Phonon States of Elements. Electron States and Fermi Surfaces of Alloys Phononenzustande von Elementen. Elektronenzustande und Fermiflachen von Legierungen, Vol. 13

LANDOLT-BijRNSTEIN Numerical Data and Functional Relationships in Scienceand Technology

New Series Editor in Chief: K.-H. Hellwege

Group III: Crystal and Solid State Physics

Volume 13 Metals: Phonon States, Electron States and Fermi Surfaces Subvolume a Phonon States of Elements Electron States and Fermi Surfaces of Alloys l

P. H. Dederichs . H. Schober +D. J. Sellmyer Editors: K.-H. Hellwege and J. L. Olsen

Springer-VerlagBerlin . Heidelberg NewYork 1981 l

CIP-Kurztitelaufnahme Znhtenrwte

der Deutschen

Bibliothek

und Funkrionen 0~s Nafunvissenscho~en und Technik/Landolt-BGrnstein. - Berlin: Heidelberg; Parallelt.: Numerical data and functional relationships in science and technology.

New York:

Springer,

NE: Land&B6msfein. . . . . PT. N.S./Gesamthrsg.: K.-H. Hellwege. N.S., Gruppe 3, Kristallund Festkiirperphysik. N.S., Gruppe 3, Bd. 13. Metalle: PhononenmstPnde, Elektronenzust%nde und FenniflBchen. N.S., Gruppe 3, Bd. 13, Teilbd. a. Pbononenmsc&nde van Elementm. Elektronmzwt%nde und FermiWhen van Legiemngen/P.H. Dederichs Hrsg.: K.-H. Hellwege u. J.L. Olsen. - 1981.

ISBN 3-540-09774-o(Berlin, Heidelberg, New York) ISBN 0-387-09774-O(New York, Heidelberg, Berlin) NE: Dederichs.

Peter H. [Mitverf.];

Hellwege,

Karl-Heinz

[Hrsg.]

This work is subject to copyright. All rights are reserved,whether the whole or part of the material is concernedspecifically those of translation, reprinting, reuseof illustrations, broadcasting,reproduction by photocopying machine or similar means,and storage in data banks. Under $54 of the German Copyright Law where copies are made for other than private use a fee is payable to ‘VerwertungsgesellschaftWort’ Munich. 0 by Springer-Verlag Berlin-Heidelberg 1981 Printed in Germany The use of registered names,trademarks, etc. in this publication does not imply, even in the absenceof a specific statement,that such namesare exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting, printing and bookbinding: Universitltsdruckerei H. Stiirtz AG Wiirzburg 216313020- 543210

Preface This collection of tables and diagrams is the first contribution to a larger programme aiming at a complete and critical tabulation of reliable data relevant to metal physics. No such complete collection exists at present, and these tables should till a long felt need of both experimentalists and theoreticians. Group III in the New Series of the Landolt-Bornstein tables deals with Crystal and Solid State Physics. Volume III/l3 to which this subvolume 13a belongs will cover all data published up to 1980 on phonon and electron states and Fermi surfaces in metals. Both experimental and theoretical results are included. To hasten publication the compilations in the subvolumes 13a, 13b and 13~ are being printed after each author has completed his manuscript. The order of the tables is thus chronological rather than thematic. A systematic survey is given on the inside of the cover. This subvolume contains two data compilations, one of phonon states in the elements, and the other of electron states and Fermi surfaces of compounds and disordered alloys. An appendix lists Bravais lattices, Brillouin zones and associated information. In general, symbols and nomenclature are those used in the literature but the reader is referred to the separate contributions for detailed information. Our most grateful thanks are due to the authors for taking on the great and most laborious task of collecting the data and critically preparing the tables in this subvolume. We are confident that their contribution will be of great value to the physics community, and we hope that the authors will find in this their reward for all their hard and careful work. Thanks are also due to the editorial staff of Landolt-Bornstein, especially to Dr. H. Seemtiller who was in charge of the editorial preparation of this subvolume and who also arranged the appendix, and to Frau D. Dolle and Frau H. Weise,for their careful checking of the manuscripts and galleys. We are also grateful to the Springer Verlag for their patient care and experienced help in the final preparation. As in the case of other Landolt-Bornstein Volumes, this subvolume is published without outside financial support. Darmstadt and Zurich, July 1981

The Editors

Contents 1 Phonon dispersion, frequency spectra, and related properties of metallic elements H.R. SCHOBER and P.H. DEDERICHS,

Institut fur Festkiirperforschung der Kernforschungsanlage Jiilich GmbH, Jtilich, Germany 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 General layout . , . . . . . . . . . . . . . . . . . 1 1.1.2 Comments on data of section 1.2 . . . . . . . . . . . 1 1.1.2.1 Phonon dispersion . . . . . . . . , . . . . . 3 1.1.2.2 Frequency spectra and related properties. . . . . 3 1.1.2.2.1Spectra . . . . . . . . . . . . . . . 3 1.1.2.2.2Debye cutoff frequencies. . . . . . . . 4 1.1.2.2.3Specific heat; Debye temperature . . . . 4 1.1.2.2.4Debye-Waller factor . . . . . . . . . 5 1.1.2.3 Theoretical models (commentsand referencesonly) 5 1.1.3 List of frequently used symbols and abbreviations . . . . . . 7 . . . . . 1.2 Data. ......... ‘129’ Ru . . . Ag ........ 7 131 Sb . . 61 Hf . . Al.. ...... 11 132 SC . . 63 As.. ...... 17 Hg . . 135 Sn . . 64 Ho . . Au. ....... 18 141 Sr . . 67 . . In Ba ........ 22 141 Ta . . 68 . . K Be ........ 23 144 Tb . . 76 La . . Bi ........ 25 147 Tc . . 76 Li . . Ca ........ 29 148 Th . . 82 Cd ........ 30 Mg . . 151 Ti . . 86 MO . . Ce ........ 34 154 Tl . . 92 Na co ........ 37 157 Tm . . 96 Nb . . Cr ........ 39 158 . 104 u . Ni . . cs ....... .40 160 v . . 108 Pb . . cu.. ..... .44 164 w . . 112 Pd . . Dy ........ 52 167 Y . . . . 117 Pt . . Er ........ 52 169 Zn . . . . * , 122 Rb . . Fe ........ 53 . 128 Zr . . . . . . . 175 Re . . Ga ........ 57

: 60’

Gd.

1.3 Referencesfor 1.1 and 1.2 .

. . .

9

.

. . . . . 180

. . . . . . . . .

2 Band structures and Fermi surfaces of metallic compoundsand disordered alloys D.J. SELLMYER, Behlen Laboratory

of Physics, University of Nebraska, Lincoln, Nebraska, U.S.A. . . . . . . . . . . . . . . .

2.1 Introduction ................... 2.1.1 Preliminary remarks and organization ...... 2.1.2 Theoretical and experimental methods ...... 2.1.3 List of frequently used symbols and abbreviations . 2.1.4 General references(handbooks and reviews) ... 2.2 Metallic compounds ................ 2.2.1 sp-metallic compounds ............ 2.2.1.1 Survey ................ 2.2.1.2 Data .................

. . . .

. . . .

. . . .

. . . . .

192 192 192 194 196

. . . .

196 197 197 198

2.2.2 Transition metal compounds .............................. 2.2.2.1 Survey .................................... 2.2.2.2 Data. .................................... 2.2.3 Quasi one- and two-dimensional compounds ....................... 2.2.3.1 Survey .................................... 2.2.3.2 Data. .................................... 2.2.4 Rare-earth, oxide, and other compounds ......................... 2.2.4.1 Survey .................................... 2.2.4.2 Data. .................................... 2.25 Referencesfor 2.2. .................................. 2.3 Disordered alloys ..................................... 2.3.1 q-metallic alloys ................................... 2.3.1.1 Survey .................................... 2.3.1.2 Data. .................................... 2.3.2 Noble metal alloys .................................. 2.3.2.1 Survey .................................... 2.3.2.2 Data. .................................... 2.3.3 Transition-metal alloys ................................ 2.3.3.1 Survey .................................... 2.3.3.2 Data. .................................... 2.3.4 Intermediate phases,hydrides, and amorphous alloys ................... 2.3.4.1 Survey .................................... 2.3.4.2 Data. .................................... 2.3.5 Referencesfor 2.3 ...................................

222 222 225 300 300 302 332 332 334 377 386 386 386 388 398 398 399 414 414 415 431 431 431 441

Appendix Bravais lattices (conventional unit cells), primitive unit cells, reciprocal lattices and first Brillouin zones 1 2 3 4 5 6 7 8 9

Body centred cubic metals .................................. Face centred cubic metals .................................. Hexagonal close packed metals ................................ Body centred tetragonal metals ................................ Face centred tetragonal (nearly fee)metals ........................... Rhombohedral metals .................................... Rhombohedral (triatomic hexagonal) metals .......................... Basecentred orthorhombic metals ............................... Simple cubic and simple tetragonal alloys. ...........................

References. .........................................

447 448 449 450 452 453 454 455 457 458

Table of conversion factors

Quantity

atomic units *)

CGS

SIU

Miscellaneous units

Length, 1 Energy, E

la.u.=a,= 1 au. = h2/&a~ =

5.291772. lo-’ cm= 4.35982+10-” erg=

5.291772.10-” m= 4.35982.10-l8 J=

0.5291772A 2 Ry (Rydberg) =27.2116 eV**)

Reciprocal length, I- ’ Reciprocal area,A

la.u.=a;‘= 1a.u.=aG2=

1.8897. lo8 cm-‘= 3.5711. 1016cmT2=

1.8897. lOlo m-l = 3.5711.lOzo mm2=

1.88978-l 3.5711A- *

*) a,: Bohr radius; m,,: electron rest mass **) 1VAs=1J=107erg=6.24115~1018eV=2.3006~10-4kcal

Energy and equivalent quantities

Quantity:

E

U= Efe

v=E/h

i;=E/hc

Unit:

J

V

Hz, s-l

cm- 1

1

6.24115.1018

1.50916.1O33

5.03403.1022

1v

=* =A

1.60219. lo- lg

1

2.41797.1Ol4

8.06547. lo3

ls-‘=lHz-

;

6.62619. lo- 34

4.13550.10-‘5

1

3.33564. lo- ”

1 cm-l

=.

1.98648. lo- 23

1.23979. 1O-4

2.99792.10”

1

1J

1.1 Introduction

Ref. p. 1801

1 Phonon dispersion, frequency spectra, and related properties of metallic elements 1.1 Introduction 1.1.1 General Layout In this contribution both experimental and theoretical results of the phonon dispersion and spectra of metallic elementsare collected. Since the development of neutron diffraction facilities around 1950, the phonon dispersion of most elements has been studied in detail, including, for some elements,studies of the temperature aehaviour and other anharmonic properties. At present, theoretical models cannot match the experimental accuracy.The emphasisof this compilation lies therefore on the experimental data. The theoretical models serve :ither as a tool to parametrize and extrapolate the experimental data or to test microscopic models. In cases where no experimental data are available, models are used to predict the phonon data by extrapolation from Jther similar materials. For the readers’ convenience, the elements are ordered alphabetically according to their chemical symbols rather than their position in the periodic table. When possible, the available data for each substance are subdivided into three subsections:1. phonon dispersion, 2. frequency spectrum and related properties, and 3. special referencesconcerning theoretical models used.The contents of thesesubsectionswill be discussedbelow. Further, Foreach element a subheading gives the crystal symmetry, the lattice constants and angles taken from [67Sml]. Phonon data published until about mid 1979 have been included. Lattice dynamics are treated in most standard text books on solid state physics. The most detailed monograph on lattice dynamics, restricted to the harmonic approximation, however, is [71Mal]. Detailed reviews also including the anharmonic properties, are to be found in [74Hol]. The compilation of the data was facilitated by two bibliographies on neutron scattering data [74Lal and 76Sal] covering the period up to 1974. The more recent literature was searchedwith the help of the literature serviceof the central library of the Kemforschungszentrum Jtilich. Acknowledgements The authors gratefully acknowledge the help of Mrs. Spatzekin preparing the manuscript and of the literature service of the central library of the Kernforschungszentrum Jtilich in the literature search.

1.1.2 Comments on data of section 1.2 All data are compiled separately for each element.

1.1.2.1 Phonon dispersion The major experiments, the employed method and the temperature are listed in the first table. The most accurate measurementsare obtained by inelastic neutron scattering (neutron diffraction) using either triple axis (TAS) or time of flight (TOF) spectrometers.X-ray diffraction measurementsare handicapped by the necessity to correct for higher order scattering and for incoherent Compton scattering and so cannot match the neutron measurements.They are only important for substancesfor which no neutron diffraction measurementshave been possible, due to a too small coherent scattering cross section or a too high absorption. Accurate values for the optical modes at the I-point have been obtained for somematerials by Raman scattering. In a short summary the measurementsare compared and typical features of the dispersion pointed out. Additional measurementsof special properties not tabulated (in the first table) are mentioned in this summary. The measuredphonon dispersion is shown in figures. Reducedwavevector coordinates, [, related to the reciprocal lattice dimensions (seecompilation of reciprocal lattices and Brillouin zones in the appendix of this volume), are used as x-axis and frequencies,v, in THz as y-axis. The symmetry points and directions are labelled according to the figures of the Brillouin zones in the appendix of this volume. The symmetry labels of the phonon branches have been worked out by Watson [68Wal]. Otherwise the corresponding referencesare mentioned. When available, the measuredphonon frequencies are also presented in a table. The error limits refer to the statistical errors. Schoher/Dederichs

1

1.1 Einleitung

[Lit. S. 180

Anomalies

For several materials anomalies in the dispersion have been found, i.e. deviations from a smooth variation of frequency with wavevector as one would expect from a not too long range coupling. The most common ones are Kahn nnomalies.These are kinks in the phonon dispersion curves which arise when the phonon wavevector q equals an extremal chord of the Fermi surface(for free electrons q=2k,). In thesecases,the phonons are strongly affectedby the singularity of the dielectric function c(q) at thesevalues of q. Related to this are anomalies caused by higher order terms in an expansion in powers of the electron ion potential function [74Brl]. Another anomaly, a changein slope at low q-values,is attributed to a changefrom first sound at low frequencies(collision dominated regime)to zero sound at higher frequencies(collision free regime).The differencebetweenthe two sound velocities is normally of the order of 1%, but can becomelarger in somecases. Besidestheseanomalies,there is a number of other anomalieswhich are not yet fully understood. To determine the origin of a given anomaly without ambiguity, a careful study of its temperature behaviour and also of the phonon widths is required. Anharmonicity

In a real solid, the lattice vibrations are not harmonic and cannot rigorously be resolved into independent normal modes(phonons). For most substancesthe anharmonic interaction terms can be treated as a perturbation giving rise to a width, r, of the phonons and a shift, A, of the frequencies,with temperature and pressure.This is causedby the deviation of the interatomic potential from a parabola form and also by the dependenceof the potential itself on temperatureand pressure.The shift of the phonon frequencies,not their widths, can be described in the quasiharmonic approximation where one takes a purely harmonic but temperature dependent coupling. The temperature and pressuredependenceof the phonon frequenciescan then be expressedin terms of a mode Grirncisen parometer yas(q, a)=

- “naY;(qT

u);

(1)

UP

in which u denotesthe polarization and .suP the strain tensor. For cubic crystals and homogeneousstrain (pressure) equ.(1) is reduced to Y(q,+ -“‘n,;4;6’. (2) The temperature dependenceof the frequenciesis thus given by v(q,fl,n=+?,a,T,)

l-y(q,a)

1

g

(T- To)];

(3)

with TOas referencetemperature. Born-von Karman

parameters

Born-von Karman coupling constants are the most common way to parametrize the experimental dispersion :urves of metals. The major limitation of this approach is the large increasein the number of fitting parameters with increasing interaction range. This makesan unambiguous determination of the parametersfrom the experinental data impossible, particularly for the more complicated structures. The coupling parametersdetermined by such a fit have not necessarily any direct physical meaning. They only provide a merely phenomenological description of the dispersion. A determination of the real physical couplings involves not only an exact knowledge of the phonon frequencies but needs also information on the polarizations of off-symmetry phonons [71Lel]. Keeping the above restrictions in mind, one can, however, determine sometrends for the near-neighbour couplings with temperature, etc. In the most general case,the only restrictions imposed on the coupling constant matrices are due to the point symmetriesof the lattice (tensor force constant model). To reduce the number of fitting parameters,often further restrictions are imposed on the coupling matrices. In the axially symmetric mode/ one assumesthat there is only one central and one transversal force constant, f, and f, respectively. Such a model can be thought of as derived from a central pair potential V(R). In that casethe two force constants are related to the derivatives of V: fr=r;

a’V(R) (4)

,=I WR) ’ RF; 2

Schober/Dederichs

1.l Introduction

Ref. p. 1801 The coupling matrix takes then the form:

~J=(f,-f,)R~Rf/(Rm)2+f,6ij=KR~R~/(Rm)Z+C6ij.

(5)

For hexagonal metals a modified axial/y symmetric model is often employed in which the constant C is different for directions in the basal plane and for those perpendicular to it. When sufficiently accurate Born-von Karman fits exist, they have been included in the tables.

1.1.2.2 Frequency spectra and related properties The data compiled under these headings comprise the spectra and some quantities related to them, namely: Debye cutoff frequencies,lattice specific heats,Debye temperatures,and Debye-Waller factors. The results derived from phonon data have been included primarily, results from other measurements,e.g. thermal data, are given for comparison only.

1.1.2.2.1 Spectra The frequency spectrum (density of phonon states as a function of frequency) for vibrations in direction i can be formally defined as: gi(v)=fC&v;I!

dq h(v-v(q, 0)) leih, 4’;

(6)

where r is the number of atoms per unit cell, e denotesthe polarization branches and e is the polarization vector. The directionally averagedspectrum is g(v)=3(g,(v)+g,(v)+g,(v)). (7) For a cubic crystal one has g(v) = gi(v). The spectra are normalized to unity:

oSg(v)dv=l.

(8)

Phonon spectra can be measuredeither by incoherent inelastic neutron scattering or by coherent scattering on polycrystalline samples.Another method would be via superconducting tunneling. Since this method involves, however, the largely unknown electron phonon coupling, such measurementshave not been included. The most common way to determine the phonon frequency spectrum is to calculate it from equ. (6) by Brillouin zone integration using either a model fit or the measuredand intrapolated phonon frequenciesdirectly. Thesecalculated spectraare in general more detailed than the directly measuredones. The spectra are presentedin the form of figures and in some casesalso as tables.

1.1.2.2.2 Debye cutoff frequencies Various experimental data can be expressedin terms of moments of the spectrum. Since the actual moments cover many orders of magnitude it is convenient for such calculations to define “Debye cutoff frequencies”, v,, [65Sal]. These are gained by equating the nth moment of the real spectrum g(v) with the corresponding moment of a Debye type spectrum g,,(v) =$ vz : with

(9)

,= ,“n (v”),= rv”g(v)dv

(104

0

(v”)~~== rv”gJv)dv= 0

jv’+dv. 0 ”

(10’4

From equ. (9) we obtain for the Debye cutoff frequencies: v” = yorvn

g(v) dv]l’“,

n*O, n>-3.

Schober /Dederichs

(11)

1.1 Einkitung

[Lit. S. 180

The cutoff frequency for n = -3 where the integrals in equ. (10) diverge at v=O can be defined by equating the divergent parts. g(v) can be expanded for low frequencies as g(v)=u, v*+a, v4+.**

and hence

v-s=(3/u,)j

=I

0,

(T=O)

(12)

where 19, is the Debye temperature defined below. For n=O equ.(ll) is undefined. A cutoff frequency can, however, be gained by building the limit n-+0 leading to: v,=exp 3+ rg(v)lnvdv . I [ 0

(13)

For most metals the v, are shown in the form of figures.

1.1.2.2.3 Specific heat, Dehye temperature The lattice entropy per mole is given in the quasiharmonic approximation as: S,(T)=Rrdvg(v) where

0

R is the gas constant

I

~n(v,T)+In(l+n(v,T)}

I

n(v, T)= [exp(h v/kT)-1-J-l

(14)

(15)

and v is the frequency measuredat the temperature T. The lattice specific heat at constant pressurecan be derived from the entropy: CL= T ($=Rordv

g(v) n(v, T) [n(v, T)+l]

(;)1{

1 - (gi}

(16)

in which the first term on the right hand side gives the (quasi) harmonic expression, Ch, and the second term the lowest order anharmonic correction. In the harmonic approximation, the specific heats at constant pressure Cf, and at constant volume CL are equal, due to the absenceof a lattice expansion in that approximation. To obtain the total specific heat of the crystal an electronic contribution Cc has to be added: in which one approximates

c=c’+c

(17)

C’=y, T.

(18)

The lattice specificheat is usually related to the one obtained by replacing g(v) by a Debye spectrumg,(v)=(3/4 yielding

v* (19)

with (20)

k@,=hv,.

The Debye temperature 0, which depends on the temperature is defined by requiring CD(T)= C’(T).

(21)

This “caloric” Debye temperature is very sensitive to small changes of Cr. An error of 1% in C’ gives rise to an error of 3% in 0, at TzOD/2 and of 12% at T=@,. Besides this “caloric” definition of a Debye temperature which is the most common one, various other definitions with other quantities are used in the literature. In thesetables we refer only to the one defined above.

1.1.2.24 Dehye-Wailer factor The thermal mean square displacement in direction i of a lattice ion is given, again in the harmonic approximation, by: (I$),=~&-&

coth (gT)

g,(v)dv.

(22)

In scattering experiments the intensity is determined by the Debye-Wller factor e-2R’=e-<(Qu)*)~.

4

Schober /Dederichs

,

(23)

1.l Introduction

Ref. p. 180

2 W is the Debye- Wailer exponent. Only for cubic crystals it is independent of the direction of the scattering vector Q. A convenient way to plot Debye-Waller data is to devide 2 W by the recoil frequency va of the free ion. 2w/vs=a~+coth

& (

1

g(v)dv;

in which h va= h* Q2/2m is the recoil energy of the free ion. Another, related definition is the Debye- Wailer coefficient: L(r)=kTa~&coth‘($)

g(v)dv.

For most metals Debye-Waller data are representedin the form of figures.

1.1.2.3 Theoretical models (comments and references only) A detailed description of the many theoretical models would have gone far beyond the scope of this contribution. The theoretical papers were therefore classified according to a rather crude schemeand in most cases only a list of referencesis given. The distinction between “references” and “further references” was done on the basis of the quality of the fit to the experimental data. Reviews of the details of the various models can be found in [68Jol, 74Hol-j. The most common models fall in the following categories.For other models we refer to the original papers. Born-von Karman and equivalent models

Born-von Karman models and all models which can, in regards to the phonon properties, be expressedby short range real spaceforce constants. Short range forces plus a simple electronic contribution

The dynamic matrix of thesemodels is split into two parts, one stemming from short range (Born-von Karman like) couplings DR and the other one simulating the responseof a free electron gas DE Wd=DR(q)+DE(q);

D:(q)=A

1 (q+Q)i(q+ Qh qrs, tq + Q)) Q

Qff;2

(q+Q)*+k:

G(rst

Q).

c

Here Q are the reciprocal lattice vectors, k, and G are a screening constant and function, r, is the radius of a sphere containing one electron and A is a parameter depending on the effective charge and the electronic compressibility. A typical model of this kind is the one due to Krebs [64Krl]. These models can be regarded as a simplified version of pseudopotential theory. Often the model is further simplified by omitting all terms with Q+O. The model violates then the translational invariance. A typical model of this kind is due to Sharma and Joshi [63Shl]. Pseudopotential models

For simple metals pseudopotential perturbation theory provides a well founded framework. These theories have been treated extensively in many textbooks and articles, e.g. [7OCo3, 74Brl]. First principle pseudopotentials are nonlocal electron ion potentials. In actual calculations one assumesmostly a specific form, local or nonlocal with parameters determined either by first principles or by experiment. For transition metals the schemehas to be modified by adding some extra terms to treat the d-band electrons, e.g. transition metal model potentials.

1.1.3 List of frequently used symbols and abbreviations *) a, b, c

Cm1

A cij

C, [J mol-’ K-l] C, [J mol-’ K-l] Ch [J mol-’ K-r]

lattice constants accustical branch elastic constants molar specific heat at constant pressure molar specific heat at constant volume molar lattice specific heat (harmonic approximation)

*) Seealsotable of conversionfactors. Schober/Dederichs

5

1.1 Einleitung C’ [J mol-’ K-‘1 C’ [J mol-’ K-‘1 C [Nm-‘1 Dij

k CNm-‘I f, INm-‘j yJ s:“z- 1 k [J K-‘1 k,. k, Cm-‘]

K [Nm-‘1 L f71Fg atom- ‘1 z [kg mol-‘1 0

q Cm-‘1 p K1 kR Cm1 R”,R; [m] T

T, r2

V Cm31 w

0 WI

lattice angles constant in electronic specific heat mode Grtineisen constant phonon width centre of Brillouin zone Kronecker delta (=0 if i+j, =1 if i=j) dielectric function Debye temperature Debye-Wailer coefficient frequency Debye cutoff frequency recoil frequency of free atom reduced wavevector coordinate force constant matrix susceptibility polarization index circular frequency (= 25rv)

II 1

parallel (to basal plane) perpendicular (to basal plane)

BvK BZ KE TAS TOF

Born-von Karman Brillouin zone Kohn effect (or Kohn anomalies) triple axis spectrometer time of flight spectrometry

[x,/I Y C&-w1 y [J mol-’ Km21 Y

I- WI l6ij E(4)

Q, WI W-1

Y [Hz]

*‘n WI “R WI 3 @; [N m-‘1 x(4) a

6

molar lattice specific heat molar electronic specific heat force constant in axially symmetric model dynamical matrix polarization vector radial (longitudinal) force constant tangential (transversal) force constant frequency spectrum Planck constant Boltzmann constant Fermi momentum force constant in axially symmetric model longitudinal branch atomic mass lattice index: coordinates of typical neighbours molecular mass optical branch wave vector wave vector, reciprocal lattice vector radius of a spherecontaining one electron lattice vector vector to lattice point characterisedby m, coordinate i of R” transversebranch transversebranch in [O[fl direction with polarisation [Oli] transversebranch in [Ora direction with polarisation [loo] volume Debye-Wailer coefficient (2W: Debye-Wailer exponent)

Schober/Dederichs

[Lit. S. 180

1.2 Phonon states: Ag

Ref. p. 1 SO]

1.2 Data Information for each element from Ag through Zr is given in the order : 1. Phonon dispersion; 2. Frequency spectra and related properties; 3. Theoretical models

Ag

Silver

Lattice: fee, a= 408pm=4.08 A. BZ: see p. 449. 1. Phonon dispersion Table 1. Ag. Measurements. Method

Fig.

Ref.

1. Ag

Kamitakahara, Brockhouse [69Kal] Drexel [72Drl, 71Drl]

TKI neutron diffraction (T-Q) neutron diffraction (TOW

296 293

The overall agreement between both measurementsis good with maximum deviations of about 4 % in the [Oil] Ti branch. The lattice dynamics of Ag is similar to Cu and Au. The mean frequency ratio (v(Ag)/v(Cu)) = 0.675with a standard deviation of 0.016for a single deviation from the mean. This agreeswell with the criterium for homology of forces (v(Ag)/v(Cu)) = ((~I~u~)~~~/(Mu*)~~)=0.679. The phonon dispersion can be well described by axially symmetric force constants. Table 2. Ag. Phonon frequenciesat 293 K measuredby pseudostatistical time of flight technique (Drexel[72Drl]). r

v [THz]

r

2,21 (16) 3.66(24) 4.71 (24) 4.54 (24) 5.08(32)

0.228 0.325 0.501 0.878 0.923

corr1 L 0.138 0.230 0.437 0.782 0.878 0.923 0.924 0.967 0.983

1.59 (8) 2.71 (16) 3.96(24) 3.66(24) 3.53(24) 3.41(16) 3.55(11) 3.31(16) 3.31(16)

2.29 (16) 3.18(24) 3.21(24) 4.42 (40) 4.89 (24) 4.92 (24)

v [THz]

r

v [THz]

1.16 (8) 1.67 (8) 2.23 (16) 3.20(24) 3.34 (24)

CNrl 5

COKITl 0.322 0.372 0.404 0.437 0.667

Crirl L 0.150 0.199 0.225 0.330 0.401 0.408

r

CO’XIT

Co@3L 0.260 0.520 0.728 0.773 0.990

v [THz]

1.38 (8) 1.69 (16) 1.85 (16) 1.96 (16) 2.75 (24)

0.129 0.212 0.239 0.346 0.386 0.411 0.451 0.478 0.497

0.92 (8) 1.53 (16) 1.59 (16) 2.48 (13) 2.58 (24) 2.72 (14) 2.94 (13) 2.91 (24) 3.26(11)

0.563 0.644 0.681 0.699 0.727 0.754 0.805 0.828

3.52(32) 3.93(32) 4.11 (32) 4.38 (24) 4.41 (24) 4.49 (24) 4.58 (24) 4.58 (24)

C5151 T 0.124 0.150 0.199 0.244 0.278 0.323 0.443 0.419

0.78 (8) 0.89 (8) 1.27 (8) 1.46 (16) 1.72 (16) 1.72 (18) 2.16 (24) 2.23 (24) Schoher/Dederichs

7

1.2 Phononenzustkde : Ag

[Lit. S. 180 A6' THZ

-E

ZX

root1

W

XK

WI i

I

!

ram

Fig. 1. Ag. Phonon dispersion at 296 K measured by TAS (o experimental points, solid line: 4th neighbour Born-von Karman tit) [69Kal]. Anomalies in the dispersion curves

The [O[fl Tr branch shows a positive dispersion for small [ values. With increasing temperatures,this anomalous behaviour becomesmore pronounced [78Bul], Fig. 2 Ag. A similar behaviour is also observedin Cu [76Lal] where it was interpreted as transition from first to zeroth sound [77Lol]. In Ag however such a transition cannot fully explain the measurements.At least part of the positive dispersion seemsto be causedby the inter-ionic forces and is present already at 0 K [78Bul].

Born-von Karman models

Table 3. Ag. Born-von Karman coupling constants, 0;. T

296 K

293 K

293 K

Ref.

69Kal

72Drl 71Drl

72Drl 71Drl

m

ij

110

xx zz XY

10.71(17) 1.75 (20) 12.32(32)

-0.4 (5)

12.3(5)

10.4 (3) -1.0 (9) 12.1 (6)

200

xx

0.06 (29) -0.23(19)

2.1 (6) -0.7 (3)

-1.6 (10) -0.5 (6)

211

xx YY YZ

0.52 (10) 0.21 (9)

-0.3 (3) -0.2 (1)

-0.1 (5)

0.05 (12)

0.1 (1) 0.1 (1)

YY

220

xz

0.30 (7)

xx

-0.13 (10) -0.14 (15) 0.01 (18)

YY zz 8

qj wrn-‘1 10.4(2)

0.3 (2) 0.0 (2)

1.0 (2) 0.3 (3) -0.3 (4) 0.2 (4) Schober/Dederichs

0

0.1 0.2 0.3 ox 0.5 0.t t-

Fig. 2. Ag. Measuredphasevelocity

Iv/c) of the [OcflT, phonon branct for various temperatures. u =mea,

sured first sound velocity [66Chl] dashedline: theoreticalcurvefor tirsl to zeroth sound transition

Ref. p. 1801

1.2 Phonon states: Ag

2. Frequency spectrum and related properties

0

4 THz 5 3 YFig. 3. Ag. Phonon frequency spectrum at 296 K calculated from the 4th neighbour Born-von Karman model by [69Kal]. 2

1

Table 4. Ag. Phonon frequency spectrum at 296 K calculated from the fourth neighbour Born-von Karman model [69Kal] of Table 3 Ag. v [THz]

g(v) CT=- ‘I

v [THz]

g(v) CTHz-‘1

v [THz]

g(v) CTHz-‘1

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05

0.000 0.000 0.000

1.75 1.80 1.85

0.106 0.117 0.128

0.001

1.90 1.95

0.141 0.156

3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00 4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00 5.05 5.10

0.259 0.256 0.253 0.250 0.245 0.241 0.235 0.229 0.222 0.213 0.203

0.001 0.002 0.003 0.003 0.004 0.005 0.007 0.008

0.010 0.011 0.013 0.015 0.017

0.019 0.022 0.025 0.027

1.10

0.031

1.15 1.20 1.25 1.30 1.35

0.034 0.038 0.042 0.046 0.051 0.056 0.061 0.067 0.074

1.40 1.45 1.50

1.55 1.60 1.65 1.70

0.081 0.089 0.097

2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40

0.172

0.191 0.213 0.241 0.277 0.351 0.373 0.372 0.373 0.374 0.374 0.374 0.374 0.374 0.374 0.374 0.374 0.375 0.375 0.375 0.375 0.376 0.376 0.376 0.376 0.340 0.315 0.297 0.281

Schober/Dcderichs

0.192 0.178 0.162 0.140 0.117 0.150 0.208 0.281 0.389 0.424 0.455 0.487 0.516 0.549 0.415 0.331 0.276 0.230

0.190 0.148 0.103 0.047 0.0

9

1.2 Phononenzustkinde: Ag 5.2 1Hz

230 K I 220

5.0 48 I ’ a.6

0

10

20

190 0

30

60

0

100

200

90 120 150 K 180 IFig. 5. Ag. Debye temperatures 0, calculated from the 3rd neighbour Born-von Karman model by [72Drl] compared with experimental values.

a.4 4.2 -10

[Lit. S. 180

30

/I-

Fig. 4. Ag. Debye cutoff frequencies Y. calculated from the 4th neighbour Born-v. Karman model by [69Kal].

Fig. 6. Ag. Debye-Wailer exponent 2 Wdivided by the recoil frequency of the free atom, va, calculated from the 4th neighbour Born-von Karman model by [69Kal].

300

400 K 500

6 cal Kmole I 4 G

2

0

30

60

90

120 150 180 210 K 240 IFig. 7. Ag. Specitic heat C, calculated from the 3rd neighbour Born-v. Karman model by [72Drl] compared with experimental values.

3. Theoretical models Due to their simple shape the dispersion curves are described adequately by most phenomenological models. A truly microscopic description however meets with rather great difficulties due to the hybridization of s and d electrons.

Born-von Karman and equivalent models: see Table 3 Ag and [72Drl], further references:[73Shl]. Short ranged forces plus a simple electronic contribution: [75Cll, 76Ma1, 77Kh2, 78Ku1, 77Sal], further references:[71Shl and 71Sil; 70Be1, 70Sh1, 74Go1, 75Be1, 77Di1, 73Shl].

Shell model: [78Si4]. Local model pseudopotential calculations: [71Drl, 72Drl,76Nal], 78Ku2-J.

further references:[71Nil, 76Kul,78Shl,

Models incorporating electronic d-Band terms: [73Sil, 75Lal], further references:[72Mol, 73An1, 72Pr1, 75Si3]. 10

Schoher/Dederichs

Ref. p. 1801

Al

1.2 Phonon states: Al

Aluminium

Lattice : fee, a = 404 pm = 4.04 A. BZ : see p. 449.

1. Phonon dispersion Table 1. Al. Measurements. Method

Fig.

Ref.

1. Al

Stedman and Nilsson [66Stl, 65St2]

L neutron diffraction VW neutron diffraction PAS) neutron diffraction (TOW

80 300 300

Yarnell et al [65Yal]

293, 932

Larsson et al. [60Lal]

Further measurements: [57Cal, 58Brl]. The overall shape of the phonon dispersion in Al is simple, typical for fee metals. The phonons soften in average 7.5 ‘A in the temperature range from 80 K to 300 K and further 15 y0 up to 930 K. Anomalies have been studied very carefully and a large number of Kohn anomalies have been identified. There is also evidence of anomalies of higher order in the electron ion interaction. The initial slopes are in excellent agreement with the measured elastic constants

6-

0 T =300K

0

0.2

0.4

0.6

0.8

I0

0.5c I-J 0

I p P R P It

0.2

0.4

9 $19

0.6

, % I ,t

0.8

1.0

Fig. 1a-c. Al. Measured phonon dispersion curves at 80 K and 300 K. The corresponding phonon widths r [66Stl] are shown underneath.

Schober/Dederichs

11

1.2 PhononenzusCinde: Al

[Lit. S. 180

Table 2. Al. Measured phonon frequencies v at 80 K and 300 K (unpublished data of [66Stl]. T=80K c

T=3OOK v [THz]

c

T=80K v [THz]

6.

T=3OOK v [THz]

L-WI L 0.175 0.200 0.225 0.250 0.275 0.300 0.401 0.448

2.85 (3) 3.20 (3) 3.60 (3) 4.04 (3) 4.38 (3) 4.74 (3) 5.98 (3) 6.51 (3)

0.504 0.600 0.700 0.750 0.800 0.850 0.900 0.950 1.000

7.08 (3) 7.93 (3) 8.53 (3) 8.85 (8) 9.04 (5) 9.37 (10) 9.53 (5) 9.68 (6) 9.69 (5)

0.200

1.75 (2) 2.58 (2) 3.29 (2) 4.06 (2) 4.62 (2) 5.05 (2) 5.43 (2) 5.76 (3) 5.79 (3)

3.12 (3)

0.300 0.400

4.63 (3) 5.87 (3)

0.500 0.600 0.700

6.89 (3) 7.72 (3) 8.34 (5)

0.800

8.91 (6)

0.900

9.42 (5)

1.000

9.64 (5)

0.194 0.294 0.400 0.500 0.600 0.700 0.800 0.900 1.000

1.64 2.47 3.21 3.95 4.50 4.92 5.32 5.63 5.65

COCCI L 0.071 0.108 0.141

1.62 (3) 2.47 (3) 3.25 (3)

0.175 0.216 0.251 0.283 0.322 0.354 0.386 0.424 0.457 0.494 0.530 0.562 0.601 0.634 0.672 0.706 0.745 0.780

4.03 (3) 4.84 (3) 5.47 (3) 6.02 (3) 6.56 (3) 6.97 (3) 7.35 (3) 7.72 (3) 8.09 (3) 8.39 (3) 8.56 (3) 8.67 (8) 8.63 (5) 8.53 (5) 8.20 (5) 7.94 (3) 7.64 (3) 7.27 (3)

12

v [THz]

rorr1 L

COO;1 T 0.202 0.301 0.398 0.497 0.599 0.699 0.797 0.898 0.988

r

0.153

3.47 (3)

0.181 0.218 0.218 0.279 0.322 0.355

4.09 (5) 4.82 (3) 5.36 (5) 5.83 (5) 6.53 (6) 6.91 (3)

0.432 0.457 0.491 0.532 0.559 0.598 0.636

7.70 (6) 7.93 (8) 8.21 (5) 8.34 (6) 8.40 (5) 8.40 (8) 8.21 (8)

0.709 0.739 0.770

7.73 (5) 7.48 (5) 7.19 (5)

0.815 0.861 0.892 0.928 0.955 0.998

6.96 (3) 6.54 (3) 6.29 (3) 6.02 (3) 5.92 (3) 5.78 (3)

0.810 0.863

6.86 (3) 6.41 (3)

0.922 0.959 1.000

5.94 (3) 5.75 (3) 5.63 (5)

CWI Tl 0.144 0.180 0.215 0.251 0.284 0.316 0.354 0.386 0.424 0.457 0.496 0.528 0.564 0.598 0.632 0.672 0.710 0.745 0.780 0.823 0.853 0.887 0.921 0.958 1.000

1.75 (2) 2.23 (2) 2.70 (2) 3.23 (2) 3.64 (2) 4.07 (2) 4.55 (2) 4.95 (2) 5.46 (2) 5.87 (3) 6.37 (3) 6.72 (3) 7.10 (3) 7.42 (5) 7.77 (3) 8.05 (3) 8.32 (3) 8.61 (3) 8.88 (3) 9.17 (5) 9.28 (8) 9.47 (3) 9.57 (3) 9.66 (3) 9.69 (5)

0.141

1.67 (2)

0.211

2.56 (2)

0.283

3.55 (3)

0.354

4.49 (3)

0.424

5.40 (3)

0.495

6.21 (3)

0.566

6.91 (3)

0.636

7.58 (3)

0.707

8.13 (3)

0.778

8.71 (5)

0.849

9.18 (3)

0.919

9.52 (5)

1.000

9.64 (3)

crrr1 L 0.101 0.151 0.200 0.252 0.275 0.301 0.325 0.350 0.400 0.449 0.498

Schober/Dederichs

2.96 (3) 4.36 (3) 5.55 (3) 6.72 (3) 7.26 (3) 7.80 (3) 8.28 (3) 8.66 (3) 9.20 (3) 9.57 (3) 9.69 (10)

0.100 0.150 0.200 0.250

2.83 (3) 4.19 (3) 5.46 (5) 6.62 (5)

0.298

7.69 (5)

0.351 0.415 0.450 0.512

8.53 (3) 9.25 9.42 (6) 9.53 (6)

(continued)

Ref. p. 1801

1.2 Phonon states: Al

Table 2. Al. (Continued) T=80K

T= 300 K v [THz]

c

v [THz]

l

T=300 K

T=80K v [THz]

I

CO551 Tz

Ci’iCl T 0.151 0.204 Cl.250 0.300 0.353 0.400 0.425 0.450 0.500

2.15 (2) 2.83 (3) 3.37 (2) 3.76 (2) 4.04 (2) 4.20 (3) 4.19 (3) 4.19 (3) 4.19 (3)

v [THz]

5

0.149 0.200 0.250 0.300 0.350 0.400

2.04 (2) 2.64 (2) 3.20 (2) 3.61 (2) 3.90 (2) 4.06 (3)

0.450 0.500

4.04 (3) 4.04 (3)

0.212 0.283 0.354 0.423 0.496 0.566 0.633 0.707 0.779 0.847 0.915 0.991

0.141 0.212 0.284 0.357 0.424 0.492 0.566 0.636 0.707 0.777 0.846 0.916 1.000

2.42 (3) 3.18 (2) 3.74 (2) 4.20 (2) 4.65 (2) 4.98 (2) 5.28 (2) 5.54 (2) 5.71 (3) 5.83 (3) 5.84 (3) 5.76 (3)

1.45 (2) 2.26 (2) 3.02 (2) 3.66 (2) 4.06 (2) 4.47 (2) 4.87 (3) 5.16 (3) 5.40 (2) 5.62 (3) 5.70 (3) 5.75 (3) 5.71 (3)

Anomalies in the dispersion curves Detailed measurements have been made in order to locate anomalies in the dispersion [70Wel, 65Stl]. At least 11 Kohn anomalies were found, see e.g. Fig. 2 Al, which allow to determine Fermi-surface diameters with uncertainties of about 1 %. Brovman and Kagan [74Brl] made a detailed theoretical study which allows to identify most of the anomalies. In addition to Kohn anomalies at least two anomalies were shown to arise from third order effects in the ion-electron coupling. They also observed in places a partial cancelling of Kohn anomalies by these higher order effects. -A

0.5

0.4

A-

0.3

-c

0.2

0.1

0

0.2

0.4

-C

0.6

e-

0.8

1.0

0.8

0.6

0.4

0.2

0

-s

Fig. 2. Al. Anomalies in the dispersion curves. Experimental points: [70Wel], broken line: second order pseudopotential calculation, full line: third order pseudopotential calculation. Kohn anomalies are denoted by K whereas the numbers refer to singularities in the three point coupling [74Brl]. Schober/Dederichs

13

1.2 Phononenzustbde: Al

[Lit. S. 180

Anharmonicity

The anharmonic linewidth and frequency shifts measuredin Al have been investigated using effective interionic potentials and perturbation theory [70Kol, 71Gil,69Hol, 67Bjl-J.The overall agreementbetweenexperiment and theory varies from 5 % to 25 %. Born-von

Karman constants

Although the first neighbour coupling is generally believed to be dominating a good lit requires force constants to at least the seventhneighhours. Theoretical pseudopotential calculations indicate a much longer effectiverange. Table 3. Al. Born-von Karman force constants, @c. Ref.

74Co2")

65Gi2 b,

65Gi2 b,

Ref.

74Co2")

65Gi2 b,

65Gi2 b,

T

80K

80K

300K

T

80K

80K

300K

m

110

200 211

ij

6~: [Nm-‘1

xx zz XY

10.4578 -2.6322 10.3657

10.107 -1.337

xx YY

2.4314 -0.1351

2.452 -0.529

xx YY Y=

220

11.444

0.0986 -0.2366 -0.2862 -0.1819

X2

2.494 -0.515

310

xx

XY

222

xx XY

- 0.167

0.321 -0.050

fl

YY zz

-0.439

0.271

0.1854 0.3753

ij

11.424

-0.625 -0.182 -0.148 -0.296

0.1363

xx zz XY

9.808 -1.616

m

321

YY zz

0.027 0.465 -0.438

YZ xz

‘) Fit to elastic constants and phonons including off symmetry directions. b, Axially symmetric fit to elastic constants and phonons in symmetry directions.

0.461 0.227 0.198 0.088

0.518 0.141 0.094 0.141

-0.1412 0.1990

0.142

-0.109

0.076 -0.061

- 0.0747 0.0397

XY

400

-0.3003 0.1842 0.2603 -0.3239

0.1828 -0.2207 -0.0173 -0.0214

xx

-0.091 -0.182

[N m-‘1

-0.0681 -0.0202

xx YY

- 0.064 - 0.094 -0.111 0.012 0.018 0.036

- 0.049 -0.065 - 0.074 0.006 0.009 0.019

- 0.534 -0.116

-0.756 -0.063

2. Frequency spectra and related properties Frequency spectra for AI at 80 K were calculated by direct interpolation of the measuredphonons [67St3] and from the Born-von Karman parameters[74Co2], Fig. 3 Al. Both calculations give very similar distributions. The spectra derived from the Born-von Karman constants fitted to symmetry-direction phonons [65Gi2] differs only slightly, Fig. 4 Al. 0.4

1Hz'

Al

t 0.3

0

2

6

6

8

I

1Hz II

0

2

k

6

8 1Hz 10

Y-

Fig. 3. Al. Phonon spectrum at 80 K calculated from an eighth neighbour Born-von Karman model fitted to elastic constants and symmetry and off symmetry phonons (Table 3 Al [74Co2]). 14

Fig. 4. Al. Phonon spectrum “,,Gnd 300 K calculated from eighth neighbour Born-von Karman models fitted to elastic constants and symmetry phonons (Table 3 Al [65(X2]).

Schober/Dederichs

1.2 Phonon states: Al

Ref. p. 1801

Table 4. Al. Phonon spectra at 80 K and 300 K derived from Born-von Karman force constants. T

80K

80 K

300 K

T

80K

80K

300 K

Ref.

74Co2

65Gil

65Gil

Ref.

74Co2

65Gil

65Gil

-

v [THz]

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90

0.000 0.000 0.000 0.001 0.001 0.002 0.002 0.003 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.013 0.014 0.016 0.018 0.019 0.021 0.023 0.025 0.028 0.030 0.033 0.036 0.039 0.043 0.046 0.051 0.055 0.061 0.067 0.074 0.082 0.092 0.105 0.123 0.156 0.202 0.202 0.203 0.203 0.204 0.206 0.207

g(v) D-Hz- ‘1

v [THz]

0.000 0.000 0.000 0.001 0.001

5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 1.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20 8.30 8.40 8.50 8.60 8.70 8.80 8.90 9.00 9.10 9.20 9.30 9.40 9.50 9.60 9.70 9.80

0.002 0.002 0.003 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.014 0.015 0.016 0.018 0.020 0.022 0.024 0.026 0.028 0.031 0.033 0.037 0.040 0.043 0.048 0.052 0.058 0.064 0.071 0.080 0.090 0.104 0.125 0.167 0.238 0.230 0.225 0.221 0.218 0.215 0.212

0.000 0.000 0.000 0.001 0.001 0.002 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.013 0.014 0.016 0.018 0.019 0.021 0.023 0.026 0.028 0.031 0.034 0.037 0.040 0.044 0.048 0.053 0.058 0.064 0.071 0.079 0.088 0.100 0.115 0.136 0.170 0.237 0.236 0.233 0.231 0.228 0.225 0.222 0.219

Schober/Dederichs

g(v) C-J=- ‘1 0.208 0.210 0.213 0.216 0.220 0.225 0.233 0.245 0.290 0.206 0.183 0.166 0.153 0.142 0.131 0.112 0.112 0.113 0.113 0.113 0.112 0.111 0.111 0.109 0.107 0.104 0.100 0.095 0.088 0.078 0.065 0.073 0.097 0.126 0.163 0.219 0.310 0.313 0.331 0.214 0.173 0.145 0.122 0.101 0.082 0.063 0.037 0.014 0.0

0.209 0.207 0.205 0.204 0.204 0.205 0.208 0.212 0.227 0.204 0.178 0.164 0.153 0.145 0.137 0.131 0.125 0.120 0.115 0.111 0.106 0.101 0.096 0.095 0.095 0.094 0.093 0.092 0.091 0.089 0.086 0.083 0.078 0.078 0.106 0.146 0.209 0.271 0.305 0.361 0.238 0.189 0.156 0.129 0.103 0.076 0.045 0.032 0.017

0.217 0.214 0.213 0.212 0.212 0.214 0.219 0.229 0.201 0.172 0.156 0.144 0.135 0.127 0.120 0.113 0.107 0.097 0.094 0.093 0.092 0.091 0.090 0.089 0.087 0.086 0.084 0.082 0.079 0.075 0.071 0.071 0.094 0.127 0.170 0.279 0.276 0.298 0.332 0.218 0.176 0.147 0.122 0.099 0.075 0.045 0.031 0.012 0.0

15

1.2 Phononenzustkde:

Al

[Lit. S. 180

9.2 1Hz I

8.8

;

1 fg-, I= EOK165Gi21 1

-10

0

10

20

30 370 0

Fig. 5. Al. Debye kzequencies, v., calculated from the spectra of Figs. 3. Al and 4. AI.

0

100

200

300

300 400 K 500 TFig. 6. Al. Debye temperatures 8, calculated from the spectra of Fig. 6 Al (full lines) compared to specific heat data [65Gi2].

400 K 500

100

200

Fig. 7. Al. Debye-Wailer exponent 2 W divided by the recoil frequency of the free ion, va, calculated from the spectra of Figs. 3 Al and 4 Al.

T-

3. Theoretical models Apart from the small anomalies the dispersion of the phonons in Al is very simple and can therefore be easily reproduced by phenomenological models.Al belongs to the simple (s-p bonded) metals and has been a favourite test substancefor pseudopotential calculations. In their lessambitious form, such calculations are able to reproduce the dispersion with one or two fit parameters. More ambitious calculations are hampered by the unsuficient knowledge of the exchange and correlation interactions, Fig. 8 AI. Using plausible functional forms for these interactions one can describe the phonons, electronic structure etc. quite well from first principles, e.g. [75Hal]. The positions of singularities predicted by theory correlate well with the experimental results, Fig. 2 AI.

-A

cK I

7: 1Hz 6

0 0

0.2

0.4

0.6

0.8

X

Al

1.0 0.8 0.6 0.4 0.2

Fig. 8. Al. Phonon dispersion curves calculated by pseudopotential theory using different screening functions. Experimental frequencies of [66Stl] (o longitudinal, l transverse branches) [75Hal]. 16

Schober/Dederichs

Ref. p. 1801

1.2 Phonon states: As

Born-von Karman and equivalent models: see Table 3 Al, further references: [63Sql, 64Sq1, 73Shl]. Breathing shell model: [71Hal]. Models comprising short ranged forces plus a simple electronic contribution: [69Krl], further references: [66Shl, 71Gi1, 77Di1, 77Kh1, 78Kul]. Local model pseudopotential calculations: [71Ha2, 72Pr3, 73Ka2], further references:[66Anl, 66Sc1,68Sc1, 69Jo1, 69Wa1, 70Gu1, 70Wa1, 71Scl]. Nonlocal pseudopotential calculations: [7OCol, 74Ha2, 74Ra1, 75Ha1, 75Rel].

As

Arsenic

Lattice: a-phase, rhombohedral (A7) a = 559pm = 5.59A, LX=84” 36’. BZ: seep. 453.

1. Phonon dispersion Table 1. As. Measurements. Method

Fig.

Ref.

1 As

Reichardt and Rieder [76Rel]

TKI neutron diffraction PAS)

297

Further measurement: [76Lel]. The dispersion curves of a-arsenic resemble those of Bi and Sb though there are several major differences. The group theory for theseelementswas reported in [67Sml]. The optical frequenciesat the r point are in good agreementwith values measuredby Raman scattering [75La2]. A ten neighbour, 27 parameter Born-von Karman model gives a good overall description of the dispersion but does not reproduce the fine structure of the optical modes.

I-

X

K 10 n’eighbours

0

0.2 0.4 0.6 0.8 1.00 it-

0.2 0.4 0.6 0.8 1 0 t-

II

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0 P-c

Fig. 1. As. Measuredphonondispersionin a-arsenicat 297K. The solid line representsa tenth neighbourBorn-vonKarman tit with 27 parameters[76Rel]. Schoher/Dederichs

17

Ref. p. 1801

1.2 Phonon states: As

Born-von Karman and equivalent models: see Table 3 Al, further references: [63Sql, 64Sq1, 73Shl]. Breathing shell model: [71Hal]. Models comprising short ranged forces plus a simple electronic contribution: [69Krl], further references: [66Shl, 71Gi1, 77Di1, 77Kh1, 78Kul]. Local model pseudopotential calculations: [71Ha2, 72Pr3, 73Ka2], further references:[66Anl, 66Sc1,68Sc1, 69Jo1, 69Wa1, 70Gu1, 70Wa1, 71Scl]. Nonlocal pseudopotential calculations: [7OCol, 74Ha2, 74Ra1, 75Ha1, 75Rel].

As

Arsenic

Lattice: a-phase, rhombohedral (A7) a = 559pm = 5.59A, LX=84” 36’. BZ: seep. 453.

1. Phonon dispersion Table 1. As. Measurements. Method

Fig.

Ref.

1 As

Reichardt and Rieder [76Rel]

TKI neutron diffraction PAS)

297

Further measurement: [76Lel]. The dispersion curves of a-arsenic resemble those of Bi and Sb though there are several major differences. The group theory for theseelementswas reported in [67Sml]. The optical frequenciesat the r point are in good agreementwith values measuredby Raman scattering [75La2]. A ten neighbour, 27 parameter Born-von Karman model gives a good overall description of the dispersion but does not reproduce the fine structure of the optical modes.

I-

X

K 10 n’eighbours

0

0.2 0.4 0.6 0.8 1.00 it-

0.2 0.4 0.6 0.8 1 0 t-

II

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0 P-c

Fig. 1. As. Measuredphonondispersionin a-arsenicat 297K. The solid line representsa tenth neighbourBorn-vonKarman tit with 27 parameters[76Rel]. Schoher/Dederichs

17

1.2 Phononenzusttinde: Au

[Lit. S. 180

2. Frequency spectrum and related properties 400

a5 THz-'

K I 320

04 -colc.176Re11

0.3 I -s -G 02

62210 160 0

100

150

200

250

300 K 350

Fig 3. As. Debye temperature 9, of a-arsenic calculated from the experimental spectrum of Fig. 2 As compared to experimental values [74Sal].

0.1

0

50

12

3

L Y---r

5

6

7 1Hz 8 24

Fig. 2. As. Phonon spectrum in a-arsenic at 297 K. (Histogram: Born-von Karman model of Fig. 1 As; dashed curve: inelastic coherent scattering result on polycrystalline As [74Sal], A: experimental resolution) [76Rel].

I 16 d8

0

50

100

150

200

250

300 K 350

Fig. 4. As. Debye-Waller coefficient 1.of a-arseniccalculated from the experimental spectrum of Fig. 2 As [74Sal].

3. Theoretical models Born-von Karman model: seeFig. 1 As. The values of the constants have not been published.

Au

Gold

Lattice: kc, n = 407 pm = 4.07 A. BZ: see p. 449.

1. Phonon dispersion Table 1. Au. Measurements. Method

Fig.

Ref.

1 Au

Lynn et al. C73LYll

TKI Neutron diffraction F-AS)

296

The frequenciesof gold lie up to 30 % higher than those scaled from the other two noble metals, copper and silver by the ratios of (Ma 2)1’2. An analysis of the data shows that contrary to Cu and Ag a general tensor force is required for the first neighbour interaction whereas for the neighbours, beyond the first, either general tensors or axially symmetric forces give an excellent fit to the data. The [([O] T, branch shows a positive dispersion. Kohn anomalies have not been reported. 18

Schober/Dederichs

1.2 Phononenzusttinde: Au

[Lit. S. 180

2. Frequency spectrum and related properties 400

a5 THz-'

K I 320

04 -colc.176Re11

0.3 I -s -G 02

62210 160 0

100

150

200

250

300 K 350

Fig 3. As. Debye temperature 9, of a-arsenic calculated from the experimental spectrum of Fig. 2 As compared to experimental values [74Sal].

0.1

0

50

12

3

L Y---r

5

6

7 1Hz 8 24

Fig. 2. As. Phonon spectrum in a-arsenic at 297 K. (Histogram: Born-von Karman model of Fig. 1 As; dashed curve: inelastic coherent scattering result on polycrystalline As [74Sal], A: experimental resolution) [76Rel].

I 16 d8

0

50

100

150

200

250

300 K 350

Fig. 4. As. Debye-Waller coefficient 1.of a-arseniccalculated from the experimental spectrum of Fig. 2 As [74Sal].

3. Theoretical models Born-von Karman model: seeFig. 1 As. The values of the constants have not been published.

Au

Gold

Lattice: kc, n = 407 pm = 4.07 A. BZ: see p. 449.

1. Phonon dispersion Table 1. Au. Measurements. Method

Fig.

Ref.

1 Au

Lynn et al. C73LYll

TKI Neutron diffraction F-AS)

296

The frequenciesof gold lie up to 30 % higher than those scaled from the other two noble metals, copper and silver by the ratios of (Ma 2)1’2. An analysis of the data shows that contrary to Cu and Ag a general tensor force is required for the first neighbour interaction whereas for the neighbours, beyond the first, either general tensors or axially symmetric forces give an excellent fit to the data. The [([O] T, branch shows a positive dispersion. Kohn anomalies have not been reported. 18

Schober/Dederichs

Phonon states: Au

Ref. p. 1801 Z-

A-

-E w

X

-l-

5,

X

A-

K

r I

L I

IO511 I I I I 3 0.2 0.4 0.6 0.8 0.8 0.6 0.4 0.2 CFig. 1. Au. Phonon dispersion relations in the principal symmetry directions according to [73Lyl]. The solid curves represent both the fourth neighbour general force model (Ml) and the fifth neighbour axially symmetric model (M2) of Table 3 Au. The dotted line in the C direction is corresponding to the velocity of sound appropriate to the [O<[] Tr branch.

For Table 2 Au, see next page.

Born-von Karman models Table 3. Au. Born-von Karman force constants, @‘;.

T

295 K

295 K

Ref.

73Ly1, Ml

73Ly1, M2

m

ij

~3; [Nm-‘1

110

xx zz XY

16.43 (9) - 6.54 (10) 19.93 (14)

16.61 (10) -6.65 (15) 19.93 (16)

200

xx YY

4.04 (17) -1.27 (11)

3.95 (22) -1.13 (12)

211

xx YY YZ xz

220

xx zz XY

310

xx YY zz XY

0.80 0.39 0.16 0.54

(5) (5) (6) (4)

-0.75 (5) -0.14 (9) -0.36 (11)

1.00 (5) 0.28 (4) 0.24 0.48 -0.57 -0.21 -0.36

(5) (7)

-0.17 (6) -0.02 (3) 0.00

-0.06

(M 1 general model, M 2 general 1st neighbour force

constants and axially symmetric 2nd to 5th neighbour force constants) Schober /Dederichs

19

[Lit. S. 180

1.2 Phononenzust6nde: Au Table 2. Au. Phonon frequencies at 296 K measured by triple axis neutron spectrometry [73Lyl]. r,

v D-Hz1

4

0.50(7) 0.75 (6)

1.00

4.61 (5)

1.00 (5) 1.50(5) 1.65 (4) 2.25 (4) 2.32 (4) 2.85 (6) 3.30(8) 3.70 (8) 3.88 (6) 4.17 (6) 4.40(6) 4.58 (6)

r

0.10 0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90

1.00

0.41 (4) 0.58 (3) 0.75 (3) 0.91 (3) 1.09 (3) 1.46 (3) 1.84(3) 2.16 (3) 2.41 (4) 2.61 (5) 2.73 (5) 2.75 (4)

0.058 0.074 0.109 0.12 0.17 0.201 0.28 0.35 0.45 0.50 0.60 0.66 0.70 0.75 0.80 0.90

1.00

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.90 0.95

1.00

0.31(2)

0.10

0.56 (3)

0.48 (2) 0.63 (4) 0.83 (3) 1.02 (3) 1.23 (3) 1.44 (3)

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

1.10 (3)

1.00

4.61 (5)

1.64(5) 1.79(4) 1.96 (5) 2.08 (5) 2.18 (5) 2.34(5) 2.43 (5) 2.55 (8) 2.70 (6) 2.74 (6) 2.75 (4)

0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.51 (3) 0.76(3)

1.00 (2) 1.23 (2)

0.035 0.06 0.12 0.176 0.23 0.26 0.30 0.35 0.40 0.45 0.50

0.00 0.10 0.20 0.30

1.44 (2)

0.40

1.63 (3) 1.74(3) 1.85 (4) 1.86(4)

0.50 0.60 0.70 0.80 0.90

1.00

4.61 (5) 4.57 (5) 4.46 (5) 4.23 (5) 3.92 (5) 3.63 (5) 3.30(5) 3.06 (6) 2.90 (7) 2.80 (7) 2.75 (4)

0.60(8) 1.00 (8) 1.80 (9) 2.40 (10) 3.00 (15) 3.40(15) 3.75 (6) 4.14 (6) 4.45 (5) 4.64 (5) 4.70 (4)

CWll A

COUIfr

Ciiil T 0.10 0.15

1.63 (4) 2.15 (5) 2.71 (5) 3.25 (5) 3.77 (7) 4.26(7) 4.53 (7)

0.75 (8) l.OO(7) 1.25 (8) 1.50 (10) 2.00 (8) 2.40 (6) 3.00(5) 3.50(5) 3.85 (10) 3.96 (6) 3.84 (5) 3.70 (5) 3.55 (6) 3.34 (5) 3.15 (5) 2.83 (5) 2.75 (4)

Ct;Kl L

Wiil T, “1

CO;;1-T ‘1

v [THz]

coir1 L

COOiT

COKIL D.052 Ml92 0.116 0.179 0.206 0.293 0.30 0.40 0.47 0.552 0.60 0.70 0.80 0.90

v [THz]

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

1.00

2.75 (4) 2.74(5) 2.73 (5) 2.70 (4) 2.63 (3) 2.63 (3) 2.64 (3) 2.69 (4) 2.73 (5) 2.74(5) 2.75 (4)

‘) The polarization vectors for the [Oi[] Tr and T2 branches are parallel to [O[iJ and [[OO], respectively. 20

Schober/Dederichs

Ref. p. 1801

1.2 Phonon states: Au

2. Frequency spectrum and related properties 0.6 THz-' 0.5 0.4 t T. 0.3 < 0.2 0.1 4 THz 5 3 YFig. 2. Au. Frequency distribution calculated from the fourth neighbour general force constant model (Ml) of Table 3 Au. 0

1

2

Table 4. Au. Phonon frequency spectrum at 295 K calculated from the 4th neighbour Born-von Karman model [73Lyl] Ml of Table 3 Au. v ~Hz]

g(v) [THz-I]

v [THz]

g(v) [THz- ‘1

v [THz]

g(v) [THz- ‘1

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60

0.000 0.001 0.002 0.003 0.005 0.007 0.009 0.012 0.015 0.019 0.022 0.026 0.030 0.035 0.040 0.045 0.051 0.056 0.062 0.069 0.076 0.083 0.091 0.100 0.109 0.119 0.129 0.141 0.153 0.167 0.182 0.200

1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20

0.220 0.243 0.270 0.305 0.357 0.446 0.446 0.447 0.448 0.448 0.448 0.448 0.447 0.446 0.444 0.443 0.441 0.439 0.436 0.433 0.390 0.329 0.281 0.248 0.242 0.237 0.231 0.224 0.218 0.211 0.203 0.195

3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00 4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70

0.187 0.177 0.167 0.155 0.142 0.127 0.109 0.086 0.088 0.122 0.158 0.196 0.237 0.288 0.362 0.454 0.475 0.496 0.517 0.537 0.407 0.323 0.267 0.224 0.187 0.152 0.117 0.078 0.027 0.0

\ Schober /Dederichs

21

1.2 PhononenzustZnde: Ba

-10

0

10 n-

20

[Lit. S. 180

165

30

160

Fig. 3. Au. Debye cutoff frequencies, Y,, calculated for the spectrum of Fig. 2 Au.

155 0

20

40

60

80

K

100

Fig. 4. Au. Debye temperature Q,(T) calculated for the spectrum of Fig. 2 Au [73Lyl].

5 .10'2 S 4

3

I

9

52 1

0

100

200

300

400

K 500

Fig. 5. Au. Debye-Wailer exponent 2 Wdivided by the recoil frequency of the free atom, vs, calculated from the spectrum of Fig. 2 Au.

I-

3. Theoretical models Due to their simple structure the dispersion curves are normally described adequately by phenomenological models. A truly microscopic description however is still outstanding due to the difficulties stemming from the electronic s-d hybridization Born-von Karman tits see Table 2 Au. Short ranged forces plus a simple electronic contribution [75Cll, 76Ma1, 77Kh1, and 77Kh2], further references:[70Bel, 71Sh1, and 71Si1, 71Sw1, 74Go2, 75Ti1, 75Be1, 76Ca1, 77Pi1, 78Ku1, 67Gul]. Shell model : [78Si4]. Local model pseudopotential calculations [76Nal], further references[76Kul, 75Na1, 72Pr1, 78Pr1, 78Sh1, 77Upl]. Models incorporating electronic d-band terms [73Sil, 75La1, 74Khl], further references: [72Mol, 78Ku2, 75Si3].

Ba Barium Lattice: bee, a=501 pm=5.01 A. BZ: see p. 448. The phonon dispersion curves have not been measured in barium. Model calculations suggesta dispersion similar to that of the alkaline metals, seeFig. 1 Ba. Model pseudopotential calculations, [73Prl, 73Pr2, 67An1, 72Mo2, 71Gu3]. 22

!Schober/Dedericbs

1.2 PhononenzustZnde: Ba

-10

0

10 n-

20

[Lit. S. 180

165

30

160

Fig. 3. Au. Debye cutoff frequencies, Y,, calculated for the spectrum of Fig. 2 Au.

155 0

20

40

60

80

K

100

Fig. 4. Au. Debye temperature Q,(T) calculated for the spectrum of Fig. 2 Au [73Lyl].

5 .10'2 S 4

3

I

9

52 1

0

100

200

300

400

K 500

Fig. 5. Au. Debye-Wailer exponent 2 Wdivided by the recoil frequency of the free atom, vs, calculated from the spectrum of Fig. 2 Au.

I-

3. Theoretical models Due to their simple structure the dispersion curves are normally described adequately by phenomenological models. A truly microscopic description however is still outstanding due to the difficulties stemming from the electronic s-d hybridization Born-von Karman tits see Table 2 Au. Short ranged forces plus a simple electronic contribution [75Cll, 76Ma1, 77Kh1, and 77Kh2], further references:[70Bel, 71Sh1, and 71Si1, 71Sw1, 74Go2, 75Ti1, 75Be1, 76Ca1, 77Pi1, 78Ku1, 67Gul]. Shell model : [78Si4]. Local model pseudopotential calculations [76Nal], further references[76Kul, 75Na1, 72Pr1, 78Pr1, 78Sh1, 77Upl]. Models incorporating electronic d-band terms [73Sil, 75La1, 74Khl], further references: [72Mol, 78Ku2, 75Si3].

Ba Barium Lattice: bee, a=501 pm=5.01 A. BZ: see p. 448. The phonon dispersion curves have not been measured in barium. Model calculations suggesta dispersion similar to that of the alkaline metals, seeFig. 1 Ba. Model pseudopotential calculations, [73Prl, 73Pr2, 67An1, 72Mo2, 71Gu3]. 22

!Schober/Dedericbs

Ref. p. 1801

1.2 Phonon states: Be A-

0

-F

-A

c-

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0 t---c f-

Fig. 1. Ba. Theoretical dispersioncurve calculatedfrom a two parameterpseudopotential[73Pr3].

Be Beryllium Lattice: hcp, a=229 pm=2.29 A, c= 358pm= 3.58A. BZ: seep. 450.

1. Phonon dispersion Table 1. Be. Measurements. Method

Table 2. Be. Measured phonon frequencies v at 80 K and frequency shift from 80 K to 300K. The labelling corresponds to Fig. 1 Be, [76Stl].

Fig.

Ref.

1 Be

Stedmanet al. [76Stl] Schmunk et al.

TKI neutron diffraction (TAS) neutron diffraction (TW

80 300

[62Sc1,66Scl]

Further references: [71Thl, 73Ro1, 74Da2].

For Fig. 1 Be, seenext page.

