Physica status solidi: Volume 24, Number 1 November 1 [Reprint 2021 ed.] 9783112494905, 9783112494899

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plxysica status solidi

VOLUME 24 • N U M B E R 1 . 1 9 6 7

Classification Scherno 1. Structure of Solids 1.1 Alloys. Metallurgy 1.2 Solid-State Phase Transformations 1.3 Surfaces 1.4 Films 2. Non-Crystalline State 3. Crystallography 3.1 Crystal Growth 3.2 Interatomic Forces 4. Microstructure of Solids 5. Perfectly Periodic Structures 6. Lattice Mechanics. Phonons 6.1 Mossbauer Investigations 7. Acoustic Properties of Solids 8. Thermal Properties of Solids 9. Diffusion in Solids 10. Defect Properties of Solids (Irradiation Defects see 11) 10.1 Defect Properties of Metals 10.2 Photochemical Reactions. Colour Centres 11. Irradiation Effects in Solids 12. Mechanical Properties of Solids (Plastic Deformations see 10)see 10.1) 12.1 Mechanical Properties of Metals (Plastic Deformations 13. Electron States in Solids 13.1 Band Structure. Fermi Surfaces 13.2 Excitons 13.3 Surface States 13.4 I m p u r i t y and Defect States 14. Electrical Properties of Solids. Transport Phenomena 14.1 Metals. Conductors 14.2 Superconductivity. Superconducting Materials and Devices 14.3 Semiconductors ' 14.3.1 Semiconducting Films 14.3.2 Semiconducting Devices. Junctions (Contact Problems see 14.4.1) 14.4 Dielectrics 14.4.1 High Field Phenomena, Space Charge Effects, Inhomogeneities, Injected Carriers (Electroluminescence see 20.3; Junctions see 14.3.2) 14.4.2 Ferroelectric Materials and Phenomena 15. Thermoelectric and Thermomagnetio Properties of Solids 16. Photoconductivity. Photovoltaic Effects 17. Emission of Electrons and Ions from Solids 18. Magnetic Properties of Solids 18.1 Paramagnetic Properties 18.2 Ferromagnetic Properties 18.3 Ferrimagnetic Properties. Ferrites 18.4 Antiferromagnetic Properties (Continued on cover three)

physica status solidi B o a r d of E d i t o r s P. A I G R A I N , Paris, S. A M E L I N C K X , Mol-Donk, V. L. B O N C H - B R U E V I C H , Moskva, W. D E K E Y S E R , Gent, W. F R A N Z , Münster, P. G Ö R L I C H , Jena, E. G R I L L O T , Paris, R. K A I S C H E W , Sofia, P.T. L A N D S B E R G , Cardiff, L. N É E L , Grenoble, A. P I E K A R A, Warszawa, A. S E E G E R , Stuttgart, F. S E I T Z, Urbana, O. S T A S I W , Berlin, M. S T E E N B E C K , Jena, F. S T Ö C K M A N N , Karlsruhe, G. S Z I G E T I , Budapest, J . T A U C , Praha Editor-in-Chief P. G Ö R L I C H Advisory Board M. B A L K A N S K I , Pans, P. C. B A N B U R Y , Reading, M. B E R N A R D , Paris, W. B R A U E R , Berlin, W. C O C H R A N , Edinburgh, R. C O E L H O , Fontenay-aux-Roses, H.-D.DIETZE,Saarbrücken, J . D . E S H E L B Y, Cambridge, P. P. F E O F I L O V , Leningrad, J. H O P F I E L D , Princeton, G . J A C O B S , Gent, J. J A U M A N N , Köln, E. K L I E R , Praha, E. K R O E N E R , Clausthal-Zellerfeld, R. K U B O , Tokyo, M. MATY AS, Praha, H. D. M E G A W , Cambridge, T. S. MOSS, Camberley, E. N A G Y , Budapest, E. A. N I E K I S C H , Jülich, L. P A L , Budapest, M. R O D O T , Bellevue/Seine, B. V. R O L L I N , Oxford, H. M. R O S E N B E R G , Oxford, R. V A U T I E R , Bellevue/Seine

Volume 24 • Number 1 • Pages 1 to 390, K1 to K102, and Al to A34 November 1, 1967

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phys. stat. sol. 24 (1967)

Author Index D . SH. ABDINOV R . Z. ABDULOV O . ADAMETZ E . ADY G . K . AFANASEVA . . . . K . S . ALEKSANDBOV . . . G . M . ALIEV S . AMELINCKX J . ANTULA H . ABEND J . ABENDS J . AULEYTNER B . AUTIN G . M . BARTENEV . . R . BAUBINAS D . BÄUERLE G . L . BELENKY V . A . BESFAMILNAYA Y u . A . BOGOD W . BOLLMANN Y . N . BONDARENKO . D . N . BOSE H . BÖTTGER A . BOUBBET H . P . VAN DE BBAAK R . BRUGIÈRE R . A . BUBMEISTEB J R . K . H . J . BUSCHOW. . W . J . CASPERS V . I . CHEREPANOV T . H . CHEUNG K . CLAUSECKER M . V . COLEMAN T . A . T . COWELL N . CROITORU D. A. L. G. A. H. G. R. A.

DAUTREPPE S . DAVYDOV DELAEY J . DIENES M. VAN D I E P E N . DIEPERS DÖHLER DUPREE DYMANUS

V . F . ELESIN I . B . ERMOLOVICH L . ERNST J . P . PAST

.

.

K23, K145 385 583 K65 K61 K61 . . K23, K145 . . . 195, K 1 2 1 . . . . 89, 95 K69 K129 K107 501 443 K91 207 543 757 K49 K141 543 K165 65 K173 733 K77 683 715

. .

.

.

.

.

.

.

. .

. .

. . . .

. . . .

. . . .

733 51 509 721

.Kill K37 K17

K173 373 K103 461 715 . 235, 623 331 . 275, 525 487

. . . . . .

.

.

.

. . . .

.

.

.

.

757 543 177 715

Y A . S . FEIZIEV E . R . FITZGERALD . P . A . FLINN C. T . FOBWOOD A. FOURDEUX A. A . FRIDMAN B . FRITZ F . FRÖHLICH R . GEICK J . GEIGER R . GEVERS L . GHITA W . GLAESER A . GLASNER P . GOBIN H . GÖTZ E . B . GRANOVSKII . R . GRIGOROVICI E . F . GROSS P . GROSSE P . GUETIN H . - E . GUMLICH I . GUTZOW C. HAMMAR H . L . HARTNAGEL . H . W . DEN HARTOG D. HEESE B . HEIMANN D . HEITKAMP B . HEJDA 0 . HENKEL G. HENSEL K . H . HERRMANN . G. HILDEBRANDT . M. HÖHNE B . W . HOLLAND J . HÖLZL A . HUBEBT R . A . HUGGINS H . IMGBUND I . V . IOFFE Y A . A . IOSILEVSKH V . A . IVANOVA

.

.

.

.

.

.

.

. .

. .

. .

. .

. .

. .

.

.

99 457 195 K175 K5 695 K167 K95 45 K17 K107 K33 K29 K13 349 531 K133 K117 281 651 341 659 K153 535 289 245 591 275 651 669 301

. .

B . K . JONES M . S . KAGAN S . G. KALASHNIKOV N . KAWAI A . L . KAZAKOV

.

K157 37 263 525 195 45 207 . 535, 583

.

K5 K9 749 K23, K145 297

.

.

.

