960-2Special

Publication

960-2

The Fundamentals

of Neutron Powder

Diffraction

John R.D. Copley

NIST recommended

practice guide
i

The Fundamentals

of Neutron Powder

Diffraction

John R.D. Copley

Materials Science and 

Engineering Laboratory

November 2001

U.S. Department of Commerce

Donald L. Evans, Secretary

Technology Administration

Phillip J. Bond, Under Secretary for Technology

National Institute of Standards and Technology

Karen H. Brown, Acting Director

UNITED STATES OF AMERICA

DEPARTMENT OF COMMERCE

NIST Recommended Practice Guide

Special Publication 960-2
ii

Certain commercial entities, equipment, or materials may be identified in this

document in order to describe an experimental procedure or concept adequately.

Such identification is not intended to imply recommendation or endorsement

by the National Institute of Standards and Technology, nor is it intended to

imply that the entities, materials, or equipment are necessarily the best available for the purpose.

______________________________________

National Institute of Standards and Technology

Special Publication 960-2

Natl. Inst. Stand. Technol.

Spec. Publ. 960-2

40 pages (November 2001)

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iii

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

I.1 Structure and Properties . . . . . . . . . . . . . . . . . . . . . . . . .1

I.2 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

I.3 The Meaning of “Structure” . . . . . . . . . . . . . . . . . . . . . . .1

I.4 Crystals and Powders . . . . . . . . . . . . . . . . . . . . . . . . . . .2

I.5 Scattering, Diffraction and Absorption . . . . . . . . . . . . . . . .2

II. METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

II.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

II.2 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

II.3 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

II.4 The BT1 Spectrometer at the NIST Research Reactor . . .8

II.5 Scattering of Neutrons and X-rays . . . . . . . . . . . . . . . . .10

II.6 Absorption of Neutrons and X-rays . . . . . . . . . . . . . . . . .11

II.7 Incoherent Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .14

II.8 Atomic Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

II.9 Magnetic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

II.10 Rietveld Profile Refinement . . . . . . . . . . . . . . . . . . . . . .19

III. EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

III.1 High Temperature Superconductors . . . . . . . . . . . . . . . .21

III.2 Colossal Magnetoresistors . . . . . . . . . . . . . . . . . . . . . . .22

III.3 Residual Stress Measurements . . . . . . . . . . . . . . . . . . .27

III.4 Carbon-60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34

Table of Contents ♦

1

I. INTRODUCTION

I.1. Structure and Properties

Information about the organization of the atoms in materials is essential to any

attempt at a detailed understanding of their physical, chemical and mechanical

properties. Richard Feynman put it this way (Feynman et al., 1963): “If they

will tell us, more or less, what the earth or the stars are like, then we can figure

it out. In order for physical theory to be of any use, we must know where the

atoms are located.” Consider the example of diamond and graphite. Diamond

is one of the hardest materials known whereas graphite is so soft that it is used

in pencils and as a lubricant, yet both materials are pure carbon. The reason

for this dramatic difference is that the atoms are arranged very differently. In

diamond they assemble into a rigid three-dimensional network but in graphite

they are arranged in parallel sheets and a small force is sufficient to cause the

sheets to slip. Examples abound in other fields such as protein chemistry,

where "the understanding of mechanism is based on knowing the structures

and properties of the reagents and the kinetics and thermodynamics of their

reactions, as well as how the kinetics and other properties change when the

structures of the reagents are modified and the reaction conditions are altered."

(Fersht, 1999)

I.2. Diffraction

The most widely used (and the most powerful) method of studying atomic

structure is known as diffraction. Electrons are used to examine structure in

thin sections of material or close to a material surface, whereas x-rays and

neutrons are the probes of choice for looking at the bulk. X-rays are normally

used because laboratory instruments are readily available from various manufacturers. Furthermore high brilliance sources, such as the Advanced Photon

Source at Argonne National Laboratory, may be used when intensity is a problem. Despite the widespread use of x-rays there are situations where thermal

neutrons are the better choice, as well as situations where the combined use of

x-rays and neutrons is beneficial. In sections II.5 and II.6 we shall compare

and contrast x-rays and neutrons.

I.3. The Meaning of "Structure"

The positions of the atoms in a material define its atomic structure. In a gas or

liquid the atoms move throughout the available volume, and the structure is

described using one or more "pair distribution functions" which are probability

distributions of instantaneous distances between pairs of atoms. The atoms in a

solid behave quite differently, for the most part moving back and forth about

Introduction ♦
2

their equilibrium positions, and pair distribution functions can again be used to

describe the structure. The pair distribution approach is most useful when

applied to non-crystalline solids such as glasses, in which the mean (timeaveraged) positions of the atoms lack long range order. In crystalline solids the

mean positions of the atoms have a regularity that persists over large distances

in all directions, and the structure is specified by the mean positions of the

atoms (together with information that describes site occupancies and mean

displacement amplitudes).

I.4. Crystals and Powders

Crystalline solids may be classified according to how the individual crystallites within the material are organized. At one extreme is the single crystal,

which consists of one or more crystallites with virtually the same orientation.

At the other extreme is the polycrystalline sample in which all possible orientations of crystallites are presumed to be equally likely. A third possibility is

the intermediate case of a polycrystalline sample in which the distribution of

crystallite orientations is neither uniform nor strongly peaked. This latter type

of sample is often described as a sample with preferred orientation or "texture," and an important sub-field of applied neutron diffraction is the study of

texture (Prask and Choi, 1993, and references therein). In what follows we

shall be concerned with experiments on "powders," i.e., polycrystalline

samples which are presumed to have a uniform distribution of orientations.

I.5. Scattering, Diffraction and Absorption

When a beam of neutrons strikes a material several things can happen. Some

will be absorbed, others will emerge in a new direction with or without a

change in energy, and the rest will pass through the material unaffected. Those

that emerge in a new direction are described as "scattered" neutrons, and the

investigation of materials by measuring how they scatter neutrons is known as

neutron scattering. Neutron "diffraction," as opposed to "scattering," generally

implies that there is no attempt to determine the neutron's change in energy

when it is scattered. Neutron diffraction (Bacon 1975) is a powerful and popular technique, which is primarily used to determine the structures of crystalline

materials. It is also used to determine magnetic structures and to study pair

distribution functions in non-crystalline solids, liquids and gases.

