple systems which are characterized by a clearly dominant strong interaction. In the case of a magnetic system this means we

want an exchange interaction that is large compared

to dipolar or crystal field effects. From this point

of view the lattice becomes an inevitable complication and we now discuss briefly what can be done

about it. Fortunately the problem has been investigated in detail for the case of MnF2 by Stout and Catalano [4] and by Hofman, Paskin, Tauer, and Weiss [5]. Stout and Catalano use a corresponding state method to deduce the lattice heat capacity of MnF2 from

that of isomorphous ZnF2 . Hofman et al., use a method of matching two Debye entropy functions

to the experimental entropy of MnF2 at high temperatures, having first subtracted an amount

R In (2S + 1). That is, they find that for two Debye

temperatures, 0=220 °K and 0' = 575 °K.

SMnFz(T) -R In 6 = SD(dlT) + 2SD(d'IT)

for temperatures above ~ 2 T\. This procedure

when applied to ZnF2 gives a two parameter Debye

curve that fits the observed specific heat to within

1/2 percent in the temperature range 35 to 250 °K.

The results obtained by the two methods are in excellent agreement.

It is important to stress that the apparent character of the singularity depends only weakly on

the slope of the lattice background: changing this

slope by a factor of two produces only faintly per- ceptible kinks in the highest |AT| data presented

here. The apparent magnitude of the magnetic

specific heat, that is, the value of 5, is more directly

influenced by the lattice subtraction, but in view of

the extensive work by the above authors it seems

unlikely that an error larger than 10 percent is introduced in the MnF2 results by incorrect lattice subtraction. In the case of EuO our results are only tentative. The diamagnetic isomorphs CaO and BaO are ex- tremely hygroscopic, and we have not been able to make specific heat measurements on reliably dry

samples, nor can we find reference to these measure- ments in the literature. The curve fitting method of Hofman et al., must then be relied upon, and it is here we wish to interject a word of caution. In order to obtain the entropy from the measured

specific heat the usual conversion

cv = cp(l —AcpT)

is required, where the Nernst-Lindeman coefficient

kpc-„

is assumed to be temperature independent. The

value of this coefficient strongly affects the estimate

of lattice background since this estimate is based

only on high temperature data. (Clearly, accurate

specific heat data is also essential.) The choice of

thetas is strongly influenced by the slope of the

entropy curve because at high temperatures the

Debye functions are becoming independent of the

thetas. Since remarkably little expansivity or compressibility data exists for compounds it would

be hazardous to guess a value for ,4, and we believe recourse to experiment is necessary. We are

grateful to Professor R. Stevenson of McGill University for providing us with compressibility data prior

to publication. We have obtained the expansivity

from our own x-ray diffraction study [6], The ex- perimental value for A for EuO, accurate to about

20 percent, is found to be 7 X 10~6

. For comparison, the value for MgO is 1 X 10~6

. Tentative theta

values of 175 and 560 give fair fits to the entropy

above 150 °K, and the full scale numerical fitting

program will be completed shortly. Again we

emphasize that the uncertainties in lattice subtraction have very little effect on the apparent form of

the specific heat in the critical region. The magni209-493 0-66-5 51
tude of the magnetic specific heat is, of course, very

sensitive to the lattice subtraction, and we refrain from quoting a number for this magnitude until a detailed numerical analysis has been completed.

3. Experimental Technique

We use the standard discontinous heating

method in these experiments. The sample is sur- rounded by a radiation shield the temperature of

which could be held constant to about 1 mdeg by

referencing a platinum thermometer against a Mueller bridge. Sample and shield are inside a vacuum jacket surrounded by double Dewars of

liquid nitrogen. The temperature of the sample is monitored by a thermister of the type used to sense

the level of liquid nitrogen [7]. The thermister

comprises one arm of a constant amplitude 100 c/s

resistance bridge stable to 1 part in 107

. The

overall sensitivity was limited to about 30 X 10"6 °K

by Johnson noise of the bridge detector. The

thermister is calibrated in situ against the platinum

thermometer on the shield, equilibrium to within

5 X 10~4 °K being accomplished by condensing

liquid nitrogen into the vacuum chamber. The

platinum thermometer is calibrated against the

vapor pressure of the condensed pure nitrogen, and

the less than ideal arrangement for this part of the calibration is the cause of the relatively large

systematic uncertainties quoted below. The function ZU(T) is used for interpolation [8]. Our ap- paratus records drift rate and heating interval data

directly on punched cards, and all the data analysis

including addendum corrections and lattice subtraction is done in a minute or two of computer time

[9].The importance of single crystal calorimetry is indicated in figure 2. The powder results are shown

as the solid and dashed histograms, the width of the horizontal steps being equal to the heating in- terval. The single crystal results are shown by the

circles. The rounding of the powder results is not a consequence of temperature gradients since

the calorimeter is fairly small (25 cm3

), the monitoring power is only 10/xW, the drift rates are on the

order of 1 mdeg per hour, and the calorimeter re- sponse time is about 3 min as compared with an

overall discontinuous heating rate of 100 mdeg per

hour. While a similar experiment with the same

calorimeter loaded with a powder sample, the anti- ferromagnet RbMnF3 (r.v = 83 °K), produced similar

52

5011 1 1 1 r

66 67 68 69 70 71 72 TEMPERATURE - DEGREES KELVIN

Figure 2. Comparison of specific heat data of two powder

samples (histograms) with the single crystal (circles) o/EuO.

results, we believe that this indicates a similarity

in the rounding of the transition in two different powder samples and not the presence of some un- accountable systematic temperature gradient.

4. Transition Rounding

While it may not be surprising to find a smearing

out of the critical point in a powder sample, we

have two further remarks about transition rounding.

(We feel that "transition truncation" is an expression to be avoided.) The first is an inference as to the effect of impurities on single crystals. The

position of the transition in the EuO crystal is noticeably on the low side of the distribution indicated by

the powder histogram. This crystal (1.7 g) con- I tained filamentary inclusions of tantalum metal,

amounting to as much as 2.0 percent by volume,

which were observed by microscopic examination

of the polished surfaces. The rounding of the

transition which is suggested in figure 2 and is dis- I tressingly evident in figure 3 may well be correlated

with this impurity. A new method of growing crystals has been found to avoid this problem, and hopefully we will have more to say on this point in the I future. The second remark on transition rounding is con- | cerned with our results on MnF2 . These experi- 1 ments were performed on two single crystals. One

crystal (0.02 mole, closed circles in fig. 4) of Bell I Telephone Laboratories origin, was obtained from

Heller and Benedek and is the same one used by
|t-tm | ck)

|l-(T/Tc )|

FIGURE 3. Logarithmic plot of the magnetic specific heat of

the EuO crystal.

|T-TH |(°K)

IP"* KT" 10^ K)'

FIGURE 4. Logarithmic plot of the magnetic specific heat of MnF> near the Neel temperature.

Open circles. IBM crystal: closed circles. Heller and Benedek crystal. The closed squares are the specific heat density of argon. The dashed curve is for a = 0.1.

them in their classic NMR experiments [10]. The

other crystal (0.06 mole, open circles) was *'on the

shelf at our laboratory and is of lost commercial

pedigree. The character of the singularity is seen

to be identical in both crystals, but we found a sig- nificant difference in Neel temperatures. The Heller and Benedek crystal had 7\ = (67.33 ±0.01) °K

in agreement with their result, while the IBM

crystal had Ts = (67.23 ± 0.02) °K. (The error limits represent confidence in the absolute calibration of

the thermometers in each of the two runs.) The

difference is well outside experimental error and

also well outside combined uncertainties in locating

Ts- The discovery of such a large difference be- tween two high quality crystals is remarkable. The

transition in both crystals was sharper than our experimental resolution of about 10 mdeg K. and we are therefore unable to comment on the rounding

which Heller found to amount to about 15 mdeg.

The difference in Neel temperatures between the

two crystals is not so large as the difference in Curie

temperatures between the EuO crystal and the

"best" powder, so that the suggestion made above

about correlation between rounding and depression

of the critical point need not be invalidated.

5. Results for EuO and MnF.

We turn now to the results themselves. The EuO

data is shown in figure 3. The rounding is much

more pronounced on the low side of the transition

(cf. fig. 6). The Curie temperature was taken as the

apparent vertical tangent as seen on a linear plot of

the data on the high temperature side. This tan- gent was apparent to an accuracy of 10 or 20 mdeg,

so that the linearity of the high temperature data is not the result of choosing a fortuitous Tc within the

range of the rounding. The slope of the logarithmic

divergence for T > T( is given in table 1. We recall that EuO is a rock salt structure and that the dominant exchange interaction is between nearest neighbor Eu ++ ions [11]. The appropriate coordination number is therefore 12. The coefficient A+

for this crystal of EuO is seen to be noticeably larger

than that for MnF2 for which the coordination number is 8. Such a large difference is not expected

theoretically; the somewhat disappointing character

of this data may be, however, license for not taking

it too seriously.

The experimental results for MnF2 are shown by

the open and closed circles in figure 4. A logarithmic form of divergence is clearly indicated over about three decades from 10 ~4 to 10 _1 T\. The

striking resemblance to the configurational specific

heat density of argon in the same region is shown by

the closed squares which are taken from Fisher's

analysis [12] of the results of Voronel' and coworkers

[13]. Our data covers almost an additional decade

in | A77 7* | and has a rather better signal-to-noise

53
Tabi.E 1. Logarithmic parameters for magnetic and confifiarational specific heats

The specific heat is in units of R and the logarithm is to the base 10.

FuO MnFo 1YJ.11X °£ A reran ? I 11 gun 1 attir*** crtic^ 1 j Cl l llv I ii, do CoClo^HoO1

Afi — 1.15 1.15 1.1 (U.2/)

A + 1. 0.6 (0.5) 0.5 0.27

A-IA+ 2.0 (2.3) 2.3 (1.0)

B- 0.82 1.16 0.1 0.56

B+ .14 (-0.3) -.3 -.015

6. Results for Co02*6H2 0

We turn now from our work on three dimensional

Heisenberg-like systems to the work of Friedberg

!

and coworkers on CoCl2 -6H2 0, which has a transi- tion temperature of 2.29 °K. Since detailed ac- count of this work has been published elsewhere

[1, 14] we mention here only the results of specific

heat measurements, which are shown in figures 5j

and 6 and in table 1. The principal point estab- lished is that this salt resembles a two-dimensional

Ising antiferromagnet. Now that we have results on a three-dimensional antiferromagnet this re- i semblance becomes all the more impressive.

CoCl2 -6H2 0 forms a layer lattice with the spins i at Co + + arranged in planar tetragonal nets. The;

structure and exchange interactions in CoCl^F^O:

have been described by Shinoda et al., [15]. The

Co + + form a tetragonal net, the plane of which

also contains the CI". The exchange interaction in the plane is .//A: = 2.03 °K while that between

planes, which are separated by waters, is found to be J' Ik = 0.39 °K. Shinoda et al., obtained these

results by careful theoretical analysis [16, 17] of the high temperature heat capacity tail as measured

earlier [18]. (They have also analyzed their ownl

results for CoCl2 -2H2 0 (TN = 17.20 °K) and find it to be a chain structure; the resolution of their ex- perimental results is unfortunately such that they can only be said to indicate a logarithmic singularity.) In addition to the sheet-like structure of

CoCl2 -6H2 0, there are two other reasons to suggest

its similarity to the Ising model. First, the effective spin of the Co++ is S = l/2 at these temperatures and the moment is highly anisotropic. Secratio than the argon results. The parameters that

result from fitting the data with a logarithmic divergence of the form C± /R = A ± \\ogu)\AT/T\\ -\- B± are summarized in table 1. Comparison is made with

the argon experimental results and with the theoretical estimates for a lattice gas. In reference 12

the results of various approximations are listed along with detailed references to their origins. We

have tabulated here only rounded values. It is clear that while the present experiments reveal the

character of the singularity, the signal-to-noise ratio

is not good enough to warrant a discussion of the

theoretically rather small effects of lattice structure.

It is important to consider the possibility of a stronger than logarithmic singularity, particularly on the high temperature side. For purposes of comparison a divergence of the form C+ = A \hT/TN

\

a with a = 0.1 is plotted as the dashed curve in figure

4. Deviation of the experimental points from the

logarithmic straight line, i.e., that for which a = 0,

is seen to be not inconsistent with a small positive

a. Notice that shifting TN down by 5 mdeg would

improve the linear fit for T <TN but increase the

upward trend of the points for T > TN . The assignment of some definite value to a would place unjustified weight on the noisiest experimental points,

those closest to TN. We feel that these data can

only be said to suggest that 0=£a:<0.1. Subject

to the physical reservations expressed in sec. 7, a much better value of a is not inaccessible to ex- periment. Although complicated behavior and/or

rounding of the transition might be expected in the next decade closer to TN, improved precision in the

present range would allow us to set improved limits on a.

54
4.4-

4.3- 42-

4.1- 4.0- 5 3.9- O 3.8 y 3.6 ^ 3.5 < 34 ~ 3.3 ° 3.2-

3.1 3.0 ; Z9-

2.8- 27-

2.6- 2.5- 2.4- 2.2600

f i

/ I

/ I

/ I / I

/.

/ / /

/

\

'":.-\

\

\

_L 2.2700 2.2800 T (°K) 2.2900

Figure 5. The specific heat of CoCl2-6H2 0 after Skalyo and

Friedberg.

|T-TJ CK)

4.5

4.0

3.5

2 3.0 o

2o 25

!3 2o

<£ 1.5

1.0

0.5

0.0

10 10"

TN

= 2.2890°K

V t<tm CoCI 2 -6H20

FRIEDBERG 8 SKALYO

T>TN

H;!:,v

t,:V+H

10 10 10 10 |I-(T/TN )|

Figure 6. Logarithmic plot of the specific heat of CoCl2-6H 20.

Note that the horizontal scale is the same for all three logarithmic plots.

ond, the earlier cp data [18] reveal a spin entropy

gain of very nearly (1/2) R log (2S+ 1) above the

Neel temperature.

The critical point specific heat measurements,

made on large aggregate specimens of single crystals, are difficult to interpret due to the fairly severe

rounding. T\ is chosen in such a way as to produce

a symmetric logarithmic singularity as is expected

in two-dimensions [16, 19]. The magnitude of the

logarithmic coefficient is comparable with the values

calculated for the two-dimensional Ising model,

and can even be gotten quantitatively by a suitable

choice of anisotropy. While the data fall on a

straight line above Ts for about 1.5 decades, not a great deal of confidence may be derived from the

data below TN . However, the specific heat results are strongly supported by their comparison with

magnetic susceptibility measurements [14]. Good

confirmation of the theoretical relation, [20] C — A 8(x\\T)/dT, is found for experimental points in a range of ±0.1 TN. A good case is thus made for believing that outside of the complicated, rounded

region, the actual specimens measured approximate

rather well the behavior of an ideal crystal.

7. Lattice Stability

We have not mentioned in the discussion above

the problem of lattice stability at the critical point,

since there was no evidence of its being relevant to the experiments reported. Clearly, if the magnetic

transition is accompanied by a crystallographic

transition or even a lattice parameter anomaly the

rigid lattice approximation no longer applies and

the interpretation of experimental results becomes

complicated. This problem has been considered

in three dimensions by Bean and Rodbell [21] using

the molecular field approximation, and more re- cently Garland and Renard [22] have studied the

two-dimensional Ising model. It is not surprising

that in the presence of a strain dependent exchange

interaction the magnetic transition may become first- order unless the lattice is "clamped." Extensive

experiments [23] on MnAs have demonstrated these

effects, and Wangsness [24] pointed out the evi- dence for structure change in MnO on the basis of

specific heat data. In the case of MnF2, the c-axis undergoes a pronounced anomaly which has been

carefully studied by Gibbon [25]. It has been

pointed out [26] that this anomaly could lead to a

first-order transition, and that, furthermore, the

existence of this sort of incipient first-order transi- tion could complicate the interpretation of a more

rapid than logarithmic divergence, particularly for temperatures very close to 7V

As an example of what can happen we would like to present some results of a perovskite antiferromagnet, KMnF:i . The temperature variation of the

lattice parameters is shown in figure 7. and the

specific heat [27] in the same range is shown in

55
+1.0

100 200 TEMPERATURE, °K

300

FIGURE 7. Lattice parameters o/KMnF.i after 0. Beckman and

K. Knox, Phys. Rev. 121, 376 (1961).

14

5 dT,

10

KMnF, DRIFT METHOD AT TN

AT»0.27°KFlGURE 9. Qualitative measurement of the specific heat of powder sample of KMnF:) near TN

.

150

50

—1 1 1

KMn F3 —i

186 "K

TN

- 88.0 °K ^^"^

/ '^—TETRAGONAL

i i i

CUBIC —

50 100 TCK)200 250

Figure 8. The specific heat of KMnF.,.

figure 8. The instability of this crystal is presumably due to a consequence of the low Goldschmit

tolerance factor (t = 0.94). The situation may be

described as follows: the pseudocubic unit cell is made up of Mn++ on the corners and F

- on the

edges. The size of the cell is dominated by the

Mn-F-Mn bonding, and the K +

fits rather loosely

in its position at the body center. This "looseness"

has been invoked to explain the anomalously low

thermal conductivity of KMnF.{ [28]. A qualitative

investigation of the specific heat of a powder sample

near the Neel temperature is shown in figure 9.

It seems remarkable that the character of the nucleation on warming and cooling would be so clearly

displayed in a powder sample. The much larger

lattice parameter hysteresis was observed in the

x-ray study of a tiny single crystal. In contrast to the x-ray and specific heat results, however, a careful study of the antiferromagnetic resonance near T\ produced no evidence of hysteresis larger than

56

about 0.05 °K [29]. Furthermore, no hysteresis was

observed in neutron scattering experiments per- formed under conditions of even higher resolution

[30]. Perhaps the fact that there is no anomaly in

cell volume accounts for the apparent stability of

the magnetic transition. It is tempting to identify

the lower specific heat peak with the Neel temperature, but we lack the absolute accuracy necessary

to make such an identification positively.

For EuO. the hardness of this material, especially

in comparison with the fluorides, makes questions

of stability less important. No structure change at low temperatures was found using standard powder

x-ray techniques, [6] and Dillon [31] observes cubic

anisotropy in ferromagnetic resonance. Since the

rock salt structure is stable for the whole set of europium chalcogenides we would hardly expect any

structure change; the hardness argument should

therefore be adequate assurance that first-order effects will be small.

8. Conclusion

We have shown that an antiferromagnet makes a good gas — perhaps even better than argon. A

logarithmic divergence of the specific heat is clearly

found on both sides of the Neel temperature of

;

MnF2 . while the possibility of a more rapid diver- i gence on the high temperature side cannot be ex- | eluded. The specific heat of the ferromagnetic

insulator EuO. though rounded from below, shows

logarithmic divergence on the high side of the Curie

temperature. We have reviewed the work of Friedberg and collaborators on CoCl.;-6H;>0 which makes
a convincing case for the resemblance of the magnetic ordering of this crystal to the two-dimensional

Ising model of antiferromagnetism. The experimental results reported are thus in substantial

agreement with existing theory.

It is worthwhile to comment briefly on what is

yet to be done experimentally on the specific heat

of magnetic systems in the critical region. Clearly

better results should be obtained on EuO because of

the ideal nature of this material; these results should

also be extended to include finite applied fields. A

ferromagnetic metal should be studied. Nickel

would be a good case because of the amount already

known about it, but because the difficulties of specific heat measurements above about 20 °K increase

with at least the third power of temperature, a more

likely candidate may be found among the rare earth

metals. As for antiferromagnets, MnF2 should be

measured more accurately and better limits put on

a. Another ideal antiferromagnet is available for

study. RbMnF:! . This perovskite structure remains

simple cubic at low temperatures so that it is unlikely that lattice stability will be a problem. Good

results for this system will give some insight into the problem of coordination number as well as contribute to the general validity of the experimental

conclusions.

We are grateful for the stimulating and enlightening discussion with P. Heller, G. Benedek, M.

Fisher, T. Yamamoto, and R. Nathans. The expert

preparation of material by M. Shafer and C. Guerci

have made it possible to study EuO. The en- couragement of J. S. Smart and the collaboration

of V. L. Moruzzi are greatly appreciated. The

technical assistance of J. S. Leddy has made it possible to concentrate on these experiments, and

the programming talents of Miss E. Bergonzi have

been indispensible.

9. References

[1] J. Skalyo, Jr., and S. A. Friedberg, Phys. Rev. Letters 13,

133 (1964).

[2] C. F. Guerci, V. L. Moruzzi and D. T. Teaney, Bull. Am.

Phys. Soc. II. 9, 225 (1964).

[3] V. L. Moruzzi and D. T. Teaney, Solid State Comm. 1, 127

(1963). [4] J. W. Stout and E. Catalano. J. Chem. Phys. 23, 2013

(1955).

[5] J. A. Hofman, A. Paskin, K. J. Tauer and R. J. Weiss, J. Chem. Phys. Solids 1, 45 (1956).

[6] D. T. Teaney, V. L. Moruzzi and J. E. Weidenborner, to be

published.

[7] We are indebted to Dr. H. B. Sachse of the Keystone Carbon Company for samples of these thermisters and advice on

their characteristics.

[8] J. M. Los and J. A. Morrison, Canadian J. Phys. 29, 142

(1951).

[9] The computer program, expressly written to cover a wide

range of laboratory circumstances, is available to in- terested experimenters.

[10] P. Heller and G. Benedek, Phys. Rev. Letters 8, 428 (1962).

[11] B. A. Calhoun and J. Overmeyer, J. Appl. Phys. 35, 989

(1964) . [12] M. E. Fisher, Phys. Rev. 136, A1599 (1964).

[13] M. I. Bagatskii, A. V. Voronel, and B. G. Guzak. Zh Eksperim

i Theor. Fiz. 43 , 728 (1962); English translation, Soviet Phys. JETP 16, 517 (1963).

[14] J. Skalyo, A. F. Cohen and S. A. Friedberg. Proceedings of the Ninth International Conference on Low Temperature

Physics, Columbus, Ohio, 1964.

[15] T. Shinoda, H. Chihara and S. Seki, J. Phys. Soc. Japan 19,

1637 (1964).

[16] L. Onsager, Phys. Rev. 65, 117 (1944).

[17] A. J. Wakefield, Proc. Cambridge Phil. Soc. 47, 419, 799

(1951).

[18] W. K. Robinson and S. A. Friedberg, Phys. Rev. 117, 402

(1960).

[19] See, for example: M. E. Fisher, J. Math. Phys. 4, 278 (1963)

[20] M. E. Fisher, Phil. Mag. 7, 1731 (1962).

[21] C. P. Bean and D. S. Rodbell. Phys. Rev. 126, 104 (1962).

[22] C. W. Garland and R. Renard. to be published.

[23] R. W. De Blois and D. S. Rodbell. Phys. Rev. 130, 1347

(1963). [24] R. K. Wangsness, Science 116, 537 (1952).

[25] D. F. Gibbons, Phys. Rev. 115, 1194 (1959).

[26] P. Heller, G. Benedek, and T. Yamamoto, private communication.

[27] D. T. Teaney and V. L. Moruzzi, Bull. Am. Phys. Soc. II, 9, 225 (1964).

[28] Y. Suemune and H. Ikawa, J. Phys. Soc. Japan 19, 1686

(1964). [29] D. T. Teaney, Thesis, University of California, 1960, (un- published).

[30] R. Nathans, private communication.

[31] J. F. Dillon, Jr., and C. E. Olsen, Phys. Rev. 135, A434

(1964).

57
Nuclear Resonance Studies of Magnetic Critical Fluctuations inMnF2 *

P. Heller

Brandeis University, Waltham, Mass.

Atomic nuclei with magnetic moments can serve

to probe the local fields within magnetic materials.

By studying the nuclear resonance frequency and

line width in the critical temperature region near the

magnetic transition, it is possible to obtain precise

and detailed information [1, 2] on the establishment

of magnetic order. Here we shall be concerned

with the effect of the critical fluctuations on the

NMR line width: associated with the approach to the critical temperature Tc the nuclear resonance

lines in ferro- and antiferromagnets exhibit a remarkably rapid broadening. A very detailed study of

this effect has been made for the case of the antifer- romagnet IVMs. (Tc = 67.336 °K, also denoted by

Tn). These experiments shed light on the behavior

of the electron spin correlation functions in the

critical region. In this account we show that the

experimental data implies a "slowing down" of the

spin fluctuations in the critical region. A simple

physical model for the time dependence of the correlation functions will then be presented. This model

yields a theory which is in satisfactory accord with

the observed broadening of the F

19 resonance in MnF-2 as T approaches Tc from above.

The line width anomaly at the critical point

for the F

19 resonance in MnF2 is shown in figure 1. We shall be largely concerned with the situa- tion in the paramagnetic state T > Tc . The detailed behavior of the line width in the region

Tc + 0.006 °K < T< 7V+10 °K is shown by the

experimental points in figure 2. (The drawn lines correspond to the theory to be outlined below.)

The line -width is anisotropic with respect to the

direction of the externally applied d-c field H>. For Ho along c (the antiferromagnetic axis), the

line broadens by about a factor of ten as Tc is approached. In the region of severe broadening,

the ratio of the line width for H0 along the c-axis

to the line width for H0 along the a-axis is very

nearly 2 to 1, dpJ8vn = 2. (1)

"This work was supported by the Advanced Research Projects Agency under Contract SD-90 at the Massachusetts Institute of Technology. Also reported in part at the

International Conference on Magnetism, Nottingham, England, 1964.

*0.S

1

1 1 1

o PRESENT WORK

1

fi SHULMAN and JACCARINO

PHYS- REV 10.S (1219) 1957 - SHULMAN and JACCARINO

PHYS, REV [07 (1196) 1957 H0 ALONG C AXIS

1 1 l 1 l 50 100 150 200 TEMPERATURE IN OEGREES KELVIN

Figure 1. Temperature dependence o/F19 nuclear resonance

line width in MnF2 from 4 °K to 300 °K.

Applied field along c-axis.

3000

3 0 L^J '

' '

I

'

' 'I OO 10 0.030 0.100 0.300 1.00 3.00 10.0 T -T NE

- EL IN DEGREES

COMPARISON OF THEORY WITH EXPERIMENT, T>TN

x = 2.7 x 10 12 sec."

1

Figure 2. Experimental F

19 line widths in MnF-> just above Tc . Drawn curves represent theoretical results with \ = 2.7 X 1012 sec

-1

.

Just above Tc , the line widths level off at the values

8vc(Tc)= 1100 kc/sec

8i/,X7'c)=550kc/sec. (2)

One observation, made at 200 mdeg above Tc , showed that the line width increases monotonically

as Ho is turned from the a to the c direction. At

58
77 °K, where the line is relatively narrow, the line width is essentially isotropic.

We would like to understand these results. One

possible mechanism for the line width broadening

will now be stated and then shown to be incompatible with the experimental facts. Suppose that as Tc was neared, the electron spins in the sample

became permanently ordered to a degree which

varied from one region to another. This would

result in a variation AH in the time average local

field seen by different nuclei. We assume the direction of antiferromagnetic spin alinement within each

region to be along the c-axis, the direction of anti- ferromagnetic alinement below Tc. The direction

of AH will then also be along c. The nuclear reso- nance would then be inhomogeneously broadened.

This broadening would be severe if the applied field H( i were colinear with AH. On the other hand, if Ho were perpendicular to AH, the observed broadening would be of the second order in |A//|. A

simple calculation shows that if this were the

mechanism responsible for the line width, the anisotropy factor 8vcl8va would be very much greater than

2, in contradiction with the experimental results. The line width is therefore to be attributed to fluctuations in the local hyperfine fields seen by

the nuclei; it can then be related to the electron

spin correlation functions in a way that may be

expressed quantitatively as follows. If the inter- action between a nucleus and an electron spin has

the form ^f—Al-S, where A is the hyperfine coupling constant, and if z is the direction of the

average local field at the nucleus and furthermore

if the correlation time for the spin fluctuations is short compared with the nuclear Larmor frequency,

then [3, 4]

8v1 (4

7T V3 W Jo dt < {8Sz(t)8Sz{0)} >

{8Sx(t)8Sx(0)} > + l< {8Sy(t)dSy(0)} > (3)

Here 8S(t) — S(t) — < S > denotes the departure of

the spin from its thermal average value and {AB} =

f(AB + BA).

We now show that in order to explain the line width anomaly within the framework of the for- mula (3), it must be assumed that the time dependence of the spin correlation functions becomes

"slow" near Tc . Let

G^t)-

< {&SM ( f )8SM (0)} >

<[SSM(0)]

2 >

'

where /x = 1 , 2, 3 denotes a Cartesian component,

and define correlation times rM by writing

tm = G»{t)dt.

Jo

Then (3) becomes

8p = 1 (A

tz < (8SZ ) 2 >

+

\

Tr < (8SX ) 2 >+\ry < (8Syf > (4)

The experimental fine width was found to be iso- tropic above Tc + 10 °K. We then surmise that in

this temperature range the spin fluctuations are

isotropic and tx = Ti, = tz = t. If the correlation

times were constant as T—> Tc we would have

< {8Sz f >

+ ^<(§5J.) 2 > + ^<(SS,y ) 2 >

But for T>TC , <S > = 0, and the following identity holds

< (8SX ) 2 > + < (8Sy) 2 > + < (8SZ) 2 >=S(S + 1) =

5 7

2 2

From this it may be seen that the temperature de- pendence of the time independent correlations

< (SS,x)2 > cannot be responsible for the tenfold

increase in line width observed as T^> Tc . The

correlation times evidently increase in the critical region.

The observed fine width anisotropy is now inter- preted as follows. Denoting the crystal axes by

59
c, a, and a', the line widths for the average local

field along c and a are given respectively by

and

TC < (8SC ) 2 >

+ |T« < (8S„)2 > +| Ta< < (SS„>)2 >

ra < (8Sn ) 2 >

+ |rc < (8SC ) 2 > +| TV < (8Sa <) 2 >

Therefore, the factor of 2 anisotropy actually observed just above Tc implies that at those

temperatures

TC <(8SC ) 2 > >Ta<(dSn f>. (5)

A theory of the electron spin correlation functions

in the critical region was given by Moriya [5]. The physical content of the theory may be expressed

in the following terms. Just above Tc there exists a short range antiferromagnetic correlation between

the spins. Roughly speaking, the spin arrangement

at any instant of time consists of clusters of anti- ferromagnetically alined spins. The time average

value of the spin at any site is zero, however, and

so these clusters are constantly forming, dissolving

and reforming. The average size and lifetime of the clusters increases as Tc . This latter point, which corresponds to a "slowing" of the

spin fluctuations is an essential part of the theory.

The direction of antiferromagnetic alinement within

the clusters is mainly along the c-axis; this is re- sponsible for the inequality (5) and the consequent

fine width anisotropy.

On the basis of this picture it is clear that we

should rewrite eq (3) in terms of the Fourier trans- form spin deviations 8Sk(t). These are defined in terms of the ionic spin deviations SSj(i) as follows

(6)

The theory is greatly simplified if we assume that the correlation functions for the Fourier transformed

spin deviations decay exponentially with time, i.e.,

< {ds^wdstfl)} > = < |ss^(0)| 2 > e-ryTM.

(7)

Using eq (7) and (6) in eq (3) it follows that

8vc =

1 (AV 1_ ta/3 w n

< |gS,-c(0)|

2 >

rc(Ao

1 < |5S/,-„(0)| 2 > 1 < loSA-q'(0)| 2 >

2 ra(k) 2 ra,(k) (8)

when the average local field is along the c-axis.

If the average local field is along the a-axis one

interchanges c and a.

The tenfold increase in the fine width that is observed to occur as T—> Tc from above can now be understood as follows. As 7

1 —>• Tc , the quantities < |8Sa c(0)|

2 > become very large

while the corresponding decay rates rc(&) become

very small for wave vectors k near the antiferro- magnetic made ko- This corresponds to the physical picture of increased correlation range and

cluster lifetime near Tc . To make a quantitative theory of these effects we thus need to know the temperature dependence

of both the quantities < | 8Sam | 2 > and T^k). In

his theory, Moriya used the fluctuation-dissipation

theorem to relate <|8Sam | 2 > to the wavelength

dependent d-c susceptibilities. The relation is

< |8SA

- M | 2 > =

kB T

(9)

where ks is the Boltzman constant, [An the Bohr

magneton, and g—2. The susceptibilities Xm(^) can be calculated on a molecular field model. The

results are given by Moriya [5]. To calculate the decay rates V^k), Moriya employed a technique developed by Mori and Kawasaki [6] for describing spin diffusion in ferromagnets.

Here, however, we shall employ a very simple

physical model. According to this view [7], the

decay of a staggered spin configuration is inhibited

by the staggered molecular field generated by that configuration, an effect which leads to a particularly slow decay near Tc . On this model the decay

rates are related to the wavelength dependent d-c 60
susceptibilities as follows Thus the "staggered susceptibility" is effectively

r /; \ \ Xcurie(T)

l ^(K) — A - =r, XM«, T)

(10)

where A is a temperature independent parameter,

and

Xcurie(^) : g

2 ^2 S(S+l)

SkBT

(11)

is the Curie susceptibility.

The motivation behind eq (10) is as follows. Imagine that at some temperature T > Tc we arti- ficially produced a "staggered magnetization" Ms by applying a staggered field Hs. If H$ is turned

off, Ms eventually relaxes to zero. Consider the

situation at the instant that Hs is turned off. According to the viewpoint of the Weiss theory, where

the coupling between the spins is replaced by an

effective field, the spins would find themselves

in a staggered "molecular field." We suppose this

field to be strong enough to produce a staggered

magnetization M* at equilibrium. Evidently

M* =f{T)Ms where/fT) is a fraction which becomes

equal to unity at Tc , that being the temperature at which the magnetization becomes self-sustaining.

As a model to describe the decay of the staggered

magnetization we now write the following differ- ential equation

dMs

dt = -m*-M*) = -\[\-ffl\Ms

This leads to an exponential decay Ms =Ms (0)e rs' where

r,=A[i-/(7U (12)

Now let us calculate the staggered magnetization

Ms produced at equilibrium by a staggered field Hs . In the absence of the molecular field we would have ms = HsXcurie where Xcurie is given by (11). In the

presence of the molecular field Ms = HsXCune+ AT)MS

or

Ms

HsXc,

Xs = Xcurie i-/oy

Comparing (13) with (12) we find

\ XCurie

(13)

Xs

The same argument may be applied to relate the

relaxation of spin arrangements of arbitrary wavelength to the corresponding susceptibilities. We

thus obtain eq (10). In applying this relation to a

calculation of the time dependence of the correlation functions, we are assuming that the time decay

in the Ath Fourier component of the magnetization

which occurs as a result of spontaneous fluctuations

is similar to that which characterizes the return to equilibrium following an artificially produced nonequilibrium state. The line width may now be expressed in terms

of the wavelength dependent susceptibilities and

the parameter A. Using (9), (10), and (11) in (8) we

find

8vc =

1 1 A2 1 kB T

XirVzH2 \g2 fJig' S(S+1)N

1

2Mi(*)+xi(ft)]

k

(14)

8va = -

1 1 A

2 (

k

B T

X7TVSt2 \g2 fJLp S(S+l)N

xP)+§xI.W

where x\\(k) an^ XJ.W denote the components of the wavelength dependent susceptibilities respectively parallel and perpendicular to the c-axis. The discussion so far has assumed that each

fluorine nucleus is coupled to a single manganese

spin. In actual fact, there are three spins to which

the coupling is important [8]. We denote these

by S, S', and S", their positions in the lattice by

r, r', and r" and the corresponding hyperfine coupling tensors by A, A' and A". We then replace the

coupling AI S in the preceding discussion by the

coupling I-(A-S + A'-S' +A"-S"). The line width

formulae (14) then become revised in a way which

can be expressed compactly as follows. For each

61
wave-vector k construct the tensor Also

A(k) = Ae iAw' + A'eikr' + A"e"oik-r" (15) XAko+q)-

For the present purposes, the principal axes can be taken to be along the c = [100], x = [110]

and y= [ lTO] directions. Then the line widths can

be written

1 ti-z ( kB T

A

8vc = 7= I 77V3\

1 g

2 ^/ S(S+1)N

\A cc(k)\ 2xn(k) +

\A xAk)\*+\A yy(k)\ 2 X

2M

(16)

s

1 (kB T

AW3 V S(S +

\A cc(k)\ 2x\\(k)+2

3 \A xAk)\ 2 +\A tw(k)\ 2 Xl(k) 2 A±

Since the modes k = k0 make the most significant

contribution near Tc , we can replace A(k) by A(&0). The numerical values of the tensor components can

be obtained from the electron spin resonance

experiments [8] on Mn++ in ZnF2 and may be ex- pressed as follows

|

|J cc(A:o)| = 160 Mc/s

; g

2/4S(S+l)

3knTc

1

8 + 8anis +ka2 (<?„+ ql>) + c

2 q2

]

(19)

where 8anis . = 0.02 is roughly the ratio of the dipolar

anisotropy energy to the exchange energy. Note

that X-l(^o) does not diverge at Tc , while xil(^o) does.

Using (17), (18), and (19) in (16), the temperature

dependence of the line widths is found to be

asymptotically

8vc —

14

A3 | S(S+1)J«

5 /r

A3 V3

1 4 2 8^ = 7^ \/4 S(S+l)

[S-^ + O.lSfS + Sa^)- 1 ' 2 ]

\Acc(ko)\ 2

[- 8-^ + 0.26(8 + Sanis)- 1/2 ]- (20)

The quantity A, which we regard as an adjustable

parameter, is now chosen to fit the line width data

for 7V + 0.015 °K < T < 7V +1 °K. We find

s_y\AxAko)\ 2 +\Ayy(k0)\ 2 =m Mc/s)2 (17) h ] 2

where S — 5/2. To obtain line width formulae which are asymptotically correct as f-^ Tc , it will be sufficient to express the wavelength dependent susceptibilities

X(k) in a form valid near the antiferromagnetic mode

k0 . Letting k = k0 + q where \q\ < h, the molecular

field approximation yields the following results above Tr :

X||(&o + <7) = g

2 nlS(S+l) 1 MbTc

8 + l[a2(q

2 + q2

,) + c

2 q2]

(18)

Here a and c refer to the unit cell dimensions, and

8 =

T-Tr

A = 2.7X 1012 sec"1

. (21)

The fit is shown in figure 2, the solid lines corre- sponding to eq (20). Some deviation in the fit is apparent above 7V+1 °K. This is attributable to the use of the approximate expression (18) and (19) and the replacement of A(Zr) by A(&o) in (16). Both

approximations tend to underestimate the line width at temperatures where contributions from

modes k not very close to ko become important.

A more serious defect in the theory occurs in the immediate vicinity of Tc where eq (20) predicts

divergent line widths proportional to S

-1 ' 2

. This

contrasts to the constant experimental values re- ported in eq (2). The infinity in the theory arises from the divergence at Tc of the staggered susceptibility XI|(^o, T). We now suggest that in reality

this quantity does not become infinite at Tc , but is merely very large over some narrow range of temperatures centered on Tc . In other words we suggest that the critical point is not perfectly sharp,

62
there being a distribution of critical temperatures

of width 8TC . Such a situation might well arise when on account of crystalline strains, different parts of the sample have different Curie points.

We do not wish to assert that this is really the source of the broadening, however. In any case,

a distribution of Curie points should lead to two

other experimentally observable effects near Tc :

(1) There should be a coexistence region in which

nuclear resonances corresponding to both the para- magnetic and antiferromagnetic phases are simultaneously observable.

(2) The NMR lines observed in the antiferro- magnetic state T < T, should be inhomogeneously

broadened. The line width should be maximum

when the average local field is along c and minimum when along a. Furthermore, the anisotropy

factor 8vmaJ8vmin should be greater than the 2:1

ratio associated with fine broadening by fluctuating

fields. Both of these effects were observed and both were consistent with a Curie point distribution of width

8T( = 17 millidegrees. (22) We now show that such a distribution would

result in a finite observed line width at the (average) Curie point Tc , and we calculate this width.

At that temperature, the observed line corresponding to the paramagnetic phase will be due to those

members of the distribution which have Curie

points 7^ somewhat below Tc . The larger the dif- ference Tc — Tc , the narrower and hence the stronger

will be the contributed NMR line. Roughly speaking, the observed line should then be due to those

members of the distribution for which Tc — Tc = 2 8TC . The observed line width should then

be given by (20) by inserting the value

S = *|?£=1.3 xlO-4.

When this is done, using for A. the value given in

(21) we find

oVc =1200 kc/s

8v„ = 600 kc/s

(23)

just calculated. Thus the divergent theoretical

formulae, when modified for the experimentally

observed Curie point distribution, yield a good

account of the line width broadening above Tc . Let us see whether our model eq (10) for the

relaxation rates is still valid well above the critical point. For T>TC , X(k) =XCuTie and hence F{k) = \

for all k. Then the decay rate for the single-ion

correlation functions is also A.. This enables us

to compute A. from the F

19 line width observed at high temperatures [9],

The result is

8vx =170 kc/s.

k= 1.8 X 1012 sec" 1

. (24)

Comparing this with the result (21) deduced from

the critical fluctuations, we see that the model

eq (10) gives a satisfactory picture of the fine width

behavior above Tc . We now turn briefly to the situation below Tc , where the line width data is shown by the experimental points in figure 3. In the antiferromagnetic

state, the average local field at the nucleus HNUCL

is the resultant of the applied field and the field produced at the nucleus by the antiferromagnetically ordered Mn++ spins:

19 nucl— H0 + HMn++ .

Most of the observations were made with H0 and

hence HNUCL along c. Some observations were made with H0 so oriented that HNUCL was along a.

MEASUREMENTS

- THEORY

INHOMOGENEOUS BROADENING FOR

H NU(. L I|C DUE TO ASSUMED NEEL

TEMPERATURE DISTRIBUTION

I I

.

I I , I I 0.003 O.OI 0.03 0.10 0.30 1.0 3.0

in agreement with the experimental values reported

in eq 2. The theoretical line width curves in figure 2 have been "leveled off" at the values (23)

Figure 3. Experimental F19 line widths below Tc

. Curves drawn through experimental points represent smoothed data. Other drawn curves represent theoretically calculated line width for HNm. along a and inhomogeneous broadening with HNm. along c, calculated for a Curie point distri- bution of widht 6T, = 17 mdes.

63
200

1 r— 1 'I i

1

1

i 1

_ T< T H SMOOTHED DATA CORRECTED FOR

-

FOR INHOMOGENOUS BROADENING

- T>TM \

SMOOTHED DATA \ X.

1 i i i i i il i 1 i i i i ul i 0.003 0.01 003 0.1 0.3 10 3.0 100

I T-T„ I IN DEGREES KELVIN

Figure 4. Line widths for HNlcl . along c above and below Tc

. Line widths for T< T, have been corrected for the inhomogeneous broadening due to I he Curie point distribution.

When this condition was satisfied the NMR lines were quite narrow. This is because the inhomogeneous field AH arising from the distribution

of Curie points is along c; only a second order broadening results with HNUCL perpendicular to AH.

An upper bound on the second order inhomogeneous broadening for HNUCIj along a was calculated

by postulating that all of the observed broadening

for HNUCL along c was inhomogeneous. This upper

bound was considerably smaller than the line width 8v„ observed for HNUCl along a. We conclude

that the 8v„ line widths shown in figure 3 do not

reflect the inhomogeneous broadening. They are due presumably to fluctuations.

It was possible roughly to correct the data for HNLJC | along c for the inhomogeneous broadening.

This was done in two ways as follows:

(1) The corrected value of 8vc was set equal to twice 8v„ at those temperatures where the 8v„

data was available.

(2) Assuming that the 8vc line widths observed

just below Tc were entirely inhomogeneous, the

width of the Curie point distribution was found to be about 17 mdeg. Then the inhomogeneous part

of the broadening was calculated as a function of

temperature and "subtracted off' from the observed

line width.

The estimated line width for HNlJCL along c, cor- rected for the inhomogeneous broadening is shown

as a function of \TC — T\ in figure 4, together with

the actual 8vc line widths for T > Tc . Note that the corrected line width is continuous at Tc. The theory described above, which gave a satis- factory account of the behavior above Tc, can be

used to calculate the line widths for T < Tc. Using

the molecular field model to calculate the wavelength dependent susceptibilities, the theoretically

calculated line width is found to fall off much more

rapidly with decreasing temperature than the ex- perimentally observed width. This discrepancy is illustrated in figure 3, where the theoretical line width for HNLn . along a is compared with the data. At present, therefore, we do not understand the

line width behavior below Tc.

The author is deeply grateful to Professor G. B. Benedek for many important comments and suggestions, to Professor T. Moriya for making avail- able his theory prior to publication, and to Dr. Y. Obata, Dr. T. Izuyama, and Dr. K. Kawasaki for many helpful theoretical discussions.

References

[1] P. Heller and G. B. Benedek, Phys. Rev. Letters 8,428(1962).

[2] P. Heller and G. B. Benedek, Phys. Rev. Letters 14, 71 (1965).

[3] R. Kubo and K. Tomita, J. Phys. Soc. Japan 9, 888 (1954).

[4] T. Moriya, Prog. Theor. Phys. 16, 23, 641 (1956).

[5] T. Moriya, Prog. Theor. Phys. 28, 371 (1962).

[6| H. Mori and K. Kawasaki, Prog. Theor. Phys. 25, 723 (1962).

[7] P. Heller, Thesis, Harvard University 1963 (unpublished).

[8] A. M. Clogston, J. P. Gordon, V. Jaccarino, M. Peter, and

L. R. Walker, Phys. Rev. 117, 1222 (1960).

[9] R. G. Shulman and V. Jaccarino, Phys. Rev. 108, 1219 (1956).

Discussion

G. S. Rushbrooke: May I ask Dr. Domb how summing over further graphs besides ring graphs differs fundamentally from applying

say the Percus-Yevick equation to the Ising model, which one knows is not very good?

C. Domb: The approximation which I propose attempts to find an asymptotic representation of polygons and other graphs on

the lattice. In terms of Mayer theory it corresponds to summing

all cluster integrals which can be formed using all the bonds

of these configurations on the lattice. Thus even in lowest order when it is concerned only with polygons it still takes into account

a much more complex set of integrals than those of simple ring

type. In this respect the lowest order approximation is the generalization to infinity of the ring approximation of Rushbrooke and

Scoins. It seems to me that approximations like those of Percus-Yevick are derived by summing Mayer ring integrals, and make no serious attempt to estimate asymptotically the number of polygons on the lattice and the corresponding excluded volume effect._

M. E. Fisher: I would like to make a comment on the point that Dr. Rushbrooke raised. It is in the spirit of the Van der Waals

equation to take into account the short range repulsion exactly. The short range repulsion with excluded volume effects is also a

difficult problem, but the point is that it is possible to split the problem this way. This approach using the data from lattice models represents it, one hopes, in a better way.

C. N. Yang: Dr. Domb said he believed that for some three dimensional lattice models the specific heat singularities below

64
the critical temperature are logarithmic. What is the constant in front of the logarithm?

C. Domb: Unfortunately, I have not got the figures at this moment, but I can easily provide them at a later stage.

B. R. Livesay presented measurements of the magnetization of

thin nickel films near the Curie temperature [1].

J. C. Eisenstein: I would like to report some measurements on NdCla, which have been carried out by Hudson and Mangum

here at the National Bureau of Standards [2]. This substance

has highly anomalous magnetic properties. It has two extremely

sharp spikes in its susceptibility at approximately 1.0 and 1.7 °K. With regard to the dependence of the susceptibility on the temperature in the vicinity of the 1.7 °K peak, we found that the coefficients y and y' are approximately the same. However, they are not 1.3, but approximately 0.69. Also the susceptibility of CeCl3 was measured [3]. It was found that the

susceptibility of this substance has a very sharp spike at 0.345

°K. We think CeCl3 has a second susceptibility peak at a lower

temperature, but it has not been located yet. For CeCl3 the value of y on the high side of the transition is about 0.61. The

measurements on NdCl3 extended from about 5 to 55 mdeg on

either side of the transition. For CeCl3 the range was 10 to 100 mdeg.

E. Callen: There exists some theoretical analysis of the Heisenberg Hamiltonian. If one looks for the asymptotic behavior, any cluster theory or Green's function theories give the value j8=V2 . That is in M = D(\-TITr f (1)

As the averages are calculated by some kind of approximation

the two methods do not lead to the same result. However it is possible to make a guess as to which of the two methods is better

at low temperature and which one better at high temperature. One therefore introduces a parameter and says that the total Sz is a linear combination of (2) and (3), the ratio being (V2S) (S2/SK. This has been investigated for various values of the exponent x. At first we [5] had carried out the method for x= 1 but at the meeting in Kansas City in March 1965 Copeland

and Gersch indicated that they had investigated the method for general x, and that for x = 3 the theory leads to excellent results

[6J. For the shape of the critical isotherm, Copeland and Gersch

reported 8 = 4.22, whereas various experiments that have been

reported here give 8 = 4.2 ±0.2. For the coefficient of the susceptibility above the Curie temperature they reported y — 4/a quite accurately. As my brother told you, all Green's function

theories give /8 =1/3, but only at some distance below the Curie temperature (much further below than Dr. Benedek found).

In fact, one finds that the value of (i changes from V2 close to the Curie temperature to fi= lh some distance away from the

critical temperature. Incidentally this is very reminiscent of the data in He3 that Dr. Sherman presented, where they found

experimentally a narrow region where the coexistence curve

follows a V2-power law going over into a V3-power law beyond

this region. Finally, I remind you that the coefficient D in (1)

is correctly predicted to within 2 percent. Anonymous: How is the value 3 chosen for the exponent xt

M. E. Fisher: By comparison with experiment?

H. B. Callen: Copeland and Gersch find the magnetization to be double valued if one takes x in the range 1 < x < 3. The

selection of x = 3, rather than x > 3. is on the ad-hoc basis of the

results cited.

the behavior of M asymptotically at T=TC is given by f3=V%. On the other hand, however, if one is content with some sort of a numerical procedure, it turns out that many results are al- ready given by simple models. In particular, if one carries out

the variational procedure as described by Dr. Livesay, in which

one considers dMjdT as a function of /3 and takes the value of

j3 which gives the best fit to the magnetization curve, the Green's

function theories give /3 =1/3 for a broad temperature range. Also the cluster theory including nearest-neighbor and next- nearest-neighbor interaction leads to the V3 power [4]. Further- more, by taking the nearest and next-nearest exchange constant from experimental data without using any adjustable parameter,

the cluster theory predicts the observed value of the coefficient D in (1) within 2 percent. Although these theories are not correct asymptotically at T=TC , numerically one obtains rather good results from the present existing theory away from Tr . Perhaps only higher and higher sophistication will extend this range of validity to temperatures closer and closer to T,-.

H. B. Callen: I would like to amplify that. There has been a

great deal of discussion, it seems to me, about rigorous theories. Nevertheless, the molecular field theory, the cluster theory and

i other phenomenological theories have historically turned out

to be very convenient. These theories to my mind have not

1 been considered sufficiently in the discussion so far. As my

brother just observed, they are in fact considerably more sucI cessful than they have been given credit for. I would like to review very rapidly some of the successes these theories have

had, choosing specifically a form of Green's function theory.

If one asks for the equation of motion of a spinwave, one has

1 to take into account the interaction between two spinwaves.

This interaction is described by the appearance of Sz in the

, Hamiltonian [5]. Now one can represent Sz in either of the

,

< following two forms [5]:

S* = S-S-S+

(2)

or

! SZ = »/2(S +S--S-S+

) (3)

G. E. Uhlenbeck: I would like to make a comment with regard

to the remarks of Dr. Callen. I am not in favor of saying that the theories you mentioned have not led to interesting results. However, I do not think it is right to treat them on the same level as the classical theory. The classical theory corresponds to a

definite model namely long range forces, while the theories you mentioned contain what I would call uncontrolled approximations.

H. B. Callen: At the risk of a rebuttal which may call forth an- other rebuttal, I remark that the molecular field theory only

became an exact model many years after it was phenomenological. That is, the model for which it is exact was not discovered

until long after the theory has been presented. I imagine that

for any so called phenomenological model, one can find, if one is clever enough, some obscure model for which it is a rigorous

solution. In the case of the molecular field solution one contrives

a model in which every spin interacts equally strongly with every

other spin. It seems to me to be a technicality to call that a physical model, and to make a distinction between that theory and other phenomenological theories.

L. Tisza: The internal field theories can be discussed from two

points of view and these are not in conflict if properly distin- guished. We may search for models or types of interactions

for which such a theory is more or less adequate, or else, we may

consider the method as an approximation to the real state of matters and observe its point of breakdown. I wish to make

a remark from the second point of view. It is possible to for- mulate the statistical thermodynamics of equilibrium in a fashion

that a finite system appears to be coupled with its infinite en- vironment. The system is specified by additive random vari- ables the distribution of which contains the intensities of the environment as parameters. The theory is still valid for small systems, but it fails for critical points where an important Jacobian vanishes [7]. This argument justifies the use of an

internal field as the intensity of the environment of a small open system, independently whether the interaction is of short or long range. However, the theory fails near the critical point and here more incisive methods have to take over.

65
O. K. Rice: I would like to say something about the accuracy

with which one can measure ft. As was mentioned by Dr. Rowlinson, Thompson, and I [8] measured the coexistence curve of the binary liquid system perfluoromethylcyclohexane-carbontetrachloride. We had a temperature control to about 0.0001 °C and we measured a curve which extended to about 0.02 °C from the critical temperature. We obtained data which looked

as good as the magnetic data under discussion. However, we

tried to distinguish between the value /3 =1/3 and the value

/3 = 5/i6 that Fisher suggested. For this purpose we plotted our data both ways. We figured we did not know the critical temperature better than within 0.0001 °C, so that we had a small range in which the critical temperature could be adjusted. We

found in doing this that one cannot distinguish between these two exponents. Therefore, I wonder whether in the case of the magnetic data the critical temperature is really definitely fixed. Perhaps with a slight shift of the critical temperature, one might

get a fit of the data with exponents that are slightly different. In other words, I wonder whether even in the best cases, the data

are really good enough to fix the exponent /3 to be Va.

P. Heller: In response to that last question, it should be re- marked that there is no uncertainty about the value of /3. If you plot the cube of the magnetization as a function of the temperature you get a straight line. If you plot the magnetization

to a power slightly different from 3, you do not get a straight

line. In this procedure of finding the exponent that rectifies the observed data, one does not need to know the temperature

of the Curie point.

D. H. Douglass: I wish to point out, in addition to the magnetic and liquid-gas systems, that superfluid liquid helium is another system for which the exponent V3 probably appears. The quantity in liquid helium which corresponds, in the magnetic case, to the magnetization is p'J- (I get this from analogy with superconductivity). I have used the data of Dash and Taylor [Phys. Rev. 105, 7 (1957)] to determine if, near the A-point, the superfluid

density follows the relation pl^aiTy—Ty. I find that y = 0.30 ±0.03. which includes the value V3 (and 5/i6, too).

C. Domb: With regard to the paper of Dr. Wolf, I would like

to make two comments.

Firstly, I think the first order phase transition may be sig- nificant and is worthy of further investigation. The antiferro- magnetic transition is the analog of an order-disorder transition with unequal ratios of the constituents. I believe the experi- mental data on the order-disorder transition of this kind in Cu3 Au are also thought to represent a first order transition. We might well expect something different to happen when the symmetry of equal ratios of constituents is removed in an order- disorder transition, and when the field is nonzero for an anti- ferromagnet. I would think this is a result worth following up. Secondly, I do not think one can really say any one shape of specimen is better than another for testing the theory. I really think we just have to do more theory. In fact what has held

us up with long range dipole forces is the terrifying nature of the calculations. However, if the dominating term is the short range interaction, and the dipole force, although it produces a shape dependent effect, is nevertheless a relatively small cor- rection, I think the calculations may be manageable, and then we should be able to get a better comparison.

W. P. Wolf: Let me make that point quite clear. In our anti- ferromagnet the energy of three cells of nearest neighbors ac- counts for the major part of the energy. However, when you

apply a magnetic field, the change in dipolar energy, the de- magnetizing energy, that is, is comparable with the energy due

to the applied magnetic field, and in a large field it can become

rather comparable with all the other energies. I didn't give any numbers for this but for DAG, the effects are in fact all about comparable.

C. Domb: Well, perhaps you get cancellation in the higher order terms. W. P. Wolf: Yes, when it is antiferromagnetic. However, not when you magnetize the system and polarize the spins.

C. Domb: Nevertheless, I think the susceptibility might possibly be manageable.

P. W. Kasteleyn: I have been asked to report some measure- ments which have been performed by Dr. W. J. Huiskamp,

Dr. A. R. Miedema and Mr. R. F. Wielinga in the Kamerlingh Onnes laboratory in Leiden. Measurements were carried out on two groups of salts, one group representing the Ising system

[9] and the other group representing the Heisenberg model [10]. The first group consists of cobalt potassium sulfate and cobalt cesium chloride, which have very anisotropic g-factors and which

are neat Ising systems also in other respects. It was found,

e.g., that for the first salt y, the index for the susceptibility, is about 1.22, whereas the theoretical conjecture is y=1.25 for Ising systems and y = 1.33 for Heisenberg systems. For T> Tr the specific heat seems to diverge with a very high value of a in those cases. Namely, for the first salt a = 0.57 ±0.06 and for the second salt even a = 0.68 ±0.05 which is rather surprising. In most other salts measured up to now this coefficient of diver- gency was found to be much smaller. For T > Tr it seems that the experiments can be fitted very reasonably by a logarithmic

divergence. That these systems are Ising systems can also be

seen from the difference in entropy S at infinite temperature and at the Curie temperature. This difference was found to be

(S x —Sr )IR = 0.14 and 0.125 for the two salts mentioned, whereas

for a simple cubic spin V2 Ising system one finds 0.133 and for a Heisenberg system 0.25. In the second series of salts they took potassium chloride, rubidium chloride, ammonium chloride, and ammonium bromide. The structure is nearly body centered cubic. {S x —Sr )/R = 0.222, whereas for the Ising body centered cubic model this should be

0.107 and for the Heisenberg model about 0.25. The value of y

turned out to be 1.36, which is close to the value 1.33 expected

for the Heisenberg system. It seems that the specific heat data can be fitted accurately by a logarithmic type of divergence both below and above the Curie temperature. Only above the Curie temperature the curve is much lower. The specific heat data

plotted as a function of the reduced temperature T/Tr fit very

closely the same curve for all four salts. This gave the authors reason to think that these salts are fairly ideal systems, because

all deviations from ideality would be expected to be different

for the various salts. As the data lie on the same curve, these

deviations are apparently small. The data can be represented

by a logarithmic curve over a broad temperature range from

0.1 to 0.001 °K from the Curie temperature. The authors also

tried to fit the data to some power law, but it turns out that this can only be done for a much smaller temperature range, namely, between 0.01 and 0.001 °K from the Curie temperature. So

it seems that the logarithmic shape is really the best one.

S. A. Friedberg presented experimental data for the specific heat and susceptibility of the antiferromagnet CoCL>-6H:;C) near

its Neel point [11] drawing attention to the fact that these data

satisfy Fisher's relation C = Ad(\T)ldT. It was further noted

that the coefficient A yields a value for the curvature of the antiferro-paramagnetic phase boundary in the H-T plane at H~0 in good agreement with experiments.

E. Callen: I want to mention a quantity that one can measure

and that gives essentially the same information as the specific

heat. This is the magnetic contribution to the thermal expan- I, sion coefficient. This quantity is easier to obtain and in any

| case would allow an independent measurement of the same

thing. W. Marshall: Commenting on the paper of Dr. Heller, it is worth

noting that the neutron scattering results of Turberfield [12] on manganese fluoride showed again explicitly the fact that the

correlations in the a and b direction are more erratic than in the c direction as we would expect.

I have only one comment to' make about the theory you de- scribed which, of course, pulls out all the essential physics but which is not actually strictly correct. This is because in the fluc- tuations which matter for you, there is a fluctuation near the superlattice vector and at the superlattice vector at high temperatures your response function does not follow an exponential

decay. In fact you can show that its Fourier transform has a

66
Gaussian shape. Therefore, for example, the discrepancy between the A you needed and the A which was observed at high temperature may be due to a temperature dependence. However, a more important correction is probably due to the

fact that the shape changes from Lorentzian to Gaussian as you

go to high temperatures.

P. Martin: I am a little surprised that your theory agrees so

well with experiment. We have been hearing that the molecular field theory is not adequate near T, where it gives incorrect

values for the temperature dependence of the susceptibility. Since the theory you have presented is basically a time dependent version of the same molecular field theory and the quantity you are calculating so closely related to the susceptibility, why

does your theory work so well?

P. Heller: I agree that the molecular field theory should not be very accurate. In principle we could use the line width data

to determine the actual temperature dependence of the susceptibilities in the critical region. I think that both the data and the theory are too crude for this purpose. The fact that theory and experiment agree as well as they do may be fortuitous. However,- I feel that the qualitative nature of the line width anomaly has been correctly described by the theory.

References

[1] B. R. Livesay and E. J. Scheibner, J. Appl. Phys. 36, 3240

(1965).

[2] J. C. Eisenstein, R. P. Hudson, and B. W. Mangum, Phys. Rev. 37, A1886 (1965).

[3] J. C. Eisenstein, R. P. Hudson, and B. W. Mangum, Appl.

Phys. Letters 5, 231 (1964).

[4] E. Callen and H. Callen, J. Appl. Phys. 36, 1140 (1965).

[5] H. B. Callen, Phys. Rev. 130, 890 (1963).

[6] J. A. Copeland and H. A. Gersch, Bull. Am. Phys. Soc. 10,

304 (1965).

[7] L. Tisza and P. M. Quay, Ann. Phys. (N.Y.) 25,48 (1963).

[8] D. R. Thompson and O. K. Rice, J. Amer. Chem. Soc. 86,

3547 (1964).

[9] A. R. Miedema, R. F. Wielinga, and W. J. Huiskamp,

Phys. Letters 17, 87 (1965).

(10] A. R. Miedema, R. F. Wielinga, and W. J. Huiskamp, Physica 31, 1234 (1965).

[11] J. Skalyo and S. A. Friedberg, Phys. Rev. Letters 13, 133

(1964). J. Skalyo, A. F. Cohen and S. A. Friedberg, Proc.

9th Int. Conf. Low Temperature Physics, Columbus,

Ohio, 1964.

[12] A. Okasaki, R. W. H. Stevenson and K. C. Turberfield, Proc. Intern. Conf. on Magnetism, Nottingham, 1964.

209-493 0-66—

6

67

LOGARITHMIC SINGULARITIES

Chairman: C. N. Yang

69

The Lambda Transition in Liquid Helium

W. M. Fairbank

Stanford University, Stanford, Calif,

and

C. F. Kellers

Wells College, Aurora, N.Y.

This talk is based on work done at Duke University several years ago by C. F. Kellers, M. J. Buckingham and myself [1]. A major part of the work

constituted the Ph.D. thesis of Dr. Kellers [lb]. Much of what I will say has been previously reported

[1]. Still, when Dr. Green asked me to give this

talk I consented because I believed a review of the

lambda transition in liquid helium is particularly

pertinent to the present conference on critical phenomena. Not only were the Ehrenfest rela- tionships for a second order transition enunciated

first for the lambda transition in liquid helium, but

liquid helium also served, through the experiments

which I will describe, to first call attention to the

logarithmic character of the lambda transition which

is so much a part of the discussion of this conference.

The lambda transition in liquid helium is, of

course, of particular interest because of the unique

momentum ordering that takes place below the

lambda point which makes it perhaps similar but

certainly different from all other transitions. In

addition, however, of all the cooperative transitions,

it is the one which offers the best chance for experimental observations very close to the transition. Because liquid helium does not have any crystal

structure, because it is a pure liquid changing at the lambda point only in its momentum ordering, it

is apparently possible with sufficiently good thermal

equilibrium to make measurements as close as experimental techniques permit to the lambda point

without observing any of the broadening characteristics of other transitions. Ever since London suggested that liquid helium

represents in its superfluid phase a long-range order

in momentum space, physicists have been puzzled

by exactly what this means. London, Landau,

Supported in pari by the National Science Foundation and the Army Research

Office Durham.

Onsager, and Feynmann have suggested that liquid

helium as a superfluid is required to be irrotational.

If the helium rotates, it rotates according to the

Onsager-Feynmann theory, with quantized vortex motion [2]. Helium rotating slowly enough is pre- dicted to rotate in one single quantized state of

rotation with each helium atom having angular

momentum fi regardless of the size of the bucket in which it is contained [2, 3]. The theoretical suggestion is then the following: If a bucket of helium no

matter how large is cooled through the lambda

transition into the superfluid state while rotating

slowly enough, each superfluid helium atom will attain the same angular momentum. For the

slowest stage the superfluid will stop rotating and

the bucket, if suspended freely, will rotate faster

[4]. This experiment has never been performed.

Some physicists, including a graduate student

George Hess at Stanford, are trying to perform this experiment [5]. This then is the type of order transition which is predicted for liquid helium, longrange order in momentum space of a very special

kind.

Blatt, Butler, and Schafroth have taken another

approach to the problem [6, 7, 8, 9]. They questioned whether a real liquid, as contrasted with an

ideal Bose gas can exhibit arbitrarily long-range

order or correlation in momentum space. They

suggest that there may be a finite correlation length

of approximately 10~5 cm over which helium atoms

are correlated in momentum space, but beyond

which there is no correlation at all. In particular,

they developed a theory of a Bose liquid in which

helium atoms are correlated over a distance of about 10 -5 cm, the correlated atoms representing

the superfluid. By calculating the amount of this correlated phase p, they calculate the ratio of uncor- rected to correlated phase (normal to superfluid,

Pnlps) as a function of temperature, the correlation

length being an adjustable parameter. With a cor71
relation length of 10~5 cm, they were able to accomplish something which to my knowledge had not been accomplished before. They were able to derive p n lp.i as a function of temperature in agree- ment with experiment up through the lambda point

and with the same theory, to derive the specific heat

at the lambda point in quite good agreement with

experiment both as to shape, magnitude, and temperature of the transition. Their theoretical curves

are shown along with the then existing experimental

data in figures 1 and 2. One characteristic new

feature of this specific heat transition is that it is a

quasi-transition. Instead of being a second-order

transition with a discontinuity in the specific heat at the lambda point, the specific heat is rounded at the lambda point for a temperature interval of ap- proximately 10"3 deg which is a direct result of a

finite correlation length. This rounding could not have been observed in the then existing data. It also follows from their theory that curl vs = 0 is not an equilibrium property of superfluid helium II. Feynmann [10] took issue with the premise that the correlation length in liquid helium could be

cutoff at some finite distance, and suggested that the lambda point in liquid helium is not only sharp

at one-thousandth of a degree from the lambda

point but it is sharp at a hundred-thousandth of a degree. He believed that even though the correlation in liquid helium II gets weaker and weaker as the distance gets bigger, it still remains finite. Thus, no matter how large the bucket, liquid helium

would still rotate in a single quantum state if the

rotation were sufficiently slow. At the expense of bringing you a free quote which was only told to me by Feynmann and therefore is not quoted from the literature, I am going to quote an exchange between Feynmann and Blatt because

I think it illustrates the problem of long-range order

very clearly for this conference. "Blatt, 'according

to Feynmann', said to him, 'Feynmann, how is your

intuition so good that you can say that there is a correlation between helium atoms separated by the

size of the universe.' 'Feynmann said he replied',

'Blatt, how is your intuition so good that you can say

a correlation which decreases as 1/r2 and is down by

something like 1070 at the edge of the universe,

doesn't exist'." That is just the point, this correlation falls off very rapidly. If the helium is to stop

rotating in a rotating bucket when the bucket is cooled below the lambda point, then the rotational

speed must decrease as 1/r2 as the size of the bucket

becomes larger and larger. 72

10 r

8

-

0 I i i i

1

i i I 1.95 2.0 2.1 2.2 2.3

FIGURE 1. The specific heat of liquid helium versus temperature

in the neighborhood of the k-point. The curve is theoretical, the experimental points are taken from Keesom: Helium (Amsterdam, 1942), Table 4.20. The theoretical curve is a continuous and differenti- able function of temperature everywhere, represented by the same theoretical expression throughout the temperature region covered in ref. [8].

01 i ' 1 ' I i 1 sH- 195 2.0 2.1 2.2 2.3

FIGURE 2. The normal fluid concentration pjp in liquid helium

versus temperature, in the neighborhood of the k-point. The curve is theoretical, the experimental points are based on the second soun

data of Pellam. The theoretical curve is a continuous and ditferentiable function o temperature everywhere, represented by the same theoretical expression throughou

the temperature region covered in the ref. [9].
These conflicting theoretical predictions served as an incentive for us to perform a specific heat measurement to at least 10~5 deg of the lambda point.

This experiment was performed by Buckingham,

Kellers, and Fairbank, and has been reported in reference 1. Ultimately Cs was measured to within

10-6 deg of the lambda point T\, with equal precision

both above and below 7\. I would like to recall here just a few highlights of the experiment and

comment on the possible significance of the data. Many previous measurements have been made

on the lambda point, including an unpublished

master's degree thesis from our laboratory. The

difficulty with all these earlier experiments was that reasonably large amounts of liquid helium was used

in containers which were connected to the outside

world. To obtain the needed high temperature resolution it is essential that the attainment of equilibrium

be unaffected by the drastic change of thermal

conductivity of liquid helium at the lambda point, or by the onset of the creeping film. Both of these

requirements were met by permanently sealing the

helium (0.0587 g) in a cooper container (200 g), the

inside of which was in the form of fins so placed

that the helium was everywhere within 0.003 in. of the copper surface (fig. 3). With a heat input of

10 erg/s temperature equilibrium of greater than

10" 6 deg could be obtained, even in the helium I region. In order to eliminate the need for removing

exchange gas, the sealed container was suspended

in a vacuum and a mechanical heat switch provided

for contact with the bath when required.

Measurements were made by means of a carbon

resistor with a minimal detectable change representing 2.10~ 7 deg. It was possible to make measurements both while increasing and decreasing the

temperature.

Figure 4 shows the data. These of course are not new data just taken, but the reason I was asked to present them is that perhaps they are very similar

to data which are now being obtained on other transitions. The specific heat, C.s , is plotted as ordinant

versus 1 7^ — 7"^ | . In order to show the nature of the

transition very near the lambda point, the data are shown on successively expanded temperature scales. To aid in a visualization of the very large amount of expansion of each successive curve, a small vertical line has been drawn just above the origin.

The width of the line indicating the fraction of the curve which is shown expanded in the curve directly

to the right. The ratio of the expansion between

Figure 3. Schematic diagram of adiabatic chamber for specific heat measurements [7c]/ (A ) stainless steel wire for closing heat

switch; (B) brass cap on filling capillary; (C) wires connecting

to heater and resistors on sample; (D) cotton plug dyed with carbon black as radiation trap; (E) indium "0" ring; (F) filling capillary; (G) Kovar seal used as thermal short for wires to sample; {H) radiation shield and thermal short for heat switch;

(/) nylor cord; (J) three prongs of heat switch {copper); (K) indium coating and suspension for sample; (L) temperaturesensitive resistor; (M) heater; (N) sample cavity; (O) temperature-sensitive resistor; (P) copper shield over resistor; (R) calorimeter wall.

Figure taken from ref. (lc].

the first and last curve is about 5 X 104

. Thus if the projected slide of the first diagram is 10 ft,

it would have to be expanded to 100 miles to obtain

a 10 ft projection of the third figure.

It can be seen that as the specific heat is displayed on a more and more expanded scale, it maintains

the same geometric shape. There is certainly no

indication of a rounded quasi-transition as predicted

by the theory of Blatt, Butler, and Schafroth.

Figure 5 shows the data in a form with which you

are all now familiar. Cs has been plotted as ordi- nant against T—T\ in degrees Kelvin on a logarithmic scale. It is seen in this kind of a plot that near the lambda point on each side there is a factor

of 10 4 in T—TK over which the data fall in two

73
-1.5 -1.0 -.5 0 .5 1.0 1.5 -4 -2 0 2 4 6 -20 -10 0 10 20 30

( T - Tx

) degrees (T-Tx

) millidegrees (T-Tx) microdegrees

Figure 4. Specific heat of liquid helium versus T — T* in °K [1]. 0 represent data cif Kellers, Fairbank and Buckingham [1]. <8> represent, above 1.5 °K, data of Hill and Lounasmaa

[20] and Lounasmaa and kojo [18] and. below 1.5 °K, data of Kramers. Wasscher, and Gorter |21[. Solid line near

\-point represents eq (1). Width of small vertical line just above origin indicates portion of diagram shown

expanded (in width) in the curve directly to the right. Figure from ref. [lc].

Figure 5. The specific heat of liquid helium versus log\T—Tx | [1]. 0 represent data of Kellers, Fairbank and Buckingham 1 1 ]. # represent, above 1.5

3K

data of Hill and Lounasmaa [20] and Lounasmaa and Kojo |18] and, below 1.5 °K, data

of Kramers, Wasscher, and Gorter [21]. Two parallel straight lines represent eq (1). The broken line represents formula shown in the figure. Figure from ref. [lc],

parallel straight lines which are branches of the

expression

Cs = 4.55-3.00 log 10 |:T-7\|-5.20A (1)

where A = 0 for T < TK , and A = 1 for T > 7\. One

reason that this was very interesting and exciting

at the time, is that Onsager had published the exact

solution of the two-dimensional Ising model [11]-

The two-dimensional Ising model gives a logarithmic

transition but only for an exact solution.

I l I l l I 1 -8 -4 0 4 8 T-Tx (MICRODEGREES)

Figure 6. Plot of Cs versus T — Tx very near \-point.

Data taken from one single run randomly selected from five runs where there were, no disturbing influences. Figure from ref. [lb].