Phonon

~(80

l-3+

20.28 (4) 13.73 (3) 17.79(4) 17.45 (5) 16.73 (3) 12.41 (3) 16.90 (3) 11.86 (3) 18.80(5) 15.97 (5) 14.65 (5) 14.59 (3) 14.90 (3) 10.72 (4) 19.04(4) 15.07 (3) 14.23 (7) 19.10 (4) 17.30 (4) 12.30(3)

G+ W MT W W M; W

Kl K3 KS K.5 Al A3 L: Li L HI H: H;

K) [THz]

~(80K)-~(300 K) [THz] 0.08 (4) 0.06 (5) 0.12 (5)

0.06 (4) 0.07 (7) 0.06 (4) 0.13 (6)

The different measurementsagreewell. Contrary to the other group II hcp metal Mg three body interactions are important for the dispersion of beryllium. This can be seen from the violation of the sum rule for the frequencies at the K symmetry point: 2v2(K,)=v2(K,)+v2(K3) which applies for pair forces,but does not hold in Be, seeTable 2 Be. Group velocities corresponding to the lowest acoustic frequencies agreewithin 3 % with the measuredsound velocities. Anomalies have not been identified unambiguously. The temperature shift from 80 K to 300 K is small. Schober/Dedericbs

23

Ref. p. 1801

1.2 Phonon states: Be A-

0

-F

-A

c-

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0 t---c f-

Fig. 1. Ba. Theoretical dispersioncurve calculatedfrom a two parameterpseudopotential[73Pr3].

Be Beryllium Lattice: hcp, a=229 pm=2.29 A, c= 358pm= 3.58A. BZ: seep. 450.

1. Phonon dispersion Table 1. Be. Measurements. Method

Table 2. Be. Measured phonon frequencies v at 80 K and frequency shift from 80 K to 300K. The labelling corresponds to Fig. 1 Be, [76Stl].

Fig.

Ref.

1 Be

Stedmanet al. [76Stl] Schmunk et al.

TKI neutron diffraction (TAS) neutron diffraction (TW

80 300

[62Sc1,66Scl]

Further references: [71Thl, 73Ro1, 74Da2].

For Fig. 1 Be, seenext page.

Phonon

~(80

l-3+

20.28 (4) 13.73 (3) 17.79(4) 17.45 (5) 16.73 (3) 12.41 (3) 16.90 (3) 11.86 (3) 18.80(5) 15.97 (5) 14.65 (5) 14.59 (3) 14.90 (3) 10.72 (4) 19.04(4) 15.07 (3) 14.23 (7) 19.10 (4) 17.30 (4) 12.30(3)

G+ W MT W W M; W

Kl K3 KS K.5 Al A3 L: Li L HI H: H;

K) [THz]

~(80K)-~(300 K) [THz] 0.08 (4) 0.06 (5) 0.12 (5)

0.06 (4) 0.07 (7) 0.06 (4) 0.13 (6)

The different measurementsagreewell. Contrary to the other group II hcp metal Mg three body interactions are important for the dispersion of beryllium. This can be seen from the violation of the sum rule for the frequencies at the K symmetry point: 2v2(K,)=v2(K,)+v2(K3) which applies for pair forces,but does not hold in Be, seeTable 2 Be. Group velocities corresponding to the lowest acoustic frequencies agreewithin 3 % with the measuredsound velocities. Anomalies have not been identified unambiguously. The temperature shift from 80 K to 300 K is small. Schober/Dedericbs

23

[Lit. S. 180

1.2 Phononenzustkinde: Be A-

-E

It001

I

U-

PK

l24 THZ 22

L

HM

b roog1 6’

20

A2

18

‘a

16 I 14 P '2 10 e E 4

i c 0.4

0.3 0.2

0.1

0

-t

s-

S-

0

0.5 0

0.5

0.5 0 f-

I-

t-

-R

Fig. 1 a-c. Be. Measured dispersion curves at 80 K. The lines show connectivity only [76Stl].

2. Frequency spectrum and related properties Fig. 2. Be. Phonon spectrum measured by inelastic neutron scattering for various mean values of the incident neutron energy, curve a: 4.54meV, curve b: 3.359 meV, curve c: 2.48 meV [76Si5].

24

Schober/Dederichs

0

4

8

12 Y-

16

1Hz

Ref. p. 1801

1.2 Phonon states: Bi

3. Theoretical models Comparison of theoretical results with experiment is hampered by the fact that the earlier measurementsdid not include enough information. The later measurements[73Rol, 76Stl] showed the importance of three body interactions. Beryllium is a simple metal where standard pseudopotential perturbation theory should be applicable. The usual second order calculations however, yield only pair forces. A third order calculation was done by Bertoni et al. [75Be3]. The general shapeof the curves is in good agreementwith the experimental data. There is a large difference between the second and third order results by these authors which seemsto indicate a slow convergence of the perturbation series, see Fig. 3 Be. No adequate fit of the latest experimental data has been published so far. T-

T’-

r

T-

$I 28

0.5 i

0 5-

00 -f

0.5

5-

e-

-l

b-

Fig. 3 a, b. Be. Theoretical phonon frequencies in high symmetry directions for modes polarized perpendicularly to the basal plane. a) second order pseudopotential perturbation theory b) third order theory. The experimental points are taken From[66Scl] and [73Rol], [75Be3].

Born-von Karman and equivalent models: [65Del, 66Sc3,71Srl], further references:[69Mel, 71Be1,71Tr2, 78Ku5]. Short ranged forces plus a simple electronic contribution: [73Kul], further references:[62Scl, 70Sh2,71Kul, 73Bo1, 73Ra2, 73Up1, 78Mil]. Pseudopotential calculations: [75Be3,69Brl, 75Ha2], further references:[66Sal, 69Gil,69Kil, 70Pr2,75Mal, 76Dal].

Bi

69Sal,70Kil,

Bismuth

Lattice: rhombohedral (A7), a = 654pm = 6.54A, a = 87“ 34’. BZ: see p. 453. 1. Phonon dispersion

Table 1. Bi. Measurements. Method

Fig.

Ref.

kl neutron diffraction 75 1 Bi (T-49 neutron diffraction 75 (TAS) neutron diffraction 296 (T-W Further measurements: [64Yal]. Schober/Dederichs

Macfarlane [71 Mall Smith [67Sml] Sosnowski et al. [68Sol]

25

Ref. p. 1801

1.2 Phonon states: Bi

3. Theoretical models Comparison of theoretical results with experiment is hampered by the fact that the earlier measurementsdid not include enough information. The later measurements[73Rol, 76Stl] showed the importance of three body interactions. Beryllium is a simple metal where standard pseudopotential perturbation theory should be applicable. The usual second order calculations however, yield only pair forces. A third order calculation was done by Bertoni et al. [75Be3]. The general shapeof the curves is in good agreementwith the experimental data. There is a large difference between the second and third order results by these authors which seemsto indicate a slow convergence of the perturbation series, see Fig. 3 Be. No adequate fit of the latest experimental data has been published so far. T-

T’-

r

T-

$I 28

0.5 i

0 5-

00 -f

0.5

5-

e-

-l

b-

Fig. 3 a, b. Be. Theoretical phonon frequencies in high symmetry directions for modes polarized perpendicularly to the basal plane. a) second order pseudopotential perturbation theory b) third order theory. The experimental points are taken From[66Scl] and [73Rol], [75Be3].

Born-von Karman and equivalent models: [65Del, 66Sc3,71Srl], further references:[69Mel, 71Be1,71Tr2, 78Ku5]. Short ranged forces plus a simple electronic contribution: [73Kul], further references:[62Scl, 70Sh2,71Kul, 73Bo1, 73Ra2, 73Up1, 78Mil]. Pseudopotential calculations: [75Be3,69Brl, 75Ha2], further references:[66Sal, 69Gil,69Kil, 70Pr2,75Mal, 76Dal].

Bi

69Sal,70Kil,

Bismuth

Lattice: rhombohedral (A7), a = 654pm = 6.54A, a = 87“ 34’. BZ: see p. 453. 1. Phonon dispersion

Table 1. Bi. Measurements. Method

Fig.

Ref.

kl neutron diffraction 75 1 Bi (T-49 neutron diffraction 75 (TAS) neutron diffraction 296 (T-W Further measurements: [64Yal]. Schober/Dederichs

Macfarlane [71 Mall Smith [67Sml] Sosnowski et al. [68Sol]

25

1.2 Phononenzust&de:

4.0r

THz a

X

K I [

Bi

Bi

[Lit. S. 180 l-

1

binary

2.5 I 2.0 h 1.5 1.0 0.5

n &2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 -g bK LOX I 1Hz b bisectrix I

0 0.2 0.4 0.6 0.8 1.0 Ir

lY trigonal

1

,

I 2.5 * 2.0 1.5 1.0

0 1.0

0.8

0.6

0.4

0.2

-t

0 0.2 0.4 0.6 0.8 1.0

I-

Fig. la, b. Bi. Measuredphonon dispersiopcurvesat 75K. The initial slopes of the acousticmodeswerecomputedfrom the elasticconstants[6OEcl]. The smoothcurvesshownare only guidesto the eye[71Mal]. The dispersion curvesof bismuth resemblethose of As and Sb.The group theory for theseelementswas reported in [67Sml]. The optical frequenciesat the r point are in good agreementwith the values obtained from Raman

xattering [73Rol, 75La2]. The decreaseof these frequencieswith increasing temperaturesis small v(T=80 K)= v(T= 300K) = 0.05THz for (r5) and 0.09THz for (l-l), respectively. A five neighbour Born-von Karman model gives a good overall description of the dispersion, whereas it is estimated that an accurate fit would need forces to the 25th neighbours (at least 50 parameters).

26

Schober /Dederichs

1.2 Phonon states: Bi

Ref. p.1801 _ -

Table 2. Bi. Measured phonon frequencies. The estimated random error in any particular frequency is 2.. .4 %. 64Yal

Ref. T

75 K

300K

75 K

300K

5

75 K

300 K

75 K

300 K

AI(LA)

AI&O)

&Q-O)

0.99 1.23 1.40 1.54 1.66 1.75 1.80

3.02 3.01 3.06 3.14 3.20 3.25 3.26 3.28 3.26 3.23 3.25

2.99 3.01 2.98 3.07 3.14 3.18 3.26 3.21 3.20 3.17 3.21

v [THz] T+T A,(TA)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.56 0.69 0.83 0.96 1.06 1.12 1.16

A,VA)

A,(W

MW

0.51 0.70 0.83 0.95 1.05 1.10 1.11

2.33 2.26 2.40 2.56 2.12 2.85 2.99 2.99 3.04 3.06 3.04

2.16 2.24 2.40 2.56 2.12 2.83 2.93 2.96 2.98 3.04 3.01

(trigonal) 4&N

0.99 1.22 1.41 1.57 1.66 1.77 1.80 67Sml

Ref. T

75 K v [THz]

I

T-K--+X branch 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

MA)

1.10 1.50 1.67 1.64 1.48 1.27 1.15

-h(O)

&(O)

2.23

3.02

UN

&(A)

UO) 2.23

0.53 0.64

2.85 2.94 2.99 2.93 2.97 2.67 2.66

0.95

(binary)

0.89

0.89 0.80

3.02

0.72 0.84 0.95 0.94

2.86

1.02

2.86

1.03

2.86

[continued on next page)

2. Frequency spectra and related properties Frequency spectra of bismuth at 77 K and 296 K: see Figs. 2 Bi and 3 Bi. 2.5 arb. units 2.0 I

1.5

r; -G) 1.0

4 Fig. 2. Bi. Phonon frequency spectrum at 77 K measured

0.5 0

0.5 1.0

1.5 2.0 Y-

25

3.0

3.5 THz It.5

by coherent inelastic neutron scattering of a polycrystalline sample. The spectrum calculated from a Born-von Karman model of [67Sml] folded with the experimental resolution [74Sal] is also shown.

Schober/Dederichs

27

[Lit. S. 180

1.2 PhononenzustCinde: Bi Table 2. Bi (continued) 67Sml

Ref. T

75 K

i

v [THz] l-+X

branch 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

a,(A)

dA)

1.21 1.45 1.53 1.46 1.34 1.16 1.05 1.03

0.59 0.73 0.83 0.94

1.0

0.95

a,(O)

2.23 2.39 2.67

3.02

2.33

2.99

2.51

2.91 2.94 2.90

2.98

0.64

2.63

2.94

0.75

2.67

2.85

3.02 2.94 3.02

0.78

2.69

0.80

2.66

2.86

2.5 orb. units

2.0

%(TO)

dTN

a,(O)

180 K ---BvK

168Brll

l&O

I

I 1.5

0 100 0

'; 0, 1.0

60

0.5

20

O

5

YFig. 3. Bi. Phonon frequency spectrum at 296 K measured by coherent inelastic neutron scattering of a polycrystalline sample. Two spectra calculated from Born-von Karman models folded with the experimental resolution [74Sal] are also to be seen.

IFig. 4. Bi. Debye temperatures 0, calculated from the experimental spectrum of Fig. 2 Bi compared to experimental values [74Sal].

32 24 I 16 t-t 8

0

20

40

60

80 I-

100

120

140 K 160

Fig. 5. Bi. Debye-Wailer coeflkient 1.of bismuth calculated from the experimental spectra of Figs, 2 Bi and 3 Bi [74Sal].

3. Theoretical models The models reproduce only roughly the overall shape of the dispersion curves. Born-von Karrnan models: [67Sml, 73Cz1, 68Brl]. 28

Schober/Dederichs

1.2 Phonon states: Ca

Ref. p. 1801

Ca

Calcium

Lattice: fee, a=557pm=5.57A.

BZ: seep. 449.

For calcium, no measurements of dispersion curves exist, as it has so far been impossible to grow single crystals Iwing to an fee-bee phase transition at 448 “C!. A crude picture of the phonon spectrum was obtained by Gompf :t al. [72Gol] by coherent inelastic scattering of slow neutrons from polycrystalline samples, Fig. 1 Ca. Unfortunately, the experiment does not give sufficient information to distinguish between the theoretical models. 411models suggest a simple dispersion of the shape typical for fee metals, see e.g. Fig. 2 Ca. Models including short ranged forces and a simple electronic term [73Swl]. Local model pseudopotential calculations: [72Mo2, 73Pr3, 67An1, 71Gu3]. ?Tonlocal model pseudopotential calculations: [76Ta2].

m-b. units

c; 20

0

.

I

.

12

3

L

n

5

6

Fig. 1. Ca. Phonon spectrum measured by coherent inelastic neutron-scattering from polycrystalline samples, circles: experimental points, full line: nearest neighbour axially symmetric Born-von Karman model [72Gol].

SO

150 I-

hr

x

K-001

Ca

100

200

250 K 3

Fig. 3. Ca. Debye temperature O,(T) calculated from the experimental spectrum of Fig. 1. Ca. [72Gol].

-E

Ar 6-

190 0

7 THz 8

L

@CC1

Ki501

1Hz

0

1.0

0.8

0.6

OA

f-

0.2 -c-

0

0.5

Fig. 2. Ca. Dispersion curves calculated by a two parameter pseudopotential model [73Pr3].

Schober/Dederichs

29

1.2 Phononenzustgnde

Cd

: Cd

[Lit.

S. 180

Cadmium

Lattice: hcp, n = 297pm = 2.97A, c = 561pm = 5.61A. BZ: see p. 450. 1. Phonon dispersion Table 1. Cd. Measurements. Method

Fig.

Ref.

1 Cd

Chernysov et al. C79Chl-J

TKI neutron diffraction U-AS)

77

Further measurements[75Chl]. Due to the strong neutron absorption of natural cadmium the neutron diffraction measurementswere done on a crystal of the isotope l*°Cd. The results of an earlier X-ray measurement[72Tol] overestimatethe maximum frequency by about 2THz. The structure of Cd has maximal anisotropy c/a=189 as compared to the ideal ratio of 1.63.Severalabrupt changesofgroup velocity asa function ofwavevector have beenidentified asKohn anomalies. The initial slopesof the accoustic branches agreewell with the measuredelastic constants.

Table 2. Cd. Measured values of the phonon frequency of cadmium at 77 K [79Chl].

r

\I ~Hz]

v [THz]

Ir

l

Branch 1 3-G KS01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25

30

0.208 (1) 0.295 0.368(2) 0.447(1) 0.501(1) 0.546(1) 0.593(2) 0.635(1) 0.670(2) 0.703(2) 0.725(2) 0.749(2) 0.767(2) 0.790(1) 0.804(2) 0.821(3) 0.842(3) 0.855(2) 0.871(3) 0.884(2) 0.896(4) 0.919(2) 0.946(3) 0.973(3)

0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50

v [THz]

I r

v [THz]

Branch C3 [[OO]

1.012(3) 1.041 (3) 1.075 (3) 1.106 (3) 1.147 (3) 1.188 (3) 1.224 (3) 1.258 (3) 1.280 (4) 1.295 (4) 1.307 (4) 1.308 (3) 1.320 (4) 1.363 (3) 1.441 (2) 1.524 (2) 1.583 (2) 1.632 (2) 1.669 (3) 1.699 (2) 1.728 (2) 1.749 (4)

0.0250 0.0375 0.0500 0.0675 0.0750

0.163 (1)

0.0875 0.1000

0.444 (1) 0.488(1) *)

0.1125 0.1250 0.1375 0.1500 0.1625 0.1750 0.1875 0.2000 0.2125 0.2250 0.2375 0.2500 0.2625

0.499(1) *) 0.539(1) 0.584(2) 0.620(2) 0.651(1) 0.679(1) 0.704(1) 0.729(2) 0.752(2) 0.773(2) 0.797(2) 0.815(2) 0.837(2) 0.856(2)

1.762(3) 1.769(3) 1.776(4)

Schober/Dederichs

0.232(1) 0.294(1) 0.394(1) 0.399(2) *) 0.413(1) *)

0.2750 0.2875 0.3000 0.3125 0.3250 0.3375 0.3500 0.3625 0.3750 0.3875 0.4000 0.4125 0.4250 0.4375 0.4500 0.4625 0.4750 0.4875 0.5000

0.875(1) 0.888(3) 0.909(2) 0.932(2) 0.958(2) 0.986(2) 1.014(2) 1.048(2) 1.082(2) 1.110 (2) 1.147 (2) 1.181 (2) 1.212 (2)

1.243(2) 1.272(2) 1.300(2) 1.322(2) 1.333(2) 1.342(3) (continued)

1.2 Phonon states: Cd

Ref. p. 1801 Table 2. Cd. (continued) 5

v [THz]

v [THz]

I

Branch A‘1, 0.150 0.200 0.250 0.300 0.350 0.375 0.400 0.425 0.450

0.611(6) 0.890 1.079(6) 1.272(6) *) 1.314(8) *) 1.442(3) *) 1.483(7) *) 1.566(10) 1.654(8) 1.754(8) 1.837(8)

v [THz]

( I

v [THz]

Branch A5, A6 [OOl] 0.475 0.500 0.525 0.550 0.600 0.650 0.700 0.800 0.900 1.ooo

1.952(6) 2.069(6) 2.174(4) 2.278(6) 2.424(5) 2.593(7) 2.730(7) 2.957(12) 3.110(11) 3.174(11)

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55

0.304(5) 0.433(2) 0.556(1) 0.656(2) 0.760(1) 0.843(2) 0.928(2) 0.992(1) 1.055(2) 1.109(2)

0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

1.156(2) 1.197(2) 1.230(2) 1.266(3) 1.279(3) 1.295(3) 1.307(4) 1.310(3) 1.310(3)

Branch u, wr1 0

*) The two values representthe overlap betweenthe measurements for the lower freauencies and the higher frequencies,respectively, using different experimental setups.

0.05 0.10 0.15 0.20 0.25

1.388(13) 1.384(14) 1.340 (9) 1.276(10) 1.260 (7) 1.299 (7)

I

0.30 0.35 0.40 0.45 0.50

1.351(6) 1.370(4) 1.384(4) 1.392(7) 1.412(6)

I

A-

Fig. 1. Cd. Measured phonon dispersion at 77 K. The lines are drawn as guide to the eye to show connectivity [79Chl].

Anomalies in the phonon dispersion

Contrary to the usual behaviour the transversebranches T, and C, which are polarized in the [OOl] direction have group velocities which at small q decreaserapidly as q increases,Fig. 2a, b Cd. Furthermore severalabrupt changesof the group velocities are evident, Fig. 2 Cd. These can be interpreted as Kohn anomalies. Their positions agree quite well with those predicted by second order pseudopotential perturbation theories. Additional anomalies derived from third order terms could not be identified unambiguously. Schober/Dederichs

31

1.2 Phononenzustbde: Cd

[Lit. S. 180 C-

1’ -

l-

M2.0 1Hz

Cd IttO T=77K

1.5 I a 1.0

-A

A-

1

M

1.7 IHz 1.5 I P 1.3

0

0.2

0.4

0.6

0.8

1.0

0.5

0

f-

f-

Fig. 2a-d. Cd. Upper parts: dispersion curves (full circles) and group velocities dw/dq (open circles) at 77 K. The full circles at q =0 represent the group velocity obtained from the elastic constants [60Gal]. The line is a guide to the eye. Lower parts: Theoretical predictions of the group velocities. The full line is calculated in second order pseudopotential perturbation theory; the broken line including third order terms in the potential. The numbers refer to predicted Kohn anomalies. a) T, -Ti branch b) Z, branch c) A, and A, branches d) U, branch according to [79Chl].

32

Schober/Dederichs

Ref. p. 1801

1.2 Phonon states: Cd

Temperature dependence

The phonon frequency shift from 90 K to 300 K was measuredfor the Ta, Tj branch, Fig. 3 Cd. From an X-ray measurement of forbidden reflexions, Merisalo et al. [78Mel] derive a value of c1a3 =1.3 (2) . lOlo Nm-’ for the cubic force constant in a one particle potential formalism for the anharmonic vibrations. This value is a factor of 10 larger than the one given by the same authors for Zn [78Mel]. T-

T’-

4.0[

nrh ua,v.

units

3.5 3.0

t 2.5 c; 7n

Fig. 3. Cd. Measured relative shift in the phonon frequencies for a temperature change from 300 K to 90 K. The point at q=O was obtained from measurements of the elastic

constants[60Gal] (accordingto [75Chl]).

0

1

2

3

4

5

6 THz 7

Y-

2. Frequency spectra and related properties

Fig. 4. Cd. Measured phonon frequency spectrum [76Erl].

226 K 200 I

180

560 140 120

-ccalc.

VI

I76 Er 11

80 100 120 140 K 160 TFig. 5. Cd. Debye temperature On calculated from the measured spectrum of Fig. 4 Cd compared to the results of calorimetric measurements[76Erl]. 20

40

60

0

50

100 150 200 250 300 350 K 400 I-

Fig. 6. Cd. Debye-Waller exponent 2W divided by the recoil frequency of the free atom, va, calculated from the measured spectrum of Fig. 4 Cd [76Erl].

3. Theoretical models Most theoretical calculations up to date were aimed at reproducing the earlier X-ray measurementsand are :hereforeunreliable. Reasonablefits using pseudopotential perturbation theory are given by [76Na4] and [75Chl]. Schober/Dederichs

33

1.2 PhononenzustEnde: Ce

Ce

TLit. S. 180

L-m

Cerium

Lattice: y-phasefee,a= 516pm= 5.16A. BZ: seep. 449. 1. Phonon dispersion

Table 1. Ce. Measurements. Method

Fig.

Ref.

1 Ce

Stassiset al. [79St1]

TKI neutron diffraction (TW

295

The measureddispersion curves in Ce are in general softer than one would expect by comparison with those of Th taking into account the differencesin mass,lattice parameter,and melting temperature.This relative softening is more pronounced in those branches whose initial slopes involve the elastic constants cI1 and cIz which enter into the bulk modulus. Thus it seemsthat premontory effectsof the y-x phasetransition are present in the room temperature dispersion curves. A-

0

ZX

-E W

1

1.0 0.8 0.6 0.4 0.2 0

1.010

b-

hl-

XK

-l

5-

0.5 s-

Fig. 1. Ce. Measuredphonon dispersionof y-Ce at 295K. The solid lines wereobtainedfrom the Born-von Karman fit parametersgiven in Table3 Ce [79%1-j. Table 2. Ce. Measured phonon frequenciesin y-Ce. T= 295 K, [79Stl]. 5

v CTHzl

C

0.1

1.0 34

5

KW -I-

Cwl L 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

v P-1

0.35 (2) 0.52 (2) 0.76 (3)

0.1 0.2 0.3

0.29 (1) 0.60(2) 0.97 (3)

1.04(6)

0.4

1.27 (2)

1.35 (6) 1.70(7) 2.04 (7) 2.45 (5) 2.72 (8) 2.94 (8) 3.04 (7)

0.5 0.6 0.7 0.8 0.9

1.50(4) 1.74(4) 1.99 (5) 2.04 (6) 2.10 (9) 2.05 (6)

1.0

v D-Hz1

C

0X4(4) 1.20(5) 1.55 (6) 2.27 (8) 2.66 (8) 2.75 (7)

6’

CCC<1 T

crrn L 0.1 0.15 0.2 0.3 0.4 0.5

v D-Hz1

COCCI L

0.1

0.43 (1)

0.1

0.15 0.2 0.3 0.4 0.5

0.60 (1)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.76 (2) 0.99 (3) 0.83 (3) 0.75 (3)

v [THz]

1.0

0.58 (2) 1.28 (3) 1.74(3) 2.09 (3) 2.20 (5) 2.20 (4) 2.19 (6) 2.15 (5) 2.04 (6) 2.05 (6)

(continued) !Schoher/Dederichs

Ref. p. 1801

1.2 Phonon states: Ce

Table 2. Ce. (continued)

v CTHzl

r

C

COCCI ‘L 7

i

0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.27 (1) 0.40 (1) 0.54(l) 0.83 (2) 1.09 (3) 1.30 (3) 1.43 (5) 1.62 (7) 1.79 (5) 1.85 (6) 2.05 (6)

v CT=1

v [THz]

I

Fml A

iY4’ll -L “1

0.47 (1) 0.70 (1) 0.95 (1) 1.41 (2) 1.79 (2) 2.13 (4) 2.30 (6) 2.59 (7) 2.82(7) 2.90(7) 3.04 (7)

0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

v [THz]

0.05 0.1 0.2 0.35 0.50

CWI n:

1.87 (8) 1.89 (6) 1.82(6) 1.71 (6) 1.75 (8)

0.2 0.4 0.6 0.8 1.0

2.78 (6) 2.54(6) 2.16 (6) 1.99 (6) 2.05 (6)

“) The polarization vectors for the [OYC]TI and T, branches are parallel to [Occ] and [[OO], respectively.

Table 3. Ce. Born-von Karman force constants, @!. T=295 K, [79Stl].

m

ij

110 xx zz XY 200 xx YY 211 xx YY YZ xz

@t [Nm-‘1

m

ij

4.3726(700) 220 xx zz -0.2264 (900) 4.5798 (1000) XY -2.3562(1600) 310 xx 0.0773 (800) YY zz 0.2058 (600) XY 0.3169 (400) -0.0547 (200) -0.0496(300)

@! [Nm-‘1

m

0.1231(400) 0.0114 (700) 0.1505(900) -0.0525 (600) -0.0992 (400) -0.1044 (500) 0.0193 (300)

222 xx XY 321 xx YY zz YZ xz XY 400 xx YY Constraints:

CD; [Nm-‘1

ij

-0.3316(400) -0.2194 (700) 0.1057(400) -0.1138(400) 0.0263(200) 0.0050(200) 0.0763(200) -0.0068 (200) -0.0009 (20) 0.2219(800) 8@g” = 9@g” -

@Jo

&$~;“=3@~10-3@~o 3@,321= YZ

@321=2@321

w

xz

2. Frequency spectrum and related properties -1.00

3.0 THz

THz-' 0.75

2.8

I 2 0.50

I s

0.25

0

0.5

1.5 2.0 2.5 THz 3.0 VFig. 2. Ce. Phonon frequency spectrum for y-Ce at 295 K :alculated from the Born-von Karman parameters of rable 3 Ce. 1.0

2.6

2.4

IO 20 30 ffFig. 3. Ce. Debye cutoff frequencies v, calculated from the spectrum of Fig. 2 Ce.

Schober/Dederichs

-10

0

35

1.2 PhononenzustZnde: Ce

[Lit. S. 180

136 K 132

IO .10-v S

128

8

t,126 0 120

I

6

x 1 N

4 2

116 112 0

50

100

150

200

0

250 K 300

50

I-

Fig.4. Ce. Debye temperature @,(‘I) calculated from the Born-von Karman parameters of Table 3 Ce [79Stl].

100 150 200 250 300 350 400 T-

K

500

Fig. 5. Ce. Debye-Wailer exponent 2W divided by recoil frequency, vR, calculated from the spectrum of Fig. 2Ce.

Table 4. Ce. Phonon frequency spectrum of y-Ce at 295 K calculated from the Born-van Karman model of table 3 Ce. v [THz]

g(r) [THz- ‘1

v [THz]

g(v) CTHz-‘I

v [THz]

gb) D-Hz- ‘1

0.0250

0.0001

0.0500 0.0750 0.1000 0.1250 0.1500 0.1750 0.2OOQ 0.2250 0.2500 0.2750 0.3OOil 0.3250 0.3500 0.3750 0.4OcHI 0.4250 0.4500 0.4750 0.5000 0.5250 0.5500 0.5750 0.6000 0.6250 0.6500 0.6750 0.7000 0.7250 0.7500 0.7750 0.8000 0.8250 0.8500

o.ooo3 0.0007 0.0013 0.0021 0.0029 0.0044 0.0055 0.0067 0.0085 0.0102 0.0121 0.0143 0.0165 0.0191 0.0217 0.0246 0.0276 0.0309 0.0346 0.0386 0.0425 0.0469 0.0513 0.0563 0.0613 0.0669 0.0731 0.0796 0.0963 0.1159 0.1346 0.1532 0.1751

0.8750

0.2009

0.9000 0.9250 0.9500 0.9750 l.OOCQ 1.0250 1.0500 1.0750 1.1000 1.1250 1.1500 1.1750 1.2000 1.2250 1.2500 1.2750 1.3000 1.3250 1.3500 1.3750 1.4000 1.4250 1.4500 1.4750 1.5000 1.5250 1.5500 1.5750 1.6000 1.6250 1.6500 1.6750 1.7000

0.2303 0.2764 0.3298 0.3366 0.3471 0.3548 0.3630 0.3726 0.3821 0.3914 0.4017 0.4124 0.4231 0.4344 0.4460 0.4583 0.4703 0.4835 0.4963 0.5098 0.5236 0.5368 0.5511 0.5641 0.5769 0.5887 0.5987 0.6051 0.6114 0.6125 0.5692 0.5291 0.5057

1.7250 1.7500 1.7750 1.8000 1.8250 1.8500 1.8750 1.9000 1.9250 1.9500 1.9750 2.OOMl 2.0250 2.0500 2.0750 2.1000 2.1250 2.1500 2.1750 2.2000 2.2250 2.2500 2.2750 2.3000 2.3250 2.3500 2.3750 2.4000 2.4250 2.4500 2.4750 2.5000 2.5250 2.5500

0.4907 0.4799 0.4719 0.4665 0.4628 0.4619 0.4620 0.4650 0.4715 0.4786 0.4915 0.5147 0.6120 0.6275 0.6484 0.6737 0.6966 0.7073 0.6956 0.6653 0.6171 0.5585 0.4808 0.3723 0.3167 0.3529 0.4450 0.5255 0.5957 0.6619 0.7221 0.7862 0.8454 0.9131 (continued)

36

Schober/Dederichs

1.2 Phonon states: Co

Ref. p. 1801 Table 4. Ce. (continued) v [THz]

g(v) CT=-- ‘1

v [THz]

g(v) CT-- ‘1

v [THz]

g(v) [Ttiz-

2.5750 2.6000 2.6250 2.6500 2.6750 2.7000

0.9869 0.8234 0.5813 0.3211 0.1724 0.1498

2.7250 2.1500 2.7750 2.8000 2.8250 2.8500

0.1310 0.1148 0.1012 0.0893 0.0783 0.0679

2.8750 2.9000 2.9250 2.9500 2.9750 3.0000

0.0584 0.0491 0.0400 0.0301 0.0168 0.0

‘1

3. Theoretical models Born-von Karman fit: see Table 3 Ce.

Co

Cobalt

Lattice: cc-Co hcp a=251 pm=2.51 A, c=407pm=4.07A;

/?-Co fee a=354pm=3.54A.

1. Phonon dispersion

l-

Table 1. Co. Measurements. Method

T WI

Fig.

Ref.

neutron diffraction WS) neutron diffraction PAS) neutron diffraction

472...721

1 Co

Frey et al. [79Frl]

L

-A

A I

-A

Shapiro and Moss [77Shl] Svensson et al.

77,300

297

AA-

L5f

BZ: see p. 450.

2co

[79Svl]

PAS) b

Fig. 1. Co. Measured temperature dependenceof the TA/TO branches along (r-A),,,=(r-L),,, in Co [79Frl].

A-

-r

R-

-E X

9 THz 8

K

L

r

7 6 5 I 54 3 2 1 0

0

0.2

0.4 t-

0.6

0.8

1.0

0.8

0.6 -t

0.4

0.2

0

0.5 5-

Fig. 2. Co. Measured phonon dispersion curves in fee Co,,,,Fe,,,s. solid lines represent the Born-von Karman tit of Table 3 Co [79Svl]. Schoher/Dderichs

The

37

1.2 Phonon states: Co

Ref. p. 1801 Table 4. Ce. (continued) v [THz]

g(v) CT=-- ‘1

v [THz]

g(v) CT-- ‘1

v [THz]

g(v) [Ttiz-

2.5750 2.6000 2.6250 2.6500 2.6750 2.7000

0.9869 0.8234 0.5813 0.3211 0.1724 0.1498

2.7250 2.1500 2.7750 2.8000 2.8250 2.8500

0.1310 0.1148 0.1012 0.0893 0.0783 0.0679

2.8750 2.9000 2.9250 2.9500 2.9750 3.0000

0.0584 0.0491 0.0400 0.0301 0.0168 0.0

‘1

3. Theoretical models Born-von Karman fit: see Table 3 Ce.

Co

Cobalt

Lattice: cc-Co hcp a=251 pm=2.51 A, c=407pm=4.07A;

/?-Co fee a=354pm=3.54A.

1. Phonon dispersion

l-

Table 1. Co. Measurements. Method

T WI

Fig.

Ref.

neutron diffraction WS) neutron diffraction PAS) neutron diffraction

472...721

1 Co

Frey et al. [79Frl]

L

-A

A I

-A

Shapiro and Moss [77Shl] Svensson et al.

77,300

297

AA-

L5f

BZ: see p. 450.

2co

[79Svl]

PAS) b

Fig. 1. Co. Measured temperature dependenceof the TA/TO branches along (r-A),,,=(r-L),,, in Co [79Frl].

A-

-r

R-

-E X

9 THz 8

K

L

r

7 6 5 I 54 3 2 1 0

0

0.2

0.4 t-

0.6

0.8

1.0

0.8

0.6 -t

0.4

0.2

0

0.5 5-

Fig. 2. Co. Measured phonon dispersion curves in fee Co,,,,Fe,,,s. solid lines represent the Born-von Karman tit of Table 3 Co [79Svl]. Schoher/Dderichs

The

37

1.2 Phononenzustgnde: Co

I-Lit. S. 180

Due to a high absorption and low inelastic scattering cross section, measurementson pure Co are difficult. The measurements[77Shl] and [79Svl] were therefore done on fee Co,,,,Fe,,,, crystals whose phonons should be similar to the ones of fee /~-CO.The measurement[79Frl] was restricted to the r-A(A) direction of hcp a-Co, respectively, the T-L(A) direction of fee /~-CO.This investigation was aimed at a study of the martensitic phase transition at about 700 K. A softening of the whole branch but no soft mode behaviour was observed. Table 2. Co. Measured phonon frequencies in fee Co,.,,Fe,,,, p-co, [79Svl]. c

v D-Hz1

I

v [THz]

at 297 K which should roughly correspond to

0.142(6) 0.200 0.202(5) 0.300 0.300(5) 0.400 0.426(5) 0.500 0.504(7) 0.600 0.700 1.ooo

0.94 (6) 1.50 2.00 (6) 2.00 2.96 (5) 3.00 3.80 (5) 4.00 4.44 (5) 4.50 4.99 (6) 5.28 (8) 5.80(10)

0.120 0.193 (6) 0.200 0.273 (7) 0.350 0.360 (7) 0.400 0.423 (9) 0.496(12) 0.570(15) 0.660(15) 0.705(20) 1.000

1.83 (5) 3.00 3.06 (6) 4.00 4.87 (8) 5.00 5.42(12) 5.60 6.25 6.75 7.25 7.50 8.1 (4)

0.096(5) 0.100 0.162(5) 0.200 0.234(5) 0.300 0.334(7) 0.400 0.500

1.25 1.31 (5) 2.00 2.42 (4) 2.75 3.26 (7) 3.50 3.72(12) 3.87 (5)

0.200 0.237 (7) 0.295 (8) 0.364 (6) 0.400 0.425(15) 0.492(10) 0.645(15) 1.000

1.71 (6) 2.00 2.50 3.00 3.24 (8) 3.50 4.00 5.00 5.80(10)

0.098 (4) 0.100 0.132 (5) 0.140 0.200 0.202 (4) 0.300 (5) 0.364 (6) 0.400 0.505 (7) 0.565 (8) 0.640(12) 0.684(13) 0.736(20) 0.800 1.000

1.50 1.50 (5) 2.00 2.01 (6) 2.96 (6) 3.00 4.25 5.00 5.32 (10) 6.25 6.75 7.25 7.50 7.75 7.88 (15) 8.10(40)

0.075 0.081 (3) 0.127 (5) 0.188 (6) 0.227 (6) 0.314 (7) 0.370(12) 0.432(25) 0.772(15) 0.851(13) 1.000

1.81 (4) 2.00 3.00 4.25 5.00 6.25 6.75 7.25 7.00 6.50 5.80(10)

0.100

Table 3. Co. Born-von Karman force constants, @z, of fee Co,,,,Fe,,,,, m

ij

110

xx zz XY xx YY

200

38

@7 mm-‘] 15.71(27) 0.27 (56) 18.77(36) 0.32 (68) 0.27 (37)

rrrn L 0.100 0.113 (4) 0.143 (3) 0.184 (3) 0.228 (3) 0.272 (4) 0.308(12) 0.330(12) 0.376(18) 0.500

T=297 K, [79Svl].

m

ij

q

211

xx YY YZ XY xx YZ XY

0.36 (35) -0.20 (17) 0.38 (15) 0.27 (13) 1.52(19) 0.09 (30) 1.36(25)

220

Sehober/Dederichs

v [THz

5

C6CSl T

CWI L

C’WI ‘I-

v [THz]

r

[Nm-‘1

3.04 (5 3.50 4.25 5.25 6.25 7.00 7.50 7.75 8.00 8.10 (25

1.2 Phonon states: Cr

Ref. p. 1801

2. Frequency spectrum and related properties

0

1

2

3

4

5

6

7

.,,% I

THZ

Fig. 3. Co. Phonon frequency spectrum of Co,,,,Fe,,,s calculated from the Born-von Karman parameters of Table 3 Co [79Svl].

v5

oc1.0 -? 52 N 0.5 0

0

100

200

300

400

Fig. 5. Co. Average Debye-Waller exponent 2W of Co,,,,Fe,,,, divided by the recoil frequency of the free atoms, vR, calculated from the spectrum of Fig. 3 Co [79Svl].

500 K 600

Fig. 4. Co. Debye temperature 0, of Co,,,,Fe,,,, lated from the spectrum of Fig. 3 Co [79Svl].

200 400 600 800K 1000

calcu-

3. Theoretical models Born-von Karman fits: seeTable 3. Co and [77Shl, 79Svl], further reference: [78Ra6]. Short ranged forces plus a simple electronic contribution: [78Ra7], further reference: [74Si2].

Cr

Chromium

Lattice: bee,a=288 pm=2.88 A. BZ: seep. 448. 1. Phonon dispersion Table 1. Cr. Measurements. Method

T L-W

Fig.

Ref.

neutron diffraction (TAS) neutron diffraction (T-W neutron diffraction VW

300

1 Cr

Shaw and Muhlestein [71Sh3] Muhlestein et al. [72Mul]

300,403 503 78,200

Muhlestein et al. [73Mul]

Further reference:[65Mol]. The dispersion curves of chromium are similar in shape to the ones of the other group VI metals MO and W. Four regions of a pronounced anomalous behaviour are found. Schober /Dederichs

39

1.2 Phonon states: Cr

Ref. p. 1801

2. Frequency spectrum and related properties

0

1

2

3

4

5

6

7

.,,% I

THZ

Fig. 3. Co. Phonon frequency spectrum of Co,,,,Fe,,,s calculated from the Born-von Karman parameters of Table 3 Co [79Svl].

v5

oc1.0 -? 52 N 0.5 0

0

100

200

300

400

Fig. 5. Co. Average Debye-Waller exponent 2W of Co,,,,Fe,,,, divided by the recoil frequency of the free atoms, vR, calculated from the spectrum of Fig. 3 Co [79Svl].

500 K 600

Fig. 4. Co. Debye temperature 0, of Co,,,,Fe,,,, lated from the spectrum of Fig. 3 Co [79Svl].

200 400 600 800K 1000

calcu-

3. Theoretical models Born-von Karman fits: seeTable 3. Co and [77Shl, 79Svl], further reference: [78Ra6]. Short ranged forces plus a simple electronic contribution: [78Ra7], further reference: [74Si2].

Cr

Chromium

Lattice: bee,a=288 pm=2.88 A. BZ: seep. 448. 1. Phonon dispersion Table 1. Cr. Measurements. Method

T L-W

Fig.

Ref.

neutron diffraction (TAS) neutron diffraction (T-W neutron diffraction VW

300

1 Cr

Shaw and Muhlestein [71Sh3] Muhlestein et al. [72Mul]

300,403 503 78,200

Muhlestein et al. [73Mul]

Further reference:[65Mol]. The dispersion curves of chromium are similar in shape to the ones of the other group VI metals MO and W. Four regions of a pronounced anomalous behaviour are found. Schober /Dederichs

39

1.2 Phononenzustkde:

Cr

-F

A-

-A

H

-L

[Lit. S. 180

P

-t

f-

Fig. 1. Cr. Phonon dispersion curves at 300 K. The solid line represents the fourth neighbour Born-von Karman fit (Table 2 Cr) [71SH3].

Anomalies

in the dispersion runes

Strong anomalies have been found in four regions: i) a striking depression of the [Oi<] T2 branch near the symmetry point N (0, f, f) Fig. 2 Cr. ii) a depression in both branches near the symmetry point H (0, 0,l) Fig. 3 Cr. L branch is observed around {=0.25.

iii) an abrupt change of slope in the dispersion of the [[[c]

iiii) an abrupt change in frequency is observed near the symmetry point P (f f i) in the [[[[I

T branch.

The first two anomalies are related to the structure of the Fermi surfacein agreementwith band structure cal:ulations [71Sh3]. At 311 K, chromium undergoes a first order paramagnetic to antiferromagnetic transition. The anomalies are very similar in both phases,Fig. 4 Cr. The sameanomalies were also observedin a single - Q antiferromagnetic Cr sample at 78 K and 200 K below and above the spinflip temperature of T&=123 K. The anomaly is more pronounced for the lower temperature. A large anisotropy between the phonons, parallel and perpendicular to the spin density wave vector Q is seen,Figs. 5 Cr and 6 Cr. At higher temperatures(503K) the rnomaly is somewhat reduced, Fig. 4 Cr. A-

-F

h2 THZ 5.9

I 5.6 1 5.3 I.2 5.0 0.30

7.0y,, 0.35

OAO

O.L5

0.7

0.50

bFig. 2. Cr. Section of the measured

[O;[] T2 phonon branch near the symmetry point N. The lines are drawn as guide to the eye [72Mul].

40

Illtl

loot1 0.8 t-

0.9

1.0

I

I

0.9 -l

0.8

Fig. 3. Cr. Section of the measured phonon dispersion at 300 K near the H symmetry point. The lines are drawn as guide to the eye [71Sh3].

Schober/Dederichs

1.2 Phonon states: Cr

Ref. p. 1801 -F

A8.2 THz

6.8 I

A-

H

Cr

8.0

rooe1

I

I

15551

I

I

THz

I

I

0.9 1.0 0.95 0.90 0.85 0.80 0.8 f-!Z Fig. 4. Cr. Section of the measured phonon dispersion near ihe H point for various temperatures [72Mul].

Cr I=200K

6.8, 0.6

0.7

0.7

0.8

0.9

1.0

t-

Fig. 5. Cr. Section of the [OOr;]T branch of the phonon dispersion of single Q-antiferromagnetic chromium measured near the H-symmetry point at 200 K with q parallel (full circles) and antiparallel (open circles) to the spin density wave vector Q [73Mul].

AH

8.0 THz

7.87.6Born-von Karman fit

Apart from the anomalies, a fourth neighbour fit reproduces the dispersion within 3 %. The strong maximum in [OOC]L direction indicates a strong second neighbour force constant. Table 2. Cr. Born-von Karman force constants, @E, T= 300 K, [71Sh3]. m

ij

111

xx XY

13.526 6.487

200

xx

35.915 -1.564

YY 220

xx zz XY

311

xx YY YZ XY

~0; [Nm-‘1

I 1.LL 7.27.0Y 6.8 0.6

I 0.8

I 0.7

I 0.9

1.0

CFig. 6. Cr. Section of the [004’] T branch of the phonon dispersion of single Q-antiferromagnetic chromium measured near the H symmetry point at 78 K with (I parallel (full circles) and antiparallel (open circles) to the spin density wave vector Q [73Mul].

2.042 -0.050 2.871 -1.257 0.432 0.516 0.007

Schober /Dederichs

1.2 Phononenzusttinde: Cr

[Lit. S. 180

2. Frequency spectrum and related properties Table 3. Cr. Phonon frequency spectrum at 300 K as calculated from the Born-von Karman parameters of table 2 Cr.

v ITHzl 0.0 0.10

g(v) [THz- ‘1

v [THz]

g(v) IYz- ‘I

v D-Hz1

gb9 CTHz-‘I

0.0

3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60

0.020 0.022 0.024 0.025 0.027 0.029 0.032 0.034 0.036 0.039 0.042 0.045 0.048 0.052 0.055 0.060 0.064 0.069 0.075 0.081 0.087 0.095 0.104 0.113 0.125 0.140 0.162 0.198 0.200 0.202 0.205 0.207 0.210

6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20 8.30 8.40 8.50 8.60 8.70 8.80 8.90 9.00 9.10 9.20 9.30 9.40 9.50 9.60 9.70

0.214 0.218 0.222 0.227 0.232 0.239 0.248 0.258 0.272 0.293 0.335 0.363 0.354 0.303 0.254

0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.001 0.001

0.20 0.30 0.40

0.50 0.60 0.70 0.80 0.90

1.00 1.10

0.001 0.002 0.002 0.002 0.003 0.003 0.004

1.20 1.30

1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30

0.004 0.005 0.005 0.006 0.007 0.007 0.008 0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016 0.018 0.019

0.160 0.087 0.089 0.120 0.158 0.228 0.249 0.265 0.280 0.295 0.304 0.323 0.184 0.095 0.017

0.0

as TH2-l

13 THz

a3 12 I

0.1

0

10

2

6

4

8

9 -10

1Hz 10

Y-

Fig. 7. Cr. Phonon frequency spectrum at 300 K calculated from the Born-von Karman coupling constants ofTable 2 Cr. 42

0

10 n-

20

30

Fig. 8. Cr. Debye cutoff frequencies,Y,, calculated from the spectrum of Fig. 7 Cr.

Schober/Dederichs

Ref. p. 1801

1.2 Phonon states: Cs

Fig. 9. Cr. Debye-Waller exponent 2 W divided by the recoil frequency of the free ion, vR.

I

0

100

200

300

400 K

500

3. Theoretical models A first principle calculation of the phonon dispersion has not been done so far. The results of the various phenomenological models are in general inferior to the Born-von Karman model ones. Born-von Karman and equivalent models, see Table 2 Cr, further references: [77Pal]. Short ranged forces plus a simple electronic contribution: 77Gul,78Gu2,78Khl, 78Pal,78Ra2].

[76Brl, 76Si2], further references: [74Pa2, 75Pr1,

Shell model: [76Jal]. Model pseudopotentials:

Cs

[73Anl, 75Sil].

Cesium

Lattice: bee, a=605 pm=6.05 8, at 92 K. BZ: see p. 448. No measurements are available for cesium. Extrapolating from the lighter alkali metals, one expects pseudopotential theory to work well. Theoretically predicted phonon dispersion: see Fig. 1 Cs.

I

0 0.2 0.4 0.6 0.8 1.0

f-

I

0.8

I

I

I

0.6

-c

0.4

I

I

0.2

I

I

0

5--’

I

I

I

I

0.50 0.2 0.4 0.6 0.8 1.00.5

5-

I

-Z

I

1

0

Fig. 1 Cs. Theoretically predicted phonon dispersion for a one-parameter local pseudopotential. The heavy solid lines represent initial slopes derived from measured elastic constants. A self-consistent dielectric screening was used which gave the best results for Li, Na, and K [70Prl]. Theoretical models Models comprising short ranged forces plus a simple electronic contribution [75Sil, 72Sil]. Local pseudopotential 68Hol,71Gu3].

models: [70Prl,

[76Sil], further references:

73Pr2, 74Sh2, 77Va1, 77Prl], further references: [61Tol,

Nonlocal pseudopotential calculations: [72Be2,77Sol,

66An1,

78Sol].

Schober/Dederichs

43

Ref. p. 1801

1.2 Phonon states: Cs

Fig. 9. Cr. Debye-Waller exponent 2 W divided by the recoil frequency of the free ion, vR.

I

0

100

200

300

400 K

500

3. Theoretical models A first principle calculation of the phonon dispersion has not been done so far. The results of the various phenomenological models are in general inferior to the Born-von Karman model ones. Born-von Karman and equivalent models, see Table 2 Cr, further references: [77Pal]. Short ranged forces plus a simple electronic contribution: 77Gul,78Gu2,78Khl, 78Pal,78Ra2].

[76Brl, 76Si2], further references: [74Pa2, 75Pr1,

Shell model: [76Jal]. Model pseudopotentials:

Cs

[73Anl, 75Sil].

Cesium

Lattice: bee, a=605 pm=6.05 8, at 92 K. BZ: see p. 448. No measurements are available for cesium. Extrapolating from the lighter alkali metals, one expects pseudopotential theory to work well. Theoretically predicted phonon dispersion: see Fig. 1 Cs.

I

0 0.2 0.4 0.6 0.8 1.0

f-

I

0.8

I

I

I

0.6

-c

0.4

I

I

0.2

I

I

0

5--’

I

I

I

I

0.50 0.2 0.4 0.6 0.8 1.00.5

5-

I

-Z

I

1

0

Fig. 1 Cs. Theoretically predicted phonon dispersion for a one-parameter local pseudopotential. The heavy solid lines represent initial slopes derived from measured elastic constants. A self-consistent dielectric screening was used which gave the best results for Li, Na, and K [70Prl]. Theoretical models Models comprising short ranged forces plus a simple electronic contribution [75Sil, 72Sil]. Local pseudopotential 68Hol,71Gu3].

models: [70Prl,

[76Sil], further references:

73Pr2, 74Sh2, 77Va1, 77Prl], further references: [61Tol,

Nonlocal pseudopotential calculations: [72Be2,77Sol,

66An1,

78Sol].

Schober/Dederichs

43

1.2 Phononenzustkde:

Cu

[Lit. S. 180

Cu

Copper

Lattice: fee,a = 361pm = 3.61A. BZ: see p. 449.

1. Phonon dispersion Table 1. Cu. Measurements. Method

Fig.

Ref.

296

2 Cu

49 298 296 673 973 1336

1 Cu

Svenssonet al. C67Svl-J Nicklow et al. [67Nil] Larose and Brockhouse [76Lal]

TKI neutron diffraction F-AS) neutron diffraction U-AS) neutron diffraction U-AS)

neutron diffraction O-AS)

80

neutron diffraction W’F)

300

2 Cu 2Cu 2cu 2Cu

Nilsson, Rolandson [73Nil] Sinha [66Sil]

Further references: [71Mil, 70Bu1, 55Ja1, 6OCr1, 61Crl,62Sol, 63Mal,65Vil, 63Sil]. Of all metals, Cu is the one whose phonon dispersion has been investigated most often. The earliest measurements,by X-ray diffraction in 1955,were rather unreliable. Neutron diffraction experiments were done from 1960 onwards. The agreementbetween the various measurementscompiled in Table 1 Cu is generally good, seeTable 2 Cu. The phonon dispersion has been measuredin the temperature range from 49 K to 1336K, i.e. 20 K below the melting point. Even at the highest temperaturethe phonon lines were clearly identifiable well abovethe background intensity [76Lal]. The temperature dependenceof the dispersion curves is quite strong, Fig. 2 Cu. The absolute changeis about the samefor both transverseand longitudinal branches.The longitudinal frequenciesare softened in averageby about 10 % and the transverse ones by about 15 %.

n-

Cl

1.010 f-

f-

1.0 0.8 0.6 OX 0.2 -t

0

0.5 t-

Fig. 1. Cu. Phonon dispersioncurvesat 49 K. The continuouslines representa sixth neighbouraxially symmetricBornvon Karman fit to the experimental points [67Nil]. 44

Schoher/Dederichs

Ref. p. 1803

1.2 Phonon states: Cu A-

i-

. n

. . v

1336Kj 296K I67Svll C9K 167Nill I

I

0.22 0.4

0.6

0.8 I.010

0.2

C-

I

I I

0.4 C-

I

I

0.6

0.8

1.0

0.8

0.6 -6

0.2

0.4

0

Fig. 2. Cu. Temperature dependence of the phonon dispersion relation. The lines joining the points are only guides to the eye [76Lal].

Table 2. Cu. Measured phonon frequencies, (additional and 66Sil].

values of off symmetry phonons are given in [73Nil

T

49K

80K”)

296K

298K

673 K

T

49 K

8OK”)

296K

298 K

673 K

Ref.

67Nil

73Nil

67Svl

67Nil

71Mil

Ref.

67Nil

73Nil

67Svl

67Nil

71Mil

1.60 (2)

1.48 (3)

r

v [THz]

v CTYzl

CWI L 0.1 0.15 0.2 0.25 0.3 0.40 0.5 0.6 0.65 0.7 0.75 0.8 0.85 0.9 1.0

1.25 2.43 (4)

2.46

3.56 (4) 4.56 (5) 5.44 (7) 6.15 (10)

3.55 4.52 5.42 6.16

6.66 (10) 6.66 6.90 (15) 7.08 7.18 (17) 7.28 7.25 (20) 7.38

CW’I T

1.90 (7) 2.42 (7) 3.02 (8) 3.56 (6) 4.47 (7) 5.32 (7) 6.05 (8) 6.37 (8) 6.60 (8) 6.77 (12) 6.99 (13) 7.14 (14) 7.17 (12) 7.19 (12)

2.44 (4)

2.38 (6)

3.56 (4) 4.50 (5) 4.34 (4) 5.42 (7) 6.14 (10) 5.83 (10) 6.66 (10)

0.15 0.2 0.25 0.275 0.3 0.35 0.4 0.45 0.5 0.55 E5

1.65 (2)

1.68

2.40 (2)

2.41 b,

3.09 (2)

3.12

3.69 (3)

3.71 b)

4.22 (3)

4.23 b,

4.64 (5)

4.64 b,

6.90 (10) 6.74 (14) 7.15 (17) 7.20 (20) 6.99 (16)

0.7 0.75 0.8 0.9 1.0

Schober/Dederichs

4.94 (10) 4.96 b, 5.08 (10) 5.10b) 5.13 (15) 5.16

1.17 (4) 1.56 (4) 1.92 (4) 2.12 (4) 2.30 (4) 2.64 (4) 3.01 (4) 3.30 (4) 3.62 (4) 3.88 (4) 4.15 (5) 4.34 (5) 4.54 (5) 4.73 (6) 4.86 (7) 5.02 J7) 5.08 (8)

2.35 (2) 3.05 (2) 3.67 (2) 4.17 (3)

4.00 (3)

4.59 (5) 4.88 (10) 4.67 (4) 5.07 (10) 5.09 (15) 4.94 (7) (continued) 45

1.2 Phononenzust%nde: Cu

[Lit. S. 180

Table 2. Cu (continued) T

49 K

8OK”)

296K

298 K

673K

T

49 K

8OK”)

296K

298 K

673 K

Ref.

67Nil

73Nil

67Svl

67Nil

71Mil

Ref.

67Nil

73Nil

67Svl

67Nil

71Mil

Y [THz]

I

cwl 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.75 0.8 0.9 1.0

2.01 (2) 3.73 (3) 5.10 (7) 6.05 (10) 6.42 (15) 6.39(15) 6.18 (10)

1.98 3.68 5.06 5.99 6.50 6.51b) 6.45b)

5.80 (15) 6.94 7.29 5.13 (15) 5.16

v [THz]

r

crm

L

2.03 (10) 3.70 (8) 5.11 (7) 5.97 (8) 6.36 (10) 6.38(12) 5.91 (10) 5.73 (8) 5.51 (7) 5.19 (7) 5.08 (8)

1.96 (2) 3.67 (3) 5.03 (7) 6.00 (10) 6.35 (15) 6.35(15) 6.14 (10)

1.88 (6) 3.54 (5) 5.76 (10) 5.97(9)

5.71 (15) 5.19 (10) 5.09 (15) 4.94 (7)

0.05 0.075 0.1 0.125 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

1.22 (3)

1.24 (6) 1.86 (7) 2.49 (5) 2.46 b, 2.46 (7) 2.99 (6) 3.59 (5) 3.59 (6) 4.61 (5) 4.60b) 4.54(6) 5.51 (10) 5.43 (7) 6.21 (10) 6.24 6.14 (7) 6.80 (15) 6.67 (8) 7.12(15) 7.16b) 7.06(13) 7.27 (20) 7.25 (13) 7.30 (20) 7.44 7.40 (13)

C’Xi-1T) 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.75 0.80 0.9 1.0

0.71 1.37 1.79 (3) 2.12(4) 2.79 (3) 3.43 (4) 3.93 (7) 4.40(5)

4.80(6) 4.80 5.07 (10) 5.11 5.13 (15) 5.16

2.03 (4) 2.70 (4) 3.34 (4) 3.89 (5) 4.34 (5) 4.55 (5) 4.75 (7) 5.03 (8) 5.08 (8)

1.26 (3) 1.73 (3) 2.07 b, 2.73b) 3.39 b, 3.85 (7) 4.33 (5)

2.55 (4) 3.70 (3) 4.37 (4) 4.54(5)

4.70(6) 4.93 (10) 5.09 (15) 4.94(7)

0.075 0.1 0.125 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.79 (4) 1.02b) 1.01 (5) 1.23 (6) 1.53 (6) 1.47(6) 1.90 (6) 1.91 1.87 (6) 2.35 (4) 2.29 (5) 2.72 (5) 2.72 2.66 (6) 3.03 (6) 2.97 (6) 3.24 (7) 3.23 b, 3.17 (7) 3.39 (8) 3.34 (7) 3.42 (10) 3.41 b, 3.37 (7) 1.05 (6)

1.210(15) 1.20b) 1.11 (3) 1.69 (4) 2.385 (20) 2.36b) 2.27(4) 2.82 (4) 3.46 (3) 3.43b) 3.37 (4) 4.39 (3) 4.39b) 4.30 (5) 5.16 (6) 5.16b) 5.07 (6) 5.85(7) 5.85b) 5.71 (6) 6.04 (6) 6.40 (12) 6.31 (7) 6.54 (10) 6.84 (15) 6.80 (11) 7.05 (20) 7.13 (15) 7.25 (20) 7.19 (15)

1.120(15) 2.345 (20) 2.17(3) 3.41 (3) 4.33 (3) 5.12 (3) 5.76(7),

4.10(4) 5.55(7)

6.39 (12)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

4.40(6) 5.94 (8) 6.87(9) 7.28 (10)

0.98 (6) 1.44’) 1.89b) 2.30b) 2.69 b, 2.98 b, 3.21 b, 3.37b) 3.41 (10)

7.19 (12) 7.17(15) 7.07 (14) 6.80 (9) 6.44 (9) 6.10 (8) 5.77 (8) 5.47 (8) 5.27 (8) 5.13 (8) 5.08 (8)

6.79 (15) 6.52 (8) 7.08 (20) 7.20 (20) 6.99 (16)

‘) Average estimated error Av=0.04 THz. Individual errors may be about twice as large. b, Value obtained by interpolation. ‘) The polarization vectors for the [O[[] T, and T2 branches are parallel to [O;n and [[WI, respectively. 46

4.57 (5) 5.46 (10) 6.18 (10) 6.71 (15) 7.01 (15) 7.16 (20) 7.29 (20)

1.81(4) 2.55 (5) 3.01 (6) 3.21 (6)

COc-~l~

C0;i-l T;) 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.65 0.7 0.75 0.8 0.9 1.0

2.35 (8)

CC4’4’1 T

1.35 (4)

2.06 2.75 b, 3.39 3.93 4.40

L

7.21 (9) 6.76 (12) 6.08 (8) 5.85 (6) 5.51 (6) 5.01 (6) 4.94 (7)

car11A 0.0 0.1 0.2 0.3 0.4 0.5

Schober/Dederichs

5.13 (15) 5.11 (10) 5.08 (10) 4.98 (10) 4.96 (9) 4.95 (9)

5.08 (8) 5.03 (8) 4.99 (7) 4.97 (8) 4.89 (9) 4.89 (9)

5.09 (15) 5.05 (15) 5.02 (10) 5.00 (10) 4.97 (9) 4.89 (9)

4.94 (7) 4.72 (6) 4.59 (6) 4.51 (7)

Ref. p. 1801

1.2 Phonon states: 0.1

Anomalies in the dispersion curves The [O{C] T, branch shows around 5=0.2 a positive dispersion[67Svl]. With increasing temperaturethis anomaly becomeslarger, seeFig. 3 Cu. The small wave vector segmentO< [<0.2 softens considerably more than the rest of the spectrum.Whereasa similar anomaly in Pd could be explained in terms of the Kohn effect,the apparent enhancementwith temperature of the anomaly in Cu seemsto forbid this explanation: The observed lineshapes do not show any anomaly and thus do not indicate any unusual damping behaviour [76Lal]. Loidl [77Lol] explained this anomaly as transition from zero to first sound. Whereas,normally, the change between zero and first sound elastic data is of the order of lx, in Cu the changein (c11-c12)/2 corresponding to the [O[[] Ti branch is much larger since the changesof err and ci2 are of opposite sign and since (cl1 - cl,)/2 itself is relatively small. For the samereason the transition between zero and first sound should be shifted far into the Brillouin zone since it occurs at frequencies corresponding to the average lifetime of the thermal phonons. The theory of Loidl gives a good fit to the experimental data, Fig. 3 Cu. Kahn anomalies were studied by Nilsson [68Nil] and Nilsson and Rolandson [74Nil]. They found about twenty very weak anomalies which they interpreted by points on the Fermi surface.Fig. 4(a) Cu shows four frequencies

of transverse phonons measured in the (110) sphere parallel to the [[[O] direction. The line representsa Bornvon Karman fit, which cannot follow the rapid variation of v with [ in the vicinity of the anomalies. The positions

of the anomalies agree well with results of high-precision DHVA measurements[74Nil].

2

THz 6.E THz

0

6.EI-

6.i

I-

>-

I-

4.6 5.2 I-

4.9 0

004

0.08

5Fig. 3. Cu. Anomalous dispersion of the [OQJ TI phonon. Temperature and wave vector dependence of the phonon phase velocity ]v/[l. The full symbols are mean values from energy loss and energy gain measurements, the open symbols represent neutron energy gain data only [76Lal]. Full curve: theoretical values [77Lol].

0.12

0.16

0

0.04

0.08

0.12

0.16

f-

Fig. 4a, b. Cu. Kohn anomalies [74Nil]. a) Four series of transverse phonons measured in the (Oil)-plane parallel to the [Or51 direction. The magnitude of the estimated error is given by the size of the dots. The full line shows the dispersion curves for a Born-von Karman model [73Nil]. b) Differences between mean slopes from measurements (dv/d[), and from the model @v/d [)avK for the curves in (a). Positions of anomalies are indicated by arrows.

Schoher/Dederichs

47

1.2 PhononenzustZnde:

Cu

Eit. S. 180

Born-von Karman constants

Nearly all experiments were fitted in terms of Born-von Karman parameters.The coupling parametersshown in Table 3 Cu show that the nearestneighbout longitudinal force constantft= @z,‘”+ @i,!”dominates,which arises from the large overlap of d-electrons. All other interactions are lower at least by one order of magnitude. One can obtain a reasonablefit even with a general nearest neighbour model [73Nil]. An axially symmetric model works well for Cu, however, according to Nilsson and Rolandson [73Nil] a general model gives slightly better results. At high temperatures,the longitudinal spring constant ff softenswhereasthe corresponding transversespring constants increasesslightly. The other constants show no systematictrend with temperature. Table 3. Cu. Born-von Karman force constants, @E. T

49 K”)

80 Kb)

296 K ‘)

673 K d,

973 K d,

1336 Kd)

Ref.

67Nil

73Nil

67Svl

76Lal

76Lal

76Lal

m

ij

q

110

xx zz XY

13.278 -1.351 14.629

13.570 -1.078 15.542

13.102 -1.417 14.820

12.275 -1.321 14.062

11.682 -1.424 14.384

11.718 -1.787 13.653

200

xx YY

- 0.041 -0.198

0.199 - 0.209

0.361 -0.238

0.696 -0.355

1.554 0.074

0.238 - 0.279

211

xx YY YZ xz

0.742 0.284 0.153 0.306

0.442 0.315 0.113 0.217

0.642 0.315 0.190 0.385

0.744 0.246 0.185 0.331

0.658 0.193 0.155 0.310

0.325 0.095 0.076 0.153

220

xx z.2 XY

0.350 -0.327 0.677

0.112 -0.100 0.226

0.104 - 0.284 0.396

-0.174 - 0.024 -0.150

- 0.490 - 0.024 - 0.466

-0.246 0.093 - 0.339

310

xx YY zz XY

-0.195 - 0.006 0.017 - 0.071

- 0.223 - 0.020 -0.186 0.084

-0.137 0.009 -0.016 - 0.055

-0.190 - 0.080 - 0.067 - 0.041

-0.120 - 0.063 -0.055 - 0.022

- 0.074 -0.092 -0.095 0.007

222

xx YZ

-0.137 -0.135

-0.141 -0.126

-0.138 - 0.232

321

xx YY zz YZ xz XY

0.022 0.100 - 0.031 - 0.006 - 0.040 0.034

400

xx YY

0.016 0.123

mm-‘]

‘) Axially symmetric model, fit includes isothermal elastic constants. b, General forces determined largely from off symmetry phonons. ‘) Axially symmetric 5th neighbour force constants. d, General force for next neighbour, otherwise axially symmetric model.

48

Schober/Dederichs

Ref. p. 1801

1.2 Phonon states: Cu

2. Frequency spectra and related properties

THz 6.8 I 6.4 s'

n-

t

(

(

( ---

0.1 - / b&Y cl0

1

2

1336K

3

Fig. 6. Cu. Debye cutoff frequencies Y, obtained from g(v) for 49 K and 298 K g(v) of Fig. 5(a) Cu, for 973 K and 1336 K g(v) of Fig. 5(b) Cu.

(

\\\$11 !,\\ ’II u \‘, II

4

5

6

340

7 THz 8

Y-

Fig. 5 a, b. Cu. Frequency spectra at different temperatures obtained from Born-von Karman constants a) for 49 K and 298 K coupling constants from [67Nil] b) for 296 K, 973 K, and 1336K coupling constants from [76Lal].

II 330 00 00 320

300@ 00

50

100

150 T-

200

250 K 3OC

Fig. 7. Cu. Comparison of the calculated Debye temperatures O,(T) with experiments. Curve A and B refer to the spectra for 49 K and 298 K [67Nil]. Curve C represents the results for the 80 K general force constant model [73Nil].

8 - Cd Kmr

- _ _ _ ___ _

A

T . cp A cp -c( ---ch 0

100

200

300

400

K

500

‘ig. 8. Cu. Debye-Waller exponent 2Wdivided by the recoil ‘equency of a free copper atom, va, calculated for the spectra f Fig. 5 Cu.

0

200

400

expt. I41 Gi 11 expt. [ 65 Pa 1 I BvK [71MillBvK [ 71 Mi I]

600

800 K IC100

T-

Fig. 9. Cu. Specific heat, calculated to lowest order in the phonon anharmonicity (C, specific heat at constant pressure, C, total lattice specific heat, C, harmonic lattice specific heat) according to [71Mil].

Schoher/ Dederichs

49

1.2 PhononenzustZnde:

Cu

[Lit. S. 180

Table 4. Cu. Phonon frequency spectraat various temperaturescalculated from Born-von Karman forceconstants. T

49 K

298K

296K

673K

1336K

T

49 K

298K

296K

673K

1336K

Ref.