♦ Introduction
3

II. METHODS

II.1. Definitions

A low energy (non-relativistic) neutron has energy E, wave vector k, velocity

v, and wavelength λ. These quantities are related as follows:

(1)

where h is Planck's constant and m is the mass of the neutron. Approximate

conversions are as follows:

The most commonly used units of neutron wavelength and energy are the

Ångström (Å) and the milli-electron volt (meV) respectively. Note that 

1 Å=0.1 nm and 1 meV ≈ 8.1 cm-1 ≈ 11.6 K ≈ 0.24×1012 Hz ≈ 1.6×10-15 erg.

Hence 1 kJ/mol ≈ 10.4 meV/molecule. Much more exact conversion factors

can be derived using the recommended values of the fundamental physical

constants (see, e.g., http://physics.nist.gov/cuu/Constants/index.html).

In a scattering event (fig. 1a) the energy of the incident neutron is Ei and that

of the scattered neutron is Ef. (The subscripts "i" and "f" mean "initial" and

"final" respectively.) Similarly the incident and scattered neutron wave vectors

are ki and kf. The "energy transfer" = Ei – Ef is the energy transferred to

the sample by the neutron when it is scattered. Similarly the "wave vector

transfer," or "scattering vector," is Q = ki – kf . The cosine rule, applied to

the (Q, ki, kf) scattering triangle (fig. 1b), gives Q2 = ki

2

+ kf

2 – 2ki kf cos (2θ),

where 2θ is the "scattering angle" (the angle between the incident and scattered neutron beam directions). For the special case of "elastic scattering,"

i.e., scattering in which the energy of the neutron does not change, Ei = Ef ,

= 0, ki = kf , and Q = 2kisin θ = 4πsin θ/λi where λi is the incident 

wavelength.

II.2. Basic Theory

In order to understand neutron powder diffraction, and to compare neutron

diffraction with x-ray diffraction (Warren, 1969), we shall discuss the basic

theory of the method. Detailed treatments are found in books by Bacon (1975),

Squires (1978), and Lovesey (1987).

Methods♦

k = = == 2 1

2 2

2

2

2

π

λ λ λ ;; ,

h

mv

E mv h

m

E meV [ ] ( [ ]) ≈ 82

2 λ Å v mm s [ /] [ ] μ λ ≈ 4

Å

, .

hω

hω
4

♦ Methods

Fig. 1. A scattering event is illustrated (a) in real space: (b) shows the

corresponding scattering triangle. The symbols are defined in the text.

Fig. 2. Unit cells (a) without a basis and (b) with a basis. Two-dimensional

examples are shown for simplicity. Real space basis vectors a and b are

shown. With this choice of a and b the fractional coordinates of the atoms

represented by open squares, in (b), are (0.75,0.25) and (0.25,0.75).
5

The atoms in a crystalline material are periodically arranged. Their positions

can be generated by specifying the size and shape of a "unit cell" and the positions of the atoms within the unit cell. The complete structure is obtained by

repeating the unit cell many times in all directions. This is illustrated in fig. 2.

In a few cases (those of some but not all elements) there is only one atom in

the unit cell so that all atomic positions are equivalent and the environment of

every atom is the same (fig. 2a). In the vast majority of cases there are several

atoms (indeed any number from 2 to many thousands) in the unit cell (fig. 2b).

In all cases the crystal structure is described by three real space basis vectors

a, b and c, that define the size and shape of the unit cell and by the (equilibrium) positions of the atoms within the cell, which are generally expressed in

fractional coordinates (fig. 2).

Neutrons are scattered by nuclei, and by atomic magnetic moments due to

unpaired electrons. For now we shall only consider the nuclear scattering.

We shall also assume (for the time being) that all unit cells are identical (even

to the extent that the atoms do not move), and that complicating effects, such

as extinction, absorption, inelastic scattering and multiple scattering, can be

ignored. 

The scattered intensity in a neutron diffraction experiment is proportional to

I0NS(Q) where I0 is the incident beam intensity and N is the total number of

atoms in the crystal. The structure factor S(Q) is given by

(2)

where the sum is over all atoms, ri is the position of atom i, and bi is the

"scattering length" or "scattering amplitude" for atom i, which is a measure

of the strength of the interaction between the incident radiation and the atom.

(Values of bi are readily available, e.g., on the Web at http://www.ncnr.nist.gov/

resources/n-lengths/). In a crystalline material the periodicity of the lattice

implies that S(Q) is zero except at specific values of Q, such that the phase

factors exp (iQ.ri

) constructively interfere. These special values of Q, plotted

in a 3-dimensional space called "reciprocal space," form a regular grid of

points known as the reciprocal lattice. Vectors among these points, called

"reciprocal lattice vectors," may be written in terms of three reciprocal space

basis vectors a*, b* and c*:

(3)

Basic Theory ♦

S

N

b i i i

i

N

( ) exp( . ) , Q Qr = ∑1

2

h * k * . Ghkl = a + b +lc*
6

The "Miller indices" h, k and l are integers which characterize the reflection

that occurs when Q = Ghkl . These reflections are known as Bragg reflections,

and the peaks observed in diffraction experiments (both with crystals and with

powders) are generally called Bragg peaks.

The reciprocal space basis vectors are related to the real space basis vectors a,

b and c: a* = (2π/V) b × c, with cyclic permutations, where V= a.(b × c) is

the volume of the unit cell.

The periodicity of the lattice (with the assumption that N is very large) enables

us to write equation (2) somewhat differently:

(4)

The first sum is over all reciprocal lattice vectors; the delta function implies

that S(Q) is only nonzero when Q is a reciprocal lattice vector. The second

sum is restricted to the n atoms in a single unit cell.

In a single crystal measurement using a mono-directional monochromatic

(single wavelength) incident beam, the incident wave vector ki is well defined.

Since scattering only occurs when Q = Ghkl , and since kf = ki - Q (fig. 1a),

the scattered neutrons leave from the sample in discrete directions.

In the case of a powder the structure factor must be averaged over all possible

crystallite orientations. As long as the dimensions of all crystallites are much

greater than those of the unit cell we obtain a simple function of scalar Q,

which vanishes except when Q is equal to the magnitude of one of the reciprocal lattice vectors Ghkl . Since (for elastic scattering) Q = 4πsin θ / λi (section

II.1), the scattering from a powder sample in an idealized experiment with a

mono-directional monochromatic incident beam only occurs at scattering

angles 2θ such that sin θ = Ghkl λi / 4π , i.e., λi = 2dhkl sin θ (which is the

familiar Bragg equation), where the distance between diffracting planes is

dhkl = 2π / Ghkl. Because all crystallite orientations are present the scattered

neutrons leave from the sample in "Debye-Scherrer" cones.

