UNITED STATES DEPARTMENT OF COMMERCE

Alexander B. Trowbridge, Secretary • NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Calculation of the Properties of

Vacancies and Interstitials

Proceedings of a Conference

Shenandoah National Park, Va.

May 1-5, 1966

Supported in part by the

Advanced Research Projects Agency

National Bureau of Standards Miscellaneous Publication 287

Issued November 17, 1966

For sale by the Superintendent of Documents, U.S. Government Printing Office. Washington, D.C 20402

Price $2.50
Abstract

This is the Proceedings of a Conference on the Calculation of the Properties of Vacancies and

Interstitials, Skyland. Virginia, USA, held on May 1-4, 1966. The Conference dealt with the theory and techniques of calculation of the properties of point defects in metallic and nonmetallic crystals. The contributed and invited papers divided about evenly among three major topics: (1) static-lattice calculations of the energies and configurations of simple vacancies and interstitials in, mainly, metals and ionic crystals; (2) electronic states at and near point defects in metals, rare gas solids, and insulators (/-centers, electron traps); and (3) vibrational states at point defects. The report of a panel dis- cussion on each topic is also included. The emphasis is on the theory of the properties of isolated, simple defects rather than on the statistical properties of defect assemblies. The Conference attempted

to examine the point defect theory and calculations critically, from the standpoint of general theory, rather than simply compare results with experiment.

Key Words: Calculations, electronic states, energies of formation, energies of motion, interstitials, point defects, theory, vacancies, vibrational states.

Library of Congress Catalog Card Number: 67-60032
Foreword

The Institute for Materials Research of the National Bureau of Standards has as a major responsibility the task of ensuring that basic, urgently needed data on the properties of materials are available to meet the requirements of the Nations' scientists and engineers. Data in this sense is a rather broad term. Not only does it imply experimentally measured

numbers, but it also includes theoretical data, such as the calculated properties of point defects in crystals, which formed the subject of this Conference.

In an immediate sense, this responsibility is met in part through measurements done in Institute laboratories, in part through compilation and publication of critically evaluated data gathered from the literature. It is also necessary for the

Institute to take a long-range view, to look further ahead, and to try to help in stimulating fields from which data of the future

will come. The Conference on the Calculation of the Properties of Vacancies and Interstitials was part of that long-range

effort. The field is a rapidly growing one, containing diverse elements, related to each other in subject and often in method,

but being pursued to some extent independently. No conference bringing together these diverse elements had previously been held on this topic. Thus it appeared that the present Conference could perform an important function in providing an opportunity for a critical examination of the whole field, and a forum for interchange of ideas and discussion of problems

at a most opportune time for such an interchange.

The Institute presents these Proceedings of the Conference in order to make the results available to a wider audience. The Institute would like to express appreciation to the participants, whose time, energy, and thought are embodied in these Proceedings, and to the Advanced Research Projects Agency, whose sponsorship made the Conference possible.

Gordon K. Teal

Director,

Institute for Materials Research

iii

Introduction

The Conference on the Calculation of the Properties of Vacancies and Interstitials was held at Skyland, on the Skyline

Drive in the Shenandoah National Park, Virginia, on May 1-4, 1966. It brought together approximately 70 scientists from

the United States, Great Britain, Japan. West Germany, France, Italy, and the Netherlands, to discuss theoretical problems

concerning isolated point defects in crystalline solids. Prior to this conference, the theory and calculation of the properties of point defects had not been the sole topic of such

a gathering. The field, however, had become an important one, and seemed ready to benefit by the concentrated discus- sions a conference of this kind makes possible. Several apparently unrelated currents of research bearing on this topic have developed in recent years, and it appeared desirable to try to bring their practitioners together. Essentially static-lattice calculations for metals and nonmetals have been quite successful in obtaining estimates of defect formation energies, and to a lesser extent, migration energies as well. These calculations have passed the initial rough-approximation state, and rather sophisticated problems involving interatomic potentials, electron redistributions, and atomic configurations about the defects are now being attacked. Calculations of electronic states in the vicinity of a point defect are also making progress. Those working on color centers in ionic crystals are struggling to find ways of dealing with polarization effects and with problems presented by the overlap of the defect wave function with the neighboring

ions. The problem in metals is the collective electron one in the presence of a sizable perturbation in the periodic crystal

potential. Rather similar methods are being developed to handle vibrational and electronic states induced by the presence

of point defects in otherwise periodic lattices; in both cases scattering theory is proving to be of great value. These three general topics, static-lattice energies and configurations, electronic states, and vibrational states, formed

the program of the conference. The object was to try to assess and compare the reliability of calculations of these properties from a theoretical point of view; not simply to compare the results with experiment, but to examine the calculations

critically, from general theory. The conference concentrated on the theory of isolated defects rather than of statistical properties of assemblies of defects, so that topics like radiation damage or diffusion do not appear as such. As the Table of Contents shows, the conference program was actually subdivided into three sections corresponding to these three topics. Within each section, contributed papers on current research plus several invited talks were presented.

Then a panel discussion was held on the topic at hand, in which an attempt was made to summarize and comment upon

what the contributed papers and their discussion had revealed. It was hoped in this way to provide some synthesis, some

overview, of the topic. These proceedings consist of the contributed papers, arranged by topic, and followed by a report on the panel discussion on the topic. These reports were prepared by the several panel chairmen, and each of course bears the stamp of its author's approach to the problem presented. We believe this use of panels to have been an interesting experiment in communication among scientists, and hope these reports will provide guidance and stimulation to others involved in similar experiments.

All conferences require a good deal of work on the part of many p'eople. The editor would like to take this opportunity

to call attention to the indispensable efforts of those whose work made the conference possible. The brunt was borne by

R. E. Howard (Institute for Materials Research, NBS) and R. F. Wood (Oak Ridge National Laboratory), and important

contributions were made by A. B. Lidiard (United Kingdom Atomic Energy Research Establishment, Harwell). C. McCombie

(Reading University), A. Seeger (Max Planck Institut fur Metallforschung, Stuttgart), G. Vineyard (Brookhaven National

Laboratory), and J. A. Krumhansl (Cornell University). Messrs Lidiard, McCombie, and Krumhansl served as panel chair- men as well.

V

Contents

Foreword

Introduction

I. Energies and Configurations

Interatomic Potentials and Defect Calculations in Ionic Crystals* M. P. Tosi The Energy of Formation of Schottky Defects in Ionic Crystals

/. M. Boswarva and A. B. Lidiard Recent Calculations Concerning Point Defects in Alkali .Halides , „

G. ./. Dienes, R. D. Hatcher, R. Smolchowski, and

W. D. Wilson Energy of Formation of Vacancy Pairs in KC1

K. Tharmalingam

Vacancies and Monovalent Cation Impurities in the

Alkali Halides M. Doyama and M. P. Tosi Born — Model Defect Energies for CaF2 A. D. Franklin

Lattice Distortion Around Defects

T. Kurosawa

Calculation of Lattice Distortion Around Point Defects by Lattice Statics H. Kanzaki

Calculation of Migration Energies of an Interstitial

in Germanium and Silicon R. R. Hasiguti

Calculation of The Physical Properties of Defects in Crystals by Approximate Self-Consistent-Field Methods

E. B. Moore, Jr. Interatomic Potentials for Defect Calculations in Metals* H. B. Huntington

Calculations for Defects in an F.C.C. Lattice R. A. Johnson

Atomistic Calculations of Interstitials in F.C.C. Metals M. Doyama and R. M. ./. Cotterill Morse Potential Calculations of Vacancy and VacancyType Defect Properties in F.C.C. Metals

R. M. J. Cotterill and M. Doyama

A Two-Body Central-Force Interatomic Potential for Anharmonic Solids R. Chang and L. J. Graham

A Re-examination of the Born-Mayer Potential for Ion-Core Interactions in Copper

S. S. Jaswal and L. A. Girifalco The Interactions Between Point Defects

J. S. Koehler The Energies and Configurations of Point Defects

in Solids: Report of The Panel I Discussion

A. B. Lidiard

II. Electronic States Vacancies. Phonons, and Dielectric Screening in Close-Packed Metals*

N. H. March and B. Oli

Calculations on Point Defects in Metals Allowing for

Page

iii

11

15

17

19

21

23

27

31

35

39

43

47

53

55

59

61

69

'Invited Paper.

Realistic Fermi Surfaces and Scattering Mechanisms 75 A. Seeger and E. Mann

Electrical Resistivity of Impurities and Vacancies in Aluminum 97

Y. Fukai

Structure Factor Approach to Energies of Migration 99

H. B. Huntington and M. D. Feit Perturbed Bloch Functions Obtained From KosterSlater Perturbation Theory 101

B. Mozer

Residual Resistivities of Dilute Aluminum Alloys 109

F. J. Blatt and H. R. Fankhauser

Point Imperfections in Solid Rare Gases Ill A. Gold The Use and Misuse of Models in The Calculation of Energy Levels in Insulators 115 W. B. Fowler Methods for Calculation of The Electronic Structure

of Defects in Insulators 119

U. Opik and R. F. Wood

A Model to Treat Lattice Distortions and The F- Center Electron Consistently 123 H. S. Bennett

/-Matrix Theory of Localized Electronic States Due

to a Vacancy With Application to Diamond 127 K. H. Bennemann

Energy Levels Associated With Isolated Vacancies

in Silicon 131

J. Callaway and A. J. Hughes

Electron Trapping as a Function of Normal Modes 133 A. M. Lemos

Electronic Properties of Point Defects in Solids. Report of the Panel II Discussion 135

J. A. Krumhansl

III. Vibrational States Phase Shift Analysis of The Scattering of Lattice Vibrations by Localized Perturbations 141

G. Toulouse and J. Friedel

Influence of Defects on Elastic Constants 151 W. Ludivis A Calculation of Force-Constant Changes for the U

Center 157 R. F. Wood and R. L. Gilbert Local Modes and Resonance Scattering of Lattice Waves Due to Point Defects: Optical and Thermal

Properties of Imperfect Crystals 161

G. Benedek and G. F. Nardelli Comment of The Paper of G. Benedek and G. F. Nardelli 179 Wm. M. Hartmann

One-Phonon Infrared Absorption in a Rare Gas Crystal Induced by Rare Gas Defects 181 Wm. M. Hartmann

Vibrations in Molecular Lattices 185 K. Delt man n and W. Ludivig

The Use of Morphic Effects for The Study of Vibrational and Optical Properties of Impurity Atoms in Crystals 189

A. A. Maradudin, S. Ganesan, and E. Burstein The Vibrational Properties of Point Defects in Solids. Report of The Panel III Discussion 195

C. W. McCombie

vii

I. Energies and Configurations

Interatomic Potentials and Defect Calculations in Ionic Crystals*

M. P. Tosi

Argonne National Laboratory, Argonne, 111. 60440

Lattice calculations of point defect properties in crystals started in 1938 with the classical work of

Mott and Littleton [l]1 on the energies of formation

and migration of a vacancy in alkali halide crystals.

A defect is a rather severe disturbance of a crystal

lattice: it disrupts the equilibrium conditions of

the perfect crystal, causing atomic displacements

to new positions where the net force on each atom

vanishes, and produces rearrangements in the

electron distribution. The theory describes the

billiard-ball part of the problem by means of empirical interatomic potentials, and takes care of the

electron redistribution in ionic crystals by the di- polar approximation, introducing appropriate ionic

polarizabilities. The scheme is basically analogous

to that adopted in lattice dynamics calculations

and is justified to the extent that the cohesive

properties and the dielectric properties of the

perfect crystal can be understood on the basis of

such concepts.

According to the analysis of the dielectric properties of ionic crystals given by Szigeti [2], the

polarization mechanisms of an ion in a crystal

include the electron-shell response to an electric

field, as well as mutual distortions of the electron

shells of neighboring ions accompanying relative displacements of the nuclei. The work of Eshelby

[3] and Kanzaki [4], on the other hand, shows that the lattice distortion has long range character even

in the absence of electric fields. The long range

distortion and polarization around a charged point

defect in an ionic crystal have generally been

described by means of quasi-continuum solutions

appropriate to a dielectric in a nearly uniform

electric field and to an elastic medium, with an

atomistic treatment of the immediate neighborhood of the defect. In this approximation, the

relevance of the polarization mechanism of Szigeti

and of the long range elastic relaxation in the

vacancy formation problem in the alkali halides

have been investigated by Kurosawa [5] and by

Brauer [6] and Fumi and Tosi [7], respectively; a

unified discussion of these effects is being given by

Boswarva and Lidiard [8]. An atomistic treatment

*Based on work performed under the auspices of the U.S. Atomic Energy Commission.

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

of a crystallite containing a vacancy has been

given recently by Scholz [9]. The approach is being extended to other solids, such as the alkaline- earth halides [10], which seem to conform closely

to the ideal ionic model[ll].

A considerable body of theoretical [12,13] and

experimental [14] evidence is available to show

that the electron distribution of an ion is distorted

in passing from the free ion state into the crystal, even in a crystal formed from closed-shell ions. A

characteristic deformation is a "loosening" and a "tightening" of the outer electron shells of the

positive and of the negative ions, respectively, that quantum mechanical calculations show to be partly

due to the Madelung potential [13]. Such distor- tions give rise to many-body contributions to the

binding in the alkali halides, although noncentral

terms in the crystal energy appear to be of minor

importance [12]. While the use of a pair-wise

central potential in the ionic model is an approximation, some of the features of the real crystal

which are a consequence of many-body effects are included empirically in the potential through the

determination of its parameters from experimental

data. This applies, in particular, to the ionic

radii [15,16]. The additional electron rearrange- ments which take place around a defect have often

dipolar character, to a first approximation, and are accounted for separately in the model.

The standard description of the short-range inter- actions in the ionic model is by means of an ex- ponentially decaying repulsive term extended to second neighbors and supplemented by a van der

Waals attractive term. The van der Waals coeffi- cients available in the literature on ionic crystals

have generally been estimated by the procedure

developed by Mayer [17] in the early thirties, which

is based on an analysis of optical absorption data

for a few alkali halides involving an assignment of

the absorption in different energy regions to the

different ions. A recent reexamination of this analysis by Lynch [18] points out that the absorption

regions of the positive and of the negative ions are in effect overlapping to some extent in the

alkali halides and intermixed in silver chloride, and

that the analysis is also affected by uncertainties in the effective field correction. As is well known, the

1
model yields very close agreement of the cohesive

energy with experiment, in the alkali halides as well as in calcium fluoride; on the contrary, the

Mayer values for the van der Waals energy in the

silver halides are insufficient to account for the ob- served binding, and sizable increases in these

values, or possibly some covalent binding, are indicated [19]. The accompanying tables present a numerical

illustration of effects of the potential on the basic

energy parameters of the vacancies in sodium

chloride and potassium chloride, for which the

experimental values are fairly well established.

The effects are analogous in the two salts, though more dramatic in sodium chloride. Table 1 reports

the values of the first and second derivative of the

first neighbor interaction energy, at the equilibrium

interionic distance in the perfect crystal, for the

various potentials. Quite independently of the

detailed functional form of the short range potential, these provide a measure of its "hardness"

in the first neighbor bond over a limited range of

interionic distance. The energies reported in table 2 (the lattice energy El, the energies of formation of a positive-ion and a negative-ion vacancy,

the Schottky defect energy Es =Ef

+ + E-~ —El, and

the energies of vacancy migration for a positive and

a negative ion along the face diagonal) have been

evaluated in the ionic model with a consistent de- scription of the crystal polarization based on the

Mott-Littleton model [20]. The values given in the first column have been

obtained with the Born-Mayer form of the short

range potential, which includes the van der Waals

interactions and the second neighbor repulsive

interactions, with the parameters obtained recently

by Tosi and Fumi [16]. In the second column we

give the results obtained by using a single exponential repulsion between first neighbors fitted to the same data. A partial cancellation between the

repulsive and the van der Waals interactions of

second neighbor pairs, and the small radius dif- ferences in the TF potential combine to reduce the net effects of the second neighbor interactions on

the defect energies. The two potentials yield

results in essential agreement and in fair agree- ment with experiment. The SE values for the

Schottky defect energy are in close agreement with

the early results of Mott and Littleton [1], but the

migration barriers are higher than their values of about 0.5 eV for NaCl.

Historically, the parameters in the Born-Mayer

form have passed through several stages of refine- ment. Many defect calculations have been based on the values reported originally by Born and

Mayer [21], who adopted the Goldschmidt values

of the ionic radii and fitted the compressibility of

the alkali halides on the average. This potential

yields rather erratic results for the vacancy formation and motion energies, and this has in fact been

a motivation for the latest steps in its refinement

[15,16]. These defect energies are reported here,

in the columns labeled BM, as an illustration of the

sensitivity of defect energy calculations to the

potential parameters. Clearly, the hardness of the

first neighbor interactions is seriously underestimated, especially in sodium chloride; and this reflects itself into exceedingly low values for the

formation energy of the negative ion vacancy and

for the Schottky defect energy [22,23] as well as

for the migration energy of the positive ion [23]. The overestimate of the chlorine-chlorine repulsion

implied by the use of Goldschmidt radii and the

large dipoles carried by the negative ions combine to keep the relaxation around a positive ion vacancy

within reasonable bounds, giving reasonable values

for [22] whereas the not unreasonable values

for E^ are largely a consequence of a cancellation of

TABLE 1. "Hardness" parameters of the various potentials a

TF SE BM FT WTF

— V'lro ) 0.81 0.77 0.57 0.79 0.87 NaCl

¥>V„ ) 2.59 2.69 1.65 2.34 2.74

-*>Vo) 0.63 0.62 0.57 0.63 0.75 KCI

tp"(r0 ) 1.95 2.00 1.69 1.94 2.22

a The units are 10" 4 dynes for ip'[ r 0 ) and 10 4 dynes/cm for <p"(r0 ).

Table 2. Effects of the short-range potential in defect energy calculations* (eV)

NaCl KCI

SE Experiment TF SE BM FT WTF Experiment

El

Ez,

8.06 4.70

5.30

1.94 0.85 0.90

8.09 4.78

5.25

1.94 0.90

0.88

8.06

4.78 4.59

1.31 0.30 0.72

8.01 4.60

5.19

1.78 0.70

0.75

7.81 4.65 4.97

1.81 0.92

1.00

'8.07

"2.12 + 0.06.

» 0.80 ±0.023,

°1.06

7.34 4.50

4.84

2.00

0.85

0.83

7.28

4.51 4.72

1.95 0.86 0.84

7.29 4.52

4.58

1.81 0.67

0.76

7.33 4.49

4.85

2.01 0.83

0.80

7.01 4.37

4.58

1.94 0.96

0.95

"2.22-2.31 "0.59-0.84

e 0.95±0.1

*The repulsive parameters in the various potentials are taken from the following sources: TF and WTF potentials, from M. P. Tosi and F. G. Fumi, ref. [16], table 1, first column; SE potential, from M. P. Tosi, ref. [19], table VIII, second column; BM potential, from M. Born and J. E. Mayer, ref. [21|, with the preexponential parameter fitted to the equation of

state; FT potential, from F. G. Fumi and M. P. Tosi, ref. [15]. table 2. first row. The Mayer values of the van der Waals coefficients (ref. [17]) are used in the TF and FT potentials, whereas the BM potential involves the use of the values reported by J. E. Mayer and L. Helmholz (Z. Phys. 75, 19 (1932)).

3 From the cohesive energy at room temperature reported by M. P. Tosi, ref. [19], after correction for the vibrational energy. "From R. W. Dreyfus and A. S. Nowick, J. Appl. Phys. 33, 473 (1962).

c From the activation energy for CI" diffusion in pure NaCl reported by N. Laurance (Phys. Rev. 120, 57 (I960)), after subtraction of \ER = 1.06 ev. "From R. W. Dreyfus and A. S. Nowick. loc. cit.; A. R. Allnatt and P. W. M. Jacobs, Trans. Faraday Soc. 58, 116(1962): P. W. M.Jacobs and J. N. Maycock, Jr.. J. Phys. Chem.

Solids 24, 1693 (1963); R. G. Fuller, Phys. Rev. 142, 524 (1966).

e From R. G. Fuller, loc. cit.
errors between underestimates of the saddle point

energy and of Ej [23]. In the FT potential [15], on

the other hand, no assumption is made on the ionic

radii, but the compressibility is still fitted on the

average — using, of course, recent data. This pro- cedure underestimates somewhat the hardness of

the potential in sodium chloride, and this is re- flected by the defect energies. The Huggins-Mayer

potential [24] represents an intermediate stage of

refinement between the BM and the FT potentials,

and yields values for the defect energies which lie approximately midway between the BM and the FT

results. As an additional illustration of the sensitivity of

defect energies to approximations in the potential,

the columns labeled WTF report the results ob- tained by the TF potential with the van der Waals

energy omitted. The crystal is moderately stiffened

by this inconsistency and the binding of an ion in the crystal is sizably weakened.

In conclusion, two more questions may be touched

upon. The first concerns possible improvements in the existing potentials based on the cohesive

properties [19]. While the electron maps provide

supporting experimental evidence for the ionic

radii, it would be desirable to carry out other tests on the second neighbor interactions. An effort at refining the Mayer values for the van der Waals

coefficients would also be desirable. Critical tests on the functional form of the potential would re- quire accurate pressure-volume data over a large

range of relative compression. Recent studies of

the equation of state of NaCl [25] indicate, however, that the perfect crystal potential is quite accurate over a sizable range. The second question is whether some defect configurations are indeed so far removed from the perfect crystal that the perfect crystal potential and the

dipolar approximation become suspect. An ex- ample may be provided by the saddle-point configuration for vacancy migration; [23] and it was

suggested, on the basis of the migration barriers

obtained by us with the BM potential and by Mott

and Littleton with the SE potential, that the perfect

crystal potential is indeed inadequate in a treatment

of this problem based on the dipole approximation.

However, as is apparent from the first two columns

of table 2, the ionic model yields reasonable values

for these barriers. This remains nevertheless an

outstanding theoretical question in the treatment

of defect configurations where ions approach very

closely. While a "molecule-in-crystal-field" method

would clearly be the correct theoretical approach in principle, the Verwey potential [26,23] is a crude

attempt at interpolating between the solid and the

molecule. In the light of the sensitivity of defect

energies to the potential parameters displayed above

in the TF and BM columns, a careful reexamination

of the interpolation scheme is needed.

Appendix

The Mott-Littleton model makes recourse to the

macroscopic polarization to fix the long range dis- placements and dipoles around a charged defect. The macroscopic polarization enters also the model

given previously by Jost [27]. The relationship

between the two models has been the subject of some discussion at the Conference, and it has been

suggested that the Mott-Littleton model is theoreti- cally even less satisfactory than the Jost model.

The writer deems it useful to give a detailed dis- cussion of this point. In a continuous dielectric medium, as considered

in the Jost method, one has no difficulty in defining

the macroscopic polarization around an extra point

charge as a mathematical function of position over

space. Integration of the field energy density of

the charge in vacuo and in the dielectric yields the

polarization energy as

'POl' 101

2 R

(1)

where R is the radius of a cavity in the dielectric. The method contains no prescription for determining this physically arbitrary limit of integration; Jost chose R equal to one-half the distance to neighboring

ions to show that the polarization energy could be

of the same order of magnitude as the rigid lattice energy, thereby reducing the defect formation

energy from 8 to 10 eV to 1 to 2 eV as experimentally

observed.

In a crystal of discrete ions, as considered in the

Mott-Littleton method, the calculation of the macroscopic polarization around an extra point charge

clearly requires an average of the ionic dipoles over macroscopic regions of space over which the field be essentially uniform. The equation

P(r)=- M'l+ M')] (2)

is the result of such an average over a region of

space centered at r in the limit r— Both sides

of this equation have no meaning close to the point

charge, and the method does not use the equation

in these conditions. Equation (2) fixes in the method

the sum of the dipoles in a lattice cell at large dis- tance from the charge, whereas the ratio of these

dipoles is fixed by the equation

fx+ (r)lfx-(r) =

a+ + a

(3)

where the displacement polarizability a is also evaluated by taking the field as essentially uniform.

The Mott-Littleton dipoles are therefore correct
only at large distance from the charge, a well known

fact. The dipoles themselves are extrapolated into region II. Let us examine the physical assumptions of the

method on the dipoles in region II. Equation (2) fixes the inverse square distance dependence of

the dipoles, that we may write in the form

Mn)=|i>oM;j| (4)

where r, is the distance of the lattice site, occupied

by either a positive or a negative ion, from the

charge. The use of eqs (2) and (3) to determine M'+ and M'_ implies then that the Mott-Littleton expression,

M'+ =±(l-l) <*± + a

(5) 4vr \ K + «

is taken directly from the atomic theory of dielectrics in a uniform field. Namely, the method is describing the response of the ions in region II to the field Q/r'f of the charge by analogy with their

response to a uniform field. The factor a± + a in eq (5) gives the response to a uniform effective field, and the other factors represent the effective field correction.

We may rederive these results in detail without

using the macroscopic polarization. We consider an extra charge in a NaCl-type crystal, with coulomb

interactions and short range interactions between

first neighbors described by a potential (p(r). The

effective field Eef((ri) at the lattice site r, induces an

electronic dipole a±Eelf(ri) and provides a driving

force ± eEe{{(ri). The restoring force is provided by

the short range interactions and is derived by Mott

and Littleton from the harmonic Hamiltonian

written as — k ^uf + ~ & ^ tne nrst term follows

from the diagonal elements of the force constants matrix and is exact, whereas the second follows

from the nondiagonal elements [28] under the

assumption that neighboring ions have equal and

opposite displacements, as would be exact in a uniform field. The force constant k is given by

Mott and Littleton as 2VV(r)|r = r0

; it is in fact the

restoring force constant for long-wavelength optical

vibrations and is related in the present model to the

compressibility by k = 6r0 /{i [29]. Force balance

yields the displacement dipole ±ein as equal to — Eeff(ri), and the total dipole is

fjL ± (n) = (a ± + a) — — (6)

e

2 where [30] a = -^r and we have written the effective

field as -jz Q/r'j, with L an effective field correction

factor. The assumption that L be independent of

r, is, again, correct only if the field is essentially

uniform. If one uses eq (6) for all the dipoles

around a vacancy, one finds for the Schottky defect

The quantity «+ + a_ + 2« gives in fact the re- sponse of a lattice cell to a uniform field, and satisfies the relationship

I (a+ + a. +2a) |=i(l-i) (8)

where L' is the effective field correction factor for a uniform field. If one takes L = L' in eq (6), as it should be by the manner in which this equation has

been derived, one finds the Mott-Littleton formula

for the dipoles in region II. We note the following:

(1) equations (8) and (2) are in fact equivalent;

(2) the Mott-Littleton formula for the dipoles is independent of the value that dielectrics theory

assigns to the effective field correction factor L' —

although modern theory accepts the full Lorentz

value L' = (K + 2)/3 for ionic crystals when ionic overlap effects are accounted for by the Szigeti

effective charge [31];

(3) the zeroth-order Mott-Littleton approximation

for the Schottky defect energy, obtained by using

eq (8) in eq (7), has the same formal structure as the Jost formula, but with the arbitrary limits of integration fixed by a lattice summation, because the

Mott-Littleton method relates the defect problem to the atomic theory of dielectrics and one is summing

over the two vacancies. The use of the macroscopic polarization in both methods does not imply

that they are based on the same physical assumptions, and in fact the two methods give structurally

different formulas for, say, the Frenkel defect

energy. It implies, however, that the two methods

will converge to the same limit as region I is enlarged.

We also note that, even under the assumptions

specified above, the harmonic Hamiltonian that we

have used is incorrect for the case of a vacancy,

since the breaking of the bonds to the first neighbors

of the vacancy implies a constant term and a modification to the linear term in the restoring force on

these ions. This fact, as well as the fact that the

above assumptions are most inappropriate for these

ions, are accounted for in the first-order approximation of the method.

The relevant question on the usual scheme which i decomposes the crystal around the vacancy into a region I and a region II to be described by approximate displacements and dipoles correct at large

| distance is, of course, the convergence of the energy 1 results as region I is enlarged. Experience shows I that the polarization energy is not as sensitive as the I
dipoles to the assumptions of the method. In- vestigations of the convergence of the scheme for a crystal of polarizable but undisplaceable ions were

carried out first by Mott and Littleton, who extended

region I to include four shells of ions, and later by

Rittner et al. [32], who included up to ten shells of

ions. Several cases corresponding to different values of the dielectric constant and of the polarizability ratio were considered, and the results depend

to some extent on these parameters. In all cases

the dipoles keep fluctuating around the MottLittleton values but the polarization energy con- verges so rapidly that practically perfect convergence

is attained after the third-order approximation.

The first-order approximation is fairly accurate, the

largest error being 4 percent of the polarization

energy; the zeroth-order approximation itself is not poor, although the polarization energy is underestimated by as much as 10 percent in the same

case. In most cases the first-order approximation

overestimates the polarization energy. The con- vergence of the Jost method was also investigated

and found to require up to eight successive approximations.

The Mott-Littleton model is therefore a quasicontinuum model whose accuracy has been tested

in the dipole approximation. Since the displacements of the ions beyond first neighbors are suffi- ciently small to be equivalent to dipoles in their electrostatic effects, one expects that the foregoing

results apply also in the real case. One also ex- pects, however, that the method will become less

reliable as the dielectric constant increases [1]. The treatment of the real case, on the other hand,

References

[1] N. F. Mott and M. J. Littleton, Trans. Faraday Soc. 34,

485(1938).

[2] B. Szigeti, Proc. Roy. Soc. (London) A204, 51 (1950).

[3] J. D. Eshelby, Solid State Physics 3, 79 (1956).

[4] H. Kanzaki, J. Phys. Chem. Solids 2, 24 (1957).

[5] T. Kurosawa, J. Phys. Soc. Japan 13, 153 (1958).

[6] P. Brauer, Z. Naturforsch. 7a, 372 (1952).

[7] F. G. Fumi and M. P. Tosi, Disc. Faraday Soc. 34, 485

(1957).

[8] I. M. Boswarva and A. B. Lidiard, this conference.

[9] A. Scholz, Phys. Stat. Sol. 7, 973 (1964).

[10] A. D. Franklin, this conference.

[11] J. D. Axe, Phys. Rev. 139, A1215 (1965).

[12] P. O. Lowdin, Phil. Mag. Suppl. 5, 1 (1956).

[13] M. I. Petrashen, L. V. Abarenkov. and N. N. Kristofel, Optics and Spectroscopy (USSR) 9, 276 (1960); Vestnik Leningrad Univ. 16, 7 (1960).

[14] H. Witte and E. Wolfel, Revs. Modern Phys. 30, 51 (1958). (NaCl, LiF, CaF2 ); G. Schoknecht, Z. Naturforsch. 12a, 983 (1957). (NaCl): R. Brill, C. Hermann, and C.

Peters, Z. anorg. u. allgem. Chem. 2 5 7, 151 (1948) (MgO); U. Korhonen and M. Linkoaho, Ann Acad. Sci. Fenn., Series A VI, No. 195 (1966) (AgCl).

[15] F. G. Fumi and M. P. Tosi, J. Phys. Chem. Solids 25, 31

(1964).

[16] M. P. Tosi and F. G. Fumi, J. Phys. Chem. Solids 25, 45

(1964).

[17] J. E. Mayer, J. Chem. Phys. 1, 270 (1933).

[18] D. W. Lynch, J. Phys. Chem. Solids, to appear. I am

indebted to Professor Lynch for a preprint of his work.

[19] M. P. Tosi. Solid State Physics 16, 1 (1964). The van der

involves a new set of difficulties already in the

first-order approximation. These are the use of a

reliable ion-ion potential and the inclusion of the

long-range elastic distortion and of the coupling

between ionic displacements and electronic de- formations. A proper treatment of these effects has had to wait for developments of other aspects of

the theory of ionic crystals. An examination of the first three columns of table 2 will show that the

difficulties associated with the potential are at least as important as the convergence question, and, moreover, have a direct bearing on it. The recent work of Scholz and of Simpson shows

that an enlargement of region I is well within present

capabilities, and the results of the first-order ap- proximation are consistent with the results of these

authors. It is nevertheless clear that even improved

calculations of the Schottky defect energy will generally be affected by uncertainties of a few

tenths of an eV. It is also clear that the uniform

field response assumption of the zeroth-order MottLittleton approximation is a simple physical assumption which allows back-of-the-envelope calculations

without requiring a fitting of parameters to the

desired result. It should. therefore be noted that it yields already very reasonable values of the Schottky

defect energy. This fact is of little help to the study

of other defect configurations.

I am indebted to Professor F. G. Fumi for helpful

suggestions and to Dr. A. B. Lidiard for informative

correspondence.

and Notes

Waals interactions and partially covalent bonds will, of course, play an essential role for the stability of the silver

interstitial [5]. [20] Details of the calculation and the comparison with experi- ment are discussed by M. P. Tosi and M. Doyama, Phys. Rev. 151, 642 (1966).

[21] M. Born and J. E. Mayer, Z. Phys. 75, 1 (1932). The BM

potential has so far been approximated in defect calculations by omitting the van der Waals interactions, the pre- exponential parameter being refitted to the thermodynamic

equations and the hardness parameter being left un- changed. These corrections are less rigorous than those adopted by Boswarva and Lidiard [8] but do not affect significantly the results. Thus these approximations

lower the vacancy formation energies given in table 2 by

about 0.1 eV, in agreement with previous results [23], but

affect the values of the Schottky defect energy and of the motion energies by a few hundredths of an eV. The

values of the motion energies given here are higher than

those reported previously [23], by 0.2 eV in NaCl and

0.1 eV in KC1, because of the more accurate treatment

of the dipole-dipole interaction in the treatment of the

local relaxation energy.

[22] F. Bassani and F. G. Fumi, Suppl. Nuovo Cim. 1, 114(1955).

[23] R. Guccione, M. P. Tosi and M. Asdente. J. Phys. Chem.

Solids 10, 162 (1959).

[24] M. L. Huggins and J. E. Mayer, J. Chem. Phys. 1, 643 (1933).

[25] D. L. Decker, J. Appl. Phys. 36, 157 (1965).

[26] E. J. W. Verwey, Rec. Trav. Chim. 65, 521 (1946).

[27] W. Jost, J. Chem. Phys. 1, 466 (1933).

[28] One may use, e.g., eqs (35.1), (38.1), and (38.2) in G. Leib5
fried, Hand. d. Phys. VII/1 (Springer, Berlin 1955) to find the nondiagonal term as

- o S u' " uj -

? * (ro) ~ 2 ' Ui) (ny ' Uj) 2 r0 if I L r« J If

Here, the index i runs over all the rons, the index j over the

first neighbors of i, and n„ is the unit vector joining the two ions. This may be used to find that the correction in the displacements falls as r~ 4

.

[29] This last result does not hold if second neighbor interactions are included. However, all the other relevant details remain correct, if one still interprets <p{r) as the first- neighbor potential.

[30] The inclusion of the Szigeti effective charge in the manner

discussed by Born and Huang [31] yields the result of Boswarva and Lidiard [8], a = e*2 /2k.

[31] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, section 9 (Oxford, 1954).

[32] E. S. Rittner, R. A. Hutner and F. K. du Pre, J. Chem. Phys.

17, 198 (1949).

6
The Energy of Formation of Schottky Defects

in Ionic Crystals

I. M. Boswarva

Metallurgy Department, Imperial College, London S.W.7

and

A. B. Lidiard

Theoretical Physics Division, U.K.A.E.R.E., Harwell, Didcot, Berks, England

The success of the Born ionic model in describing

the cohesive properties of alkali halides has encour- aged its use in calculations of characteristic energies of point defects. Several workers* have

obtained good agreement with experiment for the

formation energy of Schottky pairs, Ws , but the use

of differing repulsive potentials and modifications

to Mott and Littleton's basic description of the dis- placement field has, nevertheless, left a rather

confusing picture. We present here, some of the

main results of a systematic study made to clarify

the effects of such differing descriptions. Our aim

is thus not simply a comparison with experiment,

but is also to assess the range of variation caused

by reasonable alterations of the basic model. Further, we study 16 alkali halides with the NaCl

structure so that a broader view is possible than

when only NaCl and KC1 are considered.

The basis of our calculations is a model in which

each ion of the solid is described by its displacement from the perfect lattice and its electronic

dipole moment. The energy function of the solid contains electrostatic terms coming from interactions among the ionic charges and dipoles, along

with short-range closed-shell repulsions coming

from ionic overlap. To calculate the energy needed

to extract an ion from the perfect solid the crystal

is divided into two regions, I and II. The six nearest neighbor ions of the vacancy, which constitute region I, are considered explicitly whilst we

use a harmonic approximation for the rest of the

lattice (region II). The general equations of this method were published in the abstracts of the Inter- national Conference on the Nature of Defects in Crystals, Melbourne, August 1965, Paper P3.

Previous calculations of vacancy formation energies have all used the traditional Born-Mayer exponential form for the overlap repulsion potential and we have also made computations for this model

(BM). However, Guccione et al. [4], found it neces-

*Brauer [2]. 1 Tosi and Fumi [11], Kurosawa (6|, Scholz [8|, and Simpson [9].

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

sary to modify this to obtain sensible results for vacancy migration energies. They needed a harder

potential for ions coming close together and used

the so-called Born-Mayer-Verwey potential (a r~

12 dependence for r < anion-cation nearest neighbor

separation). This potential was also used by Tharmalingam [10] for interstitial ion migration energies.

We find that the general pattern of our results is the same with the B.M.V. potential as with the B.M.

potential but each formation energy is higher. In general the B.M. formation energies are too low

compared to experiment and the agreement with

B.M.V. is better. The variations in Ws resulting

from different choices of constants in these potentials (ionic radii etc.) are smaller than the difference

between the Ws values for corresponding B.M. and

B.M.V. potentials. We now consider various modifications to the description of region II in these

calculations.

The description of the displacement field in region

II in this model is essentially intuitive. The original

method, due to Mott and Littleton [7], uses the

electric polarization caused by the effective charge

of the vacancy to determine the displacement field and our basic calculation (set A in the table) employs

this approach. In an attempt to improve this description we consider the vacancy as an elastic singularity as well as an electrical one. The ion

at r is then displaced radially outwards an amount

£ = A:r$/r2 where r0 is the perfect anion-cation separation and k is the elastic strength of the vacancy.

If we take the displacement of region I ions as kro then for continuity X=k + M' where M'r0 is the

electrical displacement of the neighbors to the

vacancy as given by the Mott-Littleton theory. We

include this elastic strength of the vacancy in the

set of calculations labeled B in the table. Brauer

[2] who first included this elastic displacement term

in such calculations simply equated k to k. It seems to us that in principle this would over- emphasize the elastic component but we have also made computations using his assumption and these

are given in Columns C.
Table 1. Experimental and theoretical values of the energy offormation ofSchottky defects, Ws in eV, in alkali halides with the NaCl

structure

The corresponding theoretical values for the outward relaxation of the nearest neighbors to the vacancy are also given (in units of r0 the anion-cation spacing). For description

of the basis of the four different calculations A, B, C and D see text.

^(exp) A B C D

\ Ws \ W W W

LiF + ion — ion 2.20-2.68 0.0546

.0713

2 310 0 0445

.0639

2 339 0 0549

^0750

2 734

!0631

LiCl + ion — ion 2.12 0419

X>785

1 212 031

7

!0738

1 263

.0845

.0261 .07311 .070

LiBr + ion — ion

1.80 0388

.0813 0 970

.0775

.0378 .0883

1.214 .0225 .0768 0.798

Lil + ion 1.34 0358 0 607 0265 0 665 0352 — ion ^0865 ]0841 !0954 NaF + ion — ion .0596 .0650

2.818 0518

.0579

2 892 0612

!0671

3 181 0510

]0577 NaCl + ion — ion 2.02-2.20 .0461 !o693

1.912 .0373 .06441.988 0452

.0727

2 170 0347

'0642

1 881

NaBr + ion — ion

1.68 .0424

.0712

1.653 .0333

.0669

1.727 .0410

.0751

1.887 .0306

!0670

1 600

Nal + ion — ion .0384

.0752

1.300 .0295

.0723

1.372 .0367 .0804

1.506 0268

.0724

1.248

KF + ion — ion .0649

.0571 2.360 .0586.0492 2.428 .0676

.0577 2.654 .0572

'0491 2.393

KC1 + ion — ion 2.22-2.30 .0515 .0609

2.053 .0436

.0546 2.120 .0512!0621 2.279 .0420

.0547 2.066

KBr + ion — ion 2.35-2.53 .0479

.0622

1.889 .0397 !0565

1.956 .0469

.0638 2.099 .0381 .0567

1 890

KI + ion .0433

.0645

1.657 .0346

.0595

1.722 0416

.0666

1 845 0330

.0600

1 636

RbF + ion — ion .0672

.0540 2.074 .0612.0456 2.135 .0703 .0540

2.345 .0606

.0455

2.121

RbCl + ion — ion .0534

.0582 2.007 .0459

.0514

2.071 .0533

.0588 2.222 .0448

.0516 2.033

RbBr + ion —ion .0502

.0596

1.868 .0423 .05331.931 .0496

.0606 2.068 .0411 .0535

1.888

Rbl + ion — ion .0526

.0616

1.879 .0448

.0533

1.937 .0524

.0629

2.081 .0434

.0554

1.894

The second major modifications which we intro- duce (columns D) allows for the deformation of the

electron distribution of the ions resulting from their

relative motion. The basic idea is that relative motion of the positive and negative ions leads to a

distortion of the otherwise spherical charge distri- butions and this distortion can be represented as an

additional dipole, the 'deformation dipole.' In

region II the total polarizability per cell now is a+ + a- + 2(e*)2 /p where oc+, a- are the electronic

polarizabilities of the cations and anions respectively, e* is the Szigeti effective charge and p is the

force constant from Mott-Littleton theory. The

last term of this total polarizability is the sum of the

displacement dipole contribution of 2 ee*/p and the

deformation dipole contribution — 2 e*(e — e*)lp.

The deformation dipoles on the region I ions are

readily found using the analysis of Hardy [5]. These

deformation dipoles have been included as well as

elastic terms with strength k = K — M' (set D).

General conclusions and features of the results obtained using the B.M.V. potential are as follows:

1. The outward relaxations of the cations neighboring an anion vacancy are more than those of the

anion neighbors of a positive ion vacancy, except

for KF and RbF. This is because they suffer less restraint from second neighbor repulsive inter- actions.

2. The work to remove a cation is less than that

to remove an anion. The polarizability of the sur- rounding ions, which is much greater, appears to govern this feature although its effect is somewhat

reduced by the larger second neighbor repulsions.

3. Reasonable agreement with experiment is obtained for LiF, NaCl, NaBr, and KC1 with all modifications. The differences do not seem large

enough to permit an empirical choice between the

different assumptions for region II. However,

none of the calculations give JFs (KBr) > fTs (KCl) as required by experiment, and the predicted values

for the other Li halides are all too low.

4. If we use the B.M. potential the corresponding!

Ws values are reduced in most cases by 0.2-0.4 eV,

although in LiF the reduction is as large as 1 eV.|

The first neighbor relaxations are increased up to' 50 percent over the B.M.V. values.

5. An important energy term introduced by elastic distortion terms is an addition to the polarization!

8
energy of region II of (1.068-4. 731\)e2 (A-M')/r0 . The term in M' is missing in Brauer's method.

The absence of the term in M' causes both the

larger relaxations and the larger Ws values in this case.

6. Only the elastic term in the displacement field contributes to the change of volume of the crystal

due to relaxation around the vacancy. The volume

of formation per Schottky pair has been deter- mined experimentally for NaCl and KC1 by Bier- mann [1] and the results give, in both cases, the sum of the vacancy elastic strengths, k+ + k- ~ 0.05. This is considerably less than k+ + X_ (Brauer's

assumption) but greater than \+ + A_ — 2M' (our

assumption) which is 0.022 for both NaCl and KC1.

7. All the results for the alkali halides presented

here have used the first set of values of interionic

distance and compressibility compiled by Fumi and

Tosi [3] along with the ionic radii evaluated by them.

We have also considered (i) the second set of data

from Fumi and Tosi and (ii) Goldschmidt radii, both

of which give relatively small variations to Ws . Much early work used averaged values of strength

and hardness parameters in the repulsive potential,

although there is no theoretical reason for these

parameters to be identical for a whole family of

salts. We find that the traditional values (p = 0.345

A, 6 = 0.143 eV) give lower values of Ws by 0.1-0.2

eV whilst Fumi and Tosi's new values averaged

over all the alkali halides (p =0.339 A, 6 = 0.159 eV)

give results close to those obtained using the particular values appropriate to each individual salt.

References

[1] Biermann, W., Z. f. Phys. Chem. 25, 90 and 253 (1960).

[2] Brauer. P., Z. Naturf. 7A, 372 (1952).

[3] Fumi, F. G. and Tosi, M. P., J. Phys. Chem. Solids 25, 31

(1964).

[4] Guccione. R., Tosi, M. P. and Asdente, M., J. Phys. Chem.

Solids 10, 162 (1959).

[5] Hardy. J. R.. Phil Mag. 7, 315 (1962).

[6] Kurosawa, T., J. Phys. Soc. Japan 13, 153 (1958).

[7] Mott. N. F. and Littleton, M. J., Trans. Farad. Soc. 34,

485 (1938).

[8] Scholz, A., Phys. Stat. Solidi 7, 973 (1964).

[9] Simpson, J. H., Paper No. 176 Conference on Color Centers

in Alkali Halides, Urbana, 111. (1965).

[10] Tharmalingam, K., J. Phys. Chem. Solids 25, 255 (1964).

[11] Tosi, M. P. and Fumi, F. G., Disc. Farad. Soc. 23, 92 (1957).

256-724 0-67—

2

9

Recent Calculations Concerning Point Defects in Alkali Halides 5

G. J. Dienes

Brookhaven National Laboratory, Upton, N.Y. 11973

R. D. Hatcher

Queens College, The City University of New York, Flushing, N.Y. 11367

R. Smolchowski

Princeton University, Princeton, N.J. 08540

and

W. D. Wilson

Queens College, The City University of New York, Flushing, N.Y. 11367

The presence of a defect in an alkali halide lattice causes the neighboring ions to be displaced from

their normal positions, and consequently alters the

interaction energies in the crystal. Evaluation of

the energies involved coupled with the condition

that the additional energy be a minimum with re- spect to certain variable parameters enables a

solution to be found for the displaced position of

the ions in the vicinity of the defect. The differ- ence in the energy between the new configuration

and the perfect lattice gives the formation energy

and in addition, the activation energy for motion of

the defect can be obtained by considering the defect

in various positions.

The method used assigns variable parameters,

Pi, to certain ions neighboring the defect and in some problems to the defect itself. These parameters indicate arbitrary displacements from normal

lattice sites. The number of parameters vary with

the problems handled, up to eight have been used,

which when symmetry is taken into account allow

for up to 30 neighboring ions to be varied. The

energy is composed of several parts. The electrostatic terms are e,ej/r,j where r,j is the distance

U between ions i and j in their relaxed positions and

|i the charge on the ions. From this term must be

subtracted the electrostatic energy of the ions in the normal lattice. The Ewald summation method

and a procedure of separating the ions into two

groups, those that are allowed to relax, and those

that are kept fixed enable several of the terms to be evaluated from well-known expressions leaving

finite sums over the movable ions. The polarization energy is obtained from the expression — \ V

a,E,2 where E, is the electric field at ion i in its relaxed position and a, is the polarizability of the

ion. The summations here are handled by the same

procedures as in the case of the electrostatic terms. Since in the normal lattice the electric field at an

ion is zero, there is no initial polarization energy. An

additional term in the polarization series, the inter- action between induced dipoles is sometimes in- cluded. Summations are taken over sufficient terms

in the lattice that the model of a polarizable medium

is not needed. The repulsive energy between ions

is obtained from the Born-Mayer form Ae~Br U with

A and B constants that depend on the particular

ions involved. The summations required here are

not extensive because of the rapid falling off of

the interaction. In certain problems it is necessary

to consider special types of interactions, usually

of the nature of binding or covalent terms. These

are calculated either quant um-rnechanically or from experimental data. When all energy terms are computed the total energy as the sum of them is a function of the parameters E = E(P,). Initially

the parameters are set at certain values and the

energy obtained by a computer, then each parameter in turn is varied until a minimum is obtained

relative to that parameter. After this has been

done for all parameters it is run through several more times for smaller variations. The polarizabilities of the ions were taken from

the work of Tessman, Kahn, and Shockley [l]. 1 The constants entering into the repulsive terms were taken from the papers by Tosi and Fumi [2].

•Work performed at Brookhaven National Laboratory and Supported by the Atomic

Energy Commission.

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

1 1
This includes also such quantities as the radii of

the ions. For the radius of neutral ions an average

value of the radius was computed from wave functions and was suitably adjusted to compare with the

radii of the charged ions. Radii for ions considered

as sharing a charge was computed by interpolation.

Three applications will be considered here.

(a) Interstitial Cl° in NaCl. The method was

initially applied to the case of an interstitial chlorine

atom in NaCl to determine the amount of the relaxation involved and to find which position of the defect

would give the least energy. Several positions for the Cl° were investigated: the cube-center, the face- center and various positions along the cube and

face diagonals. The lowest energy configuration was found to be the body-center position. In this case four variable parameters were employed in- volving 20 movable ions. The nearest CI" ions

relaxed outward 14 percent while the nearest Na+ ions relaxed 16 percent. The energy relative to the

perfect lattice was found to be 2.65 eV. For the

face-center position six parameters were used in- volving 18 movable ions. Here the nearest CI" ions

relaxed outward 23 percent while the nearest Na+ relaxed 21 percent, the energy relative to the perfect lattice being 3.09 eV. The activation energy

for motion between these two positions is then

0.4 eV. In the placement of the CP along the cubediagonal in an asymmetrical position, the nearest Cl~ ion is considerably displaced; however a local minimum can be obtained using 9 parameters and

19 movable ions with an energy relative to the perfect lattice of approximately the same as for a cubecenter position. The minimum position of the Cl° was found to be about 20 percent along the diagonal

from the regular CI" site. A similar situation was

found in the case of an asymmetrical face-diagonal

position. Here the Cl° is found at about 32 percent

along the diagonal with again only a small difference

in energy from the face-center position. The acti- vation energy along the (111) direction is about

0.2 eV and along the (110) is about 0.05 eV. It would appear then that motion of the CP along the

diagonal is relatively easy. In these calculations

crystal forces alone were used. However, it would

appear that these forces alone might be able to push the CP into the diagonal or H-center type

configuration.

(b) The H-center in KC1. The //-center in KC1

is a CPr molecular ion oriented symmetrically along

a (110) direction. This has been verified by experiments involving the absorption of polarized light and

by electron spin resonance techniques. Using the

defect calculation methods we investigated the

problem of showing why the //-center preferred the

(110) direction rather than the (111) which on the

basis of the above work on the Cl° might seem to give rise to the most stable position. The inter- action potential between the two halves of the CPf was first chosen as an average of the CP-CP Born- Mayer interaction and the CP-CP Morse potential.

The binding energy here was 0.2 eV with an inter- ionic separation of 2.95 A. This interaction was

used for both the (110) and (111) orientations of

the //-center with 8 parameters involving 23 movable ions. The resulting configuration showed a compression to 2.6 A with the (111 ) direction having

the lower energy by 0.2 eV. A better interaction

potential was obtained from the wave function

calculations of Wahl and Gilbert [3] for the ground

state of the //-center, leading to an equilibrium

separation of 2.70 A with a binding energy of 0.93 eV. The interaction potential was put in the form of a polynomial for use in the defect work. Employing

this new potential in the same two directions as before led to similar results — again a 0.2 eV.

difference favoring the (111) direction. Corrections to the point ion model were considered first with respect to the polarization energy. Although

for a point ion at its regular lattice position the electric field is zero, if the charge is considered as spread

out there is an average electric field present, leading

to a polarization energy. Similarly if the CPf is considered as spread out along the lines of their wave function, a polarization energy is also involved.

Taking these aspects into account led to a correction

of closely the same amount in each of the two ori- entations. The next modification of the point ion

model was made by taking into account the fact that

in the cyrstal the //-center involves attractive inter- actions with the two nearest CP ions in the (110)

direction. This has the effect of lowering the energy

in this configuration. Considering that the wave

function spreads to these ions as well, and that the

hole spends 8 to 10 percent of the time on the end

ions leads to a reduction of the energy for the (110)

direction sufficient to make it the most favorable

orientation by about 0.2 eV.

(c) Off-center ions. Recent experimental results

[4] have indicated that a Li + substituted for K+ in KC1 does not end up on the lattice site but

slightly away from it. We have investigated this in two ways. First the stability of the Li+ at a

lattice site was investigated, allowing the neighboring ions to relax and calculating the terms entering

into a certain determinant [5] whose value indicates

whether the location considered is stable or not. In this case, the Li+ turned out to be unstable at a

lattice site. Second the Li+ was assigned a vari- able parameter so that it could move along the (100)

direction and four other parameters were employed

for the surrounding ions. The interaction of Li+ with CP was calculated from the Born-Mayer

expression with the addition of a barrier to guard

against too close contact. The minimum position)

turned out to be for the Li + about 12 percent of thej

distance along the cube edge; energywise this was! stable with respect to the lattice site position by

about 0.05 eV. At present positions other than thei (100) are being investigated for the Li +

; it would'

appear that the (111) direction [6] would be moslj favorable. In addition the situation with respecl

to Li + in other crystals such as KBr is being foil

12
lowed up (where initial results show the lattice

site as a stable location), and as well an investigation of the conditions necessary for additional ions

to go into off-center positions and especially the

relation of off-center ions to the interaction

potentials.

The method used above makes considerable use

of the interaction potential between ions at close

distances. An improvement in the calculation of

these energies is needed for better accuracy in our

work. Our method has so far been applied to the

case of neutral defects in the sense that there is no

net change in the total charge in the neighborhood

of the defect. The case of charged defects, as for example, a positive ion vacancy gives rise to long

range polarization effects. A method has been

devised to take care of these effects by splitting the

electric field into two parts, that due to a neutral

lattice with displaced ions and that of a single charge

at the defect. This enables the polarization energy

to be calculated without using the concept of a polarizable medium.

References

[1] J. Tessman, A. Kahn, and W. Shockley. Phys. Rev. 92,

890(1953).

[2] F. G. Fumi and M. P. Tosi, J. Phys. Chem. Solids 25, 31

(1964); M. P. Tosi and F. G. Fumi, J. Phys. Chem. Solids 25,

45(1964).

[3] Private communication.

[4] H. S. Sack and M. C. Moriarty. Solid State Comm. 3, 93

(1965); G. Lombardo and A. 0. Pohl, Phys. Rev. Letters 15,

291 (1965).

[5] R. D. Hatcher and G. J. Dienes, Phys. Rev. 134, A214 (1964).

[6] N. E. Byer, F. S. Welsh, and H. S. Sack, Bull. Am. Phys.

Soc. 11,229(1966).

13

Energy of Formation of Vacancy Pairs in KC1

K. Tharmalingam

Westfield College, London

It is a common practice in the calculation of the

energies of (complex) defect crystals to assume for simplicity a restricted type of relaxation; for example the ten nearest neighbor ions of a vacancy pair

in alkali halides are given a single relaxation parameter in the (100) directions. Such an approximation

may be partially justified if the long-range forces

are negligible. However, the electrical forces which

are long-range in character and the next-nearest- neighbor anion-anion repulsive forces contribute

significantly to determine the minimum energy configuration and the corresponding energy. Thus a more general relaxation is expected to lower the

energy of the defect crystal and we have in this report examined this particular aspect.

The ten nearest neighbor ions (to be referred to as region I, and the rest of the lattice as region II) are given six relaxation parameters £1, £2, £3, 171,

7J2, 173, as shown in figure 1. The energy of the

KC1 defect crystal (with region II held rigid — nonpolarized, nondisplaced) was minimized with respect

to the above six variables and the results are given

in table 1. Also in this table are given the corre- sponding results with the restricted type of relaxaFigure 1

tion. We note that the energy of the defect crystal,

at this stage, included coulombic, repulsive, and

monopole polarization energies; the first two arising

from changes in self-energy of region I and inter- action of I with II and the latter essentially an estiTable 1.

B.M. potential B.M.V. potential

Restricted General Restricted General

relaxation relaxation relaxation relaxation

0.065 0.034 0.045 0.026 6 .000 .055 .000 .038

.065 .031 .045 .027

11 .065 .072 .045 .052

1)2 .000 .053 .000 .029

Its .065 .061 .045 .048 M -1.23 -1.64 - 1.11 -1.40 AE (Coul) -1.80 -2.12 -1.28 -1.52 M (Rep) 1.13 1.22 0.80 0.84 £ (Pol-Mono) -0.56 -0.74 -.63 -.72

mate of the self-energy term (since region II is rigid).

The effect of dipole-dipole interaction and the contributions from region II are estimated in the Mott

and Littleton approximation.

We note from table 1 that:

(I) The energy of the defect crystal with general

relaxation is, as expected, lower than that with

restricted relaxation. The difference in the above

energies is smaller for the B.M.V. potential than for the B.M. potential and is consistent with the fact that the B.M.V. potential is harder than the B.M.

potential.

(II) The predominant contribution to the dif- ference in energies of the restricted case and general case comes from the "electrical terms."

(III) The polarization energy combined with the

next-nearest-neighbor repulsive forces ensure that the anions (in region I) are closer to the defects

than the corresponding cations.

(IV) The fact that we have 171 , 173 > 17-2 , for all cases, is consistent with the assumptions of re- stricted relaxation. This, however, is in contrast with the anions where £2 > f 1, £3. Using the present results an estimate of the en- ergy of formation of vacancy pairs in KC1 can be

obtained as follows, with the assumption that the

B.M.V. potential is the better of the two.

(a) In the zeroth-order approximation (where both

regions I and II are rigid, nonpolarized, and nondisplaced) the energy of formation in the rigid

15
lattice (for all four cases) is £7=2.64 eV.

(b) In the first order approximation where region

II is rigid and region I is allowed to relax £7=2.64

-1.40=1.24 eV.

(c) We note that so far the results are unambiguous with region II rigid. We must, however, include

the electronic dipole-dipole interaction of region I before we can set a lowest value for the upper limit of £/. Here again one can in principle evaluate

this exactly but an estimate of this is —0.1 eV. Hence the true Ef < 1.3 (4), since the effect of relax- ing region II must be a negative contribution.

(d) An estimate of the contribution due to polarization (and coulombic relaxation) of region II may

be obtained from the interaction of the Mott and

Littleton dipoles of region II with the vacancy pair

and is found to be —0.38 eV (for all four cases).

Hence an estimate of £ in third order approximation

which does not include any dipole-dipole interaction

is 1.24-0.38 = 0.86 eV.

(e) In the next order approximation, since the

dipoles are large close to the defects, an estimate

of this (from the interaction of electronic dipoles

of I amongst themselves and with those of II) is ~ 0.1 (5). Hence the £/ in the fourth order approximation becomes ~ 1 eV.

(f) We note that so far we have assumed that the

region I "relaxation energy" is independent of the

region II relaxation. An estimate of this may be

obtained by minimizing the energy of region I (with Mott and Littleton displacement dipoles in II). We find the value — 1.40 eV is now altered to — 1.32 eV giving us an estimate of the energy of formation

of vacancy pairs in KC1 as ~ 1.1 eV.

In conclusion we remark that:

I. The effect of generalized relaxation in region I

is not negligible compared with the less certain

dipole-dipole terms in crystals containing neutral

defects.

II. Due to the relatively large polarizability, and

the next nearest neighbor repulsive interactions, care should be taken in treating the anion sublattice

close to the defects on the same basis as for the

cation sublattice.

16
Vacancies and Monovalent Cation Impurities in the Alkali Halides*

M. Doyama and M. P. Tosi

Argonne National Laboratory, Argonne, 111. 60440

An accurate description of the short-range ionic

interactions is generally important in lattice cal- culations of defect parameters in ionic crystals,

and essential in the evaluation of the energies of

motion of point defects. We have re-examined

these calculations for several basic defect configurations in a few alkali halides, using the short- range potential recently proposed by Tosi and Fumi,

and adopting consistently the Mott-Littleton scheme

for the treatment of the lattice polarization.

The defect configurations that we have investi- gated are (a) an isolated vacancy at either lattice

site, (b) a substitutional alkali impurity, (c) the

saddle-point configuration for motion of both a

'Based on work performed under the auspices of the U.S. Atomic Energy

Commission.

(positive or negative) host ion and a substitutional

alkali impurity into a vacancy. The energy of

the latter configuration has been evaluated relative

to the state in which the impurity and the vacancy

are dissociated, so that the result is directly related

to the activation energy for diffusion of the impurity.

The calculations have been carried out for NaCl,

KC1, and RbCl crystals, the impurities being Na+

, K +

, and Rb + . The experimental information avail- able for these systems is fairly extensive.

Although there is clearly room for further improvements in the model, particularly in the treat- ment of the lattice polarization, the agreement of

the theory with experiment is generally semiquantitative. The main shortcoming of the model is the

inability to yield a difference in the energies of

motion of a positive and a negative host ion.

17

Born-Model Defect Energies for CaF2

A. D. Franklin

Institute for Materials Research, National Bureau of Standards, Washington, D.C. 20234

The Born Model of an ionic solid is being applied

to the calculation of the energies of formation of

interstitial ions and vacancies of both kinds in the

fluorite-structure alkaline earth fluorides, and to obtain estimates of the energy of replacement of

Ca2+ in CaF2 by Na+ and Y3+ . Two-body inter- ionic forces are assumed, including monopolemonopole, monopole-dipole, dipole-dipole, and

Born-Mayer overlap repulsive forces. Arbitrary

radial displacements and dipole moments are as- signed the ions nearest and next-nearest each

defect (region I), and their interactions with each

other and with the defect are treated explicitly.

The crystal beyond (region II) is considered to be a

dielectric and elastic continuum, and its relaxation

energy is calculated with the approximate methods

of Mott-Littleton [1]

1 and Brauer [2], suitably modified to take into account the effect of the region I ions. The energy as a function of the displacements

and moments of the region I ions is calculated and

minimized by computer to determine the equilibrium energy and configuration.

A major uncertainty arises from the parameters

used to describe the repulsive forces. The Born- Mayer form for the repulsive energy between two

ions, Uij = Aij exp (— ry/p), was used with a single

value for p. Interactions were included for all ions separated by V2a or less, where a is the

fluorine-fluorine distance, thus involving anionanion, anion-cation, and cation-cation interactions

and requiring four parameters, A+-, A —, A + + , and

p. These were obtained by fitting the model to two elastic constants ((Cn — C12) and the mean of

C12 and C44 to simulate the Cauchy relation were

used in the manner of Reitz, Seitz, and Genberg [3]), to the equilibrium relation, and to the dielectric constants using the equation of Szigeti [4J modified

for the CaF2 lattice. Other sets of parameters

have been calculated recently by Reitz, Seitz,

and Genberg; Benson and Dempsey [5]; and Axe [6]. Comparison among the results suggests that reasonable values for A + - and A-- can be found, but for A++ the uncertainty is large. The influence of A + + on the calculated defect energies in CaF9 is illus- trated in the following table:

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

Defect A ++ , eV Formation

energy, eV

Displacement,

fraction of dis- tance to origin

6, 62

Anion vacancy 0 4.38 0.034 -0.107

104 4.39 .030 -.110

Anion interstitial 0 -1.81 .057 -.046

104 -1.77 .059 -.039

Cation vacancy 0 22.54 .152 .003 101 22.36 .152 .003

Cation interstitial 0 -17.10 -.081 .075 2.5 X 103 -16.55 -.078 .084 W -14.70 -.072 .104

in which the last two columns show the displacements of the nearest neighbor (81) and next-nearest neighbor (82) ions to the defect as fractions of their distances from it. Only for the cation interstitial

is there a cation-cation distance sufficiently small

to make A + + important.

A second major source of uncertainty is the relax- ation of region II. Sufficiently far from the defect,

it seems reasonable to assume a combination of a

polarization and an elastic displacement (defined

as a fraction of the distance r to the origin) both

decreasing as r -3. The problem is to evaluate the

intensities of these responses in terms of the vari- ational parameters of displacement and dipole

moment assigned to region I ions. In the present

calculations, the polarization part of the displacement of the Ith region II ion at n was obtained from

the Mott-Littleton approximation, 8f=(Z0/Z/)

(bln) 3M'i, where Z0 and Zt are the effective valence

of the defect and the valence of the /th ion, respectively, b

3 is the volume containing one CaF^ molecular unit, and Mi a constant containing the ionic total polarizabilities and the dielectric constant. The

elastic part was written simply 8f = \/r3

. The

parameter k was evaluated by setting the sum

Sf + Sf equal to the total displacement of either of

the two kinds of ions in region I. The table below

illustrates the effect of this choice.

The choice appears to be rather unimportant for all three defects tested. In general, propagation of

82 has been chosen, but as examination of the struc- ture shows, a considerably larger region I would

19
Defect Displacement

propagated Formation

energy, eV 5, 82

Anion vacancy.... INone 4.41 0.048 — 0.100

Oi 4.41 .040 — .101

02 A 3Q f\0 A — . 107 Anion interstitial.. None 1 Q7 oco .UOO — .048

Ol — 1 OO 1.92 .1)01 — .047

02 — 1 .oi AC "7 .1)0/ — A/I/; .1)40 Cation vacancy •• None 23.24 .150 .001

Si 22.34 .190 .001 82 22.54 .152 .003

have to be taken before one could a priori have

confidence that the approximations used for region

II could have any real validity for even the outer- most region I ions. The energy of formation of a given defect can be

subdivided in a number of ways. These calculations corresponded to the following conceptual

steps: (1) creation of the defect in an otherwise

undeformable, unpolarizable crystal, (2) displacement of region I ions into relaxed positions, (3) polarization of relaxed region I ions, (4) relaxation

and polarization of region II. fhe division of the

defect formation energies according to this scheme

is set forth in the following table, in which the

elastic term in region II was derived from 82 and

A + + was set equal to zero:

Energy term Anion

vacancy interstitial Cation

interstitial

Rigid lattice, Madelung energy... 10.79 1.53 eV - 3.05 eV- Do , repulsion -1.81 1.54 5.77 Region I displacements, -3.85 -3.75 -13.60

monopoles.

Region I displacements, 1.11 .42 2.87

repulsion.

-.22 -.14 - 1.98 Region II relaxation,

polarization. -1.51 -1.56 -6.16

Region II relaxation, -.04 .15 .25 repulsion.

Refe

[1] N. F. Mott and M. J. Littleton, Trans. Faraday Soc. 34,

485(1938).

[2] P. Brauer, Z. Naturforsch. 6A, 255 (1951).

[3] J. R. Reitz, R. N. Seitz, and R. W. Genberg, J. Phys. Chem.

Solids 19, 73 (1961).

The "best" values for the formation energies of defects in CaF2 from these calculations are listed

in the table below:

Defect Formation ener£;y, eV 8, 82

Anion vacancy 4.5 ±0.2 0.034 -0.106

23.5 ±0.3 .149 .003 Anion interstitial -1.8 ±0.2 .057 -.045

Cation interstitial -15.9 ±0.5 -.080 .080

where the uncertainties shown reflect the uncertainty in A + + and in the elastic part of the region II relaxation. From these and the lattice energy the

formation energies of anion and cation Frenkel pairs

and of Schottky trios can be calculated. These are

Defect pair Formation energy

Anion Frenkel 2.7 ±0.4 eV/pair Cation Frenkel 7.5 ±0.6 eV/pair

Schottky 5.1 ±0.9 eV/trio

These energies are estimates of the enthalpies

of formation of the various possible intrinsic defect

sets. On the basis of these enthalpies alone, it would be expected that anion Frenkel pairs would

be the dominant intrinsic defects, although the

rather large uncertainties do not allow this conclusion to be definite. Generally speaking, en- tropies of formation of interstitials are found to be

larger than those of vacancies, a fact tending still further to favor the anion Frenkel pair. The for- mation energy for the anion Frenkel pair calculated

here agrees well with Ure's [7] experimental value

of 2.8 eV. This agreement lends further support

to the use of the Born model for defect calculations

in CaF2 .

rences

[4] B. Szigeti, Proc. Roy. Soc. A204, 52 (1950).

[5] G. B. Benson, E. Dempsey, Proc. Roy. Soc. 266A, 344 (1962).

[6] J. D. Axe, Phys. Rev. 139, A1215 (1965).

[7] R. Ure, J. Chem. Phys. 26, 1363 (1957).

20
Lattice Distortion Around Defects

Tatumi Kurosawa

Chuo University, Tokyo, Japan

A general procedure of determining the lattice distortion (or the electronic polarization) around a

defect is presented.

The lattice distortion may be calculated by solving

the equation of equilibrium of elasticity theory

(or of electrostatic theory) in the corresponding

continuous medium [l], 1 beyond the vicinity of the

defect. There are many solutions of the equilibrium equation which tend to zero at infinity, and the

lattice distortion should be represented by a linear

combination of them, in general. In the case of a

point defect, the displacement u(r) is given by

fhere

u(r)= 2) u„(r).

u„(r) = r

n 2 anm fnm (#,$),

(1)

(2)

D using eq (5) with a tentatively given set of the

coefficients. Namely, the position r, of the t'th atom is given by

r; = rio + u(rio), (6)

in which r,o is the position before the formation of the defect. Next, we calculate the equilibrium

position of atoms in the region C under the given

position of the atoms in D. If the values of the

coefficients are correctly given, all atoms in D as well as those in C are in equilibrium, at least approximately. If not, however, the atoms in D, particularly those bordering the region C, are not in equilibrium and suffer forces. In this case, it is convenient to the calculation to consider the

following quantities:

(7)

in which the f„ m (#,(/>) are analogous to the spherical

harmonics in the case of the electrostatic field, and some of them can be omitted by the symmetry. In

the case of a dislocation, the displacement is given

by

(3)

where

u(r) =^u„(r),

u„(r) = r

n^anm fnm(0), (4)

in which the factor r " becomes a constant or log r

if n = 0. In either case, we can write u(r) as

u(r) = £^uM

(r). (5)

The coefficients are determined through a microscopic calculation of the defect structure. However, there has been no systematic method to do this. The procedure described below may be

a reasonable way for this purpose.