67Nil

67Nil

76Lal

76Lal

76Lal

Ref.

67Nil

67Nil

76Lal

76Lal

76Lal

KHz,

f&r,

KHz,

$Iz-

0.0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70

0.0 o.ooo1 0.0003 0.0007 0.0012 0.0021 0.0029 0.0039 0.0050 0.0063 0.0079 0.0096 0.0115 0.0135 0.0157 0.0180 0.0208 0.0235 0.0267 0.0301 0.0337 0.0378 0.0422 0.0469 0.0522 0.0581 0.0645 0.0719 0.0802 0.0900 0.1009 0.1142 0.1308 0.1534 0.2008 0.2137 0.2188 0.2240

3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50

0.2291 0.2340 0.2392 0.2438 0.2486 0.2531 0.2573 0.2617 0.2658 0.2698 0.2736 0.2775 0.2432 0.2142 0.1905 0.1902 0.1886 0.1862 0.1819 0.1757 0.1670 0.1544 0.1367 0.1098 0.0925 0.1519 0.2420 0.3358 0.3794 0.4207 0.3517 0.2488 0.1899 0.1397 0.0865 0.0081 0.0 0.0

0.2337 0.2387 0.2437 0.2483 0.2530 0.2572 0.2616 0.2658 0.2695 0.2730 0.2767 0.2599 0.2303 0.2128 0.2015 0.1982 0.1939 0.1878 0.1797 0.1691 0.1548 0.1352 0.1065 0.0821 0.1201 0.1801 0.2699 0.3499 0.3862 0.4250 0.3090 0.2264 0.1721 0.1243 0.0625 0.0187 0.0 0.0

0.2454 0.2505 0.2552 0.2599 0.2645 0.2688 0.2724 0.2762 0.2795 0.2827 0.2851 0.2440 0.2118 0.1811 0.1788 0.1761 0.1726 0.1680 0.1621 0.1550 0.1460 0.1345 0.1194 0.0971 0.0954 0.1536 0.2390 0.3160 0.3523 0.3857 0.3663 0.2446 0.1863 0.1388 0.0844 0.0486 0.0014 0.0

0.2619 0.2672 0.2723 0.2769 0.2814 0.2851 0.2892 0.2914 0.2942 0.2489 0.2278 0.2078 0.1945 0.1877 0.1800 0.1715 0.1619 0.1508 0.1372 0.1200 0.0962 0.0788 0.1297 0.1910 0.3069 0.3490 0.3842 0.4192 0.2974 0.2148 0.1589 0.1059 0.0556 0.0184 0.0 0.6 0.0 0.0

0.2761 0.2804 0.2844 0.2882 0.2916 0.2557 0.2223 0.1797 0.1765 0.1727 0.1683 0.1635 0.1579 0.1514 0.1439 0.1345 0.1229 0.1078 0.0859 0.1071 0.1636 0.2604 0.3106 0.3447 0.3773 0.3683 0.2409 0.1830 0.1362 0.0861 0.0392 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 o.ooo1 0.0003 0.0008 0.0014 0.0021 0.0031 0.0044 0.0056 0.0071 0.0086 0.0105 0.0124 0.0144 0.0170 0.0195 0.0221 0.0253 0.0284 0.0321 0.0358 0.0399 0.0444 0.0493 0.0549 0.0609 0.0676 0.0752 0.0835 0.0931 0.1044 0.1179 0.1346 0.1571 0.2069 0.2182 0.2235 0.2290

0.0 0.0001 0.0003 0.0009 0.0015 0.0022 0.0032 0.0044 0.0059 0.0074 0.0090 0.0108 0.0131 0.0153 0.0180 0.0206 0.0235 0.0268 0.0301 0.0339 0.0380 0.0425 0.0472 0.0524 0.0581 0.0643 0.0713 0.0790 0.0877 0.0974 0.1087 0.1220 0.1379 0.1584 0.1901 0.2292 0.2349 0.2402

0.0 0.0001 0.0004 0.0011 0.0018 0.0028 0.0038 0.0054 0.0069 0.0089 0.0109 0.0131 0.0155 0.0184 0.0213 0.0244 0.0279 0.0316 0.0357 0.0401 0.0447 0.0501 0.0551 0.0619 0.0687 0.0762 0.0846 0.0942 0.1050 0.1175 0.1325 0.1510 0.1757 0.2319 0.2374 0.2447 0.2503 0.2565

0.0 0.0002 0.0008 0.0017 0.0030 0.0046 0.0067 0.0091 0.0120 0.0149 0.0179 0.0215 0.0255 0.0296 0.0340 0.0390 0.0443 0.0500 0.0562 0.0627 0.0702 0.0783 0.0873 0.0972 0.1088 0.1220 0.1377 0.1581 0.1891 0.2252 0.2314 0.2376 0.2439 0.2496 0.2555 0.2610 0.2663 0.2713

r,

3. Theoretical models A reasonable fit to the dispersion curves is already obtained with a general nearest neighbour force model. Born-von Karman fits including six or eight neighbour shells reproduce all measuredphonon frequenciesvery well. The other phenomenological models give in general good tits. Attempts to describethe dynamics of Cu on more microscopic terms have however met with considerable difficulties, due to the effectsof s- d electron hybridization. Local model potentials have failed to give good tits. Augmenting a local model potential by a shell model to represent the d-electrons, Filek was able to reproduce the phonons with a deviation of 2.5 % [75Fil], seeFig. 10 Cu. Recently Upadhyaya and Dagens were able to reproduce the phonon dispersion using a resonant model potential constructed for d-band materials [78Upl], seeFig. 11 Cu. 50

Schober/ Dederichs

1.2 Phonon states

Ref. p. 1801

I 0

I

0.2 0.4

I

cu

I

0.6 0.8

1.0 0.l

Fig. 10. Cu. Fit to the phonon dispersion by a combination of a local model potential, shell model, and nearest neighbour forces. The broken and full lines represent different screening functions [75Fil].

0

0.2 0.4 0.6 0.8

0.8 0.6 0.4 0.2

fFig. 11. Cu. Phonon dispersion curves calculated with a resonant model potential for d-band materials [78Upl]. The broken and full lines refer to two different approximations of the model. Experimental points from [67Nil].

Born-von Karman and equivalent models see Table 3 Cu and [67Svl, 67Ni1, 76La1, 73Ni1, 66Si1, 71Mi1, 70Bul], further references: [62Srl, 65Shl,66Del, 67Yul,68Val, 71Sh2,73Go2,73Shl]. Breathing shell model: [71Hal]. Models comprising short ranged forces plus a simple electronic contribution: [69Krl, 73Sh2,74Sh1,78Ku2, 75Ku2, 77Sal,78Si3], further references: [64Shl, 64Sh2, 67Gu1, 69Be1, 70Be1, 71Ba1, 71Pr1, 71Sh1, 73Sh2, 74Gol,74Go2,75Bel, 75Cll,76Mal, 77Dil,77Khl, 77Kh2,75Kul, 76Gol]. Plane wave Hartree-Fock calculations: [58Tol, 65Sr2,65Srl, 66Scl,66Sil, 72Prl]. Local model potential calculations: [76Kul, 77Nal,78Shl]. Local model potential calculations with an additional shell to reproduce d-electrons: [75Fil, 75Fi2]. Calculations taking the d-electrons explicitly into account: [78Upl], further references:[72Mol, 72Prl,73Sil, 73Anl,78Ku2,75Lal, 75Si3,78Ku4]. Schober/Dederichs

51

1.2 Phononenzusttinde:

Dy

Dy, Er

[Lit. S. 180

Dysprosium

Lattice: hcp a= 358pm = 3.58A, c= 565pm = 5.65A. BZ: seep. 450. The phonon dispersion of dysprosium has not yet been measured.Fig. 1 Dy shows a theoretical estimate of the dispersion. The twelve parameters of the Born-von Karman model were determined from the elastic constants and from zone boundary and zone centre frequencies gained by interpolation between the measuredvalues of Tb and Ho. The corresponding estimate of the phonon spectrum is shown in Fig. 2 Dy [73Ra4].

0

0.1

0.2

0.3

04

0.5

0

0.1

b-

0.2

0.3

0.4

0.5

1

2

THz 3

Y-

Fig. la, b. Dy. Theoretical estimate of the phonon dispersion The twelve parameters of the Born-von Karman model were titted to the measured elastic constants and to some phonon frequencies interpolated between the measured values of Tb and Ho [73Ra4].

Er

0

5-

Fig. 2. Dy. Phonon frequency distribution obtained from the same model as Fig. 1 Dy [73Ra4].

Erbium

Lattice: hcp, a= 355pm = 3.55A, c= 558pm = 5.58A. BZ: see p. 450. No measurementof phonon frequenciesis available. Fig. 1 Er shows a theoretical prediction of the dispersion obtained assuming a Lennard-Jones interaction potential between the ions. The calculated frequenciesdiffer up to 12% from the ones obtained by extrapolation from the measuredvalues of Tb and Ho. The corresponding estimate of the phonon spectrum is shown in Fig. 2 Er [77Ra5]. Further references:[73Ral]. A-

C-

3

orb. units

I

2

-2 -cl 1

0

0.1

0.2 5-

0.3

0.L

0.5

0

0.1

0.2 t-

0.3

0.L

0.5

Fig. la, b. Er. Theoretical estimate of the phonon dispersion, The ion-ion interaction was simulated by a Lennard-Jones potential [77RaS].

52

Schober / Dederichs

0

1

2

1Hz 3

Y-

Fig. 2. Er. Theoretical estimate ofthe phonon spectrum obtained from the same model as Fig. 1 Er [77Ra5].

Ref. p. 1801

Fe

1.2 Phonon

states: Fe

Iron

Lattice: CIphase bee, a = 287 pm = 2.87 A. BZ: see p. 448.

1. Phonon dispersion Table 1. Fe. Measurements. Method

Fig.

Ref.

1 Fe

Minkiewicz et al. [67Mil] Brockhouse et al. [67Brl] Van Dijk and Bergsma [68Va2] Van Dingenen and Hautecler [67Val]

TKI neutron diffraction (TAS) neutron diffraction (TAS) neutron diffraction (TAS)

295 296 296

neutron diffraction (TOF)

2Fe

296

Further references: [52Cul, 611yl,62Lol,

67Bel].

The four major measurements agree very well. A comparison of the measurements of [67Mil] and [67Brl] suggests an accuracy of about 1 ‘A. The dispersion curves show no indications of particular anomalies. They can be fitted reasonably already with a third neighbour model. For good fits forces up to the fifth neighbours have to be included. A-

G-

IOf

N I

f-

5-

%c-

D-

P

F-

A-

N

0.6

Schober/Dedericbs

dispersion Fig. 1 a-c. Fe. Phonon curves in a-iron at 295 K. ExperiDashed mental points: [67Mil]. curve: fifth neighbour Born-von Karman model (Table 3 Fe [67Mil]).

53

1.2 Phononenzustgnde: Fe

1.0

-0 P

G-

A-

N

H

P 1

lDH .lHz

[Lit. S. 180 --c N

0.8

0.6 0.1 0.2 -% Fig. 2. Fe. Phonon dispersion curves in a-iron at 296 K. Experimental points: [68Va2]. Solid curve: fifth neighbour Born-von Karman model (Table 3 Fe [68Va2]).

Table 2. Fe. Measured phonon frequencies in cl-iron at 295 K, [67Mil],

4.

Y fTHz]

I

CWI L

v [THz]

r

v

t;

v

COWT

3.34 (7) 4.79 (10) 6.04 (10) 7.08 (10) 7.78 (7) 8.27 (10) 8.58 (18) 8.66 (12) 8.70 (15) 8.56 (5)

0.091 0.182 0.274 0.365 0.456 0.50 0.60 0.70 0.80 0.90 1.0

1.21 (5) 2.47 (1) 3.63 (2) 4.66 (5) 5.63 (5) 6.07 (5) 6.94 (5) 7.59 (7) 8.07 (12) 8.56 (12) 8.56 (5)

0.064

8.44 (12) 8.12(10) 7.64 (7) 6.79 (10) 6.31 (10) 6.08 (7) 5.34 (7) 4.62 (7) 4.47 (12)

0.323 0.387 0.452 0.500

6.99 (12) 6.70 (10) 6.33 (10) 6.45 (5)

COSSI L 0.064 0.097 0.105 0.113 0.121 0.129 0.137 0.145 0.153 0.163 0.193 0.226 0.258 0.290 0.323 0.339 0.355 0.371 0.403 0.436 0.468 0.50

1.81 (7) 2.78 (7) 3.12 (5) 3.34 (5) 3.58 (5) 3.77 (5) 4.01 (5) 4.21 (5) 4.42 (5) 4.57 (7) 5.39 (7) 6.16 (7) 6.91 (5) 7.57 (5) 8.12 (5) 8.32 (7) 8.58 (7) 8.73 (7) 8.97 (7) 9.24 (7) 9.26 (10) 9.26 (12)

r

v

CTHzl

D-Hz1

0.182 0.274 0.365 0.456 0.547 0.639 0.730 0.821 0.912 1.00

0.129 0.193 0.258 0.300 0.322 0.387 0.452 0.484

[Oc[] T, branch Ref. [67Brl].

CW’~lT 0.0625 0.125 0.1875 0.25 0.375 0.50

0.80(3) 1.63 (4) 2.40(4) 3.13 (4) 4.20 (5) 4.53 (5)

D-Hz1 CNCIT, 0.100 0.150 0.200 0.250 0.258 0.290 0.323 0.355 0.387 0.419 0.452 0.484 0.500

1.91 (5) 2.88 (5) 3.72(5) 4.47 (5) 4.59 (5) 5.03 (5) 5.42 (5) 5.78 (5) 6.07 (5) 6.31 (5) 6.41 (5) 6.45 (5) 6.45 (5)

(continued)

54

Schober/Dederichs

Ref. p. 1801

1.2 Phonon states: Fe

Table 2. Fe. (continued)

l

v [THz]

5

v [THz]

r

v [THz]

5

v [THz]

0.105 0.158 0.211 0.263 0.316 0.343 0.369 0.422 0.474 0.500 0.527 0.553 0.579 0.606 0.632 0.685 0.738 0.789 0.842 0.895 0.948

3.14(10) 5.42(12) 6.77(12) 7.91 (12) 8.20 (10) 8.39 (10) 8.36(7) 8.17 (5) 7.59(7) %20(5) 6.89(2) 6.60(2) 6.26(2) 5.91(5) 5.75 (2) 5.73 (10) 6.02(12) 6.70(5) 7.35(5) 8.03(5) 8.44(7) 8.56(5)

0.105 0.158 0.211 0.263

1.91(10) 2.90(5) 3.89(5) 4.81(7) 5.66 (10) 6.23(5) 6.12(5) 7.11 (5) %35(5) 7.69(5) 7.95(7) 8.07(7) 8.34(7) 8.32 (10) 8.44 (15) 8.49(10) 8.61 (10) 8.56(5)

0.091 0.137 0.182 0.273 0.365 0.456 0.541 0.681 0.772 0.863 0.954

9.26 (10) 9.02(12) 8.90 (10) 8.56(7) 8.05(7) 7.49(7) 6.96(7) 6.21(10) 5.42(12) 4.86(7) 4.55(5)

0.588 0.681 0.112 0.863 0.954

7.11 (10) 6.96(12) 6.77(7) 6.62(10) 6.45 (10)

1.ooo

0.316 0.368 0.422 0.414 0.529 0.579 0.632 0.685 0.738 0.790 0.843 0.896 0.948

1.00

2. Frequency spectrum and related properties 0.5 THz-'

Table 3. Fe. Born-von @‘;, for a-iron.

Ref. ij

111

xx XY

200

xx

220

YY xx zz

311

XY xx

222

YY YZ xz xx

0.4 I 0.3 -2 -G 0.2

I

T

m

0

2

4

6

8

THz IO

coupling

constants,

295 K

296 K

67Mil

68Va2

~0: [Nm-‘1 16.88 15.01 14.63 0.55 0.92 -0.57 0.69 -0.12 0.03 0.52 0.007 -0.29 0.32

XY

0.1

Karman

17.86(10) 14.91(13) 14.92(25) 0.36(14) 1.24(8) -1.09(13) 0.30(12) -0.60(8) -0.06(4) 0.28(8)

0.10 (5) -0.23(7) -0.24 (10)

48C K

Y-

Fig. 3. Fe. Frequency spectrum of a-iron at 295 K calculated from the Born-von Karman force constants of Table 3 Fe [67Mil].

46C

I 0 4400 420 -

-10

0

IO n-

20

30

4 Fig. 4. Fe. Debye cutoff frequencies, v,, in a-Fe, calculated from the spectrum of Fig. 3 Fe.

4001 0

80

160

320

K

LOO

T-

Fig. 5. Fe. Debye temperatures 0, in a-Fe calculated from a fifth neighbour Born-von Karman model [67Brl].

Schober/ Dederichs

55

1.2 PhononenzusCnde: Fe

[Lit. S. 180

0.8 40‘12

s I 0.6 I -t” 0.1 B N 0.2

bb

Fig. 6. Fe. Dehyc-Wailer exponent 2A’dividcd by the recoil frequency of the free ion. \iR, calculated from the spectrum of Fig. 3 Fe.

0

100

200

300

600 K

500

TTable 4. Fe. Phonon frequency spectrum of a-iron at 296 K calculated from a fifth neighbour Born-von Karman model (Table 3 Fe, [67Mil]). v [THz-j

g(v) [THz- ‘1

\’ [THz]

g(v) [THz- ‘1

v [THz]

g(v) [THz- ‘1

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

0.000 0.000 0.000 0.001 0.001 0.001 0.002 0.002 0.003 0.003 0.004 0.005 0.006 0.006 0.007 0.009 0.009 0.011 0.012 0.013 0.015 0.016 0.018 0.020 0.022 0.023 0.025 0.027 0.030 0.032 0.035 0.038

3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40

0.041 0.044 0.047 0.051 0.055 0.060 0.065 0.070 0.076 0.083 0.092 0.102 0.124 0.130 0.135 0.140 0.147 0.153 0.159 0.166 0.175 0.184 0.193 0.206 0.205 0.178 0.170 0.167 0.167 0.170 0.177 0.190

6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20 8.30 8.40 8.50 8.60 8.70 8.80 8.90 9.00 9.10 9.20 9.30 9.40

0.229 0.220 0.211 0.204 0.197 0.191 0.184 0.177 0.165 0.157 0.151 0.147 0.145 0.145 0.147 0.150 0.158 0.172 0.198 0.237 0.259 0.383 0.435 0.255 0.195 0.154 0.121 0.090 0.058 0.009

1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20

3. Theoretical models Due to the absence of pronounced anomalies phenomcnological models describe the dispersion in general well. First principle calculations are not available. Born-von Karman and equivalent models: see Table 3 Fe and [67Brl],

further references: [76Ca2, 77Pal).

Short ranged forces plus a simple electronic contribution: [66Ma2, 69Krl,76Brl], 69Shl. 72Bal,72Bel, 74Kul,75Upl, 76Ku2,76Si2,77Gul, 78Khl]. Model potential: [73Anl]. 56

Schober/Dederichs

further references: [68Va2.

Ga

Gallium

Lattice: a-phase, orthorhombic,

A 11, a = 452 pm= 4.52 A, b= 765 pm = 7.65 A, c = 451 pm =4.51 A. BZ: see p. 456.

1. Phonon dispersion Table 1. Ga. Measurements. Method

Fig.

Ref.

1 Ga

Reichardt et al. [69Rel] Waeber [69Wa5]

k, neutron diffraction VW neutron diffraction (TAS)

77 77

Gallium has four atoms per unit cell and hence a complicated dispersion with twelve branches. A group theoretical analysis is given by Waeber [69Wa4]. The results of the two available measurements differ in some branches strongly. The unpublished results of [69Rel] are thought to be the more reliable ones due to a higher neutron flux. The gross features can be fitted with an axially symmetric model to 19” neighbours. The measured abrupt changes of frequency in some directions, e.g. in [[, 0, 0] direction, cannot be reproduced by such a model due to its short range. See Fig. 2 Ga.

Table 2. Ga. Measured phonon frequencies at 77 K [69Wa5]. v [THz]

r Branch 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.50 0.55 0.60 0.70 0.80 0.90 0.95 1.00

Tjl’

T{ZJ

T$3)

TJl’

T&2’

Ti3’

T$l’

7.45 (27)

2.78 (10)

4.62 (29)

5.75 (27)

9.67 (44)

0.82 (2)

7.59 (29)

3.50 (17)

4.42 (24)

5.66 (24)

1.64 (7)

7.47 (24)

4.59 (29)

3.99 (17)

5.44 (19)

2.37 (10)

7.28 (24)

3.63 (17)

5.20 (7)

1.64 (7) 2.08 (5) 2.54 (7) 2.88 (10) 3.02 (12) 3.09 (15) 3.14 (15) 3.14 (17)

3.24 (12) 3.87 (12)

7.25 (29) 7.25 (27)

5.80 (36) 6.65 (41)

3.14 (17) 2.68 (12)

4.91(10) 4.62 (12)

4.55 (19) 5.00 (22) 5.44 (22) 5.75 (27)

7.18 (29) 7.01 (24) 6.89 (24) 6.77 (31)

6.91 (39)

2.08 (7) 1.57 (5) 1.33 (2) 1.16 (2)

4.42 (10) 4.35 (15) 4.33 (19) 4.34 (17)

5.90 (27)

6.70 (34)

9.07 (44)

1.09 (2)

4.35 (22)

5.44 (24)

3.07 (15) 3.02 (15) 3.00 (12) 3.00 (12) 3.09 (15) 3.02 (5) 2.78 (5) 2.54 (10)

T$3’

Til’

Ti2’

8.03 (48)

3.26 (15)

2.88 (10)

0.48 (2)

3.26 (15)

2.90 (10)

0.92 (2)

3.31 (12)

2.97 (10)

1.21(5)

3.02 (7)

3.02 (12)

1.69 (7)

1.43 (5) 1.64 (5)

2.73 (10) 2.42 (10)

3.02 (12) 3.02 (12)

1.93 (7) 2.08 (10)

1.93 (5) 2.18 (7) 2.35 (7) 2.35 (10)

2.32 (12) 2.64 (10) 2.90 (10) 2.80 (10)

3.09 (15) 3.05 (19)

2.15 (12) 1.98 (10) 1.60 (7)

2.27 (12)

3.00 (15)

1.18 (7) (continued)

Ti2’

2.37 (10)

1.28 (5)

Ti3’

Table 2. Ga. (continced) v [THz-J

I Branch 0 0.05 0.1 0.2 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.0

c\”

c\*’

z(:’

c\“’

c’:’

cy

z$*’

7.45 (27)

8.03 (48)

5.75 (24) 5.83 (29) 5.92 (29) 5.92 (36)

2.88 (10)

9.67 (44) 8.95 (48)

1.16 (7)

2.78 (10) 2.90 (10) 3.05 (12) 2.97 (12) 2.93 (15) 3.02 (12) 3.14 (12) 3.24 (17) 3.34 (15) 3.60 (12) 3.80 (15) 3.87 (19) 3.94 (19) 4.16 (19) 4.40 (24) 4.81 (27) 5.20 (24) 5.39 (24) 5.44 (24)

A\“’

0.89 (2) 1.33 (2) 1.76 (5) 2.18 (5) 2.59 (2) 2.90 (5) 3.26 (10) 3.67 (15) 3.99 (12)

0.60 (2) 1.09 (5) 1.57 (5) 1.98 (10) 2.30 (10)

3.02 (19)

3.34 (12) 3.41 (12) 3.55 (15) 3.72 (12) 3.82 (15) 3.99 (15) 4.13 (17) 4.28 (19) 4.35 (19) 4.35 (22) 4.35 (24)

Branch

A’1”

A’:’

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

2.88 (10) 3.41 (15) 3.55 (12) 3.70 (12) 4.09 (19) 4.01 (12) 3.38 (12) 2.75 (7) 2.18 (7) 1.86 (10) 1.67 (7)

7.45 (27)

2.61 (15) 2.78 (15) 2.90 (17) 3.02 (17)

1.57 (5) 1.45 (5) 1.40 (5) 1.35 (7) 1.18 (5)

6.89 (34)

2.95 (29) 2.93 (12) 2.88 (7) 2.80 (7)

2y

cs”

cy

ck”

g’

1.67 (10)

3.26 (12)

9.24 (48)

0.39 (2) 0.73 (2) 1.04 (2)

0.60 (2) 0.85 (2)

1.67 (7) 1.89 (7) 2.03 (7)

3.22 (12) 3.19 (17) 3.17 (12)

4.62 (31) 5.08 (29) 5.56 (27) 5.73 (19) 5.73 (24)

9.19 (48)

1.16 (2)

0.92 (2)

2.15 (7)

3.12 (10)

5.73 (24)

2.76 (10)

1.26 (2)

0.97 (2)

2.27 (7)

3.02 (12)

5.83 (27)

1.28 (2)

0.99 (2)

2.35 (12)

2.85 (15)

5.85 (24)

6.89 (36)

2.73 (10)

9.43 (48) 9.19 (36)

7.01 (36)

2.49 (7)

8.75 (36)

1.28 (2)

1.04 (2)

2.35 (10)

2.83 (12)

5.87 (19)

6.52 (29)

2.42 (7)

8.44 (36)

1.28 (2)

1.09 (2)

2.30 (10)

2.66 (10)

5.92 (19)

6.52 (34)

8.70 (41)

1.28 (2)

1.09 (2)

2.30 (12)

2.59 (12)

5.92 (19)

6.70 (34)

2.39 (7) 2.39 (10) 2.37 (10)

9.07 (44)

1.28 (2)

1.09 (2)

2.27 (12)

2.54 (12)

5.92 (29)

A\‘+’

Ai1 ’

Ai”’

AY’

AI”’

A\”

A\“’

Ak”

Ak”

2.78 (10) 2.83 (12) 3.05 (12) 3.14 (17) 3.46 (15) 3.50 (15) 3.63 (10) 3.80 (15) 3.94 (17) 3.99 (17) 3.99 (12)

3.34 (24)

8.03 (48)

9.67 (44)

1.67 (10)

3.26 (12)

6.04 (24) 5.44 (24) 5.20 (19) 5.08 (19) 4.91 (22) 4.71 (17) 4.52 (24) 4.28 (24)

6.45 (36) 6.00 (36) 5.56 (24) 5.32 (29) 5.08 (24) 4.88 (19)

4.62 (31) 4.35 (24) 3.89 (15) 3.75 (19) 3.70 (19) 3.38 (12) 3.14 (15) 3.12 (17) 2.80 (10) 2.61 (10) 2.42 (12)

4.84 (29) 4.11 (24)

0.58 (2) 0.85 (5) 1.18 (5) 1.40 (5) 1.52 (5) .1.45 (7) 1.55 (5) 1.62 (7) 1.67 (7)

2.18 (15) 2.42 (12) 2.88 (17) 2.66 (10) 2.49 (12) 2.42 (12)

0.82 (2) 1.23 (2) 1.69 (5) 1.81 (10) 1.72 (10) 1.93 (7) 2.13 (10) 2.30 (12)

-

‘Z

Ref. p. 1SO] l-4-

x-

2-

r.l-

k-

-l--

c-

>-

-I-

‘G

1.2 Phonon states: Ga

Schober/Dederichs

1.2 Phononenzusttinde:

Gd

-Z

[Lit. S. 180 A-

x - direction y- direction z-direction

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0 0.5 -c ffFig. 2. Ga. lSh neighbour Born-von Karman lit to the measured phonon dispersion at 77 K (full line). The broken curves in [[, O,O] direction indicate the deviation from the experimental values [69Rel]. 0

Gd

Gadolinium

Lattice: hcp, n= 363 pm = 3.63 A, c= 579 pm = 5.79 A. BZ: seep. 450. The phonon dispersion of gadolinium has not been measured so far. Fig. 1 Cd shows a theoretical estimate of the dispersion. The model uses twelve parameters which represent two and three body interactions up to sixth neighbours. The parameters are determined from the measured elastic constants and some frequencies obtained by extrapolation from Tb and Ho. The corresponding estimate of the phonon spectrum is shown in Fig. 2 Cd [74Ra2]. 60

Schober/Dederichs

1.2 Phononenzusttinde:

Gd

-Z

[Lit. S. 180 A-

x - direction y- direction z-direction

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0 0.5 -c ffFig. 2. Ga. lSh neighbour Born-von Karman lit to the measured phonon dispersion at 77 K (full line). The broken curves in [[, O,O] direction indicate the deviation from the experimental values [69Rel]. 0

Gd

Gadolinium

Lattice: hcp, n= 363 pm = 3.63 A, c= 579 pm = 5.79 A. BZ: seep. 450. The phonon dispersion of gadolinium has not been measured so far. Fig. 1 Cd shows a theoretical estimate of the dispersion. The model uses twelve parameters which represent two and three body interactions up to sixth neighbours. The parameters are determined from the measured elastic constants and some frequencies obtained by extrapolation from Tb and Ho. The corresponding estimate of the phonon spectrum is shown in Fig. 2 Cd [74Ra2]. 60

Schober/Dederichs

1.2 Phonon states: Hf

Ref. p. 1801

3 arb. unit: 2 I

0

a

b

%-

2

THz 3

VW

Fig. 2. Gd. Phonon frequency distribution obtained from the same model as Fig. 1 Gd [74Ra2].

Fig. la, b. Gd. Theoretical estimate of the phonon dispersion. The twelve parameters of the model were fitted to the measured elastic constants and to some phonon frequencies obtained by extrapolation from Tb and Ho [74Ra2].

Hf

1

%-

Hafnium

Lattice: hcp, a = 319pm = 3.19A, c = 505pm = 5.05A. BZ : see p. 450. Fig. 1 Hf shows a theoretical estimate of the dispersion curves.The model usescentral fourth neighbour interaction plus a screenedCoulomb interaction term. The five parameters of the model are determined from the elastic constants.A comparison of the calculated Debye temperature to experimental values is shown in Fig. 2 Hf. The phonon dispersion of Hf at 295,800,and 1300K has beenmeasuredby neutron spectroscopy(TAS) by [80Stl, BOSt2],seeFig. 3 and 4 Hf. 1’ -

T-

-E r

0

0.2

0.1

0.3

0.4

0.5

f-

0.4

0.3 -C

0.2

0.1

0

A-

Fig. 1a, b. Hf. Theoretical estimate of the phonon dispersion using a five parameter model fitted to the measured elastic constants [74Ra3].

Schober/Dedericbs

1.2 Phonon states: Hf

Ref. p. 1801

3 arb. unit: 2 I

0

a

b

%-

2

THz 3

VW

Fig. 2. Gd. Phonon frequency distribution obtained from the same model as Fig. 1 Gd [74Ra2].

Fig. la, b. Gd. Theoretical estimate of the phonon dispersion. The twelve parameters of the model were fitted to the measured elastic constants and to some phonon frequencies obtained by extrapolation from Tb and Ho [74Ra2].

Hf

1

%-

Hafnium

Lattice: hcp, a = 319pm = 3.19A, c = 505pm = 5.05A. BZ : see p. 450. Fig. 1 Hf shows a theoretical estimate of the dispersion curves.The model usescentral fourth neighbour interaction plus a screenedCoulomb interaction term. The five parameters of the model are determined from the elastic constants.A comparison of the calculated Debye temperature to experimental values is shown in Fig. 2 Hf. The phonon dispersion of Hf at 295,800,and 1300K has beenmeasuredby neutron spectroscopy(TAS) by [80Stl, BOSt2],seeFig. 3 and 4 Hf. 1’ -

T-

-E r

0

0.2

0.1

0.3

0.4

0.5

f-

0.4

0.3 -C

0.2

0.1

0

A-

Fig. 1a, b. Hf. Theoretical estimate of the phonon dispersion using a five parameter model fitted to the measured elastic constants [74Ra3].

Schober/Dedericbs

1.2 Phononenzustiinde: Hf

[Lit. S. 180 A-

.

THz

3.5 3.0 180

0

20

40

60

2.5

80 K 100

IFig. 2. Hf. Comparison of the calculated Debye temperature (model of Fig. 1 Hf) with experimental values [74Ra3].

I 2.0 ir 1.5

1.0

0.5 0

0.2 t-

0.1

0.3

OX

0.5

Fig. 3. Hf. Room temperature and 1300K measurements of the dispersion curves of hcp Hf along the [OOl] symmetry direction [8OStl].

A-

-E

T

I a

0

0.1

0.2 0.3 0.4

I;-

-l

6-

Fig. 4. Hf. Measured dispersion curves at 295 K. The lines show a theoretical model [8OSt2].

62

Schober/Dederichs

1.2 Phonon states: Hg

Ref. p. 1801

Hg

Mercury

Lattice: rhombohedral (AlO), triatomic hexagonal cell: a= 346pm = 3.46A, c = 668 pm = 6.68A. BZ: seep. 454.

1. Phonon dispersion Table 1. Hg. Measurements Method

Fig.

Ref.

1 Hg

Kamitakahara et al. [77Kal]

TKI neutron diffraction (TAS)

4 THZ

fi , IOO%l

80

1

it001

3,

r-e 04g1

Fig. 1. Hg. Measured phonon dispersion in solid mercury. The wavevector is given in a hexagonal coordinate system in units of [(4x/l/?a), 4a/fia, 2x/c]. Only the [OOC]direction is a symmetry direction. All other directions lie in a mirror plane and there is always a pure transverse mode with eigenvector perpendicular to the mirror plane. These modes are shown as triangles. The solid line represents an eighth neighbour Born-von Karman model [77Kal].

Mercury has a one-atomic unit cell i.e. only acoustic phonon branches. Someof its transversebranches have very low frequenciesat the zone boundaries, as small as 6 y0 of the maximum lattice frequency.

2. Frequency spectrum 0.3 THz' I 0.2 3 G

0.1

0

0.5

I.0

1.5

2.0 Y-

2.5

3.0

3.5 THz4.0

Fig. 2. Hg. Phonon frequency spectrum in solid mercury obtained from the Born-von Karman fit shown in Fig. 1 Hg [77Kal]. Schober/Dederichs

63

1.2 Phononenzustiinde: Ho

Ho

[Lit. s. 180

Holmium

Lattice: hcp. n = 358 pm = 3.58 A, c= 562 pm = 5.62 A. BZ: see p. 4.50. 1. Phonon dispersion Table 2. Ho. Selection of measured phonon frequenties at room temperature [71Ni2].

Table 1. Ho. Measurements Fig.

Method

Ref.

k1 neutron diffraction U-AS)

298

1 HO

Nicklow et al. [71Ni2]

Further reference: [69Lel].

The phonon dispersion of Ho is qualitatively oeen seen.

0.1 0.2 0.3 01 %U-

5-

v [THz]

r,+ G+ A3 A, Mi M4’

1.94 (3) 3.40 (7) 1.34 (3) 2.56 (4) 1.96 (3) 1.65 (3)

M: Mi M: K6 L, (4

3.04 (3) 3.08 (5) 3.05 (5) 2.46 (5) 1.85 (7)

A-

K

0.z

0.5 -S’

1

.M

Phonon

-1

M

II

v [THz]

similar to the one observed in Tb. No major anomalies have

-1’

C-

Phonon

0.3 -l

0.2

0.1

tR-

-s

-t

PH

1 K

A

H

5-

t-

Fig. la, b. Ho. Phonon dispersion curves at room temperature The lines shown represent the eighth neighbour Born-von Karmnn model of Table 3 Ho [71Ni2] 3) along major symmetry directions b) along the boundaries of the Brillouin zone Schober/Dederichs

1.2 Phonon states: Ho

Ref. p. 1801 Born-von Karman model

Due to the similarity of the dispersion of Ho with the one of Tb the sametype of Born-von Karman model was used,i.e. an eighth neighbour model with tensor forces to the fourth neighbours. Table 3. Ho. Born-von Karman force constants, GE, T=298 K, [71Ni2]. ij

in

@; mm-‘]

xx

ah% O,cP

0, a, 0

- 2 a/j/3,0, c/2

YY 22 xz

7.054 1.055 11.517 6.766

xx YY zz XY

1.084 12.716 -0.927 2.259

xx

-1.496

YY

-0.937

zz

- 1.066 0.965 0.080 - 3.897

xz xx zz

co, c

From the fifth neighbours outward the model is axially symmetric (@~=[R;R~/(Rm)2](f, -.fJ+f,?&) m

f, CNm-‘I

f, CNm-'I

5 a/2 fi,

a/2, c/2

afi,O,O 0, a, c 0, 2a, 0

0.488 1.213 1.048

-0.011 0.318 -0.133

-0.344

0.040

2. Frequency spectrum and related properties 1.2 THz’

THz

I 0.9

3.7

-2 0.6 -G

I c 3.5

3.9

3.3

0

0.5

1.0

1.5

2.0

2.5

3.1 -30

3.0THz 3.5

0

IO

20

30

n-

Fig. 2. Ho. Phonon frequenEy=um at room temperature calculated from the eighth neighbour Born-von Karman constants of Table 3 Ho.

Fig. 3. Ho. Debye cutoff frequencies, Y,, calculated the spectrum of Fig. 2 Ho. 6 .I~-12

s

l

0

20

40

60

T = OK expU70Pa21

80 100 120 140K 160 TFig. 4. Ho. Debye temperatures 0, calculated from the Born-von Karman constants of Table 3 Ho compared to experimental values obtained from elastic constant measurements [71Ni2].

0

400 K 500 300 TFig. 5. Ho. Debye-Wailer exponent 2Wdivided by the recoil frequency of the free holmium atom, vs, calculated from the Born-von Karman force constants of Table 3 Ho.

Schoher/Dederichs

100

200

65

1.2 Phononenzustkinde: Ho

[Lit. S. 180

fable 4. Ho. Phonon spectrum at 298 K calculated from the Born-von Karman force constants of Table 3 Ho. Y DHz]

g(v) [THz- ‘1

3.04 D.08 3.12 D.16 3.20 3.24 D.28 3.32 3.36 D.40 3.44 0.48 3.52 3.56 D.60 3.64 D.68 D.72 3.76 D.80 D.84 8.88 D.92 3.96

1.00 1.04 1.os 1.12 1.16 1.20

v [THz]

g(v) [THz- ‘1

v [THz]

g(v) [THz- ‘1

0.000

1.24

0.000 0.001

1.28 1.32 1.36 1.40 1.44 1.48 1.52 1.56 1.60 1.64 1.68 1.72 1.76 1.80 1.84

0.116 0.130 0.143 0.162 0.184 0.242 0.382 0.441 0.452 0.468 0.488 0.522 0.534 0.497 0.431 0.434 0.453 0.470 0.448 0.422 0.399 0.376 0.361 0.339 0.339 0.355 0.372 0.391 0.422 0.449

2.44 2.48 2.52 2.56 2.60 2.64 2.68 2.12 2.76 2.80 2.84 2.88 2.92 2.96 3.00 3.04 3.08 3.12 3.16 3.20 3.24 3.28 3.32 3.36 3.40 3.44 3.48

0.481 0.515 0.563 0.564 0.512 0.498 0.502 0.520 0.533 0.544 0.531 0.509 0.471 0.522 0.634 0.918 0.964 1.127 0.861 0.557 0.125 0.112

0.001 0.002 0.003 0.004 0.005 0.007 0.009

0.010 0.012 0.015 0.017 0.020 0.023 0.026 0.029 0.033 0.037 0.041 0.047 0.051 0.057 0.064 0.070 0.078 0.086 0.094 0.105

1.88 1.92 1.96 2.00 2.04 2.08 2.12 2.16 2.20 2.24 2.28 2.32 2.36 2.40

0.100 0.088 0.075 0.031 0.0

3. Theoretical models No microscopic theory is available so far. The simple phenomenological modelsare only moderately successful. Born-von Karman and equivalent models: seeTable 3 Ho, further references:[76Upl, 72Mel]. Short ranged forcesplus a simple electronic contribution: [75Ca2], further references:[70Lal, 73Upl,75Up2]. Model potential calculations: [77Up2,77Si2].

66

Schober/Dederichs

Ref. D.

1801

In

Indium

1.2 Phonon

states: In

Lattice: face centered tetragonal, A6, a =458 pm =4.58 A, c = 494 pm = 4.94 A. BZ: see p. 452.

1. Phonon dispersion Table 1. In. Measurements. Method

Fig.

Ref.

1 In

Smith and Reichardt [69Sml]

&I neutron diffraction PAS)

II

Indium is a moderately high neutron absorber and measurements are therefore difficult. An llth neighbour, 19 parameter Born-von Karman model fits the measured dispersion well. C-

G-

A,-

<-

5-

b-

-A

-c

Fig. 1 a, b. In. Measured phonon dispersion curves at 77 K. The solid line represents the llth neighbour, 19 parameter Born-von Karman model of Table 2 In [69Sml].

Schober/Dederichs

67

1.2 Phononenzustiinde: K Table 2. In. Born-von Karman coupling parameters, @;, T= 77 K, [69Sml].

m

The model is of axially symmetric form:

v,o,1) (Ll,O) GO, 0) (0,0,2) (2,L 1)

~~=(~--I;)R~R~/(R”)2+j;6ij

[Lit. s. 180 f, CNm-

‘I

12.316 16.763 1.278 1.695 -0.452 -0.601 -0.423 -1.130 0.167 -0.026 0.225

(1,1,2) co, 2) (2,2,0)

(3, ho) (LO,3) (3>0,1)

/, CNm-‘I -2.064 -2.759 0.929 0.294 0.002 0.268 -0.216 0.033

2. Frequency spectrum and related properties 0.6 1Hz-l I

0.4

r x

0.2

128.8

I s”

121.6 114.4

0

2

1

3

1Hz 4

107.2

Y-

Fig.2. In. Phonon frequency spectrum at 77 K calculated from the 11 neighhour, 19 parameter Bornvon Karman model of Table2 In [69Sml].

100.0 0

8.04

16.08

24.12

32.16

K LO.20

Fig. 3. In. Debye temperature 0, calculated from second order nonlocal pseudopotential theory compared to experimental

specific

heat data [76Gal].

3. Theoretical models The phonon dispersion of indium can be fitted well by Born-von Karman and pseudopotential models. Born-von Karman and equivalent models: seeTable 2. In and [69Sml], further references:[72Ku2]. Short ranged forcesplus a simple electronic contribution: [73Sh7]. Pseudopotential models: [76Gal], further references:[75Rel, 73Gul].

K

Potassium

Lattice: bee,a= 531pm = 5.31A. BZ: see p. 448. 1. Phonon dispersion Table 1. K. Measurements. Method neutron diffraction (TAS) neutron diffraction (TAS)

T CKI 9

Fig.

Ref.

1K

Cowley et al. [66Col]

92, 215,

299

Further measurements:[77Dol, 76Mel]. 68

Schober/Dederichs

Buyers and Cowley [69Bul]

1.2 Phononenzustiinde: K Table 2. In. Born-von Karman coupling parameters, @;, T= 77 K, [69Sml].

m

The model is of axially symmetric form:

v,o,1) (Ll,O) GO, 0) (0,0,2) (2,L 1)

~~=(~--I;)R~R~/(R”)2+j;6ij

[Lit. s. 180 f, CNm-

‘I

12.316 16.763 1.278 1.695 -0.452 -0.601 -0.423 -1.130 0.167 -0.026 0.225

(1,1,2) co, 2) (2,2,0)

(3, ho) (LO,3) (3>0,1)

/, CNm-‘I -2.064 -2.759 0.929 0.294 0.002 0.268 -0.216 0.033

2. Frequency spectrum and related properties 0.6 1Hz-l I

0.4

r x

0.2

128.8

I s”

121.6 114.4

0

2

1

3

1Hz 4

107.2

Y-

Fig.2. In. Phonon frequency spectrum at 77 K calculated from the 11 neighhour, 19 parameter Bornvon Karman model of Table2 In [69Sml].

100.0 0

8.04

16.08

24.12

32.16

K LO.20

Fig. 3. In. Debye temperature 0, calculated from second order nonlocal pseudopotential theory compared to experimental

specific

heat data [76Gal].

3. Theoretical models The phonon dispersion of indium can be fitted well by Born-von Karman and pseudopotential models. Born-von Karman and equivalent models: seeTable 2. In and [69Sml], further references:[72Ku2]. Short ranged forcesplus a simple electronic contribution: [73Sh7]. Pseudopotential models: [76Gal], further references:[75Rel, 73Gul].

K

Potassium

Lattice: bee,a= 531pm = 5.31A. BZ: see p. 448. 1. Phonon dispersion Table 1. K. Measurements. Method neutron diffraction (TAS) neutron diffraction (TAS)

T CKI 9

Fig.

Ref.

1K

Cowley et al. [66Col]

92, 215,

299

Further measurements:[77Dol, 76Mel]. 68

Schober/Dederichs

Buyers and Cowley [69Bul]

1.2 Phonon states: K

Ref. p. 1801

The phonon dispersion of K is very similar to that of Na apart from a scaling factor: 1.67=(v(Na)/v(K)> N <(i14a~)~~~/(Ma~)~~~) =1.62. This is the condition for homology of the forces.A model with first and secondnearest neighbour forces describesthe dispersion qualitatively. Longer ranged forces to the fifth neighbours are only necessaryto fit the finer details. This is in agreementwith the results of pseudopotential theory. The temperature behaviour of the phonons has been studied carefully [69Bul] and the mode Grtineisen parameters have been determined [77Dol]. Kohn anomalies have not been found. A descrepancyof up to 8 % (for err) has been reported between the values of the elastic constants obtained from the slope of the dispersion curves and from ultrasonic measurements[66Col]. -F

A-

0

0.2 0.4 fC-

-0

0.6

0.8

1.0

-A

0.8

0.6 -c

0.2

0

-G

o-

0.510 0.2 OX 0.6

t-

04

0.8 1.010.5

f‘-

-5

Fig. 1. K. Measured phonon dispersion curves at 9 K. The solid curves represent the fifth neighbour axially symmetric Born-von Karman fit (Table 3 K [66Col]).

Table 2. K. Measured phonon frequenciesat 9 K [66Col]. v [THz]

r

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55

IX151T

CWI L

CO’XIT

0.66(4) 0.890(25)

0.52(3) 0.680(25) 0.82(3) 0.99(2)

1.00(6) 1.20(6)

1.15(3) 1.280(25) 1.44(3) 1.57(2) 1.69(2)

1.53(5) 1.68(4) 1.785(20) 1.91(4)

1.08(5) 1.21(3) 1.35(4) 1.53(3) 1.63(3) 1.740(35) 1.81(4)

v [THz]

I

0.77(5)

0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

CWI L

CWI T

CC54T

1.89(3) 1.96(6) 1.965(40) 2.085(30) 2.110(25) 2.15(3) 2.190(25) 2.210(25) 2.21(2)

1.790(25) 1.89(2) 1.99(2) 2.07(3) 2.110(25) 2.19(4) 2.20(3) 2.21(4)

2.00(4) 2.05(6) 2.09(3) 2.12(4) 2.16(3) 2.19(3) 2.180(25) 2.22(3) (continued]

Schober / Dederichs

69

1.2 Phononenzust5nde: K

[Lit. S. 180

Table 2. K. (continued) v [THz-J

i

v [THz]

W-l D,

[‘ift;l Jh

[WI D,

0.10 0.20 0.30 0.40

1.52(4) 1.62(4) 1.69(4) 1.72(4)

2.35(6) 2.300(35) 2.20(5) 2.01(4)

0.70(3) 0.97(3) 1.27(3) 1.55(4)

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.74(2) ’ 1.08(3) 1.41(2) 1.69(5) 1.94(3) 2.08(5) 2.250(35) 2.33(6) 2.40(4)

Born-von

Karman

0.53(3) 0.95(2) 1.34(4) 1.695(20) 1.95(4) 2.100(25) 2.19(5) 2.150(25) 2.03(6)

0.67(3) 0.93(3) 1.11(3) 1.23(3) 1.36(3) 1.440(25) 1.49(5) 1.500(25)

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

KLII G,

CCC11 G,

2.22(5) 2.14(4) 2.01(6) 1.78(3) 1.55(4) 1.31(4) 1.07(4) 0.81(3) 0.61(3) 0.53(2)

2.18(5) 2.13(4) 2.100(35) 2.020(35) 1.895(30) 1.780(25) 1.70(4) 1.57(3) 1.53(4)

r

v [THz] ClCi-l F,

0.55 0.58 0.60 0.62 0.65 0.68 0.70 0.72 0.75 0.78 0.80 0.85 0.90 0.95

1.55(4) 1.38(5) 1.24(3) 1.17(3) 1.040(20) 1.005(25) 1.02(3) 1.06(3) 1.20(3) 1.36(3) 1.47(2) 1.77(3) 1.99(3) 2.19(4)

force constants

A satisfactory fit of the dispersion curves can be already achieved with an axially symmetric model to third neighbours (6 parameters).A fully satisfactory fit involves 10 parameters. Table 3. K. Born-von Karman force constants @c, T

9K

9K

Ref.

77Dol “)

66Col”)

m

ij

111

xx XY xx YY xx zz XY xx YY YZ xz

0.7688 0.8805 0.4042 0.0296

0.786 0.895 0.432 0.029

-0.0418 0.0038 - 0.0455

- 0.041 0.012 - 0.054

0.0213 - 0.0029 0.0030 0.0091

0.002 - 0.004 0.001 0.002

xx XY

0.0091 0.0062

0.006 0.004

200 220

311

222

@; [Nin-‘-j

‘) Axially symmetricmodel.

70

Schoher/ Dederichs

1.2 Phonon states: K

Ref. p. 1801 Anharmonic effects

Up to nine-tenths of the melting temperature the temperature dependenceof the frequenciesand of the thermal expansion coeffkient was found to be in agreementwith anharmonic perturbation calculations basedon an effective ion-ion pair potential. For the lifetimes the agreementis less satisfactory, Fig. 2 K [69Bul, 73Dul]. A molecular dynamics calculation gave both frequency shifts and lifetimes in agreement with experiment [77Gll] which points to the importance of including higher anharmonic terms beyond the cubic in calculating phonon lifetimes.

A-

-F

K

Experiment . Cl a

0

0.2 0.4 0.6

0.8

1.0

0.9

0.8

0.7

0.6

0.5

0.4

Theory T = 299K -T = 215K ----T= 99K

0.3

-5 C-

-G

-F

-A

Afifl ~~~~~~~ 0

0.1 0.2 0.3 0.4 0.50

e-

0.1 0.2 0.3 0.4 0.50.5 0.41.0 0.9

C-

-t

0.8

0.7

0.6

-l

0.5

0.4

0.3

0.2

0.1

0

Fig. 2. K. Shifts d v and half width r of the phonons at three temperatures. The points are neutron scattering measurements and the lines are theoretical calculations done with an effective ion-ion pair potential [69Bul].

Schoher/Dederichs

71

[Lit. S. 180

1.2 Phononenzustkde: K

fable4. K. Measured width and shifts of the phonons. The table shows the positions in reciprocal space ZX,,‘CJ) (Q,V,Q,. QL)where the phonon was observed; 1’the frequency and l-(9 K) the full width at half height at 9 K. Thewidth taken as the resolution width of the spectrometerfor each phonon. The changesin frequency Ar and n the natural half-width AT that have been derived from the measurementsat higher temperatures Tare shown n the linal column [69Bul]. 2,

QY

Q:

v [THz]

r(9K)

T [K]

Av

AT

2.0 45 !.O 45 2.0

2.0

0.2

0.68(25)

0.13 (3)

2.0

0.3

0.99 (2)

OSO(3)

0.4

1.28(25)

0.09 (3)

-0.16(3) -0.09(3) -0.19(3) -0.12(3) -0.20(4)

0.03(3) O.OO(3) 0.04(4) 0.04(4)

2.0

299 215 299 215 299

215

-0.12 (3)

299 215 299 215 92 299 215 92 299 215 299 215 92 299

-0.23(4)

0.07(3) 0.06(5) 0.06(4) 0.08(S) 0.09(4) 0.02(4) 0.08 (5) 0.06(4) 0.04(4)

45 2.0 45 2.0 A5

2.0

0.5

1.56(2)

0.13 (3)

2.0

0.6

1.79 (25)

0.13(3)

2.0 A5

2.0

0.7

1.98(2)

0.14(3)

2.0

2.0

0.75

2.07(3)

0.14(3)

A5 2.0 A5

2.0

0.8

2.110(25)

0.22(3)

2.0

2.0

0.85

2.19 (3)

0.22(4)

215

AS 2.0 AS

2.0

0.9

2.20(3)

0.21 (3)

2.0

2.0

1.0

2.225(20)

0.16(3)

299 215 92 299

215

H I5 1.0 H I5

1.0

1.2 F3 1.3 F3 1.4 F3

1.2

2.8

2.16(3)

0.30(5)

1.3

2.7

2.11 (4)

0.36(5)

3.0

2.225(30)

0.16(4)

1.4

2.6

1.89(4)

0.39(6)

1.5

1.5

2.5

1.75(4)

0.27(5)

P4 1.5 P.4

1.5

1.5

1.79(5)

0.16(5)

1.4

1.4

1.4

1.255(30)

0.18 (4)

F, 1.35 Fl

1.35

1.7

1.7

1.35

1.7

1.05(3)

- 1.02(3)

0.19(4)

0.16 (3)

Fl

92 215 92 215 92 215 92 215 92 215 92 299 215 92 299

-0.13 (3) -0.26(5) -0.15(4) -0.04(3) -0.26(4) -0.15(4) -0.06(4) -0.23(4) -0.16 (4) -0.24(4) -0.14(5) -0.03(5) -0.26(8) -0.19(5) -0.24(4) -0.19(4) -0.07(5) -0.16(7) -0.17(5) -0.06(4) -0.19 (4) -0.10(5) -0.16(4) -0.06(4)

0.12(6) -0.04(9) -0.02(6) O.OO(8) 0.02(6) O.OO(6) -0.23(6) -0.12(6) -0.06(5) -0.24(4)

215

- 0.13(4)

92 299 215 92 299

-0.04(4)

215

-0.08 (4) -0.03(3)

92

- 0.18(4) -0.09(4) -0.05(3)

-0.10 (4)

0.11(4)

0.10(4) 0.06(4) 0.05(6) 0.08(S) O.OO(4)

0.10(9) 0.04(5) 0.06(5) 0.05(S) 0.04(4) 0.15(8) 0.21(5) 0.03(4)

0.11(4) O.lO(5) O.OO(5) 0.04(6) 0.05(6) 0.05 (7) O.OO(9) 0.08 (10) 0.09 (7) 0.09(7) 0.21 (5)

0.11(6) O.OO(5)

0.11(4) 0.13(5) 0.04(5) 0.06(6) 0.04(5) 0.03(4)

0.11(4) 0.10(4) 0.04(3)

(continued

72

Schober/Dederichs

1.2 Phonon states: K

Ref. p. 1803 Table 4. K. (continued) ?X

QY

QZ

v [THz]

IV K)

T [K]

Av

Al-

1.75 5 1.8 Fl

1.75

0.75

1.20(3)

0.12(3)

1.8

1.8

1.475(10)

0.15(3)

1.6 41

1.6

1.6

2.12(4)

0.25(4)

2.3 Al

2.3

0.3

2.13(4)

0.19(4)

2.2 4

2.2

0.2

1.68(3)

0.18(4)

2.1 4

2.1

0.1

0.94(2)

0.12(2)

1.4 Gl

1.4

1.0

0.81(3)

0.11(3)

1.5 Nk

1.5

1.0

0.53(2)

0.16(3)

2.25 Cl

2.25

0.0

1.695(50)

0.25(5)

2.3 Cl

2.3

0.0

1.93(5)

0.24(9)

2.4 Cl

2.4

0.0

2.27(4)

0.30(6)

2.5 N 0.85 x3

2.5

0.0

2.385(50)

0.31(6)

299 215 299 215 92 299 215 92 299 215 92 299 215 92 299 215 92 299 215 92 299 215 92 299 215 92 299 215 92 299 215 92 92

-0.17 (4) - 0.10(4) - 0.20(4) -0.12 (3) - 0.05(3) -0.17 (8) -0.10 (5) -0.04 (5) -0.27 (9) -0.12 (6) - 0.09(5) - 0.21(4) -0.13 (4) - 0.06(4) -0.19 (3) - 0.06(3) -0.05 (3) - 0.09(4) - 0.04(4) -0.01 (3) - O.Ol(3) O.Ol(3) -0.01 (2) -0.19 (6) - 0.11(6) -0.05 (6) - 0.19(6) -0.08 (6) -0.06 (6) -0.18 (6) -0.10 (6) - 0.05(6) - 0.06(7)

0.14(4) 0.06(3) 0.10(4) 0.08(4) 0.02(3) 0.05(7) 0.04(7) 0.00(5) 0.03(9) 0.13(6) 0.06 (5) 0.08(5) 0.07(5) 0.00(4) 0.06(4) 0.04(3) 0.02(3) 0.15(3) 0.05(4) 0.02(2) 0.05(5) 0.05(5) 0.02(3) 0.00(5) 0.00(6) 0.00(6) 0.08(8) 0.05(9) 0.00(5) 0.11(9) 0.08(9) 0.04(6) 0.00(6)

0.85

2.0

0.66(1)

0.08 (2)

0.75

2.0

1.10(2)

0.1.2(2)

0.65

2.0

1.355(20)

0.10(2)

- 0.05(2) - 0.04(1) 0.01(1) - 0.06(3) -0.11 (3) - 0.02(2) -0.17 (5) -0.12 (3) - 0.04(3)

0.06(2) 0.02(2)

0.75

299 215 92 299 215 92 299 215 92

c3

0.65 x3

0.06(3) 0.04(3) 0.02(2) 0.13(4) 0.08(2) 0.05(2)

Pressure dependence

The wave vector dependent (mode) Grtineisen parameters, y, have been measured for several modes of potassium at 4.5 K over the pressure range 0.0001 to 4.1 kbar. Comparison with theoretical model calculations shows generally good agreement for modes propagating along [c[[] and [[ll] directions but significant discrepancies occur for four other modes, Fig. 3 K. There is also a substantial disagreement for the [OO[] and [O[[J T, models between the neutron scattering data and the room temperature elastic data, which may be due, at least in part, to frequency and temperature dependenceof y [76Mel]. Theoretical calculations: [72Be2,74Shl, 76Tal]. Schober / Dederichs

73

1.2 Phononenzust.kde: K -F

b-

l.flT

[Lit. S. 180

-A

H

C-

P

I lttC1

K

IOOCI

N

T

10551

I

1.6 -

------___ 0.6 -

12

1 (I

0.1 . 0

K

1

’

1 1.0

I 0.6

I 0.8

I 0.L

I

I 0.2

I

I

I

0

0.5

-t tIFig. 3. K. Measured Griineisen parameter y as a function of reduced wave vector (o longitudinal, l transverse modes) [76Mel]. The experimental points are compared with two representative models; full curve [72Bel], broken curve C74Sh2-j.

2. Frequency spectrum and related properties 1.6, lHz-‘.‘j

I

I

I

I

I

I

I I

I

I

2.~ 1Hz

I 2.2

f 2.0

0

0.5

1.0

1.5

2.0

THz

2.5

from

the

1.8 -10

0

10 n-

YFig. 4. K. Frequency spectrum at 9 K calculated Born-von Karman it of Table 3 K [77Dol].

801 0

5

Fig. 6. K. Debye Born-von Karman

10

15

20 I-

25

I

I

30

35

temperatures (3, calculated from force constants of Table 3 K [66Col].

Fig. 5. K. Debye cutoff spectrum of Fig. 4 K.

0

K the

100

frequencies

200

20

30

v, obtained

300

400

from

K

the

500

Fig. 7. K. Debye-Waller exponent 2Wdivided by the recoil frequency of a free potassium atom, vR, calculated from the spectrum of Fig. 4 K.

!Schober/Dederichs

1.2 Phonon states: K

Ref. p. 1801

Table 5. K. Phonon spectrum at 9 K calculated from the Born-von Karman force constants of Table 3 K [77Dol]. v [THz]

g(v) CTHz-‘1

v [THz]

g(v) CT=-‘1

v [THz]

g(v) P-Hz- ‘1

0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 0.775 0.800

0.000 0.001 0.002 0.004 0.006 0.009 0.013 0.016 0.020 0.026 0.032 0.039 0.046 0.055 0.065 0.078 0.090 0.105 0.124 0.146 0.172 0.226 0.259 0.272 0.285 0.300 0.314 0.330 0.346 0.360 0.379 0.398

0.825 0.850 0.875 0.900 0.925 0.950 0.975 1.000 1.025 1.050 1.075 1.100 1.125 1.150 1.175 1.200 1.225 1.250 1.275 1.300 1.325 1.350 1.375 1.400 1.425 1.450 1.475 1.500 1.525 1.550 1.575 1.600

0.417 0.438 0.461 0.487 0.511 0.544 0.575 0.616 0.541 0.486 0.457 0.438 0.424 0.415 0.410 0.405 0.403 0.403 0.404 0.410 0.415 0.422 0.436 0.449 0.467 0.491 0.525 0.582 0.660 0.644 0.628 0.618

1.625 1.650 1.675 1.700 1.725 1.750 1.775 1.800 1.825 1.850 1.875 1.900 1.925 1.950 1.975 2.000 2.025 2.050 2.075 2.100 2.125 2.150 2.175 2.200 2.225 2.250 2.275 2.300 2.325 2.350 2.375 2.400

0.608 0.598 0.587 0.579 0.568 0.560 0.549 0.531 0.515 0.505 0.499 0.494 0.493 0.494 0.501 0.510 0.526 0.549 0.584 0.634 0.718 0.915 1.018 1.141 1.557 1.041 0.827 0.676 0.541 0.411 0.244 0.0

”

3. Theoretical models Due to the short range of the interionic forces most empirical models reproduce the dispersion well. K is like Na a typical “simple metal” where pseudopotential models are very successful. The dependence on the used screening function is similar to Na, see Fig. 5 Na. Born-von Karman models: see Table 3 K. Models comprising short ranged forces plus a simple electronic contribution: [69Krl, 74Da1, 78Ku1, 77Bo1, 76Si4], further references: [65Krl, 66Gul,7OSil, 71Trl,72Si2, 74Pa1, 74Prl,75Gol, 76S1,77Ra4,78Ra4]. Local pseudopotential models: [66Sc2, 70Ku1, 70Pr1, 74Sh1, 74Sh2, 78Pr2,77Val], [61Tol, 66Anl,68Hol, 69Bo2,71Gu3,76Srl]. Nonlocal pseudopotential calculations:

further references:

[69Prl, 7OCo1, 72Be2, 73Du1, 76Jil].

Molecular dynamics study: [77Gll, 77Sol]. Other microscopic calculations:

[78Avl].

Schoher / Dederichs

75

1.2 PhononenzusGnde:

[Lit. S. 180

La, Li

La Lanthanum Lattice: /?-La dhcp, a=377pm=3.77&

c=1216pm=12,16A;

y-La fee, a=531 pm=531 A. BZ: seep. 449.

At temperaturesbelow 583K lanthanum exists in a stable dhcp /?-phaseand a metastablefee y-phase.Apart from a slight difference in their atomic volumes, they differ only in the stacking sequenceof the atomic layers. Phonon densities of state have been measured for both phasesby inelastic neutron scattering (TOF) on polycrystalline samples,Figs. 1 La and 2 La. The averagephonon frequency of the fee phase is about 0.1THz lower than that of the dhcp phase.The spectrum of the dhcp phase does not change much from 5 K to 295 K whereas a softening of the phonons is found for the fee phase in heating from 295 K to 583 K [78Nul]. 0.7 THZ” 0.6

0

1

2

3

THz

4

Fig.1. La. Measured phonon spectrum of /l-lanthanum (dhcp phase) [78Nul].

Li

0

1

2 3 1Hz 4 YFig. 2. La. Measured phonon spectrum of y-lanthanum (fee phase) [78Nul].

Lithium

Lattice: bee, a = 350pm = 3.50A. BZ: seep. 448. 1. Phonon dispersion Table 1. Li. Measurements. Method neutron diffraction (TAS) neutron diffraction (TAS)

T IX1

Fig.

Ref.

1 Li

Smith et al. [68Sml] Beg and Nielsen C76Bel-J

98 293,

110...424

Both measurementswere done on ‘Li crystals. They agreeapart from the [[[(I T branch where the observed phonon peaksare very broad or split. For this branch, [76Bel] quotes frequenciesup to 25 % higher than [68Sml] Fig. 2 Li. The initial slopesagreewithin 10 % or lesswith the values expectedfrom the isothermal elastic constants [68Sml]. With increasing temperature [76Bel] report up to 20% higher elastic constants than measured by sound echo. A characteristic is the crossing of the [OOC]L and [OO[] T branches. The temperature dependence was studied for selectedphonons from 110 K to 424 K. Kohn anomalies have not been found. 76

Schober/Dederichs

Ref. p. 1803

1.2 Phonon states: Li -F

/I-

o

-A

0.8

1.0

f-

C-

0.6 0.4 0.2

0

-c

0.5 5-

Fig. 1. Li. Measured phonon dispersion in ‘Li at 293 K. The solid lines correspond to the fifth neighbour Born-von Karman model of Table 4 Li [76Bel].

Table 2. Li. Measured phonon frequencies in ‘Li at 98 K [68Sml]. I

v [THz]

CWI L 0.125 0.195 0.264 0.345 0.46 0.62 0.70 0.77 0.80 0.90 1.00

2.00 (15) 3.00 (15) 4.00 (15) 5.00 (15) 6.00 (18) 7.00 (20) 7.60 (20) 8.00 (30) 8.30 (30) 8.76 (30) 8.82 (40)

0.198 0.242 0.275 0.310 0.346 0.380 0.442 0.540 0.660 0.700 0.800 0.900

“) The polarization

2.50 (25) 3.00 (20) 3.50 (20) 4.00 (25) 4.50 (15) 5.00 (12) 6.00 (15) 7.00 (10) 8.00 (10) 8.30 (20) 8.74 (20) 8.70 (30)

0.10 0.30 0.40 0.60 0.70 0.80 0.90

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

1.50 (20) 4.04 (10) 5.60 (18) 8.10 (20) 8.45 (20) 8.60 (20) 8.80 (20)

v [THz]

5

c51111 L

Kit’1 ‘I-

C’XCIT “1

2.60 (10) 3.92 (10) 6.40 (15) 7.35 (20) 8.15 (15) 8.70 (20) 8.87 (20) 9.00 (40)

v [THz]

cl

CWI T

COiil L 0.10 0.15 0.25 0.30 0.35 0.40 0.45 0.50

v [THz]

5

0.108 0.208 0.350 0.545 0.700 0.830

3.50 (10) 6.50 (20) 8.37 (30) 6.50 (20) 3.60 (10) 6.50 (30)

COiil-4 “1

0.97 (10) 1.20 (10) 1.40 (10) 1.55 (10) 1.70 (10) 1.80 (10) 1.90 (10) 1.90 (10)

0.10 0.15 0.20 0.30 0.40 0.50

1.88 (10) 2.60 (15) 3.48 (20) 4.54 (10) 5.35 (10) 5.70 (20)

vectors for the [O((‘] TI and T, branches are parallel to [Ocf] and [[OO], respectively.

Schober/Dederichs

77

1.2 PhononenzustZnde: Li

TLit. S. 180

Table 3. Li. Measured phonon frequenc’es in ‘Li at 293 K [76Bel]. v [THz]

v [THz]

COKIL

CONI-I-

0.112 0.168 0.196 0.223 0.279 0.335 0.363 0.391

1.84 (7) 2.64 (12) 2.97 (12) 3.43 (12) 4.11 (12) 4.81 (12) 4.95 (12)

1.23 (7) 1.91 (10)

0.441

6.07 (24) 6.65 (24) 6.55 (24) 6.77 (24) 7.25 (19) 7.78 (19) 8.61 (15) 8.70 (24)

i

0.503 0.559 0.614 0.670 0.782 0.950 1.00 r

0.04 0.08 0.119 0.138 0.158 0.198 0.237 0.257 0.316 0.356 0.375 0.395 0.407 0.435 0.50

r

2.56 (10) 3.19 (12) 3.99 (15) 4.84 (19) 5.58 (24) 6.65 (15) 7.18 (15) 8.05 (24) 8.70 (24)

v [THz]

v [THz]

v ~Hz]

CKil L

CKil T, “1

Wiil Tzn)

1.09 (7) 2.08 (7) 3.09 (15)

0.53 (7)

1.21 (7) 1.81 (10)

0.097 0.129 0.161 0.194 0.226 0.258 0.290 0.323 0.355 0.419 0.451 0.548 0.645 0.677 0.710 0.742 0.774 0.839 1.0

v [THz]

v [THz]

cri4I L

CCiil -I-

3.14 (24)

1.55 (12) 2.03 (12) 2.18 (12) 2.83 (15) 3.07 (15) 3.67 (15) 4.42 (19) 4.84 (24) 5.68 (29) 6.29 (29) 6.70 (19) 7.08 (19)

5.03 (24) 6.65 (15) 7.69 (24) 8.05 (29) 7.33 (19) 6.09 (19) 3.99 (15) 3.63 (12) 3.43 (24) 4.11 (19)

8.10 (15) 8.22 (19)

6.60 (24) 8.70 (24)

8.70 (24)

F-

AP I

H

0.89 (7) 4.06 (15) 1.18 (10) 6.00 (15) 7.45 (15) 7.98 (12) 8.22 (12) 8.34 (15)

1.45 (12) 1.74 (12) 1.76 (15)

2.49 (12) 3.14 (12) 3.58 (12) 4.38 (12) 4.74 (12)

1.93 (15) 8.66 (12) 8.82 (12)

2.08 (15)

5.20 (12) 5.36 (12)

‘) The polarization vectors for the [O[(]T, and T, branches are parallel to [O;c] and [(OO], respectively.