II.3. Instrumentation

In a conventional neutron powder diffractometer, at a continuous source such

as the NIST Research Reactor (http://www.ncnr.nist.gov/reactor.html), a

beam of monochromatic neutrons is incident upon a sample and the scattered

♦ Methods

( ) ä ( ) b exp(i . ) . nV

2ð S( )

2 n

i

i i

G

3

Q = ∑ Q − G ∑ Q r
7

neutron intensity is measured as a function of the scattering angle 2θ (fig. 3).

The monochromatic incident beam is produced by Bragg reflection from an

appropriate single crystal monochromator. The neutron count-rate is measured

in one or more detectors. Masks and collimators restrict the spatial and angular

widths of the beam. In the absence of complicating effects such as those listed

previously, and ignoring resolution effects and other sources of unwanted

counts such as ambient background and scattering from the sample environment, the neutron count-rate is proportional to S(Q)/(sinθ sin2θ), where S(Q)

is simply S(Q) summed over all directions of Q (Squires 1978, page 43).

In an ideal experiment the incident beam would have a single wavelength and

unique direction, but there would be no intensity at the sample. In practice the

incident beam has a narrow distribution of wavelengths and a narrow distribution of directions. The intensity at the sample is roughly proportional to the

product of the widths of these distributions whereas the overall resolution of

the instrument (assuming its design is optimized) is roughly proportional to

one of the widths. Depending on the experiment the compromise between

intensity and resolution will vary.

A very different neutron diffraction technique, which we shall not discuss

further, is known as "time-of-flight diffraction" (Windsor, 1981). In this case a

pulsed source, or a "neutron chopper" at a continuous source, provides pulses

of neutrons with a broad spectrum of wavelengths. The neutrons strike a

sample and are scattered in various directions, arriving at the detector(s) at

times that depend on their wavelengths. Given a detector's scattering angle

and its distance from the source, each peak in the time-of-flight spectrum is

Instrumentation ♦

Fig. 3. A schematic drawing of a diffractometer, showing the neutron source,

the monochromator crystal, a beam monitor, the sample, and a detector.

Collimators are placed before and after the monochromator crystal and

between the sample and the detector.
8

uniquely associated with one or more Bragg reflections. As before the intensities of the Bragg reflections are used to determine structure factors S(Q).

II.4. The BT1 Spectrometer at the NIST Research

Reactor

In 1992 a third generation multipurpose powder diffractometer was installed

at the BT1 (beam tube #1) beam port of the NIST Research Reactor

(http://www.ncnr.nist.gov/instruments/bt1.html). Selecting from a choice of

three monochromator crystals and two white beam collimators with different

acceptance angles (fig. 4), the response of the instrument may be tailored to

the needs of the experiment. The resolution in wave vector transfer Q is shown

in fig. 5 for the six choices of instrumental setup. The Ge(311) monochromator,

which operates with an incident wavelength of 2.0783(2) Å, yields the highest

neutron intensity and best resolution at low Q, but Q is limited to about 6 Å-1.

On the other hand Si(531), at 1.5905(1) Å, provides the best resolution at high

scattering angles, but six times longer data collection times are required 

♦ Methods

Fig. 4. A schematic drawing of the BT1 diffractometer at the NIST Research

Reactor. The beam monitor and the detector assembly are shown in locations

appropriate for data collection using the Si(531) monochromator.
9

because the flux at the sample position is much lower. For most experiments

the Cu(311) monochromator, at 1.5402(2) Å, offers the most useful compromise between intensity and resolution, with intermediate data collection times.

The BT1 instrument is equipped with an array of 32 detectors located at 5o

intervals within a common shield. It also has a low efficiency detector, placed

in the incident monochromatic beam to monitor the beam intensity (fig. 4). In

a typical experiment the detector assembly is moved to the desired starting

position and counts are accumulated in the 32 detectors until a predetermined

number of counts has been recorded in the beam monitor. Measurements are

repeated and checked for internal statistical agreement. The assembly is then

rotated through a small angle (typically 0.05o), and counts are again accumulated, repeated and checked, using the same number of beam monitor counts.

This step/count sequence is repeated until the total angle of rotation is at least

5o, so that data from each detector overlap with data from its neighbors. The

assembly can be rotated through 13o, and measurements can be made with 2θ

from 0o to 167o. 

The BT1 Spectrometer at the NIST Research Reactor ♦

Fig. 5. The FWHM (full width at half maximum height) of the resolution

function of the BT1 diffractometer, as a function of Q, for the six possible

combinations of monochromator crystal and in-pile collimation. FWHM resolution in Q (Å-1)
10

II.5. Scattering of Neutrons and X-rays

The discussion of section II.2, with the same simplifying assumptions, also

applies to x-ray diffraction. 

In comparing the two techniques the first remark to be made is that the scattered intensity is generally much greater for x-rays than for neutrons because

the source is much more intense. Hence x-ray diffraction is an extremely

powerful technique that is frequently preferable to neutron diffraction.

Nevertheless there are many situations where the neutron technique is superior

and/or complementary in the sense that new information can be obtained by

combining the results of measurements using the two techniques.

An important difference between x-rays and neutrons is in the nature of the

interaction between probe and sample. The neutron-nucleus interaction is of

very short range (of order 10-15 m), very much shorter than interatomic distances (which are of order 10-10 m), so that scattering lengths b are essentially

independent of Q over the range of Q values accessible in a neutron scattering

experiment. On the other hand x-rays interact with electrons and since the

spatial extent of the electron cloud surrounding a nucleus is comparable with

interatomic distances the corresponding quantity, known as the "form factor,"

decreases quite quickly with increasing Q (fig. 6). This quantity, which we

shall write as f(Q), is the Fourier transform of the radially averaged density of

electrons surrounding the nucleus. The decrease in f with increasing Q can

♦ Methods

Fig. 6. The neutron scattering length of carbon, b, which is independent of Q,

and the corresponding x-ray atomic scattering factor, f(Q).
11

cause problems in x-ray experiments when there is significant background

scattering such as Compton (incoherent) scattering, either from the sample or

from the sample environment, since the ratio of signal to background is greatly

reduced at high angles. A more fundamental distinction is that x-rays provide

information about the electron density distribution in a material whereas neutrons give nuclear positions. Thus qualitatively different information can be

extracted from the two probes. It is not always the case that nuclear positions

can be derived from electronic density distributions and of course the converse

never applies. 

Another difference between x-rays and neutrons is that for x-rays the forward

scattering value of f(Q), i.e., f (0), is proportional to the number of electrons,

i.e., to the atomic number Z, whereas for neutrons there is no simple relationship between the equivalent quantity, b, and the composition of the scattering

nucleus. Indeed b varies somewhat erratically with Z, as illustrated in fig. 7.