551 551 K83 K9

766

Author Index

D . W . KEEFER V . V . KELAREV S. KIKUCHI A . I . KITAIGORODSKU V . V . KLYUSHIN J. KOIODZIEJCZAK B . KORNEFFEL E. G. KOR YAK-DORONENKO F . KOSEK H . D. KOSWIG A . A . KRALINA F . J. KRINGS K . - E . KKÖLL I . N . KRUPSKII B. I . KULIEV M. KURIYAMA E. Z. KURMABV I . A . KUROVA

217 385 K83 K61 385 323 K137 443 K69 605 K107 163 707 K53 K25 743 K43 757

H . J. LEAMY H . LEMKE A . LINENBERG P . LUCASSON W . LUDWIG

149 127 695 K77 K137, K149, K 1 6 1

M. H . VAN MAABEN R . MACH E . F . MAKAROV E . MANN V . G. MANZHELY A . MARAIS M. J. MARCINKOWSKI M. MARINA M. MARQUES F . MATOSSI H . MELL A . N . MEN T . MERCERON U. K . MISHRA D . P . MORGAN R . MOSER G.O.MÜLLE R L . E . MURR H . E . MÜSER

K125 K149 45 721 K53 635 149 K17 481 K65 183 51 635 K87 9 K13 127 135 109

Y A . N . NASIROV G. M. NEDLIN H . H . NEELY S. A . NEMNONOV R . E . NETTLETON E . NEUMANN A . E . NIKIFOROV T . V . NORDSTROM A . S. NOWICK

K157 K25 217 K43 561 K13 51 K121 461

G. T. V. N.

OELGART O. OGURTANI A . ONISHCHUK N . ORMONT

V. V.

289 301 373 757

O.STROBORODOVA

Y U . B . PADERNO J. B. PAGE JR H . PAGNIA P . PAUFLER J. PEREZ A . J. PERRY B . PIETRASS B . PISTOULET C. N . PLAVITU S. POKRZYWNICKI J. POTTHARST V . A . POVITSKII J. PRZYSTAWA J. PUNZEL

757

KLL,

K73 469 K97 77 K167 K141 571 481 361 KLL, K73 109 45 313 K L

R . K . PUROHIT

K57

B. S. RAZBIRIN H . G. REIK R . REISFELD U . RETTER A . G. REVESZ J . L . ROBERT B . ROESSLER M . ROUZEYRE

K107 281 695 605 115 481 263 399

A . SAKALAS A . SAWAOKA U . SCHRÖDER G. E . R . SCHULZE H . SCHULZE A . SEEGER M. SELDERS B. SELLE M. K . SHEINKMAN S. K . SIDOROV D. SIEBERT A . SMILGA M. J. A . SMITH O. V . SNITKO H . SPREEN A . J. SPRINGTHORPE B . SPRUSIL B . STALINSKI J . STANKOWSKI M. STASIW D . A . STEVENSON D . STRAUCH J. STUKE

K91 K83 99 77 K95 721 K33 K149 543 385 K65 K91 525 543 413, 431 K3 K41 K73 451 591 683 469 99, 183

Author Index T . L . TANSLEY

615

E . TENESCU D . J . D . THOMAS

K175

. . . .

Kill

767

P . VOSTRY

K41

A . K . WALTON

K87

S . WANG

509

S . TOSCHEV

349

P . WEISSGLAS

J . M . TROOSTER

487

K . WERNER

V . A . T YAG A I

543

H . WEVER

K5

H . W . DE W I J N

715

W . ULRICI

605

L . VESCAN

K17

J . VLSCAKAS

K91

G . VOIGT S . V . VONSOVSKII

K161 . . . .

51

531 K95

J . WOODS

K37

T . W . WRIGHT

37

R . ZEYFANG N . G . ZHDANOVA

221 . . . .

551

Contents Page

Review Article D . P . MORGAN

Helicon Waves in Solids

9

Original Papers E . R . FITZGERALD a n d T . W .

WRIGTH

Invariance of Sound Velocity Sums in Crystals

37

E . F . MAKAROV, V . A . P O V I T S K I I , E . B . G R A N O V S K I I , a n d A . A . F R I D M A N

Study of Alnico 8 by Mössbauer Spectroscopy

45

S . V . VONSOVSKII, V . I . CHEREPANOV, A . N . M E N , a n d A . E . NIKIFOROV

H . BÖTTGER

Group-Theoretical Classification of Electronic States in Crystals Composed of Pairs and Complexes of Bound Impurity Ions . . . .

51

Aufspaltung der ESR-Linie durch Spin-Phonon-Wechselwirkung an Ionen mit Spin 1

65

P . P A U F L E R a n d G . E . R . SCHULZE

J. J.

ANTULA

ANTULA

Plastic Deformation of the Intermetallic Compound MgZn 2 . . . .

77

Tunnel and Schottky Current in Dielectric Thin Films Considering Film Thickness Fluctuations

89

Method of Measuring Effective Electron Mass in Thin Insulating Films

95

A

R . GEICK, U . SCHRÖDER, a n d J .

STUKE

Lattice Vibrational Properties of Trigonal Selenium H . E . MÜSER u n d J .

A.

G.

REVESZ

99

POTTHARST

Zum dielektrischen Verhalten von Seignettesalz im Bereich der Dezimeter- und Zentimeterwellen

109

Noncrystalline Structure and Electronic Conduction of Silicon Dioxide Films

115

H . LEMKE a n d G. O. MÜLLER

L. E.

MURR

Transient Behaviour of Space-Charge-Limited Currents in p-Type Silicon

127

Study of Erbium Thin Film Oxidation in the Electron Microscope .

135

M . J . MARCINKOWSKI a n d H . J .

LEAMY

Analysis of Dislocation Loops in Superlattices F.

J . KRINGS

L. ERNST H . MELL a n d J .

149

Ferromagnetische Hochtemperaturnachwirkung an Eisen-SiliziumLegierungen bei gleichzeitiger plastischer Verformung

163

About the Fowler-Nordheim Plots of Germanium Field Emitters .

177

STUKE

Magnetoconductivity of Trigonal Selenium Single Crystals

. . . .

183

A . FOURDEUX, R . GEVERS, a n d S. AMELINCKX

Electron Microscopic Contrast Effects at the Interface between Two Different Polytypes (I)

195

D . BÄUERLE a n d B . FRITZ

Infrared Vibrational Absorption by U-Centers in N a l 1»

207

Contents

4

Page H . H . NEELY a n d D . W . KEEFER

R . ZEYFANG H . DIEPERS

Resistivity Studies of Electron Irradiated Iron

217

Der Gitterwärmewiderstand durch Versetzungen in Legierungen bei tiefen Temperaturen

221

Elektronenmikroskopische Untersuchung von Fehlstellenagglomeraten in ionenbestrahlten Kupferfolien (I)

235

G. HILDEBRANDT Röntgenwellenfeider in einem Dreistrahlfall

245

B . ROESSLER a n d P . A . F L I N N

A Theory of X - R a y Diffraction from an Ordered Alloy Containing Antiphase Domains

263

R . D U P R E E a n d B . W . HOLLAND

The Range of 3-Factors and t h e Breakdown of Motional Narrowing in Conduction Electron Spin Resonance

275

H . G. REIK a n d D . HEESE

The Small Polaron Problem with an Application to Optical and DC D a t a of Reduced Barium Titanate

281

K . H . H E R R M A N N u n d G . OELGART

B. K. JONES

Piezowiderstandseffekt an Tellur (I)

289

Measurement of t h e Magnitude of the Electronic Attenuation of Sound in Metals

297

T . 0 . OGURTANI a n d R . A . HTTGGINS

J . PRZYSTAWA

Theory of Electric Field Gradient due to Conduction Electron Charge Density Redistribution around Screw Dislocations in Metals . . .

301

The Neel Temperatures and Stability of Magnetic Phases in t h e Disturbed Body-Centered Tetragonal Lattice. Uranium Compounds. .

313

J . KOLODZIEJCZAK

Magnetic Resonance in Higher Harmonics Generated by Free Carriers in Semiconductors

323

G. DÖHLER

On Slow High Field Domains in Homogeneous Photoconductors . .