We divide the whole crystal into two regions;

the region C which is the immediate vicinity of the

defect and the region D which is distant from it. First, we calculate the position of atoms in the region

where F, is the force on the ith atom and the sum

is taken over the region D. Generally speaking,

the sum converges quite rapidly for large r, and is easily evaluated. If FM

is positive, the force on the

region D works to increase the coefficient Au , and

vice versa. Therefore, FM may be regarded as a generalized force on the /v,th distortion mode uM

. We should choose the coefficients so as to remove

these forces FM's. In practice, we retain a finite number of terms of

(5), calculate the following symmetric matrix:

dFJdAi ... dFJdA^...,

(8)

' Figures in brackets indicate the literature references at the end of this paper.

and evaluate the coefficients in the linear approximation.

The present method is useful for many problems

concerning defects. For instance;

(1) A substitutional impurity atom or interstitial atom produces a strain field around it. However,

the strength of the strain field cannot be determined

by a simple application of the Mott-Littleton

method [2]. The strain field of the distant region

21
should be known when we calculate the displacement of neighboring atoms to the defect, whereas

the former depends on the latter. A similar situ- ation is found in the case of vacancy pair in ionic

crystals, the dipole moment of which is not unambiguously determined by the usual method. The

present method is applicable to such cases.

(2) The first term is dominant in the expression

(1) or (3) when r is large. However, the effect of -u higher order terms is not small unless the size of

the region C is very large. Generally, the error a produced by the neglect of the higher order terms >^

is much larger than that by the use of the approxi- ^ mate expression (6), which replaces the distortion

of a discrete lattice by that of a continuum, for the

position of the atoms in D. The addition of the

higher order terms to the distortion in D improves

considerably the approximation. An example is illustrated in figure 1, which shows the sum 2 V |F;| 2 iED

in the case of an edge dislocation. This demonstrates how the force on the atoms in D is relaxed

by the addition of the higher order distortion modes;

the abscissa means that the sum in (3) is taken

from zero to 7imp V.

References

[1] J. D. Eshelby, Solid State Physics, Vol. 3 (Academic Press

Inc., New York, 1957).

[2] Y. Fukai, J. Phys. Soc. Japan 18,1413(1963).

-A

i.

2 The sum ^ \F,

| 2 is always zero except for calculational error.

22
Calculation of Lattice Distortion Around Point Defects By Lattice Statics

Hiroshi Kanzaki

Institute for Solid State Physics, University of Tokyo, Tokyo, Japan

In the ordinary procedure for calculating lattice distortions around point defects, the crystal containing the defect is divided into two regions: the region

in the immediate vicinity of the defect will be called

"region I"; "region II" is the region outside the

region I. It is usually further assumed, at least in the case of a simple defect such as a vacancy, that displacements of atoms in the region II can be

obtained by applying isotropic elasticity theory.

Final expressions for lattice distortion in this pro- cedure are obtained by "joining" the distortion in region II thus obtained to the distortion in region I where existence of the defect is taken into account

in the discrete crystal lattice. In the calculation of distortion around defects by

lattice statics, discreteness of the crystal lattice is considered throughout the crystal. The method

was first developed by Professor Pryce at Bristol,

applied to vacancy in solid argon by the present

writer [l],1 and later very extensively applied to the

K+ impurity in NaCl by Hardy [2]. The procedure

for obtaining displacements around defects is essentially to solve the problem of obtaining dis- placements of lattice points in the perfect lattice under a given external force representing the atomic

nature of the defect. We start by expanding the

displacements ijir 1

) in a series of plane waves

assuming the following expression in the harmonic

approximation;

£(r 0=2Q(q

).e«rr', (1) q where q is the wave vector and the expansion coeffi- cients Q(q) are the normal coordinates as in the

theory of lattice dynamics. After having solved

the equilibrium equations for Q(q) under the

existence of generalized forces G(q) representing

the defect, we can calculate the displacements

£(r') at any lattice points r' by using the eq (1). Several features of the lattice statics method can

be summarized as follows, (a) The whole crystal

containing the defect is allowed to relax at the same

time. It is not necessary to use the procedure of "dividing and joining" as in the ordinary method

described above. The results thus obtained will be more reliable compared with these previous

calculations not only for the distortion in the neighborhood of defects but for the distortion on the

macroscopic scale, (b) Consideration of image

forces due to the existence of the free surface can

be introduced easily by relaxing the rigid cyclic

lattice boundary and making the free surface the

boundary, (c) It is generally possible without too much difficulty to take account of nonharmonic

interactions in the immediate vicinity of the defect. The lattice statics method can be applied to such

types of crystals that can be represented by two- body interatomic potential functions. Ionic, van der Waals, and valency crystals are in this category [3]. Possible extension of the method to metals will be one of the future problems and will not be discussed at the present stage.

The following discussion based on the lattice statics method is concerned only with long-range

distortion far from the point defects. The problem

is essentially that of macroscopic distortion in the

anisotropic elastic continuum as was already clari- fied by Hardy [2]. It will be shown that patterns of

distortion are quite different from those expected

from isotropic elasticity theory not only around

noncubic axial defects but also around simple cubic

defects in the case of the face-centered-cubic

lattice.

1. The Model and Its Elastic Coefficients

In the following, we use the same model as in the

calculation for solid argon [1]; each site in a face- centered-cubic lattice is connected by purely central interactions of Hookeian force Constant A with

its twelve nearest neighbors.

Elastic coefficients of the crystal thus assumed

are expressed as follows;

; Cn = 2A/a, C44 = C12 =Ala,

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

where aj\2 is the distance between nearest neighbor atoms (see figs. 1 and 2). The elastic anisotropy factor in a cubic crystal is defined as the square of the ratio of velocities of

shear waves propagating in the (100) and (110)

directions, and in the case of our model, the anisotropy factor = 2C4 J(Ci i — Ci2 ) = 2. The experimental value of anisotropy factor is not available for solid argon. The value is 0.76 for NaCl and 3.3

for Cu.

23
1 <0I0>

>/\< I0>

<I00>

X

FIGURE 1. Double forces along the (001) axis and anisotropic pattern ofdisplacements in the (001 ) plane through the center.

<ioi>

r —j<

,1 —X

-a/2

FIGURE 2. Double forces along the (101) axis and anisotropic pattern of displacements in the (101) plane through the center.

2. Anisotropic Long-Range Distortion in a Cubic Crystal Due to Defects of

"Cubic" Character

The expression for this case has been already

given [1]. Displacements at distance r= (^j^

from the defect are shown in table 1. In table 1, F is the magnitude of force acting radially (along

(110)) on each of the twelve first nearest neighbors

of the substitutional defect. The results are exactly

the same for applying force V2F radially (along

(100)) on each of the six second nearest neighbors

of the defect or applying 2V2F on each of the six nearest neighbors of interstitial atom in a bodycentered-cubic position.

Table 1. Displacements at distance {2,}^ around the "cubic'

defect (a numerical error in ref. (1) has been corrected).

Direction

Displacement in units

(100) 0.079 <no> .498 (111) .670

In the case of K+ in NaCl, Hardy evaluated longrange distortions. in terms of the force F« originating

from repulsive interactions between ions. His

results [2] are duplicated in table 2.

Table 2. Displacements at distance Xr0 around K+ in NaCl

(after Hardy)

Direction Displacement in units of(/W-2

)

cm dyne' 1 (100) 1.78 XIO"5 <no> 1.24 X 10-5 <111> 0.747 X 10- 5

The difference between the anisotropies of dis- tortion shown in table 1 and table 2 can be understood in terms of the difference of elastic anisotropy

factor between the two cases.

3. Long-Range Distortion in Cubic Crystal Due to "Noncubic" Axial

Defects

There are several circumstances in which the treatment of distortion of this nature becomes necessary. (1) Substitutional impurity ions subjected

to a Jahn-Teller distortion in a cubic lattice, such as Cu2+ or Ag2+ ions in alkali halides. Self-trapped

holes in alkali halides are in a similar category.

(2) Interstitial atoms in a cubic lattice, such as crowdions, split interstitials, pairs of vacancies, and

saddle-point configurations of these and other

"cubic" defects.

It can be understood that many defects of physically interesting nature are included in this group

of defects. In the following, results of numerical

calculations are presented for two kinds of lattice distortion in our model crystal. One is that due to two equal and opposite forces along the (100) axis

and the other is due to those acting along the (110)

axis.

3.1. Double Forces Along (001) Acting at Points Distance h Apart Along the Axis

Using the coordinate system shown in figure 1, displacements at distance r are evaluated for various

directions as shown in table 3. It is interesting to notice that the value for (001) is comparable with

that obtained from isotropic elastic theory,

Also shown in figure 1 is the anisotropic patterr

of displacements in the (001) plane through th<

24
center. The pattern is always circular in an iso- tropic elastic body.

3.2. Double Forces Along (101) Acting at the Points Distance h Apart Along the

Axis

Figure 2 shows the coordinate system used and

the anisotropic pattern in the (101) plane through

the center. Displacements at distance r are shown

in table 4. It is again interesting to notice that the

axial displacement is V2~ times that of the (001)

double forces given in table 3.

3.3. Interaction Energy Between Two Anisotropic Defects

Especially interesting is the interaction energy

between two defects as a function of relative orientations. Such evaluation has been made for K+

in NaCl by Hardy [2]. In such anisotropic defects

as discussed here, we can expect large attraction

between two defects in perpendicular relative orientations and, furthermore, strong angular dependence

in the perpendicular plane as seen in figures 1 and

2. This type of calculation of energy requires a knowledge of the distortion in the neighborhood

of the defects where nonquadratic terms always

become important, as shown already for the va- cancy [1]. Here, we shall not enter into further

details of this discussion, where the choice of

interatomic potential becomes more critical, es- pecially for the case of interstitial defects, such as the crowdion, as compared with vacancies.

Table 3. Displacements at distance r due to double forces F

along (001 ) acting at the points distance h apart Displacements expressed in unit of (— ~\

\4ttCh rV

Direction (001) (100) (110) (1. 0.414, 0) ((cos J

sinf.O))

0 -0.460 -0.527 -0.501

i« 0 0 -0.527 -0.323 & 1.000 0 0 0

III 1.000 0.460 0.747 0.596

Table 4. Displacements at distance r due to double forces F

along (101) acting at the points h apart Displacements in unit of (— -)

\4ttC44 rV

Direction (101) (101) (010) (HI)

&

&

&

-1.000

0

1.000

- 0.080 0 -0.080

0 -0.456

0

-0.204

-0.303

-0.204

III 1.413 0.113 0.456 0.418

The writer thanks Miss S. Ozawa for numerical

calculations.

References

[1) H. Kanzaki, J. Phys. Chem. Solids 2,24(1957). [3] M. Born and K. Huang, Dynamical Theory of Crystal Lattices

[2] J. R. Hardy, J. Phys. Chem. Solids 15, 39 (I960): 23, 113 (Oxford University Press) 1964.

(1962).

256-724 0-67—

3

25

Calculation of Migration Energies of an Interstitial in

Germanium and Silicon*

Ryukiti R. Hasiguti

Department of Metallurgy, University of Tokyo, Bunkyo-ku, Tokyo, Japan

1. Introduction

The formation and migration energies of an atomic

vacancy and an interstitial atom of germanium and

silicon have been offering many difficult and inter- esting problems to be solved. One of the important

problems, which may be called the "triangular

paradox," is as follows. Let us take as an example

the migration energy of a vacancy. (1) The experimental results on germanium, (2) the experimental

results on silicon, and (3) the theories of germanium

and silicon, which give nearly the same but a little different calculated values for germanium and sili- con, form the triangle of paradox. The migration

energy, Em, experimentally obtained for a vacancy

in germanium is about 1 eV (refs.'[l, 2, and 3], 1 and references therein). This value agrees well

with theoretical estimations, which give the values

of 0.95 eV [4] and 0.98 eV [5]. On the other hand,

the experimental value of Em for silicon, 0.33 eV [6],

is considerably different from the theoretically

estimated values, 1.06 eV [4] and 1.09 eV [5]. This

situation may be taken as an indication that any

one of the three components of the paradox tri- angle must equally be doubted and reinvestigated

at this stage, and if any one of the components is

found to be appropriately modifiable, then there

would be a possibility of dissolution of the triangular

paradox. Recently evidence was found that a vacancy of germanium may migrate at very low

temperatures (at ~ 60 °K) [7]. This seems to make

the paradox begin to dissolve.

The situation for the migration energies of an

interstitial in germanium and silicon seems to be

exactly the same as that of the above mentioned

vacancy, although the situation for the interstitial

is much less clear than that of the vacancy, because

the experimental results for the interstitial are yet

very insufficient [1], and only one theoretical result

is available at this moment, worked out by Benne- mann [5J. The objective of the present paper is to make a contribution to the theoretical evaluation of migration energies of an interstitial in germanium and

silicon in the hope of diminishing the difficulties in the paradox. The method used here is an ex- tension of Weiser's theory [8] which was successfully used in treating the migration energies of

interstitial impurity atoms in germanium and silicon.

2. Theory and Results

We shall consider only a positive singly charged

interstitial atom which is believed to exist in germanium [1, 2] and silicon [6]. Then the migration behavior of an interstitial can be treated as was that of a monovalent interstitial impurity atom

such as lithium, copper and silver in germanium

and silicon, although there may exist some dif- ferences between the above two cases as will be

discussed later. The proposal of considering a

positive singly charged interstitial is one of the

unique features of the present theory.

The migration of a monovalent impurity atom in germanium and silicon was successfully treated by

Weiser [8] considering an attractive potential due

to the polarization of the host atoms and an ex-

*This research was partly performed at the Institute of Physical and Chemical

Research. Bunkyo-ku, Tokyo.

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

ponential repulsive potential due to the overlapping

of nonbonding electrons. This method will be,

in the present paper, extended to the case of the

interstitial migration.

As the two interstitial sites, the so-called [5, 8] tetrahedral sites (T site) and hexagonal site (H

site) will be considered, of which one will be the

stable site and the other the saddle point.

The attractive polarization energy, Uvo\, due to the

interaction of the positive singly charged interstitial ion with the dipoles it has induced in the

host atoms was calculated by Weiser using the

Mott-Littleton method [9] for the T and H interstitial sites. The details will be omitted here, be- cause we use Weiser's result without any modification as far as Uw\ is concerned. The result which we use is

Ai/po, = 0.75 eV, (1)

27
where Up0l and U%ni are the polarization energies at H and T sites, respectively. The same result is applicable for both germanium and silicon.

2.1. Repulsive Energy

In order to obtain the repulsive energy, £/rep , we

shall use the repulsive interaction of Born-Mayer

type, as was done by Weiser in the case of impurity atoms. As this is a phenomenological method, the

most important point is in the choice of Born- Mayer constants. Now we shall express the Born-Mayer potential

in the following form:

Uiep =A exp [{r0 -r)l p], (2) where A and p are the Born-Mayer constants,

r0 is the equilibrium distance between the nearest neighbor atoms, and r is the distance between re- spective pairs of atoms. While Weiser uses the Born-Mayer constants which are appropriate for sodium chloride-type

ionic crystals, the present author uses the Born- Mayer constants derived from the compressibilities of germanium and silicon using the method

described by Leibfried [10] (potential 1 in table 1 and figs. 1 and 2). In the derivation of the constants, the Madelung

constant of 1.638 for zinc blend type crystals rather

than that of 1.747 for sodium chloride type crystals was used.

Table 1 shows the Born-Mayer constants used in the present paper. Figures 1 and 2 shows the

10 u

(eV)

10'

10

10

10

10

10

1. From bulk modulus.

2. "A" reduced (Em «0>.

3. Bulk mod. reduced (E m

=0)

M. Morse potential.

1.0 20 'o 30

r(A)

10'

u

(eV)

io10'

10

I0

_

10"

10'

1. From bulk modulus.

2. "A" reduced (Em-0).

3. Bulk mod. reduced (Em=0

M. Morse potential.

1.0 2 0 r0 3.0 — r(A)

Figure 2. Plots of Born-Mayer potential in silicon.

TABLE 1. Born-Mayer constants used in the present paper

1 A(Ge) = 0.202 eV; ^(Si) = 0.195 eV

From bulk

p(Ge) = 0.206 A; p(Si) = 0.182 A modulus

A(Ge) = 0.177 eV; A(Si) = 0.150 eV

2 "A" reduced

p(Ge) = 0.206 A; p(Si) = 0.182

3 ^(Ge) = 0.267 eV; A(Si)= 0.305 eV

Bulk modulus

p(Ge) = 0.272 A; p(Si) = 0.296 A reduced

FIGURE 1. Plots of Born-Mayer potential in germanium.

plots of the Born-Mayer potentials for germanium

and silicon. The repulsive interaction energies are calculated

for both H and T interstitial sites of germanium

and silicon. The pair-wise interactions of an

interstitial with the nearest and the next-nearest neighbors are included in the calculation for H\

and T sites, respectively, as follows: U%= 6A exp [(r0 -r„)/p]

+ 8A exp [(r0 -1.56rH )lp], (3) UT

Tep = ^A exp [(r0 -rr)/p]

+ 6A exp [(ro-1.15rr)/p], (4)

where rn is the distance between the H site and its! nearest neighbor, and rj is the distance between

T site and its nearest neighbor, the latter being the! same as ro. Interactions with further neighbors!

28
are negligible. Thus the difference between H

and T sites are

Af/rep =£/?ep -t/?ep. (5)

Then the migration energy of an interstitial, Eim, is obtained

to be

Eim = At/p0 i — At/rep = 0.75 eV-At/rep (6)

for both germanium and silicon.

2.2. Results

The results obtained using the Born-Mayer potentials given in table 1 and figures 1 and 2 are shown in table 2.

3. Discussions

Now we shall make some discussion as to the

choice of Born-Mayer constants. Before doing

so, however, questions must be raised as to the

applicability of Born-Mayer type repulsive potential in the case of germanium and silicon. Objections against the use of Born-Mayer potential may arise concerning the following points:

firstly, in the diamond lattice, the covalent bonding

is important, so that the potential may be rather

expressed by the Morse function [4], which is also an empirical form of expression in the case of dia- mond lattice. However, as the Morse potential

consists of attractive and repulsive parts, it is possible to express the repulsive part of Morse potential by the Born-Mayer potential.

Table 2. Migration energies of an interstitial Em = U

(H)-U<T, in eV

Born-Mayer

potential

1 2 3

Ge -0.25 0 0

Si -0.22 0 0

Now the second objection may be expressed as

follows. In" such an open lattice as that of diamond

structure, the overlapping of the wave function of

closed-shell core-electrons may not be appreciable. If the Born-Mayer type potential is to express only the closed-shell repulsion, it may not be appropriate to use it in germanium and silicon. However, if all kinds of repulsive interactions in- cluding the one due to valence electrons are in- cluded in our expression of the Born-Mayer potential, this can be conveniently used as an empirical

expression of the repulsive part of the potentials

of germanium and silicon [11]. Thus if the Born-Mayer constants are determined

from the elastic constants or the bulk modulus of

the bulk crystal, then the above two objections are simultaneously solved. The potentials 1 for germanium and silicon shown in table 1 and figures 1 and 2 are obtained

from the bulk moduli of germanium and silicon,

respectively. These potentials result in migration

energies of 0.25 eV and 0.22 eV for germanium

and silicon, respectively.

It is quite possible that the repulsive force

exerted by a singly positively charged interstitial atom is rather smaller than that exerted by an atom at a regular site with covalent bonds, be- cause three valence electrons of an interstitial atom do not make regular covalent bonds. The

reduced repulsive force results in a reduced migration energy. The extent of reduction of the repulsive force

of an interstitial atom compared with that of a regular atom is not known at this moment. Therefore, the following approach was made; i.e., Born- Mayer potentials which give the zero migration

energy were calculated, and were compared with

the potentials 1. Now in the potentials 2 shown in table 1 and

figures 1 and 2 the constants A were reduced so

that the migration energies for germanium and

silicon will be zero. In the potentials 3 the bulk moduli, from which

Born-Mayer constants are obtained, were reduced

so that the zero migration energy for both ger- manium and silicon was obtained. Thus-obtained

Born-Mayer potentials 3 are shown in table 1 and

figures 1 and 2. These potentials correspond to the bulk moduli reduced 29 percent and 45 percent

for germanium and silicon, respectively.

Now the Born-Mayer potentials used here will be compared with the Morse potentials used by

Swalin [4] when he calculated the energies of

vacancies in germanium and silicon. In figures

1 and 2 the broken lines represent the Morse

potentials. At larger distances the Morse potentials deviate from the Born-Mayer potentials

because of the increase of attractive component.

Actually the curves go to minus infinity at r=r0 . At smaller distances, neither Born-Mayer potentials nor Morse potentials are reliable, because

they are determined at or near equilibrium separations. It is significant, therefore, that the potentials 1 and 2 agree fairly well with the Morse

potentials at intermediate distances. Moreover,

the potentials 2 agree the best with the Morse

potentials.

Now if the above discussions are accepted, the

migration energies obtained from the potentials 1 are rather the upper limits of migration energies.

Although we do not know the extent of reduction

of the repulsive force of a singly positively charged

interstitial atom compared with that of a regular

29
atom in a regular site, it was shown above that a

slight change of Born-Mayer potentials reduces the

migration energy easily to zero. This is especially

the case for the potentials 2. This suggests that the actual migration energies are somewhere be- tween 0 and the above-shown upper limit values. As shown in table 2, the migration energies have

the minus sign, which means that the T-site is the

saddle point and the //-site is the stable site. This is contrary to Bennemann's results. Finally all the available theoretical results of

formation and migration energies are collected in table 3.

Table 3. Formation and migration energies of a vacancy and

an interstitial in eV

In conclusion the author thanks Miss S. Motomiya

for her assistance in making calculations.

EVJ Evm E,i Elm Authors

Ge

2.07

1.62 ~ 2.54

1.91

0.95

0.98 0.93 0.44

0.25

0 ~ 0.25

Swalin [4] Scholtz-Seeger

[12, 13] Bennemann [5] Hasiguti

(present

result)

Si

2.32

2.13

1.06

1.09 1.09 0.51

0.22

0 ~ 0.22

Swalin [4] Bennemann [5]

Hasiguti

(present

result)

Diamond

4.16

3.68

2.02

1.85 1.76 0.85

Swalin [4] Bennemann [5]

4. References

[1] R.

[2] S.

[3] A.

[4] R.

[5] K.

[6] G.

R. Hasiguti and S. Ishino, Proc. Symp. on Radiation Damage in Semiconductors (7th Int. Conf. Phys. Semiconductors, Paris, 1964), Dunod, Paris (Academic Press, New York, 1965), p. 259.

Ishino, F. Nakazawa and R. R. Hasiguti, J. Phys. Soc. Japan, 20, 817 (1965). Hiraki and T. Suita, J. Phys. Soc. Japan 18, Suppl.

Ill, 254 (1963). A. Swalin, J. Phys. Chem. Solids 18, 290 (1961). H. Bennemann, Phys. Rev. 137, A1497 (1965). D. Watkins, Proc. Symp. on Radiation Damage in Semiconductors (7th Int. Conf. Phys. Semiconductors,

Paris, 1964), Dunod, Paris (Academic Press, New York,

1965), p. 97.

[7] R. E. Whan, Appl. Phys. Letters 6, 221 (1965).

[8] K. Weiser, Phys. Rev. 126, 1427 (1962).

[9] N. F. Mott and M. J. Littleton, Trans. Faraday Soc. 34,

485 (1938).

[10] G. Leibfried, Handbuch der Phys. edited by S. Fliigge, VII I, 1955, p. 104.

[11] The author appreciates Dr. K. H. Bennemann's private discussion on this point.

[12] A. Scholz, Phys. Stat. Solidi 3 , 42 (1963).

[13] A. Scholz and A. Seeger, Phys. Stat. Solidi 3, 1480 (1963).

30
Calculation of the Physical Properties of Defects in Crystals by

Approximate Self-Consistent-Field Methods

Emmett B. Moore, Jr.

Boeing Scientific Research Laboratories, Seattle, Wash. 98124

The rigorous Roothaan-Hartree-Fock linear-combination-of-atomic-orbitals self-consis tent-field

(RHF-LCAO-SCF) method [1, 2] 1 of calculating

the physical properties of small molecules may be

extended by the use of suitable approximations [3]

and group theory to rather large molecules, or in our case to molecules which may be regarded as small regions of a crystal containing a defect. In

this paper we outline the rigorous RHF-LCAOSCF method, we show the relation between this method and the simpler extended-Hiickel-theory

(EHT) which has been used [4] to calculate some

of the properties of defects in graphite, and we show

how the RHF-LCAO-SCF method may be approximated to permit calculations of the physical properties of defects in crystals. This approximation

of the SCF method has the advantage of being

applicable to large molecules (or regions in a crystal)

while being much more rigorous than the EHT

method and nearly as rigorous as the parent SCF

method.

In the Hartree-Fock self-consistent-field treat- ment the wave function <t> for the system is considered to be an antisymmetrized product of spin

orbitals t//",

<t>= (N\)~ ll2AP\\+%\ (1)

where each spin orbital contains one electron. There are /V electrons in the system and they are

referred to by the Greek letters /i. and v. There

are also /V spin orbitals which are referred to by the

Greek letter k. Each spin orbital is composed of two parts, an orbital part and a spin part:

(2)

There are thus N/2 orbitals <f>?K which we shall designate as molecular orbitals (subscripts i and j)

in anticipation of Roothaan's treatment. When the wave function <I> is substituted into the

Schriidinger equation the energy of the system is given by

E= » ,3, / <t><t>dr where the integration is carried out over the space

and spin coordinates of all the electrons. We omit

the complex conjugate notation because, in the

end, all our wave functions will be real. The

Hamiltonian if of the system is given by

1 Vg1 A J_ 1 A Z^Zs

9 w r^v 9 2u rab (4)

in atomic units where the terms in order are elec- tron kinetic energy, electron-nuclear attraction

energy, electron-electron repulsion energy, and

nuclear-nuclear repulsion energy. The last term

can be carried outside the Hamiltonian because

the wave function <I> does not contain nuclear co- ordinates. There are Q nuclei in the system which

are referred to by the letters A and B. Each

nucleus carries a charge Z. When the energy

expression is written out it becomes

N/2 £ = 2 £ tf, + £ (2Jij - Kij) + (NR)

where

Hi = jfrHfrdv, H = h V 2 Q 7 a

(5)

(6)

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

f d>>xd> v<b>

J-d>1' f <b^(b"(b^d)!' , , h= JlJZJZtlJdfidv, Kv=\ _ZiimZi dfjidv

and (NR) are the nuclear repulsion terms. Equations 5, 6, and 7 are included here only for completeness since we shall concern ourselves only with

Roothaan's expression for the energy rather than

the Hartree-Fock expression (eq (5)). For simplicity we drop the superscript v in eq (6) and subsequent

expressions when the integration is carried out over the coordinates of just one electron. Roothaan's valuable contribution [1J was to extend

Hartree-Fock theory (which mainly concerns atoms)

to molecules explicitly by assuming the molecular

orbitals to be linear combinations of atomic orbitals

31
X centered on the atoms in the molecule.

n2 c'«Xa (8)

The subscripts a and /3 refer to the atomic orbitals of which there are n in the system. It is customary

to take n > N/2. When the energy (eq. (5)) is minimized with respect to the atomic orbital coefficients

8c,a = 0 (9)

there results the familiar set of equations

n ^ (Ha/} — €iSaf3)Cit3 = 0, (X= 1 . . . Tl (10) 0=1

where

ffaj3 = [a|/3] +2 X Cjycjs(2[al3\y8]-[a8\yffl (11)

j ys HB]= f Xa f-|V 2 -^)^ (12)

_ f xSxjxjxj

dfidv

and

/

Sa/3= XaXfidv.

(13)

(14)

In the Roothaan treatment one evaluates all the in- tergrals explicitly, assumes a set of filled molecular

orbitals, i.e., assumes the coefficients a, calculates

Hap and then solves eq (10) for the molecular orbital energies e, and a new set of molecular orbital coeffi- cients Cj. This process is repeated iteratively until the set of coefficients from the solutions to eq (10)

becomes the same as that used to calculate Hap- This process insures self-consistency within the

framework of the method.

The energy of the system then becomes:

E= £ Hi+ £ G + (NR)

i = 1 i = l

2i = 1 y 2 £ Ci - Y(2^-£y)+(7V7?)

where

Hi=^CiaCil3[a\P}.

a/3

(15)

(16)

(17)

In principle these equations can be solved for any

size system but in practice the evaluation of the

three and four center integrals is so difficult and time

consuming that the method is limited at present to about eight atoms and thus is not really applicable

at all to the problem of calculating the physical

properties of defects in crystals. On the other

hand, the extended-Huckel-theory method [5] has

been used for 35 atoms [4] and simple application of

group theory can make this number much larger.

For instance in our graphite calculation [4] we con- sidered a single layer containing 24 carbon atoms

and 12 hydrogen atoms (to simulate boundary conditions), while use of the point group C-zv would have

permitted us to consider a three layer structure containing 96 carbon atoms and 42 hydrogen atoms.

In EHT one again assumes the molecular orbitals

to be linear combinations of atomic orbitals (eq (8)) and carries out a variation procedure (eq (9)) which

leads to a set of equations of exactly the same form

as eq (10). Again the overlap integrals {S ap) are evaluated explicitly but here the similarity ends.

The diagonal Hamiltonian matrix elements (Haa) are approximated, usually by valence state ioniza- tion energies, and the off diagonal elements {Hap) are approximated, usually by the Wolfsberg-Helmholz approximation [6]. If the EHT Hamiltonian

matrix elements turned out to be exactly the values

obtained in the last cycle of the SCF method, then

of course the EHT molecular orbital energies and

coefficients would be exactly the same as those

obtained by the RHF-LCAO-SCF method. This is one of the things we seek and a method for doing

this will be outlined in the last part of this paper. There is one further difficulty with the EHT

method and this involves the way the total energy of

the system is calculated. The EHT energy of the

system is assumed to be just the first term of eq (16),

i.e., twice the sum of the energies of the filled molecular orbitals. This includes electron kinetic

energy and electron-nuclear attraction energy but

ignores electron-electron repulsions (Ju and Kv) and

the nuclear-nuclear repulsions. The success of our previous work [4] on the calculation of the energy

of vacancy migration and Frenkel pair formation in graphite depends on the fact that these two terms

partially cancel each other as well as on the fact that the sum of the energies of the filled molecular

orbitals is sufficient in most cases to establish an energy minimum. The second thing we seek then

is a better method of calculating the energy of the

system so that we may have more confidence in the energies we calculate for defect formation and

migration etc. Newton, Boer, and Lipscomb [3] propose a refine- ment of the RHF-LCAO-SCF method (or improvement of the EHT method, depending on your point

of view) which will allow calculation of values of the

molecular orbital coefficients and energies more

nearly equivalent to the exact RHF-LCAO-SCF

values than the EHT method will allow. The

major advantage is of course that the method can

be extended to systems containing the same number

of atoms as can be handled by EHT. They have

32
compared many small-molecule SCF calculations

and have concluded that the Hamiltonian matrix

elements from small-molecule calculations can be

used to calculate the Hamiltonian matrix elements

for similar but much larger molecules. They thus

avoid the problem of calculating all the necessary

integrals but at the same time simulate what eq (10) would be for an exact large molecule calculation.

They write the diagonal Hamiltonian matrix ele- ments as:

(18)

and calculate the first term (kinetic energy) exactly.

This term is subtracted from the Haa term, taken

from small molecule calculations, to give Vaa . They

have determined that the Wolfsberg-Helmholz ap- proximation applies only to the potential energy

part of the Hamiltonian matrix elements so for the

off diagonal elements Hafj they use the expression

Hats = ( Xa(- § V2 )x^ + "KSae(Vaa + Vm)\T (19)

where the kinetic energy term is again calculated

exactly. We place the Wolfsberg-Helmholz ap- proximation in quotation marks because the exact

expression and value of K depend on which atomic

orbitals are involved. Finally the energy of the

system as calculated from eq (15) becomes

E=2j^CiaCili \Xa V2 - £ — \Xpdv

i = 1 a/3 L J \ A = 1

' A

1

+ ^€i + (NR). (20)

i = 1

This expression necessitates the calculation of

nuclear-electron attraction integrals and nuclearnuclear repulsion terms as well as the kinetic

energy integrals which were necessary to set up

the Hamiltonian matrix elements in eq (10). The

nuclear-nuclear repulsion terms are evaluated

classically and the kinetic energy and nuclearelectron attraction integrals may be evaluated with

the use of computer programs available from the

Quantum Chemistry Program Exchange [7]. Programs are also available from the same source

for solving eq (10). Lipscomb and his co-workers are using this ap- proximate SCF method with success to calculate

the physical properties of a wide variety of large

molecules for which prototype small molecule

SCF calculations are available. These calculations, our success with EHT, the close relation between

EHT and SCF theory, the impossibility of solving

the Schrodinger equation directly, and the dif- ficulty of carrying out complete SCF calculations on large systems, lead us to believe that this ap- proximate self-consistent-field method will be very

useful in calculating the physical properties of defects in crystals.

References

[1] C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951).

[2] S. M. Blinder, Am. J. Phys. 33, 431 (1965). This is a very

readable account of the Hartree, Hartree-Fock, and Root- haan-Hartree-Fock self-consistent-field methods, although

the author carries out the treatment for spin-orbitals rather than molecular orbitals and thus omits a factor of 2 in some of the expressions.

[3] M. D. Newton, F. P. Boer, and W. N. Lipscomb, J. Am. Chem.

Soc. 88,2353(1966).

[4] E. B. Moore, Jr. and C. M. Carlson, Solid State Comm.

4, 47 (1966).

[5] R. Hoffmann, J. Chem. Phys. 39, 1397 (1963).

[6] M. Wolfsberg and L. Helmholz, J. Chem. Phys. 20, 837

(1952).

[7] Quantum Chemistry Program Exchange, Indiana University, Bloomington, Indiana.

33

Interatomic Potentials for Defect Calculations in Metals

H. B. Huntington

Rensselaer Polytechnic Institute, Troy, New York 12180

Ideally a paper with this title should answer two

questions: "How are the calculation results in- fluenced by the characteristics of the potentials?"

and "What is the 'best' potential for defect calculations?" Neither answer is ready to hand at

this time and this paper is for the most part only a chronological review of the topic with a few critical comments at the end. But first some general re- marks on potentials in metals.

1. The problem is intrinsically more complex

and intractable for metals than other solids because

of the nature of the so-called metallic bond, which

is really not a bond at all in the sense of an inter- atomic force but rather a cooperative interaction

between the electrons and ions. The cohesion re- sides in such nonlocalized concepts as the Fermi

surface and the electron-electron correlation. Of

course, in principle any situation can be described

by a complete set of interatomic force constants. Only in the case of metals such a description may

converge slowly and has an artificial character in that it is difficult to assign physical meaning to the < various coupling parameters. Actually, a

potential dependent on volume only appears often

to represent metallic cohesion better.

2. The metals share with all other types of solids a repulsion between atoms at close distances of

approach as a consequence fundamentally of the

Pauli Exclusion Principle. Unfortunately, it has

not been possible to develop the magnitude of this repulsion directly from first principles, but the general form is certainly one where the repulsion

increases rapidly at close distances of approach.

The classical Born-Mayer form for this is a simple

exponential and there is some theoretical justi- fication for this at the separations of the usual

lattice spacings in that the wave function tails fall off exponentially at large distance for most atoms.

Positive potential terms varying as a large inverse

power of r are another form for this closed shell repulsion as in the Lennard-Jones approach.

3. Although there has been considerable effort

to treat the cohesive forces in metals from first principles, the repulsive forces have almost in- variably been established by semiempirical methods. Naturally semiempirical considerations have

played an important role in establishing the potentials used for all the large machine calculations. One is not likely to embark on an extensive

program without first ensuring that the model

will reproduce the most obvious parameters of the

metal concerned. The parameters most usually

chosen are the lattice constant and the compressibility, which respectively serve to establish the

distance scale and the curvature of the potential.

A third parameter to set the scale for the magnitude

of the potential is most naturally the cohesive

energy. On the other hand the potential has been

required sometimes to give agreement for the cal- culation of vacancy formation or the radiation

damage threshold for Frenkel pairs, as alternate procedures for setting the scale of the potential.

For additional parameters there are the second

order elastic constants. Since for most metals the

Cauchy relations are not obeyed, exact fitting

involves something other than two-body potentials.

The addition of. a volume dependent term as emphasized by De Launay [l]1 appears to be the

simplest way to meet such a requirement. Currently third-order elastic data are becoming avail- able [2] and it may be possible to fit such data

later into more elaborate semiempirical potentials.

The first results appear to indicate that the potentials as currently used give quite reasonable

agreement in the main with the anharmonic results so that we may anticipate that such changes as might result will be in the nature of refinements

only. Of course there is also a wealth of data on

interatomic forces available from the dispersion curves obtained, by the diffraction of neutrons and

x rays, and attempts [3] have' already been made to incorporate these results in defect calculations

via Born Lattice coupling coefficients.

4. While the earliest calculations neglected

atom motion at large distance from the defect, it was soon apparent that one should introduce an

elastic solution in this region to be realistic and to find the volume change associated with the defect. The latter could be determined from the leading

term in the expansion of the displacement in spherical harmonics about the defect:

u(r) = Crlr3

.

From which it follows that the volume change was &V=4ttC. Eshelby [4] showed that the requirement of the surface boundary condition increased

by a factor

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

35
or effectively about 50 percent. In general the

elastic energy associated with the elastic field outside the first 100 atoms is small. For a defect

with the symmetry of the lattice the higher order

terms in the expansion of the strain are quite

negligible. Defects with lower symmetry such as

split-interstitials and divacancies will act as elastic dipoles and may be detected by mechanical or magnetic relaxation. Here the enhancement from

the image force considerations may be even more

substantial [5]. 5. Consider the aspect of electron redistribution

around point defects. For interstitials this problem

has been largely honored in the breach. The single

vacancy has been reasonably well treated by several different workers but the problem certainly

does not lend itself to the treatment of defects of

lower symmetry. As part of this general picture

the Friedel oscillations may well play some role in this difficult problem.

6. As a summary item to this general introduction

it should perhaps be remarked that to some extent

different phases of the model potential are important for different applications. The part of the

potential at short range is critical in determining

energy thresholds for radiation production of defects. The fast changing repulsive potential

plays a dominant role in the motion of crowded

interstitial defects, whereas the energies for vacancy configurations appear to depend sensitively on the electron redistribution. We turn now to a chronological review of developments.

Probably the first effort to treat the defect potential in a metal was an attempt [6] that Seitz

and the author made several years ago to establish

the prevailing diffusion mechanism in a metal such

as copper. We used a Born-Mayer repulsion and

made an earnest effort to take into account the

electron redistribution about the vacancy in a

self-consistent way. This part of the work involved some serious errors which fortunately (?) to some

extent canceled, but the main result, that the

vacancy mechanism was preferred, seemed rea- sonably established. Later Zener's [7] stimulating

suggestion as to a possible ring mechanism forced

a reevaluation [8] of the original Born-Mayer potential and the introduction of a less rapidly varying

alternative. The semiempirical considerations in back of this potential were again restudied at a

still later time when the mechanism of intersti- tialcy motion was reexamined [9] as a possible

explanation of low temperature anneal of radiation

damage; it appeared to be a defect with a low

energy of motion but a high energy of formation.

We were inclined to play down certain oscillations

in the self-consistent charge density at intermediate

distance from the vacancy but Friedel pointed out

[10] that such variation in charge and hence in potential were the natural consequences of the

sharp Fermi Surface in a metal. These Friedel

oscillations have subsequently turned out to be

most important in the development of solid state

physics.

Friedel's influence in this field has made itself

felt in another important way. He showed [11] how one can calculate directly from a knowledge

of the phase shifts r)i from a partial wave scattering

problem for the wavelengths corresponding to the

Fermi Surface the charge Z displaced by the scat- tering center. The formulation of this Friedel

condition,

Z=-V(2£ +l)7,i,

77 ^

I

is a most useful tool in developing a self-consistent

solution in the neighborhood of a defect since it gives one an instant check as to when the defect is electrically neutral. While the satisfaction of this condition does not insure complete self-consistency,

it is a most important first step — and to some

extent sufficient in some applications. Fumi

[12] employed the Friedel condition to develop a general treatment of the vacancy problem, ap- plicable for any density of free electrons. This

increase in flexibility was needed since any complete self-consistent treatment would be valid for only one particular electron density. The resulting

predictions [12] for the energies of vacancy formation appear to be on the high side but to vary qualitatively in the right way with electron density.

Comparison with experiment for the alkali metals

has been limited for the most part to the activation

energies for diffusion. However, this may have

been more relevant than one would at first suppose

since Fumi believed that the energy for motion

would be small for these metals.

An even simpler approach to the electron re- distribution problem was that of Brooks [13] who

suggested that one could treat vacancy configurations as small internal voids whose intrinsic energy

consisted primarily of the energy of the extra surface

formed. Relaxation of the lattice inward about

the' vacancy then operated to minimize the total energy. The results of this method tended to give

numbers that were somewhat too large for the

formation energies of vacancy defects. The general technique of the partial wave analysis

hinged on the Friedel condition was the basis for many subsequent papers dealing with energy of point defects and their specific resistivity. While

for the most part the work dealt with defects of

high symmetry, Seeger and Bross [14] studied the

divacancy with ellipsoidal coordinates and found

for the noble metals a binding energy of about

0.06 of the Fermi Energy. Seeger [15] also made

application to the vacancy in nickel and cobalt. Around 1957 there began the somewhat simpler

game of defect treatment by use of the two-body

interaction only. The Lennard-Jones potential was used by two investigators that year, Kanzaki

[16] and Hall [17]. The former was concerned

primarily with the application to solid argon. The

36
latter introduced external pressure as a variable

and explored the displacement pattern around both

the vacancy and the cube-centered interstitial. The

anisotropic nature of the displacement function

pointed up the difficulty with the previous efforts

to fit on an isotropic, elastic displacement, particularly in the case of the interstitial. In the next year Tewordt [18] showed a reasonable way to establish the elastic field at large distances by

allowing the amplitudes of the respective harmonic

components of the displacement to be treated as parameters for the energy minimization on a par

with the displacements of the individual atoms near

the defect. Tewordt used two alternate Born- Mayer potentials to treat the interstitial, the

crowdion, and the vacancy. His work was also, I think, the first to make use of a large scale computer for defect calculations, the Illiac in Urbana.

In 1960 a much more ambitious machine program was first reported [19] in the literature, using the

GRAPE program developed bv the group at Brookhaven working with Vineyard. Here again a Born- Mayer potential was used but instead of fitting to an elastic solution at infinity the machine dealt with a finite number (1,000) of atoms, arranged in a rectangular chunk of lattice and constrained at an

equilibrizing pressure by an elastic membrane at the boundary. The membrane also had some

dissipative character so the specimen metal could

lose energy to its environment at a reasonable

rate. Vineyard and his colleagues gave considerable attention to the choice of Born-Mayer

potential. Because their primary interest was to follow defect formation during fast particle irradiation, the behavior of the potential at close distances

of approach was important. They considered the

shielded coulomb potential of Bohr, presumably

valid for r<0.lA, and also the Fermi-Thomas

modifications for somewhat greater distances. At

normal interatomic distances possible Born- Mayer functions were considered. Lattice constant and compressibility as usual were input data

but also the check with the threshold for radiation

damage displacements was considered. The one

finally chosen for the bulk of the work did not, as luck would have it, differ much from the form originally used by others in prior papers. This potential was used again more recently in a revived treatment of the static defects of the

fee lattice by Johnson and Brown [20]. In addition

to the calculations of energies and defect volumes

the stabilities of the various configurations were

investigated in a completely systematic way so

that the energy profiles of the interconfigurational

paths could be plotted.

Not all the two-body machine calculations, however, have used the Born-Mayer potential. In recent years there has been an increasing interest

in the Morse potential for such purposes. Originally the group at Lewis Research Center in Cleveland developed [21] the parameters appropriate for a large number of fee metals to put in the formula

V(r)=D[exp (-2(r- r„)/a-2 exp-(r-r0 )/a].

Heat of sublimation, lattice constant and compressibility were used to determine D, a, and r0 , and the resulting V{r) was applied to various

vacancy and interstitial configurations. Benneman [22] applied both Born-Mayer and Morse potentials to two interstitial configurations apparently without essential difference in his findings.

For reliable results with the Morse potential one

considers quite distant neighbors and some arti- ficial truncation schemes have been shown to give

falacious results. Proponents for this potential

usually feel that it offers an advantage in that

attractive as well as repulsive forces are involved

in minimizing the energy and in determining the

atom positions. On the other hand this model uses

only central forces and therefore can not be ex- pected to adjust to departures from the Cauchy

relations for the elastic constants. Recently

Cotterill and Doyama have made extensive use of

this potential in several kinds of defect problems.

A thoroughgoing treatment [23], which involves

elastic fields determined to minimize the energy,

makes possible estimates of defect volumes — but

this work will be reported directly later in this conference.

For some metals such as copper and silver, the

Born-Mayer potential plus retaining pressure has

been able to reproduce quite satisfactorily the

observed elastic constants. Iron is an example

where this is not possible and the Brookhaven

group first applied a modified Morse potential for use in a dynamic treatment [24] of radiation

damage events. Later Johnson [25] replaced the

Morse curve with a cubic expression over a region

including the range of nearest and next nearest

neighbors. The cubic was chosen to give exactly

the elastic constants minus the electronic contri- butions computed electrostatically. At large dis- tances the fitted cubic was contrived to go to zero with zero slope at a distance halfway between

second and third neighbors. This model has met

with good success in several applications.

Lastly, for a long time we have been conscious

of the influence of the Friedel oscillations for the

potential of an impurity in a metallic lattice but the

effect of these oscillations on defects calculations

has not been explicitly explored. A step in this direction appears in a recent paper by Cotterill and Doyama [26] on line and plane defects wherein

they investigate the stacking fault in aluminum

by several potentials. Among these are two oscil- lating potentials, one determined experimentally

by Johnson et al. [27], from neutron scattering

data, the other evolved theoretically by Harrison

[28] from a pseudopotential approach. Certainly

these potentials will be used also in point defect

problems [29].

37
After this chronological review a word is maybe

in order by way of critical summary and perhaps

suggested lines for progress.

First the electronic redistribution is especially

important for the vacancy configurations. Its complete omission in the two-body-only force model

leads to too small energies for formation and motion.

On the other hand results of the partial wave

analysis give somewhat too large values for sodium

and values somewhat on the small side for the noble

metals. One wonders if satisfying the Friedel

condition alone is enough to establish satisfactory

self-consistency. Also the neighbors' inward

motion on the vacancy reduces its volume ap- preciably and thereby the charge to be removed.

Clearly a thoroughgoing treatment of all these

considerations would involve a very extensive

program even for any real metal.

The application of large machines to explore

interstitial configurations governed by two-body

forces appears on the whole to give reasonably

consistent and hence on the whole believable

results. The addition of a volume-dependent

force appears to be an improvement in that departure from the Cauchy relation can be incorporated.

From the standpoint of macroscopic elasticity the use of two-body central forces plus a volume-dependent potential may suffice for a reasonably

complete description in second order and maybe

even in third order, although at this point there are only a few reliable third order measurements.

The long range oscillations of the potential are also being quantitatively explored. Whether more

refinement in this direction would significantly

alter the results of defect calculations is still unknown.

There has up to now been relatively little effort to trying to understand the differences in the results of defect calculations in terms of differences in the underlying potentials. A simple example of

such a consideration is the comparison of the

volumes for defect formation against the anharmonic

character of respective potentials. Detailed

machine calculations have been carried out for the fee lattice (Cu) using the Morse potential

[23] and the Born-Mayer potential [20]. Let the

anharmonic character of each be represented by a

dimensionless parameter yp dr3

I

f cPV] dr2 where fj is set at some convenient interatomic

distance such as the nearest neighbor distance.

From the table below one sees that the higher yp goes with the larger volume expansion with the

interstitial, the larger inward relaxation around a vacancy, and the larger volume increase for the

moving atom at the saddle point, as might be

expected. Further study of this sort should give

worthwhile insight into the influence of potential

parameters on calculated results.

Morse Born-Mayer

potential potential

&Vi for split interstitial 1.34 2.20 AVy for vacancy 0.83 0.43-0.62

.03 .19- .38 yp 8.1 13.9

(All AK in terms of atomic volume)

References

[1] J. De Launay, Solid State Physics, Vol. 2, p 283 (Academic

Press, New York, 1956).

[2] Yosio Hiki and A. V. Granato, Phys. Rev. 144, 411 (1966).

[3] A. Seeger, E. Mann, and R. N. Jan, J. Phys. Chem. Solids 23, 639 (1962).

[4] J. D. Eshelby, Solid State Physics, Vol. 3, p 79 (Academic

Press, New York, 1956).

[5] H. B. Huntington and R. A. Johnson, Acta Met. 10, 282

(1962).

[6] H. B. Huntington and F. Seitz, Phys. Rev. 61, 315 (1942).

[7] C. Zener, Acta Cryst. 3, 346 (1950).

[8] H. B. Huntington, Phys. Rev. 76, 1728 (1949).

[9] H. B. Huntington, Phys. Rev. 91, 1092 (1953).

[10] J. Friedel, Phil. Mag. 43, 153 (1952); A. Blandin and J. L. Deplante, Metallic Solid Solution p IV- 1 (W. A. Benjamin, Inc., New York, 1963).

[11] J. Friedel, Phil. Mag. 43, 153 (1952).

[12] F. G. Fumi, Phil. Mag. 46, 1007 (1955).

[13] H. Brooks, Impurities and Imperfections, pp 1—27 (ASM

Cleveland, 1955).

[14] A. Seeger and H. Bross, Z. Physik 145, 161 (1956).

[15] A. Seeger, Z. Physik 144, 637 (1956).

[16] H. Kanzaki, J. Phys. Chem. Solids 2, 24 (1957).

[17] C. L. Hall, J. Phys. Chem. Solids 3, 210 (1957).

[18] L. Tewordt, Phys. Rev. 109 , 61 (1958).

[19] J. Gibson, A. N. Goland, M. Milgram, and G. H. Vineyard,

Phys. Rev. 120, 1229 (1960).

[20] R. A. Johnson and E. Brown, Phys. Rev. 127, 446 (1962).

[21] L. A. Girifalco and V. G. Weizer, Phys. Rev. 114, 687

(1959); P. Hockstra and D. R. Behrendt, Phys. Rev. 128,

560 (1962).

[22] K. H. Bennemann, Phys. Rev. 124, 669 (1961).

[23] M. Doyama and R. M. Cotterill, Phys. Rev. 137A, 994

(1965). [24] C. Erginsoy, G. H. Vineyard and A. Englert, Phys. Rev. 133A, 595 (1964).

[25] R. A. Johnson, Phys. Rev. 134A, 1329 (1964).

[26] R. M. J. Cotterill and M. Doyama, Phys. Rev. 145, '465

(1966). [27] M. D. Johnson, P. Hutchinson, and N. H. March, Proc. Roy. Soc. A282, 283 (1964).

[28] W. H. Harrison, Pseudopotentials in the Theory of Metals, p 45 (W. A. Benjamin, Inc., New York, 1966.)

[29] H. B. Huntington and Michael D. Feit, Structure Factor! Approach to Energies of Migration, this symposium.

38
Calculations for Defects in an F.C.C. Lattice*

R. A. Johnson

Brookhaven National Laboratory, Upton, N.Y. 11973

1. Introduction

The properties of point defects in metals can be

studied by lattice calculations in which the atoms

near a defect are treated individually. In such

calculations a model is devised which matches

various known physical properties of a metal and

which, one hopes, can be extrapolated to yield

results about unknown properties of the metal,

e.g., configurations and energies associated with

lattice defects. These models normally represent

the solid by a crystallite in which the atoms interact

by two-body central forces surrounded by a boundary region which is meant to simulate the re- mainder of the lattice. An energy equation is written which includes the interatomic forces and

the boundary forces, and parameters in this equation

are adjusted to yield a reasonable approximation

of the perfect lattice. Then the energy equation

is investigated in a region of configuration space

where it represents the defect lattice, i.e., the whole

process may be thought of as very complicated

and sophisticated curve fitting to one section of

configuration space and extrapolation to another

section. There is of course no unique way of

fitting the perfect lattice: any number of quite

different models can do a reasonably good job.

Also, there is the question of whether or not the

forces which describe the perfect lattice are adequate to describe the defect state, i.e., whether

extrapolation is permissible.

Most of the work in this field has been carried

out for copper, an f.c.c. metal, and the most ex- tensive of these calculations have been reported

by Johnson [1 ,2]

1 and Brown [1], Seeger and co- workers [3-5] and more recently by Doyama and

Cotterill [6]. All these authors used well-known

potential interactions between the atoms: Johnson

and Brown used a Born-Mayer interaction, Seeger

et al., used a Born-Mayer for some calculations

and modified it with Born lattice theory inter- actions in others, and Doyama and Cotterill are using a Morse function. All these calculations

give unsatisfactory results pertaining to vacancies

[2] and the reliability pertaining to interstitials

is uncertain [7]. Johnson [8] has recently carried out corresponding calculations for a-iron with a different form of

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

'Figures in brackets indicate the literature references at the end of this paper.

potential. In a b.c.c. lattice the first- and secondnearest neighbor separation distances are similar,

while the third neighbor is at a much greater dis- tance. Therefore a short-range interatomic inter- action can reasonably be cutoff between the sec- ond- and third-neighbor separation distance.

When this is done, the elastic moduli of a-iron are such that the first-neighbor interaction must be

repulsive and the second-neighbor interaction must

be attractive. A potential curve, shown in figure 1,

2.2 2.4 2.6 2.8 3.0 3.2 3.4 RADIAL DISTANCE , A

FIGURE 1. The potential energy for the two-body central inter- action of iron atoms in either the body-centered cubic or face- centered cubic phase.

Interaction I together with an electronic contribution reproduce the elastic moduli

of a-iron whereas interaction II by itself yields these data.

was devised which fit these criteria. In a-iron,

the nearest neighbor distance is 2.48 A and the

second neighbor distance is 2.86 A. Curve I is fitted to the elastic moduli with an electronic

contribution corresponding primarily to the bulk

modulus of a free electron gas, and curve II is fitted to the full experimental elastic moduli.

The differences in the results given by these two

curves were not significant.

Although there is not very much experimental

data with which the calculated results can be

39
compared, what comparisons are available are

quite encouraging. These calculations were ex- tended to include the case of carbon interstitials in a-iron [9] and in iron martensite [10], a bodycentered tetragonal phase of iron, and the structure and energy of Fe3 C (cementite) [9] were also in- vestigated. Here too the results were encouraging.

When applied to an f.c.c. lattice structure with

a nearest neighbor distance corresponding to y-iron, 2.57 A, the elastic moduli predicted by this interaction are very similar to those of nickel,

(the elastic moduli of y-iron are not known). If y-iron or nickel are to be approximated by a short- range central interaction which fits the elastic moduli, this form of interaction is just as satis- factory as a purely repulsive interaction such as the Born-Mayer, and there is no a priori reason to choose one or the other. Since this form of interaction was required for the b.c.c. case, and since

it yields reasonable elastic moduli fox an f.c.c. structure, calculations were carried out to investi- gate the properties of vacancies and interstitials which it gives for an f.c.c. lattice. The mathematical model used in the present

f.c.c. calculations is as close as possible to being

the same as that used in the earlier b.c.c. calculations [9]. A spherical set of 531 atoms was treated

as a crystallite of independent particles and the

atoms outside this set were treated as though they were imbedded in an elastic continuum. The

changes in energy in this model were carefully

monitored as defects were introduced and the

set was allowed to relax to the minimum energy

configuration. By applying constraints on the

positions of some atoms, saddle point and inter- mediate configurations could be investigated.

2. Results

The stable vacancy configuration was that in which an atom is missing from a normal lattice site and the saddle point configuration for migration was found by moving one of the twelve nearest- neighboring atoms to the vacancy directly towards

the vacancy until it is midway between its site and

the vacancy site. The vacancy formation energy was found to be £/j)=1.49 eV, the formation

volume was Vfy — 0.85 ft, where ft is the atomic

volume, the activation energy for migration was £ik=1.32 eV, and the activation volume was

Vfv = ~ 0.05 ft. By adding the formation and

migration energies a self-diffusion energy is found;

£f/? = 2.81 eV. The results are summarized in table 1. Table 1. Summary of results for vacancies and interstitials in an f.c.c. lattice model with an interatomic interaction appropriate for nickel

The listing of experimental results is based on an interpretation of the data in which

single interstitials migrate in the so-called stage I region, divacancies migrate in stage

III and vacancies migrate in stage IV. Energies are given in eV, and EF means formation energy, Eyl means motion energy, ESD means energy for self diffusion, EB means

binding energy, and Es means energy by which a configuration is metastable.

Calc ulation Experimental Calculation

1.49 a 1.4 Eg (<111 > split) 0.16 pM

F-\v 1.32 a 1.5 EJJ«111) split) .13 Es° 2.81 8 2.9 Eg (<110> split) .02 *$, 0.90 "0.8 ((110) split) .04 E* 4.08 El 1.16 E» 0.15 "0.15 E£ 0.29

a For a summary of vacancy results see ref. (11). "See ref. [12]. The stable divacancy configuration was that in which two atoms are missing from nearest-neighboring normal lattice sites. The saddle-point

configuration has very little symmetry and involves

the jump of an atom which is in a nearest-neighboring lattice site to both halves of the stable divacancy

from its site to one of the vacant sites. The path

goes out of the plane defined by the three sites and

the jumping atom does not closely approach the center of the triangle of the three sites. The

divacancy binding energy was found to be

Egv= 0.25 eV and the activation energy for migration was ££(. = 0.90 eV. The most stable interstitial configuration was the

(100) split configuration in which two atoms are symmetrically displaced in the (100) direction from

a vacant lattice site. The migration mechanism

consisted of one end of the split jumping to form a

split at a nearest-neighboring lattice site. This

involves rotation of the axis of the split as well as migration. The interstitial formation energy was

found to be £i/ = 4.08 eV, the formation volume was

Vf! = 0J ft, the activation energy for migration was

£J^

= 0.15 eV, and the activation volume was VM = o.ia.

Another metastable interstitial was found which

gives rise to another migration mechanism — the

(111) split, in which two atoms are symmetrically

displaced in the (111) direction from a vacant

lattice site. The migration mechanism consisted

of one end of the split jumping to form a split at a nearest-neighboring lattice site. This configuration was metastable by Eft {(111) split) = 0.16 eV

and the activation energy for migration was £#«111) split) = 0.13 eV. The (110) split configuration, the so-called crowdion, was just barely metastable and migrated

very easily: £^((110) split) = 0.02 eV and E$ ((110)1

split) = 0.04 eV. The di-interstitial was found to be very tightly

bound. The configuration consists of two parallel

(100) single interstitial at nearest-neighboring

lattice sites with the line joining their centers per- pendicular to their axes. The binding energy was = 1.16 eV and the activation energy for migra-; tion was £# = 0.29 eV. Close Frenkel pairs were investigated and it was; found that the region of spontaneous interstitial;

40
vacancy recombination was quite small — 32 fi — and action at separation distances greater than several

that there was negligible interstitial-vacancy inter- lattice constants.

3. Discussion

The object of this calculation was to investigate

the consequences of using an interatomic interaction for an f.c.c. lattice with a form similar to that previously used for b.c.c. calculations. This type

of interaction is somewhat intermediate between the

purely repulsive short-range Born-Mayer and the

long-range attractive-tail Morse interactions used

in earlier calculations. The constants used should

make the interaction reasonably appropriate for y-iron or nickel, and the results are compared to the

experimental values for nickel in table 1. The

interpretation of the experimental data is not uni- versally agreed upon, but these calculations cer- tainly favor the interpretation in which interstitials freely migrate in the so-called stage I region, divacancies in stage III, and vacancies in stage IV. In

contrast to earlier point defect calculations, the

present calculations do not yield any results which

are clearly inconsistent with experimental data,

and although this certainly is not proof of its validity,

the consistency is sufficiently good that this type of

calculation warrants further study.

4. References

[1] R. A. Johnson and E. Brown. Phys. Rev. 127, 446 (1962).

[2] R. A. Johnson, J. Phys. Chem. Solids 26, 75 (1965).

[3] A. Seeger, E. Mann, and R. v. Jan, J. Phys. Chem. Solids 23, 639 (1962).

[4] G. Schottky, A. Seeger, and G. Schmid. Phys. Stat. Solidi 4, 419 (1964).

[5] G. Schottky, A. Seeger, and G. Schmid, Phys. Stat. Solidi 7, K25 (1964).

[6] M. Doyama and R. M. J. Cotterill, Phys. Rev. 137, A994

(1965).

[7] R. A. Johnson, Phys. Letters 19, 191 (1965).

[8] R. A. Johnson, Phys. Rev. 134, A1329 (1964).

[9] R. A. Johnson, G. J. Dienes, and A. C. Damask, Acta Met. 12, 1215 (1964).

[10] R. A. Johnson, Acta Met. 13, 1259 (1965).

[11] A. Seeger and D. Schumacher, in Lattice Defects in Quenched Metals, p. 15, edited by R. M. J. Cotterill, M. Doyama, J. J. Jackson, and M. Meshii (Academic Press, New York, 1965).

[12] J. Verdone, P. Peretto, P. Moser, D. Dautreppe, and J. Verdier, Compt. Rend. 260, 5209 (1965).

256-724 0-67—

4

11

Atomistic Calculations of Intcrstitials in F.C.C. Metals*

M. Doyama and R. M. J. Cotterill

Argonne National Laboratory, Argonne, 111. 60440

1. Introduction

Interstitials and interstitial-type defects are

particularly important in the study of radiation

damage and plastic deformation. It has been ap- parent for many years that the process of irradiation

damage is one of considerable complexity. When

a property such as electrical resistivity is monitored during the warm up of a metal specimen

irradiated at a low temperature, numerous annealing stages are observed. These have been classified into stages I, II, III, IV, and V, according to the temperature range in which they are observed

in a given metal: and some of these stages exhibit

substages. The point defects generated by the

irradiation can interact with each other in a variety

of different ways. Interstitials can interact with

other interstitials and also with vacancies. Moreover, each of these interactions might itself occur in several different ways. For instance, several dif- ferent stable interstitial-vacancy pairs might exist,

each with its own binding and migration energy,

and each with its own energy barrier against

transition into another form.

Stage I annealing is generally considered to be

due either to the annealing of close pairs of va- cancies and interstitials or to the migration of free

interstitials. There is an opposing school of

thought which believes that free interstitials migrate in stage III rather than in stage I. Important properties of these interstitial-type defects are the formation energy, the migration

energy (or energies, if several types of interstitials exist), and the interaction energies between

interstitials and other defects. Energies and

atomic configurations of these defects are here

studied by using an atomistic model. Central

forces were assumed, and the pair-wise inter- action between atoms was represented by a Morse

potential function.

Several possible configurations of the single

interstitial will be discussed, and it will be shown

that a Morse potential [l] 1 calculation gives essentially the same results as previous calculations

which used a Born-Mayer potential [2]. Huntington and Seitz [3] examined both the

interstitial and vacancy. For the former defect

they considered an extra atom located at the cube

center of the f.c.c. lattice (i.e., the so-called "bodycentered" type). Paneth [4] considered a configuration in which the extra atom was located in a closest-packed row of atoms, the "crowdion."

Several atomistic calculations on simple lattice defects have been published in recent years. Among these are the calculations due to Huntington

[5], Fues and Stumpf [6], Kanzaki [7], Tewordt

[8], Benneman [9], Girifalco and Weizer [10], Johnson and Brown [11], Johnson [12], and Dominigos [13]. These calculations are static calculations, and dynamical calculations [14] are not discussed here.

2. Method of Computation

The atoms close to a defect suffer displacements

which are too large to be adequately described in terms of linear elastic theory. Because of this, these atoms are treated as individual atoms, the

displacements of which are independent. This

region is called region I. Atoms lying well away

from the defect are displaced by small amounts

which are susceptible to treatment by the elastic

theory. Region I is surrounded by an array of atoms which are also regarded as discrete particles

but whose displacements are given by linear

elastic theory. This array can be divided into two

*Based on work performed under the auspices of the U. S. Atomic Energy Commission. 'Figures in brackets indicate the literature references at the end of this paper.

regions (II and III). In region III none of the

atoms interact with atoms in region I. The interaction energy of the atoms in the

crystal was represented by a Morse function.

The interaction energy, E(nj), of a pair of isolated

atoms is given by

E(ru ) =D{exp[—2a(rir

-r„) ] — 2 exp [—a(nj—ro)\},

where r,j is the distance between the atoms, D

is the dissociation energy of the pair, and r0 is the

equilibrium separation distance of the two atoms, a is a constant which effectively determines the

43
"hardness" of the potential. In the work described

here, the interactions within a radius of V9.4

times the nearest-neighbor distances were taken into account. In a perfect crystal the number of atoms

in this sphere of influence is 176. The constants were determined by the method used by Girifalco

and Weizer [10, 15], using the Born stability cri- teria [16] and known experimental values of the

lattice parameter and bulk modulus. Elastic constants calculated with the potential are in good

agreement with experimental values. The procedure for determining the equilibrium

configuration and energy of a defect was as fol- lows. All atoms were initially located on lattice

sites. Atoms were then removed from or added to the center of region I so as to produce the desired

defect. Each atom in region I was then moved in turn until the force on it was again zero. The

atoms were moved in this manner in a series of

cycles in which the atoms closest to the defect were moved more frequently than atoms more dis- tant from the defect. The volume change and the

total crystal energy were calculated after each

cycle.

3. Results and Discussion

The formation energies of five different possible

configurations of single interstitials in copper were

calculated. The configurations that were examined

are shown in figure 1. The relative formation

energies of the various configurations, when they

are artificially prevented from degenerating during

the calculations, are shown in figure 1. It will be

noted that the split-100 form has the lowest energy. When the constraint against degeneration is re- laxed, it is found that all configurations spon- taneously degenerate to the split-100 configuration.

The latter is therefore the only stable form. This is in agreement with the conclusions drawn by Johnson

and Brown [11] on the basis of their calculations,

which involved the use of a Born-Mayer potential

function. The absolute values of the formation

energies of the various configurations are given in table 1. The potential used in obtaining these

values was derived by use of the sublimation

energy. Table 2 lists the corresponding values

obtained with a potential which was derived from

RELATIVE ENERGY 0-3

VOLUME , CHANGE

1.34 a

(2.20 2)

1.34 2

(2.37 2)

1.36 a

(2.44 2)

1.32 2

(2.46 2)

TETRAHEDRAL

1.36 2

(2.58 2)

CHANGES OF ENERGY AND VOLUME

FOR INTERSTITIALS.

FIGURE 1. Changes of energy and volume for interstitials.

Dotted lines in the energy plot are the values by Johnson and Brown [11]. The

values of volume changes in parenthesis are also by Johnson and Brown.

the vacancy formation energy. The discussion

which follows refers only to the results listed in table 1. The difference between tables 1 and 2

is deferred to a later publication.

Table 1. Energies and volumes of formation associated with

interstitials (calculated with potential based on sublimation

energy)

Formation Formation Energy Type volume energy above

100-split

n eV eV

100-split 0.34 3.81 0.00

110-split .34 3.96 .15 Ill-split .36 4.07 .26 Body-center .32 4.02 .21 Tetrahedral .38 4.12 .31 Act 110 .36 3.97 .16

Table 2. Energies and volumes of formation associated with

interstitials {calculated with potential based on vacancy

formation energy)

Formation Formation Energy Type volume energy above

100-split

n eV eV

100-split 0.60 3.39 0.00

110-split .56 3.77 .38

Ill-split .78 3.96 .57 Body-center .77 3.65 .26 Tetrahedral .91 4.02 .63 Act 110 .80 3.77 .38

The volume change associated with the various

types of interstitials are also given in figure 1. Thfj

actual increase in volume when an atom is removet

from the surface and inserted into the interior o the crystal is about 0.4 atomic volumes.

Migration energies of the interstitials have als< been calculated. The results are slightly differen

from the calculations using the Born-Mayer poten;

tial [11,13]. The 100-split interstitial can migrate!

in a (110) direction with an activation energy o 0.15 eV. The 100-split interstitial can also rotate around the perfect lattice point on a {100} plane! with an activation energy of 0.15 eV. The rotatioij

is as easy as the migration.

Frenkel pairs were also studied. When th<|

44
vacancy and interstitial are nearest neighbors,

there is no energy barrier against recombination.

We may describe the various configurations of

Frenkel pairs by quoting the distance between the

vacancy and the interstitial and the angle between

the axis of the dumbbell (split interstitial) and the

line joining the vacancy to the center of the axis

of the "dumbbell." Thus the nearest-neighbor

pairs are described as the d® — 90° type (of which

there are four per interstitial) and the d0 — 45°

type (of which there are eight), do is the nearest- neighbor distance. Both of these types are unstable. There are four d0 \

/2 — 90° and two

c?0 V2 — 0° types, which are all unstable. All the

types — a are unstable, and so are all the

types 2d0 — a except the four pairs of type 2^0 — 90°. Thus the closest stable pair shows a separation

distance of 2do- Except for this last type of pair,

the rule for spontaneous recombination seems to be that this occurs if the vacancy and interstitial have a common nearest-neighbor atom. The greatest difficulty of these calculations is that the electron redistribution was not taken into account. Such effects could affect the results appreciably. The tetrahedral interstitial, for in- stance, has a high energy in the present calculations, but because of its high symmetry it may

eventually be found stable when electron redis- tribution is taken into consideration.