Figure 6 shows previously unpublished data from

a single run taken from C. F. Keller's Ph.D. thesis

[lb]. About 20 runs were taken back and forth through the lambda point over ± 10 ;u,deg pen

minute. About half of the runs were discarded

because the rate of cooling or heating at the begin

ning or end of the run was different. In about half of the remaining runs there was a big spike in the! data due to someone opening the door and so forth. Of the remaining 5 runs, the one shown in figure t

is typical and picked at random. The data is

74
averaged over 1 /udeg so the lambda point is still clearly observable. The three solid curves are placed in the di'agram to show the effect of averaging over finite temperature intervals 10" 6 deg

Kelvin and 2 X 10"6 deg Kelvin. The curve labeled

infinity is the curve given in eq (1) and represents

infinite resolution. As stated above these data were

averaged over 1 fideg interval and should corre- spond to that curve labeled 10

6

. Thus we see from the experimental data that the lambda point, instead of being a rounded quasitransition as suggested by the theory of Blatt,

Butler, and Schafroth assuming a finite correlation

length, is in fact, sharp to at least 2 orders of magnitude closer in temperature to the lambda point

than predicted by their theory. The simple logarithmic behavior of the data shown in figure 4 which

extrapolates to a logarithmic singularity of the

lambda point has stimulated more detailed experimental and theoretical investigation of the lambda

point, as witnessed so completely by this confer- ence. Although the two-dimensional Ising model

has been solved exactly by Onsager giving such a logarithmic singularity, it is obvious from this conference that an exact solution to the three- dimensional Ising model will not be easily found.

In figure 5 the curve of the form

A

| f— T>,\ ll5 ~^ ^ Tx\ + C

which has been fitted to the data for T —T\ between

2 X 10" 1 and 2 X 10"3 as shown. This is the fundamental form suggested by the Pade approximation

of the three-dimensional Ising model. It is seen

that although it fits the data rather well for [T— 7\) > 2 X 10 -3 °K, it departs very dramatically

for temperatures closer to the lambda point. Unless data are obtained close enough to the lambda

point one could not differentiate between this ex- pression and the logarithmic singularity.

Much of this conference will be taken up with the exact nature of other kinds of lambda transitions. However, I wish to compare quickly two results by

Robinson and Friedberg [12] and Skalyo and Friedberg [13] on the lambda transition in hydrated nickel

and cobalt chloride with specific heat data on liquid

helium.

Figure 7 shows the specific heat data of nickel

chloride to within 0.07 °K of the lambda point

from above, and within 0.2 °K from below the transi- tion temperature. Figure 8 s hows the same data

with Cp plotted against \og\T—Tk \. If one defines a reduced temperature by t = \ T — 7\|/7\ and'

compares the curve for nickel chloride with the

helium curve, one sees for the same value of t both curves show nonparallel straight fines. Figure 9 shows the data for cobalt chloride taken

to smaller values of f. These data show two parallel

straight lines and then a flattening out of the data.

75
Figure 8. Cp versus |T-TN | and Cp versus log|T-TN | for NiCl2 -6H2 0.

Dala by Robinson and Friedberg taken from ref. [12].

A comparison with helium indicate agreement

as to the two parallel straight lines. The flattening out at values of t closer to the lambda point would

be expected in either an experiment or theoretical

model where long-range correlation is cutoff. These

two sets of data are then seen to be consistent with

the lambda transition for helium and indicates the

possibility that other lambda points have the same

form as the helium transition. The suggestion that the specific heat of helium

might become infinite at the lambda transition

was first made by Tisza [14]. The first experimental evidence concerning the logarithmic nature

of the lambda transition was obtained by Atkins and

Edwards [15]. They suggested that a logarithmic

term could be used to derive the result of their measurements of the thermal expansion coefficient below the lambda point. Figure 10 shows the data on the expansion coefficient of Atkins and Edwards,

Chase and Maxwell, and Kerr and Taylor. It is seen that the data can be represented by two parallel straight lines from the plot of the expansion

coefficient versus log(r— T\). This curve is re- produced from the paper by Kerr and Taylor [16].

Figure 11 shows the relative molar volume of helium

in the vicinity of the lambda point as given by Kerr

and Taylor. It is seen that the lambda point comes where the slope is infinite rather than at the

point where the molar volume is minimum.

Buckingham [lc] has derived rigorously the ther- modynamic consequences of lambda transitions

characterized by the absence of a latent heat, but

at which the specific heat at constant pressure

becomes infinite. Pippard [17] had previously

considered such a transition and worked out ther- modynamic relationships based on the assumption

that the entropy surface is cylindrical near the

lambda point. The thermodynamic relationships

worked out by Buckingham and Pippard can be

4.5

4.0

3.5

UJo 3.0 S 2.5 Q\

< 2.0

o 1.5

1.0

0.5

00

10

.,s.~,^>,".v. :•>>

"V,

VX

X

V X

X.

> SKALYO AND FRIEDBERG MSl x ^* T>TJ »

•T<TN

' ROBINSON AND FRIEDBERG [|2]

L

10

-3 10 ' 10" IT-TJ CK)

I0

l 10'

Fici re 9. The heat capacity, C,„ 0/C0CI2 6H.O as a function o/ln |T-TN | for TN = 2.2890 °K.

Figure taken from ref. [13]. 76
FIGURE 10. The expansion coefficient, as

. near the \-point as a function o/log|T—

Curve from Kerr and Taylor. Figure taken from ref. (16].

zjTS 2J6 2J7 tie f!i9 T5i 2 TEMPERATURE (°K)

FIGURE 11. The molar volume of He* in the vicinity of the k-point.

Data by Kerr and Taylor. Figure from ref. [16].

used to compare the behavior of various thermodynamic properties in the neighborhood of the lambda

line [lc]. In reference lc and 16 the specific heat

and coefficient of expansion are compared using the

Buckingham relationship. The slope of the lambda

line (dSjdT)\ is taken from the data of Lounasmaa

and Kojo [18] and Lounasmaa and Kaunisto [19]. The data can be plotted in a parametric plot which

should give asymptotically a single straight fine. Two parallel straight lines are obtained by Kerr and

Taylor using the data shown in figures 4 and 10. This shows that the fundamental form of the data is correct but the details need to be refined. It is dif- ficult to obtain exactly the same experimental conditions in different experiments very close to the

lambda point. It seems worthwhile to make simultaneous specific heat and expansion coefficient measurements to within 10"6 °K of the lambda

transition. The behavior of the velocity of sound in the vicinity of the lambda point can also be predicted using

specific heat and other thermodynamic data [lc]. Here again there is approximate but not exact agree- ment. Since there will be a separate talk on this subject I will not present further data here. So in summary, the specific heat expansion coeffi- cient and velocity of sound all give evidence to the

logarithmic nature of the lambda transition in helium

although refined details of comparison still need to be worked out. I would like to end by again mentioning that liquid helium presents the unique opportunity to obtain data close to the lambda point. The

question naturally arises, is the transition in liquid

77
helium similar to lambda transition in other substances, and does liquid helium present an idealized

experimental model for all lambda transitions? At

first glance one might think that liquid helium

doesn't. I would, however, like to point out a sense

in which liquid helium does present such a model;

a sense that has been emphasized by Buckingham in private discussions.

When one goes from the normal liquid to the

superfluid in liquid helium, one gets a change from a normal system to something which is suddenly

ordered throughout the whole container. In this particular case, it is an ordering in the superfluid.

The superfluid atom in one part of the container

knows what is happening to the superfluid atom in a

different part of the container, and how long that order is depends only on the size of the container.

One can disturb the system without destroying this order throughout the entire system provided the

disturbances, for example rotational speeds, become

smaller and smaller as the size of the system

becomes larger and larger. In the case of superconductors one has a quantized magnetic flux over supposedly the distance of

the length of the wire in superconducting magnets

that may be thousands of feet long. The length

of the ordering depends only on the size of the super- conducting loop and especially in the case of super- conductors, exists in the presence of large quantities

of impurities. Now there is different kind of order

parameter in superconductors and that is the length

of the correlation between electrons in a pair. This

is only 10~4 cm, but the two electrons in a pair have

equal and opposite momentum. The momentum of each pair is zero and therefore every pair throughout

the whole superconductor has the same zero momentum. Thus there is a long-range order in momentum throughout the entire superconductor even though each pair is paired over 10 -4 cm. One doesn't have this momentum ordering in any

other kinds of transition besides superconductors

and liquid helium. But on the other hand, as Buckingham is going to point out at the end of the morning, one has a transition from a state where ordering

exists within a small cluster in a particular part of

a container, to a system where one has suddenly,

for example in a liquid gas transition, order throughout the entire container. Part of the atoms have one special density and they are all together in the

container, and all the rest of the atoms have a dif- ferent density. This sudden transition is to a state of order which depends only on the size of the j container.

I want to close with this question. Is the lambda

transition in liquid helium, except for the fact that one has a lambda line, an exact model for other

transitions when long-range correlations are not

cutoff in other experiment systems or theoretical

models?

References

[1] (a) W. M. Fairbank, M. J. Buckingham, and C. F. Kellers, Proc. 5th Int. Conf. Low Temp. Phys. (Madison, Wis.,

1957), p. 50; (b) C. F. Kellers: Thesis (Duke University,}

1960): (c) M. J. Buckingham and W. M. Fairbank, Progress;

in Low Temperature Physics, Vol. 3, edited by C. J." Gorter (Amsterdam, 1961), ch. Ill; (d) W. M. Fairbank,

Estratto da Rendiconti della Scuola Internazionale djj Fisica (E. Fermi) XXI Corso.

[2] R. P. Feynmann, Progress in Low Temperature Physics,)

ch. 2. [3] F. London, Superfluids, Vol. II, (John Wiley & Sons, New?

York, N.Y., 1954).

[4] H. London, Physical Society 1946 Cambridge Conference; Report (London, 1947) p. 48.

[5] W. M. Fairbank and C. B. Hess, Angular Momentum of} Helium II in a Rotating Cylinder (Proceedings of the IX

International Conference on Low Temperature Physics,!! P: 188 (Plenum Press. New York, 1965).

[6] J. M. Blatt, S. T. Butler, and M. R. Schafroth, Phys. Rev. 100, 481 (1955).

[7] S. T. Butler and J. M. Blatt, Phys. Rev. 100, 495 (1955).

[8] S. T. Butler, J. M. Blatt, and M. R. Schafroth, Nuovo Cimento

4, 674 (1956).

[9] J. M. Blatt, S. T. Butler, and M. R. Schafroth, Nuovojj

Cimento 4, 676 (1956).

[10] R. P. Feynmann, private communication.

[11] L. Onsager: Phys. Rev. 65, 117 (1944); G. G. Newell and1 E. W. Montroll: Rev. Mod. Phys. 25, 353 (1953).

[12] W. K. Robinson and S. A. Friedberg, Phys. Rev. 117 , 402:

(1960).

[13] J. Skalyo and S. A. Friedberg, Phys. Rev. Letters 13, 133

(1964). See also the discussion remark of Dr. Friedbergjj on page 66.

[14] L. Tisza, Phase Transformations in Solids, edited by; Smolychowski, Mayer, and Weyl (New York, 1951).

[15] K. R. Atkins and M. H. Edwards, Phys. Rev. 97, 1429,

(1955).

[16] E. C. Kerr and R. D. Taylor, Annals of Physics 26, 292-306!

(1964).

[17] A. B. Pippard: The Elements of Classical Thermodynamics

(Cambridge, 1957), ch. IX.

[18] O. V. Lounasmaa and E. Kojo, Physica 36 , 3 (1959).

[19] O. V. Lounasmaa and L. Kaunisto, Ann. Acad. Sci. Fennicae( A6, No. 59 (1960).

[20] R. W. Hill and O. V. Lounasmaa, Phil. Mag. 2, 145 (1957).! [21] H. C. Kramers, J. D. Wasscher, and C. J. Gorter, Physica

18, 329 (1952).

78
The Specific Heat of He3 and He4 in

the Neighborhood of Their Critical Points*

M. R. Moldover and W. A. Little

Stanford University, Stanford, Calif.

Introduction

Recently Bagatskif, Voronel', and Gusak [1] showed that the specific heat at constant volume

of argon exhibited what appears to be a logarithmic singularity at the critical temperature (7V) for measurements taken at a density near the critical density. This singular behavior is in sharp contrast to the predictions of the traditional view of

this phase transition by Landau and Lifshitz [2]. However, the behavior is precisely that to be ex- pected for the so-called "lattice gas" model for

the liquid-gas transition. Lee and Yang [3] have

shown that the partition function of a classical

gas of particles moving on a discrete lattice with

a repulsive force preventing double occupancy of any site, and a nearest neighbor attraction can

be mapped precisely onto the partition function of an Ising model of a spin system in an external magnetic field. The specific heat for this Ising model

in zero field exhibits a logarithmic singularity at the Curie point. The specific heat for the corre- sponding lattice gas on the critical isochore exhibits

a logarithmic singularity at the critical point.

The measurements on argon then indicate that

for a real gas the specific heat behaves in a similar manner to that of a lattice gas. We have investi- gated this point further by studying the specific heat at constant volume (Cr) of both He3 and He4

at densities close to the critical density. We

have done this for two main reasons. Firstly, to see whether the behavior observed for argon is also observed for helium, for which quantum effects should be important, and secondly, to investigate

the detailed nature of the singularity in the pressure-density plane, not only on the critical density,

but also in its immediate neighborhood. Yang

and Yang [4] have conjectured that the quantum

effects would reduce the magnitude of the singular

contribution to the specific heat in helium. Our

results confirm this view.

*Work supported in part by the National Science Foundation and the Ghee of Naval

Research.

Experimental Procedure

There are several practical advantages to using

helium rather than another noble gas for Cr meas- urements near the critical point. The low heat

capacity of metals at liquid helium temperature

permits one to use a massive calorimeter of large

surface to volume ratio. Thus the path for heat

transfer through the helium may be kept short. In addition, more nearly adiabatic conditions and

high resolution thermometry are most easily at- tained at liquid helium temperatures.

Our calorimeter was built of two OFHC copper

parts. The helium was contained in the lower

part in 50 slots. Each slot was 0.01 cm wide to facilitate good thermal contact between the calorimeter and the helium. The slots were made only

0.3 cm deep in an effort to minimize possible gravi- tational effects. [In contrast a thin stainless steel

shell 10 cm high and 4 cm in diameter containing

a magnetic stirrer was used for the work on argon

[1, 5]. This construction was necessary to obtain

a low ratio of heat capacity of the calorimeter to

its contents while maintaining constant volume at the high critical pressure of argon.] The lower part

of our calorimeter was wound with a constantan

heater and had a carbon resistor clamped and

cemented to it. The calorimeter was supported on

nylon threads in an evacuated chamber. Thermal

contact to the bath was made with a mechanical

heat switch. A stainless steel filling capillary

5 in. in length and 0.006 in. I.D. led from the calorimeter to a needle valve. The dead volume was

about V2 percent of the total volume of the calorimeter. The helium was admitted to the calorimeter via a Toeppler pump which was used to measure the volume of gas to an accuracy of about

0.2 percent.

One of the precautions taken was the measure- ment of the stray heat input to the calorimeter

before and after each data point. The approach

to temperature equilibrium of the calorimeter was

observed after each heating interval. As Tc was

79
approached, the equilibrium time increased from

a few seconds to several minutes and became one

of the limiting factors in this measurement.

A Vio W Ohmite carbon resistor of nominal re- sistance 560 fi was used as a secondary thermometer. Its resistance was measured with a 100 c/s

bridge using a lock-in amplifier. Temperature

changes of about 2 X 10 « °K at 5.2 °K and 1 X 10"fi °K at 3.3 °K could be observed. At the end of a run exchange gas was admitted to the vacuum

space and the resistor was calibrated against the

vapor pressure of the He4 bath using the T5S scale either from 4.2 to 2.9 °K or from 5.15 to 4.2 °K

depending on whether He3 (TV = 3.3 °K) or He4

(TV = 5.2 °K) was being studied. A smooth resist- ance-temperature relation was fitted. Hence

relative specific heat measurements near the

critical points are not sensitive to errors either

in the calibration or in the vapor pressure-temperature scale near Tr . Calibration was always done

as the bath temperature was reduced. A hydrostatic head correction was made. All residuals were less than 8 X 10~4 °K from a fit to the ClementQuinnell [6] formula and less than 3 X 10-4 °K from

a four constant formula. Each run, Tc was tentatively defined as the temperature at which

C, for a density near pc fell most abruptly. The

final value of Tc was then chosen to give the best

fit to |=-alo& (^)+ 6 ,1)

over the range 10"3 °K<(T( ~T)< 10"' °K. The

adjustment was less than 3 X 10"4 °K. This value

of Tc was also used for analyzing data taken during

the same run at densities far from pc . On four

separate runs with He4

, Tc was found to be 5.189

±0.001 °K. This is considerably below the value

Tc = 5.1994 °K defined on the Tf, H temperature scale. A single run with He3 also yielded a value for T, well below the accepted value.

Results

Figure 1 is a general view of the specific heats

at the critical density of He3 and He4 at low temperatures [7]. We have indicated the temperature

range covered by these measurements. In figure 2 we have plotted on both a linear and logarithmic

scale measurements of C, of He4 at a density within

0.5 percent of the critical density. In the two

o I I I i 0 1 2 3 4 5 6 TEMPERATURE °K

FIGURE 1. Cv o/He3 and He4 at their res/iective critical densities

is plotted as a function of temperature.

The solid curves are the present work.

phase region below Tc , C, exhibits logarithmic

behavior.

In the one phase region above Tc two alternatives

have been suggested: First, Cr 91 log {T— Tc ) has

been suggested by an exact calculation on a two! dimensional Ising model [9], by C, measurements

on argon [1] and oxygen [8], and by velocity of sound measurements on helium four [10]. Second,

C, ^ (T—TcY

a for some small positive a (such as

Vs) has been suggested by Fisher [11] on the basis

of approximate calculations on a three dimensional

Ising model and his analysis of the argon and

oxygen data. We do not believe the present data

permit one to distinguish between these two

alternatives. In figure 3 similar plots are given for He3 for a density of 0.985p,.. Logarithmic behavior is again

observed for T< Tc . The data for argon, oxygen

[8], and both isotopes of helium expressed in the dimensionless form (1) are summarized in table 1. We see that the coefficient a decreases from Ar to He4 to He3 in agreement with the conjecture of Yang and Yang [4]. We note that the coefficient 0.62 for He4 is very close to that observed for the

specific heat along the saturated vapor pressure curve near the A-point of He4 (0.64) by Buckingham

and Fairbank [12] and may have particular sigl

nificance in explaining the quantum transition. Yang and Yang [4] have pointed out that one may

write in the two phase region,

c'=-w(S),^(f)r *
10

( T - T ) °K

10

40

30

20

0

10

_ \- He

3

P = -98%c ^- ^ - -.37 log (

T|L

T )+I.7

t < tV-.

i

<-> 1.4

10

( T - T ) °K

10

| FIGURE 2. Cv of He4 is plotted against |T — Tc | at a density of

0.998pc

.

TABLE 1. Coefficients a and b of eq (1) for 0>, Ar, He4

, and He3

Element a b: T<TC

o2 2.4 10

Ar 1.8 8

He4 0.62 2.0

He3 .37 1.7

where p is the chemical potential. They suggest

that both ((Pfi/dT2

^- and (d2PldT2

) v become infinite

at Tc for a real gas, whereas {d2 p,ldT2

) v = Q for the two dimensional Ising model. Figure 4 shows

data on He4 taken at the densities 0.96 pc and

1.07 pc . Tc for these plots was chosen from data

(not shown) taken during the same run at 0.99 pc . The two phase region (lower temperatures) is the

upper branch of each curve. The difference between the upper branches in this region is proportional to (d^PldT2

),. It is clear that (cPP/dP)t

.

FIGURE 3. Cv of He3 is plotted against |T — Tc | at a density of

0.98. Pc60

0

10

.,...07,, ^ • p =0.96

/o

c

1 i i 10

( T - T ) °K

10

Figure 4. Cv of He4 is plotted against |T —

T

c | at densities of

1.07 pc and 0.96 pc.

does not approach zero as T approaches Tc . A

similar plor with the ordinate heat capacity/volume

rather than heat capacity/mole indicates that

(d2 p/dT2

) v also does not approach zero as T ap- proaches Tc . This is consistent with the suggestion

of Yang and Yang.

81
In the one phase region the two curves on figure 4

differ drastically. This suggests particular attention to density measurements will be required to determine behavior at the critical density. We

also note the transition to the one phase region is particularly well marked in the case of p=1.07 pc . This suggests specific heat measurements may be

a sensitive method of determining the coexistence

curve.

We thank Dr. Derek Griffiths, Dr. R. S. Safrata,

Dr. C. F. Kellers, and Dr. M. H. Edwards for many

useful discussions and Dr. Derek Griffiths and Donald Moldover for much help in analyzing data. We

thank Professor C. N. Yang for his suggestions,

interest and advice. Some of this data was pre- sented at the LT9 Conference in 1964.

References

[1] M. I. BagatskiT, A. V. Voronel', and V. G. Gusak, Zh. Eksperim. i Teor. Fiz. 43, 728 (1962). [English Transl:

Soviet Phys.-JETP 16, 517 (1963)].

During the years between 1956 and 1963 I measured the density of saturated He4 liquid and vapor

along the coexistence curve from 0.30 to 0.99 T( , by refractive index measurements with a modified

Jamin interferometer [1-3]. Between 0.95 and

0.993 Tc , 76 experimental points were taken.

Advantages of this method of density measurement

near the critical point are

1. High resolution of density to ~ 0.01 percent

is possible.

2. There are no dead space corrections to the

observed densities.

3. The density of a horizontal "slice" only 1 mm

deep is measured and hydrostatic head effects were always less than 0.05 percent in density.

4. If temperature inhomogeneities of ~ 10 4 °K

appear in the optical cell the fringes disappear

and no results are obtained.

On leave I'rnm K.>\al Militar) College, Kingston. Ontario. Canada.

[2] L. D. Landau and E. M. Lifshitz, Statistical-Physics (Addi- son-Wesley Publishing Co., Inc., Reading, Mass., 1958).

[3] T. D. Lee and C. N. Yang, Phys. Rev. 87, 410 (1952).

[4] C. N. Yang and C. P. Yang, Phys. Rev. Letters, 13, 303

(1964).

[5] A. V. Voronel' and P. G. Strelkov, PTE 6, 111 (1960). [English Transl: Cryogenics (G. B.), 2, 91 (1961)].

[6] J. R. Clement and E. H. Quinnell, Rev. Sci. Instr. 23, 213

(1952).

[7] The dashed curve for He4 was calculated from data of ref. 12; R. W. Hill and O. V. Lounasmaa, Phil. Mag. 2, Ser. 8, 145 (1957); O. V. Lounasmaa and E. Kojo, Series A., VI. Physica 36, 3 (1959); H. C. Kramers, J. D. Wasscher and

C. J. Gorter, Physica 18, 329 (1952); M. H. Edwards,

Canad. J. Phys. 36, 884 (1958); H. van Dijk, M. Durieux,

J. R. Clement, and J. K. Logan, J. Res. NBS 64A (Phys. and Chem.) No. 1, 1 (1960). The dashed curve for He3 was calculated from data of Henry Laquer, Stephen G. Sydoriak. and Thomas R. Roberts, Phys. Rev. 113, 417

(1959): R. H. Sherman, S. G. Sydoriak, and T. R. Roberts,

J. Res. NBS 68A (Phys. and Chem.) No. 6, 579 (1964); and T. R. Roberts and S. G. Sydoriak, Phys. Rev. 98,

1672 (1955).

[8] A. V. Voronel', Yu R. Chashkin, V. A. Popov, and V. G. Simkin, Zh. Eksperim. i Teor. Fiz. 45, 828 (1963) [English Transl: Soviet Phys.-JETP 18 , 568 (1964)].

[9] L. Onsager, Phys. Rev. 65, 117 (1944).

[10] C. E. Chase, R. C. Williamson, and Laszlo Tisza, Phys. Rev. Letters 13, 467 (1964).

[11] Michael E. Fisher. Phys. Rev. 136, A1599 (1964).

[12] M. J. Buckingham and W. M. Fairbank, The Nature of the

\-Transition, Progress in Low Temperature Physics,

III. P. 86 (North Holland Publishing Co., Amsterdam, I 1961).

We analyzed our data on the coexistence curve between 0.98 and 0.993 Tc by extending Landau

and Lifshitz's theory using a fit to a power series jj expansion in volume and temperature [3]. If, > as is now believed, the critical point is a singular , point, this whole expansion procedure is questionable, since the coexistence curve cannot in principle be represented asymptotically by a Taylor

series in the density about such a nonanalytic

point. A reanalysis of this same data will now|

be presented, without the use of a Taylor series;* expansion.

The coexistence curve of saturated liquid andl

vapor densities pi and ptf as a function of tempera-jj

ture, is symmetrical, not about the critical density,!

but about a "rectilinear diameter." This fine of! mean densities of vapor and liquid passes through

Pc the critical density, and for He4 would extrap-| olate linearly to 1.1 pc at T=0. The symmetry of the coexistence curve is obscured on a plot of satThe Coexistence Curve of He4

M. H. Edwards*

Stanford University, Stanford, Calif.

82
urated molar volume V\ and V,, of liquid and vapor

since the mean volume curves sharply away from

the temperature axis at lower temperatures. In examinations of the shape of coexistence curves,

it is customary to consider the quantity (p/ — pfl ) or (pi — p<j)\2p<- as a function of (Tc — T). If one

attempts to write

Pi ~ P<i 2Pr A(Tr -T)e.

(1)

where A and fi are constants, then classically (van

der Waals), /3 = V2, whereas a variety of experiments

suggest that /3 = 1/3 for a certain range of temperatures. Accurate values of Tr are needed for meaningful tests of such a relationship. Furthermore,

the slope of the "rectilinear diameter," or line of mean densities, differs from substance to substance,

so that the similarity of coexistence curves may be

obscured by comparing experimental data in that manner. M. J. Buckingham has suggested that

the shape of the coexistence curve should be

analyzed using the natural variable

X = Pi- Pa

p, + P!l V,,+ V,' 2p

Kl^P'-py

(2)

Advantages of this variable are

(1) X ranges from 1 to 0 as T ranges from 0 to Tc for any substance,

(2) there is equal symmetry using either density

or molar volume, and the effect of the slope of the

"rectilinear diameter" is entirely removed, and

(3) if we plot X", ^where n=

~j^J- against T, we

need not know Tr or pc or Vc and in fact may de- termine Tc by such plots.

Figure 1 shows how plots of X2 (or /3 = 1/2), and

Xs (or /3= V3), appear for He4 over the whole range

of measurements [1, 2, 3] from 0.3 to 0.99 Tc . Clearly, Xs is nearly linear, (or fi = Vz), above about

0.8 Tc (but not too near Tc , see later), in agreement

with many other measurements for many fluids. Note that the classical X2 is not linear over any

extended range of temperature.

M. J. Buckingham has shown [4] that the simplest

singular entropy surface which is consistent with

a logarithmic infinity in Cr at the critical point,

would imply a coexistence curve whose asymptotic

form as T—> Tc is given by

X2 \-\nX at (3)

' • •

He COEXISTENCE CURVE

vs TK

vs T,58 WHERE x: Vg-Vj _Pl -Pq P,+ Po

FIGURE 1. Temperature dependence ofX2 and X3 for He4

. X3

is linear above 0.8 7V.

where a is a constant, and t = Tc — T. The quantity

X2 l(l — \n X) lies between X2 and X3 for the whole

temperature range. The question of the asymptotic form of the co- existence curve of He4 will now be examined

using the 76 experimental points listed in table III of reference 3. All these points were taken within

250 mdeg of 5.1994 °K (the critical temperature of

He4 on the 1958 He4 scale of temperatures [5]). The temperature of each data point was obtained

directly from the measured saturated vapor pressure. Although both Vg and V\ were not often

measured at precisely the same temperature, we may evaluate an X for each experimental point by

writing

X =

2Va -i. orX=l-- 2V,

(4)

For each temperature at which either Vg or Vi was

measured, the quantity (Vg + Vi), which varies rather

slowly and smoothly with temperature, was read

by interpolation, or, within 50 mdeg of Tc , by linear

extrapolation. Thus 76 values of X are obtained

within 250 mdeg of Tc . The added uncertainty

in X produced by the uncertainty in the value of

Vy+Vi falls from 0.6 percent at £ = 35.9 mdeg to below 0.3 percent above £ = 50 mdeg.

Figure 2 shows a graphical test of the three

functional forms for the coexistence curve of He4

for all points within 250 mdeg of 5.1994 °K. The

straight line drawn on the plot of X2 is a least83

209-493 0-66—

7
He

4 COEXISTENCE CURVE

200 150 100 50 0 m deg K

Figure 2. Tests of the asymptotic form of the coexistence curve for He4 above 0.95 Tc

.

squares-fitted straight line for the 39 points within

100 mdeg of Tc . The lines shown on the plots of

X2 /{1— In X) and of X3 are least-squares-fitted fines

Table 1. Linearfits of coexist

fc

for the 65 points within 200 mdeg of Tc . Clearb over this temperature range the data are repre sented best by the function X2I(1 — In X). Note

particularly the marked curvature of X3 as T—> Tc Table 1 shows the slopes and intercepts, will standard errors, computed by least squares fits o the data in the form

f[Xi) = ati + aATc (5

where j{X) is X1

, X2 I(1 -In X), or X3, and when

£ = 5.1994 — r58 , in millidegrees, and kT= the shif in critical point implied by a linear extrapolatioi

of fiX) to zero. None of these asymptotic fit extrapolates to the value rc = 5.1994 °K assumed ii the r58 scale of temperatures. Yang and Yang [t have suggested that the T5S scale may be in erro near Tc , and that the actual critical temperatur

should be lower.

As a more sensitive test of the experimental ev dence in favor of any of these functional form

for the coexistence curve, we plot the residual

Ati = ti--f(Xi)-ATc (i I a

for each of six fits with f(X) equal toX2

, X2l(l — \n),

j and X3

, over the ranges t < 100 mdeg and

ce curve data to three functional

Range of data in fit

f(X)

t < 100 mdeg (39 points) t < 200 mdeg (65 points)

103a (mdeg K)- 1 ATC mdeg K

103a (mdeg K)" 1 ATC mdeg K

X2 1.005 ±0.004 + 4.2±0.3 0.882 ± 0.007 + 14.2±0.9

X2 \-\nX 0.527 ±0.002 -7.5 ±0.3 0.513 ±0.001 -5.9 ±0.3

X3 0.404 ±0.003 - 18.1 ±0.4 0.429 ±0.002 -20.8 ±0.4

84
I50mdeg 100

IGURE 3. Deviations from linearity, expressed in millidegrees, for linear fits of the data for t < 200 mdegfor the threefunctions

f(X) = X2

, XV(l-ln X) and X3

.

+ 2 r

4

+ 2

-2 1-inX

+ 2 —

-2

4

150 100 mdeg 50

t = (5.I994-T58)

IGURE 4. Deviations from linearity, expressed in millidegrees, for linear fits of the data for t < 100 mdegfor the three functions

f(x) = x-. x

2

/( 1 - In x), and x

:i

.

t < 200 mdeg. Figures 3 and 4 are plots of these

deviations from linearity, expressed in millidegrees.

Figure 3 shows clearly that both X2 and X3 show

systematic deviations fitted within 200 mdeg of

T. Figure 4 shows that, using only the data points

within 100 mdeg, both X2 and X2/(l-ln X) ap- pear to give almost equally good fits. However,

X2/(l — In X) gives much the best fit when all points within 200 mdeg of Tc are included.

Conclusions

The main conclusions of this analysis of the form

of the He4 coexistence curve are

1. X3 is not asymptotically linear in T above about

0.98 Tc, despite its linearity from 0.8 to 0.98 Tc . 2. X2 can be shown to be asymptotically linear

only above 0.98 Tc , with the critical temperature

4 mdeg above the T^h value. Such a shift is in the

opposite sense to that expected [6], and this functional form is inconsistent with the observed [7] logarithmic singularity in Cr below Tc3. X2I(1 —In X) is the best asymptotic form, with

the critical point lowered 6 to 8 mdeg below the r5 8 value. This shift is in the sense expected [6], and the functional form is consistent [4] with the

observed [7] logarithmic singularity in Cr below Tc.

I am grateful to M. J. Buckingham and M. R. Moldover for many discussions.

References

[1] M. H. Edwards, Can. J. Phys. 34, 898 (1956): 36, 884

(1958).

[2] M. H. Edwards, Phys. Rev. 108, 1243 (1957).

[3] M. H. Edwards, and W. C. Woodbury, Phys. Rev. 129, 1911

(1963). [4] M. J. Buckingham, These proceedings.

[5] F. G. Brickwedde, H. van Dijk, M. Durieux, J. R. Clement, and J. K. Logan, J. Res. NBS 64A (Phys. and Chem.)

No. 1, 1 (1960).

[6] C. N. Yang and C. P. Yang, Phys. Rev. Letters 13, 303

(1964). [7] M. Moldover and W. A. Little, Proc. Ninth International Conference on Low Temperature Physics (Plenum Press, New York, 1965), to be published, and These proceedings.

85
Anomalous Specific Heats Associated With Phase

Transitions of the Second Kind

T. Yamamoto, O. Tanimoto, Y. Yasuda, and K. Okada

Kyoto University, Kyoto, Japan

There are many reasons to expect a logarithmic

singularity in the anomalous specific heat accompanied with the phase transition of the second kind.

From the theoretical side one may refer to the

classical work of Onsager [1] and the recent in- tensive investigations by Domb and Sykes, Fisher

and others [2]. From the experimental side one may refer to at least three elaborate measurements

of specific heat: the first is the experiment by

Fairbank, Buckingham, and Kellers on the A.-transi- tion of liquid He4 [3], the second is that by Voronel'

and his collaborators on the liquid-gas transition

at the critical point [4], and the third is that by

Skalyo and Friedberg on the antiferromagnetic

transition of CoCl2 6H,0 [5]. We have analyzed the experimental data avail- able at present on the specific heat of many magnetic substances. The purpose is to disclose the

analytical nature of the singularity which the

specific heat has at the magnetic transition point

of the second kind. We would like to report the

results of our analyses. Our main conclusion is as follows: In the case of ferromagnetic transition

in terbium and antiferromagnetic transition in CoClj • 6H2O, if the system were ideally homogeneous, its anomalous specific heat, in the immediate

neighborhood of the transition point, could be fit- ted by a "pinched" logarithmic function with a common coefficient above and below the transition

point added to a step function with a step at the

transition point. There are many other transitions

in which the same conclusion seems to hold, but

the experimental data now available in these cases

are not sufficient to derive a conclusion as definitely as in the above two cases. We start with the following assumptions:

I. Any real system is heterogeneous and composed of a large number of subsystems. Each

subsystem is homogeneous and has its own transi- tion temperature [5]. The observed specific heat

is given by an average over those of the homogeneous subsystems:

C„{T)=\ Chomo,(T; T,)f\TK )dT,. (1)

Jo

II. The distribution of the transition points of

subsystems is Gaussian 1:

/(7\) = a -exp [-aCTx-To)2 ]. (2)

77

To is identified with the Curie or Neel point of the

real system.

III. The specific heat of the homogeneous system

consists of two parts:

CSym = a function of \T—T\ (3)

^step

f0 for T>n,

U for T<TX .

IV. Csym is a "pinched" logarithmic function:

Csym =A In (|r-rA | + 6) + const., (4)

A < 0, 8 & 0.

From these assumptions one has

C£r) = A 1(t: D) + A • F(t) + const., (5)

!

'This obvious assumption has already been adopted in, for example, Belov's book

"Magnetic Transitions," Consultant Bureau, N.Y., 1961.

86
where

I(t;D) = —^=\ e~x

'2 In (\x-t \+D)dx.

Vjt.

F{r) = —=\ e-^dx,

vVJ 7

(6)

(7)

T^V^tt, t = T-T0, and D= 8. (8)

/(t; D) is even with respect to t so that one has from

(5)^L} ^l{CIA-\t\)-Cp

(\t\)} = F(-\T\)-F(\r\).

(9)

It should be noted here that (9) is independent of

the assumption IV. The values of I(t; D) were machined — calculated

and tabulated, r runs from 0 to 1 with step of

0.1, from 1 to 2 with step of 0.2 and from 2 to 5 with step of 0.5. D runs from 0 to 0.2 with step

of 0.02 and from 0.2 to 0.7 with step of 0.05. Fig- ure 1 gives examples of the specific heat calculated

from (5).

/

v -I(7;0)+2F(r)— -*/ » . -1(7-; 0.3) + 2F(7")

/ '// \\ / / ' /

\

\ \ \ / / ! // 1

\\ \

' \

*

-in(lrl +0.3)—z_w

' \

*

\ \ \

1 \ V "X', \ ^ -I(r;0.3)

\ \

v \\ ^ W v

\ v \ \ \ \^\

A/IAI= 2

r

The process of analysis consists of the following

four steps. First the normal specific heat is subtracted from the observed values if it can be done

without danger of introducing an appreciable

error. When the transition temperature is in the

region of liquid helium temperature, we omitted

this step. In most cases we approximated the

normal part by a linear equation of temperature,

because we are interested in only a rather narrow

interval of temperature. For our purpose we do

not have to know the absolute values of the specific heat. This makes it considerably easier to find a linearized equation for the normal part with

sufficient accuracy even by inspection. The

second step of the analysis is to plot Cv or its anomalous part against log \T—Tq\. To is chosen

so that, except in the closest neighborhood of To, as many observed points as possible form two parallel straight lines, corresponding to the high- and

low-temperature sides. Generally speaking, it is not always easy to carry through this procedure

without any ambiguity. In several cases we have

succeeded in locating the transition point quite

definitely and in finding two parallel straight lines. Examples are shown in figures 2, 3, 4, and 5 [6, 7, 8]. When we could not get rid of some ambiguity, To was found such that the plot resembles the characteristic shape as much as possible in the sense

that the observed points begin to form two parallel

straight lines after 1 7^ — T^o | becomes less than

several hundredths of To (figs. 6 -and 7) [9, 10]. Once To is fixed in this way, the values of the step

A and the slope A of the fines are read from the

plot at once.

In not a small number of cases examined, the

semilogarithmic plot gave a bill-shaped figure in the closest neighborhood of the transition point.

We can proceed one step further in these cases and determine the value of a by making use of (9). Namely we plot AC/; (f)/A against log \t\ and compare

it with the plot F(— |t|) —F(]t|) versus log |t|.

The difference in the abscissae of the two figures

is just equal to V2 log a. If the figures coincide

with each other through horizontal translation,

this provides an empirical justification for the

validity of our assumptions I, II, and III. 2 Figures

8 and 9 show examples of the third step. The

-2

Figi re 1. Calculated curves for the specific heat: C„ = -I(r: D) + 2F(r).

-The anomalnus specific heat at the superconducting transition in Sn was measured

very accurately by Kellers and Fairbank. Their data can almost exactly be repro- duced by (5) with 7\-=3.7230.; °K.^ = 0 and ( V2^r,J

"

' = 2.39 X \0~\ This fact can

be supposed as another supporl for the Gaussian distribution of the local transition temperatures. 87
( cal /deg-moie )

Dy

T. = 175.3 - K ! A = -1.34 & - 5 94

*\ M Griffei, R E. Skochdopole, * and F. H. Spedding

T <T" (1956)

T > Tn N. \ "Op \ °0 0ln 16 1 IT-lil/T.

3 102

Cp(cal / deg-mole)

1 10 10 IT-Tnl('K)

Figure 3. Cp versus log\T — TN | plot of dysprosium.

2.00 T. =52,39 "K A = -0 31 Kcal/deg-mole) A = 0 730(cal/deg-mole)

o >• 1.50 T<Th

T > Tn \^

1.00

• \

• ^

0.50 - CL = 0.1402 X(T - IB) (cal/deg-mote) \o • X o

R. H. Busey and W F Giauque (1952) \°o° • \ oo,

ip 3 io

2 V IT-Jil/T.

Figure 6. C„ — C,, versus log\T — TN | plot of NiCh.

Cp(cal/ deg-mole )

FeCI,

T. = 23.55 "K A = -1.26 (cal /deg-mole) A = 2.8 ± 0.2 (cal /deg-mole)

(Cl subtracted )

CL =0.16A5X(T-K.A )(cal/deg

0. N Trapeznikowa and L.W. Schubnikow

( 1935)

1 IT-T»I/T„

Figure 4. C„-C, versus log |T-T N | plot o/FeCF.

Cp(cal/deg-mole)

CL = 0 1092X(T-K) I cal /deg-mole)

R. C. Chisholm and I W Stout ( 1962 )

1 0* 1 o' 1 0"

10' 1 0

CoCI;

T.= 24 69 "K A: -P.4A7(cal/deg-mole) A= 1 08 (cal /deg-mole)

10 IT-T>I/T»

Figure 7. C„ — C L versus log\T — TN | plot of C0CI2
FlGl'RE 8. Determination of a in terbium.

value of 8 is adjusted in the last step to reproduce

' the bill shape as truly as possible.

We have examined about 30 phase transitions of

the second kind, mainly magnetic, and have suc- ceeded to carry through the analysis up to the

final step in about one fourth of these cases.

Figures 10 [11], 11 [5]. 12 [13], 13 [14], and 14 [15]

'show examples of the final results. The agreement

between the observed and calculated values is satisfactory in at least two cases, terbium and

i

ll

1 0 CoCI2 -6H20

ACp

JoiTn = 382

0 5

1

I—F(-r)-F(r)

1 6"1 7

Figure 9. Determination of a in C0CI2 -6H:>0.

CoCl>-6H2 0. In the other cases we can't say

anything definite without reservation. Table 1 collects the numerical results of our analyses.

Now we would like to point out the following

features common to all the magnetic transitions

taken up in table 1. First, \A\/A is less than one

half. This seems to be one of the characteristics

of three-dimensionally coupled spin systems 3

:l Even when the spins substantially couple two-dimensionally, we observe that this

ratio is less than one half. It becomes larjier than unity only when the spins couple

essentially one-dimensionally.

Substance To -A

dee — mol

cal

deg— mol

^

X103 2aT0 T0 x 10 3 Reference

No.

Samarium

Terbium

CoCl,-6H,0

MnF,

NiF,

Cr,03 /3-UH3

(°K)

105.8

228.3

2.2890

67.05

73.33

306.5

172.0

(Tc)

(Tc) (TN) (TN) (TN) (TN) (Tc)

0.471

1.97

.538

.530

.643

1.25

.469

3.04

5.20

1.14

2.73

2.08

2.64

1.81

3.01

2.06

1.85

4.86

2.51

4.45

6.55

0.85

.88

.79 3.4

.71 1.9 2.8

89
Secondly, the half-width of the Gaussian distribution, divided by T0 , is always of the order of magnitude of 10"3

. Thirdly, the degree of pinching,

8/7o, also has the same order of magnitude, 10-3

, and is almost always less than (V2q: To)

-1

. This

fact has been preventing us from directly observing

the pinched logarithmic anomaly.

As regards the origin of pinching in the logarithmic anomaly in an ideally homogeneous system, we

would like to propose the following explanation.

As is well known, a transition of the second kind

is accompanied with an anomalous expansion.

Now the transition temperature 7\ is a "temperature

Cp(cal/deg-mole)

14.0- Tb

12.0-

Tt = 228.3 , K

A=-1 97cal/deg-mole

10.0- A=5 20cal/deg-mols

6/Tc=8.76X10* T<Tc Cb.C-subtacM)

8.00-

6.00-

400 T>!c\

2.00- Y= 0 cal/ mole-deg1

0.00- Jennings-Stanton-Spedding(l957)

Id

3 10* 10"' IT-Tcl/l

10''

1 10 10

! ITFigure 10. Cp —

C

L —

C

E versus log\T — Tc | plot of terbium.

The solid and dotted lines correspond to S/7V=o.76 X 10 -4 and zero, respectively.

dependent quantity," as Heller and Benedek put

it in their Phys. Rev. Letters [17]. To make the

discussion more specific, let us assume that, when we approach the transition point from the high temperature side, the sample contracts and the value of the exchange coupling increases. This means that,

if we stay at a distance from the transition point, we

overestimate the distance therefrom. When we

are in the low temperature side, we again over- estimate it for the same reason. Since the anomalous expansion is really large only in the closest neighborhood of the transition point, the shift of the estimated transition point from the real transi- tion point is almost constant, so long as we stay

outside this narrow region. We identify this constant shift with our 8. Then the thermal expansion

effect may be separated off by neglecting 8 in the

expression for the symmetric part of specific

heat at constant pressure to get that of specific heat;

at constant volume. This interpretation leads us]

to the important conclusion that, if the system isf ideally homogeneous, the specific heat at constant; volume has a symmetrical nonpinched logarithmic

singularity in addition to a discontinuity describable by a step function.

Before closing our report, we would like to add

a few words: The conclusion of the present analysis

is tentative and the possibility of some other kind

of singularity still remains in the cases other than

terbium and CoClo^rTO. To establish ouij conclusion more definitely, we have to make furthei

investigations from both the experimental anc

theoretical sides.

4.5

4.0 h

3.5

° 3.0

?2.5

•o3 2 0 a

<-> 1.5

1.0

0.5

0.0

V. T < Tn X.

"v. V

T>T- 'Nx

Skalyo Jr. and S. A. Friedberg ( 19W )

10 1.0 1.0' 1.0"'

CoCl2 6H2 0

T„= 2.2890 °K A = -0.538(cal/deg-mole) A = 1.1A(cal/deg-mole)

6/Ii=7 86 X 10

4

1 IT-TJ/T.

10 10"; 1 0 1 0 IT-ThICK)

10" 10

Figure 11. Cp versus log\T — TN | plot o/CoCl2-6H2 0.

The solid and dotted lines correspond R 8/7\=7.86x 10^ and zero, respectively.

90
Cp(cal/deg-mole)

5 00

T.

NlF2 =73.33 "K

= -0.614 cal/deg-mole

= 2.10 uU deg-mote

6/Tti.7.06 X10*

(Ci subtracted)

T<T« \

CL = 0 0840( T-20 )cal/deg- mole \. T>Tm

E Catalano and J. W. Stout (1955)

1,0'* 1,0"

3 IT-T.I/T.

1 0'2

1 0''

1 10 IT-T

Cp (cal/deg mole

)

Cr2 03

30 0 \

T. =306 5 'K "\

\

a = 2.64 {cal/deg-mole) 6/Tt.1.89X10"

3

29.0 T<T«\

280

T>T«

27.0 \

26.0

C. T. Anderson ( 1937 )

1 d" ,o

3 10'? IT-T.I/T.

1 0"' IT- T.I (K) Figure 12. Cp — C,. versus log\T — TN | plot o/NiF->

The solid lines correspond to S/7V=7.08 x 10 ^.

Cp (cal/deg-mole)

Figure 13. Cp versus log\7-TN \ plot of Cr,03 . The dotted lines correspond to 6/TN =1.89x 10~ : '.

P-UH3 Tc = 172.0 'K

\ A = -0.469 (cal/deg-mole) 3.00 A = 1.81 (cal/deg-mole) 6/Tc=2 79X10"

3

(Ci subtracted)

2.00 T<Tc

1.00 CL = 0.0257 X( T-1 00) + 6 00 (cal/deg-mole) T>Tc \

Flotow, Lohr, Abraham and Osborne ( 1959 )

1 6" 1 o"

3

1 1 10

2 /\ IT-Tcl/Tc

1 0 10 IT-Tcl('K)

Figure 14. Cp —

C

L versus log |T — Tc | plot o//3 — UH3 . The dotted lines correspond to 8/7'r =2.79 x 10"3

. References

[1] L. Onsager, Phys. Rev. 65, 117 (1944). [6] M

[2] A. J. Wakefield, Proc. Cambridge Phil. Soc. 47, 419, 799

(1951); C. Domb and M. F. Sykes, Phys. Rev. 128, 168 [7] O.

(1962) ; J. W. Essam and M. E. Fisher, J. Chem. Phys. 38,

I 802 (1963); J. Gammel, W. Marshall, and L. Morgan, Proc. [8] L. Roy. Soc. (London) A2 75, 257 (1963). G. S. Rushbrooke. [9] R.

J. Chem. Phys. 39, 842 (1963): G. A. Baker, Jr., Phys.

Rev. 129, 99 (1963); M. E. Fisher, J. Math. Phys. 4, 278 [10] R.

(1963). [3] W. M. Fairbank. M. J. Buckingham and C. F. Kellers, Proc. [11] L

5th Int. Conf. Low Temp. Phys., Madison, Wisconsin

(1957), p. 50. . [12] L

[4] M. I. Bagatskii, A. V. Voronel', and B. G. Gusak, Zh. Eksperim. i Teor. Fiz. 43, 728 (1962) [Translation: Soviet [13] E. Phys. -JETP 16, 517 (1963)]; A. V. Voronel', Yu. R. [14] C. Chashkin, V. A. Popov, and V. G. Simkin, Zh. Eksperim. [15] H.

i Teor. Fiz. 45, 828 (1963) [Translation: Soviet Phys.

-JETP 18 , 568 (1964)]. [16] J

[5] J. Skalyo, Jr. and S. A. Friedberg, Phys. Rev. Letters 13, [17] P. 133 (1964). See also J. Skalyo, Jr., A. F. Cohen and

S. A. Friedberg, Proc. 9th Int. Conf. Low Temp. Phys.,

Columbus, Ohio 1964. (To be published.)

. Griffel. R. E. Skochdopole. and F. H. Spedding, Phys.

Rev. 93, 657 (1954); J. Chem. Phys. 25, 75 (1956). N. Trapeznikowa and L. W. Schubnikow, Phys. Z. Soviet. 7,66(1935).

D. Roberts and R. B. Murray, Phys. Rev. 100, 650 (1955). H. Busey and W. F. Giauque, J. Am. Chem. Soc. 74,

4443 (1952). C. Chisholm and J. W. Stout, J. Chem. Phys. 36, 972

(1962). D. Jennings, R. M. Stanton, and F. H. Spedding. ibid. 27, 909 (1957). D. Jennings, E. Hill and F. H. Spedding, ibid. 31, 1240

(1959). Catalano and J. W. Stout, ibid. 23, 1284 (1955). T. Anderson, J. Am. Chem. Soc. 59, 488 (1937). E. Flotow, H. R. Lohr, B. M. Abraham, and D. W. Osborne, ibid. 81, 3529 (1959). W. Stout and H. E. Adams, ibid. 64, 1535 (1942). Heller and G. B. Benedek, Phys. Rev. Letters 11, 428

(1962).

91
The Logarithmic Anomaly in the Pressure Coefficient

of Helium Close to the Lambda Line*

H. A. Kierstead

Argonne National Laboratory, Argonne, 111.

Introduction

Several investigators have reported logarithmic

anomalies in the heat capacity [1], thermal expansion [2-6], and pressure coefficient [7-8] of liquid

helium in the neighborhood of the lambda transition. Unfortunately, all of these experiments ex- cept those of Lounasmaa [7] and of Lounasmaa and

Kaunisto [8] were done on the vapor pressure curve, where the anomalous behavior is confined to a very narrow temperature interval. The experiments of Lounasmaa and Kaunisto [8] at various pressures

along the lambda curve show that the anomalous

behavior is spread out over a larger temperature

interval at higher pressures. Consequently, we

have measured the pressure coefficient, {S v={dP/dT)v , of liquid helium along the isochore which crosses

the lambda line at the melting curve. This inter- section is called the upper lambda point (Ty, Py).

Experimental Procedure

The apparatus used in these experiments is essentially the same as that used by Lounasmaa [7]. Therefore, it will be described only briefly here. It

is shown schematically in figure 1. Helium gas was purified in a trap (not shown)

immersed in liquid helium, and was condensed into the piezometer G through the low-temperature valve

A, which was kept closed during measurements. G

was isolated from the liquid helium bath by the vacuum case B. Its temperature was controlled

by the heater F and by pumping on liquid helium

in E.Changes in the pressure on the sample were measured by the oil manometer K and read with a cathetometer to 0.01 mm, which corresponded to about a millionth of an atmosphere. The absolute

pressure was read to 0.01 atm on the Bourdon gage

P, which was calibrated against a dead weight tester.

A

* Based on work performed under the auspices of the U.S. Atomic Energy Commission.

92

Figure 1. Schematic drawing of the apparatus.

(A) needle valve; (B) vacuum case; (C) pumping and vapor pressure tubes (2) f(|| (E); (D) 30 percent Cu-Ni capillary tubing, 0.1 mm ID; (E) temperature and vapilk pressure compartment; (F) heater; (G) piezometer packed with fine copper wire fiJI rapid equilibrium (volume 39.83 cm3 at 1.76 °K, height 10 mm to keep hydrostatlr pressure differences small); (H) dashed lines enclose space immersed in liquid heliu

|

bath: (I) germanium thermometer; (K) differential oil manometer, 0.060 in. ID constat/

1

bore glass capillary (left arm 40 cm long, right arm 90 cm long); (L) oil reservoir aril

1 leveling device filled with Apiezon B oil; (M) to He*1 supply tank and purifier: (N) pump: (O) to atmosphere; (P) 25-cm dial test gauge calibrated against a dead weiglj

tester; (Q) ballast volume (1461 cm3

) for balancing the right-hand side of K, held constant temperature by an ice bath; & needle valves.

Small changes in the sample density could be mad

by raising the oil in the left limb of the manometer

forcing gas to condense into G. A change of 0.0' mm in the oil level caused a change of 2 X 10

g/cm3 in the density.

Temperatures were measured with a germaniun

resistance thermometer in a potentiometer circui]

using a double potentiometer. At the uppej

lambda point the thermometer resistance way

1483 fl, and its sensitivity was 636 microdegreesi I per ohm. With 50 /jlA of measuring current, thfL resistance could be measured with a precision aL

0.002 fi. The measuring current heated the thei' v mometer above the cell temperature; but this effecL was reproducible, and the same measuring curren was used in calibrating the thermometer. It wa :

, calibrated against the vapor pressure of helium in li on the T5g scale [9], using a tube separate from th< pumping tube. The change in the thermometer

resistance at the upper lambda point after warminjj

to room, temperature and recooling corresponded

to a temperature change of less than 10~4 °K. AJ
temperatures were referred to the upper lambda

point in order to correct for small changes in the

thermometer or the measuring circuit. Points on the lambda curve were found by warning the sample slowly at constant volume. Because

i)f the large change in thermal conductivity at the

ambda transition, the heating curve showed a sharp

oreak, which could be located to within a microdegree. The sample was held at this temperature

until equilibrium was established before reading

he manometer. Then the density was changed

slightly by raising the oil in the left limb of the

nanometer, and the new lambda point was found.

The break in the heating curve was especially sharp

(vith solid present in the cell, because the heat of

melting increased the apparent heat capacity.

Thus the upper lambda point could be determined

directly, rather than as the intersection of two independently measured curves. The pressure coefficient (3 v = (dP/dT)r was deter- Inined by keeping the oil level in the left limb of

j he manometer constant and measuring the height

|)f oil in the right limb at successively lower temperalures. After the lambda point was passed, as sludged by the thermal response, the temperature

Ivas raised slowly in order to determine the lambda

i>i joint exactly.

Results

FIGURE 2. Pressure coefficient ^v = (flP/ST)v of He I along the isochore V = V(T\).

Equation of line: /3, =9.39 + 7.69 log,,, l7"-7\) atm/°K.

In connection with these experiments we have

measured the change of pressure and density with

temperature along the lambda line. These experiments, which are reported in detail elsewhere [10]. can be summarized by the equations

dT k = - 55.54 - 96(7\ -TV) atm/°K (2)

(^;)a = -43.7-230(7\-7Y) mg/cm:i"°K (3)

A The pressure coefficient /3, = {dP/BT)v was meas- ured along an isochore which crossed the lambda

Mine very close to TV. Results are shown in figure 2,

l" blotted against logio (T — 7\). The equation of

' he line is

j8 r = 9.39+ 7.69 logtoOT-Tx) atm/°K, (1)

ai vhere T\ is the lambda transition temperature for

e ;he isochore. This graph contains points taken on

I>lwo different cooling runs. The lambda point for

o|»ne run was 15 /xdeg below TV, and for the other it c'lvas 5 /Ltdeg above. These small differences are not

5

' egarded as significant, but the observed lambda

i| mint was used in each run in plotting T—T\.

>a

! >ince this isochore meets the melting curve at the

>' ambda point, the measurements in figure 2 refer to Te /. One point was measured in supercooled

t( =le//, where /3 r was found to be —27.6 at T—T\

111

; = — 1.08 X 10"5

. This point is not plotted in figure

' eT, but it is approximately the value to be expected

A V T-TK = 1.08X 10-\

X = 0.7868 + 2. 8(7\-7V) mg/cm:i atm. (4)

where T\ is the lambda temperature for the pressure

P and density p, and 7Y is the temperature of the

upper lambda point, 1.7633 °K. Discussion

The logarithmic dependence of /3 r on T— T\

is similar to that found by Lounasmaa [7] and by

Lounasmaa and Kaunisto [8] at lower densities,

and is consistent with their observation that the

two constants in eq (1) both increase with increasing density. The measurements of Lounasmaa

at 13 atm pressure (7\ = 2.023 °K) are of particular

interest since he found that /3 r fitted an equation

of the same form as (1) to within 2 X 10" 5 °K of the

lambda point for both He I and He II, the constants being different for the two cases. In the same

paper Lounasmaa presents measurements of the

compressibility kt= p~l {dpldP)T along an isotherm

93
passing through the same lambda point. He

finds no evidence for anything but a slow linear

variation of kt with P — P\ to within 10~3 atm from

the lambda line (corresponding to 2 X 10 5 °K for T— T\) and a small discontinuity at the lambda

point.

Since an isochore cannot cross the lambda line with a slope greater in magnitude than that of the

lambda line, {dPldT)k is a lower bound for fi v (both

are negative); so eq (1) must break down for some

small value of T—T\. On the other hand, there

is no such restriction on kt- In fact, Buckingham

and Fairbank [11] have shown that if Cp becomes

infinite at the lambda line, kt must also. The exper- iments seem to suggest that fi v becomes infinite and

Kt does not. This situation can be examined more

closely with the help of the equation

pr = {dPldT)x -{PKT

)- x (dPldT)K , (5)

which should apply to any point close enough to the lambda line that fi v and kt depend only on the

distance from the lambda line. Solving for kt we have

kt

(dpldT)k (dPldT)K -pe

' (6)

This equation shows that kt will not vary much

until fi r becomes nearly equal to {dPldT)k . If fi r becomes equal to (dP/dT)\, as must happen if CP becomes infinite, then kt becomes infinite. If

fir were to exceed (dP/dT)\ in magnitude, kt would

become negative and a first-order phase transfor- mation would ensue. Equation (6) has been used

to calculate kt along the isotherm studied by

Lounasmaa, [7] using his equation for fi r and

Lounasmaa and Kaunisto's [8] data for (dP/dT)\

and (dp/dT)\. These calculated values are plotted

as the solid line in the lower part of figure 3. The

circles are Lounasmaa's measured points and the

dashed lines are the ones he drew through his data. Although the agreement is not very good,

it is apparent that he could not have expected to see an anomaly at the pressure resolution he at- tained. A similar calculation using the data re- ported in this paper is shown in the upper part of

figure 3. The much larger change in kt results from the larger magnitude of fi v and the smaller

magnitude of (dPjdT)x at the higher pressure.

He I 1.763 °K

-8 -6 -2 0 2 4

I0

3 (p-Px ),atm

10 12

FIGURE 3. Compressibility kt = p

l {dpldP)T of liquid helium.

Open circles: Points measured by Lounasmaa [7]. Dashed lines: lines drawn b> Lounasmaa through his points. Solid lines: calculated from logarithmic relations foi

We plan to measure Kt near the upper lambda point

in the near future. To investigate the nature of the lambda trans- formation, it is very important to study how kt goes,

to infinity and how fi r approaches (dP/dT)\ (if indeed they do) at temperatures so close to 7\ that eq (1) breaks down. This is probably only pos

sible near the upper lambda point. In Lounas

maa's experiment at 13 atm the logarithmic relation predicts that, for T—T\ = 10~6

, fi v will be less than

a fourth of (dPldt)\. In the present experiment^

fir was a fourth of (dP/dTh at T-Tk =lO-:i

- and

if eq (1) remains valid, fi v will be two-thirds ofj (dP)dT) k at T-TK = lCTl We are at present en- gaged in extending the measurements to smaller

temperature intervals.

The author expresses his indebtedness to Pro; fessor O. V. Lounasmaa of the Institute of Tech!

nology, Helsinki, who originally constructed tht1 apparatus and has contributed valuable advice;

to Professor Dillon Mapother of the University oj Illinois for fruitful discussions during the progress

of the work; and to S. T. Picraux and Z. Sungail;

for assistance in modifications to the apparatus.

94
References

[1] W. M. Fairbank, M. J. Buckingham, and C. F. Kellers.. Proc. 5th Int. Conf. Low Temp. Phys., Madison, Wisconsin (1957), p. 50: C. F. Kellers. Thesis, Duke University (1960).

[2] K. R. Atkins and M. H. Edwards, Phys. Rev. 97, 1429

(1955).

[3] M. H. Edwards, Can. J. Phys. 36, 884 (1958).

[4] E. Maxwell. C. E. Chase, and W. E. Millet. Proc. 5th Int. Conf. Low Temp. Phys., Madison, Wisconsin (1957),

p. 53.

[5] E. Maxwell and C. E. Chase, Physica 24, 5139 (1958).

[6] E. C. Kerr and R. D. Taylor, Ann. Phys. (N.Y.) 26 , 292

(1965).

[7] O. V. Lounasmaa, Phys. Rev. 130, 847 (1963).

[8] O. V. Lounasmaa and L. Kaunisto, Ann. Acad. Sci. Fenn. AVI. No. 59 (1960): Bull. Am. Phys. Soc. 5, 290 (1960).

[9] H. van Dijk. M. Durieux. .1. R. Clement, and J. K. Logan.

National Bureau of Standards Monograph 10, 1960.

[10] H. A. Kierstead. 138, A1594 (1965).

[11] M. J. Buckingham and W. M. Fairbank, Progress in Low

Temperature Physics (C. J. Gorter, ed.) Vol. Ill, ch. Ill (North Holland Publishing Company, Amsterdam, 1961).

The Nature of the Cooperative Transition

M. J. Buckingham

University of Western Australia, Nedlands, Western Australia

1. Introduction

A thermodynamic singularity arises in statis- tical mechanics only in the limit N^>°°, where N

is the number of atoms in the system. By regarding an infinite system as made up of an infinite

TM8

' number of large but finite parts, called cells, with

M atoms each, the singularity is seen to arise from

the interactions between the cells — which for M

' al large correspond to surface interactions, since the

10

' interatomic forces are short -ranged. For the same

* reason interactions between other than nearest m

neighbor cells are zero. Thus any thermodynamic

,ia11 singularity may be regarded as an example of a generalized near-neighbor lattice problem.

For very large M, one might expect the approximation of neglecting the surface interactions

11 between cells to be a very good one since it cor-

"ft responds to the neglect of only a fraction 1/M 1/3 of

all interactions. Indeed for a homogeneous stable thermodynamic system the approximation is ex- tremely good. However it is not good near a singularity or in a two phase region, since the correct,

partition function for the system is then singular,

whereas the exact solution of the zero-interactionbetween-cells problem results in some power of the

partition function for a finite system of M atoms,

which is nonsingular.

It is easy to see the nature of the solutions in the two phase region. The correct solution describes

a system with two phases separated in space by some minimum-area boundary. In the zero-interPro<

ice;

0\

action solution, each cell is properly described as a piece of one or other of the phases, but in the ab- sence of the interaction these are mixed at random.

The qualitative effect of the interactions between

cells is thus the segregation or phase separation of the cells. The entropy associated with this separation is constant (depending on composition) in the two phase region and is, of course, zero in the one phase region — except for a very small temperature interval in the immediate neighborhood of the transition temperature.

At first sight the very small magnitude of the mixing entropy — associated with but one degree of

freedom out of M, would suggest that it is entirely

negligible. However the extreme smallness of the

temperature interval (~ 1/M) 1/3 over which it can change means that, in some circumstances, it may

be far from negligible — indeed accounting for the

asymptotic form of the thermodynamic singularity.

An example is the ordinary first order transition from

gas to liquid phase, as mentioned below. The case

of the critical transition is much more subtle, but

by incorporating the assumption that the pair correlation function is just a function of r/ro where r<j is the correlation length, we show below that a system,

with finite range forces only, ostensibly possessing

a "classical" van der Waals critical point would in fact have to have a logarithmic singularity in its specific heat. The discussion that follows is not intended as a "theory of the critical point" but as a preliminary account of the viewpoint that is being

developed.

95
2. Division Into Cells

Consider a system with N atoms in a volume V, described by a Hamiltonian with finite range forces

only, and with an exact partition function Z(N, ft). Imagine a division of the system into Jf "cells"

each with M atoms (N = M.A-"') but with no restriction on the volume or energy of a cell (or of the magnetization in the case of a magnetic system). Number the cells with i = 1, . . . . Jf. In any configuration contributing to the parti- tion function, separate the interactions linking

atoms in different cells. Imagine, for convenience,

the latter all multiplied by a parameter A say. Thus

H = JH; + 2 (tfint ) + X X. {H>nt ~ (Hint >

)

i=l i.j i.j

where the first term contains all interactions link- ing atoms all of which are in the same cell; inter- actions between atoms in different cells contribute

to the last term. (1) denotes mean value. Clearly

for M large enough, interaction between atoms

linking' cells other than near neighbors can be

neglected because of the finite range of the forces,

so the sum of interaction terms can be written as

a sum over near neighbors only. Note that the exact Hamiltonian is obtained with A=l.

Let Z\(N, ft) be the partition function for the

system for some value of A.. Then Z\{N , ft) is the exact Z(N, ft) while for A = 0 we have a collection

of noninteracting cells which can exchange energy, volume, etc. Thus

z„(/v, ft)={Z(M. p>y

This relation is exact, with Z(M, ft) the exact parti- tion function for a system of M atoms. We use Z

somewhat schematically for the appropriate partition function — the constant pressure one in the case of a gas liquid system for example. We will use the fact that, considering ft=l/kT as a complex variable, Z(N, ft) is analytic 1 in the right

half-plane for /V finite, but not necessarily so in the

limit N —> oo. Even when there is a singularity in Z{N, ft) as N—><x>, the corresponding Z0(/V, ft)(k — 0) is analytic.

'Z is the Laplace transform of the nonnegative degeneracy function <ul£, V, N). which is identically zero for E less than some finite (for .V finite) ground stale energy

value and for V less than some finite value.

When Z\ is at a normal, homogeneous stable ther- modynamic point, then the operation of changing

A. from 0 to 1 has a negligible effect for large M.

However, near a point where say the specific heat

at constant volume Cy— for the A=l system,

C\ remains finite but dependent on M, for the A = 0

system, see figure 1.

It is easy to show at a first-order transition for example that whereas C,,(T) has a 8-function singu-!

larity, the corresponding (Ct,(T))M reaches a value

of order M with a width — (all thermodynamic quantities being expressed per atom). Thus near

U , • U" 8T 1 v 1 U enough (within — —, say) to a singularity the

1 c M

operation of putting A—>1, far from being negligible indeed produces the dominant effects and can

be regarded as leading for example to the infinity

1

in the specific heat. Considering now the problem of obtaining the 1 exact solution for the system from the solution

Z(M, ft) we observe the following: The solution for A = 0 represents an ensemble of cells with extensive

parameters distributed according to the partitiom

function Z(M, ft). Thus at a temperature a finite: amount above a critical temperature, say at 7V + t,i the density of cells is distributed about the meani

value (say the critical density pc) with a Gaussian!

distribution of width corresponding to the fluctua-i

T

FlGl'RE 1. Sketch of the qualitative temperature dependence o Cv near a singularity at Tr and that of Cyi\ = 0) M for twi values ofM.

96
tions of the density for cells of size M.

((p — pc ) 2

) ~ X (compressibility).

On the other hand, at a finite temperature below

the critical temperature, Tc~t, (in a 2-phase coexistent region) the possible density values will be

distributed in two Gaussians separated by the

density difference of the two phases in equilibrium,

figure 2. We can regard the A. —> 1 problem as a generalized, near neighbor, lattice problem and we can

estimate the magnitude of the effective interaction

parameter. It is the total interaction energy due

to interactions finking atoms across the boundary,

say for the interaction between cells 1 and 2. This

energy is proportional to the number of atoms within the interaction distance of the surface in cell 1 multiplied by the same number for cell 2.

i.e., Em

1. 2 M2I 3 €0 (piPi)

lie

"ii

or

ve where e0 is the strength of the interaction (eo ~ kTc)

ite and pi is the density at the surface of cell 1. In the

-. problem with A = 0, pi is typical of the density any

anj where in cell 1 so p!~pi. We now approximate

an E-mt , by dropping the superscript on p\. The effecia- 1 tive interaction parameter of the lattice-interaction

[problem is the difference between the Eint in opposite configurations: i.e.,

EintiPu pi) + Em{p-2, p2) — 2Emt(pi, p2 ). Thus the effective interaction in the lattice problem

is of order

M2/ 3e0 ApiAp2

The equivalent lattice variable is the ratio of the

extreme values of Eint to kT

^.5±M*tz x <£LZ££. t<Tc -t

kl o

i.e. Eint _j

kT M- 0 T>Tr + T

Pip)

T »Tf Pip)

FIGURE 2. Probability P(p) of finding a cell with density p,

at temperatures above and below Tc

.

Thus "switching on" the interaction between cells does nothing to the distribution of the states of

cells when T> Tc + t since the effective temperature of the lattice problem is then T^ff ~ 00 whereas

for T<Tc~t the effect is very great, Tetf ~ 0 in fact causing full "condensation" or "phase-separation" of the two types of cell — that is, those near

the liquid density or the gas density, respectively;

these are mixed at random for A — 0 but separated

for \=1. The entropy change involved is thus

In 2 —^j- (per atom) and all this is lost in a very small

temperature interval around the critical

temperature.

AC ln 2

ZV5 = —r— per atom. (2.1)

oo T < Tr—T.

3. Long Range Density Fluctuations

This entropy contribution, M_1ln 2, arising for T < Tc as sl result of the sorting of the previously

(A. = 0) randomly mixed "fluctuations of size M"

might seem at first sight completely negligible.

There are, after all, surface terms of order M213 times greater. However we note that the contribution remains constant, M_1ln 2, throughout the

temperature range T < Tc — t, and all its effect oc- curs in a small temperature interval, approaching

zero as M— about the critical temperature.

(It may be noted that if the density had a value

other than the critical one, say corresponding to a fraction / of the atoms in one component and

(1—/) in the other, then the entropy would be

-M-*{f In f+(l-f) In (l-f)} and this can describe

the smoothing, for a finite system, of the first order

transition at temperatures and pressures below

the critical values.)

This mixing entropy term corresponds, of course,

to just one degree of freedom per cell of size M,

97
but it describes the relative configuration of the

large size density fluctuations — and these are what

ultimately become correlated near enough to the

transition temperature. Thus if Im is the linear

dimension of a cell of size M, the number of Fourier

components of the density with wavelength greater

than Im is just the same number, namely 1/M of

all the degrees of freedom.

While it would seem quite possible for many

more than one degree of freedom per cell to be

involved in the cooperative transition — and in- deed this is apparently the case for the two-dimensional Ising problem, we offer a tentative proof

that, in fact, just this contribution is sufficient to make impossible the "classical" critical point —

with properties typified by van der Waals equation.

That is, a statistical system with finite range forces

(in any number of dimensions) would not be self- consistent if it possessed a "Classical" critical point. In particular, we show that a system whose

compressibility diverges as a simple pole as a function of temperature must have at least a logarithmic singularity in its specific heat.

4. Calculation of the Mixing Entropy

In order to obtain quantitative information con- cerning the interaction problem we shall invoke the

fact that the nature of the problem is independent

of the size, M, of the cells (provided M is sufficiently

large); the size only determining the scale of the

temperature interval in which the transition occurs. For this purpose we now define a sequence

of cell divisions with ever increasing size of cell. In the cell division defined above in section 2,

let us replace M by M0 and A. by A0 , and introduce

a new parameter \i multiplying the interactions

at the boundaries between cells of size M\ > M0 . Thus \i marks the interactions which would have

been marked by A. before, had we chosen Mi in- stead of M. For convenience let M\ = JfM&< V>1).

Similarly let us also introduce A2, A3 . . . \„ . . ., dividing into cells of size M->, M:i ,

. . . Mn . . ., respectively, with M„=. VM,,-\ = . Y"Mu.

Now by setting Aa- = 1 , k # n and k„ — 0, we obtain

a division of the system into cells of size M„, which we call the nth order. The exact solution can now

be regarded as that obtained in the nth order as

n —* °°, or as A.,, —* 1. We now determine the temperature interval required for the transition in the

various orders. At some very small temperature

t = T/Tc—l, above the transition temperature,

say, cells of sufficiently large size will be statisti- cally independent of each other, whereas those of

smaller size will not be. We can determine the

number of statistically independent regions or elements by examining the fluctuations.

5. Number of Statistically Independent Elements

We are concerned with the development of cor-j

relations in a particular extensive property of our system — namely the one .that is becoming longrange ordered and identifies the two components

in the two phase region. The fluctuations of this quantity determine the number of statistically in- dependent elements we require.

A number n of things each with an extensive

property x, with mean square deviation dx2

, will,

if they are statistically independent, have a mean

square deviation ((X — X)2

) = n8x2, where X =

n

x\. In a gas-liquid system x\ is the number of]

atoms in a volume element bV i.e., p8V. This

number is either 0 or 1, and has mean value pdV.

The number n of statistically independent fluctuat- ing units in a large volume with mean number N\

of atoms is then

_ p

2 n ((p-m

which reduces, as it should, to /V in the absence of any correlation, i.e., at infinite temperature. The mean number of atoms per independent element

is then

M=^=^p-N^kTpKT = X(t), (5.1) n p

z

say where Kt is the isothermal compressibility. Inl

the case of a magnetic system, the corresponding!

number of statistically independent magnetization 1 elements is the magnetic susceptibility in units of

the magnetic moment per particle.

6. Recurrence Relation

We have seen that the fluctuation of the natural!

extensive variable of the system determines thel

size M(t) of the statistically independent regions.!

Returning to the cell division of our system, welf assume that the temperature interval over which thej

nth order undergoes its transition is the temperaL

2

98
ture, tn, such that M„ ~ M(t„) = x(tn)- That is. using

the result (2.1) obtained in section 2, the nth order

solution would lead to a change of entropy

AS„ = ln 2/Mn , occurring in a temperature inter- val t„. We can now assert that for large enough cells the

functional form of this entropy change is given by

ASn(t) = In 2

f(tltn) (6.1)

where the function / is independent of n. This

assertion enables us to write a recurrence relation

which we solve for the function /, and is the main

source of our argument — it depends on the fact

that for large enough M (or small enough t) the

interaction problem has ceased to depend on any

detailed property of the interaction between atoms,

or of any finite number of atoms and has taken an asymptotic form. We are effectively assuming

the existence of an asymptotic "lattice problem"

such that the description of the nth order in terms

of the (n— l)th is the same as that of the (n+l)th

in terms of the nth. The function f(x) is continuous monotonic in- creasing and satisfies the boundary conditions

/(*)- 0, x> 1 and f(x)^> — l, x< — \.