Fig. 2. Li. DifTerence of the [((flT according to [68Sml] and [76Bel].

phonon frequencies

Born-von Karman force constants The data can be adequately fitted with fifth or sixth nearest neighbour general force constants. A lit with an eighth neighbour axially symmetric model is somewhat worse. 18

Schober/Dederichs

Ref. p. 1801

1.2 Phonon states: Li

Table 4. Li. Born-von Karman force constants, CD!,for 7Li. T

98 K

98 K

293 K

T

98 K

98 K

293 K

Ref.

76Bel “)

68Sml

76Bel

Ref.

76Bel “)

68Sml

76Bel

m

ij

@E [Nm-‘1

111 xx XY 200 xx YY 220 xx zz XY

2.271(127) 2.360(182) 0.790(376) 0.163(301) -0.368 (148) 0.058(191) - 0.657(392)

2.336(29) 2.462(38) 0.694(66) 0.140(46) -0.277 (26) 0.125(34) -0.158 (38)

m

2.114(33) 2.207(35) 0.862(79) 0.016(54) -0.268 (28) 0.055(40) -0.197 (76)

ij

@J [Nm-l]

311 xx YY YZ xz 222 xx YY 400 xx YY

0.180(41) -0.071 (62) 0.064(154) 0.079(110) 0.112(98) 0.184(220)

0.171(26) -0.126 (17) -0.122 (34) O.Oll(26) 0.148(17) -0.038 (41) - 0.282(60) 0.012(36)

0.146(24) - 0.057(12) 0.033(30) 0.052(21) 0.062(19) 0.077(47)

“) Fit to the data of [68Sml] amended by measurements for the [[[[I T branch at 110 K from [78Bel]. Temperature dependence

The measured frequency shifts and half width of the phonons as function of the temperature are shown in Fig. 3 Li. Numerical values are calculated in [76Bel]. The shift depends strongly on the c value. The maximum observedshift was about 20 %. A-

F-

0

0.2

0.4

0

0.2

0.4

E-

0.6

0.8

1.00

0.6

0.8

1.00

5-

0.5

0.50 f-

0.2 ox f-

0.6 0.8

1.0

Fig. 3. Li. Phonon frequency shifts and half width in ‘Li at three temperatures. Lines are guides to the eye [76Be2]. Pressure dependence

Theoretical calculations: [72Si2,72Be2,74Sh2], further references:[69Wa2]. Schober/Dederichs

79

1.2 Phononenzustbde: Li

[Lit. S. 180

2. Frequency spectrum and related properties 9.2 1Hz

8.8 8.4 I 2 8.0 7.6 7.2 6.8 -10

8 1Hz 10 6 Y------c Fig. 4. Li. Frequency spectra for ‘Li at 98 K and 293 K calculated from the Born-von Karman force parameters of Table 4 Li [76Bel]. 0

2

L

0

10

20

30

nFig. 5. Li. Debye cutoff frequencies vn obtained from the spectra of Fig. 4 Li.

Table 5. Li. Phonon spectrum of ‘Li calculated from the general fifth neighbour Born-von Karman force constants [76Be2] of Table 4 Li. T \’ (THz] 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 80

98 K

293 K

g(v) [THz- ‘1 0.000 0.000 0.001 0.001 0.002 0.003 0.004 0.005 0.006 0.008 0.010 0.012 0.014 0.017 0.020 0.024 0.029 0.036 0.050 0.059 0.064 0.069 0.074 0.079 0.087 0.095 0.106 0.125 0.259 0.178 0.145

0.000 0.000 0.001 0.001 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.009 0.011 0.013 0.015 0.018 0.021 0.026 0.031 0.038 0.075 0.079 0.084 0.088 0.092 0.096 0.101 0.105 0.110 0.115 0.121

T

98 K

v [THz] 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20

293 K

g(v) [THz-‘1 0.129 0.119 0.111 0.106 0.102 0.098 0.096 0.094 0.092 0.091 0.090 0.090 0.090 0.090 0.090 0.091 0.092 0.094 0.096 0.099 0.102 0.107 0.113 0.129 0.132 0.131 0.130 0.129 0.129 0.129 0.129

0.125 0.130 0.136 0.139 0.145 0.150 0.155 0.159 0.136 0.126 0.121 0.118 0.116 0.114 0.114 0.114 0.115 0.117 0.119 0.123 0.129 0.141 0.151 0.149 0.147 0.145 0.144 0.142 0.141 0.140 0.138

Schober/Dederichs

T v [THz] 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20 8.30 8.40 8.50 8.60 8.70 8.80 8.90 9.00 9.10 9.20

98 K

293 K

g(v) [THz- ‘1 0.130 0.130 0.131 0.132 0.133 0.134 0.136 0.137 0.139 0.140 0.141 0.141 0.142 0.144 0.148 0.154 0.164 0.184 0.192 0.198 0.204 0.213 0.222 0.231 0.243 0.257 0.282 0.315 0.220 0.0

0.137 0.136 0.135 0.134 0.132 0.131 0.127 0.125 0.123 0.122 0.123 0.125 0.128 0.134 0.142 0.154 0.171 0.206 0.249 0.253 0.255 0.260 0.264 0.277 0.259 0.198 0.052 0.0

1.2 Phonon states: Li

Ref. p. 1801

280 0

LOO K 500

300

200

100

r-

LOO K 500 300 I *. Fig. 7. Li. Debye-Waller exponent 2 W divided by the recoil frequency of a free ‘Li atom, ~a, calculated from the spectra of Fig. 4 Li. . 0

Fig. 6. Li. Debye temperatures 0, [76Be2]: Values calculated from the neutron data at 98 K [68Sml] (solid line) and It 293 K [76Be2] (broken line) and from specific heat measurements[59Mal] (solid circles).

200

100

3. Theoretical models The dispersion curves of Li are more complicated than the ones for Na and K. Correspondingly, theoretical nodels have met with greater difficulties, which are caused by the distortion of the electronic bands. In particular nany models fail to reproduce the crossing of the two*phonon branches in [OO[] direction. Pseudopotential

theory, both local and non local has been extensively testedfor Li. In general,the dispersion curves agreewell with :xperiment. There is however a clear dependence on the choice of pseudopotential screening function, see e.g. Fig. 8 Li.

-A

-F

A-

form factor and dielectric

P

107 THz 8

6 I a 4

0

0.2

I

I

0.4

0.6

I

0.8

1

-C

t-

Fig. 8. Li. Theoretical phonon dispersion for a one-parameter pseudopotential model (o experimental points [68Sml]. S, G, H, R different choices for the dielectric constant) [70Prl].

Schober/Dederichs

81

1.2 Phononenzustfinde:

Mg

[Lit. S. 180

Born-von Kaman models: seeTable 3 Li and [68Sml]. Models comprising short ranged forcesplus a simple electronic contribution: [69Krl, 74Dal,78Kul, 77Bol], further references:[65Krl, 70Pal,7OSil, 71Trl,72Sil, 73Sh4,74Prl, 76Si1,77Ra2, 78Ra4,77Ra4]. Local pseudopotential models: [70Prl, 71Gu1, 72Si2, 73Sr1,74Sh2,76Pel, 77Sil,77Val], further references: [66Anl, 68Hol,69Wal. 71Gu3,72Kal, 72Prl,73Bel, 75Na2,77Srl]. Nonlocal pseudopotential calculations: [69Prl, 730nl,72Be2,73Srl, 76Dal,77Si1,75Bal], further references: [69Bol, 70Gu23. Other ab initio calculations: [77Wel, 77We2].

Mg

Magnesium

Lattice: hcp, a= 321pm = 3.21A, c= 521pm = 5.21A. BZ: seep. 450. 1. Phonon dispersion Table 1. Mg. Measurements. Method

Table 2. Mg. Measured phonon frequenciesat 390K

Fig.

Ref.

Ref.

1 Mg

Pynn and Squires [72Pyl, 68Py1, 66Sql] Iyengar et al. [63Iyl, 651~11

C66Wl

WPYll

Phonon neutron diffraction (mostly TOS, TAS) neutron diffraction (T’AS)

290 300

Further references:[63Mal, 62Col].

v [THz]

G

3.70 (2)

G K, k

3.70 7.25

6.05 (15) 5.65 (15) 6.88 (4) 6.58 j4j 5.32 (10) 3.70 (3)

M: M; Mi

W

5.80 6.88 6.58 5.45 3.70 7.05

T2 (q =0.2/a)

The various measurementsare consistent with each other. The polarization branchesat the point K, Fig. 1 Mg. have not been unambiguously resolved. At any point K in the extended zone scheme,two of the modesgive rise to equal scattering cross sections. It is therefore impossible to discriminate between K, and K, and between K, and K, modes.Contrary to Be the dispersion of magnesium can be well described by axially symmetric forces, i.e. pair interactions. No evidence for Kohn anomalies has been found. l-

P-

U-

AH

6

p2

1'13‘13 b 1

1 I I I 0.5

f82

!Schober/Dederichs

6-

A

1.2 Phononenzustfinde:

Mg

[Lit. S. 180

Born-von Kaman models: seeTable 3 Li and [68Sml]. Models comprising short ranged forcesplus a simple electronic contribution: [69Krl, 74Dal,78Kul, 77Bol], further references:[65Krl, 70Pal,7OSil, 71Trl,72Sil, 73Sh4,74Prl, 76Si1,77Ra2, 78Ra4,77Ra4]. Local pseudopotential models: [70Prl, 71Gu1, 72Si2, 73Sr1,74Sh2,76Pel, 77Sil,77Val], further references: [66Anl, 68Hol,69Wal. 71Gu3,72Kal, 72Prl,73Bel, 75Na2,77Srl]. Nonlocal pseudopotential calculations: [69Prl, 730nl,72Be2,73Srl, 76Dal,77Si1,75Bal], further references: [69Bol, 70Gu23. Other ab initio calculations: [77Wel, 77We2].

Mg

Magnesium

Lattice: hcp, a= 321pm = 3.21A, c= 521pm = 5.21A. BZ: seep. 450. 1. Phonon dispersion Table 1. Mg. Measurements. Method

Table 2. Mg. Measured phonon frequenciesat 390K

Fig.

Ref.

Ref.

1 Mg

Pynn and Squires [72Pyl, 68Py1, 66Sql] Iyengar et al. [63Iyl, 651~11

C66Wl

WPYll

Phonon neutron diffraction (mostly TOS, TAS) neutron diffraction (T’AS)

290 300

Further references:[63Mal, 62Col].

v [THz]

G

3.70 (2)

G K, k

3.70 7.25

6.05 (15) 5.65 (15) 6.88 (4) 6.58 j4j 5.32 (10) 3.70 (3)

M: M; Mi

W

5.80 6.88 6.58 5.45 3.70 7.05

T2 (q =0.2/a)

The various measurementsare consistent with each other. The polarization branchesat the point K, Fig. 1 Mg. have not been unambiguously resolved. At any point K in the extended zone scheme,two of the modesgive rise to equal scattering cross sections. It is therefore impossible to discriminate between K, and K, and between K, and K, modes.Contrary to Be the dispersion of magnesium can be well described by axially symmetric forces, i.e. pair interactions. No evidence for Kohn anomalies has been found. l-

P-

U-

AH

6

p2

1'13‘13 b 1

1 I I I 0.5

f82

!Schober/Dederichs

6-

A

1.2 Phonon states: Mg

Ref. p. 1801

AM

2

8 THz

2-

15‘0’/21

d

A3 [ai’/2

1

l-

1 0 I-.--

0.5 0.4 0.3 0.2 0.1 -f

.5 0.4 0.3 0.2 0.1 -5

0

0.4

OO-

zone boundor

P

I

84

0.5

5-

zone boundr

UA

c-

sSchober/Dederichs

83

1.2 Phononenzust5nde: Mg

[Lit. S. 180

6

Fig. la-g. Mg. Measured phonon frequencies at 290 K. The lines show the dispersion calculated from the eighth neighbour Born-von Karman model ofTable 3 Mg. Experimental points: [72Pyl], directions U, P, A, R, Z: [65Iyl] according to [72Pyl]. Born-van

Karman

model

The measurements can be reproduced well with an axially symmetric model including eighth neighbour interactions. An extension of the range of interaction improved the quality of the fit only slightly [72Pyl]. Table 3. Mg. Born-von Karman T=290 K [72Pyl].

force constants,@!,

The model is axially symmetric: (i,j=x,y,z)

~~=(~--S,)(RpR;)/(Rm)'+1;6ij;

L CNm-‘I

m

I; CNm-‘I

(a/j4 0,43

10.483

-0.309

(0, a,01

10.099

-0.292

(-2an/5,0,

c/2)

-0.222

-0.246

(f&O, c)

0.305

(5012 1/?, a/2, c/2)

0.748

0.013

(0 (7

0.529

0.091

0, 0)

-0.490

(0, a, 4

-0.049

0.157

(0, 26 0)

-0.401

0.042

2. Phonon spectrum and related properties 9.0

THZ

0.6 1HZ-l

8.5

I 0.5

I 8.0

-; -G 0.2

j 7.5

0

1

2

3

L

5

6

Fig. 2. Mg. Phonon spectrumvat= calculated from the eighth neighbour Born-von Karman model of Table 3 Mg.

84

6.5 -10

1 1Hz 8

0

10

20

30

nFig. 3. Mg. Debye cutoN frequencies r, at 290 K calculated from the eighth neighbour Born-von Karman model of Table 3 Mg.

Schober/Dederichs

1.2 Phonon states: Mg

Ref. p. 1801 a50 K

n

Table4. Mg. Phonon spectrum at 290 K calculated from the Born-von Karman parameters of Table 3 Mg.

0 exot.[30CI 11

Mq

v [THz]

g(v) [THz-‘1

v [THz]

g(v) [THz-l]

0.10

3.80 3.90 4.00

0.50

0.000 0.000 0.000 0.001 0.001

0.60 0.70 0.80 0.90

0.002 0.002 0.003 0.004

1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40

0.005

0.416 0.366 0.306 0.276 0.255 0.239 0.227 0.213 0.212 0.214 0.220 0.228 0.233 0.242 0.243 0.253 0.249 0.234 0.186 0.198 0.201 0.159 0.128 0.109 0.113 0.138 0.211 0.315 0.453 0.395 0.376 0.266 01194 0.136 0.074 0.0 L

0.20 0.30 0.40 0

50

100

150

200

250 K 300

TFig. 4. Mg. Debye temperatures 0, calculated from the eighth neighbour Born-von Karman model of Table 3 Mg compared to experimental values [72Pyl].

n7L

0

100

200

300

LOO K 500

T-

Fig. 5. Mg. Debye-Wailer exponent 2W divided by the recoil frequency of the free atom, vR, calculated from the eighth neighbour Born-von Karman model of Table 3 Mg.

2.50 2.60 2.70 2.80 2.90 3.00

3.10 3.20 3.30 3.40 3.50 3.60 3.70

0.006 0.007 0.009

0.010 0.012 0.014 0.016 0.018 0.021 0.024 0.027

0.031 0.034 0.039 0.043 0.049 0.055 0.062 0.070 0.080 0.090 0.105 0.120 0.141 0.167 0.205 0.267

4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30

3. Theoretical models Contrary to Be, three body interactions seem to be less important in magnesium and the second order pseudopotential perturbation theory is therefore able to reproduce the dispersion well, see Fig. 6 Mg. The inclusion of third order in the perturbation theory gives only relatively small changes [75Be3]. Some of the simple phenomenological models also reproduce the experiment well. Born-von Karman and equivalent models: see Table 3 Mg, further references: [65Lel, 651~1, 66Sq1, 68Py1, 59Mel,70Lal, 71Srl,71Tr2,73Bol, 78Ku5]. Short ranged forces plus a simple electronic contribution: 70Sh2,70Sh3,72Ba3,73Sh6,73Upl, 75Up2,78Mil].

[73Kul, 73Ra2, 77Si3], further references: [68Czl,

Local pseudopotential calculations: [69Pil, 70Fl1, 72Br1, 75Be3], further references: [68Sc2, 69Br1, 70Pr2, 72Ba2,74Mal, 75Mal,76Na4]. Nonlocal pseudopotential

calculations: [71Kil,

74Es1, 75Sha2, 75Rel], further references: [67Ro2, 69Gil].

Schober/Dederichs

85

1.2 Phononenzustfinde:

MO

[Lit. S. 180 -P

-f

lU-

-it

-R

s-

o I

01 0

I

0.5 t-

I

I 0

K

I.5 -t

0

S-

colt. I75Ha 21 exot. I72 Pvll I I 0.1 0.2

-5

H

I 0.3

I 0.4

I 0.5

5-

Fig. 6. Mg. Phonondispersioncurvescalculatedfrom an ab initio pseudopotential[75Ha2].

MO

Molybdenum

Lattice: bee,a= 314pm= 3.14A. BZ: seep. 448. 1. Phonon dispersion Table 1. MO. Measurements. Method

Fig

Ref.

TKI neutron diffraction 296 (TW neutron diffraction 296 (TW neutron diffraction PAS)

1 MO Powell et al. [77Pol] Walker and Egelstaff [69Wa3] 88,298 Buyers et al. [72Bul]

Further references:[64Wol, 68Wol-j. The dispersion curves of molybdenum are similar in shape to the ones of the group VI metals Cr and W. The gross featurescan be described by short range forces,however very long range forcesare required to give an exact fit to the data. A number of pronounced anomalies has been found. 86

Schober/Dedericbs

1.2 Phononenzustfinde:

MO

[Lit. S. 180 -P

-f

lU-

-it

-R

s-

o I

01 0

I

0.5 t-

I

I 0

K

I.5 -t

0

S-

colt. I75Ha 21 exot. I72 Pvll I I 0.1 0.2

-5

H

I 0.3

I 0.4

I 0.5

5-

Fig. 6. Mg. Phonondispersioncurvescalculatedfrom an ab initio pseudopotential[75Ha2].

MO

Molybdenum

Lattice: bee,a= 314pm= 3.14A. BZ: seep. 448. 1. Phonon dispersion Table 1. MO. Measurements. Method

Fig

Ref.

TKI neutron diffraction 296 (TW neutron diffraction 296 (TW neutron diffraction PAS)

1 MO Powell et al. [77Pol] Walker and Egelstaff [69Wa3] 88,298 Buyers et al. [72Bul]

Further references:[64Wol, 68Wol-j. The dispersion curves of molybdenum are similar in shape to the ones of the group VI metals Cr and W. The gross featurescan be described by short range forces,however very long range forcesare required to give an exact fit to the data. A number of pronounced anomalies has been found. 86

Schober/Dedericbs

1.2 Phonon states: MO

Ref. p. 1801 I

A-

-F

-A

D

F3

0

0.2

DA

5-

0.6

0.8

1.0

0.8

0.6

0.4

0.2

-%

0 0.1 0.2 0.3 0.4 0.5

5‘-

Fig. 1a. MO. Measured phonon dispersion at 296 K. The solid lines represent a Born-von Karman tit (Table 3 MO [77Pol]). The dashed lines through the origin are the velocity of sound lines calculated from the elastic constants [72Hul] (according to [77Pol]). T

Table 2. MO. Measured phonon frequencies at 296 K [77Pol]. v [THz]

i

0.10

Fig. lb. MO Measured phonon rrequencies at 296K in the [3, 5, l]direction. The solid lines represent the seventh neighbour Born-von Karman model of Table 3 MO [77Pol] [78Wol].

0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.54 0.56

CC5ClL

C5lL’lT

3.40 (10) 4.50 (10) 5.01 (7) 5.45 (7) 5.85 (5) 6.15 (5) 6.33 (4) 6.46 (5) 6.64 (5) 6.79 (5) 6.99 (7) 7.08 (7) 7.18 (7) 7.23 (10) 7.26 (8) 7.15 (10) 7.08 (10) 6.97 (10) 6.80 (14) 6.64 (4) 6.57 (6) 6.49 (5) 6.36 (7) 6.32 (7) 6.25 (6) 6.24 (4)

1.88 (6)

3.84 (8)

5.40 (10) 5.85 (5) 6.01 (5) 6.10 (5) 6.22 (3) 6.29 (4) 6.34 (4) 6.39 (7) 6.50 (7) 6.49 (5) 6.56 (5) 6.65 (5) 6.74 (5) (continued)

Schober/ Dederichs

87

1.2 Phononenzust%nde:MO

[Lit. S. 180

Table 2. MO. ontinued)

0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.75 0.76 0.78 0.80 0.82 0.84 0.85 0.86 0.88 0.90 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00

v [THz]

v [THz]

i

6.20 (8) 6.22 (8) 6.20 (8) 6.18(10) 6.17(10) 6.15(10) 6.18 (6) 6.18 (6) 6.20 (5) 6.22 (6) 6.21 (5) 6.14 (5) 6.08 (5) 6.08 (5) 6.12 (7)

6.78(7) 6.82(8)

0.10 0.15 0.20 0.25 0.30 0.35 0.36 0.38 0.40 0.42 0.44 0.45 0.46 0.48 0.50

7.08(8)

6.90(10)

cot;51L

COCCI Tz’)

2.92(7)

1.40(4)

5.13(7) 6.00(6) 6.74(10) 7.25(10) 7.28(15) 7.42(15) 7.62(10) 7.73(10) 7.85(15) 8.00(20) 8.05(15) 8.12(15) 8.14(10)

2.84(5)

Piil T,“) 2.47(5) 3.93(8)

4.12(6) 4.59(9)

5.05(10)

4.82(8) 4.76(10)

5.70(10)

4.56(6)

5.73(6)

6.52(10) 6.17(10) 6.20 (8) 6.25 (8) 6.30 (8) 6.28(10) 6.23(10) 6.18(10) 6.00(10) 5.88(10) 5.62(10) 5.62(10) 5.52 (4)

6.20(7)

5.80(10)

5.52(4)

‘) The polarization vectors for the branches are parallel to [O;a and [(OO].

[O[[]T, and respectively.

T2

0.20 0.30 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.50 0.60 0.70 0.76 0.80 0.82 0.84 0.85 0.86 0.88 0.90 0.92 0.94 0.95 0.96 0.98 1.00

CWI L

CON’IT

4.04(4) 5.68(8) 6.07(5) 6.30(7) 6.50(5) 6.72(5) 6.80(7) 6.93(7) 7.04(7) 7.32(10) 7.61(8) 7.60(10) 7.29(9) 7.17(7) 7.04(10) 6.88(20)

1.98(4)

3.93(6)

4.64(6) 5.18(5) 5.65(7) 6.00(8) 6.05(8)

6.52(15) 6.43(20) 6.32(8) 5.95 5.75(7) 5.57(7) 5.51(5) 5.52(4)

5.92(7) 5.82(8) 5.72(8) 5.60(10) 5.59(8) 5.52(4)

Anomalies in the dispersion curves

The most pronounced anomalies are: i) a depressionnear the symmetry point H =(0, 0,l) of the longitudinal and transversebranches; ii) a depressionof the [O[(] T, branch at the symmetry point N =(0,&f). Extensive measurementsin the whole (110) plane showed 5 additional regions of anomalous frequenciesin off symmetry directions [69Wa3]. To identify theseanomalies the frequenciescalculated from the seventh neighbour Born-von Karman fit (Table 3 MO, ref. [69Wa3] were compared to the measuredfrequencies,Fig. 2 MO. In the determination of the coupling parametersareaswith especially rapid frequency changeswere omitted. 88

Schober / Dederichs

1.2 Phonon states: MO

Ref. p. 1801

0.76

G-

(001)

(t

1 2 .1. 0.

0.2t 0 (0~0,

li .2

; ..

17

/ 0.1

I 0.2 C-

I 0.3

III/ O.4

;;

T

(000)

0.1

0.2

C-

0.3

(i

Fig. 2a-d. MO. a and b. Constant frequency contours in the (017) plane calculated from a seventh neighbour Born-vor Karman model (Table 3 MO [69Wa3]) for the low and high branches, respectively (frequencies in units of 10THz). c and d. Difference of the measured and calculated frequencies given as a percentage of the measured frequency. The dashec lines are contours of constant percentage misfit (c) low branch, d) high branch) [69Wa3]. Schoher/Dederichs

[Lit. S. 180

1.2 Phononenzustkde: MO

by

The anomaly around the H point is not strongly temperaturedependent.From 88 K to 298 K the branch softens about 0.08THz in the whole range from 0.78< 4’<1.00, Fig. 3 MO [72Bul].

Born-von Kannan fit The gross features of the dispersion can be described by a third neighbour model. A detailed description necessitates long range forces.The pronounced maximum in the [OO{] L branch indicates a large secondneighbour force constant. Table 3. MO. Born-von Karman coupling parameters, q. T

296 K

296 K

296 K

Ref.

77Pol

69Wa3

64Wol”)

9: IXm-‘1

m

ij

111

xx XY

16.51(41) 11.78(54)

15.82(12) 12.00(41)

15.9 11.9

200

xx YY

44.57(89) - 2.69(42)

42.22(83) -1.54 (34)

43.9 - 3.0

220

xx zz XY xx YY YZ XY

3.60(22) 0.69(47) 1.72(37)

2.44(22) -0.57 (30) 1.92(54)

3.1 -1.1 4.2

- 2.65(42) -0.43 (26) 0.71(32) 0.26(24)

-1.55 (26) 0.14(9) 0.76(10) 0.58(21)

222

xx XY

0.57(23) 0.69(30)

‘0.74(11) 0.36(22)

400

xx YY

3.97(56) 0.85(43)

4.54(66) -0.51 (28)

133

xx YY YZ xz

-0.75 (21) 0.29(16) -0.16 (27) -0.06 (20)

- 0.84(29) 0.59(21) -0.45 (41) 0.11(7)

311

101 A-

6.0-

‘) Axially symmetric model.

2. Phonon spectrum and related properties 0.6 THZ-’

0.5

Fig. 3. MO. Measured phonon frequencies of the [OOJ L branch at temperatures of 88 K and 298 K. For clarity the abcissa scalesfor the two temperatures do not coincide. The insert is a sketch of the complete branch with the investigated region marked by cross hatching. The lower section shows the frequency differences [72Bul].

I 0.4 --$ 0.3 G 0.2 4

0

1

2

3

4 Y-

90

5

6

7

1Hz 9

Fig. 4. MO. Phonon frequency spectrum at 296 K calculated from the Born-von Karman coupling parameters of Table 3 MO C77Pol-J.

!Schober/Dederichs

Ref. p. 1801

1.2 Phonon states: MO

4 Fig. 5. MO. Debye cutoff frequencies v, calculated from the spectrum of Fig. 4. MO.

375 350

0

nTFig. 6. MO. Debye temperature 0, calculated from the spectrum of Fig. 4. MO [77Pol].

0.8 .10-1’2 I 076 s” 0.4 2 N 0.2

0

4 100

200 T-

300

400 K 500

Fig. 7. MO. Debye-Wailer exponent 2Wdivided by the recoil frequency of the free molybdenum atom, vR, calculated from the spectrum of Fig. 4 MO.

‘able4. MO. Phonon frequency spectrum at 296 K as calculated from the Born-von Karman parameters of ‘able 3 MO. v [THz]

g(v) CTHz-‘1

v [THz]

g(v) CTHz-‘I

v [THz]

g(v) CTHz-‘1

0.10

0.000

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

0.000 0.000 0.001 0.001 0.001 0.002 0.002 0.003 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.014 0.015 0.017 0.019 0.020 0.022 0.024 0.027

2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40

0.029 0.031 0.034 0.037 0.040 0.043 0.047 0.051 0.055 0.060 0.065 0.070 0.076 0.083 0.092 0.101 0.112 0.126 0.147 0.202 0.212 0.216 0.222 0.230 0.239 0.250 0.264

5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10

0.282 0.308 0.353 0.422 0.388 0.386 0.349 0.281 0.236 0.221 0.214 0.197 0.188 0.127 0.127 0.142 0.186 0.200 0.221 0.246 0.273 0.303 0.354 0.341 0.170 0.108

1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70

Schoher/Dederichs

0.0 91

1.2 Phononenzust5nde: Na

[Lit. S. 180

3. Theoretical models Although the phonon dispersion in transition metals has attracted considerable theoretical effort the question of the origin of the anomalies is still unresolved. Varma and Weber [79Val] could reproduce the anomalies in a model which splits the dynamical matrix into a short ranged part which is parametrized and a long range part which is calculated by tight binding methods, seeFig. 7 MO. Born-von Karman and equivalent models; seeTable 3 MO, further references[76Ca2,77Pa2]. Short ranged forces plus a simple electronic contribution: [76Brl, 66Ma2], further references[69Shl, 72Be3, 75Upl. 78Gu2-J. Charge fluctuation model: [77Wal]. Transition model potential: [73Anl]. Electronic calculation [79Wal] and referencescited therein.

Na

Sodium

Lattice: bee. a=428 pm=4.28 A. BZ: seep. 448. 1. Phonon dispersion Table 1. Na. Measurements. Method

Fig.

Ref.

&I neutron diffraction (TAS) neutron diffraction (TOF)

Woods et al. [62Wol] Millington and 296 Squires [71Mi2] Further measurement:[71Erl, 71Wel].

THZ k-

90

1 Na

Na rooti 1=90K

I 5

Fig. 1. Na. Measured phonon dispersion curves at 90 K. The solid lines were calculnted from an estimate of the elastic con;tants [6X’ol].

[lll]ajl)(~l]

Schober/Dederichs

1.2 Phononenzust5nde: Na

[Lit. S. 180

3. Theoretical models Although the phonon dispersion in transition metals has attracted considerable theoretical effort the question of the origin of the anomalies is still unresolved. Varma and Weber [79Val] could reproduce the anomalies in a model which splits the dynamical matrix into a short ranged part which is parametrized and a long range part which is calculated by tight binding methods, seeFig. 7 MO. Born-von Karman and equivalent models; seeTable 3 MO, further references[76Ca2,77Pa2]. Short ranged forces plus a simple electronic contribution: [76Brl, 66Ma2], further references[69Shl, 72Be3, 75Upl. 78Gu2-J. Charge fluctuation model: [77Wal]. Transition model potential: [73Anl]. Electronic calculation [79Wal] and referencescited therein.

Na

Sodium

Lattice: bee. a=428 pm=4.28 A. BZ: seep. 448. 1. Phonon dispersion Table 1. Na. Measurements. Method

Fig.

Ref.

&I neutron diffraction (TAS) neutron diffraction (TOF)

Woods et al. [62Wol] Millington and 296 Squires [71Mi2] Further measurement:[71Erl, 71Wel].

THZ k-

90

1 Na

Na rooti 1=90K

I 5

Fig. 1. Na. Measured phonon dispersion curves at 90 K. The solid lines were calculnted from an estimate of the elastic con;tants [6X’ol].

[lll]ajl)(~l]

Schober/Dederichs

Ref. p. 1801

1.2 Phonon

states: Na

The phonon dispersion in Na can be qualitatively described by first and second nearest neighbour force constants. Longer ranged forces to the fifth neighbours are only necessary to fit the finer details. Also pseudopotential theory predicts a rapidly decreasing interatomic potential. Kohn anomalies have not been found. The observed temperature shift can be explained by anharmonic lattice dynamics using an effective ion-ion potential obtained from pseudopotential theory [72Gll]. The agreement is not as good for the lifetimes. Mode Griineisen parameters for long wavelength phonons have been measured by Ernst [71Erl]. Table 2. Na. Measured phonon frequencies at 90 K [62Wol]. c

v [THz]

5

CowI L 0.20 0.30 0.40 0.45 0.50 0.55 0.60 0.65 0.68 0.70 0.72 0.74 0.75 0.76 0.80 0.85 0.90 1.00

1.43 (7) 1.94 (6) 2.44 (5) 2.68 (10) 2.78 (6) 2.91 (7) 3.01 (7) 3.19 (7) 3.14 (10) 3.24 (6) 3.25 (8) 3.31(8) 3.36 (10) 3.36 (7) 3.44 (5) 3.53 (6) 3.55 (5) 3.58 (4)

v [THz]

CSiOlL 0.10 0.20 0.25 0.30 0.35 0.40 0.45 0.50

1.25 (4) 2.32 (3) 2.77 (5) 3.17 (5) 3.46 (6) 3.67 (5) 3.75 (9) 3.82 (7)

CITOITl 0.14 0.21 0.28 0.35 0.42 0.50

0.43 (3) 0.61 (3) 0.76 (3) 0.87 (3) 0.92 (4) 0.93 (2)

KU’1 T, CWI T 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42

0.44 0.50 0.60 0.70 0.75 0.80 0.90 1.00

0.97 (3) 1.09 (4) 1.18 (4) 1.29 (4) 1.42 (4) 1.52 (4) 1.64 (3) 1.74 (5) 1.83 (4) 1.94 (5) 2.07 (5) 2.17 (4) 2.25 (5) 2.32 (4) 2.59 (5) 2.96 (3) 3.23 (4) 3.35 (4) 3.45 (5) 3.57 (6) 3.58 (4)

i

0.15 0.20 0.25 0.28 0.30 0.35 0.40 0.45 0.50

1.16 (4) 1.52 (4) 1.81 (3) 1.97 (3) 2.09 (3) 2.27 (4) 2.47 (4) 2.52 (6) 2.56 (5)

v [THz]

i

cr511L 0.10 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.62 0.64 0.65 0.66 0.68 0.70 0.72 0.725 0.74 0.75 0.76 0.78 0.80 0.82 0.84 0.85 0.90 1.00

1.53 (5) 2.72 (6) 3.16 (6) 3.38 (6) 3.44 (5) 3.42 (6) 3.22 (6) 2.88 (4) 2.48 (5) 2.06 (4) 1.90 (4) 1.78 (4) 1.74 (3) 1.71 (4) 1.67 (4) 1.68 (3) 1.78 (5) 1.77 (5) 1.89 (5) 1.94 (4) 2.04 (4) 2.22 (5) 2.43 (4) 2.64 (7) 2.82 (9) 2.87 (5) 3.28 (6) 3.58 (4)

Kiil T 0.20 0.30 0.40 0.50 0.60 0.70 0.72 0.74 0.75 0.76 0.78 0.80 0.82 0.84 0.86 0.90 1.00 Schoher/Dederichs

1.28 (6) 1.92 (6) 2.47 (5) 2.88 (4) 3.21 (6) 3.42 (6) 3.44 (9) 3.46 (9) 3.46 (5) 3.44 (9) 3.44 (9) 3.48 (5) 3.48 (9) 3.54 (9) 3.52 (9) 3.56 (5) 3.58 (4)

v [THz]

c+tn A 0 0.125 0.25 0.375 0.50 0.625 0.75 1.00

2.56 (5) 2.62 (6) 2.74 (6) 2.92 (7) 2.88 (4) 2.80 (8) 2.73 (7) 2.56 (5)

Ddrl~ 0 0.25 0.50 0.625 0.75 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00

3.82 (7) 3.57 (8) 2.88 (4) 2.38 (6) 1.81 (5) 1.28 (4) 1.23 (4) 1.14 (3) 1.07 (3) 1.01 (3) 0.96 (4) 0.93 (3) 0.93 (2)

KT11 El 0 0.125 0.25 0.375 0.50

3.58 (4) 3.33 (7) 2.50 (5) 1.49 (4) 0.93 (2)

c5111x2 0 0.125 0.25 0.375 0.50

3.58 (4) 3.48 (7) 3.14 (6) 2.75 (6) 2.56 (5)

93

1.2 Phononenzustbde: Na

[Lit. S. 180

Born-von Karman force constants The fitted force constants decreaserapidly in magnitude. The fifth neighbour constants are only about 2% of the nearest neighbour ones. Fig. 2 Na shows the dependenceof the fit on the range of the force constants. Table 3. Na. Born-von Karman force constants, @E, T=90 K [62Wol].

m 111

‘XX XY

200

xx YY xx zz XY xx YY YZ xz

220

311

1.178 (10) 1.1320 (10) 0.472 (30)

2.5

0.104(30) -0.038 (10)

t 2.0 a

-0.0004 (30) -0.065 (10)

1.5

0.052 (20) -0.007 (10) 0.003 (10)

0.014(10) 0.017(10) 0.033 (10)

xx XY

222

1Hz 3.5

@c [Nm-‘1

ij

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 t;Fig. 2. Na. Secondand fourth neighbour Born-von Karman tits to the [(ii] L branch [62Wol].

Pressuredependence Measurement: [71Erl]. Theoretical calculations: [72Be2, 74Sh2]. Further references:[67Hol, 7OCol). 2. Frequency spectrum and related properties

L.5 1Hz

1.5 1Hz’

I 4.0 1.0 I T -G

3.0 -10

0.5

0

0

10 20 30 nFig. 4. Na. Debye cutotT frequencies vn obtained from the spectrum of Fig. 3 Na. 1

2 3 1Hz 4 YFig. 3. Na. Frequency spectrum calculated from the Bornvon Karman force constants of Table 3 Na.

Fig. 5. Na. Debye-Walk exponent 2Wdivided by the recoil frequency of a free Na atom, vR,calculated from the spectrum of Fig. 3 Na.

94

0

Schober/Dederichs

100

200

300

LOO K 500

1.2 Phonon states: Na

Ref. p.1801

Table 4. Na. Phonon spectra at 90 K and 296 K calculated from the Born-von Karman force constants of Table 3 Na. v [THz]

g(v) [THz-‘1

v [THz]

g(v) [THz-‘1

v [THz]

g(v) [THz-‘1

v [THz]

g(v) [THz-‘1

3.05

0.000

0.164 0.174 0.185 0.195 0.207 0.219 0.232 0.247 0.263 0.280 0.299 0.319 0.346 0.374 0.339 0.296 0.275 0.262 0.253 0.248

2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90

0.244 0.243 0.242 0.244 0.249 0.255 0.263 0.272 0.286 0.306 0.331 0.378 0.440 0.425 0.412 0.402 0.391 0.380 0.364

0.349

0.001

1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95

2.95

D.10 D.15

3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90

0.339 0.334 0.332 0.333 0.339 0.349 0.366 0.394 0.442 0.539 0.660 0.763 1.002 0.593 0.433 0.311 0.178 0.0 0.0

0.20 0.25 0.30 0.35 0.40

0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

0.002 0.003 0.005 0.007 0.010

0.013 0.016 0.020 0.026 0.031 0.038 0.046 0.056 0.067 0.081 0.102 0.145

3. Theoretical models Due to the short range of the interionic forces most empirical models reproduce the dispersion well. Na is the typical “simple metal” and has hence been a favourite testing ground for pseudopotentials theory. A number of investigations have been concerned with testing the dependenceon the choice of dielectric screening, seee.g. Fig. 5 Na. The gained effective ion-ion potential can explain the observedshift of the frequencieswith temperature. The predicted, very small, Kohn anomalies, Fig. 6 Na, have not been observed so far. A-

6-

-A

-F

-C

Fig. 6. Na. Theoretical phonon dispersion for a one-parameter local pseudopotential model. o experimental points [62Wol]. S, G, H, R: different choices for the dielectric constant [70Prl].

Schober/Dederichs

95

1.2 PhononenzusCnde: Nb A-

-F H

T ,:f _

[Lit. S. 180

No

I~OOI

-6-8. 0

’

’

’

’

I 0.6

I 0.8

1.0

C-

-t

5-

Fig. 7. Na. Theoretical values of the group velocity for phonons, showing the positions of predicted Kohn anomalies (K) [74Brl] Born-von Karman and equivalent models; see Table 3 Na and [62Wol, 71Mi2], further references: [65Prl, 70Be2. 71Ba2,72Kol, 77Ra3]. Breathing shell model: [71Hal]. Models comprising short ranged forces plus a simple electronic contribution: [64Krl, 65Kr1, 69Kr1, 72Sh1, 75Kul. 76Shl,78Ku3,76Gol], further references: [63Sh5,64Sil, 65Srl,66Sil, 7OSil,71Sil, 71Trl,72Sil, 73Sh1, 75Gol,76Sh2.77Bol, 77Ra4,78Ra4]. Local pseudopotential models: [66Scl, 66Sc2, 68Ho1, 68Sc2, 70Ge1, 70Ku1, 70Pr1, 71Mi2, 74Sh2, 78Pr2, 77Val], further references: [58Tol, 61Tol,63Col, 64Srl,65Sh2,66Anl, 67Hol,69Pr2,70Mal, 73Bel]. Nonlocal pseudopotential calculations: [64Shl, 69Prl,7OCol, [65Vol].

Nb

72Be2,75Jil, 75Rel,77Dal],

Niobium

Lattice: bee, o= 329 pm= 3.29 A. BZ: see p. 448.

1. Phonon dispersion Table 1. Nb. Measurements. Method

T CKI

Fig.

Ref.

neutron diffraction (TAS)

296

1 Nb

neutron diffraction (TAS) neutron diffraction (TAS)

296

Nakagawa and Woods [63Nol] Sharp [69Sh2] Powell et al. [68Pol, 77Pol] Powell et al. [72Pol] Chang and Collela [77Chl]

neutron diffraction (TAS) X-ray diffraction

296

296,700, 900,103o 296

Further references: [78Wol] 96

!khober/Dederichs

further references:

1.2 PhononenzusCnde: Nb A-

-F H

T ,:f _

[Lit. S. 180

No

I~OOI

-6-8. 0

’

’

’

’

I 0.6

I 0.8

1.0

C-

-t

5-

Fig. 7. Na. Theoretical values of the group velocity for phonons, showing the positions of predicted Kohn anomalies (K) [74Brl] Born-von Karman and equivalent models; see Table 3 Na and [62Wol, 71Mi2], further references: [65Prl, 70Be2. 71Ba2,72Kol, 77Ra3]. Breathing shell model: [71Hal]. Models comprising short ranged forces plus a simple electronic contribution: [64Krl, 65Kr1, 69Kr1, 72Sh1, 75Kul. 76Shl,78Ku3,76Gol], further references: [63Sh5,64Sil, 65Srl,66Sil, 7OSil,71Sil, 71Trl,72Sil, 73Sh1, 75Gol,76Sh2.77Bol, 77Ra4,78Ra4]. Local pseudopotential models: [66Scl, 66Sc2, 68Ho1, 68Sc2, 70Ge1, 70Ku1, 70Pr1, 71Mi2, 74Sh2, 78Pr2, 77Val], further references: [58Tol, 61Tol,63Col, 64Srl,65Sh2,66Anl, 67Hol,69Pr2,70Mal, 73Bel]. Nonlocal pseudopotential calculations: [64Shl, 69Prl,7OCol, [65Vol].

Nb

72Be2,75Jil, 75Rel,77Dal],

Niobium

Lattice: bee, o= 329 pm= 3.29 A. BZ: see p. 448.

1. Phonon dispersion Table 1. Nb. Measurements. Method

T CKI

Fig.

Ref.

neutron diffraction (TAS)

296

1 Nb

neutron diffraction (TAS) neutron diffraction (TAS)

296

Nakagawa and Woods [63Nol] Sharp [69Sh2] Powell et al. [68Pol, 77Pol] Powell et al. [72Pol] Chang and Collela [77Chl]

neutron diffraction (TAS) X-ray diffraction

296

296,700, 900,103o 296

Further references: [78Wol] 96

!khober/Dederichs

further references:

1.2 Pbonon

Ref. p. 1801

states: Nb

Niobium has been of central interest in neutron diffraction studies because it shows a number of pronounced anomalies in the phonon dispersion which are typical for the group V metals (V, Nb, Ta). The measurements of Nakagawa and Woods [63Nal] and of Sharp [69Sh2] agree well with each other. Sharp measured in addition to the high symmetry directions a number of off symmetry phonons in the (1 TO) plane. Due to the anomalies, very long ranged forces are needed to obtain reasonable fits of the measured data. Table 2a. Nb. Measured phonon frequencies at 296 K [77Pol]. v [THz]

4

v [THz]

r

CWI L 0.031(4) 0.044 (4) 0.059 (5) 0.074 0.088 (6) 0.091 (4) 0.10 0.143 0.145 (6) 0.15 0.193 0.20 0.205 (8) 0.213 0.233 0.243 0.25 0.25 (7) 0.253 0.273 0.291 (7) 0.293 0.30 0.313 0.333 0.343 0.35 0.353 0.375 0.393 0.40 0.425 0.443 0.45 0.455 (12) 0.46 0.48 0.493 0.50 0.505 (15) 0.52 0.54 0.545 (12) 0.55 0.58 0.60 0.65 0.70

0.50 0.75 1.00 1.25 0.50 1.50 1.63 (6) 0.73 (4) 0.75 2.39 (8) 0.93 (6) 3.10 (8) 1.00 1.05 (4) 1.14 (4) 1.19 (6)

0.725 0.75 0.775 0.80 0.84 0.85 0.86 0.88 0.90 0.92 0.94 0.95 0.97 1.00

CSiil WJ

3.73 (8)

4.35 (8)

1.25 1.25 (6) 1.37 (6) 1.50 1.51(6) 1.54(6) 1.63 (4) 1.78 (4) 1.87 (6)

4.82 (6) 1.97 (6) 5.05 (8) 5.29 (8) 5.46 (8)

2.28 (6) 2.27 (8) 2.82 (8)

5.60 (7) 3.00 5.63 (10) 5.72 (10) 5.71(7)

3.38 (8) 3.55 (8) 3.50

5.81 (8) 5.79 (7) 4.00 5.82 (10) 5.75 (8) 5.71 (8) 5.55 (10) 5.50 (10)

4.57 (10) 5.04 (10)

5.50 (12) 5.59 (8) 5.68 (10) 5.70 (10) 5.92 (12) 6.03 (10) 6.10 (12) 6.15 (10) 6.29 (10) 6.32 (10) 6.37 (10) 6.40 (12) 6.43 (8) 6.49 (8)

0.031 (6) 0.048 (6) 0.067 (6) 0.070 (6) 0.10 0.117 (5) 0.15 0.166 (5) 0.20 0.25 0.275 0.30 0.32 0.34 0.35 0.36 0.375 0.40 0.425 0.44 0.45 0.46 0.475 0.48 0.50 0.52 0.525 0.54 0.55 0.56 0.575 Schober/Dederichs

CWI T 5.90 (10) 6.23 (12)

6.42 (10)

6.47 (10) 6.49 (8)

Crrll T(h)

0.83 1.33 0.83 1.83 2.46 (8) 3.51 (8) 4.34 (8) 5.04 (8) 5.42 (8) 5.69 (6) 5.90 (8) 6.00 (10) 5.91 (6) 5.97 (12) 5.76 (8) 5.54 (8) 5.32 (8) 5.11 (10) 5.17 (8) 5.11 (8) 5.12 (8) 5.10 (6) 5.03 (6) 4.86 (8) 4.93 (8) 4.69 (8) 4.69 (8) 4.58 (8) 4.46 (8)

1.33 1.71 (6) 1.83 2.11 (6) 2.62 (6) 3.23 (8)

3.88 (6)

4.39 (6)

4.82 (8)

5.03 (6)

5.07 (8) (continued) 97

1.2 Phononenzust5nde: Nb

[Lit. S. 180

Table 2a. Nb. (continued) C

c 4.35 (8) 4.20 (8) 3.94 (6)

0.58 0.60 0.62 0.625 0.64 0.65 0.66 0.675 0.68 0.70 0.72 0.725 0.74 0.75 0.76 0.775 0.78 0.80 0.82 0.84 0.85 0.86 0.88 0.90 0.95 1.00 i

0.05 0.075 0.10 0.125 0.15 0.20 0.25 0.30 0.338 0.35 0.40 0.45 0.50

98

v [THz]

v [THz]

0.00 0.125 0.20 0.25 0.30 0.375 0.40 0.50 0.625 0.75 0.875 1.00

5.10 (10) 5.05 (20)

3.76 (6) 3.68 (6) 3.62 (6) 3.51 (6) 3.50 (5) 3.50 (5) 3.60 (6) 3.66 (8) 3.72 (6) 3.80 (8) 3.79 (6) 3.95 (8) 3.94 (8) 4.14 (8) 4.27 (8) 4.77 (8) 4.99 (12) 5.05 (8) 5.42 (8) 5.80 (10) 6.29 (10) 6.49 (8)

EI

1.13(6) 1.62 (6) 2.12 (6) 2.55 (8) 3.00 (8) 3.83 (8) 4.32 (8) 4.86 (8) 5.05 (8) 5.12 (8) 5.26 (12) 5.47 (16) 5.66 (12)

5.11 (10)

5.20 (10)

5.45 (12) 5.60 (16)

0.00 0.10 0.125 0.15 0.20 0.25 0.30 0.35 0.375 0.40 0.50

5.80 (16)

6.12 (10)

6.30 (12) 6.41 (10) 6.49 (8)

5.07 (10) 4.73 (8) 4.52 (10) 4.39 (8) 4.44 (8)

5.66 (12) 5.70 (10) 5.56 (10) 5.27 (14)

4.84 (8) 5.04 (8)

5.04 (8) 4.57 (8) 4.26 (8) 3.93 (6) 3.93 (6)

CSUI G,

CCC11 G,

KC’11G,

6.49 (8) 6.50 (20)

6.49 (8) 6.07 (10) 5.82 (14) 5.54 (10) 4.76 (10) 4.20 (10) 4.07 (8) 4.04 (8) 3.97 (10) 3.96 (8) 3.93 (6)

6.49 (8) 6.20 (12)

6.10 (14) 5.80 (12)

5.68 (12) 5.66 (12)

5.64 (10)

5.16 (10) 5.07 (10)

v [THz]

0.73 (8) 0.93 (8) 1.19 (10) 1.76 (10) 2.39 (8) 3.17 (10)

0.78 (6) 1.04 (6) 1.31 (6) 1.57 (4) 2.15 (4) 2.71 (6) 3.24 (6)

3.82 (10) 4.53 (10) 4.92 (10) 5.07 (10)

3.57 (6) 3.80 (6) 3.88 (10) 3.93 (6)

Table 2b. Nb. Measured phonon frequencies at 296 K [77Pol, 78Wol]. v [THz] off symmetry branches, not identified 0.1, O&O.7 O.l,O.l, 0.5 0.2,0.2,0.4 0.25,0.25,0.5 0.3,0.3,0.6 0.4,0.4, 0.8 0.2, 0.2, 0.6 0.3,0.3,0.4 0.4,0.4,0.2 0.1, 0.1,0.75 0.2,0.2, 0.75 0.3,0.3, 0.75 0.4,0.4, 0.75

!khober/Dederichs

4.67 (15) 3.77 (10) 3.52 (9) 3.85 (10) 3.81 (10) 4.05 (15) 4.04 (11) 3.82 (9) 3.85 (9) 5.03 (8) 4.04 (10) 3.61 (8) 3.84 (8)

5.90 (15) 5.74 (10) 5.48 (9) 6.03 (10) 5.77 (16) 4.81 (12) 5.99 (10) 5.99 (11) 5.86 (15) 6.05 (10) 5.76 (8) 5.03 (10) 4.71 (8)

Ref. p. 1801

1.2 Phonon states: Nb

Table 3. Nb. Phonon frequenciesat several temperatures [72Pol]. T

296 K

6.

700 K

900K

1030 K

1

v [THz]

296 K

T

700 K

900 K

I

1030 K

v [THz]

CWI L 0.30 0.40 0.50 0.60 0.70 0.80 0.90

1.00

4.35(8) 5.27(8) 5.71(7) 5.71(8) 5.50(10) 5.70(10) 6.29(10) 6.49(8)

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.50 0.60 0.70 0.80 0.90

1.00

0.10 0.15 0.20 3.25 3.30 3.35 D.36 0.38 0.40 0.42 0.44 0.45 0.46 0.48 D.50 0.52 0.54 0.55 0.56 0.58 0.60 0.62 0.65 0.70 0.75

1.25(6) 1.54(6) 1.97(6) 2.27(8) 3.55(8) 4.57 (10) 6.23(12) 6.42(10) 6.49(8)

2.46(8) 3.51(8) 4.34(8j 5.04(8) 5.69(6) 5.91(6) 5.97(12) 5.54(8) 5.11(10) 5.17(8) 5.11 (8) 5.10(6) 5.03 (6) 4.86(8) 4.69 (8) 4.69(8) 4.58(8) 4.35(8) 4.20(8) 3.94(6) 3.68 (6) 3.50(5) 3.80(8)

0.80 0.85 0.90

4.37(6) 5.27(7) 5.79(8) 5.77(10) 5.59(12) 5.79(10) 6.19 (10) 6.49(8)

0.62(3) 0.85(3) 1.12(3) 1.42(3) 1.76(4) 2.16 (4) 2.55(4) 3.60(4)

4.65(5)

0.91(3) 1.22(3) 1.50(4) 1.86(4) 2.24(4) 2.67(4) 3.65(5) 4.61 (4)

5.50(4) 6.04(9) 6.32(9) 6.49(8)

6.02(7) 6.33(9) 6.42(18)

5.51(6) 5.35(9) 5.25 (8) 5.24(8) 5.13(9) 5.06(7) 4.94(7) 4.83 (7) 4.68 (6) 4.63 (6) 4.51(7) 4.34(7) 4.14(4) 3.81(8) 3.69 (6) 3.82(15)

T

0.64(3) 0.88(3) 1.21(3) 1.51(3) 1.85(3) 2.22(4) 2.64(4) 3.57(5) 4.56(4) 5.08(4) 5.98 (7) 6.31(8) 6.37(12)

5.87(10) 5.77(10) 5.72(7) 5.56(9) 5.33 (7) 5.20(9)

5.77(U)

5.11 (8) 4.96(7) 4.92(6) 4.81(5) 4.58 (8)

5.12(8) 5.00(8) 4.83(7) 4.80(8) 4.64(8)

4.44(9) 4.35(6)

296 K

5.58(8) 5.34(9) 5.23 (8)

3.98(8) 3.78 (7)

4.51(7) 4.34(7) 4.26(7) 4.00(6) 3.83(6)

3.96(9)

4.01(9)

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90

4.45(7) 4.95(13) 5.76(7) 6.49 (8)

6.42(18)

700 K

5.74 (10) 6.37(12) 900 K

v [THz]

5

2.47(12) 3.63(12) 4.65(9)

3.52(U) 5.09(9) 5.76(9) 5.82(9)

1.00

4.14(8) 4.99 (12) 5.80(10) 6.49(8)

1.00

1.71 (6) 2.11 (6) 2.62(6) 3.23(8) 3.88(6) 4.39(6) 4.82(8) 5.03(6) 5.07(8) 5.10(10) 5.11 (10) 5.20(10) 5.45(12) 5.80(16) 6.12(10) 6.30(12) 6.49(8)

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

3.00(8) 3.83(8) 4.32(8) 4.86(8) 5.12(8) 5.26(12) 5.47(16) 5.66(12)

1.20(3) 1.73(4) 2.28(4) 2.82(4) 3.37(4) 3.92(3) 4.41(5) 4.72(4) 4.91 (8) 4.94(6) 5.00(7) 5.12(8) 5.17(7) 5.42(7) 5.72(7) 6.06(6)

1.25 (4) 1.81(4) 2.35(4) 2.89(4) 3.40(4) 4.35(5)

5.06(9) 5.21(9)

6.49(8)

5.72(7) 5.90(7) 6.16(15) 6.42(18)

2.98(8) 3.80(9) 4.51(9) 4.97(9) 5.18 (10) 5.46(8) 5.68(12) 5.81(6)

4.96(10) 5.24(11) 5.45(12) 5.67(14) 5.84 (10)

CWil -L 0.10

0.73 (8)

0.13 0.15 0.20 0.25 0.30 0.40 0.45 0.50

1.19 (10) 1.76(10) 2.39(8) 3.17(10) 4.53 (10) 4.92 (10) 5.07 (10)

Schober/Dederichs

0.85(3) 1.17(3) 1.34(3) 1.88(4) 2.42(4) 3.14(3) 4.81 (4) 4.97(8)

0.86(2) 1.17(2) 1.34(2) 1.87(3) 2.46(3) 3.14(3) 4.35(3) 4.77(5) 4.98 (9) 99

1.2 Phononenzustiinde: A-

Nb

[Lit. S. 180

G-

-0

P I

N

7r HZ

I

6-

I I

I

I

0.2

0.4

0.6

I

I

I

I

I

I

(1.51.0 0.8

0.8 1.0

11;; 1

I

I

I

0.6

0.4

0.2

[111 I 222

0 [

0

1Hz 6

. TH: Nb

5 13&l 4 1 a3 2

1 0 0.2 0 0.8 0.6 0.4 1.0 l0011=l1111 -t Fig. 1 a. Nb. Dispersion curves at 296 K. The solid lines represent the eighth neighbour Born-von Karman model ofTable Nb [63Nal] (according to [63Nal]). Fig. 1b. Nb. Measured phonon frequencies at 296 K in the [3,X1]direction. The solid lines represent the seventh neighbour Born-von Karman model of Table4 Nb [77Pol] [78\\'01].

Anomalies

The principle anomalies in the high symmetry directions are, see Fig. 1 Nb:

i) There is a crossover of the longitudinal and transverse branches in [OOfl direction near c=O.7. The two transversebranches in [Oifl direction cross near (=0.3. ii) The COO;]L branch has an additional maximum and minimum leading to two additional critical points in the spectrum. iii) The [OO;] T branch is clearly below the velocity of sound line at 3=0.2 and rises above for larger ( values. The [O;Q T2 branch shows a similar behaviour. At higher temperaturesthe anomalies tend to disappear, Fig. 2 Nb and 3 Nb. loo

Schober / Dederichs

Ref. p. 180

1.2 Phonon states: Nb

H

If THz

6-

I

Y._l

THz

Nb [DOfIT

5-

.

4I * 3-

P’

I

5.5-

5 5.0-

. P

4.50 T= 296K

1 I

0 T= 296K T=1030K

l

4.0-

T=l030K

l

I

Nb [~LLII

6.0-

,$t @”

l

01

F-

h-

A-

I

I

I I I I I I 3.51 0.30 0.35 0.40 0.‘+5 0.50 0.55 0.60 0.65 bFig. 3. Nb. The [((S] L (AI, FJ phonon branch in Nb at

I

0.2 0.4 0.6 0.8 1.0 fFig. 2. Nb. The [OO(] T (A5)phonon 0

branch at 296 K and 1030 K. The modes represented by full squares have frequencies at the two temperatures which cannot be distinguished [72Pol].

296 K and 1030 K in the region of the anomaly [72Pol].

c=O.46

Born-van Karman fit

Due to its anomalous behaviour Nb requires very long ranging force constant models. The values of the long range constants depend critically on the range allowed in the fitting procedure and no physical significance can be attributed to them. Omitting the force constants to the eighth neighbours in the fit [63Nal] would even lead to an elastically unstable crystal (cd4x0). Table 4. Nb. Born-von Karman force constants, @y. T

296 K

296 K

296 K

T

296 K

296 K

296 K

Ref.

63Nal

77Pol *)

69Sh2**)

Ref.

63Nal

77Pol *)

69Sh2**)

In

ij

111

xx XY xx YY xx zz XY xx YY YZ XY

200 220

311

@t [Nm-‘1 14.14(9) 8.84(12) 14.16(9) - 3.64(20) 2.27(7) - 6.38(11) 0.79(13) 3.61(9) - 0.75(6) -0.95 (8) 1.26(5)

13.79(20) 9.06(21) 12.22(56) -1.68 (31) 2.23(16) - 6.89(27) 0.52(20) 3.98(21) -1.08 (13) -1.48 (11) 1.57(11)

13.95 8.96 11.15 -1.71 2.24 - 5.41 0.92 3.62 -0.52 -1.16 1.36

m

ij

222

xx XY xx YY xx YY YZ XY xx YY zz XY

400 133

420

@z [Nm-‘1 -1.16 (6) -1.33 (12) - 7.08(25) 1.32(14) -0.03 (8) -0.10 (5) 0.37(8) -0.17 (5) 0.51(6) -0.27 (11) 0.81(9) - 0.06(7)

- 0.96(13) -0.77 (15) - 5.53(31) 1.80(15) 0.75(13) -0.15 (8) -0.05 (10) - 0.20(9)

-0.89 -1.09 - 5.48 1.00

*) In off-symmetry directions a maximum deviation of 10 % and an average deviation of 3.7 % from the measured iequencies was found for this model [78Wol]. **) There seems to be an error in the eighth neighbour node1 of Sharp, since it leads to a wrong value for the ela.stic constant cd4.

Schober/Dederichs

101

[Lit. S. 180

1.2 Phononenzusttinde: Nb 2. Frequency spectrum and related properties 1.25

THT

6.2 1Hz

I 1.00

6.0

0.75 T < 0.53

I 5.8 i

0.25

5.6

0

1

2

3

I

5

6 1Hz 7 n-

Fig. 4. Nb. Frequency spectrum at 296 K calculated from the eighth neighbour Born-von Karman fit parameters of Table 4 Nb [63Nal].

Fig. 5. Nb. Debye cutoff frequencies, Y,, at 296 K, calculated from the spectrum of Fig. 4 Nb.

Table 5. Nb. Phonon frequency spectrum at 296 K calculated from the eighth neighbour Born-von Karman model [63Nal] of Table 4 Nb. v DHz]

g(r) DHz-‘1

v [THz]

g(v) [THz-I-J

v [THz]

g(v) DHz-‘1

v [THz]

g(v) [THz-‘-J i

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55

0.000 0.000 O.ooO

1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40

0.055 0.058 0.060 0.063 0.064 0.067 0.069 0.071 0.074 0.076 0.078 0.082 0.084 0.086 0.089 0.092 0.096 0.098 0.102

3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00 4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00 5.05 5.10

0.197 0.211

5.15 5.20 5.25 5.30 5.35 5.40 5.45 5.50 5.55 5.60 5.65 5.70 5.75 5.80 5.85 5.90 5.95 6.00 6.05 6.10 6.15 6.20 6.25 6.30 6.35 6.40 6.45 6.50

0.155 0.159 0.169 0.181 0.198 0.221 0.253 0.298 0.374 0.565 0.836 0.622 0.573 0.644 0.500 0.161 0.129 0.112

1.60 1.65 1.70

102

0.001 0.001 0.002 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

0.011 0.012 0.014 0.016 0.018 0.020 0.023 0.025 0.027 0.030 0.032 0.034 0.037 0.040 0.042 0.044 0.047 0.049 0.051 0.053

0.104 0.109 0.112 0.116 0.120 0.125 0.130 0.134 0.139 0.146

0.152 0.159 0.167 0.176 0.184

Schoher/Dederichs

0.236 0.297 0.400 0.395 0.404 0.422 0.456 0.390 0.385 0.294 0.261 0.240 0.226 0.218 0.213 0.211 0.213 0.220 0.229 0.228 0.226 0.224 0.219 0.217 0.212 0.211 0.207 0.206 0.203 0.200 0.198 0.160

0.100 0.090 0.083 0.075 0.068 0.063 0.056 0.048 0.034 0.000

1.2 Phonon states: Nb

Ref. p. 1801

0 Fig. 6. Nb. Debye temperatures On calculated from an eighth neighbour Born-von Karman model [69Sh2].

100

200

300

400 K 500

Fig. 7. Nb. Debye-Waller exponent 2Wdivided by the recoil frequency of the free atom, vs, calculated from the spectrum of Fig. 4 Nb.

3. Theoretical models The phonon anomalies and their relation to superconductivity have attracted a lot of attention by theorists. The origin of the anomalies is however still unresolved. Varma and Weber [79Val] could reproduce the anomalies in a model which splits the dynamical matrix into a short ranged part which is parametrized and a long range part which is calculated by tight binding methods, seeFig. 8 Nb. A8

-F

H

THz Nb

-A

P

I

2 0 0

1.0 0.8 0.6 0.4 0.2 0 0.5 f-c fFig. 8. Nb. Phonon dispersion calculated from a model with a parametrized short range interaction and a long range term calculated from a tight binding calculation. Experimental points from [68Pol] (according to [79Val]). Born-von Karman and equivalent models: seeTable 4 Nb, further references:[69Sh2,77Pal].

Short ranged forces plus a simple electronic contribution: [75Si2], further references: [72Bal, 75Up1, 76Br1, 79Gol]. Charge fluctuation model: [77Wal, 77AllJ Model potential calculation: [73Anl, 72Ko2].

Schober/Dedericbs

103

1.2 Phononenzustfinde: Ni

[Lit. S. 180

?Ji Nickel Lattice: fee, D= 352pm = 3.52A. BZ: seep. 449. I. Phonon dispersion Table 1. Ni. Measurements. Method

T WI

Fig.

Ref.

neutron diffraction (TAS) neutron diffraction (TAS)

296

Birgenau et al.

298 576 1 Ni 676’

neutron diffraction (TOF)

296

DLzl?d Brockhouse [68Del] Hautecler and Van Dingenen [67Hal]

Further measurements:[64Hal, 65Vi2,75All, 71Frl].

The agreementbetween the two TAS measurementsis very good in averagewithin 0.2% whereasthe TOF measurementgives 5 % higher frequenciesat the zone boundary. The measurementsat 576 K in the ferromagnetic phase and at 676 K in the paramagnetic phase show a softening of the phonons with temperature comparable to the one observed for Pd. The average ratios of the frequencies at equivalent positions in reciprocal space are 191676 K)‘\$298 K)=0.962 (3) and ~(576K)/v(298 K)=0.976 (4) with a standard deviation of a single ratio from the mean of 0.023. No abvious change of the dispersion at the magnetic phase transition was observed. The frequency shift with temperature arises, therefore, probably from anharmonicity and not from magnetic effects. Contrary to Pd, a large broadening of the phonons by about 45% against the width at room temperature was observedat 576K and 676 K. Fourier analysis of the dispersion shows that relatively short ranged interatomic rorces(at least up to third nearest neighbours but not necessarilybeyond fifth) provide a satisfactory description. dtnomalieshave not been reported. A-

0

-E

1.0

1.010

t-

0.8

1-

0.6 -c

0.L

0.2

0

0.5 b-

Fig. 1. Ni. Phonon dispersion curves at 298 K and 676 K. The lines represent the corresponding fifth neighbour Born-von Karman tits of Table3 Ni [68Del] [68Del]).

104

Schober/ Dederichs

Ref. p. 1 SO]

1.2 Phonon

Table 2. Ni. Phonon frequencies at 296 K measured by triple axis neutron spectroscopy [64Bil].

5

v [THz]

5

1.71(10) 3.12(g) 4.42 (11) 5.58(12) 6.54(13) 7.34(12) 7.94(14) 8.34(13) 8.50(24) 8.56(18) 8.65(20) 8.55(13)

0.2 0.3 0.4 0.5 0.6 0.7 0.75 0.8 0.85 0.9 0.95 1.0

CCC<1 T 0.1 0.15 0.2 0.25 0.3 0.35 0.375 0.4 0.425 0.45 0.475 0.5

1.33(4) 1.89(5) 2.47(5) 2.99(5) 3.37(5) 3.76(5) 3.90(7) 4.02(6) 4.10(8) 4.26(6) 4.24(8) 4.24(6)

1.96 (5) 2.81 (8) 3.62(9) 4.36(8) 4.98 (10) 5.59(12) 5.97(13) 6.26(15)

v [THz]

2.03 (4) 2.99(6) 3.83 (6) 4.49(8) 5.12(9) 5.67(11) 5.83(12) 6.01 (12) 6.07(12) 6.23(13) 6.24(14) 6.27 (10)

crrr1 L 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.425 0.45 0.475 0.5

wrr1 I. 0.1 0.2 0.3 0.4 0.5 D.55 3.6 3.65 3.7 3.75 3.8 3.85 3.9 1.95

2.34(9) 4.44(11) 6.08(10) 7.25(17) 7.63(20) 7.69(27) 7.68 (18) 7.47 (19) 7.39(17) 7.30(24) 6.85(17) 6.74(25) 6.54(17) 6.36(15)

I.5 I.6 I.7 0.8 0.9

6.21(15) 6.20(12) 6.40(14) 6.36(15) 6.32(16)

3.05 (5) 4.39(7) 5.60(g) 6.61 (10) 7.44(14) 8.14(16) 8.53(17) 8.61(24) 8.79(18) 8.58(25) 8.88(17)

Table 3. Ni. Born-von Karman force constants, @t. T

296 K

296 K

298 K

Ref.

64Bil

64Bil “)

68Del b, 68Del b,

ij

110

xx zz XY

17.178 -0.026 19.316

17.720 -1.015 18.735

17.319 -0.436 19.100

16.250 -0.970 19.390

200

xx YY

0.880 -0.519

1.148 -0.998

1.044 -0.780

1.070 0.056

211

xx YY YZ XY

0.626 0.320 0.453 -0.173

0.940 0.182 0.505 0.253

0.842 0.263 -0.109 0.424

0.963 0.449 -0.391 0.458

220

xx zz XY xx YY zz XY

0.275 -0.160 0.424

0.459 -0.153 0.612

0.402 -0.185 0.660

0.115 -0.457 0.222

-0.363 0.100 0.158 -0.174

-0.085 0.007 0.018 -0.035

-0.256 -0.063 -0.040 -0.072

310 0.1 0.2 0.3 0.4 0.5 0.6 0.65 D.7 0.75 3.8 3.9

1.28 (5) 2.76 (10) 4.14(14) 5.50(18) 6.15 (12) 6.85(20) 7.22(18) 7.67(15) 7.93(23) 8.13 (14) 8.52(17)

676 K

m

Co~LI’b “1

COYllA

C’XCIT, 7 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

I

COOiT

cooi1 L 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.85 0.9 0.95 1.0

v [THz]

states: Ni

@f [Nm-l]

All fits use the measured elastic constants “) axially symmetric forces, “) general force up to fourth neighbour, axially symmetric force for fifth neighbour.