Thus neutrons can sometimes be used to distinguish among elements that are

close to one another in the periodic table and therefore barely distinguishable

in x-ray experiments; an example occurs in the study of zeolites containing

Al (Z=13, b=3.5×10-15 m) and Si (Z=14, b=4.2×10-15 m). Furthermore it is

frequently easier to detect light atoms in the presence of heavy atoms using

neutrons rather than x-rays; light elements with significant scattering lengths

include the common elements H, C, N and O.

The neutron-nucleus scattering length also depends, sometimes dramatically,

on the isotope whereas f(Q) does not since x-rays are scattered by electrons.

The best known example of the isotope dependence of b involves H and D,

whose scattering lengths are –3.8×10-15 m and +6.7×10-15 m respectively,

whereas the value of f(0) is 2.8×10-15 m (for both isotopes). The isotope

dependence of b has been used to particular advantage in small angle scattering

studies of polymers and biological materials and in studies of partial distribution functions in liquids and dense gases. There have been a few reports of the

use of isotope substitution in powder diffraction studies.

II.6. Absorption of Neutrons and X-rays

With very few exceptions, x-rays are much more strongly absorbed than neutrons. For example the 1.8 Å x-ray and neutron linear absorption coefficients

for aluminum are ≈ 21 mm-1 and ≈ 0.0014 mm-1 respectively. Thus at this

wavelength a 150 mm thick aluminum plate has about the same transmission

for neutrons as that of a 10 μm thick plate for x-rays (≈ 80%). 

Scattering of Neutrons and X-rays ♦
12

♦ Methods

Fig. 7. The real part of the neutron scattering length b for the naturally

occurring elements.

Fig. 8. The absorption cross section for 2200 m/s neutrons for the naturally

occurring elements. Notice that the ordinate is plotted on a log scale.
13

Absorption probabilities are written in terms of microscopic or macroscopic

absorption cross sections, denoted by σabs and Σabs respectively. The probability of neutron absorption by nuclei of a given element, within an infinitesimal

distance x, is simply ρσabsx or Σabsx, where ρ is the macroscopic number density of nuclei of that element. (Note that the macroscopic number density of

nuclei is the product of the crystallographic number density and the powder

packing fraction, which is typically of order 65 %.) The microscopic cross section σabs is the more fundamental quantity since it is a cross section per atom;

the macroscopic cross section Σabs = ρσabs is a cross section per unit volume

so it depends on the material in which the element is found. The macroscopic

absorption cross section of a material is simply the sum of contributions from

each of its constituent elements. Microscopic cross sections are normally

expressed in barns (abbreviation "b"), a unit that is "temporarily accepted for

use with the [International System of Units,] SI" (Taylor 1995); 1 b = 10-24 cm2

= 10-28 m2. The microscopic absorption cross sections of the natural elements

are shown in fig. 8. Once again there are wild fluctuations from one element

to the next: neutron absorption cross sections span roughly six orders of

magnitude. They also vary, sometimes greatly, from one isotope to another.

For example σabs = 940 b/atom for 6Li but only 0.045 b/atom for 7Li.

Neutron diffraction experiments on materials containing elements with a large

absorption cross section (e.g., Li, B, Cd, Gd, Sm, Eu, ...) are generally more

challenging because the scattering is greatly reduced and a significant absorption correction may be necessary. In addition there are safety concerns because

both prompt and delayed γ-radiation are produced; a notable exception is

lithium. If the sample contains a strong absorber extra shielding may have to

be placed to protect personnel, and it may be necessary to let the sample's

activity decay for days, if not weeks or months, before it can be removed from

the site without special handling. All such matters must be discussed with the

appropriate authorities in advance of the proposed experiment. Potential problems can sometimes be avoided by using samples containing elements that

have been depleted of the offending isotope. For example a sample made with

112Cd (σabs = 2.2 b/atom) or 114Cd (σabs = 0.34 b/atom) is clearly better than

a sample made with natural Cd (σabs = 2520 b/atom), but it may be very

expensive. On the other hand many isotopes are separated for medical purposes and some of these isotopes may be relatively inexpensive, so it is prudent

to research prices before dismissing an isotope substitution experiment out of

hand.

A comprehensive list of absorption cross sections may be found on the Web at

http://www.ncnr.nist.gov/resources/n-lengths/.

Absorption of Neutrons and X-rays ♦
14

Absorption cross sections depend on the energy of the neutron whereas the

scattering cross sections σC and σI, to be introduced in the following section,

do not. For most elements and isotopes σabs is directly proportional to the

neutron wavelength but there are important exceptions such as the strong

absorbers Cd and Gd. When single values of σabs are tabulated they are most

commonly (but not always) quoted for the canonical thermal neutron speed of

2200 m/s, which corresponds to a wavelength of approximately 1.8 Å.

The ability of neutrons to penetrate deep within materials can be an important

advantage. In studies of materials with compositional gradients, such as cast

ingots and coatings, neutron diffraction yields a structure that is representative

of the bulk material whereas x-rays provide less useful information, regarding

the crystal structure very close to the surface. The ability of neutrons to penetrate materials also opens up the possibility of measuring residual stresses

within thick samples; we shall briefly return to this subject in section III.3.

A second important consequence of the low absorption of neutrons is that sample environments for neutron experiments are relatively easy to design, since

neutron beams readily penetrate the various structures that surround the sample in devices such as cryostats, furnaces and even high pressure cells; some

examples are shown in figs. 9-11. Information about sample environments

available at the NIST Center for Neutron Research may be found at

http://www.ncnr.nist.gov/equipment/ancequip.html.

II.7. Incoherent Scattering

In our discussion of the basic theory we assumed that all unit cells were identical. To the extent that this is untrue the expressions given in section II.2 are

not entirely correct. We shall use the word "disorder" to describe situations

that cause unit cells to differ from one another, and in this section we shall

discuss two types of nuclear disorder. Various kinds of atomic disorder will

be discussed in section II.8.

Consider a material that contains one or more elements with more than one

stable isotope. Since neutron scattering lengths depend both on Z and on the

isotope, the unit cells in this material are not identical and our expression for

S(Q) must be modified. For example eq. (2) becomes

(5)

♦ Methods

b exp (i . ) [ ] b b , N

1 S( )

N

i

2

i

2

i

2 N

i

i i

















Q = ∑ Q r +∑ −
15

where angle brackets denote averages over isotopic distributions. The first

term in this equation is the Bragg scattering, which is "coherent" and of course

strongly dependent on Q. The second term, which is independent of Q, is

"incoherent" scattering. The strength of the incoherent scattering due to any

given type of atom is proportional to the variance of its scattering length.