4. Conclusions

Interstitials and Frenkel pairs in copper were

studied. The formation energies and volume

changes due to single interstitials are given in table 1. The 100-split interstitial was found to be the only stable interstitial. This interstitial can migrate into another 100-slit interstitial posi- tion with an activation energy of 0.15 eV, or it can

rotate around the center of mass of the "dumbbell"

with an activation energy of 0.15 eV.

The authors acknowledge the continuing in- terest and encouragement of T. H. Blewitt, H. H. Chiswik, F. G. Foote and O. C. Simpson. It is a pleasure also to acknowledge helpful discussions

with K. H. Benneman, T. H. Blewitt, L. A. Giri- falco, H. B. Huntington, R. A. Johnson. J. S. Koehler, F. Seitz, O. C. Simpson, L. Tewordt, and

M. P. Tosi. The work described here was greatly

facilitated by the help and cooperation of C. G. Le Vee and the scheduling, consulting, and op- erating personnel of the CDC 3600 computer at Argonne.

4. References

[1] P. M. Morse, Phys. Rev. 34, 57 (1929).

[2] M. Born and E. Mayer, Z. Physik 75, 1 (1932).

[3] H. B. Huntington and F. Seitz, Phys. Rev. 61, 315 (1942).

[4] H. Paneth, Phys. Rev. 80, 708 (1950).

[51 H. B. Huntington, Acta Met. 2, 554 (1954).

[6] E. Fues and H. Stumpf, Z. Naturforsch. 109, 136 (1955).

[7] H. Kanzaki. J. Phys. Chem. Solids 2, 24 (1957).

[8] L. Tewordt, Phys. Rev. 109, 61 (1958).

[9] K. H. Benneman, Z. Physik 165 , 445 (1951).

[10] L. A. Girifalco and V. G. Weizer, Phys. Rev. 114, 687

(1959).

[11] R. A. Johnson and E. Brown. Phys. Rev. 127,446(1962).

[12] R. A. Johnson, Phys. Rev. 134, A1329 (1964).

[13] H. Dominigos, Bull. Am. Phys. Soc. II, 11, 250(1966).

[14] For example: J. B. Gibson, A. N. Goland, M. Milgram and

G. H. Vineyard. Phys. Rev. 120, 1229 (1960).

[15] L. A. Girifalco and V. G. Weizer, NASA, Report No. R-5

(1959).

[16] M. Born, Proc. Cambridge Phil. Soc. 36, 360 (1940).

Morse Potential Calculations of Vacancy and Vacancy-Type

Defect Properties in F.C.C. Metals*

R. M. J. Cotterill and M. Doyama

Argonne National Laboratory, Argonne, 111. 60440

1. Introduction

Atomistic models and the central force approximation have been used in numerous studies of

lattice defects in recent years. Among these are

the calculations due to Huntington [1] \ Fues and

Stumpf [2], Kanzaki [3], Tewordt [4], Benneman

[5], Girifalco and Weizer [6], and Johnson and

Brown [7]. This paper describes calculations on

vacancies and vacancy clusters, and the method of

computation involves many of the features used by

these authors. The interatomic potential was in

all cases represented by a Morse [8] potential

function. Generally speaking, the atomic configurations

of vacancy clusters are simpler than their interstitial counterparts. This is because the vacancy

must of necessity be located at a lattice site, whereas the interstitial atom might be located at any other point in the Wigner-Seitz cell. The configuration of the divacancy is also obvious; two

vacancies separated by a nearest-neighbor dis- tance. The trivacancy is more complicated be- cause it can exist in four different forms, each

having a different binding energy. The tetra- vacancy is even more complicated. In this paper

we examine all of these vacancy clusters in some

detail. Also examined, but in less detail, are several larger clusters. Large aggregates of vacancies usually find it energetically favorable to exist in the form of

collapsed defects bounded by dislocations. This

fact has been established by numerous experiments performed over the last eight years, the

first direct evidence of this in metals being re- ported by Hirsch et al. [9], for quenched aluminum.

More recently, however, open voids also have been

seen [10]. If the aggregates are relatively large, their crystallographic nature can be determined

quite precisely by transmission electron microscopy.

The nature of the simplest clusters, such as the

divacancy and trivacancy, is also well understood.

It is the intermediate-sized clusters which now pose

the main difficulty. Their energies and atomic

configurations are not yet well known. A knowledge of their properties might permit a more complete interpretation of the quenching process and

also perhaps the related processes of irradiation

damage, diffusion, and mechanical deformation.

2. Method of Computation

The atoms immediately adjacent to a point

defect, or a cluster of these defects, suffer dis- placements which are too large to be adequately

described in terms of linear elastic theory. Because of this, these atoms must be treated as

discrete particles. Atoms lying well away from

the defect, on the other hand, are displaced by

small amounts which are susceptible to treatment

by the elastic theory. Because of this, the crystal

containing the defect under examination was considered to consist of various regions. In the

region immediately surrounding the defect, the atoms were discrete particles and their displace- ments were treated independently. The displace- ment of each atom in this region was determined

by allowing it to interact with surrounding atoms

"r!a>ed on work performed under the auspices of the U.S. Atomic Energy Commission. 'Figures in brackets indicate the literature references at the end of this paper.

according to a pair-wise central force law. This

inner region was surrounded by an array of atoms

which were also regarded as discrete particles but

whose displacements were given by linear elastic theory.

The procedure for determining the equilibrium

configuration and energy of a defect was as follows.

All atoms were initially located on lattice sites. Atoms were then removed from or added to the center of region I so as to produce the desired

defect. Each atom in region I was then moved in turn until the force on it was again zero. The atoms were moved in this manner in a series of cycles in which the atoms closest to the defect, which suffered

the greatest disturbance, were moved more fre- quently than atoms more distant from the defect. After this variational process had been completed,

the outer "elasto-atomic" region was displaced in such a way that equilibrium was restored. The

47
whole process was then repeated several times until the minimum energy condition was obtained. The

number of iterations which were required to bring

about complete stability was typically of the order

of 25 to 50. The formation energy of a cluster of k point

defects is Efv = E'xv + \Es -Ep , (1)

where E'xv is the energy of the crystal containing

the X point defects (in the desired configuration),

E s is the energy of an atom on the surface of the

crystal, and Ep

is the energy of the perfect crystal

with the A. atoms restored to their original positions.

The binding energy against breakup into A. individual

point defects is E& = \Ef-E&. (2)

3. Calculation of the Constants of the Truncated Morse Potential

We follow the general method outlined by

Girifalco and Weizer [11] by relating the potential

to certain physical properties. The interaction

energy, Einj), of a pair of isolated atoms is then

given by a Morse function,

E(nj) = D {exp [- 2a(ru

- - r0)] - 2 exp [- afoj - r0 )]},

(3)

where ry

is the distance between the two atoms, D

is the dissociation energy of the pair, r0 is the equilibrium separation distance of the two atoms, and a is a constant which effectively determines the

"hardness" of the potential.

The energy of a crystal containing N atoms is

E =

\ N°

J% { exP ( ~~ 2a[r

j ~ r° ]) -2exp(-a[rj -r0])}, (4)

where the factor \ is required to allow for the double

counting of bonds. J is the total number of atoms

which fall within the sphere of influence of a given

atom, rj is related to the lattice parameter by

£/o(ao)=£(ao). (6)

The Born stability criteria [12] require that

\da)a=ao ' ^

and the compressibility at zero temperature and

pressure, K0o, is given by

Km

d

2E

dVVa (8)

=a0

where Vqo is the atomic volume at zero temperature

and pressure and V0 is the volume at zero temperature. Because the volume per atom in the f.c.c.| structure is 2a3

, eq (8) may be written

1 1 Koo ISNao \da2 d

2E

(9)

rj = I 2 a? + m)a2 + n]a2 =M]a2

, (5)

where a is the half-lattice parameter, lj, mj and nj are integers, and their sum is always even for the

f.c.c. lattice. The sublimation energy at zero temperature and pressure is given by

Equations (6), (7), and (9), when used together, are| sufficient to determine the three parameters D, a,\

and r0 in the Morse potential. Equation (4) is used

throughout in such a determination of constants,

and the latter will clearly depend upon the value of J. In the work described here, J was taken to be 176j

and the constants were then r0 = 2.91295 A,o do! = 2.54756 A, D = 0.32265 eV and a =1.28663 A~li Elastic constants calculated with the potential are in good agreement with experimental values.

4. Results

4.1. The Single Vacancy

The nearest-neighbor atoms of a single vacancy

are found to relax inwards by a distance of 3.0 per- cent of the nearest-neighbor distance. The second

neighbors relax outwards slightly by about 1.5 percent. The formation volume of the vacancy was

found to be 0.83 fl for copper, where Cl is the atomic

volume. This compares with the experimental

value for gold, which ranges from the value of

De Sorbo [13] (0.57 O), to that of Simmons and

Balluffi [14] (0.45 Q). Theoretical values of t&F

v

have been reported for copper by Tewordt [4 (0.47-0.55 fl), Seeger and Mann [15] (0.71-0.91 O).

Benneman [5] (0.60 fl), Johnson and Brown [7 (0.43-0.62 fl), and Schottky, Seeger, and Schmid

[16] (0.76-0.98 fl). All these values were obtained

by use of a Born-Mayer potential. The vacancy

formation energy was found to be 3.2 eV. The

activation energy of vacancy migration is found tc be 0.69 eV. This compares with previous values

of Ef! calculated for copper by Huntington and Seit2j [17] (1 eV), Fumi [18] (0.6 eV), Bartlett and Dienes

[19] (0.97 eV), Damask, Dienes, and Weizer [20

48
(1.3 eV), and Johnson and Brown [7] (0.43 eV).

Direct experimental values of E\! have been re- ported by Schiile et al. [21], and by Hasiguti et al. [22]. They obtain 1.08 eV and 1.06 eV, respectively. The activation volume of self-diffusion was found to be 0.86 ft in the present study, and

activation volume for vacancy migration is therefore

0.03 ft. Johnson and Brown [7] found a value = 0. 19-0.38 H. Schottky, Seeger, and Schmid

[16] find 0.02-0.10 ft. Emrick [23] has reported an experimental value of 0.15 ft in gold.

4.2. Divacancies

The present calculation gives a divacancy binding energy in copper of 0.53 eV. Bartlett and

Dienes [19] calculated 0.3 eV, and Weizer and

Girifalco [24] give 0.64 eV. No experimental

values of E$v in copper have been reported. For

gold it is greater than 0.3 eV [25]. For silver it is 0.38 eV [26]. The activation volume associated

with the formation of a divacancy was found to be

1.53 ft. The volume change associated with the

formation of the divacancy from two isolated vac- ancies is 0.13 ft. The binding energy for the loosely

bound divacancy in which the two vacancies are separated by a second-nearest-neighbor distance was found to be 0.19 eV, and the volume change as- sociated with this configuration is virtually the same

as for two isolated vacancies.

When a divacancy migrates, the saddle point was

found to be located at the center of the equilateral

triangle defined by the original positions of the two

vacancies and the position of the moving atom. The saddle point lies very close to the plane of that

triangle. This saddle point is therefore equivalent

to a triangular arrangement of three two-thirds

vacancies surrounding the migrating atom. The

energy barrier against motion was found to be 0.02 eV. This value is, of course, much lower than

what would be expected. There are no experimental values of Ei'y for copper, but this quantity

has been measured in gold. Bauerle and Koehler

[27] give 0.6 eV approximately.

4.3. Trivacancies

There are four possible types of trivacancies.

They are the 60°, 90°, 120°, and 180° types, where

the angles refer to the angles which the two outer vacancies subtend at the middle vacancy. The

60° type was investigated by Damask et al., [20]. They find that the atom located immediately above

the three vacancies is able to relax downwards

towards the center of the triangle and that its final equilibrium' position is exactly at the center

of the tetrahedron defined by the positions of the

three vacancies and the moving atom. This configuration is equivalent to a symmetrical arrange- ment of four three-quarter vacancies surrounding

an interstitial atom. Figure 1 shows the relative

RELATIVE ENERGY

(eV)

VOLUME

CHANGE CHANGES OF ENERGY AND VOLUME

FOR TRIVACANCY. FIGURE 1. Relative energies and activation volumes for

trivacancies. binding energies of various trivacancies obtained

in this study. It may be noted that the 60° is much

more tightly bound than the other types. The

actual formation volume of each defect is the

number given in the figure plus 3 ft, where ft is the atomic volume. As might be expected, the

60° type shows the smallest contraction.

4.4. Tetravacancies

Calculations for some tetravacancy configurations have been reported by Vineyard and Gibson

[28]. The configurations shown in figure 2 are of

particular interest because they are quite compact. The relative binding energy of each of the

FIGURE 2. Configurations of the three tetravacancies consid- ered in the present study.

49
only stable or metastable configurations is given

in figure 3. The results do show that the rhombic

form is the most stable. The rhombic form is re- lated to the 60° trivacancy discussed earlier. It

o

-

RELATIVE ENERGY

(eV)

TYPE

VOLUME

CHANGE

TETRAHEDRAL

-0.29 il

RHOMBIC

CHANGES OF ENERGY AND VOLUME

FOR TETRAVACANCY.

Figure 3. Relative energies and activation volumes for

tetravacancies.

permits two of the relaxations which occur in that trivacancy, since it is effectively two adjacent 60°

trivacancies joined along a common side. It has

been shown that the low energy of the rhombic form

may indicate collapse of vacancy clusters to dis- location loops at an early stage of growth.

4.5. Larger Clusters of Vacancies

Calculations are now under way for larger

clusters of vacancies. The smallest octahedral

void, consisting of six vacancies, is stable. There

is a vacant space of about two atomic volumes in the interior of the defect. This could be the

nucleus of the large uncollapsed octahedral void

observed by Yoshida et al., [10]. Larger clusters

of vacancies on {111} planes are also being ex- amined, and it is being found that the role of the

60° trivacancy as the "building block" of collapsed

clusters, which was previously discussed in con- nection with the rhombic tetravacancy, is continued. The next larger octagonal void, which

contains 19 vacancies, has also been examined

recently. It too is found to be stable against

spontaneous collapse. This result is intrinsically

dependent upon the use of a potential which has an attractive term. A calculation which employed

a Born-Mayer potential would not give stability.

5. Postscript

When a Morse potential is used to calculate the

formation energy of point defects, the absolute

values of the latter are always rather high. Electron redistribution around the defect must account

for part of the discrepancy. Another source of error may arise from the fact that eq (6) implicitly assumes that the electron environment which applies when (4) is used is the same as the environment

which prevails during the measurement of £/0 . This

Table 1

Defect

Single vacancy

Single vacancy motion...

Divacancy

Divacancy motion

Next-nearest neighbor divacancy

60° trivacancy

90° trivacancy

120° trivacancy

180° trivacancy

Tetrahedral tetravacancy Rhombic tetravacancy

Properties calculated by use of sublimation energy

Formation volume

0.83

.86

1.53

1.47

1.66

2.50

2.26

2.38 2.39

3.71

2.90

Formation

energy

eV

3.19

3.88

5.85

5.88

6.21

7.33 8.09

8.54

8.46

11.73 8.96

Properties calculated by use of vacancy

formation energy

Formation volume

2.49

2.59

2.67

2.65

3.85

3.42

Formation

energy

eV

3.05 3.10

3.14

3.14

3.70

3.82

is probably not true because the evaporation of a

neutral atom from the surface requires first the

localization of an electron at the atom which is to be removed. Because of these difficulties, an

alternative approach could be used which links

the Morse potential to the experimental value of the vacancy formation energy, rather than to Es,

the sublimation energy. The hope would then be

that the absolute values of the binding, migration,

and formation energies might also be more reliable. Table 1, which summarizes the energies obtained

in the present calculations, includes entries for energies calculated by this alternative approach.

The authors acknowledge the continuing interest and encouragement of T. H. Blewitt, H. H. Chiswik.

F. G. Foote, and O. C. Simpson. It is a pleasure

also to acknowledge helpful discussions with K. H. Benneman, T. H. Blewitt, L. A. Girifalco, H. B. Huntington, R. A. Johnson, J. S. Koehler, F. Seitz

O. C. Simpson, L. Tewordt, and M. P. Tosi. The

work described here was greatly facilitated by the help and cooperation of C. G. Le Vee and the1 scheduling, consulting and operating personnel o; the CDC 3600 computer at Argonne.

SO
5. References

[1J H. B. Huntington, Acta Met. 2, 554 (1954).

[2] E. Fues and H. Stumpf, Z. Naturforsch. 109, 136 (1955).

[3] H. Kanzaki. J. Phys. Chem. Solids 2, 24 (1957).

[4] L. Tewordt, Phys. Rev. 109, 61 (1958).

[5] K. H. Benneman, Z. Physik 165 , 445 (1961).

[6] L. A. Girifalco and V. G. Weizer, Phys. Rev. 114, 687

(1959).

[7] R. A. Johnson and E. Brown, Phys. Rev. 127 , 446 (1962).

[8| P. M. Morse, Phys. Rev. 34, 57 (1929).

[9] P. B. Hirsch, J. Silcox, R. E. Smallman, and K. H. Westmacott, Phil. Mag. 3, 897 (1958).

[10] M. Kiritani, J. Phys. Soc. Japan 19, 618 (1964); M. Kiri- tani and S. Yoshida, Japan J. Appl. Phys. 4, 148 (1965).

[11] L. A. Girifalco and V. G. Weizer, NASA Rept. No. R-5

(1959). [12] M. Born, Proc. Cambridge Phil. Soc. 36, 160 (1940).

[13] W. De Sorbo, Phys. Rev. 117, 444 (1960).

[14] R. O. Simmons and R. W. Balluffi, Phys. Rev. 125 , 862

1 1962).

[15] A. Seeger and E. Mann, J. Phys. Chem. Solids 12, 326

(1960). [16] G. Schottky, A. Seeger and G. Schmid, Phys. Stat. Solidi

4, 419 (1964).

[17] H. B. Huntington and F. Seitz, Phys. Rev. 61, 315 (1942).

[18] F. G. Fumi. Phil. Mag. 46, 1007 (1955).

[19] S. H. Bartlett and G. J. Dienes, Phys. Rev. 89, 848 (1953).

[20] A. C. Damask, G. J. Dienes and V. G. Weizer, Phys. Rev. 113, 781 (1959).

[21] W. Schiile, A. Seeger, D. Schumacher and K. King, Phys.

Stat. Solidi 2, 1199 (1962).

[22] R. R. Hasiguti, Y. Nakao, and H. Kimura, J. Phys. Soc. Japan 20, 553 (1965).

[23] R. M. Emrick, Phys. Rev. 122, 1720 (1961).

[24] V. G. Weizer and L. A. Girifalco, Phys. Rev. 120 , 837

(1960).

[25] J. W. Kauffman and M. Meshii. Lattice Defects in Quenched

Metals, p. 77, edited by R. M. J. Cotterill, M. Doyama,

J. J. Jackson, and M. Meshii (Academic Press. New York,

1965).

[26] M. Doyama and J. S. Koehler. Phys. Rev. 127, 21 (1962).

[27] J. E. Bauerle and J. S. Koehler, Phys. Rev. 107, 1493

(1957).

[28] G. H. Vineyard and J. B. Gibson, Bull. Am. Phys. Soc. 6, 158 (1961).

51

A Two-Body Central Force Interatomic Potential for

Anharmonic Solids

Roger Chang and Lloyd J. Graham

North American Aviation Science Center, Thousand Oaks, California 91360

A prerequisite to the application of numerical

methods with high speed computers to studying

defects in solids is the availability of reliable inter- atomic potentials between the atoms. Previous

investigations of the two-body central force in- teratomic potential in solids have not touched upon

the effects of anharmonicity. Since nearly all solids have some degree of anharmonicity, it is the

purpose of this paper to find an empirical approach

to obtain an interatomic potential which takes into consideration the anharmonic effects. The theory of anharmonic effects in crystalline

solids has been discussed in considerable detail by Liebfried and Ludwig [l]. 1 For a two-body central force model, explicit relationships exist between the second, third and higher order elastic constants of a solid and the interatomic potential,

the details depending on the symmetry of the crystal

and the number of shells used in computing the

interactions. In this paper the body-centered

cubic iron is used as an illustration where nearest and next-nearest neighbor interactions only are considered. If <^(r) is the potential between a pair

of atoms of separation r, n the nearest neighbor

separation and r2 the next-nearest neighbor separation, the relationships between the second and

third order elastic constants and the interatomic

potential are, using the Voight notation [2],

Cn =

tl,2 =

3r2

_2_

3r2

2_

3r,

+ <p"(n) + 3<p"(r2 )

' 1

Atp'in) ,, — 'A(p'(r-2 Yip (n)

-YP

r,

2<p'(ri)

+ <p"(n)+

3<p'(r2 )

'

/2 + P

<p"'(ri) <p"(ri) 3V3

8<?'(n) . „ ,,„ , 6<p"(r2) + Zip (r2

)

3V3

V3r, r?V3

<pVi)

| 4<p'(ri) ; 2<p'(r2 V3r, rfV3 r§

-SP

+ P

C123 —

<p"'(ri) y?"(ri) 2<p'(ri) 2y?'(r2 3V3 V3r, rfV3 rf -P (11

where P is the external pressure and <p'(ri), <p"{ri)

and ip"'(ri) are the first, second, and third derivatives of the potential tp(r) with respect to r and

evaluated at n, respectively.

We express the potential tp(r) as a Taylor's expansion up the the fourth power in r {tp in eV, r in Angstroms),

ip(r) = A+Br+Cr2 + Dr5 + Er* (2)

Although higher orders in r may be included in eq (2), we feel that the expression to the fourth

power in r is enough for the present purposes. The

constant A in eq (2) goes out when it is fed into eq (1). Thus there are only four independent

constants to be evaluated in eq (2). Since six ex- perimental elastic constants are available for comparison, there is sufficient redundancy to check

out the consistency of the computation.

Differentiating eq (2) with respect to r, one finds,

tp'(r) =B + 2Cr + 3Dr* + 4Er^

iP "(r)=2C + 6Dr+l2Eri <p"'(r)=6D + 24Er.

(3)

We choose P in eq (1) to be equal to (C\ > — Ch)/2,

where Cv> and C44 are the experimental elastic constants. Substituting eq (3) into eq (1), one obtains a set of computed elastic constants which

can be compared with the experimental values. The fitting is done numerically with the aid of computers. A reasonably good fit yields the follow- ing constants for eq (2), B=- 12.2404 eV (A)->

C= 1.24594 eV (A)~2 0 = 0.815729 eV (A)-3

1 Figures in brackets indicate the literature reference at the end of this paper. £ = -0.156140 eV (A)-4

.

53
The computed elastic constants using the above

parameters are compared with the experimental

values in table 1, showing reasonably good agree- ment. The force-distance plot is compared with

those from three other potentials, the Morse [3], Lennard-Jones [4] and Johnson [5] potentials in figure 1. The comparison shows that our potential

is very close to that of Johnson's. It is noted that the method yields the potential only within a region

between and shortly beyond the nearest and next- nearest neighbor distances. Further information

is needed to obtain a potential which will cover the

full range. This is beyond the scope of the present

paper.

Table 1. Comparison of computed and experimental elastic constants (in units of 1012 dynes/cm2

)

Calculated Experi- Refer-

(this paper) mental ence

Cu 2.323 2.33 [6] 2.28 [7]

C,2 1.327 1.35 [6] 1.32 [7] c„ 1.148 1.18 [6] 1.165 [7] C,„ -24.40 -33.5 [8] Cm -2.673 -3.3 [8]

Ci23 -2.494 -3.1 [8]

This paper

Johnson potential (5)

Morse (3)

Lennard - Jones (4)

_i i i i I i i i i_ 2 5 3 0 3.5 Distance-r (A)

FIGURE 1. Comparison of force-distance plots: body-centered

cubic iron.

References

[1] G. Leibfried and W. Ludwig, Solid State Physics 12, 275

(1961).

[2] R. A. Coldwell-Horsfall. Phys. Rev. 129, 22 (1963).

[3] L. A. Girifalco and V. G. Weizer, Phys. Rev. 114, 687 (1959).

f4| R. Fiirth, Proc. Roy. Soc. London A183, 87 (1944-45).

[5] R. A. Johnson, Phys. Rev. 134, 1329 (1964).

[6] J. A. Rayne and B. S. Chandrasekhar. Phys. Rev. 122, 1714

(1961).

[7] A. E. Lord and D. N. Beshers, J. Appl. Phys. 36, 1620 (1965).

[8] A. Seeger and O. Buck, Z. Naturforsch. 15A, 1056 (1960).

54
A Reexamination of the Born-Mayer Potential for Ion-Core

Interactions in Copper*

S. S. Jaswal 1 and L. A. Girifalco

University of Pennsylvania, The School of Metallurgical Engineering and Laboratory foi

Research on the Structure of Matter, Philadelphia, Pa. 19104

Ion-core short-range interactions between nearest neighbors in copper have been approximated by

the Born-Mayer potential by Huntington and

Seitz [l],2 Huntington [2], and others [3]. Huntington

used the experimental values of the bulk and shear

moduli, the free electron approximation with effective mass for valence electrons and Fuchs's cal- culations [4] reduced by one half for the electrostatic contribution to the shear moduli to compute

the parameters in the Born-Mayer potential.

Fuchs computed the electrostatic contributions

to the shear constants arising from distorting a

lattice of positive point charges immersed in a uniform distribution of conduction electrons. As

pointed out by Huntington it is hard to determine

what these contributions really are in a case such

as copper. The values, as computed by Fuchs, are

listed in table 2. Since this contribution to C44

is significant, the uncertainty in C44 in large.

According to Fuchs [5] the contribution of the

ground state energy to the bulk modulus is negligible and it is only through the Fermi energy

that the conduction electrons contribute to the

bulk modulus. Huntington's calculations are based on this idea.3 When a number of potentials of the Born-Mayer

type were compared in radiation damage calculations [6], a potential which is close to that suggested

by Huntington turned out to be quite good. However the contribution of this potential to (Cn — C12), as listed in table 2, shows large disagreement with the experimental value [7] even when

one takes into account the electrostatic contri- bution as computed by Fuchs. Since the electrostatic contribution to C44 is quite significant, due to the uncertainty in this contribution, it is difficult

to make any meaningful comparison for this quantity.

'Supported by ARPA. A more detailed form of this paper has appeared in J. Phys Chem. Solids 28, 457 (1967).

' Present address: Physics Dept.. Univ. of Nebraska. Lincoln, Neb. 68508.

; Figures in brackets indicate the literature references at the end of this paper. 'In referenced there is an error in equation (4). The number 5 in the denominator

in first term on right-hand side should he replaced by 3.

In present calculations we use the experimental

values of cohesive energy, lattice parameter and

bulk modulus to determine the Born-Mayer parameters and hence avoid the uncertainties in theoretical estimates of the shear constants. Since we

use the cohesive energy and its volume dependence, we must take into account all the volume dependent

terms in the energy rather than just the Fermi

energy. The various terms considered in the interaction

energy besides the Born-Mayer potential are the

ground state energy of the conduction electrons

based on a simplified theory due to Frohlich [8], and the Fermi, Coulomb, exchange and correlation

energies of conduction electrons based on the free electron model in the effective mass approximation. In Frbhlich's theory, the potential in the outer part of the unit cell is assumed to be hydrogenic and the conduction electrons are treated as

free. An advantage of this theory is that one has

an analytical expression for the ground state energy. Other terms in the energy mentioned above are computed on the assumption of a uniform conduction electron density. The correlation energy used

is the interpolation result due to Noziefes and

Pines [9]. The interaction energy per unit cell can be writ- ten as:

£"int = Eo + Ek.e + Eejc + EC0 + E' + Ec -c (1)

where

E0 = ground state energy

=-3/r, +p§//f

with p

2

, being an unknown parameter:

Ek.e = Fermi energy = 2.21/[ (/n*/m)r 2

]

55
with m* being the effective mass and m, the free electron mass;Eex — exchange energy = -0.916/rs ; Eco = correlation energy = -.115 + 0.031 In rs :

£" = coulomb energy = 1.2/rs ;

and

Ec -c = contribution of core-core interactions = 6a exp [(r0 — r)p/r0 ]

with r0 being the equilibrium separation of ions and a and p are the parameters to be determined.

In this expression rs , the radius of the equivalent

sphere, is expressed in atomic units and energy

in Rydbergs.

Eint can also be written as Eint = Ei — Esub (2)

where E\ is the ionization potential of copper atom

and Esub is the sublimation energy of copper.

The equilibrium condition for the lattice gives

(dEiJdrs) |o = 0 (3)

where | 0 stands for the value at equilibrium.

Also the second derivative of the interaction

energy is related to the compressibility ft by

(^£int/rfrf)|o^l2Wj8. (4)

Equations (2), (3), and (4) are used to determine

the unknown parameters p

2

,, a and p. Experimental values used in our calculations

are as follows;

E/ = -7.724 eV; £sub = 80.6 kcal/mol

rs = 2.66 a.u.; l//3= 14.4 X 1011 dynes/cm2

and m* = 138m as given by specific heat measure- ments [10]. TABLE 1. Parameters of the present and some of the earlier Born-Mayer. potentials

Potential xt (eV) P

0.0958 13.34

(2) Huntington's .053 13.9

(3) Gibson et al's .051 13.0

Solving eqs (2), (3), and (4) for a and p we get a = 0.0958 eV„p= 13.34 and p0 = 0.47 a.u. We

compare our results with some of the earlier potentials in table 1 and figures 1 and 2.

i.o

0 I

\\

\\ ---

-Present Calculations (m = m)

-Huntington

_

-

\\\ \ \ \ \\

V

\\\

-Gibson et. al

- \

\\

-

\A

\\\

V

1 1 1

1 \\ l\ 1 O 5 10 1.5 0 2.0 2.5 3.0 r(A)

FIGURE 1. Born-Mayer potential in eV as a function of core- core separation in A when effective mass is equal to that of a

free electron.

The contributions due to the Born-Mayer potential to shear constants are given by [4]

(C„ - C12)c-C = (H2)N[r2E'c

'_ c + 7 rE'c _c ]r0 (5)

and

(C44 )c-c = ( l/2)/V[r2E"c _c + 3rE'c _c ] r0 (6)

where TV is the number of ions per unit volume and

primes indicate the derivatives with respect to r. Using eqs (5) and (6) we have computed Born-Mayer

contributions to shear constants and results are given in table 2. Table 2. Shear constants* in units of 10u dynes/cm2

(C„-C,2 ) C44

Potential 1 5.55 9.06

Potential 2 3.49 5.51 Potential 3 2.8 4.5 Experiment, 0 °K 5.28 8.35

Electrostatic contribution 0.57 2.57

*Firsl three lines list the contribution of Born-Mayer potential to shear constants. Last line list the electrostatic contribution as computed by Fuchs.

56
r (A)

Figure 2. Born-Mayer potential in eV as a function of core- core separation in A when effective mass is 1.38 times that of

a free-electron.

Keeping in view the uncertainties in electrostatic

contributions to the shear constants, agreement

between our calculations and experimental results

is quite good.

The potential based on the Thomas-Fermi-Dirac

approximation (TFD) as computed by Abrahamson

[6, 11] is considered reasonable around ion-core

separation of about 0.5 A. Our potential at such

separations is in fair agreement with TFD.

Thus we have computed the Born-Mayer parameters which are of the same order of magnitude as have been used earlier by various people. However we have improved upon the earlier parameters

by computing them in a self consistent manner

within the approximations mentioned above and

the agreement with shear constant (Cn — C12) is much better in our case.

References

[1] H. B. Huntington and F. Seitz, Phys. Rev. 61, 315 (1942).

[2| H. B. Huntington. Phys. Rev. 91, 1092 (1953).

[3] C. Zener, Acta Cryst. 3, 346 (1950): G. J. Diennes. Phys.

Rev. 86, 228 (1952).

[4] K. Fuchs, Proc. Roy. Soc. (London) A153, 622 (1936).

[5] K. Fuchs, Proc. Roy. Soc. (London) A151, 585 (1935).

[6] J. B. Gibson, A. N. Goland and G. H. Vineyard. Phys. Rev. 120, 1229 (1960).

f7j H. B. Huntington. Solid State Phvsies 7 (19581

[8] H. Friihlich, Proc. Roy. Soc. A158, 97 (1937): H. Brooks. Supp. Nouvo Cimento 7, 236 (1958).

[9] Nozieres and Pines. Phys. Rev. Ill, 442 (1958).

[10] W. S. Corak, M. P. Garfunkel, C. B. Satterthwaite, and

A. Wexler, Phys. Rev. 98, 1699 (1955); J. A. Rayne,

Australian J. Physics, 9, 189 (1956).

[11] A. A. Abrahamson, thesis, New York University, 1960

(unpublished): A. A. Abrahamson. R. D. Hatcher, and

G. H. Vineyard, Bull. Am. Phys. Soc. 5, 231 (1960).

256-724 0-67—

5

57

The Interactions Between Point Defects*

J. S. Koehler

Department of Physics and Materials Research Laboratory, University of Illinois,

Urbana, 111. 61681

If it is assumed that the interaction between point

defects in solids can be considered as a rapidly

convergent series of two body interactions, three

body interactions and higher order terms, then if one supposes that three body interactions and

higher order interactions are negligible, information can be obtained concerning the strength of the

two body forces provided their variation with dis- tance and with the orientation of the defect separation relative to the crystal axes is known. The

experimental information used is the formation

energy of various point defects and the binding

energy of various defect clusters. In addition the

experimental stacking fault energy of both intrinsic

•Supported by the U.S. Atomic Energy Commission under Contract AT(1 1 — 1

)— 1 198.

and extrinsic stacking faults provide valuable data. From the fact that interstitials do precipitate in the form of low energy stacking faults it is shown

that either interstitials attract one another or that three body and other multibody forces are of importance. A similar statement can be made for

lattice vacancies. The equivalence of vacancies

and interstitials for producing a given platelike

precipitate is shown. The present formulation

when it includes only two body interactions can be

checked against experiment provided that enough

experimental data is available. Although not enough data exists at present for any material, such

a check would be valuable since it would determine

the magnitude of the three body and higher multibody forces.

59

The Energies and Configurations of Point Defects in Solids

A. B. Lidiard

Theoretical Physics Division, U.K.A.E.R.E., Harwell, Didcot, Berks, England

(Panel Members: R. Bullough, H. Kanzaki, A. B. Lidiard, A. Seeger, M. P. Tosi and G. H. Vineyard)

1. Introduction

The topics included for discussion by Panel I were the energies and configurations of defects —

especially those questions raised by the papers

presented in the first two sessions. It was possible

to group these questions as follows.

1.1. Ionic Crystals

(i) General relations of the new calculations pre- sented at the conference (papers by Boswarva and

Lidiard, Diencs et al., Franklin, Tosi and Doyama

and Tharmalingam) among themselves and to pre- vious calculations.

(ii) Specific questions of the method of treating

the long-range displacement and polarization fields around charged defects. Here one had the papers

by Kanzaki and Kurosawa and the discussion re- marks by McCombie and Mullen among others.

(iii) Comparison of the interionic potentials used

in these calculations, e.g., choice between Born- Mayer and Born-Mayer-Verwey forms, and the significance of van der Waals terms for these calculations. This aspect of the subject gave rise to a rather varied discussion.

1.2. Covalent Semiconductors

The discussions and contributions of the confer- ence to this class of materials were disappointingly

few. This remains an area where there appears

no very consistent approach to defect problems.

The most general approach is that of Bennemann

but this has not yet been related to more conventional methods, e.g., those of Hasiguti and of Moore

and Carlson.

1.3. Metals

(i) The whole question of the use of 2-body potential models for metals is important. While a general justification can be found in the method of

pseudopotentials it is not evident that one can in this way find a justification for the particular forms

of potential convenient for computer calculations

e.g., Morse potentials, polynomials, etc. as used in the papers by Cotterill and Doyama and by Johnson. The panel also had before it two papers on

the empirical determination of potentials by Chang

and Graham (a-Fe) and by Jaswal and Girifalco (Cu).

(ii) The complexity of some of the larger computer

calculations now being made means that, as with

experimental work, the results cannot be verified by inspection in the way that theoretical arguments

usually can be checked. The repetition of these

computations by other workers is therefore equally

as valuable as the verification of the experimental

results of one laboratory by another. This sort of

comparison was made at the conference for a-Fe

(Beeler, Chang, and Johnson) and for Cu (Doyama,

Johnson, and Seeger). Many of these calculations are made for model crystallites containing several

hundred atoms and such comparisons are there- fore valuable not simply to avoid program mistakes,

truncation errors etc. but also to assess the sensitivity of the results to the size of the model.

We shall report on these topics in the way they were dealt with during the panel discussion but we also include a number of pertinent remarks

made from the floor. 1

1 In presenting this report 1 have had the benefit of written contributions from

Drs. Beeler. Hatcher. McCombie,.Tosi. and Vineyard and 1 have not hesitated to use

these as submitted where appropriate. The extensive remarks of Dr. Vineyard form

the basis of section 3 on metals. However I have edited all this material substantially and accept responsibility for the correctness or otherwise of the views and comments

presented.

2. Ionic Crystals

2.1. Interrelations Between Different

Calculations

The defects one is concerned with in ionic crystals often have an effective electrical charge, e.g.,

ion vacancies and interstitial ions. As was shown

long ago by Jost, the polarization of the lattice by

these charges is important in lowering the energy

of the defects, but the long range of electrical inter- actions makes it difficult to deal with them as one

61
deals with short-range forces. One cannot, for example, set up crystallite models on a computer

in the same way as has been done for metals.

Owing to the presence of these long-range electri- cal interactions it has been customary, following

the original paper of Mott and Littleton, to divide

the crystal into two regions, I and II; the first in- cludes the defect and as much of its immediate

surroundings as is judged desirable, while the sec- ond is the rest of the crystal. The idea is that in the outer region II the displacements and electronic

dipole moments of the ions are small enough that a harmonic theory can be used; this is important

as it allows the incorporation of information from

other studies, e.g., dielectric constants and lattice dynamics. In region I on the other hand the har- monic approximation may not be adequate and the

ionic interactions are represented explicitly by an assumed potential function (electrical plus overlap

interactions). This division of the crystal into two

regions is followed in the papers presented by Bos- warva and Lidiard, Franklin and Tosi and Doyama,

as well as in a number of previously published

papers [l],2 which also broadly follow the Mott

and Littleton method.

Let us look more closely at the formal aspects

of this division into regions I and II. It is basic

to all models that we can specify the state of the

system by the nuclear coordinates and the elec- tronic moments of the ions. In regions I and II we

specify the positions of the ions collectively by x, g and the moments by m, fx respectively. Then

the potential energy function of the defect solid is written

V=Vi(x, m) + V2 (x, m\ /x) + V3 (£, /*). (1)

£ and fx are assumed instead. This applies to the greater part of the work in this field including the papers presented by Boswarva and Lidiard, Tosi]

and Doyama and Franklin. A convenient way toj solve eq (4), which exploits the quadratic form of V3 {^, fx) was originally described by Kanzaki [2

J

and applied by him and by Hardy [3] to electrically

neutral defects. It has now been applied to charged

defects [4] and the long-range displacement andj

polarization fields so deduced provide an objective way of assessing the Mott-Littleton and re- lated assumptions (see below).

The dependence of V upon m is also generally

assumed to be quadratic so that eq (3) is equivalent

to the elementary relation m = (polarizability

X electric field). Polarizabilities of many ions are! believed to be well known, but it is not obvious

that this linear relation between field and moment

still holds at the large fields existing near a defect

(e/rfj—lO8 V/cm). In addition as pointed out by;

Mullen these polarizabilities might also be changed

in the vicinity of the defect due to the mechanical

stresses there. Neither of these effects has yet

been evaluated.

These assumptions of harmonic dependence!

upon m, /x, and £ enable V to be put in a more con-jj

venient form, but here a difference arises between some of the calculations. Equilibrium values of m, fx, and £ can be obtained for any configurations,

x, of region I; call these m(x), /x{x) and g(x). By|j eqs (3) and (4) we then have

V(x, m: ~jx) =-^V(x, m: fx) \n,T,H = 0 (5)

It is convenient to group terms so that V3(g, /x) contains only quadratic terms in £ and fx, i.e., so

that it is the potential energy of a distorted and

polarized region II filled with a perfect undistorted

unpolarized lattice region I. The condition of

equilibrium is that V should be a minimum with re- spect to variations in x, m, and fx, i.e.,

dV

dx

dV

dm

0 (I),

= 0 (I),

dV=

dV

d{ d/x

(II).

(2)

(3)

(4)

Equations (4) are not solved explicitly in methods

derived from Mott and Littleton, but physically

plausible approximations to the equilibrium values

'- Figures in brackets indicate the literature references at the end of this paper.

as equivalent equations for the equilibrium x. In the papers of Boswarva and Lidiard and of Frank- 1 lin the first member of (5) is used whereas in the

paper of Tosi and Doyama the second is used, as

in the earlier work of Kurosawa. If the assumptions

made about m(x), fx(x) and £(x) were exact then the

calculated values of the equilibrium V would be

identical, whichever path is followed. Any di- ji vergence may therefore be a measure of the inaccuracy of these assumptions. However, the

divergence in the numerical results presented to the conference by Boswarva and Lidiard on one j side and by Tosi and Doyama on the other is not due principally to such inaccuracies but comes

from different choices of potential. When the

Tosi-Doyama model is used in the Boswarva-Lid- j iard program the results are nearly coincident [5]. 1 The inclusion of van der Waals terms in the potential function is important for this agreement,

however.

It is worth noting that the form of the equilibrium

polarization energy is — §m • F(1) summed over all the ions; in region II we can replace the moment |

62
m by the sum of fx with the moment resulting from

the displacement of the ion charge. In both regions,

however, is due only to the effective monopoles.

in region I, e.g., the missing charge on the vacancy.

The moments m, of course, depend on all the dipole sources as well (field F*2 '). It is not correct to represent the polarization energy as — ^a|F| 2 with

F = F(1) +F(2)

. This would count dipole-dipole

interactions twice. Such a double counting may

lead to large errors; the original results of Dienes

[6] on the activation energy for vacancy pair migration are very low, apparently for this reason [7]. But Scholz presented to the conference results for the formation energy of Schottky defects in NaCl

which showed that use of the incorrect form gave

energies only about 0.1 eV too high — 2.55 eV compared to 2.44 eV. These numbers were obtained

using the small crystallite method with the Tosi

and Fumi potential [8]. They stand rather high

relative to the values coming from a Mott-Littleton

approach, but this is almost certainly because the

crystallite is too small — the additional polarization

e

2 / 1 energy of an infinite crystal would be ~ —tt^ I 1 2R \ e where R is the equivalent radius of Scholz' crystallite and e is the dielectric constant. For NaCl

this is about —0.4 eV per vacancy.

2.2. The Displacement and Polarization

Fields

As noted above the use of the Kanzaki method

to solve eq (4) enables an independent assessment

of the assumptions of methods derived from Mott

and Littleton. A recent study for the NaCl struc- ture has determined the displacement and polarizations at large distances from the defect [4]. The

displacements of the ions are made up of an elastic term £elas , of the same sign for anions and cations,

and an electrical polarization term £e i ec which is opposite for anions and cations. This electrical term can be written

felec ee ex + 2

(6)

where e* is the effective ionic charge of Szigeti, oi(

is the frequency of_the longitudinal optic mode at long-wavelengths, M is the harmonic mean of the masses of the anions and cations (M+M-)I(M+ + M-)

and ex is the high frequency dielectric constant,

related to the electronic polarizabilities by the

Clausius-Mosotti formula. If the Szigeti charge

e* = e and if only nearest neighbor closed-shell re- pulsions are significant then one can show that (6) reduces exactly to the Mott-Littleton assumption,

i.e.,

selec (a+ + a- + 2otc ) 47r i)

where v is the molecular volume, eo is the static dielectric constant and ote is the displacement polarizability as defined by Mott and Littleton. On the

other hand if e* 4= e then (6) reduces to the formulae

used in the paper by Boswarva and Lidiard for the

'deformation dipole' model. In all cases the electrically induced electronic moments on the ions

are as assumed in Mott-Littleton theory. However, with the deformation dipole model e* 4= e there are additional electronic dipoles (on the

anions) caused by the relative displacement of the

ions and the form assumed for these by Boswarva

and Lidiard agrees with that obtained from the ex- amination by the Kanzaki method.

Finally, this examination shows that for an elastically isotropic solid the elastic term is

£elas 4ttCu r

3 (8)

(7)

where Cn is the elastic constant and G is proportional to the force on the near neighbors of the

vacancy (actually to that product which in a dynamical problem would be the virial of these forces).

This differs from the proposal of Brauer [1] and the

modification suggested by Boswarva and Lidiard

in fixing the elastic strength of the vacancy by the

force on the near neighbors rather than by their displacement. The magnitude of the strength,

however, appears generally to be less than one obtains following Brauer's suggestion.

Mullen criticized the Mott-Littleton type of calculation for the way it assigns the moments and dis- placements to the ions of the crystal. While this assignment was originally intuitive the above re- sults obtained by the Kanzaki method provide a rigorous basis for it and therefore answer this criticism. A more important aspect of the criticism

is the use of the forms valid at large distances from

the defect right up to the first shell of neighbors,

i.e., that a region I containing only the vacancy and

its six nearest neighbors is not large enough. The

work of Scholz on a crystallite model shows how one can make computations with a much enlarged region

I — although as previously noted it is not correct

to neglect region II altogether. The importance

of examining and establishing the validity of the

Mott-Littleton approach is that if one is confident

of its correctness for simple defects one can then

proceed to more complex ones, e.g., vacancy

clusters. At present the study of unsymmetrical

and complex defects in ionic crystals is well behind

that in metals.

It was recognized that the elastic strength of the

defect may present problems especially in these more complex cases e.g., elastically anisotropic

crystals (nearly all) or unsymmetrical defects such

as dumbbell interstitials or vacancy pairs. Some

results of Kanzaki's lattice statics method for these

63
cases were contained in the paper he put before

the conference.

An alternative approach is contained in the paper

of Kurosawa and was independently emphasized

by McCombie. The idea here is that whenever

one considers all degrees of freedom of a few shells of atoms round the defect but forces the behavior

of all more remote atoms to conform to a continuum

form specified by only a limited number of parameters (e.g., elastic strength) there is a problem of

matching the solutions in the two regions. Be- cause of the limitations imposed on the long range

displacements and polarizations it will not be possible to choose the local displacements and polarizations and the long range parameters in such a way

that every atom has zero force on it and also has

exactly the dipole moment required by the electric

field on it. A reasonable procedure (which avoids attaching

excessive weight to the exact equilibrium of the

first shell of the continuum region) is to choose the

local coordinates, local polarizations and long range

parameters in such a way as to minimize the sum

of the squares of the "error forces" and "error

polarizations" for the local atoms and for a few

shells of atoms in the region treated as a continuum.

One can check that the results obtained are rea- sonably insensitive to the number of shells of the

continuum region taken into account.

Slater and McCombie in Aberdeen some years ago

applied essentially this method to investigating low

frequency spherically symmetric vibrations round

a vacancy in diamond. For a given low frequency

they determined the amplitudes of vibration of the

first few neighboring shells and of the parameters

describing the long range displacement by minimizing the sum of the squares of the error forces in the equation of motion of the neighboring shells and of the first few shells of the continuum region.

The linear equations embodying the least squares

condition were easily set up and solved on a computer. The results were satisfactorily insensitive

to both the number of neighboring shells treated

as local, and the number of shells of the continuum

region included with them in the least squares

calculation. It would be very interesting to see

this method applied to static defect calculations.

2.3. Potentials in Ionic Crystals

The most widely used closed-shell interaction

potential is the Born-Mayer exponential form

exp { — rjp). We would intuitively expect this form

from the exponential decay of charge density on

the periphery of an atom or ion, and Hartree-Fock

calculations for the interactions of two Ne atoms

support the expectation. Even the Thomas-Fermi

model leads to an interatomic potential which is exponential over a fair range of separations (9).

In ionic crystals the potential is generally written

in an expanded form, e.g.,

Vij(r) = by exp {n + rj — r)/p,

for the interaction between ions i and j; n and r, are

ionic radii. The preexponential coefficient is generally fixed by a relation due to Pauling. The paper

by Franklin on CaF2 drew attention to this relation

and indicated that it may be misleading. It is, faute de mieux, almost always used for the alkali halides.

If one tries to develop a detailed theory of defects

in the framework of the ionic model, there are some

conditions that the potential adopted ought to

satisfy. One is on the radii, which ought to be

consistent with the electron maps. The other is consistency with the cohesive properties, i.e., the first and second derivative of the potential must

fit the interionic distance and the compressibility.

Under these conditions, one might expect that calculations on vacancies would not be very sensitive to the fundamental form of the potential, on

which, of course, the cohesive energy gives only

broad indications. The results of Tosi and of Bos- warva and Lidiard, however, show that the varia- tions may be important if one desires accuracy of, say, 10 to 20 percent.

A number of defect calculations in the alkali halides which included second-neighbor interactions

have been based on the Born-Mayer potential (BM),

of an exponential form with parameters determined

by Born and Mayer by fitting the interionic distance

and (on the average) the compressibility of the

alkali halides under the assumption that the Goldschmidt radii are representative of the ionic sizes

in crystals. This potential in both NaCl and KC1

gives only a small restoring force on the positive

ions neighboring a negative ion vacancy. This

leads to incorrectly low energies of formation for the negative-ion vacancy and for the Schottky pair.

The Born-Mayer-Verwey potential (BMV) which

stiffens the interionic repulsions at distances

smaller than the equilibrium nearest-neighbor

distance, was used by Tosi et al. [10], to test empirically the sensitivity of migration barriers to the hardness of the potential. It has led to reasonable results in activation energy calculations; this success is however achieved at the expense of having an unphysical discontinuity in the second

derivative of the short-range interaction energy

at the equilibrium interionic distance.

Tosi has also emphasized that the r~ 12 form of the BMV potential varies much more rapidly than

is allowed by the compressibilities of the alkali halide crystals. Thus fitting the exponent n in an assumed form r~ n leads to values between about

6 for LiF and 11 for Csl. Probably the BMV potential should now be discarded. There are, how-
ever, situations where the r~" form is still useful,

but for these the appropriate values of n should be

obtained from crystal data. The Tosi-Fumi potential [8] retains the BM form;

it involves a careful determination of the parameters from recent data, and yields values of the

ionic radii in crystals in agreement with the elec- tron maps. When used in defect calculations the van der Waals energy should however also be

included. This may be avoided, as Boswarva and

Lidiard have done, by adopting the Tosi-Fumi values

of the ionic radii and refitting the other parameters

of the potential to the interionic distance and the

compressibility. Nevertheless, as noted above,

explicit inclusion of the van der Waals terms leads

to Small but significant increases in the calculated

formation energies of Schottky defects. They ap- pear to be more significant in defect calculations

than previously supposed.

Further work is needed on the vacancy migration

problem in the alkali halides. So far, it has not been clear that one could obtain a reasonable barrier for the direct path along the face diagonal with

a potential appropriate to the perfect crystal. The

results presented by Tosi and Doyama assuming a

direct path show a near equality of anion and cation

vacancy activation energies for NaCl, KC1, and

RbCl. Haven pointed out that the experimental

differences between anion and cation activation

energies may be too large due to the predominance

of multivalent cation impurities in even highly

pure crystals. Nevertheless the best experimental

values are considerably farther apart than tfie cal- culated ones. Haven also pointed out that one

needs now to investigate indirect paths through

the body of the cell. Tosi believes that indirect

paths are favored in the case of large alkali impurities: in particular, for Rb+ in NaCl, for which the

impurity diffusion measurements of Arai and Mullen

[11 1 indicate a moderate energy and a considerable

entropy of diffusion. Hatcher drew attention to the difficulty of dealing

satisfactorily with the small impurity ion e.g., Li+

in KC1 and KBr. In KG the equilibrium position

of the Li + ion is known experimentally to be dis- placed from the exact cation site, whereas in KBr

it is not. The small Li + ion can polarize the anion

by moving towards it and the induced moment

attracts the Li+. According to Hatcher the TosiFumi form of BM potential cannot account for the

different behavior in KBr and KC1. Tosi regards

this as a severe test of the potential function since

the energy differences between the displaced and

undisplaced positions are only —0.01 eV. However, from the paper of Dienes et al.. it is clear that this system tests the model in a more

drastic way. Thus the small Li+ ion can move

sufficiently close to the anion that the attraction

resulting from the induced polarization, which in the point dipole model varies like r~ 4

, can outweigh

the BM repulsion (~ exp — r/p) and the model breaks

down completely. Scholz reported that in some

crystals he found related divergences of the dis- placement of the ions around a vacancy. These

occurred when a BM potential was used although

not with the BMV form. The r- 12 form of the BMV

potential clearly prevented the r~ 4 attraction from

taking over. Dienes et al., also modified the BM

potential to guard against this effect in their cal- culations. These difficulties are of the same kind as

originally led Verwey to propose his modification

of the BM potential in order to account for the properties of both alkali halide molecules and crystals

with the same ionic model. This problem, however,

remains one worthy of further attention.

It will be clear that with the BM potential there

may always be a risk of these divergences occurring.

Even if actual divergences do not occur ions may

approach too close together so that the energy is too low. As Tosi noted this is very likely to happen

when the BM form is used in conjunction with Goldschmidt radii which are small for cations. Tharmalingam reported that the use of Goldschmidt

radii in a BM potential gave a very low energy of

formation for the vacancy pair in NaCl, only 0.07 eV. whereas the use of the Tosi-Fumi crystal radii gave a sensible value —0.91 eV including the van

der Waals Term.

We may perhaps summarize this section by say- ing that fundamental knowledge of interaction

potentials is rather meager. The choice of forms

available to anyone setting out on defect calculations is still as it was 30 years ago, i.e., Born-Mayer

and r~". However there is no doubt that the particular BM potential given by Tosi and Fumi,

supplemented by a van der Waals term, is very

superior to the original Born-Mayer form. This is apparent from the manner of its derivation and also from the results which it gives for defect formation

and migration energies. However, it is evident

that there are difficulties with the model, stemming

from the point-dipole approximation, at small

separations which are avoided by using r

- " potentials. Although the color center calculations were

not the subject for the present panel it is nevertheless worth noting that BM and r~" may lead to rather different predictions especially for the

relaxation occurring in excited electronic states (papers by Bennett and by Opik and Wood). We

note that little attempt to derive interionic potentials from fused salts seems so far to have been

made.

3. Metals

Several calculations of defects in metals were ton gave an expert history of the subject. These

discussed at this conference, and Professor Hunting- calculations all start from the common assumption

65
that the Born-Oppenheimer approximation is valid,

which is to say that the complete dynamical problem of a crystal can be separated into a calculation

of the eigenstate of the electrons for arbitrary, fixed

positions of the nuclei, /?2 • • • Rn, and a calculation of the effective potential for nuclear configurations associated with this eigenstate, V(R\, . . . Rn )- This potential for nuclear configurations determines

the energies of formation and the energies of migration of the various lattice defects. Once Vis known,

the calculation of these energies reduces to a mathematical exercise in finding extremal values of a function of very many variables. Although this exercise is by no means trivial, numerical solutions

to satisfactory accuracy can be found with modern

computers, and this part of the problem is now

relatively satisfactory.

The papers presented at this conference ex- pounded two different approaches for estimating

the function V(Ri, . . . /?„), either over all of configuration space, or at least at several points in

this space. In what may be called the fundamental

approach, one attempts to calculate the potential

properly for selected nuclear configurations by

solving the wave equation for all the electrons of

the metal in the nuclear field. In the other method,

which might be called the phenomenological

method, one deduces a plausible function V(Ri,

. . . Rn) by using many diverse bits of information, some theoretical and some empirical.

Historically, as pointed out by Huntington, the

first serious attempts to calculate properties of

defects in metals proceeded by the fundamental

method, and for many years this was the only

method of any importance [12]. The fundamental

treatments that have been given have the obvious

advantage that they start from a well-grounded

position, and use respectable quantum mechanical

procedures throughout. Because of the complexity

of the quantum mechanical problem, however, they

only yield definite answers after various approximations. There are so many ways in which these

approximations may be arranged that it is impossible to describe the situation adequately briefly.

Professor Seeger described some new methods of

this sort which may well provide important improvements, and Dr. Moore opened some new avenues

with his numerical calculations for a quasi-molecule

representing a portion of a crystal containing a

defect. However, detailed consequences of the

approach of Seeger and Mann are still being worked

out, and the greater quantum mechanical rigor of

Moore's calculations has been purchased at the

price of limiting the quasi-molecule to a very small

number of atoms. In all of the fundamental cal- culations that have been made, one has been limited

to defect configurations of high symmetry, with

relaxations of neighboring atoms either assumed

small, or neglected altogether. In principle, these

relaxations could be perfected in a self-consistent

way, but the process is most laborious. These

remarks are meant not to disparage the various

fundamental calculations which have been made, I but to put them in some perspective.

The phenomenological procedures have come

upon the scene relatively recently. They have I been prompted, at least in some part, by the power

of modern computers to give elaborate and detailed i results about defects once a potential has been

assumed. An older view would have been that such potentials can only be deduced sensibly for van der Waals crystals, and that the attempt to apply them to metals reveals less about defects

in metals than of the judgement of he who applies

them. This view is too harsh, even as the literal j acceptance of all energies calculated with phenomenological potentials would be naive. It should

be mentioned that the pseudopotential approach

(see e.g., the paper of Huntington and Feit) allows one to represent the energy of the solid as a sum of

three terms (1) a free-electron volume-dependent

term, (2) the coulomb interaction energy of the bare I ions, and (3) a band structure term which is re- ducible to a sum of two body potentials which ought | to tie up with the phenomenological potentials.

The volume dependent term is left out in some

treatments, e.g., the Morse potential treatment of

| Doyama and Cotterill; this is a step which has not been justified by those taking it. In other treat- ments simple volume-dependent forces are in- cluded—a situation colloquially described as putting the crystallite in an elastic bag. In general

these treatments have not included the long-range

oscillations known to occur in the band-structure

term. Bullough, pointed out the need to include

| these oscillations in some calculations, e.g., stack- ing fault energies, where interactions at long range j are important. This might also be true for interstitial vacancy close-pair interactions.

3.1, Phenomenological Calculations

Vineyard presented the general case for phenomenological methods as useful and fruitful [! complements of more fundamental methods as

follows. In setting up a phenomenological potential to describe any crystal, whether metal or insulator,

the essential considerations are as follows. One

knows that the crystal will be at equilibrium in the configuration of a perfect lattice of a certain

form, with a certain lattice constant and a certain

binding energy. Thinking in terms of the many

dimensional configuration space of the system

(with a finite number, n, of atoms), one thus knows

that the potential function V(RU . . . Rn) possesses

minima at the perfect lattice points of configuration space, and one knows the depths of these minima.

Now a good deal is also known about Vin the neighborhood of such a minima. First, the elastic constants determine a small number of second order

coefficients in the Taylor expansion of ^(considered

in a suitable set of axes). Moreover, if one can find
all the Born-von Karman coupling constants of the

lattice through techniques such as neutron inelastic scattering, one can determine all of the Taylor

coefficients of the second order. Higher order

elastic coefficients and anharmonic coupling constants determine higher order Taylor coefficients. Besides the minima, V possesses poles at all points

where two or more nuclei coincide. Information on V in the neighborhood of these poles is obtainable in other ways. Two-body interactions clearly

dominate here, and the strength of two-body inter- actions is found from gas scattering experiments

and from direct calculations, either fully quantum

mechanical, or on the basis of statistical models.

Moreover, sputtering and radiation damage experiments shed light on the problem. Threshold

studies of radiation damage are particularly sensitive indicators of the strength of two-body inter- actions. Finally all other available information on special configurations should be used. March

pointed out the value of the pair correlation function

for liquids in providing information about the

hard-core part of the interatomic potential. Bullough mentioned the stacking fault energy of f.c.c. metals for its sensitivity to the long-range part of

the potential and reported that in Al the oscilla- tions in this potential were vital to the accurate

calculation of stacking fault energies. There was

some argument between Bullough and Cotterill about the speed of convergence of the lattice sums

occuring in the stacking fault problem: the size of the computer model is an important feature in the accuracy of these calculations with oscillatory

potentials. Owing to the different structure of

b.c.c. metals their stacking fault energies depend

also on near neighbor interactions; they thus pro- vide a check on the potential at short range as well. Out of all this one constructs a function V

{R\ . . . Rn ) which goes as far as possible toward

matching all of the data. Clearly such functions

are not unique. Their construction is essentially an interpolation in configuration space. In this way various potentials have been deduced and

have been employed for calculating the properties

of defects. An important advantage of the phenomenlolgical

approach is that it allows far more complicated

defects to be studied than will soon be handled on

anything approaching a fundamental basis. At

worst, one is studying defects on a model, which

has some resemblance to a real crystal. Thus

many clusters of point defects, and even realistic three-dimensional dislocations have recently been

subjected to study. At this conference, papers were presented by Doyama and Cotterill and by

Johnson that fall into this category. Such work in progress was also described in discussion by Beeler

and by Bullough.

A rather large number of independent calculations with somewhat different forms of V have pro- duced concordant results on several defects, and

it seems these results must be right. The small

relaxation of the neighbors of the vacancy, and

the (100) split form of the interstitial, with larger

lattice distortion, seem well established in f.c.c. lattices. Likewise the (110) split form of the

interstitial in b.c.c. lattices is rather strongly supported. The low migration energy of the interstitial compared with the vacancy is a common

feature of all calculations, even though actual values

of these migration energies are not very certain.

The conference was not, unfortunately, able to get to grips with the long standing problem of the

divergence between the well-known calculation of

Seeger, Mann, and von Jan [13] for the Cu interstitial (is,,, = 0.6 ev) and other calculations which

have, as in the paper of Doyama and Cotterill,

generally led to much lower values (—0.1 eV).

Johnson has been unable to reproduce the value

of Seeger et al., but it appeared from the discussion that the original calculations have not been

exactly repeated. This point is an important one

for the subject and should be followed up further —

quite independently of the question of experimental

interpretations. In the calculations reported at this meeting, certain interesting features deserve emphasis. These

features are unquestionably valid for the model

potentials employed, even though they may or may not prove to be true for the real crystals simulated. In general, the great complexity of possible

point defects and their migration processes is, for the first time, becoming clear. Frenkel pairs ap- pear to have very different properties in their dif- ferent configurations, and a number of close pairs

appear to be unstable. In addition, more distant

pairs may exhibit reduced activation energies for recombination, or show binding, or even trapping.

The interstitial in the f.c.c. lattice appears to migrate in a complex way in which it moves to a neighboring site in its plane and changes its axis

in so doing. A consequence of this migration proc- ess is that, without reorientation, which may in- volve a higher activation energy, a given interstitial can only reach three out of the four types of inter- penetrating simple cubic lattices into which the

f.c.c. lattice may be resolved. Furthermore, Johnson's model shows four metastable configurations

of the interstitial, in addition to the stable form,

and the differences in energy between these are

surprisingly small. Plots of lattice energy versus

a reaction coordinate for the migration of the interstitial prove to be complicated by intermediate

saddle points in addition to the principal saddle

point. With combinations of point defects more

and more kinds of stable or metastable clusters are being found as more calculations are being done.

Although, as has been emphasized, these results are demonstrably valid only for model crystals,

there seems no reason to believe that defects in real crystals are less complex. As more calculations are carried out, the apparent complexity will probably continue to increase.

67
One important practical point in these calculations is the size of the crystallite which is simulated on the computer. Beeler drew attention here

to the differences between vacancy and interstitial calculations. The calculations on a-Fe show that the relaxations around interstitials are large enough

that the continuum elastic region surrounding the

discrete crystallite exerts a significant constraint

and may not be ignored. It is less important for vacancies and vacancy clusters. There is a proper balance between the degree

of sophistication of a model and the extent of the

numerical calculations that are worth carrying out. There is now a danger that the crudities of the

phenomenological model will be forgotten in the

enthusiasm to exploit the computer in more and more detailed investigations. The practitioner of

these methods should feel the obligation to perfect

his phenomenological potentials by making them

harmonize with as many of the various separatt

bits of information, outlined above, as he can get

his hands on and can trust. As more data becomes

available, more considerations of this kind must

be taken. Finally, these calculators should hav<

a proper scepticism of their own results, and shoulc

feel a responsibility to prevent their results from

being taken too literally by nonspecialists. In the present stages, calculated activation energies

of motion surely do not have significance to two

decimals — possibly not to one! Nevertheless, the range of defect phenomena which the calculations

suggest, and their broader features, are sufficiently

plausible that they warrant extensive experimenta

testing, and also cross-checking by fundamenta

calculations to the fullest extent that is feasible.

4. References

[1] See N. F. Mott and M. J. Littleton, Trans. Far. Soc. 34, 485

(1938); P. Brauer, Z. Naturforsch 7a, 372 (1952); F. G. Fumi and M. P. Tosi, Disc. Far. Soc. No. 23, 92 (1957); T. Kurosawa, J. Phys. Soc. Japan 13, 153 (1958): K. Tharmalingam, J. Phys. Chem. Solids 25, 255 (1964).

[2] H. Kanzaki, J. Phys. Chem. Solids 2, 24 (1957).

[3] J. R. Hardy, J. Phys. Chem. Solids 15, 39 (1960); ibid. 23,

113 (1962).

[4] J. R. Hardy and A. B. Lidiard, Phil. Mag. 15, 825 (1967).

[5] I. M. Boswarva and A. B. Lidiard, Phil. Mag. (in press).

[6] G. J. Dienes, J. Chem. Phys. 16, 620 (1948).

[7] K. Tharmalingam and A. B. Lidiard, Phil. Mag. 6, 1157

(1961).

[8] F. G. Fumi and M. P. Tosi, J. Phys. Chem. Solids 25, 3

(1964); M. P. Tosi and F. G. Fumi Ibid 25, 45 (1964).

[9] A. A. Abrahamson, Phys. Rev. 133, A990 (1964).

[10] R. Guccione, M. P. Tosi and M. Asdente, J. Phys. Chem

Solids 10, 162 (1959).

[11] G. Arai and J. G. Mullen, Phys. Rev. 143, 663 (1966).

[12] H. B. Huntington, Acta Met. 2, 554 (1954); E. Fues and

Stumpf, Z. Naturforsch 109, 136 (1955); L. Tewordt

Phys. Rev. 109, 61 (1958).

[13] A. Seeger, E. Mann and R. v. Jan, J. Phys. Chem. Solids 12, 314 (1960).

68
II. Electronic States

Vacancies, Phonons, and Dielectric Screening in Close-Packed Metals

N. H. March

Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14850

and

B. Oli

Department of Physics, The University, Sheffield, England

The role of dielectric screening in determining the electronic configuration round a vacancy in close-packed metals is emphasized. In particular (i) Mukherjee's relation between vacancies and phonons is interpreted in a first order theory, the essential link being the response of the Fermi gas, described by its wave number dependent dielectric constant, to the two different perturbations,

(iil The Fermi energy E/ times the valency Z is shown to be a natural unit in which to measure the vacancy formation energy Ev . For simple metals, the experimental results then vary systematically with Z. and yield Fumi's value (4/15) for the limit of small Z. (iii) Utilizing the semiempirical method

of Kohn and Vosko, vacancies in Cu and Al are shown to lead to displaced charge densities which correlate fairly well with results for impurities in these same matrices. Again, the concept of a dielectric constant seems entirely meaningful, if not quantitative for the stronger scattering centers. The dielectric constant should also describe the interaction between a pair of ions. Some difficulties are en- countered for Al.

1. Introduction

Since the classic studies of Huntington [l], 1 it has been clear that the electronic wave functions

of the conduction electrons are markedly distorted

around vacant lattice sites in metals. That the

displaced charge around the vacancy also oscil- lated with distance from the vacancy was apparent

in the work of Huntington, and was perhaps especially clear in the later work of March and Murray [2], In spite of this, it is perhaps fair to say that no

very direct experimental evidence has been forth- coming about the relevance of such oscillatory

screening of the vacant lattice site. Nevertheless,

without explicit appeal to dielectric screening (see

2, below), Fumi [3] recognized the importance of

the Fermi energy £/, and of complete shielding,

inserted through the Friedel sum rule, in the problem of calculating vacancy formation energies in monovalent metals. Another step forward, which

we believe will prove of considerable importance, came with the observation by Mukherjee [4], that there is an accurate empirical relation between the

vacancy formation energy Ev and the Debye temperature 9 for close-packed metals. A first-order theory based on linear screening has been given

by one of us (March [5],) and a more general formulation has been considered recently (see especially

Enderby and March [6],). The essential argument

is briefly summarized in 3. In 4, some evidence

is brought forward that a dielectric constant can

be defined which is relevant for a variety of charged

perturbations in Cu and Al. In particular, dis- placed charges and corresponding potentials for the vacancy correlate well with those for impurities

in these matrices. Some difficulties, however,

are apparent for Pb, which have not as yet been

resolved.

2. Wave-Number-Dependent Dielectric Constant

Before proceeding to discuss the Mukherjee relation, it is worth stressing that, in the work of

Huntington and the later studies of March and

Murray [2] on the electronic redistribution round

the vacancy, individual electronic wave functions

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

were calculated explicitly. Such an approach

makes the problem appear quite complex.

