Coherent inelastic neutron scaterring in lattice dynamics

102 Pages·1982·1.739 MB·English

by  Dr. Bruno Dorner (auth.)

Springer stcarT ni Modern Physics 93
Editor: .G HShler
Associate Editor: E.A. Niekisch
Editorial Board: S.FlOgge H.Haken J.Hamilton
.H Lehmann .W Paul
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19 Structural Studies of Surfaces
With contributions by .K Heinz, .K MOiler, .T Engel, and .K .H Rieder
92 Single-Particle Rotations in Molecular Crystals By .W Press
39 Coherent Inelastic Neutron Scattering in Lattice Dynamics yB .B Dower
49 Exciton Dynamics in Molecular Crystals and Aggregates With contributions by
.V .M Kenkre and .P Reineker
* denotes a volume which contains a Classified Index starting from Volume .63
.B Dorner
Coherent Inelastic
Neutron Scattering ni
Lattice Dynamics
With 74 serugiF
galreV-regnirpS
Berlin Heidelberg weN York 2891
Dr. Bruno Dorner
Institut Max von Laue-Paul Langevin, B.P. 156, Avenue des Martyrs
F-38042 Grenoble, Cedex, France
Manuscripts for publication should be addressed to:
Gerhard H6hler
Institut for Theoretische Kernphysik der t&tisrevinU ehurslraK
hcaftsoP 6380, D-7500 Karlsruhe ,1 Fed. Rep. of Germany
Proofs and all correspondence concerning papers in the process of publication
should be addressed to:
Ernst A. Niekisch
essartsnidruobuaH 6, D-5170 hci10J ,1 Fed. Rep. fo Germany
ISBN 3-540-11049-6 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-11049-6 Springer-Verlag New York Heidelberg Berlin
Library of Congress Cataloging in Publication Data. Dorner, .B (Bruno). Coherent inelastic neutron scattering in lattice
dynamics. (Springer tracts in modern physics; 93). Bibliography: p. Includes index. .1 Lattice dynamics. 2. Neutrons-
Scattering. l. Title. ll. Series. QC1.$797 vo1.93 QCI76.8.L3 539s 530.4'1 81-14458 AACR2
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ehT aim of this book is to present the state of the art of coherent inelastic neu-
tron scattering as far sa the investigation of lattice dynamics is concerned. As-
pects of the experimental technique are discussed in much detail. Particular atten-
tion is payed to questions of resolution, intensity, focussing, dna finally, optimi-
zation of the experimental setup. ehT treatment of the latter subject sah especially
benefited from numerous discussions with scientists at the Institute Laue-Langevin,
Grenoble.
The symmetry operations contained in the space groups of the crytals under inves-
tigation play an important role in the performance of the experiment. Their influ-
ence on the analysis is discussed on experimental grounds, using examples which
avoid complicated mathematics. In several simple cases it sah been possible to mea-
sure phonon dispersion curves without having to first calculate the lattice dynam-
ical model. Yet as the number of atoms per unit cell increases, model calculations
emoceb more dna more important, dna even necessary. Besides the Born-von namraK
force constant concept, particular models for ionic, metallic, dna molecular crys-
tals are presented.
The discussion of experiments starts with the information obtained from a pre-
cise determination of phonon frequencies (peak positions), and continues with a
qualitative intensity analysis of phonon peaks dna an extended description of the
quantitative intensity analysis. Using the latter method, which is often called
a dynamical structure determination, the eigenvector of a particular phonon mode can
be extracted. Knowledge of eigenvectors provides a more microscopic insight into
lattice dynamics than knowing the frequencies of the dispersion curves alone does.
Several investigations of anharmonic effects follow. Generally speaking, anhar-
monic effects manifest themselves in the phonon lineshape dna in the temperature
dependence of phonon frequencies. The usual observation is a decreasing frequency
dna an increasing linewidth at higher temperatures. enO particular anharmonic
effect is the soft mode observed in connection with displacive structural phase
transformations. In several cases the soft mode is accompanied by a central peak
near the phase transition. Finally, the surprising observation of a double peak for
a one-phonon response at 4.2 K is interpreted by frequency-dependent damping.
The intention of this book is to provide general information no the basis of
a detailed analysis of measurements no a restricted number of substances.
