Physica status solidi: Volume 22, Number 1 July 1 [Reprint 2021 ed.] 9783112494868, 9783112494851

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plxysica status solidi

V O L U M E 22 • N U M B E R 1 • 1 9 6 7

Classification Scheme 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 o. 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 Impurity 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 Thermomagnetic 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)

phys. stat. sol. 22 (1967)

Author Index S . G. ABDINOVA SH. SH. ABELSKII P . AGULLÓ-LÓPEZ A . ALDEA M. I . ALIEV K.H.ANTHONY S . ARAJS J . ARENDS G . E . ARKHANGELSKII B . AUTIN

741 309 483 377 153, 7 4 1 667 737 131 289 K135

D . J . BAILEY M. BALARIN J . BARTHEL P . BAUDUIN CH. L . BAUER C. BAUMBERGER U . BERTELSEN K . - H . BERTHEL H . BILGER A . BIVAS A. Y A . BLANK R . BOGDANOVIÓ V . L . BONCH-BRUEVICH A . J . F . BOYLE A . BRAGINSKI P . BRÄUNLICH H . BROSS F . R . BROTZEN F . BROUERS

607 123 K151 K135 199 K67 59 K151 683 K155 47 603 267 K131 K127 391 667 9 213

J . E . CAFFYN 0 . CAHEN B . CHELUSTKA E . W . CLAFFY L . N . COJOCARU M . V . COLEMAN P . COSTA

549 K135 K95 71 361 593 349

A . DALL'OLIO R . V . DAMLE G. DASCOLA W . DEGRIECK J . DEMNY K . DETTMANN I . DÉZSI E . DIEULESAINT 1. DIMA 2. DIMITRIJEVIC B . DREYFUS V . A . DROZDOV

365 K63 365 177 K1 423,433 617 K135 K79 K55 77 K109

O. N . EFIMOV U . EICHHOFF

297 K91

N. L. V. P. K.

A . EISSA ELSTNER V . EREMENKO R . ERRINGTON R . EVANS

617 541 65 473 195, 6 0 7

V . L . FALKO L . FIERMANS H . J . FISCHBECK R . M. FISHER W . F . FLANAGAN W.FRANZ V. FREI G. N . F U R S E Y

319 463 235, 649 473 195, 607 K139 381 39

J . A. GALLOWAY A. GEORGIEVA L . GERWARD A . M. DE GOER T . L . GOODFELLOW D . GOUVERNELLE P.W.GRAVES J . B . GRÜN G. GRÜNER G. D . GUSEINOV E . GUTSCHE

491 415 659 77 549 K135 499 K155 KLL K117 229

H . W . DEN HARTOG F . HÄUSSERMANN J . O. HENNINGSEN W . HENRION J . HEYDENREICH A . HUBERT G. P . HUFFMAN

131 689 441 K33 93 709 473

Y A . A . IOSILEVSKII Y U . P . IRKHIN M. Z. ISMAILOV N. R . IVANOV V . G. IVANOV G. JACOBS S . C. J A I N K . JEGES H . L. JETTER B . JOUFFREY H . - G . KAHLE H . KALBFLEISCH W . KAMPRATH E . A . KANER E . M. KERIMOVA L . KESZTHELYI

255 309 K117 279 39 177 505 K7 K39 349 537 537 541 47,319, 333 K117 617

748 R . Y U . KHANSEVAROV E . KTERZEK-PECOLD J . M. KNUDSEN J . KOLODZIEJCZAK P . KORDO§ B . P . KOTRUBENKO E.J.KRAMER S . KRASNICKT H . KROGH H . KRONMÜLLER L . S . KUKUSHKIN D . KULGAWCZUK V . G. KURDYUMOV K . LAL H . LANGE V . N. LANGE J . LECIEJEWICZ G. LEIBFRIED R . LEVY I . LICEA A. LODDING E . MAJERNIKOVA M. M. MARKUS J . L . MARTIN T . V . MASHOVETS A. MASYROWICZ M. MATYAS E . V . MATYUSHKIN GH. MAXIM K.MEYER E . MÖHLER B . MOLNÄR D.J.MORGAN Z. L . MORGENSHTERN L . M. MORGULIS L . A. MOZGOVAYA A. MUKERJI B . MUKHERJEE A. MÜRASIK D . NENOW V . B . NEUSTRUEV S . NIKITINE H . NORDEN D. E. H. K. A.