The partition function corresponding to the

choice k,, = 0, \a- = 1, k n, i.e., the nth order, is a valid approximation to the exact one for t > tn , since regions of size M„ are then statistically independent. Similarly, except for the constant mixing

entropy, it is valid for t < — t„. The exact solution

A.ji=1, contains the additional contribution to the

entropy given by (6.1).

using the result (5.1). The derivative of /has been

written /'. For a "Classical" critical point xW ^ but

for other suggested 2 singularities x ^ f~ y where

y > 1. Let us write

X{t)- l = a'ty y*?l (7.2)

eq (7.1) becoming

CJ$-CAfi) = atl-y%tltn), (7.3)

where a is written for a

1 In 2. The cases y= 1 and

y > 1 are different and we discuss first the case y=l.

8. The "Classical" Critical Point,

y=l

The equation for the singularity becomes, for the case y=l,

Cjt)-Cn(t) = af'(tlt ll ) (8.1)

It is easy to see that the function/7 ^) cannot be

finite at x = 0. For let such finite value of the right

side of (8.1) be Co. Then C„,(0) - C,(0) = 0 for any m, n and yet Cx (0) — C„(0) = C0 ^ 0. We find below that the asymptotic solution requires J'ix)

In x as x^ 0. Taking the difference of (8.1) for n and n + r, we

obtain

Cn+At)-Cn(t)=a{f'(tlt,) -/'(, I "'*/«„)}

7. Difference Equation for Specific

Heat

Let C»(t) be the specific heat corresponding to the partition function with X„ = 0. C„{t) will have

the shape of the exact specific heat except that the

singularity will be smoothed out over an interval ~ tn , with C„(t) finite for all t. We can now write,

using 7V as the unit of temperature,

Cx (t)-Cn (t) = ^±fltltn) M„ at

ln2

f'(tltn)

Xitn)tn

(7.1)

where tn\tn-v\ —Jf- Setting x = tjt n we write this equation as

Cn+r(xtn)-Cn(xtn) =a{f'(x)-f% I "'*)} (8.2)

where the right-hand side is independent of n and

can be written ag(Jrr

, x) say. Now Cn {t) is finite for all t and by putting r—\ and, repeatedly, n + 1 for n, we find

Cn+M~ CM= ar g{yf, 0) = ar g() ln Jf

where gfo is a constant (independent of r, and n).

- See for example: M. E. Fisher. J. Malli Pliys. 5,944(19641. The Iwn dimensiimal

lsin<i pmhlem has y=7/4.

99

209-493 O-66--8
Thus

C„+i(0) - C„(0) = ago In JIT, (8.3)

so that, increasing the order n by one increases

C„(0) by a constant independent of n, and thus C„(0)

increases linearly with n. Now = Jr ~ 1 t„-i = • = Jf~nT say, so that - In (tjT) n - InJf

"

Thus

Cn(0) = Co + nago In yT,

becomes

C„(0) = Co — a go In U„/t).

Thus we see that the specific heat becomes infinite

like the logarithm of the temperature. We can

easily find the asymptotic form of the function f'(x), since by (8.2) and (8.3) we can write

f'(x)-f'(.\x) -> g0 lnJT. (8.4) x->0 Now the equation

F(x)-F(bx) = C

has the solution

F(x) = —.— h-Fo

In o

where F0 is an arbitrary constant. Hence the

asymptotic solution of the eq (8.4) is

/'(*) -> -g0 In x+f0. (8.5)

t —»

o

The eq (8.1) whose asymptotic solution we have

found contains an arbitrary multiplicative constant which should be determined from the normalization

condition given in section 6. We can see from

(8.5) together with this condition that g0 will be a number of order unity.

Summarizing the results of this section we assert

that a system whose compressibility diverges like a simple pole as T—» Tc cannot possess the "Classical" specific heat temperature dependence, but

that it must have a logarithmic divergence and

furthermore, the coefficient of the logarithm, a in eq (7.2), is proportional to the inverse of the coefficient of (T—Tc)~ l in the compressibility (5.1). That is we expect

k-iCv —a In fcp) + C0

where a ~ In 2 pMT— T< )k t, (8.6)

A

- being Boltzman's constant, kt the compressibility

and pc the critical density.

9. Nonclassical Critical Point, y > 1

We do not give a detailed discussion of this case, since it is not hard to show that the solution of eq

(7.3) for y > 1 is not singular. This suggests, in terms of the discussion in section 3 above, that a

singularity with y > 1 involves more than just the one degree of freedom per cell which describes

the spacial arrangement of the large scale

fluctuations.

It is not a pleasing circumstance that the simplest

possibility in the present theory appears inadequate to account for the behavior of the two dimensional Ising problem, but further discussion of

these points must be left to another occasion.

10. Further Properties of the

Transition

We have discussed at some length the nature

of the singularity to be expected at the critical point of a system which could be said to have a

"Classical" critical point in the "zeroth order"

approximation. We have seen that the specific

heat diverges logarithmically with a characteristic

coefficient. The singularity is not necessarily

symmetric however. Certainly the constant Co in

(8.6) is not the same on the two sides of the transi- tion temperature. In fact it differs by approximately the amount of the discontinuity in the original

"Classical" specific heat. We have obtained a

first approximation in which the coefficient, a, is different above and below the transition, but there

are indications that this is not the final asymptotic

form and that the latter may well be symmetric.

The preliminary results of this approximation lead

to a coexistence curve different from the classical one, which is a quadratic relationship between the

100
density difference of the two phases and the temperature difference from the critical temperature.

This could also be described as a quadratic relation

between the density and the energy difference from

the critical energy — the energy being taken on

the critical density line. This relation between

the extensive parameters is preserved in our result, but the relation between energy difference

and temperature is no longer linear — in fact E — Ec^ t In t. This results in a coexistence curve which can be represented:

Ap = p,-p9

0C (~t In t) 1'2

or, equivalently,

— In Ap

It is useful to express this in terms of the natural

variable x, in this case x — pi~ pjpi + py, and in the case of a magnetic system, x = M/Mo where M is the spontaneous magnetization and Mo the satu- ration magnetization. The natural variable ranges

between 0 and 1 as t = 1 — T/Tc, ranges from 0 to 1. In this variable the expression for the asymptotic

coexistence curve is

x

2 oc i — In x

but this is not very suitable for comparison with

experiment because of the spurious singularity

at x—l. An expression with the same asymptotic

form, but with the correct limit at T=0 is

x

2 oc i 1 — In x

This expression (in which the coefficient unity

multiplying In x is quite arbitrary) has been compared with his measurements of the coexistence curve for liquid helium by Dr. M. Edwards who

has presented the comparison to this conference.

It is interesting that he obtains an agreement

rather better than with the usually accepted cubic

relation. Concerning the experimental test of the

predicted relation (8.6) between the coefficient of

the logarithm and the compressibility, we have

unfortunately not been able to obtain data with

which to make a useful comparison but it is to be

hoped that some will soon be available. In passing we note that the van der Waals equation would

yield for the value of a in (8.6) the number 9U In 2, (for T > Tc) which is near values that have been

observed.

This work was carried out with the enthusiastic

and able assistance of J. D. Gunton. Much of

it was performed during the tenure of a National

Science Foundation Senior Foreign Scientist Fel- lowship at Stanford University, which is gratefully

acknowledged.

Discussion

C. N. Yang: We are very grateful to Dr. Fairbank for presenting

his most interesting results. One cannot emphasize too much

that one of the most important results of recent years is the veri- fication that in some quantum mechanical systems there exists a new long range order which is macroscopic as is demonstrated

by this experiment. In 1961 Dr. Fairbank was also involved in an experiment which demonstrated the existence of quantized

flux in a superconductor showing the macroscopic effects of long range order [1]. These together with the recent discovery

of the Josephson effect [2J, have proved once and for all that quantum mechanical systems at very low temperatures often

exhibit correlations which are different from crystalline correlations, but which also extend to infinite distances. Dr. Fairbank

said that this experiment was brought about on a fifteen dollar bet from theoreticians. Let us hope that if we make a bet at the

thirty dollar level, we will obtain more results. W. M. Fairbank: I would like to point out that Fred Kellers

carried out the experiment on the X point I just described.

M. E. Fisher: It is perhaps worth pointing out that, although you

do have a X-line in liquid helium and not a X-line in the ferro- magnet, in the antiferromagnetic transition you do have a very

close analog of the X-line. In fact, in a spin flop antiferromagnet,

I think, you have something rather closely analogous to the

off-diagonal long range order which Dr. Yang was talking about

[3].

L. Tisza: The X-"point" in a ferromagnet is a point only in the same sense as the melting "point" is. Both refer to atmospheric

pressure and a variation of this parameter yields 7\ (P), a line

in the P-T plane. This is in contrast with the critical point in fluids.

J. S. Rowlinson: Even at the liquid-vapor critical point of fluids you do have a X-line if you have a two component system, where

there is an extra degree of freedom.

L. Tisza: This is correct. The prediction of Gibbs' phase theory

is that the dimensionality of critical states in the thermodynamic

space of intensities is 8 = c— 1 where c is the number of inde- pendent components. The difficulty in interpreting X-lines as

critical states is that this would mean 8=1 for c=l, a contradiction to this rule. Ehrenfest [4] sought to remedy the situation by assimilating the X-lines to lines of ordinary phase equilibrium.

(I confine this remark to the case c= 1.) His problem was then

to explain the absence of discontinuities in the first derivatives

of the Gibbs function that are characteristic, say of melting.

He did this by postulating higher order transitions for which

nth derivatives of the Gibbs function exhibit discontinuities along a line in the space of intensities, while the lower derivatives

101
are continuous. Shortly thereafter Landau showed that such

transitions can be indeed expected to occur on the basis of the thermodynamic theory of stability. The seemingly satisfactory

situation was upset by Onsager's discovery of the singularity of the Ising model, particularly since experiment showed that logarithmic singularities provide good descriptions also of real systems. These developments posed two problems for the student of Gibbsian thermodynamics: (i) identify the flaw in Landau's argument; (ii) generalize Gibbs' phase rule to account

for X-lines. It can be indeed shown that [5] (i) at critical states the discriminant of the so-called stiffness matrix vanishes and

this invalidates Landau's power series expansion; (ii) at critical states there are two modifications that become identical to each

other. In the classical case these modifications differ energetically (liquid and vapor). At \-points we have, say, the two polar- ized states of the Ising model which are energetically equivalent and which transform into each other by a symmetry operation; in

this case the dimensionality of singular points becomes 8 = c and the above mentioned difficulty is removed. The nature of

this symmetry is evident in all cases involving solids where the mechanism underlying the X phenomenon is understood. A

remarkable exception is the X-line of liquid helium. I find myself incapable of refuting the argument that this system

exhibits some elusive intrinsic symmetry. This is interesting

also in view of the recently discovered singularity of Cr at

fluid critical points which seems to point into a similar direction

(see the contribution of Dr. Chase on Wednesday afternoon). A precise formulation of this symmetry may well be at present

the most challenging problem of the theory of critical points.

P. W. Anderson: For a study of fluctuation phenomena in super- conductors, perhaps the best direction there is not to go to in- creasingly pure systems but to increasingly impure systems, because the amount of the critical point fluctuations will increase when one shortens the correlation length. So one would hope to bring out the critical point fluctuation phenomena by decreasing

the path, thus by making an increasingly impure system, which,

however, should be as homogeneous as possible.

/. Rudnick reported measurements of the sound velocity near the

X-point [6] in liquid helium.

M. J. Buckingham: With regard to the paper of Dr. Rudnick, I would like to point out that if one would find a violation of the Pippard relations, this means either that the system is not in thermodynamic equilibrium or that the singularity is not qualitatively the same all along the X-line. The latter property is used

in deriving the Pippard relations [7].

E. Helfand: I want to mention something which surprisingly has not been mentioned yet. Perhaps, it may be important here. There has been some thought recently about whether some of these apparent second order phase transitions really should not be first order phase transitions. The analysis has been made

mostly in magnetic systems, where one takes into account the volume dependence of the exchange coupling. An analogous

effect may enter into fluids, namely, by introducing many body

potentials one may get an effective density dependent potential which may destroy the second order nature of the phase transition except for one particular density. This would imply that you have to do constant volume measurements in order to get the proper second order phase transitions.

C. W. Garland: With respect to the comment of Dr. Helfand, I wish to point out that in the paper On an ultrasonic investigation

of ammonium chloride, to be presented later at this conference, we shall show in some detail that for an Ising ferromagnet such

a first order transition does occur across an instability region.

C. N. Yang: Could I ask Dr. Edwards what happens to the critical temperature if you make a plot of the variable X2 I(\ -\nX) that Buckingham suggested?

M. H. Edwards: I forgot to point out that on this basis the best

straight line goes through a temperature which is 0.006 °K below

the critical temperature on the r58 scale. I would therefore suggest that one should treat the plots made to fractions of a millidegree from the critical temperature with reservation, if this is indeed that uncertain. In view of the thermometry in- volved and my inability to put a thermometer into the actual

optical path, I do not know whether this difference is precisely 0.006 °K, but I think that there is a significant difference. It is conceivable that my whole curve is shifted somewhat. I should have said that for the A^-plot this would require a shift of 0.020 °K, which is certainly beyond the realms of possibility. The A^-plot leads to a critical temperature somewhat above the listed one depending on where you put the tangent.

M. J. Buckingham: I would like to make two remarks. The

first is a plea for the use of the proper natural variable in studies

of the coexistence curve of gas-liquid systems. This is the var- iable X = {pi — p„)(pi + pg) = (va — vdivy + vi), rather than the often used (pi — pg )l2pc . The variable X ranges from 0 to 1 as t=l — 777V ranges from 0 to 1 and is indifferent to the choice of number of atoms or volume as the basic extensive variable. The

other variable and X are equal if the "rectilinear diameter" is not only straight but constant as in the case of the Ising cell model. It has other advantages for plotting experimental re- sults as mentioned by Dr. Edwards. Concerning the particular form of the suggested coexistence

function, namely X2I(\ — In X), I should say that the theoretical argument which I will mention in my talk later, merely indicates the asymptotic form — X2 j\n X for X —* 0. The factor (—In A')-1

in this particular expression is not useful because of the spurious

infinity at X= 1. One is led therefore to consider the factors

( 1 — n In X)~* which have the required asymptotic form and which

for all n are zero at X = 0 and unity at X= 1 where the slope is n. These properties are shared with the set of functions X"; indeed

their general shape is remarkably similar to tjjie corresponding

simple power function. We can now emphasize the great difficulty of obtaining re- liably the correct asymptotic behavior of empirical results however accurate, over a limited range of the variable. As Dr. Edward's illustrations showed, the function X2/{1 — In X) is over

his liinge of parameters, numerically much closer to the function X:< than X1

, although the corresponding value for Fisher's

coefficient /3 would be V2 as for X'2

, rather than V3.

L. Tisza: Even if the Buckingham variable X is "natural" for the analysis of the coexistence curve, it is not a substitute for p

used with its conjugate intensity p. (see paper of Chase and Williamson). In fact, X is defined only on the coexistence curve and has no meaning for the homogeneous fluid.

J. S. Rowlinson: Dr. Moldover gave us some information on the

variation of C, with the density in the two phase region which he

said implied (<Pp.ldP)v * 0. Does this imply that - NT{d2 p.\dT%

becomes infinite? Or do your C, data imply that the curvature

of the vapor pressure line (d2pldT2

)\ becomes infinite?

M. R. Moldover: So far as I can see in the region I showed both terms are at least not decreasing and one of them is surely

increasing.

R. E. Barieau: Dr. Moldover, did you say that the quantity

-NTid2

fildTl )v is positive?

M. R. Moldover: Yes, the sign of this quantity does not change

in the vicinity of the critical point.

R. E. Barieau: Recently, I have analyzed the data of Hill and Lounasmaa [8] for helium in this way to obtain the quantities

(d2 p/dT2

)\ and (cPpldT2

)^. Now ordinarily below the critical point (d2p,/dT2

)t is negative, so that the two contributions to C, namely — NT(d2p.ldT2 )v and VT{d2pldT2

)\ are both positive down

around the boiling point. Now when the critical point is ap- proached, the data of Hill and Lounasmaa indicate that — NT(d2 u.ldT2

)i- is first positive and then becomes negative.

102
M. R. Moldover: I believe the data of Hill and Lounasmaa were

taken at intervals of 0.050 °K or larger and were not taken particularly at the critical density.

R. E. Barieau: That is correct. They made the measurements

at different densities and the measuring points were not chosen

to get these two quantities with the maximum accuracy. However, it is possible to deduce some figures for these quantities and there are. two series of experiments between 4.2 and 4.5 °K

in which the densities are far apart so that one can obtain these

quantities with considerable accuracy.

M. R. Moldover: Yes, there is a conflict between your conclusion from the Hill and Lounasmaa data and mine. With respect to the data of Hill and Lounasmaa in the vicinity of the critical point it should be remarked that they used a large chamber in which

complete thermal equilibrium is difficult to realize.

M. H. Edwards: They also smoothed their data.

M. S. Green: 1 would like to ask Dr. Edwards and Dr. Moldover how much the density dependence on the height in the fluid would affect their experiments. I think Dr. Edwards had some

control of this height. Is that correct?

M. H. Edwards: Yes. Of course, this effect is not important

further away from the critical point. At least at 5 °K the cor- rection due to the compressibility, which incidentally, I have measured in this range, is less than 0.05 percent in any of the

individual density measurements. The liquid had only a helium

liquid pressure between 1 mm and perhaps in the extreme case

4 mm above it. This correction was completely trivial even

while the extrapolated compressibility is going to infinity.

M. R. Moldover: Most of the helium in my cell is in the slots which are 3 mm deep. There will be a distribution of densities

in this helium column as you approach the critical point. Dr. Edwards measured the compressibility far from the critical point and also gave an equation of state, which he derived in the earlier days when one used a power series expansion. Using this I calculated that in my cell the distribution of densities is less than

2 percent at the temperatures closest to the critical point. So

on the plot I presented this gives a contribution to the curvature.

M. S. Green: In computing this correction one should probably

use the nonanalytic character of the density dependence as Dr. Larsen pointed out yesterday. It actually makes a significant difference. One cannot be satisfied to use an analytic expansion

in computing this correction.

M. E. Fisher: I want to ask Dr. Edwards: what is the actual

value of the parameter X in your experiments? In their experi- ments on xenon, Weinberger and Schneider [9] approached the

critical point to within A7'/7'(

. = 3 X 10""5

, while you essentially approached the critical point only to within 2 percent. So

Weinberger and Schneider approached the critical point a few

decades closer, but even as close as that this parameter X was

0.07. In other words, the curve is so flat that the density dif- ference pt — fig is still very large. Dr. Rowlinson made the point, which I think is a relevant one, that you have to approach the

critical point considerably closer.

M. H. Edwards: Yes. this is a curious feature. I went to within

2 percent from the critical temperature. My corresponding .Y values and consequently my density differences pi — pu, however,

are smaller. In other words, their xenon data look flatter than my helium data. After studying the data of Weinberger and

Schneider. I concluded that they did not adequately guard against

the gravitational gradients effe 'ts. They had a sample that was

1 cm deep and the compressibility correction is 15 times what

it would be for helium in the corresponding case.

M. E. Fisher: Well, I do not want to defend the measurements

of Weinberger and Schneider either, but they did a series of experiments to make a special study of the density effect. (The

experiments on xenon, hydrogen. He 4 and He3 have since been

analyzed systematically with a suggestion as to the nature of the quantum deviations [3].)

M. H. Edwards: Yes. but only down to 1 cm. They had a cell that was approximately 1.2 cm by approximately 10 cm. normally used in the vertical position. They rotated to the horizontal

position to change the gradient from a height of 10 cm to a height

of 1 cm. but not further. I claim that with a smaller slice of the fluid I got a different result.

G. B. Benedek: I wonder whether we could clear up the question

not of the difference in flatness between your data and those of Schneider, but between your data and those of Moldover.

Dr. Moldover made the point that he had evidence that his density curve was flatter than a dotted curve of yours.

M. H. Edwards: Not entirely clarified. On the density versus temperature plot Woodbury and I had results that were extrap- olated to go through a temperature which corresponded to the

T^h value of the critical temperature. That is the temperature

to which he is comparing it with. My evidence now suggests

that the critical temperature should be reduced. If I now fit the data not with the power series but with the logarithmic function, which calculation I have not yet completed, and extrapolate, the curve becomes flatter. However, the real answer is that our experimental data do not

yet overlap so that the disagreement is between an extrapolation based on a misconception and a clear experiment. I would say

that the weight of evidence is in favor of the direct measurements

of Dr. Moldover with respect to this point. As he pointed out, he is measuring a very large quantity.

R. Renard: I would like to point out that Dr. Garland and I have developed a theory in which not only the spin contribution but

also the lattice contribution to the specific heat of a crystal is taken into account. For an Ising model in two dimensions it can be shown that the system is mechanically unstable around

the critical temperature, leading to a hysteresis. This must also be true for a three dimensional system if it is correct that C, goes to infinity at the critical temperature for this model, where

one has to replace dTJdp by dT/dp along a line of instability points. It is predicted that C,, should go to infinity but that C, would not. References

[1] B. S. Deaver and W. M. Fairbank. Phys. Rev. Letters 7, 43

(1961).

[2] B. D. Josephson, Phys. Rev. Letters 1, 251 (1962).

[3] See, M. E. Fisher. Phys. Rev. Letters 16, 11 (1966).

[4] P. Ehrenfest. Leiden Comm. Suppl. 75b (1933).

[5] L. Tisza, Ann. Physics (N.Y.) 13, 1 (1961). See also L. Tisza "Phase Transformations in Solids." Smoluchowski. Mayer. and Weyl. eds.. ch. 1 (John Wiley & Sons. Inc.. New York,

N.Y., 1951).

[6] I. Rudnick and K. A. Shapiro. Phys. Rev. Letters 15, 386

(1965).

[7] M. J. Buckingham and W. M. Fairbank, Progress in Low

Temperature Physics, ed. by C. Gorter (North Holland

Publ. Comp. Amsterdam. 1961. Vol. Ill, p. 80).

[8] R. W. Hill and O. V. Lounasmaa. Phil. Trans. A252, 357

(1960).

[9] M. A. Weinberger and W. G. Schneider. Can. J. Chem. 30,

422(1952).

103

SCATTERING: ELASTIC

Chairman: P. Debye

Introductory Remarks

P. Debye

Cornell University, Ithaea, N.Y.

Since Dr. Montroll will not talk, I will take the

opportunity to speak two minutes to give you my

conception of what he would have said. I think he

would have said that, if you want to look at what

radiation is going to do when it falls on a medium

with irregular fluctuations, you have to characterize

these fluctuations first in a phenomenological way.

He would have started (don't believe what I say;

this is only my conception) by characterizing these

fluctuations by a correlation function. Then he

would have insisted that this correlation function

has to be a correlation function in space as well as

in time. Then he would have said that from this correlation function you can calculate the intensity

of the scattered radiation directly by a simple

Fourier transformation. Then he would have

said that you have to look at two things in the

intensity distribution. First, how is the intensity

distributed over the angles? This geometrical

feature of the intensity gives you an opportunity to characterize a length. Then he would have said

that you can also make a spectral investigation of

the scattered light, which would be characteristic

for time correlations. Then he would have said that

this is characterized by a correlation time or by a

relaxation time, if you want to call it that way de- pending on the circumstances. This would be

completely phenomenological.

I was prepared to tell him that this is a linear

theory, which you generally call a Born approximation. If you come in the neighborhood of the

critical point, the correlations become bigger and

bigger and bigger, so that probably your linear

theory will not work anymore. If you want to tell something about the angular distribution of the

frequency distribution of the scattered radiation

derived from actual observations, you have to intro- duce the right theory. The linear theory may not be correct at the critical point, because you get

such big fluctuations. So I had hoped to induce

him to say something about the second approximation and about what we are going to see there at least in a qualitative way.

Now, this is, of course, a phenomenological

background. If you want to know anything about

your correlation function, you have to build that upon a molecular theory. I hope that we will hear something about this molecular background.

What about it, Dr. Fisher? From you?

107
Theory of Critical Fluctuations and Singularities

M. E. Fisher

King's College, London, Great Britain 1

(1) In the previous lectures the behavior of the

macroscopic thermodynamic functions of systems

in equilibrium near critical points has been dis- cussed. Notations for the various exponents

describing the critical singularities have been in- troduced and the connections between them have

been described [1]. In this talk we enter the

microscopic domain by studying the correlation

functions and their relation, on the one hand, to the thermodynamic properties and, on the other

hand, to the phenomena of critical fluctuations as observed through critical opalescence and scat- tering experiments.

Perhaps the most important correlation function

is the net pair correlation function, namely, in a fluid system

G(r) = g(r)-\ = v

2 n2 (r)-l (1)

and

f/maSnetic ^-^yWHr), (6)

r

where (f(r) is the pair potential, with Fourier trans- form <p(k), and J{r) is the exchange energy coupling

the spins. The specific heat, in which we expect

a critical point anomaly, can, of course, be derived

by differentiating the energy with respect to temperature.

It is convenient to introduce the Fourier

transforms

X(k) = 1 + pG(k), G(k) =

j

e'^G(r)dr, (7) = l + ]>Vk T(r), (8)

r=t=0

and in a magnetic system in zero field

r(r)=r^(r)=<s*s*>/gS(S + i) (2)

both of which reduce to (s Sr) in a simple Ising

spin system (s = ± 1) or a lattice gas at critical density. In terms of these functions the compressibility and magnetic susceptibility of the systems are given by

which are directly related to the fluctuations of the

Fourier coefficients of density and spin deviation

through

<|8p k ! 2

) (|8Sk

2

| 2

) = X(k) Vp /V<|8So| 2

) (9)

The thermodynamical formulas (3) to (6) may then

be rewritten as

1 + G(r)dr

xr=xVdeai 1+2 T(r)

(3)

(4)

Furthermore the internal energy can also be found

since

^coring— hp<p(0) + hp j

<p(r)G(r)di (5)

' Present address: Cornell University, Ithaca, New York.

KT\ Z^63 '

1 = x(0) - Xt/Xt

eal

, (10)

and

<p(-k)x(k)dk « At/ a ~j j(k)x(kWk. (ID

Furthermore the scattering intensity observed with

light, x rays or neutrons at wave number k = (4>irlX) X sin (6/2) is given (in first Born approximation) by

/(k)//0(k) = x(k), (12)

where 7o(k) is the intensity which would be ob- served from the ideal, noninteracting, system.

108
These results show that x(k) is the quantity of

most fundamental and direct theoretical and ex- perimental interest. In as much as a critical point is associated with an infinite divergence of

Kt and \t we see from (3) and (4) that the correlations must become of increasingly long range near

a critical point. This, in turn, implies some sin- gularity in the behavior of x(k) for small k as T

approaches Tc . Indeed a rapid increase of x(k)

for small k is the mathematical expression of the

observed critical opalescence.

(2) Let us review briefly the "classical" approximate theories of the behavior of the correlations near a critical point. The first theory was that due

to Ornstein and Zernike [2] for a fluid; in many

ways it is still to be preferred since its theoretical

assumptions are clear and easy to appreciate.

Firstly, the direct correlation function C(r) is intro- duced via the relation for its Fourier transform.

X(k)=l/[l-pC(k)]. (13)

It is then argued, or assumed [3], that C(r) is of

short range and remains so up to and at the critical point. If this is so one may expand its transform

in a convergent Taylor series as

C(k) = C(0)-rik2 + . • • , (14)

since C(k) is analytic in k

2 even at the critical point.

The "interaction length" r0 is constant (or slowly

varying) and remains finite at the critical point.

Truncation yields the well-known result xW-roW+K2

), (15)

where k is related to C(0) and turns out to be the

inverse range of correlation. From (15) a plot of the experimentally observed inverse scattering

intensity I~ l {k) should be linear in k

2 at all temperatures approaching Tc as shown in figure 1. This Lorentzian line shape is a central prediction

of the theory.

As k approaches zero we see from (10) that KT , Xt~ lim x(k) = x(0)~ 1/k2

, (16)

so that k must vanish as the critical point is ap- proached. We may assume, rather generally, that k(T)~(T-TcY as T-*TC,

( i 7)

FlGl RE 1. Plot of inverse scattering intensity versus k

2 near the

critical point to be expected on the basis of the Ornstein- Zernike and equivalent theories.

along the critical isochore or in zero field. It fol- lows that a plot of 7

_1 (k) should intersect the origin

when T=TC (see fig. 1). Inverting the transform (15) for a (/-dimensional

system yields:

(a) for fixed T > Tc and r—> °°

C(r) ~ e-^/rW- 1 *'/ 2

, (18)

(b) for T= Tc and r^> °o

G(r) = Gc(r)~ l/^"2

, (19)

where l/r° corresponds to logr foro?=2. In general

these results can be combined into a formula of the

form

G(r)**D(erKrli*-')[l + Q(Kr)'], (r-> «) (20)

where

()(*)-»• 0 as *->0, (21)

and

1 + (?(*) ~jC(d - 3) / 2 as x-^°°. (22)

If we assume a van der Waals equation of state we

have [1] from (16) y=l and v=h, (23)

but more generally according to Ornstein and

Zernike we would have only the relation

y=2v. (24)

109
We may cite briefly many later alternative treat- ments that are essentially equivalent as regards

their consequences (but in which the nature of the

basic assumptions are frequently somewhat obscure). For fluids there are the semithermodynamic or phenomenological theories based on the

introduction of gradient terms into the free energy

due to Rocard, Klein and Tisza, Debye, Fixman,

and Hart. Yvon has argued that C(r) could be re- placed by the Mayer function fir) i.e., by its leading

term in a virial expansion. In a somewhat similar

way the Percus-Yevick equation always predicts a short range C(r) if the potential <p(r) is of short range. Brout's self-consistent field approach

yields, with p=HkT,

X(k) - XH.c.(k)/ [1 - )8p£'(k)xH.c.(k)] , (25)

where H.C. denotes the corresponding pure hard

core system and <£'(k) is the Fourier transform of ^WgH.c.M- This is again equivalent near Tc . (Some of these theories have been reviewed in more detail recently [3].) For binary alloys one may mention work by Landau, Krivoglaz, Zernike,

and Cowley amongst others. For magnetic sys- tems the phenomenological theory of Van Hove

was modeled on the Ornstein-Zernike and van der

Waals theories and is quite analogous. The generalized mean field treatments of Brout, and Elliott and Marshall [4], etc., yield results of the form

X(k)~l/[l-£/(k)], (26)

which, for short range exchange potentials, are

essentially equivalent. The more recent Green's

functions treatments of Tyablikov and others,

Mori and Kawasaki, and Kubo and Izuyama (for the itinerant electron model) lead again to virtually

the same results; in particular the direct correlation function is always of short range. The spherical model of Berlin and Kac is also similar but

with v—1 rather than v=\ as in most of the other

theories, since y — 2 in that case.

(3) I would now like to point out a conclusion

drawn from many of these theories 2 concerning the nature of the specific heat singularity (CV or C«=o).

Firstly note that one can hope to justify most of these approximate expressions for G(r) at fixed

T> Tc only as r—> °o [3]. If, however, we assume

they are also valid for small r (specifically of order

of the range of direct interaction) we may calculate

the specific heat as T—> Tc . If, as is often fashionable, we perform the calculations entirely in terms

of Fourier transforms it is quite easy to overlook

this implicit assumption since the "derivation" runs roughly as follows: For example for a magnet,

assuming (15) and using (11), we have

J(k)dk

+ k

2

)2

' (27)

Thus putting k — KX, we find as k^O (i.e., T—> Tc )

so that from (17) the exponent for the specific

heat [1] would be

a=l-(d-2)v. (29)

For d— 3 and the usual approximation, v— i, we

thus find C„~~1I(T-Tcyl*, (30)

which is a surprizingly common prediction in the

literature! The analysis, however, is quite unjustified as is rather easier to see if one works in

real space using (18) or (20) and (5) or (6) and explicitly inserts the short range character of the potential by cutting off the integral or sum at some r= ri [5] . In fact the specific heat is determined by

the variation of G(r) or T(r) with T as 7

1 —* Tc for small, fixed r (i.e., by Q(x) for small x in (20); see

also below). Thus although in a general sort of

way the long range behavior of G(r) is a direct "indi- cator" of the transition it is not sufficient to deter- mine the specific heat singularity.

(4) We have listed the "classical" approximate

theories which concur with the predictions (15)

to (20) of Ornstein and Zernike. Recently, however, other answers have been proposed, in the

first place by M. S. Green [6] on the basis of the

hypernetted chain integral equation. Green argued

that certain irreducible diagrams might be neglected

at the critical point and thereby obtained an equation for Xc(k)- As observed by Stillinger and

Frisch [7] his argument may be applied in d dimensions and yields [3]

2 But not all! See the discussion comment by Marshall concerning an honorable

exception [4]. Gc{r)^DI^\ (T-Tc) (31)

110
with

r) = 2-(d/3). (32)

Taking the Fourier transform of (31) with 17 > 0

leads to the important prediction, contrasting with

that of the O-Z theory, that a plot of inverse scat- tering intensity versus k

2 at T= Tc should be curved

upwards according to

/oU) ~ 1/Xc(k) ~ (33)

(see fig. 2). This point will be returned to. More recently another expression for x(k),

namely

X(k) ~ l/(A + k

1^), (k -> 0), (34)

has been presented by Abe [8] who was inspired

by the recent paper of Patashinskii and Pokrovskii

[9] on the lambda anomaly in helium. This latter work has attracted a great deal of attention since

it claimed to predict theoretically the logarithmic

specific heat anomaly which is observed in the

beautiful experiment described by Professor Fair- bank this morning. Indeed the formula (34) is "designed" to predict a logarithmic specific heat

by just the same false argument we outlined above!

[In the analysis of Patashinskii and Pokrovskii

and of Abe it is assumed that A(T) varies as (T— Tc) to the power unity. However some other power

would not alter the logarithm but only its amplitude.] Even if (34) is correct, we cannot, therefore, believe in the "derivation" of the logarithmic

specific heat. Actually, as we will see below, (34)

is not correct for the Ising model and its analog

for helium, in terms of the one-body density matrix,

is equally to be doubted. From (34) we find

Gc(r) « D/r'W (35)

which is equivalent to (31) with

T) = 2-hd. (36)

Note that in significant contrast with all other theories, G(r) will still have a long-range tail (1/r3' 2

for d = 3) when T exceeds Tc according to (34) since

X(k) is still singular at k

2 = 0. (Patashinskii and

Pokrovskii were more circumspect on this point!)

(5) Now I will turn to some more rigorous results on the behavior of the correlation functions which

will enable us to judge the approximate theories.

Firstly, as to the question of the exponential decay

of the correlation functions away from the critical point (in the case envisaged of short range inter- actions). With fair generality one may define the

inverse range of correlation through [3]

K(D = -lim sup (1/r) log |G(r)|, (T>TC ). (37)

r —* 00

The existence of this limit with k > 0 implies the

asymptotic exponential decay of G(r). For the

Ising model the matrix method provides expressions for the correlation functions from which (37)

is an immediate consequence in a system of finite cross-section (but infinite length). Onsager's [10]

explicit solution for d=2 shows this remains true

for a system of infinite width above Tc . For a continuum classical gas one may construct a generalized integral kernel for which one similarly

finds an exponential decay, the parameter k being

determined, as in the matrix case, by the ratio of

the two largest eigenvalues.

The convergence of the activity and virial ex- pansions for gases recently established rigorously

by Penrose, Lebowitz, and Ruelle implies a similar

result for G(r) in the domain of convergence — and,

by continuation along the physical axis, up to some

singularity which would have a physical significance

(probably the critical or condensation point). We

can thus expect rather generally that x(k) should

be analytic for small k

2 above Tc (in disagreement

with Abe's result but in agreement with the O-Z

theories).

The vanishing of k as T approaches Tc according

to (17) is confirmed rigorously by Onsager's result,

exp(-Ka) =v(l + i;)/(l-v), » = tanh UlkT), (38)

for the d=2 Ising model. This yields the nonclassical value v=\ for the exponent defined in (17). Further details of the decay of correlation above

Tc for the simple Ising model follow from an analysis

of the matrix method [3, 11] which confirms the

O-Z result (18). For the more general integral

kernels an appropriate form of perturbation theory

reveals a spectrum of eigenvalues from which a

similar conclusion will follow. The analyses of nonuniform fluids by Lebowitz and Percus in terms

of the direct correlation function also serve to justify the asymptotic validity of the O-Z theory

away from the critical point [3].

1 1

1
The most important rigorous result for our present

purpose, however, follows [3, 7] from the work of Onsager and Kaufman [12] on the pair correlation

functions of the d=2 Ising model at the critical point. Their result shows that

Gc(r) « fl/ri/4, i.e., v= 1/4, (d=2). (39)

This is in clear contradiction with the predictions

of Ornstein and Zernike (tj = 0) and all other theories! [See eqs (32) and (36).] To summarize these considerations, we can still hope generally to describe the correlations near

the critical point in terms of two lengths, the

"range of correlation" I/k(T) which diverges as T^>TC and a "range of direct interaction" ro{T)

which remains finite. However, generalizing (20), we must expect for all small k that

G(r) = D(e-Kr

lrd~2 ^) [1 + Q{kt)\ , (r -> oo) (40)

where D = dnlprl +r> and Q(x) satisfies (21) and, in place of (22),

1 +Q{x) -BrfOMV***, as x -» oo. (41)

If we neglect Q (which is valid near Tc) the Fourier

transform of (40) is well approximated by

X(k)~DI(K2 + k

2y^\ (42)

which extends (33) to temperatures above Tc - (See

also fig. 2.) In view of (39) we might expect the

exponent 17 to have a relatively small but positive

value. Letting k approach zero and using (10)

and (17) shows that the divergence of the susceptibility and compressibility (along the isochore)

[1] is now given by

y={2-ri)v (43)

in place of the Ornstein-Zernike formula (24). As

observed some time ago [13] this result applied to the d = 2 Ising model using (38) and (39) yields

y = 7/4 which is just what is found by careful

extrapolation of the series expansion [14, 15] as explained yesterday by Professor Domb.

(6) In view of my remarks on the impossibility

of determining the specific heat singularity from the

long-range part of the pair potential 3

it is worth

pausing to ask how Onsager's famous result that the specific heat of the d = 2 Ising model diverges

as log \T—TC \ fits into the picture. Accepting the

form (40) as the leading contribution to G(r) for all

/' and using (6) and (38) one can easily see that the

logarithmic singularity in the specific heat means

that

Q{x) ~ Bx log x, as x —» 0. (44)

As explained, it is only the behavior for small x which is relevant. More generally if a + 2v>l

(as is probably always the case) a specific heat

exponent a > 0 would follow from (40) if

Q(x) = x-bx(1 ~a)lv +. . . asx^O. (45)

(7) In the absence of exact results for the three- dimensional Ising model (or for the Heisenberg

model) one must return to approximate methods of

calculation. However, by developing systematic

successive approximations, notably sufficiently

long series expansions, one may hope, as in the case of the thermodynamic functions, to draw re- liable conclusions about the critical behavior. I will report on progress along these lines in work

undertaken in collaboration with Mr. R. J. Burford.

Firstly, for the d = 3 Ising model one may develop

a series expansion for the range of correlation k in powers of i; = tanh (J/kT). In the standard Ising

problem the partition function can be expanded

in terms of configurations of (closed) polygons on

the lattice (each bond being associated with a factor

v) subject to the condition that an even number of

bonds meet at each vertex. To calculate e~K" (where a is the lattice spacing) one needs in addition

to the polygons a single chain of bonds that stretches

right across the lattice (in the direction of interest).

By enumerating these configurations one finds for the propagation of correlation parallel to a principal

axis of the simple cubic lattice

e-*a = v + 4v2 + 12ir3 4. 36t;4 + iQgv5 + 356|;6

+ 1204t;7 + 4420i;8 -fl6124i;9 + .... (46)

3 The specific heal divergence can, however, be found from the long range part of the fourth order correlation functions.. See remarks in the discussion following tins

talk.

112
Evaluation of this series with the aid of Pade ap- proximants enables accurate plots of k(T) to be

drawn which reveal clear deviations from the pre- dictions of the simpler approximate theories [16]. The series is not, however, sufficiently regular to yield a firm estimate of the exponent v although one may conclude that it lies in the range 0.60 to 0.7. Since y=1.25 for the simple cubic this is consistent with some small value of tj in (43) but evi- dently cannot decide for or against the OrnsteinZernike prediction 17 = 0. A more fruitful approach has been to calculate

the series expansions for the correlation function

G(r) at all values of r on the lattice. The relevant

configurations consist again of polygons plus a chain

but the chain is now of finite length and simply

runs from the origin to the lattice point at r (these

being the only points at which an odd number of

bonds may now meet). Appreciable assistance

in the calculations may be gained from the known

series for the susceptibility and from available

numerical data on self-avoiding lattice walks.

From the series for G{r) one may form the moments

txs {T) = p j

r*G(r)dr,

or = £ rT(r) = 2 r

s (^r ). (47)

r r

Of course the zeroth moment is just proportional to the susceptibility itself while the second moment

is directly related to the initial slopes of plots of the

inverse scattering versus k'2 (and to the so-called Debye "persistence length," see ref. 3). By using

(40) and (43) we find immediately that

P-iiT) « M2 /k4 -" ~ ll(T- Tc )^y. (48)

Consequently estimates of the critical behavior of

fM2 (T) will yield estimates of v and 17. For the simple

cubic the following terms have been obtained

(while one more is available for the plane square

lattice)

P> = 6v + 72w2 + 582i/3 + 4032t;4 + 25542y5

+ 153000?;" + 880422*;7 + 4920576v8 + 26879670v9

This series and the corresponding ones for the

plane square lattice and other three-dimensional

lattices are very regular and may be extrapolated

by the ratio and Pade approximant techniques. As

a check on the consistency of our analysis let us

first look at the two-dimensional series for p,-i. Using

the exactly known critical point and the result y = 7/4 the Pade approximants to (v — vr )(d/dv) log p-i yield the successive estimates:

7) = 0.2495, 0.2532, 0.2485, 0.2511, 0.2507, (d = 2).

These are evidently approaching the exact limit

7] = V4 rather rapidly which confirms the validity of our approach. For the simple cubic a corre- sponding sequence of estimates (taking 7= 1.250

and using the accurately known critical point) is:

r] - 0.0696, 0.0684, 0.0621, 0.0583, 0.0580, (d = 3).

These values seem to be converging to a limit in the

vicinity of 0.058 to 0.060 and studies of related series

lead to a similar result. We may conclude with fair confidence that

7]=0.059±0.006, i^=0.644±0.002, (d=3). (50)

The results for the B.C.C. and F.C.C. lattice are

quite similar although less precise as the series are shorter. The value (50) for r) is consistent with

the rather natural conjecture r\ = Vig = 0.06250

(implying v = 20/3i = 0.64516 . . .) but the true value

might well be lower, say, t> = Vis = 0.05555 (implying

v — 9/i4 = 0.64285 . . .). (It is interesting to note

that the values of the analogous exponent for the

self-avoiding walk or excluded volume problem are

j]' = 2/9 and Vis in two and three dimensions,

respectively.)

One may also use the series for G(r) to estimate

Gc(r) = Gc(x, y, z) directly for small values of r although this cannot be very accurate owing to the

singularity of G(r) as a function of T which is similar to that in the energy i.e., roughly {T—Tc ) log

(T— Tc ). For the nearest lattice points to the origin we estimate:

(0, 0, 1) 0.328, (0, 0, 2) 0.160, (0, 0, 3) 0.11;

+ 144230088^°+ .... (49) (0, 1, 1) 0.206, (0, 1, 2) 0.137; (1, 1, 1) 0.163.

113
These values are quite consistent with (31) and (50)

i.e., with Gc(r) ~ 1/r106

, but they do not yield an independent estimate of r\. For the Heisenberg model fewer terms can be cal- culated and their behavior is less regular especially

for low spin values. The most favorable case is the F.C.C. lattice for S = °° where we obtain the

sequence of estimates:

v = 0.786, 0.760, 0.724, 0.710, 0.702.

From this and other sequences we feel able to estimate

t} = 0.075 ±0.035, v = 0.692 ±0.012. (51)

The values of r\ and v (as those of y) do not appear

to depend significantly, if at all, on the value of the

spin (or on the lattice structure).4

(8) It seems clear from these calculations that tj has a positive value for three-dimensional Ising

models and for the Heisenberg models. The value

of 7) is, however, rather small although it is interest- ing to note that it is apparently slightly larger for the Heisenberg model. (This is in accord with

the larger deviation of the exponent y from its classical value of unity.) I feel 17 might be somewhat larger still in a continuum fluid with molecules of a nonsimple shape where "excluded volume

effects" would be expected to play a larger part. In as much as 17 is positive we must expect a plot of the inverse scattering intensity versus k

2 to be

curved at and (by continuity) near Tc for small k

as shown by (33) and (42). Since the expected

value of r\ is so small, however, this curvature (even

if present) might not be very evident. In fact,

the main effect (as may be verified by plotting curves according to (42) for small values of tj) is to cause

the scattering plots "to hang up" so that the inter- cepts appear not to approach zero as 7

1 —»7V, and

even at Tc the curve appears to extrapolate linearly

with A

2 to a nonzero value. These effects are indicated in figure 2 where, however, a relatively

large value of 17 has been assumed for the sake of

clarity.

What do the experiments say? Well, we will hear the latest situation from some of the following

*Note added in Proof. However, recent work on the susceptibility for 5= \ indicates

that the value of y is probably appreciably larger than the S — °° value y — 1.33 (al- though rj may not change much); see G. A. Baker. H. E. Gilbert. J. Eve. and G. S. Rushbrooke. Physics Letters. 20, 146 (19661.

114

k*

FIGURE 2. Plot if inverse scattering to be expected with a positive value of r) (larger than that estimated for the three-dimensional Ising or Heisenberg models) illustrating the difference between the apparent linear intercept f0(T) and the true

intercept £(D-l/x(0).

speakers but perhaps I should finish by reviewing

very briefly a few relevant experiments which, I think, give some indications of small deviations from

the Ornstein-Zernike theory. These deviations are

in the expected direction although, I should stress, the problem is still very much "in the air"! Walker and Keating in their recent neutron

scattering experiments on the binary alloy /3- brass [17] observed that when T approached Tc the intensity of the superlattice line (which is analogous to Kt in a fluid system) increased more

rapidly than expected on the classical theories.

Their observations were, in fact, consistent with

y=1.25. They also observed a slight narrowing of

the line relative to a Lorentzian line shape near Tc which would be expected from (42) with a small

positive value of rj. More recent experiments on

/3-brass by Dr. Als-Nielson and Dr. Dietrich will be reported tomorrow.

r Jacrot's original neutron scattering experiments on iron yielded, on reanalysis, the estimates y = 1.30

to 1.33 for the behavior of l/x(0) near Tc . This is in agreement with direct measurements of the susceptibility as has already been reported (and with

the Heisenberg model predictions). Passell and

coworkers have also found a value of y — 1.33 from

their more recent experiments on iron. They seem,

in addition, to find evidence that the intercepts of

the scattering curve are "hung up" in a way consistent with a small value of 17; but this is very

tentative and I will leave Dr. Passell himself to de- scribe his very careful and difficult experiments.
Thomas and Schmidt [18] have made x-ray experiments on argon in the critical region; they observed

no deviations from Ornstein-Zernike theory!

However, they did not measure along an isochore nor did they go very close to the critical point.

It is to be hoped that these experiments will be

extended both with x rays and with light, since the

behavior of (presumably) simple fluids like xenon

and argon is obviously of cardinal interest. The most extensive critical scattering measure- ments have been made on binary fluid systems

where in most cases one or both species are comparatively complicated organic molecules. This

makes me. as a theoretician, somewhat apprehensive when it comes to applying the (relatively)

simple theories we have been discussing! Professor Brumberger will review this field in detail

later this afternoon so let me just sketch some of the

recent work. Professor Debye and his collaborators working in particular with polystyrene-cyclohexane solutions have found little or no evidence

of deviations from the classical theories — even as regards the value of y for x(0) which appears to be

quite close to unity. However, the experiments

are quite difficult to perform, especially near Tc , and Mclntyre. Wims. and Green, in experiments on

ithe same system, have indeed found deviations from

the classical expectations close to Tr roughly of

the sort to be expected with a positive value of 17. Brady and Frisch [19] have also observed somewhat

similar nonclassical anomalies in the system

perfluoroheptane in iso-octane.

More recently Dr. Chu in a series of very careful

experiments on «-dodecane in dichloroethyl ether

has found significant deviations from linearity in

the inverse scattering plots at temperatures very

close to critical. I do not want to anticipate Dr. Chu's lecture later this afternoon but let me say

that the value of rj that seems to fit his excellent

data, namely rj — 0.35, is much larger than our simple theoretical models would suggest. Maybe

this is an effect of steric hindrance due to the shape

of his molecules — I do not know! Whatever the answer, it is clear that there remains much scope for experimental and theoretical work before our understanding of these subtle but fundamental questions

will be complete.

References

[1] See the associated notes on "Critical Point Singularities." these Proceedings page 21.

[2] L. S. Ornstein and F. Zernike, Proc. Acad. Sci. Amsterdam

17, 793 (1914): Physik. Z. 19, 134 (1918).

[3] M. E. Fisher. J. Math. Phys. 5, 944 (1964): reprinted in The

Equilibrium Theory of Classical Fluids, eds. H. L. Frisch and J. L. Lebowitz (W. A. Benjamin. Inc. 1964).

[4] R. J. Elliott and W. Marshall. Rev. Mod. Phys. 30, 75 (1958).

[5] This has been pointed out by others, e.g.. in M. Fixman.

Adv. in Chemical Phys. 4, 175 (1964), although it seems not

to be generally appreciated.

[6] M. S. Green. J. Chem. Phvs. 33, 1403 (1960).

[7] F. H. Stillinger and H. L. Frisch. Physica 27 , 751 (1961).

[8] R. Abe, Progr. Theoret. Phys. 33, 600 (1965).

[9] A. Z. Patashinskii and V. L. Pokrovskii, Soviet Phys. JETP

19, 677 (1964).

[10] L. Onsager. Phys. Rev. 65, 117 (1944).

[11] L. P. Kadanoff. to be published.

[12] B. Kaufman and L. Onsager. Phys. Rev. 76, 1244 (1949).

[13] M. E. Fisher, Physica 25, 521 (1959).

[14| C. Domb and M. F. Sykes. J. Math. Phys. 2, 63 (1961).

[15] G. A. Baker. Jr.. Phys. Rev. 124, 768 (1961).

[16] See the figure due to R. J. Burford and M. E. Fisher in Proc.

Inc. Conf. on Magnetism. Nottingham. Sept. 1964 (Phys.

Soc. London) page 81.

[17] C. B. Walker and D. T. Keating. Phys. Rev. 130, 1726 (1963).

[18] J. E. Thomas and P. W. Schmidt. J. Chem. Phys. 39, 2506

(1963).

[19] G. W. Brady and H. L. Frisch, J. Chem. Phys. 35, 2234

(1961); 37, 1514 (1962).

209-493 O-66—

9

115
Scattering of Light and X-rays from Critically Opalescent Systems

H. Brumberger

Syracuse University, Syracuse, N.Y.

Introduction

The experimental investigation of critical phe- nomena has become an actively pursued area of research in recent years for two major reasons —

a

renewal of interest in the statistical-mechanical

theory of these phenomena (for reviews see, for instance, references 1—5), and a considerable in- crease of experimental sophistication coupled with

a better understanding of the experimentally signi- ficant variables.

The review will attempt to survey, briefly, the] experimental literature dealing with light and x-ray scattering from one- and two-component systems

near critical points, beginning in 1950 and concluding in March 1965. Magnetic phenomena will not be considered, but attention confined to gas-liquid,

liquid-liquid, and solid-solid critical points. The

review will not be completely exhaustive, but

attempt to compare the most important results

critically, and to extract any common observational

features. Some criteria for comparison will be

discussed below. The systems reported on are) summarized in tables 1—3.

Table 1. One-component fluid systems

System Technique References / a (A)

Argon X ray 6 (1942)

7 (1951)

8 (1963) 5.4 near

transition.

Neon X ray 9 (1962)

Nitrogen X ray 10 (1950)

11 (1964) 6.0 near

transition.

Ethylene Light 12 (1950)

Carbon dioxide Light 13 (1960)

Methane X ray 14 (1957)

Ethane X ray 14 (1957)

Light 15 (1950)

Propane X ray 14 (1957)

Butane X ray 14 (1957)

Ether X ray 16 (1960)

Benzene X ray 16 (1960)

Debye "range of intermolecular forces.

116
Table 2. Fluid binary systems

System Technique References /a(A)

reriiuorometnylcyclohexanecarbon tetrachloride Light 1/ (195U) 14. / Laroon disulfide- A ray lo (i95o)

memanoi Light ZU (1954)

Phenol-water A ray in nc<;\ 19 (1950)

Light ZU (1954)

lsobutyric acid-water Light on 1 1 o c a \ ZU (1954)

r nenoi-nepiane T irrVit j_iigni zu90 HQ^Zh ^iyo4| ) Methanol-cyclohexane Light on noc/n ZU (1954)

Light Zl (196Z) 12.7

T ' L. Light 22 (1963)

Methanol-hexane Light on / 1 n c /i \ ZU (1954)

Nitrobenzeneisopentane Light on nnc/i\ ZU (1954)

Aniline-cyclohexane Light on nnc/i\ ZU (1954)

Light oi /in/;o\ Zl (1962) ~ u

Light 25 (1965—1964)

1 nethylamine-water Light on nnc/i\ ZU (1954)

p, p -Uichloroethylether-n-decane Light Z41 (1V041 ) ~ lo

l-wciane-pernuoroheptane vA ray oc. no/tit Zo (1V01)

A ray ZO (1901) vA ra) Li (iyoz)

A ray 2o (1904)

U 1 2, 6-Dimethylpyndinewater Light Z9 (1904) Q 7

Nitrobenzene-n-heptane X ray ouo.n nOA^ (iyoo) O.D

Light 0.1 nQA/1\ 51 (19041 ) Pertiuorotributylamine-isopentane T a s r Lk 4- Light oL (1904> IOC. 1Z.O

A ray 52 (1964) i /i n 14.U

Polystyrene-cyclohexane Light oo (190U)

54 (196U) r rom Z /—06

or noAO'i 55 (190Z) depending on

50 (190Z) average molecular

0.7 nOA9\ oi (iyoz) weight

o.q no^n^ oo (ivou)

Av ray OV (.IVODJ 07/ m — 14.700^

and Light

p\ /8'-Dichloroethylether-rc-dodecane Light 4U (1965)

a Debye "range of intermolecular forces."

117
Table 3. Solid systems

System Technique References

a-fi Quartz Light 41 (1957)

Aluminum-zinc X ray 42 (1958)

General Criteria

The observations reported in the tabulated publications must be subjected to critical examination

from two viewpoints: (1) Are the reported data

reliable — i.e., were proper precautions taken to insure purity of materials, precise temperature

control, etc., and were other experimental factors kept to a minimum or corrected for? These might

include interfacial reflections, dust, inconstancy

of light source, cell geometry, multiple scattering,

refraction corrections, beam divergence, monochromaticity, etc.. for light-scattering experiments;

and collimation errors, background corrections,

absorption corrections, monochromaticity. etc. for x-ray experiments. (2) Are the data sufficiently

extensive (i.e., do they extend over a sufficient range of angles and temperatures), do they confirm

theoretically predicted behavior, and does the col- lective body of observations indicate some trends

of sufficiently general nature to rule out the possibility that they are experimental artifacts? This

is a particularly important point for critical phenomena, which apparently are subject to a variety

of experimental effects not usually observed under

noncritical conditions (for instance, gravitational

effects). Most of the recent theoretical treatments of criti- cal opalescence have been based on an extended,

and' sometimes reformulated. Ornstein-Zernike

(O-Z) Theory, and on various lattice gas calculations [3. 4, 5]. Direct experimental tests of the

scattering behavior predicted by classical O-Z

theory as reformulated by Debye are

1. The inverse of the relative scattered intensity

should vary linearly with s

2

, where s = 4-n-A-1 sin (6/2),

A. is the wavelength in the medium, 9 the scattering

angle.

118

2. i~ 1 {s) versus s

2 curves obtained at different temperature distances AT{AT=T—TC ) from the

critical temperature should be essentially parallel.

3. The extrapolated zero-angle intercepts of these curves should vary linearly with AT and go to zero

at TV. Further analysis of the approximations in the

O.-Z. theory [4] indicates the possibility that the

i~ 1 (s) versus s

2 plot might show downward curvature near T^> T, for small values of s

2

, and that the

i -I

(0) versus AT plot might no longer be linear. Miinster and coworkers [43, 44] have recently

shown that a more rigorous result for i -1

, on the

basis of a modernized version of the O.-Z. theory, is

i-l(s)< 1 k

2 + s2 4npA (Di

4npA

where k•—> 0 as T^> Tr . p is an "'effective density."

A is a constant. (1) yields the O.-Z. relation as ani asymptotic approximation for small s. and would,

predict long-range curvature in the i~ l (s) versus\

s

2 plot.

Debye. in recent publications [32]. has also* stressed the fact that his original theory [1] is anj

approximation taking only the average square of; the local gradient of density or composition flue-j

tuations into account, and that inclusion of higherorder terms in the series expansion for the inverse!

intensity would introduce curvature into O.-Z. plotsj

when carried over a sufficient range of s

2

. Nevertheless, the approximate Debye theory is useful in correlating experimental observations, and yields

the characteristic parameters / and L. defined by

L

2 = (persistence length) 2 f4-TTi~G(r)i-(lr

/47r/-2 6'( r)dr

(2)
and L

2 =

/2

1 (3)

where t = T]Tc . I. the "range of intermolecular

forces." is predicted to be constant for O.-Z. behavior; where this parameter has been reported,

its value is given in tables 1-3.

One-Component Systems

One observation which turns out to be generally

true is that a differential of about 10 years be- tween publications dealing with the same system

is frequently good reason to discard the earlier work.

For argon and nitrogen, several x-ray studies are

available in the 1950-1965 period. For argon, the

1951 paper by Graham and Lund [7] is based on much earlier data obtained by Eisenstein and

Gingrich [6] which were taken to a minimum scat- tering angle of 2° only. A correlation function is obtained by direct Fourier inversion of the extrapolated scattering curves and used to calculate

(r2 )=L2

yielding values of 6-8.5 A, which would indicate

that measurements were not extended to sufficiently

small angles. In view of this situation, only the recent work on argon bv Schmidt and Thomas [8] can be used to test the O.-Z. predictions.

For the x-ray investigations of nitrogen [10], paraffins [14], ether [16], benzene [16], and neon

[9] somewhat similar remarks apply. In the Russian paper on ether and benzene [16], insufficient experimental information is given: it is difficult to credit the observation of several x-ray diffraction

maxima within 2 deg 6 for benzene, and to accept

the authors' analysis of their data as indicating

density fluctuations of certain discrete sizes only. For the other systems, the measurements were

not extended to sufficiently small angles or ATs.

nor were collimation corrections applied. Inter- pretation of these data was generally attempted by means of a Guinier approximation (assuming

spherical inhomogeneities) and invariably yielded

very small radii (3-14 A) for the inhomogeneities.

The recent nitrogen work of Schmidt and Thomas

[11] must remain the major source of useful data

for this system. Schmidt and Thomas' observations on argon and nitrogen are in general agree- ment with the classical O.-Z. theory. The zero- angle inverse intensity is linear with AT for both

vapor and liquid over several degrees near the

critical temperature Tc . A meaningful /-value 5 A near Tc — is obtained in both cases. No

deviations from the Ornstein-Zernike plot— i~\s) versus s

2 — are observed at small angles: this may

be because the smallest AT was 0.05 °K and the

smallest scattering angle about 2xl0~3 radians

(MoKa radiation for argon, CuKa for nitrogen). All three of the light-scattering measurements —

carbon dioxide [13], ethane [15], and ethylene [12] —

were made at scattering angles of 90° only. In the case of carbon dioxide, the critical temperature was only approached to within ~ 0.2°, and the

authors report generally good agreement with Rayleigh's equation except nearest Tc , where the

exponent of k is 3.2 instead of 4. The experiments

on ethane and ethylene which were very precise

as far as temperature was concerned (±0.001 °C)

but much less precise with respect to pressure

control (±1 psi), indicated four regions: (1) T > Tc , k~4 law holds, (2) a region where intensity increased,

X-dependence decreased, within about a degree

above T( , (3) a reversal in intensity with close

approach to Tc (ascribed to multiple scattering and

absorption), (4) and a rapid, erratic increase (as- cribed to condensation droplets) which showed

hysteresis phenomena on temperature reversal. The authors conclude that the O.-Z. theory, which

predicts a change in wavelength dependence pro- portional to A. -2

, does not apply at temperatures

below the temperature of maximum scattering, i.e., very near Tc . Nevertheless, they calculate "cluster

diameters"' from the O.-Z. intensity expression

(for the region where this applies) to be of the order

of 1000 molecular diameters closest to Tr . While

it is not possible to translate these observations

directly into the language of this paper, it appears

that the O.-Z. theory yields meaningful parameters—comparable to values for L found from the

Debye treatment — within a 1-deg region of Tc , but

not in the immediate neighborhood of the critical point.

The general conclusion for the one-component

systems evaluated is that the x-ray observations

show good agreement with O.-Z. theory, the lightscattering observations reasonable agreement ex- cept very near Tc , where results are seriously

afflicted by multiple-scattering and turbidity

problems.

1 19
The possibility of gravitational effects was examined by Mason and coworkers [45-49] , who report

definite effects in one-component but not in two- component systems. These are probably not seri- ous for the above cases.

Fluid Two-Component Systems

Because of considerable difficulties in controlling

pressures precisely for one-component systems near

the critical point, more experiments have been done

with two-component systems, which show analogous

behavior near the extrema of the consolute curve. The temperature variable is more easily controllable, but purification of the samples becomes

harder.

Most of the work on two-component systems has

been done by light-scattering, and for several

systems a series of recent publications is available.

The x-ray observations by Kirsh and Mokhov on

the carbon disulfide-methanol and phenol-water

systems [18, 19] neither approach Tc sufficiently

closely nor extend to sufficiently small scattering

angles. It is difficult to understand the series of

five small-angle maxima reported within a scatter- ing angle of one degree at temperatures 22.5 and

28.5 °C from Tc for the CS2 -MeOH system.

An extensive series of binaries was investigated

by Chow Quantie [20] using light-scattering.

Results fell generally into two categories — systems

showing strong opalescence, a wavelength exponent

between 3.2 and 4.3, and a maximum scattered

intensity at about y (CS 2 -MeOH, <POH-H,0,

<t>N0 2 -isopentane, <J?NH 2 -cyclohexane); and systems

showing weak opalescence, a wavelength exponent

between 2.2 and 2.5, strong dissymmetry, but no maxima (MeOH-cyclohexane, isobutyric acid-H2 0,

MeOH-C6 H 14 , Et 3 N-H2 0). It is likely that the

angular intensity distributions for the strongly

opalescent systems are distorted in the forward

direction by multiple scattering effects; this is borne out by Quantie's observation that even the

weakly opalescent systems showed a forward dropoff (and corresponding development of a maximum)

a? the wavelength was shortened. Similar observations, and a shift of the intensity maximum to larger scattering angles with smaller k are reported

for the aniline-cyclohexane system. Attempts to account for multiple scattering were made by

Quantie.

120

Replotting of the Quantie data for the aniline- cyclohexane (strong) and isobutyric acid-water

(weak) systems shows the following: for isobutyric

acid-water runs at 0.02 and 0.07 °C from Tc , nearly

parallel O.-Z. fines are obtained, both showing a

slight downward curvature at small angles. For

aniline-cyclohexane (at 0.04 and 0.10 °C), parallel

but curved O.-Z. lines result, with upward curvature!

at both smallest and largest angles. The highangle behavior seems fairly typical for strongly!

opalescent systems and is also observed for nitro-i benzene-heptane, etc.

It is interesting to compare these results to the! light-scattering observations of Debye and co-, workers [21, 22] for cyclohexane-aniline and methanol-cyclohexane. The latter system, according

to Quantie, exhibits weak opalescence and similar,

light-scattering behavior to isobutyric acid-water.

( Slightly different concentrations and critical tem|

peratures are reported by the two investigators:

it is probable that the purity of Debye's samples isj better. In the methanol-cyclohexane case, I "normal" O.-Z. plot is found within 0.68 °C from Tc There are no obvious deviations from linearity

except a slight upward drift at small and large

angles. O.-Z. curves are nearly parallel. These

measurements were only extended to a scattering

angle of 30°, however. The zero angle intercepts:

somewhat doubtful because of the limited lowl

angle range, are linear with AT". For cyclohexane!

aniline, the Debye group reports nearly horizontai O.-Z. curves bending upward substantially at higf

scattering angles and slightly at small ones, anc becoming more erratic at larger A7"s. Intercept: are no longer linear in AF. The similarity betweer

Debye's and Chow Quantie's data close to Tc h

striking.

More recent and very careful light-scatterinj

experiments by Mclntyre and Wims [23] have

brought several new and interesting discrepancies

to light for the cyclohexane-aniline system. A

definite downward curvature of the O.-Z. plots is observed at AT < 0.01 °C and at small scattering

angles, using sample cells of thickness down t<j 0.5 mm. For smaller cells (—0.1 mm), nev phenomena appear — for instance, a change ii separation temperature by as much as 0.5 °C

Whether this is due to surface or other effects ij not known; nevertheless it is a graphic illustratioi that attempts to reduce one difficulty (multipl

scattering in this instance) may introduce a host o I new ones in the neighborhood of the critical point!
Among the remaining fluid binary systems

e investigated by means of light-scattering alone are

er perfluoromethylcyclohexane-carbon tetrachloride

l [Zimm, 17], 2, 6-dimethylpyridine-water [Pancirov

and Brumberger, 29], rc-decane-/3, /3'-dichloroethyl

ether and rc-dodecane-/3, /3'-dichloroethylether

[Chu, 24, 40]. Of these, two [17 and 24] apparently

show "normal" O.-Z. behavior, the latter to within

0.04 °C of Tc . The 2, 6-dimethylpyridine-water

system can largely be described by the OrnsteinZernike-Debye theory, but shows typical upward

deviations from O.-Z. linearity at high scattering

angles and the beginnings of a downward deviation

at small angles and near Tc (Ar=0.02 °C). The

dodecane-dichloroethyl ether system, on which particularly careful measurements were made by Chu

very near Tc (AT — 0.002 °C), and at several wavelengths, shows a gentle concave-downward curvature of the O.-Z. lines over the entire range of s.

It is significant that these results cover a consider15 able range of s and that multiple scattering effects

1S were minimized by choosing a system of small An

and superimposing scattering curves for several

f wavelengths. It is not, of course, unexpected that

a large-angle deviation of the O.-Z. plot from lineier.

|||S.

arity appears, since the O.-Z. correlation function

is an asymptotic approximation for large r and

should therefore hold best at small s. An x-ray investigation exclusively has been made

by Brady and coworkers on the perfluoroheptanei-octane system [25-28]. Here also, a downward

deviation of O.-Z. curves is found at the smallest

scattering angles, the O.-Z. curves typically showing

jan inverted S-shape; it must, however, be noted

^ that the curves were obtained no closer than 0.16

fen

'

, I C from the separation temperature at the composilit . ii. " jtion used, which was hot the critical composition.

More recent precision experiments on this system

""fare reported to be in progress [50].

13,e The remaining fluid binaries to be discussed

probably represent the most important experimental

^ set, since composite light and x-ray scattering data

ts isfare available for them, and they thus extend over a

>nnfvery much greater range of s-values. Included are

1 '"'nitrobenzene-n-heptane [Pancirov, Farrar, and

ffl Brumberger, 30, 31], perfluorotributylamine-

'"'isopentane [Debye, Caulfield, and Bashaw, 32], ^'and polystyrene-cyclohexane [Debye et al., Mclntyre

let al., Eskin, Chu, 33-39]. Experimental difficulties

jiiooi are considerable in trying to overlap the two regions.

P^This must primarily be done by extending the x-ray

ill measurements to extremely small angles, which

necessitates particularly accurate intensity meas- urements and collimation corrections. In the case of nitrobenzene-n-heptane, while an

overlap has still not been achieved, the change in slope with Ar for the O.-Z. curves based on x-ray

intensities is several times greater than for light-scattering curves in the same temperature

interval; further, an /-value nearly three times as

large is obtained from the light-scattering results. These observations imply that there is long-range curvature in the O.-Z. plot and that zero-angle x-ray

intensity extrapolations are likely to be most

unreliable unless the light-scattering range is substantially overlapped.

The perfluorotributylamine-isopentane measure- ments reported show just adjacent curves which

are not, however, taken at exactly the same temperatures. The x-ray results alone show a distinct curvature in i _1 versus s

2

; a downward curvature at large s

2 and no curvature at small s

2 are indicated,

Ar=0.08 °C. Light and x-ray data yield a consistent picture in terms of the Debye parameter I. Here also, the high-angle behavior is not un- expected.

Where the use of critical opalescence data of

polymeric systems is concerned, the author feels that some caution is indicated because interpretation of the observations is rendered highly complex

by the considerable range of molecular weights in even the most carefully fractionated polymer

sample, and by the anisotropy of the molecules

themselves. Nevertheless, Debye and coworkers

have found that the light-scattering observations

agree well with the Ornstein-Zernike-Debye theory.

Mclntyre, Wims, and Green, on the other hand,

report that the extension of measurements to small angles reveals a severe drop-off of O.-Z. curves near Tc ; extrapolated i _1

(0) values versus A71 deviate upward from linearity. Chu has

achieved a composite light- and x-ray scattering

O.-Z. curve which apparently shows no small- angle deviations but does drop below the O.-Z.

line at large s. Here, again, each curve is a composite of data taken over a range of temperatures

as large as 0.32 °C. All of the polystyrene observations are strongly dependent on the average molecular weight of the dissolved fraction. Solid Systems

Phenomena analogous to those observed in fluid systems appear in the neighborhood of certain

phase-transitions in the solid state.

121
The aluminum-zinc system exhibits greatly

enhanced small-angle x-ray scattering at temperatures above the separation temperature of the

solid solution phase, 351.5 ±0.4 °C. A careful

study by Minister et al. [42, 43, 44], indicates that

a treatment of the data is feasible by means of an

extension of the Ornstein-Zernike theory to eq (1). Here, again, curvature in the i~ l (s) versus s

2 plot

is observed.

The available light-scattering evidence for the

a-(3 transition of quartz [41] is limited to measure77 ments at — and was found to obey the X

-4 law.

Partial depolarization (about 6%) of the scattered

light is reported. The presence of an opalescent

strip of finite width between the two phases is taken

as evidence that the transition is not first-order.

General Conclusions

It is clear that a search for deviations from the

predictions of the classical Ornstein-Zernike-Debye

theory must for the moment center on observations

dealing with binary systems. There is no clear

deviation apparent in any of the one-component

studies so far undertaken. This is unfortunate,

since our theoretical understanding is greater for the simpler systems. Conversely, it is not unex- pected that deviations from classical behavior occur

in systems of greater complexity, where the scat- tering involves not just the intermolecular potential e(r) , but tn, e-22. and 612 [1]. For binary systems investigated by one tech- nique only, the results on the whole are inconclusive. Nearly all the light-scattering observations

have certain features in common: reasonably good

fit to the O.-Z. curves is found, but deviations appear

at largest and smallest scattering angles. Those

at large angles are usually upward, the others are sometimes downward for temperatures quite near Tc , but do not appear regularly; this may be a function of the minimum scattering angle accessible

in each experiment. Extrapolated zero-angle inter- cepts are proportional to AT for some systems but

not for others and in any case rather suspect since

few small-angle light-scattering data exist. The

observed anomalies unfortunately often occur at the limits of experimental accessibility, and un- doubtedly are at least to some extent rendered

uncertain by multiple scattering effects. Perhaps

the best light-scattering results are those for the

122

rc-dodecany —/3,/3'dichloroethyl ether system, for which long-range O.-Z. curvature is reported for AT -0.002 °C [40] and where multiple scattering

is apparently negligible, and those for cyclohexaneaniline [23] indicating downward small-angle curvature of the O.-Z. plots for AT < 0.01 °C. Small-angle x-ray measurements alone are less conclusive; the small-angle drop-off reported for isooctane-perfluoroheptane is in the range where

collimation corrections become very significant. In any case only one or two experimental points

are shown lying below the O.-Z. curves. Zero-angle

extrapolations are almost certainly misleading here. Additional evidence that the theoretical predictions of the O.-Z. theory are inadequate comes

from the combined light-scattering and x-ray scat- tering data of references 30-39. All of these report curvature over a large range of s, but the observations are not sufficient to permit general conclusions

to be drawn.

The total weight of evidence indicates that there

are real deviations from the Ornstein-Zernike pic- ture, and that these are in the direction suggested

by newer theories. What is uncertain is the precise nature of the deviations. This suggests quite

clearly what needs to be done:

1. Future experiments should be a composite of

light- and x-ray scattering to cover the maximum

possible range in 5. To this end, small-angle lightscattering studies should be included, and a range

of wavelengths should be used. With the avail- ability of laser sources, much greater control overi

the monochromaticity of the radiation and the scat-! tering geometry can be exercised.

2. The critical point must be approached more

closely (AT ~ 0.001°) though this may introduce new;

difficulties which are not now apparent.

3. A serious attempt (theoretical and experimental) needs to be made to account for multiple

scattering, since this problem will be aggravated as the critical point is approached more closely.

4. It is likely that, at least for some time, the most

precise experiments will be done with binary sys- tems. It would be of great value to concentrate on relatively few of these which display deviations

and also are good experimental systems, and to! test them with a much greater variety of experi-i

mental approaches — scattering, transport properties, etc. It is quite apparent that many of the, phenomena occurring near critical points are poorlyjl

understood and others still unknown. In spite ofjj the concentration on binary systems, precision!
experiments on one-component systems of spherical molecules are still desirable. It is to be ex- pected, however, that deviations from classical

behavior will be harder to find, and will occur in regions where the experimental difficulties are very great.