331

D. HEITKAMP

On the Thermodynamics of Thermal Diffussion in Substitutional Alloys

341

S . TOSCHEV a n d I . GUTZOW

C. N. PLXVITU

Derivation of Nucleation Kinetics in Solids by Examination of Plane Sections through the Samples

349

On the Phonon Drag Effect Contribution to Thermomagnetic Phenomena

361

A . S . DAVYDOV a n d V . A . ONISHCHUK

The Dielectric Permeability of Molecular Crystals

373

V . V . K E L A R E V , S . K . SIDOROV, V . V . K L Y U S H I N , a n d R . Z . ABDULOV

Neutron-Diffraction Study of Antiferro-Ferromagnetic Transition in a System of Ordered F e ( P d I P t i _ x ) 3 Alloys

385

Contents

5 Page

Short Notes J.

PUNZEL

Temperature Dependence of the Surface Structure of Ion-Bombarded Si Single Crystals

K1

A . J . SPRINGTHORPE

The Rhombohedral Distortion in NiO W . GLAESER, H . IMGRUND u n d H .

K3

WEVER

Zur Präge der reproduzierbaren Restwiderstandsmessung an hochreinem Eisen

K5

I . V . I O F F E a n d A . L . KAZAKOV

On the Theory of Ferroacoustic Resonance

K9

Y u . B . PADERNO a n d S. POKRZYWNICKI

Magnetic Properties of Some Heavy Rare E a r t h Tetraborides . . . K 1 I H . - E . GUMLICH, R . MOSER, a n d E . N E U M A N N

On the Mn-Mn Interaction and the Structure of Optical Spectra of ZnS(Mn)

K13

R . GRIGOROVICI, N . CROITORU, L . V E S C A N , a n d M . M A R I N A

Hole Injection in Junctions between Amorphous Ge Layers and n-Type Ge Single Crystals K17 V . A . IVANOVA, D . S H . A B D I N O V , a n d G . M . A L I E V

Some Structural, Electrical, and Thermal Parameters of VCr 2 Se 4 , FeCr 2 Se 4 , and NiCr 2 Se 4 Ternary Compounds K23 B. I. KULIEV a n d G. M. NEDLIN

The Types of Magnetic Ordering in Crystals Having the Structure of R b N i P j K25 P.

GUETIN

Note on the Density-of-State Effective Mass in GaAs

K29

P . GROSSE a n d M . S E L D E R S

The p-Band of Tellurium at Very Low Temperatures

K33

T . A . T . COWELL a n d J . WOODS

Thermally Stimulated Cuurrents and Infra-Red Lminescence in CdS Crystals K37 B . SPRUSIL a n d P . VOSTRY

Reply to "On Quenching Experiments in Gold" by F. J . Kedves . . K41 S. A . NEMNONOV a n d E . Z . KURMAEV

Band Structure and Superconductivity of A 3 B-Type Intermetallic Compounds with ß-W Structure K43 Yu.

A . BOGOD

On the Resistance of Bi in High Magnetic Fields

K49

I . N . KRUPSKII a n d V . G. MANZHELY

Thermal Conductivity of Solid Argon R. K. PUROHIT GaP-GaAs n-p Heterojunctions

K53 K57

G . K . AFANASEVA, K . S . ALEKSANDROV, a n d A . I . KITAIGORODSKII

Elastic Constants of Anthracene

K61

D . SIEBERT, E . A D Y , a n d F . MATOSSI

Magnetoresistance of High-Ohmic Semiconductors by a Capacitive Method K65 F . KOSEK a n d H . AREND

On High Temperature Conductivity of BaTiO a

K69

6

Contents

YU. B. Paderno, S. Pokrz ywntckt, and B. Stalinski Magnetic Properties of Some Rare Earth Hexaborides

Page

K73

R. BRUGiiRE and P. Lucasson On the State I I I Recovery of Electron Irradiated Aluminium. . . K77 A. Sawaoka, N. Kawai, and S. Kikuchi Change of the Magnetic Anisotropy Constant K t of Manganese Ferrite and Manganese Zinc Ferrite under Hydrostatic Pressure K83 U. K. Mishra and A. K. Walton Infra-Red Faraday Effect in p-Type InAs and Si

K87

R. Baubinas, A. Sakalas, A. Smilga, and J . Viscakas p-Type CdSe Single Crystals

K91

H. Gotz, H. Schulze, and K. Werner Overhauser Effect in Semiconducting Crystalline TCNQ-Complexes. H. Pagnia Slow Photoeonduction in InSb Films

K95 K97

Pre-printed Titles and Abstracts of papers to be published in this or in the Soviet journal „OnCHKa TBepaoro T e j i a " (Fizika Tverdogo Tela)

A1

Contents

7

Systematic List Subject classification:

Corresponding papers begin on the following pages (pages given in italics refer to the principal subject classification) :

1.1 1.2 1.3 1.4 2 3.1 4 5 6 6.1 7 8 9 10 10.1 10.2 11 . 12 13 13.1 13.3 13.4 14.1 14.2 14.3 14.3. 1 14.3.2 14.4 14.4.1 14.4.2 15 16 17 18 18.1 18.2 18.3 18.4 19 20 20.1 20.3 21 21.1 21.1.1 21.2 21.6 21.7 22 22.1.1 22.1.2

263 349, K25 Kl 115, 135 115 349, K91 195,263, K3 135, 245, K23 37, 65, 99, 207, 221, 281, 361 45 37, 297, K9 221, K53 341 149, 195 77, 149, 163, 195, 221, 235, 301, 341, K41, K77 207, K37 217, 235 K l , K77 37, K61 9, 281, K25 95, 183, 289, 297, 323, K29, K33, K43, K87 177 51, 65, 301, K13, K37 9, 217, K5, K41, K49, K77 K43 9, 183, 289, 323, K23, K37, K65, K91 115, K97 89, 95, 115, K17, K57 373 89, 95, 127, 331, K17 109, 281, K69 361, K23 331, K91, K97 177 K25, K87 Kll, K73 45, 163, 313, 385, K5 K83 313, 385, K3 65, 275, 323, K9, K95 9, K87 99, 207, 281, K33 K13, K37 9, 77, 275, 297, 341, K77 9, 195, 221, 235, 263, 275, 297, 301 45, 149, 163, 217, 385, K5 9, 275, 297 9, K41 K49, K73 313, 323, 349, K l l 9, 177, K17 127, K l , K87

8 22.1.3 22.2. 1 22.2. 2 22.2. 3 22.4. 1 22.4. 2 22.5.2 22.6 22.8 22.9 23

Contents 99, 183, 289, 331, K33 9, K29, K57, K65, K87 K57 9, K97 K13, K37 K91 207 51, 65, 95, 115, 135, K3 313, K23 K61, K95 K53

Contents oi Volume 24 Continued on Page 393

Review Article phys. stat. sol. 24, 9 (1967) Subject classification: 13 and 20; 14.1 j 14.3; 21; 21.1; 21.2; 21.6; 22.1.1; 22.2.1; 22.2.3 Department of Electrical Engineering, University College London

Helicon Waves in Solids By D . P . MORGAN

Contents 1.