The first clear demonstration that a simpler

point of view could be adopted was in fact given

earlier by Mott [7], who stressed that an excellent

physical description of a Zn impurity in Cu, say, was to view the Zn ion as 'dressed' with a very

69
localized screening cloud of electrons; the dis- placed charge. Mott's model was linear and the

screened potential energy V(r) around a point

charge Z (Z being the excess valence, i.e., 1 for Zn in Cu)2 was

V{r) — exp (—qr):

, 4Av q

2 =~T, (2.1)

7T

where the inverse screening radius q was deter- mined solely by the properties of the unperturbed

Fermi gas, that is the Fermi wave number kj. The

corresponding displaced charge density p(r) — po, po being the unperturbed constant density in a simple free electron description, was given by

p(r)~pb=£F(r). (2.2)

The charge piles up around an attractive impurity,

of course, but is repelled from a vacant site. In addition to this important idea of an ion or

defect, 'dressed' with a localized electron cloud

which screens it perfectly, the Mott theory also implies that the screening of the defect center is simply to be described by a dielectric constant. Thus, introducing the Fourier transform of the

screened potential

V(r)e ik r dr, (2.3)

we may write

y\k) =

-4ttZ

k

2e(kY (2.4)

where e{k) is the wave number dependent dielectric constant. From (2.3) it follows that

e(k)

k

2 + q2 (2.5)

A more correct result was given by Lindhard as -Ml)

where

e(A)=-

g(x)=2 +—L ln 1-x

l +x

(2.6)

(2.7)

This leads to Friedel 'wiggles' in V{r).

But now the question arises: are the defects or impurity centers of interest in metal physics suf- ficiently weak probes so that we can describe their

effect by such a dielectric constant? For point

charges, Boardman and March [8] have calculated

V(k)' beyond a linear theory and the failure of the

dielectric constant picture is evident when Z is 3 or 4 in a Fermi gas of density po appropriate to Cu metal (see especially the curves of figure 1 of

their paper).

The pseudopotential theory, on the other hand,

contends that with core electrons present, the effect of the ions on the conduction electrons is much

weaker, and that the dielectric constant picture

therefore is still valid even for large Z. We pre- sent some evidence below which suggests that pseudopotential theory should be a useful starting

point in practice, particularly if the valence Z is not too large. In the case of the vacancy, of course, the picture

is that in a zeroth order approximation, we pluck

a 'dressed' ion (Ziman [9] terms it a pseudoatom) out of the metal, and the electron redistribution is thereby almost accounted for. We obviously should include relaxation, and once again,

in another paper of this Conference, Huntington

has led the way by showing how ideally suited is Harrison's approach, using pseudopotentials, to this problem. The Dexter model, to which we now turn, is a very simple example of such an idea,

and was indeed developed precisely for point-ions

by Corless and March [10]. However, the approach

of 3 will have, of course, to be generalized eventually to include core electrons (and therefore explicit pseudopotentials: see some further remarks at the end of sec. 3).

3. Phonons and Vacancies

It might seem, at first sight, surprising that there

should be a simple relation between the Debye

temperature 6 and the vacancy formation energy

Ev . But, as Mukherjee [4] emphasized, the experimental results are well represented by

9- (3.1)

where M is the atomic mass and Cl the atomic vol- ume. The constant C is the same for eight close1 We use atomic units in which e = 1, 1,#=1.

packed metals, for which experimental results are available. These include however, Ni and Pt, which are not at all appropriately described by the

considerations below. We have in mind then,

specifically the noble metals, Mg, Al and Pb.

In the simplest model of a vacancy, due to Dexter, we represent the vacant site as a 'dressed' charge — Ze in a Fermi gas, where Z is the valency of the

metal, in complete accord withjhe ideas outlined

in Z. The Fourier components V(k) of the screened

potential V(r) are given by eq (2.4), with the dielectric constant of eq (2.6). Now following essentially the argument of Fumi [3], we calculate the

70
shift in the individual electronic levels using perturbation theory. Obviously, in a metal of volume V,

the first-order perturbation energy is given by

Table 1

Ifr) dre V(r)e iV (3.2)

Obviously, eq (3.2) is independent of k and from

eq (2.3) is simply V~\ multiplied by the long-wavelength limit V(0) of V{k). This is immediately obtained from eqs (2.4) and (2.6), and if we add up

the electronic energy change over N electrons and

use the free electron relation between the electronic

density and the Fermi energy, we find the result - ZEf. But, if we neglect relaxation, the wave

ofunctions now extend into an extra atomic volume,

when we remove an atom from the interior of the

metal and place it on the surface. Hence the

kinetic energy is lowered by a calculable amount

(Fumi, [3]), which for free electrons is simply

2- ZEf. We have then as our starting approximation

4

(3.3)

Lidiard and Tharmalingam (private communication) have looked at the numerous corrections to

this result for the noble metals, and while the

various energies arising from electron-electron

interactions tend to cancel, the errors look somewhat serious. But taking the simplest picture, and

recalling that (3.3) is obtained from first-order perturbation theory, it would seem that the natural

unit in which to measure the vacancy formation

energy is ZEf. Following Enderby and March

[6], we have collected in table 1 the experimental

results for Ev in these units. It is readily shown

from the last column that the result of extrapolating

back to Z = 0 is quite consistent with the value —4

of (3.3). Following Fumi [3], if we correct for

the fact that (with his model square-well potential

now) the s waves are strongly perturbed and therefore their phase shift rjo is not adequately given by

the Born approximation employed in (3.2), then we

find a reasonable estimate for the slope of Er/ZEf

against Z at the origin. This is gratifying and therefore further calculations of the phase shifts of the

p, d, etc. waves for larger valencies are now being

carried out by S. Baranovsky and one of us (N.H.M.),

to see whether Fumi's model can reproduce the

experimental data in table 1, which flatten out in a very striking way around Z = 3. This aD seems to indicate, therefore, that we

can indeed obtain a sensible account of the vacancy

Measured vacancy Valency multiplied by free £D in units Metal formation energy Valency electron value of Fermi o!ZE,

Ec in eV energy £/ in eV

0.94 1 5.5 0.17 Ag 1.09 1 5.5 .19 Cu 1.17 1 7.0 .17 Mg 0.89 2 10.4 .09 Al .75 3 33 .02 Pb .53 4 36 .01(4)

formation energy in the simpler metals by summing

the one-electron eigenvalues, though the self- energy of the displaced charge and the other inter- action energies considered above may perhaps play

a more important role in the polyvalent metals.

But the situation is even more interesting in the

light of the Mukherjee relation. For, in the Dexter

model, the screened potential scattering the elec- trons is simply the negative of the electron-positive

ion scattering potential in the linear theory. And

this latter potential gives us, after multiplication

by Ze, the pair potential <£(r) according to the theory

of Corless and March [10]. Thus, since the ion-ion

interaction determines the phonon spectrum, and

hence the Debye temperature, it is entirely reasonable in a linear theory that it is connected with

the vacancy formation energy. Thus, from ele- mentary Debye theory (cf. Mott and Jones [7]), we

have for the Debye temperature,

vs ( 3 Y

3 h

(3.4)

The velocity of sound vs is given, in the most ele- mentary theory, by the Bohm-Staver result (cf. Bardeen and Pines [11]),

Vs (3.5)

where vj is the Fermi velocity. Eliminating ZEf

between (3.3), (3.4) and (3.5), Mukherjee's relation

(3.1) follows immediately. The constant C is given

to much better than order of magnitude and no more could be expected since the Debye temperature is calculated from (3.4) assuming a common

velocity for transverse and longitudinal waves. But while this proof of (3.1) is strictly valid only for small Z, the Mukherjee result has wider generality.

The reason for its validity appears from the studies

of Dr. S. H. Vosko and one of us (N.H.M.) to be

clearly related to pseudopotential concepts, the

essential quantity in the theory being a generalization of (2.4) in which V{r) of the point ion model is replaced by a pseudopotential (see Vosko, Taylor,

and Keech [12], for a detailed discussion of the

phonon problem). We believe that Mukherjee's

relation is direct experimental evidence of the pic- ture of a metal built up as a superposition of

'dressed' ions. Such a superposition result leads

directly to a description of ion-ion interactions in

71
terms of pair potentials (Corless and March [10], Johnson and March [13]), a result widely employed

in defect calculations reported in this Conference

(see, for example, the contribution by R. Johnson).

Finally, we might stress here that in some cases a

calculation of the properties of the 'dressed' ion,

that is the single-center problem, may necessitate

transcending the Born approximation. Even then,

it may be possible still to build up the metal, to a good approximation, by superposing such 'dressed'

ions. The work of section 4 below bears on this argument.

4. Dielectric Constant and Vacancies in Cu and Al

Though we have indicated that the argument of

section 2 may be generalizable to include pseudopotentials, as presented in this paper it treats the

conduction electrons as though they were free, and

furthermore neglects core electrons. At this point, we shall therefore consider the

scattering of the Bloch waves representing the con- duction electrons by a vacancy, following the semiempirical method of Kohn and Vosko [14]. Since

this theory has been widely applied, we shall give

only a brief summary before presenting some new

results for vacancies in Cu and Al. The idea is that we can represent the scattering

of Bloch waves by a spherical defect potential V(r) in terms of only two phase shifts for the 5 and p

waves. If i/> k(r) is the perturbed wave function and

4>k(r) is the unperturbed Bloch function, then we may write

«// k

(r) = (/> k (r) + [/"(k, k')/r]c/> k

, (r) (4.1)

for the asymptotic form of \\i (r). Then we can expand /, assuming it depends only on |k| and the

scattering angle 9 between k and k' ^k' = |k' the

initial and final wave vectors, in the usual way as

/fc (0) =2^5(2/+l)[e2i^-l]P/(cos 0). (4.2)

If we write

i//k

(r) = f/k

(r)e' kr

, (4.3)

then the displaced charge Ap may be written

Ap(r)=- (~^){exp (2ikfr)\Ukf{r)\%fkf(7T)

+ complex conjugate (4.4)

The method hinges now on the fact that if we use

(a) the Friedel sum rule, and

(b) the measured residual resistivity,

then we can evaluate the 5 and p phase shifts, assuming the other phase shifts are negligibly small. We

discuss these phase shifts below, but to avoid the

complication of graphically representing (4.4), thus

requiring knowledge of Uklr), we have put C/*y(r)=l.

Roughly then, we can consider that we are getting

an envelope, or a spherical average, of the displaced

charge. To make closer contact with dielectric screening, we have then put the results in terms

of a 'screened potential,' using Poisson's equation.

We want to stress of course, that for some pur- poses (especially the interpretation of nuclear magnetic resonance intensity) the retention of Uk(r) in

(4.4) is crucial. But our main interest is to compare

results for different defect centers in the same

matrix, and for this the present representation

proves convenient.

We now turn to the results for the vacancies in Cu. Table 2 shows the phase shifts obtained for Table 2

Phase shifts for vacancies in Cu

1o 1i 12

Imicro-ohm-cm per atomic %)

1.0 -0.52 -0.35 0 Semiempirical 1.25 -.71 -.29 0

1.5 -.80 -.26 0

Theoretical Lee and March [16] 0.94 -.61 -.10 -0.03

Theoretical Jongenburger [17] 1.25 -.73 -.23 -.01

various choices of the vacancy resistivity Apt, per

atomic percent vacancies. The values of 172 calculated on theoretical models are very small, indicating

that the approximation of neglecting 172 in the semiempirical method is probably a good one. We show the form of the vacancy potential there- by obtained in figure 1. We have plotted results

for both kpv = 1.0 and 1.5, and for comparison, we

have plotted, with a sign change of course, the

results of Kohn and Vosko for Z =+ l. The pointion result is also shown for comparison. The overall agreement among the results is good, and

indicates that, to a useful first approximation we

can regard these results as reflecting the dielectric function of Cu metal, though some differences in phase are evident^from figure 1. We have also carried out calculations for Mg, Al,

and Pb, but only for Al was a result known to us for the vacancy resistivity at the time the calculations | were carried out. However, it seems clear that

| already the perturbation is such that the phase

shift 172 may be beginning to play a significant role. We plot the results for Al in figure 2, for a vacancy,

and for Zn, Mg, and Ge impurities. The curves, scaled with Z of course to be directly comparable,

show a scatter which is a good deal larger than for Cu. Presumably, the smallest perturbation (Mg) I

72
0 02

Screened

potentials

in av.

001

-001-

FlGURE 1. Asymptotic form of 'screened potentials' round

vacancy and positively charged impurity in Cu.

Curve (1) Vacancy Ap,.= 1.5 n ohm cm per atomic percent. Curve (2) Kohn and Vosko result for Z= 1. Curve (3) Vacancy Apr= 1.0 y. ohm cm. Curve (4) Point-ion shielding (cf. eqs (2.4) and (2.6)).

should best reflect the dielectric function of the Al

matrix. The situation for Pb is even worse and

indicated that we may have to use a somewhat

different approach in this case for a vacancy. The

perturbation appears strong. Other evidence that

this is indeed so seems to be contained in the

marked curvature with Z of the vacancy formation

energy, evident in table 1. It seemed of interest

finally to plot in figure 2 the ion-ion interaction

energy for liquid Al, derived by Johnson, Hutchinson, and March [15] from the liquid structure data. There is some agreement with the one-center

potentials, but the use of a common dielectric function for one and two-center problems seems rather

rough. This may be connected with the remarks

at the end of section 3. Nevertheless, we suggest that the considerations

of the present paper lend weight to the view that the vacancy, as well as impurities which do not

010-

0 08-

Screened

potentials

Figure 2. Asymptotic form of 'screened potentials' round

vacancy and various impurities in Al.

Curve (1) Vacancy in Al. Curve (2) Mg in Al (Z = - 1 ). Curve (3) Zn in Al (Z = - 1). Curve (4) Ge in Al (Z = + 1) Curve (5) Liquid metal ion-ion interaction.

strain the lattice too greatly, can be usefully de- scribed in terms of a dielectric constant characteristic of the Fermi gas. Whether the description can be made entirely quantitative may seem in some doubt, particularly for the polyvalent metals.

But Mukherjee's relation, on the other hand, sug- gests that with appropriate choice of pseudopotentials, a quantitative relation between forces characteristic of the perfect crystal and the vacancy

should emerge.

Thanks are due to Drs. J. E. Enderby and S. H.

Vosko, and to Messrs. S. Baranovsky and J. Worster

for valuable discussion and correspondence. One

of us (N.H.M.) wishes to acknowledge that his contribution to the present paper was partially

supported by the U.S. Atomic Energy Commission.

5. References

[1] H. B. Huntington, Phys. Rev. 61, 325 (1942).

[2] N. H. March and A. M. Murray, Proc. Roy. Soc. (London)

A256 , 400 (1960).

[3] F. G. Fumi, Phil. Mag. 46, 1007 (1955).

[4] K. Mukherjee, Phil. Mag. 12, 915 (1965).

[5] N. H. March, Phys. Letters 20, 231 (1966).

[6] J. E. Enderby and N. H. March. "Phase Stability in Metals and Alloys," Editors, Rudman, P. S.. Stringer. J.. Jaffe, R. I., McGraw Hill (New York), 1967. p. 479

[7] N. F. Mott, Proc. Camb. Phil. Soc. 32, 281 (1936); N. F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys (Clarendon Press, Oxford, 1936).

[3] A. D. Boardman and N. H. March. Proc. Kyoto Conference

on Crystal Lattice Defects, Vol. II, 80 (1962).

73

[9] J. M. Ziman. Adv. Phys. 13, 89 (1964).

[10) G. K. Corless and N. H. March, Phil. Mag. 6, 1285, 1961 and 7, 1765 (1962).

[11] J. Bardeen and D. Pines, Phys. Rev. 99, 1140 (1955).

[12| S. H. Vosko, R. Taylor, and G. H. Keech, Can. J. Phys. 43,

1187 (1965).

[13[ M. D. Johnson and N. H. March, Phys. Letters 3,313(1963).

[14] W. Kohn and S. H. Vosko, Phys. Rev. 1 19, 912 (1960J.

[15] M. D. Johnson, P. Hutchinson, and N. H. March, Proc. Roy.

Soc. (London) A282, 283 (1964).

[16] P. M. Lee and N. H. March, Phys. Rev. 118, 138 (1960).

[17] P. Jongenburger, Appl. Sci. Research 3B, 237 (1953).

256-724 0-67—

6

Calculations on Point Defects in Metals Allowing for Realistic Fermi

Surfaces and Scattering Mechanisms

A. Seeger and E. Mann

Institut fur Physik am Max-Planck-Institut fur Metallforschung, Stuttgart, Germany

1. Introduction

The majority of the calculations on point defects in metals have used either one of two ap- proaches [1]:

1 (i) The model calculation or (ii) the free electron theory calculation.

(i) In the model type of calculation a model for the interactions between the atoms or ions of

the metal is adopted, the characteristic parameters of which are usually fixed by using empirical

data such as elastic constants, force constants, cohesive energies, etc., and/or calculations on

individual ions, e.g., for determining the ion-ion interaction potentials. Typical properties of

point defects for the calculation of which this kind of model is usually employed, include the

configurations of the atoms surrounding a point defect, defect entropies, and the energies of

migration of point defects.

(ii) The electron theory calculations are used to calculate the "electronic properties" of point

defects [2], such as the electrical resistivity and the thermoelectric power. The majority of these

calculations have been carried out for free electrons, which has the advantage of simplicity and

generality. Calculations based on pseudopotentials, which have recently become fashionable,

will be discussed separately (sec. 14). Up to now these two approaches have remained rather unrelated to each other. It is true that

in the more advanced theoretical treatments of some point defect properties, e.g., energies of

formation and binding, both types of theories have been combined, but even here they have re- mained essentially unrelated, the results of the different calculations simply being superimposed

on each other. From the point of view of a theoretician who seeks a fundamental understanding of the properties of point defects in metals, it is clearly an unsatisfactory state of affairs that quite different

approaches have to be used to treat different properties of point defects. The use of simple

models and the restriction to easy-to-carry-out free electron theory calculations was a necessity

when the computational work had to be done on small electronic computors or even on desk

machines. With the present capacity of large electronic computors available, one should be able

to do better and to make some progress towards a unified approach to the problem.

A number of years ago we have started a program of research, which, we hope, will finally

unite the two theoretical methods mentioned above and permit reliable calculations of point

defects properties in specific metals to be carried out. It appears still to be outside the realm

of present possibilities to base such an approach entirely on first principles. What does seem

possible, however, is to develop a quantum-mechanical framework which enables us to incorporate

in an unambiguous manner the kind of empirical information which at present is used in the

model calculations.

Such an approach will of necessity have to be developed along similar lines as the theory of

perfect metal crystals. This means that at one stage we shall have to arrive at a one-electron

Hamiltonian for the perturbed crystal, and that from this the one-electron wave functions and the

one-electron energies (or rather their variations caused by the presence of the defects) will have

' Figures in brackets indicate the literature references at the end of this paper.

75
to be calculated. In order to be able to account for the individualities of the various metals, this stage must not be based on the free electron picture of a metal but on realistic electron energy

surfaces and wave functions. The calculation of changes in wave functions, electron energies,

and other electronic properties due to arbitrary localized perturbations for (within reasonable

limits) general energy surfaces and wave functions of the perfect metal is central to the whole

problem. Most of our efforts have gone into this question. The present account is mainly a summary on the results so far achieved in this field, an early progress report having been given a number of years ago [3]. The next stage will have to be the numerical evaluation of the general expressions for individual

metals. This requires a detailed knowledge of the energy surfaces and the wave functions of the

perfect crystals. Due to the experimental studies of the Fermi surfaces and the large theoretical

efforts in band calculations, the present information on the electron energies is satisfactory for quite a few metals. Unfortunately, this cannot be said of our knowledge of the wave functions.

It is hoped that the calculation of wave functions will see enough progress within the next few

years to provide the background information necessary for the defect calculations.

The final step, leading to the merger of the approaches (i) and (ii), will be the numerical deter- mination of the one-electron Hamiltonian of the perturbed crystal. The potential acting on the

electrons will, among other things, depend on the configuration of the atoms surrounding the point

defect. This configuration, however, depends on the electron redistribution caused by the point

defect. Clearly, a problem of self-consistency arises here, and the many-body aspects of the

problem must be included. At this stage it may well be found expedient to determine some of

the parameters of the problem empirically or semiempirically. Some work on these questions

has already been carried out, but in the interest of the coherency of the presentation we shall not

report on it here. In sec. 14, we shall comment, however, on one proposal made in the literature

in order to solve this self-consistency problem for metals, the method of "pseudoatoms" [4].

2. Basic Formulas

In a perfect crystal, the one-electron energies £ n(k) are periodic functions of the wave vector

k. They may therefore be written as a Fourier series ( n = band index) [5]

£,( (k)=2 e »( Rs)<r

,k' R!

' (2.D

R,

where the summation extends over the Bravais lattice of the crystal. The Fourier coefficients may be expressed as matrix elements between the Wannier functions an (r — R;) (a* = complex

conjugate of a)

e„(Rs) = j a*(r-Rs )H0(r)an (r)dr. (2.2)

The Wannier functions a n (r— R,) are localized around the lattice site R; and are related to the

Bloch functions ^„jk (r) by the equations

a,i (r-Ri ) = ^=y e- !kRV (r); <p M, Ue^a^-^ (2.3)

The Bloch functions satisfy the one-electron Schroedinger equation

H0 (r)<p (r)=En(k)<p (r), (2.4)

n, k n , k

76
where Ho is the one-electron Hamiltonian operator of the ideal crystal.

The Schroedinger equation of the perturbed crystal is written in the form

(H0 +V(r))iP(r) = Eijj(r), (2.5)

where the "perturbation" V(r) may be an "ordinary potential" or an operator. It enters into the

following expressions usually through its matrix elements

Vnm(Ri, Rj) =

j

<(r-Ri)F(r)am(r-R,-)dr. (2.6)

The solution t//(r) of the Schroedinger eq (2.5) of the perturbed crystal is expanded in terms of

Wannier functions:

<//(r)= 2 Un (Ri)an (r-Ri). (2.7)

n. It

The coefficients C/„(R,) satisfy the equation [6]

Un(Ri)- £ Gn ,E(Ri-Ri)Vnm(Ri, Rj)£/m (Rj) = 0, (2.8)

m, R„ R;

where

G».,(R,-R,)=i2|;^ (2-9)

may be denoted as Green's function of a crystal with periodic boundary conditions.

The condition for the solubility of the system of homogeneous linear equations (2.8) is the

vanishing of the determinant

D(E) = |5,„„8 0 -2 Gn , E (Ri-Ri)Vnm(Ru Rj)\=0. (2.10)

R,

Equation (2.10) is the equation from which the (discrete) energy values E = Ek of the perturbed

problem are to be determined.

3. Change in Energy

The energy change associated with an energy value E„(k) = Ek of the ideal crystal is given by

(Ek — Ek), where Ek reduces to Ek when the perturbation V is "switched off." In a metal (with

compensated spins) the total change in one-electron energies is given by (Im = imaginary part)

A£ = 2 j±(Ek -Ek) =

^ J' arc D(E)dE =

^ JSm log D(E)dE, (3.1)

where the summation extends over all eigenvalues up to the Fermi surface, which is characterized

by the Fermi energy £. In eq (3.1) the transition to a continuous distribution of energies has been

made. This has the consequence that the Green's function (2.9) has to be replaced by the Green's

function for a continuous energy spectrum (Vb = volume of the Brillouin zone)

1 f e'

k R G„(E, R)= lim jr -=— dk. (3.2)

e -o ' b J E — E„(k) + ie

77
D(E) is defined by eq (2.10) with G„. £ (R) replaced by G„{E, R). Bound states (outside the bands of

continuous energy values) are included in eq (2.10), but not in the last two expressions of eq (3.1). They have to be added separately in order to obtain AE.

Equation (3.1) may either be derived by a heuristic argument [7], or by a technique based on

the theory of functions that was first employed by Wentzel [8] in his meson pair theory. The

application of this technique to electrons in metals [7] differs from that in the theory of lattice vibrations [9] in that in the metal case the integration in eq (3.1) ends at the Fermi energy, whereas

in the lattice vibration case one always deals with "full bands."

4. Symmetry Considerations

We call a lattice site R, "affected by the perturbation F(r)," if F,im (R,, Rj) is different from zero

for any Rj. The order of the determinant D in eq (3.1) is equal to the number of energy bands to be taken into account, times the number of sites affected by the perturbation. The latter number

will be the smaller the more localized the perturbation is. The method is therefore suitable for localized perturbations, e.g., for the treatment of point defects. Nevertheless, the order of D may

be relatively large even in simple problems. Consider as an example a vacant site in an f.c.c. crystal and assume that in addition to the vacant site (R, = 0) only the twelve nearest neighbor sites are affected. When only one band is taken into account, the order of D is 13. Fortunately, the secular problem eq (2.7) and (2.10) may be considerably simplified and re- duced to a set of smaller problems, if the point symmetry of the problem is taken into account.

In particular, the determinant D may be represented as a product in the form

D = \\(D^)."» . (4.1)

Here Z) Mis the secular determinant associated with the irreducible representation fx. The superscript mM denotes the multiplicity with which the irreducible representation fx occurs in the problem

and is equal to the order of D^. The multiple occurrence of the irreducible representation /a may

either be associated with shells containing a relatively large number of sites, or with different

shells. The quantity mM depends therefore on the number of shells affected by the perturbation.

The exponent n M is equal to the dimension of the irreducible representation /x. For a problem

involving a fixed number of sites, the order of the determinant D of the total problem is given by

In the example considered at the beginning of the section, eq (4.1) takes the form (employing

the usual notation of the cubic point group 0 h [10]) [7]

D = D\x (D\-jHD^ )3(Z)fis )3(D},

5 )3. (4.2)

The symmetry eigenvectors of the problem are easily obtained from the basis functions of the

irreducible representations.

5. Phase Constants

Defining generalized phase constants a^E) associated with the irreducible representations

ixby

aJM) = ^ arc D^E), (5.1)

78
we may write the total energy change eq (3.1) as

A£ = 2 j"a(E)<lE. (5.2)

where

a(E)=Y « Ma M (£) = -arc D(E) . (5.3)

77

Equations (5.2) and (5.3) are analogous to the expression of the energy change of . a gas of free electrons perturbed by a potential of spherical symmetry [11, 12]

AE iTM = --f(2l+l)l\l (E)dE (5.4)

in terms of the phase shifts i)i of the partial waves. This demonstrates that the generalized phase

constants are indeed generalizations of the quantities —- t}i(E).

TT Since in problems of point symmetry in crystals the number of irreducible representations is limited, the present formulation of the phase-shift problem comprises only a finite number of terms

[12a. 12b], in contrast to the free electron case, where on account of the higher symmetry the

number of phase shifts is infinite.

6. Scattering Problems

In section 5 we have introduced the generalized phase constants aM by considering the effect of the perturbation on the spectrum of discrete energy eigenvalues. They may also be introduced

through the scattering problem, i.e., in a formulation based on a continuous spectrum of energy

eigenvalues.

In such a formulation one considers a fixed energy value £\,(k) and writes the total wave

function

«Kk(r)=2 2 £/m

k (RJ )am(r-Rj ) (6.1)

as a sum of the unperturbed function <p n , k (r) (see eq 2.4) and the so-called scattered wave. Subdividing the expansion coefficients Um k (Rj) in the same way gives us

U* \Rj ) = e

,kRj Snm + fC k(Rj )• (6.2)

By a procedure analogous to that of section 2, the following equation for Um k (Rj) is obtained [13, 14]:

U ",; k (R( ) = Vg 1 /2c'kR

'Snm +

2) X G^XRt-RdVnn+RuRjWbHRj). (6.3)

'"' R,.R/

Here

. , iV. -(R/-R))

CfnJk,(R, - Ri)= lim - -

c

. dk' (6.4) e-o VB J E»(k) — Em (k )+l€

79
is the Green's function of the scattering problem.

In matrix notation eq (6.3) may be written as

18 k=e k -t-G £ Fu k . (6.5)

The solution of eq (6.5) is

u k =(M£ )-ie

k

, (6.6a)

where

ME =l-GE V. (6.6b)

The determinant of ME is equal to the determinant D introduced in section 3, i.e.,

det ME = D(E). (6.7)

Inserting eq (6.6) into the right-hand side of eq (6.5) gives us

u k=e k+G*r*ek , (6.8a)

where

TE =V(ME )~ l (6.8b)

is the so-called transition matrix.

The preceding treatment is exact. From now on we consider the asymptotic behavior of the

scattered wave function. Provided R; lies sufficiently far outside the region affected by the

perturbing potential, we may write the expansion coefficients

£C

k (R; ) ^VE ^(e'

kR'8nm + ^S:;t, (6.9)

where

S

nm

kK=gm , K

A"m

kK (6.10)

is the scattering amplitude.

The coefficients [15]

_47r2 e-« | gm,K~— |V£m(k£)| \Km(K)\ m (6 ' U)

are entirely determined by the energy functions Em (h.) of those energy bands that contain the

energy E under consideration; the contributions of the other bands are negligible in the asymptotic

region.In eq (6.9) to (6.11) the wave vectors k|, are defined by (see figure 1, with r replaced by R<)

Em (K) = E and VEm{K) parallel to Rj. (6.12)

80
k

FIGURE 1. Relation between an incident Block wave {wave vector k) and the waves (wave vectors ) scattered into the

direction r: S; and s xc denote the directions of the incident and the scattered particle fluxes.

The subscript v enumerates the wave vectors that fulfill eq (6.12). Xm (k'„) is the Gaussian curvature of the energy surface £OT (k) at the point U.'„. \m(ls.'v ) is 1 if the curvature at k^, is elliptic concave when looking in the direction of R/, — 1 if the curvature is elliptic convex, and 0 if the curvature is hyperbolic.

As we shall see below,

A

nm\: =V-B

' ^e-'XR, T&n (Ru Rj)e**> (6.13)

Ri, Rj

has the physical meaning of the transition amplitude from a state with wave vector k in the nth

band to a state with wave vector k[, in the mth band. In eq (6.13) T^fR;, Rj) denotes an element

of the matrix TE introduced in eq (6.8b). As shown in detail elsewhere [16J. the transition amplitude may be written as (for a similar

expression, however without introducing phase shifts, see [16a]) = 2 Qm. k. Msin (6.14)

Qm, k;, M

is a quantity which, in addition to the functional dependencies indicated, depends also on

the matrix V of the perturbing potential. For free electrons and spherically symmetric potentials

the transition amplitude for the transition from k to k' reads

A

(Cw=

~*

S (2l+l)Pi{cw0isinvie% (6.15)

7T{27T)3 n(E) ifn

where n(E) denotes the density of states, 8 the angle between k and k', and Pi, as usual, the

Legendre's polynomial of degree /. The r)t are the phase shifts already introduced in section 5. As in section 5, we recognize again the close analogy between the a^E) and the quantities —- r)i(E).

TT By inserting eqs (6.10) and (6.14) into the expression (6.9), we see that the contribution of the

scattered wave to the coefficients U"m

k(Ri) is made up of a linear combination of terms of the form

exp R/ — rra^) \/Ri. This shows that the aM have indeed the physical meaning of phase shifts of partial waves.

HI
7. Neutrality Condition

From the exact solution of the scattering problem given at the beginning of section 6 we may

calculate the change of the electron density per unit energy

Ap(r;£) = 2]rf ([«M(r)f»-

V

2 k. (7.1)

n JE = const

where the integration extends over an energy shell in k-space and the summation includes all bands containing E. By integrating eq (7.1) over all space, one obtains the change in the number

of electrons per unit energy, i.e., twice the change, uiE), of the density of states. Carrying out these integrations [14] gives

aiE) = —— Im trace (GETE GE

), (7.2)

7T

which can be transformed into

o{E) = —j^ f- Im log det M£ I. (7.3) dE \7T

By virtue of eq (6.7) and the relations given in section 3, eq (7.3) is equivalent to

<T(E) = —^a(E). (7.4)

Equation (7.4) establishes the relation between the change in the density of states cr(E), defined for a continuous energy spectrum, and the relative energy change a(E) in the picture of discrete

energies.

Going back to the original meaning of 2a(E) as a change in the electron density per unit energy,

Sbands = 2

J (T(E)dE (7.5)

gives the total change in the number of electrons within the bands caused by the perturbation.

The change in the electron density decreases sufficiently rapidly to zero at large distances (sec. 8) for Sbands to be equal to the number of electrons introduced within the bands in the immediate

environment of the perturbation. An additional number s bound of electrons may be localized in bound states at the perturbation. The sum s = s bands + s bound has therefore the meaning of the

screening charge Z (in elementary charge units) of the perturbation.

It can be shown that fully occupied bands give a zero contribution to 5. It suffices therefore

to extend the integration in eq (7.5) upwards from the minimum energy Emin of the bands containing the Fermi energy, provided bound states split off at the bottom of these bands are properly

allowed for. Integrating the right-hand side of eq (7.4) gives us

Sbands = -2[aU)-a(£min)]. (7.6)

In the one-electron picture, in which all bound states below Emin are occupied by two electrons,

the relation

5 bound = -2a(£'min ) (7.7)

82
holds [14]. We have finally

2a(0, (7.8)

which is one of the formulations of Friedel's condition for the neutrality of a perturbation in a metal.

The present derivation of eq (7.8) has the advantage over the usual derivations that the supposition of the conservation of the Fermi energy is not required.

8. Asymptotic Charge Density

By inserting the asymptotic expressions fort//

n k

(r) into eq (7.1) and integrating over E, one

obtains for the change in the electron density at large distances r from the perturbation (Re = real

part)

gm,W,gm\KA >n,K ' <Pm- K ( r ) V1"' <K ( r) Ap(r) Re

7T 2

i i r

3 |V£m (k;)| \ve„ak)\

III', v'

(8.1) E = i

where the U.[ are to be taken at the Fermi surface. Equation (8.1) was first derived by Blandin [17], although in a way different from that outlined here. Equation (8.1) exhibits a characteristic vari- ation of Ap as a function of r, containing the product of r~ a times sinusoidal terms. This variation

is brought out more clearly by averaging the additional charge density over the extension of the

Wannier functions. The result is [14]

, m, — k'.

<> i ir - A ,, -((k' + k'.) • R," Ap(R,)

7T(277) : Re I Rf ^ \VEm(K)\ |V£m (k;,.)|

(8.2) E=i

As the simplest special case of eq (8.2) we consider one band only and confine the perturbation to R, = Rj = 0. This means that we have only one nonvanishing matrix element Voo= Vnn(0, 0). Making use of the relation (6.14) between the transition amplitudes and the generalized phase constants a^iE), we obtain

8K #k

7r-(27r) ,, n(4) 2

i +i

giUK+k',,) R,-i7a,,(£))

S7E(K)\ |V£(kV)|

(8.3)

In the special case considered here we have

«(£) = «,,(£), (8.4)

so that for a substitutional atom of valency two in a monovalent metal the neutrality condition

takes the form

(8.5)

giving a phase shift in eq (8.3) of

Figures 2 to 4 show the results of a numerical evaluation of the asymptotic average charge

density according to eq (8.3). The parameters describing the shape of the energy surfaces have

83
been calculated by fitting the Fermi surface of Cu as determined by Roaf [18J to eq (2.1), taking

into account one band only. Figures 2 and 3 give the lines of constant Ap in the {100}- and {110 }

planes. One can see a very pronounced anisotropy of Ap. Figure 4 gives polar plots of Ap in the {100}- and {110}-planes for two different radii |R/a|= const.

[001]

FIGURE 2. Lines of constant electron density Ap in a {100}- FIGURE 3. Lines of constant electron density Ap in a {110}-

plane, calculated according to eq (8.3) for a substitutional plane ( for details cf. fig. 2). atom of valency two in copper. The figure near the origin gives the {110}-section through the BriUouin zone and the R denotes the lattice vector, a is the lattice parameter. The figure near the origin Fermi surface, shows a { 100 }-section through the first Brillouin zone and the Fermi surface.

[wo] 0'0]

FIGURE 4. Polar plots of the change in electron density Ap in the {\00}-and {110}-p/anes for two different radii K = const. The radii of the plots give the additional electron density Ap in units 10-s electrons per atomic volume.

For directions R; that according to figure 1 are associated with points k',,on the Fermi surface

with K(k'v),=0 eqs (8.1) to (8.3) lead to an infinity in Ap (R,) (comp. eq 6.11). In such situations

these equations are no longer applicable. A more detailed investigation shows that such points

of parabolic curvature give rise to an asymptotic contribution to the scattered wave function (6.9)

84
whose amplitude decreases with distance more slowly than —. If in a point K(h'„) = 0 only one

ft; of the radii of curvature is infinite, and if the next higher derivative of the surface associated with

that curvature is finite, the amplitude varies as i?~ 5/fi

. Even slower variations are possible, if the

preceding conditions are violated. This means that in the special directions indicated above the

amplitude of Ap (R/) decreases more slowly than Rj~ 3- At sufficiently large distances from the

defects the average charge density Ap (R;) will be larger for directions associated with K(U.[,) = 0

than for others.

If we had not carried out the ^-integration mentioned at the beginning of the section, we would

have obtained for the change of the average electron density per unit energy, instead of eq (8.3), r,D r\ 0

sin 7mi,(£) T Ap (R/: E) =^—2 Im

i>, v k,. k„<

'{(k|,+ k;,)- R|-7rai,<£)} (8.6)

Characteristic of (8.6) and all the other asymptotic expressions associated with Ap (r: E) is that it varies with r as r~2 times oscillating terms, i.e., that it descreases more slowly than the total electron

density.

9. Interactions Between Defects

The expressions of sections 3 and 7 remain valid if the potential matrix elements F„,„(R( , Rj)

describe several defects separated in space. It is possible to obtain from them the electronic

interaction energy AE, between the defects as a function of their relative positions. We give the

results for two equal defects characterized by the matrix elements

Vnn (0, 0) = V,M. R/) = F00 , (9.1)

assuming the separation R; between the defects to be large enough to allow the application of the

asymptotic formulas.

The one-electron interaction energy is defined as

A£, =- Pare D>(E)dE-2- Tare D1 {E)dE,

7T J IT J (9.2)

/here

D1 (E) = l-G(E, Q)V'M

describes an isolated defect, whereas

DAE) =

1-G(£,0) Ko — G(E, R,)K0

-G(E, R,)K0 l-G(E, o)v;0

(9.3b)

is the determinant in the presence of both defects. The values V[m and FJ,',, of the matrix elements

Too are to be determined from the neutrality conditions, eqs (5.3) and (7.8),

| arc D,U) =2a,,(£)=-Z, (9.4a)

| arc D->(0 =-2Z. (9.4b)

85
For large distances R7 one obtains

Ko « ^. + 7/77777 I" [C2 U, B,)r*«r.«>] (9.5)

wiith

1 p iK R< (9.6)

The one-electron interaction energy becomes

2 H sin2 Tra\\{E)

AEi7T3 (27T) 6

f

£ sin2 (E)

dE Ii 2 e f

,i{(K+K) • R,- 2TM, (£)}-i Rf E = i (9.7)

and shows that the dependence on the separation is (apart from a different phase shift) that of Ap (R/; £") and not that of the total charge density Ap (R/). This at first sight surprising result is directly related to the fact that the charge neutrality condition has been applied to both defects

taken together and not to each defect individually. It means that the electronic interaction

between two point defects in metals is rather far-reaching and should definitely be taken into account in considerations of long-range interactions between defects. With regard to the dependence of the interaction energy on the direction of the line joining

the defects, we have to expect a similarly strong anisotropy as has been discussed in section 8

for the asymptotic change in the charge density.

10. Extended Defects

We have seen that the change in electron density associated with a localized perturbation

extends rather far out. The effect of a point defect on the self-consistent potential of a metal

will therefore also extend rather far out from the site of the defect. Elastic strains associated

with the defects have a similar effect, too. However, the perturbing potential at the more distant

sites will in general be small enough to be treated as a small perturbation. This suggests the fol- lowing approach:

The large matrix elements Vnm(Ri, Rj) for a limited number of sites in the immediate en- vironment of the defect are taken into account exactly by the methods outlined in the preceding

sections. The remaining small matrix elements are allowed for by a small perturbation of that exact solution, using the standard techniques of perturbation theory.

We shall not discuss this approach in detail here. As an example, its application to the calculation of the energy may be found in reference [7].

11. Transition Probability

In order to treat the effect of a perturbing potential on transport properties, the transition

probability W(k, k') for the transition from state k to state k' is required (see sec. 12 ). (From now on we shall confine ourselves to one band — the conduction band — and drop all band indices.)

W{h, k') is related to the ratio of the scattered particle flux (built up from Bloch waves with wave

vector k') to the particle flux of the incident Bloch waves of wave vector k [19] (compare fig. 1). When calculating the particle fluxes from the wave function i^ k

(r), it suffices to consider the asymptotic solution. This brings the scattering amplitudes S w into the expression for W(\t, k'). When

the geometry of the Fermi surface is taken into account properly, it is seen that the final expression

does not contain the scattering amplitudes but the quantities A £ (defined by eq (6.13)), which

86
for non-spherical energy surfaces have a k'-dependence different from that of S^ . The formula

for the transition probability is [16] (Vc = volume of the crystal)

r(k, k') =^-\AtmE(k) -E(k')), (11.1)

justifying the name "transition amplitude" for A

k

. The 8-function results from the fact that during the scattering the energy is preserved. It is convenient to introduce the function

w(K V)={-^-\A kw\\ dl-2)

which is related to the transition probability by

F(k, k') =w(k, k')S(£(k) - £(k')). (11.3)

From the definition of A^ (eq (6.13)) we see that w (k, k') may be written in the form

(k, k') = £Sa(k)Mk'), (11.4)

i.e., as a linear combination of products of functions u«(k), depending on k, and vJU.'), depending

on k'. The number m of terms in this decomposition is equal to the square of the number of sites affected by the perturbation.

12. Electrical Resistivity and Thermoelectric Power [16]

We shall base the treatment of the effects of localized perturbations on the electrical resistivity and the thermoelectric power on the Boltzmann equation

(D +(!) =0. (12.1) \at/ fi e ids V"/scattering

The rate of change of the distribution function of the electrons (occupation probability of the

k-states) /(k, r) due to fields is given by

(f) =~l ^Eib) Vr /(k, r) ~l F • Vk /(k, r), (12.2)

V**/ fields " "

where ~h is Planck's constant and F = — eE is the force exerted on an electron by the electrical

field E. The rate of change of the distribution function due to the scattering of the electrons may

be written as

(%) =7^ |V(k'> k)/(k')[l-/(k)]-^(k, k')/(k)[l-/(k')]}rfk'. (12.3) W /scattering (Z7T) J J

where W(lt, k') is the transition probability introduced in section 11. If only the scattering by

static imperfections is taken into account (very low temperatures), or if the temperature is higher

than the Debye temperature, the scattering is to a very good approximation elastic. In this case,

87
to which we shall confine ourselves in the present paper, the transition probability may be written

in the form of eq (11.3). It is then convenient to write the distribution function as

/(k, r)=/o(k)-H F*- X(k), (12.4)

where

/o(k)=/o(£(k))

1

e

(£(k)-i)/Ar_j_ i

is the Fermi distribution function in the absence of external fields and

(12.5)

F* = F-^-r^ VrT(r). (12.6)

Vr T(r) is the temperature gradient. Linearizing the Boltzmann equation in F* leads to the fol- lowing integral equation (inhomogeneous Fredholm equation of the 2d kind) for the unknown vector

function X(k):

X(k) |

r(k, k')dk'-j r(k, k')X(k')^k' = -^|^ \ (12.7a)

or, employing eq (11.3),

X(k)- f K(k, k')X(k')d2 k' = g(k)

JE'=E(k)

(12.7b)

with d

2 k' = dSk ./| V£(k')| (dS k = surface element).

Here

g(k) =

(2t7) 3 V£(k)

' u»(k, k')d2 k'

(12.8a)

#(k, k')

w(k, k')

L

u;(k, k')d2 k'

(12.8b)

The electrical conductivity tensor g and the tensor of the absolute thermoelectric power S are given in terms of the solution X(k) of eq (12.7) as

<Z = e

2 A(£),

s--

3e

A-! (£) ME)

dE ~

E=i

7T2k

2 T

3e dE —

(12.9)

(12.10)

where

=(£)= 7TT^ I V£(k)X(k)rf2 k

(12.11)
is a second rank tensor. The integral equation (12.7b) may be solved simply in the two limiting cases already mentioned,

namely either at very low temperatures, where the scattering is caused by the crystal defects only, or at temperatures above the Debye temperature, where the scattering by the static imperfections

is small compared with that due to thermal vibrations and where the latter may to a sufficient approximation be described by a relaxation time, Ttn . In the low-temperature case B^(k, k') and u>(k, k') are given by the equations of sec. 11. The

kernel of the integral equation (12.7b) may therefore be written in the form

HI K(k,k') = Ju,(kK(k'), (12.12)

i.e., it is degenerate. In such a case, it is well known [20, 21] that the solution of eq (12.7b) may

be expressed as

X(k) = g(k)+2Mk)Y„, (12-13)

where

Y, = j^„(k')X(k')d2 k' (12.14)

has to satisfy the set of linear inhomogeneous equations

Y„-]£ [

i;„(k')uM (k')d2 k' YM =[ ^,(k')g(k')d2 k'; v = \ . . . m. (12.15)

^ JE JE

In the preceding equations, the integrations have to be carried out over surfaces of constant energy

in k-space, as indicated by the suffix E. The order of the system (12.15) is equal to the square of the number of sites affected by the

perturbation V(r). In many cases of practical interest the problem of solving this equation can

be reduced to smaller problems by use of symmetry considerations. The fact that the linearized

Boltzmann equation may be solved exactly is a substantial advantage of the present formulation

of the scattering problem. In marked contrast to this, the usual formulation of the scattering of

free electrons by anisotropic defects does in general not permit the exact solution of the transport

problem and necessitates the use of approximations or of the variational principle [22]. In the other limiting case, that of high temperatures, the scattering term (12.3) of the Boltz- mann equation is rewritten in the form

m =(|) .12.16,

\dt/ scattering V" /defects Tth

where (df/dt) defects is given by (df/dt) scattering °f the low-temperature case just discussed. Intro- ducing again the vector function X(k) by eq (12.4), the integral equation

I V£(k)- —X(k)+-p- |V(k,k')(X(k')-X(k))<*k' = 0 (12.17) n Ttn (27r)'! J

is obtained. Since the second term of eq (12.17) is much larger than the third, the solution of eq

(12.17) may be written as

X(k)=5! V£(k)+X,(k), (12.18) n 89

256-724 0-67—

7
where Xi(k) is small enough to be treated by perturbation theory. This leads to X,(k)=^^^ £«,(k, k')(V£(k')-V£(k))^k'.

Inserting X(k) into eqs (12.9) to (12.11) gives us

<x = ojh + Acr,

2e2 Tth (2tt) 3

ft2 J£=4

V£(k)V£(k)d2 k,

(12.19)

(12.20)

(12.21)

Ao== (2tt) 3

ft2 (2ir) E=E'=i w(k, k')[V£(k)V£(k')-V£(k)V£(k)]d2 kd2 k'. (12.22)

It is convenient to express the effect of the lattice defects on the transport properties in terms

of the change Ap of the electrical resistivity p and the change AS of the absolute thermoelectric

power S^. Making use of Ao- <(Tth , one obtains with pt h — o^1

Ap = p-pth =

-fftl/^th1

-

The general expression for the change in the thermoelectric power is

AS=-

2 k2 T

3e

(ojh + Ao-) - 1 — ( o:th + Aa) - o-j-:

1 — ath dE v =in

' th dE=l

or, making use of the tensor generalization of an expression first given by Friedel [23], W [\, / , d

(12.23)

(12.24)

E=i

(12.25)

In the high-temperature case, the preceding exact expressions may be replaced by

ir2k

2 T

7T2A:2 r

Ao;- 1 — Aoj — dE — A

dE

3e (^ A

£)

A£

_, -(^£th )£'

(12.26)

(12.27)

The last expression for A!5 shows that at high temperatures the thermoelectric power is (as is the

electrical resistivity) proportional to the concentration of defects. (This does not hold for low temperatures.) As long as the high-temperature resistivity is proportional to the absolute temperature

T, the high-temperature value of AS_ due to the defects is independent of temperature. At low

temperatures the electronic contribution to AS^is proportional to T. Table 1 gives numerical examples for the electrical resistivity and the change in thermoelectric power for vacancies and divacancies in the noble metals as computed by Thierer [24] from the

preceding formulas. They are based on the energy surfaces of Roaf [18] and on the assumptions

that only the following matrix elements are different from zero:

(a) Single vacancy: Voo, Vu=V(Rn, R»)=j5 ^oo

90
(Rn = nearest neighbors of the vacant site)

(b) Divacancy: V00 =V(RS , Rs ), Vn = V(Rn , R»)=^ V00

(Rs = vacant sites,

R„ = nearest neighbors of both vacant sites).

The absolute magnitude of the matrix elements was determined by using the neutrality condition

(sec. 7 ). In the results on the electrical resistivity the energy function E(U) enters only through

the shape of the Fermi surface, which is well known from experiments. For the thermoelectric

power, however, the magnitude of Vis(k) at the Fermi surface must be known. It has been fixed by using a one-band expression, eq (2.1), and giving to the Fermi energy £ its free electron value. For comparison with experiments, the results of Huebener on the effect of vacancies on the thermoelectric power of gold [25] are given in parentheses.

Table 1. Electrical resistivity Ap and change in thermoelectric power AS at low (1) and high (h) temperatures for monovacancies (lv) and divacancies (2v) in noble metals {preliminary results) The values fur divacancies are averaged over all direclions. The differences beiween Ap' and Ap* are negligibly small. T= absolute temperature. exp= experi- mental values.

Cu Ag Au

Apjv'fp-n cm/at. %] Apy/Apy

AS(,,[io-v''7n<)2

]

ASi,.[10-V^/(°K) 2

] ASf,.[p.*7°K at.%] ASyAS{„

1.47

1.78 -0.28 T

-0,30 T

-0,98

1,86

1,73

1,75 - 0,36 T

-0,38 T

-1,31

1,89

1,80

1,82 -0,53 T

(exp -0,5 T) -0,56 T

-1,47

(exp - 1,67) 1,85

13. Magnetoresistivity

As a tool for studying the anisotropy of Fermi surfaces, the magnetoresistivity at low temperatures has proven very important [26-29], The scattering theory outlined in the present paper

allows us to treat not only the anisotropy of the energy surfaces, but also the anisotropy of the

scattering mechanism in an exact way [30]. As usual, we base our treatment on the Lorentz-force formula and the Boltzmann equation.

The wave-number space is described by the coordinates e, kz and (f>, where e(k) is the energy,

kz the component of the wave vector parallel to the magnetic field H, and

ch fdk

<P = c„-j- (13.1)

a phase variable measuring the position of a point in k-space in its orbit in a plane kz = const.

[31, 32]. The quantity v± is the component of the electron velocity v=^Ve(k) perpendicular

to H, and dk is a path element in k-space. The cyclotron frequency wh is given by the integral over a closed orbit

2tt cH fdk c ^dA(e, k,)

o)h eti J v± eti de

where A(e, kz ) is the area in k-space enclosed by the orbit.

91
Writing the distribution function as in eq (12.4), the Boltzmann equation for elastic scattering

reads to first order in X(k) [31]

0>HTT-v = 77^77, I JP(k, k')X(k')dk'-7^X(k) I r(k,k')dk'.

d(j) (2tt) 3 (13.3)

Inserting the expressions of section 11 for the transition probability and evaluating the integrals over k' gives us for e= £

co„ 3X(

f;

y-v=ftt^; &,)Y,,-X(«fr ft*) V ft*);*,, (13.4)

Here

(27r) a j e = 4 v*(k')d2k'; Y„= X(k>„(k')d%\

where the integrals extend over the Fermi surface e= £. If we introduce the function

(13.5)

uA4>'; kz )d<j)', (13.6)

we may write the solution of eqs (13.3) and (13.4) as

exp [— F(<ft; ftg)] X(0; ft,

r 4>" = <b Y m _ —

J <*" = = L „ exp [F(^";ftz )W- (13.7)

The constants Y„ are obtained by solving the following system of linear equations:

YA=

J

v

k

e) exp [-F<«M«)]{J § ((/)"; A:,)Y,+ v exp [F(^";^w}rf2 k; (13.8)

/u.= 1 . . . m.

Inserting the solutions of eq (13.8) into eq (13.7) completes the solution of the Boltzmann equation.

The tensors of the electrical conductivity and the thermoelectric power are obtained by inserting

this solution into the expressions (12.9) to (12.11).

These solutions are valid for all magnetic fields below the quantum oscillation region. Fol- lowing the procedure of Mertsching [32], the quantum oscillations may also be included in the treatment by allowing for the density of states of the energy levels quantized in the magnetic field. From eqs (13.7) and (12.9) to (12.11) the magnetoresistivity, the Hall effect, and the analogous

thermoelectric effects at high and intermediate fields can be obtained in a convenient form by

expanding the exponential functions in powers of (aj,1

.

14. Pseudopotentials and Pseudoatoms

In recent years the theory of pseudopotentials has been repeatedly applied to the calculation

of defect properties in metals [4,33]. The basic idea of the pseudopotential approach, viz., sup- planting the condition of orthogonality of a state of high principal quantum number to the states of lower quantum numbers by a short-range repulsive "pseudopotential," is a fairly old one [34, 35]. Its first application to the theory of crystals appears to have been made by Fues and Statz [36]. Recently, the idea has been extended to the concept of "pseudoatoms" [37]. A pseudoatom is an

ion in a metal plus its screening charge; on account of the large spatial extension of the screening

92
charge in a metal (see sec. 8) it is considerably more extended than a free atom.

In applying the pseudoatom idea to defect calculations, the assumption is made that not only

the perfect metal but also a perturbed metal may be built up from pseudoatoms carrying their

screening charges with them while they are displaced to off —lattice-sites positions. This implies

that the pseudopotentials associated with each ion superimpose without perturbing each other,

which in turn means that they must be well localized within each atomic cell. In actually solving

the Schrddinger equation of the perturbed crystal, the further assumption is made that the pseudopotentials are so weak that they may be treated by second-order perturbation theory, starting from

plane waves. Although this additional assumption is not inherent in the physical concept of

pseudoatoms, it has so far been a practical necessity in the applications of the pseudopotential

method to defects in metals, since the pseudopotential is usually formulated in terms of matrix

elements between plane waves.

Let us first examine whether the pseudopotential approach as outlined above, based on the

superposition principle, the quasi-free electron picture, and the use of second-order perturbation

theory can be expected to give reasonable results for point defect calculations. Unfortunately

it will turn out that the basic assumptions made in this treatment, which is the one almost ex- clusively adopted in the recent book by Harrison [33], are incompatible with each other. This can be seen as follows: In order to permit the linear superposition of the pseudopotentials of

neighboring pseudoatoms, each of these pseudopotentials must be well concentrated within the

atomic cell. In order to hold the screening charge of one electron per atom in monovalent metals

(or even of two or more electrons in polyvalent metals) this potential must at the same time have

sufficient strength. The more concentrated the potential is, the deeper has the potential well to be in order to attract a given screening charge. From the calculations on vacancies and interstitials based on the free electron approach (see, e.g., [3, 2]) it is well known that even in monovalent

metals a potential repelling or attracting one electron is much too strong to be treated by perturbation theory, if localized within the unit cell. A physical measure of the failure of the second-order

perturbation theory is the phase constant <p? in the asymptotic charge density

Ap(r)^-j^cos(2kF r+<pF) (14.1)

around a defect localized at r— 0. The parameters aF and <pF are given in terms of the free electron

phase shifts 17/ by [17]

aF sin ipF = ^ (- 1)'(2/ + 1) sin2 j)AEF) (14.2)

aF cos <Pf=^ (- 1)'(2/+ 1) sin 2V,(EF ). (14.3) z

1

In the perturbation theory treatment we have <pF = 0 [38]; the expression for aF is invalid if aF 1

is not fulfilled. As numerical examples for the exact evaluation of eqs (14.2) and (14.3), we consider a potential based on the Hartree-Fock solution for a Cu-ion determined by Stehle [39| for vacancy calculations, and an attractive square-well potential used by Seeger and Mann [40] for

interstitial atoms in monovalent metals. In order to make the situation least unfavorable for the

pseudopotential theory, we choose the radius for the square-well potential as large as possible

within the framework of the pseudoatom approach, namely equal to the radius rs of the WignerSeitz sphere. The numerical results are in the first case

a* =-0.478, ^=-|+0.198=- 1.313

and in the second case a,=0.159, yj F = -0.733.

93
We see that in none of these examples cpF is negligible. In typical situations in monovalent

metals \<pp\ will lie between 77-/4 and tt/2. For polyvalent metals the asymptotic phase <Pf will be even larger. Phase shifts \<pf\ comparable with or larger than 7r/4 will completely invalidate the

theory of interactions between ions based on pair potentials derived from the pseudopotential

theory. This can be seen by comparing the pair interactions for Al determined experimentally from

liquid metal scattering data with the theoretical curve [41]. While these data confirm the existence

of the long-range charge oscillations (compare sec. 8), they demonstrate at the same time that the

phase of these oscillations differs from the one calculated by the pseudopotential theory by ap- proximately tt, in agreement with our predictions. This difference in phase is extremely serious

whenever the charge oscillations enter into the final result of pseudopotential theory. In the case

of Al it leads to a reversal of sign in the interaction potential. Any agreement of pseudopotential

calculations of such properties as interaction energies between point defects, stacking-fault

energies etc., must therefore be considered as fortuitous. We cannot therefore support Ziman's claim [4] that in calculations on defects in metals the

usual pseudopotential approach is superior to the free electron calculations. The latter have at least the advantage that, once a potential describing the defect has been adopted, the results follow in a mathematically reliable way from the model, whereas the pseudopotential approach

makes assumptions which are certainly not justified and which become more and more serious, the

higher the valency of the metal. On the other hand, the polyvalent metals are just those for which

the pseudopotential theory is claimed to be best applicable. It appears to us that the only way in which the success of the quasi-free electron picture in describing the Fermi surface of aluminum

can be utilized for defect calculations is to start from an exact free electron solution of the defect

problem (if obtainable) and to allow subsequently for the small deviations of the Fermi surface

from sphericity by perturbation theory.

15. References

[1] A. C. Damask and G. J. Dienes, Point Defects in Metals (Gordon and Breach, New York, 1963).

[2] H. Bross, in: A. Seeger, Moderne Probleme der Metallphysik, Vol. I (Springer, Berlin-Heidelberg-New York, 1965).

[3] A. Seeger, J. Phys. Radium 23, 616 (1962).

[4] J. M. Ziman, Adv. in Phys. 13, 49 (1964).

[5] J. C. Slater, Handbuch der Physik. Vol. XIX (Springer, Berlin-Giittingen-Heidelberg, 1956); Quantum Theory of Molecules and Solids, Vol. 2 (McGraw-Hill Book Co., New York, 1965).

[6] G. F. Koster and J. C. Slater, Phys. Rev. 95, 1167 (1954).

[7] E. Mann, Phys. Stat. Sol. 11, 753 and 767 (1965).

[8] G. Wentzel, Helv. Phys. Acta 15, 111 (1942).

[9] A. A. Maradudin, E. W. Montroll, and G. H. Weiss, Solid State Physics, Suppl. 3 (Academic Press, New York and

London, 1963).

[10] H. Jones, The Theory of Brillouin Zones and Electronic States in Crystals (North-Holland, Amsterdam, 1960).

[11] J. Friedel, Adv. in Phys. 3, 446 (1954).

[12] F. G. Fumi, Phil. Mag. 46, 1007 (1955). [12a] F. Gautier and P. Lenglart, Phys. Rev. 139, A705 (1965). [12b] E. Mann, Phys. Letters 18, 110 (1965).

[13] G. F. Koster, Phys. Rev. 95, 1436 (1954).

[14] E. Mann, Phys. Stat. Sol. 13, 293 (1966).

[15] I. M. Lifshitz, Suppl. Nuovo Cimento [X] 3, 716 (1956).

[16] E. Mann, A. Seeger, and G. Thierer, Z. Physik 193, 295 (1966). [16a] J. Callaway, J. Math. Phys. 5, 783 (1964).

[17] A. Blandin, J. Phys. Radium 22, 507 (1961).

[18] D. J. Roaf, Phil. Trans. Roy. Soc. (London) A255, 135 (1962).

[19] L. M. Roth, Thesis, Harvard University, Cambridge, Mass., 1957.

[20] W. Schmeidler, Integralgleichungen I (Akademische Verlagsgesellschaft, Leipzig, 1950).

[21] S. G. Mikhlin, Integral Equations (Pergamon Press, London-New York-Paris-Los Angeles, 1957).

[22] H. Bross and A. Seeger, J. Phys. Chem. Solids 4, 161 (1958).

[23] J. Friedel, J. Phys. Radium 14, 561 (1953).

[24] (i. Thierer, Diplomarbeit, Technische Hochschule Stuttgart. (1966).

[25] R. P. Huebener, Phys. Rev. 135, A1281 (1964).
[26] I. M. Lifshitz, M. Ya. Azbel', and M. I. Kaganov, Zh. eksper. teor. fiz. 30, 220 (1955); 31, 63 (1956).

[27] I. M. Lifshitz and V. G. Peschanskii, Zh. eksper. teor. fiz. 35, 1251 (1959); 38, 188 (1960).

[28] I. M. Lifshitz and M. I. Kaganov, Uspekhi fiz. nauk 69, 419 (1960).

[29] R. G. Chambers, The Fermi Surface, p. 100 (W. A. Harrison, M. B. Webb, eds.), (John Wiley & Sons, New York-London

19601.

[30] A. Seeger, Phys. Letters 20, 608 (1966).

[31] J. M. Ziman, Electrons and Phonons (Clarendon Press, Oxford 1960).

[32] J. Mertsching, Phys. Stat. Sol. 14, 3 (1966).

[33] W. A. Harrison, Pseudopotentials in the Theory of Metals (Benjamin, New York and Amsterdam, 1966).

[34] H. Hellmann, Einfuhrung in die Quantenchemie, sec. 8 (Deuticke, Leipzig und Wien, 1937).

[35 1 P. Gombas, Die statistische Theorie des Atoms und ihre Anwendungen, sec. 19 (Springer, Wien, 1949).

[36] E. Fues und H. Statz, Z. Naturforschung 7a, 2 (1952). [37 1 J. M. Ziman, in: Physikertagung 1965 Frankfurt (M)-Hoechst, Plenarvortrage, Deutsche Physikalische Gesellschaft (Teubner, Stuttgart, 1965).

[38] A. Blandin, in: Alloying Behavior and Effects in Concentrated Solid Solutions, Metallurgical Society Conferences.

Vol. 29 (Gordon and Breach, New York, 1965).

[39] H. Stehle, Dissertation, Technische Hochschule Stuttgart, 1957; cf. also A. Seeger [3|.

[40] A. Seeger and E. Mann, J. Phys. Chem. Solids 12, 326 (1960).

[41] cf. W. A. Harrison [33], fig. 8-10, p. 319.

95

Electrical Resistivity of Impurities and Vacancies in Aluminum

Yuh Fukai

Department of Physics, Chuo University, Bunkyo-ku, Tokyo, Japan

Electrical resistivities due to impurities and vacancies in Al have been calculated in a pseudopotential formalism. Comparison with experimental values suggests that the atomic displacement is outward around a vacancy in Al in contrast to noble metals.

1. Introduction

Although most of the existing calculations of

defect scattering have been based on an approximate phase shift analysis, recent progress in the

pseudopotential approach now allows us to attack

this problem in a more reliable way, provided a screened lattice potential is weak enough to be

treated by perturbation theory. Now we have a

potential of Al, well examined both theoretically

and experimentally; also we have good reason to expect that calculations could be more reliable in Al than in noble metals.

2. Pseudopotentials Adopted

Considering only elastic scattering of electrons £lj€i(g) , , on the Fermi surface, k->k+q, where |k| = |k+q| 9

fto€o(<7) = kp, we need a Fourier component of the potential

w(q) in the range 0 =£ q =£ 2Rf for the calculation of

electrical resistivity. For this we use "experi- The ratio of atomic volumes Cli/Clo comes from the

mental" values corresponding to a few lattice wave different normalization volume of wave functions,

numbers deduced from de Haas-van Alphen experi- and the ratio of the SCF dielectric function €i(q)/eo(q)

ments on Zn [1J,

1 al [2J, and from optical experiments from different screening by conduction electrons of on Si and Ge [3]. Potential curves calculated by impurities embedded in the Al lattice from that in Animalu and Heine [4] were used for the inter- their own lattice. The validity of this simple

polation of experimental points. transformation was examined by Harrison theoreti- Potential curves of atoms as solutes in the Al cally [5], and also by our NMR experiments on

lattice, Wi(q), are constructed from the ones in their Al-alloys [6|. own lattice Vj(q) in the manner,

3. Electrical Resistivity Due to Impurity Atoms

The electrical resistivity is calculated, treating

the lattice pseudopotential as a first order perturbation. The formula is given by

. 1 37Tflo f

2

. , N Ap ~4^8 les; Jo tow-*^

+ NAS(x)wo{x)¥x3dx fJLfl cm./at.%,

where x = qjkF, Wi(x) and Wo(x) are pseudopotentials

of the impurity and the host atoms, respectively.

AS(x) is the change in the geometrical structure

factor due to lattice distortion. The Fermi energy

£V and potential m/s are written in Rydbergs.

In the absence of any exact knowledge of lattice distortion, we shall here calculate Ap originating

1 Figures in brackets indicate the literature references al the end of this paper.

from the difference in pseudopotentials only. It turned out that approximate inclusion of distortion as described in the next section does not affect our

results by more than 10 percent in the case of

impurity scattering, if we use lattice distortion as estimated from lattice-parameter changes. Results

of calculations are listed in table 1, together with

experimental ones.

Table 1. Electrical resistivity due to impurities in Al

l/iOcm/al.% solute)

Solute APclc. Ap„c. <ip<-.e./<V>calc.

Zn 0.16 0.24 1.5o Si .41 .60 1.4. Ge .59 .79 1.3,

97
The agreement seems gratifying in view of ap- become even closer to experimental ones when we

proximations involved. Theoretical values will proceed to second order perturbation.

4. Electrical Resistivity Due to Vacancies

The same formula is applied to vacancies, putting

Wi(jc) = 0. Neglecting, as before, the effect of lattice distortion, we obtain Ap = 0.66 /jlQ. • cm/at.%, which

is much smaller than the experimental value Ap = 2.2 ±0.6 n£l • cm/at.% [7]. Let us tentatively assume that our calculation on

vacancies is reliable to the same extent as on impurities, and take (1.3 ~ 1.5) X 0.66 /xfl • cm as a

theoretical value, then we may estimate the amount

of lattice distortion by comparison with experimental

values ascribing the remaining discrepancy to distortion effects. For this purpose, we calculate

a geometrical structure factor as a function of frac- tional displacement of nearest-neighbors, assuming

the isotropic elastic solution for the atomic

displacement:

Comparison with experimental value yields

0.09

-0.06

+ 0.11.

The positive sign indicates that displacement is outward.

The outward displacement thus deduced, though

apparently incompatible with our knowledge on

noble metals, should not be taken as improbable.

The same conclusion was reached by Harrison [8] on vacancies in Zn. We may rather regard it as

fairly common to metals with small ion cores where

repulsive forces are of long range character as

distinct from noble metals where short range core

repulsion is dominant.

The present work is of course roundabout as an approach to lattice distortion, and more direct

calculation is necessary in drawing any definite conclusions. This is possible, in theory, within

the same pseudopotential formalism, but we cannot

be too optimistic about it. Very strong cancellation

of electrostatic and band-structure energies in Al [9], as well as the long range, oscillating character

of interatomic interactions via conduction electrons

[10, 11] will make such calculations quite unreliable. This is the very reason why we have made a detour and have left a more orthodox way of ap- proach for future studies.

5. References

[1] W. A. Harrison, Phys. Rev. 126, 497 (1962). [2jvN. W. Ashcroft, Phil. Mag. 8, 2055 (1963).

[3] D. Brust: Phys, Rev. 134, A1337 (1964).

[4] A. O. E. Animalu and V. Heine, Phil. Mag. 12, 1249 (1965).

[5] W. A. Harrison, Phys. Rev. 131, 2433 (1963).

[6] Y. Fukai and K. Watanabe, To be published in J. Phys. Soc. Japan.

[7] J. Takamura, Lattice Defects in Quenched Metals, p. 521,

(Academic Press, Inc., New York, 1965). The value is corrected for deviations from Matthiessen's rule.

[8] W. A. Harrison, Phys. Rev. 129, 2512 (1963).

[9] W. A. Harrison, Phys. Rev. 136, A1107, (1964).

[10] W. A. Harrison, Phys. Rev. 129, 2503 (1963).

[11] M. H. Cohen, in Metallic Solid Solutions, (J. Friedel and A.

Guinier, ed.) (W. A. Benjamin, Inc., New York, 1962).

98
Structure Factor Approach to Energies of Migration

H. B. Huntington and Michael D. Feit

Rensselaer Polytechnic Institute, Troy, New York 12180

Several crystal lattice defects move with hardly

any activation volume. Examples of such are the

interstitialcy, the crowdion, the dislocation, and

possibly some interstitial impurities. Presumably

their energy of motion is largely the difference in the ordering of the atoms between two rather similar

configurations. The individual environment of each

atom changes relatively little in going from one

configuration to the next as far as its ambient

electron density or cell volume. One would like a method of attack where energy could be tied very

directly to structure. For this purpose Harrison's

recent technique [l] 1 growing out of his pseudopotential approach [2] seems made to order. According to the Harrison approach the energy

of any configuration can be represented by the sum

of three terms: one of which is volume dependent

and will be avoided in these applications, another

of which is just the coulomb energy of the undressed

ionic charges (as might be found by a modified

Ewald method), and the third is the so-called band

structure term, which has the following form,

Ebs = Nz^\S(q)\*F(q).