Bibliography

[1] Debye, P., J. Chem. Phys. 3 1 , 680 ( 1959).

[2] Kac, M., Uhlenbeck, G. E., and Hemmer. P. C, J. Math.

Phys. 4, 216 (1963).

[3] Fixman, M., Adv. in Chem. Phys. VI, (1964).

[4] Fisher, M. E., J. Math. Phys. 5, 944 (1964).

[5] Miinster, A., article on "Critical Fluctuations," Fluctuation Phenomena in Solids, p. 180, R. E. Burgess, ed., (Aca- demic Press, N.Y. 1965).

[6] Eisenstein, A., and Gingrich, N. S., Phys. Rev. 62 , 261

(1942).

[7] Graham, W., and Lund, L. H., J. Chem. Phys. 19, 1380

(1951).

[8] Thomas, j. E., and Schmidt, P. W., ibid. 39, 2506 (1963).

[9] Stirpe, D., and Tompson, C. W., ibid. 36 , 392 (1962).

[10] Wild, R. L., ibid, 18, 1627 (1950).

[11] Thomas, J. E., and Schmidt, P. w\, in press. Tl2j Cataldi, fl. A., ancTDrickamer, H. G., J. Chem. Phys. 18,

650(1950).

[13] Skripov, V. P., and Kolpakov, Yu. D., Kriticheskiye Yavleniya i Flyuktuatsii v Rastvorakh, Moskov. Gosudarst.

Univ., Trudy Soveshchaniya, Moscow, 1960: p. 126.

[14] Buttrey, J. W.,J. Chem. Phys. 26, 1378(1957).

[15] Babb, A. L., and Drickamer, H. G., ibid. 18, 655 (1950).

[16] Mokhov, N. V., and Labkovskii, Ya. M., ref. 13, p. 81.

[17] Zimm, B. H., J. Phys. and Coll. Chem. 54, 1306 (1950).

[18] Kirsh, T. V., Zh. Khim. Fiz. 32, 1116 (1958).

[19] Mokhov, N. V., and Kirsh, T. V., ibid. 30, 1319 (1956).

[20] Chow, Quantie, Proc. Roy. Soc. 224A, 90 (1954).

[21] Debye, P., Chu, B., and Kaufmann, H., J. Chem. Phys. 36,

3378(1962).

[22] Chu. B.. J. Phys. Chem. 67, 1969 (1963).

[23] Mclntyre. D., Wims, A., private communication.

[24] Chu. B., J. Chem. Phys. 41, 226 (1964).

[25] Brady. G. W., and Petz, J. I., ibid. 34, 332 (1961).

[26] Brady, G. W., and Frisch, H. L., ibid. 35, 2234 (1961).

[27] Frisch, H. L., and Brady, G. W., ibid. 37, 1514 (1962).

[281 Brady, G. W., ibid. 40, 2747 (1964).

[29] Pancirov, R., and Brumberger, H., J. Am. Chem. Soc. 86,

3562(1964).

[30] Brumberger, H., and Farrar, W. C, Proc. Interdisciplinary Conf. Electromag. Scatt. p. 403 (Pergamon Press, New

York, 1963).

[31] Brumberger. H., and Pancirov, R., J. Phys. Chem. 69,4312

(1965).

[32] Debye. P.. Caulfield, D.. and Bashaw, J., J. Chem. Phys. 41,

3051 (1964).

[33] Debye, P.. Coll. H.. and Woermann, D.. ibid. 32 , 939 (1960).

[34] Debye, P.. Coll, H.. and Woermann, D., ibid. 33, 1746 (1960).

[35] Debye, P., Woermann, D., and Chu, B., ibid. 36, 851 (1962).

[36] Debye, P., Chu, B., and Woermann. D., ibid. 36, 1803 (1962).

[37] Mclntyre, D., Wims, A., and Green, M. S., ibid. 37, 3019

(1962).

[38] Eskin, V. E., Vysokomolekul. Soedin. 2, 1049 (1960).

[39] Chu. B., J. Chem. Phys. 42, 426 (1965).

[40] Chu, B., Paper presented at APS Meeting, Kansas City, March 1965.

[41] Yakovlev, I. A., and Velichkina, T. S., Uspekhi Fiz. Nauk. 63,411 (1957).

[42] Miinster, A., and Sagel, K.. Mol. Phys. 1, 23 (1958).

[43] Miinster, A., and Schneeweiss, ch., Z. physik. Chem. (NF)

37,353(1963).

[44] Miinster. A., and Schneeweiss, ch., ibid. 37, 369 (1963).

T45] Murray, F. E., and Mason, S. G., Can. J. Chem. 30 , 550

(1952).

[46] Whiteway, S. G., and Mason, S. G., ibid. 31, 569 (1953).

[47] Murray, F. E.. and Mason, S. G., ibid. 33, 1399 (1955).

[48] Murray, F. E., and Mason, S. G., ibid. 33, 1408 (1955).

[49] Murray, F. E., and Mason, S. G., ibid. 36, 415 (1958).

[50] Mclntyre. D., Wims, A., and Brady, G. W., Proc. Conf. Small-Angle X-Ray Scatt. Syracuse, 1965; to be published.

Experiments on the Critical Opalescence of Binary

Liquid Mixtures: Elastic Scattering*

B. Chu

University of Kansas, Lawrence, Kans.

The purpose of this paper is two-fold. Firstly,

it tries to provide a careful review of many of the

pertinent experimental studies on the elastic scattering of visible light as well as on the smallangle x-ray scattering of critical binary liquid mixtures. It is hoped that the review portion of the

article, although in the main argumentative in

•Elastic scattering here refers to the unresolved scattered intensity in the absence

of a spectrometer or an interferometer. This differs from inelastic scattering where

the frequency spectrum of the scattered light has been resolved into central. Brillouin, and Raman components by means of optical beat frequency technique, interferometry and spectrometry.

character, will clear up some of the confusion

which many readers inevitably get in trying to in- terpret scattering data on critical opalescence

without investigating the experimental details. Secondly, preliminary work on visible light scatter- ing of a binary fluid mixture, n-dodecane-/3,/3'

dichloroethyl ether, at small values of s/A. (5 = 2

sin (9/2, A = wavelength in the medium) and very

close to its critical mixing point does demonstrate

a breakdown of the Ornstein-Zernike [1] and the

Debye [2] theories. Further theoretical work

becomes necessary.

123
Experimental Validity of Elastic Scattering Measurements

Accurate experiments are not easy to perform

near the critical point and the interpretation can be confused by many experimental effects, such as multiple scattering, impurities, and temperature

fluctuations. Qualitatively, the observed data on

binary fluid mixtures seem to confirm the classical

theories [1, 2]. On closer examination, one finds numerous "anomalies," especially from data pubfished in the past 10 years or so. Perhaps, one of the reasons for us to find so many "anomalies"

is because we expect to see deviations from the

classical theory on the basis of simple analysis in terms of its theoretical limitations and approximations. Needless to say, the reported deviations

tend to coincide with some of the theoretical expectations and vice versa. However, I am reasonably

certain that many observed data are tinged with

experimental effects [3]. It would be desirable to have more accurate and extensive data to substantiate my suspicions. Indeed, several systems

have been and others are being reinvestigated.

For the time being, I am only able to provide mostly

indirect experimental evidence. Reviews on both

theory [4, 5] and experiment [3, 6] have been pre- sented elsewhere. Data are quoted for illustrative purposes only.

1. Maximum scattered intensity (light scattering):

For the system aniline-cyclohexane, Chow [7] observed a pronounced peak at a scattering angle

9 of about 90° from a plot of scattered intensity versus 6 at intermediate temperature distances

from the critical solution temperature (T—Tc < 0.3°). Debye, Chu, and Kaufmann [8] reinvesti- gated the system and concluded from measure- ments at one concentration that the maximum

scattered intensity reported by Chow was an experimental effect. It should be noted that, in critical opalescence studies, scattering measurements at one concentration seldom provide definitive answers. This is even more so for the system

aniline-cyclohexane partly because aniline cannot

be purified easily and it undergoes photochemical

decomposition; and partly because the system scatters light very strongly near its critical mixing

point. As the shape of the coexistence curve changes markedly in the presence of impurities,

it is not certain whether the results reported by

Debye, Chu and Kaufmann [8] was indeed measured

at the critical solution concentration. Furthermore,

the horizontal curves in reference 8 of figure 3 may partially be the result of an over correction on the alinement of the light scattering photometer

as indicated by the slightly negative slope of LUDOX

in the same figure [3]. It seems that further investigation on the system

aniline-cyclohexane becomes necessary. Such

experiments have been performed at Kansas with

better instruments [3, 6, 9]. We have measured

the critical opalescence of the aniline-cyclohexane

system at three concentrations, 39.6, 40.6, 41.6 vol. percent of aniline. The corresponding phase

separation temperatures are respectively 29.89g,

29.89^, 29.8% which are about 0.5° above the values

reported in the literature [10]. A typical plot of

reciprocal relative scattered intensity versus sin2 9/2

at k() — 578 m/x is shown in figure 1. For such a

39.61 vol. % Aniline in Cyclohexane

40 61 vol. % Aniline in Cyclohexane

AT = 03I

01 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9

sin

!(f)

FIGURE 1. Plot of reciprocal relative excess scattered intensity

2 0

versus sin*

39.6, and 40.6, vol. percent aniline in cyclohexane. Observed phase separation temperature= T„= 29.89 °C. / excess (6, T) = 1(6, T)-/(0, T-Tp ~ 10°). X„=578 m/i.

124
appreciable with cylindrical cells of about 8 mm in inside diameter even at relatively large temperature

distances. For this reason, we used an incident

wavelength of 578 m/u, and performed our measure- ments not very close to the critical solution temperature so that the effects of multiple scattering were

reduced. The results illustrate that the scattering

behavior of this binary fluid system is similar to that of other binary liquid mixtures [3, 6]. It is quite certain that both the maximum scattered

intensity and the lack of dissymmetry [8] for this system are experimental artifacts. The Debye

/-parameter was estimated to be in the same range

as those of other binary liquid mixtures [3, 6].

2. Upward curvature {light scattering): Virtually

all binary fluid mixtures in the strongly opalescent

region exhibit upward curvatures at large angles in a plot of reciprocal scattered intensity versus

^jj (or sin2 912). Such a plot is sometimes referred to as an OZD plot. Closer inspection shows

that the cylindrical scattering cells are either so

large (e.g., 10 mm diam) that multiple scattering

becomes appreciable or so small (e.g., 2 mm diam)

that reflection and refraction problems are no longer

avoidable. The only exception was Zimm's classi- cal work [11] in which a different geometry for his photometer was used. The most convincing test

for multiple scattering consists of a tedious pro- cedure involving collimation and detector slits of

various cross sections, scattering cells of different diameters and geometry, and composite scattering curves by means of both light and small-angle

x-ray scattering [12, 13]. With those parameters,

especially from a composite curve of light and small- angle x-ray scattering of polystyrene in cyclohexane

[13], we have shown that the scattering curves continue from the visible light to the x-ray region

without abrupt bends. Similarly, the same conclusion can be drawn from a superimposition of the

scattering curves using three wavelengths (Ao = 365, 436, and 578 m^t) in the visible region [3,

6]. Therefore, the upward sweep has to be an

experimental effect. In addition, the scattering

due to fluctuations seems to behave differently

from that due to particles, as a binary fluid mixture

could show an upward curvature in an OZD plot

indicating the presence of multiple scattering

while a LUDOX (colloidal silica) solution of comparable scattering power may reveal only a hori- zontal straight line in the same plot [14].

3. Attenuation correction (light scattering): For

very intense opalescence, the extinction of both

incident and scattered light as the beam goes

through the critical fluid mixture has to be taken into account [6]. Multiple scattering usually changes

the shape of the scattering curves, such as an up- ward sweep in an OZD plot. Even in the absence

of multiple scattering, attenuation correction be- comes important especially when we want to know

the temperature dependence of the extrapolated

zero-angle scattered intensity over large temperature ranges. According to the Debye theory, [2] we may assume

that in the visible region, the scattered intensity

/ at constant wavelength has the form:

j = A(T) + B sin2 1

(1)

while, in fact, we have for the measured scattered

intensity I* after correction for effective scattering

volume:

A(T) + B sin2 (2)

F'(T) is an experimental correction factor based

on the temperature dependence of the observed

extinction coefficient and the size of the scattering

cell. We have assumed that F'(T) is independent

of the scattering angle. In an OZD plot, we should

not get parallel straight lines corresponding to different temperature distances since, according to eq (2), the measured slope is F'(T) • B. Instead, a small increase in slope may be detected for strongly

opalescent systems as the critical solution temperature is approached. This could be one of the rea- sons which may contribute appreciably the changing

slopes reported by R. Pancirov and H. Brumberger

[15] and by D. Mclntyre, A. Wims, and M. S. Green

[16] since neither articles have discussed this ex- perimental effect. The same correction was also ignored in Zimm's work [6, 111.

4. Downward bend in an OZD plot (light scattering): Eskin [17] found a downward bend at relatively large temperature distances from the

phase separation temperature (7^) for the system

polystyrene-cyclohexane away from the critical solution concentration, while Mclntyre, et al. [16], found a downward bend at only small temperature

distances from Tp . Furthermore, the pronounced

downward dips reported by Mclntyre, et al., appear

only after the presence of upward sweeps for which

125
multiple scattering is responsible. With unaccountable effects due to polydispersity, it should be

emphasized that only binary liquid mixtures of small molecules or one component systems are appropriate if we are interested in testing theories.

On the other hand, polymer fluid mixtures are of

interest in terms of the Debye /-parameter which is related to the cohesive energy densities. We have

shown that, in an OZD plot, at very small temperature distances, say T— Tp — 0.002°, the system

rc-dodecane and fi, ft' dichloroethyl ether, [18] indeed

exhibits a gentle downward curvature virtually

extending over the whole range of s/\. More ex- tensive studies are still in progress.

5. Temperature dependence of extrapolated zero- angle scattered intensity: Plots of reciprocal extrapolated zero-angle scattered intensity versus temperature are slightly concave upwards and tend to level off or to extrapolate to a nonzero value at Tc. For a binary liquid mixture, it is very difficult to obtain this extrapolation in view of the fact that the

measured scattered intensity depends not only on

the attenuation correction factor but it is also a sum of two terms, one due to concentration fluctu- ations, and the other due to density fluctuations.

When we want to compare the observed data with

theory, it is the concentration term that is of interest. The density term is usually, but not always, negligible over large temperature ranges. A plot of the

reciprocal extrapolated zero-angle scattered intensity due to concentration fluctuations,

| versus temperature shows that the straight line behavior, predicted by the approximate Debye

theory, holds over short temperature ranges [6].

It is conceivable that deviations from the form

j (n. =C(T—Tp) exist over large ranges of tempera-

/ f(o) ture, i.e., over decades of the temperature distance

from Tp

. We have identified the maximum phase

separation temperature on the coexistence curve as the critical solution temperature. Under these

conditions, a correction term for density fluctuations, in addition to the attenuation correction,

should be taken into account. So far, all the re- ported data on the temperature dependence of extrapolated zero-angle scattering, except for the

system n-decane and dichloroethyl ether [6], have ignored both corrections.

6. Extrapolation to zero-angle from OZD plots

(small angle x-ray scattering): It may be justified

to extrapolate the reciprocal of scattered intensity

126

to zero angle by means of fight scattering where the

s ° values of — go down to about 1 X 10~4 A

-1

. The

A.

validity of this extrapolation by means of small- angle x-ray scattering often becomes questionable

at small temperature distances from the critical solution temperature. It is my belief that the ab- normally small I (~ 2.6 A) for the system nitrobenzene-n-heptane [19] is in error.

7. Constant intensity (small-angle x-ray scattering): The usual critical opalescence behavior clearly

contradicts the constant intensity results of G. W.

Brady and H. L. Frisch [20]. The same system,

ra-perfluoro-heptane and isooctane, has been rein- vestigated by fight- and small-angle x-ray scatter- ing [21]. Our light scattering data, although partially qualitative in nature because of multiple

scattering, showed no region of constant intensity.

Instead, they were found to be similar to the be- havior observed in other binary liquid mixtures.

Plan of Attack

Once we know the difficulties, it takes a little planning before we should start our experiments.

In principle, we want to select a critical binary

fluid mixture which satisfies the following conditions:

1. Purification: Both components could be purified by preparative gas chromatography.

2. Stability: Both components are stable and do

not undergo photo-chemical reactions.

3. Temperature control and measurements: The

phase separation temperature by means of visual

observation could be determined to about 0.001°

without excessive patience. The same temperature limit applies to its controls (proportional temperature controller) and measurements (G-2 Mueller

bridge). Therefore, it is reasonable to aim at a (T—Tp) value of 0.001°. We also try to select a system which has a critical mixing temperature

near the room temperature.

4. Multiple scattering: For strongly opalescent

systems, multiple scattering invariably becomes the

j most serious and difficult obstacle to overcome. As we have mentioned earlier, one way is to reduce

the sample cell size and the effective scattering

volume. A cleverer approach (in my opinion) is to select a system which is not strongly opalescent.

Even then, we are limited to cylindrical cells no

larger than about 6-10 mm in diameter. A small
value of An, which implies a small (—], is a suit- able criterion. The symbols n and c refer to the

index of refraction and the concentration of the

fluid mixture respectively.

5. Large electron density differences: We want a system which scatters x rays as strongly as possible.

It will be helpful if the two components possess a

large difference in their electron densities.

6. "Gravitational" effect: In one component

systems, the effect of gravity very near the critical point has to be taken into account. This effect

is negligible for critical binary fluid mixtures where

the scattering due to concentration fluctuations is predominant. Since, in many respects, the behavior of two-component systems is closely

analogous to the condensation of simple fluids, we choose to begin our investigation on two- component systems. One-component systems are much more satisfying in terms of theory, although

they are experimentally more difficult. In view of the above conditions which I have

imposed, we start looking for a two-component

system in which both components can be purified

by preparative gas chromatography, and are stable; has a convenient critical solution temperature, a small difference in the index of refraction of the

two components, and a large difference in their

electron densities. Fortunately, we have found not one two-component system but a homologous series

of saturated paraffins as one of the two components

with /3, /3'-dichloroethyl ether or chlorex as the

other component. The paraffin-chlorex systems

satisfy all of the imposed conditions. In addition,

the homologous series provides us a series of sys- tems with increasingly smaller refractive index

differences as the molecular weights of the alkanes

increase. Therefore, it is within reason to expect

that experimental data on critical binary liquid

mixtures very close to the critical point by means of

light scattering are accessible with existing tech- niques. We may also take advantages of the homologous series for molecular interaction studies in terms of the Debye theory.

Experimental Details

Detailed descriptions of the light scattering photometer, [9] purification of materials, preparation

of solutions, phase separation temperature determinations are discussed elsewhere [6, 18]. Only

the changes will be specified.

FIGURE 2. A schematic diagram of the modified constant tem- perature bath and slit systems in the light scattering photometer.

Further insulation on the thermostat-tank inside

the light scattering photometer was necessary. The

modifications are shown in figure 2. We favored

the immersion tank technique because the bath

had a reasonably large heat capacity, and the

immersion oil, which matched the index of refraction of glass, reduced both refraction and reflection

difficulties of a small light scattering cell. The bath

temperature was controlled to better than 0.0005°.

All metal parts in the tank were insulated, and there was a metallic shield around the light scattering

cell to insure uniformity of temperature over

the whole fluid mixture. The phase separation

temperature of the sample was measured in a separate constant temperature bath and inside the

photometer by means of a telescope. The observed

phase separation temperatures agreed to within

0.001° for each liquid mixture.

Results and Discussion

The Debye theory provides an excellent repre- sentation of experimental facts in the intermediate

temperature distances over large ranges of s/k. Even deviations from the straight fine behavior

in plots of reciprocal scattered intensity versus

(s/A)2 (or (d/k)2

), by means of small-angle scattering

of x rays, has been interpreted in terms of the Debye

theory [12, 13, 22]. On the other hand, deviations

from the straight line behavior in an OZD plot at small values of s/k and very close to the critical solution temperature do exist [18]. Figure 3 shows

a plot of reciprocal scattered intensity versus

(s/k)2 using a cylindrical cell of approximately 6 mm

127
—1—I—I—I—I—I—I—I—I—I—1—I

I I

40.6 wt% n-DODECANE IN 0,?- DICHLOROETHYL ETHER

Tp

= 33.502

°C CYLINDRICAL CELL DIAMETER = 6.1 mm

(SA);

20 X I0 8 A"

FIGURE 3. Plot of reciprocal relative scattered intensity

versus HjJ

Values of scattered intensities are denoted by hollow squares, solid circles and

hollow circles for Kn values of 365, 436, and 578 m/x respectively.

in inside diameter. The curve with AT — 0.002°

has been reproduced, and results from cells of

different diameters and incident light of different wavelengths agree. A careful examination ol fig- ure 3 reveals that the curvature at AT= 0.046° is almost as large as that at AT= 0.002°. This could

partially be attributed to the fact that only the

scattered intensities measured at A. = 3023 A had

been corrected for dust and stray light, while

measurements from the other two wavelengths

(A. = 2516 A, 4027 A) were not corrected for these

effects. Note that the three hollow circles for the curve at AT— 0.046° for the smallest values of

(s/A.)

2 were much lower than the general trends

of the other points belonging to the same curve. When the system is more strongly opalescent, contributions due to dust and stray light become

negligible as shown by the first few hollow circles

at small values of s/A (A = 4027 A) for the curve

at AT = 0.002°. We believe that, for the system

n-dodecane and /3,/3'-dichloroethyl ether, there

exists a gentle downward curvature in an OZD

plot at small values of s/A. and at very small temperature distances from the phase separation temperature. However, the magnitude of the curvature

is yet uncertain as the measured scattered intensity

has to be corrected for both attenuation and density

fluctuations. Furthermore, we do not know the

range of s/A. in which any of the modified theories [5] are expected to hold. According to the equation [5] - = D0 [k2 + (s/A.)2

] '~ 7,/ -, where D0 , k, and i) are assumed to be three adjustable parameters, J] is estimated to be in the range 0.2-0.3 when T—Tp

is small. It is expected that r\ should be smaller

after the intensities have been corrected for at- tenuation and density fluctuations. The temperature dependence of the extrapolated zero-angle

scattered intensity is sketched in figure 4. It

is quite certain that —— is approximately proportional to the temperature distance. Deviations do

exist and seem to concave upwards. On the other

hand, there is perhaps an uncertainty on the crit- ical solution concentration of about 1—2 wt percent

in our preliminary measurements of the coexistence curve for the n-dodecane-chlorex system. Further

scattering experiments with several concentrations near and at the critical solution concentration are

in progress. We must also remember that, in an OZD plot, extrapolation to zero scattering angle

is permissible only in the absence of very long

range correlation, i.e., no abrupt change in curvature in the extrapolated region is present. For

this reason, measurements at even lower angles

by means of "long-wavelength" light scattering

become desirable.

FIGURE 4. Schematic variation of the inverse extrapolated zero- angle scattering due to concentration fluctuations estimated from preliminary results on the system n-dodecane and /3, f3' dichloroethyl ether. 128
This work was supported by the National Science

Foundation (GP-3922). The author wishes to thank W. P. Kao and J. S. Lin who performed

the measurements.

References

[1] L. S. Ornstein and F. Zernike, Proc. Acad. Sci. Amsterdam

17, 793 (1914): Physik. Z. 19, 134 (1918): ibid. 27, 761

(1926).

[2] P. Debye, in Non-Crystalline Solids, ed. by V. D. Frechette

(John Wiley & Sons, Inc., New York, 1960) p. 1: J. Chem.

Phys. 31, 680 (1959): in Electromagnetic Scattering,

ed. by M. Kerker (The Macmillan Company, New York,

1963) p. 393.

[3] B. Chu, J. Am. Chem. Soc. 86, 3557 (1964).

[4] M. Fixman, Advan. Chem. Phys. 6, 175 (1964).

[5] M. E. Fisher, J. Math. Phys. 5, 944 (1964).

[6] B. Chu, J. Chem. Phys. 41, 226 (1964).

[7] Q. Chow, Proc. Roy. Soc. (London) A22 4, 90 (1954).

[8] P. Debye, B. Chu, and H. Kaufmann, J. Chem. Phys. 36,

3373(1962).

[9] B. Chu, Rev. Sci. Instr. 35, 1201 (1964).

[10] O. K. Rice and R. W. Rowden, J. Chem. Phys. 19, 1423

(1951): D. Atack and O. K. Rice, Discussions Faraday

Soc. 15, 210 (1953), J. Chem. Phys. 22, 382 (1954):

F. R. Meeks, R. Gopal, and O. K. Rice, J. Phys. Chem. 63,

992(1959).

[11] B. Zimm, J. Phys. Colloid Chem. 54, 1306 (1950).

[12] B. Chu, J. Chem. Phys. 42, 2293 (1965).

[13] B. Chu, J. Chem. Phys. 42, 426 (1965).

[14] P. Debye, H. Kaufmann, K. Kleboth, and B. Chu, Trans. Kansas Acad. Sci. 66, 260 (1963).

[15] R. Pancirov and H. Brumberger, J. Am. Chem. Soc. 86,

3562(1964).

[16] D. Mclntyre, A. Wims, and M. S. Green, J. Chem. Phys. 37,

3019(1962).

[17] V. E. Eskin, Vysokomolekul. Soedin. 2, 1049 (1960): V. N.

Tsvetkov, V. E. Eskin, and V. S. Skazka, Ukr. Fiz. Zh. 7 , 923 (1962).

[18] B. Chu and W. P. Kao, J. Chem. Phys. 42, 2608 (1965).

[19] H. Brumberger and W. C. Farrar, in Electromagnetic Scattering, ed. by M. Kerker, (The Macmillan Company, New

York, 1963), p. 403.

[20] G. W. Brady and H. L. Frisch, J. Chem. Phys. 35, 2234

(1961).

[21] B. Chu, M. Pallesen, W. P. Kao, D. E. Andrews and P. W.

Schmidt. J. Chem. Phys. 43, 2950 (1965).

[22] P. Debye, D. Caulfield, and J. Bashaw, J. Chem. Phys. 41,

3051 (1964). Discussion

F. H. Stillinger: Dr. Fisher has very lucidly pointed out that the long range behavior of the critical pair correlation function

doesn't tell one very much about specific heat anomalies. I would like to point out that there is, however, an analogous

quantity which gives direct information on the specific heat anomalies by virtue of its long range behavior in the critical region. It is well known that the specific heat C, is a fluctuation

of potential energy. If one writes out the fluctuation formula

in terms of spin distribution functions for an Ising model, one

encounters a combination amounting to a quadruplet distri- bution function for two pairs of spins, the spins in each pair being close together, minus the product of the two pair correlation functions. The implication of a logarithmically divergent

specific heat is that the critical point long range distance behavior

of this combination would have to be as r~ d

, where d is the dimensionality. Here the long range distance r is the distance between

the centers of the two pairs.

If one goes ahead naively and attempts to evaluate this combination of distribution functions using a Kirkwood superposition approximation, or something roughly equivalent to it, one

arrives at the conclusion that the long range distance behavior

should vary as the square of the critical pair correlation function.

Since we have seen that the exponent 17 is positive but small

relative to unity this leads to a rather remarkable conflict with

the requirement that this combination should vary as r~'' as mentioned above. It is rigorously known, of course, that the specific heat for an Ising model in two dimensions diverges logarith- mically. I think the situation in three dimensions is in a state of flux. One would have at worse a small positive index a, perhaps of the order of V10. In three dimensions, therefore, there could be some slight deviation of the index d from 3 for

this quadruplet distribution function long range behavior, but

it certainly would not become equal to the long range behavior

of the square of the pair correlation function. M. E. Fisher: May I make one comment on that? What Dr.

Stillinger said is, of course, correct. If I look in the Ising model

at saS\, where 0 and 1 are nearest neighbors, and at s,s, +8, where

r and r + 8 are near neighbors, then the behavior of

(s„s,SrS r +&) — (soSi) (srSr+s) for large r should tell me how the specific heat diverges. John Stephenson, a student of mine, has actually evaluated this correlation function explicitly which is not

too difficult to do in the Ising model [1]. He finds in fact, as one

might anticipate, that the behavior is as e'2Krlr2 in this case. His correction factor [1 + (>4 (kt)] satisfies Q4 (x) —* o as x —* o and

1 + Q.i(.x) —> 2

as x—» ». Once you can express the problem in terms of the long range behavior and you have a good theory for the long range behavior, it is fine. However, as Dr. Stillinger pointed out again, although formally all the information is contained in this quantity (indeed one could also get the pair correlation function from it), it is a different set of asymptotic limits that are required.

G. S. Rushbrooke: May I make a comment which is due to Dr. Widom? It concerns the things that Dr. Fisher talked about,

but below the critical temperature rather than above. Widom

simply takes the equation which relates the compressibility to the density fluctuations. Suppose the system is separated into two phases. Then in a small volume of dimensions of 1/k, the range of the correlation function, we ought to have fluctuations in the density equal to the width of the phase boundary. Now,

if one just introduces this idea, one obtains the relation: y'=v'd-2p (1)

where y' is the exponent of the susceptibility below the critical point, v' the exponent of k (T) below the critical point, /3 the exponent for the phase boundary and d the number of dimensions. This works exactly for the Ising model in two dimensions. In three dimensions it depends what you take for v' . If you take

/3 and 8 as given by Fisher and introduce the relation y' = /3(8 — 1) into (1), you get exactly v' =0.646 below the critical point, equal

to the value for v above the critical point as given by Dr. Fisher. So for the Ising problem it looks as if these ideas which work

above the critical point might also work below the critical point.

I would like to ask Dr. Fisher whether he thinks this is reasonable. The other point I want to ask Dr. Fisher is how far he thinks

the conclusions about exponential decay depend on short range

forces. This is established for the Ising model. However,

quite a lot of people argue that the direct correlation function goes off like the pair potential. This is not a very strong argu- ment. However, when the pair potential and consequently

C(r) decays as l/r", you cannot have the direct correlation function decaying more rapidly than the total correlation function, if the Ornstein-Zernike theory is true. M. E. Fisher: Well, that last question is one which I suppose I have to answer since it was asked. However, I think it is a rather

difficult question!

If one thinks of Coulomb forces, then of course, they do not make sense unless one has some counter-charge. But then we

have the phenomena of screening and the correlations are presumed to decay more rapidly. This seems to be a property

of the fact that Coulomb forces obey the Poisson equation. This is, I think, one way of interpreting it. But if you try and work the arguments which would give you that result with

a force which is like a Coulomb force and even has counter- charges, but which decays a little faster than 1/r, you would not

gel out complete screening. Now in as far as the Van der Waals

129
forces can be thought of as originating through the electro- magnetic interactions of matter, it seems to me still possible,

in principle, that you have some overall screening effects and

that you still do have an exponential damping. But if you just take a completely radially symmetric fixed 1/r6 potential without any corrections, then I agree you cannot expect pure exponential

damping since the direct correlation function would presumably

go down in this way. So I am thinking here of models which

have a potential that cuts off either exponentially or goes strictly

to zero, if one really wants to prove it. But I would think that

for real systems it is something of an open question.

W. Marshall: Dr. Fisher mentioned that the old theories can

give very misleading results about the specific heat because

you essentially take the long range behavior and extrapolate

that into small distances. This, of course, is correct but one

must be rather careful about the precise meaning. It is not a necessary consequence of the theories that you must get the wrong answer. In particular in the Elliott-Marshall theory

[2] you get a difference equation for the correlation function and that difference equation happens to be valid at small dis- tances as well as large distances. Therefore, that particular

theory just reproduces the usual Bethe-Peierls result for the energy.

P. Debye: I would like to go back to some experimental facts. For instance the theoretician says, "Now I am at the critical temperature." Well, I do not know the critical temperature. Nobody can determine what the critical temperature really is. So how is there going to be an experiment at the critical temperature? Any experiment is a little bit away from the critical temperature and I cannot even define how far it is away from the

critical temperature.

M. E. Fisher: Well, for example, some of the experiments dis- cussed yesterday on the magnetic systems give you Tc very

accurately.

P. Debye: I don't know of an equivalent experiment in liquids. M. E. Fisher: Well, liquids are more difficult. The coexistence

curves, I think, will tie down the critical points sufficiently ac- curately. Within a few millidegrees, to which some experiments seem to pin it down, I would not worry about the differences.

P. Debye: I would like that the theoretical people tell me when

I am so and so far away from the critical point, then my curve

should look so and so.

I would like to make two remarks in connection with the paper

of Dr. Brumberger.

One remark is that I am not so pessimistic as Dr. Brumberger.

If one wants to find deviations which appear in the immediate

neighborhood of the critical point, one should not use x rays because the relevant variable is not s = 2 sin (0/2), but s/X. Thus

when X is very small, one must go to extremely small angles 6

in order to obtain some information. Thus if one wants to find deviations in the neighborhood of the critical point, one has to use light where s/X can be small for reasonable values of 9. In the paper of Dr. Chu, it is shown that there are probably deviations from the Ornstein-Zernike theory near the critical point. However, if one goes to large values of s/X, one should also find deviations from a straight line in an Ornstein-Zernike plot. The

straight line is only a first approximation. So if one measures for large values of s/X, one obtains information on the second moment

of interaction and, if one goes farther, on the fourth and sixth moment of the interaction. Thus from the molecular point of view it is also interesting to go to large values of s/X and there one

has to use x rays.

It is really true, as Dr. Brumberger said, that you should connect the light experiments and the x-ray experiments with each other. This has been done at several places and we have done it too. For this purpose, I present two curves for /"' as a function of (s/X)2 in figure 1 and figure 2 for a polymer, which

Dr. Brumberger does not like. I do like polymers, as one obtains information about their interaction, which cannot be predicted

easily. If you ask me what the interaction of two argon atoms is.

Figure 2

I know approximately the answer. However, if you ask me what

the interaction between two polymers is which go through each other, I do not know the answer and I must do the experi- ment. Figure 1 shows I~' as a function of (s/X)2 X 10fi

. One

sees small deviations of a straight line if you go to large values of

slk. And if one really wants deviations which give the value for the fourth moment of interaction, one has to go to considerably

larger values of s/X as shown in figure 2. From these data one

can obtain the second moment and the fourth moment but not the sixth moment. The maximum value of s/X possible with light

is 5 X 10"4 A

-1

; the x-ray measurements covered the range from

2.6 X 10"3 A" 1 to 2.5 X 10"2 A"'. Over this range the scattered

intensity changed by a factor of 56.

Dr. Brumberger made a nice remark when he said that other experiments of another kind should be done. It has been said and also Uhlenbeck made this remark at the beginning of this conference that in the neighborhood of the critical point the system can be disturbed rather easily from the outside. In

this con ction I remark that we studied the effect of an electrical

field on the critical opalescence of a binary liquid mixture. For the details we refer to the publication [3]. Also here it is apparent that the Ornstein-Zernike theory is only correct if one

is not too near to the critical point. So I want to emphasize that one has a straight line I~ l as a function of (s/X)2 over a large distance, but if you go to very large values of s/X, you get deviations as has to be expected. If you

go to small values, which you can only do with light, then you can expect to find the things which are the subject of this after- noon.

130
W. Marshall: Dr. Brumberger explained the difficulties that you

frequently get with light and x rays. I would like to emphasize

the fact that in looking at these systems many of the difficulties are very much eliminated if you did the experiment with neutrons. This is a technique about which many people who have worked

traditionally in this field do not think of because it is something

new. Indeed Egelstaff and Enderby [4] have already verified

that the structure factor of simple liquid metals as given by the

x rays is not quite correct. The correction is quite important

in the region of small s/\. I would support very strongly, as Dr. Brumberger said, that neutron experiments also must be

done. It would be nice to do it on the same material which means that almost all systems which Dr. Brumberger discussed have to be ruled out because they contain hydrogen and they all have to be deuterated. M. S. Green: I would like to ask Dr. Marshall: What are the smallest values of k that you can reach with neutrons as compared with

light and witlucrays?

W. Marshall: That depends on which experimentalist you talk

to. It is important to go from one to the other and get them into competition. The usual kind of A-vectors that are easy are those

[ covered by x rays. But there is no reason why one should not

build a special small angle scattering apparatus like that which Lowde or Jacrot have used [5]. Then you can get very small

A-values. I think Jacrot can probably tell us exactly what it would be.

B. Jacrot: 1 think you can go to 10~:! A

-1 or maybe slightly lower. W. Marshall: Thus then the neutron experiments would be as good as light experiments in this respect.

A. Arroll: I want to ask Dr. Debye what is meant by the critical region. When the susceptibility or compressibility obeys a

4/3 power law already for ( T— T, )jT,- = 0. 1, should you not say

that you are already in the critical region when the temperature

is 7V/10 away from the critical? In discussing possible deviations from the Ornstein-Zernike theory you consider what happens at temperatures hundredths or thousandths of a degree- away from the critical temperature. However you did not mention what is found for the power law of 7~'(0) which is inversely

proportional to the compressibility: I~'{0) means the inverse extrapolated zero angle intensity.

P. Debye: It is hard to find a deviation from linearity for /~'(0) as a function of T, because you are not certain that the extrapolation is correct.

, H. Brumberger: The behavior of I~'{0) as a function of temperature does deviate from linearity in many cases over a wider

temperature region.

I M. S. Green: I think we have a real problem here. Although I

, don't know whether Dr. Debye would agree, I believe that light

j scattering experiments do in fact show a curvature for /~'(0) as a function of temperature many degrees away from the critical temperature. On the other hand, any deviations from the Ornstein-Zernike theory (one should also say Debye theory) do

not occur unless the temperature is within 0.01 °K from the

j critical temperature. Perhaps this means there are two critical regions. I don't know the answer, but it is a problem which

needs to be resolved. Perhaps it is related to the fact that the logarithmic singularity of the specific heat does not show up

j until one comes within a hundredth of a degree of the critical point, although the behavior of Cv is nonclassical in a much

wider region.

is " B

e D. Mclntyre: I see no essential disagreement between Dr. Chu's new results for the cyclohexane-aniline system and our results

a with the same size cell in the same temperature range. I per-

! e sonally believe on the basis of our data that the plot of /~'(0)

|U against AT is gradually curved upwards over the whole tempera-

* ture range. Now I would like to enter a slight quibble. I am

a little surprised at the form of your uncorrected scattering

curves. There are some wiggles in your curves at 0.1 to 0.2 °C

above Tc . They are sometimes straight, sometimes a little erratic, and even seem to have a little s shape.

B. Chu: The data on the aniline-cyclohexane system as I have shown in figure 1 of my paper demonstrate the experimental

difficulties which one encounters in critical opalescence studies whenever multiple scattering becomes appreciable. These measurements on such a strongly opalescent system (aniline- cyclohexane) were performed at relatively large temperature

distances {AT = 0.2° to 1°) and at a longer wave length (\0 =578 nuz

rather than the usual blue or green lines of a mercury arc) so

that the effect of multiple scattering remains negligible. On the other hand, the wiggles in figure 3 of my paper refer to the scat- tering of a weakly opalescent system, namely, ra-dodecane and

/3,^-dichlorodiethylether. Furthermore, the plot is reciprocal scattered intensity versus (s/X)2

, and thus, the experimental

"fluctuations" in the data seem to have been magnified.

D. Mclntyre: But even 0.1 °C away from Tc ?

B. Chu: Yes. The critical scattered intensity for the system,

re-dodecane and /3,/3-dichlorodiethylether at 0.1° away from Tc is quite weak, especially when we have used very narrow slit systems. However, our main interest here was to try to deter- mine the shape of the scattering curve very near the critical mixing point, not at temperature distances 0.1° away from the

critical mixing point. In fact, such wiggles disappear whenever we increase the sensitivity of our detection scheme.

D. Mclntyre: Examining your papers, I find that you suggest

that you have corrected in some manner for the multiple scattering. Is this so? We have always found that what we thought

to be multiple scattering in this range of AT was exhibited as a convexity to the abscissa at the ends of a conventional O-Z plot of a light scattering curve. In your systems I should think that you would get the same behavior.

B. Chu: No. I do not believe that with our experimental setup, multiple scattering is appreciable for this weakly opalescent

system. We have tried to correct for attenuation and to avoid

multiple scattering. I do agree with you that whenever multiple

scattering does become appreciable, it invariably destroys

the linear behavior in an O-Z-D plot.

D. Mclntyre: In one of your curves it appeared.

B. Chu: This is not multiple scattering.

D. Mclntyre: What is it?

B. Chu: I do not believe that the curvature in figure 3 is due to multiple scattering because we have performed our experiments

with cylindrical cells of two different diameters, and the shape of such curves can be superimposed. This may not be fool proof. We also hope to carry our studies further with more variations

in the geometry of our optical system. I would also like to make

another remark. It is difficult to determine the behavior of

7~'(0) as a function of temperature. For one thing, most investi- gators have neglected correcting for the attenuation of light as it goes through the sample cell. Since the turbidity of the sample

depends strongly on temperature as we approach the critical temperature, a plot of the extrapolated reciprocal zero angle

scattered intensity (without attenuation corrections) versus temperature does not represent the true behavior of 7"'(0) versus

T. Now, all I can say is that the curvature in a plot of I~'(0) versus T for our system is very small if there is any. As far as I can see, it is very straight as shown in figure 4 of my paper. Yet, we must remember that I remarked on its nearly straight

line behavior over a temperature range of about only 0.3°. The

nonlinear behavior over a whole temperature range (say 10°) remains to be examined.

131

209-493 0-66— 10
D. Mclntyre: I think that there is actuaEy experimental con- currence in the fact that when measurements are made at small A7*s and low angles a downward curvature in the O-Z plot, such as yours, will be found. So I think that the scattering

experiments made by various people are in agreement on this

point.

P. Debye: I think everybody is now convinced that there is curvature in the immediate neighborhood of the critical point and that the curvature is in the expected direction. A. Isihara: Could you explain why the/"" 1 versus (s/\)2 plot curves downwards for small values of s/A?

M. S. Green: 1 would like to comment on this question. As Dr. Fisher emphasized, the essential approximation of the Ornstein- Zernike theory is that the direct correlation function is short ranged which means it has a finite second moment. The second moment of the direct correlation function is essentially the slope

of the versus (s/A)2 curve. The fact that this curve turns downwards means that the second moment is increasing. The

truly significant prediction would be that the second moment of the direct correlation function not only increases but becomes

infinite at the critical point. Nobody can actually show that, but this seems to be indicated by the experiments.

M. E. Fisher: The value of tj estimated by Dr. Chu is much bigger than I would have expected. As far as I am concerned there

are still some puzzles, because the curve of /"'(()) versus temperature does not go to zero if you extrapolate to the critical temperature. I think if you had done P-V-T measurements you would have observed that the fluctuations-of-compositioncompressibility would go up to infinity while your /~ l (0) still seems to stick to a small finite value. Maybe there is some

reason why the real critical temperature is lower, but from what

Dr. Chu has said, I don't see it!

A. Arrott: Returning to my previous question, it seems that for a binary mixture the critical region is much smaller than for a magnetic system. Why is there such an order of magnitude

difference when you consider the critical region for a binary mixture and that for a magnetic system?

E. Helfand: I believe the size of the critical region is related to the range of the forces. This conclusion emerges in the study

of systems with long range forces. The shape of the particles

in the binary systems about which we are speaking may cause

these systems to be less in corresponding states with the Ising or Heisenberg models than one component fluids or magnets.

M. S. Green: I don't think your answer is correct. I don't think we really have a narrow critical region in a binary system. The

phenomena which Dr. Chu showed take place within indeed a range of one hundredth of a degree. However, there are other phenomena in the two phase region which occur over a much

wider temperature range. For instance, the (AT)113 power law

for the phase boundary curve, which is a nonclassical phenomenon, is just as much true for these systems as it is for a magnetic

system or for a one component fluid. I guess the question of the

4

/3 power law for the compressibility is still a matter of controversy, but in my opinion (and I think Dr. Mclntyre agrees with me), there are deviations from the straight line for the compressibility as a function of temperature in a much larger temperature range than we are talking about here.

E. Helfand: The degree of the coexistence curve is not as clearly related to the distance parameter of the correlations as are phenomena above the transition such as the compressibility

(or its analogue for mixtures or magnets) and the critical scat- tering curves.

P. W. Anderson: To answer the question of Dr. Arrott, is not the reason why the critical region appears to be so terribly narrow,

132

that you are studying terribly long range effects. A light scat- tering experiment is sensitive to distances between 4000 A anc 10,000 A and the temperature differences T—Tc , which corres- sponds to such ranges are expected to be in the millidegree range.

If one were to look in the intermediate range between light and

x rays, I would guess one would see curvature in a range T—Tc of one degree. In the x-ray results one sees curvature in the 10 deg range. Thus I think you have to look in the temperature

range that corresponds to the range of the wavelength used in the scattering experiment. Unless the experiments show other

wise, this interpretation answers the question of the size of the

critical range. j

P. Debye: It one wants to investigate something characteristic | for the immediate neighborhood of the critical point for a giver angle 0, one has to go to longer and longer wavelengths Because

the characteristic variable is not s, but s/A. P. Anderson: I am not talking about s/A, but I am talking about

a physical range, in particular, about the correspondence be- tween ranges in temperature and ranges in space. If you were

1 deg away from the critical point, you would find curvature^al much shorter wavelengths.

P. Debye: No.

M. E. Fisher: I will attempt not to add anything to the confusion!

First of all, I think it is a question of pure semantics. Unless you have some theory of the critical point you cannot talk about

the critical region except in terms of deviations from this theory. We are talking in terms of the Ornstein-Zernike theory here, where there is a ^-dependence and a temperature dependence

to be considered. Now, if we look at the magnetic case, there I is no doubt that for the susceptibility deviations can be seen

very far away from the critical point. I find it puzzling that one does not seem to see these deviations in the binary system. As I far as one does not see these deviations here, one has not really | understood the difference between these two systems. However, looking at the ^-dependence, it is perfectly possible

that you can have an Ornstein-Zernike /r-dependence absolutely

j correct right up to the critical point even if the thermal behavior

is not correct. The index tj I find theoretically is very small. Because 17 is small you would expect, I think, to see deviations only in a small region. So I think to talk about the critical region

is not too helpful. What Dr. Debye said is correct, namely,

that when one is talking about some deviation one must attach

to it a magnitude and a range of temperatures where it is to b< observed, which can be different for one phenomenon compare(

to another. When these ranges differ from one system to anothe

for the same phenomenon, we have a real challenge.

I Dr. Sette: I want to point out that the results of sound absorption measurements for a liquid vapor critical system and a binary: ( mixture are different. In a one-component system the absorp- / tion increases sharply only in a temperature range of one degree | from the critical temperature. In the case of a binary mixture , the absorption increases rapidly starting at T— 7'c = 30 °C. This indicates that the two systems behave in a different way.

I believe that a comparison between the magnetic system and I the binary system should be made with some precaution.

References

[1] J. Stephenson, preprint, Ising-model spin correlations o: the triangular lattice. II. Fourth order correlations, to b

published.

[2] R. J. Elliott and W. Marshall, Rev. Mod. Phys. 30, 1 (1958

[3] P. Debye and K. Kleboth, J. Chem. Phys. 42, 3155 (1965).

[41 P. A. Egelstaff and J. E. Enderby, to be published.

[5] B. Jacrot and T. Riste, Thermal Neutron Scattering, ch. VI,

ed. P. A. Egelstaff, (Academic Press, 1965).
SCATTERING: INELASTIC

Chairman: K. S. Singwi

133

Critical Scattering of Neutrons by Ferromagnets

W. Marshall

Atomic Energy Research Establishment

Harwell, Great Britain

1. Introduction

Recently a number of authors have discussed

the theory of critical phenomena in magnetic materials and produced results differing from the conventional and simple theory as described in many

textbooks [1-4]. Some predictions of the new

theories have already been verified experimentally.

Neutron scattering techniques can be used to give

a thorough investigation of critical phenomena

and this paper discusses the theory needed in the

interpretation of the experiments.

In section 2 the cross section formulae are

written down in various forms and it is shown how

the cross section is related to the generalized wave

vector dependent susceptibility. In section 3

special attention is given to the time dependence

of fluctuations near the critical point and it is shown that the conventional theory fails. A modified theory is described in section 4. Because of

the difficulty of the problem this theory is only

qualitative but it points to the existence of damped

spin waves, or some similar phenomena, even above

Tc . Section 5 mentions two other experiments

indicating the existence of spin waves above the

critical temperature. Section 6 gives conclusions.

2. Formulae and Definitions

The neutron has initial wave vector k and initial energy E. It is scattered into a state of wave

vector k' and energy £". The wave vector change

and energy change are, respectively:

K=k-k

na> = E-E' = (h 2l2m0W~ k'2

). (2.1)

The cross section is then

2 (j ^N/eWk

where

^(K, (o)=-^- r dt y eiK R^(Sg(0)S|(0). (2.3)

Only the terms with a and /3 equal contribute to

(2.2) if the sample has only scalar interactions be- tween the spins. Throughout this paper we shall assume this, i.e., ignore dipolar interactions and

anisotropic exchange.

Define

S(K, t) =Jeat' k&(t) (2.4)

then

fxx(K, fo)=A; f

* dte-m(Sx(-K, 0)Sx{K, t)) ZttN J- x

(2.5)

with similar definitions for the y and z-directions. Below the critical temperature the correlation

function in (2.3) has a nonzero value at infinite time and this gives the Bragg scattering contribution &f(K, a)).

y^(K, o)=i 8(6>)<S«(-K)><S0(K)>. (2.6)

The remainder is the "diffuse" contribution

=

J*

dte->«>{(S«( - K, O)S0(K, t)>

-(S«(-K)>(SW)}. (2.7) We notice that

I:

(2.2)

da>ytf(K, w)=- {<S°(-K, 0)S*(K, 0))

i /V

<S«(-K,0)><S*(K,0)>} (2.8)

135
and in the limit A^O this is proportional to the mean square fluctuation in magnetization which,

by elementary thermodynamics, is proportional to the magnetic susceptibility. Hence

lim da&ff(K, a) = kTx*ls?n* (2.9) K^U J -x

where the susceptibility tensor x

ali is defined in the

usual way. For the correct choice of principle axes, x"'

3 has two values x

zz and X

xx ~Xyy below

T,-, and one value, X

zz=z X

xx ~Xyy- above T,. The form of (2.9) suggests that we seek a more

general relation, valid at all K, by introducing the

quantity x^K) which describes the response to a

field of wave vector K. The idea of this is well

known but, so far as the author is aware, the first formulation for an explicitly neutron scattering

problem was given in a recent paper by Mori and

Kawasaki [5]. Following the latter paper we

introduce a relaxation function

Rali {K, t)= I dk(e"*Sa(- K, 0)e-^(K. t)) Jo

-/3<S«(-K, 0)) (S0(K, 0>- (2.10)

It can then be shown that

yw»(K, aj)-y^(K, oo) Bragg

= i^^\ye

~iatRa^t) (2J1)

and also that

X

aTO = ^/^(K, 0). (2.12)

Hence

y^(K, a>)-y*(K, a>) Bragg

where

= f-,r^) X^^F^K^) (2.13)

F^(K, w) = y- [

* dte-**y*»(K, t) (2.14) " 77" J — x 136

and

(2.15)

By definition /°^(K. 0) is unity and hence, from

(1.14), i

do)Fa^{K, &>)=!. (2.16)

Therefore (2.13) gives

AT f x 1 — % X

a/3(K)= dor

1

{^(K, w)-y«f(K, 6>) Bragg}- (2.17)

This is an important relation because it is the generalization of (2.9) and because it unambiguously

relates the observed cross sections to a quantity

of theoretical interest. Whether or not we understand the full a> dependence of the cross section we can deduce x

a/3 (K) provided the experimental

results are known with sufficient accuracy to evaluate the integral on the right-hand side of (2.17). Frequently it is more convenient to measure the ; total count rate at a fixed scattering angle instead j of the full 5^(K, <w); to deduce x(K) from such meas-|

urements involves two approximations. First,!

the integral in (2.17) should be performed at fixedt K not at fixed scattering angle. The error intro- duced by this is small if the incident neutron energy

is large compared with the energy changes, 7f(o. Second, the count rate is not weighted with the) factor {1 — e~nwti } l~H(i>{5 and therefore we require

that iTui kT. If this last condition is satisfied the factor becomes unity and the experiments give;

X^(K) directly.

The molecular field theory for a ferromagnet

above Tc predicts x<K) to have the form:

X-rx(K) = x

Z2(K) =

r?(KT + q

2

) (2.18

diere

K=r+q (2.19
and t is the nearest reciprocal lattice vector,

X, = ^/AS(S+l)/3/,T(

.. (2.20)

r? gives a measure of the range of the exchange

interaction

.-, i it 1-

6 2^ (2.21)

and Ki is an inverse correlation range defined above

7V by

Xc T-Tc x"(0) 7V

(2.22)

Other simple theories of ferromagnets give results

qualitatively similar. These predictions have been tested by Passell

et al. [6], by neutron experiments on metallic iron. They conclude that the general form (2.18) is correct within the limits of experimental error but that rffcf does not have a linear variation with T—Tc , as predicted by (2.22) but instead

T1

K

2 ~ (T-Tcyl* ioxT-Tc (2.23)

This 4/3 law is in agreement with recent theoretical

work by Domb [1] and by Gammell, Marshall, and

Morgan [2] who found a 4/3 law for the divergence

of x

Z2(0) using the mathematical technique of suc- cessive Pade approximants.

Below Tr the molecular field theory again gives

the general form (2.18) but with

rV2 = Tc -T

for T<TC

«ix =0 (2.24)

Because this simple theory gives the incorrect

power law above Tc we would expect it also to be

incorrect below Tr but, to date, no experimental

results of sufficient accuracy are available below

Tc. The theory due to Elliott and Marshall [7] gives results virtually identical with (2.18) to (2.24)

with one important difference, they predict a nonzero k\x below Tr . The precise reason why the theories differ in this way is not yet understood; noninteracting spin waves also give k

2x = 0 and therefore it appears that in the theory of Elliott and

Marshall a diffusive-like behavior is built into the

perpendicular correlation function (Sx

(0)S${t))

.

We will return to this point in the next section.

3. The Inelasticity of the Scattering

To consider this problem we concentrate attention on F XX(K, co). Above Tc this is identical to F'

ZZ

'(K., co) and therefore gives a complete description of the co-dependence, while below Tc it gives

the single spin wave scattering. We are not able

to calculate FXX(K, co) rigorously but we can get some information about the moments of it. We

first note that

jt R xx(K, 0 = 1

<[S*(-K, 0), S

X(K, t)]>. (3.1)

Hence

a

2

and

d -Rxx(K,t) = l dt

S*(-K,0), fS

X(K, t) dt

S'(-K, 0),£^Sx(K, t)

(3.2)

It is easy to show that FXX(K, co) is even in w so only the even moments are nonzero. Then

(w2 )=|" da)(x)2Fxx(K, co)

|r-/~(K, o

-3^»i£*"<B

-?>L

(3 ' 3)

and similarly

<«,<>

If we now assume a nearest neighbor model these

two expressions can be evaluated. We find, using

(3.2) and (3.3), < 0)2 > =

aff(K) (7° " yK)(SyS

" + SZS« } (3 - 5)

137
where

P

(3.6)

matic effect, shows that long wavelength fluctuations are slow. At infinite temperature the discussion is straightforward. From (2.18) and (2.22) we get

The expression for (to4

) is very long and will not be

repeated here. It is of a similar form to (3.5) with

a denominator \

XX(1Q and a numerator involving

correlation functions between four spins which are

close together. This numerator is also proportional to (yo — jh•), as is (3.5). From these remarks we can now reconstruct the

general character of the moments (to") as a function

of n.

(a) (a)0

) is unity.

(b) (to

2 ), (to4

) and all higher moments are in- versely proportional to x

xx(K).

(c) (to

2 ), (to4

) and all higher moments are pro- portional to (jo — Jx)- This follows as a general

rule from the remark that /^(K, t) is independent

of t in the limit /£—» 0.

(d) (to

2 ), (to4

) etc., are proportional to correlation functions between spins which are close to- gether. This follows from (3.1), (3.2), which involve

a commutator, and therefore reduce, at t = 0, to short range correlation functions.

We now note that effect (d) cannot produce

important qualitative effects near the critical temperature. For example we note that (3.5) can

be rewritten, above Tc , as

(3.7)

where e(7) is the thermal energy for a typical pair

of spins

e(D = -27<S-Sn >. (3.8) We know that the specific heat has a weak singularity (probably logarithmic) at Tc, and hence e(T)

is continuous, but with an infinite slope, at Tc . This is a weak singularity compared to the divergence in x(0)- this essential difference is because

any short range correlation function can have only

weak singularities whereas x(K) involves long range

correlation functions. We conclude that effect

(d) produces numerical factors slowly varying with T. Effect (b), the thermodynamic slowing down of

fluctuations, shows that the scattering becomes

more nearly elastic near Tc . Effect (c), the kine138

X(K) -> Xr Tr IT=^^S(S + l)/3/,T= x„ (3.9)

where we use x<> to stand for the Curie susceptibility.

The short range correlation functions appearing

in the moment formulae are also inversely proportional to T and a tedious calculation gives the

results

<&>

2 >^m 3 S(S+ l) (y0 -yK) (3.10)

(<o4

) y)

4 |{S(S+l)}2

( 7o -7A)

7 3 , « Jo ~ « 7a: - 2- 3

4S(S+1)

(3.11)

In an earlier discussion de Gennes [8] evaluated

these moments at infinite temperature working

directly from the correlation function (2.5). His

discussion is rather more difficult than that given

here because he had to take four commutation

operations with to evaluate the fourth time

derivative. In the present approach (3.2) shows

that the fourth moment involves only three time

derivatives and therefore three commutation operations with furthermore the final commutation

simplifies the formulae, de Gennes quoted only

(to4

) for the simple cubic lattice averaged for a

polycrystal. For that special case (3.11) gives the same answer except for the sign of the term 3/4

S(S+ 1) which it appears de Gennes gave incorrectly.

The subsequent discussion given by de Gennes is

still valid because this term is very small. In the forward direction we notice, since both

(to2 ) and (to4

) are proportional to K

2

, that (to4

) is much larger than (to2

) 2

. It follows that although

the half width at half height of F(K, to) must be

small the shape must have large wings to it, i.e. F(K, to) must be something like a cutoff Lorentzian

We therefore assume

F(K, to)

1 7Tto2 + r

2

=o

for I col < s

for \to\ > s

(3.12
where

77 r<fa>

2

)

3

i

l/2

and

s =

<"2

>

1/2 (3.14)

are chosen to give the second and fourth moments

correctly. Substituting from (3.10) and (3.11) gives,

to leading order in K

2

,

Ta=^|(7()-7a)

4J

S(S+1)

{/• - l-3/16S(S+l)}

1/2(3.15)

s=-^r [5(5+l)] 1 ' 2 [/— l-S/ieStS+l)] 1 ' 2

(3.16)

where r is the number of nearest neighbors. If we

write

TA = AA2 atT=* (3.17)

and use molecular field theory to give J in terms of

fc , we can now evaluate A. For Fe we find

2m0A/£= 19 atT=* (3.18)

[The factors on the left-hand side of (3.18) are convenient and give a dimensionless number.] (3.18)

is only a rough estimate because we have used a nearest neighbor Heisenberg model to represent

metallic Fe and furthermore we used molecular

field theory to relate J to Tr . Nevertheless it should be of the correct order of magnitude. It should be noted that the only significant singularities of (3.12) are poles on the imaginary axis, at w = ±/FA . At low temperatures the discussion is also straight

forward. In a ferromagnet the last factor of (3.5) becomes S

2 and both molecular field theory and

spin wave theory give

(K) =

2J(y„-yA )

11 ence < w2 >^4J2 S

2 (>-yA W atT = 0

(3.19)

(3.20)

The general expression for < w4 > can be used to give

< to

4 > = <(o2 >2 at T= 0. (3.21)

The only distribution which satisfies (3.21) is a S-function. Hence (3.20) and (3.21) together give

FXX (K, oj) = ^ 8(a-uh) + h 5(oj + co a ) rr = 0]

(3.22)

where

u)K = 2JS{yo — yK)lh: (3.23)

is the correct spin wave frequency. As the temperature is raised the spin waves get damped and

the S-functions of (3.22) become broadened into cutoff Lorentzians. i.e., the singularities of FXX(K., w) move slightly off the real axis. As the temperature is raised still further the spin wave interactions result in an energy renormalization. It is well known that at low temperatures this energy renormalization follows a T5l~ law. Up to, say 0.87V, it is probably reasonable to use (3.5) directly to give a rough estimate of the spin wave

energy. Hence, very roughly,

(ok ~ 2 J (7o " 7a ) { <S»S}> + S*S?, >V'2 lt

for 0 < T < 0.8 Tr . (3.24)

In the neighborhood of T, we expect

< SuS" > ~ < SI'S' >

5(S+1)

3(r-l)"

Henc<

where

A K2

(3.25)

(3.26)

A ~(2.A/V/6fi){2S(S+l)/3(r-l)} 1/2 - (3.27)

Substituting for J as before we find for Fe [S=l,

r — 8] the result

2m*A\n ~ 30. (3.28)

This last estimate is rough because we have made

no real attempt to calculate the renormalized oj h within, say. a factor of 2. Nevertheless within

this kind of uncertainty it should be satisfactory

139
up to about 0.87V and beyond that temperature

the spin wave interaction problem is too difficult

to discuss here. We now notice by comparing (3.17) and (3.28) that the separation between the spin wave peaks at about 0.87V is roughly the same as the width of

the scattering curve at infinite temperature. The

shape of F•"'(!£, o>) is very different at the two temperatures but the scale of the dependence on o» is roughly the same. We now turn to a discussion of F-rj (K, oj) near

TV. Several authors [5, 9, 10] have discussed this or a similar problem using various theories; all of these theories are, however, basically the same

as that which we will now summarize. Because of

the slow T dependence of effect (d), near but just

above TV we may estimate moments as

<g>2 >k,t~ <oj2 > A ^ Xo/x(K,r) (3.29)

< oj4 > k,t ~ < a)4 > A>x Xo/x(K, T)

where Xo is the Curie susceptibility. By comparison

with the discussion of F(K, oj) at infinite temperature we see this has the effect of replacing K2 by

K2 Xo/x(K, T). The ratio < to

4 >/< or > 2 is now

even larger than at infinite temperature because

X(K, T) is large near TV. In the absence of any

other information about F(K, oj) we assume it is a

cutoff Lorentzian just as at infinite temperature;

by analogy to (3.17) we get immediately

rK=^X%FAK*ii(Ki + **)TclT. (3.30)

From either (1.22) or (1.23) we see that t\k\ is small near TV and in typical neutron experiments

the observations are made at angles such that K2 ~ k

2

. Thus, for an experimental situation such

as that used by Passell et al. [6], or Jacrot et al. [11], the expression (3.30) is some two orders of

magnitude smaller than the appropriate expression

for infinite temperature, (3.17). Hence the theoreti- cal prediction is that there is a very substantial

narrowing of the scattered neutron distribution as the temperature is reduced towards Tc . However, in both the experiments by Passell

et al., and by Jacrot et al., none of the features of

(3.30) are observed. In particular the experiments

suggest:

(1) The order of magnitude of YK is given by an

expression like (3.17) with

2moAtft ~ 11.4 [Experiment T~ 7V]. (3.31)

This is about two orders of magnitude larger than

(3.30)

(2) T K is approximately independent of temperature. This is in contrast with (3.30) which varies

sensitively with T because of the r\i<\ dependence.

(3) TA is proportional to K2 within experimental

error. No term proportional to K*, as predicted by \ (3.30), is observed.

We are forced to conclude that the conventional

theory as we have just summarized it, fails substantially. Therefore, in the next section, we look

for an improvement.

4. Theoretical Discussion for T ~ Tr

The discussion of section 3 has shown that the

inelasticity near TV is hard to understand and thereJ fore in this section we shall discuss the qualitative

features which it now appears a good theory of the

critical region must have.

We first note that the calculation for infinite temperature (3.18), the spin wave calculation at about Ij 0.8 TV, i.e., (3.28), and experiment near TV, (3.31),

all give results roughly in agreement as regards

the dependence of Fx,r (K, oj) on oj. This suggests

that the oj scale of Fxx (K, oj) does not in fact change

significantly from 0.8 TV up to infinite temperature

but that the shape of Fxjc (K.,oj) changes considerj ably from two distinct spin wave peaks below TV j into a single Lorentzian at infinite temperature,

j But calculations which rely upon moment calculations are quite unable to give sensitive information on shapes and therefore the theory as described

in section 3 is suspect.

We now examine more carefully the behavior

of (3.29). Using (2.18) gives

< co

2 > a, r = < oj2 > a. x (r? k\ + r\K2

) TJT (4.1) j < 0>*>K.T=<ai > k. x (r? k\ + riK2

) TJT (4.2)

j

(4.1) is proportional to K2 at infinite temperature,! 5

because < oj2 >k.* is proportional to K2 but as the

'

temperature is lowered (4.1) becomes smaller and

j becomes proportional to K4 at temperatures sufficiently close to Tc that k\< K2

. If we assume

that FXX(K, co) remains a cutoff Lorentzian we are

140
immediately led to (3.30). But the above behavior

is quite consistent with a gradual change of shape

in FXX (K, to) from a Lorentzian at infinite temperature to one which is dominated by two peaks at

o) = ±AK2 for temperatures such that k\ < K2

. These peaks would make a negligible contribution

themselves to < to

4 > and the higher moments.

Such peaks, if they existed, would have the obvious

interpretation of quasi-spin waves above Tc . However we notice that a rectangular distribution

bounded by o) = ±AK2 would serve equally well

as a dominant distribution near T< - We also recall

that, above Tc , Fxx {K,(o) and F2Z (K, to) are identical and we cannot possibly associate single spin wave processes with F"(K,a>). Nevertheless

the general point remains valid, it is quite consistent

with (3.32) and (3.33) for any curve whether double

peaked or not, to be sharply confined to limits

(o = ±AK2 provided it also has weak "tails" which

can give large values to < oj4 > and higher moments.

This argument can be summarized as follows:

(i) The moment calculations are sensitive to the

tails of the distribution Fxx (K.,oj) whereas the neutron experiments measure only the central portion

of Fxx (K,a>).

(ii) This central portion has a width, ~ A K2

, roughly independent of T from 0.8 Tc to infinite temperature but with a shape varying from a double

peak (representing spin waves) at the lower limit

to a single peak at high T. This central portion

always contains the major part of the area of

Fxx (K,co).

(Hi) The tails of the distribution FXX (K, to) are

sensitive to T and vary so as to give agreement with

the moment calculations. They are difficult to observe experimentally.

(iv) Above Tc , Fzz (K,o>) is identical to Fxx (K,o))

but below Tc does not show distinct spin wave peaks.

However the co-scale is roughly similar. The above conclusions appear to be dictated by

the experimental results which were obtained for the general range K~~K\. For either K<K\ or K > Ki the neutron experiments give no information and therefore it is possible that in these regions

the to dependence is qualitatively different. The clearest way of describing the situation is in terms of the singularities of FXX (K, to) in the complex to plane. We have already remarked that the high temperature theory, (3.17), gives two

significant poles at ±iAK2

; the other singularities of r

? -rx (K, to), [which are needed to describe the

cutoff procedure for |to|>s], are distant ~~ Jjfi

from the origin. The conventional theory, (3.30), near Tc tells us that the two significant poles have

moved along the imaginary axis to± i A/C2 xo/x(K", T)

and the other singularities of FXX (K., to) remain

at a long distance from the origin. However, the neutron experiments insist that this is not correct,

there must be singularities of some type distributed

at distances ~ AK2 from the origin; the poles

predicted by (3.30) may "or may not exist but if they do exist they must have a weight small compared to those at distances ~ AK2

. The form of F XX(K, to) below Tc is of special in- terest. Well below Tc we know that FXX(K, to)

is dominated by two spin wave peaks centered at ±(i)k, where coa is the renormalized spin wave

energy, and with a width dependent on the spin wave lifetime. As the temperature is raised does

FXX(K., to) retain this form or does a distinct dif- fusive mode appear in addition? We may speculate that an analogy to the classical fluid may exist. A classical fluid, at small K, may be described by

the usual equations of hydrodynamics. In the

absence of any dissipative effects, viscosity or thermal conductivity, the relaxation function

[equivalent to F XX(K, to)] has two 8-function peaks

at ± csK where c.s is the velocity of sound. The

introduction of viscosity or thermal conductivity

damps these sound waves and simultaneously a

diffusive mode also appears in the relaxation function. In other words, in the complex to plane, as soon as the sound wave poles move off the real axis, a new pole appears on the imaginary axis. In suit- able cases therefore, the relaxation function as measured by experiment as a function of real to, can have three maxima, one at to = 0 and two symmetrically placed either side at the appropriate

renormalized frequencies [12, 13]. In the magnetic

system the same phenomena might take place

as the spin waves are damped. This possibility

is intimately linked with the problem of the energy

renormalization of spin waves very close to Tc and

in view of the extremely speculative nature of this particular discussion we shall not pursue it further

other than to remark that it is also connected with

the behavior of k\x below Tc (this problem was

mentioned in sec. 2). The discussion of this section is most clearly

summarized in figure 1 which shows the positions

of the singularities of FJ

' r(K, to) in the complex to plane according to the conventional theory and also two possibilities which would be consistent with

the neutron results. In this figure singularities

141
T~0 8TC

T » T c

-X X- K X-

) :

oK

X X X x

x X

X X X

:

X X

t

X

IAK ! r

, 2 (k 2 + K,

2 )tc /T /

"spin wave" singularities remain within a distance

- v ^ A K

2 FROM ORIGIN r -

\

\ y

1 / J

- / —

AK 2

, t AK2 > 'lAK2

: LAK2

: ; : > c

(0

FIGURE 1. Possible singularities o/Fx *(K, &>) in the complex co-plane.

Figure la shows the results of conventional theory. Figure lb indicates the qualitative nature of the singularities of the analogy to hydrodynamics holds. Figure lc indicates an alternative possibility in which poles do

not collapse to the origin at 7V. In all diagrams the singularities at distances — Jlfl from the origin are ignored. which are a long distance —Jlfi from the origin are not shown. Figure la gives the conventional

theory, figure lb the theory we would expect by

analogy to hydrodynamics and figure lc the theory we might expect if no "diffusive" mode appears

below Tc . At the present time we are not able

to tell whether possibility, (b) or (c) is correct.

5. Further Experiments

In this section we shall list briefly other experiments which give information on the form of FXX(K, w).

(a) In experiments on Fe.j04, Riste [14] observed

that spin waves existed above Tc and plotted their lifetime as a function of temperature. His results can be taken as experimental evidence in favor

of the existence of spin waves above T,.. Unfortunately his experiments did not cover a wide

enough range of scattering surfaces to distinguish

between figures lb and lc.

(b) Although all the discussion of this paper has

concerned ferromagnets we would naturally ex- pect to carry over some results to antiferromagnets.

In particular if spin waves exist above Tc in a ferro- magnet we would expect them to exist above the

Need temperature of an antiferromagnet. It is therefore worth noting that Murthy et al. [15], have observed spin waves in MnO above the Need

temperature. As we would expect, these spin waves peaks are broad. MnO is an unusual anti- ferromagnet in that TN ~ 120 °K but 0 ~ 600 °K; therefore at the Need temperature we expect an

exceptional amount of short range order to exist. In the presence of this large short range order we

would expect the spin waves to be better defined

than in the typical antiferromagnet and therefore

in figures lb and lc the "spin wave" singularities

would approximate closer to simple spin wave poles.

This is probably why these neutron experiments

observed them whereas experiments on other

antiferromagnets, for example, Turberfield et al. [16], on MnF2 do not see distinct spin waves above

142
6. Conclusions

The theory of neutron scattering from ferromagnets has been reviewed and it is shown how the

experiments give the wavelength dependent susceptibility unambiguously. The experiments show

that the conventional theory of the time dependence

of the fluctuations near Tc , is incorrect. It is too

difficult to construct an alternative theory immediately but to explain the results it is necessary

that some remnant of "spin-wave motion" remains

above Tc . Two possibilities are described in terms

of the singularities of the relaxation function in the complex co-plane.

7. References

[1] C. Domb and M. F. Sykes, Phys. Rev. 128, 168, (1962).

[2] J. Gammel, W. Marshall and L. Morgan, Proc. Roy. Soc. 275, 257 (1963).

[3] M. E. Fisher and M. F. Sykes, Physiea 28, 939 (1962).

[4] M. F. Sykes and M. E. Fisher, Physiea 28, 919 (1962).

[5] H. Mori and K. Kawasaki, Prog. Theor. Phys. 25,723(1962).

[6] L. Passell, K. Blinowshi, P. Nielsen and T. Brun, Proc.

International Conference on Magnetism. 1

[7] R. Elliott and W. Marshall, Rev. Mod. Phys. 30, 75 (1958).

[8] P. G. de Gennes, J. Phys. Chem. Solids 4, 223 (1958).

[9] L. Van Hove, Phys. Rev. 95, 1374 (1954).

[10] P. G. de Gennes and J. Villain, J. Phys. Chem. Solids 13,

10 (1960).

[11] B. Jacrot, J. Kostantinovic, G. Perette and D. Cribier. 2

[12] L. D. Landau and E. M. Lifshitz. Fluid Mechanics, Per- gamon Press.

[13] W. Marshall, Lectures on Critical Phenomena, A.E.R.E.-L.

144 (1964).

[14] T. Riste, Journal Phys. Soc. Japan 17, Supplement Bill, 60 (1962). Proc. of International Conference on Magnetism and Crystallography.

[15] N. S. Satya Murthy, G. Venkataraman, K. Usha Denig, B. A. Dasannacharya and P. K. Iyengar, Symposium on In- elastic Neutron Scattering, Bombav, 1964.

[16] K. C. Turberfield. A. Okazaki, and R. W. H. Stevenson, Proc. Phys. Soc. 85,743(1965).

' Nottingham 1964. 99.

2 Symposium on Inelastic Scattering of Neutrons in Solids and Liquids, Chalk River,

1962.

Critical Magnetic Scattering of Neutrons in Iron

L. Passell

Brookhaven National Laboratory, Upton, N.Y.