Since the neutron has a spin of 1⁄2, the interaction of a neutron with a nucleus

with nonzero spin I depends on the spin state of the compound nucleus, which 

is either I+1⁄2 or I-1⁄2. To the extent that the distribution of nuclear spin directions in a material is random, this produces an additional type of incoherent

scattering, which is again independent of Q. Equation (5) still applies, but the

angle bracket averages now include nuclear spin states as well as isotopes.

The quantities σC = 4π +b,2 and σI = 4π [+b2, – +b,2] are called coherent and

incoherent scattering cross sections respectively. In analogy with our earlier

Incoherent Scattering ♦

Fig. 9. A popular low temperature setup, enabling sample temperatures down

to of order 10 K. The sample is bolted to the cold stage of a closed cycle

helium refrigerator. There are two heat shields and an outer vacuum vessel,

all made of aluminum.
16

discussion of absorption the probability of incoherent scattering by atoms of a

given element whose incoherent scattering cross section is σI, within an infinitesimal distance x, is simply ρσIx. On the other hand no such simple expression applies to coherent scattering. The coherent scattering probability depends,

inter alia, on the structure of the material and on the incident neutron wavelength. Beyond the so-called Bragg cutoff wavelength λc, which is such that

the Bragg equation λi = 2dhkl sin θ cannot be satisfied for λi > λc, there is no

coherent scattering whatsoever (neglecting atomic disorder). Furthermore the 

coherent scattering probability for a single crystal clearly depends on its orientation with respect to the incident beam; there is no coherent scattering unless

Q is equal to a reciprocal lattice vector.

Most elements are predominantly coherent scatterers. The most important

exceptions are hydrogen and vanadium. The spin-incoherent scattering cross

section of hydrogen, which is ≈ 80 b/atom in a hypothetically bound material,

is a nuisance in diffraction studies since it contributes a large constant

background to the measured intensity. For this reason deuterated samples are

♦ Methods

Fig. 10. A schematic drawing of a high pressure cell, made of a high

strength aluminum alloy and capable of operating pressures up to ≈ 620 MPa

(≈ 6100 atmospheres).
17

strongly recommended. Since vanadium (at natural abundance) scatters almost

completely incoherently, it is frequently used to make sample containers for

powder diffraction experiments; the weakness of its Bragg peaks, due to its

very small coherent scattering cross section, is a major advantage which

outweighs any disadvantage associated with the incoherent scattering

background. On the other hand it is difficult to determine the positions of V

atoms in a sample because of its very small coherent scattering cross section.

II.8. Atomic Disorder

At least one type of atomic disorder is never absent, no matter how carefully

the sample is prepared. This is the thermal disorder that is associated with the

motion of atoms about their mean positions. The effect on the scattering is to

shift a certain amount of intensity from the Bragg peaks to scattering that is

continuous in Q (whereas Bragg scattering occurs at discrete values of Q). The

continuous scattering is sometimes called "thermal diffuse scattering." If the

Incoherent scattering ♦

Fig 11. A schematic drawing of an 1800 °C furnace, designed at the Institut

Laue-Langevin, Grenoble, France. At beam height the heater consists of two

concentric 0.02 mm thick Nb foils, the eight heat shields are also

0.02 mm Nb foils, and the outer vacuum jacket is made of ≈1.5 mm thick

aluminum.
18

thermal motion is isotropic Bragg intensities are modified by replacing bi with

bi exp(–1⁄2Q2<ui

2>), where the exponential expression is the so-called "temperature factor," or "Debye-Waller factor," for atom i; <ui

2> denotes the atom's

mean square displacement. The temperature factor may also be written as

exp(–Bi[sinθ / λ]2) , where Bi is the "displacement parameter" for atom i.

In many cases the thermal motion is anisotropic, in which case temperature

factors are considerably more complicated (Willis and Pryor, 1975).

Displacement parameters are routinely obtained in the analysis of powder

diffraction experiments.

Other kinds of atomic disorder modify the scattered intensity in a diffraction

experiment (Guinier, 1963). They include local effects such as compositional

disorder, vacancies and interstitials, as well as more extended phenomena such

as dislocations, grain boundaries, clusters and pores. The larger scale structures are generally better studied using the small angle scattering technique

whereas smaller scale objects show up better in diffraction experiments.

Defects generally modify the Bragg scattering and contribute to the diffuse

scattering between the Bragg peaks. The Bragg peaks may be modified in

intensity, position and/or width, and the different ways that these modifications

depend on the reflection can be used to distinguish among the various phenomena. For example line broadening can be due to the small size of the

crystallites in the sample, in which case it is independent of Q, or it can be due

to the effects of strains, in which case it increases with increasing Q.

II.9. Magnetic Scattering

We have seen that neutrons interact with nuclei. Because of their magnetic

moment they also interact with magnetic moments on atoms with unpaired

electrons (Furrer 1995). The magnetic interaction of neutrons is more complicated than the nuclear interaction, and we shall only consider the simplest of

situations. For a detailed discussion see Izyumov and Ozerov (1970).

A crystalline material with an ordered magnetic structure (e.g., a ferromagnetic

or anti-ferromagnetic material), placed in a neutron beam, produces magnetic

Bragg peaks in addition to the nuclear Bragg peaks previously discussed. If

the beam is unpolarized the nuclear and magnetic structure factors, S(Q) and

SM(Q), add in the following way:

(6)

where α is the angle between the magnetization vector M and the scattering

vector Q. The expression for SM(Q) is similar to equation (2), except that bi is

♦ Methods

S SS tot M ( ) ( ) ( )sin , QQ Q = + 2 α
19

replaced with the quantity pi(Q) = Aifi

M (Q) where Ai is proportional to the

magnetic moment of atom i and fi

M (Q) is its magnetic form factor. For a

powder the coefficient sin2 α is replaced by its average value, which is 2/3. 

In some cases (depending on the crystal symmetry), powder diffraction experiments with unpolarized neutrons can be used to determine magnetic structures

as well as conventional crystal (chemical) structures. Magnetic moments can

also be determined.

II.10. Rietveld Profile Refinement

The standard method of analyzing the results of a neutron powder diffraction

experiment, known as "Rietveld profile refinement" and named for Hugo

Rietveld (who developed the method), is to fit the parameters of a model to

the measured "profile," which is the intensity measured as a function of

scattering angle 2θ. A typical measurement and the fitted profile are shown in

fig. 12. The Rietveld technique is discussed in Young (1993). See

http://www.ncnr.nist.gov/programs/crystallography/ for information about its

implementation at the NIST Center for Neutron Research.