Here /V is the number of ions in the lattice and z is their valence. The summation is over all the points

of wave number space. The S(q) is just the struc- ture factor as usually defined as in diffraction

and F (q) is called "the energy wave-number

characteristic" with units of energy per electron.

It is this function which embodies the particularities

of the band structure of each metal and must be

separately evaluated in each case via application

of the pseudopotential technique. Harrison has

carried out this evaluation [1] for sodium, magnesium, and aluminum. These functions are

large and negative for small wave number, exhibit

minima in magnitude where the form factor changes sign, and go rapidly to zero at large wave

number. Defect calculations based on this formulation of the total energy would involve the elec- trons and their redistribution to a greater extent

than prior approaches.

The formulation is, however, reducible to the use

of a two-body potential. For multibody forces

the perturbation development would need to be

carried to higher order. The connection between

F(q) and the equivalent two-body potential V(r)

is easily demonstrated [3]. The latter exhibits the

familiar Friedel oscillations at large distance since

in developing F(q) account is taken of the sharp

Fermi cutoff in the dielectric response of the metal.

In our application we want to compare the

energies of two related defect configurations in- volving several atoms on a nearly equal basis. Although the more recent defect calculations have

used an elastic displacement function at large

distances, the explicit evaluation of the structure

factor in our method made it necessary to treat only

a finite number of displaced atoms. With rj for the displaced positions of these atoms and r, for their original locations the structure can be

written as a small number of terms

N >t NS(q) = ^ e-'i 'l = ^ (e -«q r;

i i — e fqn )±e">' r + N8qh

Here n stands for the number of atoms in the specimen that are displaced. The second term repre- sents the lattice defect, + for interstitial and — for vacancy. The final term is now expressed as an

integral plus a sum

Ebs —

zQ,q

J F(q) ^ (<? -'«' r '' - e

~ i(i ri)(e"i '"j — e

i(> rd)

± 2 {cos q(rl — rd ) — cos q(n — rd)} + 1 d

3q

+ 2z^F(K) £(cos K-n- cos K-r,)±cos Kr

As a test case, to give experience with this ap- proach we calculated the restoring force for the

motion of a single atom in an otherwise perfect

aluminum lattice. In energy units of milli-rydbergs

and with d the lattice constant we found for the

energy of the atom displaced by 8,

Electrostatic energy

Integral term in band structure energy

Discrete Sum term in band

structure energy

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

46.3 (Sid)2 - 108.3 (8/d)2

70.6 (8/d)2

8.6 (8/d)2

99
which is equivalent to an Einstein temperature of

about 580 °K. Note the high degree of cancellation

making it important to work to several significant

figures.

We have applied this procedure so far only to a comparison of the "split interstitial" and cube

center interstitial in the face-centered cubic lattice. In each case the same number of displaced atoms was used {n ~ 87) and since the outer atoms were

held fixed, the volume was the same for both

defects. For our configuration we have chosen

the displacements found by Johnson and Brown [4] using a Born-Mayer type potential. The sensitivity

of defect energy to atom positions near equilibrium

is usually small but we intend to explore this point more later. Preliminary results from this calculation favors the split configuration. The energy

difference between the two configurations has been

calculated for aluminum and hypothetical fee metals

of magnesium and sodium at their normal density.

The results are given below for Ecc ~ESr in eV. Note that the electrostatic contribution dominates

and serves as an upper limit in these cases. This

Aluminum Magnesium Sodium

U. iWO U.uiOO

Integral term .0328 .0099 .0038 Discrete sum term -.1278 - .0441 - .0129

1 otal 0.254 0.106 0.020

energy difference is associated with, but exceeds,

the motion energy for the free interstitialcy.

The method is of interest for several reasons. It may supply insight into how the energy difference

between two configurations depends on the trans- form of the interaction potential. It attempts to take into account the energy of the electrons and

of their redistribution in the presence of a defect. Once developed for the comparisons of a particular

pair of defects the result can be quickly extended

to any other material of the same crystal structure

for which F (q) is known.

References

[1] W. A. Harrison, Phys. Rev. 136A, 1107 (1964).

[2] W. A. Harrison, Pseudopotentials in the Theory of Metals (W. A. Benjamin, Inc., New York, N.Y. 1966).

[3] Ref. [2] p. 43; M. H. Cohen, in Metallic Solid Solutions. J.

Friedel and A. Guinier, editors (W. A. Benjamin, Inc., New York, N.Y. 1962).

[4] R. A. Johnson and E. Brown, Phys. Rev. 127, 446 (1962).

100
Perturbed Bloch Functions Obtained from the Koster-Slater Perturbation

Theory*

Bernard Mozer

Brookhaven National Laboratory, Upton, New York 11973

Approximations for Bloch functions for a crystal

containing an impurity can be obtained from ap- proximate solutions of the Koster-Slater equations.

A well-known solution of the Koster-Slater equations

for a single matrix element of the perturbation

potential at the impurity site will yield perturbed

Bloch functions whose scattered wave part corre- sponds to summing in a simple way an infinite number of terms in the perturbation series and

hence a simple result for the scattering matrix or

T matrix. Immediate comparison with the results

of the Friedel method can be made as well as with

results obtained for a 8 function potential using a formal scattering theory solution. The question

of the role of phase transformations on the un- perturbed Bloch functions is answered and shown

not to fix the perturbation potential but to help

choose the most localized Wannier functions which

allows reasonable truncation of the Koster-Slater

equations.

1. The Koster-Slater Equations

We shall derive the Koster-Slater Equations [l] 1 for a crystal containing a single impurity. We make

use of the formal theory of scattering [2] to obtain an expression for the perturbed Bloch waves and

then transform the formal solution of the scattering

problem to a Wannier representation. Our ap- proach is the same as that first put forth by Koster

and Slater and others [1] except for the introduction

of the Green's function to discuss the electron spin

or charge density. Consider that we have a oneelectron Hamiltonian H0 which is a good first approximation for describing the band electrons in the perfect crystal. H0 can be determined from a

self-consistent potential. The Bloch eigenfunctions

<|>„.k have eigenvalues E„,u. of Ho satisfying the

unperturbed Schrddinger equation

H0<p„,k = E„,h.(fn.V.- (1) We shall denote by V the perturbation potential

which is the difference between the imperfect

crystal potential and the perfect crystal potential.

V can be determined in a self-consistent manner when we are interested in spin and correlation

effects as well as charge difference as shown by

Wolff [3] and Clogston [4J. The new eigenfunctions

We have eigenvalues E determined from the solution

of the perturbed Schriidinger equation

H^e = E^e . (2)

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

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

Formal solutions of are Ve = {<Pn,v } + (E-Ho)- 1 VVE (3a)

or VE = {<Pn, k } + (£-#)" 1 F {<Pn,k }, (3b)

where E is an energy in the allowed bands of the

perfect crystal and k) are the set of solutions

of the homogeneous equation or unperturbed

Schriidinger equation. We have specified a set of

unperturbed {</?,,, k} all having energy En,w=E, but one could have chosen a single member of the set which we shall do in the future and leave the possibility of choosing a linear combination of ^'s or

<p's satisfying some symmetry property of the crystal

Hamiltonian. When the energy E does not lie in any of the allowed bands of the perfect crystal, then

the formal solution for ^ is «W=(E-ff«)-W*. (4) The Green's functions that are used to describe the

response of the system are in operator form

G°(E) = (E — Ho)' 1 (for the unperturbed system)

(5)

and

G(E) = (E — H)~ l , (for the perturbed system).

(6)

101
The perturbed Green's function can be obtained

in terms of the unperturbed Green's function or a3 a Dyson equation by matrix manipulation to yield

G(E) = [1 - G°(E)V]- l G

0(E) = G°(E)[1 - VG°{E)Y l

(7)

or

G(E) = G°(E) - G°(E)VG, etc. (8)

It is preferable to continue on in the Bloch representation and rewrite (3a) and (4) as follows: VE = <Pn,U+ V (<Pn.u>\V\VE ) (9)

and

*E =^

E-'L <y*i*T**>- (10)

n. k

(Vnkl^l^e) is a matrix element of the T matrix and

as we shall see that working in the Wannier repre- sentation or in the Koster-Slater equations gives us

various approximations to the T matrix for the impurity problem. We can now transform our eqs (9) and (10) to a Wannier representation where we

define the Wannier functions as follows:

f> — ik Xj a„(x-X0 = 2 —^"^"•k<x) (H)

k and

^"k =

Z.^Kir a >^-Xi). (12)

There are other possible choices for the Wannier

function for complicated bands as discussed by

Blount [5] but we shall work with the simple trans- formation here. The perturbed wave function can be expanded in the Wannier representation as

follows: VE = X Uni(E)an(x-Xi)

n,Xt (13)

o»(x-Xi) = 2 UniEfViE). (14)

E The transformation coefficient U are elements of a unitary matrix as are the transformation coefficients

e lkX>

yyi/2

in (11) and (12).

If we make use of the orthogonality relations for the Wannier functions,

(a„(x — Xi)|am(x — Xj)) = §„m§ij, (15)

we find the following expression for (9) and (10)

in the space spanned by the transformation

coefficients: E in the band

p ik X, p ik' Xi TJ.(E) = - h-V- V e ~ ik'- x ' V(sOTf(E).

k' n> K m, s, t

(16)

E not in the band

1 P ik' x;

2 <•"• V,m(st)U„($). (17)

m, s, f, The above are the Koster-Slater equations. The

matrix elements of the potential are

Vnm\st) = < O, (X - X.) | V\ am (X - X,) > . (18) One should note that there are no restrictions on

the number of Wannier functions required to specify

the Bloch functions and no restrictions on the per- turbation potential. In a straight-forward manner

one can write similar Green's functions for the t/'s as was introduced in (5) and (6) for the <p's and ^'s:

p

ik-(X,-Xj) G\j;mi (E) =N~1 y F F

8nm , (19)

G n, m,(E)=y Umi{E

J'

)

U

/

E»\ (20)

and the solutions for U are U=U° + G°VU (21)

or U=U° + GVU° (22)

where U° refers to the coefficients

e' k *

Nil*

'

Algebraic manipulation of (21) and (22) will allow one to write the Green's function equations anal- ogous to (7) and (8), G(E) = [l-G°(E)r}-1G°(E) (23)

or G(E)=C0 (E)-G°(E)VG(E), (24)

discussed by Goodings and Mozer [6].
2. Lowest Order Solutions to the Koster-Slater Equation

One can now see how the perturbation series in the Koster-Slater equations can yield perturbed

Bloch functions. For the first example we consider

the very simplest problem of using a single matrix

element for V in a single band. All other matrix

elements of V in this band and other bands are con- sidered small and to be treated in higher order. This is a problem which has been studied in great

detail, yet it will provide insight into more complicated problems. We shall drop all band indices

and call the matrix element of the perturbation

potential at the impurity site Vo3 The KosterSlater equations become

UdE)

ik x, N

for energies in the band and

, ik X,

k k UoiE)

for energies outside the band. U0 is given by

Uo=[l-V0C(E)]->

(25)

(26)

(27)

for energies in the band. G has real and imaginary parts

Re G(E)

(P means principal value) and

Im G(E)=-i7Tg(E),

:f(E) (28)

(29)

where g(E) is the density of states in the band and we have added a small imaginary part to E which

vanishes in the usual limiting procedure. The

condition for the existence of_a state outside the

band is the vanishing of 1 — VG(E) corresponding to a pole in the Green's function 6" of (23). Uo can be

obtained for the state outside the band by imposing

the normality condition on the Vs,

2\UX (E)\*=1,

and we find

\U0 (E)\ 2 = Vl-E

GiE)

(30)

(31)

The Vs at other lattice sites are not so easy to obtain unless one knows the energy versus k in the

Brillouin zone for the unperturbed crystal. Numerical techniques have been developed for obtaining the Vs at other sites if the energy versus k

is known.

3 We take the impurity site at the origin of the coordinate system.

It is worthwhile at this stage to transform back

to the Bloch representation. The resulting approximation for ^ we call

<P k' T(E) (32)

diere

T(E) = V0 [1-V0G(E)] (33)

and

Fo=<ao(*)|FM*)>=/V-' Y (<p |*V>.(34)

k. k The state outside the band is

iIr(D =AM y —

E — E, T{E). (35)

We see that the Koster-Slater equations have given

us an approximation for the T matrix which is fairly

simple in form as it depends only on the energy E

but not on

follows:

or T(E) can be rewritten as

T(E) = V0 \l-V0G(E)\-1e

i \E),

where

k(E) = tan- 1 TTVog(E)

[l-Vof(E)]

(36)

(37)

and is in the familiar form of a phase shift that one

expects from scattering theory except there is only one phase shift and not a sum whose terms contain

the product of a spherical harmonic and the phase

shift. An expression similar to (26) can be obtained

from the formal solution (36) for yi

r

, which we rewrite

as follows

(38)

The approximation one uses in evaluating terms

such as (G°V)' in (38) to get a similar expression

for given by (26) is to replace the operator G°V

by

GW\^ >

><<pJV\<p. >

E-E, ={V} ^E^

(39)

where { V } is some suitable average of Vwk . In other words, the approximation consists in saying

that Vkk' is a very slowly varying function of k and

k'-or Ek and E k compared to the energy denomina103
tor and the density of states Ek' and we replace Vkk> by a mean value and sum the series to yield (32). The mean value of Vkk is judiciously chosen by the

approximation made in the Koster-Slater equations;

it is an average over the various k, k' states given

by (34). Such approximations are common in perturbation theory but one usually makes the

approximation only on the energy shell, Ek,~E,

and not for states both on and off the energy shell which is what we did here. One can argue that such approximations are very reasonable for problems in solids except for the very unusual accident

of a real metal whose electrons are completely

described by single plane waves. For any metal

where the electrons are not described by plane waves, the one-electron wave function could be

represented by a tight binding wave function

<P k = e

lk x J Xi

j

and the matrix elements of, say, a coulomb potential of the impurity will vary with k and k' as

V cc 1 kk' k-k'| 2 + K

2 + 0 (40)

where k is the minimum inverse decay length of

the tight binding functions. The approximation

considered above is reasonable only if k > max |k — k'| and then perturbation theory could describe

the fluctuations of the potential about the mean

value. If one includes the correlation of the band

electrons, the coulomb potential of the impurity

is screened and (39) is then a better approximation

as k in (40) contains the screening parameter of

the electron gas. One should be at least a little careful in using the argument given above if bands

have configuration mixing as then the Xj contain

functions of k, e.g., s-p mixing in the alkalis, and

if the overlap of the tight binding functions is large

then additional terms must be evaluated in (40).

It seems what one would prefer to show is that the

Wannier functions for most systems are quite local- ized and then taking only one or several matrix

element of the potential in the Koster-Slater equations is sufficient to describe the impurity problem.

For a discussion of the localization of Wannier

functions we refer the reader to Blount's work [5], Kohn's work [7], or an extensive discussion by des

Cloizeaux [8] which contains some remarks on tight

binding functions that are of interest. The perturbed Bloch functions have several

interesting properties. One most important property is that the perturbed functions are orthonormal

since the unitarity requirement was imposed upon

the transformation coefficients U; it should be emphasized that if a bound state exists, the orthogonality conditions are still satisfied. Another

interesting property is that the wave functions

are approximate eigenstates of the Hamiltonian

H in the sense that for states in the band

($E AH\fa.)=E.+iyd -Vo\ (41)

where Vd is the diagonal matrix element of the

perturbation potential between two unperturbed

Bloch states. (if h \V\ip

k

). We have evaluated

matrix elements of V occurring between

%T=eJ k k {E) and <

or the quadratic term in T(E) by using the average

value for V. The term in parenthesis is the cor- reaction to the energy Ek and is equal in magnitude

to the average fluctuation of the potential about

the mean value determined by (34). The shift given by this term is the same for all states E k and

hence the band is displaced by this amount. Since

the perturbed states in the band are labeled by Ek , a one-to-one correspondence can be made between

the ifk and t// k which is especially useful for counting

states. If a bound state occurs because of the

strength of the perturbing potential, the matrix

elements of H are

(\jjn\H\{lfH )=EB , (42)

where En is the energy of the bound state. The

density of states of the system will not be per- turbed until a finite concentration of impurities

is considered; local densities of states or local charge density vary as pointed out previously [6]. Our results for the first approximation for the

perturbed Bloch wave should be compared with

results obtained for a 8 function potential and a well or barrier. We immediately obtain the follow- ing result for a 8 function potential using (3) and

considering only one band,

WW)

and solving for i//£<0) we find

^A-(O)

<//>?>;

MO) l-VsG'(EY

/here

\<p (0)l

5 C'(£) = 2-3E-E,

The T matrix is simply

*V*(0)n,(0) Tkk 1-VS G'(E)

(43)

(44)

(45)

(46)

The poles in the T matrix, when 1 — VG'{E) = 0, define the bound states, which is very similar to the

104
condition for a bound state as determined from the

Koster-Slater equations. We see one significant

difference in the T matrix for the first approximation

to the wave function given by the Koster-Slater

equations and (46). This difference occurs in the

definition of G(E) in (28) and (29) and G'{E) in (45). The latter is evaluated only for those states <p k non- vanishing at the impurity. The T matrix has only

matrix elements for nonvanishing states and hence

the scattered wave must consist of onlys-like waves. Only if the <pk are plane waves, can the T matrix for the 8 function potential have the same value as the

T matrix for the first approximation in the KosterSlater equations and generally the amplitude and

phase shift of the scattered waves are vastly dif- ferent for the two approximations. If one considers

other bands, the 8 function potential will force the

T matrix to have elements between states nonvanishing at the impurity and again only s-wave scattering

whereas the Koster-Slater equations allow one to take reasonable matrix elements for the potential

for both interband and intraband transitions and the

symmetry of the scattered waves depends on the

symmetry of all the wave functions in the bands.

The scattering of electrons in a crystal when the

impurity potential is represented by a well or barrier

has been considered in detail by Friedel and

coworkers [9]. The general solution for this well or barrier problem is given by the following:

fa = <pk + X jzir V{<p (47)

k' k' where (^Jt/u) represents the integral over the

extent of the well or barrier. If we consider plane wave states of all possible energy k, and a spherical

well or barrier, and ask for the scattering solution

corresponding to outgoing waves, (47) becomes

$E = e

ik - 1 J (2n + 1) ^sin 8„e'8 «P„(cos 0), (48)

n

where 8„ is the phase shift of the nth partial wave

with angular dependence given by the Legendre

polynomials Pn - The T matrix, (<p \V\4>), is re- quired only on the energy shell (when E = Ek ) for the solution to the scattering problem and can be

written for each partial wave as T„ = 2i sin 8„e'8 ». The phase shifts are functions of the size of the well

or barrier and its depth (height). To insure that the system is electrically neutral, the Friedel sum

rule is invoked to fix the parameters of the well,

AZ = |^(2n + l)8„(£», (49)

where AZ is the difference in charge between an

impurity ion and lattice ion, and Ef is the Fermi

energy of the system. There are large differ- ences between (47) for Bloch waves and (48) for plane waves and greater differences between

(47) or (48) and our first order expression obtained

from the Koster-Slater equation which has only one phase shift. It is not at all apparent that if one needed only a single phase shift in (47) or (48) that this phase shift would disagree with the KosterSlater result. Rather than go into detailed dif- ferences in the phase shifts, one can compare the

two methods of approach. In the Friedel method,

or a generalization of it where the potential is not restricted for example to a well or barrier (see

W. Kohn and S. Vosko [10]), the formal solution or Green's function solution (3) is evaluated for a scattering problem using Bloch waves for the

unperturbed crystal functions. As in most scat- tering problems, the Green's function is evaluated under boundary conditions for outgoing waves asymptotically far from the impurity and

on the energy shell. One can then express the

solution for i|/ in terms of phase shifts where for most cases only a few phase shifts are needed to describe the problem. In order not to restrict oneself to a particular potential, a set of finite linear equations can be given to relate the phase

shifts to known or experimentally determined

quantities, i.e., the Friedel sum rule is invoked to conserve total charge, and the residual resistance

expressed in terms of the phase shifts, etc. Thus

with a few parameters one could account for a

variety of phenomena in dilute alloys. Using the

truncated Koster-Slater equations, one can obtain

approximate perturbed Bloch waves with the fol- lowing properties. The wave functions are ortho- normal and thus one does not need the Friedel sum rule to conserve total charge. The first approximation to the T matrix yields a single phase

shift [11] which supposedly is completely deter- mined by the matrix element of the potential

using the Wannier functions at the impurity site, or (34) if the Bloch waves are known, and the density of electron states for the perfect crystal.

Since not too much is known about Wannier

functions, potentials, Bloch waves, or density of

states, one can do the same thing as for the Friedel

method and determine the parameter V„ using

the residual resistance and its relation to the phase

shifts. One has then a perturbed wave function

which can be used to describe transport phenomena

and additional phenomena in dilute alloys associated with the impurity itself, for example the Knight

shift, Mossbauer shifts or structure, and effects arising from local spin and charge density since

in the Koster-Slater method one has a wave func - tion defined over all the crystal and the T matrix

both on and off the energy shell. Truncating the

Koster-Slater equations so as to include additional terms involving matrix elements of the per- turbation potential for Wannier functions off the

impurity site will produce a T matrix with additional phase shifts and of course more parameters to describe the impurity. One needs the

E versus k relations for the perfect crystal in addition to the density of states and Wannier func105

256-724 0-67—
tions but one has considerable information on E

versus k relations and hence the method is not

so restrictive. If interband matrix elements are important, then both methods can be modified

accordingly.

It appears from the preceeding paragraph that working with the truncated Koster-Slater equations would be more advantageous than the Frie- del method or pure scattering solutions which

require more parameters. There is a problem with

using the truncated Koster-Slater equations that was first brought to the author's attention by V. Hiene in connection with the x-ray emission problem

in lithium [12]. The problem is brought about when

one considers general phase transformations of the

Bloch waves. One wants to consider general phase

transformations in order to achieve Wannier functions which are very localized. It is just such

localized Wannier functions that make the KosterSlater method acceptable since one would only need

a few matrix elements of the perturbation potential

to adequately describe a problem. The latter notion would be valid for even a moderately screened

coulomb potential. Consider a phase transformation e ")< k>that produces the most localized Wannier

functions a,j(x — X,):

a;(x-X0=2^e^(k- nVk>n . (50) We know then that <p and ip can be expanded in the

quantities a',

^.„ = E^e~"(k

' " >a ';(x ~ Xi) ' (51)

«Jto-=2 f/;i-(^)e^ I " ,k-" ) a;( (x-X,). (52)

n, E One can obtain the Koster-Slater equations for the U' and the only modification coming in is a new

definition of the matrix elements V'nm (st),

Km (st)= (a^x.-Xs)\V\a^(x-Xt)). (53)

Truncating the Koster-Slater equations yields the

following approximation to the perturbed Bloch wave

<p. ,^;fa(k')-T)(k)] VP= % + S E-Ek

, T

' iE)

' (54)

where

T'(E) = V'0 [\-V&(E)Y^ (55)

with

V^(a'(X)\V\a'(X))=±2<<\ VK^ < 56> k, k'

where <p' = e"

)(kVk- It is possible to view the result obtainecTin (54) in several ways: (1) We have ob- tained a new T matrix, T— e

i7)(k

' ) 7"(£')e-!T,(k)

, with

matrix elements determined by the choice of phase

for the Bloch waves in defining the a's; or (2) we

have started out with a general phase transformation

e

"> (k) such that \]t'k= e"> <k)

i/> k and <p k = e"> (kVk and a

solution

K=K+lp^J-T'(E). (57)

k £ ^

New phase shifts of scattered waves appear because

of the phase transformation used in defining the a's and, depending upon what use is made of the i//', one could expect results different from those found

for ifj. Another obvious difference seen to occur

as a consequence of the phase transformation is the

strength of the potential required to produce a bound

state, when 1 — VoG(E) = 0. Thus in order to get

any consistency in the method of using the KosterSlater equations for continuum states or bound

states one can require

|Fo-^| =7V-i| J

(l-e-<h<k>-^k

'>])Fkk,| <e (58)

k, k where e is small compared to V0 or V'q. This means

that some variation is allowed in the phase choice

but according to (58) one would require that the

Bloch functions that have been determined for the

perfect crystal produce Wannier functions that are almost as localized as the phase transformed Bloch

functions. It is the opinion of the author that one

should use the most localized Wannier functions in the Koster-Slater method in order to have a tractable

expression involving the truncated equations and

thus, to first order, use (54), (55), and (56). If the

Bloch functions are well represented by tightbinding functions such as to justify the approximation in (39), the inequality (58) will be satisfied and

estimates of the matrix elements of the perturbation

potential in (34) can be made using atomic functions.

In the previous paragraph we have shown the

role of the phase transformations that should be

used in conjunction with the Koster-Slater equations

to obtain perturbed Bloch waves. Recently Turner

and Goodings [13] have discussed the importance

of phase transformations in what they call the local- ized approximation of the Koster-Slater theory.

Their version of the localized approximation places

such restrictions on the Koster-Slater theory as to bring into question the validity of its use. The

reason this occurs is that they impose restrictions on the potential that can be used in the KosterSlater theory. According to their expression (5) the localized approximation means in our notation

Vnm{st) = Vnm 8so8to, (T~G 5)

a potential having only matrix elements in the

Wannier representation for Wannier functions at

106
the impurity. Consequently they conclude that the localized approximation in the Koster-Slater

theory fixes the phase of the Bloch waves. They

further remark that a potential yielding matrix

elements such as (T—G 5) is hard to understand

physically. We agree completely with this last statement and cannot imagine a physically realiz- able potential satisfying those requirements.

What is even more peculiar about the restriction of the potential according to (T—G 5) is that when

one wants to extend the truncation of the KosterSlater equations one has to introduce a new potential

whose properties are completely different from the

potential leading to (T-G 5). It is important then

to restate what in the Koster-Slater theory is a local approximation [14]. For simple potential scattering, the local approximation means a potential which

falls off fast enough with distance, e.g., a screened

coulomb, and localized Wannier functions so that one can comfortably truncate the Koster-Slater

equations using a few matrix elements of the potential and regard the remaining potential matrix

elements as small and to be treated later by perturbation theory. This is the context in which Koster

and Slater first proposed the method. As we have

seen in this article phase-transformed Bloch waves

are used to define localized Wannier functions in order to use the Koster-Slater theory to get an

approximate expression for the T matrix which

describes the effect of an impurity potential, and

not to fix or define the potential being used.

Another comment on the paper of Turner and

Goodings [13] can be made with respect to the

approximate wave functions they suggest for the

perturbed problem. If we consider as they do the

first approximation for the wave function (32), then we may rewrite this as

<jj k = <pk P(E) + iTr[g(E)<pv - ]T <pk

. 8(E -

w

) )T(E)

f(E)<p-P

K dk' E-Ew

T(E) (59)

/here

P(E) = T(E)IV0 = [1-V0 G(E)Y

Turner and Goodings suggest using only the first term in (59),k<p P(E), for the perturbed wave function, but this must be justified for each particular

problem depending upon what use is made of i// k

. If one can show that the second term in (59) is small for a particular problem, the third term will also be small because of the Kramers-Kronig relation

between the parts on and off the energy shell of the

quantity ^ £ — £

' ^ ^ ls we^ approximated

by a tight binding function, then the Knight shift for the impurity is given by the expression in Turner and Goodings,

Kimp = 8nl3giMlN(\vk (0)\z)EF -\P(E)r \*g(E) F .

(T. G-21)

For other problems, say the x-ray emission problem, one might find it difficult to approximate i|) k by

(p kP(E) even in the tight binding approximation,

and the correction terms have to be estimated.

One can also look at the correction terms in another

way by expanding <p k in Wannier functions; the

first quantity in brackets is

ITT ^ a(x — X,) ik'Xjg(E)

(> j'k'-Xj

k'

1 T(E), (60)

and here again if the a's are localized the correction

terms can be small in a region confined to the

impurity site. (60) is modified in a straightforward manner for phase-transformed v?'s,

y\n/2 2 a'(x-Xi)

i 0 0

e' kX'e r' tk)g(E)

\? e

ik x, e;[„(k')-„(k)]8(£_£ k ) T'(E). (61)

As a final remark one can point out that the approximation i|/ k ~ <p h P(E) must be used with care since

to assume that it represents the perturbed wave

function is incorrect because normalization of i// k would require replacing the factor P(E) by unity

and furthermore for scattering problems where i// is needed far from the impurity the correction terms

are most important.

In conclusion, we can say that the Koster-Slaterequations can yield useful estimates of perturbed

Bloch functions for the impurity problem and that the role of phase transformations on the unperturbed

Bloch waves should be to provide localized Wannier

functions which allow a reasonable truncation of

the Koster-Slater equations. This first approximation can be used in deriving a self-consistent

potential where higher order effects of the scattering

have been approximately taken into account. From

our discussions of the simplest solution of the

Koster-Slater equations one can easily see what the

perturbed Bloch wave looks like when truncating

the Koster-Slater equations so as to include matrix

elements of the perturbation potential at sites in the

neighborhood of the impurity. The T matrix is more complicated and has additional phase shifts; the perturbed Bloch waves show new effects in which scattered waves are introduced with an additional phase dependence corresponding to transla- tions of the Bloch wave to sites in the neighborhood

107
of the impurity. Multiband effects can also be

treated in the Koster-Slater theory, as is well known,

and the result of their inclusion in the theory pro- duces additional modifications of the approximate

Bloch waves. A fuller discussion of the multiband

effects and the effects of extending the truncated

Koster-Slater equations to sites in the neighborhood

of the impurity will be given elsewhere.

3. References and Notes

[1] G. F. Koster and J. C. Slater, Phys. Rev. 95, 1167 (1954); G. F. Koster, Phys. Rev. 95, 1436 (1954); and G. F. Koster and J. C. Slater, Phys. Rev. 96, 1208 (1954). For generalized Green's function approach see ref. 6.

[2] We intend to use the Green's function approach that was

introduced for scattering problems (L. F. Schiff, Quantum

Mechanics, p. 110 (McGraw-Hill Book Co., New York,

N.Y., 1949) and more general developments in M. L. Goldburger and K. M. Watson, Collision Theory (John Wiley & Sons, Inc., New York, N.Y., 1964) but can also be used for other problems by a proper choice of boundary

conditions.

[3] P. A. Wolff, Phys. Rev. 124, 1030 (1961).

[4] A. M. Clogston, Phys. Rev. 136, A1417 (1964).

[5] E. I. Blount, Solid State Physics Vol. 13, pp 306-370 (Aca- demic Press, New York, 1962).

[6] D. A. Goodings and B. Mozer, Phys. Rev. 136, A1093 (1964).

[7] W. Kohn, Phys. Rev. 115, 809 (1959).

[8] J. des Cloizeaux, Phys. Rev. 135, A698 (1964).

[9] Friedel's work and that of his school is described in Metallic

Solid Solutions (W. A. Benjamin, Inc., New York, 1963).

[10] W. Kohn and S. H. Vosko, Phys. Rev. 119, 912 (1960). See also A. Blandin and J. Friedel, J. Phys. Rad. 21, 689

(1960).

[11] One should note that this does not imply only s-wave scat- tering unless one has completely spherical bands. If the bands are not spherical, then scattered waves of various symmetry appear, but a peculiar nature of the first KosterSlater approximation is that all the scattered waves have

an identical phase shift coming from the potential and

perhaps different shifts coming from the nonspherical

behavior of the energy. One should really talk about the

scattered waves in cubic harmonics for the systems considered here but we will ignore this modification of the problem. Choosing the Wannier functions via phase

transformed Bloch waves brings in additional phase shifts even if only one matrix element of the perturbation potential is used in the Koster-Slater equations.

[12] Interpretation of the soft x-ray emission spectrum of lithium

metal, B. Mozer and D. A. Goodings, BNL # 7965 (unpublished); D. A. Goodings, Proc. Phys. Soc. 86, 75 (1965). Heine's remarks concern the symmetry (parity) of the Wannier functions under phase transformations of the Bloch waves when the inversion operator and time, re- versal operator commute with the perfect crystal Hamiltonian. Arguments about the symmetry of the Wannier

functions become rather subtle when one has overlapping bands or when free atom energy levels are close together.

[13] R. E. Turner and D. A. Goodings, Proc. Phys. Soc. 86, 87

(1965).

[14] In ref. 6 we erroneously used V„m(st)= Vnm 8so 8to instead of considering the truncation of the matrix V as we have seen

here is the correct view of the process. Unfortunately

this has led to confusion regarding the generality of that paper. The method as introduced in that paper is quite

general but the approximate solutions of the Green's

functions should be considered only as a truncation of the perturbation potential. What is hidden in the KosterSlater equations is that the theory provides a method for obtaining impurity energy levels, changes in density functions such as spin, charge, or energy states, but does not provide the behavior of the perturbed wave function which

is also of great interest in the impurity problem. One

uses the Green's functions to provide the coefficients in the Wannier expansion of the perturbed Bloch waves.

108
Residual Resistivities of Dilute Aluminum Alloys *

F. J. Blatt and H. R. Fankhauser

Physics Department, Michigan State University, East Lansing, Mich. 48823

We have performed calculations of the residual

resistivities of several aluminum base alloys using

the approach described by one of us some years

ago [l]. 1 Briefly, we represent the scattering potential by a square well (or barrier) whose range is given by

r' = rs [\+(8VI[l)yi\ (1)

where rs and (1 are the radius and volume of the

Wigner-Seitz sphere of aluminum and 8V is the

increase in the volume of the cell containing the

impurity atom. Values for 8V were obtained from

measured lattice parameter changes [2]. (See

table 1.) The strength of the potential, Vo, was

adjusted so as to satisfy the modified Friedel

condition

- f (2/+l)8, = V= Z,-3[l + (8F/n)], (2)

where Zj is the valence of the impurity atom. The

correction term, 38V/CL, is intended to account, at least roughly, for the partial screening (or anti- screening) due to local lattice distortion. The

phase shifts were calculated numerically and

residual resistivities determined from the relation

Ap(/u, ohm-cm/atomic %)

S 1626 *> = -^-T 5 </ + D sin2 (8,-5, + (3)

where ne{{ is the effective number of free electrons

per atom and kp the wave vector at the Fermi

surface of aluminum in reciprocal Angstroms.

The phase shifts and Friedel sums are listed in table 2. The calculated resistivities, experimental

data, and calculated results obtained by Harrison

[3], who used a pseudopotential method, are given

in table 3. In Tables 2 and 3 we also present the

calculated parameters for vacancies in aluminum,

obtained under the assumption that this defect

demands a screening charge N'=— 3, i.e., we have

neglected local distortion since it is not known.

The calculated residual resistivities are consistently too small if we take rceff = ZA1 = 3. This

choice of neff may well be inappropriate since in aluminum the first BZ is completely filled and the

principal contribution to the current presumably

derives from electrons in the partly filled second

zone. Two independent observations support a choice of neff ~ 1. First, Sondheimer oscillations

in thin aluminum films [4] are in excellent agree- ment with theory provided the Fermi momentum is

TABLE 1. Lattice distortion due to impurities in aluminum

Solute 2, ha_

a n

N'

r' (Angstrom)

Li 1 -0.014 -0.0278 - 1.9166 1.57 Cu 1 -.122 -.253 - 1.241 1.43 Ag 1 .0 .0 -2.0 1.58 + .101 + .201 - 1.603 1.68 Mg 2 + .063 + .125 - 1.375 1.64 Zn 2 -.0035 -.0069 -0.9793 1.58 Ge 4 + .035 + .0694 + .7918 1.62 + .042 + .0833 + .7501 1.62

Si 4 -.045 - .0903 + 1.2709 1.53

-Y (2,+ 1)6,

7T

- 1.9167 - 1.2411 -2.0003

-1.6031 -1.3748

-0.9792 + 0.7918 + 0.7501 + 1.2706 -2.9994

that appropriate to neff =l. Second, and more

significant as regards resistivity calculations, is the

Table 2. Range and strength of square well potentials

Phase shifts and Friedel Sum.

Solute r' K„(eV) So S, 8* «3 S,

Li 1.57 3.711 -0.9240 -0.4392 -0.1234 -0.0193 -0.0017 Cu 1.43 3.086 - .6782 -.2909 -.0669 -.0082 -.0006 Ag 1.58 3.822 - .9585 -.4566 -.1300 - .0207 -.0019 Mg 1.68 2.377 -.6271 - .3758 -.1190 -.0211 -.0022

1.64 2.172 - .5555 - .3269 -.0986 - .0164 -.0016 Zn 1.58 1.697 -.4106 -.2388 - .0665 -.0100 -.0009 Ge 1.62 -1.152 + .2316 + .2082 + .0645 + .0093 + .0008

1.62 - 1.093 + .2129 + .1971 + .0609 + .0088 + .0008

Si 1.53 -2.127 + .3790 + .3509 + .0969 + .0115 + .0009 Vacancies 1.58 6.8 - 1.4831 - .6587 -.1970 -.0336 -.0032

•Research supported in part by the National Science Foundation.

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

109
fact that the calculated ideal resistivity of aluminum,

using a free-electron model, leads to a result about

3.6 times too small if ne{{ — 3, although the same cal- culation when performed for the alkali metals is in reasonably good agreement with experiment.

If, then, we take neff— 1 we obtain results in fairly

good agreement with observation. The notable

exception is Ge in Al. It would appear that in

this instance the pseudopotential method, which

should be superior to our crude model in any event, can come rather close to the measured result. It must be recognized, however, that in this instance

the strength of the repulsive delta function

in the point-ion potential for germanium was ad- justed to optimize agreement with experiment.

TABLE 3. Residual resistivities of aluminum alloys —/xfl-cm/ atomic percent

Solute Apcalc

nc[r = 3 n cfr = 1 Harrison [3]

Li 0.94 a 0.438 1.314 Cu 0.75 a

, 0.8" .248 0.744 0.76 Ag 1.1 ", 1.2 b

, 1.5 c .468 1.404 .58 Mg 0.45 a

, 0.33 b, 0.4' .218 0.654 .78 .172 .516 Zn 0.22 0.24" .097 .291 .1 .050 .15 Ge 0.79 0.98" .045 .135 1.0*

Si 0.7 \ 0.56 d .148 .444 Vacancy 2.5 + 0.7 e 1.00 3.0

*Parameters adjusted for optimum agreement with experiment.

a F. Pawlek and K. Reichel, Z. Metall 12, (1), 1(1958).

h Quoted by W. A. Harrison, Pseudo-Potentials in the Theory of Metals, p. 150 (W. A. Benjamin, New York, 1966).

c Y. Fukai. private communication.

d Quoted by Y. Fukai (see preceding paper).

e J. Takamura: Lattice Defects in Quenched Metals, p. 521 (Academic Press, Inc., New York, 1965).

The value selected, /3Ge = 60, corresponding to j (jSce - /3ai) = 8/3 = 23, is somewhat different from

that deduced from preliminary results of Heine

and Animalu (quoted by Harrison), for which

8/3 — 17. Since in the pseudopotential formulation

8/8 and also (S/3)2 appear in the expression for Ap, varying this parameter may lead to significant

changes in Apcalc. We are well aware of the gross inadequacy of our procedure, both as regards the choice of scat- tering potential as well as the selection of nei{ , and undertook these calculations primarily because we were curious to see if the rough manner by which we successfully incorporated the influence of lattice distortion in calculations of residual resistivities of

noble metal base alloys, might perhaps reproduce

the corresponding trends in aluminum, a metal

which exhibits much larger local strains about

imperfections than do the noble metals. In the case of monovalent and divalent impurities in aluminum, the indications are that lattice strains do

provide some local screening; for example, the

experimentally observed large difference between

Ap due to copper and silver in aluminum reappears

in the calculated values, and

rAp(Mg)i rAp(Mg)i

_ Ap(Zn). exp. _ Ap(Zn). calc.

We are, however, unable to account at this time for the wide disparity in the case of germanium impurities.

References

[1] F. J. Blatt, Phys. Rev. 108, 285 (1957). [3] W. A. Harrison, Pseudo-Potentials in the Theory of Metals,

[2] W. B. Pearson, A Handbook of Lattice Spacings and Struc- p. 150 (W. A. Benjamin, Inc., New York, 1966). tures of Metals and Alloys (Pergamon Press, New York, [4] K. Forsvoll and I. Holwech, Phil. Mag. 9, 435 (1964)

1958).

110
Point Imperfections in Solid Rare Gases*

Albert Gold

Institute of Optics, University of Rochester, Rochester, N.Y. 14627

The low-lying electronic states of impurity atoms

in rare gas crystals are well described by a tightbinding model in which the atomic wave functions

of the impurity are Schmidt-orthogonalized to those

centered on neighboring host atoms. It is necessary to account for the second order, van der Waals,

energy differences between ground and excited

states. A simple approximation is quite successful.

Applications to the lowest transitions of hydrogen

in argon and of argon in neon have been made.

The extension of the model to include so-called charge-transfer states is discussed. The line shape

for the Ar:H transition has been studied in detail,

considering both the usual linear interactions with

the lattice and the quadratic interaction with the

local modes. The shape and associated isotope

effects can be well understood provided that the

quadratically coupled modes are treated fully

quantum mechanically.

1. Introduction

Within the one-electron approximation the cal- culation of the electronic energy levels of impurities

in rare gas crystals is, in principle, an entirely

straightforward matter to be prosecuted in complete

analogy with the computation of the eigenstates of

any other molecular or solid state system. One

may write the many-electron wave function for the

ath level as ^„=2 Ca A (1)

where the <J) M are the members of any arbitrary

complete orthonormal set and the caii are coeffi- cients obtained variationally by solving the appropriate secular equation. The Hamiltonian to be

diagonalized is, in the Born-Oppenheimer approximation,

+irKr^r+*.....) ,2)

where i and j run over electron coordinate r,, / and

J run over the coordinates R/ of the nuclei which

*Research supported in part by the National Science Foundation.

carry charge Z/ and the final term schematically

indicates the spin-orbit interaction of the ith electron. Unfortunately, filling this prescription as given seems both computationally unfeasible and intel- lectually unrewarding. Hence, one usually begins

by choosing the <I> Mto be some particular set that experience, preference or prejudice indicates is convenient. Specifically, one hopes to make a choice such that only a few c's are substantially

different from zero. However, even this limited

task is often beyond practicality and instead one

attempts to "model" a wave function; i.e., he

chooses a specific <!>« 553 Wa and, assuming that it diagonalizes the Hamiltonian, proceeds to calculate

levels, or at least excitation energies. The ap- propriateness of a model wave function to represent

a given state of a system rests primarily on its empirical success; theosophical arguments over

various descriptions can usually be expected to be

devoid of physical content, even though they may

be of some pictorial or heuristic value. Different

states of the same system may, in fact, be best

understood using vastly different models! The

effective mass approximation seems to work well

for don^r states in valence semiconductors. The

tight-binding approximation, with which we shall primarily concern ourselves, seems adequate for molecular crystals and the low lying states of rare gas systems. We shall also look briefly into the use

of charge transfer states and pseudopotential tech- niques. We will find that sometimes apparently

different models lead to equivalent or nearly equivalent descriptions.

Ill
2. The Tight Binding Model [l] 1

In the Heitler-London picture, the electronic wave functions for the ground and excited states are taken, respectively, to be of the forms ^g =^na

(TAa) II (3a)

l.i *a

and

=MeAa (VAa) J2_ ^(r/.). (3b)

/, i #a

Here si is the antisymmetrization operator, the

double subscript indicates the ith electron belonging to the Ith nucleus and the letters Aa are reserved

for the optical electron bound to the impurity A.

In the ground state all of the one-electron functions,

(|/, are usually (but not necessarily) taken to be the

corresponding ground state wave functions for the

free atom. In the rare gases, overlap of atomic wave functions centered on different atoms, as measured by the overlap integrals

S,m= HftWjdT,(I * J) (4)

are sufficiently small to be neglected without

substantially affecting normalization of The

excited state is composed of the same ground state atomic functions for the host atoms and the cor- responding excited state functions for the impurity.

However, the overlaps of the excited optical electron

are not generally negligible (they may be ~ 0.5) and

hence, to obtain a normalized excited state, it is

necessary to orthogonalize the excited one-electron

function to those centered in the neighbors. A

simple Schmidt procedure suffices, and so we take

<Wa = (<P%a ~ XSAaIi </$) (l ~ £ (SAali)2) ^ (5)

* li ' ^ li

'

where tp%a is the atomic excited state wave function

for the optical electron. Insertion of (5) into (3b)

gives an excited state wave function normalized to order S

2

. Excitation energies are now obtained by computing the difference of the expectation values

<¥e |#|¥e> and (%\H\%). The resulting ex- pression is complicated, involving coulomb,

exchange and three-center interactions. Terms

through second order in overlap are consistently

retained. The form of the results will not be re- produced here. Suffice it to say that numerically

they yield a small (< 10%) correction to the corre- sponding atomic excitation energy but that this correction is composed of terms of both signs which

are individually large in magnitude compared to that of their algebraic sum.

In undergoing excitation the impurity atom also

suffers a considerable increase in polarizability.

Recalling that the cohesion of rare gas systems is entirely due to van der Waals forces, it is not sur- prising that this in turn leads to an appreciable

negative contribution to the excitation energy. No

rigorous methods of including the van der Waals

term in a tight-binding calculation has yet been

given, but apparently satisfactory approximate

estimates can be made [2]. We shall not go beyond

mentioning their existence here.

3. Impurities in

Impurity states in solid rare gases have been the

subject of recent study both experimentally [3-7]

and theoretically [1, 2, 8-10]. Absorption spectra

have been obtained for dilute alloys of rare gases

[5], for hydrogen atoms in argon and for alkali atoms

in various rare gas matrices [6, 7]. Calculations

have been made for Ne : Ar [1, 2, 10] and Ar : H

[8, 9]. The last is perhaps the simplest and most

successful. The quantity of machine-performed

numerical computation engenders unavoidable

uncertainty in the calculation, but the predicted

value of 10.6 ±0.2 eV for the ls-2p transition is in

4. Extensions of the

Recently Webber, Rice, and Jortner [10] have

extended the Heitler-London model by including

charge transfer states in the wave function for the

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

Rare Gas Crystals

excellent agreement with Baldini's measurement of 10.56 eV. Further, the excitation energy has been

calculated as a function of the breathing mode

configuration coordinate and the use of these results together with a careful quantum treatment of inter- actions with the quadratically coupled local mode

[9, 11] gives good fits to the line shape and the ob- served differences in width and peak position for Ar : H and Ar : D. Calculation for the lowest, *P

and 3 P states of Ar in Ne also agree well with

experiment [10].

Tight Binding Model

excited electron. The wave function is taken to be

of the form

(6)

112
where t|^„ is defined by eq (5), ip'jj are wave functions

corresponding to the "extra" electron of the negative rare gas ion on the Jlh neighboring nuclear

sites orthogonalized in the same manner as i//fla and

the C/'s are variational parameters. They have

applied this technique to Ne : Ar. The sum on J

has been carried to third neighbors and the 4s ex- cited state orbital of neutral Ne has been used for the "ionic" wave function. Analytic fits are used

for all atomic functions. These authors find that a noticeable lowering of energy (~ 0.5 eV) takes place

for i//C( compared to that which they obtain for \\i eAa alone [12]. Though these results certainly reflect the su- periority of a truly variational calculation, the very

large increase computational labor and attendant

necessary approximations somewhat obscure their

quantitative significance. Some of the simplification made include neglect of the distinction

between 'P and 3 P states and the use of the Mulliken approximation in evaluating three center

integrals. No estimated computational uncertainty is given; it seems likely that it is comparable

in magnitude to the 0.5 eV difference found.

These same workers have also studied the k = 0

exciton state of pure solid Ne [13]. It is interest- ing to compare \\) eAa and \\)ct for the case, as shown

in figure 1, taken from their paper. The resemblance of the orthogonalized tight-binding function to the charge-transfer function is striking.

One is tempted to conjecture that the elaborate

variational calculation has served to confirm the

accuracy of the simpler model.

0 4

03

0.2

0.1

00

-0.1

• 3s (Not Orthogonalized

)

3s (Schmidt Orthogonalized)

c A3s (Full Charge Transfer)

4 c)

c :

Ne*

FIGURE 1. Comparison of 4fXa and iffa for the k = 0 exciton state of pure solid Ne.

5. Summary

The lower lying excited states of impurities in rare gas crystals seem reasonably well accounted

for by a simple Heitler-London model. Both

excitation energies and line shapes are well predicted. Addition of charge-transfer states shows

promise for somewhat improving results while

generally confirming the original predictions. The

higher excited states, however, seem to fall well outside the region of validity of these treatments. Hydrogenic series are observed at short wavelengths in both pure rare gases [14] and their

alloys [3, 5]. They seem to correlate well with an effective mass model of the states [3, 5J. An attempt to unify the theory of the entire

spectrum has been made through use of a pseudopotential model [10]. Application was made to Ne:Ar. Unfortunately no prediction of excitation energies were made. Only the absolute

value of the excited state was calculated. Thus,

though a single theory of the excited states of defects in rare gas crystals would be at least aesthetically pleasing, none is yet in sight. It is still necessary to apply models a posteriori to de- scribe different portions of the spectra. On the

other hand, this situation is not entirely unique.

For example, one is still content to say that dif- ferent excited states of the same atom may be

best described in different coupling schemes

ranging from LS to jj.

113
6. References

[1] A. Gold, Phys. Rev. 124, 1740 (1961). References to earlier work on the tight binding model are contained in this paper.

[2] See, for example, A. Gold, J. Phys. Chem. Solids 18, 218

(1961).

[3] G. Baldini and R. S. Knox, Phys. Rev. Letters 11, 127

(1963).

[4] G. Baldini, Phys. Rev. 136, A248 (1964).

[5] G. Baldini, Phys. Rev. 137, A*508 (1965).

[6] W. Weyhmann and F. M. Pipkin, Phys. Rev. 137, A490

(1965).

[7] B. Meyer, J. Chem. Phys. 43, 2986 (1965).

[8] T. H. Keil and A. Gold, Phys. Rev. 136, A252 (1964).

[9] T. H. Keil and A. Gold, Phys. Rev. 140, A906 (1965).

[10] S. Webber, S. A. Rice and J. Jortner, J. Chem. Phys. 42,

1907 (1965).

[11] T. H. Keil, Phys. Rev. 140, A601 (1965).

[12] The differences found between these results for tfaa anc' those of reference 1, appear to be due to a numerical

error in the latter. [13] S. Webber, S. A. Rice and J. Jortner, J. Chem. Phys. 41,

2911 (1964).

[14] G. Baldini, Phys. Rev. 128, 1562 (1962).

114
The Use and Misuse of Models in the Calculation of

Energy Levels in Insulators*

W. Beall Fowler

Lehigh University, Bethlehem, Pa. 18015

There exist a number of models which have been used to correlate and interpret data on energy

levels in insulators. In this note we demonstrate by means of three examples that a model is not necessarily correct just because it appears to agree with experiment. The three phenomena considered have all been "explained" in terms of particular models; we show that for each of these a completely

different model leads to an equally good "explanation."

1. The A and C Bands of Tl+ in Alkali Halides

Ghosh [1] 1 compiled the available room-temperature data on peak positions of the so-called A and C

absorption bands associated with T1+ in the alkali halides. He then found that the following expressions fit rather well the observed energies (with the

exception of the simple cubic cesium halides):

EA = 0.306r+ - 1.789r_ + 7.845 eV,

(1) Ec = 0.244r+ - 2. 722r_ + 10.925 eV. Ea and Ec are the peak positions of the A and C

bands, respectively, while r+ and r_ are the Goldschmidt radii of the alkali and halide ions in ang- stroms. Ghosh's fit is such that if 8 is the difference

between theoretical and experimental energies for a given alkali halide, then the sum of S

2 for all 9

alkali halides considered is 0.0021 for Ea, 0.052 for Ec

We have performed a least-squares fit in a similar

attempt to optimize the values of Ea and Zsein terms

of ionic polarizabilities. We obtain

EA = 0.098o+ - 0. 1991 ou + 5.4452 eV,

(2) Ec = 0.029o+ - 0.3036a- + 7. 1307 eV. a+ and a_, in (A3

), are the polarizabilities deter- mined by Tessman, Kahn, and Shockley [2]. For

EA the sum of S

2

is 0.0030, for Ec it is 0.056. Our

fit is thus for all practical purposes as good as Ghosh's.

There may well be little theoretical content in either Ghosh's or our results. However, if one were to formulate a picture of what is going on in terms of the former result, he would probably guess

that the presence of ionic radii implies that the

energies of the T1+ depend mainly on the size of

the "box" in which it finds itself. Things are

actually a bit more complicated than this, because

the interionic distance does not enter explicitly into eq (1). In terms of our result a quite different picture

might emerge. The presence of polarizabilities

implies that perhaps in the excited state the elec- tron and hole of Tl+ are fairly well separated and are

polarizing the neighboring ions. Thus Ghosh's

result leads one to think in terms of a highly local- ized state, our result to a more diffuse state. The fact that both pictures fit is simply a result of the fact that the larger an ion is, the more polarizable it is. The relationship is not generally linear,

but it is sufficiently monotonic for data-fitting ap- proaches to be unable to distinguish from among

the two extreme possibilities or any intermediate

ones.

2. Exciton Binding Energies

Several authors [3], including Fischer and Hilsch

[4], have attempted to fit the observed binding

energies of n — l excitons in alkali halides by means

of an effective-mass model. In this model the

binding energy Gn of the rath exciton state is Eopi Egx — Gn _ 2 (3)

*Work supported in part by the Advanced Research Projects Agency under Contract SD-131 at the University of Illinois.

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

where n is the principal quantum number, K0 the

optical dielectric constant, and m* the reduced

electron-hole effective mass. Eopi . the optical band

gap, is the least energy required to excite an electron

from the valence band into the conduction band.

Ee'x the energy to create the nth exciton. In an attempt to test the validity of eq (3), we

have plotted in figure 1 K0Gi versus l/K0 for a number of alkali halides. The values of Gt have

been obtained from various experimental sources

115
2.0

CD

1.0

0.2 (Lil) 0.3

3.0

2.0

FIGURE 1. Exciton binding energies for a number of alkali

halides. The crosses and the circles represent data plotted according to the effective-mass model and the transfer model, respectively. The lines refer to the results of the latter model. Also shown is the value of 1/Ko for Lil, whose binding energy is not known.

See the text and reference 5 for further discussion.

and are tabulated in reference 5. The data fall into such a pattern that one might try to fit them with

three lines, one for each halogen. One might then

try to argue that the existence of three lines instead

of one implies that there is a core correction, dif- ferent for different halogens, which has been

ignored. The different slopes might come from

different effective masses for different halogens.

The preceding arguments are rather weak, but

they become extremely suspect if we recognize

that in most of the alkali halides the n — 1 excitons

are so small that the effective-mass approximation

should not work at all [5], and that there is really

very little dielectric polarization between or around

the electron and hole when in this state. We then search for another description of the

binding energies, particularly their dependence on

dielectric effects. Such a description is obtained

when we recognize, with Mott and Gurney [6], that when one excites an electron from the valence band

into the conduction band, one is creating not "bare"

particles but rather electronic polarons, with large

self-energies associated with the electronic polarization surrounding the hole and the electron. Else- where [5] we have shown that the classical MottLittleton [7] approach is valid for calculating these

self-energies in most of the alkali halides.

A simplified picture of the situation is then as

follows: one creates an n=l exciton by placing an electron and a hole into a state in which they are

confined to the same unit cell; these are "bare"

particles and little polarization is involved. But

exciting the electron into the conduction band does

involve large polarization effects; the binding energy, being the difference of these two energy states, thus depends on electronic polarization.

With very little quantitative justification, we

describe this picture in terms of a transfer model.

For discussions of the features and shortcomings

of this model, see reference 8. According to the semiclassical transfer model,

the energy required to take an electron from a negative ion and place it into a state in which it spends its time on positive ions a distance R away is

E(R) 2R+ Hse (R) + E' (4)

The coulomb energy of the electron and hole is (— e

2 /R ); the factor | comes from the virial theorem

and takes account of the kinetic energy. Hse (R) is the polarization energy discussed above. E' contains a hosl of other important contributions,

including ionization and Madelung energies, but it is assumed independent of R. We now assume

that placing the electron into the conduction band

requires E("x), while creating an h — 1 exciton

requires E(a), a being the nearest neighbor distance.

We further assume that Hse (a) is zero and write

Hse (°°) as just Hse . Then G\ is given by

la

(5)

Now, it turns out that the quantities Hse for all of the alkali halides are given very accurately by

the expression [9]

14.6 L5

K0 1.43 eV. (6)

Here a is in angstroms. Note that Hse may be as

large as ~ 5 eV.

Qualitatively, then, this model says that G\ should

vary as fl/a] [(1/A^0 ) + const], and it suggests that

if we plo L aG \ versus 11 we should obtain a straight

line. Such a plot is given in figure 1 (circles), and we do indeed obtain 3 straight lines whose separation may again be due to core corrections. The

slopes of the iodide and bromide lines even turn out to be very close to that predicted by eq (6). Although these results may be quite fortuitous, we feel that the physical picture described is more

accurate than that provided by the effective-mass

model. Note that it is not surprising that both

models work; empirically, a tends to be a monotonic, sometimes linear, function of K0 .

116
3. The Relaxed Excited F Center

Spinolo [10] has shown that the thermal ionization

energies Et of excited F centers in a number of

alkali halides are fit quite well by an effective-mass

model similar to that of eq (3): ET ~ 1/ K 2, where K is the static dielectric constant. In figure 2 we plot

0.5

FIGURE 2. Thermal ionization energies for the excited F center

in a number of alkali halides.

The c rosses and the circles correspond to large-orbit and small-orbit models, respectively. The data for Et were obtained from reference 10. KEt versus 1/A^, using Spinolo's data. A rather

good straight fine obtains. Such a fit is in the spirit of the "large-orbit" picture of the relaxed excited

state developed by the author. In this picture

there exists some lattice polarization between elec- tron and vacancy which leads to an "effective"

dielectric constant whose value is between K0 and A.. Actually, it is possible to take account of the

change in the lattice self-energy accompanying the

thermal process and do a bit better than l/K 2

. We

neglect the effect of self-energy due to electronic

polarization, assuming it to be the same in the

excited state as in the ionized state. Mott and Gurney [12] have shown how to compute

this difference of lattice self-energies El in a con- tinuum approximation. The result is

EL =

2R

1

^eff

(7)

R is the "radius" of the electronic state, K the static dielectric constant, and Ke(( the "effective" dielectric constant when the electron is in the excited

state.

The optical ionization energy is

E =op 2RKf (8)

The thermal activation energy Et equals E0i> — El, or

«2

In the effective-mass theory [11] for the F-center 2p state, R approximately equals 4Kefr /m*, and so we have

„ 13.6 m*

ET = —— e V

4 KK F = ^OP

eff

13.6 m*

4 K2 eV. (10)

Using the parameters m* = 0.6, Ke({ —^.2 derived

by the author [11] for NaCl, we find that Et equals

0.09 eV and that Fop equals 0.12 eV. Since Ke{{ will tend to vary linearly with K, we see that the

expression for Et is consistent with Spinolo's law. Now suppose that the state were not diffuse but were compact, and that R ~ a, the nearest neighbor

distance. Then

Et —

2Ka (11)

2KR (9)

In fact, this fits Spinolo's data about as well as does l/K2 (see fig. 2). The slope even turns out

to be about correct, although the experimental

line does not extrapolate to (0, 0). The reason

that both models "work" is again that K tends to be a monotonic function of a. In the small-orbit model, if we assume Kef[ to be 2.3 for NaCl, we find that the difference between

Et and Eop is 0.7 eV. This is quite different from

the 0.03 eV obtained for the large-orbit model;

however, the thermal activation energies ET turn out to be about the same for both models. Thus

the thermal ionization energy is apparently not a very sensitive indicator of the nature of impurity

states. If the state is compact and the optical

binding energy large, there will be a large lattice energy El which tends to make Et small. If the

state is diffuse, Fop is already small, and Et will also be small. In the case of the relaxed excited F center, there

are good reasons for thinking that the large-orbit

model is valid [11]. The point stressed here is that the behavior of Et does not by itself lead un- ambiguously to this conclusion.

As often happens, improvements in the amount

and quality of experimental data seem to lead to poorer agreement with either model than indicated

in figure 2. Lowndes [14] has recently measured

static dielectric constants for a number of alkali halides. while Podini and Spinolo [15] have re- ported values of Et for several more alkali halides.

The situation in NaF is apparently more complex

than earlier recognized [16]. The updated values of K,' Et, and a, plotted as

in figure 2. indicate more scatter than the older

values shown. In particular, Rbl appears to have

a value of Et too small by nearly a factor of 2. The

need for more data is clear: of the crystals for which

both ET and new values of K are known, only CsBr

and NaCl have values of K which are outside the

117
range 4.6 to 4.8. For curve-fitting purposes this

is not a desirable situation. D. L. Dexter has told the author of another case

in which two models have been used to fit the same

data. The Mollwo-Ivey law [17], in which F-center

absorption energies vary as a~n

, where n is about 2, has long been known to agree rather well with

experiment. The physical interpretation of this model is that the F center in absorption behaves

as a particle in a box. Wood has recently investigated the theoretical validity of this model [18]. In a note published ten years ago, Levy [19] ex- hibited another relationship, Eabs = 3e2l8K0R, where

R is the cation radius. This seems to agree with

experiment to within 10 to 20 percent. The physical interpretation of the Levy model is not clear;

it appears to contain some features of a continuum

treatment, some features of a localized treatment. But it does work fairly well.

4. References

[1] A. K. Ghosh, J. Chem. Phys. 43, 2978 (1965).

[2] J. R. Tessman, A. H. Kahn, and W. Shockley, Phys. Rev. 92, 890 (1953).

[3] J. Hermanson, Phys. Rev. 150, 660 (1966). Hermanson

has made an extensive study of the corrections to eq (3) and has shown that in some cases these corrections

cancel.

[4] F. Fischer and R. Hilsch, Nachr. Akad. Wiss. Gottingen,

II Math.-Physik. Kl. 8, 241 (1959); F. Fischer, Z. Physik

160, 194 (1960).

[5] W. Beall Fowler, Phys. Rev. 151, 657 (1966).

[6] N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals (The Clarendon Press, Oxford, 1948), 2d ed., reprinted by Dover Publications, Inc., New York, 1964.

[7] N. F. Mott and M. J. Littleton, Trans. Faraday Soc. 34, 485

(1938).

[8] R. S. Knox, Theory of Excitons (Academic Press, Inc., New

York, 1963).

[9] F. K. Du Pre, R. A. Hutner, and E. S. Rittner, J. Chem.

Phys. 18, 379 (1950).

[10] G. Spinolo, Phys. Rev. 137, A1495 (1965).

[11] W. Beall Fowler, Phys. Rev. 135, A1725 (1964).

[12] Ref. 6, p. 162.

[13] To be consistent with our earlier discussion of electronic polaron effects, and our omission of them here for the reason stated, we should assume that R >a in this case, and R < a in the exciton problem. This distinction will not substantially affect the curve-fitting approach which we use.

[14] R. P. Lowndes, Physics Letters 21, 26 (1966).

[15] P. Podini and G. Spinolo, to be published.

[16] P. Podini, Phys. Rev. 141, 572 (1966).

[17] E. Mollwo, Nachr. Ges. Wiss. Gottingen Math.-Physik. Kl. p. 97 (1931); H. F. Ivey, Phys. Rev. 72, 341 (1947).

[18] R. F. Wood, J. Phys. Chem. Solids 26, 615 (1965).

[19] M. Levy, Nature 177, 241 (1956).

118
Methods for Calculation of the Electronic Structure of Defects

in Insulators*

U. Opik** and R. F. Wood

Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 38730

The comments which we shall make in this paper

apply most directly to calculations of the electronic structure of the F center in alkali halides. We

expect them, however, to be valid for a large number of defects in insulators, such as the U, M, and

R centers in ionic crystals, the hydrogen interstitial

in CaF>, an yttrium ion in CaF2, etc. In calculating the electronic structure of the F

center in the alkali halides two methods of repre- senting the defect wave function, i//d, have generally

been used. In the LCAO-DO method the defect

orbital (DO) is expressed as a linear combination

of atomic orbitals (LCAO) on the neighboring ions. Thus, ^= 2c,.j</>j(r-R„), (1)

where <fo(r— R„) is the jth unoccupied atomic

orbital on the" vth ion. This leads to a secular

equation

\(v,j\h\fjL, k)-(v,j\fi, k)e\=0 (2)

for the one-electron energies, e. The difficulty

with this method is that the functions </>„, j are quite

diffuse, whereas will usually be rather well local- ized and hence the sum in eq (1) must contain several

orbitals on each ion in order to give the proper de- gree of localization. In addition to the orthogonality

problems encountered in evaluating (v, j\fi, k) one

has a somewhat related problem in calculating the

matrix elements of the Hamiltonian, (v,j\h\fx, k), in that they may have large contributions from regions

of space in which itself is practically vanishing.

The point which we wish to make here is neither new nor surprising, but perhaps needs reemphasizing occasionally. Stated very simply, it is that

it is not economical to express a localized function

in terms of a basis set of diffuse functions, and

conversely. Perhaps the most familiar case of this principle occurs in band theory where the OPW

method circumvents the need to express the highly

localized core states in terms of plane waves.

In calculating for defects in alkali halide

•Research sponsored by the U.S. Atomic Energy Commission under contract with Union Carbide Corporation. "Visiting scientist from Queens University. Belfast, North Ireland.

crystals, it seems more useful and natural to use what we shall call the defect centered (DC) method.

In this method, ifjd is written as Ur) =N[Ur)-^ (0\v,j)<j>vJ(r)] (3)

j

in which </>o(r) is a fairly smooth function which

may be chosen in various ways. The sum over

v, j ensures that (//<, will be orthogonal to the occupied ion orbitals, /r) = </>j(r — /?„). Equation

(3) assumes, in effect, that the <f> v j do not overlap

with each other, and this simplification must be

investigated in each case. If these overlaps are not negligible, it might prove convenient to first use the method of symmetrical orthogonalization

to transform to a set of mutually orthogonal core

functions.

Expressing tya as in eq (3) and evaluating (»/>,/ \h\ipti), one obtains in addition to the familiar point ion

approximation a contribution from exchange with

the core orbitals, a coulomb term due to penetration of $0 into the core region, and a contribution,

arising from the overlap terms in eq (3), which we

shall refer to as the overlap energy. Table 1 shows

the magnitude of these contributions in a number

of cases [l]. 1

Table 1. Defects in KC1: contributions to the total energy, in eV

These are in addition to the point-ion energy.

Defect and state Exchange Penetration Overlap Total

F center ground -3.016 - 1.715 + 6.993 -4.721 F center excited -4.720 -3.178 + 10.934 - 1.978 -1.422 -0.828 + 4.391 - 28.734

Since these terms are not negligible and do not even approximately cancel each other, we now wish

to consider methods of calculating them. In fact,

at the present level of approximation, only the

exchange terms present a really diffic ult problem.

A part of the one-electron Hamiltonian is the ex- change operator, hex , defined by

/ ft v jl / \ f

</>-M-(r')u(r' ) , ftexa(r) = 2] 2 J \r-r'\ { *

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

119
where all wave functions are assumed real. To

understand the difficulty, consider that each of the

<\> v , i in eq (4) is supposed to be a Hartree-Fock free- ion orbital. The preferred method of calculating

these orbitals now is to expand them in terms of

Slater orbitals with nonlinear variation parameters

[2]. This expansion may contain six or seven such

orbitals in the case of positive alkali ions. In order

to calculate the exchange integrals between (f)o and

4>„j accurately, it is customary to use elliptic co- ordinates. This method, though quite accurate, is very time consuming, involving as it does the calculation of the exchange between the defect electron

and charge distributions given by products of the

individual Slater orbitals. Furthermore, since (f)o may also be expressed as a sum of orbitals

</>o

: 2 dk^Ok,

k

(5)

the number of two-center, two-electron integrals

which must be calculated at each stage in the

minimization procedure, can become very large.

Even on a fast computer the computational time

involved in this method becomes a very serious

factor. Therefore, in the remainder of this paper we shall sketch a method which circumvents this

difficulty and then end by giving a few comparisons

of accuracy and time using the two methods.

Let

u„,(r, <9, <f>) =- P(nl\r)Ylm(0, 4>) (6)

be the orbital of an electron outside the spherically

symmetrical closed shells of an ion, r, 6, <fi being

the spherical polar coordinates centered at the

nucleus, / and m being the azimuthal and orbital magnetic quantum numbers, respectively, and Yi,n being a spherical harmonic. The Hartree-Fock

equation for the radial function P(nl\r), belonging

to an energy value E [3], is

2Zp

(r) r

1(1 +1)

P(nl\r)

2 2/' + l + - X =Hp 2 2 CiVkYk{nl, n'l'\r)

r ? 2

X P(n'r\r)-^ en i. „'iP(n'l\r) = -2EP(nl\r), (7)

n'

where

Yk(nl, n'l'\r) = r- k

\ r'kP(nl\r')P(n' l'\r')dr'

+ r

k + i r'-K-iP(nl\r')P(n'l'\r')dr' ,

Zp{r) is Hartree's "effective nuclear charge for potential," primed quantities refer to the core

orbitals, the constants Cwk are defined by Condon

and Shortley [4] (sec. 9

6

), and the so-called nondiagonal parameters, e„;, n'i, are determined so as

to make the function P(nl\r) orthogonal to the core functions P(n'l\r).