Abstract

Measurements of the angular and energy dis- tributions of 4.28 A neutrons scattered at small

angles from iron at temperatures above the Curie

temperature are described. The results are inter- preted in terms of Van Hove's theory of critical magnetic scattering and yield information on the

range of spin correlations and the dynamics of the

spin ordering process. For the, dimensionless

parameter, 2mA/7?, which describes the time de- pendence of the spin fluctuations, we obtain the

value 11.0 ±0.6 at T-Tc = 2 and 18 °C. The zero

field magnetic susceptibility, as determined by

the parameters Ki and rx (which represent the range

and strength respectively of the spin correlations),

is found to vary as (T— Tc)~ l - 30±0- 04

. Near the Curie

temperature there was sufficient intensity to measure the ratio of the coefficients of the K4 and K

2 terms of the angular distribution. The value of

this ratio, 29 A

2

, is related to the existence of long

range couplings within the spin system. Details

of certain recent modifications of the theory of

critical systems are discussed and compared with

the experimental results. This work is described in detail in a paper by L.

Passell, K. Blinowski, T. Brun and P. Nielsen,

Phys. Rev. 139A, 1886 (1965).

143
Critical Neutron Scattering From Beta-Brass

O. W. Dietrich and J. Als-Nielsen

Research Establishment Riso, Roskilde, Denmark

Introduction

Beta-brass, which is an approx. 50 percent to 50

percent alloy of Cu and Zn, forms at low temperatures an ordered B.C.C. lattice. A B.C.C. lattice can be thought of as composed of two S.C. lattices —

so-called superlattices, denoted respectively A

and B — displaced half a cube diagonal with respect

to each other. Perfect order means that all Cu

atoms are at A sites, all Zn atoms at B sites. The

order gradually decreases with increasing temperature; at the critical temperature Tc (Tc — 468 °C)

the average occupation of say an A site is entirely

random, i.e., the long range order (abbr. LRO) has

disappeared. Above Tc , however, local order still exists: If a certain A site is occupied by say a Zn

atom there will be an excess probability over ran- domness that the nearest site — a B site — is occupied

by a Cu atom and so on. This local order or short

range order is described by a pair correlation function (abbr. pcf.) In a diffraction experiment the LRO will give

rise to narrow diffraction peaks (Bragg scattering

peaks), whereas the SRO will give rise to broader

and less intense peaks (critical scattering) provided

that the radiation is scattered sufficiently different

from a Cu atom and a Zn atom. This criterion

make neutrons more suitable than x rays in case

of /3-brass. Walker and Keating have previously

reported on a neutron scattering experiment from

a /3-brass crystal [1], which was furthermore

enriched with Cu65 to enhance the difference in scattering amplitudes. Due to rapid Zn-evaporation

from their crystal, it is however not possible to draw detailed quantitative conclusions from this experiment. From our present experiment, where

a /3-brass single crystal with normal isotopic abundance was used, the following can be deduced:

(1) The temperature dependence of LRO in the

close vicinity of Tc ,

(2) the absolute value and the temperature de- pendence of the SRO correlation range, and

(3) some conclusions concerning the shape of

the pair correlation function.

Cross Section

The changes in occupation of lattice sites are believed to be so slow that a static approximation is rigidly valid in deriving the relevant cross section.

The cross section for a scattering vector #c close

to a superlattice point t in the reciprocal space is da

dCt bV LR02 8(k

+ Er pcf (R)e-« r (1)

a and b denote respectively the scattering amplitudes for Cu and Zn nuclei, R is a direct lattice vector and the Debye-Waller factor e~ 2W accounts

for the influence of the thermal vibrations on elastic scattering.

For T < Tc the temperature dependence of (LRO)2

is deduced from the Bragg peak intensities. For

T > Tc this kind of scattering disappears and only

critical scattering remains. It will later be shown

that the Ornstein-Zernike pcf ~ e~K ^ R/R fits the

A

critical scattering data excellently, and with this shape of the pcf the critical scattering cross section becomes:

(da

\dCl, cnt

1 1

r

1 \k — t| 2 + k

2 (2)

Ki is called the inverse correlation range, n the

strength parameter.

144
(3 - Brass

Single X-tal Sphere 1" Diameter Encapsuled

in Steel

Ideal Situation in Reciprocal Space

Be(002)

Figure 1. Experimental setup in the (1, 0, 0) reflection. The angle between vertex and (0, 0, 1) is approx. 12°. whereby the occurrence of multiple scattering is prohibited.

The Experimental Setup

Monochromatic neutrons were extracted from the

polyenergetic reactor beam by (0,0,2) — Bragg

reflection in a Be single crystal (mosaic spread

5' s.d.). Higher order neutrons were filtered out by means of a mechanical velocity selector (see

fig. 1). A rather long wavelength of 2.70 A was

chosen to safely avoid parasitic multiple Bragg

reflections in the /3-brass single crystal, which in the (1,0,0) reflection was oriented with (0,0,1) approx. 12° from vertex. Also the instrumental

resolution could be made quite narrow at this wavelength, which is essential for the interpretation of the critical scattering as shown below.

The reactor beam was only collimated by the beam

tube dimensions, the monochromatic beam was

cylindrically collimated (30 mm diam per m) and

the scattered beam was horizontally collimated

by Soller slits (3.5 mm per m). The monochromatic beam intensity was 2.5 -105 neutrons/sec.

The counting time was mastered from a preset

count on a monochromatic beam monitor (abbr.

BMc in figures).

The spherical crystal (1 in. diam) was tightly

encapsuled in a thin stainless steel container

to avoid Zn evaporation. A 5 percent variation

of the Zn content changes Tc 14 °C, [2] but in the six- months duration of the experiment Tc remained

constant within a few tenth's of a degree. The

crystal was mounted in an oven of sufficiently wide

diameter so neutrons scattered from irradiated

parts of the walls of the heating coil could not possibly hit the BF.s-detector. The crystal table could

be turned around a horizontal and a vertical axis; the latter could also be coupled in the ratio 1:2

1 : 2 Arm Scan Rock Scan

Real Situation in Reciprocal Space

M(T-T0 ) J(K-XJ

FIGURE 2. Upper part: Wavevectors in reciprocal space cor- responding to figure 1 (idealized resolution). Lower part: Definitions of distribution function with finite resolution. The contour of J(k-k0) indicates f.i. a horizontal cut in the surface of half widths.

to the spectrometer arm. Two different scan types

through a superlattice reflection in reciprocal

space is sketched in figure 2.

Instrumental Resolution

Due to instrumental resolution the endpoint of the scattering vector k is smeared, which is in- dicated by the distribution function JKo ( k). In the .limited region in reciprocal space, where

JK<) { k) will be used in the calculations below it is assumed that J is only dependent on the difference

k — K0 , i.e., JKQ ( k) =J( k — Ko). Also the endpoint of the f-vector is smeared due to the mosaic

spread, indicated by the distribution ' function

M (t~ t0) (see fig. 2). The intensity 7(k0 , to) with the

spectrometer and crystal set at the average values

K0 and to of the scattering vector and T-vector re- spectively is found as the 6-dimensional convolution

integral:

/(Ko, to) = j" jj( k - k0)-M{t -to)

•ct(k, t) d K(h, (3) da

where the cross section -j^ (k, t) has been abbreviated to ct(k, t).

145
For Bragg scattering ct(k, t) — 8( K — t) and

/ Bragg* KO — To) =

J j

J ( It Ko) M(T — T0 )

8(k -r)dKdr. (4)

For critical scattering ct(k, t) = (r(fc — t) =

cr(e)S( k -r-€)de

and

/

/crit (K0 — r0 )= I (T(e)de I I 7( k - K0 )M(T —ro)

•8(k — r — e)d Kdr

k*-(ko-€))M{t-to)

S(k* —T)d K*c?r

/crit( Ko —T() ) —

j

/ Bragg( «o — « — T„)or(e)f/c (5)

22-10

20

18

16

CD

N 10

D

<_> 6

4-

2-

Critical Scan : T-Tc= 4.4°

o Measured intensities

Lorentzian, X, = 4. 50 *10~2 A" 1

Resolution

Folded Lorentzian, best fit

Background

0

Crystal Setting A

Figure 3. A rock-scan through (1, 0, 0) 4.4° above Tc

. The excellent fit of an Ornstein-Zernike correlation function is apparent.

The measured critical scattering is thus the

3-dimensional convolution integral of a measurable

function 7Bratw ( Ko —rQ ) and the cross section o\

and an exact convolution can be performed without

any assumptions of the shape of the distribution

functions J{k — k0 ) and M(t— t0 ), apart from the

already mentioned approximation J k „(k) — J(k—Ko).

Analysis of Critical Diffraction

Peaks

In practice, the unfolding calculation was per- formed in the following way with the aid of a

digital computer:

Three mutual perpendicular scans through the

Bragg peak were measured below Tc and ap- proximated by three normalized Gaussians. The

product of these was used as 7Bragg (k0 —

t

0 ). In

f.i. a rock scan measurement the computer calculated the ratio between ar(Ko,r0) from (2) and

/ (ko, to) from (5) for different directions of t0 and for a certain value of Ki. For this value

of Ki the sum of weighted, squared deviations

between /(ko,t0 ) and the measured intensities was calculated, and by iteration in Ki -values this sum was finally minimized.

An example is demonstrated in figure 3, where a scan perpendicular to (1, 0, 0) was measured 4.4 °C

above Tc. The circular points are measured

intensities as function of crystal setting. The

narrow curve is the corresponding normalized

Bragg reflection peak, measured below TP , and

the dashed curve is the Lorentzian (2) for the best

K\ -value. These curves are drawn upon a constant background, which was measured at several

points far from the superlattic point (1,0,0). A

scan at a high temperature (70° above Tc ) showed

the same constant background, when corrected

for the remaining small amount of critical scattering. This justifies to some extent a constant background subtraction, which was used throughout

the analysis of all critical data.

It is seen that the Ornstein-Zernike pair correlation function fits the data excellently. The

scatter of measured points around the folded

Lorentzian (solid line) is random and in statistical agreement with the uncertainties on the count rates.

Best Fit for Different Correlation Functions

Pair Correlation Function 2.0* 5.7* 12.1'

Ornstein Zernike

e-X,R

Ft Xi 2.38 1<T2 4.54-10" 2 5.12-10-2 6.82 >10" 2

Fisher

e -XiR Xi 2.38-10" 2 4.54 .10-2

5 . 12 - 10" 2 6.82« 10"2

V 0.0000 0.0000 0.0000 0.0000

Hart

Xi 2.38-10"2 4.54-I0"2 5. 12 « 10"2 6.82. 10"2

3 1.00-106 1.08-10 6 2.97-10 5 2.32. 106

146
Temperature Dependence of LRO

and SRO

Theoretically (LRO), which is proportional

to the Bragg peak intensity, is proportional to (Tr-Tf3 with 0 = 5/16 = 0.3125 for the Ising

Model [5]. Figure 4 shows the peak intensity from the

(1,1,1) reflection as function of the temperature

from 10° below the critical temperature to a little above this point. When the intensities are plotted

as function of AT = Tc — T'm a double log plot, it is seen that a power law is valid near Tr and that

(3 = 0.325. The intersection of the long-range order curve

with the abscissa gives the thermocouple voltage

when the crystal is in the critical state. The temperature dependence of all the parameters to be

discussed is expressed by the temperature differ- ence AT and this is measured as the corresponding

difference in thermocouple voltage multiplied with

the scaling factor 0.103 °K//xV. The uncertainty

of the intersection contributes the main part of the uncertainty of 0.1 °K on AT. A possible temperature gradient can be found by

measuring similar curves with reduced beam hitting

different parts of the crystal. The currents in the heating coils in the oven were adjusted until

all curves intersected the abscissa at the same

thermocouple voltage, and it was then concluded

that temperature gradients could be neglected.

Figure 5 shows similar curves for the (1,0,0) reflection. (3 is here found to be 0.27 which

differs significantly from the value 0.325 obtained

from the (1, 1, 1) reflection. This might be due to a

different degree of extinction in the two cases. The extinction correction will be relatively larger

for big values of AT than for small values, since the

cross section and thereby the beam attenuation

through the crystal increases with increasing values

of AT. The extinction correction will therefore

qualitatively have the effect of increasing the slope

of the line in the double log plot. At present we

have not investigated this problem in detail, and

the figures show uncorrected Bragg peak intensities. The tail of the curve for temperatures above the

critical point is the peak intensity of the critical scattering, and the difference in order of magnitude

from the Bragg peak intensity should be noted.

Counts

5x105 BMc

-6x10 3 5 ^^.^

A3

•2

1 \

1uV-0.103*C

i

. v.-

3800 3850 Temperature

3900 pV

Counts

i —

5x105 BMc

5x10 3

2

• 1

.3 .5 1.0 2.0 5.0 10.0°K Temperature (Tc -T)

Bragg peak intensity versus

(/, /, 1) reflection. temperature

Counts

10 6 BMc 30.000

20.000

10.000

luV -0.103° C

3800 3850 Temperature

3900 uV

Counts 106BMc

5 10 20 K Temperature ( Tr - T ) —»-

Figure 5. Bragg peak intensity versus temperature for the

(1, 0, 0,) reflection.

147

209-493 0-66— 1

1
Also the temperature dependence of the SRO has

been calculated in the Ising Model. For a ferro- magnet it has been found that the susceptibility

(T— T~\y

X obeys a power law being proportional to

with y equal to 1.25 [5]. It is a well known fact that the susceptibility x ls related to the pair cor- relation function through (x ^ 2k pcf (/?)). On the other hand the peak intensity 70 for T > Tc is also proportional to this sum as seen from the

cross section formula (1). Recalling the close

analogy of the order-disorder transition in alloys

and magnetic materials, it is therefore expected that

/"J

1 — f \-y

h —Tfr^) y can therefore be determined

from the measured temperature dependence of the

peak intensity, independently of any shape that the pair correlation function might have, provided

of course that the resolution correction is correct. Figure 6 shows the peak intensities of critical scattering from the (1, 0, 0) and from the (1, 1, 1) reflections for temperatures up to about 35° above

Tc . The intensities are corrected for reduction

due to resolution and the curves shown in the lower

part of the figure show the corrections. The peak

value reduction amounts to as much as a factor of

2 at the lowest temperatures and since the corrected

intensities for both reflections obey the same power

law, this gives confidence in the calculations on

resolution corrections.

The difference between peak intensities for the

(1, 0, 0) and the (1, 1, 1) reflections is due to dif- ferent Debye-Waller factors. The measured ratio

is seen to be 1.28 and, using Chipmann's data for the Debye temperature around 750 °K [6], the ratio of the Debye-Waller factors is 1.32. The weighted average value of the power y is 1.11 ±0.01, which should be compared with the

theoretical value of 1.25. Figure 7 shows the values of K\ obtained at dif- ferent temperatures in different scan types through

different superlattice points in the reciprocal

lattice. It is worthwhile to note the consistency

of the results. Each point represents one diffrac- tion peak, which takes, on an average, 48 hrs of measuring time.

For comparison with theory, the value of K\ at 75° above Tc calculated from the theory of Elliott and Marshall is also shown.

The peak intensity 70 is with the OrnsteinZernike pcf proportional to • Ki)~2

.

r, is theoretically expected to decrease slightly

with increasing temperature [7]. This implies

that twice the slope of the least square fitted line

to the Kj-values is an upper limit of the parameter

y already discussed. It is concluded therefore

that y is less than 1.18 ±0.02, in consistency with

the value found from the peak value runs. On the other hand, the measured values of

Ki together with the experimental result that

Peak Intensities Correlation Range M<

1

2JD0 - 1.50 >£

1.00

<f - 1.104 i 0.018

X - 1,1161 0.011

1.113* 0.009

Counts 2*10 7 BMc

5000

2000

1000

200 ~

3 Jzk

5 10 20 Temperature

50*K T-Tr

FlGURE 6. Critical peak intensities, corrected for resolution

reduction, for (1, 0, 0) and (1, 1, I) reflections.

41:2 scan (1.0.0 K \-k Theoretical value

o Rock scan(U).O) L i|

Rock scand.1.1 )

*<1.18»0.03

50x10"U" 1

. 20

- 10

1 1 ' ' I ' ' ' "

'

10 20 50X10"3 "S^-

I

5 10 20 Temperature50'K T-Tc

FIGURE 7. Inverse correlation ranges k, versus temperature.

Each point represents the result of a scan as shown in figure 3.

148
(n • Kij -2<r T-Tc\~

, can be used to deduce the

temperature dependence of r:

. In a linear approximation with rf = rfc (l — a—jrV it was found that a

was equal to 3.6±1.2. which is compared with the

temperature dependence of ri given by Elliott and

Marshall [7], who found that a — 1.2. After completion of these measurements it was

concluded that the most accurate information could

be obtained from rock-scans through the (1, 0, 0) reflection. A series of such scans were therefore

measured, but the inverse correlation ranges were now found to be 20 percent lower than the previous

data over the entire temperature region as shown

on figure 8. The (1, 1, 1) reflection data, which was obtained

at the end of the previous run, tends toward the

data of the latest (1, 0, 0) reflection run. The crit- ical temperature, as well as the Bragg peak intensities, were still exactly reproducible. Also the

temperature dependences of Ki and Io as well as

Io(k\) are in agreement with the previous data.

It must therefore be concluded that some propso i 10* Counts Vi 2x107

20-

JT r 1.10 i 0.03

o =Rock scan (1.0.0) A =1:2 scan(I.O.O)

X

10-10-2

^5

20 5'o-l6- 3'{'I=Ic

)

5 10 Temperature20 50°K T-Tr

FIGURE 8. Later results of critical scattering around the (1,0,0)

reflection. These results are not consistent with the results in figure 6 and figure 7. Note, however, that 7V, /3, y, and /o(k,) are the same.

erties of the crystal have changed during the long

period at elevated temperatures, but we are unable

to give any detailed explanation of the change.

Future experiments on other /3-brass crystals might

shed fight on the problem.

Conclusion

The following conclusions can be drawn from the

present experiment:

(1) The Ornstein-Zernike correlation function is in excellent agreement with the data. A complete

Fourier analysis to obtain all SRO parameters is not possible due to the very limited regions of the

reciprocal space where the critical scattering is sufficiently intense for experimental access.

(2) The absolute value of the short range order

correlation range is roughly in agreement with

theory, but could possibly be an individual property of a given crystal.

(3) The temperature dependence of LRO is not

in contradiction to the latest Ising model results,

but definitely outrules the results of older theories

^f.i. Bethe-Peierls theory, P =~^j-

(4) The temperature dependence of SRO is in

slight but definite disagreement with the Ising

model in the nearest-neighbor interaction approximation.

Note added in proof. Later measurements have

indicated that the /3-brass crystal contains a few

percentages of y-phase. Recent measurements on

a pure /3-brass crystal gave an exponent in the temperature law for the susceptibility of y= 1.25 ± .02.

References

[1] Walker, C. B., and Keating, D. T., Phys. Rev. 130, 1726

(1963).

[2] Hansen, M., Constitution of Binary Alloys (McGraw-Hill

Book Co., New York., N.Y., 1958).

[3] Fisher, M. E., Physica 28, 172 (1962).

[4] Hart, E. W., J. Chem. Phys. 34, 1471 (1961).

[5] Essam, I. M., and Fisher, M. E., J. Chem. Phys. 38 , 802

(1963).

[6] Chipman. D. R.. MRL Report No. 67, Watertown Arsenal, Massachusetts (1959).

[7] Elliott, R. I., and Marshall, W., Rev. Mod. Phys. 30, 1 (1958).

149
The Spectrum of Light Inelastically Scattered by a Fluid

Near Its Critical Point*

N. C. Ford, Jr.,1 and G. B. Benedek

Massachusetts Institute of Technology, Cambridge, Mass.

If a light beam passes through a perfectly homogeneous medium we expect no scattering of the

light in directions other than the forward direction.

This conclusion results from the fact that waves

scattered from each point in a homogeneous me- dium interfere destructively in every direction

other than the forward direction. Of course, if the wavelength of the radiation is of the order of the interatomic distances then on this scale the

medium will be nonhomogeneous and scattering can be expected to occur in conformity with the

Bragg conditions. However, for light waves the

Bragg equations cannot be satisfied. Despite this argument there is considerable light

scattering even from the most highly purified liquids.

This scattering grows stronger as one approaches

the liquid-gas critical point. In fact, in this critical region the scattering is so impressive that it is called "critical opalescence." Von Smoluchowski

[1] was the first to suggest that in a real liquid

the density is not perfectly uniform but that over

regions of the order of the wavelength of light there were fluctuations in the density of the liquid. At

each instant of time, he suggested, the liquid is not

of a single density, but instead there is a distribution of density around the average density. The

light is then scattered off the fluctuation in the

density around the average. Stimulated by Von Smoluchowski's suggestion,

Einstein [2] undertook a quantitative calculation

of the scattering of light from the fluctuations in a liquid with special attention to the situation near

the critical point. If we take the wave vector of the incident light beam as k, and that of the scat- tered beam as k', then Einstein was able to show

that the electric field scattered a distance R away

from the illuminated region is given by

E(R.K,t)

(1)

where:

K = k -k (2)

and where E() is the amplitude of the incident plane wave, oio is the angular frequency of the incident

radiation, sirup is the angle between the direction

of polarization of the incident light and the direction

of k'. 8e(K, t) is the fluctuation of the dielectric constant having the wave vector equal to the scat- tering vector K. Equation (1) signifies that scatter- ing in the direction k' is produced by a fluctuation

in dielectric constant having wave vector K = k' — k. This corresponds classically to a Bragg reflection

off the fluctuation having wave vector K. Quantum

mechanically eq (1) and (2) signify that constructive

interference of the scattered wavelets occurs only

when the momentum is conserved between the

photons and the scattering fluctuation. In order to determine the intensity of the scat- tered light Einstein then went on to calculate the mean squared scattered field:

<|E ( fi.K,„p>= £S (^)

4 ^,2^(|8,(K,„|^

The fluctuation in the dielectric constant can be

taken as arising out of the fluctuation of the density

8p viz:

8e(K, t) de8p(K, t) (4)

''This research was supported by the Advanced Research Projects Agency under

Contract SD 90.

1 Present address: University of Massachusetts. Amherst. Mass. and Einstein evaluated the mean squared amplitude

of the fluctuation in the density for K = 0 using the

150
Boltzmann principle which weights a fluctuation

by the work required to produce it. In this way

Einstein found for K = 0

B7 (5)

where V is the volume of the illuminated region,

p0 is the mean density, BT is the isothermal bulk

modulus, kT is Boltzmann's constant times the

absolute temperature. The phenomenon of crit- ical opalescence follows simply from (5) since as

T

7 —> Tc the isothermal bulk modulus goes to zero,

or perhaps more strikingly, the compressibility

grows very large indeed and hence (|8e(0, t)\ 2

) gets very big. Physically the enormous increase

in the scattering results because large density

fluctuations become energetically very easy to produce since the compressibility is very large.

The large density fluctuations in turn produce large^

fluctuations in the dielectric constant and hence

a great deal of light scattering.

The work of Einstein was later extended by Ornstein and Zernike [3]. They noted that eq (5) was

a good approximation for 8e(K, t) provided that one was not too near Tc . As 7

1 —» Tc , however, it be- comes necessary to evaluate (|8e(K, t)\ 2

) for K ^ 0. Their result, which has also been obtained in other

ways [4, 5], has the form for small K

(277)3 <|8e(K, t)\ 2

) BT + (aK)'< (6)

where a is a length of the order of the interatomic

distance. Thus, the scattered light intensity grows

extremely large only in the forward direction K = 0. For other directions the scattered intensity will not in fact diverge.

Following the work of Einstein, and Ornstein

and Zernike, much experimental effort was devoted

to a study of the temperature dependence and the

K dependence of the scattered light intensity.

Work continues to the present day and in fact is being reviewed at the present conference. Despite

this activity, there exists no experimental infor- mation on the spectrum of the scattered light in the critical region of a fluid. The first person to investigate theoretically the

spectrum of light scattered from a liquid or solid was L. Brillouin. His first investigation [6] of this subject was in 1914. This was followed by a more

complete analysis in 1922 [7]. The spectrum of the

scattered light depends upon the form of the time

dependence of the fluctuation in the dielectric constant. In fact, the spectrum is obtained if the

time correlation function (E(K, t'+r) • E*(HL, t))

is known [8]. This correlation function is related

to the temporal fluctuations in the scattered field by

<£(K, t + r) £*(K, f)>

= 1TM 2T J_,

E(K,t + r) E*(K, t)dt (7)

where we have dropped the distance R in the designation of the functional dependence in E. The

correlation function for the scattered field is re- lated to the correlation function for the fluctuations

in the dielectric constant by

(E(K, t + r)

~^(27r)3 <Se(K, t + T)8e*(K. t)>, (8)

and the spectrum of the scattered light is given by

S(K, w') =7T- f < E< K- ' + T>

• E *< K- t))e iui

' T dT (9) Ztt j and the total intensity under the spectrum S is just

(\E(K, t)\ 2

) = s(K(o')d(o' . (10)

The question thus arises, what is the time cor- relation function for the fluctuations in the dielectric constant? To answer this we must have some

model for the origin of the fluctuations in e. Bril- louin pointed out that one such origin was to be

found in the thermally excited sound waves which

continuously travel through the medium. These

sound waves are fluctuations in pressure at constant

entropy. The pressure fluctuations in turn modulate the dielectric constant and so produce scattering. To calculate the modulation of the dielectric constant produced by the thermally excited sound

waves, we observe that:

(Se(K, t))s

- ^) (8p(K, t))s dp/s

(11)

where 8p(K, t) is the Fourier amplitude for the pres- sure fluctuation in the sound wave having wave

vector K. Since the sound wave travels as a wave

with frequency ±fl(K) and has a finite lifetime

151
1/HK) we may expect that the correlation function

for the pressure fluctuations is

<8p(K, t + T)8p*(K, t)> = (\8p(K, 0|

2 )e

±,n,K,Te-rr

. (12)

From this it follows at once that the spectrum of

the scattered radiation is given using equations

12, 11, 9, 8, by

r(;

, K , n 9 /Wv2 sin2

<p Lo(IV, (O ) J sound waves ^0

\ c J (4<jfJ{)2

(27r)3(^ <|8p(K, t)\ 2

) U E(K) 1 7rl[a) '-(wo±n(K))] 2 + r

2(K)J

(13)

Thus the spectrum of light scattered by the sound waves in liquid consists of a doublet. Each component of the doublet is shifted away from the in- cident frequency cu0 by an amount ±fi(K). This

shift is simply the frequency of the sound wave

which was responsible for the scattering. The

Brillouin doublets can be regarded as a Doppler

shift in the frequency of the incident light as a

result of a reflection off the moving sound waves. Or equivalently the condition

w'=aj„±a(K) (14)

can be looked upon as a conservation of energy

between the incident photon, the emergent photon,

and the scattering phonon. The plus sign corre- sponds to phonon annihilation and the negative sign

to phonon creation.

The existence of the Brillouin doublets for light

scattered by a liquid was first shown by the experiments of Gross in 1930 [9, 10]. This subject has

since expanded considerably and the experimental

information on liquids has been well reviewed by

I. Fabelinskii [11]. The experiments showed,

even in the most carefully purified liquids, that the

spectrum also contained fight whose frequency was

equal to that of the incident radiation. That is, in addition to the Brillouin doublets, there is a central component in the spectrum of the scat- tered fight. The existence and relative magnitude

of the central components and the Brillouin doublets was explained by Landau and Placzek [12, 4]. They

pointed out that the fluctuations in the dielectric constant are produced not only by the adiabatic

pressure fluctuations (the sound waves) but also by the isobaric entropy fluctuations. Putting the matter very explicitly, the dielectric constant can

be regarded as a function of the entropy (s) and

the pressure (p) and fluctuations in either of these

quantities will cause fluctuations in the dielectric constant. Thus

8e(K, t) =

(^j (&(K, t))p +

^

(8p(K, t))s . (15)

The second term on the right hand side of eq (15) was considered by Brillouin. Landau and Placzek

stressed the importance of the first term. In

order to obtain the time dependence of the entropy

fluctuations they argued that the fluctuations in entropy were proportional to those of temperature

through

T(8s)p = cp(8T)p. (16) But the time dependence of a temperature fluctuation is governed by the Fourier heat flow equation

viz:

4sr(r,f) = 4v2 SHr, t) (17) at c* where c* is the specific heat per unit volume, A. is the thermal conductivity, 8T(r, t) is the temperature

fluctuation at the point r and time t. To obtain the

time dependence of the temperature fluctuations

with wave vector K, we compute the Fourier trans- form of eq (17) and find that

4- 8T(K, ;)==4r^ 8T(K, t). (18) at Thus the temperature fluctuations do not propagate, they are diffusion like in their decay, and a

fluctuation will die away exponentially with time.

Thus we have that the correlation function for the

entropy fluctuation will have the form

(8s(K, t + T)8s*(K, i)) = (\8s(K, t)\ 2 )e- r^

(19a)

where

re(K) = (19b)

152
Since the entropy fluctuations are independent

of the pressure fluctuations we may calculate the

spectrum of light scattered by the entropy fluc- tuations alone and we obtain using 8, 9, 15, and 19

that (S(K, o/))entropy

fluctuation

top fsin2 if

X

'p

] Te (K)

tt l(o)'-ft)0)2+ri:(K;

(20)

Since the entropy fluctuations do not propagate,

they appear in the spectrum with the same fre- quency as the incident light frequency. The width

of this central component gives the lifetime for the entropy fluctuation with wave vector K.

The entire spectrum of the scattered light is the sum of the spectrum due to the sound waves

and the entropy fluctuations, i.e.,

S(K, <u)total = (s(K, &>))entropy + (s(K, Co))sound (21)

fluctuation waves.

By evaluating <|8s(K, t)\ 2 )/( |6>(K, t)\ 2

) for K~0

Landau and Placzek could show that the intensity

in the central component is greater than that in the Brillouin components by the ratio (c,, —

c

r)lc r. As we can see from eqs (13) and (20), the spectrum

of the light scattered by a liquid contains a great

deal of information of importance in the critical region. In particular we should mention the follow- ing points. The width of the Brillouin components

provides the sound wave lifetimes as a function of temperature and wavelength. The spectral position gives the sound velocity as a function of wavelength and temperature. Note that since the

adiabatic bulk modulus approaches zero as T—> T, the Brillouin doublets should move in toward the center as one approaches Tv. The intensity of the

Brillouin components gets large like the adiabatic

compressibility. The central component contains

the following information. The width gives the

lifetime for decay of the nonpropagating fluctuations

of the entropy. The precise shape of the central

component gives the time dependence of the correlation function. The intensity gives (cp — cv ) as a function of the temperature.

Despite the information contained in the spec- trum, until very recently, there have been no observations of the spectrum of light scattered near the

critical point of a fluid. The reason for this is

that the line width of the conventional light sources were so broad that its spectrum spread out well beyond the Brillouin doublets thus preventing

resolution of the spectrum near the critical point.

The invention of laser light sources with high

intensity and very narrow spectral width has made

it possible to study [13, 14] the spectrum of light

scattered from a normal liquid with much greater

resolution than was previously possible. In view of the information contained in the Brillouin components we undertook a- study of the spectrum of

light scattered from the fluid sulfur hexafluoride

(SF6 ) in the vicinity of the critical point Tc = 45.55

°C. The critical pressure and density of this fluid of nearly spherical molecules is pc = 37.11 atm and

pc = 0.730 g/cm3

. We used as a spectrograph a high resolution Fabry-Perot interferometer which

could resolve lines spaced as closely at 50 Mc/s.

The expected splitting of the Brillouin components was about 500 Mc/s. The light source was a single

moded, helium neon laser, whose frequency was

constant to within about 10 Mc/s. We searched

along the critical isochore over a temperature range

0.1 °K < T- Tc < 20 °K without finding any resolved

Brillouin doublets. While this experiment must

be repeated with greater sensitivity, we feel that the negative result may indicate that the widths of

the Brillouin doublets in this critical region are much broader than the displacement of each fine from the central component. In other words, the

sound wave lifetimes for sound waves with a fre- quency of ~ 500 Mc/s is actually shorter than the

period of the wave. These considerations are supported by an co

2 extrapolation of measurements

of the low frequency sound wave attenuation in SF6 near the critical point [15]. As a result of

these observations it appears that the most strik- ing feature of the spectrum is the central component which grows in intensity and gets extremely narrow as Tc - Very recently Alpert, Yeh, and Lipworth [16]

have reported the observation of the width of the

central component in the spectrum of light scat- tered from a critical mixture of cyclohexane and

aniline. They used an elaborate "homodyne spec- trometer" [17] which analyzed the beat between

the scattered light and part of the incident light.

We report here the first measurements of the

width of the central component for light scattered

from a pure fluid, SF6 , near its critical point. We

use for this study the very simple detection scheme

shown in figure 1. 153
Laser \s -$4

Photomultiplier

from the entropy fluctuations is given by

Figure 1. Block diagram of "self beating" or "square law"

spectrometer.

This spectrometer provides the frequency spectrum of the fluctuations in the scat- tered electromagnetic held.

The light scattered in the angular range between

6 and 0 + A.6 is collected by a lens and focused

onto the surface of a phototube. The phototube

current is proportional to the square of the inci- dent electric field. Thus the fluctuations in the

output current of the phototube gives the fluctua- tions in E(t)E*(t). This fluctuating current is ana- lyzed into its spectrum using a spectrum analyzer

and the spectrum is displayed on a recorder.

The spectrum, as obtained by this "self beat"

or square law detection system, gives the spec- trum of the fluctuations in |£"(/)| 2 - We may

relate this spectrum to the spectrum of E(t) by

making use of the fact that if E(t) is a Gaussian

random variable, the correlation function for j E{t)\-

is related to the time dependence of E(t) by the

equation

<|£(K, r+r)| 2 |£(K, f)| 2

> = ((|£(K, 0| 2 ))

2

+ (E(K, f + T)£(K, t)) <£*(K, t+r)E*(K, t)>

+ \(E(K,t + r)EUK,t))\ 2 (22)

where we have explicitly written K the scattering vector in the expression for the correlation functions to emphasize that these functions depend on

the scattering vector. We now use the fact that the time dependence £(K, t) for the scattering

<£(K,n 'entropy cooy sin <p e

ik R

4tt R

( 2 ttW^I) (8s(K,r) )„e- iioQt 23)

With this time dependence of E(K., t) it follows

that the second term on the right-hand side of eq

(22) is zero and the correlation function for the

output of the phototube is determined entirely by

the correlation function of the scattered field viz:

Re(T) — (E (K, t +t)E* (K, t) > entropy

-)e\ 2

ii

sin2 <p /0

(\8s(K,t)\ 2 ).<>-i«o' e

-r.(K)T (24)

where we have used the Landau- Placzek assumption, eq (19), to obtain the form of the correlation function for the entropy fluctuations. Thus

the correlation function for the output of the phototube has the form:

(\E(K,t +t)| 2 |£(K,0| 2

> = (/U0))2 +\Re(T)\ 2 (25)

and the spectrum of this fluctuation, i.e., the spec- trum of the input to the spectrum analyzer has the

form S^K, a.*):'

Sb(K,»)=^- f

X (E(K,t+r)

E*(K,t))e+il» T dT (26)

1 = Rm \8(a>) + -

TT 2Te (K)

(cu2 + (2r,(K))2 (27)

The first term on the right side of eq (27) corre- sponds to the d-c photocurrent. This d-c current does not enter the spectrum analyzer as it is re- moved with an RC blocking filter as shown in figure 1. Thus the power spectrum of the current

fluctuations entering the spectrum analyzer is given by the second term on the right side of eq

(27). The spectrum analyzer uses a linear full wave rectifier as its detector element. As a result,

154
it can be shown [8] that the output current I(a>) as a function of frequency a) is directly proportional to the square root of the input power spectrum. /(o») is displayed on the recorder, and from the discussion

above we see that I{co) is given by:

/( cj)— (const)

RAO) 2TP (K)

tt^ \(af2 +(2T(,(K))2

:

1/2

(28)

We can therefore use eq (28) to analyze the shape

of the I(co) spectrum and thereby determine the

decay rates of the entropy fluctuations re (K). Of

course, if the output spectrum I(oj) does not have

the form predicted by the Landau theory we can

obtain from 1(a)) information on the form of the

time dependence of the correlation function for the entropy fluctuation.

We show in figure 2 the spectrum analyzer out- put I(<o) as observed at T= 45.78 °C, about 0.2 °C

from the critical point, and near the critical iso- chore. The scattering angle was about 8.7°. We

estimate that the half width at half height of this spectrum is about 350 c/s. This measurement is remarkable in that it shows that lasers now make it possible to detect shifts in the frequency of the scat- tered light as small as a few hundred cycles per

second out of the incident light frequency of

5 X 10 14 c/s. In figure 3 we show similar spectra

taken at fixed temperature for different scattering

angles to check the K

2 dependence of the half width

of the line. The results of this investigation are shown in figure 4 for two different temperatures.

Within the error limits imposed by these early

measurements we see that the width of the scat- tered spectral line is indeed proportional to K2 In figure 5 we show the temperature dependence

of the half width for fixed scattering angle in order

to examine the temperature dependence of A./c*.

500 1000 1500 Frequency (c/s)

FIGURE 2. The spectral distribution of the light scattered by

the nonpropagating entropy fluctuations in SF6 very near the

critical point.

Note that the width of this spectral distribution is only a few hundred eycles per second. The incident light frequency is 5 X 1014 c/s. The resolving power of the spectrometer (f/Af) is about 1 X 10'-.

T = 45. 317 °C

(A-), = 200c/s XV

9 = 8.7° ^^Cl<><iSs

I

I I I 1

Frequency (c/s FIGURE 3. The spectral distribution of light scattered from SF6 near its critical point for different scattering angles.

Observe t he broadening at the larger scattering angles.

sin

2 (9/2)

FIGURE 4. The dependence of the half width of the scattered spectrum on the square of the scattering vector K = kn sin 6/2 for two different temperatures near the critical point.

Finally we may inquire as to whether the value of rr(K) as deduced from 1(a)) and eq (28) is in agree- ment with the formula of Landau. Along with

J. V. Sengers [18] we have estimated the temperature dependence of A. and c* for SFB at the critical isochore and thus obtain a theoretical estimate for the temperature dependence of TP(K). This esti155
FIGURE 5. The temperature dependence of the half width of the scattered spectrum of lifiht from SF<; for a density near the

critical isorhore.

The critical point is al 45.55 °C.

mate is in a good qualitative agreement with the

present observations [19]. In conclusion we wish to stress the following

points. First, that the present experiments demonstrate that it is possible to study the spectral dis- tribution of the light scattered from entropy fluctu- ations near the critical point. These studies can be

carried out using a very simple "self beating" or "square law" spectrometer whose operation has

been described. The spectrometer, which uses a

laser beam as a light source, permits a measurement

of the spectral distribution in the scattered light

which is as narrow as a few tens or hundreds of

cycles per seconds out of the incident light fre- quency of 5 X 10 14 c/s. By studying the detailed

shape of the spectrum and its temperature depend- ence we can obtain information on the time and temperature dependence of the correlation function for the entropy fluctuations in the critical region.

Finally, we have presented the first measurements of

these fluctuations in a pure fluid (SF6) near its critical point.

[9 [10

[11 [12

[13

[14

[15

[16

[17

[18

[19

References

M. Von Smoluc-howski. Ann. d. Fhys. 25, 205 (1908). A. Einstein. Ann. d. Physik 38, 1275 (1910). L. F. Ornstein and F. Zernike. Proc. Acad. Sci. Amster- dam 17, 793 (1914-1915).

L. Landau and E. M. Lifshitz. Electrodynamics of Contin- uous Media, pp. 393ff. (Addison Wesley. Reading. Massachusetts. 1960). M. E. Fisher. J. Math. Phys. 5, 944 (1964).

L. Brillouin. Comptes Rendus de l'Academie des Sciences 158, 1331 (1914).

L. Brillouin. Ann. de Physique 17, 88 (1922). W. B. Davenport and W. Root. An Introduction to the Theory of Random Signals and Noise (McGraw-Hill Book Co.. New York. N.Y., 1958). Gross. Nature 126, 201. 400. 603 (1930). Gross. Zeit. f. Phys. 63, 685 (1930). Fabelinskii. Usp. Fiz. Nauk 63, 355 (1957). Landau and G. Placzek. Phvsik Z. Sowjetunion 5, 172

(1934). Benedek. J. B. Lastovka. K. Fritsch. and T. Greytak.

.O.S.A. 54, 1284 (1964). Y. Chiao and B. P. Stoichieff. J.O.S.A. 54, 1286 (1964). G. Schneider. Can. J. Chem. 29 , 243 (1951). Alpert. Y. Yeh. and E. Lipworth. Phys. Rev. Letters 486 (1965). Cummins. N. Knable and Y. Yeh. Phvs. Rev. Letters 150 (1964). Senders, private communication.

s. 14,

Z. 12,

V. N. C. Ford and G. B. Benedek, Phys. Rev. Letters 15, 649

(1965).

156
Time-Dependent Concentration Fluctuations Near the

Critical Temperature*

S. S. Alpert

Columbia University, New York, N.Y.

The title of this talk which is printed in the con- ference program as "Time-Dependent Density

Fluctuations Near the Critical Temperature"

is incorrect and should be altered to read "TimeDependent Concentration Fluctuations Near the

Critical Temperature." This is readily apparent

when one considers the differential element of scattered energy given by the expression

dEsc dfl — OC Io A

4 B(6, A).

Here dEsc is the differential element of scattered

energy which is normalized to the incident intensity

/0 ; dCl is the element of solid angle into which the

light of wavelength A is scattered. The factor 5(0, A) gives the dependence of the scattered energy on the scattering angle 6 and on the wavelength A.

It is seen that the scattered intensity depends on

the change of the dielectric constant e of the me- dium with respect to both the density p and the

concentration c. For a binary system such as aniline-cyclohexane, which is the system studied,

there is reason to believe that the mean square

fluctuation of concentration 8c2 becomes very large

as the critical temperature is approached. This

point will be discussed later in this talk. However,

since the compressibility of the binary system is nonsingular in the region of the critical temperature, the mean square of the density fluctuation

remains finite and small: hence it is reasonable

to assume that it is the concentration fluctuations

which are responsible for the light scattered from

binary systems near the critical temperature.

In the preceding talk, Professor Benedek clearly

outlined the nature of the light scattered from a pure

liquid system. The scattered light from a binary

mixture is a triplet where the outer two components,

the Brillouin doublet [1], result from thermally

excited acoustic modes and where the central

unshifted component [2] results from fluctuations

*This work was supported in part by the Army Research Office — Durham under

Contract DA-31-124-ARO-D-296. and in part by the Joint Services Electronics Program under Contract DA-28-043 AMC-00099 (E).

of concentration and temperature at constant pressure. It is this central unshifted fine which my

colleagues, Dr. Y. Yeh and Dr. E. Lipworth, and

myself have experimentally observed, using a laser homodyne spectrometer of high resolving power.

Before discussing the results of our experiment,

I should like to digress for a few minutes and to sketch for you the general functioning of the laser homodyne spectrometer developed at Columbia

University by Cummins, Knable, and Yeh [3] in the early part of 1964. The laser homodyne spectrometer [3] has a re- solving power of one part in 1014 and is capable of

scanning the spectral profile of the central component of the scattered triplet. The essential functional components of this spectrometer are the

laser light source, a divided optical path, the ex- ternally driven Bragg tanks, and the detection ap- paratus and associated data-processing electronic

equipment. (See figs. 1 and 2). These functional

components are presented in outlined form below:

1. Laser source. As previously noted in this re- port it was desired to observe the spectral profile

of the central component of the scattered light

triplet. This profile was expected to be on the

order of a few hundred cycles per second. Clearly

the spectrometer fight source must be inherently more narrow than the desired spectrum. The

only available light source fulfilling this requirement at the present time is the He-Ne laser. Jaseja

[4] has shown that under highly controlled conditions and for short times, the fine width of a single

mode of the He-Ne laser is less than 1 c/s. This was the reason underlying the use of a He-Ne laser capable of a power output of about 15 mW with an intermodular spacing of 60 Mc/s.

2. Divided optical path. Although the frequency

output of the He-Ne laser is instantaneously quite narrow, over any reasonable time interval of ob- servation (~ 1 sec) the fine width is broadened

to the bandpass of the optical cavity which is about

10 kc/s. This broadening results from thermal

vibration effects on the mirrors and also any

microphonic effects present. It thus becomes

157
6328 A He - Ns LASER

COLLIMATING

LENS

SAMPLE CELL

BRAGG SSB MODULATOR

6328 . NARROW BAND 1 FILTER

FIGURE 1. Schematic arrangement of optical components of

laser homodyne spectrometer.

ANTENNA

input n^-

2.5 Mc

XTAL OSC

PHOTO MULTIPLIER S-20

COMMUNICATIONS RECEIVER

501,150 c/s XTAL OSC.

STRIP CHART

RECORDER

PASS BAND

FILTER LOW LEVEL MIXER

LOW

PASS

FILTER

NARROWBAND

AMPS

SWEPT L.O.

1550 ± 200 c/s

FIGURE 2. Block diagram of electronic equipment used in laser homodyne spectrometer.

essential to avoid this cavity broadening by a "beating" or heterodyning technique. This is accomplished by dividing the laser light into two separate

optical beams. One beam falls directly on the

photocathode detector, together with light from the

other beam which is first scattered off to sample

being studied.

3. Bragg tanks. The heterodyning technique

suggested under the previous heading is difficult

to detect because the resulting "beats" occur at ultraviolet frequencies and at dc. For practical

detection a "beat" should occur at radio frequencies. The "beat" can be made to occur at radio

frequencies by shifting the frequency of each arm of the optical path by known amounts. This was done

by the use of two Bragg tanks [5] filled with water

and acoustically driven with quartz transducers at 18 Mc/s and at 30 Mc/s.

4. Detector and data handling electronics. The

12-Mc/s "beat" falls on an S-20 photocathode and

appears in the photomultiplier output current (fig.

2). The photomultiplier output which contains the

spectral-profile information centered about a fre- quency of 12 Mc/s is fed into a radio receiver tuned

to that frequency. The resulting i.f. signal is fur- ther "mixed" in two sequential stages so that the

resulting signal is greatly reduced in frequency.

The last mixer is provided with a mechanically

scanned local oscillator input. The output of the

final mixing stage is passed through a narrow-band

amplifier centered at a frequency of 400 c/s. Since

the narrow-band amplifier will only amplify in this range and since the local oscillator input to the final mixer is swept, the final output will yield a frequency

profile of the original 12-Mc/s beat. The instru- mental bandwidth of such a system was 13 c/s. The system studied in the present experiment was

a cyclohexane-aniline mixture, 53 percent cyclohexane and 47 percent aniline by weight. The

mixture was contained in a cylindrical glass cell in a constant-temperature enclosure, the temperature of which could be adjusted and held constant

to about 1 X 10~3 °C over extended periods of time.

Some characteristic spectra are shown in figure 3. These are typical of the spectra observed with the

laser homodyne spectrometer; I might add that the

word "typical" is not used here in the sense that Miss America is the "typical" American girl, but

to indicate the ordinary performance of our spec- trometer. The spectra shown in figure 3 are normalized to give the same magnitude on our final re- corder. In actuality the signals become less intense

H h50

T-T,=.0695°C

id)

T-T,=.I54°C T-T,=.400°G

FIGURE 3. Characteristic spectra observed at a scattering

angle of 20.5° and at varying temperatures.

Signals are normalized to give recorder traces of approximately the same magnitude.

158
Figure 4. The halfwidth at half-intensity versus T —

T

c

. Included in observed halfwidths is an instrumental error of 13 c/s which can he

directl) subtracted out. Note that at T—T,* the extrapolated halfwidth does not vanish.

as the temperature above critical is increased, as can be deduced from figure 3, by the increasing

presence of noise superposed on the signal. As

the sample approached its critical temperature from

above, a definite narrowing of the width of the

scattered profile was observed as shown in figure 4

where the scattering angle was 20.5° from the for- ward direction; T, was defined as the temperature

at which a meniscus was first seen to appear

visually. It is to be noted that at T= Tc, the

halfwidth of the scattered spectrum is 38 c/s. This includes a 13-c/s instrumental line width

which can be directly subtracted out if the scat- tered spectrum is Lorentzian. which seems to be

the case. Thus for T= T, . a half width of 25 c/s was observed for the corrected spectrum.

A question to be answered is what is the line shape of the observed spectra. At first glance it would seem (see fig. 3) that the lines are Lorentzian.

The curves do. indeed, match an assumption of

being Lorentzian as is shown in figure 5. Such

an assumption would predict a concentration

fluctuation decay of the form V^c2 e~

t,T which would

correspond to a 7 = 0.013 sec for our sample at the critical temperature.

The scattering angle 6 was varied between 20°

and 40° and the width at half intensity was observed.

The results are shown in figure 6 where it is seen

a

that ^ sin 2 -. The intercept corresponding

to 0 = 0 represents the instrumental line width

which can be directly subtracted out.

(»-»0

) 2 + r

2

(c/.)2

FIGURE 5. Log-Log plot for Lorentzian fit.

Since it is assumed that

S(K, «l =

Kly\ . the log of

v)

is plotted versus Inji (» — vo)-+y*. Here y is the halfwidth at halfintensity.

3mm SAMPLE CELL

T-T, =.0I4"C

0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0

I0 2 XSIN2 |

-«-

FIGURE 6. Plot of Av versus sin2 {6/2) for a 3-mm sample cell at T — Tc = 0.014 °C.

To indicate that our overall results are reasonable,

I would like to invoke the following argument

based on published work of Landau and Lifshitz [6]. From hydrodynamical considerations these authors

deduce the following coupled equations for a binary

system:

y8c = Dfo*8c + j v2 sr

(1)

159
and

at cp \dcJp.T at *

where D is the diffusion constant kr is the thermaldiffusion ratio, x ,s tne thermal diffusivity, and ix

is a suitably defined chemical potential. The other

symbols have been previously defined. Landau

and Lifshitz [7] have also shown that for a binary

mixture, — AT

8c

2 =

\dc Jp.T.

(2)

As the critical temperature is approached, the

chemical potential and its first and second derivatives with respect to concentration vanish. Thus

near the critical temperature 8c2 goes to a l arge

value and since 8T remains nonsingular 8c2 > 8T2

. Under these simplifying assumptions we derive

from eqs (1) the single diffusive expression

t-8c = Z)V2 8c. at

The solution of this equation is

1 Sc = - exp

(47rDf)3/2 Wt

which has a Fourier transform of the form

T(K, a>) =

2DK2 (3) (DK2

)2 + oj2

where K is -^-sin0/2. Hence we would, indeed,

expect a Lorentzian line shape at a fixed scattering

angle and also a line width dependence on sin2 0/2

at fixed temperature.

Dr. Raymond Mountain [8] of the National Bu

reau of Standards has solved the eqs (1) using

Fourier and Laplace transform techniques and gets

the same results for the line shape in expression (3). In closing I would like to point out that in the

past, researchers have only been able to measure

scattered intensities. We now have available

techniques as demonstrated in this paper of measuring spectral profiles. Hopefully, as our tech- niques become more refined and our results more

quantitative, studies of the spectral profile of scattered fight such as we have performed can be

used as a new experimental basis with which to construct theoretical models of the nature of

liquids.

References

[1] L. Brillouin, Ann. Phys. (Paris) 17, 38 (1922).

[2] E. Gross, Nature 126, 201 (1930).

[3] H. F. Cummins, N. Knable, and Y. Yeh, Phys. Rev. Letters 12, 150 (1964).

[4] T. S. Jaseja, A. Javan, and C. H. Townes, Phys. Rev. Letters 10, 165 (1963).

[5] H. F. Cummins and N. Knable, Proc. IEEE 51, 1246 (1963).

[6] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (AddisonWesley Publishing Co., Inc., Reading, Mass., 1959).

[7] L. D. Landau and E. M. Lifshitz, Statistical Physics (Addi- son-Wesley Publishing Co., Inc., Reading, Mass., 1958).

[8] R. Mountain, J. Res. NBS 69A (Phys. and Chem.), No. 6, 523

(1965).

Discussion

B. Jacrot: I would like to comment on Dr. Marshall's explanation

of the inelastic neutron scattering by assuming that the spin waves persist even close to the Curie point. Indeed the line width which I observed experimentally is in'good agreement with

the contribution estimated from spin waves as shown by Dr. Marshall. At this time I think it is just a coincidence, but the explanation may well be correct.

E. Callen: I am uncomfortable with the application of the beautiful analysis of Dr. Marshall to metals, because I see no evidence

of a large line width mechanism, the complex susceptibility arising from excitation of electrons among Landau states. W. Marshall: I certainly agree with you. that we must be cautious

in applying the theory to metals, which, of course, is the only

thing I have done. I would answer your remark in the following way. What I am really trying to say is that as soon as you stop damping the spin waves a diffusive mode will appear on the imaginary axis. There are various ways in which you could

describe this. You may prefer to describe it in terms of spin waves and the damping of spin waves. That is perfectly satis- factory. But we have also to look, in this particular formulation, at what this pole is on the imaginary axis, and on the effect of this diffusive motion. In the 'case of a liquid we know that the sound waves are damped by thermal conductivity and vis- cosity. Similarly we can explain our results in terms of the damping of spin waves. But as soon as we put in the thermal

conductivity, a third pole appears in the expression on the imaginary axis. And that pole becomes largest in the proximity

of the critical temperature.

H. B. Callen: A simple analogy may make this diffusive mode a

little less mysterious. The excitations of a noninteracting Fermi

gas at very low temperatures are oscillations of the Fermi surface which are coherent excitations of the whole system which occur

at high frequency. I think they are the analog of the spin waves. When the temperature becomes large enough so that the relaxation term becomes appreciable these high frequency excitations disappear, being replaced by one-particle excitations (at very low frequencies). These one-particle excitations seem to cor- respond to the diffusive mode. In the transition region the oneparticle and the collective mode may coexist, although the col- lective oscillations of the Fermi surface are then damped.

160
R. Nathans presented measurements of neutron scattering in antiferromagnetic potassium manganese fluoride [1J.

B. Jacrot: In connection with the work of Dr. Nathans. I want to remark that the inelasticity for an antiferromagnetic system like manganese fluoride should be ten or twelve times larger than for iron due to the absence of kinematieal slowing down. This has been confirmed qualitatively by some unpublished results ob- tained a few years ago at Saclay. New experiments are now in progress.

R. Nathans: Well, that may be possible. We have only seen

the data of Turberfield in manganese fluoride, which indicate a small inelasticitv in the critical region.

M. S. Green: Dr. Passell. why did you expect to observe elastic scattering before you started the experiment?

L. Passell: Because the parameter A which describes the time dependence of the spin fluctuations should be inversely pro- portional to the susceptibility as indicated by Dr. Marshall. This was the argument which Van Hove used in his original formulation of the theory. On this basis we assumed that we

were going to have essentially quasi-elastic scattering at the ordering temperature.

M. S. Green: Did not Van Hove predict a finite inelastic scattering

for finite ^-values?

L. Passell: Yes, that is right. But that would give only a small

correction.

W. Marshall: May I comment on this point? First, if you use what I call the conventional theory and which Van Hove gave, then of course the scattering is inelastic. And',*hs is gives you

the difference between say a neutron experiment and a N.M.R.

experiment. However, when you ask theoretically how inelastic the scattering is, you discover that so far as neutrons are con- cerned the spectral distribution looks like a sharp 8-function. So if that were true, we could say that for experimental purposes

the scattering is as elastic as we could possibly want it to be.

Secondly, Dr. Passell was much more polite in describing the

situation than I would have been. We knew about Jacrot's

results and that he had found the scattering to be inelastic, but we did not believe it could be true. So we felt that there probably was some experimental flaw in these experiments; I now apologize to Dr. Jacrot for thinking that to be the case.

I also like to make another point. It is true, of course, thai we would expect proportionally very much greater inelasticity

in an antiferromagnet than in a ferromagnet, simply because we do not have the kinematic narrowing. However, we also have the fact that the Curie temperature of iron is 1042 °K compared to 80 °K for manganese fluoride, so that the higher ex- changes energies in iron compensate the difference to a certain

extent.

J. Villain: As a theoretician, J would be very interested in the temperature dependence of the inelasticity parameter A. In

tact, de Gennes 12] has predicted a thermodynamic slowing down

which must be added to the kinematic slowing down. It is pre- dicted that the line width close to the critical temperature is smaller than at the temperatures higher than the Curiev

temperature. I think that the original treatment of de Gennes and

myself [2] is not correct and that this thermodynamic slowing

down is perhaps weaker, but it must exist. Our present

theory [3] to account for this effect is in good argeement with

the results of Jacrot for the temperature dependence of A.

However, the results of Jacrot do not allow a detailed comparison

with the theory. I think it would be very interesting to go also

to higher temperatures. Perhaps, it is necessary to work with

a higher precision for this purpose.

L. Passell: May I make a brief comment on that? When we

tried to do an energy analysis, we ran out of intensity very rapidly as we went up in temperature. With the facilities we had in Denmark, we could never hope to get much at temperatures

higher than 30 °K above the Curie temperature. We do not have a sufficient intensity for that purpose. On the other hand,

I believe that in Saclay you do have the possibility to cover a larger temperature range, because you have a cold neutron

source. We could not see any difference between the value of the parameter A at 2 "K above the Curie temperature and that at 18 °K above the Curie temperature. It may be that there is a small change of A with temperature, which begins to occur at higher temperatures. Would you care to comment on this, Dr. Jacrot?

B. Jacrot: I agree that it is extremely difficult to observe any

effect of the temperature. In the curve which we first published, we showed a temperature effect for A. However, that were

just crude data and after a careful analysis with the computer,

it turned out that A is practically the same at the Curie point and at 30 °K above the Curie point. This means that we have to carry out experiments with much higher accuracy. It is very doubtful that this is possible. In order to observe the variation

of A with temperature it is better to do the experiments on nickel where the effect is bigger.

P. Debye: In connection with the paper of Dr. Ford and Dr. Benedek, I want to remark that already 30 years ago the spectral

distribution of light scattered in liquids was photographed by

Ramm using an echelon [4]. I think this work should not be for- gotten in any historical review.

P. Debye gave also theoretical arguments as to why the line width of light scattered by a binary mixture in the vicinity of the critical point, as observed by Dr. Alpert and coworkers,

should

a. increase proportional to 5

2 = 4 sin 2 {012) with increasing

scattering angle 6; b. increase proportional to the temperature distance T—T,.

The substance of this discussion remark has been published in Phys. Rev Letters 14, 483 (1965).

W. Marshall: I wonder why the line width observed by Dr. Alpert does not go to zero when the temperature approaches

the critical temperature.

S. S. Alpert: There is a possibility that we were not exactly at the

critical point. W. Marshall: That is one suggestion which I was going to make. However. I want to bring up another suggestion. We are used

to the assumption that the scattering from a particular molecule can be taken as a constant. We associate with each molecule

a given scattering power which we allow to vary with angle. However, suppose that the scattering power of each molecule is slightly dependent on its neighboring molecules. Then the

scattering power would depend on short range order. You may

then get a term which is in fact not divergent. This might be something that would have to be taken into account in these very

delicate experiments. It is really a question; I do not know

whether it is sensible or whether the orders of magnitude are

right.

161
Commenting on the neutron scattering from /i-brass. I would

like to congratulate Dr. Dietrich and Dr. Als-Nielsen on two facts:

(1) They did some very beautiful experiments which other people have attempted and have failed to do.

(2) They were very good politicians because on the last day of

their experiments, they managed to destroy their crystal, which was probably the only good crystal of /3-brass in the world. We

may have to wait a long time before we get any better results.

I would like to make also some comments which are stimulated by Dr. Benedek's talk. He commented and in my talk I referred to the fact that as soon as the sound waves get damped

a diffusive type mode appears. Some of you may think that sound waves are not only damped by the thermal conductivity

but also by the viscosity, which, of course, is true. You may

then ask why the viscosity did not appear in the formula for the

line width of the central line. The reason for that is that the

viscosity and thermal conductivity appear quite differently in the hydrodynamic equations. Indeed there is another mode

which appears as soon as you introduce damping through vis cosity. but it is a mode which stays on the imaginary axis and

never comes down to the origin even at the critical point. Therefore, it does not produce any appreciable effects at the kind of

wave vectors we are talking about. If we indeed go to higher

/-values in these systems which seems possible by pushing the

laser technique further, and certainly when we use neutrons where we have higher /-values, it is very important to modify

the discussion of the fluid to include rigidity. At the higher fre- quencies the liquid is just as rigid as the solid. In EgelstafT

s

neutron experiments on liquids already you can see that you must

include rigidity to get the right results. Thus the rigidity of the

liquid will affect the shape of the curve for the higher /-values. Whether the laser will reach that region, has to be found out in the future.

References

fl] R. Nathans, M. Cooper and A. Arrott, Proc. Conference or Magnetism, San Francisco, 1965, to be published.

[2] P. G. de Gennes and J. Villain, J. Phys. Chem. Solids 13, 10

(I960).

[3] J. Villain. J. Phys. 26,405 (1965).

[4] W. Ramm, Physik Z. 35, 111, 756 (1943); E. H. L. Meyer and W. Ramm, Physik Z. 33, 270 (1933).

162
TRANSPORT AND RELAXATION PHENOMENA

Chairman: R. W. Zwanzig

209-493 0-66-12

163
[.L.
Behavior of Viscosity and Thermal Conductivity

of Fluids Near the Critical Point

J. V. Sengers

Institute for Basic Standards, National Bureau of Standards, Washington, D.C.

1. Introduction

This paper reviews the experimental information on the behavior of the viscosity and thermal conductivity of fluids near the critical point. Section

2 presents some general remarks on experimental

complications associated with the measurement

of these properties in the critical region. Experimental work on the viscosity of one component

fluids near the liquid-vapor critical point is reviewed

in section 3. The thermal conductivity of one component fluids is considered in section 4. The

behavior of the viscosity and thermal conductivity

in binary liquid mixtures near the critical mixing

point is discussed in sections 5 and 6, respectively.

Section 7 is a summary of conclusions based on

these experimental results.

2. Some General Remarks

In addition to the usual difficulties associated

with experiments in the critical region, measure- ments of the transport coefficients are hampered

by the fact that they are nonequilibrium quantities.

Most accurate measurements have been carried

out by introducing macroscopic gradients into the

system. In order to obtain the transport coeffi- cients the hydrodynamic equations must be solved

under certain imposed boundary conditions. Exact solutions of these equations are known only

for fluids with constant properties in idealized

geometrical situations. However, in the critical region a number of physical properties like local density and specific heat vary appreciably with

position as a result of the gradients. This complicates the analysis considerably. Close to the

critical point even the validity of the Navier-Stokes

equations [41] and of the linear laws has been

questioned [38, 42, 59]. If in a fluid near the liquid-vapor critical point

the temperature is not uniform, gravity can easily

generate convection currents and convective

heat flow which must be distinguished from the

irreversible fluxes associated with the thermodynamic forces by the linear laws. This is the most

serious difficulty encountered in measuring the

thermal conductivity and will be further discussed

in section 4. As is well known, equilibrium or a steady state is approached very slowly by a physical

system near a critical point. Consequently, some

authors have observed hysteresis effects in their experimental observations of the viscosity [2, 6, 50]. A prerequisite for the study of the behavior of

any physical property in the critical region is a knowledge of the values of the appropriate thermodynamic variables at which the property is measured. Near the liquid-vapor critical point the temperature and density and near the critical mixing

point of a binary liquid mixture the temperature and

concentration should be known. If a P-V-T re- lationship is known, one can calculate the density

from the measured pressure and temperature.

Close to the critical point, where the isotherms

become very flat, this procedure becomes inac- curate. This inaccuracy is greatly increased by

uncertainties in the absolute value of the temperatures at which either the transport coefficients or the isotherm data were measured. Uncertainties

in the temperature also hamper a comparison with

other properties near the critical point. Such a comparison for a binary liquid mixture is complicated by the fact that the critical parameters

are very sensitive to impurities.

If one measures the transport coefficients as a function of pressure and temperature, care should

be taken to work at small enough pressure and

temperature intervals as otherwise the whole critical region is easily overlooked. In experimental work

on transport coefficients this has led to considerable

confusion, which will be discussed further in section 4 in connection with thermal conductivity

measurements. It is equally important for the

study of any other property near the liquid-vapor

critical point.

Another procedure consists in following an iso- choric method where the temperature is varied

165
while the total amount of gas is kept constant. However, this method also has its pitfalls. First,

if one lowers the temperature at constant density,

phase transition occurs at a certain temperature

which is equal to the critical temperature at the

critical density. Below this transition temperature one measures either in the vapor or in the

liquid, depending on the experimental arrangement.

As a consequence the density, at which the properties under consideration actually are measured,

does change rapidly as a function of temperature,

although the density averaged over vapor and liquid

phase is kept constant. The temperature at which

this transition occurs should be determined ac- curately. Moreover, even above this temperature

a large density gradient is developed close to the

critical temperature so that the local density at a certain position in the vessel is still not independent

of the temperature. Ultimately, close to the liquidvapor critical point a knowledge of the local value of

the density is required.

The influence of critical phenomena is not re- stricted to the immediate vicinity of the critical point. For instance, an anomaly in the specific

heat can be detected at temperatures considerably

higher than the critical temperature. In this paper

the qualitative properties of viscosity and thermal

conductivity regarding the existence of similar

anomalies in the critical region will be discussed.

However, the precise mathematical character of

such anomalies can only be obtained from measure- ments very close to the critical point. In view of

the difficulties mentioned above an analysis of the

latter kind is not yet possible for the transport

coefficients.

3. The Viscosity of a One Component

Fluid Near the Liquid-Vapor Critical Point

Accurate measurements of the viscosity of compressed gases have been mainly carried out either

by a transpiration method or by the oscillating disk

method. In the first method the gas is caused to flow through a capillary and the viscosity is ob- tained from the flow rate. In the latter method the

damping of an oscillating disk due to the viscosity

of the surrounding medium is determined. Both

methods have been used to measure the viscosity

in the critical region. Close to the critical point

the oscillating disk method has the advantage that

it allows the measurement of the viscosity at a certain horizontal level in the fluid and consequently, in principle, at a certain local density. In the capillary flow method the density varies along

the capillary.

An example of the "normal" behavior of the vis- cosity outside the critical region is shown in figure

1; the viscosity coefficient of argon [4, 7] at three

temperatures, high relative to the critical temperature {tc =— 122.3 °C), as a function of density. One

amagat unit of density is the density at 0 °C and

1 atm. 1 At a given temperature the viscosity coefficient increases monotonically with the density;

(dr]/dp)r and (d 2r}jdp2)T are both positive. The

temperature derivative (d7]/dT) p

is positive up to a density well beyond the critical. At very high densities (dr)ldT)p becomes negative [3, 8], but this is not relevant for our present discussion. Sometimes the excess viscosity 17 — 170, where 170 is the

viscosity coefficient in the dilute state at the given

temperature, is considered to be only a function of the density. This is, however, not true in general.

Most measurements of the viscosity and thermal

conductivity in the critical region have been carried out for carbon dioxide because of its easily accessible critical temperature.2 The discussion will therefore be mainly based on the phenomena ob- served for CO^. The few investigations on other

substances sustain the conclusions obtained for CO2

The experimental investigation of the viscosity of CO2 in the supercritical region started with the work

of WARBURG and VON BABO in 1882 using a capillary flow method [15]. Even that early these

authors were aware that knowledge of the density is essential and were careful to determine the density

as well as pressure and temperature. They realized

that the pressure difference along the capillary

should be kept small and applied corrections for the variation of the density along the capillary. In an illuminating discussion of this experimental work

M. BRILLOUIN [1] remarked that these corrections are practically canceled by the variation of the

viscosity associated with the variation of the density. The data of Warburg and von Babo at 32.6 °C

1 One amagat unit of density for argon corresponds to 0.0017834 g/cm3

. 2 In this review the values given by Micheis et ai. [60], are used as critical parameters I ofCOz :

(, =31.04 °C. P, = 72.85 atm. pr = 0.468 g/cm 3 = 236 amagat. An alternative possible set of critical values was given by Schmidt and Traube [65]:[

(,. = 31.00 °C. />,. = 72.86 atm. pr = 0.468 g/cm 3 = 236 amagat.

One amagat unit of density for CO* corresponds to 0.0019764 g/cm 3

.

166
ARGON O 75°C

A 25°C

• -50°C

P- 5

300 400 500

DENSITY. AMAGAT

600 700

r IGLiRE 1. The viscosity of argon as a function of density at temperatures high relative to the critical.

C02 O 32.6°C WARBURG a VON BABO

tc

= 31.04 °C

fl = 236 AMAGAT

3-

100 200 300

DENSITY , AMAGAT 400

FIGURE 2. The viscosity of CO , at 32.6 °C according to Warburg

and von Babo.

are shown in figure 2. They also measured vis- cosity isotherms at 35.0 and 40 °C which have not been included in figure 2 since the temperature

effect turned out to be small. Since the viscosity

isotherms at 35 and 40 °C are nearly parallel to the one at 32.6 °C. the authors conclude that the vis- cosity does not show any anomaly at supercritical

temperatures down to 32.6 °C (about 1.6 °C above

the critical temperature). Although the absolute

accuracy of these old measurements may be limited,

the relative precision is sufficiently good to warrant

this conclusion.

After some inconclusive work on the subject by

PHILLIPS [11], CLARK [2], and SCHROER and

BECKER [13], an experimental investigation of the

viscosity near the critical point of CO2 was undertaken by NALDRETT and MAASS [10] using an

oscillating disk [6]. The critical region was ap- proached by the isochoric method. One viscosity

isotherm at 31.14 °C was also determined. The

authors conclude that the viscosity coefficient shows

a small anomalous increase close to the critical point. At the critical density this anomalous in- crease is about 1 percent at 32.0 °C and about 10

percent at 31.1 °C. The results confirm the conclusion of Warburg and von Babo that at 32.6 °C any

anomaly is smaller than 0.5 percent. The authors

did not consider the influence of the variation of the

density as a function of the height in the fluid. A

cell with a height of about 13 cm was used, the

oscillating disk being located in the lower end of

the vessel, and an estimate from the density versus

height relationship published in the literature [65,67]

shows that this effect could have caused an apparent

increase of the same order as the anomaly observed.

Therefore, the conclusion that the viscosity exhibits an anomalous increase near the critical point

is not justified.

A pronounced anomalous increase in a wider temperature range and with the viscosity increased by a

factor 2 at 31.1 °C was reported by MICHELS, BOTZEN, and SCHUURMAN [8]. Their data were

obtained with a capillary flow method which was known to be accurate outside the critical region.

As pointed out by the authors, the method is not

reliable in the critical region due to the large density

gradient in the capillary resulting from the use of a pressure difference of about 0.5 atm. Nevertheless, they conjectured that the anomalous

behavior observed was essentially real [8, 9]. In order to resolve the discrepancy between these

results, new measurements of the viscosity of CO2

were carried out by KESTIN, WHITELAW, and

ZIEN using an oscillating disk method [5]. They ap- proached the critical region along different iso- chores as did Naldrett and Maass. Their results show convincingly that any anomalous increase is much smaller than that reported by Michels et al. Close to the critical point they find an anomalous

increase of the same order of magnitude as that

167
reported by Naldrett and Maass. The data obtained by Kestin et al., at 31.1 °C and at 34.1 °C

are shown in figure 3a, b as a function of density.

For comparison the behavior of the viscosity iso- therm at 50 °C is indicated and data obtained by

Naldrett and Maass and a few data of Michels et al., outside the critical region are included in the figure.

Recently the viscosity of CO2 was also measured

by BARUA and ROSS with a capillary flow method

[12]. By using a horizontal capillary and extrapolating to zero flow rate [4], they were able to avoid the

difficulties inherent in the viscometer of Michels

et al. The analysis of their experimental data has

not yet been completed, yet it seems that their results are in agreement with those of Kestin et al.

8

2

E

co2 J

. t=3l.l°C

a NALDRETT, MAASS

° MICHELS. BOTZEN, SCHUURMAN

• KESTIN, WHITELAW. ZIEN /

t = 50 °C // //

, 1 !

(a)

4

2

1 1 1

C02

t - 34 l°C (° MICHELS • BOTZEN , SCHUURMAN - " V KESTIN, WHITELAW, ZIEN t=50°C /

J?

- —

1 1 .! 1

(b)

100 200 300 400

DENSITY. AMAGAT FIGURES 3a. b. The viscosity 0/ CO; iis (i function of density near the critical temperature.

Close to the critical temperature they find an in- crease of 1 to 2 percent on one isotherm, which

is of the order of the measuring accuracy in this

difficult region.

From the experimental work described, we can draw the following two conclusions.

1. Measurements of the viscosity of CO2 with

the oscillating disk method [5, 10] as well as meas- urements with the capillary flow method [12, 15]

have shown that the viscosity coefficient does not show any anomalous behavior at supercritical

temperatures down to a temperature (T—.TC )ITC = 1 percent from the critical.

2. The significance of the small anomalous in- crease at temperatures closer to the critical is not

clear at present. A correction should be subtracted from the data of Naldrett and Maass in this region which in magnitude might be comparable to the anomaly reported. Using the same

apparatus Mason and Maass did not find an anomalous increase of the viscosity of ethylene near the

critical point, although the data are somewhat

preliminary. It is interesting that the difference

between the data of Kestin et al., and the uncor- rected data of Naldrett and Maass is also of the same order as the anomaly. Kestin et al., have

discussed the possible influence on their measure- ments of the variation of the density with height.

Because their oscillating disk is located at a posi- tion somewhat higher than the center, the varia- tion of density leads to a correction in the opposite

direction from that to be applied to the data of Naldrett and Maass and therefore tends to enlarge

the anomalous effect. The authors point out that

their data become less reliable in this difficult region and they are taking the attitude that their data prove only the absence of the pronounced

anomaly suggested by Michels et al., and that any

anomaly is smaller than 20 percent. The reality

of the small anomaly can be decided definitely

only if the local value of the density is determined

simultaneously and the effect of a variable density on the interpretation of the viscosity measurements

is carefully analyzed.