Introduction

2. Theory

for waves in an infinite

3. The rotating

coordinate

system

medium

— local

case

and the two-component

plasma

3.1 Introduction of the rotating coordinate system 3.2 Extension to two-component plasma 4. Boundary

conditions

and experimental

results

4.1 Resonances in rectangular samples 4.2 Other experimental techniques 5 . The non-local

case — Boppler-shifted

6. Helicon-phonon 7. Open-orbit

cyclotron

absorption

interaction effects

References

List of Symbols A, B, C a = +1

B '0o = — (0, 0 , B o ) b B = B0 +

b

~ rj C =

COS 0

Co = (fioi"o)"1/2 —e

E í(r, u, t) fo fl =

f - f o

J = —Nev k = (kx, ky, kz) kl — £j £Q (¿0 ft)a

sample dimensions mode parameter ( + 1 for propagating mode, —1 for damped mode) static magnetic field wave magnetic field total magnetic field value of B0 at Kjeldaas edge velocity of light in free space electronic charge (e > 0) wave electric field electron distribution function equilibrium distribution function perturbation distribution function current density wave vector; k = (k% + k'fj + kt)l¡2

10

D . P . MORGAN

m electron effective mass M ion mass N electron number density n l t n2, % odd integers R = 1/N e Hall constant r = (x, y, z) position vector s = sin 6 s sound wave velocity (Section 6) u electron velocity U ion velocity v average electron velocity fF Fermi velocity iwm iu> + 1/T a = "W + 2 = e^l ft = a h% — i to fi0 e = a/t a> e0 + £] effective dielectric constant £, lattice dielectric constant £ lattice ion displacement 0 angle between wave vector k and B0 v = 1/r p = mjN e2 r dc resistivity o conductivity r electron collision time 0 angular coordinate for electron velocity u cd wave frequency wc = e BJm cyclotron frequency o)p = (N e2lm e0)1/2 plasma frequency X 0Jr resonant frequency for a>c x = oo o>a resonant frequency for ac exctiation cot resonant frequency for transient excitation 1. Introduction The ability of a magnetised plasma to support electromagnetic wave propagation is well known, and waves in laboratory plasmas and in the geophysical ionosphere have been the subject of much study. The existence of propagating electromagnetic waves in such high conductivity media is due to the inhibiting effect of the magnetic field on the motion of the charges. In 1960, Aigrain [1] and Konstantinov and Perel [2] suggested that such propagation should be possible in solid state plasmas, and since then numerous studies of waves in solids have been made. This paper is concerned with the waves obtained in the low frequency limit in an uncompensated plasma. Aigrain gave then the name "helicon", referring to the helical configuration of the electric field. The low frequency limit here requires that W )2 + col

GXV — o^w Gzz

=

(i co+vf + col'

i CO + v '

with v = 1/t, and the other components zero. W e see that a obeys Onsager's relations Oij(B0) =

¡i0

ft)

£\S0 E

from equation (2.17) we have

18

D . P . MORGAN

Thus the helicon wave velocity must be small compared with the velocity of electromagnetic waves in the absence of the electrons, a condition easily satisfied in practice. Expressed in terms of more basic parameters, the condition is m to. e 0

•

me, mh, ve, vh are the effective masses and collision frequencies for electrons and holes. The dispersion relationis obtained from equation (3.3), using cd ix ~ 2 i In the region co is real and k+ complex. It is clear from (4.1) that resonances will occur for co = coa, and from (2.12) this is equivalent to the condition = n3 njC. (The resonances are actually slightly displaced from o)a by the interaction of other terms in the expansion (4.1), but this effect is negligible unless a>c r < 1.) Alternatively, one can study the transient response of the system by applying a steady field with the drive coil and switching this off at t = 0 , observing decaying oscillations in the detecting coil voltage for t > 0. In this case Jc+ is real and the resonant frequencies are given by equation (2.11) with k+ = n3 ji/C: (X>i=

A2clwe \ C )

l

i \ ^ wax) •

This gives a decaying signal whose frequency is the real part of cot. In practice, the signal observed is a sum of modes with different values of n 3 [22], the relative amplitudes being given by a Fourier series expansion for the driving field waveform (taken as a step function). cot is related to coa by coc r + i (1 + cot *2)1/2 and substitution of co = ft>t into (4.1) gives byjbi x = oo, a result which is expected since for t > 0 the driving field is absent. These relations are brought out more clearly by introducing a frequency cor at which the resonance occurs if

23

Helicon Waves, in Solids

there is no damping (coc r = oo). This has the same value for both the ac and transient cases, namely

The ac case gives

=

(l+o>Sr»)i/2 c^r

(4 3)

'

and the transient case gives for the resonant frequencies. Equations (4.3) and (4.4) thus give the corrections to be made for finite values of a)0 r. For the ac case, the following can be deduced from equation (4.1): a) The resonant frequencies are in the ratios of n\, i.e. 1, 9, 25, . . . . b) The effect of damping is to change the values of the resonant frequencies, but not their ratios. c) Both the sharpness (Q) and heights of the resonances increase with wc r , and the heights of the resonances decrease with w3. Legendy [12] has done a more thorough analysis for the infinite sheet, taking into account the fact that the field outside the sample includes a component produced by the sample as well as the driving field. The driving field is written as an incident plane wave, and for normal incidence the above theory is valid with = na TtjC at resonance. If the waves are travelling at an angle a similar formula is obtained, but as well as the complication introduced by non-zero kx and ky the value for kz at resonance is higher than n3 TtjC. Klozenberg et al. [19] have also taken account of the vacuum field produced by the sample to calculate the dispersion for a cylindrical sample of infinite length with its axis in the B 0 -direction. The above theory gives rigorous solutions for a sample infinite in the x- and y-directions, and approximate solutions if the x- and i/-dimensions A and B are much greater than C. To interpret accurate experimental measurements a theory is required for a rectangular block, valid at least in the regime C < A, C C, B^> C, equation (4.6) can be regarded as a version of equation (4.2) corrected for the effect of finite sample size. Helicons were first detected by Bowers et al. [23] in 1961 using the transient technique mentioned above with a cylindrical sodium sample with its axis perpendicular to B0. I t is not easy to determine the resonant frequencies by this method unless coc r is large, since the signal amplitude decays by a factor e - 1 during coc r/2 it cycles, and most investigators have used the ac method. Chambers and Jones [18] have checked experimentally the form of A = B) and taking equation equation (4.1) using square plate samples (C (4.6) as the form for the resonant frequencies. The sharpness of the resonance

Helicon Waves in Solids

25

magnetic field BA, frequency, and temperature are a rough indication only, to be comprehensive. 2 resonance method with sample enclosed by coils (Section 4.1) Frequency 30 Hz

10 GHz 2-500 Hz 200-800 Hz 1 0 - 60 Hz 200 kHz 200 kHz 20-30 MHz 0.1- 1 kHz 700 Hz 1 - 3 MHz 1 - 6 0 MHz 70 Hz 35 GHz 35 GHz 0.1, 3 MHz

Temperature (°K) 4 4 300 77, 300 4 1.5 4 4 4 1.2 4 1.5-4 4 4 4 4 4 1.5,4 77, 300 4 4

Technique 2 2 2 2 2 2 2 2 1 1 1 2 1 1 2

2 1

Remarks

Measurement of R Measurement of R

Measurement of R Non-local Non-local Quantum effects Helicon-phonon interaction Fluxmeter Open-orbit effects Measurement of R Non-local Kjeldaas edge by surface impedance Measurement of R Quantum effects Microwave interferometer bridge Superconducting sample Non-local

curve can be used to deduce the value of A>C T (which need not be 1), and since cjj O)JO>P = B0 RHJ,0 the resonant frequencies give the value of the Hall constant R. The value of coc R gives the mobility /u and resistivity q, since coc r = B0 FI = = R BJQ. Chambers and Jones measured R and Q for a wide range of materials (indicated in Table 1), although their values of R are subject to the validity of equation (4.6). Taylor et al. [20] have extended the technique by using a phase sensitive detector to separate the in phase and in quadrature components of the secondary signal. Since the phase changes as a> passes through a resonant frequency (as shown by (4.1)) a sharper resonance plot is obtained by this method and measurements of R and Q can be made on samples with lower values of „ r 1, in contrast to the resonance method described above in which this limitation is absent. F u r d y n a [27] has developed a sensitive microwave interferometer based on this method. I t is also possible to observe resonances in the sample with this sort of apparatus, and Grimes et al. [28] have done this with a disc-shaped sample of silver at frequencies below 1 kHz. Multiple reflections are involved, and the resonances are similar in character to Fabry-Perot fringes. Libchaber and Veilex [10] in one of the earlier experiments demonstrated helicon propagation in a disc of InSb a t 10 GHz. Using a circular waveguide and circular polarisation of the incident field they showed t h a t the transmission depends markedly on the sense of rotation of the fields. 5. The Non-Local Case — Doppler-Shifted Cyclotron Absorption We shall now return to the infinite medium case and relax the local limit used in preceding sections. The local limit is an expression of the fact t h a t the