OBRIKAT M. OLYMBIOS OPPERMANN OTSUKA OTTO

N . PASCHOFF J . PASTRNÄK

Author Index . . . .

K95, K103 K147 59 K147 K59 K15 199 K55 59 683 65 617 K75 141 229 K15 517 423,433 K155 147 157 113 K15 349 K95, K103 K155 K141 65 K113 K123 K49 617 491 289 K75 K109 K19 K131 517 415 289 K155 631 K123 549 K151 559,579 401 83,

93 407

M. PAULUS M. PAUNOV B . PEGEL J . PELLEG D . J . PERRIN B . PERSSON V . PETRESCU F . POLLY I . PRACKA H . PUSZKORSKI F . RAGA A. M. RAMAZANZADE R . RATTKE B. RAY P . REIMERS 0 . P . REVOKATOV F . E . ROBERTS J . L . ROUTBORT M. RÜHLE H . S . SACK G. I . SAFARALIEV K . SCHROEDER G. Y A . SELYUTIN L . A. SHUVALOV W . SIMONET Y . G . SKOBOV 1. L . SOKOLSKAYA G. D . SOOTHA H . N . SPECTOR M. O. SPEIDEL P . M . SPENCER E . STEIN M . - P . STOLL S . STONKUS J . STURE L . SVOB G. A . TANTON J . TAVERNIER B . TAVGER D . J . D . THOMAS J . TODOROVIÓ K . TOMPA F . TÓTH R . TROÓ

K87 103 223, K 4 5 K83 549 631 K113 K123 K147 355 K155 K117 123 371 K27 K91 373 203 689 203 741 423,433 K23 279 K87 333 39 505 185 K71 371 537 163 K3 K49 K141 K19 K135 31 593 K55 KLL KLL 517

L . M . UTEVSKII

K75

J . VAITKUS V . VARACCA G. VASILIU J . VENNIK J . VISCAKAS

K3 365 K79 463 K3

Author Index V . VÎTEK N . A. VITOVSKII A . N . VOLOSHINSKII V U DINH K Y R . VU H U Y DAT A. C. R. G.

WANIC M. WAYMAN K . WEHNER WEISE

453 K95, K103 309 729 K67

559,

K55 579 527 K151

749

M. WLLKENS J . E . WILLIAMS V . WITT H . C. WOLF G. WOLFRAM

689 K19 245 K39 245

S. A. G. A.

153 123 K49 517

A. ZEINALOV ZETZSCHE ZIMMERER ZYGMUNT

Centre de documentation sur les synthèses christallines Crée en 1965, sous les auspices du C.N.R.S. et avec l'aide du C.E.A. le centre a établi un répertoire des noms des laboratoires industriels ou d'université où sont fabriqués des monocristaux (à l'exception des cristaux organiques). Un fichier de cartes perforées permet un tri rapide pour le cristal demandé ; 250 laboratoires environ, en France et à l'Etranger, sont inscrits dans nos listes, et une première publication de documents a été faite en Mars 1967, dans le Fascicule d'Information Technique du C.E.A. Pour tout renseignement concernant un cristal s'adresser à Professeur Mlle Vergnoux, Départment de Physique Cristalline, Faculté des Sciences, 34 — Montpellier (France)

Documentation Centre for Synthesis of Crystals The centre which was founded in 1965 under the sponsorship of the "Centre National de la Recherche Scientifique" (C.N.R.S.) and with support of the "Commissariat à l'Energie Atomique" (C.E.A.) has issued an index of the industrial and university laboratories in which single crystals (except organic crystals) are produced. A system of punch-cards allows one to find out very quickly the crystalline material asked for; nearly 250 labs in France and abroad are contained in our lists, and the first issue of documents has appeared in March 1967 in the "Fascicule d'Information Technique du C.E.A.". For all information concerning a crystalline material apply to Prof. Vergnoux, Départment de Physique Cristalline, Faculté des Sciences, 34 — Montpellier (France)

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, 0 . 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 , Paris, 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. D I E T Z E , Saarbrücken, J.D. E S H E L B Y , Cambridge, P. P. F E 0 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. M A T Y A S , 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 22 • Number 1 • Pages 1 to 340, K1 to K66, and Al to A34 July 1, 1967

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Contents Page

Review Articlc F. R.