This picture of the viscosity in the critical region is sustained by the work on other substances. STAR-,

LING, EAKIN, DOLAN, and ELLINGTON have

measured the viscosity of ethane, propane and n- butane with a capillary flow method [14]. Although

the authors did not analyze their measurements as aj

function of density, they have measured at suffi-l ciently small pressure intervals to conclude that

168
the viscosity does not show a pronounced anomaly

in the critical region. However, near the critical point neither the density at which the viscosity was determined nor the precision of the values for the viscosity coefficient is sufficiently well known

to decide whether a small increase, as observed

by Kestin et al., exists. Recently DILLER reported that the viscosity of

parahydrogen does not show any anomaly near the

critical point [3]. Measurements at temperatures

for which T— Tc is ten times smaller than in the case of CO2 are required, because the critical temperature of parahydrogen is only 32.976 °K. Consequently, only the measurements at one iso- therm (T— 33.000 °K) are relevant for our present

discussion. Indeed, the data on this isotherm do

not show any appreciable anomaly near the critical density. However, this conclusion is based only on a few experimental points for which calculations of the density from the P-V-T data are diffi- cult. An analysis of the viscosity data on adjacent

isotherms shows that the precision is only a few

percent. Therefore, no decisive conclusion can

be drawn from this work regarding the existence of a small anomaly.

4. The Thermal Conductivity of a One

Component Fluid Near the LiquidVapor Critical Point

Outside the critical region the behavior of the

thermal conductivity is very similar to that of the

viscosity described in the previous section. However, there exists a serious controversy in the

literature whether the thermal conductivity, contrary to the viscosity, exhibits a pronounced anomaly

in the critical region. An introductory survey on the subject has been presented by Ziebland [37]. The thermal conductivity is sometimes studied

either as a function of pressure at constant temperature or as a function of temperature at constant

pressure, without considering the dependence on

the density. The region around the critical density

is restricted to very narrow pressure and temperature intervals in this procedure, because both

(c)pj()p)T and (tipjtiT),, become infinite at the critical point. Measurements at the critical pressure are

not equivalent to measurements near the critical density unless the temperature is equal to the

critical temperature within a tenth of a degree.

For instance, considering the density dependence

along an isobar, in CO-2 at a temperature 0.15 °C

from the critical temperature the density is only

185 amagat at the critical pressure, whereas the

critical density is 236 amagat. Consequently

extrapolation of data at constant pressure as a function of temperature easily overlooks the critical region entirely and therefore does not reveal any

anomaly in the critical region. Along an isotherm

the density is similarly sensitive to a small change

in pressure. Another procedure sometimes used to obtain a thermal conductivity value associated with the

critical point is based on the assumption that the excess thermal conductivity \ — A. ( >, where Xo is the thermal conductivity at 1 atm at the temperature considered, is only dependent on the density

and not on the temperature. However, this rela- tion actually has been verified outside the critical region only and therefore cannot be used legitimately

in the critical region. "Critical thermal conductivity values," obtained in this way, are often used as reduction parameters for correlation of the thermal

conductivity of different substances [66]. It must

be kept in mind that these reduction parameters

do not necessarily represent the actual thermal conductivity near the critical point. Similar pro- cedures have been used to obtain critical values

of the viscosity. The reduction parameters for the viscosity obtained in this way are closer to the

actual viscosity in the critical region, because any

anomaly in the viscosity near the critical point is small. With few exceptions, values for the trans- port coefficients in the critical region which are based on data outside the critical region are not discussed in this review.

Two methods for the measurement of the ther- mal conductivity of compressed gases have had

wide acceptance, the concentric cylinder and the

parallel plate method. In the first the fluid is en- closed between two coaxial cylinders, either in horizontal or in vertical position, and the heat flow

through the gas layer is determined as a function

of the temperature difference between the cylinders. In the latter method the fluid is contained between

two horizontal plates and the heat flowing from the

upper plate to the lower plate is determined as a function of the temperature difference. A special case of the concentric cylinder method is the hot

wire method. This method is not suitable for the

critical region because the heat transfer analysis

is complicated, too large temperature differences

are usually involved and, most importantly, convection cannot be avoided.

169
Papers reporting an anomalous increase of the

thermal conductivity in the critical region are generally considered with much skepticism due to the high probability of convection. An exact theoretical analysis of convection is very complicated

and has been carried out only in very special cases. We do not go into the details here, but remark that the presence of convection can be detected by

varying the thickness of the gas layer or the temperature difference across this layer. The heat

transfer coefficient or so-called apparent thermal

conductivity coefficient, which includes a contribution from convection, depends on the geometry of the apparatus and of the boundary conditions

involved.

Heat transferred by convection is usually related

to the Rayleigh number R defined as

R = gap2 cpd

3ATIk7] (1)

where g is the gravitation constant, a the expansion

coefficient, p the density, cp the specific heat at constant pressure, d the thickness of the gas layer,

Ar the temperature difference, X the thermal conductivity and n the viscosity coefficient. In the

critical region the Rayleigh number becomes

extremely large [26, 30]. By the use of many

approximations and assuming that the convection

is small, the error introduced by convection can

be derived explicitly: ——X' — X d = C/? 7

sine (2)

where A.' is the apparent thermal conductivity coefficient as measured, / the length of the gas layer,

e the angle between the temperature gradient and

the vertical, and C a proportionality constant. In order to obtain (2) one neglects the variation with

temperature of the fluid properties other than the

density, linearizes the hydrodynamic equations, assumes a spatially uniform temperature gradient

and considers the limiting solution for y^lWithout these approximations the error may well

depend on d and AT in a more complicated way.

The absence of convection is verified, nevertheless,

if the measured thermal conductivity coefficient

at a given density and temperature does not change

under sufficient variation of AT or d.

If convection is present, one sometimes tries to obtain the correct value of X by extrapolating to

Ar=0 from the experimental data obtained for X' using various values of AT. In the critical region

the contribution of convection becomes very large

and such an extrapolation can diminish the accuracy

considerably even when care is taken that only

values for \' corresponding to the same average

density and average temperature are used. In some cases, such as a horizontal fluid layer heated

from below, convection only seems to start if the

Rayleigh number becomes larger than a certain

critical value [64]. This has also been reported

for a concentric cylindrical layer in horizontal

position [23, 58]. When this instability occurs,

extrapolation to AT=0 is not legitimate and can

lead to overcorrections.

The controversy over the existence of a thermal

conductivity anomaly started in 1934. KARDOS

attempted to measure the thermal conductivity

of CO> using a hot wire method [21]. He found a tremendous increase of heat transfer in the critical region, which was later recognized as an effect of

convection [28]. SELLSCHOPP measured the

thermal conductivity of CO2 with two concentric

cylinders in horizontal position [29]. He observed an increase in the apparent thermal conductivity near the critical density at supercritical temperatures, but his measurements were also influenced by

convection. The whole effect was attributed to convection and it was assumed, without experimental justification however, that in the critical region the thermal conductivity was only dependent

on the density and not on the temperature [18, 29]. TIMROT and OSKOLKOVA [33] reported that the thermal conductivity of CO2 did not show any

anomaly in the supercritical region, but their measurements obtained with a hot wire method

were actually carried out too far away from the

critical point.

COMINGS et al., measured the thermal conductivity of several gases using two concentric cylinders

in horizontal position. Anomalies in the supercritical region were reported for ethane, propane,

and butane [22] but the reality of the effect could

not be established again due to the possibility of convection. An anomalous increase of the thermal

conductivity of argon at 0 °C, originally reported

by the present author among others, later turned

out to be entirely caused by convection [24]. The first critical study of the existence of m

thermal conductivity anomaly was carried out by

GUILDNER [19]. He measured the thermal conductivity of CO2 using a vertical cylindrical gas

170
layer with a thickness of 0.68 mm. Convection

could not be avoided in this apparatus so that A. was determined from measurements at various

temperature differences by extrapolation to AT = 0 at constant average temperature. He con- cluded that the thermal conductivity itself exhibits a pronounced anomaly. This anomalous increase

is already 30 percent at 40 °C at the critical density.

He approached the critical temperature to within

1 °C. Although at 75 and 40 °C the influence of convection was small, closer to the critical temperature the influence of convection became large diminishing the accuracy of the data. On the other hand, VARGAFTIK stated that the

thermal conductivity did not exhibit an anomaly

in the critical region [34]. This assertion was based

on experimental wotk of SHINGAREV, which unfortunately, has not been made available and therefore

cannot be critically examined. According to the

discussion given by Vargaftik, these conclusions were drawn after applying corrections for convection by extrapolation to zero temperature difference.

It is clear that measurements in the critical region

free of convection are highly desirable. For this pur- pose a detailed study of the thermal conductivity of

C02 was carried out by MICHELS, SENGERS, and

VAN DER GULIK using a parallel plate method [25,

CO2

25.1 °C + 30.0°G © 31.2°C A 32.1°C ® 34.8°C O 40.1°C • 49.6°C X 75.I°C

26, 30]. Contrary to the concentric cylinder

method the horizontal flat layer method has the

advantage that the angle between temperature

gradient and the vertical can be made very small to suppress convection. The experimental results obtained for the thermal conductivity as a function

of density are presented in figure 4. The thermal

conductivity shows indeed a pronounced anomaly

at the critical point. An anomalous increase can

be detected as much as 20 °C above the critical temperature which becomes 250 percent at 1 °C

above the critical temperature. Convection was

avoided by the use of a small distance between

the plates (d=0A mm) and small temperature

differences (AT varied from 0.03 to 0.25 °C). The

absence of convection was verified experimentally;

in the whole critical region the thermal conductivity values measured were independent of the

temperature difference used. Of course, one

should compare measurements with different AT,

but at the same average density and temperature.

For the details we refer to the previous publication. The reality of the anomaly was independently

confirmed by a few measurements with a larger

plate distance. An attempt was made to include

a measurement of the thermal conductivity iso- therm at 31.2 °C using even smaller temperature

differences. These results are shown in figure 5. Obviously the values at this temperature are less

cmsec°C

7

400 Am

• 3I.200°C O 31 .217 °c C 31 .27 °c A 32. I °C

FIGURE 4. The thermal conductivity of CO> as a function of

density. Figure 5. The thermal conductivity of CO > at 31.2 and 32.1 °C

as a function of density.

171
accurate than those of figure 4, since the experimental verification of the absence of convection

becomes more difficult due to the inaccuracy in the density.

Due to the variation of temperature, the density

variation in the pressure vessel is very complicated. It is therefore difficult to approach the

critical region by the isochoric method. The

density must be calculated from the experimental

pressure and temperature, which may introduce

systematic errors at each isotherm. Thus the

detailed behavior along the critical isochore. particularly the character of the divergence of the

thermal conductivity, cannot accurately be de- duced from the data.

It is notable that at 75 and 40 °C. where the

amount of convection in the apparatus of Guildner was reasonably small, there is a satisfactory agree- ment with the data of Guildner. confirming the

anomaly of 30 percent at 40 °C. It turned out, however, that closer to the critical temperature

the anomalous increase is considerably smaller

than that reported by Guildner.

The fact that the thermal conductivity measured was independent of the temperature difference

used not only guarantees the absence of convection, but also verifies the validity of the law of Fourier in the temperature range considered. The

validity of the law of Fourier in the critical region was questioned by SIMON and ECKERT [32], From

interferometric studies of laminar free convection

near a vertical wall a quantity k was derived, which

they identified with the thermal conductivity coefficient of the fluid. This quantity k turned out

to increase with increase of heat rate and it was

conjectured that the anomaly in the thermal conductivity would disappear in the limit of zero heat

rate. However, an analysis of their data shows

that this quantity k remained dependent on the

heat rate outside the critical region as well, where

the validity of the law of Fourier cannot reasonably

be questioned. Therefore, this quantity k should

not be identified with the thermal conductivity

coefficient. The existence of an anomaly in the thermal conductivity near the critical point was denied by

Abas-Zade [16| and by Amirkhanov and Adamov

[17]. These authors do not base this assertion on measurements in the supercritical region, but on

data obtained at temperatures below the critical which they claim are associated with the coexist- ence fine. ABAS-ZADE performed his measure172

ments using a hot wire apparatus at various temperatures, but at constant pressure. As mentioned

before data at constant pressure as a function of

temperature do not approach smoothly the critical density. Consequently, the measurements were

not carried out near the critical density. The in- terpretation of Abas-Zade that he followed the

coexistence line up to the critical temperature

is erroneous. Very careful thermal conductivity measurements

were carried out by AMIRKHANOV and ADAMOV

using both a horizontal parallel plate method and a

vertical concentric cylinder method [17], Using the

parallel plate apparatus, a series of measurements was carried out with plate distances of 0.3 mm and

of 0.6 mm, and the temperature difference was

varied from 0.07 to 1 °C. With the smaller plate

distance the data turned out to be independent of

AT", which guarantees absence of convection.

Data obtained with the larger plate distance and

also those obtained using the concentric cylinder

apparatus were dependent on Ar showing the

presence of convection. The results obtained with

the smallest value of AT agreed for all 3 series of measurements. The values obtained for the

thermal conductivity therefore seem to be very

reliable. As has been pointed out repeatedly

before, it is equally important to verify whether

the measurements were carried out sufficiently

close to the critical density. Unfortunately, the

authors do not describe their experimental pro- cedure. They do not report whether and how either

pressure or density was determined. In addition,

although they measure the temperature difference

AT carefully, no device was presented for measuring the absolute average temperature of the gas

layer, which obviously differs from the temperature

of the surrounding thermostat. It is not clear at which temperatures and densities the thermal conductivity actually was determined. If the measure- ments were carried out at the coexistence line, one would expect a large variation of the density when

the temperature difference was varied from 0.1

to 1.0 °C, at the liquid side even leading to evaporation of the liquid. On the contrary, their results indicate that even up to a few hundredths of a degree below the critical temperature the density

did not change when AT was varied from 0.1 to

1 °C. Thus it seems that the measurements were

not carried out near the critical density.

From the data given in figure 4 it is evident that the first indication of the existence of the thermal
conductivity anomaly can be noticed from the occurrence of intersections of the thermal conductivity isotherms as a function of density at temperatures considerably higher than the critical temperature. It is interesting therefore to study

as a function of density data given in the literature somewhat remote from the critical point. As an example, we considered the experimental data obtained by ZIEBLAND and BURTON [35] for the X

of nitrogen. These data were originally plotted

along isobars and consequently did not reveal any

anomaly. However, if the data are plotted as a function of density the thermal conductivity iso- therms shown in figure 6 are obtained. 3 A pro- nounced increase in the thermal conductivity can

be noticed at — 140 °C (7 °C above the critical temperature). This observation is based on only two

experimental points, but the crucial experimental

value at —140 °C, showing an anomalous increase

of 35 percent, was obtained with two different values

of A7\ which excludes the possibility of convection.

This is a clear illustration of how one can be misled

if the density dependence of X is not considered

explicitly.

3 One amagat unit of N2 corresponds to 0.0012502 g/cm3

.

NITROGEN

Ol I I ! I

I O IOO 200 300 400 500

density, amagat

Figure 6. The thermal conductivity of nitrogen at — 125 and — 140 °C as a function of density.

A similar result is obtained [31] if one analyzes

the thermal conductivity data obtained by IKENBERRY and RICE for argon [20]. However, since

these authors did not adequately examine the ab- sence of convection, we cannot rely on their results as decisive. In view of these results NEEDHAM and ZIEBLAND studied the thermal conductivity of ammonia

in more detail [27, 37] than before [36]. The data were obtained using an apparatus consisting of two concentric cylinders with a gap of 0.2 mm.

The results obtained at two supercritical temperatures are shown as a function of density 4 in figure 7. An anomalous increase can be noticed near the

critical density even at 157.1 °C. The absence

of convection was verified by measurements with

different values of A7\ At 138.8 °C the thermal

conductivity goes through a maximum. The densities were estimated by Needham and Ziebland

from P-V—T data published in the literature. However, at 138.8 °C these P-V—T data are vir- tually unknown, so that the densities cannot be

calculated with sufficient accuracy to conclude

that the maximum does not occur at the critical density.

We conclude that the existence of a pronounced

anomaly in the thermal conductivity has been

verified by convection free measurements obtained

both with a parallel plate method and with a concentric cylinder method.

4 One amagat unit of NH3 corresponds to 0.0007714 g/cm3

.

DENSITY . AMAGAT

FIGURE 7. The thermal conductivity of ammonia as a function of density.

173
Table 1. Comparison of the thermal conductivity with the specific heat along the critical isochore of CO2 (pc — 236 amagat)

t A/X75

k/cv k/Cp

(klCi^Ti V* v / ^ V/ 4 O (klCr^T^ V / ^y/t 0

°c

75 1.0 1.0 1.0

50 1.1 1.0 0.5

40 1.3 1.0 .3 34 8 1 6 1

J. • 1X I 32.1 2.5 1.3 m

31.2 6.1 1.6 .02

An interesting question is whether the anomaly

in the thermal conductivity is related to that in the specific heat. For this reason the variation of X of CO) as a function of temperature at the critical isochore is compared with the variation of c, and

cp

in table 1. For c,- the experimental data [62] are supplemented with values calculated from the

P-V-T data at temperatures where no experimental

data for cv are available [61]. The values for cv are obtained by adding the calculated difference

cp — cr to the values for cv obtained as mentioned

above. It is evident that c,, increases much faster

in the critical region than A., so that the ratio A/c,,, decreases by two orders of magnitude when the

temperature is varied from 75 to 31.2 °C. On the

other hand the variation of k/cr is an order of magnitude smaller than that of X itself. Very close to the critical point cv as well as a is not accurately

known. A relationship between the anomaly in X and c, is well possible. In view of the recent work on the cv anomaly, it is natural to ask whether

A diverges logarithmically. Such questions about

the detailed behavior cannot be decided at present

and further experimental research is required.

According to the classical theory the line width of the central component of the spectrum of the

scattered light is determined by the thermal dif- fusivity X/pcp [57, 63]. The line width to be ex- pected for CO> from the static measurements of A

is presented in figure 8 for k—105 cm, which

174

100

-11= <

001

1 1 1 1 1

co 2 T

c

= 31 04 °C

pc

= 2 36 AMAGAT

k = 10 5 cm"1

-_IL!£——-

/^--^__50JC__^-^^^^? 25 °C

\\\34.8°C///'

V\32l°C/y

\/3l.2 0C

1 100 200 300 400 500 600

DENSITY, AMAGAT

FIGURE 8. Expected line width of the central component of

light scattered by CO > for k = 10s cmT1

.

corresponds to light from a He-Ne laser scattered i through an angle of 60°. Because cf) diverges much | faster than A, the thermal diffusivity [31] and conj sequently the line width of the central component

decreases rapidly in the critical region.
5. The Viscosity of a Binary Liquid

Mixture Near the Critical Mixing

Point

Many binary liquid mixtures exhibit a critical mixing point near room temperature. The vis- cosity of a number of different mixtures of this type has been studied. There are not enough

data obtained by different authors for the same

substance. Moreover, the critical parameters

are very sensitive to the purity of the components.

Although some attention has been given to the

influence of a third component on the viscosity

in the critical mixing region [43, 47, 50, 51, 53], the influence of any impurity on this anomaly is not clear. It seems that the effect depends on

the binary system as well as on the character of the impurity.

Most viscosity measurements for binary liquid

mixtures have been carried out with some version

of the capillary flow method. We have not made a

critical analysis of the experimental procedures

used by the various authors. However, there is sufficient agreement to establish the existence

of an anomaly in the viscosity near the critical mixing point for systems with either an upper

or lower critical solution point [39, 40. 43-53].

In the beginning of the century, an anomalous

increase of the viscosity near the critical mixing

point was reported by various authors, among others

by FRIEDLANDER [43] for the systems isobutyric

acid-water and phenol-water, by SCARPA [48] for phenol-water, by DRAPIER [40] for nitrobenzenehexane and cyclohexane-aniline, and by TSAKALOTOS [52] for isobutyric acid-water, trimethylethylene-aniline, triethylamine-water and nicotine- water. For simple binary liquid mixtures the anomalous

increase of the viscosity is of the order of 15 to 25 percent at the critical mixing point.

REED and TAYLOR observed an anomalous in- crease in the viscosity of the mixtures isooctane-perfluorocyclic oxide, n-hexane-perfluorocyclic oxide,

isooctane-perfluoroheptane [46]. The effect was

most pronounced in the mixture isooctane-perfluoroheptane; their results for this system are shown in figure 9. The viscosity exhibits an anomalous increase up to 25 percent. According

to these data the anomalous behavior can be de- tected as far away as 10 °C from the critical temperature. However, other authors found that for some systems the anomaly occurs only at temperatures much closer to the critical.

40 80 VOLUME % i -C8 H I8 AT 25 °C

FIGURE 9. The viscosity of the mixture isooctane-perfluoroheptane as a function of concentration according to Reed and

Taylor [46]. SEMENCHENKO and ZORINA investigated the

viscosity of the systems triethylamine-water and

nitrophenol-hexane [49]. The first mixture has

a lower critical mixing temperature. For both

systems an anomalous increase of the viscosity

up to 20 percent was found. The anomalous

phenomenon only occurred at temperatures within

1.0 or 1.5 °C from the critical and in a concentration

range of 10 mole percent.

Recent measurements of the viscosity of the

nitrobenzene-isooctane mixture obtained by PINGS

et al., confirmed that for this system also a viscosity

anomaly occurs in a very small temperature range

[45]. No anomaly could be detected at 4.5 °C above

the critical temperature, but an anomalous in- crease of 5 to 7 percent was noticed at 0.45 °C

above the critical temperature.

We conclude that the viscosity does show an anomalous increase near the critical mixing point,

but for a number of simple mixtures the anomalous

behavior occurs only at temperatures within 2 °C

of the critical temperature.

6. Thermal Conductivity of a Binary

Liquid Mixture Near the Critical

Mixing Point

Experimental investigations of the thermal conductivity of binary liquid mixtures near the critical 175
mixing point are very scarce. This is surprising

since the experimental complications are somewhat

less difficult than those encountered in the meas- urement of the thermal conductivity near the

liquid-vapor critical point.

3.00 -

19.5 20.0 2 0.5 TEMPERATURE, °C

FIGURE 10. The thermal conductivity of the system nitrobenzene- n-hexane according to Gerts and Filippov [54].

(63.0 wi % GiR-.NO-.)

I C 6 H 5 N0 2 -n-C7H l6

5.9 — O

19.0 19.5 20.0 TEMPERATURE, °C

FIGURE 11. The thermal conductivity of the system nitrobenzene- n-heptane according to Gerts and Filippov [54]. (Curve 1: 62 wt <7, Cs H-,N02 : curve 2: 52.3 wt % C6H5 N02 .)

The most extensive study was carried out by

GERTS and FILIPPOV, who measured the thermal

conductivity of a number of binary liquids using a hot wire apparatus [54]. Measurements of the

thermal conductivity of the system nitrobenzene-rchexane were carried out with four different temperature differences: 0.01, 0.05, 0.15, and 0.3 °C and

found to be independent of this temperature dif- ference. In addition the systems nitrobenzene-rcheptane, methanol-n-hexane and triethylaminewater were investigated. The results obtained by

Gerts and Filippov for the nitrobenzene-/<-hexane

mixture are shown in figure 10 and those for the

nitrobenzene-rc-heptane system in figure 11. It is evident that, contrary to the behavior near the

liquid-vapor critical point, the thermal conductivity of these binary liquid mixtures does not show

any pronounced anomaly near the critical mixing

point. Similar results were obtained for the other

two systems mentioned. An anomaly even as small as that found for the viscosity of binary mixtures seems to be absent. The data were not obtained at enough different concentrations to study the behavior in detail and the results cannot

be plotted as a function of concentration as was

done for the viscosity in figure 9. The absence of a pronounced anomaly near the critical mixing

point seems to be established. This difference

between the behavior of the thermal conductivity near the liquid-vapor critical point and that near

the critical mixing point was noticed before by

SKRIPOV [56]. Additional experimental data are highly desirable to verify this conclusion and to study smaller details. On the other hand an anomalous increase of the

thermal conductivity was reported by OSIPOVA for the phenol-water system [55]. The data were obtained with a parallel plate method. The uncertainty in the temperature difference and the scatter of the experimental data suggest that these results should be considered with some reservations. No

experimental verification for the absence of con- vection was presented.

7. Summary

The experimental information on the viscosity

and thermal conductivity in the critical region can

be summarized as follows:

1. Measurements of the viscosity with the oscil- I lating disk method as well as with the capillary flow

176
method have shown that the viscosity does not show any anomaly near the liquid-vapor critical point if the temperature is not within (T—TC)TC = 1 percent of the critical temperature. Whether

an anomalous behavior of the viscosity occurs at temperatures closer to the critical point is not clear

at present. If such an anomaly does occur it is probably smaller than 20 percent.

2. Measurement with the parallel plate method

as well as with the concentric cylinder method

have shown that the thermal conductivity exhibits

a pronounced anomaly near the liquid-vapor critical point. Further experimental work is required to study the precise character of the anomaly and to investigate whether the anomaly of A. is related to the anomaly of the specific heat cv .

3. The viscosity of binary liquid mixtures shows

an anomalous increase near the critical mixing

point. For simple systems this increase is between

15 percent and 25 percent at the critical point.

For a number of liquid mixtures, however, this phenomenon is only observed at temperatures

i within 2 °C of the critical temperature.

4. It seems evident that the thermal conductivity of a number of binary liquid systems does

not show an appreciable anomaly near the critical

: mixing point. Additional experimental information

is highly desirable, since this assertion can only

be based on the results of one experimental

investigation.

The author is indebted to Dr. H. Ziebland who

presented the thermal conductivity data for ammonia shown in figure 7. The author also ac- knowledges the stimulating interest of several

colleagues at the National Bureau of Standards

and in particular that of Dr. M. S. Green.

r References

Viscosity of One-Component Fluids

[1] M. Brillouin, Legons sur la viscosite des liquides et des

gaz. I., pp. 184-195 (Gauthier-Villars, Paris, 1907).

[2] A. L. Clark, Trans. Roy. Soc. Canada III 9 (1915) 43; 18

(1924) 329; Chem. Revs. 23 (1938) 1. [3] D. E. Diller, J. Chem. Phys. 42 (1965), 2089.

[4] G. P. Flynn, R. V. Hanks, N. A. Lemaire and J. Ross, J. A Chem. Phys. 38(1963), 154.

[5] J. Kestin, J. H. Whitelaw and T. F. Zien, Physica 30 (1964),

(l 161.

[6] S. G. Mason and O. Maass, Can. J. Research 18B (1940), 128.

I- [7] A. Michels, A. Botzen and W. Schuurman, Physica 20

(1954), 1141.

[8] A. Michels, A. Botzen and W. Schuurman, Physica 23

(1957) . 95.

[9] A. Michels, Proc. Intern. Symposium Transport Processes

in Stat. Mech., Brussels, 1956, p. 365.

[10] S. N. Naldrett and O. Maass, Can. J. Research 18B (1940). 322.

[11] P. Phillips, Proc. Roy. Soc. (London) A87 (1912), 48.

[12] J. Ross, private communication.

[13] E. Schroer and G. Becker, Z. physik. Chem. A173 (1935), 178.

[14] K. E. Starling, B. E. Eakin, J. P. Dolan and R. T. Ellington, Progress Intern. Research Thermodynamic Transport

Properties, A.S.M.E., Princeton, 1962, p. 530. K. E. Starling, Critical region viscosity behavior of ethane, propane and ^-butane, Master thesis, Institute of Gas

Technology, Chicago, 1960.

[15] E. Warburg and L. von Babo, Ann. Phys. Chem. 253 (1882), 390. Thermal Conductivity of

One-Component Fluids

[16] A. K. Abas-Zade. Doklady Akad. Nauk SSSR 68 (1949) 665; 99 (1954) 227. A. K. Abas-Zade and A. M. Amuraslanov. Zh. Fiz. Khim. 31 (1957) 1459.

[17] Kh. I. Amirkhanov and A. P. Adamov. Teploenergetika

10, No. 7 (1963) 77: Primenenie Ultraakustiki k Issle- dovan Veschestva 18 (1963) 65. Kh. I. Amirkhanov.

A. P. Adamov and L. N. Levina. Teplo i Massoperenos. Pervoe Vses. Svesch.. Minsk 1 (1961) 105.

[18] E. Borovik. Zh. eksp. teor. Fiz. U.S.S.R. 19 (1949) 561.

[19] L. A. Guildner. Proc. Nat. Acad. Sci.. 44 (1958) 1149: J. Res. NBS 66A (Plus, and Chem.) No. 4. (1962) 333. 341.

[20] L. D. Ikenberry and S. A. Rice. J. Chem. Phys. 39 (1963)

1561.

[21] A. Kardos. Z. gesamte Kalte-Ind. 41 (1934) 1. 29: Forsch. Gebiete Ingenieurwesens 5 (1934) 14. [22] F. R. Kramer and E. W. Comings, J. Chem. Eng. Data 5

(1960) 462. D. E. Leng and E. W. Comings, Ind. En- Chem. 49 (1957 ) 2042. J. M. Lenoir. W. A. Junk and

E. W. Comings. Chem. Eng. Progress 49 (1953) 539.

[23] J. M. Lenoir and E. W. Comings. Chem. Eng. Progress 47

(1951) 223.

[24| A. Michels. A. Botzen. A. S. Friedman and J. V. Sengers.

Physica 22 (1956) 121. A. Michels, J. V. Sengers and L. J. M. van de Klundert, Physica 29 (1963) 149.

[25] A. Michels. J. V. Sengers and P. S. van der Gulik. Physica

28 (1962) 1201. 1216.

[26] A. Michels and J. V. Sengers. Physica 28 (1962) 1238.

[27] D. P. Needham and H. Ziebland, Int. J. Heat Mass Transfer 8 (1965) 1387.

[28] R. Plank and O. Walger, Forsch. Gebiete Ingenieurwesens 5 (1934) 289.

[29] W. Sellschopp, Forsch. Gebiete Ingenieurwesens 5

(1934) 162.

[30] J. V. Sengers, Thermal conductivity measurements at elevated gas densities including the critical region, thesis, van der Waals Laboratory, Amsterdam, 1962.

J. V. Sengers and A. Michels, Progress Intern. Research Thermodynamic and Transport Properties. A.S.M.E.,

Princeton, 1962, p. 434.

[31] J. V. Sengers, Int. J. Heat Mass Transfer 8 (1965) 1103.

[32] H. S. Simon and E. R. G. Eckert. Int. J. Heat Mass Transfer 6 (1963) 681.

[33] D. L. Timrot and V. G. Oskolkova, Izvestia V.T.I. No. 4

(1949).

[34] N. B. Vargaftik. Proc. joint Conference Thermodynamic

Transport Properties Fluids. London, 1957, p. 211.

[35] H. Ziebland and J. T. A. Burton, Brit. J. Appl. Phys. 9

(1958) 52.

[36] H. Ziebland and D. P. Needham, Progress Intern. Research Thermodynamic and Transport Properties, A.S.M.E.,

Princeton, 1962, p. 441.

[37] H. Ziebland, N.P.L. Thermal Conductivity Conference,

Teddington, England 1964.

177
Viscosity of Binary Liquid Mixtures

[38] W. Botch and M. Fixman, J. Chem. Phys. 36 (1962) 310.

[39] P. Debye, B. Chu and D. Woermann, J. Polymer Science Al (1963) 249.

[40] P. Drapier, Bull. Acad. Belg. CI. Sci. 1911, p. 621.

[41] M. Fixman, J. Chem. Phys. 36 (1962) 310.

[42] M. Fixman, Adv. Chem. Phys. 6 (1964) 175.

[43] J. Friedlander, Z. physik. Chem. 38 (1901) 385.

[44] W. Ostwald and H. Malss, KoU. Z. 63 (1933) 61.

[45] C. J. Pings, private communication.

[46] T. M. Reed III and T. E. Taylor, J. Phys. Chem. 63 (1959)

58.

[47] V. Rothmund, Z. physik. Chem. 63 (1908) 54.

[48] O. Scarpa, Nuovo Cimento (5) 6 (1903) 277; J. Chim. Phys. 2 (1904) 447.

[49] V. K. Semenchenko and E. L. Zorina, Doklady Akad. Nauk

SSSR 73 (1950) 331; 80 (1951) 903; Zh. Fiz. Khim. 26

(1952) 520.

[50] V. K. Semenchenko and E. L. Zorina, Doklady Akad. Nauk

SSSR 84 (1952) 1191.

[51] V. P. Solomko and S. M. Smiyun, Dopovidi Akad. Nauk

Ukrains'koi RSR 1961, 649.

[52] D. E. Tsakalatos, Bull. Soc. Chim. France [4] 5 (1909) 397;

Z. Physik. Chem. 68 (1910) 32.

[53] E. L. Zorina and V. K. Semenchenko, Zh. Fiz. Khim. 33

(1959) 523, 961. Thermal Conductivity of Binary

Liquid Mixtures

[54] L. G. Gerts and L. P. Filippov, Zh. Fiz. Khim. 30 (1956) 2424.

[55] V. A. Osipova, Doklady Akad. Nauk Azerbaidzhan S.S.R. 13 (1957) 3. [56] V. P. Skripov, Conference Critical Phenomena and Fluctuations in Solutions, Akad. Nauk SSSR, Moscow, 1960,

p. 117.

Miscellaneous

[57] N. C. Ford and G. B. Benedek, Conference Phenomena

Neighborhood Critical Points, N.B.S., Washington, D.C.,

1965.

[58] H. Kraussold, Forsch. Gebiete Ingenieurwesens 5 (1934)

186.

[519] I. R. Krichevskii, N. E. Khazanova and L. S. Lesnevskaya,

Inzh.-Fiz. Zh., Akad. Nauk SSR 5 (1962) 101.

[60] A. Michels, B. Blaisse and C. Michels, Proc. Roy. Soc. (London) A160 (1937) 358.

[61] A. Michels and S. R. de Groot, Appl. Scient. Research Al

(1948)94.

[62] A. Michels and J. C. Strijland, Physica 18 (1952) 613.

[63] R. D. Mountain, Rev. Mod. Phys. 38 (1966) 205.

[64] A. Pellew and R. V. Southwell, Proc. Roy. Soc. (London)

A176 (1940) 312.

[65] E. H. W. Schmidt and K. Traube, Progress Intern. Research Thermodynamic Transport Properties, A.S.M.E., Princeton, 1962, p. 193.

[66] L. I. Stiel and G. Thodos, Progress Intern. Research Thermodynamic Transport Properties, A.S.M.E., Princeton, 1962, p. 352.

[67] S. A. Ulybin and S. P. Malyshenko, Advances Thermophysical Properties at Extreme Temperatures and Pressures, A.S.M.E., Purdue Univ., 1965, p. 68.

Nuclear Magnetic Resonance Measurements Near the Critical Point

of Ethane

M. Bloom 1 2

Harvard University, Cambridge, Mass.

In this talk I shall discuss the application of

nuclear magnetic resonance (NMR) techniques to the measurement of two transport properties of

fluids near the critical point. The two transport

properties we shall discuss are the nuclear spinlattice relaxation time T\ and the sell-diffusion constant D of the fluid. Both of these observables

are easy to measure using the NMR pulse tech- nique of E. L. Hahn [1]. The Hahn pulse technique: A simple version of

the NMR pulse technique is illustrated schematically in figure 1. Suppose that a time inde1 Alfred P. Sloan Fellow.

2 Permanent address: Department of Physios, University of British Columbia,

Vancouver, British Columbia, Canada. On sabbatical leave 1964-65 while holding

a John Simon Guggenheim Memorial Fellowship.

0 T 2T

FlGURE 1. Drawing of an oscilloscope display amplitude of the detected r-f signal as a function of time for the two-pulse nuclear magnetic resonance experiment.

Following the pulses at times 0 and t indicated by shaded regions are induction

signals of maximum amplitude A(0) and A{t) respectively. The echo has a maximum

amplitude A{2t).

pendent magnetic field H = Ht> k + Gzk-GA (1)

178
is applied to the nuclear spin system such that the

gradient G is small so that

Gx, Gz < Ho (2)

over the entire volume of the sample, i and k

are unit vectors in the x- and z-directions respectively. The spin system is assumed to be in equilibrium before the first of the two r-f pulses applied

at times t = 0 and t = r respectively. Assume that the r-f pulses are sufficiently intense and that their

frequency is sufficiently close to the Larmor

frequency oio =yHo, where y is the nuclear gyro- magnetic ratio, that the lengths of the pulses can

be chosen such that the z-component of magnetiza77 tion is rotated by an angle - by each of the pulses.

Then, if the z-component of the nuclear magnetization relaxes towards its equilibrium value ex- ponentially with a time constant 7\, the maximum

amplitudes of the detected nuclear induction signals

,4(0) and A(t) following the pulses are related by [1]

tional degrees of freedom of the system so that one

can usually write

^Ra+Rb+Rc (5)

1 1 Ra is the relaxation rate due to intramolecular di- pole-dipole and/or electric quadrupolar interactions

[6]: i.e., those interactions which transform as F2m (fl,), where fl, is the orientation of a vector

fixed in the molecule i. RB is the relaxation rate due to dipole-dipole interactions between nuclear

spins on different molecules, i.e., those interactions

which vary as Y-> m(0.ij)l r

3

^, where ry =(ry, fly) is the vector joining a pair of spins on different molecules [7]. Rc is the relaxation rate due to the inter- action between the nuclear spins and the rotational

angular momentum J,- of the same molecule [8] (the spin-rotation interaction).

Of the three terms contributing to Tr1 in eq (5), a molecular theory is available only for Rb [7], which is expressed in terms of Fourier transforms

of the correlation functions

A(0)-A(t)

A(0)

exp

T Ti (3)

If the x and y components of the nuclear magnetization relax towards their equilibrium value of

zero exponentially with a common time constant

Ti and if the spatial motion of the spins in the sample

is adequately described by the solution to the dif- fusion equation, the maximum amplitude of the

"spin-echo" at time 2t is given by [2]

A(2t) a exp

2t_2

T2 3 y

2 GW (4)

Equation (4) correctly gives relative values of A(2t)

in terms of To, G, D and r for any given angles of

rotation by the r-f pulses [3]. For systems with

strong NMR signals, it is easy to measure T\ and

D to an accuracy of ± 5 percent and with care an

accuracy of ± 1 percent can be attained.

Note that the self-diffusion constant measured

here is the "spin self-diffusion constant." The

interpretation of this type of self-diffusion constant

has been given by Emery [4] and by Fukuda and

Kubo [5]. Interpretation of T\ measurements: For most

simple systems there are three main types of inter- actions which enable the nuclear spin system to exchange energy with the translational and rotaJ(q))= dvdr e

,WT g(r, r , t) —p p-3—dr

(6) where g(r, r', t) is a time dependent pair distri- bution function for the system; g{r, r', t) drdr' is the probability that the vector joining a pair of molecules is between r and r + rfr initially and

between r' and r' + dr' at a time r. For a system

of identical nuclear spins RB is expressed in terms

of J(o)o) and J{2u>o). Note that typical values of

O)o are ojo ^ 108 sec^ 1 which is an extremely low

frequency insofar as the dynamical motions of

fluids are concerned. The other important point

to notice about eq (6) is that J (to) gets its major

contribution from pairs of molecules which are very close together, because of the r~

3 dependence

of the dipole-dipole interaction.

Physically, one can say that RB gives a measure

of a specific type of inelastic scattering cross section corresponding to collisions between pairs of molecules in which the total nuclear Zeeman energy

is changed. The low frequency o>o corresponds to the fact that the energy exchanged between the

spin system and the translational degrees of free- dom is very small (< 10"7 eV).

There are no detailed molecular theories for Ra and Rc in dense fluids, except for liquid H2 [9], and this has inhibited the interpretation of the many

Ti measurements already performed on liquids

179

209-493 0-66— 13
in molecular systems. Ra and R( are related to low-frequency Fourier transforms of correlation

functions of Y-z„,{fli(t)) and J,-(f). Since the inter- actions responsible for molecular reorientation are the anisotropic intermolecular interactions, the

correlation functions of Y2m(Cli(t)) and J,-(f) are each

related to correlation functions of these (short

range) anisotropic interactions in a manner which

has been defined for one system [9], that of H2 .

It is very likely, in the light of the theory for H2 , that RA and Rc can be expressed in terms of Fourier

transforms of correlation functions similar to that of eq (6), for interactions of shorter range than r~

3

. The behavior of T\ near the critical point; From

the preceding discussion, we can see that the quantity of interest in the interpretation of T\ is g(r, r', t) at short range, i.e., for r, r' gE Ro, the range of the

miermolecular interactions. Defining j(r, r', w) by

;'(r, r, co)- e~i<u7g<r. r'T)dr (7)

the quantity of interest in the low-frequency limit

is y(r, r', 0), i.e., the area under the time dependent

pair distribution function for r, r' £; R0 . This problem is analogous to the calculation of the area under

the correlation function for relative orientations

of neighboring spins near the ferromagnetic Curie

temperature. The interpretation of T\ near the

critical point may be compared with the inelastic scattering of neutrons, which is dominated by the

long range correlations. Unlike the short range

parts of ,Mr, r', r) of interest here, the long range

parts can be studied explicitly near the critical point by the methods of macroscopic fluctuation

theory [10]. Experimental results for T\: The simplest type

of system in which to investigate Tx near the critical point is a system such as He;i in which Ra = R( =0.

However, because of the smallness of Rh near the

critical point, special precautions would have to be

taken to eliminate spin relaxation due to collisions with the walls of the container and such a study has

not yet been carried out. For all the systems

studied near the critical point thus far Ra + Rc ^> RbThe first study of 7\ near the critical point was

conducted on SF6 by Schwartz [11]. Since then

several substances have been studied including

benzene and some of its derivatives [12], HC1

and HBr [13], and ethane [14] (C 2 H6). The results obtained are qualitatively similar for all the cases

studied, but the study of ethane by Noble is the only one in which care was taken to ensure that the

sample was in thermodynamic equilibrium near

the critical point. He usually kept the sample at constant temperature for 24 hrs before making T\

measurements and then made a second set of meas- urements after waiting for an additional 24 hrs. The temperature gradient across the sample was

less than 0.004 °C.

As may be seen from figure 2, Noble's critically

loaded samples were in vertically mounted glass

tubes and he made measurements on two regions,

3"

A

6 mm

COPPER

CYLINDER

VAPOR COIL

LIQUID COIL

SAMPLE

HOLDER

FIGURE 2. Drawing of sample holder used by J. D. Noble show

ing location of r-f coils.

180
one in the lower half and the other in the upper

half of the tube so that none of his measurements

actually corresponds to the critical point. The

amplitude of the NMR signal for each of the coils

is proportional to the density of the sample within

the coil. The effect of gravity on the ratio of densities in the two coils PlIpv, corresponding to the ratio of the liquid to vapor densities below the critical temperature Tc, is important as may be seen from

figure 3. pdpv in the pure sample is compared with

those obtained by other techniques [15]. The

measurements in the impure sample will be used

later in interpreting the measurements of D made

in that sample.

It should be noted that for the critically loaded

samples, the meniscus separating the liquid and

solid phases below Tc is observed to remain in a

fixed position midway up the sample tube in the

region of temperature not too far below Tc- Thus,

in this temperature range, one can say that pt

+ pv 555 2pc independent of temperature, where

pc = 0.207 g/cm3 is the critical density. Therefore,

the measurement of PlIpv is sufficient to determine

the densities p/, and pv in the region of temperature

in which the meniscus is constant.

It was found that for the region T < Tc, the value

of Ti obtained for the liquid was^ always longer than

that obtained for the vapor in equilibrium with the

liquid. This indicates that the time required for molecular reorientation in the liquid is always

shorter than for the less dense gas. The results

are qualitatively similar to those obtained for other

systems [11, 12, 13] and have been obtained over

, i , i , , J , i , . . , i_

31 32 Tc 33 TEMPERATURE CO

Figure 3. The ratio pjpv of the average density of the ethane

within the lower coil to the average density within the upper

coil as measured by the relative amplitudes of the NMR signals

using each of the coils. Measurements are shown lor pure ethane and for ethane with 1 percent 02 added. The measurements for pure ethane are in good agreement with those of Palmer (13].

a wide range of temperatures [14]. Figure 4 shows

T\ for the sample in each of the two coils in the

vicinity of 7*c- Experimental results for the diffusion constant D:

Noble has also measured the diffusion constant in ethane with approximately 1 percent 02 impurity

added [14, 16] (see fig. 3) The O2 was added to reduce T\ and thereby enable more accurate results

to be obtained for D since the experiment could be

repeated more often. The measurements of D as

a function of temperature near Tc are shown in figure 5. The estimated maximum error in the

measurements is ±5 percent including systematic

10 t_—l—I—I—!—I——l-J—I—1—1—I—1—1——

I

0 10 20 30 40 50 60 TEMPERATURE CO

Figure 4. Ti as a function of temperature in a pure sample

of ethane which was critically loaded and mounted vertically

.

Values are shown for the region of temperature near Tc for nuclei in the upper and

lower parts of the sample tube respectively.

18

14

o

A A VAPOR

SELF -DIFFUSION IN ETHANE

A

a a

«

<p GAS

• • • LIQUID

•

• LIQUID COIL VAPOR COIL

10 20 30 40 50 60 TEMPERATURE PC)

FIGURE 5. Self-diffusion constant D as a function of temperature in ethane with approximately 1 percent 0> added as an

impurity.

The sample was critically loaded and mounted vertically. Values of D are shown

for nuclei in the upper and lower parts of the sample tube respectively.

181
5.2

4.8

4.4

| 4.0- Eu

•q 3.2- x

«. 2 2.8 c

-I

2.4

2.0

1.6

100 TEMPERATURE <°K)

150 200 250 300 350

L//> v

Experimental

o Dota taken from reference 6/

• present work

Figure 6. Plot of\n (Dp) versus In T.

The points below 0 °C are based on the smooth curve drawn through the diffusion constant data of reference 17.

errors, while the measurements of p/Jpv are accurate to ±2 percent.

Interpretation: Experiments performed at 32

and 40 °C in noncritically loaded samples showed

that D °c p" 1 in the range pc/2 < p < 2pc . Thus

Dp is a function of temperature only in this density

range far from the critical point. A plot of In (Dp) versus In T from just above the melting point at 89.9 to 333 °K approximately 30 °K above the criti- cal point, is shown in figure 6. The results of Gaven, Stockmayer, and Waugh [17] far from the

critical point are included. The resulting straight

line gives Dp T2 - 9±0A . This is an empirical

relationship which does not necessarily hold for other substances. Figure 7 gives a plot of Dp

versus T in the temperature region within a few

degrees of 7c. A comparison with the results of

figure 6, as given by the solid line of figure 7, shows

that Dp goes through a pronounced minimum for the

liquid near TV with its minimum value about 20

percent below the solid line. A similar but smaller

effect is observed for the vapor but, unlike the data

for the liquid, we cannot say definitely that the

vapor minimum is not due to a systematic error, which we estimate to be less than 5 percent. A

single measurement of D for a pure sample at 24 °C,

having an error of ± 10 percent agrees with D for the impure sample.

The interpretation which we propose for the data

1.15

1.10-

1.05-

i

o Tjj o VAPOR

o'- < fi • LIQUID

cji A NON -CRITICAL SAMPLES

/ ! SOLID LINE ^29=8.02xl0-12

j

I ( <?M V /

jTc

v SEC-CM "K"'

25 27 29 31 33 35 TEMPERATURE CK)

37 39

FIGURE 7. Plot of Dp versus temperature nearTc

. Measured values of p/./pr are also shown. The samples used were loaded to the

critical density. Measurements of D„ based on the slopes of \ID versus p plots at 32 and 40 °C using noncritically loaded samples are also shown. The dotted line at 31.9 °C labeled 7Y is an upper limit for the critical temperature since it is the lowest temperature at which p/./pr = I, within experimental error.

of figures 6 and 7 is that whereas, for systems not

too close to the region of the critical point Dp is a function of temperature only, the diffusion constant decreases anomalously near the critical point. In figures 6 and 7, TV for the impure sample

used has been nominally given as 31.9 °C as compared with 32.32 °K. This actually represents an upper limit for Tt since it is the temperature at which pijp\=\.0 within experimental error. Since we do not know the equation of state for the

impure sample, we cannot take into account the

influence of gravity in a reliable fashion, but an examination of figure 3 indicates that the true value of 7c (or region of 7V) is probably close to the

temperature at which Dp goes through its minimum value. As far as we know these are the first measure

ments of a self diffusion constant near the critical point. Dr. B. Jacrot has, however, drawn our

attention to some published measurements of the

diffusion constant of Iodine in CGv near the critical ! point [18], where it is found that the diffusion con-I

stant is practically zero. This is consistent with|

the anomalous decrease observed by Noble for[ Dp for ethane near 7V as shown by figures 6 and 7. Obviously, it would be desirable to make furtherl measurements of D near the critical point for samples having better geometry (flat, horizontal

samples) and a higher degree of purity, e.g., SFb

- would be a good system to study since the relaxa-i tion time T\ of pure SFh is reasonably short [ll].l

182
References

[1] E. L. Hahn, Phys. Rev. 80, 580 (1950).

[2] H. Y. Carr and E. M. Purcell. Phys. Rev. 94, 630 (1954).

[3] B. H. Muller and M. Bloom, Can. J. Phys. 38, 1318 (1960).

[4] V. J. Emery, Phys. Rev. 133, A661 (1964).

[5] E. Fukuda and R. Kubo. Prog. Theor. Phys. 29, 621 (1963).

[6] A. Abragam. The Principles of Nuclear Magnetism, pp. 298-

300. 313-315 (Oxford University Press, 1961).

[7] H. C. Torrey, Phys. Rev. 92, 962 (1953): M. Eisenstadt and A. G. Redfield, Phys. Rev. 132, 635 (1963): reference

6, pp. 300-305, 458-466: I. Oppenheim and M. Bloom,

Can. J. Phys. 39, 845 (1961).

[8] P. S. Hubbard. Phys. Rev. 131, 1155 (1963).

[9] T. Moriya and K. Motizuki, Prog. Theor. Phys. 18, 183

(1957): T. Moriya, Prog. Theor. Phys. 18, 567 (1957): M. Bloom and I. Oppenheim. Can. J. Phys. 41. 1580

(1963).

1 10] L. van Hove. Phys. Rev. 95, 249 (1954).

(11] J. Schwartz, Ph. D. Thesis, Harvard University, (1958), unpublished.

[12] D. K. Green and J. G. Powles, Proc. Phys. Soc. 85, 87

(1965).

[13] J. S. Blicharski and K. Krynicki, Acta Phys. Pol. 22, 409

(1962); K. Krynicki and J. G. Powles, Phys. Letters 4, 260 (1963).

[14] J. D. Noble, Ph. D. Thesis, University of British Columbia

(1965).

[15] H. B. Palmer, J. Chem. Phys. 22, 625 (1954).

[16] J. D. Noble and M. Bloom, Phys. Rev. Letters 14, 250

(1965).

[17] .1. V. Gaven. W. H. Stockmayer and J. S. Waugh, .1. Chem.

Phys. 37, 1188, (1962).

[18] I. R. Krichevskii, N. E. Khazanova and L. R. Linshitz, Dnklady Acad. Nauk. SSSR 141, 397 (1961) (in Russian).

Ultrasonic Investigation of Fluid System in the

Neighborhood of Critical Points

D. Sette

Instituto di Fisica, Facolta' Ingegneria, University of Rome, Rome, Italy

1. Introduction

It is the intention of this paper to examine the

research that has been so far carried out in fluid

critical systems by means of sound waves. Since

1940 several authors have attempted to study the

propagation of sound waves in order to find infor- mation on the nature and the behavior of these

particular media: various indications and suggestions have been obtained, which have contributed to a better definition of the problems

concerning critical systems. The solution of these

problems is however far from complete.

The quantities which are usually measured are

the velocity and the sound absorption coefficient: they may be determined as functions of temperature, frequency and, in the case of mixtures, of composition. It must be observed that the infor- mation available at present for the systems in which

sound measurement have been performed, is only

partial because it usually refers to only one of these

quantities (i.e., velocity or absorption coefficient)

determined in a limited range of one of the vari- ables, temperature, frequency, and composition.

Strong similarities exist between liquid-vapor

and liquid-liquid critical systems as well as some

dissimilarities. It is preferable for a clearer ex- position to review separately the work made on

each ol the two types of systems, though this

procedure will require some repetition. We

consider first the case of liquid-vapor critical media.

2. Liquid-Vapor Critical Systems

Table I, although it is not complete,1 indicates

the substances which have been the object of the more extensive experiments. Together with pure

liquids, few binary or ternary mixtures have been

investigated in the hquid-vapor critical region.

The experiments have been performed either at a

single frequency or in a rather limited frequency

range. Moreover velocity and absorption coeffi- cient determinations have usually not been made on

the same system.

Figure 1 gives the results of Tielsch and Tanneberger [5] for the sound velocity in gaseous CO2

as a function of pressure at constant temperature.

In the examined frequency range (0.4-1.2 Mc/s)

and in the limits of error (0.2%) no dispersion has

been found. The same conclusion was reached by

Parbrook and Richardson [3] in the range 0.5-2

Mc/s. The velocity goes through a minimum which

becomes sharper and sharper as the temperature

approaches the critical one: the minimum velocity

1 In particular much research of Russian authors is included in thesis not easily available: some results are reported in quotation [9).

183
Table I. Sound Propagation in Liquids-Vapor Critical Systems

Substance

Helium

Xenon

Carbon dioxide

Ethylene

Hydrogen

chloride N2 0 Nitrous

oxide n-hexane

rc-heptane Methyl acetate

Ethyl acetate

Propyl acetate Methyl alcohol Ethyl alcohol Isobuthyl

alcohol Sulphur hexafluoride

Water

Benzene

methyl alcohol Benzene methylalcohol-toluenq

5.1994 °K

16.74 °C

31.4 °C

9.7 °C

51.4 °C

36.5 °C

234.8

266.8 °C

233.7 °C

250 °C

276.2 °C 240-240.5

243.1 283.4

45.6 °C

374 °C

239 °C for 10%

benzene 240-320 °C

1718 mm Hg

58.2 atm

•73 atm

50.9 atm

81.6 atm

71.7

29.5 atm

26.8 atm

46.3 atm

37.8 atm

32.9 atm

78.7 atm

63.1 atm

36 atm

217.5 atm.

Temperature

7V-0.03 p-40

7V + 0.04 °K pc — 35 mm Hg

15-19 °C

19-38 °C up to 100 atm

28-38 °C 5-98 atm

25-48 °C 35-120 atm

7-18.7 °C up to 100 atm

9.7-23 °C 35-75 atm

38-62 °C

52-62 °C

32-37 °C 55-150 atm

along the saturation curve

along the saturation curve

along the saturation curve

along the saturation curve

along the saturation curve

along the saturation curve

along the saturation curve

along the saturation curve

along the saturation curve

along the saturation curve

Frequency

1 Mcjs

0.25-1.25

0.5-2

0.270 0.410

0.5-2 0.27-0.6

0.27-0.6

1.9 1-9

0.96

2-3 2-3

2-3 5-13

2-3 5-9

2-3

2-3 2-3

2-3

0.6

Quantity measured

c, a

Author

Chase-Williamson-Tisza [1]. ChynowethSchneider [2]. Parbrook- Richardson [3]. Herget [4). Tielsch- Tannaberger [5|. Parbrook- Richardson [3j. Herget [4]. Breazeole [6]. Breazeole. Noury [7].

Nozdrev [8]. Nozdrev.

Nozdrev.

Kal'ianov [10]. Nozdrev [9]. Nozdrev [8]. NozdrevSobolev [11]. Nozdrev [8]. Nozdrev.

Nozdrev.

Nozdrev.

Schneider [12].

NozdrevOsadchii- Rubtsov [13]. NozdrevTaraitova [14]. Grechkin- Nozdrev [15].

184
70 80 90 PRESSURE (ATM)

FIGURE 1. Sound velocity in C0» versus pressure, at constant temperature

(H. Tielsh and H. Tanneberger [5]).

200

5 180

>-t 160 oo

_l

UJ>

140 o

zo

<" 120

.LIQUID PHASE

SUPERCRITICAL GAS

44 46 TEMPERATURE, °C

48

FIGURE 2. Sound velocity for the liquid and the vapor phase

of sulfur hexafluoride (W. G. Schneider [12]).

,0.4

o0.Z

0.0

LIQUID PHASE

VAPOUR PHASE

Si r, °°

42 44 46 TEMPERATURE, °C

50

Figure 3. Sound absorption for the liquid and the vapor

phase of sulfur hexafluoride (W. G. Schneider [12]).

measured at 31 °C is about one half of the value in normal conditions.

Figures 2 and 3 give Schneider's results (600

kc/s) on velocity and absorption in sulfur hexafluoride, a nonpolar compound with spherically symmetric molecules [12]: the data are in this case

available for the liquid phase as well as for the vapor on both sides of critical temperature {Tc ). 2 The

velocity in the liquid and in the gas seems, in the

limits of experimental errors, to reach a common

minimum at a temperature which coincides with

Tc ; the absorption coefficient in both phases in- creases very rapidly approaching Tc . 3 The last circumstance has been observed in all systems

examined and the various researchers have not succeeded in making a measurement at Tc . Figures 4 and 5 give the results of Chynoweth

and Schneider in xenon [2]. The absorption coefficient was measured at 250 kc/s and it shows a behavior similar to that observed in sulfur hexafluoride when T approaches Tc . The velocity in- stead was measured in the range 250 kc/s- 1.250

Mc/s: a dispersion has been detected whose magnitude is largely higher than the experimental error

(0.2-^-0.7%). We will discuss the inferences of

this result shortly. At present we call attention

to the fact that Chynoweth and Schneider have not noticed any tendency of velocity towards a discontinuity or a sudden drop to zero when T approaches

Tc - Until recently, this has been a general conclusion of experiments. Nozdrev [8], for instance,

who has made velocity measurements using light diffraction by sound waves, claims to have been

able to visually follow the continuous change of

sound velocity through the critical point.

The fact that the velocity of sound remains finite at the critical point is in agreement with standard

thermodynamics [16]. In this state, the derivative

of pressure respect to volume at constant tempera-

(

dP\

ture vanishes: | —I =0.

av Consequently, the specific heat at constant pressure {Cp) is infinite while

Cv , the adiabatic compressibility and the sound

velocity stay finite.

It may be of some interest, at this point, to mention some results that have been obtained by Nozdrev [8, 9] who has performed extensive velocity

2 The full and broken lines above Tc in figure 3 correspond lo measurements in the

supercritical gas performed respectively near the sound source and far from it.

3 The velocity and absorption curves for liquid and vapor as obtained by Schneider

cross at about 0.6 °C below Tc . This fact, which has no simple explanation, has not been found in any other case although some systems have been purposely investi- gated [8].

185
\

T EMPERATURE (°C)

Fi(;i'RE 4. Sound velocity versus temperature in xenon (A. G. Chynqweth and W. G. Schneider [2 J).

15 0 16.0 17.0 18.0 19,0 TEMPERATURE i'C)

FIGURE 5. Absorption per wavelength (a -A) versus temperature

in xenon (A. G. Chynoweth and W. G. Schneider [2]).

measurement in critical fluids along the saturation

curve. He has found, in the limits of error of his instrument, that the velocity variation along the

saturation curve in both vapor and liquid near the

critical temperature can fairly well be expressed

by using van der Waals equation of state in the case

of organic unassociated substances. This conclusion is not true for the liquid phase of associated

liquids (e.g., ethyl alcohol) which proves that

186

associations may persist up to the critical point.

Moreover, in the absence of dispersion and in the

limits of experimental errors, Nozdrev establishes an empirical simple relation between the arith- metical average of sound velocities in the gas

(c,) and in the liquid (c/,) and (T—Tr ) ^^= cCT +a(Tc -T) (1)

being ccr the common velocity at Tc and a a constant. This relation would be valid in the range between

Tc and a temperature 15 to 30 °C below.

The standard thermodynamical treatment of

critical state from which the conclusion on C,„

Cv , and c have been deduced, derives from some

assumptions on the behavior of thermodynamical

functions near the critical point: 4 in particular

it is assumed that the thermodynamical quantities

(as function of V and T) do not show mathematical

singularities along the curve which limits the region

in which the substance can not exist as a homogeneous phase either in stable or in metastable

equilibrium. Only in such case the abovementioned limit curve is determined by (tt~) =0 at \BvJT the critical point alone. The truth of this treatment has been seriously

questioned recently. Some experimental results on Cv in argon [17] and in oxygen [18] have been

obtained which indicate a logarithmic temperature

dependence of such a specific heat near the critical point. If this behavior remains unchanged up to the critical point, C,- and the adiabatic compressibility would be °° and c = 0 at T=TC [19]. To obtain further information, Chase, Williamson, and Tisza [1] have performed sound velocity

measurements in helium much more carefully

and precisely than used in previous investigation near the critical point. The temperature was reg- ulated to ± 10~4 °K (instead of ±2.10 3 °K in Chyno-j

weth and Schneider measurements in Xenon), \ the pressure to ±0.1 mm Hg; the velocity measure- ments were performed with a phase sensitive

method having a resolution of 0.01 percent (the error in the Chynoweth-Schneider setup was ±0.2 — 0.7%). Figure 6 gives the results obtained fori the velocity both as a function of temperature (atj a pressure />=1718±1 mm Hg practically coinci-l

' Quotation [16] p. 262.
-40 -30 -20 -10 0 10 20 30 40

P-Pc (mmHg)

FIGURE 6. Sound velocity as a function of temperature (a) and

pressure (b) in the critical region of helium (C. E. Chase. R. C. Williamson, and L. Tisza [1]).

dent with pc ) and as a function of pressure (at temperature 5.200 ±0.0002 °K practically coincident

with rc = 5.1994 °K). The velocity decreases rapidly approaching the critical point. High absorption

has not allowed measurements over a region of about 2.10~3 °K or 2.5 mm Hg.

The fast decrease of c approaching the critical point in the range of temperature or pressure

examined makes Chase, Williamson, and Tisza

think the limit zero for c at Tc to be possible. The

adiabatic compressibility Ks as a function at T or p

has been calculated making suitable approximations for density: in the range examined, Ks seems

to have a logarithmic singularity similar to that found for Cv in argon and oxygen.

This recent research, having questioned the

standard thermodynamical treatment of critical states, opens a new exciting field of investigation.

It must be observed, however, that the experimental

results so far obtained, although they have caused

serious doubts on the exactness of traditional ther- modynamical treatment have not yet definitely

proved it wrong. The measurement of c in helium

for instance, shows a fast decrease of velocity

when T approaches Tc , but at a separation of 10-3 °K

from Tc the velocity has still a value quite different

from zero. The possibility of a leveling off at a much lower but finite value has not yet been

excluded.

The behavior of velocity, C, and adiabatic compressibility at the critical point remains a very

interesting and exciting problem, which may clarity

in depth the nature of systems in the critical state:

it requires further experimental as well as the- oretical research.

Closely related to the nature of liquid-vapor

critical systems is also the enormous absorption

that these media show. Two mechanisms have

been proposed: 5 scattering and relaxation. A. G. Chynoweth and W. G. Schneider [2] have

studied sound propagation in xenon in the hope of

finding out which is the more important dissipative process. On the basis of an approximate calculation of scattering losses produced by inhomogeneities due to density fluctuations in the medium, they

conclude that these losses do not appear sufficiently

large to explain the experiment. They believe

therefore that, as Schneider had already suggested

in the case of sulfur hexafluoride, the more important dissipative mechanism is structural relaxation

caused by sound-induced perturbation of equilibrium among molecular aggregates (clusters) present

in the medium. The sound dispersion found in xenon (fig. 4) gives support to this explanation of

high absorption. The relaxation phenomena would

be characterized by a distribution of relaxation

times.

Different conclusions were reached by H. D. Parbrook and E. G. Richardson [3] studying CO>

and by M. A. Breazeale [6] in hydrogen chloride.

The last author was unable to detect dispersion in the range 1 to 3 Mc/s; the absorption results could

be interpreted as produced by relaxation processes

with a wide distribution of relaxation times, but it seems also possible to explain them on the basis

of a scattering mechanism if correlations between

density fluctuations in adjacent volumes are taken

into account. To evaluate the scattering losses,

Breazeale, as already Chynoweth and Schneider

had indicated, chooses the Libermann-Chernow

[20, 21] formula, which was developed to deal with

sound absorption caused by temperature inhomogeneities in the ocean. In the calculation of this formula the validity of an exponential autocorrela5 It seems that losses due to finite amplitude id* waves cannot have importance, as experiments have shown that the signal detected by a sound receiver depends exponentially on the source-receiver distance.

187
tion function is assumed. The absorption coefficient due to scattering by inhomogeneities of diameter a is

ots

: k

4 a

3

1 + 4a2 F

(2)

where k=~z~ and — is the fractional change of

k c sound velocity. In order to make a numerical

calculation a was chosen 10"4 cm, i.e., approximately 103 molecular diameters as suggested by

light scattering in various substances near the criti- cal point; moreover, the value 0.25 was used for Ac

Figure 7 gives Breazeale's experimental results

c

at various temperatures; the flags indicate the limits of errors. These results seem to depend upon the

square of the frequency (dotted lines). The solid

line on the same figure gives as calculated from (2) Ac

with above mentioned values of a and —: the correct

c order of magnitude of a is obtained; according to Breazeale the deviation of the experimental curves

52°C

4 6 8 10 FREQUENCY (Mc/s)

Figure 7. Absorption coefficient versus frequency at different temperature in the super critical region of hydrogen chloride (M. A. Breazeale [6]).

from the Libermann-Chernow curve, which essentially corresponds to a dependence of a from i>4

, could be explained by using an adequate correlation between density fluctuations in neighboring

volumes. Breazeale is therefore of the opinion

that, while relaxation may be important at lower

frequencies, scattering is most likely the main

source of absorption in the range 1 to 9 Mc/s.

3. Mixtures of Partially Soluble

Liquids

Table II indicates the few systems which have

been studied in some way in the critical region of

solubility. Almost all research has been limited

in each system to only one mixture with a composition close to that of the maximum (or minimum) of

the solubility curve. Figure 8 is relative to a mixture having 47.6 wt

percent of n-hexane in aniline [2]. The velocity was measured at 600 kc/s. The values of c in the two separate phases approach each other as T tends to Tc . On this basis, Chynoweth and

Schneider formulated the tentative conclusions

that the velocity in the three phases gradually

tends to the same value as T approaches Tc and

that no anomalous behavior is present. A dif- ferent indication, however, emerges from the work

of M. Cevolani and S. Petralia [23] in the aniline- cyclohexane mixture (49 wt % of cyclohexane)

(fig. 9). More research is. needed to determine the exact behavior of velocity in the three phases in the

immediate proximity of Tc ; the use of apparatus

today available with smaller errors than those so

far used in this kind of experiment should allow

precise information.

No great effort, so far, has been made to investi- gate dispersion in the critical region of solubility

and the few indications available [26] do not allow

to reach any conclusion on the subject.

A common experimental finding is the high at- tenuation of sound waves in the critical region.

Figure 10 [22] shows the typical behavior of the

absorption coefficient per wavelength (a k) as a function of temperature through the critical region

of solubility: it refers to the triethylamine-water

system (44. 6 wt % of amine) at 600 kc/s. Figure a . 11 [25] gives the parameter — lor mixtures nhexane-nitrobenzene as a function of composition;

188
TABLE II. Sound Propagation in Mixtures of Partially Soluble Liquids

System Composition Tc Quantity

measured

Temperature

range Frequencyrange Author

°C °C Mc/s

Aniline 47.6% w rc-hex 68.3 c, a 55-74 0.6 Chynowethrc-hexane 15-55

Schneider [22]. Aniline 51% w an. c, a 3 Cevolani- cyclohexane Petralia [23]. Do 20-63% mole an 30.7 c, a scattering 29-34 1.5-5 BrownRichardson [24]. Nitrobenzene 0-100 21.02 °C a 25 8 Sette [25]. n-hexane

53.2 w% nitrob. 23.2 23-28 1-9 AlfreySchneider [24]. Water- 44.6 w% am. 17.9 c, a 10-28 0. 6 Chynowethtriethyla- Schneider [22]. mine

44.6 w% am. a 15 7-54 Sette [25]. 34 w% am. 19.5 10-20 1-9 AlfreySchneider [24]. Water- 34 w% Ph. 66 c, a 51-80 3 Cevolani- phonol Petralia [23].

I —i 1 1 1 I I u I I L

56 58 60 6 2 64 66 68 70 72 74 TEMPERATURE CO

FIGURE 8. Sound velocity through the critical region of the aniline-hexane system

(A. G. Chynoweth and W. G. Schneider [2]).

189
FIGURE 9. Sound velocity through the critical region of the FIGURE 10. Sound absorption per wavelength (oc-k) versus aniline-cyclohexane system (M. Cevolani and S. Petralia [23]). temperature for the water-triethylamine system {A. G. Chynoweth and W. G. Schneider [22]).

aou

600

200

n-HEXANE - NITROBENZENE

absorption

ility

—

() T

0.2 0.4 0.6

Mol. fraction nitrobenzene

0.8 1.0

FIGURE 11. Sound absorption versus composition and coexistence curve for n hexanenitrobenzene mixtures (D. Sette [25]).

the experiment was performed at 25 °C, i.e., at a temperature 4 °C higher than the consolute temperature. The comparison of the results in figures

10 and 11 with those obtained in the liquid-vapor

critical region (figs. 3 and 5) shows that in the criti- cal solubility case the processes which give rise

to additional losses operate in a wider range of temperature around the critical one. This fact

indicates an essential difference between the

two types of systems.

Various attempts have been made to find out what the process responsible for the largest part of losses in these systems is. The dissipative causes which have been examined are viscosity,

relaxation, and scattering. Let us briefly review

the research made on this subject.

190
3.1. Viscosity and Thermal Conductivity

Different from the case of liquid-vapor critical media, the viscosity coefficient (17) of mixtures

formed by partially soluble liquids undergoes a large increase when temperature approaches the

critical one. However, the classical losses due to

this fact, i.e., calculated with Stokes' formula

for an homogeneous medium, would be still in- adequate to explain the experiment: the total viscosity absorption coefficient would amount to a few per mille of the observed value. The fluid however is not homogenous. R. Lucas

[27] has calculated the sound absorption coefficient when density fluctuations are present in the liquid.

Lucas assumes that the density fluctuations are time independent and isotropic and that the sound

velocity does not vary along the direction of propagation. Moreover he maintains Stokes' relation

between the two viscosity coefficients. The ab- sorption coefficient then results

a$ = Av2 + B (3)

where

3 poc

3 \ tJo /

y PoC T Jo \dxj

and po the average density; p the local density;

m = —; s, the condensation, given by p = po (1 + s); P

di—dxdydz the volume element of the liquid. A

and B depend upon the properties of the liquid

and the density fluctuations in it, but they are frequency-independent. As a consequence of the

presence of density fluctuations, the expression

of the absorption coefficient has two corrective

terms for Stokes' expression

8772 V

ar> = — ~ (4) 3 poC*

The first corrective term has the same frequency de- pendence of the Stokes' term, the second one is constant with frequency. The parameter — therefore

decreases with frequency. Such behavior is gen140

100

20

1 /»

iTE:r TRIETHYLAr^ INE

CHt NC WE rH -sCHNEIC ER

** —

P

1 1 1—I 1—U 1 I I l_l r* « I I I 0.1 I 10 100

v (Mc/s)

FIGURE 12. a/v2 versus frequency far a water-triethylamine

mixture at 15 °C (D. Sette [25]).

erally found in these kinds of mixtures; figure 12

[25] gives some results for a mixture water-triethylamine with 44.6 wt percent of amine. Unfortuca nately, however, a similar decrease of — with

v frequency can be expected also in the case of other

dissipative effects, mainly of the relaxation type.

A calculation using the Lucas' formula has been

carried out for the water triethylamine mixture

for which data on the thermodynamic behavior

of the system are available. Rough approximations were introduced; the results seem to indicate

however that, though the viscosity losses in the

inhomogenous medium are higher than those

given by Stokes' formula and may become of some

importance at low frequencies, they are still in- sufficient to explain the largest part of the observed

attenuation.

Another cause of losses is the existence of thermal

conductivity. The corresponding contribution to absorption coefficient in normal liquids is much

smaller than the viscosity term. Although thermal

conductivity determinations in critical region are missing in the literature, it seems unlikely that thermal conductivity substantially contributes to the large absorption found in the critical media.

3.2. Scattering

A rough description of the medium is obtained

by thinking of the system as formed by a mother

phase in which clusters of various sizes and compositions are dispersed. The presence of in- homogeneities may lead, as in the case of liquid191
vapor critical systems, to scattering of sound

energy. On the importance of this process the

opinions do not entirely agree. The dimensions of individual clusters as deter- mined by light scattering experiments are of the

order of wavelength of visible light (0, 5T0-6 m), i.e., much smaller than the sound wavelength (~ 10"3 m

at 1 Mc/s) in the frequency range of experiments:

if the clusters acted as individual scatters the contribution of scattering to sound energy loss would

be very small. It is however necessary to assume,

as in the case of liquid-vapor critical systems, that there actually are strong correlations between

fluctuations in adjacent volumes. If one considers

Liebermann's theory in which an exponential

autorrelation is assumed one finds a dependence

of a on frequency which is not much different from

proportionality to the fourth power of v, i.e.,

should increase with frequency. This is contrary to some experiments where this parameter

has been found decreasing within the range ex- amined: figure 12 refers to water-triethylamine

mixtures and similar results have been obtained

in nitrobenzene-n-hexane (5 to 95 Mc/s).

Liebermann's theory therefore does not seem

satisfactory for an explanation of the experimental

results, and it is at the present difficult to see whether by using an adequate correlation function,

it is possible to explain the entire increase of ab- sorption in the critical mixture by means of

scattering.

Some experiments however have definitely shown

the existence of scattered energy in a critical mixture when sound goes through it. Recently A. E. Brown and E. G. Richardson [26] have shown this effect in aniline-cyclohexane mixtures. These

authors have measured the energy received by a probe whose axis forms an angle 6 with the sound

beam radiated from the source. At 1.5 and 2.5 Mc/s a broadening (4^-5° at 2.5 Mc/s) of the beam

radiated has been observed when the source is in the critical mixture. At 5 Mc/s, while the intensity

falls to zero (first minimum) at an angle 6 of 1.5° if the medium does not contain inhomogeneities when

the radiation takes place in a critical mixture one

obtains patterns of relative intensities at various

angles 6 as those shown in figure 13. From the

position of maxima and minima the authors cal- culate dimensions of effective scatters, which are

of the order of magnitude of 100 to 400/i, (compared

with \ = 500|U,). Table III gives the experimental

192

i o

0-6

0.4

Q.2

08

0.6

0-4

02

22. S °/s ANILINE

X 28. 1° • 29.2°

o 30.0°

fl 3 I • 9°

33. 1°

43.8 °/c ANILINE

7 29.3°

a 30.S0 o 31.8°

• 33. 3°

X 34.7°

33.3 °/0 ANILINE

53-9 °/0 ANILINE

x 29-7°

• 30.9°

o 33.3°

fi 36-3°

FIGURE 13. Angular distribution of scattered sound at 5 Mc/s

in aniline-cyclohexane mixtures (A. E. Brown and E. G. Rich- ardson [26]).