Helicon Waves in Solids

27

current J(r) at a point r is dependent only on the field E(r) at that point; for this to be valid the electron motion during one cycle of the wave must be much less than the wavelength. Under these conditions the equation of motion can be used, with the stipulation that the elementary volume d F must have dimensions much less than the wavelength. In a hot plasma the random electron velocities are higher, the electrons move distances comparable to the wavelength during one cycle, and the plasma becomes non-local. The current J(r) at r now depends not only on the field E(r) but also on the fields encountered by the electrons at previous points on their trajectories, which are slightly perturbed cyclotron orbits. The equation of motion is therefore invalid in this case. I t is possible to calculate the current J{r) from an integral over the trajectory involving the perturbation due to the electric field [29], and we shall see below that a similar integration results from direct application of the Boltzmann equation. The current J(r) so obtained is found to be proportional to the value of E(r) at the same point, so that it is possible to define a conductivity o as for the local case, although here we find that o depends on the helicon wave vector fc as well as the frequency to. I t also depends on the form of the carrier distribution function /„, in contrast to the local case. Cyclotron resonance is one of the more prominent features of a hot plasma. For helicons, propagating in the regime co a>c, the effect can be observed by virtue of the fact that the wave frequency as "seen" by the electrons is Dopplershifted, hence the term "Doppler-shifted cyclotron resonance". The condition for the resonance is that the wave frequency as "seen" by an electron should be equal to coc, and for a wave travelling in the «-direction with wave number kz this apparent frequency is a> — kz vz, with vz the z-component of electron velocity. The condition for resonance is therefore

kz vz = co — ft)c . Since ft)c > co, this condition is satisfied for electrons travelling in the opposite direction to the wave. Landau damping, in contrast, occurs when kz vz = a>, i.e. the electron velocity is the same as the wave velocity. I t has been studied with reference to helicon waves by Lampert et al. [30] and is only of interest for waves travelling at an angle to the magnetic field B0, since only then is there a component of electric field in the direction of the wave normal. Since vz can take any value in the range — v v . (5-1) When this equation is satisfied, some of the electrons experience a wave frequency equal to coc and pick up energy which is dissipated by collisions with the lattice. Absorption of the wave therefore occurs for values of a>c less than the value given by the equality condition in equation (5.1) and this onset of absorption is known as the Kjeldaas edge. As in the local case, the dispersion relation is obtained by inserting the conductivity a into the wave equation (2.4). The conductivity has been calculated by Cohen et al. [29] and by Kjeldaas [31]; the latter used the Boltzmann equation to study the case of waves travelling parallel to B0. Allis and Buchsbaum [6] also give an account of Boltzmann theory. For completeness the derivation is outlined here, following most closely the method used in [6]. We take the electron distribution function to be isotropic, and consider waves travelling only in the direction of B 0 , so that Landau damping will not appear. The Boltz-

28

D . P . MORGAN

mann equation is given by equation (2.6) with F

= - e (E

+

uxB)

,

the Lorentz force. The distribution function / is written / = /„ -f- /15 where /0 is the distribution function for the equilibrium case when the wave is absent and is a small perturbation. The uxB term then gives a term in ttX0/o/0tt which is zero because /„ is isotropic. As before, B is written as B = B0 + 6, and the approximations B 0 ^ > b , allow further terms to be eliminated. These approximations linearise the equation. The velocity u is transformed into cylindrical polar velocity coordinates with ux = wr cos 0, uv = ut sin 0, uz = uz. This leads to the identity (V « x B

0

0£ 0B03>

) ^ = -

8U

since B0 acts in the z-direction. The collision term on the right-hand side of equation (2.6) is written as — (/ — /0)/r = — v fv The time and space variation of ft is the same as for J, so that 0/81 = i m, 8/0r = - i k. With fc = (0, 0, kwe obtain

with p.

i co — i uz kz +

v

eE

df0

m coc 8M

Hence

h = f q ' — oo

exp [ - P '

(0

-

0 ' ) ] A0'

,

where the primed parameters are obtained by substituting 0' for 0; u becomes u' = (uT, 0', uz) and /0 becomes = f0(u'). The lower limit of the integration is determined by the condition that /x is single-valued at any point and therefore periodic in 0. The current J is given by

e0 + ej." ci ka — Cq h"~{)

[ (O (T-j-

with o±

=

(5.2)

a coc — co + i v (S2 — 1) In ( l ± i j + 2 ó

kz vv a (o,. — co + iv

This is identical to the dispersion relation obtained for the local case, equation (3.4), except for the inclusion of F(d). The conductivity t — k • r)] and fc and B0 are in the z-direction. Using the rotating coordinate system with £ ± = + 1£y a n < l similar definitions of E± and F ± we obtain (k*±Qea>)C± =

(6.2)

where Qc = e BJM. The average collisonal force F is taken to beTO(V — f / ) / r where v is the average electron velocity. The total current density J', due to both electrons and ions, is given by J ' = — N e (v — U) so t h a t J ' = — N e r F/rn and equation (6.2) becomes + —

\

c /

=

M

+

MN er

(6.3) '

I n the absence of phonons, the electron equation of motion is given by equation (2.5). When phonons are present the equation is unchanged except t h a t the collisional force becomes — m (v — V)/r instead of —mv/r. W e define J = = — N e v as the electron current density and substitute R and o as before to obtain co to _ _ ,, mV This is the same as the constitutive relation (2.7) except for the extra t e r m on the right-hand side, and it follows t h a t

where a is the local conductivity tensor obtained in the absence of phonons, as given in Section 2. This equation is also valid for a non-local plasma if t h e relevant form for c is used [29, 33], b u t only t h e local case will be considered here. The total current density J' = J + N e U is given by J'±=o±(E±

-

+

(6.4)

where cr± = oxx + i axy as in Section 3. Equations (6.3) and (6.4) are relations between E,J', and A f u r t h e r relation is given by Maxwell's equations V X E = — QB/dt and X/ X B = /j0 ./' (neglecting displacement currents). Elimination of B yields fcx (kxE) = i a> fi0 J' and using k = (0, 0, k) this leads to k2 E±

=

— i co fi0 J'± .

(6.5)

Eliminating J'± and E'± in equations (6.3), (6.4), and (6.5) we have [(*• -

P + Í3Cc co) + + ^ fl + " l ) l C± = 0 , '\ff0 ifflftiT,/ Mr \ io>n0o0/\o0 JJ —

where a0 = N e2 r / m is the dc conductivity. This is the dispersion relation, since t ± 4= 0. We shall consider the limits a>0 r oo, co eT^> 1, this leads to the following relation: U>po o>c T

Here copc and a) po refer to the plasma frequencies for closed-orbit and open-orbit electrons, and coc is the cyclotron frequency for closed-orbit electrons. The wave has w oc k2 as for a helicon, but since k2 is imaginary, it is heavily damped. For a fixed value of k, we have co oc B\ in contrast to the helicon for which w oc B0. Grimes et al. [28] have demonstrated open-orbit effects using a disc-shaped single crystal sample of silver with its axis along (001). The experimental arrangement was similar to the interferometer described in Section 4.2, but a low frequency was used and resonances were observed. The relation between a) and B0 for fixed k was investigated by varying o> and B0 in such a manner as to maintain the fields at a particular resonance. Damping of the transmitted wave was observed when B0 was in directions giving open orbits, and for these directions the above dispersion relation was qualitatively verified. Acknowledgement

I wish to express my gratitude to Professor A. K. Jonscher for his advice and encouragement during the preparation of this paper in its original form. References [1] P. AIGRAIN, Proc. Internat. Conf. Semiconductor Physics, Prague 1960, (Czech. Acad, of Sci., Prague 1961 p. 224). [ 2 ] O. V . KONSTANTISTOV a n d V . I . PEREL, Z h . e k s p e r . t e o r . F i z . 3 8 , 1 6 1 ( 1 9 6 0 ) ;

Soviet

Phys. - J. exper. theor. Phys. 11, 117 (1960). [ 3 ] R . BOWERS a n d M . C. STEELE, P r o c . I E E E 5 2 , 1 1 0 5 ( 1 9 6 4 ) .