BROTZEN

Emission of Exoelectrons from Metallic Materials

9

Original Papers B . TAVGER

Transport Phenomena in Semiconducting Thin Films

G . N . F U R S E Y , I . L . S O K O L S K A Y A , a n d V . G . IVANOV

Field Emission from p-Type Germanium

31

39

A. YA. BLANK and E . A. KANER

Waves with Discrete Spectrum in Metals in the Vicinity of the Cyclotron Resonance Frequency

U . B E R T E L S E N , J . M . K N U D S E N , a n d H . KROGH

Mossbauer Effect in FeF 3

E . V . MATYTJSIIKIN, L . S . KUKTTSHKIN, a n d V . V .

E.

W . CLAFFY

PASCHOFF

59 EREMENKO

Peculiarities of the RbMnF 3 :Nd 3 + Crystals Decay Kinetics due to the Migration of Excitation Energy

65

Impurity Absorption Bands in Thermoluminescent LiF

71

A. M. DE GOER a n d B .

N.

DREYFUS

y-Irradiation Effect on the Thermal Conductivity of A1203 with Cr or Mn Impurities at Low Temperature

77

Wachstum und Struktur dünner, aus schmelzflüssigen Filmen auskristallisierter Germanium-Kristalle

83

N . PASCHOFF u n d J .

HEYDENREICII

Zur Versetzungsanordnung in dünnen, durch Kristallisation gewonnenen Germanium-Folien

M . PAUNOV

E.

MAJERMKOVÄ

47

Feldelektronenmikroskopische keimbildung

Untersuchungen

der Quecksilber-

Faraday Effect and Spin-Orbit Interaction of the Lowest Exciton States in Molecular Crystals

M . BALARIN, R . R A T T K E u n d A . ZETZSCHE

Zur Auswertung isochroner Ausheilkurven

H . W . DEN HARTOG a n d J .

ARENDS

93 103

113 123

F-Centers in SrF 2

131

K. LAL

Dielectric Properties of Interstitial Impurity Cations in Alkali Halide Crystals

141

I.

Electrical Conduction of Semiconducting Thin Films in Strong Electric Fields

147

LICEA

S . A . ZEINALOV a n d M . I . A L I E V

A. l*

LODDIKG

Investigation of the Thermal Conductivity of InSb-GaSb Solid Solutions "

153

Thermotransport and the Associated Isotope Effect in Solids . . .

157

4

Contents

M.-P.

The Use of the Magneto-Optieal K e r r Effect for the Determination of Magnetization Curves

STOLL

Page

163

W . DEGRIECK a n d G . JACOBS

H. N.

SPECTOR

Luminescence and Photoconductivity of the B-Centre in Additively Coloured KCl:Ag Crystals

177

Optical Absorption in Electric and Magnetic Fields

185

K . R . E V A N S a n d W . F . FLANAGAN

An Analysis of the Cottrell-Stokes Law

195

E . J . KRAMER a n d CH. L . B A U E R

Variations of Young's Modulus throughout the Mixed State in Niobium

199

J . L . ROÜTBORT a n d H . S . SACK

F.

BROUERS

B. PEGEL,

Low Temperature Internal Friction of Aluminium a t 1 Hz . . .