TABLE m

/== 1 -5 Mc/s 2-5 Mc/s 5-OMc/b Temperature °c c a a. c a <*• c

22-5%

aniline 34 0 057 0 011 1262 0-145 0-11 1260

33 0 070 0 030 1262 0 15 011 1260 32 0 080 0 035 1262 015 011 1262

31 0 097 0-023 1262 0155 011 1264

30-5 0115 0 023 1262 017 012 1266 30 0140 0011 1262 0185 014 1268

29-25 0 105 0 06 1262 0-26 0145 1270

33-3%

aniline 34 0042 0006 1281 009 0-055 1282 0-23 0-20 1288

33 0 045 0 007 1284 0 105 0 055 1282 0-23 0-20 1291 32 0 057 0 011 1288 0130 0 055 1289 0-25 017 1288

31 0 060 002 1204 0160 0 055 1289 0-40 017 1294

30-5 0 057 0-007 1206 0 132 0 055 1296 0-25 0-20 1297

43-8%

aniline 34 0 050 0035 1312 0 080 0 063 1318 0-24 017 1310

33 0 057 0 035 1317 0 086 0 063 1319 0-27 0-20 1316

32 0 092 0-035 1323 0-130 0 070 1322 0-30 015 1322

31 0 083 0 04 1323 0160 0 046 1319 0-38 0 05 1333

30-5 0 077 0 052 1323 0140 0 046 1318 0-32 0 03 1342

53-9%

aniline 34 0 040 0 035 1355 0 070 0 052 1360 014 0 023 1366

33 0 040 0 035 1355 0 075 0 046 1360 016 0 023 1361 32 0 040 0 035 1356 0 087 0 043 1361 018 0 023 1360

31 0 055 0 030 1359 014 0 058 1371 0-23 0 036 1369

30-5 0 054 0 030 1360 0115 0 050 1375 0 23 0 029 136S 30 0-080 0 023 1360 0 092 0 058 1379 0-26 0 036 1376
absorption coefficient (a) and the coefficient that Brown and Richardson determine as due to scat- tering (as) for mixtures of the composition examined

and at the frequencies used. These results in- dicate a substantial contribution of scattering to the total losses: in this case, scattering appears to be the most important dissipative process. The

authors observe however that, in the limits of ex- perimental error, scattering does not seem satis- factory to explain the totality of losses experimentally found. Using the roughly approximate

procedure of subtracting the scattering absorption

coefficient from the experimental one, an unexplained part remains which has the same order of

magnitude as the absorption coefficient in pure

liquids. ( — = 50 X 10~ 17 sec2 cm"1 in aniline and

V

\ 200 10 17 in cyclohexane I.

3.3. Relaxation Processes

Various researchers have thought that relaxation

processes of various kinds may be involved in the

propagation of sound waves in critical mixtures and

could be responsible for the large absorption experimentally found.

The existence of clusters of various compositions

dispersed in a mother phase induces the presence

of a great number of equihbria which may be per- turbed by the temperature variations due to the

sound wave. The kinetics of the system is to be

described by reactions which take place either

among clusters of different kinds or between clusters and molecules of the mother phase to generate

clusters having a new composition. The situation

is similar to the one found in binary mixtures of

the water-alcohol type when a maximum of the

absorption coefficient versus composition is present:

in such case, most of the absorption increase seems

connected with equihbria which exist among various

nolecular associations of the two components, and

are altered by temperature changes produced by

ound waves. The relaxation processes which arise are of the ame kind as those found in liquids where a low

ate chemical equilibrium is present and can be

escribed by means of a specific heat function of

requency. Owing to the great number of equihbria

resent among clusters of various sizes and compoitions, a very broad distribution of relaxation times

s to be expected. This indication is in agreement

ith experimental findings: in the case of the

water-triethylamine mixture the experimental

results (fig. 12) when interpreted on the basis of the

relaxation theory, clearly show this fact; a very

rough estimation of the range of relaxation frequencies indicates that it extends at least from 2 to 30

Mc/s.

M. Fixman [28, 29, 30] has recently considered

in some detail the manner in which the characteristic fluctuations of critical mixtures may cause

a frequency dependence of one or more of the

thermodynamical quantities which are of impor- tance in sound propagation; he too has reached the

conclusion that the specific heat is the quantity

most hkely to be involved in the observed behavior.

In an early attempt [28], he considered a fluctuating

heat capacity associated with density fluctuations;

adjacent volumes of the liquid reach different temperatures when passed through by a sound wave: sound energy absorption would appear as a conse- quence of the heat flow in the equalization of temperature gradients. Such a process however can produce a much smaller absorption than observed

experimentally and therefore can not be responsible

for the high losses observed. The dissipative

effect just mentioned is described in the nonlinear

equation of motion and energy transport by a term

linear both in the temperature variation produced

by the sound wave and in the density or composition

fluctuations; in Fixman's early treatment other

terms which are either quadratic or of higher order

in local fluctuations, were neglected.

In a later study Fixman has found the way to handle the quadratic terms and he was able to show

that they are strongly coupled with local temperatures; these terms originate an anomalous static heat capacity in agreement with experiment [31]

and a frequency dependent heat capacity which

could be responsible for the sound absorption in- crease in the critical region. In this calculation,

the composition is expressed as a function of position by means of a Fourier series or integral, with

suppression of all Fourier components having a wave number (k) greater than some k max. It is then possible to obtain a fluctuating entropy density (85) associated with critical composition

fluctuations which is perturbed by the temperature

variation produced by sound and gives rise to an

excess heat capacity per unit volume

193
In the expression of (85) enters the radial distribution function g(r) at time t expressed by means of

the Fourier components up to kmax. The dynamics

of the long range part of g(r) is assumed to satisfy the diffusion equation

J=/! [K2 V

2g-V2V2^] (6)

being h a diffusion constant which may be related

to more usual parameters and k the reciprocal of an "indirect" correlation length. For a mixture

at the critical composition k can be expressed by means of a short-range correlation length /

7 _ j

Fixman's analysis led him to conclude that k

2 is strongly coupled with local temperature and that

this circumstance can be the reason for both the

increase of static heat capacity and the appearance

of a frequency dependence of specific heat. Therefore he substitutes k

2 with

«+($« (8)

and supposes 8T to be sinusoidal function of time.

Fixman obtains for the complex dynamic heat

capacity per unit volume an expression and fur- nishes a numerical integration of the real and

imaginary part close to the critical point.6 This expression can be used to calculate the

complex speed of sound from which velocity and

absorption are derived. The comparison of these

theoretical values with experiment requires the

knowledge of: (1) the correlation distance 1; (2) a

friction constant, /3, which enters in the expression

of the diffusion constant; (3) the specific heat per

mole in the absence of fluctuation C°p

; (4) the ratio C°

of specific heat y0 = —11 in the same conditions.

Cg

The results of the comparison between theory and

and experiment in the case of the mixture n-hexaneaniline (47.6 wt % of hexane) studied by Chynoweth

and Schneider are indicated in figure 14. C°

and yo were approximately evaluated by the Author

while /3 = 9.1013 sec

-1 and / = 4 A were assumed

as reasonable values. It appears that the numerical

values of the observed absorption as well as its

"The function has been tabulated by Kendig, Bigelow, Edmunds, and Pings [32].

T-Tc

FIGURE 14. Absorption per wavelength versus (T —

T

c ) for

aniline-n hexane critical

Solid line theoretical, dashed line experimental. IM. Fixman (29j).

variation with temperature could be explained by

Fixman's theory. The theory should be tested

for the variation of absorption coefficient with

frequency.

Fixman's calculation predicts a very small

velocity dispersion and this result could be in agreement with experiment because no dispersion

has been so far detected in the limits of errors. The variation of velocity with temperature indicated

by the theory (3% for a temperature increase of

5 °C above the critical point) is about twice the

change observed. Fixman's theory, which could

be extended to a liquid-vapor critical system supports the idea that relaxation processes may be an

important, if not the most important, dissipation

process in the critical systems.

4. Conclusions

At the present status of research, most of the

high absorption found in liquid-vapor and liquidliquid critical systems is most likely due to one or both of the two processes: (1) scattering by inhomogeneities due to density or composition fluctuations,

(2) relaxation in the cluster equilibria perturbed by

the temperature variations induced by sound waves.

It is not possible, at present, to go much further in specifying what the most important process may be

in particular cases.

194
It is to be mentioned at this point that recently

A. S. Sliwinski and A. E. Brown [33] have stressed

the importance of a strong difference existing between critical mixtures and liquid-vapor critical systems. In the first case, in fact, the equalization

of composition fluctuations must proceed through a

diffusion mechanism which is particularly slow

near the critical point; instead, the merging of two

or more density fluctuations to form a uniform

region in a liquid-vapor critical medium does not

require diffusion. Such a difference could play an important role when the sound wavelength is either smaller or of the same magnitude than the

linear dimension of assemblies: this is the normal

case when ultrasonic waves interact with critical systems. The sound wave produces a variation of

the pressure and temperature of assemblies around

values which are near the critical ones: in the case

of liquid-liquid system, as a consequence of the

slowness of diffusion in the short time in which the

interface between assemblies disappears, the pas- sage of sound does not strongly affect the form and

j size of assemblies; in the case of liquid-vapor the

I assemblies lose their identity when the surface

tension disappears and they may reappear sucjj cessively in a form quite different from the original

in

m

'S.

:d

of

le

Id

Pin

m

ie

dor

0-

s,

>y

s.

in

)t

configurations. 7 To support this point of view,

Sliwinski and Brown have considered some other

experiments which indicate the different ways of

interaction of sound waves with a liquid-liquid

critical mixture and a liquid-vapor critical system.

The striations which can be observed with a Schlieren setup in a medium at rest are not significantly changed when sound is applied in the

liquid-liquid case, while they are broken in liquidvapor systems as though a pulverization of assemblies had taken place. A similar effect is evident

when one follows the light diffraction patterns pro- duced by a sound wave in the medium with time

after sound application. In figure 15 the light dif- fraction pattern in an aniline-cyclohexane mixture

at 31 °C, in carbon dioxide at 3.5 °C and 72 atm,

and in ethylene at 9.6 °C and 52 atm are shown at V12 sec intervals from the instant when sound (1 Mc/s) is switched on (arrow). While the patterns

remain unchanged in the liquid-liquid critical mixture: they rapidly become blurred in the liquidvapor case.

7 The existence of substantia] differences between assemblies in the two types of

critical systems is also shown by the temperature dependences of a experimentally found (figs. 3. 5. and 101.

209-493 O-66— 14
It would seem therefore that sound has a tre- mendous effect on the structure of assemblies in the liquid vapor case while it does not practically

affect the configuration of clusters in critical mixtures. Sliwinski and Brown think that this could

be an indication of the fact that scattering is particularly important in critical mixtures while it must be less significant in liquid vapor media: in the last case structural relaxation effects would

be the main cause of absorption.

The survey of the work so far carried out in critical systems has shown how the problems

associated with the propagation of sound waves

in critical media have not yet been solved. At

present, they seem to be posed in clearer terms

than some years ago, but much theoretical and

experimental research is needed. The important

areas of research are essentially two: (a) the lowfrequency sound velocity at the critical point;

(b) the analysis of absorption and possibly, of dispersion of sound waves. The progress of experimental techniques should today allow more

precise determinations of sound velocity and of the absorption coefficient: moreover, it is important to extend the ranges of temperature, frequency

and composition in which the measurements are made. From the theoretical side, we expect

progress in the thermodynamical treatment of critical systems, in scattering theories adequately

considering correlation between fluctuation in adjacent volumes, in separating the contributions

to absorption due to relaxation and scattering proc- esses when they are of comparable importance.

The profit of these researches will surely be re- warding in establishing the characteristics and the

behavior of critical systems since sound propagation is so intimately connected with the particular nature of these media.

5. References

[1] C. E. Chase, R. C. Williamson, and L. Tisza, Phys. Rev.

Letters 13, 467 (1964).

[2] A. G. Chynoweth and W. G. Schneider, J. Chem. Phys. 20,

1777 (1952).

[3] H. D. Parbrook and E. G. Richardson, Proc. Phys. Soc. (London) B65, 437 (1952).

[4] C. M. Herget, J. Chem. Phys. 8, 537 (1940).

[5] H. Tielsch and H. Tanneberger, Z. Physik. 137, 256 (1954).

[6] M. A. Breazeale, J. Chem. Phys. 36, 2530 (1962).

[7] J. Noury, Compt. Rend. Acad. Sci. Paris 233, 516 (1951).

[8] V. F. Nozdrev, Soviet Phys. Acoustics 1, 249 (1955).

[9] V. F. Nozdrev. Application of Ultrasonics in Molecular

Physics (Gordon and Breach, New York, 1963).

[10] B. I. Kal'ianov and V. F. Nozdrev, Soviet Phys. Acoustics 4,

198 (1958).

[11] V. F. Nozdrev and V. D. Sobolev, Soviet Phys. Acoustics 2, 408 (1956).

[12] W. G. Schneider, Canadian J. Chem. 29, 243 (1951).

[13] V. F. Nozdrev, A. P. Osadchii, and A. S. Rubtsov, Soviet Phys. Acoustics 7, 305 (1962).

[14] V. F. Nozdrev and G. D. Tarantova, Soviet Phys. Acoustics

7, 402 (1962).

[15] V. I. Grechkin and V. F. Nozdrev, Soviet Phys. Acoustics

9, 304 (1964).

[16] L. D. Landau and E. M. Lifshitz, Statistical Physics (Per- gamon Press, 1959).

[17] M. I. Beyatskii, A. V. Voronel, and V. G. Gusak, Soviet Phys. JEPT 16, 517 (1963).

[18] A. V. Voronel, Yu. R. Chashkin, V. A. Popov, and V. G. Simkin, Soviet Phys. JEPT 18, 568 (1964).

[19] C. N. Yang, and C. P. Yang, Phys. Rev. Letters 13, 303

(1964).

[20] L. Liebermann, J. Acoust. Soc. Am. 23, 563 (1951).

[21] L. Chernov, Wave Propagation in a Random Medium

(McGraw-Hill Book Co., New York, 1960).

[22] A. G. Chynoweth and W. G. Schneider, J. Chem. Phys. 19,

1566 (1951)'. [23] M. Cevolani and S. Petralia, Atti Accad. Naz. Lincei 2, 674 (1952).

[24] G. F. Alfrey and W. G. Schneider; Disc. Far. Soc. 15,

218 (1953).

[25] D. Sette, Nuovo Cimento 1, 800 (1955).

[26] A. E. Brown and E. G. Richardson, Phys. Magazine 4, 705

(1959).

[27] R. Lucas, J. Phys. Rad. 8, 41 (1937).

[28] M. Fixman, J. Chem. Phys. 33, 1363 (1960).

[29] M. Fixman, J. Chem. Phys. 36, 1961 (1962).

[30] M. Fixman, Advance in Chemical Physics Vol. VI (Inter- science Publ, New York, N.Y., 1964).

[31] M. Fixman, J. Chem. Phys. 36, 1957 (1962).

[32] A. P. Kendig, R. H. Bigelow, P. D. Edmonds, C. J. Pings,

J. Chem. Phys. 40, 1451 (1964).

[33] A. S. Sliwinski, A. E. Brown, Acustica 14, 280 (1964).

196
Ultrasonic Investigation of Helium Near its Critical Point

C. E. Chase* and R. C. Williamson**

Massachusetts Institute of Technology, Cambridge, Mass.

The recent discovery of a logarithmic singularity

in the specific heat at constant volume C\ of fluids

at the critical point [1-3] has stimulated renewed

interest in this region. A consequence of this singularity that was soon demonstrated [4, 5]

is that the adiabatic compressibility ks must like- wise be singular and the velocity of sound, given by

u

2 =l/pKs , must go to zero. We have accordingly

measured the sound velocity in He4 at a frequency

of 1 Mc/s in the critical region. Some of the results

of these measurements are reported herein.

Figure 1 shows the velocity as a function of temperature along an isobar and as a function of pres- sure along an isotherm passing approximately

through the critical point. These measurements

have been reported previously [5, 6] (together

with a description of the experimental techniques,

which will not be repeated here). However, we

have subsequently found that, as a result of slight

hysteresis and lack of reproducibility in the pressure

gauge used in the experiment, the pressure and

temperature of this isobar and isotherm should

probably be increased by about 4 mm Hg and 0.003

°K, respectively. We are currently making meas- urements along families of closely spaced isotherms

passing through the critical region, and find that the shape of an isotherm is substantially independent of its temperature over a range of several

millidegrees. We therefore believe that our conclusions will not be seriously affected by the abovementioned correction, but final judgement must be

reserved until our measurements are complete.

It can be seen from figure 1 that the general

behavior is in agreement with expectation: the

velocity falls rapidly as the critical point is ap-

*National Magnet Laboratory, supported by the U.S. Air Force Office of Scientific Research. "Department of Physics.

1 1 5 i 1 \> 1

1

'

1 i i i

1 10 P= 1718 ± 1 \

MM HG _

105

_J00 o

LUw 95

o INCREASING T h

• DECREASING T \

>- 90

8 85

_i

LlI > t 10

1 1 1 1 1 1 1 1 1 40-30 -20 -10 0 10

10 3 (T-TC

)(°K)

T = 5.200 ± 0.002 °K

20 30 40

105

100

95

90 o INCREASING P \ X

• DECREASING P \ 5 85L_ii i i i at i i l i

i l I si i i 1 1_ -40 -30 -20 -10 0 10 20 30 40

P-PC (MM HG)

FlGl'RE 1. Sound velocity as a function of temperature (</) and

pressure {b) in the critical region.

proached from any direction, and it is not unreasonable to suppose that it goes to zero at the critical point itself. Unfortunately, the large attenuation

in this region makes it impossible to approach the

critical point more closely with the present ap- paratus. We can thus only draw inferences about

the behavior closer to the critical point by inter- pretation of the data we have: this is the subject

of the following paragraphs.

We note first that a plot of 1/u2 as a function of

log \T— Tr \ or log \P — P,-\ is very nearly linear over the range of our data; there thus appears to be

a logarithmic singularity in this quantity at the crit- ical point. This fact is of interest as an experimental result independent of any theory. However,

197
in order to make a more meaningful comparison

with the results of specific-heat measurements and

with theory, we would like to know the adiabatic

compressibility. This requires knowledge of the

density, which has never been measured close to the critical point in He4

; the only measurements in

this region are those of Edwards and Woodbury

[7] along the coexistence curve, which . xtend to within 36 mdeg of Tc in the liquid and 50 mdeg in the vapor. It is therefore necessary to find an

appropriate equation of state from which values

of the density can be calculated.

Inasmuch as classical treatments such as the van

der Waals equation are only approximate, and more

recent approaches have not yet succeeded in yielding an equation of state suitable for our pur- poses, we have resorted to the technique of expanding in power series about the critical point. This

procedure has to be used with caution because of the nonanalytic character of the thermodynamic

functions at the critical point [8]; nevertheless we

believe that the following developments could be

used as a point of departure to be corrected by a singular function becoming significant in the immediate neighborhood of the critical point. In

particular, this procedure has shed light on the

proper choice of thermodynamic variables in the

critical region, and has disclosed previously un- reported inconsistencies in the standard treatment

of the critical point given by Landau and Lifshitz

[9] and modified by Edwards and Woodbury [7]. Landau and Lifshitz expand the slope of an iso- thermal as a power series in t — T— Tc and v' = V— Vc according to the equation

N-\d2 F/dV2

) T, x = -(dP/dV)T = At + Bv' 2 +. . .

(1) where A and B are positive constants. This pro- cedure leads to a coexistence curve symmetrical

about the critical point when plotted as a function

of the volume:

v'(; =-vi=i-SAt/Byi2

. (2)

The fact that the exponent of the coexistence curve

is 1/2, as is nearly always the case with such

expansions, does not present a problem in the

present case; in contrast to the behavior of most

other substances (for which an exponent of 1/3

is observed), He4 obeys the 1/2-power law experimentally from the above-mentioned closest ap198

proach to within 110 mdeg of Tc [7, 10]. However,

eq (1) can be criticized on other grounds: e.g., the

coexistence curve is actually much more nearly

symmetric in the density than in the volume (for He 4 departures from perfect symmetry in density

amount to only 2% per degree Kelvin). This

difficulty was noted by Edwards and Woodbury

[7], who eliminated the spurious volume symmetry

of eq (1) and obtained agreement with their measured orthobaric densities within 110 mdeg of Tc by

adding two higher terms to the expansion, obtaining - ( dPId V)r = At + Bv' 2 + Ctv' + Dt2

. (3) There are, however, further difficulties that are not eliminated, but rather compounded, by this procedure. This can be shown by integrating

eqs (1) and (3) to obtain families of isotherms.

For T < Tc , the resulting temperature-dependent

constant of integration can be chosen to yield the

observed P—T relation along the vapor-pressure

curve; above Tc it is plausible to choose this constant so that the critical isochore lies on the extrapolated vapor pressure curve. The results of such

a calculation are shown in figures 2 and 3. The

calculated isotherms are observed to cross in both cases; moreover this crossing is worse (in the sense of occurring closer to the critical point)

for the Edwards-Woodbury expansion than for that of Landau and Lifshitz. The anomalies are so pronounced that they could scarcely be removed

by any reasonable adjustment of the parameters

Pc , Tc , Vc , or the constants A-D. These expansions are thus unsuitable for quantitative calculations. The way out of this difficulty is found by a simple

interchange of the role of the variables N and V in eq (1). This leads to a description in terms of the I conjugate pair of variables /x, p (where p. is the

J chemical potential), as opposed to the traditionally 1 preferred pair — P, V, and in fact represents a I nontrivial change in the definition of the thermodynamic system. The equation which is the analog y of eq (1) is now

V-Hci 2Flf)N2

) T. v = (d/xldp)r=at + ftP

' 2 +. . .

(4)

where p' = p — pc , and the slope of the isothermals

is given by

(c)P/c)p)r =p(c)pLlf)p)T = p(at + /3p' 2

). (5)
FIGURE 2. Isotherms and coexistence curve calculated from the Landau-Lifshitz expansion (eq (1)) with A=7.95 X 104 dyn mol cm'5 deg-', B = 62 dyn moP cm'", (dP/dT)s „, = 1.7 X 10s dyn cm'2 deg~', Pc = 2.288 X 10e dyn cmT2

, Tc = 5.200 °K, Vc = 57.628 cm3 mol''.

FIGURE 3. Isotherms and coexistence curve calculated from the Edwards-Woodbury expansion (eq (3)), with c = 8J6x 103 dyn mol2 cm'8 deg~' , D = 7.78 X 105 dyn mol cm'5 deg'2

, A, B, (dP/dTW, Pc , Tc , Vc as in figure 2.

-10 3 t/Tc Figure 4. Graph of {p'

LlpQ Y and (p'cJpc ) 2 against — t/Tc from

data of reference 7. Equation (6) predicts a single straight line of slope 3a/^3 + (p'Jpc)

2 Oip'Jpc)2

-

By development exactly parallel to that of reference

9, we now find that the coexistence curve is symmetric in the density:

p; = -PG = <-3a;//3)1/2

. (6)

To evaluate a and /3 we note that a plot of p'

L

2 and

P'q against — t should he on a single straight hne

through the origin of slope 3a//3. Such a plot,

based on the data of Edwards and Woodbury [7], is shown in figure 4, from which we find a/j8 = (1.03

±0.02) X 10~4 mole2 /cm6 °K. Note that the range

of agreement with experiment is as wide as that achieved by Edwards and Woodbury with their four-parameter expansion [11]. As in reference

[7], fi can be evaluated approximately in terms of

the measured density and isothermal compressibility [12] at 5 °K, yielding 0 = (6.86 ± 1.2) X 1013 dyne-cm10/mol4

. The resulting isotherms in the

fx, p plane are shown in figure 5. In this case the

temperature-dependent integration constant that determines the separation of the isotherms was

evaluated from the relation Ndp=VdP — SdT,

using entropy values along the critical isopycnal

derived by numerical integration of the specificheat data of Moldover and Little [3]. Isotherms

in the P, p plane, calculated with the same assumptions used in deriving figures 2 and 3, are shown

in figure 6. In neither case are inconsistencies

evident.

Although this expansion was derived for purely

practical reasons, we believe it has significance

beyond mere curve-fitting. In particular, the great

improvement resulting from the proper choice of

variables, and the natural way in which the sym199
Figure 5. Isotherms and coexistence curve in the p., p plane

calculated from the density expansion (eq (4)), with a=7.06

X10" dyn mo/-- cm4 defC' , j8 = 6.86 X 10l:

< dyn mot4 cm'", other parameters as in figure 2. Fur method of calculating spacing of isotherms see text.

P?P

C

Figure 6. Isotherms and coexistence curve in the P, p plane

calculated from the density expansion (eq (5)),- all parameters

as in figure 5.

metry of the problem enters when these variables are used, appear to us important. These features

should survive even in a more sophisticated treat- ment that includes a proper handling of the

singularity.

Values of the density along the critical isobar

and isotherm, calculated in the same way as those

plotted in figure 6, have been combined with our

200

Figure 7. Adiabatic compressibility (a) as a function of log |T —

T

c | and (b) as a function of log |P —

P

c | in the critical I region, calculated from the results offigure 1 and the density

expansion (eq (5) and fig. 6).

velocity data to yield k.s , which is plotted logarith- !| mically as a function of \ T— Tc \ or |P—

P

c | in figure 7. This figure is similar to one previously given, j

'

[5, 6] but the differences resulting from use of our 1 improved expansion for the calculation of the

density are significant. Results in the liquidlike

modification (P > Pc or T < Tc) are virtually un- changed, and the presence of a logarithmic singu- |j

larity in k.s over more than a decade is clearly

indicated. However, in the gaslike modification 1]

k.s now appears to be almost constant, and if a I singularity is present at all, it must only show up

within the last 4 or 5 mdeg (or millimeters) away

from the critical point.

Velocity measurements have also been made

along a number of isotherms above Tc . Some of these measurements are shown in figure 8. These i curves all show pronounced minima as a function

of pressure. It is perhaps important to point out j I that all these minima are rounded rather than cusp- shaped, even within about 10 mdeg of Tc (the closest

J point at which the velocity could be observed con- I tinuously through the minimum). The fact that j the locus of these minima in the P, T plane falls approximately on the extrapolated vapor pressure
a* I I II I i I I I 1

1 1700 1800 1900 2000 2100 P(MM HG)

FIGURE 8. Velocity as a function of pressure along isotherms above the critical temperature.

2000

1 800

HG) 1600

5

5 1400

Q- 1200

1000

800

4 2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 T(°K) FIGURE 9. Locus of velocity minima along the isotherms of

figure 8 in the P. T plane.

Full line: vapor pressure curve. + critical point.

curve, as is shown in figure 9, allows us to reach

I some further conclusions. According to Yang

and Yang [4], the velocity of sound along the critical isopycnal above Tc should be given by

u

2 = p-2(dPldT)lt(TICv), (7)

where Cy is taken along the same isopycnal and

(dP/dT)sat is the slope of the saturated vapor pres- sure curve just below Tc . Starting from the ther- modynamic relation

(dP/dV), = (dPldV)T -(dPldT)UTICv), (8)

i

; it is easy to see that eq (7) can be obtained from the exact expression

u

2 = (dPIdph + PcHdPldmTICv) (9)

by neglecting (dP/dph (which vanishes as T-^>TC) and assuming that (dPjdT)y ,(t>tc

) = (dP/dT^jTKT^-

T-Tc (°K)

FlGl RE 10. Plot of u%i,i against T/C\ from data of reference 3.

In this approximation a plot of u

2 versus TjCy should

be a straight line of slope Pc

2(dPldT)2at. This is shown by the solid line in figure 10; the dotted line

is a correction for the nonzero value of (dPldp)r

derived from our density expansion (eq 5). The

circles represent our values of u^in at various temperatures plotted against the specific-heat measure- ments of Moldover and Little [3]; use of umin is consistent with the assumption (BPI dT)y — (dP/ dT)sat and the fact that the observed minima he on the

extrapolated vapor pressure curve; i.e., we are assuming that the velocity minima he along the

line p = pc- The agreement of these points with

Yang and Yang's prediction, plus the small cor- rection term (dPldp)r, is reasonably good, although

all the points he about 2 to 5 percent too high [13]. We ascribe this small systematic difference to errors either in the choice of (dP/dT)sat , in the measured values of Cy, or in the temperature scales of

the two experiments.

Since eq (7) ought to become more accurate as the critical point is approached, we can now esti- mate how low the velocity ought to fall along iso- therms very close to Tc . As can be seen with the

aid of the temperature scale at the top of figure 10,

the minimum velocity along an isotherm just 10~4 °K

above Tc would be —55 m/sec, about 2/3 of

the lowest value we have observed. Thus, even in an ideal experiment in which measurements could

be made very close to the critical point in spite of

the large attenuation there, it would hardly be

possible to observe much lower values of u than we have done.

201
The authors are extremely grateful to Laszlo

Tisza for suggesting this experiment, for playing

a major part in the derivation of the density expansion and the elucidation of its significance, and for continued helpful advice and encouragement

throughout the course of this work. Most of the

numerical computations were carried out at the

M.I.T. Computation Center. References and Notes

[11 M. I. Bagatskii. A. V. Voronel', and V. G. Gusak, Zh. Eksperim. i. Teur. Fiz. 43 , 728 (1962) [translation: Soviet Phys.-JETP 16,517(1963)].

[2] A. V. Voronel". Yu R. Chashkin, V. A. Popov, and V. G. Simkin. ibid. 45, 828 (1963) [translation: Soviet Phys.-

JETP 18,568(1964)].

[3[ M. Moldover and W. A. Little. Proc. 9th Intl. Conf. on Low Temp. Phys., Columbus. Ohio. 1964 (to be published).

[4] C. N. Yang and C. P. Yang. Phys. Rev. Letters 13, 303

(1964).

[5] C. E. Chase. R. C. Williamson, and L. Tisza, Phys. Rev.

Letters 13, 467 (1964).

[6] C. E. Chase and R. C. Williamson. Proe. 9th Intl. Conf. on Low Temp. Phys., Columbus. Ohio. 1964 (to be published).

[7] M. H. Edwards and W. C. Woodbury. Phys. Rev. 129,

1911 (1963).

[8| L. Tisza, Phase Transformations in Solids (John Wiley &

Sons, Inc., New York, N.Y., 1951). p. 15; M. E. Fisher, J. Math. Phys. 5, 944 (1964).

[9] L. D. Landau and E. M. Lifshitz, p. 259 ff. Statistical Physics, (Pergamon Press, New York. 1958).

[10) The deviations from a 1/3-power law observed by Edwards

and Woodbury appear to be much too large to ascribe

to experimental error. If these deviations are real, they

might be attributed to quantum corrections to the law

of corresponding states (J. de Boer, Physica 14, 139, 149 (1948); J. de Boer and R. J. Lunbeck. Physica 14,

509 (1948)), and similar behavior ought to be observed in He3

. Such appears to be the case (see paper by R. H. Sherman in this conference).

[11] The line in the figure actually has a finite intercept of +4.6

millidegrees on the abcissa. This may indicate flatten- ing of the coexistence curve close to the critical point, or may be the result of errors in the measured temperatures in reference 7 or the assignment of Tc . [12] M. H. Edwards and W. C. Woodbury, Can. J. Phys. 39,

1833 (1961).

[13] Note that if the velocity minima did not fall exactly on the

line p = pc the agreement would be worse, for a smaller

value of (/ is required to fit the theoretical curve.

Ultrasonic Investigation of the Order-Disorder

Transition in Ammonium Chloride

C. W. Garland and R. Renard

Massachusetts Institute of Technology, Cambridge, Mass.

Introduction

The lambda transition in solid ammonium chloride was discovered by Simon [1] from heat-capacity

measurements. An x-ray analysis by Simon and

von Simson [2] showed that ammonium chloride

has a CsCl-type cubic structure both above and

below the critical temperature; thus the transition

could not be due to a change in lattice type. Hettich

[3J found that ammonium chloride is weakly piezoelectric in the low-temperature modification,

whereas it is not piezoelectric above the critical temperature. From this he concluded that the

tetrahedral ammonium ions are oriented parallel

in the low-temperature modification. Menzies

and Mills [4] came to the same conclusion on the

basis of the Raman effect. Pauling [5] suggested

that the transition was related to the onset of free

rotation of the ammonium ions, and quantitative

refinements of this idea were carried through by

Fowler [6] and by Kirkwood [7]. In opposition to

this, Frenkel [8] interpreted the transition as being

of the order-disorder type, involving the relative orientations of the tetrahedral ammonium ions;

it was assumed that in both the ordered and the

disordered states the ammonium ions liberate

about an equilibrium position. This idea has been

confirmed by many subsequent experiments.

In particular, Levy and Peterson [9] have carried

out a neutron-diffraction determination of the hydrogen positions in both the low- and high-temperature

modification, and Purcell [10] has analyzed the nmr

data to obtain the relaxation time for an NH£ ion

to move from one orientation in its cubic cell to the

other orientation. Both these investigations show

that free rotation does not occur. Indeed, in many respects NH4 C1 is an ideal

crystal for studying cooperative order-disorder

phenomena. The ordering is completely analogous

to that for a simple-cubic ferromagnet in a zero external field. The change in interaction energy

between parallel and antiparallel NH| ions is almost completely due to octopole-octopole terms between nearest and next-nearest NHJ" neighbors

202
[11], and thus NH4 C1 is quite a good example of an

Ising lattice. Furthermore, the ordering process

should have little effect on the dynamics of such

an ionic lattice. We wish to discuss here a variety of ultrasonic

measurements which have been made on singlecrystal ammonium chloride. Longitudinal and

transverse (shear) acoustic velocities have been

measured [12, 13] over a wide range of frequencies

(5 to 60 Mc/s), temperatures (150 to 320 °K) and

pressures (0 to 12 kbar). The attenuation of longitudinal ultrasonic waves has been measured [14]

at 1 atm for frequencies between 5 and 55 Mc/s

and temperatures from 200 to 270 °K. In all these

investigations, special emphasis was given to the

"anomalous" behavior near the lambda transition. As shown in figure 1, the transition temperature is a fairly strong function of pressure, increasing from - 242 °K at 1 atm to ~ 308 °K at 10 kbar. (Note

the pronounced curvature of the lambda line and

the hysteresis observed at low pressures.) At a constant pressure of 1 atm, the ultrasonic velocities were originally measured by a pulse-echo method

as a function of frequency and temperature, and

no dispersion was observed [12]. Recently, the

McSkimin pulse-superposition method has been

used at a fixed frequency of 20 Mc/s. The temperature dependence at 1 atm has been restudied,

and the pressure dependence of the velocities were

12.000

10,000

- 8000

UJ§ 6000

4000 —

2000

250 270 290 310 330

T (°K) FIGURE 1. Phase diagram for NH4 CI (the high-pressure, low- temperature field corresponds to the ordered phase).

The solid data points were obtained from the abrupt "break" in the ultrasonic shear

velocities at the A transition point (see figs. 2 and 3). The vertical bars indicate static measurements of Bridgeman [15|. The light lines numbered 1 through 7 represent

various isochores.

measured at five fixed temperatures spaced between

250 and 308 °K. Thus, we have very precise

velocity data (±0.005%) on both the ordered and

disordered phases. This experimental data will be described elsewhere [13] in detail, but is pre- sented here in a graphical form.

All the results given below are in terms of the

variation of one of the three elastic constants Cn,

C44, and C =(cn — Ci2)/2. Third-order elastic constants are not used, and at high pressures the

quantities cn, C44, and C are in a sense "effective"

elastic constants. The constants Cu and C44 were

calculated from the experimental longitudinal and

transverse velocities (U) in the [100] direction with

the relations pUf = cu and pUf = c44 ; the constant C

was calculated from pUf> — C", where Ur is the velocity in the [110] direction of a shear wave polarized

perpendicular to the [001] axis. Experimental Results

The variations of the shear constants C44 and C" and of the compressional constant cn with pressure

at five different temperatures are given in figures

3000 6000 9000

PRESSURE (bar)

12.000

FIGURE 2. Dependence of c44 on pressure at Ti=250, T> = 265 T:l = 280. J, = 295. and T5 = 308 °K.

3000 6000 9000

PRESSURE (bar)

12.000

FIGURE 3. Dependence of C on pressure at various temperatures (see fig. 2 legend).

203
3 0 3000 6000

PRESSURE (bar)

_J

9000 12,000

FIGURE 4. Dependence of Cu on pressure at various temperatures (see fig. 2 legend).

Dashed portions of the curves indicate regions where data are less accurate or are missing due to high attenuation.

150 200 250 300 350 TOO

FIGURE 5. Variation of C44 with temperature at 1 atm [open

circles) and at V, =34.00, \, = 34.15. V3 = 34.27, V4 = 34.43, V5 = 34.51, V„ = 34.77, V7 =34.93 cm:< mole' 1 (crosses). The transition temperature is known for each of these volumes, and thus the location of the change in slope for each of the constant-volume lines is fixed.

1.52

z 150

200 250

t(*k)

FIGURE 6. Variation of C with temperature at 1 atm (open

circles) and at various constant volumes (see fig. 5 legend).

3.50

A

8 $

e #

8

!

100 150 200 250 300

FIGURE 7. Variation of Cw with temperature.

Curve A: data at 1 atm (solid circles = pulse superposition method [ 1 3 J. open circles = corrected pulse-echo data [12]). Curve B: calculated curve at constant volume

J/2 =34.15 cm3 mole' 1 ; corresponds to at 280 °K and to the volume at 1 atm and

191.25 °K (marked by cross).

2, 3, and 4. Data points are not shown on these

figures since a variety of corrections must be applied

to the data, which requires some smoothing. Each

isotherm is based on at least 30 experimental points

which show very little scatter. Note that C44 is much more sensitive to pressure than C" but that the character of the anomalous variation near the

lambda line is very similar for both these shear

constants. Indeed, there is a marked reduction

in the rapid variation just above the transition as the lambda line is crossed at higher pressures. This same reduction in anomalous behavior is also seen in the cn data. The adiabatic compressibility

(3s can be calculated from

1/^=3 (c» + 2c«) = cii-|c' (1)

and it is clear that /3s goes through a sharp maximum

at a lambda point.

Experimental data points are shown in figures 5,

6, and 7 for the constants C44. C, and Cn as functions

of temperature at 1 atm. In addition, the variation

of C44 and C with temperature is given for seven

different constant volumes, and the behavior of

Cn at a single constant volume is also shown.

204
These constant-volume curves were obtained by

determining the pressure required at a given temperature to give the desired volume (Bridgeman's

data [15] and our own were used) and then reading

the appropriate elastic constant values from

figures 2 to 4. For the shear constants, one can immediately

see that most of the temperature change near 7\

at 1 atm is due to the rapid change in volume in the transition region. Our data permit us to completely analyze the effects of volume changes and

temperature changes so that one can see the effect of ordering at a constant volume and temperature.

For longitudinal waves, volume changes do have a

significant effect on the velocity, but the anomalous

dip in Cn is essentially due to ordering effects. In

figure 7, the shift in the minimum between curves A

and B is merely a result of the value chosen for the constant volume; the important difference between

these curves is their shape near the minimum.

For shear waves there is no indication of excess

ultrasonic attenuation near the lambda line, but

for longitudinal waves there is a very sharp maximum in the attenuation coefficient a at the transi- tion point. Quantitative measurements of a for longitudinal waves along [100] have been made at

1 atm as a function of frequency and temperature

[14]. The results are shown in figure 8 for 15,

25, 45, and 55 Mc/s. Data have also been taken

at 5 Mc/s, where a rather large (0.5 dB cm-

'), temperature-independent background attenuation

is present (presumably due to beam-spread losses).

200 220 240 260 280

T,"K

FIGURE 8. Attenuation coefficient a at various ultrasonic

frequencies.

Data obtained on two samples with different path lengths were in excellent agreement

at 15 and 25 Mc/s. Scatter in the 45 and 55 Mc/s data is caused by high a values and weak initial pulses (9th and 11th harmonics of 5 Mc/s transducer).

Further experimental work on attenuation is cur- rently in progress. Since there is no excess attenuation near the

lambda point for shear waves, it is possible to follow the behavior of the shear velocities through the

transition region. Both c44 and C showed a small

hysteresis, the magnitude of which is indicated

in the insert on figure 1. We have not shown data

points in this region on figures 5 and 6 since equilibrium was very difficult to achieve (slow changes in the shear velocities occurred even after 45 min).

The presence of this hysteresis and the very abrupt

change in shear velocities near Tk are significant

properties of our measurements at 1 atm. (Similar

indications of hystersis and abrupt changes near 7\ are seen from volume measurements and cooling

curves.)

Discussion

Equilibrium data. Because of the formal anal- ogy between the ordering of NH4 C1 and of a ferro- magnet in the absence of an external field, it is appropriate to consider the mechanical behavior

of an Ising lattice. If the coupling between the

lattice system and the "spin" system is sufficiently

weak, an Ising lattice will become unstable in the

immediate vicinity of its critical point and a first- order transition will occur. Furthermore, hysteresis

will be expected in the critical region. This in- stability has been demonstrated for a two-dimensional square lattice [16, 17], where there is an

analytic solution to the Ising problem. The important conditions for this instability are (1) weak

lattice-spin coupling, (2) a finite compressibility

for the disordered lattice, and (3) an infinite Cv for the "spin" system at the critical temperature. The

effect will be large only when (a) the lattice is soft,

(b) the critical temperature is large, and (c) dJ/dV

is large (where J is the usual Ising interaction

energy). Such an instability has been proposed

previously for other models [18, 19] and discussed

in general thermodynamic terms by Rice [20], but

little attention seems to have been paid to this problem. It would appear that NH4 C1 is a much

more appropriate system for observing these effects than most ferromagnetic solids would be. To illustrate the instability and the possible

hysteresis in order-disorder transitions, let us look

at the two-dimensional case [16]. Figure 9 shows

a plot of pi and —p<ti against the area cr at several

temperatures 7\ < T-z . . . T6 < T-t

. The pressure

205
FIGURE 9. Plot of p/ and — pdi versus the area cr for a two- dimensional Ising lattice. The family of curves pi were calculated at seven evenly spaced temperatures from

T\ up to 7V The lamily of lines— pai were drawn to represent a disordered lattice with typical compressibility and thermal expansion coefficients. The encircled numbers 1 and 7 indicate the spin and lattice isotherms at 7*1 and 7V

Pdi is that of the completely disordered lattice,

and pi represents the attractive (negative) pressure

due to the ordering of an Ising spin system. An

intersection of the two appropriate isotherms

(pi and —pdi) will correspond to p = p, + p,u=0 and

will give an equilibrium area cr if the stability condition {c) 2A/cta 2 > 0) is satisfied. The stability

condition requires that the slope of —p,n be greater

than that of p,; e.g., the system is unstable at a point such as 4". Now consider the change in area with T for a system under zero external stress

(p — 0). As the temperature increases from T\ to 7 5, cr can increase continuously from cr, to o-5 (points 1 to 5 on fig. 9) but as 7

1 —> T5 from below the

system becomes unstable (d 2A/da2 = 0) at point 5 and there must be a first-order change in area from

o"5 to cr.y. On further heating, cr increases continuously from ov to cr7 . However, on cooling

from T7 to T3 the area can decrease smoothly from

cr7 to cr3 '. As T^>Tz from above the instability occurs at point 3' and there is a first-order change

from cr3

' to cr3 . Below T3 , a decreases smoothly on cooling. Thus, there can be a hysteresis loop near the critical point with a first-order jump in

cr at T5 on heating and a first-order drop in cr at T3 on cooling. The values T3 and T5 determine

the maximum width of this loop since the system

becomes mechanically unstable at points 5 and 3'. Naturally, there is a temperature T4 for which the

free energy at point 4 equals that at point 4'; thermodynamic equilibrium would give a first- order transition at T4 and no hysteresis. The region

2Q6

between 4 and 5 on heating or 4' and 3' on cooling

is only metastable. Although the shape of the

pi curves will be different for a three-dimensional

Ising lattice, the instability will still occur if Cv (and thus dp,/dV)^^ at the critical point. Thus

the hysteresis described above can still occur. As seen in figures 2, 3, 5, and 6, the shear constants of NH4CI are sensitive functions of volume.

Thus the experimental hysteresis in Cu and C

can indicate the presence of the proposed hysteresis

in V. Note that experimentally the hysteresis

decreases for a transition higher up along the

lambda line (i.e., at some finite external pressure).

This would be expected from our analysis of the

Ising lattice since the only difference between working at zero pressure and at some large constant pres- sure is in the lattice stiffness. For high external pressures, the slope of — pai will be greater and the

maximum width AT of the hysteresis will be less. Moreover, a semiquantitative estimate of the magnitude of AT at 1 atm is in reasonable agreement

with experiment. Completely analogous considerations can be developed for an instability and hysteresis on compressing an Ising lattice at constant temperature. An elaboration of the effects of

hysteresis and instability on measurements at either constant pressure or constant volume will be given in detail elsewhere [16]. In addition, our shear data strongly suggest a sluggish first-order transition in the critical region. For points taken

at temperatures more than 1 °K away from 7\ and

at pressures more than 100 bar away from p\,

equilibrium was achieved within about 15 min after the temperature or pressure was adjusted. In the immediate vicinity of a critical point, very slow

changes in velocity were still observed 45 min

after the temperature or pressure was adjusted,

as would be expected in a metastable region. Also

the changes in C44 and C in this narrow hysteresis

region are extremely abrupt, even when compared

with the very rapid variations observed in the

ordered phase near the transition line. Leaving aside the important but relatively narrow

range where instability may occur, it is possible

to discuss the effect of ordering at constant volume

on the elastic constants in terms of an Ising model.

We have carried out a general stress-strain analysis

of the two-dimensional Ising lattice (using Onsager's

analytic solution for the case J^J') and have ob- tained explicit formulae for the isothermal stiff- nesses Cu, C44, and C = (cu — Ci2)/2. The details

will be reported elsewhere [21], but the essential
features can be seen from the simpler formula for the isothermal volume compressibility fS T

: W

= =

(w)<n ~J~

C/(0, H) (w)

where C/(0, H) and Ui(0, H) are the Ising heat ca- pacity and internal energy at zero field as a function

of H =J/kT. The subscript dl denotes the appropriate quantity for a completely disordered lattice

(for which C/ and Ui equal zero). The elastic constants dj are given by {d 2A/dXjdxj)r where the x's are the appropriate strains. Thus the expression

for Cn will have a form similar to eq (2), and the

dominant term in determining the shape of Cu near

T\ will involve C/. For C = {c u — c v>)/2 one re- quires a difference between two such expressions,

and the singular part involving C/ will cancel out. The final expression for C is complicated in its dependence on V and T, but it does predict a smooth

variation through the transition region. For c44 the

term containing Ci disappears identically since

dJldy = 0 (y is the change in angle during a c44 shear), but d

2Jldy2 # 0 and the shape of C44 near

7a is determined by Ui. In general terms these

predictions are all fulfilled by our NH4CI data as functions of temperature at constant volume. In the case of Cn one must take account of the sig- nificant difference between the adiabatic and the

isothermal constants near 7\, but even so the shape

m Cn is strikingly related to the shape of the heat

capacity curve. In addition, the shape of c 44 is what one would expect from an internal energy

curve. No detailed discussion of C will be given

here, but in crude terms the effect of ordering is very small for this shear mode and this indicates

that the "anomalous" variations in Cw and Cr> almost

completely compensate each other. A final remark about the equilibrium properties

of NH4 C1 can be made in connection with the

Pippard equations. It is known that these phenomenological relations are fairly well obeyed [12] in NH4 C1, but a generalization of them in terms of

stress-strain variables [22] shows that there are some unexplained difficulties. We have recently

discussed the Pippard equations [23] on the basis

of a three-dimensional Ising model and can now

give a better insight into the reasons for their

behavior.

Dynamical data. In the past few years there

has been increasing interest in the theory of the

dynamical properties of substances near a lambda

point. Landau and Khalatnikov [24] were the first to treat this problem by combining the thermodynamics of irreversible processes with an approximate equilibrium solution to the Ising problem.

This approach has been extended and improved

by others [25, 26] and a somewhat simplified result

for the relaxation time t for long-range ordering can be given in the form

T+--=A+>-l\T-Tc \ (3)

where + and — denote temperatures above and

below Tr . The phenomenological constants A + and A~ depend on the substance involved but have

a fixed ratio of A

+ /A~ — 2. A more general statis- tical-mechanical treatment of this problem has

been given by Kikuchi [27]. For a temperature

range close to the lambda point one can show that Kikuchi's results for the long-range relaxation times

are still of the form given in eq (3). However, the constants A + and A~ now depend on microscopic

transition probabilities and the ratio A + /A~ near Tc is ~ 1.6 rather than 2. An essential feature of our ammonium chloride

data which requires explanation is the fact that the

attenuation a has its maximum at T\. One would

expect a to vanish at 7\ if t becomes infinite there. However, the theories mentioned above deal with

Ttv, the relaxation time for ordering at constant temperature and volume. The experimental conditions for measuring attenuation are adiabatic

rather than isothermal, and although the process is irreversible one can consider it to be essentially

isentropic for small sound amplitudes. In this case the appropriate relaxation time is Tsv, which

can be related to Ttv by

J_=J_+ M7-Y (4)

where A: is a rate constant, C, is the specific heat at infinite frequency ("frozen" specific heat), and £ is a progress variable characterizing the internal (nonequilibrium) state of the system at a given V and T. The detailed behavior of (dS/d^)2 is not known,

but it seems likely that this quantity varies only

slowly with temperature near 7\. We shall assume

below that the quantity kT(dS/r)^)2 ICr has a constant value (call it B) over the temperature range of

207
interest. With this assumption and eqs (3) and (4), we find that

TSV

(5)

for the relaxation time associated with long-range

ordering, where AT=\T— T\\ and the constant A

has different values above and below 7\. According to eq (5), TS v will have a finite maximum value

at Tx and if this value is reasonably small (i.e.,

co

2T2 <l) the attenuation will have its maximum

For a mechanism involving a single relaxation

time, we have the usual expression

Wcu 1 + o)2t|,

(6)

Unfortunately the variation of (eft — c§i)/2Ucp with

temperature is not known, but it is unlikely that

this could be an important factor in determining

the temperature dependence of a. One would

probably expect only a small and slowly varying

dependence of eft — c?i on temperature. We shall assume (as did Chase in his analysis of helium [28]) that (eft — c?i)/2{/cn has a constant value throughout the transition region. We shall also assume

(and can subsequently justify) that o»

2r| r

<^ 1 for all temperatures near T\. With these assumptions

and eq (5), we can rewrite eq (6) in the form

ora 2Ucu 1 =2Ucu /AT

' ?x-c°n\A

' li (7)

If all our assumptions are valid, then oj 2 /a should

be independent of frequency and should vary lin- early with the temperature on both sides of 7\.

Figure 10 shows a plot of a>

2/a, in units of cps2 N

_1 cm, versus temperature for all of our data close

to T\. Various details concerning the correction

for background attenuation and the justification of oj2t

2 < 1 are given elsewhere [14], but in a general way the data shown in figure 10 seem to agree

with eq (7). The value of co

2 /a at its minimum

(at ~ 242.0 °K) is 0.52 X I0ut

: the slope above TK is 0.46 X 1016 and the slope below 7\ is 0.66 X 101B

. These slopes give a ratio A+ /A~ = 1.43, as compared

to ~ 1.6 expected on the basis of Kikuchi's model.

When the analysis is extended to data obtained

at higher frequencies and at temperatures further

away from 7\, the deviations from eq (7) are not

• 5 Mc (a -0.53 db/cm) O 15 Mc (a -0.15 db/cm) A 25 Mc a

FIGURE 10. Plot of ar/a versus temperature, close to the lambda

point at p = 1 atm.

Data a! 45 and 55 Mc/s were not available over this temperature range. Note t hat the a values tor 5 and 15 Mc/s have been corrected for a temperature-independent background attenuation (beam-spread losses).

quite what one would expect [14] and new experimental work along these lines is currently in progress. Also if NH4CI undergoes a first-order transition very close to the "lambda point," then one would expect r to undergo a discontinuous

change between two finite values. This may

supersede the explanation given above for the

behavior of a [29]. References

[1] F. Simon, Ann. Physik 68, 4 (1922).

[2] F. Simon and C. von Simson, Naturwiss. 14, 880 (1926).

[3] A. Hettich, Z. Physik. Chem. A168, 353 (1934).

[4] A. C. Menzies and H. R. Mills, Proc. Roy. Soc. A148, 407

(1935).

[5] L. Pauling, Phys. Rev. 36, 430 (1930).

[6] R. H. Fowler, Proc. Roy. Soc. A149, 1 (1935).

[7] J. Kirkwood, J. Chem. Phys. 8, 205 (1940).

[8] J. Frenkel, Acta Physicochimica 3, 23 (1935).

[9] H. A. Levy and S. W. Peterson, Phys. Rev. 86, 766 (1952).

[10] E. M. Purcell, Physica 17, 282 (1951).

[11] C. W. Garland and J. S. Jones, J. Chem. Phys. 41, 1165

(1964). [12] C. W. Garland and J. S. Jones, J. Chem. Phys. 39, 2874

(1963).

[13] C. W. Garland, and R. Renard, J. Chem. Phys. 44, 1130

(1966).

[14] C. W. Garland and J. S. Jones, J. Chem. Phys. 42, 4194

(1965). [15] P. W. Bridgeman, Phys. Rev. 38, 182 (1931).

[16] C. W. Garland and R. Renard, J. Chem. Phys. 44, 1120

(1966). [17] C. Domb, J. Chem. Phys. 25, 783 (1956).

[18] C. P. Bean and D. S. Rodbell, Phys. Rev. 126, 104 (1962).

[19] D. C. Mattis and T. D. Schultz, Phys. Rev. 129, 175 (1963).

[20] O. K. Rice, J. Chem. Phys. 22, 1535 (1954).

[21] R. Renard and C. W. Garland. Phys. Rev. J. Chem. Phys.

44,1125(1966).

[22] C. W. Garland, J. Chem. Phys. 41, 1005 (1964).

[23] C. W. Garland and R. Renard, J. Chem. Phys. (in press).

[24] L. D. Landau and I. M. Khalatnikov, Akad. Nauk. S.S.S.R. 96, 469 (1954).

[25] I. A. Yakovlev and T. S. Velichkina, Sov. Phys. Uspekhi 63,

552 (1957).

[26] T. Tanaka. P. H. E. Meijer, and J. H. Barry, J. Chem. Phys.

37, 1397 (1962).

[27] R. Kikuchi, Ann. Phys. (N.Y.) 10, 127 (I960).

[28] C. E. Chase, Phys. Fluids 1, 193 (1958).

[29] C. W. Garland and C. F. Yarnell, J. Chem. Phys. (in press).

208
Discussion

E. H. W. Schmidt presented some experimental observations on

convective heat transfer near the critical point of CO> [1) emphasizing the difficulties which arise in measuring the thermal conductivity in the critical region.

J. J. M. Beenakker: I have a problem concerning the dimensions

of the apparatus in which one tries to measure the transport

properties near the critical point. As soon as one expects that any anomaly in viscosity or thermal conductivity is related to the long range correlations or fluctuations, care has to be taken

that the dimensions are large compared to the magnitude of the

correlation length.

M. E. Fisher: It seems to me there is no need to worry about

the dimensions relative to the correlation length because all the light scattering and neutron scattering experiments show

that even when you have what you call long range correlations, they are still of the order of 1000 A when you come very close

to the critical point. And that is for the pair correlation function. Most of these transport coefficients involve higher order correlation functions which one would expect to die out more rapidly. Thus although in principle one could get difficulties very close to the critical point, I don't think one has to worry about this in practice.

J. J. M. Beenakker: In the case of an oscillating disk viscometer one has to consider a penetration depth between oscillating plates and the gas layer. As in the critical region the density is changing rapidly with height, I wonder whether you can make a state- ment of the absence of an anomaly in the viscosity close to the

critical point.

J. V. Sengers: That was one of the reasons why I made the state- ment that any anomaly in the viscosity is absent at temperatures more than 1 percent away from the critical temperature. At

present I would rather leave open the question of the behavior

of the viscosity closer to the critical point.

J. S. Rowlinson: I would like to raise the question whether the Cr of a binary mixture has an anomaly near the critical mixing

point. There is certainly an anomaly in Ct „ but as far as I know

there is no anomaly in C, in this case.

J. V. Sengers: I don't know. I looked for experimental data for C, near the critical mixing point of binary mixtures in the litera- ture, but I did not find any.

C. J. Pings: Dr. Sengers, did you not also run your experiments

with two different distances between the plates? You seem to get the same results which proves the absence of an effect from

convection.

J. V, Sengers: At the isotherms close to the critical temperature

(31.2, 32.1, 34.8 °C) measurements were taken using only one

plate distance, because it is difficult to do those measurements

with a larger plate distance. However, the anomalous behavior

is already present at higher temperatures where measurements

with two different plate distances were indeed carried out, which

give an additional proof of the reality of the anomaly.

M. S. Green: I want to point out that the difference in behavior between viscosity and thermal conductivity is theoretically very

surprising. Both the viscosity and the thermal conductivitv can be expressed as the time average correlation of relevant

fluxes. As the flux for the thermal conductivity and the flux for the viscosity have a similar form, it is hard to imagine why

there would be any difference in behavior.

R. Zwanzig: If you analyze these correlation function formulas

for fluids which have internal degrees of freedom interacting

weakly with the translational degrees, it turns out that the shear

viscosity does not show any particular effect from the internal degrees of freedom. On the other hand the thermal conductivity shows some effect and the bulk viscosity even a bigger effect [2]. Thus there exists a counter example to Dr. Green's suggestion

that the correlation function formulas should show roughly the same behavior.

E. Helfand: I think the anomaly in the thermal conductivity

arises from the thermodynamic force involved. Any molecular model would speak in terms of energy transport. However,

the thermal conductivity is related to a temperature gradient. Therefore to go from the energy gradient to the temperature

gradient one has to introduce the specific heat. This is similar

to the case of the diffusion coefficient where one has to transform

a gradient of the chemical potential into a concentration gradient. For this reasion a connection between the anomaly in the thermal

conductivity and in the C, is not surprising.

H. Zwanzig: 1 would suggest to the experimentalists to study

the dependence of the friction on the frequency with the oscil- lating disk viscometer as well as observing slow oscillations.

If there were nonlocal effects involving a penetration length and a correlation length, one would perhaps be able to see from

the frequency dependence of the friction whether any peculiar behavior would occur.

J. Kestin: One can regard all these experiments as having been performed at zero frequency. When one wants to go to higher

frequencies, one must consider the possibility that hydrodynamic

instabilities will be encountered. As Dr. Sengers pointed out. I think one should measure the two quantities: density and viscosity or density and thermal

conductivity, simultaneously. This does not seem to be impossible.

N. J. Trappeniers presented experimental data for the spinlattice relaxation time oi deuterated methanes [3].

N. J. Trappeniers: We have also made diffusion measurements

in the critical region, but I did not present jhose. because one

has to be extremely careful to find the density. I think that one

cannot say much, if one does not have accurate density measurements. In the critical region the density varies extremely

rapidly. Therefore I would like to ask Dr. Bloom the following: You make your measurements of the diffusion coefficients with an accuracy of about 5 percent in the critical region. If you would make an error in density of 5 percent or even 10 percent, which is easily made if one does not have accurate P-V-T dald, would it not be possible to accumulate the errors by plotting the product Dp so that the scattering in your points thus obtained would explain the anomaly which you think to detect in the

critical region?

M, Bloom: I agree with you that it is dangerous. We have made

density measurements in three different samples (these were

the impure samples) and I showed the ratio pi.lpv of the liquid and gas densities. These were obtained by making measure- ments at the top end of the tube and at the bottom end of the tube, while the density gradient was in the middle of the tube. We

measured the ratio pi.lpv to within 2 percent. The diffusion

coefficient was measured within approximately 2 percent. An

error of 5 percent has been assigned to D to take into account

systematic errors and that is a conservative estimate. The

anomalous decrease in Dp which we observed is about 20 per- cent. But I agree with you that it is dangerous. I think one

should do measurements with a flat horizontal sample to eliminate gravitational effects.

C. J. Pings presented experimental data for the sound absorption

in the nitrobenzene-iso-octane system [4].

209
L. Tisza: To Drs. Chase and Williamson should go the credit for their experiments. However, I have to assume the responsibility for the prediction that the sound velocity tends to zero as

the critical point is approached. The argument is based on

thermostatics and can be applied only to the real part of the sound

velocity in the low frequency limit. Since the absorption of sound need not decrease, one can expect that the sound velocity would drop until the signal is lost. This seems to be the experi- mental finding.

R. Zwanzig: I would like to make one remark about the general

interpretation of ultrasonic data. Many years ago Dr. Tisza

pointed out that ultrasonic absorption data could be explained

formally by the introduction of a volume viscosity or bulk vis- cosity into the hydrodynamic equations [5]. However, as far as

1 know, nobody ever does this in connection with ultrasonic

relaxation, especially near the critical point. There is one good

reason for this, namely that there is hardly any way of measuring the bulk viscosity. The only way you do it nowadays is to measure the sound absorption, so that it seems redundant. At

present, however, there are theoretical expressions for the bulk

viscosity (derived first, in fact, by Dr. Green) and there is at least a handle by which one can investigate theoretically what the magnitude of the bulk viscosity ought to be. Very little is known

about this either, but one does not have as yet a good reason to disregard the classical hydrodynamics with the bulk viscosity.

K. F. Herzfeld: The situation is not so simple. It is completely

correct, of course, that one can formally describe the sound

absorption via a bulk viscosity. In a polyatomic gas the bulk

viscosity is arising mainly from the relaxation of the internal degrees of freedom to their equilibrium distribution. In liquids, however, there are more processes contributing to a bulk vis- cosity. At low frequencies one can describe the relaxation of

the different processes with one "bulk viscosity constant." However, at higher frequencies one has to take into account

the fact that the different processes relax with different relaxation times and show resonances at different frequencies. Consequently the bulk viscosity is dependent on the frequencies and

one has to analyze the different processes involved. The situa- tion is analogous to that regarding the dielectric constant which

only can be considered constant at low frequencies.

R. Zwanzig: That is right. The only reason why I raised the

point is the fact that there are presumably rigorous statistical mechanical expressions for the frequency dependent viscosity. The question arises whether the ultrasonic absorption can be

described by these formulas or whether the ultrasonic absorption

is due to something more complicated like scattering by density

fluctuations.

O. K. Rice: In connection with the paper of Dr. Garland and

Dr. Renard, I might say that about 12 years ago I wrote a paper

in which I showed by purely thermodynamical means that for a compressible lattice the transition will change into a first- order transition [6|.

C. W. Garland: We agree that this can be proved in a phenomenological way, but we considered that doing it for an explicit model, in which the shape of the curve is given analytically by

the Onsager solution, provided an additional way of looking at the problem with which people who like models can perhaps

feel more confident.

M. E. Fisher: In this connection I would also like to draw attention to the work of Domb {7J, Bean and Rodbell [81 and Mattis and Schultz [9] who reach similar conclusions to those first discussed by Dr. Rice but from a more microscopic view

point. It seems to me that the general validity of all these treatments is open to doubt in respect of the conclusions drawn

about the changed nature of the transition point. The physical reason is the neglect of the fluctuations in the lattice expansion and the consequent "renormalization" of the original interaction responsible for the phase transition. This difficulty can be seen

explicitly in the case of a linear chain model which never has a phase transition (as shown by an exact solution) but for which

the treatments referred to would predict a first order transition

for certain parameter values. This has been demonstrated

by Mattis and Schultz [9] (although the arguments they give as

to why this might not matter in three dimensions are

unconvincing).

N. J. Trappeniers presented experimental data for the C- of ammonium chloride as a function of pressure along the \-line.

References

[1] E. Schmidt, Int. J. Heat Man Transfer 1, 92 (1960).

[2] R. Zwanzig, J. Chem. Phys. 43, 714 (1965).

[3] N. J. Trappeniers. C. J. Gerritsma, P. H. Oosting, Physica

31, 202 (1965); Phys. Letters 16, 44 (1965).

[4] C. J. Pings and A. V. Anantaraman, Phys. Rev. Letters 14,

781 (1965).

[5] L. Tisza, Phys. Rev. 61, 531 (1942).

[6) O. K. Rice, J. Chem. Phys. 22, 1535 (1954).

[7] C. Domb, J. Chem. Phys. 25 , 783 (1956).

[8] C. P. Bean and D. S. Rodbell, Phys. Rev. 126, 104 (1962).

[9J D. C. Mattis and T. D. Schultz, Phys. Rev. 129, 175 (1963).

210
SUPPLEMENTARY SESSION

21 1 209-493 0-66— 15

Effect of Gravity on the Equilibrium Mass Distribution of Two-Component

Liquid Systems. The Observation of Reversed Diffusion Flow

H. L. Lorentzen and B. B. Hansen

University of Oslo, Oslo, Norway

At the conference the similarity between the

coexistence curves of one-component and two-component systems has been stressed. The similarities between the two systems have to be extended to the equilibrium mass distributions in vertical containers. In one-component systems it is well known that close to the critical point the mass distributions at equilibrium are represented by sigmoid curves in the height versus density diagrams. This is caused

by the effect of gravity. The same applies to the

two-component systems. Thus in a vertical container filled with mixture at critical conditions,

homogeneous mixtures do not represent equilibrium.

Due to the effect of gravity — also in the two-component systems — equilibria in the critical region are represented by sigmoid curves in the height versus composition diagrams. (See fig. 1.) In the experimental part of this paper will be

shown exposures of curves of mass distribution,

which spontaneously developed in two-component

liquid systems. They are representing equilibrium mass distributions. One of the observations which

support this is the observation of transport of material through the layers of the equilibrium distributions. The transport was towards greater composition differences.

It will be shown in three different ways that the

equilibrium mass distribution in the critical region

cannot be the homogeneous mixtures, but distribu- tions represented by sigmoid curves in the height versus composition diagram. The developments

are founded on the low values of the gradient of

the chemical potential (d/x/dc) in the critical region.

According to classical thermodynamical view the

gradient of the chemical potential is zero at the

critical point:

Furthermore, according to modern views the gra- dient of chemical potential is the virtual force

producing diffusion. Therefore, when the gradient

of the chemical potential is zero, also the diffusion

coefficient will be zero. Bj0rn Bergsnov Hansen [1] and I determined

the diffusion coefficient at critical density as function of the temperature distance from the critical for the two systems: succinonitrile-ethanol and

triethylamine-water. We found the diffusion coefficients to be close to linear functions of the

temperature distance from critical, AT—T—TC , and approaching zero at the critical temperature.

Determinations of the diffusion coefficients as function of the composition difference from critical Ac = c —

c

c , as well as the temperature difference

from critical AT has shown the following relation- ship to be basically correct.

D = k>AT+ \k2Acn

\ (2)

where k\ and k> are constants (k-z not the same on

both sides of the critical composition) and n is close to 3.

Composition

FIGURE 1. Schematic drawing of equilibrium mass distributions

in the height versus composition diagram.

213
This is in agreement with theoretical considerations which Professor Joseph Mayer had pointed

out to me [6]. When the diffusion coefficient approaches zero

at the critical point the thernutl diffusion ratio ap- proaches infinite. In a binary system a temperature

gradient leads to a flow which will ultimately separate the liquid in two layers. However, by this flow a composition gradient is built up and conse- quently a mass diffusion flow will counteract the

effect of the temperature gradient. At the

"Endzustand" the two effects are in balance.

At the critical point the diffusion coefficient ap- proaches zero and thus the composition gradient

dc

—jjj will approach infinite.

Studies of the thermodiffusions in the critical region of the system aniline-cyclohexane here have

shown the Endzustande to be sigmoid curves with

the greatest gradients at critical composition.

The gradient increases where the temperature

of the critical composition approaches the critical,

and the gradient may well become infinite at this temperature. When this is so, it is obvious that,

when the temperature gradient diminishes towards

zero (T= constant) and the Endzustande goes

towards that of an equilibrium, the mass distribu- tions at this equilibrium will also be sigmoid curves

in the height versus composition diagram.

In a binary mixture the molecules of the components will have different buoyancy relative to the

mixture. Therefore, the molecules of one of the

components will have a tendency to move upwards

and the molecules of the other to move downwards

in the liquid. Consequently, a composition gradient

will develop and an inhomogeneous equilibrium

distribution will be established, whereby these

tendencies are balanced by diffusion flow towards

homogeneity.

Because of the overwhelming influence of the

diffusion flow such effect has not been observed

at normal conditions. In the critical region, however, where the diffusion coefficients are very small,

the effect will have to be remarkable, and at the

critical point the effect will result in an equilibrium

distribution of an infinitely great composition

gradient. In a liquid the potential energy increases with the

height. In liquid mixtures — at equilibrium — the

difference of potential energy of the layers at the

various heights will be balanced by a change of the

composition of the mixtures with height.

214

In a binary mixture the equilibrium condition

will be governed by the equation

dx_ Mi —pVt

dh dfJL\ldx g (3)

where x is the mole fraction n\/n\ + n>, h is the

height, M is the molecular weight, p is the density

of the mixture, V is the partial molar volume, \ \ = {dV/dn, the chemical potential and g

the acceleration of gravity. When in the critical]

region dfx/dx is very small, dx/dh will be great.

Reversed diffusion flow. It was mentioned in

j the introduction that diffusion flow towards greater j composition differences has been observed. Fick's ! law is founded on the observation that at normal ! conditions the Endzustande of the diffusion process

is that of a homogeneous mixture. When, how1

ever, homogeneous mixtures do not represent || equilibrium, the diffusion process will not go to- i'( wards homogeneity, but towards the equilibrium j \ condition.

Fick's law then may have to be rewritten in the form:

J = -D* dc

~dh

dc\°

dh

where h is the direction of the diffusion flow in a

vertical container, dcjdh is the concentration

gradient of the liquid with respect to the height;

dc/dh approaches the equilibrium distribution

characterized for the temperature and composition

in question by the gradient (dc/dh)0 at equilibrium.

I dc fdc\"\ The term yj^— \rrr\ j of eq (4) will be negative

when (dc/dh)° is greater than dc/dh. The flow./ is then negative, and we will observe a "reversed

diffusion flow." Reversed diffusion flow can also be observed

independent of the effect of gravity. If a mixture

is left at a temperature of the coexistence curve

the mixture may be split in two phases divided by

a surface. This is schematically shown on figure 2. The lines AB and B'A' indicate equilibrium phases.

When now the temperature is changed in the di- rection away from the critical temperature, then

the compositions of the phases at the new temperature will be outside the compositions corresponding

to the former temperature (dotted lines CD and

D'C)

Generally the formation of the new phases will be

introduced by fog formation. In some cases,

(4) :
+>

1

i A'

Composition

FIGURE 2. A schematic drawing of the mass distribution at

I two temperatures Ti and T>.

T, is the temperature closest to I he critical, and the lines AB and A'B' indicate the two phases at equilibrium at 7Y The temperature difference between T, and T± is very small, and the dulled lines DC and D'C indicate the equilibrium distribution

1 at T,. When the temperature is changed from 7*, to T± no fog is formed. A diffusion How

of material passing the surface DD' will be observed as mass distribution curves of

( the form AD and A'D'.

however, when the phases remain "undercooled,"

the formation of fog does not take place, and a

diffusion flow of material will take place. The

diffusion flow is crossing the surface border and

is a "reversed diffusion flow" towards greater dif- ferences in composition.

The two arched curves at the surface together

with the straight lines at greater distance from the

surface show the distributions observed after a period of time at the new temperature.

I have not been able to make an exact determination of the form of the arched curves. The form

appears, however, to be close to the error function curve observed by free diffusion. Experimental Procedures

The average compositions of the mixtures were

close to the critical compositions.

The liquid mixtures were sealed in all-glass con- tainers— ca. 15 mm high and of about 5 mm2 cross

section area. The liquid mixture almost co ipletely

filled the container.

The container was placed in a thermostat which

allowed the temperature to be controlled with an accuracy better than ± 1 mdeg [2]. The temperature was determined by means of a platinum resist- ance thermometer.

The compositions of the mixture at the various

heights inside the container were determined by means of an integral optical system [3]. This

allowed visual observations and photographic re- cordings, which actually are diagrams with the

height in the container as ordinate and the optical

density as abscissa.

In the region of the critical solution points, one

observes that the mass transports towards equilibrium distributions are using the formation of fog as vehicle. Thereby droplets of compositions

at densities different from that of the mother

liquid are formed and move by buoyancy upwards

or downwards in the liquid mixture. After a period

of time the fog has disappeared and a new mass

distribution can.be observed.

When the temperature of a nearly homogeneous

mixture of a composition close to the critical, and

at a temperature outside the coexistence curve, is changed to one of the temperatures of the co- existence curve, the material of the sample is then

brought to a state of unattainable instability. Two

phases must then develop and they do so probably

instantaneously by the formation of fog.

Figure 3 shows an example whereby a nearly

homogeneous mixture of succinonitrile-ethanol was

cooled to about 50 mdeg below the critical. When

the fog had disappeared the composition of both

phases were inhomogeneous and had the form of

two arms of a sigmoid curve.

Figure 3. System: Succinonitrile-ethanol. Ordinate: Height in container. Abscissa: Composition.

Straight vertical and horizontal lines are orientation

lines.

Figure 3 shows the composition which appeared when the near homogeneous mixture

of succinonitrile-ethanol had been cooled from a temperature above the critical down

to a temperature of about 50 mdeg below the critical and the fog had disappeared.

215
Figure 3 is representative for a great number of

observations also with other liquid systems which

have been studied here. Figure 3, however, does not represent the sole form which the phases may show when the fog dis- appears. In some cases only one of the phases may

be smoothly sigmoid and the other closer to homogeneous. This is explained by convection flow in this phase during the establishment.

When the temperature of the mixture is not brought to the temperature of the coexistence curve,

a redistribution of the material can not be enforced.