[4] A. K. JONSCHER, Brit. J. appl. Phys 15, 365 (1964). [5] A. A. VEDENOV, Uspekhi fiz. Nauk 84, 533 (1964); Soviet Phys. -

Uspekhi 7, 809

(1965).

[6] W. P. ALLIS, S. J. BUCHSBAUM, and A. BERS, Waves in Anisotropic Plasmas, M.I.T. Press, 1963. [7] J. A. RATCLIFFE, The Magneto-Ionic Theory and Its Applications to the Ionosphere, Cambridge University Press, 1959. [8] S. J. BUCHSBAUM, Proc. 7th Internat. Conf. Physics of Semiconductors, Paris 1964, Vol. 2 (Plasma Effects in Solids), Dunod, Paris 1965 (p. 3). [9] R. BOWERS, Proc. 7th Internat. Conf. Physics of Semiconductors, Paris 1964, 1. c. (p. 1 9 ) . [ 1 0 ] A . LIBCHABER a n d R . VEILEX, P h y s . R e v . 1 2 7 , 7 7 4 ( 1 9 6 2 ) . [ 1 1 ] P . COTTI, A . QUATTROPANI, a n d P . W Y D E R , P h y s . k o n d e n s . M a t e r i e 1 , 2 7 ( 1 9 6 3 ) . [ 1 2 ] C. R . LEGENDY, P h y s . R e v . 1 8 5 , A 1 7 1 3 ( 1 9 6 4 ) .

[13] J. K. FURDYNA, Phys. Rev. Letters 16, 646 (1966). [ 1 4 ] W . P . DRUYVESTEYN, G. J . VAN GURP, a n d C. A . A . J . GREEBE, P h y s . L e t t e r s

(Ne-

therlands) 22, 248 (1966). [15] C. C. GRIMES, Proc. 7th Internat. Conf. Physics of Semiconductors, Paris 1964, 1. c. (p. 87). 3

36

D. P. MORGAN: Helicon Waves in Solids

[16] C. A. NANNEY, A. LIBCHABER, and J . P. GARNO, Appi. Phys. Letters 9, 395 (1966). [17] J . L. DELCROIX, Introduction to the Theory of Ionised Gases, Interscience Publishers, New York 1960. [18] R. G. CHAMBERS and B. K. JONES, Proc. Roy. Soc. A270, 417 (1962). [19] J . P . KLOZENBERG, B . MCNAMARA, a n d P . C. TIIONEMANN, J . F l u i d M e c h . 2 1 , 5 4 5 (1965) [20] M . T . TAYLOR, J . R . MERRILL, a n d R . BOWERS, P h y s . R e v . 1 2 9 , 2 5 2 5 ( 1 9 6 3 ) . [21] F . E . ROSE, M. T . TAYLOR, a n d R . BOWERS, P h y s . R e v . 1 2 7 , 1 1 2 2 (1962).

[22] J . R . MERRILL, M. T. TAYLOR, a n d J . M. GOODMAN, P h y s . R e v . 131, 2499 (1963). [23] R . BOWERS, C. LEGÉNDY, a n d F . ROSE, P h y s . R e v . L e t t e r s 7, 3 3 9 ( 1 9 6 1 ) .

[24] T. AMUNDSEN, Proe. Phys. Soc. 88, 757 (1966). [25] G. N . HARDING a n d P . C. THONEMANN, P r o c . P h y s . S o c . 8 5 , 3 1 7 (1965). [26] J . R . HOUCK a n d R . BOWERS, R e v . sci. I n s t r u m . 3 5 , 1170 (1964).

[27] J . K. FURDYNA, Rev. sci. Instrum. 37, 462 (1966). [28] C. C. GRIMES, G. ADAMS, a n d P . H . SCHMIDT, P h y s . R e v . L e t t e r s 15, 409 (1965). [29] [30] [31] [32]

M. M. T. P.

H . COHEN, M. J . HARRISON, a n d W . A . HARRISON, P h y s . R e v . 1 1 7 , 9 3 7 (1960). A . LAMPERT, J . J . QUINN, a n d S. TOSIMA, P h y s . R e v . 1 5 2 , 6 6 1 ( 1 9 6 7 ) . KJELDAAS, P h y s . R e v . 1 1 3 , 1 4 7 3 ( 1 9 5 9 ) . B . MILLER a n d R . R . HAERING, P h y s . R e v . 1 2 8 , 126 (1962).

[33] J . J . QUINN and S. RODRIGUEZ, Phys. Rev. 133, A1589 (1964). [34] J . C. MCGRODDY, J . L . STANFORD, a n d E . A . STERN, P h y s . R e v . 1 4 1 , 4 3 7 ( 1 9 6 6 ) . [35] M. T . TAYLOR, P h y s . R e v . 1 3 7 , AL 145 (1965). [36] P . M . PLATZMAN a n d S. J . BUCHSBAUM, P h y s . R e v . 1 4 4 , 5 3 4 ( 1 9 6 6 ) .

[37] J . L . STANFORD a n d E . A. STERN, P h y s . R e v . 144, 534 (1966). [38] M. T. TAYLOR, J . R . MERRILL, a n d R . BOWERS, P h y s . L e t t e r s (Netherlands) 6, 159 (1963). [39] A . W . OVERHAUSER a n d S. RODRIGUEZ, P h y s . R e v . 1 4 1 , 4 3 1 ( 1 9 6 6 ) .

[40] A. W . OVERHAUSER, P h y s . R e v . L e t t e r s 13, 190 (1964).

[41] E. A. STERN, Phys. Rev. Letters 10, 91 (1963). [42] G. AKRAMOV, Fiz. tverd. Tela 5, 1310 (1963); Soviet Phys. - Solid State 5, 955 (1963). [43] C. C. GRIMES a n d S. J . BUCHSBAUM, P h y s . R e v . L e t t e r s 1 2 , 3 5 7 (1964).

[44] C. C. GRIMES, Proc. 9th Internat. Conf. Low Temperature Physics, Columbus 1964, P a r t B, Plenum Press, New York 1965 (p. 723). [45] E. FAWCETT, Adv. Phys. 13, 139 (1964). [46] S. J . BUCHSBAUM and P. A. WOLFF, Phys. Rev. Letters 15, 406 (1965). (Received

May

27,

1967)

Original

Papers

phys. stat. sol. 24, 37 (1967) Subject classification: 7; 6 ; 12 The Johns Hopkins

University,

Baltimore,

Maryland

Invariance of Sound Velocity Sums in Crystals By E . R . FITZGERALD a n d T . W . W R I G H T

A relation between t h e lattice dissociation energy D of a monatomic crystal, t h e atomic mass m, and a mean crystal sound velocity c s recently suggested b y Fitzgerald in connection with his particle-wave description of non-elastic deformation in solids is reviewed. T h e cited relation D = m c¡ implies t h a t certain sound velocity sums in crystals are invariant a n d this is shown to be t h e case for crystals in which small atomic displacements can be described b y linear elasticity. The proposed relation between lattice dissociation energy a n d a mean squared sound velocity is therefore entirely consistent with classical elasticity t h e o r y although, of course, n o t predicted b y it. Eine Beziehung zwischen der Gitterdissoziationsenergie D eines monoatomaren Kristalls, der Atommasse m u n d einer mittleren Schallgeschwindigkeit c s wird diskutiert, die kürzlich von Fitzgerald in Verbindung mit einer Partikelwellenbeschreibung der nichtelastischen Deformation in Festkörpern vorgeschlagen wurde. Die angeführte Beziehung D — m c | beinhaltet, daß gewisse Schallgeschwindigkeitssummen in Kristallen invariant sind. E s wird gezeigt, daß dies f ü r Kristalle der Fall ist, in denen atomare Verschiebungen durch eine lineare Elastizitätstheorie beschrieben werden können. Die vorgeschlagene Beziehung zwischen Gitterdissoziationsenergie und einer mittleren quadratischen Schallgeschwindigkeit ist deshalb vollständig konsistent mit der klassischen Elastizitätstheorie, obwohl sie natürlich durch diese nicht vorhergesagt wird.