203

Plasmon Satellites of S o f t X - R a y Spectra

213

On the Isotope Effect in Interstitial Diffusion

223

E . GUTSCHE a n d H . L A N G E

Electroreflectance of CdS and CdSe Single Crystals a t the Fundamental Absorption Edge

229

Theory of Weakly Bound Bloch Electrons in a Magnetic Field (I)

235

H . J . FISCHBECK

G . WOLFRAM a n d V . W I T T

Kinetics of the Thermal Mutual Conversion of F and Z 2 Color Centers in KC1: Sr Crystals

245

Y A . A . IOSILEVSKII

Some Model Problems on the Dynamics of Two-Component Crystals

255

V . L . BONCH-BRUEVICH

On t h e Theory of Electrical Domains in H o t Electron Semiconductors

267

L . A . SHUVALOV a n d N . R . IVANOV

The Second Ferroelectric Phase Transition in NaH 3 (Se0 3 ) 2 Crystals and their Physical Properties

279

G . E . ARKHANGELSKII, Z . L . MORGENSHTERN, a n d V . B . NETJSTRUEV

O. N.

EFIMOV

Colour Centres in R u b y Crystals

289

Contribution of Thermal Vibrations to the Anomalous Transmission of X - R a y s

297

Y U . P . I R K H I N , A . N . VOLOSHINSKII, a n d S H . S H . A B E L S K I I

On the Theory of Spontaneous Hall Effect a t Low Temperatures

.

309

E . A . K A N E R a n d V . L . FALKO

Magnetoacoustic Magnetic Fields

Resonance

Effects in

Metals under

Inclined 319

E . A . K A N E R a n d V . G . SKOBOV

On the Possibility of Existence of Quantum Electromagnetic Waves in Metals

333

Contents

5 Page

Short Notes J . DEMNY

Stacking Faults in Tungsten

KL

J . VISÖAKAS, J . VAITKUS, a n d S . STONKUS

K . JEGES

Influence of Heat Treatment on the Spectral Dependence of Photoconductivity in CdSe Crystals

K3

Elektrolumineszenz von Sn0 2 -Einkristallen

K7

K . TOMPA, F . T Ö T H , a n d G . GRÜNER

Susceptibility of MnO Measured by the NMR Method

Kll

B . P . KOTRUBENKO, V . N . LANGE, a n d M . M . MARKUS

Chemical Bond and Structure of the Ternary Compound InAsTe

. K15

A . M U K E R J I , G . A . TANTON, a n d J . E . WILLIAMS

X-Ray Induced F-Centers in CaF 2

K19

G. YA. SELYUTIN

On the Shift and Intensity of the Mössbauer Line in Metals

. . . K23

P . REIMERS

Änderung der Eindringkurven bei Diffusionsmessungen durch Verdampfung des Diffusionsmaterials K27

W . HENRION

Reflectivity Measurements on Trigonal Selenium Single Crystals in the Spectral Region between 1.6 and 6.0 eV K33

H . L . J E T T E R u n d H . C. W O L F

Zur Frage der Davydov-Aufspaltung des ersten Singulett-Anregungszustandes von Anthracen-Kristallen K37 B . PEGEL

On the Isotope Effect in Interstitial Diffusion II. Extended Model .

K45

E . MÖHLER, J . STUKE, a n d G . ZIMMERER

Some New Features of the Reflectance Spectrum of Trigonal Selenium Single Crystals K49 2 . D I M I T R I J E V I C , S . K R A S N I C K I , J . TODOROVIC, a n d A . W A N I C

Neutron Investigation of Magnons in Magnetite P . KORDOS

R.

V.

DAMLE

K55

Heat Treatment of InSb Crystals with Different Dislocation Densities K59 Lead Zirconate by Coprecipitation

K63

Pre-printed Titels and Abstracts of papers to be published in this or in the Soviet journal ,,®K3Hna T B e p a o r o T c ; i a " (Fizika Tverdogo Tela)

A1

6

Contents

Systematic List Subject Classification:

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

1.2 1.4 3.1 3.2 5 6 6.1 7 8 9 10 10.1 10.2 11 12 13 13.1 13.2 13.3 13.4 14.1 14.2 14.3 14.3.1 14.4 14.4.1 14.4.2 15 16

199, 279 83, 93 83, 103 K15 255, 297, K15 223, 255, 297, K 4 5 59, K23 319 77 157, 223, K27, K45, K59 77, 93, 123, 141, 255, K59 195, 199, 203, K1 131, 177, 245, 289, K19 77, K19, K37 K37 147, 213, 267, 309, 333 31, 47, 185, 229, 235, 319, 333, K33, K49 113, 229, K33, K37 9 65, 131, K 3 309 199 267, K59 31, 147 141 185, 229, 267 279, K63 31 177, K3