COilI n I.1 I.2 I.3 0.4 0.5 0.6 0.7 0.8 0.9

8.52(20) 8.39(15) 8.16(16) 7.83(14) 7.49 (14) 7.11 (13) 6.80(12) 6.47(12) 6.40 (11)

“) The polarization vectors for the TI and T2 modes propagating along the [Oii] direction are parallel to [Oc[] and [COO],respectively.

Schoher/Dederichs

105

1.2 Phononenzustkinde:

Ni

[Lit.

S. 180

2. Frequency spectrum and related properties

0

1

2

3

‘4

5

6

7

8THz 9

Y-

Fig. 2. Ni. Phonon frequency spectra at 298 K and 676 K calculated from the corresponding fifth neighbour Born-von Karman tits of Table 3 Ni [68Del].

Table 4. Ni. Phonon frequency spectra at 298 K and 676 K calculated from the 5th neighbour Born-von Karman models of Table 3 Ni. T

298 K

676 K

T

298 K

676 K

T

298 K

676 K

1’ DHz]

g (4 DHz-r]

go9 [THz- r]

V

m [THz- l-j

gb9 [THz- ‘1

V

[THz]

[THz]

l?(d [THz-r]

gw [THz-‘-j

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00

0.000 0.000 0.000 0.001 0.001 0.001 0.002 0.002 0.003 0.004 0.004 0.005 0.006 0.007 0.008 0.010 0.011 0.012 0.014 0.016 0.018 0.019 0.022 0.024 0.026 0.029 0.032 0.035 0.038 0.042

0.000 0.000 0.000 0.001 0.001 0.002 0.002 0.003 0.004 0.004 0.005 0.007 0.008 0.009 0.010 0.012 0.013 0.015 0.017 0.019 0.021 0.023 0.026 0.029 0.032 0.035 0.038 0.042 0.046 0.051

3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00

0.046 0.050 0.055 0.061 0.066 0.073 0.081 0.090 0.100 0.113 0.130 0.169 0.174 0.179 0.183 0.187 0.191 0.195 0.199 0.203 0.207 0.212 0.216 0.219 0.224 0.227 0.231 0.235 0.238 0.243

0.056 0.061 0.067 0.074 0.082 0.091 0.101 0.114 0.132 0.166 0.179 0.183 0.187 0.192 0.197 0.202 0.206 0.211 0.215 0.220 0.224 0.229 0.233 0.237 0.241 0.245 0.248 0.252 0.229 0.200

6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20 8.30 8.40 8.50 8.60 8.70 8.80 8.90

0.246 0.249 0.162 0.157 0.157 0.155 0.154 0.150 0.146 0.140 0.133 0.123 0.110 0.089 0.087 0.137 0.208 0.287 0.317 0.347 0.378 0.254 0.197 0.156 0.119 0.069 0.048 0.024 0.0

0.171 0.169 0.165 0.161 0.155 0.148 0.139 0.129 0.117 0.101 0.077 0.069 0.102 0.145 0.208 0.264 0.287 0.311 0.334 0.248 0.190 0.151 0.117 0.076 0.064 0.053 0.042 0.029 0.010

106

Schoher/Dederichs

Ref. p. 1801

1.2 Phonon states: Ni

9x THZ 9.1 I 8.8 if 8.5 8.2

410

7.9 7.6 -10

37c 0

IO n-

20

30

50

100

150

200

250

K

300

7-

Fig. 3. Ni. Debye cutoff frequencies Y, at 298 K and 676 K calculated from the spectra of Fig. 2 Ni.

Fig. 4. Ni. Debye temperature 0,. Solid line: calculated from the fourth neighbour Born-von Karman force constants of Table 3 Ni [64Bil]. Broken line: experimental heat capacity data [56Dyl] (according to [64Bil].

1.0 THZ' 0.8 t 0.8 0.6 -5 1 N OA 0.2

0

100

200

300

LOO K

500

T-

Fig. 5. Ni. Debye-Waller exponent 2 W divided by the recoil frequency of the free ion, vR, calculated for the spectra of Fig. 2 Ni.

3. Theoretical models Due to their simple structure the dispersion curves of Ni can be described very well by phenomenological models. A microscopic description is still outstanding. Born-von Karman and equivalent models: seeTable 3 Ni, further references:[67Sil, 73Sh1,77Sa1,75Pal]. Short ranged forces plus a simple electronic contribution: [67Hal, 67Sh1,69Kr1, 78Kul], further references: [66Pal, 71Prl,71Sil, 73Pa2,75Be2,77Cll, 77Dil,77Khl, 78RalJ Breathing shell model: [71Hal]. Shell model plus short ranged forces plus simple electronic contribution: [75Fil]. Model pseudopotential: [76Na3,75Si4]. Transition metal potential: [73Anl].

Schober/Dedericbs

107

1.2 Phononenzustiinde: Pb

Pb

[Lit. S. 180

Lead

Lattice: fee,a=495 pm=4.95 A. BZ: seep. 449. 1. Phonon dispersion

Table 1. Pb. Measurements. Method

T [K] Fig.

neutron diffraction (TAS)

loo

neutron diffraction (TAS) neutron diffraction (TOF)

Ref.

1 Pb

Brockhouse et al. [62Brl], [61Brl] Stedmanet al. [67Stl] Furrer and Halg [70Ful]

80,300 5,80, 290

Further measurements:[67St3]. The two TAS measurementsagree very well. The TOF measurementgives systematically higher frequencies for the long wavelength phonons. The general shapeof the dispersion curves is for lead much more complex than for most other fee metals,e.g.copper. There is a much larger number of cross avers betweenthe various branches and a higher number of turning points. Fourier analysis shows that the interionic forces are very long ranged, probably beyond eighth neighbours. Lead was the first material for which Kohn anomalies have been reported. Thefrequency shifts between5 K and 290 K could be explained by anhannonicity by a calculation using an effective ion-ion potential. The superconducting phasetransition around 7.2 K has only a negligible influence on the phonon spectrum. A shift of the phonon spectrum with applied pressure has been measured by Lechner and Quittner [66Lel]. A-

h-

Z-

I

0.8

0.6 -t

I

I

0.L

0.2

Fig. 1. Pb. Measured dispersion curves at 100 K. The older measurements [61Brl] are shown by open and closed circles the more recent ones [62Brl] by crosses.The straight lines through the origin give the initial slopes of the curves as calculated from the elastic constants [60Wal]. Anomalies in the dispersion curves

The dispersion curves show a large number of pronounced irregularities which have been interpreted as Kohn anomalies [61Br2, 62Br1, 67St2], seeFig. 2 Pb. The phonon widths on the other hand do not show anomalies

which could be attributed to the Kohn effect [70Ful]. Whereas the slopes of the dispersion curves determined bystedman et al. [65Stl] agreewith the ones.expectedfrom theelasticconstantsmeasuredby ultrasonic techniques, Furrer and Hllg [70Ful] find larger values.They attribute this differenceto the differenceof first and secondsound velocities. 108

Schober/Dederichs

1.2 Phonon states: Pb

Ref. p. 1801

Table 2. Pb. Measured phonon frequencies at 100 K, [62Brl]. be found in [67St3 and 70Ful-J). v [THz]

5

CWI L

5

v [THz]

CW’I T

0.20 0.30 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

0.87 (3) 1.27 (3) 1.61 (3) 1.71(4) 1.83 (4) 1.91 (3) 2.00 (3) 2.07 (3) 2.14 (4) 2.16 (2) 2.15 (2) 2.14 (2) 2.05 (3) 1.94 (4) 1.86 (3)

(Additional

I

v D’ffzl

r

0.20 0.30 0.40 0.45 0.50

0.47 (4) 0.73 (3) 0.90 (2) 0.96 (3) 1.04 (2)

0.60

1.115 (20)

0.70

1.10 (3)

0.80

1.03 (3)

0.90

0.95 (2)

1.00

0.89 (2)

0.15 0.20 0.23 0.25 0.265 0.28 0.30 0.32 0.335 0.35 0.37 0.39 0.42 0.46

CN61T, "1

cr5r1 L

0.1 0.2 0.3 0.4 0.5 0.6 0.68 0.75 0.83 0.88 0.905 1.0

0.14 0.21 0.28 0.35 0.42 0.49 0.575 0.65 0.71 0.79 0.86 0.93 1.0

0.08 0.14 0.19 0.25 0.30 0.325 0.35 0.375 0.40 0.425 0.45 0.475 0.50

0.53 (3) 0.75 (3) 0.95 (3) 1.17 (3) 1.37 (3) 1.57 (3) 1.78 (3) 1.92 (4) 2.02 (3) 2.02 (3) 2.02 (3) 1.93 (4) (1.86 (3))

“) The polarization vectors for the [Oil] Tl and T, branches are parallel to [O[[] and [COO],respectively.

v [THz]

CNll L

CWilL

CoCClT~“1 0.21(3) 0.41(3) 0.56 (3) 0.76 (3) 0.91(3) 1.12 (3) 1.20 (3) 1.25 (3) 1.15 (3) 1.06 (4) 0.96 (4) (0.89 (2))

values of off symmetry phonons can

0.49 0.55 0.60 0.64 0.70 0.75 0.80 0.83 0.86 0.88 0.90 0.93 0.97 1.0

0.99 (4) 1.26 (4) 1.40 (4) 1.48 (3) 1.545 (20) 1.605 (20) 1.64 (2) 1.665 (20) 1.675 (30) 1.715 (20) 1.76 (2) 1.795 (20) 1.85 (2) 1.92 (2)

C5CTlT ’ 0.11 0.15 9.20 p:23 0.26 0.29 0.32 0.35 0.38 0.40 0.43 0.46 0.50

0.82 (5) 1.24 (4) 1.51 (4) 1.76 (3) 1.91 (3) 1.97 (3) 2.00 (3) 2.05 (3) 2.085 (20) 2.09 (2) 2.08 (3) 2.16 (3) 2.185 (20)

0.35 (4) 0.44 (3) 0.55 (3) 0.61 (2) 0.66 (2) 0.73 (2) 0.77 (2) 0.79 (2) 0.805 (20) 0.835 (20) 0.86 (2) 0.88 (2) 0.89 (2)

CK I

2xT

lHz

Pb

ax

l *

It-@

0 I 0

Schober / Dederichs

X

I

E501 L

2,0- T=lOOK

b Fig. 2. Pb. Anomalies in the phonon dispersion. The experimental results (as in Fig. 1 Pb) show an anomaly at 2 Iqj=O.45 and probably also ate lqI=1.24 [62Brl].

2.01(2) 2.10 (2) 2.09 (2) 2.04 (3) 1.925 (40) 1.75 (4) 1.54 (4) 1.39 (3) 1.30 (3) 1.24 (3) 1.185 (30) 1.07 (3) 0.885 (30) (0.89 (2))

-**I

1 t x’pfg 0 (2,2.2)to (3,3.0) I.24 x (1.1.1) toi2,2,11 1 l A (2.2,rl)to ll.l,O) 0 10.975:0.975;O) 1 I I I c 1.4 1.2 0.2 OX 0.6 0.8 1.0 &lqlI I I , I I 0.2

0.1, t-

0.6

0.8

1.0

109

1.2 Phononenzustiinde: Pb Born-von Karrnan

[Li!. S. 180

constants

The interionic forcesin Pb are very long range and Born-von Karman tits are therefore only of limited value. spectrum calculated from an eighth neighbour fit to the symmetry phonons [65Gil] differs considerably from ihe one obtained directly by interpolation of the measuredphonon frequencies[67St3]. Utilizing the off symmetry phonon frequenciesto lit the force constants the spectrum can be fairly well reproduced [74Col], seeFig. 3 Pb. Even such a lit cannot reproduce the phonon dispersion over the whole range, seeFig. 4 Pb. The

Table 3. Pb. Born-von Karman force constants, @!, ‘I-= 80 K [74Col]. m

ij

s; [N m- ‘1

m

ij

0; [N m-r]

110

xx

4.3243 -2.4881 4.6730

310

xx YY zz

-0.5136 - 0.1424

XY xx

-0.1920 0.0423

XY xx YY zz

0.4359 0.2604 -0.2002 0.0078

YZ xz XY xx YY

-0.0082 -0.1973 0.0805 -0.0881 -0.0900

2. Frequency spectrum and related properties zz XY

2.5

IHz-’ 200

xx zz

211

220

1.4083 0.0719

xx YY YZ xz

0.2764 -2.996 -0.0434 0.0722

xx zz XY

0.7414 0.2446 0.4117

222 321

400

1.0

1.5

2.0 IHz

2.5

Y-

Fig. 3. Pb. Phonon frequency spectrum. The solid line is obtained from an eighth neighbour Born-von Karman calculation [74Col] and the dashed line represents the spectrum obtained by interpolation of the measured phonon frequencies [67St3]. A-

l-

Y

h-

-C

1

K

lttll

0

0

0.2

04

0.6

0.8

1.0

0.8

0.6

0.4

0.2

05

Fig. 4. Pb. Born-von Karman fit to the phonon dispersion. The lines represent the calculated values. The points are the experimental values of [67Stl] (according to [74Col]).

110

Schoher / Dederichs

0.3011

1.2 Phonon states: Pb

Ref. p. 1801

Table 4. Pb. Phonon frequency spectrum at 80 K calculated from the 8th neighbour Born-van Karman model of Table 3 Pb. v [THz]

g(v) CT=- ‘1

v [THz]

g(y) CT=- ‘1

v [THz]

g(v) CT=- ‘1

3.0

0.0

3.025 3.050 3.075 0.100 0.125 3.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750

0.0002 0.0008 0.0019 0.0034 0.0052 0.0077 0.0101 0.0135 0.0162 0.0203 0.0246 0.0297 0.0338 0.0402 0.0462 0.0533 0.0602 0.0690 0.0780 0.0875 0.0991 0.1115 0.1258 0.1419 0.1599 0.1820 0.2062 0.2352 0.2703 0.3124

0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975 1.000 1.025 1.050 1.075 1.100 1.125 1.150 1.175 1.200 1.225 1.250 1.275 1.300 1.325 1.350 1.375 1.400 1.425 1.450 1.475 1.500 1.525

0.3681 0.4423 0.5444 0.7764 0.7537 0.7319 0.7150 0.7387 0.7617 0.7951 0.8418 0.9113 1.0469 1.3224 1.0076 0.9608 0.9332 0.7780 0.7085 0.6609 0.6251 0.5950 0.5694 0.5483 0.5279 0.5111 0.4954 0.4791 0.4641 0.4474 0.4263

1.550 1.575 1.600 1.625 1.650 1.675 1.700 1.725 1.750 1.775 1.800 1.825 1.850 1.875 1.900 1.925 1.950 1.975 2.000 2.025 2.050 2.075 2.100 2.125 2.150 2.175 2.200 2.225

0.4043 0.4048 0.4053 0.4047 0.4033 0.3986 0.3936 0.3829 0.3674 0.3439 0.3017 0.2662 0.2603 0.2713 0.3009 0.3618 0.4630 0.5927 0.7642 1.0132 1.5148 2.4729 0.9954 0.8094 0.7803 0.7185 0.6630 0.0

18

.10-1’2

2.2

S

THZ

12 I cc $9 B N 6

2.1 I 2.0 i 1.9 1.8 -10

3 0

IO n-

20

0

30

100

200

300

400

K

500

I--

Fig. 5. Pb. Debye cutoff frequencies v, calculated with the Born-von Karman force constants of Table 3 Pb.

Fig. 6. Pb. Debye-Waller exponent 2W divided by recoil frequency of the free atom, va, calculated from the Born-von Karman force constants of Table 3 Pb.

Schober/ Dederichs

111

[Lit. S. 180

1.2 Phononenzust6nde: Pd 3. Theoretical models

Due to the complexity of the phonon dispersion in lead neither the usual phenomenological modelsnor pseudopotential theory have so far been able to describeit adequately. A phenomenological description of the dynamical matrix in q-spaceutilising symmetry can reproduce the dispersion within about 3 y0 [78Tal]. The spectrum gained thus agrees in general with the one obtained by direct interpolation of the phonon frequencies[67St3]. Born-von Karman and equivalent tits: seeTable 3 Pb and Figs. 3 Pb, 4 Pb, further references:[65Gil, 68Ng1, 73Cal-J. Direct fit of the dynamical matrix: [78Tal]. Short ranged forcesplus a simple electronic contribution: [69Krl], further references:[71Gu2,77Ral, 78Gul]. Pseudopotential calculations: [77Sol, 78Sol], further references:[74Hal, 73Bel,68Scl, 72Bel,7OCol, 73Pr1, 73Kal,71Gul, 65Tol].

Pd Palladium Lattice: fee,a=388pm=3.88 A. BZ: seep. 449.

1. Phonon dispersion Table 1. Pd. Measurements. Fig.

Method

Ref.

TKI neutron diffraction (TAS)

85 120 296

1 Pd

673

1 Pd

Miiller and Brockhouse [68Mil, 71Mil]

With the exception of an anomaly in the [Oc[] TI branch, the dispersion curves closely resemblethose of nickel and copper,differing by little more than a scalefactor (for the frequencies)of approximately the inverse squareroot If their massratios.

AlH2

w

X

8

Pd

I

Iootl

I

I

I

I

I

III

I

Q2 0.1 0.6 0.8 1.010 0.2 0.1 0.6 0.8

I

IJ.8- -, 0.6

0.4

0.2

,

n5

Fig. 1. Pd. Dispersioncurvesin the four majorsymmetry directions at 120 K and 673 K according to [71Mil]. The straight lines through the points r give the initial slopes of the dispersion curves at 120 K as calculated from the measured elastic constants [6ORal]. 112

Schober/Dederichs

[Lit. S. 180

1.2 Phononenzust6nde: Pd 3. Theoretical models

Due to the complexity of the phonon dispersion in lead neither the usual phenomenological modelsnor pseudopotential theory have so far been able to describeit adequately. A phenomenological description of the dynamical matrix in q-spaceutilising symmetry can reproduce the dispersion within about 3 y0 [78Tal]. The spectrum gained thus agrees in general with the one obtained by direct interpolation of the phonon frequencies[67St3]. Born-von Karman and equivalent tits: seeTable 3 Pb and Figs. 3 Pb, 4 Pb, further references:[65Gil, 68Ng1, 73Cal-J. Direct fit of the dynamical matrix: [78Tal]. Short ranged forcesplus a simple electronic contribution: [69Krl], further references:[71Gu2,77Ral, 78Gul]. Pseudopotential calculations: [77Sol, 78Sol], further references:[74Hal, 73Bel,68Scl, 72Bel,7OCol, 73Pr1, 73Kal,71Gul, 65Tol].

Pd Palladium Lattice: fee,a=388pm=3.88 A. BZ: seep. 449.

1. Phonon dispersion Table 1. Pd. Measurements. Fig.

Method

Ref.

TKI neutron diffraction (TAS)

85 120 296

1 Pd

673

1 Pd

Miiller and Brockhouse [68Mil, 71Mil]

With the exception of an anomaly in the [Oc[] TI branch, the dispersion curves closely resemblethose of nickel and copper,differing by little more than a scalefactor (for the frequencies)of approximately the inverse squareroot If their massratios.

AlH2

w

X

8

Pd

I

Iootl

I

I

I

I

I

III

I

Q2 0.1 0.6 0.8 1.010 0.2 0.1 0.6 0.8

I

IJ.8- -, 0.6

0.4

0.2

,

n5

Fig. 1. Pd. Dispersioncurvesin the four majorsymmetry directions at 120 K and 673 K according to [71Mil]. The straight lines through the points r give the initial slopes of the dispersion curves at 120 K as calculated from the measured elastic constants [6ORal]. 112

Schober/Dederichs

1.2 Phonon states: Pd

Ref. p. 1801

Table 2. Pd. Measured phonon frequenciesv [68Mil, 71Mill. T

120K

296 K

673K

853 K

v [THz]

5

T

120K

0.67(2) 1.31(2) 1.64(2) 1.97(2) 2.60(2) 3.205(20) 3.73(2) 4.135(30) 4.44(4) 4.58(5) 4.64(5)

0.66(4) 1.29(3) 1.63(2) 1.93(3) 2.555(30) 3.16(3) 3.65(3) 4.10(5) 4.28(6) 4.36(4) 4.54(6) 4.56(6)

CN51-L “1 0.635(20) 0.62(2) 1.25(2) 1.24(3) 1.90(3) 2.47(3) 3.01(5) 3.53(4) 3.93(3)

1.83(4) 2.42(3) 2.96(4) 3.45(4) 3.79(4)

4.23(7) 4.35(9) 4.40(5)

4.07(5) 4.27(4) 4.31(10)

CWI L 0.1 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2.24(6) 3.19(4) 4.06(7) 4.84(5) 5.56(5) 6.09(9) 6.48(9) 6.72(10) 6.72(12)

1.18(6) 2.21(3) 2.72(6) 3.15(5) 4.01(6) 4.78(6) 5.48(7) 6.06(7) 6.39(9) 6.66(9) 6.70(9)

2.12(4)

2.05(6)

3.06(4) 3.93(5) 4.65(6) 5.34(11) 5.86(10) 6.14(11) 6.36(11) 6.47(8)

3.05(5) 3.86(5) 4.63(5) 5.29(8) 6.04(12)

1.28(4) 1.64(5) 2.04(5) 2.46(5) 2.85(5) 3.10(4) 3.28(4) 3.34(4)

0.84(3) 1.22(3) 1.59(4) 2.01(4) 2.42(4) 2.76(4) 3.02(5) 3.18(7) 3.21(8)

1.20(6) 1.57(6) 1.95(6) 2.26(5) 2.58(7) 2.82(7) 2.95(8) 2.96(10)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.75 0.8 0.9

1.83(5) 2.76(2) 3.62(3) 4.49(3) 5.28(4) 5.94(7) 6.38(8) 6.66(10)

0.88(2) 1.84(2) 2.70(3) 3.56(3) 4.43(4) 5.15(5) 5.80(6) 6.05(7) 6.32(7) 6.64(10)

0.92(2) 1.78(2) 2.67(3) 3.52(3) 4.32(3) 5.00(3) 5.65(8)

0.91(2) 1.76(3) 2.63(4) 3.49(3) 4.23(4) 4.95(4)

6.08(9) 6.30(10)

CNCIT 7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.75 0.8 0.875 0.9

0.59(2) 1.09(2) 1.47(5) 2.07(4) 2.835(30) 3.45(3) 3.99(4) 4.35(4) 4.62(7)

0.575(20) 1.055(20) 1.475(30) 2.135(30) 2.835(30) 3.42(3) 3.90(4) 4.09(4) 4.29(4) 4.44(7) 4.47(4)

0.56(2) 1.025(20) 1.56(3) 2.14(3) 2.79(3) 3.34(4) 3.77(6) 3.95(5) 4.17(6)

0.525(20) 1.005(20) 1.525(30) 2.13(3) 2.71(3) 3.20(4) 3.63(4)

4.39(10)

4.20(5)

3.97(5)

corr1 L

Crlll T 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

853 K

v [THz]

r

CWI T 0.1 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.75 0.8 0.9 1.0

673 K

296 K

1.09(7) 1.54(7) 1.87(6) 2.18(7) 2.44(6) 2.75(6) 2.87(6) 2.90(6)

0.1 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.75 0.8 0.875 0.9

1.72(6) 2.75(7) 3.37(6) 4.67(4) 5.60(4) 5.92(10) 5.88(11) 5.47(11) 5.04(11) 4.69(10)

1.64(6) 2.46(6) 3.23(5) 4.58(4) 5.48(5) 5.85(10) 5.75(8) 5.41(7) 5.22(10) 4.96(11) 4.72(8) 4.63(8)

2.41(5) 3.19(5) 4.47(6) 5.34(5) 5.73(10) 5.64(10) 5.05(9) 4.92(10) 4.78(10)

1.62(7) 2.38(8) 3.16(6) 4.43(5) 5.27(7)

4.54(8)

Crrrl I. 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

2.18(7) 3.18(7) 4.11(7) 4.92(5) 5.64(4) 6.12(g) 6.59(10) 6.87(10) 7.02(11)

2.13(7) 3.12(g) 4.00(11) 4.84(10) 5.55(13) 6.16(12) 6.65(13) 6.85(10) 6.86(13)

2.07(8) 2.97(9) 3.92(7) 4.73(8) 5.46(8) 6.02(8) 6.40(9) 6.62(11) 6.63(13)

2.03(9) 3.05(7) 3.89(7) 4.66(8) 5.42(7) 6.25(10)

Pm1 A 0.5 0.6 0.7 0.75 0.8 0.9

4.25(6) 4.26(S) 4.36(7) 3.44(6) 4.62(5)

4.20(6) 4.22(7) 4.29(6) 4.35(9) 4.44(5) 4.53(6)

4.07(6) 4.11(8) 4.16(6)

3.99(6) 4.00(5) 4.08(7)

4.34(8) 4.39(5)

4.20(6) 4.26(5) (continued)

“) The polarization vectors for the [O[[] TI and T, branchesare parallel to [O[f] and [[OO], respectively. Schober /Dederichs

113

1.2 Phononenzustiinde: Pd

[Lit. S. 180

f’able 2. Pd. (continued)

r

120K

613 K

296 K

853 K

v [THz] /

COilI n I.1 I.2 I.25 I.3 3.4

6.66(8) 6.53(9)

3.7

6.28(9) 6.01(7) 5.69(6) 5.42 (6) 5.08(S)

3.75 3.8 3.9

4.86(4) 4.69(6)

I.5 3.6

6.67 (6) 6.51(9) 6.37(8) 6.27(6) 5.94(3) 5.56(7) 5.26(6) 4.98 (5) 4.86(6) 4.14(6) 4.62(8)

I.U4

I Pd I 6.42(8) 6.23(12)

6.25(10) 6.14(8)

6.01 (5) 5.69(6) 5.36(7) 5.08(6) 4.19 (4)

5.89(7) 5.59(9) 5.18(8) 4.98 (4) 4.69(5)

4.62(9) 4.43(5)

4.46(5) 4.34(6)

,! 1.02 G g 1.00 z 0.98 '

0.96 0.9L 0

100 200 300 LOO 500 600 700 K 900 IFig. 2. Pd. The mean ratio of the frequencies as a function of temperature [‘IlMil].

femperature dependence From 120 K to 853 K the phonons soften by about 7 %. The mean frequency ratio (V (T)/v (296K)) is shown in Fig. 2 Pd. Anomalies

in the dispersion curves

The [Oifl T, branch exhibits an anomalous wiggle about the line, representing the velocity of sound. This anomaly is strongly temperature dependent [75Mil], see Fig. 3 Pd. The anomalous frequency shift becomes rapidly smearedout in wave vector and is possibly weakenedwith increasing temperature.The anomaly has been correlated with vktual Kohn transitions across the ‘heavy’ hole Fermi sheet formed by the fifth band electrons. A broadening of the phonon lines which could arise from the corresponding real Kohn transitions has also been observed.

Fig. 3. Pd. Temperaturedependence of the frequenciesand slopesalong the T, branch in palladium.The solid straight lines representthe velocitiesof sound calculatedfrom the elastic constants [60Ral J. The arrows indicate positions of possible Kohn anomalies. The different symbols represent different experimental setups. The small ellipses define the half maximum contours of the experimental resolutions [75hIil].

114

22 ‘j 40

Schoher /Dederichs

u 0

”

0.2

”

”

0.4 I-

0.6

”

0.8

’

0 1.0

I

Ref. p. 1SO]

1.2 Phonon states: Pd

Born-von Karman models Table 3. Pd. Born-von Karman force constants, Qt.

T

120K

296 K

673 K

853 K

Ref.

71Mil (MPI)

71Mil (MPZ)

71Mil (MP3)

71Mil (MP4)

m

ij

110

xx zz XY

19.760 (274) -2.511(455) 23.194(442)

19.337(233) -2.832(410) 22.423 (434)

17.599(256) -2.412(434) 21.350(441)

17.383(483) -2.877(576) 20.766(842)

200

xx YY

0.919 (363) 0.416(205)

1.424 (346) 0.210(222)

1.390(364) 0.044(214)

2.001(850) 0.098(320)

211

xx YY YZ xz

0.907(307) 0.134(157) 0.609 (193) 0.912(113)

0.744(255) 0.249(133) 0.163 (200) 0.708(096)

0.661(281) 0.388(145) 0.039(204) 0.784(111)

0.875(362) 0.116(252) 0.373(415) 0.339(283)

220

xx zz XY

-1.041 (107) -0.128 (186) -1.865(431)

-1.142 (115) -0.223 (173) -1.370(387)

-0.720(114) -0.177(188) -1.283(416)

-0.546(166) 0.078(354) -0.026(416)

310

xx YY zz XY xx YZ

0.086(296) -0.227(117) -0.266(159) 0.118 (143)

-0.006(236) -0.207 (110) -0.232(144) 0.076(119)

0.217(264) -0.228(107) -0.284(143) 0.167(128)

-0.263 (493) -0.148(140) -0.134(200) -0.043(220)

0.219 (192) 0.154(172)

0.154 (090) 0.330(158)

-0.063(095) 0.116 (190)

-0.233(174) -0.015(263)

321

xx YY zz YZ xz XY

-0.094(155) -0.051(163) 0.041 (060) -0.022(017) -0.033 (025) -0.066(046)

0.070(125) 0.067(131) -0.020(051) -0.022(016) -0.032(023) -0.065 (042)

-0.139 (141) 0.155(144) 0.012(053) -0.030(019) -0.045 (028) -0.090(053)

-0.057(274) 0.032(209) 0.051(074) -0.021 (032) -0.032(048) -0.063(093)

400

xx YY

0.528(259) -0.113(106)

0.072(145) 0.006(126)

0.071(197) 0.028(166)

0.223(225) 0.002 (183)

222

@t [N m- ‘1

The frequencies predicted by the models for off symmetry phonons differ by up to 4 % from the measured ones.

2. Frequency spectrum and related properties 0.5 THz-' 0.4 1 0.3 -2 s 0.2

0

12

3

4

5

6 THz 7

Fig. 4. Pd. Phonon frequencyVspzat 120 K, 296 K, and 853 K calculated from the Born-von Karman force constants of Table 3 Pd. Error: dashed-dotted line for T=296

Schober/Dederichs

115

1.2 PhononenzustInde: Pd

[Lit. S. 180

1

Table 4. Pd. Phonon frequency spectra at various temperaturescalculated from the Born-von Karman constants of Table 3 Pd. T

296 K

120K

v ~Hz] 0.0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90

2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40

853 K

n(v) TTHz- ‘1 0.0 0.000 0.001 0.001 0.003 0.004 0.006 0.008 0.011 0.014 0.017 0.021 0.026 0.030 0.035 0.040 0.046 0.051 0.056 0.062 0.068 0.074 0.080 0.087 0.094 0.101 0.109 0.119

0.0 0.000 0.001 0.001 0.003 0.004 0.006 0.008 0.011 0.014 0.018 0.022 0.026 0.030 0.035 0.040 0.045 0.049 0.055 0.060 0.066 0.072 0.079 0.086 0.094 0.102 0.111 0.122 0.134 0.148

0.129

0.140 0.154 0.170 0.191 0.220 0.267

0.282

120K

v [THz] 0.0 0.000 0.001 0.001 0.003 0.004 0.007 0.009 0.012 0.015 0.019 0.023 0.028 0.033 0.038 0.043 0.049 0.055 0.062 0.069 0.076 0.085 0.094 0.104 0.115 0.128 0.143 0.162 0.187 0.226 0.266 0.277 0.287 0.297 0.306

0.165 0.186 0.217 0.271

T

3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00

296 K g(v) [THz- ‘1

0.277 0.286 0.294 0.302 0.308 0.313 0.317 0.321 0.281 0.259 0.242 0.221 0.187 0.178 0.169 0.161 0.152 0.143

0.133 0.121 0.108 0.090 0.077 0.122 0.180 0.272 0.305 0.338 0.369 0.404 0.247 0.178 0.123 0.071 0.042 0.0

0.294 0.304 0.314 0.322 0.329 0.333 0.334 0.303 0.260 0.236 0.216 0.197 0.187 0.177 0.167

0.157 0.147 0.136 0.124 0.109 0.091 0.077 0.124 0.181 0.275 0.300 0.325 0.349 0.375 0.233 0.177 0.136 0.093 0.054 0.030 0.0

6.8 THZ 65 I 6.2 s- 5.9 Fig. 5. Pd. Debye cutoff frequencies v, calculated from the spectra of Fig. 4 Pd. n-

116

!Schober/Dederichs

853 K

0.312 0.315 0.316 0.314 0.309 0.277 0.243 0.223 0.204 0.194 0.190 0.186 0.180 0.173 0.164 0.151 0.135 0.112 0.082 0.103 0.154 0.216 0.317 0.413 0.435 0.347 0.242 0.181 0.126 0.077 0.046 0.0 0.0 0.0 0.0 0.0

1.2 Phonon states: Pt

Ref. p. 1803

7

2.0

8

7

.10-12

Clll

Pd

K mole EI-

s

1.5 I

--

I 4 c?

$1.0 N

0 Cp expt.147 Cl 11 o overage CP expt. [ 36 Jo 11 --Cl tot01 lattice sp.ht.ot conkpress. ---Ch harmonic lattice sp. ht.

2

0.5

L

0

100

200

300

400

K

tjO0

IFig. 6. Pd. Debye-Waller exponent 2 Wdivided by the recoil frequency of a free atom, va, calculated for the spectra of Fig. 4 Pd.

200

II

400

600

800

K

I[:

T-

Fig. 7. Pd. Specific heat, calculated to lowest order in the phonon anharmonicity (C, specific heat at constant pressure, C, total lattice specific heat, C, harmonic lattice specific heat) according to [71Mil].

3. Theoretical models Apart from the anomaly in the [O([] Tr-branch the dispersion curves can be reproduced well by empirical models. A successfulmicroscopic description has not been given so far. Born-von Karman and equivalent models: seeTable 3 Pd, further references:[73Shl]. Breathing shell model: [71Hal]. Short ranged forces plus a simple electronic contribution: [73Pal, 77Cl1, 78Ra3], further references:[70Brl, 71Brl,71Pal, 74Go2,75Be2,76Pal]. Short ranged forces plus a simple electronic contribution plus a d-electron shell: [78Sil, 78Si2]. Models incorporating electronic d-band terms: [73Anl].

Pt

Platinum

Lattice: fee,a = 392pm = 3.92A. BZ: seep. 449. 1. Phonon dispersion Table 1. Pt. Measurements. Method

T TKl

neutron diffraction 90 473 PAS) neutron diffraction 300 VW

Fin.

Ref.

1 Pt Dutton et al. [72Dul] Ohrlich and Drexel [680hl]

The agreement between both measurementsis mostly within 5%. The general shape of the dispersion is the one typical for an fee metal. The phonons soften in averageby about 2% in heating from a temperature of 90 K to 473 K. There is a large anomaly in the [OOQT, branch.

Schober /Dederichs

117

1.2 Phonon states: Pt

Ref. p. 1803

7

2.0

8

7

.10-12

Clll

Pd

K mole EI-

s

1.5 I

--

I 4 c?

$1.0 N

0 Cp expt.147 Cl 11 o overage CP expt. [ 36 Jo 11 --Cl tot01 lattice sp.ht.ot conkpress. ---Ch harmonic lattice sp. ht.

2

0.5

L

0

100

200

300

400

K

tjO0

IFig. 6. Pd. Debye-Waller exponent 2 Wdivided by the recoil frequency of a free atom, va, calculated for the spectra of Fig. 4 Pd.

200

II

400

600

800

K

I[:

T-

Fig. 7. Pd. Specific heat, calculated to lowest order in the phonon anharmonicity (C, specific heat at constant pressure, C, total lattice specific heat, C, harmonic lattice specific heat) according to [71Mil].

3. Theoretical models Apart from the anomaly in the [O([] Tr-branch the dispersion curves can be reproduced well by empirical models. A successfulmicroscopic description has not been given so far. Born-von Karman and equivalent models: seeTable 3 Pd, further references:[73Shl]. Breathing shell model: [71Hal]. Short ranged forces plus a simple electronic contribution: [73Pal, 77Cl1, 78Ra3], further references:[70Brl, 71Brl,71Pal, 74Go2,75Be2,76Pal]. Short ranged forces plus a simple electronic contribution plus a d-electron shell: [78Sil, 78Si2]. Models incorporating electronic d-band terms: [73Anl].

Pt

Platinum

Lattice: fee,a = 392pm = 3.92A. BZ: seep. 449. 1. Phonon dispersion Table 1. Pt. Measurements. Method

T TKl

neutron diffraction 90 473 PAS) neutron diffraction 300 VW

Fin.

Ref.

1 Pt Dutton et al. [72Dul] Ohrlich and Drexel [680hl]

The agreement between both measurementsis mostly within 5%. The general shape of the dispersion is the one typical for an fee metal. The phonons soften in averageby about 2% in heating from a temperature of 90 K to 473 K. There is a large anomaly in the [OOQT, branch.

Schober /Dederichs

117

1.2 Phononenzustkde: Pt Table 2. Pt. Measured phonon frequencies at 90 K [72Dul]. For the T, branch see Table 3 Pt.

i

v [THz]

r

COKIT 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.80 0.90 1.00

0.50 (3) 0.75 (2) 1.00 (2) 1.23 (2) 1.48 (3) 1.71 (2) 1.98 (2) 2.22 (2) 2.45 (5) 2.70 (3) 2.93 (3) 3.09(3) 3.30 (3) 3.57 (4) 3.76 (4) 3.84 (5)

Pii1 -I0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.70 0.80 0.90 1.00

CWI L 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

1.99 (5) 2.79 (4) 3.52 (5) 4.19 (5) 4.77 (3) 5.18 (3) 5.56 (4) 5.73 (5) 5.80 (8)

KiCl -f 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.50

0.76 (2) 1.06 (3) 1.40 (3) 1.72 (3) 2.08 (3) 2.38 (4) 2.63 (5) 2.90 (3)

0.10 0.15 0.20 0.30 0.40 0.50

1.76 (5) 2.64 (5) 3.44 (6) 4.77 (4) 5.60 (4) 5.85 (5)

118

v [THz]

0.70 (2) 1.05 (2) 1.38 (2) 1.73 (3) 2.06 (3) 2.36 (4) 2.71 (3) 3.10 (3) 3.47 (3) 3.85 (3) 4.20 (3) 4.87 (3) 5.36 (5) 5.66 (7) 5.80 (8)

Coirl L 0.10 0.15 0.20 0.25 0.30 0.325 0.40 0.50 0.60 0.75 0.90 1.00

1.49 (4) 2.14 (3) 2.82 (4) 3.40 (5) 3.94 (5) 4.20 (8) 4.77 (7) 4.98 (7) 4.95 (7) 4.30 (7) 3.89 (6) 3.84 (5)

0.10 0.30 0.40 0.50

3.72 (10) 3.50 (10) 3.36 (5) 3.25 (8)

0.10 0.20 0.30 0.40 0.50 0.60 0.75

5.70 (9) 5.58 (7) 5.28 (7) 4.95 (7) 4.65 (7) 4.44 (8) 4.03 (5)

[Lit. S. 180

Table 3. Pt. Phonon frequencies for the branch at 90 K, 296 K, and 473 K [72Dul].

T

90K

296 K

[Ocfl TI

473 K

v [THz]

c

UKCI T, 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.700 0.800 0.900 l.ooO

Schober/Dederichs

0.58 (2) 0.835 (20) 1.05 (2) 1.255 (20) 1.345 (20) 1.43 (2) 1.51 (3) 1.54 (3) 1.65 (3) 1.77 (3) 1.895 (20) 2.02 (2) 2.255 (20) 2.51 (2) 2.81 (2) 3.08 (2) 3.27 (3) 3.57 (3) 3.76 (4) 3.84 (5)

0.51 (2) 0.66 (2) 0.78 (2) 0.90 (2) 1.005 (20) 1.125 (20) 1.21 (2) 1.30 (2) 1.39 (2) 1.485 (20) 1.57 (2) 1.665 (20) 1.775 (20) 1.90 (2) 2.025 (20) 2.165 (20) 2.29 (2) 2.40 (2) 2.53 (2) 2.66 (2) 2.785 (20) 3.025 (20) 3.23 (2)

0.495 (20) 0.635 (20) 0.75 (2) 0.86 (2) 0.97 (2) 1.085 (20) 1.18 (2) 1.27 (2) 1.375 (20) 1.47 (2) 1.56 (2) 1.67 (2) 1.78 (2) 1.90 (2) 2.045 (20) 2.155 (20) 2.275 (20) 2.39 (2) 2.515 (20) 2.62 (2) 2.75 (2) 2.86 (2)

1.2 Phonon states: Pt

Ref. p. 1801

-E

A.r 6 THZ

W

X

X

ht

r

K

5

X L 0,”

4

I ir 3

2

1

I I

0

0 t-

1.0

0.8

0.6

0.4

0.2

0.5

0 5-

-f

s-

Fig. 1. Pt. Dispersion curves at 90 K. The crossesrepresent the experimental points. The circles represent a sixth-neighbour Born-von Karman tit to the frequencies.The slopes of the solid straight lines are the velocities of sound in the various directions.

, Anomalies in the dispersioncurves The [O[Q TI branch shows a large anomaly which lessensas the temperature increases,seeFig. 2 Pt. This anomaly is similar to the one observedin Pd. As in the caseof Pd the anomaly is thought to arise from Kohn transitions acrossthe “heavy” hole Fermi sheetformed from the fifth band electrons [72Dul].

Fig. 2. Pt. Temperature dependence of the anomaly in the [Oc[] Tr branch. The arrows indicate the positions of possible Kohn transition vectors [72Dul].

0

Schober/Dederichs

0.2

0.4 t-

0.6

0.8

1.0

1.2 Phononenzustlnde:

Pt

[Lit. S. 180

Born-von Karman constants A fit to the experimental data requires force constants to the fourth neighbours. Weaker forces probably extend to at least the sixth neighbours. Extending the fit to eighth neighbours improves the goodness of the fit only marginally.

Table 4. Pt. Born-von Karman force constants, q.

T

90 K

90K

Ref.

C72Dul-J

[72Dul]

m

ij

110

xx zz XY

25.681 (168) - 7.703 (251) 30.830 (303)

25.834 (177) - 6.918 (262) 29.787 (341)

200

xx YY

5.604 (329) -1.337 (194)

3.935 (374) - 0.916 (204)

211

xx YY YZ

2.685 (121) 0.085 (93) 1.498(118) 1.410 (69)

1.732 (158) 0.237 (110) 0.819 (193) 1.335 (74)

220

xx zz XY

-1.935 (91) 0.337 (163) - 3.285 (121)

- 2.490 (105) - 0.467 (200) -2.331 (248)

310

xx YY 22 XY

0.640 (75) 0.021 (25) - 0.056 (29) 0.233 (29)

222

xx XY

0.468 (94) -0.263 (129)

0;

X2

[Nm-‘1

0.6 THz-’

Pt

The force constants to the fifth neighbours are axially symmetric.

2. Phonon spectrum and related properties

0

1

2

3

4

5 1Hz

Fig. 3. Pt. Frequency spectrum at 90 K obtained from the sixth neighbour Born-von Karman model of Table 4 Pt.

5.6 1Hz 5.4

5.2 I

a? 5.0

0

10 n-

20

30

Fig. 4. Pt. Debye cutoff frequencies v, obtained

Fig. 3 Pt.

120

from the spectrum

of

0

100

200

300

400

K

500

Fig. 5. Pt. Debye-Wailer exponent 2 W divided by the recoil frequency of a free Pt atom, vR,calculated from the spectrum of Fig. 3 Pt.

Schober/Dedericbs

1.2 Phonon states: Pt

Ref. _D. 1801 -

Table 5. Pt. Phonon frequency spectrum at 90 K calculated from the sixth neighbour Born-von Karman force constants of Table 4 Pt. v [THz]

g(v) [THz- ‘1

v [THz]

g(v) D-Hz- ‘1

v [THz]

g(v) CTHz-‘1

0.0

0.0

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95

0.000 0.000 0.001 0.001 0.001 0.002 0.003 0.004 0.005 0.006 0.008 0.009 0.011 0.013 0.015 0.017 0.019 0.022 0.025 0.028 0.031 0.035 0.039 0.043 0.047 0.052 0.057 0.062 0.067 0.072 0.077 0.083 0.088 0.094 0.099 0.105 0.112 0.117 0.124

2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95

0.130 0.137 0.144 0.151

4.00 4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.15 4.80 4.85 4.90 4.95 5.00 5.05 5.10 5.15 5.20 5.25 5.30 5.35 5.40 5.45 5.50 5.55 5.60 5.65 5.70 5.75 5.80 5.85 5.90 5.95

0.191 0.183 0.175 0.167 0.160 0.152 0.144 0.137 0.129 0.120 0.110 0.100 0.090 0.087 0.093 0.117 0.152 0.186 0.224 0.280 0.325 0.344 0.360 0.377 0.392 0.407 0.422 0.435 0.301 0.246 0.207 0.175 0.148 0.124 0.101 0.076 0.048 0.016 0.0 0.0

0.159 0.167 0.175 0.184 0.194 0.205 0.217 0.230 0.245 0.262 0.283 0.309 0.344 0.401 0.519 0.428 0.404 0.391 0.381 0.315 0.370 0.368 0.367 0.360 0.317 0.298 0.283 0.272 0.262 0.253 0.245 0.239 0.231 0.224 0.211 0.200

3. Theoretical models Apart from the anomaly in the [OILJTI branch the measured dispersion curves are mostly reproduced by phenomenological models. A microscopic model is still outstanding. Born-von Karman and equivalent models: seeTable 4 Pt and [72Dul], further references:[680hl, 71Ko1, 75Cal]. Short ranged forces plus simple electronic contribution: [69Krl, 77Cl1, 77Kh3], further references:[74Go2, 75Bel,76Ral, 77Ral,76Ku2]. Short ranged forces plus shell model: [78Sil, 78Si2,78Ral]. Model potential: [77Ku2].

Schober /Dedericbs

121

1.2 Phononenzustiinde: Rb

[Lit. S. 180

Rb Rubidium Lattice: bee,a= 569pm = 5.69A at 120 K. BZ: seep. 448. 1. Phonon dispersion Table 1. Rb. Measurements. Method

T WI

Fig.

Ref.

neutron diffraction P-AS)

12, 85 120,205

1 Rb... 3 Rb

Copley and Brockhouse [73Col]

Further measurement:[74Co3]. , The dispersion curves of rubidium are similar in shapeto those of sodium and potassium.The forcesare homologous: 1.59= (v(K)/v(Rb)) u ((M a2),$,z/(A4a2~‘z)= 1.58. The splitting of the two branches in [OOfl direction is somewhat larger than in the lighter alkali metals. The interionic forces obtained from Born-von Karman fits are short ranged, with the force constants to the fourth neighbours less than 3 % of the nearest neighbour force constants. The temperature dependenceof the phonons was measured for constant pressure [73Col] and for constant volume [74Col]. -

A-

-F

-A P

Rb

roocl

Ibftl

1=120K

NL 00 ’ ’ ’ ’

0.25

b-

0.5 1.0

-c

I

0 0.5

.-!z

I

0.8

I

I

0.6

0.4

0.2

I-

zig. 1. Rb. Measured phonon dispersion curves at 120 K. The solid lines represent the appropriate sound velocities, nterpolated to 120 K [73Col]. ?or Fig. 2, see p. 124.

122

!Schober/Dederichs

1.2 Phonon states: Rb

Ref. p. 1801 G-

-0

---c

A-

-F

-A

I I I I I I I I 0 “I’ 0 0 0.5 1.0 0 0.51.0 -L f-co 1‘-f‘ Fig. 3. Rb. Phonon dispersion at various temperatures (o experiment, [73Col], solid line: pseudopotential calculation [75Ro2]) [75Ro2].

Schober /Dederichs

123

1.2 Phononenzust5nde: Rb G-

-0

--c

[Lit. S. 180

AH

IOO~I Rb

[fftl

i

1-120K I “i

0 ' 0

' ' 1 I 0.5 0

IL l.u

t-

-t

1-

1 I I I I I I I I 0.8 0.6 0.1 0.2

b-

-f

Fig. 2. Rb. Born-von Karman fourth neighbour fit (dashed line) to the 120 K measured phonon frequencies (circles) and a pseudopotential Iit of [70Prl] (solid line) [73Col]. Table 2. Rb. Measured phonon frequencies at four different temperatures [73Col]. T

[12K

3 I

120K

85 K

v D-Hz1 twl

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.6 0.65 0.7 0.75 0.8 0.9 1.0

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.5 0.6 0.8 0.9 1.0 124

205 K

0.535(20) 0.64(2) 0.74(2) 0.820(15) 0.930(15) 1.010 (25) 1.075(20) 1.23(3)

0.495(20) 0.61(3) 0.71(2) 0.82(2) 0.910(25)

1.34 (5) 1.385 (15)

1.350 (15)

0.795(15) 0.96 (1) 1.12 (2) 1.325(15)

0.305 (15) 0.405 (10) 0.49(2) 0.575 (10) 0.66(2) 0.76 (1) 0.93 (1) 1.08 (1) 1.280(15)

1.385(15)

1.350(15)

0.59 (1)

112K

r

I

85 K

120K

205 K

v D-Hz1

L

COCI;Ll O.SOO(25) 0.62(2) 0.72(2) 0.82(2) 0.880(25) 0.98(2) 1.075(30) 1.225 (50) 1.23 (4) 1.23 (4) 1.27 (4) 1.275 (30) 1.305 (20) 1.32 (2)

0.470(25) 0.565(25) 0.690(25) 0.87(2)

0.1 0.2 0.3 0.4 0.5

0.88 (2) 1.235(20)

0.840 (15) 1.17(2)

1.50 (2)

1.480 (25)

0.1 1.20 (4)

0.41 (2) 0.82 (3) 1.185(25) 1.415(20) 1.465 (20)

0.395(20) 0.785(20) 1.125(20) 1.34(2) 1.41 (5)

rornT

1.05(2)

0.2 0.3 0.4 0.5

0.34(2)

0.34(3)

0.110 (25) 0.20(2) 0.265(20) 0.315(20) 0.320(25)

1.24 (3)

0.215 (10) 0.41 (1)

T

0.40 (1) 0.47(2) 0.57(l)

0.355 (10)

0.755(20) 0.895(25) 1.065 (15) 1.270 (15) 1.300(15) 1.32 (2)

0.705(20)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.750(15)

0.96(3)

0.735(15)

0.95 (2)

0.285(20) 0.395(20) 0.525(15) 0.66(3) 0.735(20) 0.79(15) 0.85 (15) 0.89 (15) 0.885(20)

0.46 (1) 0.66 (1) 0.785 (15) 0.840(25)

1.01 (2) 1.24 (2) 1.24 (3)

Schober/Dederichs

(continued)

Ref. p. 1801

1.2 Phonon states: Rb

Table 2. Rb. (continued) T

12K

205 K

0.595(20) 1.065(15) 1.330(15)

0.580(25) 1.03 (2) 1.325 (15)

1.345(20)

1.335(20)

1.130(15)

LOS(2)

0.775(20)

0.74(2)

0.605(30) 0.90

1.385(15)

1.350 (15)

0.525(30) l.Ol(3) 1.28 (3) 1.310(25) 1.305(25) 1.24(4) 1.10 (2) 0.935(30) 0.72(3) 0.64(3) 0.60(3) 0.685(25) 0.87(3) 1.20(4) 1.285(30) 1.32(2)

1.25(3) 1.03 (3) 0.680(25)

0.0 0.1 0.15 0.2 0.25 0.3 0.35 3.4 3.5

0.45(2) 0.68(3) 1.00 (4)

0.435(25) 0.69(3) 0.93 (5)

1.130(15)

1.08(2)

1.32(4)

1.33(3)

1.385(15)

0.96(3)

1.350(15)

0.95 (2)

0.235(30) 0.35(i) 0.48(2) 0.71(3) 0.92(3) 1.02(3) 1.10 (2) 1.20(4) 1.26(4) 1.325(20) 1.32(3) 1.33(5) 1.32(2)

0.885(20) 0.895(25) 0.920(35) 0.945 (40) 0.975(30)

12K

85K

120K

205 K

v [THz]

1.50(2)

0.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.480(25)

1.39(2) 1.130 (15)

1.08(2)

0.785(15)

0.34(2)

0.34(3)

1.465(20) 1.42(2) 1.34(2) 1.250(25) 1.10 (2) 0.965(15) 0.770(15) 0.590(15) 0.405 (15) 0.320(25)

1.41(5h

1.03(3)

0.55(2)

Err11A 0.805(15) 1.lO (4) 1.24(3)

CL-K1T 0.1 0.15 0.2 0.3 0.4 0.45 0.5 0.55 0.6 0.7 0.8 0.9 1.0

T

r

v CrHzl

5

0.1 0.2 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.9 0.95 1.0

120K

85K

0.42(4)

1.03(3) 1.12(4)

0.0 0.1 0.2 0.3 0.4 0.45 0.5

1.385 (15)

1.350 (15)

1.32 (2) 1.27 (2) 1.04 (3) 0.78 (3)

1.24 (3)

0.490(25) 0.34(2)

0.34(3)

0.37 (3) 0.320(25)

err114 0.0 0.1 0.2 0.3 0.4 0.5

1.21(4) 1.24(3)

0.0 0.1

0.840(25)

0.2 0.3 0.4 0.5

1.385 (15)

I !

1.50(2)

1.350(15)

1.480(25)

1.32(2) 1.335 (20) 1.36(5) 1.42 (3) 1.47 (3) 1.465 (20)

1.24(3)

1.41 (5)

c11114

1.385 (15)

1.350 (15)

0.96 (3)

0.95 (2)

1.32 (2) 1.290(25) 1.205(20) 1.07(2) 0.94 (3) 0.885 (20)

1.24(3)

0.840(25)

l.O7d(15)

1.130(15)

1.08 (2)

1.035(30) 1.08(6) 1.10 (2)

1.03(3)

Schober /Dederichs

125

1.2 PhononenzustZnde: Rb

[Lit. S. 180

Born-van Karman force constants

The dispersion curvescan be fitted with a fourth neighbour model. Increasing the number ofparameters changes the fit only marginally. Table 3. Rb. Born-von Karman force constants, @c. T

12K

85

Ref.

73Col

73Col

m

ij

111

xx XY

200

xx

220

YY xx

120K

205

73Col

73Col

K

@;mrn-‘1 0.669(11) 0.788 (11) 0.397(28) 0.022(17) -0.037(9) -0.009 (14) 0.043(14) 0.018 (8) -0.004(5)

7.2 XY 311

K

xx

YY YZ xz

-0.010 (7) -0.009(4)

0.657 (18) 0.746(26) 0.400(37) 0.007(27) -0.013(14) -0.025(14)

0.740(S) 0.456(19) 0.012 (11) -0.034(6) -0.003 (10)

0.618(8)

0.680(21) 0.438(39) -0.035(20) -0.014 (11)

0.591(16)

-0.011 (39)

-0.061 (11)

-0.013(16) -0.003(4) -0.027(9) -0.002(14)

-0.003(7) O.OoO(4) 0.004(6) 0.003 (3)

-0.075(20) -0.016 (11)

0.010(22) -0.011(7) 0.000(18) 0.007 (10)

remperatureandpressuredependence

The measured temperature dependence(for constant pressure)agreesvery well with the values calculated from pseudopotential theory [75Ro2] in the quasiharmonic approximation. Deviations were found for the highest measuredtemperature 205 K (2/3 melting temperature) which indicate anharmonic effects,Fig. 3 Rb. The temperature dependenceof the phonons at constant volume is shown in Fig. 4 Rb. Only one phonon :00,7 T shows a definite frequency change with temperature at constant volume. 0.68 lH7.

1.03 THz

0.61

099

0.60

0.95

0.56

0.91

0.52

0.87

0.58 I ir 0.16 THz

0.83 I ir 0.37 THz

0.14

0.35

0.52

0.33

0.10

0.31

0.38

0.29 0.27

0.36

200 250 K 300 100 150 IIFig. 4. Rb. Measured temperature dependence of four phononmodesfor three values of the lattice parameter a. Typical error bars are shown.The lines are drawn to guide the eye. Note the different ordinate scales[74Co3]. 100

126

150

200

250 K 300

Schober/Dederichs

Ref

1.2 Phonon states: Rb

?-I inni

In Table 4 Rb, the zero pressure mode Grlineisen parameters, y (q, a) = - 8 In v (q, 0)/8 In v computed from the :xperiment, are compared to theoretical values. The agreement is quite good for most values. I’able 4. Rb. Zero pressure mode Grtlneisen parameters. [0.2,

Phonon

co.2, 0.2, O]L

0, OIL

[0.2,

0, O]T

[0.2, 0.2, 0] T,

Y(rl, 4 3xpt. 74co3

2.15 (23)

1.65(7)

0.99(13)

1.31(25)

72Be2 73Co2 74Sh2

1.75 2.1 1.1

1.4 1.7 1.0

0.9 1.1 0.65

1.5 3.4 1.0

2.

Frequency spectrum and related properties

0

0.4

0.2

0.6

0.8

1.0

1.2

THz

1.6 n-

Y-

Fig. 5. Rb. Frequency spectra at 12 K and 120 K calculated from the Born-von Karman force constants of Table 3 Rb.

Fig. 6. Rb. Debye cutoff frequencies v, obtained from the spectra of Fig. 5 Rb.

6L K

60 IO"2 s

I 60 0" 56

40 I $ 1 N 20

52 48 0

IO

20

30

40

K

50

0

T-

0 T-

Fig. 7. Rb. Debye temperature On as a function of temperature calculated from a local pseudopotential using the lattice parameters at 12 K and 120 K [75Ro2].

Fig. 8. Rb. Debye-Waller exponent 2 Wdivided by the recoil frequency of a free Rb atom, va, calculated from the spectra of Fig. 5 Rb.

Schober/Dcderichs

127

1.2 PhononenzustZnde: Re

[Lit. S. 180

Table 5. Rb. Phonon spectra at 12 K and 120 K calculated from the Born-von Karman force constants of Table 3 Rb. T

12K

v D-Hz1

120K

sO3 D-Hz- ‘I

0.0

0.0

0.0

0.0200 0.0400 0.0600 0.0800 0.1000 0.1200 0.1400 0.1600 0.1800 0.2ooo 0.2200 0.2400 0.2600 0.2800 0.3ooo 0.3200 0.3400 0.3600 0.3800 0.4000 0.4200 0.4400 0.4600 0.4800 0.5000

0.0024 0.0097 0.0173 0.0296 0.0429 0.0594 0.0802 0.1003 0.1257 0.1523 0.1849 0.2280 0.3003 0.3238 0.3451 0.3679 0.3907 0.4108 0.4338 0.4565 0.4793 0.5021 0.5238 0.5493 0.5733

0.0007 0.0030 0.0072 0.0117 0.0204 0.0288 0.0400 0.0539 0.0680 0.0867 0.1075 0.1355 0.1653 0.2054 0.2569 0.3450 0.4299 0.4590 0.4894 0.5237 0.5582 0.5996 0.6351 0.6844 0.7383

T

12K

v [THz] 0.5200 0.5400 0.5600 0.5800 0.6000 0.6200 0.6400 0.6600 0.6800 0.7000 0.7200 0.7400 0.7600 0.7800 0.8000 0.8200 0.8400 0.8600 0.8800 0.9000 0.9200 0.9400 0.9600 0.9800 1.0000 1.0200

120K

g(v) [THz- ‘1 0.5980 0.6219 0.6487 0.6777 0.7041 0.7312 0.7601 0.7905 0.1729 0.6972 0.6689 0.6531 0.6451 0.6461 0.6494 0.6585 0.6792 0.6971 0.1245 0.7655 0.8199 0.9270 1.0327 1.0033 0.9749 0.9527

0.1931 0.8718 0.9555 1.0794 0.8905 0.7925 0.7409 0.7084 0.6893 0.6757 0.6682 0.6667 0.6703 0.6819 0.6975 0.7183 0.7530 0.8027 0.8197 1.0468 1.0144 0.9885 0.9662 0.9430 0.9278 0.9073

T v [THz] 1.0400 1.0600 1.0800 1.1000 1.1200 1.1400 1.1600 1.1800 1.2000 1.2200 1.2400 1.2600 1.2800 1.3000 1.3200 1.3400 1.3600 1.3800 1.4000 1.4200 1.4400 1.4600 1.4800 1.5OW 1.5200 1.5400

12K

120K

gb9 P-Hz- ‘I 0.9273 0.9079 0.8851 0.8629 0.8404 0.8025 0.7712 0.7603 0.7469 0.7425 0.7438 0.7519 0.7693 0.8052 0.8692 0.9947 1.2526 2.2704 2.2126 1.5189 1.0900 0.8350 0.6403 0.4638 0.2592 0.0

0.8918 0.8752 0.8571 0.8310 0.8012 0.7801 0.7681 0.7615 0.7619 0.7717 0.7941 0.8309 0.8981 1.0153 1.3238 1.5980 1.8688 1.5527 1.1873 0.9684 0.7793 0.5813 0.3221 0.0 0.0 0.0

3. Theoretical models Due to the short range of the interionic forces most empirical models reproduce the dispersion well. Rb is like Na and K a typical “simple metal” where pseudopotentials are very successfull,seee.g.Figs. 2 Rb and 3 Rb. The dependenceof the calculations on the used screening function is similar to Na (seeFig. 5 Na). Born-von Karman models: seeTable 3 Rb. Models comprising short ranged forces plus a simple electronic contribution: [69Krl, 74Dal,78Kul, 79Bol], further references:[7OSil, 72Sil,74Prl, 76Sil,77Ra4]. Local pseudopotential models: [70Prl, 73Co2, 73Pr2, 74Sh2, 75Ro2, 77Va1, 77Prl], further references: [61Tol, 66Anl,68Hol, 71Gu3,76Srl]. Nonlocal pseudopotential calculations: [69Prl, 72Be2], further references:[77Sol].

Re

Rhenium

Lattice: hcpn=276pm=2.76&~=445pm=4.45A.BZ:seep.450. Experimental values of the phonon frequenciesof rhenium are not available in the literature so far. Fig. 1 Re shows a theoretical estimate of the phonon dispersion [77Ra7]. The eight parameters of the Born-von Karma] type model were fitted to the measuredelastic constants. The corresponding estimate of the phonon spectrum is given in Fig. 2 Re. Further references:[72Ba3,72Kul].

128

Schober/Dcderichs

1.2 PhononenzustZnde: Re

[Lit. S. 180

Table 5. Rb. Phonon spectra at 12 K and 120 K calculated from the Born-von Karman force constants of Table 3 Rb. T

12K

v D-Hz1

120K

sO3 D-Hz- ‘I

0.0

0.0

0.0

0.0200 0.0400 0.0600 0.0800 0.1000 0.1200 0.1400 0.1600 0.1800 0.2ooo 0.2200 0.2400 0.2600 0.2800 0.3ooo 0.3200 0.3400 0.3600 0.3800 0.4000 0.4200 0.4400 0.4600 0.4800 0.5000

0.0024 0.0097 0.0173 0.0296 0.0429 0.0594 0.0802 0.1003 0.1257 0.1523 0.1849 0.2280 0.3003 0.3238 0.3451 0.3679 0.3907 0.4108 0.4338 0.4565 0.4793 0.5021 0.5238 0.5493 0.5733

0.0007 0.0030 0.0072 0.0117 0.0204 0.0288 0.0400 0.0539 0.0680 0.0867 0.1075 0.1355 0.1653 0.2054 0.2569 0.3450 0.4299 0.4590 0.4894 0.5237 0.5582 0.5996 0.6351 0.6844 0.7383

T

12K

v [THz] 0.5200 0.5400 0.5600 0.5800 0.6000 0.6200 0.6400 0.6600 0.6800 0.7000 0.7200 0.7400 0.7600 0.7800 0.8000 0.8200 0.8400 0.8600 0.8800 0.9000 0.9200 0.9400 0.9600 0.9800 1.0000 1.0200

120K

g(v) [THz- ‘1 0.5980 0.6219 0.6487 0.6777 0.7041 0.7312 0.7601 0.7905 0.1729 0.6972 0.6689 0.6531 0.6451 0.6461 0.6494 0.6585 0.6792 0.6971 0.1245 0.7655 0.8199 0.9270 1.0327 1.0033 0.9749 0.9527

0.1931 0.8718 0.9555 1.0794 0.8905 0.7925 0.7409 0.7084 0.6893 0.6757 0.6682 0.6667 0.6703 0.6819 0.6975 0.7183 0.7530 0.8027 0.8197 1.0468 1.0144 0.9885 0.9662 0.9430 0.9278 0.9073

T v [THz] 1.0400 1.0600 1.0800 1.1000 1.1200 1.1400 1.1600 1.1800 1.2000 1.2200 1.2400 1.2600 1.2800 1.3000 1.3200 1.3400 1.3600 1.3800 1.4000 1.4200 1.4400 1.4600 1.4800 1.5OW 1.5200 1.5400

12K

120K

gb9 P-Hz- ‘I 0.9273 0.9079 0.8851 0.8629 0.8404 0.8025 0.7712 0.7603 0.7469 0.7425 0.7438 0.7519 0.7693 0.8052 0.8692 0.9947 1.2526 2.2704 2.2126 1.5189 1.0900 0.8350 0.6403 0.4638 0.2592 0.0

0.8918 0.8752 0.8571 0.8310 0.8012 0.7801 0.7681 0.7615 0.7619 0.7717 0.7941 0.8309 0.8981 1.0153 1.3238 1.5980 1.8688 1.5527 1.1873 0.9684 0.7793 0.5813 0.3221 0.0 0.0 0.0

3. Theoretical models Due to the short range of the interionic forces most empirical models reproduce the dispersion well. Rb is like Na and K a typical “simple metal” where pseudopotentials are very successfull,seee.g.Figs. 2 Rb and 3 Rb. The dependenceof the calculations on the used screening function is similar to Na (seeFig. 5 Na). Born-von Karman models: seeTable 3 Rb. Models comprising short ranged forces plus a simple electronic contribution: [69Krl, 74Dal,78Kul, 79Bol], further references:[7OSil, 72Sil,74Prl, 76Sil,77Ra4]. Local pseudopotential models: [70Prl, 73Co2, 73Pr2, 74Sh2, 75Ro2, 77Va1, 77Prl], further references: [61Tol, 66Anl,68Hol, 71Gu3,76Srl]. Nonlocal pseudopotential calculations: [69Prl, 72Be2], further references:[77Sol].

Re

Rhenium

Lattice: hcpn=276pm=2.76&~=445pm=4.45A.BZ:seep.450. Experimental values of the phonon frequenciesof rhenium are not available in the literature so far. Fig. 1 Re shows a theoretical estimate of the phonon dispersion [77Ra7]. The eight parameters of the Born-von Karma] type model were fitted to the measuredelastic constants. The corresponding estimate of the phonon spectrum is given in Fig. 2 Re. Further references:[72Ba3,72Kul].

128

Schober/Dcderichs

1.2 Phonon states: Ru

Ref. p. 1801 A-

Z-

b-

f-

Fig. 1 a, b. Re. Theoretical estimate of the phonon dispersion using an eight parameter Born-von Karman type model fittc to the measured elastic constants [77Ra7].

0.4 orb. units I 0.3

-5 G 0.2

0.1 b Fig. 2. Re. Estimate of the phonon spectrum using the model of Fig. 1 Re [77Ra7].

Ru

0

.A’ 1

2

3

4

5

6

7

8 THz

Ruthenium

Lattice: hcp, a = 270 pm = 2.7 A, c =427 pm = 4.27 8. BZ: see p. 450. A-

l. Phonon dispersion Table 1. Ru. Measurements. Method

T [K]

Fig.

Ref.

neutron diffraction VW

e300

1 Ru

Smith et al. [76Sml]

Fig. 1. Ru. Measured phonon dispersion at room temperature in the extended zone scheme(acousticbranches from left, optic branches from right). The lines are guides to the eye only [76Sml].

Schober/Dedericbs

0 0

0.2

0.4 0.6 5-

0.8

129

1.2 Phonon states: Ru

Ref. p. 1801 A-

Z-

b-

f-

Fig. 1 a, b. Re. Theoretical estimate of the phonon dispersion using an eight parameter Born-von Karman type model fittc to the measured elastic constants [77Ra7].

0.4 orb. units I 0.3

-5 G 0.2

0.1 b Fig. 2. Re. Estimate of the phonon spectrum using the model of Fig. 1 Re [77Ra7].

Ru

0

.A’ 1

2

3

4

5

6

7

8 THz

Ruthenium

Lattice: hcp, a = 270 pm = 2.7 A, c =427 pm = 4.27 8. BZ: see p. 450. A-

l. Phonon dispersion Table 1. Ru. Measurements. Method

T [K]

Fig.