Magnetic Scattering ♦

Fig. 12. The high angle part of the neutron powder diffraction pattern of cubic

LaBa2Fe3O9-δ, measured at 295 K. The incident wavelength was 1.54 Å,

obtained using a Cu(311) monochromator. Notice that the widths of the peaks,

which are very largely due to the resolution of the instrument, depend on the

scattering angle 2θ. Points represent the data and the line through the data

represents the result of the profile refinement. The difference between

experiment and calculation is shown at the bottom, and the small vertical lines

indicate the positions of the calculated peaks.
20

♦ Methods

The model parameters in a Rietveld refinement can be divided into several

categories, according to which aspect of the profile they describe. First of all,

and most importantly, there are the parameters that determine the positions and

integrated intensities of the Bragg peaks. Unit cell parameters determine peak

positions whereas atom positions within the unit cell, site occupancies and

displacement parameters collectively determine integrated intensities. Next

there are parameters that describe the so-called "background," which is the

relatively smooth scattering that lies between and below the Bragg peaks.

This can include diffuse elastic coherent scattering and inelastic and/or

incoherent scattering from the sample and its environment, plus the "true"

background, which includes contributions due to air scattering, electronic noise,

and other unwanted sources. The Bragg peaks, which correspond to elastic

scattering that occurs at well-defined scattering angles, can usually be

separated from the background because the background scattering varies rather

slowly with scattering angle whereas the Bragg scattering is highly structured.

A third set of parameters describes the shapes of the peaks. Line broadening

is in part due to the resolution of the instrument itself, which is generally a

well-parameterized function of 2θ for a given instrumental configuration.

There may also be contributions to the broadening due to size and strain

effects. The results of neutron diffraction experiments at a reactor are generally

easier to refine than results from other sources because peak shapes tend to be

simpler and fewer corrections need be applied.
21

III. EXAMPLES

We now discuss some examples of applications of the neutron diffraction

technique to materials of industrial interest. We conclude with a brief

discussion of neutron experiments on C60. 

III.1. High Temperature Superconductors

It goes without saying that there is great interest in the development of high

temperature superconductors for commercial applications. Neutron diffraction

has played a very important role in elucidating the structures of these materials,

in large part because of the crucial part played by the oxygen vacancies and

the greatly improved visibility of oxygen in the presence of heavy elements

when compared with x-ray diffraction. Furthermore the ability of neutrons to

penetrate bulk samples, with very little absorption, simplifies the data analysis

considerably. Structural information from neutron scattering experiments has

helped us improve our understanding of the crucial role of the oxygen defects

in high temperature superconductors.

The mercury superconductors HgBa2Can–1CunO2n+2+δ have generated considerable interest in the superconductivity community. The oxygen dependence of

the structure of the single layer material HgBa2CuO4+δ, with 0.04≤δ≤0.23, and

its effect on the superconducting phase transition, have been studied using the

neutron powder diffraction method (Huang et al., 1995). The measurements

were performed using the BT1 diffractometer at NIST (section II.4). The

Cu(311) monochromator was used with 15' in-pile collimation (cf. figs. 4

and 5). The powders, in amounts of the order of 4 g, were contained in thinwalled (0.15 mm) vanadium cans which were mounted in a closed cycle

helium refrigerator (cf. fig. 9). Counting times were roughly 8 h at each temperature. The crystal structure (fig. 13) remains tetragonal over the whole

range of δ, and at all temperatures from 10 K to 300 K. The excess oxygen is

found to occupy so-called O(3) sites in the Hg layers. The displacement

parameters for the Hg and Ba atoms are highly anisotropic, such that the

corresponding real space displacement ellipsoids (which characterize the

probability of finding an atom at a particular location relative to its equilibrium

position) are respectively oblate (squashed) and prolate (egg-shaped). This

finding suggests that the positions of the Hg and Ba atoms are strongly influenced by the presence or absence of an oxygen atom at the O(3) site. With

regard to the Ba atoms, the conjecture is confirmed by results for the distances

of the Ba atoms from various oxygen sites in the structure, and by the success

of a model that postulates two partially occupied Ba sites at slightly different

distances from the Cu planes, such that the sum of the occupancies is unity.

Examples ♦
22

The ratio of the Ba displacement parameters B33(Ba) and B11(Ba), respectively

normal and parallel to the tetragonal planes, is shown in fig. 14(a) as a function of δ. This curve is remarkably similar to a plot of the superconducting

transition temperature Tc as a function of δ (fig. 14b), suggesting a correlation

between Tc and the displacements of the Ba atoms.

III.2. Colossal Magnetoresistors

The resistivity of a magnetic material changes when it is placed in a magnetic

field. The effect is generally small, and best observed at low temperature. In

some materials the effect is anisotropic, i.e., the resistance changes when the

current flowing through the material is changed from being parallel to the

internal magnetization to being perpendicular to it. Typically the change in

resistance on changing the current direction is of order 1 % or 2 % at most. A

good example is permalloy (Ni0.8Fe0.2), where the anisotropic magnetoresistance effect is used in device applications such as read heads for computer

hard disk drives. 

♦ Examples

Fig. 13. The tetragonal crystal structure of HgBa2CuO4+δ. Doping occurs at

the partially occupied O(3) sites.
23

About 10 years ago a much larger effect, dubbed "giant magnetoresistance"

(GMR), was discovered in an iron-chromium magnetic superlattice. GMR

materials have alternating layers of ferromagnetic and non-magnetic metals

deposited on an insulating substrate, and the resistance parallel to the layers is

largest (its value is R0) when the magnetic moments in the ferromagnetic layers alternate from one layer to the next (in the absence of an applied field). On

the other hand the resistance is smallest (R∞) when the layers are parallel to

one another (when the field saturates the magnetization). In the past few years

GMR research has advanced rapidly, with new materials being developed; the

current record for the resistance ratio R0/R∞ is ≈3.2. Much of the recent GMR

Colossal Magnetoresistors ♦

Fig. 14. (a) The ratio of the out-of-plane (B33) and in-plane (B11)

displacement parameters for the Ba atoms in HgBa2CuO4+δ. (b) The

dependence of the superconducting transition temperature Tc on δ for

HgBa2CuO4+δ. Also shown is the dependence of Tc on the hole 

concentration PTEP (the number of carriers per Cu atom), from measurements by Xiong et al. (1994).