17 18

9,39 113, 185, 199, 235

18.1

Kll

18.2

59, 163, 309

18.3 18.4

K55 59, K l l

19

47, 131, 319,

20

113, 163, 213, 333

Kll

20.1

71, 185, 229, 289, K33, K37,

20.2

289

20.3

65, 71, 177, 289, K7

K49

Contents 21 21.1 21.1.1 21.2 21.6 22 22.1.1 22.1.3 22.2.1 22.2.3 22.4.1 22.4.2 22.5 22.5.2 22.5. 3 22.6 22.8 22.9

9, 47, 103, 199, 203, 213, 319, 333, K l , K23 195 163 157, 213 K27 65, 157, 267 39, 83, 93, 267, 297 K33, K49 267 153, K59 229 229, K3 59 71, 141, 177, 245 131, K19 77, 289, K7, K l l K15, K63 113, K37

Contents of Volume 22 Continued on Page 343

7

Review

Article

phys. stat. sol. 22, 9 (1967) Subject classification: 17; 13.3; 21 Rice University, Houston,

Texas

Emission of Exoelectrons from Metallic Materials By F . R . BROTZEN

1.

Introduction

2. Recent

2.1 2.2 2.3 2.4 2.5 2.6

Contents

developments

Emission after abrasion of metals Emission from freshly evaporated metals Emission during and after plastic deformation Emission after quenching Emission after irradiation Electron emission during transformations

2.7 Chemiemission 3.

Conclusion

References

1. Introduction When the Exoelectron Conference took place in Innsbruck in September of 1956, about six years had passed since the first comprehensive reports of delayed emission phenomena had been published by Kramer [1], Considering the relatively short time that had elapsed since that event, the Conference revealed in an impressive manner the different ways in which exoelectron emission can be stimulated and proposed several hypotheses by which the phenomenon might be explained. Though ten years have passed since the Conference, definite answers to some of the most fundamental questions concerning exoelectron emission are still lacking. Specifically, by 1956 it had become quite clear that in the case of metallic materials the emission was connected with surface layers, such as oxides or adsorbed gases. The possibility that an interaction of the metal surface with oxygen was the fundamental cause for emission had already been discussed by Haxel, Houtermans, and Seeger [2], had been formulated by Roubinek and Seidl [3], and described by Seeger [4]. Y e t , the precise mechanism of ejection of the electrons and the exact role played in the process by the oxide or the adsorbed layer were not clearly understood. The state of affairs in 1956 was briefly like this: Grunberg and Wright [5] found an emission maximum under irradiation with light of 4700 A in abraded, in strained, and in evaporated alu-