Defining 8Z p by

oZ„(r)

1 f2/' + l P(nl\r)£,{ 2- 2

8ivreni, nn\P(n'V\r), (8)

we may call — 8Zp (r) jr and effective exchange potential energy, and rewrite the Hartree-Fock equation as Zp (r)+8Zp

{r) 1(1+1] + 2E\P(nl\r)=0.

(9)

In agreement with what has been previously

found by other authors (see, e.g., Biermann and

Liibeck [5], and Stone [6]), we find that the function

8Zp (r) depends strongly on the azimuthal quantum

number / but only slightly on the energy, E, provided

that the energy E is well above the energies of the core orbitals. We can thus replace the operator

-r-iZp {r)+he (10)

by an operator U, defined so that for an eigenfunction Vi(r) of the square of the orbital angular mo- mentum belonging to an eigenvalue 7i 2 l(l+ 1),

fMr) = - r-i [Z„ ( r) + SZ^(r) . (11)

8ZjP being the function defined by (8) for that value

of /. Having tabulated the functions SZj," for each

of the ions, we can use expansions of (f>ok and 0oain spherical harmonics about the ionic nuclei to evaluate the matrix elements,

<0O*|tf|0O*'>, (12)

thus avoiding the need for performing the timeconsuming calculations of the matrix elements of

hex coming from eq (4). In tabulating the functions

8Zp (r), we have to smooth out the singularities that arise at the zeros of P{nl\r). The functions

8Zp (r) are so insensitive to the energy E that a rough knowledge of E is sufficient to enable us to calculate them with the required accuracy. To summarize, the advantage of the present

method is that, in determining the functions 8Zp (r), we in effect calculate the exchange integrals once and for all, instead of having to recalculate them

every time they are needed.

120
It should be realized that our procedure replaces

the original one-electron Hamiltonian A by a new

one, say hef{, such that only those eigenvalues of h

which are above the core eigenvalues, and the eigenfunctions belonging to these, are practically indentical to the corresponding eigenvalues and eigenfunctions of he!f. This is not true of the core

eigenstates of the two Hamiltonians, and it can be

shown that in carrying out the orthogonalization

procedures, we now have to use the core eigenvalues and eigenfunctions of he(! and not those of h. This method requires the expansion of each of

the "smooth" trial functions fok in spherical har- monics about the nucleus of each of those ions

that are treated as extended ions, and this can in

itself be very time consuming. It was therefore

decided to gain speed by sacrificing a certain

amount of mathematical accuracy. A computer

program was written which expands an arbitrary

function in spherical harmonics about an arbitrary

center, on the assumption that terms of degree

l> L, where L 12, are negligible. Instead of

numerical integration, the program uses summations over a set of judiciously chosen points within

a sphere (whose radius is input data) outside which

the expansion will be inaccurate. No spherical

harmonics are evaluated during the expansion

process: instead, the spherical harmonics at the

required points, multiplied by suitable weight factors, were stored as constants of the program when

the program was written. The method turned out

to be more accurate than had been anticipated.

Depending on circumstances, the expansion of a

typical Slater-type function of not too high an

azimuthal quantum number takes between 0.5 and 4.0 sec on the CDC 1604A computer.

These new methods have been tested on the F

center in KC1, by comparing the results with those

obtained by calculating the exchange and overlap

integrals by more conventional methods (ref. 1). The energies agree to within 0.03 eV, but the energy

difference between the ground and the first excited

state was found to differ from that obtained by the

conventional methods by only 0.005 eV. Most of

the 0.03 eV inaccuracy in the energy probably

arose from our failure to achieve complete self- consistency in solving the Hartree-Fock equation

(7) for the free K+ ion when we determined SZp

(r). According to some indications, the errors arising

from our method of expansion in spherical har- monics are probably not greater than 0.003 eV. These are small sacrifices if one considers the gain

in speed: a calculation which previously took over

2 hr can now be done in less than 5 min, provided

that the functions hZlp have been previously de- termined; but this probably overestimates the

gain, because in the early calculations by the conventional methods we had used some quantum

chemistry routines which had been written for problems in which much greater accuracy was re- quired, and, moreover, we probably did not use

these routines in the most efficient way. The

determination of the functions bZ'p

, which we do

separately for six values of /, takes about 7 to 10

min for each ion: once determined, these functions can be used any number of times.

References

[1] Taken from unpublished calculations of R. F. Wood and

H. W. Joy on the F center and R. L. Gilbert and R. F. Wood (to be published) on the U center. See also R. F. Wood and H. W. Joy, Phys. Rev. 136, A451 (1964).

[2] The method used is that of C. C. J. Roothan, Rev. Mod.

Phys. 23 , 69 (1951) and 32, 179 (1960), with Slater orbitals as a basis set.

[3] D. R. Hartree, The calculation of Atomic Structures (John Wiley & Sons, Inc., New York, N.Y.. and Chapman and

Hall. London, 1957).

[4] E. U. Condon and G. H. Shortley, The Theory of Atomic

Spectra (University Press, Cambridge, 1935).

[5] L. Biermann and K. Liibeck, Z. Astrophys. 25, 325 (1948).

[6] P. M. Stone, Phys. Rev. 127, 1151 (1962).

256-724 0-67—

121

A Model to Treat Lattice Distortions and the F-Center

Electron Consistently*

Herbert S. Bennett t Department of Physics and Materials Research Laboratory,

University of Illinois, Urbana, 111. 61801

In this note we present a model of the lattice which is as simple as possible and yet which contains many of the features of an exact treatment. But most important we require that the same lattice model be able to accommodate either a spatially compact or diffuse vacancy-centered F-electron wave

function. We feel that this last condition has not received sufficient consideration in the few previous

consistent calculations of the F-electron wave function and the lattice distortions.

The F-center lattice defect in ionic crystals

consists of an electron localized about a vacant

negative ion site. Many calculations of the electronic structure of the F center appear in the

literature [1,2]. 1 They all have one feature in common. Namely, the potential energy term in the electronic Hamiltonian is a constant for dis- tances less than a distance of the order of the

nearest neighbor distance. This constant potential

(the potential well depth) contains at least one term

which depends upon the electrostatic crystal potential at the vacant site (the Madelung potential).

Most of the calculations in references [1] and [2] consider the ground state and low lying excited

states of the center in an undistorted lattice. Lattice distortions are present and may change the

potential well depth. As far as we know, only a few authors have attempted to include the effect

of the lattice distortions in a consistent manner

[2|. The emission spectra of the F center in the

alkali halides indicates the importance of lattice distortions. The emission band exhibits a large

Stokes shift and the first excited state has a life- time which is greater than the lifetime one expects

from atomic spectra [3]. We shall take the view that the nature of the wave function of the F electron may change substantially between emission and absorption [3,4]. The F center originally in its ground state, |a,0),

becomes excited into the state, | a,n ), which is assumed to be a quasi-stationary state with an

electronic wave function calculated from the same

crystal potential as that for the state, | a,0 ). The

lattice then relaxes and thereby the crystal potential which the F electron experiences changes.

The excited electronic state, | e,n ), calculated

from the relaxed crystal potential may differ from

the excited state, | a,n ). The continuum model

"This work was supported in part by an United States Atomic Energy Commission

Contract AT(U-1)-1198.

t Present address: National Bureau of Standards. Washington. D.C 20234.

' Figures in brackets indicate the literature references at the end of this paper.

of W. B. Fowler [1] suggests that the relaxed

excited state, \e,n), from which emission occurs, may be very diffuse; i.e., the wave function,

( r | e,n ), may have its maximum at four to five nearest-neighbor distances while the ground

state, | e,0 ), into which emission occurs remains

rather compact with a wave function, ( r | e,0 ), confined mostly to the region within the nearest- neighbor distance.

The electronic part of the Hamiltonian contains

the self-consistent potential which the electron

experiences and which is a function of the lattice configuration. All calculations thus far represent

this potential for distances less than the nearest- neighbor distance as a potential well whose depth is determined in part by the Madelung potential.

We compute the Madelung potential by viewing

the lattice as a collection of point ions and this suggests that we use the point ion approximation

as a zeroth order approximation to the lattice. The

electronic Hamiltonian yields an electronic wave

function which may be either spatially diffuse or compact. This F-electron wave function gives

rise to a charge density

pF (r)=-^*(r) t//(r).

The brevity of this note forces us to consider here

only the two extreme cases; namely,

diffuse: p/.(r)=0,

compact: pr(r) =— e8:! (r)

Even though we shall also neglect polarization and

defect self-energies in our discussion below, such

aspects are easily incorporated into the model.

We want to compute the change in the lattice energy due to replacing an anion with a F-center

electron. We first create a vacancy at the anion

site ro = 0 of charge Zo and permit no lattice re- laxation. This fictitious lattice state will serve as

123
the reference energy for the lattice. Classical

ionic lattice theory gives the lattice energy as a sum of the electrostatic (Coulombic) energy of the

lattice, Ee , and the repulsive energy of interaction

between the ionic cores of the lattice, Er \ EL = Ee +Er, (1)

where

Ee^^ZiZjln-rjl-K-Zo ^Z^l"1

, (2)

i.j j*0

and

Er =\%<pr(.ri-Vi)-%<Pr{r))- (3)

We may interpret the second term of Ee by saying

that we create a vacancy by adding an ion of charge — Zo at r0 = 0 or equivalently by saying that we re- move the electrostatic "bonds" involving the

point i — 0. The second term in Er means that we

remove the repulsive "bonds" involving the point

i = 0. The repulsive energy (pr (ri — rj) between

the tth and jth ions take the Pauli exclusion principle between the tth and 7th cores into account

and is a short-ranged function of r/j=|r, — r,-|. The

function <pr (nj) usually has the empirically deter- mined Born inverse power law form,

tpAru) = ary\ X > 0, (4)

the Born-Mayer exponential form,

(pr ( nj ) = bbibj exp (— rijlp), (5)

or some combination of the two forms (Verwey).

The quantities a and \, and b, bi, bj, and p are determined from experiment and are given in the

literature [5]. Forms (4) and (5) assume no elec- tronic polarization of the ionic cores and because

the experiments used to determine their parameters involve only small ion displacements, we must

at the least be cautious whenever the distance be- tween ions becomes substantially less than the sum of their ionic radii. We now allow the n nearest neighbors of the

vacancy (defect ions) to move radially inward or outward to the sites r,

! = r/(l — cr) for 1 =S i n. All other ions are to remain at their perfect lattice

sites in this model. The change in lattice energy

then assumes the form,

AEl (vacancy, distortion) = E'L — El = AEe + AFr , (6) where

AEe = El +E2 + E3 + E4 , (7)

E1 = nZ1 ^{In'-rjh-lri-rjj-1 }, (g) &=-nZ, £ Zjiln'-rjh-ln-r^hh (9)

^=4 ZiZjUn'-r/l-i-ln-vjl-i}, (10)

E4 = -Z0 £ ^{Ir/I-1 -!^-1 }, (ID:

and

AEr = Er >-Er . (12)

The lattice configuration obtains at that value of cr for which the change in the lattice energy is a minimum.

If we assume that the F electron has negligible

repulsive interaction with the core electrons, then

the limit of a very diffuse F-electron state cor- responds to removing both the electrostatic and

repulsive core "bonds" involving the point i = 0;

i.e., we minimize LEda). On the other hand, the

limit of a very compact F-electron state corresponds

to removing only the repulsive core "bonds" in- volving the point i = 0; i.e., we minimize AEdv)

-£4(0-). Past authors [2] have assumed that the ions

experience repulsive core interactions with only

their nearest neighbors. Using a computer to minimize AEL and AE1—E4 for NaCl, we find that

for the range — .06 2s crd

Ss — .20, AEd(Td) is a minimum depending upon which form for (b r we choose

to describe the nearest-neighbor-only repulsive core interactions. The same examination of AEda-) — E4 ((r) yields that A£/Jcrc ) — Ea (<tc ) will be a minimum only for values of crf

. greater than + 0.30. This large inward motion of the ions to accommodate

a very compact F-electron state results in the n

defect ions, which are next nearest neighbors of each other, approaching within a distance roughly

equal to the sum of their ionic radii. This means

that we must include the next-nearest-neighbor core repulsions, particularly for the compact

F-electron states. A qualitative examination of the terms in AEl

and in AEL — E4 as functions of cr also reveals the same results for the nearest-neighbor-only core

interactions. To facilitate this qualitative examination, we summarize the contents of the terms Ei

to F4 . The term Fi is the change in electrostatic energy which occurs when a neighboring catior moves in the background of a perfect point ior lattice potential. The quantity F2 represents the change in the electrostatic energy which occurs|

when one of the n defect ions moves in the poim

ion potential of the remaining (n — 1) defect ions al undistorted lattice sites. The term F3 is the change

124
in the electrostatic energy of the n defect ions when

all n defect ions move radially inward or outward.

And finally, En represents the change in the electrostatic energy between the effective charge ( — Zo)

and the lattice due to the motion of the defect ions. To be specific we use the form (4) for which

A ~ 7.0-9.0. The form (5) also leads to the same

conclusions. Recall that a diffuse state corres- ponds to adding an effective charge — Z0 at i = 0

(remove electrostatic "bonds" to i = 0). But — Zo

has the same sign as the nearest-neighbors to i = 0. Hence, the electrostatic part of A£"/., E 4 , and also Ei and £3, decrease for cr < 0. Only the term Ei

increases for cr < 0. In addition we have A >1 and

a sufficient number of repulsive nearest-neighbor

"bonds" which decrease in length for cr < 0. These

conditions combine to bring about an accompanying increase in A£V which offsets the decrease in AEe before |cr| becomes too large. However, such

a balance does not obtain for the case of a compact

state, which is the limit of removing only the re- pulsive "bonds." In this case we consider AEl — E4. The electrostatic terms E\ and Ez decrease

for cr > 0. Only the term E3 increases for cr > 0.

Thus, the electrostatic forces tend to. move the n

defect ions inward, cr > 0. But most important,

because there are no nearest-neighbor repulsive

"bonds" which decrease for cr > 0 the defect ions move excessively inward until the electrostatic

term E3 increases sufficiently to offset the decrease

in the remaining electrostatic part AEe — E3 — E4

and in the repulsive term AE, . Notice that when the next-nearest-neighbor re- pulsive "bonds" are included, those "bonds"

among the n defect ions of i = 0 decrease for cr > 0. In fact, upon minimizing AEL — E4 with next-near- est-neighbor repulsive "bonds" we obtain a much

smaller inward distortion for a compact F-electron

state and about the same outward distortion for the diffuse F-electron state as before. The quantity

crc is in the range 0.1 > crc > 0.01 and the quantity

<Td is in the range — 0.2 < crd 0.05 depending

upon the form and parameters of <pr . Hence, we

state that one must include next-nearest-neighbor core repulsions when considering a compact Felectron wave function in a model lattice in which

only the nearest-neighbor ions to the defect move

in the so-called breathing mode.

Refe

[1] S. I. Pekar, Uspekhi Fiz. Nauk. 50, 197 (1953); T. Kojima,

J. Phys. Soc. Japan 12, 918 (1957); B. S. Gourary and

F. J. Adrian, Phys. Rev. 105, 1180 (1957); B. S. Gourary

and A. E. Fein, J. Appl. Phys. 33, Suppl., 331 (1962); W. B. Fowler, Phys. Rev. 135, A1725 (1964): and J. K.

Kiibler and R. J. Friauf, Phys. Rev. 140, A1742 (1965).

[2] R. F. Wood and J. Korringa, Phys. Rev. 123, 1138 (1961);

rences

and R. F. Wood and H. W. Joy, Phys. Rev. 136, A451

(1964).

[3] R. K. Swank and F. C. Brown, Phys. Rev. 130, 34 (1963).

[4] W. B. Fowler and D. L. Dexter, Phys. Rev. 128, 2154 (1962).

[5] G. C. Benson and E. Dempsey, Proc. Roy. Soc. A266, 344

(1962); and M. P. Tosi, Solid State Physics 16, 38 (1964).

125

T-Matrix Theory of Localized Electronic States Due to a Vacancy

With Application to Diamond

K. H. Bennemann

Institute for the Study of Metals, University of Chicago, Chicago, 111. 60637

1. Introduction

The problem of determining the electronic state

of a vacancy in semiconductors, particularly in diamond, has found some interest in the past and

due to the remarkably sensitive derivative tech- niques used presently with great success in studying

optical properties of solids, there is good reason to believe that this problem will soon become active

again. First, I would like to describe briefly the

work which has been done in the past on the va- cancy problem. Then I would like to advertise a new method for determining the electronic structure

of a vacancy and to apply the theory to diamond.

Finally, I would like to introduce a new aspect on

charged vacancies.

In the past Coulson and Kearsley £1]\ Yamaguchi

[2], and more recently Stoneham [3] have made

detailed studies of localized electronic states at neutral and (positively or negatively) charged vacancies in diamond. In all these studies the electronic structure of the vacancy has been determined by

applying a molecular-orbital technique using oneelectron wave functions centered at the vacancy or using linear combinations of atomic orbitals (LCAO). The molecular orbitals were constructed

from the tetrahedrally hybridized 2s and 2p atomic

orbitals. Then from such "vacancy"-electronic

orbitals Slater determinantal wave functions were

formed representing a special electronic vacancy

configuration taking into account properly lattice symmetry of the vacancy and electron spins.

Finally several Slater wave functions with the same

spin multiplicity and space symmetry were linearly

combined. The spectrum of energies for the localized electronic states were then determined from

the appropriate secular determinant including

configuration interaction and interaction among the

localized electrons and among the localized elec- trons and lattice atoms. The results obtained

differed mainly due to uncertainties in determining

the interatomic electronic coulomb interaction

matrix elements.

Instead of improving these previous methods

which are all very much in the spirit of "dangling

bonds," we would like to propose a very different

method for treating vacancies in semiconductors.

Recently the pseudopotential theory as developed

by Kleinman, Phillips, Cohen, Heine, et al., which

replaces the crystal potential by an effective potential, in many cases much weaker and smoother, has

been successfully applied to semiconductors such

as C, Si, Ge, etc. Using such an effective crystal

potential permitted one to treat the valence elec- trons approximately as quasi free-electrons. This

suggested that we determine the electronic structure

of defects in semiconductors using scattering theory;

this is in a way an extension of Friedel's theory of

defects in metals.

Therefore using /"-matrices a theory of the elec- tronic structure of vacancies in semiconductors is developed and applied to diamond. This theory

unifies the effective mass approximation, which is adequate to treat shallow localized states (Kohn,

Luttinger, et al.), and the localized perturbation

theory (Lifshitz, Clogston-Slater et al.), which is adequate to treat deeply bound states.

2. Theory

The most remarkable feature of the diamond-type

lattice is that each atom sits at the center of a regular tetrahedron formed by its four nearest neighbors to which it is covalently bound. Therefore we have four broken covalent bonds around

the vacancy. This gives rise to a strong change in the valence electron distribution in the vicinity

of the vacancy and leads to a lattice distortion

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

around the vacancy. Therefore, it is not reasonable

to determine the redistribution of the valence

electrons around the vacancy by perturbation theory

using the perfect crystal as the unperturbed state. Instead, we determine self-consistently the elec- tronic distribution and crystal potential in the

vicinity of the vacancy by scattering theory. By

including multiple electron scattering involving

different lattice atoms we take explicitly into ac- count the atomic configuration of the lattice. A

127
vacancy means that an atomic scatterer is missing

in the lattice. In detail the crystal potential is obtained as follows: after replacing the crystal

potential V by the appropriate pseudopotential Vs , resulting from using orthogonalized wave functions, we introduce the one-electron Green's function

G and the ^-matrix T by

and

G — Go + GoTGa

T=V+VGoT.

(1)

(2)

The scattering matrix T describes the electron

scattering by the self-consistent crystal potential

V. The atomic configuration of the crystal comes

in through T as is seen by the following expansion:

T(k,q,E) = Jj Tl (k,q,E)e'i ' k)r, (3)

with

Ti{k, q,E) = ti{k, q,E)+ 2 f tl (k,p,E)G0 (p,E)

TAp,q,E)e^- k^'dp3 +. ... (4)

The higher terms result from multiple scattering

due to 3, 4, and more atoms. T\ gives the electron

scattering associated with the atom I. If a vacancy

is present, then l=£V, where V denotes the vacant

lattice site, ti results from V\ neglecting the

crystalline environment of the ion /, and Ti results

if the multiple electron scattering involving the

Atom I 3 Atom 5 6 7

(Vacancy)

FIGURE 1. Valence electron charge density near a vacancy in diamond, (111) axis.

Curve "a" refers to the perfect crystal, and "b" to a crystal with a vacancy. Distance is measured in terms of the cubic lattice parameter, a.

T~i~r ~ i Atom 12 3 4 5 Atom

(Vacancy

)

FIGURE 2. Valence electron charge density near a vacancy in diamond, (1—10) axis.

Curve "a" refers to the perfect crystal, and "b" to a crystal with a vacancy. Distance is measured in terms of the cubic lattice parameter, a.

surrounding lattice atoms of the atom / is take

into account. From T the electron distribution in the vicinit of the vacancy is determined. Figure 1 shows t valence electron charge density along a (lll)-axi

and figure 2 shows the charge density at the vacan

along a (1, — 1, 0)-axis. The lattice relaxation h

been neglected (which probably leads at most

about 10% corrections). These calculations she some light on the validity of the often used concep

of dangling covalent bonds, of stretching covalen

bonds, and of reforming covalent bonds from th

four electrons in the broken covalent bonds. Th<

results indicate the strong deviation from spherica

symmetry of the potentials at the nearest neighbo:

lattice sites of the vacancy and suggest that a poin

ion approximation and a dielectric screening treat ment, neglecting local field corrections, are bounc

to be poor approximations for such strong lattict perturbations as given by a vacancy.

After having determined the crystal potential the localized states in the resulting crystal potential a the vacancy are given by the poles of the T-matrix

To obtain the localized states within the forbidder

gap we look for the poles of

T(E)=VGo(E)T(E) (5;

In determining the poles of Ti (E) it is necessary

to take into account the poles associated with the other three Ti (E) which appear for the samt

energies E for symmetry reasons. Such a self consistent determination of the poles is probably

equivalent to configuration interaction. However

contrary to configuration interaction such a self consistent procedure does not lift the degeneracy

of the localized states at the four nearest neighbors

This might happen via Jahn-Teller lattice distortion

If the wave functions associated with the before

mentioned localized states do not overlap, then it is also possible that this degeneracy just persists

128
(see degeneracy of Is core electrons, e ]s same for each atom). It follows that the spectrum of bound

states is approximately given by

l-^ViJk,p)G0(p,E)

-~^e^-^,Vr (k, p)Go(p,E) = 0 (6)

if (a) the potential ^is highly localized in real space

and if (b) the Fourier transformed potential is sharply peaked, which case is adequate for shallow

bound states described by the effective mass

approximation. Using eq (6) the spectrum of bound

states is determined. The results are given in figure 3. Notice please, that the top of the valence

band refers to the crystal point symmetry r25 and

the bottom of the conduction electron band to r\5 . The symmetry of the bound states is Ei(l = 0): E2(l=l); E3 (l = 2); £4 (Z = 3).

3. Charg

The vacancy might trap electrons in the localized

states and also might release electrons from the

broken covalent bonds to the conduction band and

become charged in this way. It would be nice to get these charged vacancy states from the theory,

without introducing explicitly wave functions for the electrons in the localized states. This can be

done if one performs a self-consistent (bootstraplike) calculation taking carefully into account the

localized states in determining the crystal potential which gives rise to the bound states. Charged vacancies and also neutral vacancies

might be in paramagnetic states. They act then

like paramagnetic impurities having an exchange

interaction with the conduction electrons. Therefore paramagnetic vacancies should give rise to anomalous conduction electron scattering (Kondo

Effect) possibly exhibited in resistivity, magnetoresistivity, magnetic nuclear relaxation rate, etc. Possibly the observed anomalies in the conduct- ance of semiconductor-oxide-metal junctions are due to anomalous electron exchange scattering

Conduction Electron Band

E 4

= 0.38 eV

E

3

=2.01 eV

E

2

= 4.88 eV

E = 6.51 eV

Valence Electron Band

FIGURE 3. Energy spectrum of bound states at a vacancy in diamond.

d Vacancy

from vacancy-like paramagnetic defects in the

interface of semiconductors and oxide. As will be shown in detail in another paper these para- magnetic defects in semiconductors and their interaction with conduction electrons can be

treated very elegantly by using Anderson's Hamiltonian used in connection with localized magnetic

moments in metals. With respect to the optical

properties of the vacancy one should notice that the exchange scattering of the electrons involves

spin flips and therefore the selection rule AS = 0

for optical transitions is no longer valid for para- magnetic vacancies.

It would be of considerable interest to include

electron-phonon interaction in the theory. This

will be of importance, for example, in determining

the lifetime of the higher excited bound states at defects in semiconductors.

It is hoped that the large contrast between the

previous theories and the one proposed here will help in the study of the shortcomings and limita- tions of the existing theories.

4. References

C. A. Coulson and H. J. Kearsley, Proc. Roy. Soc., A241, [2] T. Yamaguchi, J. Phys. Soc. Japan 17, 1359 (1962). 433 (1957). [3] Stoneham. preprint (Harwell. 1966).

129

Energy Levels Associated with Isolated Vacancies in Silicon*

Joseph Callaway

Department of Physics, University of California, Riverside, Calif. 92507

and

A. James Hughes

Applied Research Laboratory, Aeronutronic Division of Philco Corporation,

Newport Beach, Calif. 92660

The energy levels of bound electronic states as- sociated with defects in semiconductors can be

studied through the use of methods based on scat- tering theory [1]. The approach is most useful for the consideration of levels lying deep within the

band gap for which the usual effective mass theory

is not valid. As an example of the use of this technique, we have investigated the bound states associated with vacancies in silicon. The energy levels of interest are the roots of the

equation det(/ — GV) = 0 which lie in the band gap,

in which / is a unit matrix, V is the matrix rep- resenting the perturbing potential, and G is the

matrix (E — Ho)' 1 usually called the Green's function matrix. These matrices are referred to a

basis of Wannier functions characterized by band

'Supported in pan by the Air Force Cambridge Research Laboratories.

and site indices. The Green's functions are obtained by numerical integration using energy bands

calculated by Brust [2]. The lowest eight bands

were considered. The perturbing potential is taken to be the negative of the contribution to the

usual pseudopotential from a single atomic site. Formulas have been developed to enable the cal- culation of matrix elements of the perturbation

directly from results of a pseudopotential band

calculation. As a byproduct of this work some

interesting results concerning the symmetries of

Wannier functions have been obtained.

References

[1] J. Callaway, J. Math. Phys. 5, 783 (1964).

[2] D. Brust, Phys. Rev. 134, A1337 (1964). We are indebted

to Dr. Brust for furnishing us numerical tables of energy

bands.

131

Electron Trapping as a Function of Normal Modes

A. M. Lemos*

Adelphi University, Garden City, New York 11530

Within the framework of the "adiabatic-harmonic" approximation there exists a unique set of

normal lattice coordinates corresponding to each

bound F-center electron state. This is due to the one to one correspondence between electron state and lattice equilibrium configuration. The various

sets of normal coordinates are related to one an- other by means of unitary transformations. The

transformation matrices depend on the lattice equilibrium configurations which in turn are functions of the F-center electron state. In the first portion of this paper, the matrices

defining the transformation between two arbitrary,

different sets of normal coordinates are developed.

The unitary properties are then established. From

these results it can be seen that given a set of

normal coordinates corresponding to a particular F-center state, one may generate the spectrum of

different sets by successive applications of unitary

transformations. In this sense the ground state normal coordinates can be used as a basis set. In the second part of this paper, the transformation matrices developed in part 1, are used to express the electron trapping energy [1]

1 for any

bound state F-center transition in terms of phonon

eigenfrequencies and a basis set of normal coordinates. This formulation is quite general. It includes degenerate and nondegenerate modes and

does not require the solution of Feynman-Hellman

[2,3] integrals. This is in contrast to previous

work [1]. As a result of bound state F-center electron

transitions, phonons are stored in the various

lattice modes. From the trapping energy the

number of phonons stored in these modes is de- termined. Only Frank-Condon transitions are considered. General expressions are then de- veloped for the Huang-Rhys [4] factors in terms

of the number of stored phonons. It is shown that

in general the Huang-Rhys factors are not equal to the number of stored phonons. The factors for ab- sorption and emission of light by deep electron

traps are compared with the corresponding expressions in a recent publication [5]. The factors de- veloped here are quadratic polynomials in A, the

distortion parameter. The zeroth order term is exactly the form previously published. Therefore

*Work supported in part by the National Aeronautics and Space Administration.

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

the present formulation for the Huang-Rhys factors

includes the previous ones as a special case. The

last item developed in this section is a general

expression for the Stokes Shift. The Shift is shown to be a function of stored phonons and

phonon eigenfrequencies.

In the third part of the paper, the general ex- pressions developed in the previous two sections

are specialized to the tight binding model of the F-center in the alkali halides. We have assumed

that both the adsorption and emission F-bands are

essentially due to electronic transitions between

two bound states. The states are "s-like" in the

ground state and "p-like" in the excited state. The local lattice symmetry is O/, in the "s-like"

state and D4h in the "p-like" state. The unitary

transformation matrices are displayed for this system. The recent experimental results of Gebhardt and Kuhnert [6] and the theoretical results of Wood and Joy [7] have been used in carrying

out the numerical calculations. From their work

and our analysis it was found that an effective mode

approximation was in order. This was the A]g mode of the Oh group (for the "s-like" state) and

the AlR mode of the D4h group (for the "p-like"

state). Since the F-center electronic charge dis- tribution is a function of state, then clearly the

electron-phonon interaction should also depend

on the F-center electron state. In a conventional one dimensional configuration coordinate scheme,

this fact is not considered. In an attempt to use

the one dimensional (single mode approximation)

scheme and yet to account for the difference in electron-phonon interactions we have introduced

effective ion masses for each state. The formulation is completely consistent and in addition has

precedence in the work of Williams [8|, Klick [9|,

and Luty [1()|.

On this basis we have calculated numerical

results for the number of phonons stored in the

"p-like" and "s-like" states of the lattice, the Huang-Rhys factors for absorption and emission

and the Stokes Shift. The specific alkali halides

considered were NaCl, KC1, and KBr. These were

the only crystals for which compatible absorption,

emission and "distortion-parameter" data are avail- able. The results of the computation are displayed

in table 1. In this table the first column, Retf., is the

ratio of the ground state to the excited state effective

alkali ion mass: the second column is the sum of

133
Table 1

«ctr SAWJVcitP) Sca ,(a) S„i(e) Afcal A£»x » A eK eV

NaCl 1.97 0.659 20.7 28.8 27.8 30.6 42.5 25.4 1.085 1.695 KC1 2.81 .664 21.4 24.6 33.2 39.0 45.0 25.3 0.900 1.098 KBr 2.00 .710 25.3 28.0 34.2 37.8 41.6 28.2 0.846 1.148

the squares of the equilibrium displacements as measured in the "p-like" and "s-like" state; the

third and fourth columns C/Vcal (p) and Nexp (p)) are

the calculated and experimental values for the

number of phonons stored in the "p-like" state of

the lattice: the fifth and eighth columns (S caI (a) and

Sca] (e) are the calculated values for the emission

and absorption Huang-Rhys factors; the sixth and

seventh columns (Ncai (s) and Nexp (s)) are the cal- culated and experimental values for the number of

phonons stored in the "s-like" state of the lattice: the ninth and tenth columns A£caI and &Eexp rep- resent the calculated and experimental values for the Stokes Shift. The experimental results are obtained from the recent work of Gebhardt and

Kuhnert. Since there appears to be an inconsistency in their interpretation of the Huang-Rhys

factors, we did not include in the table any experimental results for S. However the values that appear in most publications is 30 ±5. With the

exception of the Stokes Shift the agreement with

the experimental results is surprisingly good. The

treatment while sophisticated in its general form,

is quite crude in the actual numerical calculation.

While a tight binding approximation might be

reasonable for the "s-like" ground state, it is certainly suspect in the case of the relaxed "p-like"

excited state. This is because in the "p-like"

state the electronic charge distribution apparently

spreads over several lattice parameters. Hence

one would expect that the localized approximation

of the F-center electron-phonon interaction would

break down. In addition we have used a one-mode

scheme which we do not believe is correct. Recent

work supports this contention. It is our opinion

that the basic reason for the discrepancy in the

Stokes Shift is due to the fact that not all of the

significant modes were accounted for. This means that "in effect" we were using a one-dimensional configuration coordinate approximation. In such a scheme there seems to be no consistent

way of interpreting both absorption and emission

data. Wood and co-workers have been reformulating some of their previous work. A significant variation

in this calculation of the distortion parameters

would introduce other modes into our calculation.

Although we have restricted our numerical cal- culation to the F-center, the general equations we

have developed hold for any deep electronic or hole trap in the alkali halides. One problem in which we are presently interested in the determination of the entries in table 1 for the system KC1:T1.

Although the Thallium ion is large, apparently its charge distribution is quite localized. This means

that this system is probably amenable to our tight

binding scheme of approximation.

References

[1] Markham J. J.. Rev. Mod. Phys. 31, 956 (1959).

[2] Feynman R. P., Phys. Rev. 56, 340 (1939).

[3] Hellman H.. Einfuhrung in die Quanten Chemie, p. 285. Teuber. Leipzig (1937).

[4] Huang K. and Rhys A., Proc. Roy. Soc. (London) A204,

406 (1950).

[5] Lemos A. M. and Markham J. J., J. Phys. Chem. Solids 26,

1937 (1965).

[6] Gebhardt and Kuhnert H., Phys. Letters (Netherlands) 11,

15 (1964).

[7] Wood R. F. and Joy H. W.. Phys. Rev. 136, A451 (1964).

[8] Williams F. E., Phys. Rev. 82, 281 (1951).

[9] Klick C. C, Phys. Rev. 85, 154 (1952).

[10] Luty F. and Gebhardt W., Z. Physik 169, 475 (1962).

134
Electronic Properties of Point Defects in Solids.

Report of the Panel II Discussion

J. A. KrumhansI

Department of Physics, Clarendon Laboratory, Oxford University, Oxford, England

(Panel Members: F. Blatt, W. B. Fowler, N. H. March, and R. Wood)

1. Introduction

The topics discussed by this panel dealt primarily with the theory of and concepts regarding

electrons, considerations which underlie much of our understanding of the properties of defects. The panel I discussion presented in some detail how defect structures and their configurational

dynamics may be determined from ionic and elec- tronic properties, so reference to specific systems was not the primary concern of panel II. Rather,

it seemed appropriate to develop some perspective

regarding the limits or relevance of current descriptions of electronic properties. In subject matter, the material falls naturally into

divisions: materials (metals or insulators), individual

defects, defect interactions. In examining the

electronic properties (particularly in nonmetals) one

frequently studies the excited (electronic) states with the atomic configuration fixed. This, it is to be recognized, is different than the primary con- cern in studying "energies and configurations" —

for in this latter case it is almost invariably implicit

that the electronic configuration assumes its lowest

energy. This differentiation is worth emphasizing as it is quite possible that an approximation method suitable for one purpose may not suffice for the other; we probably can always do much better in computing elementary excitations from some ground state than we can in determining the absolute energies

or configurations starting from some remote standard state in many body configuration space. Again, one should recall the comments by Vineyard [sec. 3, Panel I] on this configuration space description; the

point of the present remarks is that there obviously

are not only the nuclear coordinates Ri, /?2, • - but also all of the electronic coordinates r_i, £2, . . . . It is remarkable that frequently [e.g., in the case of

electronic polarization] one can rather successfully

account for the electronic system by relatively few

parameters such as the electronic moment fx or Fermi wave number Kp in the computation of configurations and energy. The principal reason for

this must be that in many systems the electronic structure is strongly determined by factors other

than the specific ion locations, e.g., core potentials

in ionic insulators, or valence electron-densities in good metals. Based on this observation one may

expect that by contrast the situation in covalent

small band gap semiconductors or semimetals is rather more complex; this seems to be the case.

2. Particular Problems

The subject of electronic properties is vast; what

is of concern in the present summary are particular

topics which received discussion in relation to defects. For nonmetals these were: electronic

contributions to ion pair interactions, the ground

electronic and excited electronic states of point

defects, and the coupling of lattice motion and electronic configurations. In the case of metals the

principal topics were: ion-electron interactions,

defect-electron gas interactions, charge distribution

around a point defect, scattering by a point defect,

energy of formation of defects, and defect-defect

interaction. We proceed to note some of the main

points.

3. Nonmetals

The electronic contributions to ionic interactions

have been discussed at length over the years, and

tests of the better known models appeared in many

of the conference papers, particularly those of

Boswarva and Lidiard, Franklin, Tosi, and Doyama.

Since this topic has already been discussed extensively by Lidiard in reporting on Panel I, and has

received particularly thorough examination in relat135
ing theory and experiment for phonon spectra we

are now reasonably convinced that in the regime of

small displacements from the normal equilibrium

positions the electronic polarization contributions

to potential, particularly at long range, are well

described by a few "electronic" degrees of free- dom, which might for example be the dominant

terms in a perturbation expansion. However, it is unlikely that the short range potentials can be developed similarly from first principles beyond the

Slater-Morse work. Nor is it likely that an extensive machine calculation of ion pair electronic

configurations at short range could give much more

than qualitation information for obviously the symmetry of the interactions must play an important

role. Thus when defects are not at symmetry sites the use of simple phenomenological potentials is something of an act of blind faith; perhaps this effect already appears in Franklin's findings for CaF2 . However, no further light was shed on this question in conference discussion.

The electronic states of defects in nonmetals

have received most attention in alkali halides and

especially for color centers. The papers by Wood

and by Fowler, together with their panel remarks,

comprise complementary views. On the one hand

it must be recognized, as demonstrated by the Wood

and Gilbert calculation of the effective force constants for motion of the neighbors around a t/-center, that it is in fact possible to compute successfully

a substantial amount about the electronic excitation

and lattice coupling for these centers. To this extent the last few years have led to physically

meaningful improvements in previous computations—including some ideas on autoionization of

excited states and related questions for tightly

bound excitons. On the other hand it is not amiss

to recognize that one must always examine and

reexamine how much credit should go to the calculation and how much to the appropriateness of the

physical assumptions underlying the computer

approach. The nonuniqueness begins already

with the models, as emphasized by Fowler in his paper on the "use and misuse" of them. In the

panel discussion he summarized his views, and I will attempt to paraphrase them.

Fowler again draws attention to the fact that there

are two types of approximations in theoretical

physics, (1) physical, and (2) mathematical. By

"physical" is meant approximations which a theorist

feels will not do too very great injustice to nature. [The role of close contact between experiment and

theory is all important in this respect.] Once the

physical chaff is eliminated the solution of the re- sulting problem invariably calls for some mathematical approximations, and here is where large

computational capacity has allowed new gains.

But it is all too easy to abuse the physics then and

carry 10 percent models to five decimal place ac- curacy. One could not help but have a feeling

that this danger did exist in many of the present

day calculations.

I do not interpret these views, which are mine

also, to mean that no more calculations on Fcenter systems should be done, but rather careful

attention should be given to whether it is more important to call in a computer for extensive Hartree-Fock computations or to focus attention on the

simpler of the known many body contributions to correlation and polarization. Ideally an analysis

which involves extensive computation should make

direct contact both with other theory and with ex- periments. This is what appeals to me about the

Wood and Gilbert work; in this case both the theory

of localized lattice vibrations, and experimental

studies of optical properties of f/-centers have led

to estimates of the force constants —the results have considerable value.

4. Metals

In passing on to metals, it is useful to compare

the dominant electronic factors with those for nonmetals. The essential point is, of course, that in the electronic phenomena just treated a "localized"

basis for electron states was implicit, and usually

only a few electrons were involved at one time

[e.g., in localizing electronic polarization, ion by

ion]. By contrast, in metals the electronic be- havior, frequently collective, of the conduction elec- trons dominate.

Again there is a long history. The summary by

Huntington together with March's paper accompanying his Panel III remarks cover the traditional

approaches quite completely, showing that a number of point defect properties are given quite well

in leading approximation by one-electron theory

in the Hartree-Fock sense, neglecting band struc- ture effects. However, developments in three

aspects have supplemented the traditional quantum

mechanical methods and may be examined in the

context of the point defect problem: (i) dielectric response properties of the interacting electron

gas, (ii) psuedopotential methods, (iii) introduction

of real band structure corrections to the isotropic

electron gas. March discussed various aspects

of (i); unfortunately there was too little discussion

of (ii), but (iii) received considerable discussion by

Seeger and Mann, and by Mozer. Again, we at- tempt here briefly to develop relations between

these several aspects.

First of all, there are good physical reasons why

a number of the properties in metals are inter- related. This has become apparent from dielectric response concepts and psuedopotentials. From

these studies we recognize that two complex parts

of the electron dynamics, namely, the collective

properties of the electron gas and the strong in- teractions between the conduction electrons and

the ion cores (both many electron effects), may for many purposes be replaced by effective one-elec136
If tron potentials. If at the same time the ion core —

} ion core interactions are represented by a suitable

phenomenological potential, the following may be

ls seen:

le

1 (1) When a point test charge is introduced into the lattice, the conduction electron gas responds

[e and a particular charge or potential distribution

a

] is set up around the defect [Friedel; Vosko et al.,

j v March]. A prominent feature is the oscillating

S l character of the potential.

e. (2) Closely related is the effective potential which

d ithe conduction electrons see around a lattice ion

ie for forming one electron Bloch states, in other

c- words the pseudopotential. Regarding the ion

e core as a charge immersed in a responding electron

;e gas, the long range (small q) part of the potential

s. may be closely approximated from (i). The short

id range part reflects the physical difference between

a test charge and the electrons of the core which

ig may exchange with the conduction electrons; in

il the Hartree-Fock approximation this leads to the

famous cancellation effects developed by Cohen and

w Heine. Thus the pseudopotential and the potential

•

I found in (i) may differ inside of an ion core radius

il I (or at large q in momentum space), but obviously

i {over much of the range the pseudopotential and

r- i test charge potential in the electron gas are similar.

if (3) Taking this view one step further one may

u develop an effective interaction potential between

i a pair of ions by simply computing the energy of

f one charged ion in the polarized charge distribution

s due to the other. We may expect this to be most

if in error for close approach of the ions. In at least

7 two applications this effective potential seems to

A be a sensible one: first, the structure factors in

I liquid metals as analyzed by March et al., are in

s fair agreement with this theory; second; phonon

frequencies as estimated by this method [Cochran:

Harrison; Shorn and Ziman; Vosko, Taylor and

Keech] show very good consistency. While not

exactly given by the pseudopotential for electrons

ic the relationship is sufficiently close that electron

in band structure information may thus be used to

il estimate this ion — ion interaction, except at short

ic ranges.

is (4) Still another obviously related quantity is

ii that contribution to a point defect formation

iv energy which derives from a rearrangement of

it- the conduction electron gas. Neglecting band

n structure effects, and to first order in perturbation

theory, this may be shown again to depend on the

iv long wavelength limit of the polarization potential;

r March discusses this in some detail in his paper,

c- Of course, what the contributors to this viewpoint

in have emphasized is that the electron gas responds

Is to probes in a characteristic manner whose features

it are very similar whether the probe be a test charge,

ii an ion core, an ion pair, a point defect, or a phonon

id displacement of the ions. Any one of these probes

nr may thus be used to characterize the electron gas

c- to a very good first approximation. Moreover,

there are some cross-relations to be expected; one example is that between Debye temperature

and vacancy formation energy as proposed by

Mukherjee and interpreted by March.

To a large extent these considerations, though

relatively recent, were assumed by the conference

as established and were not really discussed at great length. However, in several aspects we know

that improvements are needed.

First of all there are the explicitly many-body

corrections to the dielectric response function;

these are very difficult to handle beyond the random

phase approximation; fortunately, for small q they may perhaps be small enough to neglect. No

discussion of this matter was attempted, since it seems apparent that other effects will produce the more important corrections to the linear screeningperturbation approximation. For instance, it is probable that the Born approximation fails seriously

as the defect charge AZ increases, and band struc- ture effects require consideration as well. As long as the effective mass dynamics are ignored one may still go beyond the Born approximation by doing a phase shift analysis along the

lines developed by Friedel, care being taken to conserve charge through an appropriate sum rule. Indeed, it is to be expected that as AZ increases

neither the defect resistivity nor the formation

energy will vary linearly with AZ.

Mathematically, the phase shift method is computable in principle for arbitrary AZ, for a spherical

potential. However, as soon as effective mass

dynamics are introduced the differential equation

defining the electron wave function no longer has

spherical symmetry and a partial wave analysis

becomes intractable. It is here that the integral

equation method, i.e., electron Green's functions

developed by Mozer and by Seeger and Mann, can

be applied. Not only can one easily go beyond the

Born approximation, but also the explicit effects of anisotropy appear. In fact, with regard to the

latter the results of Mann and Seeger show that in some crystalline directions the charge around a defect falls off quite differently than predicted for the perturbed free electron gas. That this should

happen for some electrons is not surprising, but is indeed expected. However, that the portions of

the electron energy surfaces giving rise to such

effects should have sufficient measure to modify

the asymptotic radial dependence of the net charge

in an essential way is really surprising. Further

assessment of these anisotropic effects is in order,

both experimentally and theoretically.

However, returning to the strong defect properties, whether it is eventually found that the mathematics is best done by phase shift or by electron

Green's function methods, it seems that clear that an extension beyond the linear screening theory is necessary in such cases as vacancies in polyvalent metals, etc. I am indebted to Frank Blatt

for his detailed comments on this point and now

quote them.

137

256-724 0-67 — 10
"The formalism developed by Seeger and Mann

and phase-shift calculations are not entirely unrelated; indeed the connection between the two

approaches was indicated by Professor Seeger in his talk. The Friedel sum on the phase shifts reappears in the Seeger method as a similar condition imposed on the matrix elements. Although

at first glance it may appear that the Seeger method

allows for fewer adjustable parameters than the

phase-shift calculations, the freedom in the selection of a scattering potential (subject to the Friedel

sum) in the latter, corresponds to a freedom in the

choice of the relative magnitudes of the matrix

elements injhe former.

"As regards the calculation of Fukai, it seems to me highly improbable that the relatively large dif- ference between calculated and observed resistivities due to vacancies in aluminum could be ascribed

entirely to lattice distortion. In this connection I shall first permit myself a trivial comment con- cerning the formal expression used in Fukai's

calculation. It is only in the free-electron approximation that the Fermi energy appears as indicated

by him. What should properly enter is the Fermi

velocity, that is, one should use

df

where e is the energy relative to the Fermi energy

and Vf the Fermi velocity.

"Second, we must bear in mind that although a scattering potential derived from the pseudopotential method is undoubtedly superior to various crude

alternatives that have been employed in the past,

the calculation relies on the Born approximation.

E. Stern, in a recent paper, showed that a perturbation approach is appropriate only if the screening

charge about the imperfection is small compared

to the ionic charge, and that when the charges are

of equal magnitude, perturbation theory cannot be

justified. In the aluminum problem, a perturbation

approach may be expected, therefore, to yield

reasonably good results for divalent and quadrivalent impurities. In the case of vacancies in aluminum, however, the perturbation method is unreliable. Though past experience indicates

that the Born approximation over — rather than

underestimates the scattering cross section, it is conceivable that in the present instance the situa- tion might be reversed.

"Finally, in the one case where Fukai as well as Harrison have calculated residual resistivities,

namely Zn in Al, the results differ by 60 percent.

This clearly points out the sensitivity of the calculated results to the details of the assumed pseudopotentials (Harrison and Fukai used different pseudopotentials). There now exists a variety of

data on the influence of impurities on electronic

properties of metals, for example, residual resistivity Ap, change in thermopower AS, and Knight

Shift A//. Since the scattering phase shifts appear

in linearly independent combinations in the expressions for Ap, AS, and it would be interesting to approach the problem of impurity potentials empirically by requiring that the phase shifts conform

with the experimental data. One could then construct an impurity potential of reasonable shape

which is consistent with these phase shifts and

compare this empirical potential to that derived

from the pseudopotential method. This is a pro- gram in which we are currently engaged."

Similar conclusions are reached by March, on

different evidence [see his discussion], so we con- clude that the matter of charge distributions and

potentials around strong defects is still an open

question. Perhaps this was the most quantitatively

supported result in Panel III.

5. A Survey of Quantum Formalism for Solids

It may serve a useful purpose to walk through

the paths which one follows in applying quantum

theory to the problems we have just discussed.

We begin with the Hamiltonian

H = HP Hinteraction electrons-nuclei

then a series of steps follow:

(1) Nuclear and electronic motions are separated

via the Born-Oppenheimer (B.O.) approximation:

be coupled back to the electronic excitations as the B.O. separation is only approximate. Thus

vibrionic states at defects and electron-phonon

interactions generally are subsequently introduced

as corrections to B.O.

(2) Proceeding with He , the next filtering is that the optimum reduction to a one-electron language

is carried out via the Hartree-Fock method:

self-consistency —[

H B. O.

to electron methods

to phonon methods, etc.

Hartree

Fock He Application

Subsequently Hn may be used to describe phonons, defect vibrational modes, configuration

coordinates, etc.; these excitations must eventually

Collective,

Many body

dielectric response,

quasi-particle

Pseudopotential

138
The collective and many body effects may be partly

accounted for by dielectric response functions,

and through the device of quasi-particle generalization of the Hartree-Fock excitations.

(1) The most direct way for dealing with the

electrons in the normal solid seems to be through

the OPW method. The inner shells, i.e., core

states, are understood to be filled using atomic

orbitals. By orthogonalizing plane waves (OPW)

to these core states and reintroducing some elec- tron gas polarization, i.e., screening, corrections a pseudo-Hamiltonian //^'pseudo is formed

Free electron

Orthog. to _ 1 W(l)

, W 11 e, pseudo Core

mathematics

screening

(2) There is now a choice of mathematics: (a) The plane wave approximation; it yields no band

structure but can be used as first approximation

to energetics, resistivity, etc. for good metals,

(b) The linear combination of OPW, i.e., LCOPW,

which has been developed extensively by Harrison,

Heine, and Cohen; this yields Bloch functions and

band structure behavior. Thus the story for the

normal solid is usually completed:

Mathematics;

•lowest approx-plane waves next approx- LCOPW

Bloch Functions.

(3) The above is next used as a basis for treating

a point defect. In the case of insulators the subsequent steps are somewhat better defined than

for metals, and are summarized in Wood's work.

In the case of metals the choice of methodology

is not so clear, and in fact is just where questions

are currently open, so we indicate what might be

done for a point defect in a good metal.

LCOPW

Block

Back

to -*

Defect

Exact: Integral equation

Differential equation

Approx: Perturbation

Variation

Self consistent

Hartree-Fock corrections

The several approaches possible are:

(a) If the . free electron basis is used a partial wave analysis may be used to study the charge

around the defect i.e., phase shifts, Friedel sum rule,

etc. This approach has been explored much by

Friedel and by March and their co-workers.

(b) Perturbation methods may be used in either

basis if the defect is not a strong scatterer, i.e., the

first Born approximation is valid.

(c) If the defect is a strong scattering center for the conduction electrons, in view of the wide use and development of Green's function methods

there seems to be no excuse for not using these as Mozer, and Seeger and Mann have indicated. Not

only is this integral equation approach easily capable of dealing with the strong scattering case, it also furnishes a direct way of dealing with the

anisotropy of real Bloch states. Of course as yet we do not have nearly as extensive experience

in making the always-necessary approximations

here, as in the differential form of the Schrbdinger

representation: no doubt this will develop rapidly

as more calculations on real metals are done.

In conclusion, I take pleasure in acknowledging

the comments and views of the other panelists,

and make the usual apologies for unintended in- clusions or omissions. One last detail — I have

not referenced this conference report as all of the

references are either well known or to be found in the conference papers.

n

139

III. Vibrational States

Phase-Shift Analysis of the Scattering of Lattice Vibrations

by Localized Perturbation

G. Toulouse and J. Friedel

Laboratoire de Physique des Solides associe au C.N.R.S., Faculte des Sciences,

91-Orsay, France

The scattering of lattice vibrations by localized perturbations is considered here in its similarity with the scattering of electrons in metals. A phase-shift analysis enables one to obtain in a direct way the perturbed energy density of modes, and consequently, the additive thermodynamic functions such as the zero-point energy [l]. 1 Further, the vibration amplitudes and phases on the defect are shown to be simply related to the asymptotic phase shifts of formal scattering theory. The relevance of this analysis to the interpretation of optical properties of defects in ionic solids (vibrational structure, isotope shifts) when one wants to go beyond the simple "one configuration coordinate approximation" is particularly stressed, but these methods might also be of value for the discussion of other physical properties of defects. The one-component linear chain is first considered, as it permits one to get various general results on a fully tractable model. Considerations concerning the electron-phonon coupling contribution to elastic terms in the vicinity of defects are developed (1). The extension to three-dimensions is made for simple cubic lattices with nearest neighbor central and noncentral forces, which enables one to separate the polarization complication from the "extended

defect" complication. By "extended defect complication," we mean the complication which arises from the fact that one irreducible representation of the point group of the defect may appear several times in the reduction of the perturbation matrix (2). Application is made of the general concepts to the mass defect and to the vacancy problems in very simple models (3).

1. The One-Component Linear Chain [2]

The model which will be discussed consists of a

chain of atoms of identical mass M and force constant k containing an impurity atom of different mass A/(l + /3) coupled to its two neighbors by

different force constants, A.(l + y). A Ml + y) Ml + y) A M M M(l + /3) M M

The unperturbed "crystal,"" with nearest neighbor

forces, has the following dispersion relation

. qa . , , 4A , „ _

oj = 0)i. sin with tof — — and 0 < qa < it: 2 M

the unperturbed energy density of modes per atom

is given by:

9 n0(to) =

1.1. Eigen Modes

The eigen modes of the perturbed system can be

separated by symmetry into even and odd modes,

with the following asymptotic expressions for the

displacements of the nth atom which are exact in the unperturbed region of the chain:

iu„=y4 sin (q\n\a-\- 8o) — — u-„, n > 0

1 7T \v„ = A sin (q\n\a— — + 5i)= v. n .

After some algebra, one finds for the phase shifts 80 and Si

TT 0)L COS

qa

tg 80

:

tg S, = tj

y sin (qa)

1 + y cos (qa)

qa (l+/3)(ycos (go)-+l)— (1 + y)

'Figures in brackets indicate the literature references at the end of this paper.

y(l+j3)(l-cos (qa))+{l + yY

The energy of a perturbed mode is still given by

a> = O)/. sin j, with (qa) to be determined by

141
boundary conditions. For instance, with the

boundary condition un, Vn = 0, we get: N(qa) + 80 = 77s, \N(qa) + 81 = 775 + (5 is an integer.)

The phase-shifts So and Si are zero for the unperturbed crystal.

The vibration amplitude of the "pulsating mode"

on the defect has a simple dependence on 80: -1 0 +1

2A

u 1 — U\ = 2 A sin {qa + 80) = —sin So. y

The vibration amplitude of the impurity atom

depends on 81:

v0 = - 1 + /3 cos Si + sin 81 cotg

1 + B . I qa) sin | —

The normalization condition ^ M\un \ 2 —1 detern mines the value of A [3].

The perturbed energy density of states v(co) — Vo(a)) is related to the derivative of the phase

shift [4]:

v(a>) — v0((o) = — ~- , with 8 = So + 81

.

77 aoj

1.2. Behavior of the 8o(qa) andSi(qa)

Phase-Shifts

From the Formula for tg 80 , it appears that (see

fig. 1): (1) 80 is independent of /3; (2) for y > - 1 or \' = k(l + y) > 0, 80 has the sign

opposite to that of 7, and

180 = 0 for qa — 0,

So = 0 or — 7T for qa — tt

according as there is or there is not a localized mode

of the same symmetry; the number of localized

modes is given by the sum rule No — — [So(0) — 80(77)];

77 (3) a localized mode appears only for a finite positive value of y(y > 1); (4) when there is a localized mode i.e., when

8o(0) — 80(77) = 77, the phase shift necessarily takes

77 the value — for one energy of the band continuum

FIGURE 1. Phase-Shift 80 for one-component linear chain.

and there is a scattering antiresonance for this energy;

77 (5) for y=l, 80(77) = — because a localized mode

is just appearing. By continuity, it is counted for half in the band continuum, for half as a localized

state, and the sum rule is still valid;

(6) for 7 < — 1, 8o(0) = 77 and it appears a localized

mode of negative energy; the situation is unstable

as it is obvious by direct inspection.

The phase-shift Si depends on both (3 and 7 (see

figs. 2 and 3):

(1) 8i=0 for <?a = 0; 81(77) = — ^ or +

f"> according as there is or there is not a localized mode

(pushed to a) = 30 for /3 —— 1);

FIGURE 2. Behavior regions for phase-shift 81 for the one- component linear chain.

142
2

i

@

7T qc?

J®

@

FIGURE 3. Phase-shift 8 t for one-component linear chain.

(2) one can define in the plane /3, y (unphysical

regions /3 < — 1 and y < — 1 excluded) six regions,

according to the following criteria (see fig. 2):

81(77) — ± — (localized mode).

81 can or cannot take the value 0 in the band,

•j.

' if 81 can or cannot take the value — in the band

(resonance);

(3) there are scattering resonances which can

be very sharp, even in the region /3 < 0;

(4) for y = 0 (mass defect), we are in regions (II), (III), there is no resonance and no zero. The six regions are summarized in the following

table:

S,(7r)/7r/2 a zero Resonance

I Yes No

II No No

III + No No

IV + No Yes V + Yes Yes

VI + Yes No

"The fact that 5,( 7r) = half integer number of rr is general for the modes of that symmetry in one dimension. This is due to the fact that modes of even and odd symmetry alternate in such a way that a mode of symmetry 1 constitutes the edge of the band. Hence, the smallest perturbation (5 can substract this mode from the band and make it localized. In the sum rule language, these modes are counted for one half.

1.3. Scattering

A plane wave is scattered by this localized per- turbation:

y4 €

-iq\n\a

j4€

iQ\n\a

_

ge

iq\n\a fe

'q\n\a

If we define

D0 = —2— and Dl ~ —

2—

then we obtain

-j- = + ie tSo sin 80 , ~£~ ie sin

and we derive:

B—= jgitSo + s,] sin (81— 80), A

F

A

G = A+F

A A

=e

iis+s1 ] C os (8, -80)- 1,

The vibration amplitude for the "pulsation" in this stationary scattering wave is

2Z)° 2i /is * u- 1 — u \ = y- = — y-Ae'" 0 sin 80,

whereas the amplitudes of vo and Ui + u-i are

A

v0 =

2A .s y

cos 81 + sin 81 cotg (^2~^j

l + y

1 + J8

+ cotg

COS 81

sin 81 — cos 81

If fi = — 1 (i.e., in the "vacancy case") u\ + u~i = 2A

cos 81 . One can notice here that the phases of the various

vibration amplitudes on the perturbed sites are

shifted with respect to the incident wave; these

mechanical phase-shifts are the phase-shifts which

characterize the asymptotic form of the scattered waves and, thereby, the perturbed density of

states. This appears to be valid as well in the 3

dimensional system. Another general fact which

should be noticed is the homographic dependence

of the phase shift's tangents on the perturbation

constants, here (i and y; hence the phase shifts are monotonic functions of these variables.

143
1.4. Localized Modes (Bound States)

We have seen how the existence of localized

modes can be inferred from the knowledge of the

phase shift in the band continuum. We shall show now how one can extract quantitative information on localized modes from the phase-shifts. In the scattering problem, the plane wave Ae iqna gave rise to a scattered wave:

Be

-iqna with - = ie'^ + 8 J sin (Si -So)

A localized mode is a scattered wave without

ingoing wave: this implies

e

i5 oo or tg S = — i.

The energy of the localized mode will be solution of

Q

tg 8=f(qa)=f{7T + id) = —i, with oj — a)L cosh -.

The phase-shifts S0 and Si take the form

y sinh 6

tg 80 = — i

tgS,=-

1 — y cosh 0'

tanh 0/2

(l+/3)(l-y cosh 9)-(l + y)

y(l + £)(! + cosh 0) + (l + y)

1.5. Zero Point Energy

The phonon zero-point energy variation due to the defect, which is its self-energy, is given by:

t ft 1 AE = --—\ 8((o)do) + y-Jt(o)i-mL ) ZTTJ 0 "7 2.

where the u> t are the pulsations of the eventual

localized modes. One obtains this formula directly

by noticing that — for a localized mode, the pulsation

variation is (oi — ojl since the mode is substracted

from the edge ojl of the band — for the continuum

modes, the q variation due to the perturbation is

dq 1 5(6)) Na ' (1.1)

1 the mode density between q and q

+ dq is —Nadq,

IT and the corresponding energy variation is

vJ0 2 dq \ Na ) tt 2tt Jo

8{a>)d(t).

The quantity which is measured in an isotope

shift optical experiment [5] is related to

GAMAy = ^==j x AMAy, dm dy

where AM is the mass variation of impurity (or host)

ions and Ay is the force constant modification (supposed small) due to the electronic excitation. 2 We shall take as an example the "F center"

defect model, with M' = M(1 + /3) = Q. The lo- calized level of infinite energy corresponding to the disparition of the Uo variable will have no

influence; taking account of the fact that 8 depends

only on —, one finds

dM 2M '

d

2 AE 1 dAE %

dMdy 2M dy 4ttM Jo " dy

L d8

doj

is negative, for any y; if one admits that the

force constant must be softer in the excited state, one finds a "red shift" contribution of the self- energy to the isotope shift when the host ions mass

is diminished.

Moreover in the "F-center" model,

dy

d80

' dy

1 y

2 sin qa

X sin 2 So.

The contribution of the frequency band (w, co + da>)

to the self-energy shift is

G((o)AMAy withG(<u) = 4ttM

sin 2 80 y

2 sin qa

G(oj) is simply related to the vibration amplitude

of the "pulsating mode" and to the one-phonon

structure intensity which is considered now.

1.6. Vibrational Structure

As usually done, we explain the vibrational struc- ture of optical transitions by introducing an electron- phonon interaction term linear in the ions' variables.

This term shifts all the coordinates of the normal

modes without changing their energy and makes

nonorthogonal the n-phonon ground state wave

functions and the rc'-phonon excited state wave

functions.

z One can notice that

7>[,-»»S («.-«,.)] 2dF

Trie-en] X dy

where u, a is a first neighbor of the central site, F the free energy, and (A > the thermal

average of A. In particular, at absolute zero:

nd the quantity GAM&y may be pictured as:

GMUt^y =

I Jj [( £ (u0 - U , „)*] AMAy.

144
We write the adiabatic potential for the mode K,

as:

to

with the normalization relation ^M\u„\ 2 = 1, which

determines A — —r , and with

. . 2A sin 80 2 sin 80 x-x — x\ = (u-i — Ui) qK = qK = — —7

<7a, /T y

/here

2(A)K (aK +a£).

Then V can be transformed into

CO 2F sin 80 * (q°K Y

2 with a»ltf=-^

2 y

We shall call S(o>) the relaxation energy — iq%)

2

expressed in ~hb>K units. SM 2 x

shVAo.

Then we can rewrite G(oj) as /rWco£

G(cu) =

32 /T2 x ^-rco(ft>) S(w).

Taking account of the fact that Jf2 — NM for a 2N

atoms chain,

C(a>) — ?oj a;/, 2

3F

X (i/0 (<o) S(a>)).

The value of F depends on the electronic state of

the defect, so one defines

The one phonon structure intensity is given by [6]

I (a>) =V0 ((d) Seg {(o),

where o> is measured from the zero-phonon line energy, and I{(o) is relative to the zero-phonon line intensity.

One gets then the relation

c/2 A£

dMdy l2HFe -Fg ) X/(a>),

which relates in a simple way the contribution to the isotope shift from the frequency band (to, a> + dw)

to the observable quantity

In the first Born approximation, valid for y small,

d

2 AE

Seg(o)) and are found to become independent

of the perturbation constant y:

s M=_

2_ (Fe _ jF)2x 5Hii^

9 i tta 3 9) Jf 2

' d2 t±E ha>L

. dMdy 6nM

1.7. Electron-Phonon Interaction

The phase-shift analysis allows us to calculate

the so-called "linear" and "quadratic" effects. We

give here a short discussion of the electron-phonon

interaction terms. Let us call qg and q u the even and odd coordinates

("F-center model")

qg = u-i — u\ q u = ui + u-i

and let us develop the electron-phonon interaction

up to second order:

Hint = Agqg + A u q u +Aggq2g + A gu qgqu+AuuqlWe are interested in the diagonal matrix elements

(*li\H

nt\iji) where an electronic wave function,

is by itself even or odd:

<^„|^>=0, (MAg\^)^0.

(This is why only the 80 phase-shift appears in the

calculation of Seg(o)).) As for the second order terms:

{^\A ug \^) = Q,

which means there is no mixing of even and odd

modes up to second order, and

(As we only consider symmetry properties, we may

think of Agg and A uu as effective terms including

second-order perturbations coming from the first- order terms.)

The Agg term gives an energy contribution in u_i)2 and the equation of motion of atom

1 becomes

Agg{U

M<ji 2 U\ = k[ui — u2 ] + Agg [ui — U- 1 ]

instead of, as in our previous model.

Mw2 ui = k[ui — u2 ]+ Ml + 7)["i — «o],

145
which, with Uo = — —-, becomes

(1 + y) M(o2 ui = X[ui — u2 ] + A. —-— [ui — u-i\.

the 81 phase-shift becomes

ix cos (qa) + 1 tg 8,

fx sin (ga)

so the Ann term was included in A

i + y

; however

the term, which is symmetry allowed, was not considered. It may be necessary to include these

quadratic terms in order to explain the observation

of isotope shifts of both sign. If one defines

Auu = A —

g—[»i + "-iJ,

J81 1

/i. 2 sin (qa) cos 2 81.

The sign of the contribution of this term will depend

on the sign of the A/u variation between the excited

and ground states.

2. Three-Dimensional Crystal

A. We restrict our study to a one-component,

simple cubic lattice with nearest-neighbor forces. In this model, the various polarizations are conveniently independent [9]. The integral equation of scattering is now:

\u)=\u°) + G$P\u)

where \u) is a column vector whose components

are the x displacements of atoms, G$ = —5

j——

or — L0 + ie

is the Green's function, Lo is the dynamical matrix

and P is the perturbation matrix. The unperturbed vectors \u°) are solutions of

(o)2 -L0)|«°> = 0.

This equation is similar to the Slater-Koster equation, where the displacement of atom / takes the

place of the coefficient of the Wannier function

centered at / in the electronic wave function. To

describe the scattering process, we introduce the

usual T+ matrix by its integral equation [8]:

T+ = P + PGtT+.

The stationary scattering wave | u } is obtained from

the incident wave \u°) by application of the X+- matrix

\u)=X+ \u°) with X+ = l + G$PX+ and T+ =PX+.

Finally, the phase-shift matrix will be given by:

tt8T+

tgrj = - 1 - iTT8T+ where 8 = 8(a)2 —

L

0 ).

B. For a point defect interacting with its 6 nearest neighbors, the P matrix is a 7 X 7 matrix; and

so is the T+ matrix; the 7-component vector \ur)

is a basis for a reducible representation Tr of the

symmetry point group G (here D4 h); the R index will designate vectors or matrices restricted to the

perturbed subspace.

The reduction of the Fr representation will give:

^, 0. a

where 7)% is the number of times the irreducible

representation appears in the reduction of Tr. In our model, the reduction gives [11] YR=Aig + B2u + Eg + 3A2u. As G$ and P possess the G symmetry, Xr and T+ have nonzero matrix elements only between basis vectors of the equivalent representations Tf, a . C. The case where 7^=1, i.e., the case where

appears just once, is easier to handle and we

shall deal with it in this paragraph. Indeed, it is the case for the A\g representation, corresponding

to the "pulsating" motion which is most important

for electron-phonon interaction.

We define:

Gfr = 2

l<K|g)| w

2 -o)2 (K) +ie

where |K) is an unperturbed plane wave:

J (o —oj l

means that the Cauchy principal part must be

taken)

77I^(fa>) 2co

ve (a>) =2 l<K|£>| 2°(w-w(K)

,

Remembering the definition of the phase-shift

matrix, we may write

1-PeRt

or 7jf = -arg (1 - P( G+( )

146
The and Xfe matrix elements can be conveniently expressed as

e

,r> ( sin 7)e y+ e

,ri f sin T) ( at — — • •RC

The £-type vibration amplitude on the defect is then derived as [10]

<£|«> = <f|*+

|f> (£\u°)=Xi(€\K).

So we have extended the linear chain results con- cerning the mechanical phase shift and the homographic dependence in P of the tangent of the

phase shift. The T/f phase-shifts govern also the asymptotic

behavior of the scattered wave [11]; writing the

stationary wave \u) as

\u) = *+ |K° ) =|Ko> +\us

) =|Ko> + G^r+ |Ko),

then the amphtude in / of the scattered wave can

be written

</|^> = 2</|C0+

|o-> <o-|7+|Ko>,

where the |cr) vectors span the perturbed subspace.

We can extract the £ partial amphtude from the

sum,

Introducing the first Born approximation for the

scattered wave

| u|) = G+P|Ko>,

(l\u°Bh=(l\G+(Z)P{ (£\Ko),

we derive also:

e">f sin rjf ak>f=-

PfVf

{l\u%) (.

D. In order to obtain the phase-shifts corre- sponding to the representations IY a , we should

like to diagonalize the phase-shift matrix; however a difficulty comes from the fact G+R and P do

not have to be diagonal together. The total £ phase

shift is given by ^ = 2>f.« = -Argdet (l-PGi)e

hence Arg detf T+ = Arg detf X+ = + 7](

E. Finally, in our model, the total phase-shift y) t is the following sum over partial phase-shifts.

rj, = n (Aig) +n (B2U ) + 2r) (Eg ) + 17 (A2U ).