Grenoble, July 1981 Bruno Dorner
stnetnoC
1. Introduction ..........................................................
1.! Reciprocal Space and Normal Coordinates ..........................
1,2 Momentum and Energy Transfer of the Neutron ......................
1.3 Time-of-Flight and Three-Axis-Spectrometer Techniques ............
2. Experimental Technique with Three-Axis Spectrometers ................. 8
2.1 Reflectivity of Monochromators and Resolution .................... 8
2.2 Higher Order Contaminations ...................................... 01
2.3 Resolution and Focussing ......................................... 01
3. The Scattering Function and Symmetry Operations in the Crystal ........ 61
3.1 Polarization and Symmetry of Eigenvectors ........................ 81
3.2 Intensity of Phonons in Different Brillouin Zones ................ 12
3.3 Extended Zone Scheme for Non-Symmorphic Space Groups ............. 23
4. Lattice Dynamical Models ............................................... 25
4.1 Ionic Crystals: AgBr and CuC1 .................................... 26
4,2 Metals: Cadmium .................................................. 3I
4.3 Molecular Crystals: Naphthalene and Anthracene ................... 93
5. Analysis of Phonon Intensities ......................................... 46
5.1 Electron-Phonon Interaction in Cadmium ........................... 46
5.2 Anticrossing of Dispersion Branches and Exchange of Eigenvectors in
Naphthalene ...................................................... 53
5,3 Eigenvector Determination ........................................ 56
5.3.1 Eigenvectors and Lattice Dynamical Models ................. 57
5.3.2 Exchange of the Transverse Mode Eigenvectors in AgBr at the
L point ................................................... 58
5,3.3 Eigenvector Exchange of owT Modes with Varying Temperature
in Quartz ................................................. 95
5.3.4 Eigenvector Determination of the Soft Mode in Tb2(Mo04) 3 .. 68
VIII
6, Analysis of Phonon Line Shapes ........................................ 73
6.1 Frequency Shift and Damping in AgBr .............................. 74
6.2 Structural Lattice Instabilities ................................. 77
6.2.1 Soft Mode in Tb2(Mo04) 3 78
6.2.2 The Central-Peak Phenomenon ............................... 18
6.3 Frequency-Dependent Damping in CuCl at 5 K ....................... 83
7. Final Remarks ......................................................... 88
References ............................................................... 19
Subject Index ............................................................. 95
1, Introduction
Condensed matter appears in different states such as liquid, amorphous, and crystal-
line. There are substates - phases - such sa superfluid liquids, the different phases
of liquid crystals, amorphous states having different histories, and a very large
variety of crystal structures classified into 032 space groups. There are crystalline
substances which retain the same structure from lowest temperature to melting. Others
undergo phase transitions from one crystalline ordered structure to another ordered
one by varying, for example, the temperature. There yam eb partial disorder of atom
positions and molecule orientations no a microscopic scale at a given temperature,
such that only the averaged position or orientation is compatible with a periodic
lattice. Order yam appear at a lower temperature. Generally it is a question of tem=
perature, pressure, fields, etc., dna sometimes history which phase a particular -am
terial is found in.
The different phases dna the transitions between them appear as a consequence of
the interactions between the atoms.
There are many different techniques to study these atomic interactions. gnomA
them, inelastic scattering of thermal neutrons has the unique advantage that thermal
neutrons have wavelengths comparable to atomic distances dna energies comparable to
excitations in condensed matter. ehT technique is described in detail in Chap. 2.
The investigation of atomic interactions exhibits a many-body problem because all
atoms are coupled and their displacements are not independent variables. This fact
sekam understanding of liquid dna amorphous states extremely difficult. ehT analysis
of inelastic neutron scattering intensities is limited to two-particle correlations
as the intensity represents the squared mus over the scattered amplitudes of the
different atoms. In the case of crystalline solids the many-body problem is reduced
due to the periodicity of the lattice. In well-behaved crystals (away from phase
transformations) translational symmetry allows restricting consideration to the
smallest periodic volume, the unit cell. Additional symmetries (rotations, mirrors,
etc.) facilitate the analysis of the atomic interactions further. emoS basic aspects
of symmetry operations and their effects in inelastic neutron scattering are dis-
cussed in Chap. 3.
In the following ew will restrict ourselves to lattice dynamics in crystals,
leaving out liquid and amorphous materials sa well as phase transformations in solids.