10

F . R . BROTZEK

minum. In an approach similar to that suggested by Nassenstein [6] and Bohun [7], they attributed this unusual photoelectric sensitivity to the creation of color centers in the aluminum-oxide surface layer. Grunberg and Wright considered specifically F'-centers, i.e., oxygen vacancies at which two electrons were trapped. During deformation the required vacancies could be formed mechanically in the oxide layer in anion-cation pairs. The emission thus takes place through photo-excitation of these centers, which give up their energy to nearby shallow surface centers from where electrons are emitted. The decay of the emission is accelerated by increased oxygen pressure, as the anion vacancies essential for the process are occupied by oxygen. After careful investigation of emission from copper and iron films under oxidizing conditions, Seidl [8] suggested that electron traps were created by the adsorbed oxygen. Bohun, Karpiskovâ, and Duskovâ [9] also considered the emission to be connected with the dissociation by photo-excitation of color centers in the oxide layer. A corroborating feature to this proposition was the similarity of luminescence and exoelectron emission which Kramer [10], Bohun [7], Kantûrek [11], and Lepper [12] had found in insulating crystals. In terms of a comprehensive band picture, Hanle [13] formulated this analogy clearly at the Innsbruck Conference. On the other hand, Muller and Weinberger [14] proposed that deformation, abrasion, or evaporation tended to expose clean metal to oxygen adsorption. The adsorption lowers the photoelectric work function while progressive formation of the oxide layer tends to create the opposite effects. Their ideas were supported by measurements of threshold values during decay after deformation of aluminum single crystals. Lohff and Raether [15] and Lohff [16], having worked with numerous metals in high vacuum, found also impressive evidence for emission through chemisorption or oxidation of the clean, freshly abraded surfaces. Yet, in the case of straining zinc single crystals in high vacuum the emission was independent of the oxygen pressure [17], The role that surface layers play in the delayed emission from metals after irradiation had become quite clear as early as 1940 through the work by Tanaka [18]. Furthermore, Kusterer and Bruna [19] showed that nickel oxidized in air emitted readily after X-radiation, while vacuum-heated nickel failed to produce emission. Working with tungsten and using an electron-multiplier tube, Seeger [20] demonstatred that cold work of the specimen surface was necessary to produce emission after bombardment of the foil with electrons of an energy of 1000 eV. Annealing in vacuum at 1000 °C deactivated the foil, and emission after electron bombardment could be restored only through oxidation in air and abrasion. Seeger referred to Nassenstein [6], who had suggested that traps in the oxide were lattice defects which produced electron states of sufficiently high energy to make thermal emission near room temperature possible. The reoccupation of the vacated traps would be accomplished through electron or photon irradiation. At the Conference in 1956, Muller [21] demonstrated in a conclusive way the significance of lattice defects through the observation that cuprous-oxide films, whose X-ray pattern revealed a strongly disturbed lattice, were far more sensitive to emission after X-radiation than films with a nearly perfect structure. Emission during phase transitions had been observed by Kramer [1], who linked it to the exothermic nature of the transformations. A comprehensive study of the lead-tin system was carried out by Futschik, Lintner, and Schmid [22]. They found emission peaks upon cooling of pure lead and pure tin at the

11

Emission of Exoelectrons from Metallic Materials

melting points as well as at the eutectic and solvus temperatures of the alloys. Brotzen [23] also found peaks during solidification in cadmium and tin but was unable to detect any anomalous emission during solidification of a lead-tin alloy of eutectic composition. He noted that emission during solidification was observable only whenever a visible oxide layer covered the melt. The work by Bathow and Gobrecht [24] removed any doubt concerning the need for an oxide layer for successful emission during solidification of a number of metals. Research at this stage showed the unquestionable need for thorough further investigations of the nature of the emission from the surface layer and of possible interactions between the base metal and the surface layer. 2. Recent Developments 2.1 Emission

after abrasion

of

metals

As Kusterer and Bruna [19] had pointed out, some of the emission after abrasion may originate in the abrasive particles embedded in the metal and not in the metal itself. Lohff [25] circumvented this problem by using a steel brush on aluminum specimens at low oxygen pressures. He allowed the subsequent emission to decay for a period; when the system was suddenly evacuated, the emission dropped rapidly. Restoring the original oxygen pressure raised the emission to a level substantially lower than was registered at the time of evacuation. Lohff suggested that the internal processes taking place in the specimen in the absence of oxidation and of emission affected the subsequent emission rate. In a number of relatively recent papers from the United States abrasion effects on exoelectron emission were discussed in some detail. Ku and Pimbley [26] studied the emission from abraded beryllium, calcium, aluminum, and magnesium, at different temperatures. They suggested a mechanism for the emission in which lattice vacancies are generated in the metal by the abrasion. The vacancies diffusing toward the surface become active sites for adsorption or oxidation, in which case the energy of the vacancy is released. If the sum of the potential energy of the vacancy released by adsorption or oxidation at the surface plus the photo energy of the illumination exceeds the photoelectric work function of the metal, emission of an electron takes place. Subsequently, Pimbley and Francis [27] used this model to explain the results of measurements made with abraded aluminum under a Geiger counter. Their observations fitted an equation of the type I

= A

e - * '

(

+

Be- 1'--

(1)

1

(Fig. 1), where k t and k2 were decay constants, which varied with temperature 1000

K i g . 1. D e c a y o f e x o e l e c t r o n e m i s s i o n , a c c o r d i n g t o P i m b l e y and Francis [27], from an abraded aluminum sample at room temperature. T h e two lines are exponential-decay curves which, when added, f i t the experimental d a t a