PTEP

δ
24

research has been driven by a desire to develop a new generation of read heads

for computer hard disk drives. A continuing challenge is to manufacture

devices that can read increasingly high bit densities. Other types of devices

that are candidates for development using GMR materials include magnetic

memory chips, magnetic transistors, motion sensors, and sensor arrays for land

mine detection.

A somewhat different phenomenon, called "colossal magnetoresistance"

(CMR), was first reported in 1994. CMR materials have truly enormous

resistance ratios, some being well in excess of 1000. Indeed the change in

resistance that occurs when a CMR material is placed in a sufficiently high

magnetic field is greater than the change produced when a high temperature

superconductor is cooled through its superconducting transition temperature!

The resistance of a CMR material drops enormously when the spins order ferromagnetically, either when the temperature is lowered or when a magnetic

field is applied. This large variation in carrier mobility originates from a

metal-insulator transition that is closely associated with the magnetic ordering.

The most well-known CMR materials are the calcium-doped lanthanum manganites La1–xCaxMnO3, and the compound with x=0.33, which we may write

as La3+0.67Ca2+0.33Mn3+0.67Mn4+0.33O3, shows the most dramatic magnetoresistance effect. The undoped material, LaMnO3, is antiferromagnetic but it can

be converted to a ferromagnet in an oxidizing atmosphere. This is also true of

materials with small x, but at larger values of x the materials are ferromagnetic,

no matter how they are prepared.

Much of our understanding about the relationship between the magnetoresistive properties of these materials and their structures has come from neutron

powder diffraction experiments. The experiments that we shall briefly describe

were part of a comprehensive study of the crystal and magnetic structures of

La1–xCaxMnO3 compounds with 0<x≤0.33. In order to resolve a discrepancy

regarding the magnetic ordering in low concentration Ca-doped manganites,

Huang et al. (1997) looked at two such materials with x=0.06. The first sample, which we shall call the oxidized sample, was prepared in air at 1350 °C by

mixing appropriate quantities of La2O3, MnCO3 and CaCO3. The other sample, described as reduced, was prepared from the first sample by annealing at

900 °C in a reducing atmosphere. The reduced sample, whose chemical formula

may be written as La0.94Ca0.06Mn3+0.94Mn4+0.06O3, had all atomic sites

fully occupied whereas the oxidized sample had cation vacancies; its approximate chemical formula was (La0.94Ca0.06Mn3+0.69Mn4+0.31)0.96O3. The

reduced sample was antiferromagnetic at low temperatures, in common with

♦ Examples
25

Colossal Magnetoresistors ♦

Fig. 15. The orthorhombic crystal structure of La1-xCaxMnO3 is shown in (a)

and the tilting of the MnO6 octahedra is shown in (b). The orientations of the

Mn magnetic moments are shown in (c) and (d) for the ferromagnetic and

antiferromagnetic structures respectively.
26

♦ Examples

Fig. 16. Rietveld refinements using neutron diffraction data for the

stoichiometric reduced compound La0.94Ca0.06MnO3 at 12K. The points are

the data and the line through the points is the result from the refinement. The

Bragg peak positions are shown as short vertical lines. At the bottom is shown

the difference between the data and the calculated profile. The inset shows

the low angle data together with the calculated profile for the nuclear

contribution to the diffraction pattern; the magnetic contribution shows up in

the difference plot at the bottom of the inset.

Fig. 17. Rietveld refinements for the oxidized compound

(La0.94Ca0.06)0.97Mn0.97O3 at 12 K. See the caption to figure 16.
27

the undoped compound LaMnO3, but the oxidized sample was ferromagnetic,

since a far greater proportion of the Mn ions were in the +4 valence state. The

orthorhombic crystal structure is illustrated in fig. 15, together with the magnetic ordering in the ferromagnetic and antiferromagnetic states. Figs. 16 and

17 show refinements of the structures of the reduced and oxidized compounds.

In each of these figures the upper part of the inset shows the low angle data,

represented by symbols, and the calculated profile for the crystal structure

alone. The lower part of each inset shows the difference, which represents the

magnetic scattering. We see from these figures that the reduced compound has

peaks that are purely magnetic whereas the oxidized sample does not. The

oxidized compound is ferromagnetically ordered as in fig. 15(c), whereas the

reduced sample is antiferromagnetically ordered as in fig. 15(d).

III.3. Residual Stress Measurements

One of the best examples of an applied research area that benefits from neutron diffraction is the study of residual stresses in materials (Prask and Brand,

1997). The neutron method is non-destructive and capable of relatively deep

penetration within many materials. The non-destructive nature of the technique is especially important because the act of cutting into a material in order

to perform a destructive measurement can actually modify the stresses that the

measurement was intended to determine.

Colossal Magnetoresistors ♦

Fig. 18. A schematic representation of the experimental setup for a residual

stress measurement. The incident and scattered beams are well collimated

and spatially restricted using neutron-absorbing masks.
28

♦ Examples

Fig. 19. The actual experimental setup for the determination of residual

stresses in a slice of railroad rail, using the BT8 spectrometer at the NIST

Research Reactor. The incident beam tube is at the top and the collimation 

system for the diffracted neutrons is at the lower right of the picture.

Fig. 20. A contour map showing the residual stress σxx in the rail head as a

function of position. Stresses are in MPa. Positive values are tensile; negative

values are compressive.
29

A typical experimental setup is shown in fig. 18. The incident and diffracted

beams are not only well collimated (restricted in direction) but also limited in

lateral extent using neutron-absorbing masks. Thus the only once-diffracted

neutrons that can enter the detector are those that were scattered within a

"gauge volume" defined by the intersection of the incident and diffracted

beams. The optimum setup occurs when the scattering angle is 90o: for fixed

widths of the incident and scattered beams the irradiated volume is proportional

to 1/sin(2θ), which is minimized when 2θ = 90o. By translating the sample

within the beam, different gauge volumes can be examined.

Residual stresses are not directly measured in these experiments (nor using

any other experimental technique). Writing the Bragg equation as λ = 2dsin θ,

the mean lattice spacing within the gauge volume, d, is determined from the

angular position of the observed Bragg peak. The local strain is (d-d0)/d0 ,

where d0 is the unstressed lattice spacing. Stresses are computed using the

diffraction elastic constants for the reflection.