1. Introduction A connection between the dissociation energy of a monatomic crystal lattice and a mean sound velocity for the crystal has been presented by Fitzgerald [1, 2] such that D = mcl,

(1)

where D is the lattice dissociation energy per atom, m is the atomic mass, and ca is an r.m.s. sound velocity which can be obtained from the sum of three orthogonal sets of one longitudinal and two transverse velocities each (i.e., a total of nine sound velocities). Good agreement [2] with equation (1) has been shown for eighteen f.c.c. and b.c.c. metal crystals with atomic masses ranging from 3.82 to 38.5 X10" 23 g and dissociation energies from 1.13 to 10.3 eV/atom. Because of their symmetry cubic crystals may require a knowledge of only two sound velocities [2] to obtain c8, but as demonstrated in this article, at least four sound velocities are needed to calculate cs for hexagonal crystals. The relation between lattice dissociation energy and sound velocity described by equation (1) was originally advanced in connection with a particle wave description of deformation in solids [1,2] and, in particular, was used to calculate a fission velocity, vf, for comparison with hypervelocity impact tests in which craters are formed by a projectile striking a target at high speeds. Values of fission velocity obtained were in good agreement with the experimental data available for ten metals [2].

38

E . R . FITZGERALD a n d T . W .

WRIGHT

I n the present article ideas used to suggest the existence of the relation expressed by equation (1) are reviewed and equation (1) is shown to be entirely consistent with the classical elasticity theory of sound wave propagation in solids. T h a t is, the relation D = m c\ implies t h a t certain sound velocity sums are invariant and this is shown to be the case for a linear elastic medium. 2. Association of Phonons with Vibrational Modes The vibrational energy of a crystal containing N atoms is often taken [3, 4] as equal to the energy of a system of 3 N harmonic oscillators when attention is confined to linear elastic interactions between atoms. Following Einstein [5] the harmonic oscillators may be considered to be quantized and hence the possible energies of an oscillator are given b y [6] UQ = (q+\)hve-,

q = 0,1,2,3,...,

(2)

where ve the frequency of oscillation of a particular harmonic oscillator of the system. Assuming a Boltzmann distribution of oscillators among the available energy levels, the average energy per oscillator a t any temperature T is readily shown to be [3] +

(3)

where k is Boltzmann's constant. The first term of (3) represents the average zero-point energy of the oscillators (atoms) since this term remains when T = 0. This, of course, is in agreement with the quantum mechanical concept [6] t h a t atoms in a crystal lattice have vibrational energy even at absolute zero. W i t h each elastic or vibrational mode of frequency ve a certain number of sound quanta or phonons [3] can be associated. If h ve is taken to be the energy of each phonon, then the number of phonons ne to be associated with any vibrational frequency v0 will be _ _ e

hve

1 2

1 ehvelkT

_

1 '

W

The results expressed in (4) are suggested by analogy with light quanta or photons to which certain particle aspects are also ascribed. T h a t is, electromagnetic waves of frequency v are said to be equivalent to particles or photons of energy h v with momentum h v/c where c is the propagation velocity of light [3]. I n the same way a phonon momentum given by h vjca can be defined where cg is the elastic wave or sound propagation velocity. From (4) it is also clear t h a t lattice phonons of two types occur, viz. intrinsic (non-thermal) phonons and thermal phonons. The intrinsic phonons exist even when T = 0 and hence are always present in a constant amount (1/2 phonon per mode). B u t as stated a t the outset, each lattice atom has three vibrational modes associated with it, so t h a t each lattice atom corresponds to 3/2 intrinsic phonons. There will then be three intrinsic phonons associated with every two lattice atoms and this suggests the schematic drawing of Fig. 1 in which two lattice atoms are shown as characterized by one longitudinal and two transverse phonons with sound propagation velocities ct, and cta, ct¡, respectively [2], Although two lattice atoms are associated with the three phonons, only one atomic mass is involved since one-half of each lattice atom mass can be assigned

39

Invariance of Sound Velocity Sums in Crystals Fig. 1. Diagram showing how one longitudinal phonon (velocity ci) and two transverse phonons (velocities cta and cta) can be associated with any two-atom combination in a crystal lattice. Each atom in the combination contributes a mass, mj2, to the isolated vibrational modes, so t h a t the total phonon mass of such a two-atom system is m

Cta t

-CD--0—Ci—«©/ Ctb

i I

to the adjoining atom pair as indicated in Fig. 1. The advantage of considering three intrinsic phonons to be associated with every two lattice atoms is t h a t a distinction between longitudinal and transverse phonons (vibrations) can be maintained in this case. The 3/2 phonons of a single atom, on the other hand, cannot be easily unambiguously assigned to longitudinal or transverse modes. Three intrinsic phonons of the type depicted in Fig. 1 are sufficient to describe an isotropic material or crystals with particular symmetries, but are clearly insufficient for arbitrary anisotropic solids. Instead three mutually orthogonal three-phonon sets are needed as adduced in Fig. 2. Thus in general a linear elastic medium or small atomic displacements from equilibrium in a crystal lattice of the type under discussion can be characterized by a set of nine intrinsic phonons consisting of three longitudinal and six transverse phonons. The corresponding sound propagation velocities will then consist of three longitudinal velocities and six transverse velocities as given below: C l l , C t i a , Ctl&j' CZ2> Ct2d) C t 2 & ; C¡ 3 , Cfc3 a , C t 3 j , .

The nine intrinsic phonons in turn correspond to 18 vibrational modes of 6 lattice atoms since each atom has 3 lattice vibrational modes and each mode is equivalent to 1/2 intrinsic phonon. For an arbitrary direction in a lattice the vibrational modes (and associated phonons) will not be necessarily purely longitudinal and purely transverse, but we will continue to refer to them as such with the understanding that they m a y be actually quasi-longitudinal and quasi-transverse in many cases. There will of course be one or more specific directions in which purely longitudinal and purely transverse vibrational modes are present.

Fig. 2. Sketch showing how an anisotropic crystal must be characterized by three sets of phonons in the mutually orthogonal directions, x u x l t A mean sound velocity cB to be associated with the 3/2 phonons of each atom can be calculated f r o m the nine velocities indicated according to equation (9) of the text

40

E . R . FITZGERALD a n d T . W . WEIGHT

3. Mass-Energy Equivalence for Atoms in a Crystal Lattice From this consideration of intrinsic phonons in a crystal lattice it is possible to obtain a mass-energy equivalence for lattice atoms [1, 2]. Consider first a sufficiently symmetrical crystal (e.g., a cubic) so that only three intrinsic phonons are needed to characterize the material. Then if any two atoms are sufficiently separated by removal from the lattice or otherwise, three phonons will be destroyed. The mean squared sound velocity of the three phonons can be written as „2

C

S —

c

? + 4a + 4b Q

ft-*. 0

'

I J

As indicated in Fig. 1 and the preceding section only one atomic mass is to be associated with each three phonons. Hence the "phonon" mass per atom is m J 2 where m is the actual mass of a lattice atom. This follows since three intrinsic phonons are associated with every two lattice atoms. In any case the annihilation of three phonons with mean sound velocity c3 might be expected to result in the release of a vibrational energy per atom of -, — cl m

or

m 2ci,

where cl = c^/2. Now if a solid lattice is vaporized by supplying heat energy equal to D per atom the final result is the destruction of all intrinsic lattice phonons, and the separated atoms in the vapor have an energy of D per atom above their original energy at 0 ° K in the lattice. Hence it seems reasonable to infer that the destruction of intrinsic (zero-point) lattice phonons will result in the release of a vibrational energy per atom equal to the dissociation energy, D , per atom such that D — m cl ,

(6)

where m is the mass of a lattice atom and ct is a mean squared sound velocity given, for example, by « _

s

_ c? + 4a + 4b

~ Y ~

6

•

m

The form of equation (6) is, of course, suggested by analogy with the relation between energy, mass, and light velocity, c, from Einstein's special theory of relativity : E = mc 2.