Ref.

neutron diffraction VW

e300

1 Ru

Smith et al. [76Sml]

Fig. 1. Ru. Measured phonon dispersion at room temperature in the extended zone scheme(acousticbranches from left, optic branches from right). The lines are guides to the eye only [76Sml].

Schober/Dedericbs

0 0

0.2

0.4 0.6 5-

0.8

129

1.2 PhononenzustZnde:

Ru

[Lit. S. 180

The dispersion curves of ruthenium have been measuredonly in [OOO-direction (A). The most striking feature is the absenceof the anomalous dip of the LO branch at the r-print which is very pronounced in the neighbouring elementTc. The experimental data are reasonably well described by a Born-von Karman type model fitted to the measuredelastic constants and the frequenciesof the two optical modes at the r point [79Ra2], Fig. 2 Ru. A-

Ru

1Hz

C-

IOObl

I-300K

2 0 0

0.1

0.2

0.3

0.L

0.50

0.1

0.2

0.3

0.5

04

ttFig. 2. Ru. Phonon dispersioncalculatedfrom a sevenparameterBorn-von Karman type model [79Ra2].

2. Phonon spectrum and related properties 0.8. orb.

I

units-----

6 & moleK

R” 1=300K

Ru

-

I-

a6 I 0.5 -; 04 c, 0.3 0.2 0.1

-

@onon

o

fit

,.

2.

K 3o

rI

0

1

2

3

L

5

6

7

8 1Hz9

0

40

80

Y-

:ig. 3. Ru. Phononspectrumcalculatedfrom the modelof Gg.2 Ru (unsmoothedcomputerplot) [79Ra2].

120 160 200 l-

I

I

2LO 280 K 320

Fig.4. Ru. Lattice contribution to the specific heat at constantvolumeci.calculatedfromthespectrumof Fig. 3 Ru, comparedto experimentalvalues[79Ra2].

The high temperaturelimit of the Debye temperature was calculated from the theoretical spectrumof Fig. 3 Ru IS8,(03)=4O4 K comparable with the experimental value at 298 K of 415 K.

b. Theoretical models Sam-von Karman and equivalenf models: [79Ra2], further reference:[77Ra6].

fkhoher /Dederichs

Ref. p. 1801

Sb

1.2 Phonon states: Sb

Antimony

Lattice: rhombohedral (A7), a = 622 pm = 6.22 A, CI= 87’ 24’. BZ: see p. 453.

1. Phonon dispersion Table 1. Sb. Measurements. Method

T [K]

neutron diffraction (TAS)

295

neutron diffraction (TAS)

z 300

Fig.

Ref.

1 Sb

Sharp and Warming [71Sh4] Sosnowski et al. [72Sol]

Further measurements: [71Sol, 63Fol]. r

1r

tr

X

5 THz

5cSFig. 1. Sb. Measured phonon dispersion at 295 K. The smooth curves are given by the result of a ninth neighbour Born-von Karman model [71Sh4]. The dispersion curves of antimony are similar to the ones of Bi. The group theory for these elements was reported in [67Sml]. The optical frequencies at the r point are in good agreement with the values measured by Raman scattering [75La2]. A ninth nearest neighbour Born-von Karman model gives a good overall description of the measured dispersion and a reasonable agreement with the measured spectrum.

2. Phonon spectrum and related properties

7 orb. units

Fig. 2. Sb. Phonon spectrum measuredby inelastic coherent scattering on a polycrystalline sample compared to spectra obtained from Born-von Karman tits to the measured lispersion curves and folded with the experimental resoluion [74Sal]. (error: 71Shl should read 71Sh4).

2 1 II

1

2

3 Y-

Schober /Dederichs

4

5 THz 6

1.2 Phononenzusttinde: SC

120 0

160 200 2LO K 280 IFig. 3. Sb. Debye temperature 8, calculated from the experimental spectrum of Fig. 2 Sb compared to experimental values [74Sal]. 40

80

120

0

40

TLit. S. 180

160 200 240 K 280 IFig. 4. Sb. Debye-Wailer coeficient I, calculated from the experimental spectrum of Fig. 2 Sb [74Sal]. 80

120

3. Theoretical models Born-von Karman models [71Sh4,72Sol].

SC Scandium Lattice: hcp, a= 331 pm = 3.31 A, c= 527 pm = 5.27 A. BZ: see p. 450.

1. Phonon dispersion Table 1. SC. Measurements Method

T

Fig.

Ref.

1 SC

Wakabayashi et al. [71 Wal]

CKI neutron diffraction (TAS)

295

The phonon dispersion curves of scandium are similar to the ones of yttrium. They yield long range interactions in the basal plane but interactions which decrease rapidly in the direction normal to the basal plane. One Kohn type anomaly was found. lK RT lH;I 7SC ?,lT,,&CO]\

f-

A-

-Z M

A

I-

15001

1

-t

I

10061

I

6-

Fig. 1. SC. Measured phonon dispersion curves at 295 K. The lines correspond to the sixth neighbour Born-von Karman model of Table 3 SC [70Wal]. 132

Schober/Dederichs

1.2 Phononenzusttinde: SC

120 0

160 200 2LO K 280 IFig. 3. Sb. Debye temperature 8, calculated from the experimental spectrum of Fig. 2 Sb compared to experimental values [74Sal]. 40

80

120

0

40

TLit. S. 180

160 200 240 K 280 IFig. 4. Sb. Debye-Wailer coeficient I, calculated from the experimental spectrum of Fig. 2 Sb [74Sal]. 80

120

3. Theoretical models Born-von Karman models [71Sh4,72Sol].

SC Scandium Lattice: hcp, a= 331 pm = 3.31 A, c= 527 pm = 5.27 A. BZ: see p. 450.

1. Phonon dispersion Table 1. SC. Measurements Method

T

Fig.

Ref.

1 SC

Wakabayashi et al. [71 Wal]

CKI neutron diffraction (TAS)

295

The phonon dispersion curves of scandium are similar to the ones of yttrium. They yield long range interactions in the basal plane but interactions which decrease rapidly in the direction normal to the basal plane. One Kohn type anomaly was found. lK RT lH;I 7SC ?,lT,,&CO]\

f-

A-

-Z M

A

I-

15001

1

-t

I

10061

I

6-

Fig. 1. SC. Measured phonon dispersion curves at 295 K. The lines correspond to the sixth neighbour Born-von Karman model of Table 3 SC [70Wal]. 132

Schober/Dederichs

1.2 Phonon states: SC

Ref. p. 1SO]

JYable2. SC. Measured phonon frequencies at 295 K [71 Wall.

r

v [THz]

CON1LA@,) 0.105 0.128 0.155 0.175 0.1875 0.2 0.2125 0.225 0.2375 0.25 0.2625 0.275 0.282 0.2875 0.293 0.3 0.3125 0.325 0.3375 0.35 0.375 0.3875 0.4 0.425 0.45 0.5

0.1 0.15 0.2 0.25 0.3 0.35 0.41 0.5

1.05 (2) 1.31(2) 1.59 (2) 1.66 (3) 1.94 (3) 1.98 (3) 2.10 (2) 2.25 (1) 2.29 (2) 2.38 (2) 2.65 (2) 2.72 (2) 2.80 (2) 2.84 (2) 2.95 (2) 2.97 (2) 3.15 (2) 3.24 (2) 3.34 (3) 3.43 (2) 3.69 (3) 3.76 (4) 3.89 (4) 4.16 (2) 4.23 (2) 4.74 (3)

0.62 (1) 0.89 (1) 1.17 (1) 1.47 (1) 1.83 (4) 2.05 (1) 2.38 (4) 2.87 (2)

v [THz]

5

v [THz]

5

v [THz]

KOOILO(L)

C’WI LO@,) 0.0 0.05 0.1 0.12 0.15 0.172 0.2 0.257 0.272 0.29 0.3 0.34 0.39 0.44 0.491

5

6.91 (3) 6.89 (3) 6.83 (4) 6.69 (4) 6.66 (4) 6.51 (3) 6.58 (3) 6.29 (6) 6.08 (3) 6.09 (2) 6.01(2) 5.78 (2) 5.50 (2) 5.15 (5) 4.75 (2)

0.09 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.3 0.35 0.5

1.75 (2) 2.00 (2) 2.43 (1) 2.86 (1) 3.37 (2) 3.77 (2) 4.20 (2) 4.52 (2) 4.87 (1) 5.22 (2) 5.70 (2) 6.21 (10)

CC001 TA,@,) 0.1 0.15 0.2 0.3 0.4 0.45 0.5

1.07 (1) 1.59 (1) 2.10 (1) 3.06 (2) 3.75 (2) 4.08 (3) 3.97 (2)

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.5

Cl001TO,@,) 0.0 0.05 0.1 0.2 0.25 0.3 0.4 0.45 0.5

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

4.04 (4) 4.04 (7) 4.14 (4) 4.09 (4) 3.92 (4) 3.83 (5) 3.68 (4) 3.51(3) 3.35 (2)

1.60 (1) 2.12 (1) 2.58 (1) 2.96 (1) 3.30 (3) 3.49 (3) 3.69 (4) 3.57 (4)

CCC01 TA,(T,, T;) 0.1 0.15 0.2 0.25 0.3 0.4 0.45 0.5

Schoher /Dederichs

1.83 (1) 2.79 (1) 3.61(l) 4.32 (2) 5.00 (2) 5.86 (2) 6.16 (7) 6.23 (4)

6.91 (3) 6.70 (3) 6.58 (6) 6.59 (6) 6.51 (3) 6.38 (5) 6.39 (4) 6.11 (6) 6.23 (4)

CL’ 001TO II&)

Cl001TA,, CL,) 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

4.04 (4) 4.08 (4) 4.41(3) 4.71(3) 5.22 (2) 5.66 (2) 5.93 (4) 5.92 (4) 6.05 (2) 6.23 (5)

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.5

4.04 (4) 4.11 (3) 4.30 (3) 4.55 (3) 4.88 (3) 5.15 (4) 5.49 (2) 5.66 (3) 5.90 (2) 6.11 (4)

KC01TO,&, -G) 0.0 6.91(3) 0.05 0.1 0.2 0.25 0.3 0.333 0.4 0.45 0.5

6.87 (5) 6.85 (5) 6.41(4) 6.08 (5) 5.67 (3) 5.35 (3) 4.68 (6) 4.19 (4) 3.97 (2)

133

1.2 Phononenzustbde: SC

[Lit. S. 180

Table 3. SC. Born-von Karman coupling constants, PP;,derived by a least squares fit to selectedphonon

A-

branches and to the elastic constants, T=295 K [7OSil].

The model is of the “modified axially symmetric” Form[65Del] q=K(R:R;)/(R”)2+C,6ij; whereC,=C,,

4

(i,j=x,y,z);

for i=j=x,y

and C,=C,,

for i=j=z.

m

K mm-‘]

C [Zm-‘3

C [I?m-r]

Iah4 0, c/2)

23.198 11.793 -9.944

-2.576 2.374 2.284

-2.559 1.525 1.981

I3 a

10,a, 0) I-2afl,O,c/2) 2 Cc)

,542 r/5,4& 49 J5a,O,O)

K+C,,= 3.557 1.307

-0.610; CBx= -0.622 0.539

-0.648

-0.041

-0.115

2 .

1

0 0

Fig. 2. SC. Measured phonon frequencies of the A, branch showing the anomaly at q = (0, 0, 0.27)24~. The error bars represent only the statistical uncertainty in the peak positions [70Wal]. 4nomalies

In the A, branch a change of slope was observed near q =(O, 0,0.27) 2x/c which is close to a peak in the susxptibility x(g) estimatedfrom magnetic structure data, Fig. SC2. The peak in x(g) probably arisesfrom a “nesting” eature of the Fermi surfacein scandium along the c-axis [70Wal].

2 Frequency spectrum and related properties 5 orb. units 6

0

1

2

3

5

6

THz 7

Fig. 3. SC.Frequencyspectrumat 295K derivedfrom the Born-von Karman fit given in Table 3 SC.Some of the critical points arising from high symmetry directions are indicated by arrows [70Wal].

Schoher/Dederichs

Ref. p. 1SO]

1.2 Phonon states: Sn

K 360

I 00

345 330

J”“l

I

8

15

0

30

45

,

60 K 75

315 300

I

I

I

100

200

300

I

K

LOO

Fig.4. SC. Debye temperature 0, derived from the spectrum of Fig. 3 SC. The insert shows a comparison with experimental results [70Wal].

3. Theoretical models The available experimental data are not sufhcient to test the existing models rigorously. The phenomenological modelsreproduce the data sufficiently well. Fits using either a model with noninteracting free electron like s and p bandsand a tight binding d band or an electronic transition metal model potential deviate more than 20 % from the measuredvalues. Born-von Karman and equivalent models: seeTable 3 SC,further references:[73Ra3,78Ra5]. Short ranged forces plus a simple electronic contribution: [75Si5], further references:[73Bo2, 74Kal,75Ca2]. Transition metal model potential: [78Up2], further references:[75Si6, 76Ku4]. Noninteracting electron band model: [77Si5].

Sn Tin Lattice: white tin, /?-Sn,body centeredtetragonal, A5, a = 583pm= 5.83A, c = 318pm = 3.18A. BZ: seep. 451. 1. Phonon dispersion Table 1. Sn. Measurements Method neutron diffraction (TAS)

Fig. 110,296 1 Sn

neutron diffracx300 tion (TOF, TAS)

2 Sn

Ref. Rowe [67Ro4, 65Rol] Price [67Prl]

Further measurements:[65Bol, 65SclJ The agreementbetween the different measurementsis good. A group theoretical treatment of the dispersion :urves has been given by Chen [67Chl]. Characteristic features are the dip of the [OOQ LO@,) branch near : = 0.5 and the flat dispersion of a number of acoustic and optic branches. Schober/Dederichs

135

Ref. p. 1SO]

1.2 Phonon states: Sn

K 360

I 00

345 330

J”“l

I

8

15

0

30

45

,

60 K 75

315 300

I

I

I

100

200

300

I

K

LOO

Fig.4. SC. Debye temperature 0, derived from the spectrum of Fig. 3 SC. The insert shows a comparison with experimental results [70Wal].

3. Theoretical models The available experimental data are not sufhcient to test the existing models rigorously. The phenomenological modelsreproduce the data sufficiently well. Fits using either a model with noninteracting free electron like s and p bandsand a tight binding d band or an electronic transition metal model potential deviate more than 20 % from the measuredvalues. Born-von Karman and equivalent models: seeTable 3 SC,further references:[73Ra3,78Ra5]. Short ranged forces plus a simple electronic contribution: [75Si5], further references:[73Bo2, 74Kal,75Ca2]. Transition metal model potential: [78Up2], further references:[75Si6, 76Ku4]. Noninteracting electron band model: [77Si5].

Sn Tin Lattice: white tin, /?-Sn,body centeredtetragonal, A5, a = 583pm= 5.83A, c = 318pm = 3.18A. BZ: seep. 451. 1. Phonon dispersion Table 1. Sn. Measurements Method neutron diffraction (TAS)

Fig. 110,296 1 Sn

neutron diffracx300 tion (TOF, TAS)

2 Sn

Ref. Rowe [67Ro4, 65Rol] Price [67Prl]

Further measurements:[65Bol, 65SclJ The agreementbetween the different measurementsis good. A group theoretical treatment of the dispersion :urves has been given by Chen [67Chl]. Characteristic features are the dip of the [OOQ LO@,) branch near : = 0.5 and the flat dispersion of a number of acoustic and optic branches. Schober/Dederichs

135

1.2 PhononenzustZnde: Sn

[Lit. S. 180

v-

A-

AX

sr 1Hz

/?-%I l=llOK

l,r-

A A

IOR1 As A A A A AAA; . . . Al. . .

0.8

1.0

0 -t

f-

s-

Fig. 1. Sn. Measured phonon dispersion in /I-% at 110 K. The straight lines give the initial slopes of the curves as calculated from the elastic constants i67R041.

,

0.2 0.4 0.6 0.8

1.0

0.1

0.2

0.2

-t

e-

,

,

,

0.4

,I,

0.6

,

0.8

,

1.0

b-

-A

AX

!F IHz b

0.6

0.8

T

lttni

rtfni

Fig. 2a-c. Sn. Measured phonon dispersion in /I-Sn at 297 K. For clarity the axesin Figs. a and b have been reflected about the zone boundary with measurementsplotted to the right or left. The lines represent a twelfth neighbour Bomvon Karman fit [67Prl].

t-

136

-t

Schober/Dederichs

Ref. p. 1801

1.2 Phonon

states: Sn

Table 2. Sn. Measured phonon frequencies in white tin at 110 K [67Ro4] and 296 K [65Rol]. v [THz]

r T

110K

296 K

T

1.00

1.00 2.43 (10)

2.67 (6) 2.75 (10) 2.95 (9) 3.06 (6) 2.80 (10) 3.00 (8) 2.87 (9) 2.81(7)

2.68 (10) 2.64 (10)

2.78 (7) 2.78 (7) 2.92 (7) 2.80 (15) 2.99 (7) 3.03 (8) 2.89 (8)

2.60 (10) 2.20 (10)

2.38 (9) 2.18 (5) 1.80 (4) 1.99 1.74 1.74 1.47 (3) 1.36 (4) 1.23 (4)

0.00 0.007 0.075 0.081 0.160 0.167 0.200 0.250 0.300 0.350 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.700 0.741 0.800 0.830 0.839 0.850 0.900 0.925 0.950 0.975 1.00

0.53 (3) 0.69 (3) 0.85 (2) 0.95 (2) 1.00 (2) 1.06 (2) 1.10 (3) 1.14 (2) 1.14 (2) 1.14 (2)

0.52 (3) 0.68 (3) 0.83 (2) 0.89 (2) 0.92 (4) 0.97 (4) 1.04 (4) 1.07 (4) 1.07 (4) 1.07 (4)

296 K

1.30 (3) 1.40 (4) 1.51 (3) 1.70 (4) 2.20 (10) 2.37 (4) 2.69 (4) 2.93 (4) 3.09 (4) 3.08 (4) 2.91(6) 2.77 (6) 2.73 (5) 2.84 (5) 2.98 (5) 3.02 (5) 3.03 (6) 2.96 (5) 2.91 (5) 2.70 (4) 2.54 (8)

2.00 (10) 2.25 (8) 2.55 (5) 2.80 (5) 2.97 (7) 2.98 (6) 2.89 (8) 2.78 (9) 2.70 (6) 2.90 (8) 2.94 (6) 2.94 (6) 2.85 (8) 2.81 (7) 2.66 (6) 2.00 (3)

1.90 1.74 1.74 1.44 (5) 1.35 (3) 1.30 (3) 1.29 (3)

1.37 (4) 1.24 (4) 1.23 (4)

A, t-9 9 Cl TA, v, Cl,031-Cl

A1,CO, ‘JO TA, V, CLO, l -11 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550

110K

A., co,0, (1 LO, VI CLO,l- 51

A1 F4~,~lLA, V, C&0,1-Cl 0.09 0.250 0.263 0.333 0.345 0.400 0.425 0.450 0.500 0.550 0.575 0.600 0.625 0.650 0.667 0.675 0.700 0.750 0.833 0.844 0.850 0.900 0.903 0.907 0.925 0.950 0.997 1.00

v [THz]

r

0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.00

1.08 (2) 1.01 (3) 0.96 (3) 0.89 (3) 0.86 (3) 0.85 (3) 0.85 (3) 0.84 (3) 0.83 (3)

0.97 (4) 0.92 (4) 0.88 (4) 0.82 (4) 0.79 (3) 0.77 (3) 0.76 (3) 0.76 (3) 0.75 (3) (continued)

Schoher /Dederichs

137

1.2 PhononenzustEnde: Sn

[Lit. S. 180

Table 2. Sn. (continued)

r

v D-Hz1

r T

1lOK A, CQQtlTA,V,

0.00 0.100 0.167 0.200 0.300 0.333 0.400 0.500

4.00(7) 3.99(6)

296 K

1lOK

T

Cl,O,l-Cl

296 K

A, CO,0, Cl TA, V, CL 9 1 -Cl 0.600 0.667 0.700 0.800 0.833 0.900

3.85(10) 3.85(20)

4.04(5) 4.00(6) 3.75(10) 3.98(6) 4.03(7)

v Wzl

1.00

4.09(7) 3.90(10) 4.10(8) 4.12(9) 4.10(20) 4.22(7) 4.23(6)

4.07(8)

3.80(20)

3

v D-Hz1

r

v [THz]

r

v D-Hz1

T

1lOK

T

1lOK

T

110K

A, CLL 01T.4

A, CLL 01LA 0.10

0.89(5) 1.28(5) 1.68(5) 2.05(4) 2.39 (4) 2.73(4) 2.96(4) 3.16(4) 3.31(4)

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.10

0.35(2) 0.47(2) 0.57(2) 0.70(3) 0.76(3) 0.81 (3)

0.15 0.20 0.25 0.30 0.35

4 CLLO11-0 -

0.0

4 CLLOI LO 0.0 0.05

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

3.97(4) 3.96(4) 3.92(4) 3.80(3) 3.75(4) 3.69(4) 3.66(4) 3.61(4) 3.59(4) 3.45(4) 3.31(4)

0.10 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.350 0.400 0.450 0.500

3.97(4) 3.85(4) 3.80(4) 3.81 (3) 3.82(4) 3.83(3) 3.77(4) 3.70(3) 3.60(4) 3.46(5) 3.42(5) 3.42(5) 3.33(5) 3.19(5)

C3 CC, 0,01J-A 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

1.00

0.60(3) 0.78(2) 0.91(2) 0.93(2) 0.92(2) 0.92(3) 0.86(3) 0.83 (3) 0.84(3)

Ce CL 0, 011-O 0.0 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

3.98(4) 3.99(5) 3.96(4) 3.92(4) 3.99(4) 4.03(4) 4.08(5) 4.12(4) 4.13(4) 4.16(5)

1.0

4.19(5)

0.10

Anomalies Measurementsfor the nonsymmetry branches [qO[], Fig. 3 Sn, show that the dip at [=0.5 in the [OOfl LO(A,) branch is a local minimum and is sharpest along the symmetry line. The [[CO] LA(A,) branch shows evidence

of kinks near (=O.l and C=O.4.The complicated structure of the Fermi surface of /3-Snprecludes an assignmer of these kinks to specific electronic transitions acrossthe Fermi surface (Kohn-effect). 138

Schoher/Dederichs

1.2 Phonon states: Sn

Ref. p. 1 SO] 3.6 THz 3.4

2.8

Fig. 3. Sn. Measured phonon frequencies in j%Sn at 110 K along the lines [n, 0, t;] showing the behaviour of the anomaly in the A, -V, branches [67Ro4]. Anharmonicity The curvature of the measured Debye-Waller coeffkients, Fig. 6 Sn points toward relatively large anharmonic contributions already at low temperatures [66Brl]. An X-ray measurement of the almost forbidden reflections yields a value of t1s2+= (0.63 &0.05) 10” Nme2 for the coefficient of the cubic term in the effective potential felt by the Sn ions [78Me3].

2. Phonon spectrum and related properties 2.5 arb. units I 1.5 r; G 1.0 0.5 1

100

0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0THz4.5

Fig. 4. Sn. Phonon spectrum 28-sn 300 K measured by coherent scattering on a polycrystalline sample. The histogram was calculated from the Born-von Karman model of [66Brl] ; [68Kol].

Fig. 6. Sn. Debye-Waller exponent 2W of polycrystalline /?-Sn divided by the recoil frequency of a free atom vs calculated from a sixth neighbour Born-von Karman model [66Brl]. Schober /Dederichs

0

IO

20

30 40

50 K 60

TFig. 5. Sn. Debye temperature 0, of fi-Sn calculated from a sixth neiahbour Born-von Karman model [6&rl].

0

100

200 T-

300 K 400

139

[Lit. S. 180

1.2 PhononenzustEnde: Sn 3. Theoretical models

To reproduce the dispersion curves of j?-Snaccurately long range interactions are needed.A reasonablefit is obtained both by a 16 parameter Born-von Karman and a 5 parameter pseudopotential model.

Born-von Karman model: [66Brl] Fig. 7 Sn, further references:[67Prl]. Pseudopotential model: [67Br2] Fig. 8 Sn.

ov 0 0.2’ 0.1I 0.6I 0.8I f-

t-

-6

Fig. 7. Sn. Phonon dispersion in j-Sn calculated from a sixth neighbour 16 parameter Born-von Karman model compared to experimental results (o (67Prl], l [65Rol]) [66Brl].

v-

A-

---II

A-

it001

1 0 0

0.2

0.6

s-

0.6

0.8

-c

t-

Fig. 8. Sn. Phonon dispersion in /Mn calculated from a five parameter pseudopotential model compared to experimental results [65Bol, 65Ro1, 67Prl], [67Br2].

140

Schober/ Dederichs

1.2 Phonon states: Sr, Ta

Ref. p. 1803

Sr

Strontium

Lattice: fee,a = 605pm = 6.05A. BZ: seep. 449. The dispersion curves have not been measured.Theoretical calculations suggesta simple dispersion typical for fee materials, Fig. 1 Sr; calculated Debye temperature: seeFig. 2 Sr. Models including short ranged forces and a simple electronic term: [73Swl]. Local model pseudopotential calculations [72Mo2,73Pr3, 67Anl,71Gu3].

0

I.0

0.8

0.6

0.4

5-

0.2 ---i-f

0

0.5

Fig. 1. Sr. Dispersioncurvescalculatedby a two parameterpseudopotentialmodel[73Pr3].

160 K I 140 0 @ 120 100 0

20

40

60

80

100

120 K 140

Fig. 2. Sr. CalculatedDebyetemperatureO,(T) [73Swl].

Ta

Tantalum

Lattice: bee,a = 330pm= 3.30A. BZ: seep. 448.

1. Phonon dispersion Table 1. Ta. Measurements. Method

T [K]

Fig.

Ref.

neutron diffraction (TAS)

296

1 Ta

Woods [64Wo2]

The phonon dispersion curves of tantalum are similar to the ones of niobium as one would expect from its position in the periodic table below Nb. In the [OOQ-direction an accidental degeneracybetween the [OO[] L and [OOQT-branch occurs at [=0.7. The characteristic dip in the [OOQL branch in this region is not as pronounced as in Nb. The [OO[] T branch is at [=0.3 remarkably below the velocity of sound line and moves for larger 5 up again. Schober/Dederichs

141

1.2 Phonon states: Sr, Ta

Ref. p. 1803

Sr

Strontium

Lattice: fee,a = 605pm = 6.05A. BZ: seep. 449. The dispersion curves have not been measured.Theoretical calculations suggesta simple dispersion typical for fee materials, Fig. 1 Sr; calculated Debye temperature: seeFig. 2 Sr. Models including short ranged forces and a simple electronic term: [73Swl]. Local model pseudopotential calculations [72Mo2,73Pr3, 67Anl,71Gu3].

0

I.0

0.8

0.6

0.4

5-

0.2 ---i-f

0

0.5

Fig. 1. Sr. Dispersioncurvescalculatedby a two parameterpseudopotentialmodel[73Pr3].

160 K I 140 0 @ 120 100 0

20

40

60

80

100

120 K 140

Fig. 2. Sr. CalculatedDebyetemperatureO,(T) [73Swl].

Ta

Tantalum

Lattice: bee,a = 330pm= 3.30A. BZ: seep. 448.

1. Phonon dispersion Table 1. Ta. Measurements. Method

T [K]

Fig.

Ref.

neutron diffraction (TAS)

296

1 Ta

Woods [64Wo2]

The phonon dispersion curves of tantalum are similar to the ones of niobium as one would expect from its position in the periodic table below Nb. In the [OOQ-direction an accidental degeneracybetween the [OO[] L and [OOQT-branch occurs at [=0.7. The characteristic dip in the [OOQL branch in this region is not as pronounced as in Nb. The [OO[] T branch is at [=0.3 remarkably below the velocity of sound line and moves for larger 5 up again. Schober/Dederichs

141

1.2 Phononenzust5nde: Ta A-

IL

1tI2

la

F-

A-

C-

l

[Lit. S. 180 P

El01

IOObl

68 0 AAA

I z

8

AA0A0 A 0 n 0

L

0

0

12

I

I

I

k,,

I

0.2 0.4 0.6 0.8 1 f-

0.2

01 0.2 0.3 0.1I 5-

0.S

0.8

0.6

Fig. 1. Ta. Measured dispersion curves for the high symmetry directions at 296 K. The solid points are modes which are required to be triply degenerate by symmetry. The solid lines through the origin have been calculated from the measured elastic constants [64Wo2]. Table 2. Ta. Phonon frequencies at 296 K [64Wo2].

5 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Born-von

6.

v t3’Hzl

roan L

CWI -I- K511 L

1.23(6) 2.2%(5) 3.10(5) 3.82(6) 4.20(8) 4.25(8) 4.45(8) 4.68(8) 4.98(10) 5.03(7)

1.28(4) 1.88(4) 2.60(4) 3.37(5) 4.03(7) 4.63(8) 4.85(8) 5.03(8) 5.03(7)

2.24(5) 3.75(7) 4.38(10) 4.30(8) 3.78(6) 3.03(6) 2.70(6) 3.70(10) 4.78(10) 5.03(7)

Cllrl T 0.98(4) 1.73(4) 2.48(4) 3.28(5) 3.78(6) 3.80(10) 4.20(15) 4.48(15) 4.90(10) 5.03(7)

Km L 0.10 0.20 0.25 0.30 0.35 0.40 0.41 0.45 0.50

1.65(10) 1.10(10) 3.51(7) 3.95(8) 4.15(8) 4.24(8) 4.28(8) 4.32(8) 4.35(8)

v IT’Hz1 KS01Tl LWI T, 0.96(4) 1.95(4) 2.97(5) 3.51(5) 3.97(6) 4.24(6) 4.35(6)

2.63(8)

Karman models

As in the other group 5 transition metals extremely long range forcesare neededto fit the dispersion. A seventh neighbour general force model (19 parameters)tits the dispersion reasonably well. There is not enough experimental information available to tit a larger number of parameters. Table 3. Ta. Born-von Karman constants of, @z, T= 296 K [64Wo2] “).

m

111

ij

0; [Nm-‘1

m

ij

q [Nm-‘1

xx

16.983 11.201 1.182 1.423 3.546 - 5.427 1.943 3.577 -0.718 -1.728 0.983

222

xx XY xx YY xx YY YZ XY

-0.493 0.812 - 3.705 0.134 0.558 -0.237 0.106 -0.683

XY

200 220

xx YY xx i!Z

311 ‘) General force model.

142

XY xx YY YZ XY

Schoher/Dederichs

400 133

1.2 Phonon states: Ta

Ref. p. 1 SO]

2. Frequency spectrum and related properties 1.5

THz-’ 1.2

1

0.9

L; -G 0.6 0.3 b Fig. 2. Ta. Phonon frequency spectrum at 296 K calculated from the Born-von Karman parameters of Table 3 Ta.

0

1

2

3

4

THz 5

Y-

Table 4. Ta. Phonon frequency spectrum at 296 K calculated from the seventh-neighbour model of Woods [64Wo2], see Table 3 Ta.

Born-von Karman

v [THz]

g(v) CT=- ‘1

v [THz]

g(v) CT=-‘1

v [THz]

g(v) CTHz-‘1

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70

0.000 0.000 0.000 0.001 0.001 0.002 0.003 0.003 0.005 0.005 0.007 0.008 0.009 0.011 0.012 0.014 0.016 0.018 0.020 0.022 0.024 0.027 0.029 0.032 0.035 0.037 0.040 0.044 0.047 0.051 0.054 0.058 0.061 0.066

1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40

0.070 0.075 0.080 0.085 0.090 0.097 0.103 0.110 0.118 0.128 0.137 0.150 0.163 0.180 0.202 0.235 0.314 0.334 0.355 0.391 0.739 0.411 0.343 0.305 0.280 0.262 0.249 0.238 0.231 0.225 0.221 0.218 0.218 0.218

3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00 4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00 5.05 5.10

0.220 0.225 0.232 0.240 0.254 0.279 0.284 0.291 0.297 0.310 0.325 0.347 0.374 0.414 0.475 0.600 1.023 1.101 1.168 1.236 0.641 0.265 0.222 0.187 0.157 0.134 0.115 0.097 0.082 0.068 0.055 0.041 0.027 0.0

Schober/Dederichs

143

1.2 PhononenzustEnde: Tb

[Lit. S. 180

3.0 402 2.5

5

I 2.0 I c 1.5 1 N 1.0

4.9 4.8

0.5

4.1 4.6 -10

,

0

10

20

30

0

100

200

300

400 K 500

n-

Fig. 3. Ta. Debye cutoff frequencies,vnr calculated from the spectrum of Fig. 2 Ta.

Fig. 4. Ta. Debye-Wailer exponent 2Wdivided by the recoil frequency of the free ion, vR, calculated from the spectrum of Fig. 2 Ta.

3. Theoretical models A first principle theory is not available. Born-von Karman and equivalent models: see Table 3 Ta, further references: [77Pal]. Short range forces plus a simple electronic contribution: 75Upl,78Pal].

[76Brl].

Further references: [72Bal, 75Pr1, 75Si1,

Model potential: [73Anl]. Tight binding calculation with additional parameters: [78Lal].

Tb

Terbium

Lattice: hcp, a= 360 pm = 3.6 A, c= 570 pm = 5.7 A. BZ: see p. 450.

1. Phonon dispersion Table 2. Tb. Selection of measured phonon frequencies at room temperature, [70Hol].

Table 1. Tb. Measurements. Method

Fig.

Ref. Phonon

;I neutron diffraction (TAS)

298

1 Tb

Houman and Nicklow [70Hol]

The phonon dispersion of Tb is qualitatively similar to the one observed in Ho. No major anomalies have been seen.

144

Schober/Dederichs

v CT=1

Phonon

v P-Hz1

1.82 (3) 3.25(4) 1.30(2) 2.44(4) 1.75(3) 1.59(4) 2.85 (5) 2.90(3) 3.05(4)

W K, K2 K6 KS L, (A) I-1(0)

2.89(4) 2.91 (6) 2.86(4) 2.32(4) 2.35(4) 1.78 (6) 3.13(6) 2.25(6) 1.38(6)

L2

H3

1.2 PhononenzustEnde: Tb

[Lit. S. 180

3.0 402 2.5

5

I 2.0 I c 1.5 1 N 1.0

4.9 4.8

0.5

4.1 4.6 -10

,

0

10

20

30

0

100

200

300

400 K 500

n-

Fig. 3. Ta. Debye cutoff frequencies,vnr calculated from the spectrum of Fig. 2 Ta.

Fig. 4. Ta. Debye-Wailer exponent 2Wdivided by the recoil frequency of the free ion, vR, calculated from the spectrum of Fig. 2 Ta.

3. Theoretical models A first principle theory is not available. Born-von Karman and equivalent models: see Table 3 Ta, further references: [77Pal]. Short range forces plus a simple electronic contribution: 75Upl,78Pal].

[76Brl].

Further references: [72Bal, 75Pr1, 75Si1,

Model potential: [73Anl]. Tight binding calculation with additional parameters: [78Lal].

Tb

Terbium

Lattice: hcp, a= 360 pm = 3.6 A, c= 570 pm = 5.7 A. BZ: see p. 450.

1. Phonon dispersion Table 2. Tb. Selection of measured phonon frequencies at room temperature, [70Hol].

Table 1. Tb. Measurements. Method

Fig.

Ref. Phonon

;I neutron diffraction (TAS)

298

1 Tb

Houman and Nicklow [70Hol]

The phonon dispersion of Tb is qualitatively similar to the one observed in Ho. No major anomalies have been seen.

144

Schober/Dederichs

v CT=1

Phonon

v P-Hz1

1.82 (3) 3.25(4) 1.30(2) 2.44(4) 1.75(3) 1.59(4) 2.85 (5) 2.90(3) 3.05(4)

W K, K2 K6 KS L, (A) I-1(0)

2.89(4) 2.91 (6) 2.86(4) 2.32(4) 2.35(4) 1.78 (6) 3.13(6) 2.25(6) 1.38(6)

L2

H3

1.2 Phonon states: Tb

Ref. p. 1801

Pl

f-

-l

tU-

-

-s

S’

P-

RA

LK

H

4 THz 3 I P2 1 0 0

0.1

0.2 5-

0.3

0.4

0.5

Fig. 1 a-c. Tb. Phonon dispersion curves at room temperature. The lines shown represent the eighth neighbour Bornvon Karman model of Table 3 Tb [70Hol] a) along major symmetry directions b) along the boundaries of the Brillouin zone c) along the [CO&]direction.

Born-von Karman model

For a good fit an inclusion of at least eighth neighbours is necessary. From the zone boundary phonons one finds a strong violation of axial symmetry for the near neighbours. Off symmetry phonons (Fig. lc Tb) had to be included in the fit to determine all parameters. Schober /Dederichs

145

1.2 Phononenzustlnde: Tb Table 3. Tb. Born-von Karman force constants, @j, T=298 K, [70Hol]. m

ij

a@, 0, 42

xx YY u

- 2 a/1/5,0,

c/2

xx

5 a/2 f3,

a/2, c/2

1.225

-0.180 0.241 -0.098 0.066

1.250 0.762 -0.410

0, a, c 0, 2a, 0

I; CNm-‘I

- 0.975

YY u

-0.894 0.046 -0.032 -2.228

X2

0, 0, c

f, CNm-‘I

afi,O,O

7.286 0.954 11.416 -0.952 2.426 -1.889

the fifth neighbours

o;=CR;R~/(Rm)‘1(1,-f;)+j;6,j; m

9.901

XZ xx YY 2.z XY

0, a, 0

The model isaxiallysymmetricfrom outward

(P; [N m- ‘1 5.467 1.562

[Lit. S. 180

xx 22

2. Phonon spectrum and related properties 1.6 1HZ-l

3.3 IHZ

3.2

gI 3.1 3.0 0.5

0

1.0

1.5

2.0

2.5

2.9

3.01Hz

Y-

Fig. 2. Tb. Phonon frequency spectrum at room temperature calculated from the eighth neighbour Born-von Karman model of Table 3 Tb.

nFig. 3. Tb. Debye cutoff frequencies,Y,, calculated from the spectrum of Fig. 2 Tb.

8 .10-‘2 Tb

/

S

///,

160 K 150 I Co

0

50

100

150

200

250

300 K 350

0

I-

200

300

600

K

500

I-

Fig.4. Tb. Debye temperatures 8, calculated from the Born-von Karman constants of Table 3 Tb [70Hol].

146

100

Fig. 5. Tb. Debye-Wailer exponent 2 Wdivided by the recoil frequency of a free terbium atom, vR, calculated from the Born-von Karman force constants of Table 3 Tb.

!Schober/Dederichs

1.2 Phonon states: Tc

Ref. p. 1 SO]

Table 4. Tb. Phonon spectrum at 298 K calculated from the Born-von Karman force constants of Table 3 Tb. v [THz]

g(v) CTHz-‘1

v [THz]

g(v) CT=- ‘1

v [THz]

g(v) CT=- ‘1

0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.000 0.000 0.001 0.002 0.003 0.004 0.006 0.008 0.011 0.013 0.016 0.019 0.021 0.025 0.029 0.033 0.038 0.042 0.048 0.053 0.059 0.067 0.073 0.081 0.090 0.100 0.110 0.122 0.136 0.150

1.24 1.28 1.32 1.36 1.40 1.44 1.48 1.52 1.56 1.60 1.64 1.68 1.72 1.76 1.80 1.84 1.88 1.92 1.96 2.00 2.04 ’ 2.08 2.12 2.16 2.20 2.24 2.28 2.32 2.36 2.40

0.169 0.193 0.224 0.358 0.431 0.440 0.450 0.465 0.477 0.498 0.530 0.472 0.436 0.448 0.433 0.426 0.411 0.391 0.378 0.362 0.342 0.342 0.348 0.366 0.387 0.406 0.439 0.471 0.524 0.588

2.44 2.48 2.52 2.56 2.60 2.64 2.68 2.72 2.76 2.80 2.84 2.88 2.92 2.96 3.00 3.04 3.08 3.12 3.16 3.20 3.24 3.28 3.32

0.540 0.503 0.494 0.503 0.531 0.554 0.569 0.584 0.420 0.339 0.364 0.642 0.792 0.825 1.145 0.977 0.663 0.472 0.102 0.081 0.059 0.002 0.0

3. Theoretical models No microscopic theory is available so far. The simple phenomenological models are only moderately successful. Born-von Karman and equivalent models: seeTable 3 Tb, further reference:[76Upl]. Short ranged forcesplus a simple electronic contribution [75Ca2,76Vil, 73Up2]. Model potential calculations: [77Up2,77Si2].

Tc

Technetium

Lattice: hcp,a=274pm=2.74&~=439pm=4.39A.

BZ:seep.450.

1. Phonon dispersion Table 1. Tc. Measurements. Method

T [K]

Fig.

Ref.

neutron diffraction (TAS)

~300

1 Tc

Smith et al. [75Sml]

Technetium shows a pronounced anomaly of the [OOQ LO mode at the zone centre. This mode represents neighbouring closed-packedplanes of atoms vibrating out of phase to each other. A similar dip but somewhat smaller was observedin the dispersion curves of Ti and Zr. Schober/Dederichs

147

1.2 Phonon states: Tc

Ref. p. 1 SO]

Table 4. Tb. Phonon spectrum at 298 K calculated from the Born-von Karman force constants of Table 3 Tb. v [THz]

g(v) CTHz-‘1

v [THz]

g(v) CT=- ‘1

v [THz]

g(v) CT=- ‘1

0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20

0.000 0.000 0.001 0.002 0.003 0.004 0.006 0.008 0.011 0.013 0.016 0.019 0.021 0.025 0.029 0.033 0.038 0.042 0.048 0.053 0.059 0.067 0.073 0.081 0.090 0.100 0.110 0.122 0.136 0.150

1.24 1.28 1.32 1.36 1.40 1.44 1.48 1.52 1.56 1.60 1.64 1.68 1.72 1.76 1.80 1.84 1.88 1.92 1.96 2.00 2.04 ’ 2.08 2.12 2.16 2.20 2.24 2.28 2.32 2.36 2.40

0.169 0.193 0.224 0.358 0.431 0.440 0.450 0.465 0.477 0.498 0.530 0.472 0.436 0.448 0.433 0.426 0.411 0.391 0.378 0.362 0.342 0.342 0.348 0.366 0.387 0.406 0.439 0.471 0.524 0.588

2.44 2.48 2.52 2.56 2.60 2.64 2.68 2.72 2.76 2.80 2.84 2.88 2.92 2.96 3.00 3.04 3.08 3.12 3.16 3.20 3.24 3.28 3.32

0.540 0.503 0.494 0.503 0.531 0.554 0.569 0.584 0.420 0.339 0.364 0.642 0.792 0.825 1.145 0.977 0.663 0.472 0.102 0.081 0.059 0.002 0.0

3. Theoretical models No microscopic theory is available so far. The simple phenomenological models are only moderately successful. Born-von Karman and equivalent models: seeTable 3 Tb, further reference:[76Upl]. Short ranged forcesplus a simple electronic contribution [75Ca2,76Vil, 73Up2]. Model potential calculations: [77Up2,77Si2].

Tc

Technetium

Lattice: hcp,a=274pm=2.74&~=439pm=4.39A.

BZ:seep.450.

1. Phonon dispersion Table 1. Tc. Measurements. Method

T [K]

Fig.

Ref.

neutron diffraction (TAS)

~300

1 Tc

Smith et al. [75Sml]

Technetium shows a pronounced anomaly of the [OOQ LO mode at the zone centre. This mode represents neighbouring closed-packedplanes of atoms vibrating out of phase to each other. A similar dip but somewhat smaller was observedin the dispersion curves of Ti and Zr. Schober/Dederichs

147

1.2 Phononenzustkde: Th

[Lit. S. 180

A-

8 THZ 7

TC

IDRI

I-300K

6 5 I 4 L 3

1 4

Fig. 1. Tc. Measured phonon dispersion at room temperature. The lines are drawn as guides to the eye [75Sml].

1 0

0.1 0.2

a3

0.4

Thorium

Th

Lattice: fee, a= 508 pm= 5.08 A. BZ: see p. 449.

1. Phonon dispersion Table 1. Th. Measurements. Method

T [K]

Fig.

Ref.

neutron diffraction F-AS)

296

1 Th

Reese et al. [73Rel] h-

-E

Al4 --

W

X

X

1

r

1ct51

HZ

3L

21 l-

/ $ OL

0

0.2

0.4 5-

0.6

0.8 1.010 [OOll

0.2

0.4 f-

0.6

0.8

1.0 IO111

0.8

0.6 -6

0.1

0.2

!

b -2

[

Fig. 1. Th. Phonon dispersion curves at room temperature.The experimental errors lie within the circles unless indicated by an error bar. The smooth curves represent the seventh neighbour force constant model [73Rel]. The measurements indicate that the dispersion curves in thorium are dominated by a large non-axially-symmetric nearest neighbour interaction. There is also a residual long-range interaction extending to at least seven nearest neighbour shells. The initial slopes of the dispersion curves are compatible with the ones obtained from the elastic constants except for the [O[fl Tr branch where the measured phonon frequency is about 20 % greater than expected. 148

Schober/Dederichs

1.2 Phononenzustkde: Th

[Lit. S. 180

A-

8 THZ 7

TC

IDRI

I-300K

6 5 I 4 L 3

1 4

Fig. 1. Tc. Measured phonon dispersion at room temperature. The lines are drawn as guides to the eye [75Sml].

1 0

0.1 0.2

a3

0.4

Thorium

Th

Lattice: fee, a= 508 pm= 5.08 A. BZ: see p. 449.

1. Phonon dispersion Table 1. Th. Measurements. Method

T [K]

Fig.

Ref.

neutron diffraction F-AS)

296

1 Th

Reese et al. [73Rel] h-

-E

Al4 --

W

X

X

1

r

1ct51

HZ

3L

21 l-

/ $ OL

0

0.2

0.4 5-

0.6

0.8 1.010 [OOll

0.2

0.4 f-

0.6

0.8

1.0 IO111

0.8

0.6 -6

0.1

0.2

!

b -2

[

Fig. 1. Th. Phonon dispersion curves at room temperature.The experimental errors lie within the circles unless indicated by an error bar. The smooth curves represent the seventh neighbour force constant model [73Rel]. The measurements indicate that the dispersion curves in thorium are dominated by a large non-axially-symmetric nearest neighbour interaction. There is also a residual long-range interaction extending to at least seven nearest neighbour shells. The initial slopes of the dispersion curves are compatible with the ones obtained from the elastic constants except for the [O[fl Tr branch where the measured phonon frequency is about 20 % greater than expected. 148

Schober/Dederichs

1.2 Phonon states: Th

Ref. D.1801

In both transverse [Oc[] phonons a pronounced structure was observed.The anomalies are too broad to be ascribed to the Kohn effect.They might arise due to peaksin the generalizedsusceptibility function x(Q) [73Rel]. Table 2. Th. Measured phonon frequenciesat room temperature [73Rel]. cc

v CrHzl

0.1 0.2 0.3 0.35 0.4 0.45 0.50 0.55 0.6 0.65 0.7 0.75 0.80 0.85 0.9 0.95 1.0

0.377(15) 0.657(15) 0.998(15) 1.168(15) 1.302(15) 1.407(15) 1.500(15) 1.558(15) 1.631(15) 1.720(15) 1.808(15) 1.883(15) 2.002(15) 2.113(15) 2.202(15) 2.246(15) 2.259(15)

Ir 0.1 0.2 0.3 0.35 0.40 0.45 0.50 0.55 0.60 0.7 0.8 0.9 1.0

v [THz]

r

v I?=1

0.545(15) 1.071(15) 1.553(15) 1.744(15) 1.934(15) 2.080(15) 2.241(15) 2.463(15) 2.693(15) 3.083(15) 3.32(3) 3.45(2) 3.474(15)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.425 0.45 0.475 0.5

1.055(15) 1.502(15) 1.864(15) 2.25(4) 2.542(15) 2.841(15) 3.054(15) 3.163(15) 3.160(15) 3.3(1) 3.24(8)

0.0

0.825(10) 1.549(15) 2.06(2) 2.630(15) 3.056(15) 3.116(15) 2.80(4) 2.666(15) 2.355(15) 2.259(15)

CW’I T

cowl L 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.013(15) 1.48(1) 1.94(2) 2.417(15) 2.81(1) 3.08(2) 3.347(15) 3.45(3) 3.474(15)

0.1 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.465(30) 0.850(15) 0.98(2) 1.110(15) 1.215(15) 1.23(3) 1.25(2) 1.28(2)

A 2.259(15) 2.200(15) 2.21(5) 2.22(5) 2.213(15) 2.259(15)

0.2 0.4 0.6 0.8 1.0

lwll L 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

lwl

v [THz]

Ii

Table 3. Th. Born-von Karman force constants, @t, T= 296 K, [73Rel]. .. ij q q m mm-11

m

lJ

IYm-‘I

zz YZ xz XY

-0.021(6) 0.009(2) 0.013(2) 0.026(5)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

xz 220

xx zz XY

-0.072(11) 1 -0.013 (13) 0.096(22) 0.054(39)

The force constants are constrained to fit the elastic constants.Axial symmetry conditions are imposed on the fifth and seventh neighbour force constants. Schober/Dederichs

149

1.2 Phononenzustiinde: Th

[Lit. S. 180

2. Frequency spectrum and related properties Fig. 2. Th. Frequency spectrum at 296 K obtained from the seventh neighbour Born-von Karman fit of Table 3 Th. 4

3.1 1Hz

3.3 0

0.5

1.0

1.5 Y-

2.0

2.5

3.2

3.01Hz 3.5

I 3.1 i 3.0

Table4. ‘Ph. Phonon frequency spectrum at 296 K

00

2.9

calculated from the Born-von Karman force constants of Table 3 Th. g(v) ~Hz-‘1

v [THz]

g(v) ~Hz-‘1

0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

0.0 0.000 0.001 0.002 0.003 0.005 0.007 0.010 0.013 0.016 0.021 0.025 0.030 0.036 0.042 0.049 0.057 0.066 0.077 0.089 0.103 0.120 0.141 0.167 0.205 0.265 0.389 0.411 0.434 0.457 0.480 0.501 0.520 0.536 0.546 0.550

1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55

0.549 0.545 0.538 0.530 0.522 0.477 0.438 0.416 0.403 0.396 0.402 0.388 0.356 0.339 0.328 0.319 0.312 0.305 0.297 0.288 0.277 0.261 0.232 0.219 0.284 0.395 0.572 0.946 0.927 0.686 0.231 0.195 0.159 0.116 0.0 0.0

150

I

10

20

2.8 -10

v ~Hz]

I 0

30

nFig. 3. Th. Debye cutoff frequencies Y, obtained from the spectrum of Fig. 2 Th. 160 160 K 150 140

6130 I 6130 120 120

0

20 20

40 40

60 60

80

100

120

140 K 160

I-

Fig. 4. Th. Calculated Debye temperature 8, [73Rel].

8

.10-l? S I

6

s 4 1 N

0

100

200

300

400

K

500

Fig. 5. Th. Debye-Wailer exponent 2Wdevided by the recoil frequency of the free atom, va, calculated from the spectrum of Fig. 2 Th. !Schoher/Dederichs

1.2 Phonon states: Ti

Ref. p. 1SO] 3. Theoretical models

Even the long range Born-von Karman fit deviates in the [Ocl] A branch markedly from the measured values. The other phenomenological models including the pseudopotential calculations deviate clearly also in the main symmetry directions. Born-von Karman and equivalent models: see Table 3 Th Fig. 1 Th, further reference: [75Cal]. Short ranged forces plus a simple electronic contribution: 78Gu43. Local pseudopotential:

Ti

[77Vrl, 77Kul],

[78Gu2,77Ral,

77Cll], further references: [77Bel,

further reference: [75Rol].

Titanium

Lattice: hcp, a=295 pm=2.95 A, c=468 pm=4.68 A. BZ: see p. 450.

1. Phonon dispersion Table 1. Ti. Measurements. Fig.

Method

T CKI

neutron diffraction (TAS)

295,773, 1 Ti 1054

Ref. Stassis et al. [79St2]

The dispersion curves ot titanium are similar to the ones of hcp-Zr. The zone centre mode of the [loo] LO branch softens with decreasing temperature, an anomaly also found in hcp-Zr.

T-

0

T’-

--

0.1

0.2 0.3 0.4 0.5 0 0.5 5-C 6Fig. 1. Ti. Measured phonon dispersion curves at 295 K. The solid lines were obtained from the sixth neighbour Born-von Karman model of Table 3 Ti [79St2]. Schober /Dederichs

151

1.2 Phonon states: Ti

Ref. p. 1SO] 3. Theoretical models

Even the long range Born-von Karman fit deviates in the [Ocl] A branch markedly from the measured values. The other phenomenological models including the pseudopotential calculations deviate clearly also in the main symmetry directions. Born-von Karman and equivalent models: see Table 3 Th Fig. 1 Th, further reference: [75Cal]. Short ranged forces plus a simple electronic contribution: 78Gu43. Local pseudopotential:

Ti

[77Vrl, 77Kul],

[78Gu2,77Ral,

77Cll], further references: [77Bel,

further reference: [75Rol].

Titanium

Lattice: hcp, a=295 pm=2.95 A, c=468 pm=4.68 A. BZ: see p. 450.

1. Phonon dispersion Table 1. Ti. Measurements. Fig.

Method

T CKI

neutron diffraction (TAS)

295,773, 1 Ti 1054

Ref. Stassis et al. [79St2]

The dispersion curves ot titanium are similar to the ones of hcp-Zr. The zone centre mode of the [loo] LO branch softens with decreasing temperature, an anomaly also found in hcp-Zr.

T-

0

T’-

--

0.1

0.2 0.3 0.4 0.5 0 0.5 5-C 6Fig. 1. Ti. Measured phonon dispersion curves at 295 K. The solid lines were obtained from the sixth neighbour Born-von Karman model of Table 3 Ti [79St2]. Schober /Dederichs

151

1.2 Phononenzust5nde: Ti

[Lit. S. 180

Table 2. Ti. Measured phonon frequencies, [79St2]. T

295 K

773 K

1054 K

T

v [THz]

3

295 K

IX001TO,&,)

0.1 0.2 0.3 0.4 0.5

0.74 (3) 1.38 (3) 2.03 (3) 2.56 (3) 3.05 (3)

0.66 (2) 1.26 (3) 1.83 (3) 2.36 (2) 2.81 (8)

0.57 (2) 1.16 (2) 1.68 (2) 2.19 (4) 2.63 (4)

0.1 0.2 0.3 0.4 0.5

4.64 (8) 5.45 (4) 6.24 (6) 6.62 (8) 6.95 (12)

0.0

4.10 (15) 4.07 (15) 3.95 (10) 3.79 (5) 3.43 (5)

3.77 (12) 3.60 (5) 3.54 (5) 3.40 (5) 3.13 (6)

3.53 (12) 3.50 (6) 3.45 (8) 3.31 (4) 3.00 (6)

0.1 0.15 0.2 0.25 0.4 0.5

4.70 (6)

CWIWA,) 0.1 0.2 0.3 0.4 0.5

1.45 (5) 2.74 (6) 3.91 (5) 4.97 (6) 5.73 (5)

1.34 (3) 2.61 (3) 3.77 (8) 4.79 (4) 5.59 (4)

5.54 (15) 5.67 (15) 5.75 (20) 6.06 (20) 6.28 (12) 5.73 (5)

6.14 (15) 6.13 (12) 6.24 (10) 6.31 (10) 6.11 (10) 5.59 (4)

1.33 (4) 2.55 (4) 3.69 (6) 4.67 (6) 5.50 (6)

1.29 (3) 2.46 (5) 3.32 (10) 3.78 (10) 3.82 (10)

1.14 (3) 2.17 (3) 3.03 (2) 3.64 (6) 3.73 (8)

152

1.16 (3) 2.45 (10) 4.45 (10) 5.8 (1) 6.58 (20) 7.10 (6)

1.05 (3) 2.25 (6) 4.19 (4) 5.50 (8)

1.07 (4) 2.10 (3) 3.70 (4) 4.82 (6) 5.93 (4)

3.70 (8) 4.96 (8) 6.23 (3) 6.65 (3)

6.42 (4) 6.73 (10)

5.07 (6) 5.70 (6) 6.20 (6) 7.10 (3)

4.70 (5) 5.40 (4) 5.92 (10) 6.8 (1)

0.95 (4) 1.88 (2) 3.45 (3) 4.80 (15)

0.88 (4) 1.74 (2) 3.32 (4) 4.64 (6)

3.32 (9) 4.69 (4)

3.04 (5) 4.28 (4)

6.18 (20)

6.04 (5)

CL’WITM-L)

I 0.05 0.1 0.2 0.33 0.4

1.09 (4) 2.26 (5) 4.75 (4) 6.00 (2) 5.09 (2)

I 1.02 (2) 2.09 (4) 4.00 (4) 5.58 (8)

4.03 (6) 4.83 (5)

K’WI LA(T,) 0.1 0.2 0.33 0.4

1.03 (3) 2.01 (3) 2.90 (3) 3.50 (4) 3.65 (8)

4.21 (6) 5.03 (6) 6.04 (12) 6.4 (2) 6.82 (12)

KS01TALL) 0.05 0.1 0.2 0.3 0.4

6.33 (20) 6.44 (12) 6.46 (10) 6.40 (20) 6.06 (6) 5.50 (6)

CC001 LA@,) 0.05 0.1 0.2 0.3 0.4 0.5

7.59 (4) 7.69 (4)

I

CC001 TA,G) 0.1 0.2 0.3 0.4 0.5

6.26 (4)

I

CWI LO@,) 0 0.1 0.2 0.3 0.4 0.5

1054 K

v D-Hz1

r

CWI TN&)

0.1 0.2 0.3 0.4

773 K

0.86 (3) 2.05 (10) 4.65 (3) 5.58 (4) 4.70 (2)

0.78 (3) 1.88 (3) 4.02 (4)

Cl6.01TO,&) 0.1 0.2 0.3 0.4

5.88 (4) 5.9 (2) 5.44 (2) 4.5 (4)

5.50 (2) 5.35 (2) 5.02 (2) 4.34 (10)

5.22 (4) 5.14 (10) 4.76 (12) 4.23 (10) (continued)

6.72 (12)

!khober/Dederichs

1.2 Phonon states: Ti

Ref. p. 1 SO] Table 2. Ti. (continued) T

295 K

773 K

1054 K

T

v [THz]

I

0.1

1.13(3) 2.12(2) 2.83(4) 3.15(3) 3.40(4)

1054 K

v [THz]

I

CLW TA,,W 0.2 0.3 0.4 0.5 ~

773 K

295 K

K~ol TOI,

0.9 (4) 1.78(4) 2.55(3) 3.07(6) 3.2(8)

0.77(2) 1.63(3) 2.35(3) 3.00(3) 3.10(5)

0.1 0.15 0.2 0.25 0.3 0.33

5.41(4) 6.03(6) 6.6(4) 6.9(3) 7.1(2) 7.01(4)

4.95(6) 5.87(25) 6.20(8)

4.83(4) 5.30(3)

6.87(6)

6.84(10)

CL’001 TO,@,) 0.1 0.2 0.3 0.4 0.5

5.5(2) 5.62(15) 5.86(15) 6.12(12) 6.06(15)

6.17(15) 6.10(10) 6.05(15) 6.07(12) 5.96(15)

6.30(12) 6.20(15) 6.25(12) 6.02(15) 5.94(12)

CL’L’OI L0f-L) 0.1 0.2 0.3 0.4

5.73(10) 7.3(1) 6.59(10) 6.6(1)

5.30(4)

4.78(6)

6.42(10)

Table 3. Ti. Born-von Karman coupling constants,@G; T=295 K [79St2]. The model is of the “modified axially symmetric” form [65Del] @;=K(R~R;)/(R”)2+C,6ij where CB= C,, for i = j = x, y and C, = C,, for i = j = z m

K [Nm-‘1

C [I?m-‘1

C [I?m-‘1

b/l& 0, 42) (0, a, 0) (-wf3 @c/2)

41.38 22.28 - 8.33

- 3.03 0.171 1.10

-17.21 3.78 0.22

(0,034

K+&=12.20,

PaI2 VT 0, 49 (04 0, 0)

2.26 0.65

I 5

t&=0.68 0.17 1.51

0.02 0.23

Fig. 2. Ti. Temperaturedependence of the dispersioncurves in [OOC] (d) direction [79St2]. Temperaturedependence

Increasing the temperature the frequencies of all but the [OOl] LO branch decrease(typically about 5 % from 295 K to 1059K), Fig. 2 Ti and Table 2 Ti. The [OOl] LO branch has at room temperature a dip at the zone centre which disappearsat higher temperatures.A similar behaviour is also found in Zr. An even more pronounced dip is found in Tc at room temperature. No dramatic change in the phonon width or frequencies was observed as the hcp-bee phasetransition temperature was approached. Schober /Dederichs

153

1.2 Phononenzustkde: T1

[Lit. S. 180

2. Frequency spectrum and related properties 6 20

orb.

co1 moleK

units I 4 ,3 zl 2

I

1 0

Y

IJ 1

2

3

4

5

6

7

8lHz 9

Y-

Fig. 3. Ti. Phonon frequency spectrum at 295 K obtained kom the Born-von Karman model of Table3 Ti [79St2].

200

800 1000 K 1200 IFig.4. Ti. Temperature dependence of the lattice specific heat at constant pressure, c’,, calculated using the measured temperature dependence of the phonons [79St2]. 400

600

8.5 col moleK

7.5 In.

u 6.5 Fig. 5. Ti. Comparison of the calculated specific heat at constant pressure(c’, of Fig. 4 Ti plus an electronic contribution c;) with experimental results [793t2].

.zoo 400

600

800

1000 K 12

3. Theoretical models The theoretical models derived before the publication of the measureddispersion curves are unreliable. Born-von Karman model : seeTable 3 Ti.

T1 Thallium Lattice: hcp, a= 345pm = 3.45A, c= 551pm = 5.51A. BZ: seep. 450.

1. Phonon dispersion Table 1. Tl. Measurements. Method

T [K]

Fig.

neutron diffraction (TAS)

77,296 1 Tl 2 Tl

Ref. Worlton and Schmunk [71Wol]

Only the dispersion in the symmetry directions A and Z has been measured.No anomalies are obvious. The datacould be fitted well with thirteen parameter Born-von Karman models.The data are not sufficiently accurate :o extract a clear information on the temperature bchaviour. 154

Schober/Dederichs

1.2 Phononenzustkde: T1

[Lit. S. 180

2. Frequency spectrum and related properties 6 20

orb.

co1 moleK

units I 4 ,3 zl 2

I

1 0

Y

IJ 1

2

3

4

5

6

7

8lHz 9

Y-

Fig. 3. Ti. Phonon frequency spectrum at 295 K obtained kom the Born-von Karman model of Table3 Ti [79St2].

200

800 1000 K 1200 IFig.4. Ti. Temperature dependence of the lattice specific heat at constant pressure, c’,, calculated using the measured temperature dependence of the phonons [79St2]. 400

600

8.5 col moleK

7.5 In.

u 6.5 Fig. 5. Ti. Comparison of the calculated specific heat at constant pressure(c’, of Fig. 4 Ti plus an electronic contribution c;) with experimental results [793t2].

.zoo 400

600

800

1000 K 12

3. Theoretical models The theoretical models derived before the publication of the measureddispersion curves are unreliable. Born-von Karman model : seeTable 3 Ti.

T1 Thallium Lattice: hcp, a= 345pm = 3.45A, c= 551pm = 5.51A. BZ: seep. 450.

1. Phonon dispersion Table 1. Tl. Measurements. Method

T [K]

Fig.

neutron diffraction (TAS)

77,296 1 Tl 2 Tl

Ref. Worlton and Schmunk [71Wol]

Only the dispersion in the symmetry directions A and Z has been measured.No anomalies are obvious. The datacould be fitted well with thirteen parameter Born-von Karman models.The data are not sufficiently accurate :o extract a clear information on the temperature bchaviour. 154

Schober/Dederichs

1.2 Phonon states: Tl

Ref. p. 1 SO] E-

1’

-1

A-

R-

s'

-s

A3

f-

-6

5-

Fig. 1. Tl. Measured phonon dispersion at 77 K. The solid line represents the values calculated with the Born-von Karman parameters of Table 3 Tl [71 Wol]. T

C-

-T

M

3.0r THz +o

r&-001

E

A-

R-

-s

S

K

A

TI

KC01

T-296 K

Kt'/21

2.5

H3

2.0 pI 1.5

A3

0 0

0.5

0.4

0.3

0.2

0.1

-f

5-

f-

-C

t-

Fig. 2. Tl. Measured phonon dispersion at 296 K. The solid line represents the values calculated with the Born-von Karman parameters of Table 3 Tl [71Wol]. Table 2. TI. Measured phonon frequencies [71 Wol]. 77K

T

296 K

v [THz]

77K

296

K

v [THz]

d4nmx

C, (LA)

c, (LO)

0.4 0.6 0.8

1.21(19) 1.68(10) 2.07(12)

1.20(19) 1.66(24) 2.12(22)

1.0

2.16 (12)

2.19 (19)

0.0 0.2 0.4 0.6 0.8

1.0

1.28(36) 1.38(36) 1.63 (10) 1.99(7) 2.25(7) 2.32(7)

1.06(36) 1.33(36) 1.60(36) 1.98 (15) 2.26(19) 2.33(17)

(continued) Schoher /Dederichs

155

1.2 Phononenzustkde: Tl

[Lit. S. 180

Table 2. Tl. (continued) T

77 K

296 K

I 77K

T

v [THz]

296 K v [THz]

4hnl”X

C, @AI) 0.4 0.6 0.8 1.0

0.57(12) 0.85 (7) 1.02(7) 1.03(10)

0.59(17) 0.77(10) 1.04(10) 1.21(15) x3

0.0 0.2 0.4 0.6 0.8 1.0

A, (TO)

2.84(12) 2.82(7) 2.76(7) 2.67(7) 2.55(12) 2.40(12)

0.41(19) 0.65(15) 0.89(12) 0.97(15)

0.32(12) 0.60(24) 0.77(12) 0.81(7)

1.28(36) 1.31(36) 1.39(12) 1.50(10) 1.58(7)

1.09(36) 1.12(36) 1.16(12) 1.33(12) 1.40(10)

0.80(12) 1.20(15) 1.50(12) 1.84(10)

0.73(24) 1.11(17) 1.37(12) 1.68(12)

2.78(7) 2.61(7) 2.27(17) 2.04(10) 1.84(10)

156

0.30(10) 0.45(12) 0.58(22) 0.76(15)

~=K(R~RT)/(Rm)2+Ca6ij where C,=C, for i=j=x,y and Ca=C,, for i=j=z. The number of parameters is further reduced by the condition: C&=Q C,, with u= -1.91267 for 77 K and c = 1.56194for 296 K. T

77K

296 K

Ref.

[71Wol]

[71Wol]

K, CBx,C,, CNm- ‘1

k@, 0,cm (0,

c, 0)

( - 2 a/1/5,0, c/2) 2.72(22) 2.61(22) 2.33(29) 1.89(24) 1.68(12)

K C Bx C BZ K C BX

0.48(12) 0.61(15) 0.69(12)

KBz C BX

CBZ K

(0, O,c)

CBX CBZ

(542 0, a/2,@)

A.6(TN 0.4 0.6 0.8 1.0

The models are of the “modified axially symmetric” form [65De2]

C

A2 (LO) 0.2 0.4 0.6 0.8 1.0

Table 3. Tl. Born-von Karman coupling constants, @t, fitted to the measured phonon frequencies and elastic constants

m

A, (LA) 0.4 0.6 0.8 1.0

0.64(36) 0.69(19) 0.71(24) 0.74(15) 0.68(12)

2.80(12) 2.78(17) 2.50(15) 2.16(22) 1.91(20) 1.69(24)

z.4 (TO,,)

0.2 0.4 0.6 0.8 1.0

1.12(22) 1.11(22) 0.97(12) 0.85(10) 0.76(15)

(TO,)

C4 (TAII)

0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

(fh

0, 0)

Schober/Dederichs

K

CBX CBZ K C BX

CBZ

10.2475 -1.9278 3.6872 10.5406 - 0.0733 0.1402 -2.386 0.8603 -1.6455 -1.4887 -0.2527 0.4833 1.8162 0.0314 - 0.0601 -1.1599 0.1008 -0.1930

15.4911 - 2.4216 - 3.7824 12.8704 -1.1571 -1.8073 -0.4819 0.6088 -0.9509 -0.8846 0.0785 0.1226 -0.2955 0.1774 0.2771 -0.3212 - 0.2748 - 0.4292

Ref. p. 1 SO]

1.2 Phonon states: Tm

2. Frequency spectrum and related properties

II

0.5

1.0

1.5

2.0

0

2.5 THz 3.0

Fig. 3. Tl. Phonon spectrui %?%K calculated with the Born-von Karman parameters of Table 3 Tl [71Wol].

1.0

0.5

1.5

2.0

2.5 THz 3.0

Fig. 4. Tl. Phonon spectrum vat calculated with the Born-von Karman parameters of Table 3 Tl [71Wol].