Many residual stress studies have been undertaken over the past 15 years, and

dedicated facilities are now found at neutron scattering centers in North

America, Europe and Asia. Systems that have been studied include weldments,

ceramic coatings, composites, reference specimens, turbine blades, and various

military components. An example that illustrates the technique is a recent

study of residual stresses in a thin (6.35 mm thick) section of railroad rail

(Brand et al., 1996). The experimental setup is shown in fig. 19. Measurements

were made using a ≈2.4 mm ×2.4 mm ×2.4 mm gauge volume, at a mesh of

points that were 5 mm apart in orthogonal x and y directions normal to the

length of the uncut rail. Measurements of 5 different lattice spacings, in five

different orientations at each of the mesh points, were used to extract four

components of the residual stress tensor. A typical result is shown in fig. 20.

The results of these types of experiments are typically compared with the

results of model calculations, enabling engineers to improve their models.

They are also compared with the results of other types of experimental

investigations, some of which may have the advantage that they are relatively

inexpensive though they may not be as accurate. The ultimate objective is to

be able to predict performance with reasonable accuracy, thereby improving

economy and durability. 

Residual Stress Measurements ♦
30

III.4. Carbon-60

Buckminsterfullerene, C60, was discovered in 1985, and a method of making

macroscopic samples was reported about five years later. Since that time there

has been considerable research activity in many different fields. One of the

more fundamental problems has been to understand the structure of solid C60 ,

and some of the most useful information has come from diffraction experiments using neutrons (Copley et al., 1992). One of the reasons that neutrons

have been so useful in structure studies of C60 has to do with the very high

symmetry of the molecule. The scattering due to an icosahedral C60 molecule

appears at relatively high Q, which is where the atomic scattering factor for

x-rays, f(Q), is small so that signal-to-background becomes problematic in

x-ray experiments.

Solid C60 is crystalline, with icosahedral molecules located at face-centered

cubic lattice sites. The nearest neighbor distance between molecular centers is

approximately 10 Å. The room temperature diffraction pattern, shown in

♦ Examples

Fig. 21. The room temperature powder neutron diffraction pattern for C60.

Points represent the measurements and the solid line represents a calculation

assuming that molecules are located on a face-centered cubic lattice, 

randomly oriented. A difference plot is included near the bottom of the figure.
31

fig. 21, only has peaks at positions where peaks are predicted on the assumption that all molecules are structurally equivalent. On the other hand the low

temperature diffraction pattern, shown in fig. 22, has additional peaks that

imply that the molecules have become structurally inequivalent. A reasonable

first approximation to the crystal structure of C60 is that below the first order

orientational order-disorder transition temperature, T1≈ 260 K, each molecule

adopts one of four possible orientations such that the unit cell is a simple cube

containing four molecules (fig. 23), whereas above T1 the molecules are

randomly oriented.

The reader will have noticed that the relatively smooth "background" under

the room temperature diffraction pattern (fig. 21) is intense, and curiously

structured. It is in fact diffuse scattering which can be rather well explained as

a consequence of the orientational disorder that is postulated to explain the

Bragg peaks. Indeed the complete diffraction pattern (Bragg peaks plus diffuse

scattering) can be described by a very simple model which assumes that all

molecular orientations are equally likely. This picture of room temperature 

Carbon-60 ♦

Fig. 22. The low temperature powder neutron diffraction pattern for C60.

Points represent the measurements and the solid line represents a calculation

which assumes that each molecule adopts one of two possible orientations,

pentagon-facing with probability 0.83 and hexagon-facing with probability

0.17, such that there are four molecules in the simple cubic unit cell. A

difference plot is included near the bottom of the figure.
32

C60 is certainly oversimplified, since the icosahedral symmetry of the C60

molecule and the cubic symmetry of the lattice must at some level influence

the distribution of molecular orientations. To gain a better understanding of

what is going on requires a multi-pronged approach, involving theory (e.g.,

Michel and Copley, 1997), computer simulations and experiments. Both neutrons and x-rays have been used, and there have been studies of powder

♦ Examples

Fig. 23. A guide to understanding the crystal structure of C60. Each ball

represents a C60 molecule, with carbon atoms at each vertex connecting a

pentagon and two hexagons. At all temperatures the centers of the molecules

are located at face-centered cubic lattice sites. Above the phase transition the

molecules are almost randomly oriented. Below the phase transition there are

four molecules in the unit cell and each molecule in the unit cell has a

different orientation. Each type of orientation, be it pentagon-facing or

hexagon-facing, is obtained by rotating the molecules from their "standard"

orientations (with 2-fold axes pointing along cube edge directions, as shown in

the figure) through the same angle but about different (111) axes, as shown.

For the pentagon-facing orientation the angle is about 98°; for the hexagonfacing orientation it is ≈38°.
33

samples and single crystals. The goal is to develop an improved representation

of the interactions between the molecules in this fascinating material.

Carbon-60 ♦

Fig. 24. Relative orientations of nearest neighbor molecules in the low temperature phase of C60. Part of a neighboring molecule is viewed from within

the molecule at the origin. The pentagon-facing orientation is shown at (a)

and the hexagon-facing orientation is shown at (b). In each case the polygon

(pentagon or hexagon) faces a hexagon-sharing "double bond" on the

neighboring molecule.

(a) (b)

Fig. 25. The temperature dependence of the probability of the pentagonfacing orientation in low temperature C60 , as derived from neutron diffraction

experiments (David et al., 1992).
34

Our description of the low temperature structure is also woefully incomplete.

Powder neutron diffraction data (David et al., 1992) have shown that each molecule adopts two types of orientations (fig. 24), described as "pentagon-facing"

and "hexagon-facing," and that the probability p that a molecule adopts the

pentagon-facing orientation increases with decreasing temperature, as shown

in fig. 25. Below ≈ 86K, p no longer changes with temperature, indicating that

the system has frozen into an orientational glass.

The work described in the previous paragraph was performed using a time-offlight diffractometer at a pulsed neutron source. Such instruments are well 

suited to measurements at large Q. A good example is a study of the structure

of the C60 molecule itself, using data obtained up to Q-values as high as 50 Å-1

(Soper et al., 1992). The extended range of Q was needed in order to be able

to convert the data to a real space pair correlation function.

Acknowledgments

It is a pleasure to thank my colleagues at NIST, especially Julie Borchers,

Paul Brand, Jeremy Cook, Qing-Zhen Huang, Jeff Lynn, David Mildner,

Dan Neumann, Hank Prask, Louis Santodonato, Tony Santoro, Judith Stalick

and Brian Toby, for helpful discussions, for the use of figures, and for the use

of unpublished results.

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35

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References ♦
36

B.T.M. Willis and A.W. Pryor (1975), "Thermal Vibrations in Crystallography"

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C.G. Windsor (1981), "Pulsed Neutron Scattering" (Taylor and Francis,

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♦ References
November 2001