F. For the localized modes, the condition tg8 = — i implies

(l\u*) f = -(l\G+\0

e

,r>{ sin t}^

TTV£((I) 2

)

3. Results for

1 - P(Re G+

0 )- iP(Im G+) = 1 - PG+ = 0.

3.1. Two Examples in the Debye Approximation: the Mass Defect and the Vacancy

(F-Center)

a. Green's Functions for Small Distances

The Green's function G^00 has been calculated by

various authors [13]:

G3bo=-(l--loge —U^.A

Simple Models

b. Mass Defect

There is only one phase shift given by

where x =

Oil.'

The Green's function Cj00 has been estimated in the

long wavelength range [12]:

G200 = Gooo -—A-\-2x2 + i; a>i\ 6

)•

where Ci — 1. AmFIGURE 4. Phase-shift for a mass defect b = 0.01 = 33. m

147
The curve r)(x) is plotted in figure 4, for the parameter value 6 = 0,01 ^-^ = 33^ which corresponds to a very heavy defect. We see a very sharp increase

of rj around the value x0 = V6 = 0,l; in this region, we may take the approximate formula:

77 X

3

c. Vacancy (F-Center)

We are mostly interested in the (A ig) phase shift, as far as optical properties are concerned; then we have

a study of spin-wave impurity states, and we have

used their computed values for Gj00 and G%00.

b. Discussion

(a) The phase-shift for the mass defect if given by:

,__^(1 _^^(0,o,0,3-g)}

where (3 = ———, fl2 = —, and g+(K, m, n, E)

(Aig \\- PGt \A lg) = l-^X (G+00 - G+

00 )

,

where yi =—r^ is the relative change in central

force at the vacancy site. In the long wavelength

range [12], the phase-shift 17(^19) is then given by:

i(K+ m+ n + l) dte-*tJK ( t )Jm (t)Jn(t).

A plot of 17 for the value (3= 10 is given in figure 5, showing a marked resonance behavior at low

frequency.

772 1+ _J_

3 27l

This formula is very similar to the small x approximation for the mass defect. For the vacancy, we

expect yi — — 1, so that

3 2yi 6'

and we have the relation (keeping in mind all the

approximations in its derivation)

7j(A lg : yi=-l)=-Tj03 = 2).

The low frequency "resonance" is centered

around Xo with

x0 =

3

+

27l

for yi >-l, x 0 >0,4.

3.2. Simple Cubic Model With Equal

Central and Noncentral Forces

a. Green's Functions

Numerical values for the Green's functions have

been given by T. Wolfram and J. Callaway [14], in

2

Mass defect

Figure 5. Vacancy: breathing mode; mass defect: 6=10.

u_AM_, n

(b) In the vacancy case [15], the phase-shift for the "breathing mode" is given by

„=-arg{l-f[^(0,0,0,3-g)

-^(2,0,0,3-$]}

On figure 5, we see that 17 has a smooth behavior

which seems to indicate that the "resonance"; found above is too high in o> for the Debye approximation to be valid.

148
Appendix: Green's Function Calculation for the Linear Chain

4.1. Green's Function Derivation

We define the Green's function gi(m) by the

usual equation:

(oj2 — Lo)gi(m) =8m0 .

Then

,iKma 2N^(o2 -a)2 (K)+ie

2i

a>'[ sin (qa )

,i\m\qu

where a> = a> L sin ar>d with 0 < qa < it for

o) < (xiL, and qa =tt + id for w > a;/,.

4.2. Reduction of Perturbation Matrix

For our general defect, the equations of motion

of U-i, Uo, ii\ contain perturbing terms. The per- turbation matrix P is a 3 X 3 matrix:

/2a-/3oi2 P = \ -a

a — a

0 I, with a = ^

yA._yw = —

2

which operates on column vectors (u? ). On symmetry arguments we make the following change of

basis:

Uq= u0 ; u'i — —7=(iii + u_i); u'o — —X{u\ — U-\). V2 V2

Then P becomes P'

,

aV2

a

0

and gt becomes gt,'

,

' =go\ V2e; * V2e'*

1 + e

2 ''* 0

With go = ~ = T"

oil Sln 0

4.3. Phase Shifts

The odd or "pulsating" mode phase shift tj 0 can

be derived from vu , given by

vu =l-PvG+ = l + a — 2i

oil sm 0

(l-e2i*) = l+yei<(

and is expressed as

tg 1?0 ; y sin 0

Re{vu ) 1 + y cos 0

q.e.d.

As for the even mode, we are in the case nig) —2.

We then diagonalize the 2X2 matrix, 1 — (PC J)e , with the secular equation

£0 [-/3&r + 2a(l -<"'*)] -l + v -g0 [pa>2 V2 <•»'* + « V2 (e'*- l)

2

]

-g-oa V2 ( 1 - e'*) g-oa(c'* - 1 ) 2 - 1 + v

0,

and we get the phase shift t) x from

tgT7 1 =tg(T7| + T7'f),

with

so that

tg t?i =

tgi7i =- t 2= ^'"^ -, /4>\ (l + j8)(y cos 0 + l)-(l + 7) , g1?1

' g \2/ y(l+/8)(l-cos0) +(l+7)' q

One sees in this example, where a representation

of the point group occurs several times, that there

is only one partial phase-shift which governs the

asymptotic behavior and the energy density modification.

149
5. Appendix: Short Derivation of Useful Formulae

The energy density of states functions, v((d2) and so that v0(io2

), are given by

v(o>2 lid ) — Vo((o2

) — +

, det G

+ log

det GJ

v{oi2 )= Im trace G+

,

v0 (oj 2

) = -- Im trace G+. because G0+ = G+

(1 -PGJ). Finally, we get

However, for a resolvant function (and such is a ^ w 1/0 w77 (/a/ Green's function), with

a fdet G

+

l Trace R—

—

J^log det R, r? = -argdet (l-PGj) =arg

J6. References

[1] For a review of the work done on the influence of defects on lattice vibrations, we refer the reader to: I. M. Lif- shitz, Nuovo Cimento Supp., 716 (1956); A. A. Maradudin,

E. W. Montroll, and G. H. Weiss, Theory of Lattice Dy- namics in the Harmonic Approximation, ch. HI (Aca- demic Press, New York, 1963): and A. A. Maradudin,

Reports on Progress in Physics, p. 331, (1965).

[2] A. A. Maradudin, Phonons and phonon interactions, p. 424

(Benjamin, New York. 1964); and J. A. Krumhansl and

J. A. D. Matthew, Phys. Rev. 140A, 1812 (1965).

[3] P. G. Dawber and R. J. Elliott, Proc. Roy. Soc. A2 73 , 222

(1963).

[4] A. Blandin, Journal de Physique 22, 507 (1961).

[5] Numerous isotope shifts experiments have been performed. For instance: G. F. Imbusch et al., Phys. Rev. 136A,

481 (1964): G. Baldini, E. Mulazzi, and N. Terzi, Phys. Rev. 140A, 2094 (1965): H. R. Fetterman and D. B. Fitchen, International Symposium on color centers in alkali halides, Urbana (1965); P. Durand et G. Toulouse,

Comptes-Rendus Acad. Sci. (1966); and A. E. Hughes,

submitted to Proc. Phys. Soc.

[6] M

[7] E.

[8] T.

[9] Si

[10] D

[11] A

[12] J.

[13] R

[14] T

[15] G

. Lax, J. Chem. Phys. 20, 1752 (1952); and Y. E. Perlin, Soviet Phys. Uspekhi, 6, 542 (1964). W. Montroll. R. B. Potts, Phys. Rev. 102 , 72, (1956). Y. Wu and T. Ohmura, Quantum Theory of Scattering

(Prentice Hall, 1962). milar considerations have been developed for the alloy problem. See F. Gautier and P. Lenglart, Phys. Rev. 139A, 705 (1965). L. Mills, Colloque de Grenoble sur les phonons, Mar.

1966, has previously derived analogous results for the mass defect. Messiah, Mecanique quantique, Dunod.

A. Krumhansl, International Conference on Lattice Dynamics at Copenhagen (1963). Brout and W. Visscher, Phys. Rev. Letters 9, 54 (1962). Wolfram and J. Callaway, Phys. Rev. 130, 6 (1963).

F. Nardelli and N. Terzi, J. Phys. Chem. Solids 25, 815

(1964).

150
Influence of Defects on Elastic Constants

W. Ludwig*

Institut fur Reaktorwerkstoffe, K F A Jiilich, Institut fiir Theoretische Physik C,

Technische Hochschule, Aachen, Germany

If a defect is introduced in a lattice, even in Bravais-crystals, the atoms are no longer centers of inversion. In calculating elastic constants, one has to use the procedure of nonprimitive lattices in a generalized form; since there is no translational invariance now, the region around the defect

has to be considered as a large molecule (unit cell). Starting from this, general expressions for the elastic constants in lattices with "small" concentrations of defects can be given. In many

cases the expressions can be simplified to a convenient form. The simple case of a vacancy is considered as an example.

When a defect (impurity, vacancy, interstitial) is introduced into a crystal, the force-constants

are changed, mainly in the vicinity of the defect. The atoms are displaced from their "ideal"

positions. Even in Bravais-lattices, the atoms are no longer centers of inversion. If one wants

to calculate the change in the elastic constants due to the defect, then one has to use the procedure

of nonprimitive lattices. In the following we give the general procedure for calculating elastic constants in defect lattices, and an application to vacancies in cubic crystals.

We assume a small concentration of defects, in order to neglect the interaction of defects.

It will turn out that disregarding the interaction between defects is a good approximation if the mean distance between defects is larger than the range of interatomic forces. But the procedure can be extended immediately to cases with interaction between defects, for example to Frenkelpairs.

Every defect produces a displacement field (which includes the image-term). We divide

the crystal into volumes with radii of half the mean distance of the defects, and calculate the

change of elastic constants by a defect from its "own" volume; i.e., we calculate the contribution

of a single defect and multiply by the number of defects. We assume further, that there are no

defects within a surface layer of depth equal to the range of the interatomic forces, or rather, we

assume that we can neglect the contributions of these "surface" defects. This is always possible,

if the defects are distributed homogeneously.

We have to distinguish between different displacements:

(i) The displacements produced by the defect (static displacement field) Uf, e.g., for a defect

with cubic symmetry in an isotropic elastic medium,

M X?

£/> =1 4tt(\ + 2/a) (R m f

(M: force-moment of defect, A., fx: Lame constants). The image term has to be added.

(ii) The displacements, which are produced by exerting small external forces (stresses) to the

defect crystal, Wf. The expansion of the potential energy of the defect crystal with respect to Wf defines the force-constants of the defect lattice. The Wf can be related to the elastic strains.

(hi) Vi\j is the (infinitesimal) elastic strain tensor. But now, because no ion is a center of

inversion in a defect crystal, Vi\j does not describe all the elastic displacements; each ion may

have an individual displacement Vf too, therefore

*Present address: Institut fiir Theoretische Physik. University, Giessen. Germany.

151
Wf=^Vi\y.Xf+Vf,

j

(1)

m indicates the atoms in an arbitrary way. But we number now the atoms according to their undisplaced positions in the lattice without defect m, because the static displacements are smaller than

a lattice constant. 1 W is the potential energy of the defect lattice, e.g., between impurity and neighbors,

<I> is the potential energy of the ideal lattice, between host lattice atoms.

Positions and force-constants of the defect lattice are

Xf, . . R" . . .), <J>|?n

(. . . R" . . .).

Corresponding positions and force constants of the ideal lattice are

If, ^mn {. . . R" . . .), <bmn

(. . . Rft

. . .).

In the following we drop all those relations which are standard [1-4] and emphasize only the

steps, which are characteristic for the defect. The change in energy with deformation by small external forces ff (stresses) is 2

^-^o=| ^ . . Rh

. . .)WfW*+. . ., (2) mn, ij

where we have assumed that the defect crystal is in equilibrium before the stresses are applied:

^f = 0, all 77i, i.

With (1) and (2) [1-5]

^-^o=^ x vrXn

k v? vJ\k+\ s

vvnxTMx»vilk vjll +

.

. .. o)

mn mn mn

ij ijk ijkl

After applying the external forces ff the equilibrium condition is

ft = % *7wl = 2 *rvj + 2 vTMxkvj\* (*)

nj nj njk

The ff are surface forces (stresses) and are exerted only on surface atoms. In the interior of the

crystal (4) simplifies to

5) ¥^F;=-2 ^Tjpfo. (5)

nj njk As in the theory for ideal crystals, it is sufficient to solve this equation in the interior and to neglect

the surface terms, because there are only N2^ surface terms compared to TV interior terms. The elastic constants are defined by 3

1 We restrict the piocedure to Bravais-lattices. The application to other ones is again obvious. If there are interstitials, they must have an extra number,

which makes no difficulty.

2 The sums extend over all the places in a lattice. In case of a vacancy the vacant position has to be included and to be described by missing force constants.

If there are N atoms in the lattice, and one vacancy the sum is over (N + 1) positions in a volume V. In an ideal lattice the "corresponding" volume contains N + 1 atoms.

3 This holds only, if there are no stresses in the initial state before deformation. The generalization to initial stresses can be done according to standard pro- cedure [3-5].

152
V-Vo^V^SiwV^Vjn, (6) Z ijkl

where V is the volume of the crystal, here of the defect crystal (footnote 1). With (6) and (5) we

have from (3)

V £ SikjiVi^ = 2 nnWm\k Vj\i + X W^JI*

ijkl mn (7)

ijkl

sjl

Because ^ ^??n = 0 (conservation of momentum), Vj = tj is a solution of (5). Therefore the re- n ciprocal of ^mn does not exist. But (5) allows for a solution with another symmetric matrix Rmn

in the form

vf = ~ E RT^nri'Xk' v^- (8)

1)1' 71 rik

Inserting (8) into (7) we have

VSik.ji = X VmnXfX? - £ XfV$'TMRTM-V*fX?'. (9) mn mm'

nn'

rs

This expression has the correct symmetries, because

n n n

which hold for every set of mass-points [1—5].

As in the standard procedure, we form Sikji+Sujk because Sikji contains absolute coordinates, whereas the symmetrized expression does not. Defining

2VCij,ki = -"Z n

n

(Xk—X?)(X?-XT), (11) mn we can resolve the symmetrized expression and obtain

VSik, ji= VCij, ki+VCkj, a- VC^ji- £ Xf^Rf^nn'Xf. (12) mm'

nn'

rs We are left with the calculation of (11, 12) and the solution of (5) for the defect crystal. For

this, we divide the crystal into an atomistic region around the defect, with radius r0 , and an elastic region. The atomistic region shall be of the order of the range of interatomic forces (or larger).

The outer region can be handled according to elastic theory.

Calculation of Cij)kl . A slight modification of (11) gives

2VCij,ki=- ^ ^i ^m+h{Xf+h-Xf){Xf+h -Xf)

m<r0 h - ^

2®m y m+ h (Xf+h -x%)(xr+h -xr); (i3) m>r0 h

153

256-724 0-67— 11
m< r0 indicates the atoms in the atomistic region, m > r0 those in the elastic region. In the

elastic region we have the potential function <t> of the host lattice (range of forces smaller than r0). In the elastic region we have small and slowly varying displacement fields, therefore:

Xr=X?+Ur; Zf+ft =Zf+Z*; (14)

<J)mra+/i — (Jjmm+A -). V (j)ram+Am+j7/m+9 -|- • m > Tn ^i j i j I J k k • • • i u

g, k

Um+9=Um +XfU%ll + . . .; m>r0 .

Inserting (14) into (13) and neglecting higher terms in U% we obtain

2Vdj, kl =- x 29rfh {x^-Xf) (x?+h -x?) - 2 2#r*^ m<r0 h m>r0 h - 2 2{2*rTTi

^^

+ 2^*^^+^^)}" (15) m>r0 h ^ g s

rs Now - £ *«"+*X»,y* = 2F20C9.

; w, (16a)

ft

independent of m, is the 2d-order-Born-Huang-tensor of the ideal lattice, V0 = NVz0 its volume.

Further - 2 flg-wtfS** = 2F,oG9, rs> tt> (16b)

hg

is the 3d-order-Born-Huang-tensor. Adding and subtracting

m<ro

we have

2{Fcij .A-/-j/

0q.ifc j=- x 2 {^> m+/! (^r/, -^fc)(^r'! -^r) -%m+ft*&r?}

m<r0 h

+ 2V*o{\rs,k, + Clsfrr + Cl ks8lr} £

^ (17) m>ro

If the static displacement field has cubic symmetry, it is

>i>r0 m>r0 i

This vanishes in some cases (vacancies). Then contributions of the displacement field arise from

higher order terms in U^s. In other cases the last term in (17) contributes. Its influence can be

estimated. Sometimes it is small and might be neglected; this depends strongly on the kind of

defects and on their atomistic structure.

154
Solution of (5). We make the same assumptions as before, and divide (5) into the equation

for the atomistic region

m < r0 : £ * = ~~ 2 **m

*+ *^'*; < 18a)

ftj ftj* and for the elastic region

m > r0 : ^ &fm + hVmf

k =-^ <&Ty hXm +

h Vj\ k ; (18b)

ft? hjk

where on the right-hand side of (18b) we have used (14). Then (m > r0 )

hj jk A rl ^ hg ft

where use has been made of (10) and $«i»+*m

jf»=0. In (before introducing defects) Bravaislattices all the terms on the right-hand side of (19) vanish, because of the inversion symmetry of

Bravais-lattices:

S>mm + h = (f)0h =z(ho-h. (J) mm + hm + g = <f)0/io = —fi\tt -ft- y . Yft__y_/,

i j ^ij Y i j

* j A- ^ i j it I J t 1

j

•

Up to first order in f/ m

| ( the right-hand side is zero. Higher orders can be neglected, since they

would correspond to 4th-order elastic theory. The solutions of the remaining homogeneous equation for m > r0 are Vf + h — tj which solves also the homogeneous equation of (18a). This means

a uniform translation of the lattice and does not influence the elastic constants. The Vj are

different from zero only in the atomistic region; we have to solve (18a) only. The displacement

field in the elastic region of a defect contributes to the change in the elastic constants (i) via the

last term in (17) and (ii) via the change in volume (difference in ^and Vo). The first effect vanishes

in some cases. Example: Vacancy in a fcc-lattice with central-forces. We assume a central-force interaction

up to second neighbors. The force constants of the ideal lattice are / and af. The atomistic

region includes the second "neighbors" of the vacancy. We consider an isotropic pressure as the external force, therefore we calculate only the change in the compressibility K. The calculations can be done in a straightforward manner. It turns out that even the differ- ences between the positions in the defect lattice and those of the ideal lattice can be neglected

(contributions smaller than 1%). Also the difference in force-constants in the vicinity of the defect

makes a negligible contribution. With Vj\k = e8jk for an isotropic pressure we have for the dis- placements according to (5) or (18a) (a = lattice constant)

V"°- £ 1lxb4-{l,l,0>: V2"="iT^i^°.°>- <20)

That means, the atoms in the vicinity of the vacancy are more displaced than described by the

strainfield Vj\k- This is plausible because of the missing springs. The second term in (7) then is ^ 1 v 1, 3 , 0/.l + a/4 + a2 + 3a3/4 sw^n-^/ i+,+Wi6 : (21) mn

sjl

155
The difference in (17) is

^{VCiUJJ-VoCljj }=-6aja + a) (22)

ij and therefore

This shows, that the second term in (12), the consequence of the fact that the ions are no longer centers of inversion in defect lattices, contributes about 25 percent to the change in the elastic constants! V_Vo^_ 5 1 + 33<*/20 + 23a 2/20 + 6a3 /20

K K0 'V' /( } l + 2a+19a2 /16 + 3a3/16 ' U4)

where r is the number of vacancies. The change in volume is V-Vo = K0 M-r; 1/K0 = a

2f(l + a)l3Vz0 . (25)

Using Kanzaki's [6] values for M (van der Waals' forces for argon) we finally have for the relative change in the compressibility due to vacancies

a/K- 1,X,,K,.f {,,02 -f^^^.

The first term arises from the change in volume; it is about 1 percent and can be neglected. Also

in other cases this contribution will be small. p — r/N is the concentration of vacancies. The difference between the elastic constants can

be related to Kroner's elastic polarizabilities [7]; in our notation it is e.g., for small p

HK-1/K0 = p(3 k /Vz , (27) where fik is the bulk-polarizability. With (26) we have

fi ,__L „ L no 5 1 + 33a/20 + 23a*/20 + 6«3/201

~ — 1, 46 eV for a vacancy in Argon. This value is somewhat small, but it is comparable with the

values given by Kroner [7] for the polarizabilities. Equation (26) can be used also to calculate

the temperature dependence of the compressibility due to the equilibrium concentration of vacancies. This is given by (Ef — formation energy)

p~exp {-EflkT}, (29)

and has to be inserted into (26). More detailed calculations for other defects and better models

will be done (see also [8]). References

[1] G. Leibfried, Hdb. d. Physik 7/1, 104-324 (Springer, Berlin, 1955).

[2] G. Leibfried, W. Ludwig, Solid State Physics 12, 275 (Academic Press, 1961).

[3] G. Leibfried, W. Ludwig, Z.f. Physik 160, 80 (1960).

[4] D. C. Wallace, Revs. Mod. Physics 37, 57 (1965).

[5] W. Ludwig, Springer Tracts in Modern Physics 43 (1967).

[6] H. Kanzaki, J. Phys. Chem. Solids 2, 24 (1957).

[7] E. Kroner, Phys. kondens, Materie 2, 262 (1964).

[8] J. Melngailis, Phys. Stat. Solidi 16,247 (1966).

156
A Calculation of Force-Constant Changes for the U Center*

R. F. Wood and R. L. Gilbert**

Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tenn. 37830

It was established some time ago that a hydride

ion can replace a halide ion in alkali halide crystals.

This defect, now known as the U center, is re- sponsible for a characteristic absorption in the

ultraviolet which has been intensively and extensively investigated [l]. 1 Through this work many

of the properties of crystals containing U centers have become understood. More recently, attention has shifted to theoretical and experimental

research on the infrared absorption of the defect. Shaefer's [2] original experimental work on the

subject coincides with an increasing interest in the effects of a defect atom on the vibrational properties of the host crystals, particularly in localized

and quasi-localized modes. The U center is an

interesting example of the former case, in which a

light impurity atom produces a localized mode well

above the limit of the band of transverse optical

vibrational frequencies of the perfect crystal. The

problem of the localized mode connected with the

U center and of its interaction with the "in band"

modes has been discussed by many people [3]. It was claimed earlier that one could understand the

experimental results simply by considering the mass

difference between the H~ ion and the negative

ion it replaced. It is now generally recognized

that it is also necessary to take into account the

changes in force constants which must surely occur.

In a recent paper, Fieschi et al. [3], go into this question rather thoroughly. Last summer we carried out extensive calculations on the electronic structure of the U center,

and recently these have been taken up again. Our

original aim was simply to see how well methods

which had been employed for similar calculations on the F center [4] could give the absorption energy

of the U center. Since lattice distortions around

the defect are allowed in our model, it occurred to us that we could actually calculate the potential

energy curve for both the nearest-neighbor relaxation and the movement in a (100) direction of the

H~ ion. We were thus in a position to calculate

the force constant of the H

- ion from a fairly rigorous

quantum mechanical formulation. It is the purpose

of this paper to sketch the calculations of both the

optical and infrared properties of U centers in KC1,

KBr, and KI. Unfortunately, the calculations are

*Research sponsored by the U.S. Atomic Energy Commission under contract with Union Carbide Corporation. **Summer participant from Illinois Institute of Technology. Chicago. Illinois.

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

not yet finished, and this paper must really be con- sidered as in the nature of a preliminary report.

We follow here very closely the work in reference

4. The wave function of a crystal containing a

single U center is written as

¥(1, 2, n) = Atyv{\, 2)i//c(3, 4, ... 77) (1)

in which i//<y(l, 2) and i|/c(3, 4, . . . n) are appropriately

antisymmetrized group functions describing the

H~ ion embedded in the crystal and the rest of the

crystal respectively. A is then an antisymmetrizing

operator which interchanges the electrons among

the two groups U and C and appropriately normalizes the resulting function. So-called "strong

orthogonality" is assumed, e.g., that

^r(l,2>M3 • • • k-1, I, k+1, . . . re)dn = 0,

(2) and <|/t; and *pc are considered to be separately

normalized. The Hamiltonian for the problem can

be written as H — Hu + HCT + Hint , (3)

/here

ff„(l,2)=-±V*-|v2

2 -^-^+J-,

2 ' 2 n r2 ri2

(4)

// int (l,2,3, . . . tt) = 2j-rr^Rl+

| r \ I f1

. I \ri — K„| |ri — rv,j\ Zv Ir,-R t r 2 — r„,

(5)

HJ3 . . . ")=22{! V

2

*V i~ Rt

(6)

rv j is the coordinate of the j electron on the „th

ion, etc. The expectation value of H with respect

to tyutyc is

{\}/u^/c\H\ifiu^c) = (^u\Hu\^u)

+ (iPc\Hcr \xJjc)+ (<^C | tfint I •M'c). (7)

157
Antisymmetrization merely introduces exchange

between the groups U and C. We treat the first and third terms quantum mechanically and the

second term by classical ionic crystal theory. The

form of 0u(l, 2) is taken as

Table 1. Optical absorption energies (in eV)

The calculated values were obtained with Slater orbitals on the K+ ions.

«M1, 2) = iVl,[iMl)»M2) + ^a (2)iMl)], (8)

where

0a— 2 (a\ v> i)0".J (9)

and

0a— \/ — e~

77 (10)

An exactly similar form is taken for This type

of function, first used for the helium atom, allows

for some in-out correlation as well as giving a fairly

adequate approximation to the Hartree-Fock doublyoccupied orbital. From eqs (1) through (8) an expression for the

energy can be derived which we shall write, for purposes of discussion, as ET — Tu + EMadi u+ A.EMadj U~~^~ Epent -&ex

(11)

The first three terms are respectively the kinetic energy, the Madelung energy, and a correction to the Madelung energy due to the fact that some of

the charge density may be outside the various rings

of neighboring ions; together these three terms

give the point ion approximation as discussed by

Gourary [5]. Epent is the coulomb energy due to the penetration of the defect electrons into the cores of the neighboring ions, Eex is the exchange

energy with these core electrons, and Eov is the

energy introduced by the overlap terms (a\v,j) in eq (9). Ecr is the energy of the rest of the crystal. All of the terms in eq (11), except Tu, depend directly on the positions of the neighboring ions and on the

displacement of the H~ ion. All of the terms ex- cept Ecr depend on the variation parameters a and b appearing in 0f, and 0&. The calculation consists

in first minimizing Ej as a function of the variation parameters and the positions of the Inn ions

in a displacement. Thus, we get the distortion of the crystal in the vicinity of the defect. Holding the ions in their new positions, we then displace

the H" ion in a (100) direction in order to map out ET as a function of this displacement. According

to the Born-Oppenheimer theorem, we can obtain

in this way the effective potential energy curve for the motion of the H" ion.

Crystal Calculated Experi- Percent ment error

KC1 5.86 5.79 1.2 KBr 5.68 5.44 4.4 KI 5.32 5.08 4.7

TABLE 2. Force constants, K, in various approximations

Column 1 shows the equilibrium displacement of the Inn ions in the three crystals

in percent of Inn distance of the perfect crystal. The second column contains the

results for the force constants in the point-ion approximation, the third the results with

the Slater approximation to the 3s and 3d K

+ Hartree-Fock functions, the fourth the

results with accurate Hartree-Fock orbitals for all core functions, and the fifth the experimental values.

Crystal Cubic

distortion

io- 3 a: i n dynes/cm)

Point ion Slater Hartree-Fock Experimental

KC1 2 2.78 19.80 26.10 14.43 KBr 2.5 2.31 14.95 18.63 11.44 KI 3.0 1.83 10.10 11.57 8.94

Some of the results of our calculations thus far are given in tables 1 and 2. The optical transi- tion energies in table 1 have been obtained by

using very simple approximations to the Hartree- Fock 3s and 3p K+ core orbitals constructed according to the Slater prescription. They are

03S = /V3.s r

2e-2 - 583 '' and 0s„ = AV2e"2 - 583r cos 6

where A^3S and Ns,, are normalizing factors. Calculations employing accurate Hartree-Fock orbitals

for all core electrons give a ground-state energy

level somewhat higher but, since the excited-state

energy level should be raised by about the same

amount, we do not expect the transition energies

to change greatly when the more accurate calculation is completed. Following Gourary's earlier point-ion calculations, we have lowered the groundstate energy by 0.389 eV to include the remaining

correlation energy unaccounted for by the form of

eq (8). It is assumed that the correlation energy

is negligible in the excited state. The results of the force-constant calculations

given in table 2 show, as expected, that the point-ion

approximation is totally inadequate to account for the observed local-mode frequency. It is somewhat

surprising that the results using the Slater approximation to the 3s and 3p Hartree-Fock orbitals should

be as close as they are to the experimental values. The inclusion of the H" polarization discussed below

would make this agreement even better, but we

are forced to assume that it is fortuitous. The re- sults using accurate Hartree-Fock orbitals for all core electrons appear to be very inaccurate, but

this calculation is not yet finished. We have still to add the corrections due to orthogonalization to the core electrons in the calculation of the expectation value of l/ri2, and we have not reminimized the

energy for each displacement of the H" ion. The

first of these steps should decrease K significantly:

158
the second will probably have only a small effect. In all of the calculations of K reported in table 2, the potential curves are very nearly harmonic.

We now wish to include in the calculations the

possibility that the H" ion can be polarized during

its vibrations. By far the easiest way of doing this,

within the framework of our model, is to assume that the entire electronic shell of the H

- ion can be dis- placed relative to the proton. We can then calculate the effective force constant between the shell and the proton. In practice, we displace the H~ ion

a given distance assuming no polarization, hold the

proton at that position and allow the electronic shell

to displace until equilibrium is established. Thus, we can calculate an effective potential curve with

and without polarization.

Somewhat to our surprise, the polarization of Id- calculated in this way was negligible. The reason

for this soon became apparent. The two functions

4> (, and <}>b involved in i//(l,2) have exponential

parameters a and b whose magnitudes are about

1 and 0.5 respectively. (f) a is very nearly a free hydrogen Is function, while gives a fairly adequate representation of the F-center ground state; compared to </><,, </><, is rather compact. Hence,

when the H

- ion is polarized, we expect most of the

distortion from spherical symmetry to be attribu- table to the 4>b function. In our model this means

that 4>b is shifted relative to the proton much more

than is In fact, it appears to be a good approximation to assume that <b a is not displaced at all. Even this method of calculating the polarization may

not be adequate. A very simple calculation on the

free hydrogen atom by the method of rigid shell displacement gives a polarizability too small by a

factor of six [6]. Our own calculations give a

polarizability of H

- somewhat larger (less than a

factor of 2) than the value of 1.9 A3 estimated by

Calder et al., [7j. The calculations indicate that the effect of the H~ polarization will be to lower the

force constants by 10 to 20 percent.

Finally, we would like to mention two problems

which have arisen in this calculation. First, there

is the one of accuracy. The change in the energy

of the H~ ion as it is displaced slightly from equilibrium (say 2% of the nearest neighbor distance) is of the order of 10-3 eV. One must calculate all quantities involved to very high accuracy. The

second problem really stems from the first. In calculations of this type, accuracy is very expensive

in terms of computer time. This problem has been

somewhat alleviated .for us recently by the instal- lation of an IBM 360-75 at Oak Ridge, and we hope

to have these calculations completed sometime in the near future. We are hopeful of eventually

obtaining the force constants to within 5 to 10 per- cent of the experimental values, but this may depend

on the importance of the polarization of the K+ ions.

References

[1] For background information on the U center, see J. H. Schul- man and W. Dale Compton, Color Centers in Solids (Perga- mon Press, Inc., New York, 1962).

[2] G. Shaefer, Phys. Chem. Solids 12, 233 (1960).

[3] From a rather lengthy list we mention only the recent work of

R. Fieschi, G. F. Nardelli, and N. Terzi, Phys. Rev. 138,

A203 (1965), which seems to be the most closely connected

with our work.

[4] R. F. Wood and H. W. Joy, Phys. Rev. 136, A451 (1964).

[5] B. S. Gourary, Phys. Rev. 112, 337 (1958).

[6] We wish to thank U. Opik for pointing this out to us.

[7] R. S. Calder, W. Cochran, D. Griffiths, and R. D. Lowde.

Phys. Chem. Solids 23, 621 (1962).

IS

!)

if

It

iJ

159

Local Modes and Resonance Scattering of Lattice Waves Due to Point

Defects: Optical and Thermal Properties of Imperfect Crystals*

G. Benedek and G. F. Nardelli

Istituto di Fisica dell'Universita, Milan, Italy

The theory of the imperfect lattice dynamics is briefly reviewed and the scattering matrix is intro- duced. The scattering matrix for a single defect is analyzed according to the irreducible representations

of the crystal point group, and defect self-properties, such as entropy, thermal resistance, and optical absorption, are shown to be simply related to the numerator and denominator of the scattering matrix. A model for substitutional defects accounting for both changes in mass and local interaction is considered in NaCl-lattices as well as in monoatomic f.c.c. lattices. The dynamics of an interstitial atom in a f.c.c. lattice is also presented.

The numerical computations concern mainly substitutional impurities in alkali halides, although some applications are mentioned also for interstitials and vacancies in f.c.c. lattices. For alkali halides the definition and the calculations based on the Hardy DD model of an effective nearest- neighbor force constant are given. Good agreement is found between theoretical predictions and

experimental data on optical absorption for several point impurities and also molecular impurities in a KI crystal.

1. Introduction

Since Lifschitz's early papers, the lattice dynamics of imperfect crystals has been extensively

developed from the formal point of view and the description of the effect of imperfections on a number of crystal properties has reached a satisfactory state on the basis of well defined models

for the perturbation. The simplest of these is the mass defect model. Due to the exact knowledge

of the defect mass, this model does not present any difficulty in the physical interpretation of the

parameter characterizing the perturbation: so it has been applied exhaustively; see, for example

the excellent review by Maradudin [1].

1 However this model has a limited range of validity,

because an isolated defect actually interacts with the host lattice through a field of forces which

may differ from the field in the host lattice. Sometimes, as one goes far from the defect, the

perturbation on the force field vanishes so slowly that it involves a large number of neighbors. In

this case the solution of the dynamical problem may represent a formidable task. Fortunately,

in many cases of physical interest the model in which only the change in nearest-neighbors (n.n.) interaction is considered (beyond the mass change) represents a good approximation, provided

that the parameters which characterize the change in n.n. interaction are suitably defined [2]. Recent calculations on defect properties, such as fZ-center Raman scattering [3], isotope

[4,5], and stress effects [6] on low frequency resonances in alkali halides make use of the change

in n.n. effective force constant obtained from fitting theory to the experimental resonance fre- quencies. Nevertheless, the applicability of such a model to defects in alkali halides, as well as metals or solid rare gases, presupposes the definition of an effective n.n. force constant (which is affected by the presence of defects) on the basis of the model employed for the host lattice dynamics.

On this line of thought we have based a recent discussion of doped alkali halides [7J. In this paper we summarize the general method of the scattering matrix, in terms of which we express

thermal and optical properties of an imperfect lattice. Then we report the resonance conditions

for:

(a) Mass and n.n. force constant changes in NaCl-type lattices;

(b) mass and n.n. force constant changes in monoatomic f.c.c. lattices. The vacancy case, already studied [8], is here mentioned:

*This research has been sponsored by EOAR under Grant N. 65-05 with the European Office of Aerospace Research — U.S. Air Force.

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

161
(c) interstitial atoms coupled with n.n. in monoatomic f.c.c. lattices as a limiting case of the

point (a). We report the models (b) and (c) with the aim of applying them to the solid rare gases and

simple metals (Na-type): however we restrict ourselves to a discussion of a simplified model for calculating the interstitial resonances. As regards point (a) an effort is made to define an effective

n.n. force constant. The numerical section of this paper is devoted to the point (a), namely to the KI crystal with positive or negative impurities. Indeed many experimental results on doped

KI resonant modes [9-12] as well as a very good calculation for KI:C1~ based on the mass-defect

model [12] are available to permit a detailed comparison with our theoretical predictions. The

effective n.n. force constant here used is defined according to Hardy's DD model [13].

2. Scattering Matrix Formalism

In this section we summarize the main results of the imperfect lattice dynamics on the basis

of the scattering matrix formalism. For sake of simplicity we consider the case of low defect

concentration. Hereafter we refer to the vibrational spectrum of the perfect lattice as the vibrational continuum. Let A(w2

) be the frequency-dependent perturbation; then, the normal mode

equation reads:

[I0 +A(w2 )]i// = co

2

i//, (1)

with the orthonormality condition

L0 and M0 denote the dynamical matrix and the mass matrix, respectively, of the perfect lattice,

M, the mass matrix and i/i the normal mode of the imperfect lattice, Sx, y the Kronecker symbol.

Index A. labels the normal modes and 10 is the circular frequency. A normal mode of the imperfect

lattice can be written as

\jj=aipu + a; (2)

a is an arbitrary a priori constant (eventually a = 0), cpy a lattice wave, i.e., a normal mode of the

perfect lattice, and cr the diffused wave. By putting

Ro(z)= (Lo-z)~ l (3)

where z is the complex squared frequency, i.e., z = a»

2 + i0+

, eq (1) is seen to be equivalent to

\\i = oup<,— R() (z)A(a)-)ifj. (4)

For a =t= 0, (4) is a linear equation of inhomogeneous type; it accounts for the wavelike modes of

the imperfect lattice. Indeed, eq (4) admits solutions only for a =t= 0 when u> is one of the frequencies of the vibrational continuum and these solutions are recognized to be the wavelike modes.

For a = 0, the homogeneous case, eq (4) may admit solution only when co is not a frequency of the

vibrational continuum; then we consider the equation

a- = R(co2 + iO+)\{(o2

)<pg , (5)

which follows from eq (4) when eq (2) is taken into account. By R{z) we have denoted the inverse

matrix

[L0 + A(a)2)-z]-S

162
which is seen to play in eq (5) the role of the resolvent matrix. At a frequency a> of the vibrational

continuum R((o2 + tO +

) exhibits resonance denominators with real and imaginary parts; the imaginary part never vanishes, while the real may have one or more zeros. Resonance scattering is said to occur when the real part of this denominator vanishes. Moreover, R(z) may have singularities on the real axis at frequencies above the maximum, o»/.. of the vibrational continuum or inside

the forbidden gap. The inspection of the homogeneous case of eq (4) shows that these singularities correspond to local modes. We look for resonance denominators and singularities of the

resolvent matrix. R(z) can be split as follows:

R(z) = R0(z) - R0 (z)T(z)Ro(z), (6)

where

T(z) =J(- l)"A(co2

) [fl„(z)A(a>2

) ]" (7)

n = 0

is the scattering matrix. Thus, the scattering matrix is seen to be the solution of the equation

T(z) = A(a>2

) - T(z)R0 (z) A(w2) = A(w2) - A(a>*)R0 (z)T(z) (T)

and the diffuse wave turns out to be

a = R0 (o}2 + iO + )T((o2 + iO +

)<pg . (5')

The first term on the right side of (7') corresponds to the first Born approximation of the diffuse wave. (5') is seen to be in the form of the first Born approximation, provided that we use the

scattering matrix instead of the physical perturbation A(w2

). It appears from eq (7) that the scat- tering matrix has the same dimension and the same symmetry as the perturbation matrix. The

inspection of (6) tells us that the defect-induced resonance denominators as well as the singularities of the resolvent matrix are all contained in the scattering matrix.

The scattering matrix can be analyzed in terms of the irreducible representations (i.r.) T of the

point group pertaining to the perturbation A(o»2

). One obtains:

Tr(z)=jrr(z)IDT(z), (8)

where jVy(z) is a matrix of rank equal to the number of times the i.r. T is contained in A(w2

), and

D\ (z) is the resonance denominator. The inspection of the solution of eq (7') tells us that Dr{z)

is given by the following expression

D, (z) = det [/ + i? 0 (z)A(a>2 )] r . (9)

The subscript T on the right side of eq (9) means that we have considered the projection on the

oriented symmetry coordinates which transform according to the i.r. f. In the limit z — or + 10 + eq (9) can be written as Dr(z) = \Dr(z)\eier^'\

where 8r((o-) is the argument of the resonance denominator pertaining to the i.r. f. The resonance

condition is

Re Dr(z) = 0. (10)

163
In the same limit considered above, Ro(z) splits into real and imaginary parts as

Ro(co 2 + iO +

) = 0> (a> 2) + invito 2

), (11)

where £P{oj 2

) is the projection operator

V^--) taX¥>*1 (12)

and 9°{o) 2

) its Hilbert transform.

Remembering that the secular equation for the squared frequency fractional shift £(oj 2

) reads [8]

det {/ + ^(tl> 2 )A(a> 2 )-7r cotg (7r£)^(w 2 )A(o>2 )} = 0, (13)

except for a trivial numerical factor, the argument of the resonance denominator for a given i.r. is recognized to be nothing else than the sum of the squared frequency fractional shifts which per- tains to the same i.r., i.e.,

n(T) #r(a> 2

) = 7r]r Zrj(a> 2

). (14)

j = i

In other words, 7r£r,j(<w 2

) is recognized to be the phase shift in the scattering problem.

It is an easy matter to analyze, in terms of J^riz) and Dr(z) the effects due to resonance and local modes on crystal properties.

2.1. Thermodynamic Properties

Let F denote a thermodynamical potential of the crystal phonon gas. In a harmonic lattice F can be written as F= 2x/(o»5). (15)

We regard eq (15) as the definition of the function f((o2

). The Lifschitz trace formula [14] for the

thermodynamic self-potential reads

= lrjdr j

da> 2f'(a> 2 )Zr,j(co 2

) = Xrdr^ j <W(w2 )#r(w 2

). (16)

A prime on f((i)

2

) denotes the first order derivative with respect to co

2

; index j labels the number

of times an i.r. appears in A(oj 2

), and dr is the dimension of A. If a local modes occurs, we have

to add the term

*{/(<c )-/K)}

and a similar term for a gap mode.

2.2. Transport Properties

Let us consider a phonon gas interacting with a random distribution of defects of the same

kind at low concentration, and let the system be subjected to a constant thermal gradient. It can be shown easily [15] that the inverse phonon life time, as limited by defect scattering, can be

written as

T-i =-(7rp/ft> 9) Im (<p g \T(a>l+iO+)\<pg ), (17)

164
where p denotes the fractional concentration of defects. The long wave approximation of (17) in cubic crystals involves only the T 15 i.r. and exhibits Lorentian-shaped peaks at the resonance

frequencies like the optical absorption coefficient.

2.3. Optical Properties

As a last example, we consider the linear response of an imperfect polar crystal to the electro- magnetic field at infrared frequencies. It can be shown [7, 16] that in cubic crystals the dielectric susceptibility can be written as

X(<»)=(4-I}m)(<Po,to \[Lo+pT(a>2 + iO+)-(o*-iO+]- 1

\<p 0iTO ), (18)

where the subscript 0, TO on <p stands for g ~ 0, transverse optic branch. The quantity e* is the macroscopic effective charge associated with transverse optic mode, \x the reduced mass of

the primitive cell and v its volume. Thus, the absorption coefficient reads

a(o)) = — (47t&>/t7c) Im x(<u)> (19)

in which 17 denotes refractive index and c velocity of light. Keeping in mind eq (8) and putting

/Vr

15 U) = (^ 0. ro l^r15 (z)k 0. ro ) (20)

the absorption coefficient is seen to be given by

=

4ne*2 a) p[fl<2 W(1 >-yV<2 >,P">] ~ Aurnc (wf. -co2

) 2 [D(1) + p(to 2 -a>2 )-W1 >] 2 + [Z) (2) + p(co 2 -<o2 )- W<2

>] 2 ' (21)

By the indices (1) or (2) we have denoted real or imaginary parts and we have dropped out the

subscript Ti 5 as well as the argument cu

2 of both D and N.

3. Resonance Conditions

We consider now the frequency-dependent perturbation

A(w2

) = -/l/(7'/'- {A(I)-w2 AM}Mo 1/2 (22)

where the change in force constants of both central and noncentral types are assumed to extend

only to the nearest neighbors of the defect. A<!> is the variation of the force constant matrix <t>. As usual <P is defined by the tensor <P = VVip(r), i.e.,

where <p(r) is the interionic potential, function of the distance r, r

2 = 2a x2

, x = x(/k) — x(/'k')

and / and k denote Bravais and cell indexes respectively. A<I> must satisfy the translational

invariance condition:

XlKa^afi(lK, ZV) = 0. (24)

165
3.1. Substitutional Defects in NaCl-Type Lattices

Consider first a substitutional defect in a NaCl-type lattice; let M± and M+ be the masses of the substituted and n.n. ions, respectively, ro the lattice constant, (Og j, and e (±|g./) the frequency

and eigenvector of the host lattice for wave vector g and branch index j. The change of mass

involves only the lattice site of the defect (assumed as the origin of coordinate axes) and enters A ( to

2

) through the parameter e = — AM+/M+.

a b

FIGURE 1. Representations of the impurity atom {0) and its nearest neighbors for a NaCl- lattice (a) and a f.c.c. lattice (b).

The geometry of the perturbation on the central and noncentral n.n. force constants is de- scribed by figure la. The symmetric tensor matrix A<P is given by:

a<m±i, o)=-A<Mi, i)=-A<M-i, -i)

A/2

0

0

0

A'/2

0

0

0

A'/2

and cyclic permutations,

A#), 0) =

(A + 2A') 0

0 -(A + 2A')

0 0

0

0

-(A + 2A')

(25)

where

A$(x) = 0 otherwise,

(26)

account for the changes of central and noncentral force constants, respectively. On the whole

A</> is, in the lattice displacement representation, a 21 X 21 matrix and so is the scattering matrix T. The point group which pertains to the perturbation is 0/,: the irreducible representation (i.r.) Y of

the 0/,-group is contained in A n(Y) times, as follows:

n{Y,)=l zi(rW) = l n(r 15 <) = l

n(Y12) = l n(Y25 ) = l 7i(r15) = 3. (27)

166
The symmetry coordinates in the wave vector representation are

(Ti\gj) = (2i/V6) {ex(T | g/) sin rcg* + | gj) sin rogy + e«(-+- gy) sin r0g2 }

(Tit, t\gj) = (i/V3) {2ea.(T | gj) sin r0gx — | gj) sin rogy —

z (=P |g/) sin r0gj

(rJ2 , r\gj) = i {ex!1? | gj) sin rogx — ey(+ \ gj) sin r0gy }

(r; 5 , z\gf) = iW+lg/) si" ^ogj — ej(+ | g/) sin r0gy }

(r25, z|s/)= Ue^+lg/) sin rogj + | g/) sin r0gy }

(ris.i, z|g/)= ez(±\gj)

(ri5 ,2, z\gj) = V^FF | g/) cos r0gz

(ris,3, z|g/) = e«(+|g/){cos rogj + cos r0

g-y }

(r25 , z|g/) = 1 g/) { cos r0gj — cos r0gy }

(28)

(t = tetragonal; r= rhombic)

In the symmetry coordinate representation the perturbation matrix is diagonalized in blocks of

dimension at most 3X3 (for Ti 5) i.e.:

(Ti|A|ro=(r«|A|r«)=x/2A#T

(r». I a| r25 .) = (r25 1 a i r25) - (r15- 1 a | r«o = \'/2MT , (29)

(r, 5 |A|r15)=

e<o2 + (\ + 2\')/M ± -klV2MUM* -K'/VmTM-.

i i X'/2M T

Hereafter we drop the argument at2 from A(oj2

). For each i.r. the scattering matrix can be written

down straightforwardly; the matrix elements of (L0 — z) _1 are easily obtained by using the symmetry coordinates and are expressed by means of complex-valued integrals on the Brillouin zone

(BZ). Usually k' is an order of magnitude smaller than k and a central force constant model in which k' is neglected is a good approximation. In what follows we put A.' = 0 in our perturbation;

in this case we have only Ti, r\ 2 (Raman active) and Ti5 (optic active) resonance modes, whose

frequencies are roots of the secular equations (10).

Re { 1 + (k/M±) [&f(z) + a&f(z)]} = 0 (30.a)

with a = 2 for Ti and a=— 1 for Ti 2 , and

Re { 1 + eto2 ^f(z) + (A/A#± )[S?f (z) MMJM^%(z)

- 2VM± /M? #f(z)} + eo>

2 (X//W^)[^f (z)%i (z) - &f(z)2 ]} = 0 (30.b)

167
for Ti 5 i.r. The complex valued integrals are given by:

SP*(z)= {r%l^)%} j

dSe

2Ji+\sj)IHj-z)

<gf{z)={T%l^)%j j

dgex(±\gj)ex(+\gj) cos r0gJ((olj-z)

%%(z) = {rilWZj j

dgeli+lsf) cos

2 r^/C^-z)

^|(z) = (rg/47r3)Sj J

c?ge^+ |g/) sin2 rofo/to^-z)

(z).= ( rg/ 47r3 )Sj

J

rfgey( =i=

| g/)e2( + | g/) sin r0£y sin r0gW(w^ —z). (31)

The evaluation of the integrals S?jz(z) needs the knowledge of the eigenvectors and eigenfrequencies

of the crystal. In our numerical application to KI, given below, these dynamical data are calculated

according to the Hardy DD model: in this framework a definition of \ is given there.

3.2. Substitutional Defects in a Monoatomic F.C.C. Lattice

The geometry of the lattice is shown in figure lb. We consider the perturbation on the mass and n.n. force constant of both central and noncentral type. By using the same notation as before we have

A<M0, 0) =

-8\! 0 0

0 -8Xi 0

0 0 -8k,

A0(11, O) = A0(11, 0) = - A</>(11, 11) = -A</>(11, 11)

Ac/)(11, 0) = A(M11, O) = -A0(ll, 11) = -A(/>(11, 11) =

0

X2

0

0 0

0 \j

0 -\2

0

where

A</)(x)= 0 otherwise,

\i=(l/2)A{^"(r)+^'(r)//-}

\2 =(l/2)AV(r)-^'(r)/r}.

and cyclic permutations,

and cyclic permutations,

(32)

(33)

Here the first derivative term <p'(r)/r accounts for the change in noncentral force constant. On

the whole A<f> is a 39 X 39 matrix in the lattice displacement representation. The group analysis

in this case was already performed [8]; one has

rc(I\) = l n(r12 )=2 rc(r 15 ,)=2 n(T12 ,)-l

n{r2 ) = i ra (r25 ')=2 n (r2 ,) = i n (r25 ) = 2 *(r15 )=4. (34)

For sake of simplicity we refer to the case in which noncentral terms are neglected, i.e., \i = A. 2 . In this case the symmetry coordinates which are involved in the perturbation are

(Tilg/) = (2i/V6){ex (gj) sin agx (cos ag^ + cos ag^ + cycl. perm.}

168
(ri2,i; t\gj) = (i/V3){2(ey {gj) sin agy — ex (gj) sin agx) cos agz — (ez (gj) sin agz -ey (gj) sin agy) cos agx — (ex (gj) sin agx — ez (gj) sin agz) cos agy}

(Im.i; *|g/) =i V2{ey (g/') sin agy cos agi + e^g/') sin ag* cos agy )

(Ti5, 1; *|g/) =ex (gj)

(ri5 , 2; *|g/) = V2 ex (g/) cos ag-x (cos agy + cos agz) (35)

(Tis.s: Jf|g/) = V2 {ej,(g/) sin agy + ez (gj) sin agz} sin agx

(r2s, 1: x|g/') = V2 {ey (g/) sin agy -ez (gj) sin agz} sin agx

(r25 , 2; x|g/) = V2 ex {gj) cos agx (cos ag> — cos agy)

In symmetry coordinates the perturbation is diagonalized in the following blocks:

(r1 |A|r1 )=-\1/(2M)

W(2M) 0

(r12 |A|r12 ) = (r25 '|A|r25 ^

0 0

(r25 |A|r25 )^

(r15 |A|r15) =

Xi/4M

\,/4M

\,/4M

(36)

ear A./V2M \,/V2M 0

KJV2M -X./4M - \i/4M 0

X,/V2~M -\,/4M -X./4M 0

0 0 0 0

The resonance condition for each i.r. can be easily obtained in terms of complex valued in- tegrals on the BZ, in analogy to the preceding case. In a previous work on the vibrational properties of a vacancy in a f.c.c. lattice [8|, these integrals

have been evaluated in an approximate way and expressed in term of the zero-order spherical

Bessel function. As a general remark we note that the perturbation due to a vacancy should contain, beyond the force constant perturbation in a rigid lattice, the anharmonic contribution due to the nonnegligible elastic relaxation around the defect. An example of the evaluation of the self- entropy of a vacancy in solid Ar in such a framework is given in that paper, where a reasonable

agreement was found with experimental data.

3.3. Interstitial Atom in a Monoatomic F.C.C. Lattice

Consider now a diatomic f.c.c. lattice (NaCl-type) containing a substitutional defect. Let the

169

256-724 0-67— 12
mass M± be replaced by M i.e., e=l —M\M± and let the 1st neighbor central and noncentral force constants be changed by the amounts A/2, A'/2, as well as the 2d neighbor force constants by the

amounts Ai and A2 respectively. Take now the limit in which either M+ or M±, as well as the n.n. force constants in the whole host lattice, vanish: the first case yields the case of section 3.2, namely

the substitutional defect in a monoatomic f.c.c. lattice, with parameters e=l — M/M, Ai and A2 , being M = M+ and A = A' = 0. The second case is more interesting, leading to the configuration

of an interstitial with mass M coupled by central and noncentral force constants /= A and /' = A' to its six nearest neighbors, in a monoatomic f.c.c. lattice, with atomic mass M— M+; of course we

put Ai = \2 — 0. The above considerations are meaningful only from a geometrical point of view, because the

dynamics of a f.c.c. lattice, such as a rare-gas crystal, cannot of course be considered as the hmiting

case of a NaCl-type polar lattice, due to the substantial difference between the fields of force exist- ing in these two types of lattice. Nevertheless the analytical form of our secular equations depends

exclusively on the geometry of the perturbation, while the host lattice dynamics enters only through

the explicit values of the frequencies and eigenvectors in the BZ integrals. Thus the only care we

must take in performing these limits on the eqs (30) and (31) is to consider the three optic branches

of the original diatomic lattice as branches of Einstein type whose frequency tends to infinity, and

the three acoustic branches as the branches of the monoatomic f.c.c. lattice. By neglecting/' the

secular equations which are so obtained for the interstitial read:

l + (f/M) (a3 /4773

) £ [ dsiwjj - io2

gJ )- 1 X ( e%(gj) sin2 agx + cxe y{gj)ez{gj) sin agy sin agz) = 0, (37.a)

Mw2 -/+ Mo)2(7/M)(a3/4773

) V f f/g(wi-w

2)- 1 X e2Jgj) cos

2 agx = 0,

i=i (37.b)

where g, j, ojyj, dgj), and a refer now to the f.c.c. lattice. Notice that the present results on the

interstitial dynamics can be directly obtained starting from the equations of motion for a free atom

and a perfect monoatomic f.c.c. lattice [17]. The first step is to introduce an additional lattice

site I for the interstitial so that the new coordinate space has 3(7V+1) dimensions, N being the

number of atoms of the perfect lattice. Consider now the interstitial atom and the lattice as uncoupled systems, i.e., Laf£l, I) = La/3(I, l) = Lat3(I, I) = 0,

where L denotes the dynamical matrix in the 3(N+ 1) dimensional space; one realizes immediately

that

e(g/Ve "/ for/ =1=1

AV28e ,oe(oj) for 1 = 1

(38)

are the 3(N + 1) dimensional eigenvectors of the uncoupled-system. Indeed, they are solutions

of the eigenvector equation

Xl'a'Laa'(l, I')\\)a \ njU')— (t)ljl}ja, g

j{l),

A being a suitable real constant and Jf the normalization factor. The orthonormality condition

for the eigenvectors (38) yields:

JV = N+8s , 0 A; now we need the orthonormalized set (38) to be complete in the 3(/V+ 1) dimensional space; one

finds

170
8aa {dir-A/(AN + m) for I, I' *I

8a„v4/G4 + A0 for/=/' = I 8„avi 1 ' 2 /(/4 + W) for /=*=/' = !,

(39)

so that, by choosing A = N2 the completeness relations (39) are satisfied within terms of the order

of N-1. We suppose now the interstitial atom to be in equilibrium at the center of the cage formed

by the six n.n. atoms / = ± 1, ±2, ±3 and to be coupled only with them through central force constants. The coupling force-constant tensor matrix reads:

flM 0 0

Ad, I) = 0 flM 0 ; A(±l, I) = A(I,±1)

0 0 Jim

0 0

0 0

0 0

and cyclic permutations; A(±l, ±1) =

and cyclic permutations; A(/, /') = 0 otherwise.

flM 0 0

0 0

0 0

We are now able to express the scattering matrix by using the same group analysis as in sec. 3.3,

except that the set of eigenvectors (38) must be used. By taking into account that

(I |(L - z)" 1

1 1) = - l/o>2, (I \{L - z)-> | ± 1) = - l/N(o2 -> 0,

the resonance conditions as expressed by eqs (37) are easily obtained. In this framework the

interstitial atom presents some analogy with the substitutional defect: when M is very small, a

Tis-localized frequency appears, while a very small coupling (f~*0) gives rise to a low ri 5 -reso- nance frequency. As regards breathing resonance modes (Ti or Ti 2 symmetry) only a large

enough value of / can produce resonances; beyond this, since f> 0, resonance frequencies will occur when the BZ integral in eq (37. a) is negative, i.e., near the maximum frequency of the host

lattice or in the upper region. To give a qualitative idea about these considerations, we calculate the BZ integrals in eqs (37) for a monoatomic chain, for which

M^= 4/sin2 (ag/2),

where /is the host lattice n.n. force constant. Using the nondimensional notations

/3 = M/M, y =f/4f and x = u)

2 M/4f

the odd-mode and even-mode resonance conditions, corresponding to Fi5 and Ti (or Ti2 undifferently) modes, respectively, are

|3*-y-yj3x(l-Re Vl-l/*)=0, (40.a)

1 + y ( 1 - Re Vxl(x-l)) = 0, (40.b)

where Re denotes real part. The hyperbolas in the (/3, y)-plane which eq (40. a) represent are

plotted in figure 2 for some values of the interstitial resonance frequency x. The cases x < 1

171
0. 1. 2. 3. /J= m/m

FIGURE 2. Interstitial atom in a linear chain: odd-mode resonance frequency mapped as a function of the interstitial mass and coupling constant.

and x > 1 correspond to resonance and localized modes, respectively. Equation (40. b) for y > 0

gives only the localized frequency

*even=(l + 7)

2/(2y+l).

Do models 3.2 and 3.3 give a good picture of substitutional or interstitial defects in rare gas

crystals or metals? For a neutral impurity in a solid rare gas (say another rare gas or a hydrogen

atom) they surely do because the impurity-lattice interaction can be well described in terms of

short range potentials (like the 6-12) usually employed for the host lattice dynamics [18] which,

GAP <J|_ GAP UL GAP U) L

1.

0

1.

0

1.

0

1.

0

1.

0

1.

0

0 1. 2. 3. 0 1. 2. 3. 0 1. 2. 3.

uj dO^sec 1 )

FIGURE 3. The complex valued BZ integrals involved in the scattering matrix for defects in a Kl crystal.

Numerical values correspond to; 7'=0 oK. Only seven of these ten integrals are linearly independent.

172
in first approximation, may be cut at the nearest neighbors. Furthermore these models probably

hold also in metals. In spite of the fact that in several metals, like metallic sodium, the ion-ion

interaction is long range in character (<p(r) behaving like r~3

), the screening capability of the

electrons seems, indeed, to be large enough to prevent the perturbation on the force constants

to extend beyond the first neighbors [19].

4. The n.n. Effective Force Constant in Alkali Halides and Numerical

Results for KI

Let us now apply eqs (30) and (31) to a doped alkali halide crystal. To evaluate the BZ inte- grals a good knowledge of the host lattice dynamics is required, particularly for values of the

frequency belonging to the vibrational continuum: here we have used for KI the Hardy DD model

[13], with zero temperature input data and a grid of 4409 g-values in the BZ.

In figure 3 the ten complex-valued integrals involved in the scattering matrix and secular

equations (30) are reported. In figure 4 the fractional change in n.n. force constant is plotted versus both Ti and Ti 2 reso- nance frequencies for either positive or negative defects, according to eq (30.a). The resonance

conditions for T^-modes are shown in figure 5 and figure 6 for positive and negative defects,

respectively: here the resonance frequency cj r is mapped as a function of the fractional changes

of both mass and n.n. force constants. For a fixed value of oj r the relation (30.b) between e and

X—/eff~feu is represented by a hyperbola. We have used here the concept of the n.n. effective

force constant feSf. For a defect of given e the value of A. can be obtained by fitting an experimental resonance frequency to these plots. To obtain theoretical support for the interpretation of such

a fitted value of k we try now to define a n.n. effective force constant according to the DD model.

In the DD model the force-constant tensor matrix is

tfDD) = + ( i + Se

-i

) ^(c)( i _ e

-i ae-i^(0) - i ( i + e-i§)

(41 )

"BREATHING" RESONANCE MODES IN KI

FREQUENCY u do'

3 sec-

')

FIGURE 4. The change in n.n. effective force constant plotted versus Ti or \', < resonance frequency for positive or negative

defects in KI at T=0°K. with felf =2.270 X 10* grsec'-.

173
-6. -5. -4. -3. -2. -1. 0 1.

MASS CHANGE €=-AM/m

FIGURE 5. I"

' ^-resonance frequency is mapped as a function of the changes in mass and n.n. effective force constant for

positive defects in KI.

where 4>

(R) is the n.n. -repulsive term, </3 (c) the coulomb term, e the ionic charge matrix, a the polarizability, S the deformation dipole matrices (see ref. 13), and S the transpose of 5. Near g = 0,

(J)(dd) admits the decomposition

ftDD) : eff .MO

eff (42)

where is a coulomb-type force constant matrix with effective charge equal to e* (Szigeti effective charge) and is considered unaffected by the presence of the defect. <^>

(Jf

) is a overlap type

-2. -1. MASS CHANGE 6= • am/m

FIGURE 6. Same caption as figure 5for negative defects in KI.

The hyperbola for «r = 7.31 X lO'^sec"' corresponds to ihe experimental localized frequency for {/-center (H).

174
force-constant matrix with the n.n. effective force constant given by

/e

V I

„ 8tt /e*\ 2 A + 2B —

3 \e/ a+ + a_ + 3t;/87rj (43)

v = 2r% is the cell volume, a± the ion polarizabilities, e the electronic charge, while

A=(2vle l )[d*<pM(r)ldr*] ro

and

B=(2vle2 )[d<pW(r)lrdr] ro

(44)

are the central and noncentral force constants, respectively, due to n.n. core repulsive potential

p<*>(r).

By assuming that our decomposition (42) holds to a good approximation for all g's, we can

write

feft /eff> (45)

/eff is given by the same eq (43), where now all parameters refer to defect-n.n. interaction and r0 is replaced by the defect-n.n. equilibrium distance in the host lattice. Due to the effects of polarizability it appears from eq (43) that /eff may be negative when A + 2B is sufficiently small. It follows

that small impurity ions may be unstable, at the center of the cage formed by the n.n. ions [20, 21]. This means that for

/

eff = 0, i.e., \=—fBf[, the T^-resonance frequency must vanish. From eq

(30.b) we see that this occurs for:

-X(„=o)= {vl87T3 lj fdSo) ^[Mz1 l2eA±\si) ~ Mz^ex (+\gj) cos rog*] 2 }- 1

- (46)

The quantity — A.( (u =o) evaluated from eq (46) is found equal to

/

eff as given by eq (43) within 2 percent.

This fact well supports the present definition of /eff. In table 1 the values of the change in force constant calculated by this method are compared with the fitted values. The agreement can be

said to be satisfactory if we consider that /eff depends strongly on the n.n. elastic relaxation £

around the defect and that the Brauer method [21] we have used for evaluating £ gives only a rough

estimation of the n.n. elastic relaxation (see table 1). In these calculations the Born-Mayer form was used for (f{R) {r); for Ag+ and Tl+ van der Waals terms were included also. In the last column

of table 1 the resonance frequency used in the fitting is reported. Any other experimental reso- nance which may be activated by a given defect should be consistent with the fitted value of k. Indeed for KI:C1~ we see in figure 7 that some other resonances are predicted in the low frequency

region which agree with the experimental absorption peaks better than those predicted by the

TABLE 1. Fitted and calculated defect-nearest neighbor effective force constants

Crystal

Fitted

ftnlftn

Calculated

Calculated

n.n. elastic

relax. (%) change «

Explt. res. frequency used in

fitting (lO 13 sec"1 )

KI:Na* -0.18 0.25 -2.0 0.411 - 1.2 KI:T1- 1.27 1.33 0.6 -4.22 1.23 KJ:Ag* 0.072 0.045 -7.0 -1.77 0.33 KI:CI- .50 .58 -4.2 0.719 1.45 KI:H- .38 -5.0 .992 7.31

175
1

i

i i KI:Cr [T = 0°K]

£ =0.719

_

f efF" Tc.ti51 *v 1fl^ iu nror con sec

1.04 1.19 1.30

1 =\

10=1.01

^ /

^=

^

1.45 /

{-JJ. — cr

i 1.0 1.5 2.0 2.5 FREQUENCY u) (10

13 sec"1

)

FIGURE 7. 77<e charge in n.n. effective force constant versus V'm-resonance frequencyfor KI : CI ~ (e = 0.719). The intersections of this curve with the abscissa give the resonance frequencies for mass-defect models (compared with ref. 12). Excellent fitting to the experi- mental IR spectrum is found by assuming the CI"—

K

+ interaction to be so/rerthan the I " —

K

+ interaction: the broken line indicates the fitted fM .

simple mass defect model. However, the present model is unable to predict the weak absorption

peak which has been observed [12] at 72 cm -1(co r — 1.36 x 10 13 sec

-1

)- Note that both fitting

and theory show that for Cl~ in KI a softening in n.n. force constant occurs, in contrast with

what was argued in reference 12. For KI:H", on the basis of the fitted value of A. (which is found

to be equal to —0.51 in units of A + 2B, in perfect agreement with the value calculated previously

by the shell model [23]) two other resonances, wr = 1.15 X 1013 sec

-1 and (o r — 1.56 X 1013 sec-1

, are expected to occur: the first, localized in the gap, agrees very well with that observed by Sievers

[9]; the second has not been observed till now. On the contrary the broad absorption peak observed in KI:T1+ near the acoustic edge is not expected from the value of X fitted to the sharp peak frequency. However, as we have demonstrated extensively elsewhere [7], the infrared absorption spectrum depends on the phonon spec- trum of the host lattice in such a way that structure may appear even if the resonance condition

is not satisfied, or, vice versa, no absorption peak may be observed at a resonance frequency when

it falls in a region of high density of host lattice frequencies; thus some care must be paid in inter- preting experimental data. As a final remark it is interesting to note that the plots in figures 5 and 6 give the possibility

of assigning the experimental resonance frequencies due to "external" modes of a molecular

impurity. These external modes, which can be of vibrational or librational kind, were recently

observed as "sidebands" of internal high frequency modes [11]. The symmetry of such defects

is usually lower than Oh-symmetry but the departure from the Oh-symmetry is often small (e.g., small splitting of the degeneracy) so that the T-type irreducible representations can be used in

first approximation. Consider for example the linear molecule NCO- as a substitutional defect

in KI (« = 0.669): two sharp doublets oj ai = 1.47, (oa2—1.55 and a>6i = 2.87, &)&2 = 3.24, a broad

doublet wc i = 2.24, ajc2 = 2.51 and two other broad structures ajd=l.ll and coe = 0.59X 1013 sec

-1

are observed at T= 100 °K. From a more detailed plot than that reported in figure 6 one finds that (uai, a>6i, aid are well consistent with the value fet{> jlfe{[ = 0.80, while a>a2 , o>b2, ~ £«></ with the

value /eft,2//eff = 2.0. Thus it seems possible to assign these frequencies to translational modes of transverse type (/eff =/eff, 2) and longitudinal type (Jeti =fett, i) with respect to the molecular axis. The encouraging results we have obtained for doped KI and several other alkali halides [7] enable us to believe that a close comparison between experimental results and the theoretical

calculations based on refined dynamical models constitutes a valid instrument for the investigation

of the interatomic forces in crystals.

176
5. References

[1] A. A. Maradudin, in Theoretical and experimental aspects of the effects of point defects and disorder on the vibrations of crystals — Scient. Paper 65-9FS-442-PS-May 6, 1965—Westinghouse Research Labs. 35, Pa.

[2] We may have more than one parameter, if force constants of central and noncentral type are separately considered.

[3] C. T. Sennett, J. Phys. Chem. Solids 26, 1097 (1965).

[4] A. J. Sievers and S. Takeno, Phys. Rev. 140, A1030 (1965).

[5] L. Genzel, K. F. Renk and R. Weber, Phys. Stat. Sol. 12, 639 (1965).

[6] G. Benedek and G. F. Nardelli, Phys. Rev. Letters 16, 517 (1965).

[7J G. Benedek and G. F. Nardelli, to be submitted to Phys. Rev.

[8] G. F. Nardelli and N. Terzi, J. Phys. Chem. Solids 25 , 815 (1964).

[9] A. J. Sievers, Bull. Am. Phys. Soc. 9, 228 (1964); Phys. Rev. Letters 13 , 310 (1964) and 15, 1020 (1965) and Phys.