3. Theoretical models The experimental data can be well fitted with phenomenological models. The available data are, however, not sufficient to assessthem properly. Born-von Karman and equivalent models: seeTable 3 Tl and [71Wol, 72Me2]. Short ranged forces plus a simple electronic contribution: 75Ca2, 78Mi2].

[73Kul, 75Si7], further references: [73Upl,

Pseudopotential calculations: [75Rel, 77Si2].

Tm

Thulium

Lattice: hcp, a=352pm=3.52&

c=556pm=5.56A.

BZ: see p. 450.

The phonon dispersion of thulium has not been measuredso far. Fig. 1 Tm shows a theoretical estimate of the dispersion.The model useseight parameterswhich representtwo and three body interactions up to sixth neighbours. The parametersare determined from the elastic constants, predicted for Tm by interpolation between Er and Lu, and by approximate frequencies predicted by extrapolation from Tb and Ho. The corresponding estimate of the phonon spectrum is shown in Fig. 2 Tm [79Ral]. A-

A

4r THY a

Tm

C-

root1

f-

t-

Fig. la, b. Tm. Theoretical estimate of the phonon dispersion. The eight parameters of the model were fitted to the elastic constants and some phonon frequencies estimated by interpolation and extrapolation from Tb, Ho, and Lu [79Ral]. Schober /Dederichs

157

Ref. p. 1 SO]

1.2 Phonon states: Tm

2. Frequency spectrum and related properties

II

0.5

1.0

1.5

2.0

0

2.5 THz 3.0

Fig. 3. Tl. Phonon spectrui %?%K calculated with the Born-von Karman parameters of Table 3 Tl [71Wol].

1.0

0.5

1.5

2.0

2.5 THz 3.0

Fig. 4. Tl. Phonon spectrum vat calculated with the Born-von Karman parameters of Table 3 Tl [71Wol].

3. Theoretical models The experimental data can be well fitted with phenomenological models. The available data are, however, not sufficient to assessthem properly. Born-von Karman and equivalent models: seeTable 3 Tl and [71Wol, 72Me2]. Short ranged forces plus a simple electronic contribution: 75Ca2, 78Mi2].

[73Kul, 75Si7], further references: [73Upl,

Pseudopotential calculations: [75Rel, 77Si2].

Tm

Thulium

Lattice: hcp, a=352pm=3.52&

c=556pm=5.56A.

BZ: see p. 450.

The phonon dispersion of thulium has not been measuredso far. Fig. 1 Tm shows a theoretical estimate of the dispersion.The model useseight parameterswhich representtwo and three body interactions up to sixth neighbours. The parametersare determined from the elastic constants, predicted for Tm by interpolation between Er and Lu, and by approximate frequencies predicted by extrapolation from Tb and Ho. The corresponding estimate of the phonon spectrum is shown in Fig. 2 Tm [79Ral]. A-

A

4r THY a

Tm

C-

root1

f-

t-

Fig. la, b. Tm. Theoretical estimate of the phonon dispersion. The eight parameters of the model were fitted to the elastic constants and some phonon frequencies estimated by interpolation and extrapolation from Tb, Ho, and Lu [79Ral]. Schober /Dederichs

157

[Lit. S. 180

1.2 Phononenzust5nde: U 3 orb. units I

2

T -6 1

0

1

2

3

1Hz

Fig. 2. Tm. Phonon frequency distribution obtained from the same model as Fig. 1 Tm [79Ral].

4

Y----r

U

Uranium

Lattice: a-U, c-centered orthorhombic BZ: see p. 456.

(A20), a = 285 pm = 2.85 A, b = 587 pm = 5.87 A, c=496 pm =4.96 A.

Table I. U. Measurements.

c-

Method

T CKI

Fig.

Ref.

neutron diffraction (TAS)

x300

1 U

Crummet et al. [79Crl]

c-

A-

-A l-

IHz 3.5

3.0

2.5

irI 2.0

1.5

1.0

0.5 0

0

0.8

0.6

0.1

0.2

0

0.1

0.2

0.3

0.1

Fig. 1. U. Measured phonon dispersion in a-uranium at room temperature. The solid lines are calculated from a sixth neighbour. 27 parameter modified shell model. The vertical dashed line indicates the Brillouin zone boundary [79Crl]. 158

Schober/Dederichs

[Lit. S. 180

1.2 Phononenzust5nde: U 3 orb. units I

2

T -6 1

0

1

2

3

1Hz

Fig. 2. Tm. Phonon frequency distribution obtained from the same model as Fig. 1 Tm [79Ral].

4

Y----r

U

Uranium

Lattice: a-U, c-centered orthorhombic BZ: see p. 456.

(A20), a = 285 pm = 2.85 A, b = 587 pm = 5.87 A, c=496 pm =4.96 A.

Table I. U. Measurements.

c-

Method

T CKI

Fig.

Ref.

neutron diffraction (TAS)

x300

1 U

Crummet et al. [79Crl]

c-

A-

-A l-

IHz 3.5

3.0

2.5

irI 2.0

1.5

1.0

0.5 0

0

0.8

0.6

0.1

0.2

0

0.1

0.2

0.3

0.1

Fig. 1. U. Measured phonon dispersion in a-uranium at room temperature. The solid lines are calculated from a sixth neighbour. 27 parameter modified shell model. The vertical dashed line indicates the Brillouin zone boundary [79Crl]. 158

Schober/Dederichs

Ref. p. 1 SO]

1.2 Phonon states: U

Table 2. U, Measured phonon frequenciesin cc-uraniumat room temperature, [79Crl]. r

v [THz]

r

v [THz]

23 r, r, r,

3.12(X) 3.59(10) 2.46(15)

2: 0.05 0.08 0.123 0.148 0.18 0.20 0.25 0.30 0.40 0.45 0.50 0.55 0.60 0.614 0.65 0.70 0.75 0.80 0.85 0.90 0.95

Y6

0.57(5) l.OO(5) 1.40(5) 1.60(5) 2.00(5) 2.08 (10) 2.10(5) 2.24 (10) 2.14 (10)

1.99(5) 2.02 (10) 2.12 (10) 2.24 (10) 2.50 (10) 2.72(6) 2.72(10) 3.05(20) 2.79 (10) 2.74 (10) 2.46(15) 2.27(10) 2.28 (10)

G 3.18(15) 0.20 0.25 0.30 0.35 0.40 0.50 0.55 0.60 0.70 0.80 0.90

1.00

3.14(15) 3.18 (10) 3.17(15) 3.09(15) 3.08(20) 3.17(10) 3.23(10) 3.30 (10) 3.26(15) 3.40(20) 3.40(2) 3.34 (10)

x2

D.10 0.20 0.30 0.40 0.60 0.70 3.80 3.95

3.56(15) 3.40(20) 3.15 (15) 2.90 (10) 2.70(10) 2.68 (10) 3.05(15) 3.18 (10)

y5

3.19(15)

0.20 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.80 0.90

Y6

0.15 0.20 0.30 0.35 0.365 0.365 0.373 0.40 0.41 0.435 0.45 0.475 0.50 0.535 0.54 0.55 0.56 0.60 0.62 0.635 0.65 0.675 0.70 0.75 0.80 0.85 0.95

Y7

v [THz]

1.27(5) 1.63 (5) 1.70(5)

1.80(5) 1.91(5) 2.01 (5) 2.12(5) 2.27(5) 2.44(5) 2.59(5) 2.77(5) 2.90 (10) 2.96 (10)

0.69(5)

l.Ol(5) 1.25(5) 1.70(5) 1.50(l) 1.80(6) 1.52(3) 1.60(6) 1.40(20)

lAQ(10) 1.27(3) 1.06 (10)

1.01(10) 1.06 (10) 1.18 (3) 1.46(3) 1.30(10) 1.40 (10) 1.60(6) 1.71(3) 1.80(3) 2.00(7) 2.20 (10) 2.39(5) 2.55(5) 2.55(5) 2.50(15)

2.49(10) 2.49 (10)

0.10 0.15 0.20 0.25 0.28 0.30 0.325 0.35 0.40 0.45 0.50 0.55 0.60 0.70 0.90

Y4

2.42(10) 2.50 (10) 2.46 (10) 2.48(8) 2.20(8) 2.32 (10) 2.00(7) 1.90(l) 2.10(10) 2.37 (10) 2.59(5) 2.68(5) 2.86 (10) 3.14 (10) 3.48 (10) 3.53 (10)

A: 0.10 0.156 0.20 0.25 0.364 0.50 0.60 0.70 0.80 0.90 0.95

0.60(5) l.OO(5) 1.14(5) 1.45(5) 2.00(5) 2.50(15) 2.98(15) 3.12(15) 3.30 (10) 3.32(15) 3.30(25)

-A: 0.20 0.30 0.40 0.50 0.60 0.70 0.80

0.90

3.21(15) 3.20(15) 3.22(15) 3.10(10) 3.32 (10) 3.17 (10) 3.32(15) 3.46(15)

A: 0.30 0.40 0.50 0.60 0.70 0.80

r

v [THz]

G

2

c: 0.10

5

0.10 0.20 0.25 0.35 0.40 0.55 0.70 0.80 0.90 0.95

A.: 0.20 0.30 0.35 0.40 0.50 0.60 0.70 0.80 0.90 0.95

1.95(5) 2.28(5) 2.52(5) 2.70 (10)

0.70(5)

1.00(5) 1.12(5) 1.30(5) 1.54(5) 1.83(5) 2.00(5) 2.15 (10) 2.23 (10) 2.30 (10)

A: 0.10 0.20 0.35 0.40 0.50 0.70 0.85

2.54 (10) 2.50 (10) 2.42 (10) 2.46(15) 2.42(10) 2.38 (10) 2.40 (10)

a: 0.10 0.12

0.73 (10)

0.22 0.30 0.35 0.40 0.45

1.60(5) 2.14 (10) 2.234(5) 2.52(5) 2.38(5) 2.20(5)

-G 1.27(5) 1.65 (5)

3.55(15) 3140(20) 3.30(20) 3.25(15) 3.32(15) 3.10 (15) 3.00(20) 3.06(15) 3.06(20) 3.10(15)

1.00(3)

AT 0.10 0.20 0.35 0.40 0.45

z:

3.10 (10) 2.98 (15) 2.73 (10) 2.60 (10) 2.80 (10) 2.96(15)

(continued) Schober /Dedericbs

159

1.2 Phononenzusttinde:

V

[Lit. S. 180

Table 2. U. (continued) v [THz]

c

v [THz]

r

A:

AZ

0.20 0.30 0.40 0.45 Z2

2.50(15) 2.50(20) 2.45(10) 2.34(10) 2.06(10)

v [THz]

r

JG

0.05 0.10 0.20 0.30 0.40

0.82(2) 1.23(5) 1.55(5) 1.72(10) 1.88(10)

v [THz]

r

A:

0.20 0.30 0.40 0.45

1.02(5) 1.44(5) 1.82(5) 2.02(5)

0.10 0.20 0.30 0.40 0.45

3.66(15) 3.53(15) 3.50(15) 3.25(10) 3.16(10)

Whereasthe dispersion curves in A and A directions show no unusual featurespronounced dips were observed in X direction. Thesedips are similar to the onesobservedin high temperature superconductors[75Sml].The lower C, and C, branchessoften considerably with decreasingtemperaturee.g.the C, (LO) mode at c= 0.475softensfrom v=l.l THzat 300K to r=0.6 THz at 160 K and the& (LA) mode at [=0.5 softensfrom v=2.OTHz to v=1.4 THz. An elastic peak appears at <=(2.5, 2, 0) just above 60 K and grows in intensity as the temperature is decreased to 43 K and below. A reasonablefit of the dispersion could be attained with a 27 parameter modified shell model [79Crl].

V

Vanadium

Lattice: bee,a=303 pm=3.03 A. BZ: see p. 448. 1. Phonon dispersion Table 1. V. Measurements. Method

T WI

Fig.

Ref.

X-ray diffraction

296

1V

Collela and Batterman [7OCo3]

The dispersion relations of vanadium cannot be measuredwith neutrons sinceV is an almost totally incoherent neutron scatterer (crcoh/cincoh= 0.008).Neutron studies are therefore limited to the determination of the phonon spectrum. The dispersion curves of V show similar anomalies as the ones of Nb and Ta. E.g. the [OOfl T and [O[a T2 modesshow anomalous dispersion at low 4’values.The [OOfl L mode shows a dip at about [=0.7 which also occurs in Nb and Ta. A-

AH

gr

IOO~ll

V

THza 8-

.0

1=296K

gr 1Hz 8-

H

b

iOD511

l-

6I ir 4Interval:(200)-(2101 o between~300)ondl400~ l Ni- Co filter I 0

I

0.2

I

I

I

0.4

I

0.6

I

I

0.8

I

I

1.0

0

f‘-

160

I

0.2

t

I

I

0.4 5-

!Schober/Dederichs

I

0.6

I

I

0.8

I 1.0

1.2 Phononenzusttinde:

V

[Lit. S. 180

Table 2. U. (continued) v [THz]

c

v [THz]

r

A:

AZ

0.20 0.30 0.40 0.45 Z2

2.50(15) 2.50(20) 2.45(10) 2.34(10) 2.06(10)

v [THz]

r

JG

0.05 0.10 0.20 0.30 0.40

0.82(2) 1.23(5) 1.55(5) 1.72(10) 1.88(10)

v [THz]

r

A:

0.20 0.30 0.40 0.45

1.02(5) 1.44(5) 1.82(5) 2.02(5)

0.10 0.20 0.30 0.40 0.45

3.66(15) 3.53(15) 3.50(15) 3.25(10) 3.16(10)

Whereasthe dispersion curves in A and A directions show no unusual featurespronounced dips were observed in X direction. Thesedips are similar to the onesobservedin high temperature superconductors[75Sml].The lower C, and C, branchessoften considerably with decreasingtemperaturee.g.the C, (LO) mode at c= 0.475softensfrom v=l.l THzat 300K to r=0.6 THz at 160 K and the& (LA) mode at [=0.5 softensfrom v=2.OTHz to v=1.4 THz. An elastic peak appears at <=(2.5, 2, 0) just above 60 K and grows in intensity as the temperature is decreased to 43 K and below. A reasonablefit of the dispersion could be attained with a 27 parameter modified shell model [79Crl].

V

Vanadium

Lattice: bee,a=303 pm=3.03 A. BZ: see p. 448. 1. Phonon dispersion Table 1. V. Measurements. Method

T WI

Fig.

Ref.

X-ray diffraction

296

1V

Collela and Batterman [7OCo3]

The dispersion relations of vanadium cannot be measuredwith neutrons sinceV is an almost totally incoherent neutron scatterer (crcoh/cincoh= 0.008).Neutron studies are therefore limited to the determination of the phonon spectrum. The dispersion curves of V show similar anomalies as the ones of Nb and Ta. E.g. the [OOfl T and [O[a T2 modesshow anomalous dispersion at low 4’values.The [OOfl L mode shows a dip at about [=0.7 which also occurs in Nb and Ta. A-

AH

gr

IOO~ll

V

THza 8-

.0

1=296K

gr 1Hz 8-

H

b

iOD511

l-

6I ir 4Interval:(200)-(2101 o between~300)ondl400~ l Ni- Co filter I 0

I

0.2

I

I

I

0.4

I

0.6

I

I

0.8

I

I

1.0

0

f‘-

160

I

0.2

t

I

I

0.4 5-

!Schober/Dederichs

I

0.6

I

I

0.8

I 1.0

1.2 Phonon states: V

Ref. p. 1SO]

N

lr THz d

[0561T1

65-

Interval: (220)-(310)

R-

FP I

gT THz f

.

r5551t

8 7

6 I 5 ir 4 lnterval:(200)-(211) A I” run A 2”d run l Ni-Co filter

fi tryst. Ill01 A cryst.[llll o tryst. 11111 l Ni - Co filter 0

0.2

OX

0.6

0.8

1.0

t-

r

8 THz 7

6 5 I 4 P 3 2 Fig. 1 a-g. V. Phonon dispersion curves at 296 K measured by X-ray diffraction. The solid line corresponds to the seventh neighbour Born-von Karman fit of Table2 V, model A [7OCo3].

1

0

f-

Schober /Dederichs

161

1.2 Phononenzust%de: V

[Lit. S. 180

Table 2. V. Born-von Karman coupling constants,q, T= 296 K [7OCo2].

A”) m

ij

111

xx XY

200 220 311

222 400 133

xx YY xx zz XY xx YY YZ XY xx XY xx YY xx YY YZ XY

B”) @; [Nm-‘1

10.872(380) 7.244(390) 6.493(840) - 2.148(530) 2.994(280) -4.686 (420) 0.571(410) 1.435(360) 0.286(230) - 1.151(240) 1.216(200) 0.009(210) -0.123 (350) -1.389 (610) 0.341(310) -0.135 (120) -0.360(160) - 0.431(170) 0.086(160)

10.534(097) 6.305(141) 7.129(257) -0.955 (157) 2.730(079) - 4.494(128) 0.952(165) 1.555(090) 0.420(063) -1.527 (089) 0.977(051) -0.296 (062) 0.263(100) - 1.343(136) 0.660(110) - 0.094(039) -0.653 (062) - 0.822(049) 0.397(052)

.

‘)The constantsof modelA werederivedby a directfit to the phonondispersionwhereasthe onesof modelB werecalculatedfromfitted interplanarforceconstants.For caseB the Wing was generallypoorer.The agreementwith the measuredspectrumis howeverbetterfor modelB, seeFig. 2, V.

2. Frequency spectrum and related properties Table 3. V. Measurements of the phonon frequency spectrum. Method

T CKI

neutron diffraction 0-W neutron diffraction F-OF) neutron diffraction U-OF)

300 296

Fig.

Ref. 68Mol

2v

296

67Pal 65Gll

Further references[58Stl, 58Eil,61Tul, 63Pel,63Zel, 65Moz, 65Rul,67Rol, 63Hal]. The agreementbetween the neutron results and the ones obtained from Born-von Karman fits to the X-ray results is fair. The discrepancy between the X-ray curves of Figs. 2(a) V and 2(b) V reflects the experimental uncertainty. The parametersetwhich gives the better agreementto the neutron results for the spectrumgivesa poorer agreementfor the dispersion curves. The high energy tail of the neutron spectra between 8 and 12THz which hasbeenobservedby severalauthors can be ruled out from the X-ray data. The additional peak at low frequencies (2THz) has been observedin several of the neutron experiments. The X-ray results basedon the phonon dispersion in the high symmetry directions do not show such a peak. If it is real it can therefore only stem from anomalies in off symmetry directions. The other group V metals Nb and Ta do not show such a low frequency peak (seee.g. [67Pal]). 162

Schober/Dederichs

1.2 Phonon

Ref. D. 1801

states: V

Table4. V. Phonon spectrum at 296 K as obtained From the Born-von Karman coupling constants of Table 2 V. model A. v [THz]

g(v) [THz-‘1

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80

0.000 0.000 0.001 0.001 0.002 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.010 0.011 0.013 0.014 0.016 0.018 0.020 0.022 0.024 0.027 0.029 0.032 0.034 0.038 0.041 0.044 0.048 0.052 0.056 0.061 0.066 0.072 0.079 0.086 0.094 0.104

T: 1I

[THz]

3.90 too t.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60

g(v) [THz-‘1 0.116 0.131 0.179 0.250 0.340 0.340 0.344 0.354 0.288 0.186 0.169 0.162 0.161 0.162 0.166 0.173 0.184 0.202 0.214 0.220 0.225 0.234 0.248 0.252 0.279 0.331 0.327 0.330 0.335 0.341 0.354 0.242 0.205 0.187 0.177 0.178 0.222 0.000

5 a-b. b units model B 4 I

3-

T G2

4

6

1Hz B 12x

Y-

Fig. 2a, b. V. Phonon spectrum at 296K (dashed curve: incoherent neutron scattering [67Pal]), full curve: spectrum calculated from the Born-von Karman force constants of Table 2 V. a) model A, b) model B [7OCo3].

s I 0.9 $ 0.6 1 N 0.3

0

100

200

300

400

K

500

Fig. 4. V. Debye-Waller exponent 2 W divided by the recoil frequency of the free ion, vR, calculated from the force constants of Table 2 V.

4 Fig. 3. V. Debye cutoff frequencies, Y,, calculated from the force constants of Table 2 V, model A. n-

Schoher /Dederichs

163

[Lit. S. 180

1.2 Phononenzustiinde: W 3. Theoretical models Born-von Karman models: seeTable 2 V.

Short ranged forcesplus a simple electronic contribution: [78Ra3], further references:[71Prl, 75Upl,76Kul, 78Gu3,78Ra2,79Gol]. Model potential: [76011].

W

Tungsten

Lattice: bee,n=316.5pm=3.165,&

BZ: seep. 448.

1. Phonon dispersion Table 1. W. Measurements. Method

T [K] Fig.

Ref.

neutron diffraction (TAS)

296

neutron diffraction (TAS)

298

Chen and Brockhouse [64Chl, 64Ch2] Larose and Brockhouse [76La2]

1W

The measurementsshow a very smooth dispersion with no indication of large anomalies.The two transverse branches in [Ora direction are almost degenerate due to the near elastic isotropy ofW (2c,,=c,

I -cJ.

At [=0.45

the two branchescrossand the T2 branch showsa small dip at the N point. The generalbehaviour of the dispersion Curveas well asthis anomaly at the N point is very similar to the other group VI metals Cr and MO. The predicted [70Ril] Kohn anomaly in the [OOfl L branch around (‘=l (H-point) could not be identified unambiguously in the experiment [76La2]. A-

G-

lb001

.5 -C

0

1.0

6-

0.0

0.6 -t

0.4

0.2

Fig. 1. W. Dispersion curves at 295 K. The broken line through the points are only guides. Note the crossover of the [Ocfl T, and T2 branches near the N point [76La2].

164

Schober/Dederichs

[Lit. S. 180

1.2 Phononenzustiinde: W 3. Theoretical models Born-von Karman models: seeTable 2 V.

Short ranged forcesplus a simple electronic contribution: [78Ra3], further references:[71Prl, 75Upl,76Kul, 78Gu3,78Ra2,79Gol]. Model potential: [76011].

W

Tungsten

Lattice: bee,n=316.5pm=3.165,&

BZ: seep. 448.

1. Phonon dispersion Table 1. W. Measurements. Method

T [K] Fig.

Ref.

neutron diffraction (TAS)

296

neutron diffraction (TAS)

298

Chen and Brockhouse [64Chl, 64Ch2] Larose and Brockhouse [76La2]

1W

The measurementsshow a very smooth dispersion with no indication of large anomalies.The two transverse branches in [Ora direction are almost degenerate due to the near elastic isotropy ofW (2c,,=c,

I -cJ.

At [=0.45

the two branchescrossand the T2 branch showsa small dip at the N point. The generalbehaviour of the dispersion Curveas well asthis anomaly at the N point is very similar to the other group VI metals Cr and MO. The predicted [70Ril] Kohn anomaly in the [OOfl L branch around (‘=l (H-point) could not be identified unambiguously in the experiment [76La2]. A-

G-

lb001

.5 -C

0

1.0

6-

0.0

0.6 -t

0.4

0.2

Fig. 1. W. Dispersion curves at 295 K. The broken line through the points are only guides. Note the crossover of the [Ocfl T, and T2 branches near the N point [76La2].

164

Schober/Dederichs

Ref. p. 1 SO]

.1.2 Phonon states: W

Table 2. W. Measured phonon frequencies at 296 K [64Chl, 64Ch2-j’)

r

v [THz]

CWI L 0.125(3) 0.190(5) 0.26(1) 0.42(1) 0.50 0.60 0.70 0.80 0.90 1.0

2.0 3.0 4.0 5.5 5.95(5) 6.20(2) 6.30(7) 5.95(10) 5.55(10) 5.5(1)

v [THz]

LI

COW’1 T 0.110(5) 0.20 0.270(5) 0.40 0.60 0.80 1.0

2.0 2.30(5) 3.0 4.0 4.35(15) 4.30 5.00 5.50(5) 6.4 (2) 6.6(1) 6.75(lO)b)

0.153(40) 0.20 0.30 0.36 0.40 0.45 0.50

2.0 2.57(2) 3.65(2) 4.10(5) 4.30(5) 4.25(10) 4.15(5)

1.0

v [THz]

I

ClKl L

1.0 1.83(3) 2.50 3.53(3) 4.60(5) 5.2 (1) 5.5(1)

CNllT, “1 0.030(5) 0.080(5) 0.10 0.130(5) 0.189(4) 0.20 0.205(10) 0.26 0.30 0.40 0.45 0.50

v [THz]

I

0.20 0.30

2.36(2) 3.35(3)

0.40

0.45

4.12(3) 4.35(3)

0.50

4.40(5)

0.145(5) 0.155(10) 0.24(1) 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 1.0

4.00 4.20 5.40 5.90(5) 6.05(5) 5.50(5) 4.85(5) 4.78(5) 5.1(1) 5.25(5) 5.45(15) 5.5(1)

0

5.5(1) 5.25(10) 4.40(5)

0.25 0.50

KU1 0

0.25 0.50

0

0.20 0.50

Ci’Kl T

0.064(5) 0.125(5) 0.190(5) 0.270(5) 0.330(5) 0.40 0.50 0.60 0.70 0.80 1.00

1.00 2.00 3.00 4.00 4.60 5.05(5) 5.50(5) 5.90(15) 6.0 (1) 5.95(10) 5.5(1)

0

5.5(1) 4.73(30) 4.15(5)

0.25 0.50

E II

R3

5.5(1) 6.35(2) 6.75(2)

0

0.5 1.0

6.75(10) 5.50(5) 4.40(5)

4.15(5) 4.9 (1) 5.50(5)

“) A complete set of measured data is obtainable from the Depository of Unpublished Data, CISTI, National Research Council of Canada, Ottawa, Ontario, Canada, KIAOSZ, Ref. [76La2]. b, This value may possibly be in serious error becauseof poorly formed neutron groups. “) The polarization vectors for the [Olc] Ti and Tz branches are parallel to [Ott] and [COO],respectively.

Born-von Karman fit

The dispersion can be described reasonably well with a general third neighbour model. To reproduce the dispersion exactly, very long range forces would be needed[64Ch2]. The maximum at about [=0.7 in the [OO[] L branch indicates a large secondneighbour force constant [@z:O”). Table 3. W. Born-von Karman coupling constants,@!, T= 298 K”), [76La2].

m 111 200 220

“) Fit to the data of [64Ch2 and 76La2] including the first sound elastic constants of [63Fel]. Schober /Dederichs

ij

@t [N m- ‘1

xx XY xx YY xx

22.1(2) 18.9(3) 45.7(4) 0.7 (2) 3.7 (1)

zz XY

-1.3 (2)

5.2(2) 165

1.2 PhononenzustZnde: W

[Lit. S. 180

2. Frequency spectrum and related properties 8.3 THZ 0.6

1.9

THZ“ 7.5

I 0.4

Ic

-z T;,

p 7.1 0.2 6.7 0

12

3

1,

5

6.3

6 IHz 7

-10

0

Y-

Fig. 2. W. Frequency spectrum at 298 K calculated from the Born-von Karman parameters of Table 3 W.

10 n-

20

30

Fig. 3. W. Debye cutoff frequencies v, calculated from the spectrum of Fig. 2 W.

Table 4. W. Phonon frequency spectrum at 296 K as calculated from the third neighbour Born-von Karman parameters of Table 3 W. v DHz]

g(r) [THz-‘1

Y [THz]

g(v) [THz-‘1

v [THz]

g(v) [THz-‘1

v [THz]

g(v) [THz-‘1

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05

0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.001 0.001 0.002 0.002 0.002 0.003 0.003 0.004 0.004 0.005 0.005 0.006 0.006 0.007 0.008 0.008 0.009 0.010 0.011 0.012 0.013 0.013 0.014 0.015 0.016 0.018

1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40

0.019 0.020 0.021 0.023 0.024 0.025 0.027 0.028 0.030 0.032 0.034 0.035 0.037 0.039 0.041 0.044 0.046 0.048 0.051 0.053 0.056 0.059 0.062 0.065 0.068 0.071 0.075 0.079 0.082 0.086 0.091 0.095 0.100 0.105

3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00 4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00 5.05 5.10

0.111 0.116 0.122 0.129 0.136 0.143 0.151 0.160 0.169 0.179 0.190 0.202 0.215 0.230 0.247 0.268 0.291 0.321 0.361 0.421 0.455 0.504 0.506 0.509 0.512 0.516 0.517 0.484 0.423 0.393 0.368 0.349 0.333 0.319

5.15 5.20 5.25 5.30 5.35 5.40 5.45 5.50 5.55 5.60 5.65 5.70 5.75 5.80 5.85 5.90 5.95 6.00 6.05 6.10 6.15 6.20 6.25 6.30 6.35 6.40 6.45 6.50 6.55 6.60 6.65 6.70

0.306 0.293 0.282 0.217 0.260 0.250

1.10 1.15 1.20 1.25 1.30 1.35 1.40

1.45 1.50 1.55 1.60 1.65 1.70 166

Schober/Dederichs

0.158 0.137 0.126 0.121 0.122 0.129

0.144 0.168 0.197 0.229 0.268 0.316 0.382 0.498 0.593 0.652 0.425 0.318 0.255 0.211 0.174 0.141 0.109 0.075 0.021 0.0

1.2 Phonon states: Y

Ref. p. 1 SO] 400 K

1.2 THz-’

360

0.9

I a 320 0

$s 0.6 N

280

0.3

240

0

IFig. 4. W. Debye temperatures 0, calculated from an eighth neighbour Born-von Karman model [74CH2]. The experimental points (0) have been corrected for the electronic contribution [59Cll].

0

100

200

300 400 K 500 TFig. 5. W. Debye-Walk exponent 2 W divided by the recoil frequency of the free ion, ve, calculated from the spectrum of Fig. 2 W.

3. Theoretical models No microscopic calculation of the phonon dispersion is available. The phenomenological models are in general inferior to the Born-von Karman fits. Born-von Karman and equivalent models: see Table 3 W and [64Chl, 64Ch2, 76La2] further references: [76Ca2]. Short ranged forces plus a simple electronic contribution: [66Ma2, 76Br1, 76Si2, 77Sa2] further references: [69Shl, 72Bal,72Be3,74Kul, 76Gol,78Gul, 78Khl]. Shell model: [76Ja2]. Model pseudopotential: [73Anl].

Y

Yttrium

Lattice: hcp, a = 365pm = 3.65A, c = 573= 5.73A. BZ: see p. 450. 1. Phonon dispersion

Table 1. Y. Measurements. Method

T [K]

Fig.

Ref.

neutron diffraction VW

295

1Y

Sinha et al. [7OSi2]

f-

-f

c-

Fig. 1. Y. Measured phonon dispersion curves at 295 K. The lines correspond to the sixth neighbour Born-von Karman model of Table 3 Y [7OSi2].

Schoher/Dederichs

167

1.2 Phonon states: Y

Ref. p. 1 SO] 400 K

1.2 THz-’

360

0.9

I a 320 0

$s 0.6 N

280

0.3

240

0

IFig. 4. W. Debye temperatures 0, calculated from an eighth neighbour Born-von Karman model [74CH2]. The experimental points (0) have been corrected for the electronic contribution [59Cll].

0

100

200

300 400 K 500 TFig. 5. W. Debye-Walk exponent 2 W divided by the recoil frequency of the free ion, ve, calculated from the spectrum of Fig. 2 W.

3. Theoretical models No microscopic calculation of the phonon dispersion is available. The phenomenological models are in general inferior to the Born-von Karman fits. Born-von Karman and equivalent models: see Table 3 W and [64Chl, 64Ch2, 76La2] further references: [76Ca2]. Short ranged forces plus a simple electronic contribution: [66Ma2, 76Br1, 76Si2, 77Sa2] further references: [69Shl, 72Bal,72Be3,74Kul, 76Gol,78Gul, 78Khl]. Shell model: [76Ja2]. Model pseudopotential: [73Anl].

Y

Yttrium

Lattice: hcp, a = 365pm = 3.65A, c = 573= 5.73A. BZ: see p. 450. 1. Phonon dispersion

Table 1. Y. Measurements. Method

T [K]

Fig.

Ref.

neutron diffraction VW

295

1Y

Sinha et al. [7OSi2]

f-

-f

c-

Fig. 1. Y. Measured phonon dispersion curves at 295 K. The lines correspond to the sixth neighbour Born-von Karman model of Table 3 Y [7OSi2].

Schoher/Dederichs

167

1.2 Phononenzustkde: Y

[Lit. S. 180

The phonon dispersion curves of yttrium are similar to the onesof scandium.They yield long range interactions in the basal plane but interactions which decreaserapidly in the direction normal to the basal plane. Two Kohn anomalies were found. Table 2. Y. Measured phonon frequenciesat 295 K. [7OSi2]. v fJ-Hz]

c

CWI LA (Al) 0.2 0.225 0.25 0.275 0.3 0.325 0.35 0.375 0.4 0.425 0.45 0.475 0.5

1.37 (2) 1.54 (2) 1.71 (2) 1.90 (2) 2.05 (2) 2.21 (2) 2.35 (2) 2.52 (2) 2.66 (2) 2.82 (3) 2.96 (3) 3.05 (3) 3.20 (4)

COO,7-I-A (Ad 0.25 0.275 0.4 0.425

0.95 (2) 1.05 (2) 1.67 (4) 1.80 (5)

CC001 LA GJ

r

v CTHzl CWI LO (A,)

0.0 0.05 0.1 0.2 0.225 0.25 0.275 0.3 0.325 0.35 0.375 0.4 0.45 0.475 CWI 0.0 0.1 0.2 0.25 0.3 0.35 0.4 0.5

4.64 (5) 4.65 (6) 4.60 (5) 4.40 (5) 4.27 (5) 4.28 (5) 4.24 (4) 4.10 (5) 4.07 (5) 3.88 (4) 3.79 (3) 3.84 (4) 3.66 (2) 3.42 (3)

r

v [THz]

0.3 0.4 0.5

2.68 (3) 2.71 (5) 2.86 (3) 2.94 (2) 3.12(3) 3.23 (5) 3.33 (3) 3.53 (5) 3.66 (4) 3.72 (8) 3.83 (4) 3.78 (5) 3.99 (5) 4.08 (5) 4.10 (5)

0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.3 0.35 0.4 0.45

1.19 (2)

0.0

0.15

1.93(5)

0.1

0.2 0.25 0.3 0.35 0.4

2.48 (6) 3.19 (6) 3.45 (6) 3.80 (5) 3.83 (4)

0.2 0.3 0.4 0.5

0.45 0.5

3.91 (4) 4.02 (5)

cwx 0.3 0.4 0.5

2.03 (2) 2.29 (3) 2.30 (5)

CL-WTO, (c,) 0.3 0.4 0.5

3.53 (4) 3.90 (5) 4.04 (5)

-1-0 (AA 2.68 (3) 2.69 (2) 2.48 (3) 2.47 (3) 2.27 (5) 2.23 (5) 2.20 (2) 1.96 (2)

KU1 TA, (Ts1T;) 0.2 0.302 0.406 0.5

Cl4’01TO, (T,,-U

2.36 (2) 3.19 (2) 3.73 (3) 4.14 (4)

0.0

4.64 (5) 4.65 (6) 4.58 (6) 4.42 (4) 4.28 (4) 4.14 (4)

4.64 (5) 4.53 (5) 4.31 (3) 4.06 (2) 3.69 (2) 3.43 (2) 3.25 (2) 2.91 (2) 2.67 (4)

0.1 0.2 0.25 0.3 0.333 0.36 0.4 0.5

CWI TO, (.&I

0.1

Table 3. Y. Born-von Karman coupling constants,q, derived by a least squaresfit to selectedphonon branches and to the elastic constants, T=295K [7OSi2]. The model is of the “modified axially symmetric” form [75Del] : C$ = K(Rr Ry)/(R”)‘+

F,)

where C,=C,,

2.25(5) 2.45(3) 2.67 (4)

m

(a/f% O,cP) (- W/l

0,44

(0,0, 4 (5a/+,

aP,d4

C, dij

y and C,=C,,

for i=j=z.

F$rn-l]

F$m-l]

23.239 10.124

-1.628 1.456

- 3.641 0.150

- 6.393

1.212

1.511

K+C,,=

(Js a, 0, 0) Schober/Dederichs

for i=j=x,

K [Nm-‘1

(0, a, 0)

168

COWTA,, G)

C~OOI LO PI) 0.0 0.05

v CTHzl

5

1.392 1.856

-0.083, Car= -0.178 0.456 -0.093

-0.582 0.593

1.2 Phonon states: Zn

Ref. p. 1803 Anomalies

Two sharp dips were found in the A, (LO) branch at the positions (0, 0,0.625)2x//c and (0,0,0.775)21c/c.These peaksmight be Kohn anomalies related to maxima of the susceptibility x(q). The anomaly expectedfrom the calculated biggestpeak ofX(q) could not be found. Someother possible anomaliescould not be identified unambiguously. 2. Frequency spectrum and related properties 5

280

orb. units L

K

260 I

-5 -6

3

I

0” 240

2

220

1

0

1

2

3

200 4

THz 5

20

0

40

60

80

100 K 120

T-

Y-

Fig. 2. Y. Frequency spectrum at 295 K calculated from the sixth neighbour Born-von Karman model of Table 3 Y. Some of the critical points arising from high symmetry 3irections are indicated by arrows [7OSi2].

Fig. 3. Y. Debye temperature 0, calculated from the spectrum of Fig. 2 Y compared to experimental data [7OSi2].

3. Theoretical models The available experimental data are not sufficient to test the existing models rigorously. The phenomenological models reproduce the data sufficiently well. Fits using either electronic transition metal model potentials or a model involving noninteracting free electron like s-p-band and tight binding d-bands deviate more than 20% from the measuredvalues. Born-von Karman and equivalent models: seeTable 3 Y, further references:[70Lal, 72Mel]. Short ranged forces plus a simple electronic contribution: [74Ra3, 75SiS], further references:[72Kul, 72Ba3, 73Up2,73Vel, 74Si1,75Up2, 76Ca3]. Transition metal model potential: [75Si6], further references:[76Ku4, 78Up2]. Noninteracting electron band model: [77Si5].

Zn

Zinc

Lattice: hcp, a = 266pm = 2.66A, c = 495 pm = 4.95A. BZ : seep. 450. 1. Phonon dispersion

Table 1. Zn. Measurements. T [K]

Fig.

Ref.

neutron diffraction (TAS)

80

1 Zn

neutron diffraction (TAS) neutron diffraction (TAS)

300

Almquist and Stedman [71All] McDonald et al. [69Mcl] Millington and Squires [69Mil]

Method

300

Further measurements:[54Jol, 63Mal,65Ma2,63Bol, 68Iyl,70Cal, 76Scl). Schoher/Dederichs

169

1.2 Phonon states: Zn

Ref. p. 1803 Anomalies

Two sharp dips were found in the A, (LO) branch at the positions (0, 0,0.625)2x//c and (0,0,0.775)21c/c.These peaksmight be Kohn anomalies related to maxima of the susceptibility x(q). The anomaly expectedfrom the calculated biggestpeak ofX(q) could not be found. Someother possible anomaliescould not be identified unambiguously. 2. Frequency spectrum and related properties 5

280

orb. units L

K

260 I

-5 -6

3

I

0” 240

2

220

1

0

1

2

3

200 4

THz 5

20

0

40

60

80

100 K 120

T-

Y-

Fig. 2. Y. Frequency spectrum at 295 K calculated from the sixth neighbour Born-von Karman model of Table 3 Y. Some of the critical points arising from high symmetry 3irections are indicated by arrows [7OSi2].

Fig. 3. Y. Debye temperature 0, calculated from the spectrum of Fig. 2 Y compared to experimental data [7OSi2].

3. Theoretical models The available experimental data are not sufficient to test the existing models rigorously. The phenomenological models reproduce the data sufficiently well. Fits using either electronic transition metal model potentials or a model involving noninteracting free electron like s-p-band and tight binding d-bands deviate more than 20% from the measuredvalues. Born-von Karman and equivalent models: seeTable 3 Y, further references:[70Lal, 72Mel]. Short ranged forces plus a simple electronic contribution: [74Ra3, 75SiS], further references:[72Kul, 72Ba3, 73Up2,73Vel, 74Si1,75Up2, 76Ca3]. Transition metal model potential: [75Si6], further references:[76Ku4, 78Up2]. Noninteracting electron band model: [77Si5].

Zn

Zinc

Lattice: hcp, a = 266pm = 2.66A, c = 495 pm = 4.95A. BZ : seep. 450. 1. Phonon dispersion

Table 1. Zn. Measurements. T [K]

Fig.

Ref.

neutron diffraction (TAS)

80

1 Zn

neutron diffraction (TAS) neutron diffraction (TAS)

300

Almquist and Stedman [71All] McDonald et al. [69Mcl] Millington and Squires [69Mil]

Method

300

Further measurements:[54Jol, 63Mal,65Ma2,63Bol, 68Iyl,70Cal, 76Scl). Schoher/Dederichs

169

1.2 Phononenzusttinde: Zn

[Lit. S. 180

I1001

60” 0” /-\

I

4

\

60”

\

\ '1

120

\

'\ 180’ I I I I 0.5 0 q4moxS

-q9qmor

q/qmor -

R

S

q/qmor-

P

II

H

K

IJ’

Pj .

Of1

u3 UC U’ . k

. 6

u2

m 0

-b

1’12 u2

4

t-

1

I===

I’n’htl s

M

5-

Fig. 1. Zn. Phonon dispersion at 80 K. The experimental points (0, o) were taken from [71All]. The full and broken lines correspond to two Born-von Karman tits giving opposite relative phases R for the vibrations of the two hexagonal sublattices for the longitudinal [lOO] (T) phonons. The full curve reproduces the experimentally observed phases, Fig. 2 Zn, and corresponds to the parameters given in Table 3 Zn. The experimental points marked l were not used in the tit [74Chl].

60” 30” I -!

0 -30

b Fig. 2. Zn. Relative phase I of the vibration of the two hexagonal sublattices for the longitudinal [lOO] (T) phonons measured by inelastic neutron scattering compared to theoretical values from a pseudopotential calculation [69Gil] and a Born-von Karman fit [65Del] [74Chl].

-60” -90” -120” -150” -180” 0

0.1

0.2

0.3

q/qmor 170

!khober/Dederichs

0.L

0.5

1.2 Phonon states: Zn

Ref. p. 1801

Zinc has like Cd a strongly anisotropic lattice: c/a = 1.86as compared to 1.63for ideal packing. The dispersion curves differ markedly from the ones of Mg. A theoretical interpretation seemsdifficult due to the presenceof non centrally symmetric forces.The sum rule for the frequenciesat the K symmetry point vz (K,)+ vz(Ka) = 2v2(K,) which holds for centrally symmetric forces is not obeyed. Anomalies are relatively easyto observe,their interpretation is, however, not yet unambiguously clear. No discrepanciesbetweenthe initial slopesofthe acoustic branches and the measuredultrasonic sound velocities of [58All] were found. The temperature dependencewasmeasuredfor somephonons up to the melting point. For the longitudinal [loo] T phonons the relative phasesof the vibrations of the two sublattices were determined [74Chl] Fig. 2Zn. Table 2. Zn. Measured phonon frequenciesat 80 K *) [71All]. The branches are labelled in accordancewith Fig. 1 Zn according to the corresponding polarizations and the compatibility relations given by Warren [68Wal]. The units of q are such that the maximum values of q = q,, [at the zone boundaries) are unity. v [THz]

d4mx Branch

Ti , T; lower

T,, T; upper

T,

T3

‘L--G lower

T,,T; upper

0.10

1.26(1)

2.34(l) 2.52(l) 3.43 (2) 4.32(2) 5.03 (1) 5.52 (1) 5.76(2) 5.78 (2) 5.67(3) 5.55(3) 5.69(2) 5.97(2) 6.18(2) 6.31(3)

4.51(3) 4.35 (3) 4.13 (2) 3.84(2) 3.47(2) 3.09(2) 2.59(l) 2.17 (2) 1.89 (1)

0.59 (1) 0.94(l) 1.18 (1) 1.34(l) 1.45 (1) 1.52(l) 1.55 (1) 1.61 (1)

0.71(1) 1.25 (1)

2.50(l)

1.70(l) 2.18 (1) 2.67 (1) 3.14(l) 3.61(1) 4.08 (1) 4.48(2) 4.89(3) 4.39(2) 4.00(2) 3.75(2) 3.63(3)

3.61(2) 4.31 (2) 4.97(2) 5.43 (2) 5.65(2) 5.62(2) 5.29(2)

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

1.00 1.10 1.20 1.30 1.40 1.50

2.66 (1) 2.95 (1) 3.30 (1) 3.68 (1) 4.06 (1) 4.43 (1) 4.78(2) 5.04(3) 4.98(2) 4.62(2) 4.17(2) 3.89(2)

1.74(2) 1.78 (1) 1.86 (1) 1.97 (2)

1.69(1) 1.79(2) l.Sl(2)

2.17(2) 2.49(l) 2.65(2)

3.01(l)

S,,S; middle

S,,S; upper

2.00(2)

3.55(2)

2.49(2)

4.63(2)

2.52(2)

5.60(2)

2.38 (2)

5.77(3) 5.57(3)

2.32(2) 5.28(2) 5.63 (2) 5.88(2) 6.02(3)

2.35(2)

4.78(2) 4.38(2)

2.37 (1)

3.55(3)

6.39(2)

v [THz]

d!llnax Branch

S,,S; lower

CI

Cl

z3

x3

c4

z4

RI+&

RI+&

lower

upper

lower

upper

lower

upper

lower

upper

0.10 0.20 0.30 D.40 0.50 0.60 3.70 0.80 0.90

1.05 (2) 2.02(2) 2.87(2) 3.71 (2) 4.42(2) 5.04(2) 5.52(2) 5.82(3) 6.05(4)

2.41(1)

1.91(2)

3.39(2)

2.41(2)

4.29(2)

2.52(2)

5.33 (2)

2.43 (1)

6.10(2)

1.00

6.11(2)

2.29(2) 2.44(2) 2.88(2) 3.36(2) 3.94(2) 4.58(2) 5.24(3) 5.73(3) 6.11(3) 6.39(3) 6.44(3)

0.00

0.50 (1) 0.85 (1)

1.12(l) 1.32 (1) 1.47 (1) 1.57 (1) 1.68(l) 1.82(l) 1.94(l) 2.02(2)

4.57(2) 4.54(2) 4.41 (2) 4.25(l) 4.02(l) 3.72(l) 3.38 (1) 3.13 (1) 2.93 (1) 2.77(2) 2.70(2)

1.24 (1) 1.75(l) 2.18 (1) 2.55 (1) 2.87 (1)

3.15(1) 3.36 (1) 3.48(2) 3.52(2)

2.60(l) 2.79 (1) 3.00(l) 3.22(l) 3.45(2) 3.60(2) 3.70(2) 3.72(2)

(continued)

*) A compilation of the measured values at 300K is given in [69Mcl]. phonon widths can be obtained from the authors of [71All]. Schober /Dederichs

Additional values for 300K and measured

171

1.2 Phononenzustbde: Zn

[Lit. S. 180

Table 2. Zn. (continued) v [THz]

4knm Branch

A,

A2

0.95(l) 1.22(2)

4.56(2) 4.51 (2) 4.40(2) 4.29(2)

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

1.47(1) 1.72 (1) 1.98(2) 2.23(2) 2.59(2) 2.92(2)

45

A5

0.14(l) 0.95(l)

2.29(2) 2.28(2) 2.26(l) 2.24(l) 2.21(1) 2.16(l)

4.13(2)

1.16(l)

3.92(2) 3.71 (2) 3.46(2) 3.22(2)

1.33 (1) 1.48 (1) 1.62(l) 1.74 (1) 1.86 (1)

U,

U2

P3

PI?p3

2.46(2)

2.13 (2)

2.26(2)

4.99(3)

2.10(1) 2.03 (1) 1*94(l)

Anomalies

The measurementsat 300 K showed a number of easily observableanomalies [63Bol, 65Am2,68Iyl], seee.g. Fig. 3 Zn, which were interpreted as Kohn anomalies.However, detailed measurementsin someregions where anomalies were expectedto appear, revealed that there are few obvious Kohn anomalies, and that a major effort of theoretical analysis would be required to explain why anomalies,predicted by simple considerations oftopography of the Fermi surface,are in some casestoo weak to be observed while in other casesrelatively strong. It is also necessaryto distinguish Kohn anomalies and other irregularities attributable to other featuresof the ion electron interaction. ACM AT MT THZ

Zn

IO01 ILA

+ b'l .*

1100 1 TAl

2.5,:" "i;

0' .# 0 1 ,g:

2.0irI 1.5 -

0

1.0 -

8 00 ooo e

-

0

C' t

0 O"ol' 1'

0" 0 n On

d' I

P

1 Tf""

d1f

f@'

A

I 0.2

I 0.4

I

'$f

A

0.5 -

- 0% I

0 0

0.2

I

I

0.4 0.6 94mox-

I 0.8

l.0 0

0.6 q/q mox-

I

0.8

1.0

Fig. 3. Zn. Anomaliesmeasuredin the dispersioncurvesat 300K. The full arrowsindicatethe positionsexpectedfrom the geometryof the Fermi surface and the dashedarrowscorrespondto the positionsassignedon the basisof the measurements [68Iyl]. Temperature dependence

The temperature dependencewas measuredby neutron scattering in the region of accidental crossing of the C4optic and C, acoustic branches,Fig. 4 Zn [7OCal]. The temperaturedependenceof the lI,+ phonon wasmeasured by neutron scattering [70Cal] and by Raman scattering [76Scl], Fig. 5 Zn. In the latter measurementalso the temperature dependenceof the linewidth was determined, Fig. 6 Zn. The strong temperature decreaseof the frequencycorrespondsto a 40 % variation of the forceconstant betweenneighbouring basalplanesin the temperature interval1 80.. .670K. The analysis of the Raman data shows that anharmonicity contributes to the temperature dependenceof the phonon frequency and width. 172

Schober/Dedericbs

1.2 Phonon states: Zn

Ref. v. 180-I

C-

I 0.20 9/qmau -

I 0.25

I

Fig.4. Zn. Measured dispersion relations at 80K and 620 K for the C4 optic and Ci acoustic branches near the region of their accidental degeneracy [7OCal]. 2.5

‘Hz

Zn

1.7 0

100

200

300 T-

400

500

600

K

700

Fig. 5. Zn. Temperature dependence of the &+ phonon measured by Raman and by neutron scattering. The broken curve was calculated in the quasiharmonic approximation using a Griineisen constant y=3. Additional anharmonic contributions are included in the full curve [76Scl]. 0.8 THz

linewidth in the

Zn

nc

2 phonon

Fig. 6. Zn. Temperature dependence of the linewidth of the l-z phonon. Symbols as in Fig. 5 Zn [76Scl].

Information on the anharmonic terms was also gained by X-ray measurements of forbidden reflexions. 1, .he results are discussed in terms of the anharmonic atomic vibrations using an effective one-particle potential a is obtained [78Mel]. value for the cubic (anharmonic) term of ~1s~= - (1.5+0.3) 10”Nm-2 Schober /Dederichs

173

1.2 Phononenzusttinde: Born-von Kannan

Axially

[Lit. S. 180

Zn

model

symmetric models are not able to describe the phonon dispersion in zinc. Commonly

used is a

“modified axially symmetricmodel” [65Del]. ChesserandAxe [74Chl] have given two such modelswhich describe the phonon frequenciesin symmetry directions equally well and give nearly identical spectra but give opposite relative phases, I., for the vibration of the two hexagonal sublattices for [loo] longitudinal Fig. 1 Zn.

(T) phonon modes,

Table 3. Zn. Interatomic force constants, dsz, obtained by a fit to the phonon frequencies and the relative phase I. of the two hexagonal sublattices for the [loo] longitudinal (T) phonon modes. T=80 K [74Chl].

The model is of “modified axially symmetric” form 0; = K (R; Rr)/(R”)2 + C, 6,, whereC,=C,,for

m

i,j=x,yand

C,=C,,for

i,j=z.

K [Nm-‘1

CBr [Nm-‘1

C [I?m-I]

29.235 10.150

-3.484 -0.594

-3.347 -2.003

3.004 2.264

-0.145

(al4 0, 0) (O,a,0)

(o,qQ 0, 43

(-2a,Q/5,O,c/2)

0.803

-0.104

-0.3531

(0, 0, 4

K +CBr= -0.075, C,, =0.162

0.309

K

1.051

(Sa,f21/5, aj2, c/2)

r0Zn

340

-0.011

0

300t-o 0

-colt.

I76 Er 11

I

2. Phonon spectrum and related properties

; 2601-

40

0

80 I-

120

K

160

Fig. 8. Zn. Debye temperature 9, calculated from the measured spectrum (Fig. 7 Zn) compared with the results of calorimetric measurements [76Erl]. 3

Zn

.I042

s 2

I $ k -Jl 0

1

2

3

4

5

6 IHz 1

Y-

I

I

Fig. 7. Zn. Phonon spectrum measuredby inelastic neutron scattering (circles and heavy curve) compared to the one calculated from the Born-von Karman parameters of Table 3 Zn (light curve). The dashed curve represents the result of the convolution of the calculated spectrum with the spectrometer resolution function [76Erl].

. expt.I67Boll I 76 Er 11 I

-colt.

I

400 300 K 200 IFig. 9. Zn. Debye-Wailer exponent 2Wdivided by the recoil frequency of the free atom, vR,calculated from the measured spectrum (Fig. 7 Zn) compared to X-ray measurements [76Erl]. 0

Schober/Dederichs

100

1.2 Phonon states: Zr

Ref. p. 1801 3. Theoretical models

Due to the strong anisotropy and the presenceof non axially symmetric forces,theoretical models meet with considerable diffkulties. Also the phenomenological models are only moderately successful.For most models not enough information is given to assessthem fully. Born-von Karman and equivalent models: seeTable 3 Zn, further references:[74Chl, 65De1, 66Gu2, 69Mc1, 69Me1, 71Srl,71Trl]. Short ranged forces plus a simple electronic contribution: [73Ku2], further references:[72Ral, 73Bo1, 77Si1, 78Mi2]. Pseudopotential calculations: [69Brl, 69Gil], further references:[70Pr2,72Ba4, 75Mal,76Ku3, 77Si2].

Zr

Zirconium

Lattice: a-Zr, hcp, a=323 pm=3.23& c=5.14pm=5.14A;

P-Zr, bee, a=361 pm=3.61 A. BZ: seep. 448,450.

A. c&Zirconium (hcp)

1. Phonon dispersion Table 1. Zr. Measurements. Method

T CKI

Fig.

Ref.

neutron diffraction (TAS)

(5.5), 295, 773,1007

1 Zr

Stassiset al. [78Stl]

Further measurements:[70Be3,73Mol]. The different measurementsagree well with each other. The dispersion is similar to the one of Ti. The zone centre mode of the [OOl] LO branch softens with decreasingtemperature, an anomaly also found in Ti. T'-

T7 THz

fj-

Zr

it501

A

r

KOO1

(

T=295K hcp

A-

-E M

K

LOO51

I

5

2’

I 4 B 3 2

aI

1

0 0

0.1

0.2

0.3

0.4

0.5 0.4 0.3 0.2 0.1 0

0.5

Fig. 1. Zr. Measured phonon dispersion curves of hcp-zirconium at 295 K. The solid lines were obtained from the sixth neighbour Born-von Karman model of Table 3 Zr [78Stl].

Schober/Dederichs

175

1.2 Phonon states: Zr

Ref. p. 1801 3. Theoretical models

Due to the strong anisotropy and the presenceof non axially symmetric forces,theoretical models meet with considerable diffkulties. Also the phenomenological models are only moderately successful.For most models not enough information is given to assessthem fully. Born-von Karman and equivalent models: seeTable 3 Zn, further references:[74Chl, 65De1, 66Gu2, 69Mc1, 69Me1, 71Srl,71Trl]. Short ranged forces plus a simple electronic contribution: [73Ku2], further references:[72Ral, 73Bo1, 77Si1, 78Mi2]. Pseudopotential calculations: [69Brl, 69Gil], further references:[70Pr2,72Ba4, 75Mal,76Ku3, 77Si2].

Zr

Zirconium

Lattice: a-Zr, hcp, a=323 pm=3.23& c=5.14pm=5.14A;

P-Zr, bee, a=361 pm=3.61 A. BZ: seep. 448,450.

A. c&Zirconium (hcp)

1. Phonon dispersion Table 1. Zr. Measurements. Method

T CKI

Fig.

Ref.

neutron diffraction (TAS)

(5.5), 295, 773,1007

1 Zr

Stassiset al. [78Stl]

Further measurements:[70Be3,73Mol]. The different measurementsagree well with each other. The dispersion is similar to the one of Ti. The zone centre mode of the [OOl] LO branch softens with decreasingtemperature, an anomaly also found in Ti. T'-

T7 THz

fj-

Zr

it501

A

r

KOO1

(

T=295K hcp

A-

-E M

K

LOO51

I

5

2’

I 4 B 3 2

aI

1

0 0

0.1

0.2

0.3

0.4

0.5 0.4 0.3 0.2 0.1 0

0.5

Fig. 1. Zr. Measured phonon dispersion curves of hcp-zirconium at 295 K. The solid lines were obtained from the sixth neighbour Born-von Karman model of Table 3 Zr [78Stl].

Schober/Dederichs

175

1.2 Phononenzustkde: Zr Table 2. Zr. Measured phonon frequencies in hcp-zirconium

T

5.5 K

295 K

1

[78Stl] *),

v [THzl

5

[Lit. S. 180

v F-Hz1 773 K

1007K

T

5.5 K

295 K

773 K

1007K

0.34 (3) 0.77 (2)

0.33 (1) 0.72 (1)

4.34 (15)

4.16 (4) 4.60 (5)

4.87 (8)

1.12 (2) 1.43 (2) 1.68 (4)

1.06 (2) 1.37 (2) 1.62 (4)

0 0.10 0.20 0.30 0.40 0.50

4.75 (7)

4.71 (10)

4.85 (8)

4.68 (10)

4.86 (6) 4.86 (8) 4.79 (8) 4.71 (10) 4.62 (10) 4.46 (8)

2.16 (6) 2.08 (4) 2.00 (3) 1.93 (4) 1.80 (3) 1.62 (4)

0

2.66 (2)

0.10 0.20 0.30 0.40 0.5

CW’l TA (A,) 0.10 0.20 0.25 0.30 0.40 0.50

1.90 (4)

0.45 (1) 0.83 (1) 1.02 (2) 1.19 (1) 1.54 (2) 1.81 (2)

CWI -1-0 (As) 0.00 0.10 0.20 0.30 0.40 0.50

2.66 (2)

1.90 (4)

2.56 (3) 2.56 (3) 2.41 (3) 2.21(2) 2.04 (2) 1.81(2)

2.32 (15) 2.24 (5) 2.15 (4) 2.01 (4) 1.89 (3) 1.68 (4)

4.70 (6)

CWI LA (4) 0.1 0.2 0.3 0.4 0.5

1.01 (3) 1.94 (3) 2.81 (3) 4.20 (5)

4.15(6)

0.96 (3) 1.85 (4) 2.69 (4) 4.01(3)

crw LO @I) 0.95 (3) 1.82 (2) 2.64 (3) 3.48 (4) 4.03 (3)

CW’l LO (AA 0 0.10 0.20 0.30 0.40 0.50

4.23 (15) 4.45(10) 4.50 (10) 4.67 (10) 4.40(8) 4.20 (5)

4.61 (4) 4.58(6) 4.59 (6) 4.61 (8) 4.15 (6)

4.81(8) 4.43 (15) 4.72 (12) 4.07 (3)

4.85 (6) 4.90 (8) 4.72 (8) 4.74 (8) 4.45 (8) 4.03 (3)

C6.001 TA, @s) 0.10 0.15 0.20 0.30 0.40 0.50

0.78 (1) 1.18(l) 1.57 (1) 2.21 (2) 2.62 (2) 2.73 (4)

0.71 (2)

0.67 (1)

1.40 (2) 2.00 (2) 2.41 (3) 2.63 (5)

1.31 (1) 1.88 (1) 2.30 (3) 2.48 (5)

CC001 TA,, 6%) 0.10 0.20 0.30 0.40 0.50

0.83 (1) 1.53 (1) 1.97 (6) 2.21 (3) 2.33 (6)

Ct’OOlLA (CA 0.10 0.20 0.30 0.40 0.50

176

1.68 (3) 3.23 (3) 4.18 (3) 5.14 (4)

1.53 (5) 3.01 (6) 4.04 (10) 4.78 (10)

2.56 (3) 2.98 (2) 3.59 (4) 4.28 (5)

1.53 (2) 2.89 (4) 3.97 (4) 4.63 (6) 4.73 (8)

0

2.55 (3)

2.32 (15)

0.10 0.20 0.30 0.40 0.50

4.17 (6) 5.02 (6) 5.31 (6) 5.36 (5)

3.52 (20)

2.16 (6) 2.76 (10) 3.60 (15)

4.84 (18) 5.03 (15)

4.76 (15) 5.16 (15)

0.05 0.10 0.20 0.30 0.33 0.40 0.50

0.71 (1) 1.40 (1) 2.53 (2) 3.51 (4) 3.88 (8) 4.52 (6) 4.82 (6)

0.05 0.10 0.20 0.30 0.35 0.40 0.45 0.50

0.78 (1) 1.61 (2) 3.18 (2) 3.60 (10) 3.70 (6) 3.38 (6) 2.60 (6) 2.20 (4)

0.05 0.10 0.20 0.33 0.40 0.45 0.50

1.50 (2) 2.63 (4) 3.70 (6) 4.32 (10) 4.55 (6) 4.62 (6) 4.70 (6)

2.66 (2)

KC01TA,, U-J

Schoher/Dederichs

1.17 (2) 2.22 (4) 3.68 (6) 4.19 (10) 4.38 (6)

1.2 Phonon states: Zr

Ref. p. 1 SO]

Table 3. Zr. Born-von Karman coupling constants, @P;,for hcp-zirconium, T=295 K, [78Stl].

Table 2. Zr. (continued)

r T

v [THz] 5.5 K

295 K

773 K

1007K

The model is of the “modified axially symmetric” form [65Del] @; = K (R~R;)/(Rm)2+ C, ?iij

II@1 TO, W 0

0.10 0.20 0.30 0.33 0.40 0.50

4.62(8) 4.62(8) 4.69(8) 4.20(6) 3.92(8) 3.12(10) 2.67(6)

4.68(8) 4.73(8) 3.68(6) 2.77(6) 2.38(6)

Klol TO,, PiI 0

0.10 0.15 0.20 0.30 0.50 0

0.05 0.10 0.15 0.20 0.30 0.45 0.50

2.58(8) 3.50(4) 4.41(4) 4.94(6) 5.00(10) 5.36(5) 2.58(8) 3.05(10) 3.90(5) 4.64(4) 4.94(6) 4.64(8) 4.95(12) 5.14(4)

K [Nm-I]

m

(a/Jr,0,c/2) (0, a, 0) (-2alv3

@a,

5.03(15)

39.74 22.80

0, 43

-4.60

(0,0, 4 (5aI2fi, a/2,42)

C [I&‘]

-3.77 -0.70

-12.88 3.28

-0.39

-1.45

K+CBz=8.40; C,=O.43

0, 0)

1.95

0.28 1.71

1.48

4.75(8)

K

IT

4.78(10)

C [I?m-I]

-0.26

0.10

5.16(15)

4.73(8)

*) The symbols I and 11refer to polarizations perpendicular and parallel to the local plane, respectively.

Temperature dependence

whereC,=C,,fori=j=x,yandC,=C,,fori=j=z

THz 6-

Zr

rmJ 1

M

T

15001

hcp

5 4 I * 3 2 1

n

-0

0.1 0.2 0.3 0.4 0.5 0.4 0.3 0.2 0.1 0 5-c Fig. 2. Zr. Measured phonon dispersion curves of hcpzirconium at 295 K and 1007 K. The lines are guides to the eye [78Stl].

Increasing the temperature the frequenciesof all but the [OOl]LO branch decreaserapidly, Figs. 2 Zr and 3 Zr. The largest frequency shift observedin the temperature range of 5.5K to 1007K was 10 %. These shifts cannot be explained solely by the effect of the thermal expansion of the lattice. The frequency shift due to that effect is estimated as one seventh of the observed shift. With increasing temperature no significant changesin the phonon width were observed. Approaching the hcp-bee transformation temperature (1135K) no dramatic change in either width or frequency of the phonons was observed. Contrary to the general behaviour the zone centre mode of the [OOl] LO branch softens significantly with decreasingtemperature, and at 5.5 K exhibits a dip at the zone centre. This behaviour is similar to Ti. Tc shows a similar but more pronounced anomalous dispersion for that branch at room temperature. Schober /Dederichs

177

1.2 Phononenzustbde: Zr A-

Zr hw

[Lit. S. 180

2. Frequency spectrum and related properties orb.1

root1

IZrI

I

I

I

I

I

3

4

5

6 IHz 7

. l= 295K 0 I= 1007K &I= 6K

0

1

2

Fig. 4. Zr. Phonon frequencc s=m of hcp-Zr at 295 K calculated from the sixth neighbour Born-von Karman model of Table 3 Zr [78Stl].

7.0 cot moleF

Fig. 3. Zr. Measured phonon dispersion curves of hcpzirconium at 6K, 295 K, and 1007 K. The lines are guides to the eye [78Stl].

I

wZr

6.5

6.0

5.5 3

500

700 900 K 11 lFig. 5. Zr. Temperature dependence of the lattice specific heat at constant pressure CLcalculated from the temperature dependence of the phonon frequencies [78Stl]. 8.5 col molek 8.0

I 1.5 u” 7s

6.5 Fig. 6. Zr. Comparison of the calculated specific heat at zonstant pressurecp($, of Fig. 5 Zr plus an electronic contri)ution c’,l with expenmcntal results [78Stl].

178

6.0 2

Schober / Dederichs

400

600

800

1000 K 1200

1.2 Phonon states: Zr

Ref. p. 1 SO]

3. Theoretical models The better of the empirical models describe the dispersion quite well, whereas electronic calculations using Itransition metal) pseudopotentials are not successfull. Born-von Karman models: seeTable 3 Zr, further references:[70Be3]. Short ranged forces plus a simple electronic contribution: [75Si5, 75Si7, 76Vil], further references:[75Ca2, 72Ba3]. Pseudopotential calculations: [77Si2,78Up2]. B.

/?-zirconium(bee) Table 4. Zr. Measurements. Method

T CKI

Fig.

Ref.

neutron diffraction (TAS)

1423

7Zr

Stassiset al. [78St2]

The dispersion curves of bee-zirconium differ strongly from the onesof its neighbouring elementsNb and MO. fn particular the [OOl] L and [OOl] T branches do not cross as in the caseof Nb and no strong anomaly is observedat the H symmetry point. On the other hand /I-Zr shows a pronounced anomaly in the vicinity of the point N which is not present in the dispersion of the other bee metals. This anomaly might be related to the anomaly at the zone centre of the [OOl] LO mode of the hcp phase. The [Oil] Ti and T, branches are degeneratealmost up to the zone boundary. The most unusual feature is the anomalously low frequency of the [2/3, 213,2/3] L mode.An exact determination of this frequency was not possible,due to experimental resolution and the possibility of an admixture of w-phase Zr which would result in Bragg scattering at this C-value. -E

-F

A-

-A P

rrsr1

roof1

I

I

I

I

0.2 OX 0.6 0.8

1.0

I

I

0.8

0.6

Fig. 7. Zr. Measuredphonon dispersioncurvesof bee-zirconiumat 1423K. The solid lines areguidesto the eye[78St2].