(8)

In the most general case a crystal lattice requires nine intrinsic phonons for adequate characterization as indicated in Fig. 2. Hence a mean sound velocity Cg for the intrinsic lattice phonons can be given in terms of three sets of one longitudinal and two transverse phonons each, Cs = (c?i+ c\la + c\2a + c?2 + cf 2 a + c?26 + cfs + C?3a + c\ S6 )/9 ,

(9)

where cl = c\j2 as before and (6) then holds for the most general case when cl is calculated from (9) above. The relationship between lattice dissociation energy per atom and a mean sound velocity described by equation (6) has been presented previously [1, 2],

41

Invariance of Sound Velocity Sums in Crystals

but an additional consequence of it remains to be noted. That is, if the average dissociation energy per atom, D, of a crystal lattice is to be a constant of the material the existence of the relationship D = m c\ implies that cl is also a constant. Hence it follows that the sum of any three mutually orthogonal sets of one longitudinal and two transverse squared sound velocities (a total of nine squared sound velocities) must be constant in an elastic medium. I n the case of a cubic crystal or other solid where only three intrinsic phonons are necessary c'(a + to calculate a value for cs2, the corresponding result is t h a t the sum cf must be invariant for any direction in the crystal. Quite aside from any quantitative agreement obtained for various crystals by setting D = m cl, the proposed relation can then be checked by determining if the relevant velocity sums are invariant according to classical elasticity theory and also from actual measurements of sound velocities in various directions in a crystal. The first of these checks is presented in the next section. 4. Invariance oí Sound Speed Sums from Classical Elasticity Theory The differential equations governing the displacements u t in a linear elastic homogeneous continuum of density q can be written in cartesian coordinates as [7] CÍ j iU k J j = Q ^ ,

(10)

where C\\ is the modulus tensor and i, j,k,l — 1, 2, 3. We assume a plane harmonic wave solution for equation (10) of the form Ut = Ak e a(vj-et)

(11)

;

where Ak are components of the displacement amplitude (Al7 A2 , A3 ), c is the sound (phase) velocity in a direction of propagation specified by a unit vector a (a¡ are direction cosines of the normal to the wave front), and k = 2 Ttjk is the wave number. Substituting (11) in (10) the wave speeds (sound velocities) for the medium can be determined as eigenvalues of the acoustic tensor *) C\\ a,] at, where {C{{ ajai -Q

c 2 d\) A

= 0 .

(12)

6\ is the Kroenecker delta ( such t h a t the unit vectors aW = [100], [010], and [001] for a = 1, 2, 3 respectively. Then if the trace of the acoustic tensor is summed over the three directions a, ¿ C\i afJ ) a{«> = C¡¡ = ¿ i e(cg)» . a=l a=l/S = l Also called the Christoffel stiffnesses (cf. [7]).

(14)

42

E . R . FITZGERALD a n d T . W . WRIGHT

C\ I is invariant under changes of coordinates and is independent of the choices for aM. Hence we conclude from (14) that the trace of the modulus tensor (C\i) is equal to the density q times the sum of the nine squared sound velocities (e.g., in particular cases three longitudinal and six transverse) which can be associated with any three mutually orthogonal, but arbitrarily chosen directions in the material. Since C|| is invariant it follows directly that Í

Í

a = l /S = l

(c}f

Q

const

(15)

which is equivalent to the requirement following from equations (6) and (9) of the preceding section in which c\ = c¿lt c] = Ctu, = c t u , etc. It can be shown that the necessary and sufficient condition that the trace of the acoustic tensor (equation (13)) be independent of direction a is that it have the form = si¿eK)2(16) «=i A cubic crystal meets this requirement and thus the sum of three (orthogonal) squared sound velocities is invariant as has been pointed out, for example, by Miller and Musgrave [8] and also discussed by Hearmon [7]. That is, in a cubic crystal the independent modulii are Cu, C12, and C44 and the amplitude relations given by (12) can be written c¡Í

(al f — H)A1 + at a2g A2 + a3axg As = 0 , a 2 ax g A1 + (a¡ / — H) A2 + a2 a3 g A3 = 0 , . (17) a A A H 0 «3 i 9 i + «a «3 9 2 + («3 / - ) = > where Cn — C44 = /, C12 + C44 = g,Qc2 — Cu = H. For non-zero solutions (A1, A2, A3) of equations (17) to exist the determinant of the coefficients must vanish, and this leads to the conditions on H (and hence velocities, c) that H*-1H* + K{j + g)FH-H*(f + 2g)Q = 0, (18) where h = / — g, F = a\a\ + a\a\ + a\a\, G = a\ a\ a\. The sum of the three roots Hi (where c¿ = c¡, cta, cti>) of equation (18) turns out to be equal to /, and thus [8] cf + cL + ch = (19) e so that the sum of the squares of the three sound velocities for any direction is invariant in a cubic system. A knowledge of the elastic modulii Cu and C44 was, in fact, used to calculate cl for cubic crystals previously considered [2] in connection with equations (6) and (7). A completely isotropic material can always be characterized by one longitudinal sound velocity and two identical corresponding transverse sound velocities. In an anisotropic medium, the displacement amplitudes Au A2, A3 of the elastic waves are always orthogonal [7], but their directions are not necessarily coincident with (or in any fixed relation to) the direction ax, a2, a3. Hence the nine sound velocities in question in general refer to semi-longitudinal and semi-transverse waves. In many anisotropic media, however, it is possible to select three mutually orthogonal directions with which three entirely longitudinal and six entirely transverse waves may be associated.

43

Invariance of Sound Velocity Sums in Crystals

I n any case, the requirement that certain sound velocity sums in a crystal be invariant following from the suggested [1, 2] relationship between lattice dissociation energy, atomic mass, and an r.m.s. sound velocity (viz. D = m c\) is seen to be in agreement with classical elasticity theory. As long as our attention is confined to elastic waves described by small atomic displacements from equilibrium and wavelengths large compared to interatomic distances, the classical theory of elasticity is appropriate, and the agreement, therefore, is gratifying. 5. Conclusions The general equation for the invariant sum of nine squared sound velocities given by equation (15) can be reduced for a cubic crystal to an expression in which three squared sound velocities are invariant as described by equation (16). For particular directions in a cubic crystal (100, 010, and 001) two of the velocities are equal so that equation (19) results. That is, the invariant squared sound velocity sum can be determined from a knowledge of only two sound velocities or two elastic modulii (On and C44). For a hexagonal crystal a similar reduction in the number of required sound velocities needed to compute the nine squared sound velocities' sum is possible if the three mutually orthogonal directions chosen consist of one coincident with the z-axis and two lying in the basal (x y) plane. In this case Cn

3

i = l Cli + where At is the position index of the i-th impurity ion in the lattice, t is the ionic type (with taking into account not only the ionic specification but also its environment conditions), r is the irreducible representation of the group Gt, ¡x its row, a additional quantum numbers (distinguishing, in particular, the repeating representations F), and q combination of space and spin coordinates of all nt electrons of the »-th ion. These functions are assumed to be antisymmetric. Every element g of the group Gk has its corresponding impurity ion permutation Pg which can be always presented as a product of independent cyclic permutations g^Pg

= Pa{A1A2...Arn)-....

(7)

Here and in what follows all the remaining cycles (except for a single one A1A2... Am) are denoted by points at the end. It is clear that every cycle can be composed only of the same species of ions t (equivalent ions). Antisymmetric eigenfunctions of operator H0 can be written as A^to-i Tji«, . . . Amtxmrmiim\ .. ,(