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--

-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,-

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