Schoher /Dederichs

179

1.3 Literatur

zu 1.1 und 1.2

1.3 Referencesfor 1.1 and 1.2 lOCII IIEUI 16Jal )6Ja2 llGi1 17Cll 19Arl 5OCol 51Skl 52Bul 52Cul 52Gel 53CII S4Crl 54Del 54De2 54Jol 54Kel 55Bhl 55Jal 57Cal 57Scl 55Sml 57Wal 58All 58Brl 58Eil 58Stl 58Tol SST02 58Whl 59Cll 59Eil 59Mal 59Nal 59Ph1 6OCll 6001 60Dal 6OEcl 60Gal 60Jel 6OLal 6OMal 6ORal 60Shl 60Wa1 61Bol 61Brl 61Br2 61Crl 61Fll 61Frl 180

Clusius, K.;Vaughan, J.V.: J. Am. Chem. Sot. 52 (1930) 4686. Eucken, A., Clusius, K., Woitinek, H.: Z. anorg. allg. Chem. 203 (1931) 47. Jaeger,F.M., Rosenbohm, E., Fonteyne, R.: Rec. Trav. Chim. 55 (1936) 615. Jaeger,F.M., Poppema, T.J.: Rec. Trav. Chim. 55 (1936) 492. Giauque, W.F., Meads, P.F.: J. Am. Chem. Sot. 63 (1941) 1897. Clusius, K., Schachinger, L.: Z. Naturforsch. 2a (1947) 90. Armstrong, L.D., Craypson-Smith, H.: Can. J. Res. 27A (1949) 9. Coughlin, J.P., King, E.G. : J. Am. Chem. Sot. 72 (1950) 2262. Skinner, G.B.: PhD thesis, Ohio States University, USA 1951. Burk, D., Daniel, F. : Phys. Rev. 86 (1952) 628. Curien, H.: Acta Crystallogr. 5 (1952) 393. Geballe, T.H., Giauque, W.F. : J. Am. Chem. Sot. 74 (1952) 2368. Clement, J.R., Quinnell, E.H.: Phys. Rev. 92 (1953) 258. Craig, R.G.: J. Am. Chem. Sot. 76 (1954) 238. De Sorbo, W.: Acta Metall. 2 (1954) 274. De Sorbo, W.: J. Phys. Chem. 62 (1954) 965. Joynson, R.E.: Phys. Rev. 94 (1954) 851. Keeson, P.H., Pearlman, N.: Phys. Rev. 96 (1954) 897. Bhatia, A.B.: Phys. Rev. 97 (1955) 363. Jacobsen,E.H. : Phys. Rev. 97 (1955) 654. Carter, R.S., Palevsky, H., Hughes, D.J.: Phys. Rev. 1% (1957) 1168. Scott, J.L.: Report ORNL 2328, Oak Ridge National Lab., USA 1957. Smith, P.L.: Phil. Mag. 46 (1955) 744. Walcott, N.M.: Phil. Mag. 2 (1957) 1246. Alers, G.A., Neighbours, J.R. : J. Phys. Chem. Solids. 7 (1958) 58. Brockhouse, B.N., Stewart, A.T.: Rev. Mod. Phys. 30 (1958) 236. Eisenhauer, CM., Pelah, I., Hughes, D.J., Palevsky, H.: Phys. Rev. 109 (1958) 1046. Stewart, A.T., Brockhouse, B.N.: Rev. Mod. Phys. 30 (1958) 250. Toya, T. : Prog. Theor. Phys. 20 (1958) 974. Toya, T.: J. Res. Inst. Catal. 6 (1958) 161. White, H.C.: Phys. Rev. 112 (1958) 1092. Clusius, K., Franzosini, P.: Z. Naturforsch. 14a (1959) 99. Eichenauer, W., Schulze, N.: Z. Naturforsch. 14a (1959) 28. Martin, D.L.: Physica 25 (1959) 1193. Nash, H.C., Smith, C.S.: J. Phys. Chem. Solids 9 (1959) 113. Phillips, N.E.: Phys. Rev. 114 (1959) 676. Clusius, K., Franzosini, P., Piesbergen,U.: Z. Naturforsch. 15a (1960) 728. Cribier, D., Jacrot, B., Saint-James,D.: J. Phys. Radium 21 (1960) 67. Daniels, W.B.: Phys. Rev. 119 (1960) 1246. ’ Eckstein, Y., Lawson, A.W., Reneker, D.H.: J. Appl. Phys. 31 (1960) 1534. Garland, G.W., Silverman, J.: Phys. Rev. 119 (1960) 1218. Jennings, L.D., Miller, R.E., Spedding, F.H.: J. Chem. Phys. 33 (1960) 1849. Larsson, K.-E., Dahlborg, U., Holmryd, S.: Ark. Fys. 17, (1960) 369. Martin, D.L.: Can. J. Phys. 38 (1960) 17. Rayne, J.A.: Phys. Rev. 118 (1960) 1545. Shapiro, U.G., Shpinel, V.S. : Sov. Phys. JETP (English Transl.) 19 (1960) 1321. Waldorf, D.L.: Bull. Am. Phys. Sot. 5 (1960) 170. Boyle, A.I.F., Bunbuty, D.St., Edwards, G., Hall, H.E.: Proc. Phys. Sot. A77 (1961) 129. Brockhouse, B.N., Arose, R., Cagliotti, G., Sakamoto, M., Sinclair, R.N., Woods, A.D.B.: Inelastic Scattering Neutrons Solids Liquids, IAEA, Vienna 1961, 531. Brockhouse, B.N., Rao, K.R., Woods, A.D.B. : Phys. Rev. Lett. 7 (1961) 93. Cribier, D., Jacrot, B., Saint-James,D.: Inelastic Scattering Neutrons Solids Liquids, IAEA, Vienna 1%1, 549. Flinn, P.A., McManus, G.M., Rayne, J.A.: Phys. Rev. 123 (1961) 809. Frank, J.P., Manchester, F.D., Martin, D.: Proc. Roy. Sot. (London) A263 (1961) 494. Schober/Dederichs

1.3 References for 1.1 and 1.2 611~1 61Tol 61Tul

Iyengar, P.K., Satya Murthy, N.S., Dasannacharya, B.A. : Inelastic Scattering Neutrons Solids Liquids, IAEA, Vienna 1961, 555. Toya, T. : J. Res. Inst. Catal. 9 (1961) 178. Turberfield, K.C., Egelstaff, P.A. : Inelastic Scattering Neutrons Solids Liquids, IAEA, Vienna 1961, 581.

62Brl 62Col 621nl 62Lel 62Lol 62Scl 62Sol 62Srl 62Wol 63Bol 63Chl 63Col 63Fol 63Hal 631~1 63Mal 63Nal 63Pel 63Shl 63Sql 63Zel 64Ahl 64Bil 64Chl 64Ch2 64Frl 64Hal 64Kal 64Krl 64Lel 64Shl 64Sh2 64Sh3 64Sh4 64Sil 64Sql 64Srl 64Wol 64Wo2 64Yal 65Bol 65Del 65Dil

Brockhouse, B.N., Arase, T., Cagliotti, G., Rao, K.R., Woods, A.D.B.: Phys. Rev. 128 (1962) 1099. Collins, M.F.: Proc. Phys. Sot. (London) 80 (1962) 362. International Tables for X-ray Crystallography, Kynoch Press,Birmingham 3 (1962). Lehmann, G.W., Wolfram, T., De Wames, R.E. : Phys. Rev. 128 (1962) 1593. Low, G.G.E.: Proc. Phys. Sot. (London) 79 (1962) 479. Schmunk, R.E., Brugger, R.M., Randolph, P.D., Strong, K.A. : Phys. Rev. 128 (1962) 562. Sosnowski, J., Kozubowski, J.: J. Phys. Chem. Solids. 23 (1962) 1021. Srivastava, P.L. : Phys. Status Solidi. 2 (1962) 713. Woods, A.D.B., Brockhouse, B.N., March, R.H., Stewart, A.T., Bowers, R.: Phys. Rev. 128 (1962) 1112. Borgnovi, G., Cagliotti, G., Antal, J.J.: Phys. Rev. 132 (1963) 683. Chernoplekov, N.A., Zemlyanov, M.G., Chicherin, A.G., Lyashchenko, B.G. : Sov. Phys. JETP (English Transl.) 17 (1963) 584. Cochran, W.: Proc. Roy. Sot. A276 (1963) 308. Fouret, R.: Ann. Phys. (Paris) 8 (1963) 611. Haas, R., Kley, W., Krebs, K.H., Rubin, R.: Inelastic Scattering Neutrons Solids Liquids, IAEA, Vienna Vol. 2, 1963, 145. Iyengar, P.K., Venkataraman, G., Rao, K.R., Vijayaraghavan, P.R., Roy, A.P. : Inelastic Scattering Neutrons Solids Liquids, IAEA, Vienna Vol. 2, 1963, 99. Maliszewski, E., Sosnowski, J., Kozubowski, J., Padlo, I., Sledziewska, D.: Inelastic Scattering Neutrons Solids Liquids, IAEA, Vienna Vol. 2, 1963, 87. Nakagawa, Y., Woods, A.D.B.: Phys. Rev. Lett. 11 (1963) 271. Pelah, I., Kley, W., Krebs, K.H., Peretti, J., Rubin, R.: Inelastic Scattering Neutrons Solids Liquids, IAEA, Vienna Vol. 2, 1963, 155. Sharma, P.K., Joshi, S.K. : J. Chem. Phys. 39 (1963) 2633. Squires, G.L. : Inelastic Scattering Neutrons Solids Liquids, IAEA, Vienna Vol. 2, 1963, 55. Zemlyanov, M.G., Kagan, Y.M., Chernoplekov, N.A., Chicherin, A.G. : Inelastic Scattering Neutrons Solids Liquids, IAEA, Vienna Vol. 2, 1963, 125. Ahlers, G.: J. Chem. Phys. 41 (1964) 86. Birgenau, R.J., Cordes, J., Dolling, G., Woods, A.D.B.: Phys. Rev. 136 (1964) A1359. Chen, S.H., Brockhouse, B.N.: Solid State Commun. 2 (1964) 73. Chen, S.H.: Thesis, MC. Master University, Hamilton, Ontario, Canada 1964. Franzosini, P., Clusius, K.: Z. Naturforsch 19a (1964) 1430. Hautecler, S., Van Dingenen, W.: J. Phys. (Paris) 25 (1964) 653. Kamm, G.N., Alers, G.A. : J. Appl. Phys. 35 (1964) 327. Krebs, K.: Phys. Lett. 10 (1964) 12. Leupold, H.A., Boorse, H.A. : Phys. Rev. 134 (1964) A 1322. Sharma, P.K., Joshi, S.K., J. Chem. Phys. 40 (1964) 662. Shukla, M.M. : Phys. Status Solidi 7 (1964) K 11. Sham, L.J.: Proc. Roy. Sot. (London) A283 (1964) 33. Shimizu, M., Katsuki, A. : J. Phys. Sot. Jpn. 19 (1964) 1856. Singh, N., Joshi, S.K.: Physica 30 (1964) 2105. Squires, G.L.: Ark. Fys. 26 (1964) 223. Srivastava, R.S., Dayal, B.: Progr. Theor. Phys. 31 (1964) 167. Woods, A.D.B., Chen, S.H.: Solid State Commun. 2 (1964) 233. Woods, A.D.B.: Phys. Rev. 136 (1964) A781. Yamell, J.L., Warren, J.L., Wenzel, R.G., Koenig, S.H. : IBM Res. Dev. 8 (1964) 234. Borgnovi, G., Cagliotti, G., Antonini, M. : Neutron Inelastic Scattering IAEA; Vienna Vol. 1, 1965, 117. DeWames, R.E., Wolfram, T., Lehman, G.W.: Phys. Rev. 138 (1965) A717. Dixon, M., Hoare, F.E., Holder, T.M., Moody, D.E. : Proc. Roy. Sot. (London) A285 (1965) 561. Schoher / Dederichs

181

1.3 Literatur zu 1.1 und 1.2 iSFil iSGil jSGi2 i5Gll iSHa iSHo my1 iSKr1 iSLeI 55Mal 55Ma2 j5Mol ;5Mo2 55Pal 55Prl 55Rol 55Sal 55Scl 55ShI 55Sh2 55Sil 55Srl 55Sr2 55st1 55st2 55Tol 65Vil 55Vi2 55Vol 55YaI 56Anl 56Brl 56Bul 56Chl 66Col 66Del 66Gul 66Gu2 66Lel 66MaI 66Ma2 66Pal 66SaI 66Scl 66Sc2 66Sc3 66Shl 66Sil 66Si2 66SqI 66Stl 182

Filby, J.D., Martin, J.L.: Proc. Roy. Sot. (London) A284 (1965) 83. Gilat, G.: Solid State Commun. 3 (1965) 101. Gilat, G., Nicklow, R.M. : Phys. Rev. 143 (1965) 487. Gkiser, W., Carvalho, F., Ehrct, G.: Neutron Inelastic Scattering IAEA. Vienna Vol. 1, 1965, 99. Hautecler, S., Van Dingenen, W.: Symp. Inelastic Scattering Neutrons by Condensed Systems Brookhaven National Laboratory, USA 1965, 83. Hohenemser, C.: Phys. Rev. 139 (1965) A185. Iyengar. P.K., Venkataraman, G., Vijayaraghavan, P.R., Roy, A.P. in: Lattice Dynamics (R.F. Wallis ed.), Pergamon Press,Oxford 1965. Krebs, K.: Phys. Rev. 138 (1965) AI43. Lehmann, G.W., Wolfram, T., Dewames, R.E.: Lattice Dynamics (R.F. Wallis ed.) Pergamon Press,Oxford 1%5, 101. Martin, D.L.: Phys. Rev. 139 (1965) A 150. Maliszewski, E., Rosolowski, J., Sledziewska, D., Czachor, A.: Lattice Dynamics (R.F. Wallis ed.) Pergamon Press,Oxford 1965, 33. Moller, H.B., Mackintosh, A.R.: Neutron Inelastic Scattering, IAEA, Vienna Vol. 1, 1%5. Mozer, B., Otnes, K., Palevsky, H.: Lattice Dynamics (R.F. Wallis ed.) Pergamon Press, Oxford 1965, 63. Pawel, R.W., Stausburg, E.: J. Phys. Chem. Solids 26 (1965) 607. Prakash, S., Joshi, S.K. : Phys. Rev. 140 (1965) A 1754. Rowe, J.M., Brockhouse, B.N., Svensson,EC.: Phys. Rev. Lett. 14 (1965) 554. Salter, L.S.: Adv. Phys. 14 (1965) 1. Schmunck, R.E., Gavin, W.R. : Phys. Rev. Lett. 14 (1965) 44. Sharma, K.C., Joshi, S.K.: Indian J. Pure Appl. Phys. 3 (1965) 329. Sham, L.J.: Proc. Roy. Sot. (London) A283 (1965) 33. Sinha, S.K., Squires, G.L.: Lattice Dynamics (R.F. Wallis ed.) Pergamon Press, Oxford 1965, 53. Srivastava, P.L., Dayal, B.: Phys. Rev. 140 (1965) AlOl4. Srivastava, P.L.: Neutron Inelastic Scattering IAEA, Vienna Vol. 1, 1%5, 49. Stedman, R., Nilsson, G. : Phys. Rev. Lett. 15 (1965) 634. Stedman, R., Nilsson, G.: Inelastic Scattering of Neutrons IAEA, Vienna Vol. 1, 1%5, 211. Toya, T.: Lattice Dynamics (R.F. Wallis ed.) Pergamon Press,Oxford 1965, 91. Vintaikin. E.Z., Gorbachev, V.V., Gruzin, P.L. : Sov. Phys. Solid State (English Transl.)7 (1965)296. Vintaikin, E.Z., Gorbachev, V.V.: Sov. Phys. Solid State (English Transl.) 7 (1965)1547. Vosko, S.H., Taylor, R., Keech, C.H. : Can. J. Phys. 43 (1965) 1187. Yamell, J.L., Warren, J.L., Koenig, S.H.: Lattice Dynamics (R.F. Wallis ed.) Pergamon Press, Oxford 1%5, 57. Animalu, A.O.E., Bonsignori, F., Bortolani, V.: Nuovo Cimento 44B (1966) 159. Brovman. E.G., Kagan, Yu.: Sov. Phys. Solid State (English Transl.) 8 (1966) 1120. Biihrer, W., Schneider, T., Gllser, W.: Solid State Commun. 4 (1966) 443. Chang, Y.A., Himmel, L.: J. Appl. Phys. 37 (1966) 3567. Cowley, R.A., Woods, A.D.B., Dolling, G.: Phys. Rev. 150 (1966) 487. De Wette, F.W., Cotter& R.M.J., Doyama, M.: Phys. Lett. 23 (1966) 309. Gupta, R.P., Kishore, B.: Phys. Status Solidi 13 (1966) 261. Gupta, R.P., Dayal, B.: Phys:Status Solidi 13 (1966) 519. Lechner, R., Quittner, G.: Phys. Rev. Lett. 17 (1966) 1259. Martin, D.L.: Phys. Rev. 141 (1966) 576. Mahesh, P.S., Dayal, B.: Phys. Rev. 143 (1966) 443. Pal, S., Gupta, R.P. : Solid State Commun. 4 (1966) 83. Sahni, V.C., Venkataraman, G., Roy, R.P.: Phys. Lett. 11 (1966) 633 Schneider, T., Stoll, E.: Solid State Commun. 4 (1966) 79. Schneider, T., Stoll, E.: Phys. Kondens. Mater. 5 (1966) 331. Schmunk, R.E.: Phys. Rev. 149 (1966) 450. Shukla, M.M., Dayal, B.: Phys. Status Solidi 16 (1966) 513. Sinha, S.K.: Phys. Rev. 143 (1966) 422. Singh, N., Joshi, S.K.: Physica 32 (1966) 127. Squires, G.L.: Proc. Phys. Sot. (London) 88 (1966) 919. Stedman, R., Nilsson, G.: Phys. Rev. 145 (1966) 492. Schober/Dederichs

1.3 Referencesfor 1.1 and 1.2 67Anl 67Bal 67Bel 67Bj 1 67Brl 67Br2 67Chl 67Fll 67Gul 67Hal 67Hol 67Hul 67Mil 67Nil 67Pal 67Prl 67Rol 67Ro2 67Ro3 67Ro4 67Shl 67Sil 67Sml 67Sm2 67Stl 67St2 67St3 67Svl 67Val 67Yul 68Brl 68Czl 68Del 68Hol 681~1 68Jol 68Kol 68Mal 68Mil 68Mol 68Nil 680hl 68Pol 68Pyl 68Scl 68Sc2 68Sml 68Sol 68Val 68Va2 68Wal

Animalu, A.O.E.: Phys. Rev. 161 (1967) 445. Barron, T.H.K., Munn, R.W.: Acta Crystallogr. 22 (1967) 170. Bergsma, J., van Dijk, C., Tocchetti, T. : Phys. Lett. 24A (1967) 270. BjBrkman, G., Lundquist, B.I., Sjolander, A.: Phys. Rev. 159 (1967) 551. Brockhouse, B.N., Abon-Helal, H.E., Hallman, E.D. : Solid State Commun. 5 (1967) 211. Brovman, E.G., Kagan, Yu. : Sov. Phys. JETP (English Transl.) 25 (1967) 365. Chen, S.H.: Phys. Rev. 163 (1967) 532. Flotov, H.E., Osborne, D.W.: Phys. Rev. 160 (1967) 467. Gupta, R.P., Sharma, P.K. : J. Chem. Phys. 46 (1967) 1359. Hautecler, S., van Dingenen, W.: Physica 34 (1967) 257. Ho, P.S., Ruoff, A.L., Phys. Status Solidi 23 (1967) 489. Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., Wagman, D.D.: Selected Values of the Themodynamic Properties of the Elements, ASM 1967. Minkiewicz, V.J., Shirane, G., Nathans, R. : Phys. Rev. 162 (1967) 528. Nicklow, R.M., Gilat, G., Smith, H.G., Raubenheimer, L.J., Wilkinson, M.K. : Phys. Rev. 164 (1967) 922. Page, D.I.: Proc. Phys. Sot. (London) 91 (1967) 76. Price, D.L.: Proc. Roy. Sot. A300 (1967) 25. Roy, A.P. : Physica 34 (1967) 384. Roy, A.P., Venkataraman, G.: Phys. Rev. 156 (1967) 769. Rosen, M.: Phys. Rev. Lett. 19 (1967) 695. Rowe, J.M.: Phys. Rev. 163 (1967) 547. Shukla, M.M., Dayal, B. : Phys. Status Solidi 19 (1967) 729. Singh, S.P.: Can. J. Phys. 45 (1967) 1655 Smith, D.L. : Los Alamos Sci. Rep. La-3773, Los Alamos, New Mexico, USA 1967. Smithells, C.J.: Metals Reference Book, Butterworth, London 1 (1967). Stedman, R., Almquist, L., Nilsson, G., Raunio, G. : Phys. Rev. 162 (1967) 545. Stedman, R., Almquist, L., Nilsson, G., Raunio, G.: Phys. Rev. 163 (1967) 567. Stedman, R., Almquist, L., Nilsson, G.: Phys. Rev. 162 (1967) 549. Svensson, E.C., Brockhouse, B.N., Rowe, J.M. : Phys. Rev. 155 (1967) 619. Van Dingenen, W., Hautecler, S. : Physica 37 (1967) 603. Yuen, P.S., Varshni, Y.P.: Phys. Rev. 164 (1967) 895. Brovman, E.G.: Transl. Los Alamos Sci. Lab. Rep. La-TR-68-33, Los Alamos, New Mexico, USA 1968. Czachor, A. : Phys. Status Solidi 29 (1968) 423. De Wit, G.A., Brockhouse, B.N.: J. Appl. Phys. 39 (1968) 451. Ho, P.S.: Phys. Rev. 169 (1968) 523. Iyengar, P.K., Venkataraman, G., Gameel, Y.H., Rao, K.R. : Neutron Inelastric Seattering, IAEA, Vienna 1968. Joshi, SK., Rajagopal, A.K.: Solid State Phys. (H. Ehrenreich, F. Seitz and D. Turnbull eds.) Academic Press, New York 22 (1968) 160. Kotov, B.A., Okuneva, N.M., Plachenova, E.L. : Sov. Phys. Solid State(English Trans1.)10(1968) 402. Martin, D.L.: Phys. Rev. 170 (1968) 650. Miiller, A.P., Brockhouse, B.N.: Phys. Rev. Lett 20 (1968) 798. Mozer, B. : Neutron Inelastic Scattering, IAEA, Vienna Vol. 1, 1968, 55. Nilsson, G. : Neutron Inelastic Scattering, IAEA, Vienna Vol. 1, 1968, 187. Ohrlich, R., Drexel, W.: Neutron Inelastic Scattering, IAEA, Vienna Vol. 1, 1968, 203. Powell, B.M., Martel, P., Woods, A.D.B.: Phys. Rev. 171 (1968) 727. Pynn, R., Squires, G.L. : Neutron Inelastic Scattering, IAEA, Vienna Vol. 1, 1968, 215. Schmuck, P., Quittner, G.: Phys. Lett. 28A (1968) 226. Schneider, T., Stoll, E.: Neutron Inelastic Scattering, IAEA, Vienna Vol. 1, 1968, 101. Smith, H.G., Dolling, G., Nicklow, R.M., Ijayaraghavan, P.R.V., Wilkinson, M.K. : Neutron Inelastic Scattering, IAEA, Vienna Vol. 1, 1968, 149. Sosnowski, J., Bednarski, S., Czachor, A.: Neutron Inelastic Scattering, IAEA, Vienna Vol. 1, 1968, 157. Varshni, Y.P., Yuen, P.S.: Phys. Rev. 174 (1968) 766. Van Dijk, C., Bergsma, J. : Neutron Inelastic Scattering, IAEA, Vienna Vol. 1, 1968, 233. Warren, J.L.: Rev. Mod. Phys. 40 (1968) 38. Schober / Dederichs

183

1.3 Literatur zu 1.1 und 1.2 68Zel 69Bel 69Bol 69Bo2 69Brl 69Bul 69Cel 69Drl 69Gil 69Hol 69Jol 69Kal 69Kil 69Krl 69Lel 69Mcl 69Mel 69Mil 69Pil 69Prl 69Pr2 69Rel 69Sal 69Shl 69Sh2 69Sml 69l\\‘a I 69Wa2 69\\‘a3 69Wa4 69WaS 70Bel 70Bc2 70Be3 70Brl 70Bul 70Cal 7OCol 7oco2 7OCo3 70FIl 70Ful 70Gel 70Gul 70Gu2 70Hol 70Kil 70Kol 70Ko2 70Kul 70Lal 70hIal 70Pa I 70Pa2 70Prl 184

Zemliyanov, M.G., Mironov, S.P., Syrkh, G.F.: Neutron Inelastic Scattering. IAEA, Vienna Vol. I, 1968. 79. Behari. J., Tripathi, B.B.: Phys. Lett. 29A (1969) 313. Bortolani, V., Pizzichini, G.: Phys. Rev. Lett. 22 (1969) 840. Boffey, T.B.: Phys. Lett. 29A (1969) 167. Brovman. E.G., Kagan. Yu., Kholas, A.: Sov. Phys. Solid State (English Transl.) 11 (1969) 733. Buyers, W.J.L., Cowley, R.A. : Phys. Rev. 180 (I 969) 755. Cetas. T.C., Holste. J.C., Swenson, C.A.: Phys. Rev. 182 (1969) 679. Drexel, W., GlBser, W., Gompf, F. : Phys. Lett. 28A (1969) 531. Gilat. G., Rizzi, G., Cubiotti, G.: Phys. Rev. 185 (1969) 971. Hiigbcrg. T., Sandstr8m, R.: Phys. Status Solidi 33 (1969) 169. Johnson, R., Westin, A. : Report AE-365, Aktiebolagct Atomenergi, Stockholm 1%9. Kamitakahara, W.A., Brockhouse, B.N.: Phys. Lett. 29A (1969) 639. King III. W.F., Cutler, P.H.: Solid State Commun. 7 (1969) 295. Krebs, K., Hiilzl. K.: Report EUR-3621 e, Euratom, Ispra, Italy 1969. Leake. J.A., Minkiewicz, V.J., Shirane, G.: Solid State Commun. 7 (1969) 535. McDonald. D.L., Elcombe, M.M., Pryor, A.W.: J. Phys. C2 (1969) 1857. Metzbower. E.A.: Phys. Rev. 177 (1969) 1139. Millington, A.J., Squires, G.L.: J. Phys. C2 (1969) 2366. Pindor, A.J., Pynn, R.: J. Phys. C2 (1969) 1037. Prakash. S., Joshi, S.K. : Phys. Rev. 187 (I 969) 808. Prakash, S., Joshi, S.K.: Phys. Lett. 30A (1969) 123. Reichardt. W., Nicklow, R.M., Dolling. G., Smith, H.G.: Bull. Am. Phys. Sot. 14 (1969) 378, (unpublished). Sahni. V.C., Venkataraman, G.: Phys. Rev. 185 (1969) 1002. Sharma. P.K.. Pal, S., Gnpta. R.P.: Rev. Roum. Phys. 14 (1969) 247. Sharp, R.1.: J. Phys. C2 (1969) 421. Smith. H.G., Reichardt, W.: Bull. Am. Phys. Sot. 14 (1969) 378 (unpublished). Wallace, D.C.: Phys. Rev. 187 (1969) 991. Wallace. D.C.: Phys. Rev. 178 (1969) 900. Walker, C.B., Egelstaff, P.A.: Phys. Rev. 177 (1969) 1111. Waeber, W.B.: J. Phys. C2 (1969) 882. Waeber, W.B.: J. Phys. C2 (1969) 903. Behari. J., Tripathi, B.B.: J. Phys. C3 (1970) 659. Behari, J., Tripathi, B.B.: J. Phys. Sot. Jpn. 28 (1970) 346. Bezdek, H.F., Schmunk, R.E., Finegold, L.: Phys. Status Solidi 42 (1970) 275, Brown. J.S.: Phys. Status Solidi 39 (1970) K75. Biihrer, W.: EIR-Bericht 174, Eidg. Inst. f. Reaktorforschung, Wiirenlingen, Switzerland. Cagliotti. G., Rizzi. G., Cubiotti, G.: Solid State Commun. 8 (1970) 367. Coulthard. M.A.: J. Phys. C3 (1970) 820. Cohen, M., Heine, V., Weaire, D.: Solid State Phys. (H. Ehrenreich, F. Seitz and D. Turnbull eds.) Academic Press, New York 24 (1970). Collela. R., Batterman. B.W.: Phys. Rev. Bl (1970) 3913. Floyd. E.R., Kleinman. L.: Phys. Rev. B2 (1970) 3947. Furrer, A., HHIg. W.: Phys. Status Solidi 42 (1970) 821. Geldart, D.J.W., Taylor, R., Varshni, Y.P.: Can. J. Phys. 48 (1970) 183. Gupta. H.C., Tripathi. B.B.: Phys. Rev. B2 (1970) 248. Gupta. R.P.: Solid State Commun. 8 (1970) 991. Houman. J.C.G., Nicklow, R.M.: Phys. Rev. Bl (1970) 3943. King III, W.F., Cutler, P.H.: Phys. Rev. B2 (1970) 1733. Koehler, T.R., Gillis, N.S., Wallace, D.C.: Phys. Rev. Bl (1970) 4521. Kotov, B.A.,Okuneva.N.M.,Plachenova, E.L. : Sov. Phys. Solid State (English Transl.) ll(l970) I61 5. Kushwaha. S.S., Rajput, J.S.: Phys. Rev. B2 (1970) 3943. Lahteenkova. E.E. : Solid State Commun. 8 (1970) 947. Martinelli. L.: Nuovo Cimento 70B (1970) 58. Pal. S.: Phys. Rev. B2 (1970) 4741. Palmer, S.B.: J. Phys. Chem. Solids 31 (1970) 143. Price, D.L., Singwi, K.S., Tosi, M.P.: Phys. Rev. B2 (1970) 2983. Schober / Dederichs

1.3 Referencesfor 1.1 and 1.2 70Pr2 70Ril 70Rol 70Shl 70Sh2 70Sh3 70Sil 7OSi2 70Wal 70Wel 71All 71Bal 71Ba2 71Bel 71Brl 71Drl 71Erl 71br1 71Gil 71Gul 71Gu2 71Gu3 71Gu4 71Hal 71Ha2 71Kil 71Kol 71Kul 71Lel 71Mal 71Mil 71Mi2 71Nil 71Ni2 71Pal 71Prl 71Scl 71Shl 71Sh2 71Sh3 71Sh4 71Sil 71Sol 71Srl 71Swl 71Thl 71Trl 71Tr2 71Wal 71Wel 71Wol 72Aml 72Bal 72Ba2 72Ba3

Prakash, S., Joshi, SK. : Phys. Rev. Bl (1970) 1468. Rice, T.M., Halperin, B.I.: Phys. Rev. Bl (1970) 509. Roy, A.P., Brockhouse, B.N. : Can. J. Phys. 48 (1970) 1781. Sharan, B., Bajpai, R.P.: J. Phys. Sot. Jpn. 28 (1970) 334. Sharan, B., Bajpai, R.P.: J. Phys. Sot. Jpn. 28 (1970) 338. Sharan, B., Bajpai, R.P.: Phys. Lett. 31A (1970) 120. Singh, S.N., Prakash, S.: Physica 50 (1970) 10. Sinha, SK., Brun, T.O., Muhlestein, L.D., Sakurai, J. : Phys. Rev. B 1 (1970) 2430. Wallace, D.C.: Phys. Rev. Bl (1970) 3963. Weymouth, J.W., Stedman, R.: Phys. Rev. B2 (1970) 4743. Ahnquist, L., Stedman, R.: J. Phys. Fl (1971) 785. Bajpai, R.P., Neelakandan, K. : Physica 53 (1971) 628. Bajpai, R.P., Neelakandan, K.: Solid State Comm. 9 (1971) 167. Bezdek, H.F., Finegold, L. : Phys. Rev. 4 (1971) 1390. Brown, J.S.: J. Phys. Fl (1971) 650. Drexel, W.: Rep. KFK 1383, Kernforschungsanlage Karlsruhe, Germany 1971. Ernst, G.: Acta Phys. Austriaca 33 (1971) 27. Frikkee, H.: Phys. Lett. 34A (1971) 23. Gillis, N.S., Koehler, T.R.: Phys. Rev. B3 (1971) 3568. Gupta, H.C., Tripathi, B.B. : Phys. Status Solidi (b) 45 (1971) 537. Gupta, H.C., Tripathi, B.B. : Phys. Status Solidi (b) 45 (1971) 235. Gurskii, Z.A., Krasko, G.L.: Sov. Phys. Dok. (English Transl.) 16 (1971) 298. Gupta, H.C., Tripathi, B.B.: J. Phys. Fl (1971) 12. Hanke, W., Bilz, H.: Z. Naturforsch. 26a (1971) 585. Hartmann, W.M., Milbrodt, T.O.: Phys. Rev. B3 (1971) 4133. King III, W.F., Cutler, P.H.: Phys. Rev. B3 (1971) 2485. Konti, A.: J. Chem. Phys. 55 (1971) 3997. Kushwaha, S.S.: Phys. Lett. 37A (1971) 193. Leigh, RX, Szigetti, B., Tewary, V.K.: Proc. Roy. Sot. (London) A320 (1971) 505. Macfarlane, R.E.: The Physics of Semimetals and Narrow Gap Semiconductors (D.L. Carter and R.T. Bate eds.), Pergamon Press,New York 1971, 289. Miiller, A.P., Brockhouse, B.N.: Can. J. Phys. 49 (1971) 704. Millington, A.J., Sqires, G.L.: J. Phys. Fl (1971) 244. Nikulin, V.K., Tsarev, Yu.N.: Phys. Lett. 36A (1971) 337. Nicklow, R.M., Wakabayashi, N., Vijayaraghavan, P.R.: Phys. Rev. B3 (1971) 1229. Pal, S.: J. Phys. Fl (1971) 588. Prakash, J., Semwal, B.S., Sharma, P.K.: Acta Phys. Hung. 30 (1971) 231. Schmuck, P.: Z. Phys. 248 (1971) 111. Sharma, P.K., Singh, N.: Phys. Rev. B4 (1971) 4636. Shukla, M.M., Cavalheiro, R.: Phonons (M.A. Nusimovici ed.), Flammarion, Paris 1971, 313. Shaw, W.M., Muhlestein, L.D.: Phys. Rev. B4 (1971) 969. Sharp, RI., Warming, E.: J. Phys. Fl (1971) 570. Singh, N., Sharma, P.K.: Phys. Rev. B3 (1971) 1141. Sosnowski, J., Maliszewski, E.F., Bednarski, S., Czachor, A.: Phys. Status Solidi (b) 44 (1971) K65. Srinivasan, R., Rao, R.R.: J. Phys. Chem. Solids 32 (1971) 1769. Swaroop, A., Tiwari, L.M.: Phys. Status Solidi (b) 43 (1971) K153. Thaper, C.L., Rao, K.R., Dasannacharya, B.A., Roy, A.P., Iyengar, P.K. : Phonons (M.A. Nusimovici ed.), Flammarion, Paris 1971, 140. Tripathi, B.B., Behari, J.: J. Phys. Fl (1971) 19. Trott, A.J., Heald, P.T.: Phys. Status Solidi (b) 46 (1971) 361. Wakabayashi, N., Sinha, S.K., Spedding, F,H. : Phys. Rev. B4 (1971) 2398. Werner, S.A., Pynn, R.: J. Appl. Phys. 42 (1971) 4736. Worlton, T.G., Schmunk, R.E.: Phys. Rev. B3 (1971) 4115. Am. Inst. Phys. Handbook, McGraw Hill, New York 1972. Bajpai, R.P., Neelakandan, K.: Nuovo Cimento 7B (1972) 177. Bajpai, R.P., Ono, M.: Phys. Lett. 41A (1972) 141. Bajpai, R.P. : Physica 62 (1972) 574. Schoher/Dederichs

185

1.3 Literatur zu 1.1 und 1.2 72Ba4 72Bel

72Be2 72Be3 72BrI 72Bul 72Drl 72DuI 72Gll 72Gol

Bajpai. R.P.: Phys. Lett. 42A (1972) 163. Behari, J.: Phil. Mag. 26 (1972) 737. Bertoni. C.M., Bortolani, V., Giunchi, G.: J. Phys. F2 (1972) 833. Behari, J., Tripathi, B.B.: J. Phys. Sot. Jpn. 33 (1972) 1207. Brovman. E.G., Kagan. Yu., Kholas. A. : Sov. Phys. JETP (English Transl.) 34 (1972) 394. Buyers, W.J.L., Powell, B.M., Woods, A.D.B.: Can. J. Phys. 50 (1972) 3069. Drexel, W. : Z. Phys. 255 (I 972) 281. Dutton, D.H., Brockhouse, B.N., Miiller, A.P.: Can. J. Phys. 50 (1972) 2915. Clyde, H.R., Taylor, R. : Phys. Rev. B 5 (1972) 1206. Gompf, F., Lau, H., Reichardt, W., Salgado, J.: Neutron Inelastic Scattering, IAEA, Vienna 1972, 137.

72Hul 72Kal 72Kol 72Ko2 72Kul 72Ku2

72Mel 72Me2 72Mol 72Mo2 72Mul

Hubbel, W.C., Brotzen, F.R.: J. Appl. Phys. 43 (1972) 3306. Kachava, C.M.: Phys. Status Solidi (b) 54 (1972) K29. Kothari, LX, Singhal, U.: J. Phys. C5 (1972) 293. Korsunskii, M.I., Genkin, Ya.E., Markovnin, V.I., Zavodinskii, V.G., Satsuk, V.V.: Sov. Phys. Solid State (English Transl.) 13 (1972) 1793. Kushwaha. S., Kumar, A.: Nuovo Cimento 11B (1972) 301. Kumar. A., Sharan, B.: J. Phys. C5 (1972) 3161. Menon, C.S., Rao, R.R. : Phys. Status Solidi (b) 49 (1972) 843. Menon, C.S., Rao, R.R.: Solid State Commun. 10 (1972) 701. Moriarty, J.A.: Phys. Rev. B6 (1972) 1239. Moriarty, J.A.: Phys. Rev. B6 (1972) 4445. Muhlestein, L.D., Giirmen, E., Cunningham, R.M.: Neutron Inelastic Scattering, IAEA, Vienna 1972, 53.

72Pol 72Prl 72Pr2 72Pr3 72Pr3 72Ppl

72Ral 72Shl 72Sil 72Si2 72Sol

Powell, B.M., Woods, A.D.B., Martel, P.: Neutron Inelastic Scattering. IAEA, Vienna 1972, 43. Prasad. B., Srivastava, R.S.: J. Phys. F2 (1972) 247. Prakash, S., Joshi, S.K. : Phys. Rev. B5 (1972) 2880. Prasad. B., Srivastava, R.S.: Phys. Lett. 38A (1972) 527. Prasad. B., Srivastava, R.S.: Phys. Rev. B6 (1972) 2192. Pynn, R., Squires, G.L.: Proc. Roy. Sot. (London) A326 (1972) 347. Rajput. J.S., Kushwaha, S.S.: Phys. Lett. 38A (1972) 497. Shukla. M.M., Da Cunha Lima, I.D., Brescansin, L.M.: Solid State Commun. 11 (1972) 1431. Singh, S.N., Prakash, S.: Physica 58 (1972) 71. Singh, K.S., Srivastava, R.S.: J. Phys. Sot. Jpn. 33 (1972) 1214. Sosnowski, J., Czachor, A., Maliszewski, E.: Neutron Inelastic Scattering. IAEA, Vienna 1972, 61.

72SnI 72Tol 73Anl

73Bel 73Bol 73Bo2 73Cal 73Col 73Co2 73Czl 73Dul 73Gol 73Go2 73Go3 73Grl 73Gul 73Kal 73Kul 73Ku2 73Lpl 73Mol 186

Swaroop, A., Bohra, B.M., Tiwari, L.M.: Physica 66 (1972) 403. Toussaint, G., Champier, G.: Phys. Status Solidi (b) 54 (1972) 165. Animalu. A.O.E.: Phys. Rev. B8 (1973) 3542 and 3555. Bertoni. C.M., Bortolani, V., Calandra, C., Nizzoli, F.: J. Phys. F4 (1973) 19. Bose. G., Tripathi, B.B., Gupta, H.C.: J. Phys. Sot. Jpn. 34 (1973) 1006. Bose. G., Gupta. H.C., Tripathi, B.B.: Phys. Lett. 43A (1973) 365. Cavalheiro, R., Shukla, M.M.: J. Phys. Sot. Jpn. 34 (1973) 1002. Copley, J.R.D., Brockhouse, B.N. : Can. J. Phys. 51 (1973) 657. Copley, J.R.D.: Can. J. Phys. 51 (1973) 2564. Czachor, A., Rajca, A., Sosnowski, J., Pindor, A.: Acta Phys. Pol. A43 (1973) 37. Duesbery, M.S., Taylor, R., Clyde, H.R.: Phys. Rev. B8 (1973) 1372. Goe1,C.M.: Phys. Lett. 46A (1973) 221. Gael, C.M.: Indian J. Pure Appl. Phys. 11 (1973) 703. Gorbachev, B.I., Ivanitskii, P.G., Krotenko, V.T., Pasechrik, M.V. : Ukr. Fiz. Zh. (USSR) (English Transl.) 18 (1973) 558. Grant, W.B., Schulz, H., Hiifner, S., Peltl, J.: Phys. Status Solidi (b) 60 (1973) 331. Gunton, D.J., Saunders, G.A.: Solid State Commun. 12 (1973) 569. Kachava, C.M : Physica 65 (1973) 63. Kushwaha, S.S.: Can. J. Phys. 51 (1973) 616. Kushawa. S.S.: Physica 64 (1973) 625. Lynn. J.W., Smith, H.G., Nicklow, R.M.: Phys. Rev. B8 (1973) 3493. Moss, S.C., Keating. D.T., Axe. J.D.: Solid State Commun. 13 (1973) 1465. Schober/ Dederichs

1.3 Referencesfor 1.1 and 1.2 73Mul 73Nil 730nl 73Pal 73Pa2 73Prl 73Pr2 73Pr3 73Ral 73Ra2 73Ra3 73Ra4 73Rel 73Rol 73Shl 73Sh2 73Sh3 73Sh4 73ShS 73Sh6 73Sh7 73Sil 73Sr2 73Swl 73Upl 73Up2 73Vel 74Bal 74Brl 74Chl 74Col 74Co2 74co3 74Dal 74Da2 74Esl 74Gol 74Go2 74Hal 74Ha2 74Ha3 74Hol 74Kal 74Khl 74Kul 74Lal 74Mal 74Ma2 74Nil 74Pal 74Pa2 74Prl 74Ral 74Ra2

Muhlestein, L.D., Gtirmen, E., Cunningham, R.M.: AIP Conf. Proc. No. 10, Part 1 (C.D. Graham and J.J. Rhyne eds.), Am. Inst. Phys. New York 1973, 41. Nilsson, G., Rolandson, S.: Phys. Rev. B7 (1973) 2393. Ono, M. : J. Phys. Sot. Jpn. 34 (1973) 26. Pal, S., Singh, R.B.: J. Phys. Sot. Jpn. 35 (1973) 1487. Pal, S.: Can. J. Phys. 51 (1973) 1869. Prasad, B., Srivastava, R.S.: J. Phys. F3 (1973) 19. Prasad, B., Srivastava, R.S. : Phys. Status Solidi (b) 56 (1973) 327. Prasad, B., Srivastava, R.S.: Phys. Status Solidi (b) 57 (1973) 743. Rao, R.R., Menon, C.S. : J. Appl. Phys. 44 (1973) 3892. Rajput, J.S., Kushwaha, S.S.: J. Phys. F3 (1973) 1531. Rao, R.R., Menon, C.S.: Solid State Commun. 12 (1973) 527. Rao, R.R., Menon, C.S.: J. Phys. Chem. Solids 34 (1973) 1879. Reese, R.A., Sinha, S.K., Peterson, D.T.: Phys. Rev. BS (1973) 1332. Roy, A.P., Dasannacharya, B.A., Thaper, CL., Iyengar, P.K. : Phys. Rev. Lett. 30 (1973) 906. Shukla, M.M., Cavalheiro, R.: Nuovo Cimento 16B (1973) 63. Shukla, M.M., Closs, H.: J. Phys. F3 (1973) Ll. Shukla, M.M., Salzberg, J.B.: Phys. Lett. 43A (1973) 429. Sharan, B., Kumar, A., Neelakandan, K.: J. Phys. F3 (1973) 1308. Shukla, M.M., Salzberg, J.B.: J. Phys. F3 (1973) L99. Shukla, M.M., Closs, H.: Solid State Commun. 13 (1973) 803. Sharan, B., Kumar, A. : Phys. Rev. B4 (1973) 1362. Singh, N., Prakash, S.: Phys. Rev. B8 (1973) 5532. Srivastava, P.L., Mitra, N.R., Mishra, N.: J. Phys. F3 (1973) 1388. Swaroop, A., Bohra, B.W., Tiwari, L.M.: Physica 66 (1973) 403. Upadhyaya, J.C., Verma, M.P. : J. Phys. F3 (1973) 640. Upadhyaya, J.C., Verma, M.P. : J. Phys. F3 (1973) 1672. Verma, M.P., Upadhyaya, J.C.: Solid State Commun. 13 (1973) 779. Barron, T.H.K., Klein, M.L. in ref. 74Hol p. 391. Brovman, E.G., Kagan, Yu. in ref. 74Hol p. 191. Chesser, N.J., Axe, J.D. : Phys. Rev. B9 (1974) 4060. Cowley, E.R.: Solid State Commun. 14 (1974) 587. Cowley, E.R.: Can. J. Phys. 52 (1974) 1714. Copley, J.R.D., Rotter, CA., Smith, H.G., Kamitakahara, W.A.: Phys. Rev. Lett. 33 (1974) 365. Da Cunha Lima, I.C., Brescansin, L.M., Shukla, M.M.: Physica 72 (1974) 179. Dasannacharya, B.A., Iyengar, P.K., Nandikar, R.V., Rao, K.R., Roy, R.P., Thaper, C.L. : Pramana 2 (1974) 179. Eschrig, H., van Loyen, L., Ziesche, P.: Phys. Status Solidi (b) 66 (1974) 587. Goel, C.M., Pandey, B.P., Dayal, B.: Phys. Status Solidi (b) 63 (1974) 625. Goel, C.M., Pandey, B.P., Dayal, B.: Phys. Status Solidi (b) 66 (1974) 759. Hafner, J.: Phys. Lett. 50A (1974) 271. Hafner, J., Schmuck, P.: Phys. Rev. B9 (1974) 4138. Hardy J.R. in ref. 74Hol p. 157. Horton, G.K., Maradudin, A.A. eds.: Dyn. Properties Solids Vol. 1. North Holland Publishing Company; Amsterdam, 1974. Kaur, M., Kushwaha, S.S.: Solid State Commun. 14 (1974) 307. Khanna, S.N., Upadhyaya, J.C., Jain, A.: Phys. Status Solidi (b) 65 (1974) K25. Kushwaha, S.S. : Nuovo Cimento 20B (1974) 83. Larose, A., Vanderwal, J.: Scattering of Thermal Neutrons, A Bibliography (1932-1974), Solid State Phys. Lit. Guides, IFI/Plenum, New York Vol. 7, 1974. Marya, J.R., Srivastava, RX: Phys. Lett. 50A (1974) 297. Maradudin AA. in ref. 74Hol p. 1. Nilsson, G., Rolandson, S.: Phys. Rev. B9 (1974) 3278. Pal, S., Seth, V.P.: Nuovo Cimento 21B (1974) 109. Pal, S.: Phys. Rev. B9 (1974) 5144. Prakash, J., Pathak, L.P., Hemkar, M.P.: J. Phys. Sot. Jpn. 37 (1974) 571. Rao, P.V.S. : J. Phys. Chem. Solids 35 (1974) 669. Rao, R.R., Menon, C.S. : J. Phys. Chem. Solids 35 (1974) 425. Schoher/Dederichs

187

1.3 Literatur zu 1.1 und 1.2 74Ra3 74Sal 74Shl 74Sh2 74Sil 74Si2 75All 75Bal 75Bel 75Be2 75Be3 75Cal 75Ca2 75Chl 75Cll 75Fil 75Fi2 75Gol 75Hal 75Ha2 75Jil 75Kul 75Ku2 75Lal 75La2 75Mal 75hiIil 75Nal 75Na2 75Pal 75Prl 75Rel 75Rol 75Ro2 75Sil 75Si2 75Si3 75Si4 75Si5 75Si6 75Si7 75Sml 75Til 75Upl 75up2 76Bel 76Brl 76Cal 76Ca2 76Ca3 76Dal 76Erl 76Gal 76GoI 76Go2

Rajput, J.S., Kushwaha, S.S.: Nuovo Cimento 19B (1974) 261. Salgado, J.: Ber. Kemforschungsanlage Karlsruhe KFK 1954, Karlsruhe, Germany 1974. Shukla. M.M., Closs, H.: J. Phys. Chem. Solids 35 (1974) 123. Shimada. K.: Phys. Status Solidi (b) 61 (1974) 701. Sinha. H.P., Upadhyaya, J.C.: Phys. Lett. 5OA (1974) 37. Singh. O.N., Kushwaha. S.S.: Nuovo Cimento 21B (1974) 354. Aleshina. L.A., Shivrin, O.N. : Sov. Phys. Solid State (English Transl.) 17 (1975) 572. Bajpai. R.P., Ono, M., Ohno, Y., Toya, T.: Phys. Rev. B12 (1975) 2194. Bertolo, L.A., Shukla. M.M.: Acta Phys. Hung. 39 (1975) 4. Bertolo, L.A., Shukla, M.M.: J. Phys. Sot. Jpn. 38 (1975) 1439. Bertoni. CM., Bisi, O., Calandra, C., Nizzoli, F.: J. Phys. F5 (1975) 419. Cavalheiro, R., Shukla, M.M.: Nuovo Cimento 26B (1975) 220. Cavalheiro, R., Shukla, M.M.: Nuovo Cimento 30B (1975) 163. Chemysov, A.A., Parshin, P.P., Rumyantsev, A.Yu., Sadikov, I.P., Severov, M.N.: Sov. Phys. JETP (English Transl.) 41 (1975) 169. Closs, H., Shukla, M.M.: Physica 79B (1975) 26. Fielek, B.L.: J. Phys. F5 (1975) 17. Fielek. B.L.: J. Phys. FS (1975) 1451. Goel. C.M., Pandey, B.P., Dayal, B.: Phys. Status Solidi (b) 69 (1975) 589. Hafner. J.: Z. Phys. B22 (1975) 351. Hafner, J., Eschrig. H.: Phys. Status Solidi (b) 72 (1975) 179. Jindal. V.K.: Can. J. Phys. 53 (1975) 1507. Kumar. S., Tolpadi, S.: Solid State Commun. 16 (1975) 265. Kumar. S., Tolpadi, S.: Phys. Lett. 53A (1975) 303. Lal, S., Paskin, A., Leoni, F.: J. Phys. F5 (1975) 697. Lannin. J.S., Calleja, J.M., Cardona, M.: Phys. Rev. B12 (1975) 585. Maurya. J.R., Srivastava, R.S.: J. Phys. F5 (1975) 2256. Miiller, A.P.: Can J. Phys. 53 (1975) 2491. Nand, S., Tripathi, B.B., Gupta, H.C.: Phys. Lett. 53A (1975) 229. Nand. S., Tripathi, B.B., Gupta, H.C.: Phys. Lett. 54A (1975) 217. Pal, S. : Nuovo Cimento 30B (1975) 299. Prakash, J., Pathak, L.P., Hemkar, M.P.: Aust. J. Phys. 28 (1975) 57. Reissland, J.A., Ese, 0. : J. Phys. F5 (1975) 110. Rosengren, A., Ebbsjii, I., Johansson, B.: Phys. Rev. B12 (1975) 1337. Rosengren, A., Johansson, B.: J. Phys. F5 (1975) 629. Singh, J., Singh. N., Prakash, S. : Phys. Rev. B12 (1975) 3166. Singh. V.P., Kharoo, H.L., Prakash, J., Pathak, L.P.: Acta Phys. Hung. 39 (1975) 37. Singh, N., Prakash, S.: Phys. Rev. B12 (1975) 1600. Singh, N., Singh, J., Prakash, S.: Phys. Rev. B12 (1975) 5415. Sinha. H.P., Upadhyaya. J.C.: Physica 79B (1975) 359. Singh, J., Prakash, S.: Phys. Lett. 53A (1975) 164. Singh. S.P., Maurya, J.R., Kushwaha, S.S.: Nuovo Cimento 29B (1975) 159. Smith, H.G., Wakabayashi, N., Nicklow, R.M., Mihailovich, S.: Low Temp. Phys. LT 13 (K.D. Timmerhaus, W.J. O’Sullivan and E.F. Hammel eds.) Plenum Press,New York 3 (1975) 615. Tiwari, M.D.: J. Phys. F5 (1975) L184. Upadhyaya, J.C., Sharma, S.S., Kulshrestha, O.P.: Phys. Rev. B12 (1975) 2236. Upadhyaya, J.C., Sinha, H.P.: J. Phys. Chem. Solids 36 (1975) 975. Beg. M.M., Nielsen, M. : Phys. Rev. B14 (1976) 4266. Brescansin, L.M., Padial, N.T., Shukla, M.M.: Nuovo Cimento 34B (1976) 103. Cavalheiro, R., Shukla, M.M.: Phys. Status Solidi (b) 76 (1976) 371. Cavalheiro, R., Shukla, M.M.: Acta Phys. Pol. A49 (1976) 27. Cavalheiro, R., Shukla, M.M.: Acta Phys. Pol. A49 (1976) 445. Day, R.S., Sun, F., Cutler, P.H., King III, W.F.: J. Phys. F6 (1976) L137. Eremeev,I.P., Sadikov, I.P., Chernysov, A.A. : Sov. Phys. Solid State (English Transl.) 18 (1976)960 Gariet, D.G., Swihart, J.C.: J. Phys. F6 (1976) 1781. Gohel, V.B., Jani, A.R. : Physica 82B (1976) 333. Gorbachev, B.I., Morozov, S.I., Parfenov, V.A., Pasechnik, M.V.: Sov. Phys. Solid State (English Transl.) 12 (1976)2157.

L

188

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1.3 Referencesfor 1.1 and 1.2 76Jal 76Ja2 76Kul 76Ku2 76Ku3 76Ku4 76Lal 76La2 76Lel 76Mal 76Mel 76Nal 76Na2 76Na3 76Na4 76011 76Pal 76PEl 76Ral 76Rel 76Sal 76Scl 76Shl 76Sh2 76Sil 76Si2 76Si3 76Si4 76Si5 76Sml 76Srl 76Stl 76Tal 76Ta2 76Upl 76Vil 77All 77Bel 77Bol 77Chl 77Cll 77C12 77Dal 77Dil 77Dol 77Gll 77Gul 77Kal 77Khl 77Kh2 77Kh3 77Kul 77Ku2 77Lol 77Nal

Jani, A.R., Gohel, V.B.: Phys. Lett. 55A (1976) 350. Jani, A.R., Gohel, V.B. : J. Phys. F6 (1976) L25. Kulshrestha, O.P., Gupta, H.C., Upadhyaya, J.C. : Physica 84B (1976) 236. Kulshrestha, O.P., Upadhyaya, J.C.: Indian J. Pure Appl. Phys. 14 (1976) 253. Kulshrestha, O.P., Upadhyaya, J.C. : Pramana 6 (1976) 291. Kulshrestha, O.P., Upadhyaya, J.C.: Phys. Rev. B13 (1976) 1861. Larose, A., Brockhouse, B.N.: Can. J. Phys. 54 (1976) 1990. Larose, A., Brockhouse, B.N.: Can. J. Phys. 54 (1976) 1819. Leadbetter, A.J., Smith, P.M.: Phil. Mag. 33 (1976) 441. Machado, J.F.C., Shukla, M.M.: Acta Phys. Hung. 40 (1976) 43. Meyer, J., Dolling, G., Kalus, J., Vettier, C., Paureau, J.: J. Phys. F6 (1976) 1899. Nand, S., Tripathi, B.B., Gupta, H.C.: Indian J. Pure Appl. Phys. 14 (1976) 430. Nand, S., Tripathi, B.B., Gupta, H.C. : Lett. Nuovo Cimento 15 (1976) 146. Nand, S., Tripathi, B.B., Gupta, H.C.: J. Phys. Sot. Jpn. 41 (1976) 1237. Naumov, II., Zhorovkov, M.F., Fuks, D.L.: Phys. Met. Metallogr. (USSR) 42 (1976) 30. Oli, B.A., Animalu, A.O.E.: Phys. Rev. B13 (1976) 2398. Pal, S. : Nuovo Cimento 3SB (1976) 215. Perdew, J.P., Vosko, S.H.: J. Phys. F6 (1976) 1421. Rao, R.R., Hemkar, M.P.: Acta Phys. Hung. 41 (1976) 165. Reichardt, W., Rieder, K.H.: Proc. Conf. Neutron Scattering (R.M. Moon ed.), Gatlinburg, USA 1976 181. Sakamoto, M., Chihara, J., Nakahara, Y., Kodotani, H., Sekiya, T., Gotoh, Y.: Jpn. At. Energy Res. Inst. Rep. JAERI-M 1976 6857. Schulz, H., Htifner, S. : Solid State Commun. 20 (1976) 827. Shukla, M.M., Tejima, H.: Acta Phys. Hung. 41 (1976) 31. Shukla, M.M., Closs, H., Oliveros, M.C. : Acta Phys. Pol. A50 (1976) 767. Singh, V.P., Kharoo, H.L., Hemkar, M.P. : Indian J. Phys. 50 (1976) 60. Singh, V.P., Kharoo, H.L., Kumar, M., Hemkar, M.P. : Nuovo Cimento 32B (1976) 40. Sirota, N.N., Bulat, I.A. : Sov. Phys. Solid State (English Transl.) 18 (1976) 561. Singh, V.P., Hemkar, M.P. : Phys. Lett. 58A (1976) 409. Sirota, N.N., Bulat, LA.: Sov. Phys. Dokl. (English Transl.) 21 (1976) 41. Smith, H.G., Wakabayashi, N., Mostoller, M.: Superconductivity in d- and f-Band Metals (D.H. Douglas ed.), Plenum Press, New York 1976, 223. Srivastava, P.L., Singh, R.N.: J. Phys. F6 (1976) 1819. Stedman, R., Amilius, Z., Pauli, R., Sundin, 0. : J. Phys. F6 (1976) 157. Taylor, R., Glyde, H.R.: J. Phys. F6 (1976) 1915. Taut, M., Eschrig, H.: Phys. Status Solidi (b) 73 (1976) 151. Upadhyaya, J.C., Sharma, R.P.: Phys. Rev. B14 (1976) 2692. Vibhute, D.B., Verma, M.P.: Indian J. Pure Appl. Phys. 14 (1976) 695. Allen, P.B.: Phys. Rev. B16 (1977) 5139. Bertolo, L.A., Shukla, M.M. : Acta Phys. Pol. A51 (1977) 41. Bonelli, E., Shukla, M.M.: Acta Phys. Pol. A51 (1977) 339. Chang, S.S., Colella, R.: Phys. Rev. B15 (1977) 1738. Closs, H., Shukla, M.M.: Nuovo Climento 42B (1977) 213. Closs, H., Shukla, M.M.: Acta Phys. Pol. A51 (1977) 31. Day, R., Sun, F., Cutler, P.H.: Solid State Commun. 23 (1977) 443. Dixit, J.P., Mehrotra, K.N.: Acta Phys. Hung. 42 (1977) 127. Dolling, G., Meyer, J.: J. Phys. F7 (1977) 775. Glyde, H.R., Hansen, J.P., Klein, M.L. : Phys. Rev. B16 (1977) 3476. Gupta, O.P., Hemkar, M.P. : Z. Naturforsch. 32a (1977) 1495. Kamitakahara, W.A., Smith, H.G., Wakabayashi, N. : Ferroelectrics 16 (1977) 111. Kharoo, H.L., Gupta, O.P., Hemkar, M.P. : Z. Naturforsch. 32a (1977) 570. Kharoo, H.L., Gupta, O.P., Hemkar, M.P. : Helv. Phys. Acta 50 (1977) 545. Kharoo, H.L., Gupta, O.P., Hemkar, M.P.: Z. Naturforsch. 32a (1977) 1490. Kumar, J.: Solid State Commun. 21 (1977) 945. Kulshreshta, O.P., Upadhyaya, J.C.: J. Phys. Chem. Solids 38 (1977) 213. Loidl, A.: J. Phys. F7 (1977) L57. Nand, S., Tripathi, B.B., Gupta, H.C.: Nuovo Cimento 41B (1977) 7. Schober/Dederichs

189

1.3 Literatur zu 1.1 und 1.2 77Pal 77Pol 77Prl 77Ral 77Ra2 77Ra3 77Ra4 77Ra5 77Ra6 77Ra7 77Sal 77Shl 77Sil 77SiZ 77Si3 77Si4 77Si5 77Sol 77Srl 77Trl 77Upl 77up2 77Val 77Vrl 77WAl 77Wel 77We2 7SAvI 7SBul 7SBu2 7SGul 7SGu2 7SGll3 7SGu4 7SKhl 7SKul 7SK112

7SKu3 7SKu4 7SKu5 7Sh4el 7SLal 7SMe2 7Sh4e3 7Sh4il 7SMi2 7SNlll

7SPal 7SPrl 7SPr2 7SRal 7SRa2 7SRa3 7SRa4 7SRa5 78Ra6 7SRa7 190

Parikh. P., Hay, D.R. : Phys. Status Solidi (b) 79 (I 977) 299. Powell. B.M., Martel. P., Woods, A.D.B.: Can. J. Phys. 55 (1977) 1601. Prasad. B., Srivastava. R.S.: Phys. Status Solidi (b) 80 (1977) 379. Rathore. R.P.S.: Indian J. Phys. 51A (1977) 108. Rathore, R.P.S.: Acta Cienc. Indica 3 (1977) 145. Rani. N., Gupta. H.C.: Solid State Commun. 23 (1977) 799. Rai. R.C., Hemkar, M.P.: Phys. Status Solidi (b) 79 (1977) 289. Rao, R.R., Ramanand, A.: Acta Crystallogr. A33 (1977) 146. Rao. R.A., Ramanand, A.: J. Low Temp. Phys. 27 (1977) 837. Rao. R.R.. Ramanand, A.: J. Phys. Chem. Solids 38 (1977) 831. Sarkar, S.K., Das, S.K., Roy, D., Sengupta, S. : Phys. Status Solidi (b) 83 (1977) 615. Shapiro, S.M.. Moss, SC.: Phys. Rev. B15 (1977) 2726. Singh. R.N., Srivastava, P.L., Mitra, N.R.: Phys. Status Solidi (b) 83 (1977) 651. Singh. J., Singh, R., Prakash, S.: Physica 90B (1977) 223. Singh. R.K., Kumar, S.: Lett. Nuovo Cimento 20 (1977) 420. Sinha. H.P., Upadhyaya, J.C.: J. Phys. Chem. Solids 38 (1977) 41. Singh. J., Prakash. S.: Nuovo Cimento 37 (1977) 131. So, C.B., Wang, S.: J. Phys. F7 (1977) 35. Srivastava, P.L., Mitra, N.R., Singh, R.N.: Indian J. Pure Appl. Phys. 9 (1977) 1977. Tripadus, V., Repcanu, S., Munteanu, C., Rotarescu, G.H., Biscoveanu, I.: Rev. Roum Phys. 22 (1977) 515. Upahyaya. J.C., Sharma. O.P.: Solid State Commun. 21 (1977) 149. Upadhyaya. J.C., Animalu. A.O.E.: Phys. Rev. B15 (1977) 1867. Vaks. V.G.. Trefilov, A.V.: Sov. Phys. Solid State 19 (1977) 139. Vrati. S.C., Gupta. D.K., Rani, N., Gupta. H.C.: Phys. Lett. 64A (1977) 333. Wakabayashi. N.: Solid State Commun. 23 (1977) 737. Weilacher. K.H., Bross, H. : J. Phys. F7 (1977) 2253. Weilacher, K.H., Roth-Seefrid, H., Bross, H.: Phys. Status Solidi (b) 80 (1977) 137. Aviran. A., Weilacher, K.H., Bross, H.: Z. Physik B29 (1978) 13. Biihrer. W.: Helv. Phys. Acta 51 (1978) 15. Bulat. I.A., Makovetskiy, G.I.: Phys. Met. Metallogr. (USSR) 45 (1978) 160. Gupta. O.P., Hemkar, M.P. : J. Phys. Sot. Jpn. 45 (I 978) 128. Gupta. O.P., Hemkar. M.P.: Physica 94B (1978) 319. Gupta. O.P., Hemkar. M.P.: Nuovo Cimento 45B (1978) 255. Gupta. O.P., Kharoo, H.L., Hemkar, M.P.: Can. J. Phys. 56 (1978) 447. Kharoo, H.L., Gupta, O.P., Hemkar, M.P. : Czech. J. Phys. B28 (1978) 77. Kumar, M., Hemkar, M.P.: Nuovo Cimento 44B (1978) 451. Kulreshta. O.P., Upadhyaya, J.C.: Phys. Rev. B17 (1978) 940. Kumar. M., Hemkar, M.P.: Physica 94B (1978) 187. Kumar. J., Upadhyaya. J.C.: Phys. Lett. 67A (1978) 77. Kushwaha. M.S., Kushwaha. S.S.: Nuovo Cimento 48B (1978) 167. Merisalo. M., JHrvinen. M., Kurittu, J.: Phys. Ser. 17 (1978) 23. Lagcrsie. D., Allan. G. : Lattice Dynamics (M. Balkanski ed.), Flammarion, Paris 1978, 27. Merisalo. M., Peljo, E., Soininen, J.: Phys. Lett. 67A (1978) SO. Merisalo, M., Jlrvinen, M.: Phil. Mag. B37 (1978) 233. Mishra. S.K., Kushwaha, S.S.: Phys. Rev. B18 (1978) 6719. Mishra. S.K.. Kushwaha. S.S.: Nuovo Cimento 46B (1978) 380. Niicker. N.: Lattice Dynamics (M. Balkanski ed.), Flammarion, Paris 1978, 244. Pathak. L.P., Rai. R.C., Hemkar, M.P.: J. Phys. Sot. Jpn. 44 (1978) 1834. Prasad. B., Srivastava. R.S.: Phys. Status Solidi (b) 85 (1978) 789. Prasad. B., Srivastava, R.S.: Phys. Status Solidi (b) 87 (1978) 771. Rathore. R.P.S.: Indian J. Phys. 52A (1978) 250. Rai, R.C.. Hemkar. M.P.: J. Phys. F8 (1978) 45. Rathore. R.P.S.: Acta Phys. Pol. AS4 (1978) 283. Ramamurthy, V., Singh, K.K. : Phys. Status Solidi (b) 85 (1978) 761. Rao, R.R., Ramanand. A.: Phys. Status Solidi (b) 87 (1978) 751. Rao. R.R., Ramanand. A.: J. Low Temp. Phys. 30 (1978) 127. Rathore. R.P.S.: J. Phys. (Paris) 39 (1978) 905. Schober/Dederichs

1.3 Referencesfor 1.1 and 1.2 78Shl 78Sil 78Si2 78Si3 78Si4 78Sol 78Stl 78St2 78Tal 78Upl 78Up2 78Wol 79Bol 79Chl 79Crl 79Frl 79Gol 79Ral 79Ra2 79st1 79st2 79Svl 79Val 80Stl 80St2

Sharma, O.P. : Phys. Status Solidi (b) 86 (1978) 483. Singh, B.P., Pathak, L.P., Hemkar, M.P. : J. Phys. Sot. Jpn. 45 (1978) 484. Singh, B.P., Pathak, L.P., Hemkar, M.P. : J: Phys. F8 (1978) 2493. Singh, R.K., Kumar, S.: J. Phys. FS (1978) 1863. Singh, B.P., Pathak, L.P., Hemkar, M.P. : Acta Phys. Pol. A54 (1978) 207. So, C.B., Moore, R.A., Wang, S. : J. Phys. FS (1978) 785. Stassis,C., Zarestky, J., Arch, D., McMasters, O.D., Harmon, B.N.: Phys. Rev. B18 (1978) 2632. Stassis,C., Zarestky, J., Wakabayashi, N.: Phys. Rev. Lett. 41 (1978) 1726. Talmi, A., Gilat, G. : Lattice Dynamics (M. Balkanski ed.), Flammarion, Paris 1978, 52. Upadhyaya, J.C., Dagens, L.: J. Phys. F8 (1978) L21. Upadhyaya, J.C.: J. Phys. F8 (1978) 1873. Woods, A.D.B., Powell, B.M., Martel, P.: Transition Metals 1977, Conference series Number 39, (M.J.G. Lee, J.M. Perz and E. Fawcett eds.). The Institute of Physics, Bristol and London 1978, 158. Bonelli, E., Machado, J.F.C., Shukla, M.M. : Acta Phys. Pol. A56 (1979) 43. Chernyshov, A.A., Pushkarev, V.V., Rumyantsev, A.Yu., Dorner, B., Pynn, R.: J. Phys. F9 (1979) 1983. Crummet, W.P., Smith, H.G., Nicklow, R.M., Wakabayashi, N. : Phys. Rev. B19 (1979) 6028. Frey, F., Prandl, W., Zeyen, C. Ziebeck, K. : J. Phys. F9 (1979) 603. Goel, A., Mohan, S., Dayal, B. : Physica 96B (1979) 229. Rao, R.R., Rajput, A.: Z. Naturforsch. 34a (1979) 200. Rao, R.R., Murthy, J.V.S.S.N.: Z. Naturforsch. 34a (1979) 724. Stassis,C., Gould, T., McMasters, O.D., Gschneider, K.A., Nicklow, R.M.: Phys. Rev. B19 (1979) 5746. Stassis,C., Arch, D., Harmon, B.N., Wakabayashi, N.: Phys. Rev. B19 (1979) 181. Svensson,E.C., Powell, B.M., Woods, A.D.B., Teuchert, W-D. : Can J. Phys. 57 (1979) 253. Varma, CM., Weber, W.: Phys. Rev. B19 (1979) 6142. Stassis,C., Arch, D., Zarestky,.I., McMasters, O.D., Harman, B.N.: Sol. State Comm. 35 (1980) 259. Stassis,C., Arch, D., Zarestky, I., McMasters, O.D.: Phys. Rev. B, to be published.

Schober/Dederichs