It happened, however, that in a mixture of tri- ethylamine-water the reestablishment took place

also at a temperature outside those of the coexist- ence curve and in the temperature range reaching

from the critical temperature to about 100 mdeg

outside the coexistence curve. The system triethylamine-water was studied at the lower critical solution point of the system. At

a lower solution point the temperatures outside the

coexistence curve are below the critical temperature, and the temperatures of the coexistence curve

above.

During the general study of this system the

sample was left for one night at a temperature out- side the coexistence curve. Next day the arbitrary mass distribution curve was found to have been

partly reformed into a smoothly formed arched

curve in the diagram height versus composition.

\>—A i%

FIGURE 4. System: Triethylamine-water. Ordinate: Height in the container, scale shown on

the right side of the figure. Abscissa: Composition of the mixture, scale unit below photograph 4. a. Exposure 4.a shows ihe originally straight curve partly disrupted by layers of log Below this layer is another layer of fog. which, however, allows the light to pass and

produces background illumination at these heights. Exposure 4.b shows the composition curve after the fog had disappeared. The

curve has attained a regular arched form where the upper layer of fog had been. When making the exposure a. the horizontal orientation lines were not illuminated. The photographically recorded composition curves do not represent the entire column

of the liquid mixture.

This surprising reformation of the mass distribution was then systematically studied, and Bj0rn

Bergsnov Hansen and I subsequently found a temperature treatment of the sample — which was sealed

inside an all-glass cell — whereby we could reproduce

the phenomenon ad libitum. The temperature of the thermostat was set at a temperature below the

critical (outside of the coexistence curve). The

cell was then removed from the thermostat, warmed

to room temperature and cooled to about 10° below

the critical and carefully tilted. Thereby a certain

concentration gradient was established in the liquid

mixture. (Typical is the lower curve of fig. 5.) The container was then reinserted in the thermostat

and it was observed that the composition curve re- mained unbroken.

After periods of time of different duration at constant temperature a belt of fog was spontaneously formed at varying levels in the liquid (in

all cases but one in the upper middle section of

the liquid column). The belt was at first fractions

of a millimeter thick and gradually developed up- wards as well as downwards in the liquid.

Figure 4a shows a photograph made after the

fog had appeared. The fog consisted of two layers

separated by a sharp boundary. The upper layer

of fog entirely veiled this part of the liquid.

The lower layer of fog allowed the fight to pass

and disclosed that the mixture at these levels had

attained nearly homogeneous compositions. The

nature of this fog, however, was such that light was scattered, and at these levels a background

illumination was observed.

30 min later the fog had disappeared and, where

the upper layer of fog had been, the curve had at- tained a smooth arched shape leading downwards to a kink from which the curve by a near-vertical section continued into the original curve. (Fig. 4b.)

The smooth arched curves represent layers with

defined composition gradients. The gradient in- creases towards critical composition, and must be

regarded as a section of an equilibrium distribution curve. As mentioned the fog started as a very thin belt,

which grew broader with time. In one series of experiments the temperature was immediately

lowered just as the fog appeared. Thereby the

fog disappeared, and a comparison between the curve, before and after the fog disappeared, showed

that the fog was first formed where the original

gradient and the gradient of the developed arched

curve were equal (fig. 5) [5]. 216
FIGURE 5. System, ordinate, and abscissa as on figure 4, however, different scales. Two photographs are mounted together by dislocation along the ordinate. The

lower is the original composition curve before the development of fog. The upper curve shows the composition after the liquid mixture had been cooled and the log had thus disappeared. Visual observation had disclosed that the formation

of fog started at composition close to the left vertical orientation line, where the exposures show that the gradients of the original and the developed regular arched

curves are the same. The latter observation explains why the belt of

fog appeared at compositions somewhat different

from critical composition (in all cases but one at triethylamine richer mixtures).

The smooth arched curves which appeared after the fog had disappeared generally did not reach to the critical composition. When the system was

left undisturbed at constant temperature after the fog had disappeared, the smooth arched curve grew in both directions. The form of the originally

formed curve remained unchanged. It was, however, observed slowly to change its height in the

container.

When enough time had gone, the smooth arched

curve grew past the critical composition and passed

a point of inflexion. From this point on the composition gradient decreased with increasing distance

from critical composition. Maximum gradient was found at — or very close to — critical composition.

When the smooth arched curve was observed to grow in both directions this naturally was caused

by an increase in the height of these layers upwards

as well as downwards. This was a consequence

of a transport of material through these layers,

which transport was a diffusion flow towards in- creasing differences in composition, "reversed

diffusion flow." Figure 6 and figure 7 show exposures from two

series in which the curves mentioned above had

been established. (See legend.) In the series of

figure 6 the temperature was lowered in steps

(away from the critical). When the temperature

changed, a fine fog veiled the regular arched section of the curve for a shorter period of time, and

Figure 6. System, ordinate, abscissa and scales as on figure 4

Figure 6.a shows the regular arched curve developed In both sides of the critical composition dining 10 hr al constant temperature. 41.5 mdeg below the critical. The temperature was then lowered to S3 mdeg below critical, and figure 6.b shows

the composition curve after 7 hr at this temperature. The composition gradients had

decreased in the layers of the smooth arched composition curve. A comparison of the "lower arm" of the curve does indicate that a diffusion towards greater composition differences has taken place (towards a continuation of the regular arched curve).

+ 2 mdeg

(exp. 25) +3 mdeg

(exp. 26)

+ 5 mdeg

(exp. 27)

+ 17 mdeg

(exp. 28)

FIGURE 7. System, ordinate, abscissa and scales as on figure 4.

Pive photographs of the same series with stepwise increasing temperatures, mounted in successive sequence of time. The opening of the concentration curve in figure 7a is due to the optics of the apparatus.

In figures 7b-e two phases are formed.

217
in this section the form of the curve changed

basically by an increase of the composition gradients

dc/dh.

Figure 7 shows exposures of a series with stepwise increase of temperature. Here too a faint fog was formed as in the series above. At the temperatures of the exposures 7a and 7b the smooth

arched section of the curve changed basically at each step by a decrease of the gradients dc/dh at the various compositions.

At the step 7b the critical temperature is passed

and the concentration curve is broken close to

critical composition.

Exposure 7d is taken at a temperature 17 mdeg

above critical. The left part of the curve is arched

basically in the same way as shown on figures 1 and 3. The upper part of the right side leading to the surface dividing the phases is nearly vertical. The reason for this may be a convection flow in the lower phase during the establishment of this phase.

End Remarks

In the beginning of this paper it was said that the

effect of gravity added to the analogy which is known

to exist between the binary and the one-component

systems. This may be further illuminated:

1. The presence of a temperature gradient in the binary systems leads to the Soret effect. In

the one-component systems a temperature gradient

will lead to mass distribution curves which will constitute isobars in the temperature-density

diagram. In both cases it must however be re- membered that the effect of the temperature gra- dient in the experiments will come in addition to the effect of gravity.

It may also be mentioned that when in the one- component systems the isobars are experimentally

known, and also the gradient dp/dT at the critical density, the isotherms can be constructed. A similar relationship may be expected to hold in the

binary systems between the Soret effect and the

effect of gravity.

2. It was said that the effect of gravity in the

binary systems was a necessary consequence of

the small diffusion coefficient in the critical region,

caused by the fall of the molecules of the heavier

component towards the bottom of the container.

In the same way the known sigmoid form of the mass

distribution curves in the one-component systems I can be regarded as a consequence of the fall of

the molecules balanced by a diffusion upwards.

This leads to the conclusion that the diffusion coeffi- cients also of the one-component systems are very j small in the critical region and will apply to an equation analogous to eq (2). Hirschfelder [4] and Prigogine [5] have pointed out

that diffusion measurements are of fundamental | importance for the study of liquid mixtures on the

molecular level. By experimental determination of the equilibrium mass distributions in mixtures, preferably in fields I of increased acceleration, one should have an ! additional method for the study of liquid mixtures —

not only in the region of the critical solution points,

but also for the study of liquid mixtures in general. )

The reestablishment of the mixtures were ob- served in 1957. Professor Joseph Mayer kindly

J reviewed the reports from the experiments and I

showed that the form of the new established curves 1 were related to the gradient of the chemical poten- j

tial and consequently to the diffusion coefficient as U

function of the temperature and composition. I I

am very much in debt to Professor Mayer, who

thereby gave me the clue to the problems.

I also express my thanks to Professor Odd

j Hassel for his interest and assistance. Further- more, I express my thanks to Professor E. G. D. Cohen and Dr. J. M. H. Levelt Sengers, who have

reviewed my paper.

References

[1] Hans Ludvig Lorentzen and Bjorn Bergsnov Hansen, Acta Chem. Scand. 12, 139 (1958).

[2] Hans Ludvig Lorentzen, Acta Chem. Scand. 7, 1335(1953).

[3] Hans Ludvig Lorentzen, Acta Chem. Scand. 9, 1724(1953).

[4] Hirschfelder. Curtiss, Bird, Molecular Theory of Gases and

Liquids, p. 539 (John Wiley & Sons, Inc., New York, N.Y.,

1964).

[5] Prigogine, The Molecular Theory of Solutions, p. 51 (North- Holland Publ., 1957).

[6] Professor Joseph Mayer, private correspondence (Chicago,

j June 21, 1957).

218
Ferro- and Antiferro magnetism of Dilute Ising Model

S. Katsura and B. Tsujiyama

Tohoku University, Sendai, Japan

1. Introduction

The properties of alloys of magnetic substance

and nonmagnetic substance are investigated experimentally and theoretically as problems of dilute

magnetism. Here we consider Ising spins of S = 1/2 as a magnetic substance and the energy of the

system is assumed to be given by a sum of exchange

energies between nearest neighbor Ising spins,

positive or negative, corresponding to ferro- and

antiferromagnetic coupling. That is, the inter- action energy between nonmagnetic ions and that between nonmagnetic ions and Ising spins are neglected. This restriction can easily be removed.

The calculation based on the inclusion of the inter- action energy of nonmagnetic ions will be carried

out in future. The idealization of the preparation of the system

can be classified into two limiting cases. We

denote the spin average by ( )., and the configuration average by ( ) (

-. The free energy of the system which was heated at infinitely high temperature and was cooled infi- nitely rapidly up to temperature T is given by

FQ =-kT(ln(Z)s)c. (1.1)

Quenched random dilute magnetism has been in- vestigated by Behringer [4], Sato, Arrott and

Kikuchi [5], Brout [1], Rushbrooke and Morgan [6], Elliott and Heap [7], and Abe |8j. In experiments

ideal randomness in Brout's sense is not realized an actual "randomness" depends on the history

of the preparation of samples. Experimental situ- ation lies, more or less, between two idealized limit- ing cases. Hence it seems worthwhile to compare

physical properties of those two systems for exactly

solvable cases. On the other hand, Syozi [9] investigated a two- dimensional decorated lattice model of dilute magnetism, where Ising spins locate on lattice points

of regular two-dimensional lattice, and nonmagnetic atoms or Ising spins locate on middle points

between nearest neighbor lattice points (fig. 1). He obtained exactly thermal properties of two- dimensional lattices by reducing its partition function to Onsager's partition function. We can get

thermal and magnetic properties of the one-dimensional Syozi's model since the original Ising model

of a one-dimensional system is known in a magnetic field. We will also compare properties of Syozi's model with above mentioned two cases.

In this system the true thermal equilibrium is not

realized and it is a free energy of an idealized

quenched system in which ions are frozen randomly

at their positions.

On the other hand, if the temperature and ex- ternal parameters are varied infinitely slowly and

true thermal equilibrium is realized, the free energy

of the system is given by

FA =^ kT In (Z)s, c . (1.2)

Although the interaction energy of the nonmagnetic ions are to be taken into account, the system

corresponds to an idealized annealed system.

The distinction between these two cases has been

discussed by Brout [1], Mazo [2], and Morita [3].

1.1.1

< » i > <

> • <

r t

» O — ) • 0-

(

<

> • < >—•—

t i o —

(

m) •—

)—•—9-

FIGURE 1. Syozi's decorated lattice model of dilute Ising model. O Isinj- spins, • Ising spins or nonmagnetic atoms.

219
2. Annealed System: Formulation by

Generating Function Method

By solving D from (2.3) as a function of p and C,

D = D(K, p, Q (2.4) In this and next sections, it will be shown that the problem of the annealed system is formulated

by introducing a generating function in case of

Ising spins, and that in the case of a one-dimensional problem evaluation of the generating function

is reduced to an eigenvalue problem [10]. Moreover that eigenvalue equation is solved exactly

and the free energy, the energy and the susceptibility are obtained.

A crystal lattice whose lattice point is named

i = 1.2 .... N is considered. On each lattice point there is either an Ising spin of S = V2 or a nonmagnetic atom. If p, = 1, 0 or — 1 is the state of each lattice point, the total energy of this system

is given by

E = ~ 9 X ^3 -m%"2 p,.. (2. 1

)

can be derived. D corresponds to chemical potential of spins divided by kT in an alloy of non- magnetic atoms and spins. When (2.4) is substituted into E= ~i {^dir)

E(K- c D) (2 - 5)

and

M = m (~^) k d

=M(K, C, D), (2.6)

energy E and magnetization M can be expressed as functions of K, p, and C. This problem is equivalent to the Ising model of S = 1 with anisotropy.

Here m is magnetic moment (=\ gfxB , g'-g factor,

(Mb- Bohr magnetron) of spin, £f is external magnetic

field. J > 0 corresponds to ferromagnetic inter- action and J < 0 antiferromagnetic interaction. ^(0) denotes summations over all pairs of neighboring

lattice points. The second term is Zeeman energy. p = 1 or — 1 corresponds to Ising spin (up or down)

and p = 0 to a nonmagnetic atom, respectively.

Instead of considering the problem of the system

in which concentration of Ising spins is given by p

(that of nonmagnetic atoms by 1 —p), the generating

function Z with parameter D is calculated.

3. Annealed System — Continued.

Eigenvalue Problem in the Case of

One-Dimensional System

The formulation in 2 leads to an eigenvalue problem in the case of one dimensional system. Generating function Z is given by

Z = TrVN

, (3.1) v=v\iwyi v,vywy\ (3.2)

under a periodic boundary condition. Here

^;±1'° (2-2) vA\ 111 V2 = \

wnere

2k? kT

Since p| = 1 for spin states and 0 for nonmagnetic

atoms,

Vz = \ 1 (3.3)

If the maximum eigenvalue of V=V^WpVxV^WT

1 d In Z _ . ?

\ __ _ /js Y) C) (2 3) * s ^maxi ^ can De expressed in terms of Amax in the

N dD P PK '

' /" ' limit of 7V-> oo

220
lim Z"S =K (3.4)

The explicit expression for \max (K, C, p) which is a solution of a cubic equation is much too complicated to be differentiated analytically. It is sufficient, however, to derive Xmax exactly only up to the second order in powers of C since we are inter- ested mainly in the energy and the susceptibility

at zero field. From the symmetry of this matrix,

the eigenvalue at C — 0 and the coefficient of C

2 in the expansion of Amax can be obtained simply by

solving a quadratic equation. V can be written

as a sum of even function and odd function of C.

V =

k

2dP ch C. a ch —

,

+

k~2 d

2

c c

d ch —, 1, d ch —

r

k-'d2

, ach^, k

2cP ch C

k

2d2 sh C, d sh |,

d sh ^, 0, -dsh

0, -dsh k

2 d2 sh C

(3.5)

we have

S.7'Sr>rS,S2 = H<°>+H<»

(H<® 0 0 = o m o

o

o mi

o m

n§ m) o

(3.8)

where 6 is determined in such a way that H\V vanishes. Hm is an even function of C and HiU an odd function.

In order to obtain \max in case of C — 0, //(0) is re- garded as an unperturbed term and H(1) as a per- turbation. From second order perturbation theory

(first order perturbation term vanishes)

>w=M°i) + (3.9)

Taking C — 0 into consideration, H^, H\i§, and

Hty are expanded with respect to C,

H^} = k°° + C2k

01 + 0(C4 ),

H<$-H<®5- = C

2 \n + 0(C4 )-

(3.10)

(3.11)

where

eD = <F.

By using similarity transformation matrices

and

0

V2

0

/sin 0,

S-i = \ cos 0,

\ 0,

(3.6)

(3.7)

The term of 0(C 2

) does not appear from the third

and higher order terms of the perturbation.

Thus the maximum eigenvalue of (3.2) is obtained

exactly up to order of C

2

. When %f = 0, the relation between p and D is represented by

e- D = d' 2 =

pip — l)(k- + k--)-

(** + A-*-4)p(p-l)-l-(2p-l) V2<2 - A* - A- *)p(p- 1)+1

(3.12)

The energy E and the susceptibility \ m case

of <$f=0 is expressed in terms of d~ 2.

NJ'

1 A- + A:- 1 4k-k~ l

X 1 +

4, + k~ 2 -d-

\

/ (k 2 + k- 2 -d~ 2

)2 + 8d- 2

.

(3.13)

221
ATX =

2(X01 + X")

Nm- A

00

1

A

4 + l-d" 2 (A 2 -2)

V(A' 2 + A- 2 -J- 2

)2 + 8rf- 2 + A

2

4A-4 + 2rf" 2 +

4] (A2 + A- 2 -d-2 )(A4 ^-j+4rf-2 F

V(A-2 + A-2 -rf-2

)2 +8rf"2

3A-2 - A

2 + d~2 + V(k2 + k-2 -d-2

)2 +8d-2

(3.14)

d~ 2 in (3.13) and (3.14) is regarded as given by

(3.12) in terms of p. Substituting p—l into (3.13)

and (3.14), well-known results on the Ising model

E 1 u v ATx

Wm2 (3.15)

are reproduced.

4. Quenched System

Next we consider the quenched system in one- dimension. The partition function for a given configuration of nonmagnetic atoms is given by

(Z) s = £ exp \k £ p.m+ C£ pj (4.1)

= chPKcWiC ^ n<i

tittijthK) na +

p

A thQ

M=±l

chMKchNsC X f1 + ^i^AK+ 2 E

X(^/ pj )(^;,JLtj ,)th 2/^+. . .

x(l + S^-thC + 2SAt^th2C+ . . . .) (4.2)

Expanding the product we have

(Z) s = ch^C h'V-C £ X X {flMYjlvIM')

X (pi'" Vjffl >)fikfik' • p4n)thmi<:thnC. (4.3)

Carrying out the summation with respect to p,=± 1, we have

(Z) s = 2

N- ch^ch^C X 2 2 [ Number of graphs

»i n which have m bonds and of which the sum of the number of end points and odd junctions is n in that configuration] X th mKthn C. (4.4)

When we take the logarithm and divide by N, only

the term proportional to N in the bracket survives.

Restricting our interest in th°C and in th2 C terms

in the case of one dimension, we find that the aver- age number of graphs which have m bonds and two

end points is Npm + 1

, since the probability of finding an Ising spin at each lattice point is p. Taking the

configurational average by ( } c , we have

[n = 0, m = 0] = l,

[n = 0, to=1, 2, 3, . . .]=0

< [ii = 2, m] )c = Np"'+1

, (Ns )c = Np,

(M)c = Np\

II ence

(ln(Z)s > (

. = /Vp 2 lnchA: + /VplnchC + /Vp log

2

+ ln l+N^th2 Cp m+1 thmK (4.5)

lim - ( In ( Z ) s > f = plog2 + p

z lnchK + plnchC

AT->o=/V

Here p = ±l, in contrast to the annealed system.

Ns is the number of Ising spins. M is the number of

bonds connecting Ising spins. The summation ^ in (4.1) is carried out for only Ising spin pairs.

W

+ ^p'" + 1 th"i ^th 2 C + 0(th4 C). (4.6)

Thus the energy is given by only the first term

of the series and the series for the susceptibility

222
can be summed, giving

E

j a(ln(Z),s ), ._ NJ

2 dK —-TP~thK> ^{i+22;p-th-jc}

(4.7)

(4.8)

^ m2 Np 1 + pthX

kT 1-pthK

2 kT/ I J

I

FIGURE 2. Energy in zero magnetic field.

Solid line: the annealed system. Dotted line: the quenc hed system (0 S p S 1 1. Dashed line: Syozi's model (-J ^ p 1).

Annealed System

Ouenched System

Syozi

5 2kT/ J

5. Numerical Results and Discussions

Now we compare the numerical results of three

cases, an annealed system, a quenched system,

and Syozi's model.

Figure 2 shows the energy in zero magnetic

field. The energy of Syozi's model for p is — (p — \ ) X tanh K for \ < p < 1. Figure 3 shows the inverse

susceptibility of ferromagnetic case. Figure 4

shows the susceptibility of an antiferromagnetic case in the annealed system. Figure 5 is that for the quenched system and figure 6 shows that for Syozi's model. As the temperature goes to zero, the first tends to zero while the second and the

third tends to infinity. Figure 7 shows the concentration dependence of the antiferromagnetic susceptibility of the quenched system and the annealed

system.

In the above results the difference in the low

temperature susceptibility between the annealed

system and the quenched system in the case of

antiferromagnetic interaction is most drastic. In the quenched system the susceptibility at the

concentration p is obtained by the arithmetic mean

of each configuration corresponding to the same

I JI"X /Nn

FIGURE 3. Inverse susceptibility in the ferromagnetic case.

Solid line: the annealed system. Dashed line: the quenched system. Long dashed

line: Syozi's model. FIGURE 4. Susceptibility of an antiferromagnetic case in the annealed system.

223
IJI X/Nm2

IJ I "X /n

224
concentration p, and the paramagnetic configuration gives the dominant contribution compared with

the antiferromagnetic configuration. On the other

hand, in the annealed system, the energy of the

paramagnetic configuration is high compared with

the antiferromagnetic configuration, and hence it dies and the susceptibility tends to zero.

The authors wish to thank for valuable discussions

Professor R. Kubo, Professor M. E. Fisher, Dr.

S. Inawashiro, and Dr. T. Kawasaki. The numerical calculation was carried out using NEAC 2230

in the computing center of Tohoku University and

IBM 7090 in UNICON (University contribution

of Japan IBM).

6. References

[1] R. Brout, Phys. Rev. 115,824(1959).

[2] R. Mazo. J. Chem. Phys. 39, 1224(1963).

[3] T. Morita, J. Math. Phys. 5, 1401 (1964).

[4] R. E. Behringer, J. Chem. Phys. 26^ 1504 (1957).

[5] H. Sato, A. Arrott, and R. Kikuchi. J. Phys. Chem. Solids. 10, 19(1959).

[6] G. S. Rushbrooke and D. J. Morgan, Mol. Phys. 4, 1 (1961):

ibid. 4, 291 (1961); G. S. Rushbrooke. J. Math. Phys. 5, 1106 (1964).

[7] R. J. Elliott and B. R. Heap. Proc. Roy. Soc. A265, 264

(1962).

[8) R. Abe, Prog. Theor. Phys. 3 1, 412 (1964).

[9] I. Syozi, Prog. Theoret. Phys. 34, 189 ( 1965).

[10] H. A. Kramers and G. H. Wannier, Phys. Rev. 60, 252. 263

(1941).

The Statistical Mechanics of a Single Polymer Chain

S. F. Edwards

University of Manchester, Manchester, Great Britain

It is clear from the papers presented at this con- erence that near critical points thermodynamic

unctions and correlation functions take on forms

hich reflect only a very small amount of the deailed structure of the systems considered, and

re rather manifestations of the three dimensionality

f space and the short range of interactions. The

onfigurational statistics of polymers shares this eature and is therefore interesting as an indication

f how the critical behavior may be generated

athematically: in fact the problem is one way of

pproximating to the Ising lattice as was mentioned

y Professor Domb [1]. In this talk I shall develop

he theory applied to polymers themselves however,

nd comment on their critical behavior, since

hether the analysis will really be useful for bulk

ystems remains to be seen. A polymer without

interaction will have the well-known distribution

, (2ttLI\- 3I

'z / 3r2 \

p(r,L) = (—) exp(-—) (1.1)

for the probability that the end points are at 0, r when the length is L. This has the form

(1.2)

For the case of excluded volume i.e., a repulsion

between elements of the polymer, p has no simple

analytic form, but salient features are shown in

225
the diagram (schematic) (1.1) including the case of hard exclusions in which

2 P d

3 raL6/5

. ^9/5

(1.3)

Expressions will also be obtained for the number of configurations allowed to the polymer.

2. Mathematical Formulation of the

Problem

Firstly the problem is to be stated mathematically

and brought down to its simplest form. Suppose

the polymer consists of n identical molecules freely

hinged in a chain, and there is a potential U between molecules. If the length of one molecule

is /, and the coordinate of the end of the mth molecule

is Rm, then the probability of finding a particular

configuration P(Ri, .... Rn) is given by

P(R, R»)=^n^(R/ ' R;+,)

exp ^ £ U{R,„-Ri)

w=t=;

(2.1)

where Jf is the normalization

j

PYl(PR = l (2.2)

and p is the constraint

p(Ri, Ri+1) = 8(|Ri-Ri+i|-*)- (2.3)

(Note since normalizations are constantly occurring

• they will all be denoted by JV.) Of course, one

can put structure into the molecules making U an

integral over the molecule and one can put in bond

flexibilities, but the basic problem is already in

u |R,„-R,,| < a (2.4) = 0 otherwise.

A lattice polymer, can also be included if the defini- tion (1.3) is suitably modified, but it is again a complication not a simplification. The probability

that starting at the origin, the polymer's nth fink

will end at r, is given by

P(r, L) - R„)8(r-

(2.5)

where L, the length, is nl. A method of evaluating

p for a slightly simpler problem, that of the infinite polymers, has been given by the author (reference

[2], hereafter referred to as I) in a paper which

develops a physical picture of what is going on and

translates the physical picture into mathematics.

A more formal but precise method of approach is given in an appendix to I, and in this paper, after reviewing the physical arguments, it is this formal

development which will be given and extended to give a more complete theory of the single finite polymer.

Firstly it must be explained that the fact that there are n integrations in (1.1) is a nuisance; it is easier to consider a continuous function R(L') where

0 < L' < L and integrate over all functions R(L').

The constraint FJ p(R;, Rm) is the expression

originally considered by Weiner when he introduced

functional integration and can be expressed as

3Z JV exp ( —

y

3R

dU dL' (2.6)

so that, putting V= Ul~2

p(r, L)=JT [

R,i

' = r d(path R{L'))

J

R(0) = 0

exp

3/ BR

dL'

1 ~2kT Jo /o

V R(L')-R(L"))

dL'

dL"

(2.7)

At this point a parametric representation can be

226

!
employed. It is well known that

| e^-^i-dx i |

X e-^-dx. (2.8)

This can be generalized to

exp ( — 2 OiAijaj/2^ = I I exp ( / ^ ajXj — ^j? XiAjj?Xkl2

J-xJ-x \ j f. ft /

g-s^- ncfec, (2.9)

where is the inverse matrix to y4, and hence to the functional form

exp (-j a(a)A (ap)a(p)dad(3)

{Yldx) exp i a(a)x{a)da

x(a)A- l (aP)x(f3)dadl3

exp x{a)A- l ((x(3)x{fi)dadp Yldx

(2.10)

case one has to split V into a part with positive

Fourier coefficients and a part with negative Fourier

coefficients and parameterize them separately.

This will be equivalent to \ becoming complex

and the functional integration being over its real and imaginary parts independently. Although

physical forces always do have attractions and

repulsions it is well known from the study of the

Ursell-Mayer cluster expansion, that provided the

system remains of low density the effects of inter- particle potentials can be simulated by pseudo

potentials which will give rise to the correct virial coefficient. Thus if the excluded volume is defined

by

V =

j

[1-exp (-U/kT)]d?r (2.14)

the effective pseudo potential is v8{r), whose in- verse is v~

l 8(r). It will be assumed now that v

is positive, i.e., the mean effect of the forces is a repulsion. Then finally

I

L X(K{L'))dL'

2v X

2(S)(PS 8X8R

where A

1 is the inverse operator to A,

j

A-1 (a/3)A(/3y)dl3 = 8(a - y).

This theorem allows one to write

~2^rr v(R{L ' ) - R{L" )]dL

' dL"

= Jtr

\ 8\ exp

AT

x(R(L'))dL' ~| | d

s r(PSx{r)V-1 (r-s)Xls)

where

/

(2.11)

(2.12)

V-\rs)V(sr')(Ps = 8(r-r'), (2.13)

and j

8\ represents the integral over all functions.

It is supposed that j

_1 X i s positive definite, corresponding to V being repulsive. If this is not the

(8R for the integral over all paths). (2.15)

The theorem states that the probability p is that of

diffusion in the presence of a potential ix (a Markov

process), averaged over all potentials with a Gaussian weight. It is a well-known theorem else- where in physics. Since the evaluation of the

functional integral over 8R is equivalent to solving

the diffusion equation, one can obtain the final form

p(r,L) =jrj 8XG{r,L, [xiWdxWx (2.16)

where

r([X]) = exp(--| X

2d

3s (2.17)

^-^V2 + / X(r)l' r. / x r ,v/ (2.18)

and Jf~Jf(JS) is defined from pdA r— 1.

227

209-493 O-66— 16
3. The Solution

Two problems have presented themselves in

(2.16). Firstly one has to get G from (2.18) and

then do the integral over X- It will be argued that the mathematical tools for these problems are al- ready available, at least within reasonable stretching: the Green function can be got by the JWKB

method, and the x integration by the method of

steepest descent. Physically one can argue that,

since one knows that the polymer starts at the origin, on an average a "polymer density" will build up

having its maximum at the origin. It is this field which will be uncovered by the steepest descent

calculation, and it will be a field with a long range

effect, slowly varying in space so that the JWKB

method will be the correct method to assess its effect on the diffusion. That this mean field can

be expected to be long range is seen at once from

the noninteracting distribution (1.1). From (1.1) the probability that a very long noninteracting

polymer goes through r for some L is

'2tt

/./ exp

3

2tt/

1

where contains the angular derivatives. It will turn out that the steepest descent function is spherically symmetric, so the angular terms will be ignored for the present and </> treated as if it were <f)(r, E) and x( r) also as if it were x( r)- The

JWKB argument is now that the (dfi/dr)2 term dominates over the (Pcfr/dr term (which can be con- firmed subsequently) so that

/

2 (E+X(s)) llzds

Jo

(3.4)

and one is left with

p = Jf 8\dE exp U-X'

2 i^(E + X) ll2 ds

iEL — — X

2(Ps Zv

(3.5)

(One must watch for complications at the branch

point E + x — 0; note also that one integral is ds,

the other d

3

s). To evaluate, one looks for the

E, x which make the exponent stationary: differentiating with respect to E

If there were a very strong self-repulsion, the poly- mer would shoot out in a straight line and the aver- age would be taken over the orientation of this straight line, i.e., would give (47rr2

) -1

. These

give upper and lower bounds on the form of the mean polymer density so that one expects the x

integration to be centered about a steepest descent

function x having a dependence lying between —

and \ and in fact it will turn out to go like -jjx. r

2 r* 6

It is convenient to Fourier transform G with

respect to L

iE-^V2 + ix [ X]) = S(r) (3.1)

and to seek a solution in the form

G = e*/r.

Then

(3.2)

iL- l/6iV/2 2 W(E + Xrmds

and functionally with respect to x

1 -(-X

l2 (E+xr^

s<r

(3.6)

(3.7)

= 0}.

Therefore one may write also

I2 piE — — x(s )ds.

V Jo

Clearly X = X< i - E)

and E = E{i\ L). Make these equations clearer by introducing

5 / L \ n1 '6

(3.8)

(3.9)

(3.10)

3 V^y 2 16/

v+^w-ig-my-o ,3.3,

1 \

2' 3 /6\'/2

8tt // V/

2/3 (3.11)

228
s' = s/r

Then one has to determine e in terms of £ where

5 f

1 ds's' 2'3

p(r L)=JT(L) exp [-r^g, e)J (3.18) = ^(L) exp [ - r^Lr-5'3

)] (3.19) = yl/"(£) exp [ —Z(r, L)]. (3.20)

sav. ^ 3 J,, (tf + es' 4/ 3

)

1 ' 2

?? = (^ + e5' 4/ 3 )- 1 / 2

£ = f f

1 ds's' 2l3#(s'). J Jo The e, 5' relation is plotted below

(3.12) The function e

z

is sketched in the introduction.

Its maximum comes at dZ/dr=0, i.e.,

(3.13) 0 = ^£^+

aza£, f §z dx(s)

(3.14)

s'=o

(3.15)

Special cases are e = 0, #= 1, (£=1); e = — 00, ^e-^s'-2/ 3

; e =-oo, # = - e5 '4/3

_ Thig 5 physi. cally sensible, solution has # positive and leads to a relation like

(3.16)

One may now see that neglecting fluctuations

for the moment

p(r, L) = .#"(L) exp [0r (E + x) mds

iEL — 7- I x

2^

3s (3.17)

d£ dr J 8x(s) dr

(3.21)

Since Z is stationary with respect to E and x ^is

gives

^xVmRHy) (£ + x(o»)

1/2 (3.22)

i.e.,

X(r=rm)-

Near the peak the

6

\l/3 / „ \2/3 I 477/2 / >i'

3 (3.23)

A 1/3

t, = im i.e., L = l-j rf,{ 3 X constant. (3.24)

And the distribution looks like

JT{L) exp Lf(r-g1sl3 (IY13 J

r^t2v

(3.25)

where / and g are constants derived from (3.23), (3.13), and (3.14). For the right £ is small, e large so that £ 9? e" 1/2

.

(3.26)

This implies a distribution bA'exp (— cr72L) where

b and c are constants. Towards the origin p be- comes very small, and one is in the welbknown

trouble of trying to fit a tail of a function by working

away from its maximum. This will be considered

in detail in section 4. The fluctuations_i.e., the remaining quadratic

integrals over E — E and x~X have not yet been

229
considered. Unlike steepest descent methods in a single variable, these are of great consequence

and indeed are the substance of conventional field theory. To go straight ahead expanding <f> to order

X

2 is inadequate as can be seen physically. Consider what one expects the addition of one more

molecule to the polymer to involve. Say for the

moment one confined oneself to steps on a close

packed lattice. There would be at most eleven

sites to go on to i.e., 12, less the one it is at. But

it must have wandered around the neighborhood

before it arrived at the present site, so one expects

a further reduction bringing the number of avail- able sites to /A<11. In addition the long range

effect reduces it further to fji(l+f(n)) (where f(n)

will turn out to be (log n)/5n). If one were not using the steepest descent method by normal field theory one would argue, after Dyson, that by

studying the perturbation series, that (using (G)

for (2.16) unnormalized)

(iE -- A

2 + 2(/b, E)) (G(k, E)) = 1 (3.27)

where in the first approximation

l(k,E)= [

|F(k ~j)| V; (3.28)

the origin. The local effects should be adequately

handled by the conventional methods and they

really will involve the fine structure of V, and if one

attempts to use the pseudo potential one gets

divergences. To avoid these one must write

p(r,EAX]) = -e J J (331)

where

(V 2 + V,4VM(r) = -x(r)

(V 2 + 2V,4V)£(r, r') = 8(r-r')

(V 2 + 2V//V)C(r, r2 , r3) = Vfi(r, r2 )V£(r, r3)

I2 1 v A

477-r

2

and so on. The interference between x and the

fluctuations is analogous to the Lamb shift and a proof is needed that this does not interfere with the

functional forms discovered so far. Though this appears to be so by explicit calculation, I am trying

to obtain a general proof on the lines of the Dyson

treatment of electrodynamics.

4. The Inner Region

but it is difficult to get in general. If (G) is con-

/ A' 2 \ sidered to have a pole i.e., ( iE — /' — + 20 J (G) = 1 and 2 is expanded around it

(3.29)

E- <±±Mlp+-*!L.\ So

(G) -_1

1 + 2,

(3.30)

Here T+2~ ^ = ^' ^ S 3 "renormanzec^" ste P length,

and the 2o/(l + 22) factor represents the changed

number of available sites as expected. (Since it gives a e~loL factor in (G) it will not appear in p).

The conventional field theory fails completely to describe the long range effects and it is these which

determine the r

2

: L relation. As has been shown

the present theory suggests that these long range

effects are simulated by a potential centered at

One can see why the inner region has been so

far inaccessible. The solution has a peak near L — r'i/5 and almost all the probability lies in the

peak. This means that though the chance of the

polymer getting back close to the origin is small,

it is much larger than that suggested by the solu- tion (3.5). That solution suggests that in order to be still near r ~ 0 as L—* °o one must have stayed

all the time near r ~ 0. Clearly what will happen

is that one will go out into the high probability

region and then walk back to the inner region.

Since under the parametric integral sign one has a Markov process one may always break up G

G(r, L, [ X]) =

j

G(r — s, L — M, [x])G(s, M. [xW's (4.1)

j Note there is no integral over M and the left-hand

side should be independent of M. To modify (4.1) I to be M independent one can average along the

230
whole length to give

= ± ['' dM f dh L) 0

j

8XG(r-s, L-M[X])G(s, M, [ XW (4.2)

Now average using the JWKB for the G

p = j- j'_dM j

d?s j

Sxlr-sl-'lsl" 1 j

dEij dEi

+ (y-)'

2 T

{El + *

)1/2 ~ iEAL ~ M) ~ iElM

than one point and get a better value for this constant, but the n~915 would still come the same. 1 Having sketched out the entire function one

can now normalize and get the height of the peak

£-17/10 which clearly dominates the rest of the

function. By evaluating r2 one finds a series, the

expansion parameter being (L-1/10ir 1/3 /7/1 °) which

must be small for the above analysis to be valid. Finally, the number of walks can be remarked

upon since it is an interesting combinatorial point even if it has little bearing on polymers. As has

been noted, the dominant feature will be the local

effect of the effective phase space available per

molecule. The long range effect will come from

the unnormalized Green function which is

(4.3)

v @(|a — s|<|r — s\) + T2 |«- 5 | 2(£, + x)

1/2

d?a

(4.4)

\ / / J|a-s|<|r-s|| 5 — af

I J|a|<|s| a

2 A

In particular to get back to r = 0 one has

'6i\ 1/2 f0(|a|<|s|)

(4.5)

(4.6)

-if

(E-z + x)-

112

+

e(la-sl<| g |) \a —sr

(4.7)

+

e(|a- s |<| s |) (£i+x)

_1/2 a — s (4.8)

I won't write out the details from here on for the answer is defined dimensionally and must be

p(0, L)°c L~915 i.e., n9/5 (4.9)

The details of the integration serve to fix the constant. Presumably one could break up G at more

7^exp(-j" (L-g^rir3}

(4. 10)

Integrating over r one gets the leading term to be

p/'n 1 ' 5.

. 5. The Collapse of Polymers

What happens when the virial coefficient changes

sign? 2 Instead of the repulsion pushing the

polymer out faster than the random configurations,

it will now be pulled in. For v large and attractive

the polymer will condense into a solid ball, a state easily obtained experimentally even for a single

chain. But for v near zero a new semicondensed

state appears possible, and passage through the

Flory temperature (i.e., v — 0) will produce a phase

change for a very long chain [3]. The physical pic- ture will be of the polymer repeatedly passing

through the same region of space so that again a polymer density is set up which now traps the poly- mer. Consider this problem from the point of view

of I i.e., study the probability that, after a chain

length L'{0<L' <L), the polymer will be at r, q(r, L' , [Z,]). As in I (where the present q is called p) one can argue that the polymer field can be repre- sented by approximating

1 This value has been conjectured by Fisher, Faraday Society Discussion 25, p. 200.

2 Dr. J. Mazur has pointed out to me that t =0 may be much more complicated than

is generally supposed. The discussion of this point due to Flory assumes that chains

with r = 0 are Gaussian, but a single chain is a distinctly nonuniform entity having a high "polymer density" at its center of gravity, lending to a low value at the end points. So one could have a 'tlL)" effect from the middle to the ends. The following discussion may need modifying in the light of this comment.

231

1
Jof'TJo V(R{L')-R(L"))dL'dL"

= V(R(L')-s)q(s, L")dL'dL". (3.32)

But q{r,L') can now be argued to be independent

of L' since the self trapped polymer will repeatedly

go through every point of its density; it will, of course, stil\ be a function of L. Since V is short

range one may now approximate bv writing

V(R(L')-S)q(S, L")dL'dL"

w L CL

JLi JO JO

L

q(R(L'),L")dL'dL" (5.1)

q(R(L'))dL'

where w is Lvl~ 2

. This now gives a functional

integral in the Markov form so that one can break

the functional integral up to L at an intermediate

L' and define

q{r',L') = JT{L') j

exp -y'J jJo( [W dL

w q(R{L") )

I o dR (5.2)

which satisfies

(df7 "! V2+ ^»-C(L'))(? = §(r). (5.3)

Thus one is led to the eq (5.6) in which C now

takes the role usually taken by the eigenvalue,

and since q will not depend on L' , q can be identified

with it:

l-V2 + w(q(r)-q 2 ))q(r)=0

q(r)d3 r=l

q

2(r)d 3 r. (5.6)

By redefining r—r'w/l one brings the equation

into the final form

rather like the Schrbdinger equation with an at- tractive potential and eigenvalue q

2

. The solution

has . to be obtained numerically and clearly will have the form

/ / 3 \

3 q = Q(rPlvL)(^j

, (5.5)

where Q is a dimensionless function. Clearly in

this case

r

2 a v2L2 t~6 (5.6) which is quite different both from the L6/5 and the

Einstein Law, r

2 ^ LI. The distribution changes

abruptly as

v) 0^v = 0^v (0.

Ignoring C for the moment, the resulting differential equation will have eigenfunctions E„.

(a77 _

6

V2 + w^r>)<7» e£ "L = 0 (5.4)

(JT~Z V2 + wq(r)^g(r, L) = d(r)8(L')

g(r, L) =2 qn {r)qt (O)e^' ®(L'). (5.5)

There will be positive and negative eigenvalues.

As L' —> oo only the positive ones survive and unless

the largest positive eigenvalue tends also to zero

the whole expression will diverge i.e., this eigenvalue is brought to zero by the normalization.

I have been greatly helped by discussions with

Professor G. Gee, Professor C. Domb, Professor

M. E. Fisher and Dr. J. Mazur.

6. References

[1] Reference to the latest numerical work is contained in Domb,

C, Gillis, J.. and Wilmers, G., Proc. Phys. Soc. 85 , 625

(1965); and in a paper by J. Mazur to be published. The

relation to this problem as a run-in to the Ising model is discussed in Professor Domb's contribution to the conference.

[2] Edwards, S. F., Proc. Phys. Soc. 85, 613 (1965).

[3] The standard text particularly with reference to v — 0 is Flory, P. J., Principles of Polymer Chemistry, Cornell

University (1953). 232
Some Comments on Techniques of Modern Low-Temperature Calorimetry

I. M. Firth

University of St. Andrews, St. Andrews, Scotland

Introduction

Precision calorimetry at low temperatures started

in 1929 when a sensitive phosphor-bronze ther- mometer was made by Keesom and Van Den Ende

[1]. This led shortly to the discovery of the heat

capacity connected with the superconductive

phase transition [2] and also the electronic heat

capacity in metals [3] was discovered and found to be in reasonable agreement with the theoretical

predictions. Recently with the use of carbon

resistance thermometers [4] and a-c bridge lock-in

amplifier methods of measurement [5], a new era

of precision has been entered. This has allowed

the more exciting phase transitions at low temperatures to be checked. This era has come opportunely for at a recent meeting in Washington on "Phenomena in the Neighborhood of Critical

Points" [6] it was emphasized that to obtain agree- ment with theoretical predictions very great care and precision must be taken in experimental measurements, not only of heat capacities, but of all properties related to such critical phenomena:

e.g., magnetization, thermal conductivity, optical

and neutron diffraction. In considering the present situation in calorimetry, it is useful to review the way in which the main requirements of low temperature calorimetry were introduced. Although Gaede [7] used a metal

calorimeter isolated somewhat from its surroundings

it was Nernst [8] who recognized a few years later

in 1909, that on account of surroundings there was usually a temperature gradient inside a calorimeter which would prevent making accurate, or often

meaningful, measurements. To obviate this he

placed the calorimeter in a vacuum realizing that,

radiation being small at low temperatures, it was

possible to obtain excellent thermal isolation and

reduce the above error to a minimum.

Of course, it was necessary to change the temperature of the calorimeter and to do this he used

an exchange gas, noting in the course of his work

the drawback of remnant gas on the thermal isola- tion and, absorbed gas on the calorimeter on the

heat capacity measured. Temperature was measured by a platinum resistance thermometer the

wire of which also acted as the electrical heating

element for the calorimeter. Measurements with

this arrangement were uncertain even with the most

careful calibration of the resistance, because of irregular changes in the resistance of the platinum

wire heating the calorimeter. Nernst avoided

this disadvantage and at the same time made his method more simple, by measuring the variations

of temperature resulting from electrical heating by

using a thermocouple [9]. One junction was fixed

to the calorimeter and the other lay on a lead block

also in the vacuum chamber, figure 1. The heat

capacity of lead being large, the temperature of the

block remained nearly constant and any small drift took place slowly and uniformily so that the temperature of that junction could be corrected. A

copper shield that enclosed the calorimeter was

soldered to the lead block and this refinement was a

great improvement on the open calorimeter.

Nernst could measure temperature to 0.1 °K and

specific heat to ~ 2 percent.

It is easy to see from this arrangement how adi- abatic methods developed. The thermal insulation

of this device can never be perfect so long as there

Pb

FIGURE 1. Calorimeter of Nernst showing the lead block, copper shield, heaters, and measuring thermocouple.

233
is a temperature difference between the calorimeter

and the shield. But, by heating the shield so that the temperature difference is zero at all times, i.e., there is no voltage across the thermocouple, there

will be no heat input to the calorimeter by radiation

or by conduction along its supporting threads. All heat that is applied to the calorimeter through the

heater then goes completely towards raising its temperature and no corrections have to be made for heat losses: a correction has to be added because

of the joule heating in the heater leads but it has

been shown rigorously that half the heat generated

in the leads goes into the calorimeter [10]. In adiabatic schemes a second thermometer is re- quired to measure the temperature of the calorimeter.

Methods

Nernst used a constant heat input to his calorimeter but since then intermittent heating has often

been used in adiabatic calorimetry. However,

recently in accurate work on some phase transi- tions e.g.. Buckingham and Fairbank [11] on the

superfluid transition in He4

, Schiffman et al. [12], on superconducting transitions, a preference has

been shown again for constant heating methods.

There are two reasons for this preference. In measuring the input of heat during a pulse where

the pulse may be only a few seconds in duration,

the current through, and the voltage across, the

heater are difficult to measure accurately. The

power is often of the order of a few ergs per second

which means that currents of a few microamps

and voltages of a few millivolts have to be measured

with devices that have a short response time.

Secondly, after the pulse, the recorded temperature

will not settle down for some time, and in any case

the temperature immediately after the pulse is difficult to ascertain. It is important to know what

this temperature is after the pulse and how it is drifting with time so that meaningful extrapolations can be made to the middle of the heating pulses to measure the temperature increment caused by the

pulse. Schiffman [13] has observed that in the same temperature range, from pulse to pulse, the

calculated heat capacities of the calorimeter may

not be consistent. Where measurements on solids are carried out, "overheating" may easily occur

(as Parkinson and Quarrington [14] have described)

because of the small thermal conduction of the solid.

This gives more uncertainty about the temperature

of the sample for it will always be less than that measured for the calorimeter. Corrections for overheating can be applied but extrapolation of the

temperature drift curves on both sides of the heat

pulse are more uncertain. It should be stated,

however, that where the extraneous heat input to the calorimeter is large only the pulse method may

be useful.

It is usual to measure temperatures in the

range 0.3 to 5 °K with an a-c bridge, using a high

gain lock-in amplifier -as detector, and a carbon re- sistor as the temperature sensitive arm of the

bridge [4, 5], Commercial carbon radio resistors,

especially made to be temperature insensitive at room temperature were found some years ago to be

most usefully sensitive at low temperature. Bridges

of this type can detect changes of 10~ 7 °K, that is 0.02fi in 46 kfi when the resistor is chosen with

care. In the advanced scheme of Schiffman et al. [12], during continuous heating the bridge is repeatedly

unbalanced by very small steps in a sense that the

bridge returns to balance as the heating progresses. The time taken between successive balances is measured by electronic counters. The temperature

jump between balances is small and so the heat

capacity is related linearly to the power input and

the time intervals. There is no doubt about extrapolating the heating curve as there is in the pulse

method, for this is an inherent feature of this method.

The same method can be used to evaluate the

extraneous heat input to the calorimeter. Heating

is stopped and the temperature drift observed in the same way. It is a simple matter to examine

the stray heat input at different temperatures in the range under study. More consistent data on

the heat leak is also obtained by repeating measure- ments with varying electrical heat inputs knowing

that the heat capacity remains constant, so obtaining the heat leak from a difference equation, figure 2.

HEAT LEAK DETERMINATION / HEAT CAPACITY DETERMINATION HhQa ,,

Ql , , II II li ii ii

TEMPERATURE

FIGURE 2. Heat capacity and leak determined at two different heat inputs. 234
With careful attention to the points that follow in this discussion, leaks as low as 1-10 erg per minute,

a temperature sensitivity of 10"6 to 10~ 7 °K and

an accuracy in measuring the heat capacity of 0.05

percent can be obtained.

An interesting experimental feature of the con- tinuous heating method can be noted here. If the

continuous record of the output of the bridge is simultaneously differentiated, a direct recording

is obtained that is. to first approximation, proportional to the heat capacity. This is so because

over small increments of temperature both the

temperature response of the carbon resistance

thermometer and the response of the off-balance

voltage of the bridge, with respect to the resistance

thermometer arm. are practically linear. Directly

monitoring the heat capacity in this way is often

useful in observing a singularity in the temperature

dependence of the heat capacity that characterizes

so many critical point phenomena.

The best possible radiation shield is one that encloses the calorimeter within the vacuum space. The temperature of the shield must be monitored,

compared to that of the calorimeter and adjusted

to it by a controlled heater in adiabatic schemes.

For temperatures above 20 °K the comparison is conveniently done by two thermocouples. Below

this temperature comparison by temperature sensitive resistors is difficult because it would be most

difficult to make a. matched pair: in the case of

commercial carbon resistors, choosing a matched

pair would be almost impossible. In this temperature range, therefore, the shield very often is held at some mean temperature close to that being

investigated. This is a perfectly satisfactory tech- nique when the range of measurement is small,

so often the case in studies on critical phenomena.

In fact, where the bath temperature can be adjusted,

and adequately controlled, the shield may be dis- posed of completely, the walls of the vacuum chamber acting as the "adiabatic" shield. This omission

restricts the versatility of the apparatus for it is difficult to adjust liquid helium to above 4.2 °K,

because the vapor pressure is higher than 1 atm,

or to just above 2.2 °K where boiling takes place

vigorously.

Hence for greatest versatility an independent

shield should be used. Germanium thermometers

[15] are very useful in measuring the temperature

of the shield for they can be calibrated under

repeated cycling to room temperature to 0.05

percent and they respond to changes in temperature

in 50 msec. The temperature controlling system

must be smooth and accurate in operation, or as Nernst emphasized [9]. the heat leak will fluctuate randomly. In certain applications a temperature

guard ring from which the calorimeter is suspended

may be sufficient to reduce heat leaks: all electrical leads and other connections to the calorimeter are, of course locked to it. RF Effects

It was not realized until the report of Ambler

and Plumb [16] that there is often considerable

eddy current heating in calorimeters caused by

commercial radio transmitters, particularly those

in the television bands. Although the situation

in the U.K. is not quite as serious as that in the U.S.,

because of the greater restriction on the number of T.V. channels, there will be very few laboratories now that are not within 50 miles of some transmitter.

Television transmitters have a very small vertical

beam angle and generally have some horizontal

directivity. Consequently they have a high aerial gain so that the effective radiated power may be

as much as 500 kW. At a distance of 10 miles

the rf radiation density would be, therefore, about

104 erg sec

_1 m2

, and at 50 miles, as much as 4 10 2 erg sec

_1 m-2

. In a poorly screened system, with

many earth loops, if only a small part of this is picked up by the equipment connected to the

calorimeter there would be an overwhelming contribution to the otherwise small extraneous heat

input to the calorimeter.

It would be possible to enclose the whole ap- paratus in an rf shielded cage but this is inconvenient and expensive. In preference, other

precautions should be taken: as much equipment as possible should be operated from battery supplies;

leads should be screened with a solid, continuous

conductor — normal braided screening is effective

in keeping rf fields in. but is quite useless in excluding them; necessary removable joints in the leads

should be made by connectors that give complete

screening and force the connector pins together

with great pressure.Vibrations

This is only one of the extraneous heat inputs

that can be avoided with careful experimental planning. Others arise from vibrations of the whole ap235
paratus or of the calorimeter on its suspension

system. Good isolation of the cryostat from the

laboratory can be achieved by using a massive

frame mounted on supports whose period of oscil- lation is very low. Coil springs with a dampening

cushion of crimped knitted stainless steel mesh

is good but an air or water hydraulic suspension

will give a very long period, the pressure of the

fluid being adjusted accordingly. Where the cryostat bath is pumped to reduce its temperature,

the periodic fluctuation in gas pressure in the

pumping line caused by the periodic pumping action

of rotary pumps, can be eliminated by using a large

baffle volume or a system of plumbing that causes

destructive interference of the periodic pressure

fluctuations at the cryostat head. Vibration along

the pipes must be removed before the cryostat

suspension by massive clamping or ingenious bel- lows systems. Isolation not only reduces the

acoustical heat input but removes the experi- menter's dependence on the random noisy doings of

his colleagues!

It is customary now to use mechanical heat

switches instead of low pressure exchange gas

to cool the calorimeter and to make adjustments

of its temperature. To achieve good thermal contact over the switch at low temperatures the pres- sure between calorimeter and bath has to be large even though the contact surfaces are coated with

indium, which remains somewhat plastic even at low temperatures. Raising the calorimeter by the

supporting threads does not generally produce

sufficient contact force and upon release gives rise to wild pendulum oscillations of the calorimeter. For

an average calorimeter swinging over 1° these

oscillations add 102 ergs to the calorimeter and

equilibrium may not be established for many

minutes. The use of a metal bellows piston to

lift the calorimeter on to the heat switch enables

a large contact force to be exerted and allows controlled slow release on to the suspension threads

without causing oscillations. Heating

As only a few ergs per second are required in heating the calorimeter during heat capacity measurements, a mercury battery is a very convenient

source of stable heater current. The current and

voltage to the heater can be monitored, or if the

heater resistance can be reliably calibrated in the

working temperature range, only the current need

be recorded. Often constantan wire is used as the

heater element but in the range 1 to 5 °K this wire

shows a large variation in resistance. The exter- nal heater current control circuitry can be chosen

so that the power input is least sensitive to this change in resistance.

Accuracy

To obtain the highest long term accuracy, the important parts of the measuring equipment must

lie placed in a thermostatically controlled, room

temperature bath. The a-c bridge for temperature

measurement and the heater supply and control

circuit are the most important. Remembering

that a change of 1 part in 106 in the calorimeter

thermometer is to be measured, the other arms of the a-c bridge must he stable to this amount. Considering the temperature coefficient of standard

resistances to be 0.001 percent per °C. a local change in laboratory temperature of 0.1 °C will be

sufficient to reduce the required accuracy. The

accuracy is also hampered by the seeming impossibility of finding a switch for the other arms of the bridge that does have a reproducible contact

resistance on repeated cycling.

The accuracy of the whole experiment depends

largely on the accuracy of calibration of the ther- mometer. There have been many suggestions as

to how the original equation of Clement and Quinell

[4], representing the temperature response of carbon resistors, can be improved. The more

notable of these have been by Hoare [17] and

Moody and Rhodes [18| who introduced an expression involving a large number of adjustable parameters in the form ;j,= ^w/„(ln R)". n = 0. 1. 2. . . .

This expression allows easy recalculation to high accuracy of all the parameters after each

calibration, and removes the necessity to use cor- rection graphs. Having an exact description of the temperature response facilities subsequent

conversion of resistance to temperature in calculations to determine the heat capacity.

In this expression the best values for the

coefficients are determined by minimizing

1 1 1 77 where T, is the absolute

temperature obtained from accurate vapor pressure

measurements. Tm the temperature given by the

above expression from measurements of resistance

236
and N the number of pairs of T and R used in the

calibration. Minimizing this quantity leads to the linear simultaneous equations for a,t and does

not give undue emphasis to any region in the temperature range. Moody and Rhodes found it unnecessary to go beyond n — 3 because more

terms than this did not reduce significantly the rms deviation between experimental and calculated

values of temperature.

An important part of this calibration proceedure

is that a computer was programmed to perform the

calculations. The rms deviation was calculated

and calibration points that were above a preset

value, for a given value of u, were rejected: the re- sponse parameters were then recalibrated until the allowed deviation was not exceeded.

Automatic Evaluation of Heat

Capacity

There can be few workers now who calculate

manually the heat capacity from experimental

data: the repetitive calculation involved can be

better done, in less time, by computer. With a well designed method of undertaking the experiment, data can be recorded automatically so reducing the overall time to perform the calculations, and

the errors in recording large amounts of data. In the continuous or intermittent techniques the

methods of recording data automatically are dif- ferent, intermittent method being more difficult. Here, for each heating period the duration, and

voltages to the heater are recorded to give the energy

input and so must be the values of pairs of temperature and time during the drift periods before and

after the heating pulse. Care must be taken however, not to record any overheating for only those

parts of the drift curve that are linear will allow a computer to be programmed to extrapolate to the

middle of the heating pulse. It is clear from the

discussion of Parkinson and Quarrington [14] that

this purposeful neglect of important data cannot

lead to results of great accuracy. However, auto- matic schemes using intermittent methods have

been devised [19]. From what has been said already about the tech- niques of Schiffman et al. [12], automatic methods

could be used here to obtain meaningful records:

the energy input can be monitored by repeatedly

recording the current to the calorimeter heater:

the time between known small temperature jumps

is easily available. The temperature drift, caused

by extraneous heat input, can be recorded similarly, as can the thermometer calibration information,

except here the bath temperature would be recorded

against a definite balance condition of the a-c bridge.

The computer programme could be in three parts.

Firstly, the calibration calculation of the carbon

resistance thermometer; secondly, the heater input, extraneous heat input and temperatures of the calo- rimeter would be found; and thirdly, the specific

heat would be computed. After taking experimental results for some 12 hr, the final results should

be obtained in about 30 min of computer time.

This is to be compared with 30 to 40 hr by manual

calculations.

To take full advantage of this modification of calorimetric practice both techniques and data

recording proceedings have to be carefully considered. Precautions must be taken to ensure that necessary information is not thrown away in the

process of recording and that sufficient checks are included in the whole process, from apparatus to

final results, so that meaningful results are obtained.

References

[1) W. H. Keesom and J. N. Van Den Ende. Leiden Comm.

No. 203C (1929).

[2] W. H. Keesom and P. H. Van Laer, Physica 5, 193 (1938).

[3] J. A. Kok and W. H. Keesom, Physica 1, 175 (1934).

[4] J. R. Clement and E. H. Quinell. Phys. Rev. 92, 258 (1953).

[5] C. F. Kellers The specific heat of liquid helium near the lambda point. PhD Thesis. Duke University (1960).

[6] Phenomena in the Neighborhood of Critical Points. NBS

Misc. Publ. 273.

[7] W. Gaede. Physik. Zeitscher. 4, 105 (1902).

[8| A. Eucken, Physik. Zeitscher. 10, 586 (1910); W. Nernst. Ann. d Physik 36, 395 (1911).

[9] W. Nernst. Zeitscher. f. Elektrochein. 20, 357 (1914).

[10] W. Mercouroff, Cryogenics 3, 171 (1963).

[11] M. J. Buckingham and W. M. Fairbank. Prog. Low Temp.

Phys. 3, 80 (1961).

[12J C. A. Schiffman, J. F. Cochran, M. Garber and G. Pearsall. Rev. Mod. Phys. 36, 127 (1961).

[13] C. A. Schiffman, private communication.

[14] D. H. Parkinson and J. E. Quarrington, Proc. Phys. Soc. A

67,569(1964).

[15] F. J. Low, Advan. Cryog. Eng. 7, 514 (1961).

[16] E. A. Ambler and H. Plumb, Rev. Sci. Instr. 31, 656

(1962).

[17] F. E. Hoare, Proc. Phys. Soc. B. 68, 388 (1955).

[18] D. E. Moody and P. Rhodes, Cryogenics 3, 77 (1963).

[19] H. .). Blythe. T. J. Harvey. F. E. Hoare. and D. E. Moody.

Cryogenics 4, 28 (1964).

237

List of Participants

Alperin, H. A. Brookhaven National Laboratory

Upton, N.Y.

Alpert, S. S. Columbia Radiation Laboratory

Columbia Universit) New York. N.Y.

Als-Nielsen, J. Atomic Energy Commission

Research Establishment Risii Roskilde. Denmark.

Anderson. P. W.

Bell Telephone Laboratories Murray Hill. N.J. Andrews. D. University of Missouri Columbia. Mo.

Arrott. A. Ford Motor Scientific Laboratory Dearborn. Mich.

Baer. S. Yeshiva University

New York, N.Y.

Barieau. R. E. U.S. Bureau of Mines

Amarillo. Tex.

Barry, J. H. Ohio University

Athens, Ohio. Bashaw. J. Cornell University

Ithaca, N.Y.

Baur, M. E. University of California Los Angeles. Calif. Beckett, C. W.

National Bureau of Standards Washington, D.C. Beenakker. J. J. M.

Kamerlingh Onnes Laboratory

Leiden. The Netherlands. Benedek. G. B. Massachusetts Institute of Technology

Cambridge. Mass.

Belts, D. D.

University of Toronto

Toronto, Ont., Canada.

Bienenstock. A. Harvard University

Cambridge, Mass. Bloom. M.

University of British Columbia

Vancouver. B.C., Canada. Blume. M.

Brookhaven National Laboratory

Upton , N.Y. Boyd, M. E. National Bureau of Standards Washington. D.C. Bru m berger, H. Syracuse University

Syracuse. N.Y. Buckingham. M. J. University of Western Australia Nedlands W. A., Australia.

Buff. F. P. University of Rochester

Rochester, N.Y.

Callen. E. Naval Ordnance Laboratory

White Oaks, Md.

Callen, H. B. University of Pennsylvania

Philadelphia, Pa.

Can . H. Y. Rutgers University

New Brunswick. N.J. Caulfield. D. F. Catholic University

Washington. D.C. Chase. C. E. National Magnetic Laboratory Massachusetts Institute of Technology

Cambridge. Mass.

Chestnut, D.

Shell Development Company

Houston, Tex. Chu. B. University of Kansas

Lawrence. Kans. Cohen, E. G. D. Rockefeller Institute New York. N.Y. Cooper, Martin J. Brandeis University

Waltham, Mass.

Cooper. Martyn .1. Brookhaven National Laboratory

Upton, N.Y.

Craig. P. P. Brookhaven National Laboratory

Upton. N.Y.

Davis. B. W.

University of North Carolina Chapel Hill. N.C. Debye. P. Cornell University

Ithaca. N.Y.

Dietrich. 0. W.

Atomic Energy Commission

Research Establishment Risii Roskilde. Denmark.

' Dotnb. C. University of London King"s College London. England. De Dominicis, C. Commissariat a I'Energie Atomique

Centre d'Etudes Nucleaires de Saclay

Sacla> . France. Dorfman. J. R. University of Maryland

College Park. Md.

Douglass. D. H.. Jr. University of Chicago

Chicago. 111. Edwards. M. H. Royal Military College

Kingston. Ontario Canada

Edwards. S. F. University of Manchester

Manchester. England

Emery, V. J. Brookhaven National Laboratory

Upton, N.Y.

Fairbank, W. M.

Stanford University Stanford, Calif. Farrell. R.

Catholic University

Washington. D.C.

Felderhof. B. U.

Instituut voor Theoretische Fysica der Rijksuniversiteil Utrecht, The Netherlands.

Firth. I. M.

University of St. Andrews

St. Andrews. Scotland.

Fisher. M. E. University of London King's College

London, England

239
Fixman. M.

Yale University New Haven. Conn.

Ford. N. C.

University of Massachusetts Amherst, Mass.

Friedberg, S. A. Carnegie Institute of Technology

Pittsburgh. Pa. Gammel, J. L. Texas A and M College

College Station. Tex. Gammon, R. Johns Hopkins University

Baltimore, Md.

Garcia-Colin, L. S. Direccion del Reactor Nuclear Comision Nacional de Energi'a Nuclear

Insurgentes Sur 1079, l

er piso

Mexico 18, D. F., Mexico.

Garland, C. W.

Massachusetts Institute of Technology

Cambridge, Mass.

Gilmer, G. H. Washington and Lee University

Lexington, Va. Green. M. S. National Bureau ot Standards Washington, D.C.

Grindlay, J. National Research Council Ottawa, Ont., Canada.

Guth, E. Oak Ridge National Laboratory

Oak Ridge, Tenn. Guttman, L. Argonne National Laboratory

Argonne. 111. Hammel. E. F. Los Alamos Scientific Laboratory Los Alamos, N. Mex.

Heims, S. P. Brandeis University Waltham. Mass.

Helfand, E.

Bell Telephone Laboratories Murray Hill, N.J.

Heller. P. Brandeis University

Waltham, Mass.

Herzfeld. K. F. Catholic University

Washington, D.C.

Hill, R. M. Lockheed Research Laboratories

Palo Alto. Calif. Hilsenrath, J. National Bureau of Standards Washington, D.C.

Hiroike, K. University of Chicago

Chicago. 111. Hohenberg. P. C.

Bell Telephone Laboratories Murray Hill, N.J. Horwitz, G. Yeshiva University

New York, N.Y. Hudson. R. P. National Bureau of Standards Washington, D.C. Hugenholtz. N.

Instituut voor Theoretische Fysica der Rijksuniversiteit

Groningen, The Netherlands. Hynne. F. University of Copenhagen

Copenhagen. Denmark.

Ikenberry, L. D. University of North Carolina Chapel Hill, N.C.

Isihara, A.

State University of New York

Buffalo, N.Y.

Jackel, R. Office of Naval Research

Washington, D.C. Jackson, J. L. Howard University

Washington, D.C.

Jacrot. B. Commissariat a l'Energie Atomique

Centre d'Etudes Nucleaires de Saclay

Saclay, France.

Jalickee, J. Catholic University

Washington, D.C.

Katsura, S. Tohoku University Sendai. Japan.

Kasteleyn, P. W.

Instituut-Lorentz der Rijksuniversiteit Leiden. The Netherlands. Kawasaki, K. Massachusetts Institute of Technology

Cambridge, Mass.

Kawatra, M. P. Courant Institute of Mathematical Scie New York. N.Y.

Keating, D. T. Brookhaven National Laboratory

Upton, N.Y. Keen, B. E. Yale University

New Haven, Conn.

Kellers, C. F.. Jr. Wells College Aurora, N.Y.

Kerr, E. Los Alamos Scientific Laboratory Los Alamos, N. Mex.

Kestin, J. Brown University

Providence, R.I. Kierstead. H. A. Argonne National Laboratory

Argonne, 111. Kikuchi. R. Hughes Research Laboratory

Malibu. Calif. Klein. M.

National Bureau of Standards Washington. D.C.

Klein, M.l

Case Institute ot Technology

Cleveland. Ohio

Knobier. C. M.

University of California Los Angeles. Calif. Kouvel. J. S. General Electric Research Laboratory

Schenectady, N.Y.

Langer, J. S. Carnegie Institute of Technology

Pittsburgh. Pa. Larsen, S. Y. National Bureau of Standards Washington. D.C.

Leff, H. S. Case Institute of Technology

Cleveland, Ohio

Levitt. L. C. Georgetown University

Washington. D.C. 240
Lieb. E. Yeshiva University

New York, N.Y.

Lipsicas, M.

Brookhaven National Laboratory

Upton. N.Y.

Livesay. B. R. Georgia Institute of Technology

Atlanta, Ga.

Lorentzen, H. L. Universitetets Kemiske Institutt Blindern-Oslo, Norway.

Manchester. F. D. University of Toronto

Toronto, Ont., Canada.

f Marcus. P. M.

International Business Machines Corporation Watson Research Center Yorktown Heights. N.Y.

Marshall. W.

Atomic Energy Research Establishment

Theoretical Physics Division Harwell, England.

Martin. P. C. Harvard University

Cambridge, Mass.

Marvin. R. S. National Bureau of Standards Washington, D.C. Matsubara, T. Kyoto University Kyoto, Japan.

Mclntyre, D. National Bureau of Standards Washington, D.C.

Melrose, J. C. Socony Mobil Oil Co.

Field Research Laboratory

Dallas, Tex.

Michels. A. M. J. F. Van der Waals Laboratorium

Universiteit van Amsterdam

Amsterdam-C. The Netherlands. Misawa, S. Massachusetts Institute of Technology

Cambridge, Mass.

Moldover, M. R. Stanford University

Stanford, Calif. Montroll, E. W.

Institute for Defense Analysis

Washington, D.C.

Moorjani, K.

Catholic University

Washington, D.C.

Morita. T. Catholic University

Washington, D.C.

Moruzzi, V. L. International Business Machines Incorporated Watson Research Center Yorktown Heights, N.Y.

Mountain, R. D. National Bureau of Standards Washington, D.C.

Muenster. A.

University of Frankfurt

Frankfurt. Germany.

, Nagel, J. F. Yale University

New Haven, Conn,

i Naiditch, S. Unified Science Associates, Inc. Pasadena, Calif.

! Nathans, R. Brookhaven National Laboratory

Upton, N.Y.

Neighbours. J. R. U.S. Naval Postgraduate School Monterey, Calif. Nielsen, P. University of Copenhagen

Copenhagen, Denmark.

Nishimura. H. Catholic University

Washington. D.C. Oppenheim, I. Massachusetts Institute of Technology

Cambridge. Mass.

Papoular, M.

University of California Los Angeles. Calif. Parette. G. Commissariat a l'Energie Atomique

Centre d'Etudes Nucleaires de Saclay

Saclay, France.

Parks. R. D. University of Rochester

Rochester, N.Y.

Passell. L. Brookhaven National Laboratory

Upton, N.Y.

Piccirelli. R. A. National Bureau of Standards Washington, D.C.

Pings, C. J. California Institute of Technology

Pasadena, Calif. Potter, R. L. American Cyanamid Company

Stamford Research Laboratory

Stamford, Conn.

Rao, N. G. Catholic University

Washington, D.C. Renard. R. Massachusetts Institute of Technology

Cambridge. Mass.

Ricci, P. Universita di Roma

Rome, Italy. Rice, O. K. University of North Carolina Chapel Hill, N.C.

Roder, H. M.

National Bureau of Standards

Boulder, Colo. Ross. J. Brown University

Providence, R. I. Rowlinson, J. S. Imperial College

London, England.

Rubin, R. J. National Bureau of Standards Washington, D.C. Rudnick, I. University of California Los Angeles, Calif. Ruelle. D.

Institut des Hautes Etudes Scientifique Bures-Sur-Yvette (S. et O.). France. Rushbrooke, G. S. University of New Castle New-Castle-on-the-Tyne, England. Schmidt. E. H. W.

Technische Hochschule Miinchen Munich. Germany.

Schmidt. P. University of Missouri Columbia, Mo.

Schultz. T. D. New York University

University Heights. Bronx. N.Y.
Schwartz, L. H. Northwestern University Evanston, 111. Scott, R. L. University of California Los Angeles, Calif. Sengers, J. M. H. (Mrs.) National Bureau of Standards Washington, D.C.

Sengers, J. V. National Bureau of Standards Washington, D.C.

Sette, D. Universita di Roma

Rome, Italy Sherman, R. H. Los Alamos Scientific Laboratory Los Alamos, N. Mex.

Shirane, G. Brookhaven National Laboratory

Upton, N.Y.

Sikora, P. T. North Carolina College

Durham, N.C.

Singwi, K. S. Argonne National Laboratory

Argonne, 111. Smart, J. S. International Business Machines Corporation Watson Research Center Yorktown Heights, N.Y.

Stillinger, F. H.

Bell Telephone Laboratories Murray Hill, N.J. Storer, R. G. Courant Institute of Mathematical Sciences New York, N.Y.

Stout, J. W.

University of Chicago

Chicago, 111. Tanaka, T. Catholic University

Washington, D.C.

Taylor. T. R. University of Missouri Columbia, Mo.

Teaney, D. T. International Business Machines Corporation Watson Research Center Yorktown Heights, N.Y. Temperley, H. N. V. United Kingdom Atomic Energy Authority

Aldermaston, England.

Tisza. L. Massachusetts Institute of Technology

Cambridge, Mass.

Trappeniers, N. J. Van der Waals Laboratorium

Universiteit van Amsterdam

Amsterdam-C, The Netherlands. Lhlenbeck, G. E. Rockefeller Institute New York. N.Y.

Ulrich, D. V. Bridgewater College

Bridgewater. Va. van Kampen, N. G.

Instituut voor Theoretische Fysica der Rijksuniversiteit Utrecht, The Netherlands,

van Leeuwen, J. M. J. Instituut voor Theoretische Fysica der Katholieke Universiteil Nijmegen, The Netherlands.

Verlet. L. Universite de Paris Orsay. France

Villain, J. Commissariat a l'Energie Atomicme

Centre d'Etudes Nucleaires de Saclay

Saclay, France

Walker, C. B. U.S. Army Materials Research Agency

Watertown, Mass.

Walker. J. F., Jr. New York University Bronx, N.Y. Walker. L. R.

Bell Telephone Laboratories Murray Hill, N.J. Webb. W. W.

Cornell University

Ithaca, N.Y. Weber. L. A. National Bureau of Standards

Boulder, Colo. Weinstock, J. National Bureau of Standards

Boulder, Colo. White, J. A. University of Maryland

College Park, Md.

Wilkinson, M. K. Oak Ridge National Laboratory

Oak Ridge, Tenn. Wims, A. M.

National Bureau of Standards Washington. D.C.

Wolf, W. P. Yale University

New Haven, Conn. Yamamoto, T. Kyoto University Kyoto, Japan. Yang, C. N. Princeton University

Princeton, N.J. Yeh, Y. Columbia Radiation Laboratory

Columbia University

New York, N.Y.

Younglove, B. A. National Bureau of Standards

Boulder. Colo. Zuber, N. General Electric Company

Schenectady, N.Y.

Zwanzig. R. W.

National Bureau of Standards Washington. D.C.

U.S. GOVERNMENT PRINTING OFFICE : 1966 OL— 209-493

242