Letters 14, 271 (1965).

[10] K. F. Renk, Phys. Letters 20, 137 (1966).

[11] M. A. Cundill and W. F. Shermann, Phys. Rev. Letters 16, 570 (1966).

[12] A. J. Sievers, A. A. Maradudin, S. S. Jasval, Phys. Rev. 138, A272 (1965).

[13] J. R. Hardy, Phil. Mag. 7, 315 (1961).

[14] M. Lifshitz, Nuovo Cimento Suppl. 4, 7716 (1956).

[15] M. Klein, Phys. Rev. 141, A716 (1966).

[16] A. A. Maradudin, in Astroph. & Many-Body Probl. (Benjamin Inc., 1963).

[17] We are indebted to Dr. R. J. Elliott for having suggested this alternative way to investigate the interstitial dynamics.

[18] J. Grindlay and R. Howard in Proc. Int. Conf. on lattice dynamics, Copenhagen 1963 (Pergamon Press 1965).

[19] J. S. Langer and J. H. Vosko, J. Phys. Chem. Sol. 12, 196 (1960).

[20] G. J. Dienes, R. D. Hatcher, R. Smoluchowsky, and W. Wilson, Phys. Rev. Letters 16,25 (1966).

[21] J. A. D. Matthew, Sol. St. Comm. 3, 365 (1965).

[22] Y. Fukai, J. Phys. Soc. Japan 18, 1413 (1963).

[23] R. Fieschi, G. F. Nardelli, N. Terzi, Phys. Rev. 138, A203 (1965).

177

Comment on the Paper of G. Benedek and G. F. Nardelli

Win. M. Hartmann

Argonne National Laboratory, Argonne, 111. 60439

Professor Nardelli has pointed out that an "in- band" resonance may not be seen in the infrared

absorption spectrum if the density of unperturbed

vibrational states is high at the frequency of the resonance. In other words, when a resonance

occurs at a frequency where there is a large number

of modes of the perfect system the resonant mode

will decay rapidly and its line will be so broad that

it will not appear as a separate peak in the absorption.

There is another possible explanation for a resonance which does not appear in the absorption

spectrum. Wolfram and Callaway 1 have con- sidered the imperfect crystal density of states in

1 T. Wolfram and J. Callaway. Phys. Rev. 130. 2207 (1963).

the region of a resonance for magnetic systems.

They find that at some resonances called "anti- resonances" the density of perturbed states is less that that of the unperturbed states; a dip, instead

of a peak, should appear in the spectrum. Application of Wolfram and Callaway's idea to the phonon

problem would suggest that the occurrence of a resonance or an antiresonance is determined by the

sign of the derivative of a resonance curve, such as Professor Nardelli's figure 7, at the resonance fre- quency. The general rule still applies that both

the peak and the dip in the absorption spectrum

should be less pronounced at frequencies where

the unperturbed density of states is high. A more

detailed discussion of this idea for vibrating sys- tems with mass and force constant changes is presently being written for publication.

179

One-Phonon Infrared Absorption in a Rare Gas Crystal

Induced by Rare Gas Defects

Wm. M. Hartmann*

Argonne National Laboratory, Argonne, 111. 60439

The far infrared absorption of argon crystals

with 1 percent krypton and V2 percent xenon impurities has been measured at 80 and 55 °K respectively

[1]

1 (fig. 1). It is the purpose of this paper to begin

a theoretical discussion of the experimental results by solving a very simple model, approximating the

actual physical system only crudely, but of some

interest in itself.

20 x I 0" 3

I 8

-

I 6

14 -

1 I 2

10

1—1

1—1

1—1

1—1

1

1 1 r

I

ARGON via)

1% Kr, 80 °K j% Xe, 55 °K

0 10 20 30 40 50 60 70 80

cm~l

FIGURE 1. Experimental infrared absorption; calculated one- phonon density of states, nearest-neighbor interaction model.

moment operator if two conditions are met. The

inversion symmetry of the f.c.c. structure must be

destroyed, accomplished by the introduction of

defects, and the uncharged atoms must be made

polarizable. The atomic polarizability is introduced

theoretically by an approximate shell model [2,3]

with harmonic dynamical equations

m<D2 Uai + ^ (A + K)wafSUl>li + (B ~ K)wa$Vvn = 0

M(xi2 Val + ^ (B — K)u>ai3Ul>i3 + (D + K)u'afiVl'p = 0.

(1)

The symbols m, M and uia , via stand for the masses

and the displacements in the crth direction for core and shell of the /th atom. A, B, and D are nearest- neighbor force constants, and since cores and shells have a charge + ze and — ze, k represents the long

range Coulomb interaction. A single defect atom

of mass m' and an internal force constant equal to Buaa + c may be introduced by adding

and

[ — (c + co

2 (m — m'))ui'p + cwp] (S//-8a/3 = 8r)

[cuvfj — cvvp\St

(2)

to the right-hand sides of eqs (1). Notice that with

force constants so changed a uniform translation

of the crystal results in no force on any atom. In

addition, the charges on the shell and core of the

defect atom may be changed equally and oppositely by an amount

• | = dz \ z.

It is useful to transform to coordinates

The harmonic vibrations of an argon crystal can

be made optically active to first order in the dipole

*Present address: Solid State Science Division, Argonne National Laboratory, Argonne, 111. 60439.

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

mu + Mv

m ~M

and /jl = u — v, the center of mass and relative in- ternal displacements. Assuming M<^m, we find from (1) in matrix notation

181
mw2£ = [R + 8t o)2(m -m')]^ + Sfi

Mu>2 ix = S£ + [T+ 8t (c + (M-M')(o2

) + AkK (3)

where

R=A + 2B + D

S=-B-D

T=D + k

(4)

and where Ak results from the change dz. The

equations for the perfect crystal are obtained by

setting (m — to') = c = {M-M') = Ak = 0. These

perfect equations have solutions of acoustic and

optic forms. At acoustic frequencies the shell is considered to feel no force due to core displacements and

Sf=-7>. (5) Then substituting we find an acoustic equation for the center of mass

TOfe) '-Z = (R-ST-iS)Z = R'z. (6) The optic solution represents the motion of atomic

electrons as influenced by other atoms in the cyrstal.

If the core is too heavy to follow the electronic

motion, £ = 0 and

Mw2e ti=T(x. (7)

We approximate the electronic motion by an iso- tropic Einstein oscillator. Since the Fourier transform T(k) = V ei~k

' iri-rc) Tui is then independent of A\

v Tiv is diagonal on //' and we take Ta'ap — Mw|8»'8a/3,

where (o e —12.2 eV, an exciton frequency. This

approximation results in a great simplification of

the equations. For the imperfect crystal c and Ak

are considered to be only a lumped change c in the

diagonal elements of T. The properties of the imperfect crystal expressed

in terms of Green functions G for the type introduced

by EDiott and Taylor [4] may be calculated from

perfect crystal Green functions

Wafi{) 2 2 -a>2(k)

(8) jk

where 17 =1 or 2 for core or shell; cr's represent

appropriate eigenvectors of the dynamical matrix

equations.

In matrix form

G = P-PVG, (9)

where V is a perturbation due to the defect. For

a defect isolated in the 1 = 0 unit cell the solution

to this equation is

Giv — Pn' —P10C00P0 (10)

where C is the 6X6 matrix, which c.or a cubic

crystal is diagonal on afi:

o>'=

-<o

2 me(cP22 -l) -rfmecPH

]

(o2 mecPH c((o2 mePH - 1) J a

{1 -u*meP»-cP%+ctfme(P^-P^® }

'

(11)

e is the mass change parameter (to — m')/m.

The correction Sr_1S to R (eq (6)) is small and

may be assumed to leave the form of R' the same as

that of R, hence of S. But then using eq (5) we find from eq (8) in terms of X = — S/R'Ma)2 P12 =\(TOW2Pn -l),

P22 = k

2 m(mo)4Pn - (o2 - cu

2

) + -

(12)

. (13) M(a)2 - w!)

where w2 = ^ 2 (U1^) == J

v(oj)o)2 d(o, and v(co) is

a normalized density of one-phonon states with

maximum frequency a>m = 68 cm -1. According to the adiabatic approximation only P22 includes a contribution from the optic modes.

A specific calculation will involve the density of

one-phonon states, here calculated from a simple

nearest-neighbor interaction model (fig. 1) [5]. The

additional parameters z or M= 2.73 electrons and

\ = 2.34X10-6 g-Hr/s)- 2 may be found from the

polarizabilities of the free argon atom and the atom

in the crystal [6], the density [7], and coe [8] together

with the rules

B(k = 0) = -A(0)

B(0) = -D(0). (14) A rough guide for choosing defect parameters c and

dz may be drawn from the exciton frequencies in the imperfect crystals [9, 10]. Finally we may write an expression for the optical

absorption coefficient K in terms of the imperfect

crystal Green function for relative core-shell dis- placement, G22

. The Green function (1) to first order in the defect concentration £ is easily found

from the zeroth order Green function (0) by multiplying the defect part of G(0) by £. To first order in the

concentration, valid for this experimental system,

182
K(oj) = ( =U-) (o2z

2 Im • 7]Vc y TTl L w

+ 2i(dz) £ G<f^ + C(dz)*G$** , (15)

where tj is the refractive index of argon, vc the

velocity of light, and Fthe unit cell volume. There

is a local field correction factor A = (-^—) . The

absorption coefficient may be found by substituting

(11) with (12) and (13) into (10) and then (10) into (15). If the defect is regarded as only an atom of dif- ferent mass and shell-core charge (c = 0) only the

C11 element of C is nonzero and K((o) results from

only the defect atom itself.

K(a>) = -(t/=4.2X,10- 6)ze

2

ff £

v((o)lm

(l-^-f^^)

2 + (^(co)/2) 2

(16)

I 0 x 1 0" 1 -

Figure 2. Mass defect resonance condition,

1/e = £0 ~1 ^

- J CO (i)

where the effective charge

zeff = zdzm'a)2 k. (17)

For a light defect such as neon the local mode

results in an absorption

K((ol ) = -Uzltf C me2

(i)1 f v((o')d(t)'

J (wf-a)' 2

) 2

"- me (18)

The resonant condition for band modes is plotted

in figure 2. It may be shown that the resonance

occurs at that intersection where the slope is negative. The infrared absorption, which may be compared with experiment, is shown in figure 3.

FIGURE 3. Mass defect absorption coefficient. A calculation for c ^ 0 is best done for a given

value of e. In figure 4 is plotted that value of c which will lead to a band resonance at cu r . £(^ = 0)

is that value which effectively decouples the defect

from the rest of the crystal; the asymptotic value for large co r is that value of c which ionizes the defect. The singularities in c occur at frequencies given by

the dashed line in figure 2, close to the mass defect resonant frequencies; the conclusion drawn is that

for physically reasonable values of the polarizability

change c the resonant frequency cannot be shifted

appreciably from the mass defect value. The strength of the absorption, which now in- cludes all of the atoms in the crystal, however, will vary with c. See figure 5, which may be compared

with experimental results. The principal cause

for the discrepancy would seem to be the anharmonic lattice vibrations certainly present within 5°

183
10 x I0 5

3 6

J I L

-8 7 x I0 5

8.5x I05

0 10 20 30 40 50 60 70

FIGURE 4. Force constant resonance condition for Kr and Xe

mass defects.

of the melting point 85 °K. Whereas the experimental absorption for krypton defects is still appreciable at 120 cm -1

, no theory within the harmonic

approximation will result in finite absorption beyond

the one-phonon cutoff at 68 cm -1. The most

promising direction for further work on this problem

would seem to be towards experiments at lower

temperatures or an anharmonic theory.

3 5 x Id" 2

6 2 —

FIGURE 5. Absorption coefficient for reasonable and unreasonable force constant changes with 1/2 percent xenon defect.

: c = - 1.88 X 10* £(r/s)*, <fe = 0.366

: c = -8x 10s g(r/s) 2

. dz = -0M.

The author thanks Dr. R. J. Elliott for innumerable

helpful discussions. The problem was suggested

by Prof. G. O. Jones and the work supported by the

Rhodes Scholarship Trust.

References

[1] G. Jones and M. Woodfine, Proc. Phys. Soc. (London) 86,

101-103 (1965).

[2] B. J. Dick and A. W. Overhauser, Phys. Rev. 112, 90-103

(1958).

[3] W. Cochran, Proc. Roy. Soc. (London) 2 53, 260-276 (1959).

[4] R. J. Elliott and D. W. Taylor, Proc. Phys. Soc. (London)

83, 189-197 (1964).

[5] Our formalism actually allows any computed density of

states to be used; for example, J. Grindlay and R. Howard,

Proc. Int. Conf. Lattice Dynamics, pp. 129-134 (ed. R. F. Wallis, Pergamon Press, 1965).

[6] R. H. Orcott and R. H. Cole, Physica 31, 1779-1791 (1966).

[7] G. L. Pollack, Revs. Modern Phys. 36 , 748-791 (1964).

[8] G. Baldini, Phys. Rev. 128, 1562-1567 (1962).

[9] G. Baldini, Phys. Rev. 136, A248-251 (1964).

[10] G. Baldini, Phys. Rev. 137, A508-513 (1965).

184
Vibrations in Molecular Lattices

K. Dettmann and W. Ludwig

Institut fur Theoretische Physik, C, der T. H. Aachen, Institut fur Reaktorwerkstoffe der

KFA Jiilich, Germany

In molecular lattices there can be translational

and librational modes, which in general are coupled.

If there is a molecular defect, incoming waves of

given polarization can be scattered into another

kind of polarization. Under certain conditions

there might be resonance scattering. This scat- tering is investigated with the simple example of a linear molecular lattice, making the simplest

assumptions on the structure and the intermolecular

forces. In a preceding paper (referred to as [1])

1 we have

investigated the possible localized states in a simple

linear model of a molecular lattice. In this report we want to extend the discussion to the scattering

states in such molecular lattices. We use the same

model as in [1]: A linear arrangement of dumbbellmolecules, having an impurity-molecule at the

origin. There are translational and librational

modes. The stationary equation of motion for the ideal lattice can be written as

Cm

\ = o>

2

; D"» = (Mll Mv )- 1 l*<l>TM. (1)

The lower index p., v=\ refers to translational

motion, p., v = 2 to librational motion. Mi is the

total mass of the molecules, M2 = I is the moment

of inertia, m is the number of the molecule.

<f)TMP are the force-constants. The solution of the

ideal lattice equation is

STM(q(r) =-^=etl (q(T)ei<im; V7V

(2)

q is the wave number, cr the polarization (e.g., mainly translational or mainly librational). In

the simplest model there are four independent

force-constants [1], which we denote by

a = -401°; s =-W\%;

b describes the coupling between translational and

librational motion. We assume in the following

b<p<p + s<a; such an assumption can be justified by a central-force-model. The secular

equation is

a sin 2 q/2 — X(cr) — ib sin q

ib sin q p + s sin 2 q/2 — X(cr) J [e 2(cr)

ei(cr)]

"(4)

The squared frequency spectrum is given in figure

1 (qualitatively).

GO* P p+s o

FIGURE 1. Squared frequency distribution for a one- dimensional molecular lattice of dumbbell-molecules for a certain choice of parameters.

The wings contain mainly translational modes, the width is a mixture

of translational and librational motion.

If there is an impurity molecule in the lattice,

the equation of motion can be written as [1] £ D>TMS? - ASTM = ^ &m^S" ^

nv nv TM* describes the impurity molecule. If there is a change in mass and moment of inertia only (at the origin, e.g.,), then

p = 2D\l + 6 = 2D}§:

K. Dettmann and W. Ludwig. physica status solidi. 10, 689 (1965).

(3) ^^= ^€^8^8°'" 8°"; M'-M

M

; e 2 = '/'-/

(6)

185

256-724 0-67— 13
The solution of (5) can be given as [1] S=.S+&&S, (7) & is the Green's function which satisfies the boundary conditions. For the scattering states, where

5 is an incoming plane wave, 8F S has to represent

outgoing scattered waves. With

we have

t = (1-^)-i,

S = S+&&tS.

(8)

(7a)

In the case described by (6) a simple calculation

gives

Sm = Sm + V^mO C±H Co

M f -l-Ae^oo V

The Green's function is given in [1]:

16

(9)

&'$(k)

as + U2

lim {z—Zi)zm + 1K^{z; k)

zfii z~*Zi (z—zi)(z—Vz1)(z—z2)(z—I/z2) m > 0

(10)

Cg-mO — CgmO. ,y vv *-* vv •> = (JL^v. (10a)

The sum over z,- extends over all the roots zi, l/zi, z2 , l/z2 inside the contour of integration. This is determined by the boundary condition. Figure 2 ©

Figure 2. Choice of the path of integration in the Green-function for p + s < cu

2 < a. shows the situation for p + s < k < a. It is

1 = M+ VM*-N± V2A/2 -/V1

+ 2M VM2 -7V

(z2) ±1 = M-\/M2 -/V±V2M2-^-1-2MVm^^

M(as + 46 2

) = a(p + s)-A(a + s) (11)

7V(as + 46 2

) = 2ap + as- 462 - 2A(2p + a + s) + 4A2

b

/£M„(z; k) - P

"4z"

(1 " 2)2 - X

2z

d-22

)

From figure 2 it can be seen, that (for p + s < k < a)

4

(as + 46 2

) VW^N

_ KuviZi', k)

1 zi-l/Zl 22 K^„{Z2\ k) 1/Z2 22' (12)

Because Zi^e*?; |zi| = l, z2 = e- K <l this means,

that the first part in & represents a wave-like be- havior, whereas the second part shows a localized

behavior, decreasing exponentially with increasing m. For p + s < k < a the solutions of (5) for scat- tering states represent the following situation: There is an incoming plane wave of mostly translational character; this wave is scattered by the molecular impurity. In the scattering process a localized

state is excited, which has mainly librational character. For the mainly translational scattered part

of the wave one can define a backward and forward

scattering amplitude by the asymptotic behavior of

the waves. The second part of (12) can be dropped

asymptotically. It can be proved simply, with the use of (10, 11), that (9) can be rewritten as

for m (13)

with

F(-l) = 4X

(a5 + 46 2)VM2 -7V(zI -l/z1 )

eiKu (zu k) e2 /£22(zi;\)

0022 (13a)

F(—l) is the backward scattering amplitude which

can be related to the reflection coefficient

^=|F(-1)| 2

. A similar expression holds for the forward scattering

(transmission coefficient &~), but because

we will not give it here.

186
From (11) and figure 2 it can be seen, that contains a real (first) and an imaginary part (second

part of (12)). Therefore the scattering amplitude

shows a resonance behavior. The resonance "fre- quencies" A./; are defined by

— 4A.€i Ku(z2 ; A.)

' (as + 46 2

) VM2 -N z2 ~ ll*2

' if e2 = 0 (14a)

and

1 =

4\€2 K22U2; X) ,

(a5 + 46 2)VM2 -7V Z2-I/Z2

if e, = 0. (14b)

We discuss the cases ei # 0, €2 = 0 and €i = 0,

€2 t6 0 separately. By using the first (imaginary)

part of (12) for it can be seen immediately from

(13a) that ^?=1 at the resonance frequencies in these simple cases. Figures 3 show the resonance

frequencies ojr and the widths y of the resonances

as a function of €1 or €2, resp. It is assumed a = 12; p = s = 4; b = 2 here.

€i =

e2

:

VX(X-8)

8VX- 8

y=

XVX-8

2Vl6-(8-X)2

\3/2 2Vl6-(8-\)2 (15)

Figure 3 shows the position of the resonance fre- quencies relative to the frequency band of the ideal

molecular lattice as well as the widths. For a change in total mass only, there are resonances if

€1 > 2/V3. This means a heavy mass, therefore we have nearly total reflection and the resonances

are not very pronounced (width large). The reflec- tion coefficient is shown in figure 4a. For smaller

values of €i or for negative ones, the reflection is smaller, but there are no resonances.

Figure 4. Reflection coefficient in the band with p + s < to2 < a

as a function of oy for (a) £i=5, ei=±l/2, €2 = 0 and (b) e2 = -l/4, e2 =±l/2, e, = 0.

j 1

1 4 1

3)

1

1

b)

1

1

1

3 1 1 —-i

2

r

\\ —

1

1

1

1

1

1

\

11\

; 2345 -0/ -0.2

1

Figure 3. Resonance frequencies in the band p + s < w2 < a and widths as afunction of (a) ei with e2 = 0 and (b) €2 with d = 0.

187

x
In the case of a change in the moment of inertia of the impurity molecule, there is a pronounced

resonance, if 0 > €2 > — 0,385; this resonance has a small width.

In the scattering process of the "translational" waves the molecule is excited to a "localized"

librational motion. In the resonance this librational

motion is especially pronounced. If €2 < — 0,385, we have localized librational states with frequencies

above the maximal band frequency. The reflec- tion coefficient (fig. 4b) shows the resonance phe- nomena for €2 = — 1/4, e.g., but not for ez = +1/2 or -1/2, e.g. In general the reflection coefficient in the upper

region of the spectrum (fig. 1) can be written as

2ri,2 — Zl,2+ l/zi,2-

e2 = 0:

4\2 AMp-X--(r,-l)

(as + 462

) 2 (M2 -A0(l-r2

)

2X X +-(r2 -l)-p

(as + 46 2

) VM2 -N Vr§-1

e, = 0:

[l-e2 g2(A)] 2 + el/l(A)"

4A2

2(X)

k+-(ri -l)

(as + 46 2

) 2 (/^/ 2 -A0(l-r2

)

,

2k

a.

,

1)

(as + 46 2

) VM2 -N Vr2 -1 (16)

The following general features can be seen from

these relations. In the first case, the resonance

frequency varies from o>| = p + s = Kr for €i = °° to

(oR = a (upper limit of band) for

6r

a[a-p-5-4fe2 /q] 3/ 2

4fe 2 Va-p

>0 (18)

i.e., resonance does occur only if there is a coupling

between librational and translational modes (b 4= 0). The width y is zero for oj'r = p + s and infinite for

(oR = a. If resonance occurs, the reflection coeffi- cient has a certain finite value for a)2 = p + s, in- creases to one at cjr, then decreases (slightly) and

reaches again R — 1 for w2 = a. If there is no reso- nance, then R increases over the whole region from

a finite value to one at a>

2 = a. In the second case, there is a resonance for

0 > e2 > -- V(a-p)[a-p- s -462/a] (19)

which does not vanish for 6 = 0. Then it becomes

truly "localized", y shows the same behavior as

in the first case, but it is smaller. The reflection

coefficient is zero at w2 = p + s, passes a maximum

between these limits and is again zero for tu2 = a. The maximum value is R — l at (Or, if resonance

occurs.

If both ei and e2 are different from zero, the situa- tion is more difficult, but qualitatively similar

phenomena are possible.

This holds also, if the frequency of the incoming

"translational" wave lies between w = 0 and w

2 = p. Quantitative differences occur. So resonances of

the second kind are present if 0 < €2 < 00 . If p < co

2 < p + s, no resonance scattering is possible (for single molecular defects). But because

of the interaction it happens that an incoming wave

of given polarization is scattered into outgoing waves, one of the original polarization and one of

the other kind of polarization. Thus one has two

reflection coefficients and two transmission

coefficients ^crtr' with

J (^+ ^<T(T .) = 1.

But these states are less interesting.

188
The Use of Morphic Effects for the Study of Vibrational and

Optical Properties of Impurity Atoms in Crystals

A. A. Maradudin* and S. Ganesan*

Department of Physics, University of California, Irvine, Calif. 92664

and

E. Burstein**

Laboratory for Research on the Structure of Matter,

Department of Physics, University of Pennsylvania, Philadelphia, Pa. 19104

Morphic effects are effects induced in a crystal by the lowering of its symmetry through the appli- cation of a generalized external force. In this paper we study the effects of static electric fields and of

strains on the frequencies of localized vibration modes associated with substitutional impurity atoms

in ionic and homopolar crystals. It is shown that the splitting of the triple degeneracy of such modes

when the impurity is at a site of cubic symmetry by the application of these generalized forces can

yield valuable information about the anharmonic forces, electric dipole moment, and electronic polarizability associated with the impurity atom.

A crystal subjected to a generalized external force, such as a static electric or magnetic field, or a stress, shows several interesting effects, called morphic effects, which are the results of a general lowering of the symmetry of the crystal on the application of the force. They manifest themselves, among other ways, by lifting the degeneracies and shifting the frequencies of the normal

modes of vibration, by altering the selection rules for various physical processes, making it possible for certain otherwise forbidden processes to occur, and by modifying the strengths of various

optical transitions. The effects of static electric fields and of a strain or stress are particularly interesting, and

recently have been observed experimentally. Hayes, MacDonald, and Elliott 1 observed a splitting of the triply degenerate localized modes associated with £/-centers at the F~ sites in CaF2

(which have Td symmetry) produced by either a static electric field or a uniaxial stress, and gave a phenomenological theory of these splittings.

We present in this paper a microscopic theory of these morphic effects. Until now a theory

which relates the magnitudes of such splittings to the harmonic and anharmonic force constants

with which the impurity is coupled to the atoms of the host crystal, and to the elements of the dipole

moment and static electronic polarizability operators associated with the impurity and with its neighbors has not been available. We deal here with substitutional impurity atoms occupying

sites having either Td or Oh symmetry in nonionic dielectric crystals. The results become complicated in ionic crystals owing to the long range coulomb interactions.

First we consider the morphic effects induced by static electric fields. When an electric field E is present the interaction term in the crystal Hamiltonian is of the form 2

•Research supported by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant Number 1080-

66. "Research supported in part by the U.S. Office of Naval Research.

1 W. Hayes, H. F. MacDonald, and R. J. Elliott, Phys. Rev. Letters 15,961 (1965).

z M. Born and K. Huang. Dynamical Theory of Crystal Lattices, p. 310 (Clarendon Press, Oxford. 1954).

189
where MM and are the operators representing the electric dipole moment and static electronic

polarizability of the crystal. Since they are functions of the atomic positions, we can expand these

operators in powers of the displacements of the atoms from their equilibrium positions in the field- free crystal. The equations of motion for a dielectric crystal containing impurities in the presence

of a static field, in the adiabatic approximation, can be written in terms of the displacements of the

atoms as -Mlk£(lk)=4>a(lk) + £ Qcdlk; l'k')^{l'k')

+

l 2 S

^(/A

' : l

' k' ; l"k")m'k')Hl"k")+. ... (2) Vk'& l"klrY

Here (a(lk) is the^ a-Cartesian component of the displacement of the Ath atom in the Ith unit cell, and the various $ coefficients are the sums of contributions from the unperturbed atomic force constants, dipole moments, and static electronic polarizabilities. The first term, which is independent

of the displacements £«(//.), shows that there is a force acting on the atoms in the presence of the

field. We can write

£M) = da(lk)+ua(lk), (3)

where da{lk) is the static displacement of the atom (Ik) from its field free equilibrium position

and ua (lk) is the dynamic displacement from its new equilibrium position. In eq (3) we have

omitted terms which arise from electrostriction, as they are of higher order than the ones we consider here. The static displacement da {lk) can be determined from the condition that when all of the atoms

are situated at the new equilibrium positions there is no force acting on any atom. This means

that when we substitute eq (3) for ija(lk) in the equations of motion and reexpand them in powers

of the dynamic displacements {ua(lk)}, the term independent of ua(lk) must be zero. Using

this condition we get an expression for the effective harmonic force constants of a crystal in the

presence of a static external electric field, which, to first order in the field components, is given by

<&ap(lk; l'k')=<t>af} (lk; I'k^+^E^^ilk; l'k'), (4)

where the {<Pap(lk; l'k')} are the unperturbed harmonic force constants and

<JV. ap(lk; l'k') =-M^ aP (lk; l'k') + X S ^(lk; l'k'; l"k")

l"k irY l'"k'"S xrvs (/T; l"'k"')M^,(l"'k"'). (5)

Here <Pa0y(lk; l'k'; l"k") is the unperturbed cubic anharmonic force constant; rys{l"k"; l"'k"') is an effective inverse matrix of the matrix <t>a(}(lk; l'k'); MM , s (l'"k'") is the coefficient of the term

in the expansion of the crystal dipole moment which is linear in the atomic displacement; and

Mu., a/3(lk; l'k') is the coefficient of the term in the expansion of the crystal dipole moment which

is quadratic in the atomic displacements. Now the equations of motion in the harmonic approximation for a crystal subjected to a static external electric field can be written as

M,k u a (lk) =- ^ &af} (lk; l'k')u0(l'k'). (6)

I'k'B

190
We can solve for the eigenvectors and eigenfrequencies of the crystal in the standard manner. The dynamical matrix of the secular equation is then expanded in powers of the applied field E

Da0(lk; l'k') = D^(lk; l'k') + D§>(lk; l'k')+. . . (?)

where D%(lk; l'k') is the unperturbed dynamical matrix and D^(lk; l'k') is the perturbation due

to the field. Using degenerate perturbation theory we can calculate the splittings of the triply

degenerate localized modes associated with a substitutional impurity having Oh or Td site symmetry, and we obtain the following results.

(1) For an impurity at a site possessing Oh symmetry, there is, of course, no splitting to first order in the field, and we have to go to second order in the field to get any splitting.

(2) For an impurity site possessing Td symmetry the degeneracy is lifted to first order in the

field, when the field is applied along the [100], [110], and [111] directions.

(a) For the static electric field in the ZF-plane, the triple degeneracy is completely lifted. The

expressions for the three frequencies are

£L\ = oh) + mEl2too

0,2 — OJo

fl3 = (oo — mEI2(oo (8)

where E is the magnitude of the electric field, o)o is the frequency of the unperturbed localized mode,

and the coefficient m is given in terms of atomic force constants and eigenvectors.

(b) For the field along the [111] direction, the triply degenerate mode splits into a doublet and

a singlet. The expressions for the frequencies are

Cli = wo — m£'/(2V

/3a)o) H2 = o)o — mE\{2 V3a>o)

n3 = o)0 +mEI(VSo)o). (9) We find that in a plot of O versus E, the line corresponding to the singlet fi :) will have twice the

slope of the doublet fli, and fl-z, In the case of an impurity atom occupying a Td site in a crystal of the diamond structure, the

inversion symmetry of the two Td sites in a primitive unit cell, the result that the splitting of the

degeneracy of the localized mode is linear in the field, and the assumption that both sublattices of

the crystal contain impurity atoms, lead to the conclusion that in addition to the localized mode

frequencies given by eqs (8) and (9), there exist localized modes whose frequencies are obtained on replacing E by — E in eq (9). No new frequencies result for the field applied in the A^F-plane. However for the field in the

[111] direction, the shift in the localized mode frequencies is opposite in sign for the two Td sites and, consequently, there will be two doublet frequencies and two singlet frequencies. In their phenomenological theory of the splitting of the triply degenerate localized modes associated with U-centers at the F" sites in CaF2 . Hayes, MacDonald, and Elliott did not take into account the

existence of the two Td sites in the primitive cell of the CaF2 structure. They accordingly obtained an expression for the frequencies of the localized mode at a single Td site in an externally applied

electric field, similar to that of eq (9), which only produces a splitting into a doublet and a singlet.

Experimentally, they observed a splitting of the localized mode absorption line into two lines, rather

than into four lines which is predicted on the basis of the two Td sites. The reason for the dis- crepancy between experiment and theory is as yet unclear.

191
We now turn to the morphic effects induced by a strain. We again start with the equations

of motion of the crystal and write the displacements £; a (lk) as

ta(lk) =^ tapXpdk) + da(lk) + Ua(lk) . (10)

(3

Here the first term describes a homogeneous deformation, the second term gives the rigid body

displacement due to alteration of the force constants by the introduction of the impurity, and the

third term gives the displacement from the new equilibrium position. Now the equations of motion can be written as

-MlkUailk) =Oa (/A)+ 2 $>a0(lk; l'k')Uf}(l'k') (11)

l'k'0

where

®a (lk) = £ &ae(lk; l'k')V0 (rk') l

' k '- l"k")Ve (l'k')Vy (l"k") (12a)

I'k'p I'k'fi i"k"y ®afi (lk; l'k')=<S>afi(lk; IT) + X Qaftydk; IT; l"k") Vy(l"k") (12b)

l"k"J

and we have used the displacement

Va (lk) = ^ea0X0(lk)+da (lk). (13)

The first term on the right hand side of eq (11), which is independent of ua (lk), must be zero as there is no force on an atom in the new equilibrium position. This enables us to express da (lk)

in terms of ea/s. Using the Lagrangian finite strain parameter r) ap ( ea^ + €pa + ^ tya£yp)^

and rotational invariance conditions on the atomic force constants we get an expression for the

effective harmonic force constants. As before, we can express the effective dynamical matrix

in ascending powers of the strain parameter. Since we are interested in first order effects, we

treat the term linear in strain as a perturbation on the original unperturbed dynamical matrix, and

obtain the following results for both Td and Oh symmetries.

(a) A hydrostatic pressure does not split the degeneracy of the localized mode but merely

shifts the frequencies. The expressions for the frequencies are

ai = a2 = a3 = aj0 + (a + 26)A/(6w0 ) (14)

where a, b, c are constants given in terms of atomic force constants and eigenvectors and A is the dilation.

(b) A uniaxial strain y)ap= T}8a i8^i splits the triple degeneracy into a doublet and a singlet

whose frequencies are

Hi = coo + arjl ( 2coo) n2 = o>0 + W(2w0 ) (15)

n3 = (D0 + brjl (2w0 )

192
(c) A shear strain 7) a/3 = r)5al S/3 > + 8^ 1 8a2 completely lifts the degeneracy. The perturbed

frequencies are

fil = <t>0+ C7]/(O0

fl> = &>o (16)

fl3 = w0 — crj/wo

We next consider the experimental conditions for the observation of these effects and discuss

the orientations of the electric field vector of the incident light with respect to the static electric

field and the crystal axes which are necessary for the observation of the perturbed localized modes

in infrared lattice absorption experiments.

The localized mode contribution to the imaginary part of the dielectric tensor in the harmonic

approximation is

7rA2 3 e

(,M=-=- w sgn w £ ^JT£8(w2-ns2

) (17)

' s = 1 where

^=5) M»Jlk)V<$(lk)KMlkyi2. (18)

Ika

In these expressions o» is the frequency of the incident electromagnetic field, the light; V is the

crystal volume; V^\lk) is the eigenvector of the perturbed dynamical matrix, given, for example,

by eq (7), associated with the eigenvalue flf ; and A is a factor which takes account of the fact that the macroscopic electromagnetic field inside the crystal is not necessarily equal to the externally

applied field, due to the depolarization field. To observe the perturbed localized mode labeled by

the index 5, Jt^ must be nonzero for at least one value of /jl, and the external electromagnetic field

e?°(£) must have such nonzero components that the scalar product J(s

- £f°(i) is nonzero. On the

assumption that the localized mode is so spatially localized that only the impurity atom is vibrating, we obtain the following results.

(a) For the static electric field in the AF-plane we find that the two perturbed localized modes

will be observed regardless of the direction of the static field in the ZF-plane, provided the electro- magnetic field has a nonzero Z component. Also all three modes can be observed for c?°(t) in the

XF-plane if Ey%% ^-E$% and E$% * E$%.

(b) For the static electric field along the [111] direction we can observe all the three modes, for the electromagnetic field along the [100] direction.

(c) In the case of a hydrostatic pressure, for the observation of the triply degenerate perturbed

mode $?°{t) can be parallel to any one of the coordinate axes.

(d) In the case of a uniaxial strain, we can observe the singlet if <g% 0 and the doublet if ef°U)

is in the FZ-plane.

(e) In the case of a shear strain we can observe one localized mode if the electromagnetic field has a Z component, the second mode if the field is along the [110] direction, and the third mode if the field is anywhere in the AF-plane and not along the [110] direction.

The above theory gives us an understanding of the mechanisms responsible for the morphic

effects induced by static electric fields or strains in a crystal containing substitutional impurity

atoms. These results apply to gap modes also. We find that the frequency shift of the localized

mode induced by a static electric field can be given in terms of contributions from charge deformations and static displacements of the atoms in the crystal. The frequency shifts in this case are expressed in terms of only one parameter m. For the strain induced frequency shifts, we have

shown that they can be expressed in terms of three parameters, a, b, and c. These parameters

193
are very difficult to evaluate for a crystal containing impurities. With more experimental data

and a much simplified theoretical model, one can evaluate the strengths of the electric field or

strains necessary to produce appreciable frequency shifts. A more detailed account of the electric

field and strain induced morphic effects in crystals containing impurities will be published

elsewhere.

194
The Vibrational Properties of Point Defects in Solids

Report of Panel III Discussion

C. W. McCombie

Physics Laboratory, Whiteknights Park, Reading, Berks, England

(Panel Members: R. J. Elliott, J. A. Krumhansl, W. Ludwig, A. A. Maradudin, C. W. McCombie, and G. F. Nardelli)

1. Introduction

Panel III was concerned with the vibrational

properties of defects. Some of those who contributed to the discussion have provided short ac- counts of what they said and these accounts will form the main part (Part 3) of this report. They

are prefaced (Part 2) by an outline of the overall picture of the present state of the subject which

emerged from the Panel Session and from those of

the contributed papers which were on related

topics.

2. Overall Picture

2.1. Processes at Isolated Defects in a Harmonic Lattice

The first topic considered by the Panel was the

determination of the effect of lattice vibrations on

processes (such as the absorption of radiation) at an isolated imperfection. From the point of view

of lattice dynamics the crux of these problems

lies in taking account of the fact that the presence

of the imperfection will modify the normal modes

of vibration of the lattice. The general features of

these modifications are now well known. A few

modes may appear which are localized at the imperfection and which have frequencies outside the

ranges of perfect lattice mode frequencies. Most

modes, on the other hand, will still extend through

the crystal and have frequencies in the ranges of

perfect lattice frequencies, but their forms, and so

their coupling to processes at the imperfection,

will be changed. It may happen that modes of this

latter type in a particular fairly narrow range of

frequencies are modified in such a way as to have

their coupling to the process at the defect greatly

enhanced. In this case one talks of having an "in- band resonance" or a "quasi-localized mode," and

conspicuous features in the experimental data on

the effect being considered may be directly corre- lated with these resonances in the coupling. It

is, however, important to be able to take account

of modifications of the coupling even when the res- onances have no such direct effect or when there

are no marked resonances at all. It was emphasized (3.1) that if one can take the

changes in force constants at the imperfection to be known (and if anharmonic effects are unimportant) the coupling to the lattice vibrations, allowing

for the modification of the modes, can be reduced to machine computation. This is because the re- quired results can be expressed in terms of certain

response functions (or Green's functions) for the

perfect lattice which are easily determined if the

forms and frequencies of a sufficiently large sample

of perfect lattice modes are known. The case of

infrared absorption by a charged defect was used

as an illustration (3.1) but the method can also be

applied to determining the absorption structure as- sociated with an electronic transition, the strength

of the zero-phonon fine in the Mossbauer Effect,

the scattering of phonons by the imperfection, and

so on. The essential point is that in each case a function of frequency which determines the re- sult we require can be related directly to a response

function (or Green's function) for the imperfect

lattice. The method depends on the simple rela- tion which exists between the imperfect-lattice re- sponse functions and the corresponding perfectlattice response functions.

The relation between perfect and imperfect lat- tice response functions has been derived in many

ways by many authors. It seemed clear that the

main objective now should be to apply the methods

based on this relation to realistic models of centers

in realistic crystals, the problem becoming one of

straightforward machine computation when the

changes in force constants at the imperfection have

been decided upon (see 2.3). There seems no

reason why quite complicated centers should not be analyzed. The difficulty is greater the more

extended the imperfection and the lower its symmetry but there should be no insurmountable ob195
stacles in the way of such calculations.

The calculation of the perfect-lattice response

functions depends on determining the forms and

frequencies of the modes associated with a large

number of points in &-space. To get satisfactory

response functions (particularly at low frequencies)

a sample of something like 105 points may be desirable. It was suggested that whenever eigenfrequencies and eigenvectors for a sample of this order are determined some of the simpler response

functions (including cross response functions)

should be determined as this requires very little extra time on the computer. Some doubts were

expressed as to whether this was a practical sug- gestion but at all events it seems worth emphasizing to those concerned for any reason with large

machine computations on perfect lattice dynamics

that some cooperation with people primarily inter- ested in the dynamics of imperfect lattices may be

very fruitful. While considerable progress in the directions

indicated has been made, as is seen in the paper by

Benedek and Nardelli presented at the Conference,

much remains to be done. Alternative ways of

considering the modifications of the modes may be

important in that they may show how to get rapidly

to useful results. They may, moreover, in some

cases give better insight into what is going on. The paper presented by Toulouse on a phase shift analysis of the scattering of lattice vibrations is an example of an approach which may be useful in

this way.

2.2. Finite Concentrations of Defects in a Harmonic Lattice

Elliott pointed out (3.2) that there were problems

associated with finite concentrations of randomly

distributed defects which had not been completely

solved. One might be interested in investigating,

for example, the neutron scattering or the vibrational

contributions to the thermodynamic functions for a random alloy. The problem is essentially one

of multiple scattering. Taylor (3.3) contributed

a brief account of an approximate approach to such

problems.

2.3. Modified Force Constants at Defects

Ludwig (3.5) and Maradudin (3.6) contributed to a discussion of the treatment of the modification

of force constants in the vicinity of a defect which,

as already indicated, is a necessary preliminary to the sort of calculations discussed in 2.1. Wood

and Gilbert also contributed to this topic at the

Conference with their paper on calculations for the

[/-center. The difficulties of such calculations are formidable and there is no doubt that much work

in the immediate future will evade these difficul- ties by introducing a number of adjustable parameters which describe the force constant changes,

full use being made of symmetry considerations

to keep the number of parameters as small as possible. Where one is accounting for experimental

results which are rich in detail (as where one has

sharp-line electronic absorption accompanied by

complicated vibrational structure) one may allow

oneself one or two such parameters and still be

able to regard agreement with experiment as sig- nificant. This will hold even more strongly if one

has, for the center considered, data on a variety of

different effects involving lattice vibrations. One

might, for example, have information on both opti- cal absorption and the Raman effect associated

with an electronic transition at the center, on infra- red absorption associated with motion of the charged

ions at the center and on the scattering of phonons

by the center. A small set of force constants which fitted such a range of data for a particular center would be of great interest. The quantum mechanical calculation of the

changes in force constants can, however, be ex- pected to be a field of investigation of growing importance. Maradudin (3.6) outlines how such a

calculation might be carried through. One point

which came up in a number of places during the

Conference (cf. Ludwig's contribution 3.5) was that whenever forces such as van der Waals forces are important (as they appear to be in ionic crystals) one will have to go beyond a Hartree-Fock calculation, since these forces depend on correlations

which are ignored in the Hartree-Fock treatment.

2.4. Anharmonic Effects

Treatments which ignore anharmonic effects entirely are adequate for many purposes but there

are other effects which depend essentially on cubic

and higher terms in the interaction between ions. For example, the infrared absorption associated

with a localized vibration is frequently not a single

sharp line but is accompanied by an attendant

structure lying predominantly on the high frequency

side of the main sharp line. This is associated

with transitions in which other modes of lattice vibration are excited at the same time as the localized" mode. Such multiple transitions are possible

only because of anharmonic effects (among which

one must include nonlinear terms in the expression for the dipole moment). Anharmonic terms

are responsible also for the shift in frequency of

localized mode absorption under the action of ap- plied fields or stresses. Such effects (termed by

them "morphic effects") are discussed in the paper

by Maradudin, Ganesan, and Burstein. As dis- cussed in that paper and as emphasized by Elliott (3.4), study of these effects can form a very

interesting source of information about anharmonic

forces at the defect.

2.5. Entropy of Formation

The calculation of the entropy of formation of, say, a vacancy and an interstitial is clearly one of

196
the aspects of the dynamics of imperfect lattices which is most relevant to the central theme of the

Conference. It was discussed in the paper by

Benedek and Nardelli and some time was devoted

to it by the Panel. The calculation involves in- vestigating the changes in frequency of the modes

of vibration of the lattice consequent upon forming

the defects, since in the harmonic approximation

the entropy can be expressed as a sum over modes

of a function of frequency. These changes in frequency arise in two ways. In the first place there

is a "local" effect due to the modified masses and

force constants at the imperfections. This short

range effect can be handled by an application of

response function techniques. In addition there

is a long range effect. This arises because a defect causes a change in the bulk lattice spacing which,

because of the anharmonic terms in the interaction

potentials, changes the restoring forces on all the

atoms of the crystal. In order to calculate this long range effect one would first have to calculate

the change in lattice parameter: the change in entropy consequent upon a change in lattice parameter may then be calculated from thermodynamic

data. The calculation of both effects reduces

therefore to an investigation of the way in which the

potential energy of the system depends on the position of atoms in the immediate vicinity of the defect, so that one can calculate both the relaxed positions

of the atoms and the restoring forces returning the

atoms to these relaxed positions. Such calculations

are, of course, extremely delicate and it is here

that the whole difficulty of the problem lies. There

are additional difficulties when the vacancy and

interstitial are effectively charged as in the alkali halides. In that case the first effect, which we

termed "short range," contains a long range part

and dielectric effects have to be considered.

2.6. Diffusion

There was some discussion of the calculation of

jump rates in diffusion (the rate, for example, at which interstitial atoms jump from one site to another, or the rate at which neighboring atoms

jump into a vacancy). According to Rate Theory

one has to calculate the energy and entropy of the

system when, roughly speaking, the atom is midway

between its two equilibrium positions. Thus one

encounters again all the difficulties of the entropy

of formation calculation which has just been dis- cussed. Some doubt was expressed about the

inclusion in the saddle point entropy of the long

range part (which corresponds to dilatation of the

lattice) since, it was argued, the extent of the dis- turbance of the lattice produced by the atom during

its brief presence near the saddle point would be

very limited. Against this it was argued that the

jump process is a cooperative effect of the diffusing

atom and its surroundings and it is just as valid to think of a fluctuation in density of the lattice over an appreciable region around the defect as encouraging the jump as it is to think of the jumping atom

tending to cause the lattice to dilate: on the first of these views the time of response argument ap- pears to lose its force. The question of the status of Rate Theory in this problem was raised and considered briefly.

There was strong support from a number of people

for the validity of Rate Theory with a transmission

factor of order unity, but it was also argued that

this theory might be inappropriate if the damping

on an atom was so large that it had to be thought

of as diffusing over the barrier rather than jumping

over it. (See the paper by H. A. Kramers, Physica

Vol. 7, p. 284 (1940).)

2.7. Displaceable Ions. Jahn-Teller Effects

Krumhansl contributed a brief discussion of cases such as Li+ in KC1 where the substitutional

ion is in a potential field which has its minima,

which are shallow, at points other than the normal

lattice position. This situation (which was de- scribed as "extreme anharmonic") clearly intro- duces new features not considered in the usual

lattice dynamics.

Krumhansl's survey followed a paper by Bowen,

Gomez, Krumhansl, and Matthew in Phys. Rev.

Letters Vol. 16, p. 1105, to which reference should

be made. Most of the discussion in that paper

takes the form of a quantum mechanical treatment

of the states of the Lithium ion moving in a rigid

potential field. Many features of the experimental

results can be understood in terms of this model.

However, as pointed out by the authors, the sur- rounding ions may tend to exhibit a distortion which

follows to a greater or less extent the motion of the

Lithium ion, so that one has, strictly speaking, a Jahn-Teller type of problem to consider. One will also have to consider the way in which a center of

the type under discussion scatters phonons. Both

these topics present interesting problems for lattice dynamics. It may be remarked in conclusion that Jahn-Teller type problems for electronic states

also raise many interesting questions but these were not discussed by the Panel.

3. Individual Discussions

3.1. Response Function Methods.

C. W. McCombie (Reading)

The Green's function (or response function) ap- proach can be applied to the treatment of the effect

of defects on lattice vibrations in two rather differ- ent ways, one mathematical and the other more

physical. In the first approach the matrix equations involved in determining the normal modes

of the whole system are formally written down and

197
the matrix problem is then reduced by the intro- duction of Green's functions in the usual way.

In the second approach we start with a physical

argument, thinking of the system as a perfect

lattice with extra masses (possibly negative) rigidly attached to some atoms and extra springs

(possibly with negative spring constants) connecting certain pairs of atoms. The extra masses will correspond, of course, to the changes in masses

of any substitutional foreign atoms involved in the

imperfections and the extra springs to the changes

in force constants between the atoms in the vicinity

of the defect. To deal with an interstitial atom

one will have to consider an extra mass, not rigidly

attached to any lattice atom, but connected to several (the neighbors of the interstitial) by springs.

To deal with a vacancy one can simply add springs

which cancel out the interaction forces between

the atom to be removed and its neighbors while

at the same time adding springs between other

atoms of the perfect lattice to take account of the

changes of force between them. In some problems one will have to take care to eliminate contributions from the atom which is still present

although uncoupled to the lattice, but this is unlikely to present serious difficulties. To illustrate the procedure in this second ap- proach we shall consider the infrared absorption

by a charged substitutional atom in a monatomic

cubic lattice. The mass of the atom is supposed

to exceed the mass of a normal lattice atom by an amount AM, but the atom is supposed to interact

with surrounding atoms in exactly the same way

as a normal atom. According to the approach we

are adopting, we shall consider a perfect lattice

in which one atom carries an (extra) charge e and

also has a mass AM rigidly attached to it. Be- cause of our assumption about the interaction of

the foreign atom with its neighbors it is not nec- essary to introduce attached springs. The system

of perfect lattice with attached mass and added

charge which we have described will clearly re- spond in the same way as the actual crystal (with

its substitutional impurity) to the electric field Eeiu>t which a beam of infrared radiation causes

to act at the impurity: in particular the absorption

of energy from the beams will be the same in the

two cases. To calculate the absorption we need a response

function for the imperfect lattice which is defined

as follows. A force Fe'wl is applied to the impurity

atom and the resulting displacement of the impurity atom after transients have died away (so that the disturbance in the lattice will be an out- going wave) is denoted by Deiwt

. (The force and

displacement will be in the same direction be- cause the crystal is cubic.) The response function

Amp(^) — j<?im P (w) is defined by

D = ^{Pim^)-iQimp {(o)}F

where M, the mass of a normal atom of the lattice,

is introduced into the definition merely for con- venience. The rate of absorption of energy by

the lattice, which will be equal to the rate at which

work is done by the force Feiwt acting on a particle

with displacement De'"', is easily seen to be

§ ((o/M)Qimp \F\ 2

: replacing F by eE gives the rate of absorption of energy from the infrared beam.

A direct calculation of Pimp and (?i mp is possible

if one is prepared to investigate directly the normal

modes of the imperfect lattice: where the forms

and frequencies of the modes are known it is easy

to determine the response of each mode to a force

applied to the lattice and so to get the resulting

contribution of the mode to the displacement of

the atom of interest; the total response, which

determines the response functions, is then obtained by summing over the modes. Unfortunately ; the forms of the modes of the imperfect lattice are j difficult to determine and so the direct approach j we have just outlined is very messy and difficult

j to carry through with any accuracy.

On the other hand the corresponding response

function for a force applied to an atom of the ij perfect lattice is easily obtained since it is easy

for an electronic computer to determine the forms

and frequencies of a very large sample of perfect 1 lattice modes. If Feitot and Deiwl are the applied ! force and resulting displacement (after transients)

of an atom of the perfect lattice, the perfect lattice response function is defined by

D =

jj (PperfM — iQperf {(o))F.

As already asserted there is a simple relation

between the response functions for the imperfect

and perfect lattices, so that a direct calculation of | the imperfect lattice response function by the

method outlined above becomes quite unnecessary. . To establish the relation we suppose the force ; Fe'wt applied to the atom with the attached mass AM and consider explicitly the force exerted by

the extra mass on the lattice: by Newton's Laws

this is easily seen to be AM(o2 Deiwt where D is the |i displacement of the atom (and so, of course, of

the rigidly attached mass AM). The total force

acting on the atom of the perfect lattice is, there-

|] fore, {F + AM(D2D}eio3t and we may use the perfect 1 lattice response function to write

D = jJ {Pper{ (<o) i'(?perf(w)} {F + AM(02D}

which gives

{Ppert (<*))- iQpevf ( <» ) }

AM 1 .AM ~~ —a/fPerf(w) | + l-^-<W^perf(<«>J

198
But here De'01 ' is the displacement of the isotope

when the force Fei<o1 is applied to it and so D and

F are related by

D=^{Pimp ((o)-iQimp (<o)}F.

Comparing the last two equations gives Pimp (o>) and Oimp(w) in terms of the perfect lattice response

functions. One has in particular

(?imp(w)

<? PerfM

AM

M

This at once allows us to express the rate of ab- sorption of energy from the radiation field, which

will be | ((olM)Qimpe

2 \E\ 2

, in terms of the perfect

lattice response function. We have, therefore,

given an example of how the relation between the

response functions for the perfect and imperfect

lattices is established and how this relation is applied.

If one wishes to consider changes in the inter- action between the foreign atom and its neighbors

(which, as already explained, one does by intro- ducing attached springs as well as an attached

mass) one needs more elaborate response functions

(and cross response functions) for the perfect

lattice, and one gets in general a small number of

simultaneous equations which determine the rela- tion between the required imperfect lattice re- sponse functions and the perfect lattice response

functions. The procedure, however, still goes

through in a straightforward way and it is easy to arrange to make full use of symmetry to reduce the

complexity of the computations.

It may be worth remarking that the relation be- tween the quantity of interest and an imperfect

lattice response function is not always quite as obvious as it is in the case of infrared absorption.

Suppose, for example, that we wished to deter- mine as a function of temperature the mean square

displacement of the isotope from its equilibrium

position. One finds that this is easily obtained

provided one knows for ah the modes in each small

frequency range the sum of the squares of their

contributions to a component of displacement of

the isotope, the magnitude of the displacement in each mode being adjusted to make the potential

energy in the mode unity. But the same information

also enables us to determine the simple response

function considered previously, and, conversely, we can extract this information if the response

function is known. Thus expressing the imperfect lattice response function in terms of the perfect

lattice response function again solves the problem.

3.2. Finite Concentration of Defects.

R. J. Elliott (Oxford)

I agree with McCombie that the single defect

problem is now well understood and that future

interest will center on the solutions of models of more complex defects using one or other of the

established methods. One problem which is of

great interest which is still far from solution is that of vibrations in random alloys. At small

concentrations of defects the properties may be

discussed in terms of the single isolated defects. The density of states, for example, will be modified with peaks near resonances and impurity bands

near local mode frequencies. As the concentration of defects is increased interference between

scattering by several defects becomes important.

It is essential to include pair effects even at low

concentrations to obtain a satisfactory treatment

of impurity bands. The methods which expand

in clusters of increasing size look unlikely to be

convergent for sizeable concentrations and some

self-consistent approximation must be sought.

Some important references in this field are

J. S. Langer, J. Math. Phys. 2, 584 (1961); Phys. Rev. 131, 163

(1963).

S. Takeno, Prog. Theor. Phys. 28, 33 (1962); 29, 191 (1963); 30, 144 (1963); Supp. 23, 94 (1962).

S. Edwards and J. Beeby, Proc. Roy. Soc. A269, 518; A274,

395 (1962); A2 79 , 82 (1964); Phys. Rev. 135, A130 (1964).

J. des Cloizeaux, Phys. Rev. 139, A1531 (1965). R. J. Elliott and D. W. Taylor, Proc. Roy. Soc. (to be published).

3.3. Finite Concentration of Defects. D. W.

Taylor (Bell Telephone Laboratories)

The multiple scattering theory of Lax has been

used to obtain equations for the displacementdisplacement Green's functions for a crystal containing substitutional defect atoms. A simple

approximation for a random array of defects yields

previous results that are correct to first order in the concentration. A self-consistent procedure

using a coherent self-energy to describe the configuration-averaged system and suitable for large

concentrations of mass defects is described. It is evaluated in a simple approximation in both one and three dimensions to give the spectral function

and the density of states for various concentrations.

The method is symmetric in both defect and host

atoms and can be considered to give a good description of the coherent behavior of the imperfect

crystal. In particular it predicts that in the three

dimensional example the maximum frequency of the

host crystal phonon band is reduced as light defects

are added, a somewhat surprising result but in qualitative agreement with recent machine calculations. The nature of the approximation precludes

the appearance of the detailed structure found in the machine calculations.

3.4. Anharmonic Effects. R. J. Elliott

(Oxford)

Another aspect of crystal dynamics which can

be conveniently studied through the spectroscopy

of defect vibrations is anharmonic effects. In the

perfect lattice some information may be obtained

from third order elastic constants. These reflect

199
sums over anharmonic interatomic force constants

with a weighting in favor of those at long range. A study of very localized modes, such as those of H

ions in alkali halides and alkaline earth fluorides,

yields information about short range, anharmonic

interactions and the shape of the local potential

well. The most detailed results come from har- monics and side bands of the localized vibration

absorption.

Some important references in this field are W. Hayes et al.. Proc. Roy. Soc. A289, 1, (1965); Phys. Rev.

Letters 15, 961 (1965). M. V. Klein and T. Timusk, Phys. Rev. 141, 664 (1966).

3.5. Force-Constants and Distortions Near

Defects. W. Ludwig (Jiilich)

Straightforward quantum mechanical calculations of force constants and distortions near

defects are difficult and have scarcely been done.

R. F. Wood and R. L. Gilbert [1] 1 presented cal- culations for the [/-center in alkali halides, which

look rather promising. For alkali halides this seems to be the first attempt at calculating force- constants for defects. The main problem is to find reasonable eigenfunctions and then, using the

Hamiltonian or a suitably chosen model Hamiltonian, to calculate the energy for different positions

of the atoms. This enables one to find the equilibrium positions and the force constants. There are various steps which must involve approximations

if the calculations are to be practicable and these

approximations limit the accuracy of the results. However, this procedure seems a good starting

point for further calculations.

Often Born-Mayer [exp( — ar)] and Verwey

[(A+B/r) exp ( — ar)] potentials have been used in

alkali halides and also to other types of crystal. In a certain sense the Born-Mayer potential can be

derived from first principles (statistical models),

whereas there seems to be no justification for the

Verwey potential. It looks rather sophisticated,

and the agreement of some calculated data with

experiment cannot be looked upon as a justification.

Instead of this, there is a certain hint from statis- tical models that an exp ( — Br)/r potential might

be more realistic for small distances. It behaves

similarly to the Verwey potential for r—» 0.

It seems that a statistical calculation of inter- actions using Thomas-Fermi-Dirac-type methods

provides reasonable results, at least for closedshell interactions (Gombas, Abrahamson et al.). The difficulty here is that most investigations can

be done only numerically, at least if exchange and

correlation-effects are taken into account. So one

is forced to find analytic expressions for the potential, which are not always unique.

This method has been used also for the calculation of the mutual interaction of three atoms, result- ing in three-body-forces [2]. It should be possible

to continue with more general models. Near defects

1 Figures in brackets indicate the literature references (in this pafie.

the contribution of many-body forces (e.g., polarization effects lead to many-body forces) should be even more essential than in ideal lattices. These

many-body forces are neglected in most calculations

of force constants (perhaps apart from the electronic

volume forces) and it might turn out that the neglect

affects the results for force-constants near defects

appreciably.

In metals it should be possible to calculate the

force-constants near defects, if the electronic density

distribution is known. Then one might be able to calculate the potential using some self-consistent

method. So the task is to find some reasonable

starting point for the electronic density. For ideal

lattices the procedures of Toya [3] and Vosko [4] are promising; it should be possible to extend these

methods to lattices with defects. The situation seems to be worst for covalent

crystals. Even in ideal lattices there is little infor- mation on the true potentials between such atoms. Another possibility of studying the interatomic

forces near defects consists in the use of general

force constants. When making a model with general constants, symmetry and invariance relations

should be used to limit the number of independent

constants. The invariance relations (rotational

invariance) are not satisfied in a number of models.

Having made such a model, one has to compare with

experimental results to fit the constants. So

information about the constants, especially the

harmonic constants, can be found. But in general

this is a difficult procedure, because, for example,

the range of the forces is not known. It is also

difficult to use the constants and to construct a

potential, though this is facilitated by using also anharmonic data, for example those which can be

obtained by "morphic" effects on the localized

modes or perhaps by life-time measurements of

modes.

In any case, it seems to me very important to improve our knowledge of the interatomic forces, not only between atoms near defects, but even for atoms in ideal lattices. We know a lot of general

relations and statements, but in comparing with

experiments the knowledge of forces enters. By

fitting we can explain every experiment, but if experiment and theory do not agree, the origin of

disagreement can often not be found, because we do

not know the forces.References

[1] R. F. Wood, R. L. Gilbert, this conference.

[2] A. A. Abrahamson, R. D. Hatcher, G. H. Vineyard. Phys. Rev. 121, 159 (1961).

|3] T. Toya. Lattice Dynamics, J. Phys. Cheni. Solids Suppl. I, 91 (1965) and papers cited there.

[4] S. H. Vosko, R. Taylor. G. H. Keech, Canad. J. of Physics

43, 1187 (1965) and further papers.

3.6. Force Constants and Distortions Nearil

Defects. A. A. Maradudin (Irvine)

I should like to outline here a method for com-l

puting various properties associated with impurity

200
atoms in non-metallic crystals. These properties,

for example, can be the static distortion of the crystal about the impurity atom, the force constants

of the impurity atom-host crystal interaction, and

the effective charges on the impurity and on its neighbors in the crystal.

The basic assumption underlying this method is the validity of the adiabatic approximation in which

the potential energy for nuclear motion is the sum

of the ground state energy of the electronic system

with the nuclei displaced arbitrarily from their equilibrium positions and the potential energy of

interaction of the ion cores. By "ion core" we

mean the atomic nucleus together with the core

electrons, i.e., all of the atomic electrons not in- cluding the valence electrons. If there is no overlap between the charge distributions of the core

electrons on neighboring atoms, the calculation of

the potential energy of interaction of the ion cores can be carried out by Ewald's method. If there is appreciable overlap of the ion cores, the interaction

energy arising from core-core exchange must be

included in this _part of the potential energy as well,

This energy can be computed in various approximations [l],2 and we will regard the problem of

computing the core-core interaction energy as solved. It is the electronic contribution to the

potential energy for nuclear motion on which we

focus our attention. The starting point for our discussion is the ex- pression for the ground state energy of a system

of electons moving in a potential set up by ion cores displaced arbitrarily from the lattice sites of a perfectly periodic lattice. One of the ion cores is that of the impurity atom, which can be taken to be

at the origin of our coordinate system. The ground

state energy in the Hartree-Fock approximation

is given by ^ Tin I dxi

i- J -fA5 + »(l)

Zm «MD

otherwise. The electron number density p(l) is given by

p(D=2 M^(D<MD. (2)

Finally, we note that in writing the exchange energy

term in eq (1), we have approximated it by the free electron expression as derived by Kohn and Sham

[2].The stationarity of the ground state energy E

with respect to variations of the {ipk} yields the

following set of equations for the one-electron wave

functions *

2 2m vf + ^D+f ^e^

3 \

1/3 iMl) = E*»Ml), (3)

with the aid of which we rewrite eq (1) as

.,P(1)P(2) E = ^ n kEk ~

j

dx i jdx . 2r, 2

We now expand the electron-ion interaction in powers of the displacements of the atoms from the

lattice sites of a perfectly periodic lattice:

v(r) = "2UiK (r-x(lK)-u(lK)) = ^UtK(r-x{lK))

V ua (lK) UiK(r— x(/k))

4—1 3„

Ik<x dXa

+ dx-lfA\)p{2) —lTy>

In this equation, v(r) is the electron-ion interaction.

It is the sum of the coulomb interactions between

a valence electron and the atomic nuclei, the inter- action of a valence electron with the charge dis- tribution of the core electrons, and the exchange

and correlation interactions between the valence

and core electrons. Because the atoms are dis- placed from their equilibrium positions, the index

k merely labels the one-electron states, and is not

to be interpreted as a wave vector and band index.

«a equals unity if k is an occupied state and is zero

+ -^^u,,(Ik)^(/k)

" UlK {r-KilK))+ . . .

Ik a(i oXauXft = v w(r) + v

ii)(r) + v^(r) + . ... (5)

In eq (5) UiK (r — x(/k)) is the potential energy of interaction between a valence electron at r and an

ion at the lattice site x(/k), where / labels the unit

cells of the crystal and k labels the atoms within

a unit cell. We also expand the wave function i///,(r), the single

particle energy Ei<, and the density p(r) in powers

of the atomic displacements:

2 Figures in brackets indicate t lit- literature refe

<Mr) = <H

0, + ^

1, + </>A2, + - Ek = E^ + E\» + ES:) +

p(r) = p

(,,, + p

(1, + p( -' + .

(6a)

(6b)

(6c)

201
Substituting eqs (5) and (6) into eqs (3) and (4), we

solve for the ground state energy of the electronic

system to second order in the atomic displacements.

The result is£ =£«>) +£(D+£<*)+. . . (7a)

where

["<⁢(1) (l)p(0) (l) (7b)

£(2) =

j

dXl vW(l)P«»(l)

+

nk — m / (o)i 7.(1)1 , (0) n

x (i/^I^W)- ( 7c)

In the expression (7c) for E(2 \ the matrix element

(i// (fc0, |// (1)

|i// (, 0)

) is the solution of the integral equation

+ ^[A mnkl ~ B mnkl

kl

nk — ni

£(0) _ £(o)

x (8)

When the ground state energy of the electronic

system given by eq (7) is added to the potential

energy of interaction of the ion cores, the effective

potential energy for nuclear motion takes the form

cD = 0> 0 +X (I) « (/k)u « (/k) + |2 %

^^(Ik; Vk')

Ikcc Ikoi I'k'P

X Ua(lK)Up(l'K')+ . . (11)

to second order in the nuclear displacements. To

this order of approximation the static distortion of

the crystal about the impurity site is described by

the displacements

da (lK) = - £ Qrfilic; ZV)«MZV), (12)

while the atomic force constants are just the coeffi- cients {^q^/k; I'k')}. The /x-cartesian component

of the first order dipole moment can be written as

(13a)

with

where

hidx ^*(1)W)^2)^°>*(2)

B,

2 /SV'3

3 \8

Xe2 jdx 1

i//'«»*(l)i//'n

0 >(l)^>(l)^°'*(l)

(p(0)(l))2/3

(9a)

(9b)

The first order change in the electronic charge

density p

(1)

, which is required for the calculation

of the effective charge on the impurity atom, is given by

kl

fi * a l x ^(o)*(l)i|/(")(l). (10)

M„. a(/K) = SM«Z,fc +[ dxin^Hl), (13b)

and Zik is the charge on the ion core (Ik). The evaluation of the first and second order contri- butions to the ground state energy of the electronic

system Eiv and E(2 \ clearly presents formidable

computational difficulties. However, Green's function methods for computing p

(0) (l) have been de- scribed at this Conference by Professor Seeger, and

they should be useful for evaluating E(l) and the

first term in the expression for E(i)

. The second

term on the right hand side of eq (7c) can be evaluated without much difficulty for perfectly periodic

crystals [3]. It is hoped that this fact provides a

starting point for the evaluation of this term for a perturbed crystal.

The application of the method outlined here to the study of some simple impurity atom-host crystal

systems is now being investigated.

References

[1] S. H. Vosko, Physics Letters 13, 97 (1964). [3] L. J. Sham, Proc. Roy. Soc. (London) 283, 33 (1965).

[2] W. Kohn and L. J. Sham. Phys. Rev. 140, A1133 (1965).

202
THE NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards1 provides measurement and technical information services

essential to the efficiency and effectiveness of the work of the Nation's scientists and engineers. The

Bureau serves also as a focal point in the Federal Government for assuring maximum application of

the physical and engineering sciences to the advancement of technology in industry and commerce. To

accomplish this mission, the Bureau is organized into three institutes covering broad program areas of research and services: THE INSTITUTE FOR BASIC STANDARDS . . . provides the central basis within the United

States for a complete and consistent system of physical measurements, coordinates that system with the measurement systems of other nations, and furnishes essential services leading to accurate and uniform

physical measurements throughout the Nation's scientific community, industry, and commerce. This

Institute comprises a series of divisions, each serving a classical subject matter area: —Applied Mathematics—Electricity—Metrology—Mechanics—Heat—Atomic Physics—Physical

Chemistry—Radiation Physics—Laboratory Astrophysics2—Radio Standards Laboratory,2 which

includes Radio Standards Physics and Radio Standards Engineering—Office of Standard Refer- ence Data. THE INSTITUTE FOR MATERIALS RESEARCH . . . conducts materials research and provides

associated materials services including mainly reference materials and data on the properties of materials. Beyond its direct interest to the Nation's scientists and engineers, this Institute yields services which are essential to the advancement of technology in industry and commerce. This Institute is or- ganized primarily by technical fields: —Analytical Chemistry—Metallurgy—Reactor Radiations—Polymers—Inorganic Materials—Cryogenics2—Office of Standard Reference Materials.

THE INSTITUTE FOR APPLIED TECHNOLOGY . . . provides technical services to promote the use of available technology and to facilitate technological innovation in industry and government. The

principal elements of this Institute are: —Building Research—Electronic Instrumentation—Technical Analysis—Center for Computer Sci- ences and Technology—Textile and Apparel Technology Center—Office of Weights and Measures —Office of Engineering Standards Services—Office of Invention and Innovation—Office of Vehicle

Systems Research—Clearinghouse for Federal Scientific and Technical Information3—Materials

Evaluation Laboratory—NBS/GSA Testing Laboratory.

1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington,

D.C., 20234.

2 Located at Boulder. Colora o 80302.

3 Located at 5285 Port Royal Road, Springfield, Virginia 22151.

U S GOVERNMENT PRINTING OFFICE : 1967 OL —256-724


I