These aspects have been described by REGNIRPS (1972) dna by YESEVOL dna REGNIRPS
(1977). Lattice dynamics is concerned with a microscopic analysis of the different
forces between the atoms. The usual procedure is to produce a lattice dynamical model
with adjustable parameters which are supposed to represent the interatomic forces.
These parameters are more or less plausible. Sometimes one finds that two different
sets of parameters describe the experimental observation equally well. Thus the mi-
croscopic relevance of the parameters quite often remains an open question. But even
a non-plausible set of parameters which describes the results of inelastic neutron
scattering satisfactorily can then eb used to calculate other quantities like speci-
fic heat, heat conductivity, etc. (Chap. 4).
ehT information eno can obtain from the interpretation of the inelastic neutron
scattering intensity from phonons is presented with emos examples in Chap. 5. The
analysis of line shapes of phonon responses yields information on anharmonic contri-
butions as will eb explained in Chap. 6.
1.1 Reciprocal Space dna Normal Coordinates
sA already mentioned, the atomic displacements are not independent of each other. oT
escape the problem of coupled coordinates one uses translational symmetry to define
a reciprocal space. Points (hkl) in reciprocal space given by a reciprocal lattice
vector %.n represent the set of planes in real space which is perpendicular to !.
ehT length of I!i = 2~/d', where d' is the distance between neighbouring planes, dna
n is an integer. ehT reciprocal space is divided in many identical first Brillouin
zones around each (hkl). In the following ew will drop the definition "first" be-
cause lattice dynamics is only concerned with the first Brillouin zone. ehT second
dna further Brillouin zones play a role in electron band structure consideration.
oT overcome the difficulty arrising from the coupling of atom displacements, one
introduces normal coordinates which are plane waves in real space dna represented
by a wavevector ~ within the Brillouin zone. For one wavevector q there are 3n modes,
where n is the number of atoms per unit cell. This means there are n3 dispersion
branches for each direction, some of which yam eb degenerate. In the harmonic des-
cription these normal coordinates are orthogonal and thus uncoupled.
1.2 mutnemoM dna Energy Transfer of the Neutron
A neutron with ssam m dna velocity v sah a wavevector k = 2~/~, where ~ is the wave-
length of the neutron
~k = vm . (1)
The direction of k is the direction of the travelling neutron, e.g., of the neutron
-
o
.maeb Thermal neutrons have a wavelength of about 1.8 A, thus comparable with atomic
distances. In other words, k vectors are comparable to the dimensions of Brillouin
zones. ehT interaction of a neutron with a nucleus ,YESEVOL( 1977) is described by
a scattering length b dna a 6-function in space at the position of the nucleus. ehT
scattering length varies rapidly from element to element (even from isotope to iso-
tope, most often producing unwanted incoherent scattering). In the following ew con-
sider only the coherent scattering length d b of element d,
d = b ! wjbj (2)
where wj is the probability for the scattering length bj depending no different iso-
topes dna different spin configurations between nuclear dna neutron spin.
Incoherent scattering is considered in the following as a background which usu-
ally sah a smooth Q dependence eud to the Debye-Waller factor dna a (sometimes
disturbing) w dependence no a spectrum related to the density of states of the sample.
ehT mutnemom transfer Q (exactly )Q~ of a neutron in the scattering process is
given by
= ~I - F~ ' (3)
where ~I dna F~ are the neutron wavevectors before dna after scattering. In na in-
elastic scattering process the energy ,m~ transferred to the sample dna lost by the
neutron, is conventionally taken positive, i.e.
k# 141
mh = E I-E F =-~-
Fk~Ik
)b
ROTCAER--
NOITAMILLOC DNA
ROTAMORHCONOM
_ _ _ _ ~ --m
r
. I ~ q
o) >-cou ER
)Of~( " J)Ot ~O(I~ I " I " m ( ) 0~3
Fig, la-c. Inelastic neutron scattering: (a) path of neutrons in real space with
"black boxes" for the determination of neutron energy before dna after scattering;
(b) corresponding distribution of neutrons I V dna F V in reciprocal space around the
naem wave vectors I k dna kF; (c) mutnemom transfer-~ of the neutron in relation to
the reciprocal lattice of-the sample (vectors I) dna the phonon wave vector g.
RENROD( dna ,SEMOC 1977)