20

0

20

40

60

SO Time/mini

100

120

12

F . R . BROTZEN

as kt = k" e ~ B l k T . They found that the activation energy, E, was the same for the first and the second decay constants, and the ratio of = 4.7. The activation energy was found to be Efti 0.24 eV. In their theoretical treatment, Pimbley and Francis set up a diffusion equation for the vacancies, which was solved for the vacancy-density gradients at the surface. The emission rate was taken to be proportional to the vacancy flux across the surface. The solution indicated that the activation energy, E, should correspond to the activation energy for diffusion of vacacnies in aluminum. I t should be pointed out that, according to recent results [28], the migration energy for vacancies in aluminum is likely to be about twice the value obtained by Pimbley and Francis. Mueller and Pontinen [29], on the other hand, demonstrated that freshly abraded aluminum specimens and those that were abraded and kept at room temperature for prolonged periods yielded similar emission results when their surfaces were removed by etching. They concluded from this that the transport of lattice defects, such as vacancies, to the surface could not be of importance 343 °K

365'K

——

0.28m Torr 10ß Torr

r i

i

0.28m Torr

10)i Torr

—

i

10 15 Time (mini—»

10 15 Timelmin!—»-

381°K

400°K 10u Torr

10fi Torr 0.28 m Torr

10 15 Time (mini —-

'

0

0.28 m Torr

10 15 Time (mini—-

5 m-m'K

0.28 m Torr —

Timeimin/-

010fj Torr

10 15 Time (min)—«•

Fig. 2. Photoelectric emission from abraded aluminum for various temperatures and two different pressures, from Ramsey and Garlick [30]

13

Emission of Exoelectrons from Metallic Materials

to the emission process, since the defects in the aged specimen had ample time to migrate to the surface or to internal sinks. Mueller and Pontinen, therefore, favored the concept that an interaction between impurities in the counting gas, such as oxygen, and the surface led to emission. In England, Ramsey and Garlick [30] abraded aluminum specimens with a steel brush under controlled conditions of pressure and temperature. At a pressure of 10~ 3 Torr and a temperature of 195 °K the photostimulated emission rose rapidly after abrasion, quickly reached a peak, and decayed rapidly. The maximum emission at 195 ° K was always much smaller than that at room temperature, which suggested an inhibition by some thermal activation barrier. At pressures lower than 10" 3 Torr the rates of rise and decay were hardly dependent on pressure. At higher temperatures, 343 to 473 °K, the decay was supressed (Fig. 2). Ramsey and Garlick explained their findings in terms of an oxidation model proposed some time ago by Cabrera [31]. Chemisorption was presumed to take place immediately after abrasion, starting at preferential nucleation sites. The oxide film was thought to contain a great many defects, notably vacancies, which aid in the diffusion of metal ions and in the growth of the oxide layer. Electrons from the metal may be trapped at the vacancies lying about 1 eV below the oxide conduction band (cf. [6]). Electrons thus trapped in the oxide are removed by light absorption at photon energies lower than the clean-metal threshold (Fig. 3). In a recent paper Ramsey [32] reviewed some of these and other findings. Moreover, he presented further evidence for the influence of gas pressure on the emission from abraded aluminum and zinc [33]. Ramsey's results, which resembled those by Lohff [16], revealed an emission maximum with time after abrasion. The time required to reach this maximum increased at lower gas pressures. Ramsey, rather than attributing this behavior to the formation of an adsorption layer which could lower the work function, believed that the emission is associated with a process that is slower than adsorption, viz., the initial stages of oxidation. Lewowoski [34] pointed out that the moisture content of the counter gas may also be an important variable in the emission process. He demonstrated that the decay of the emission rate from aluminum abraded and then tempered at 400 °C in air was more rapid when the moisture content was high. The effect on the emission of a temporary submersion of abraded aluminum specimens in different liquids had already been discussed by Lewowski and Snjak [35]. These investigators considered the possibility of a catalytic reaction on the metal surface resulting in products which, in turn, could affect the oxidation of the surface. In an investigation of the effect of counter-gas composition on exoelec-

Conduction band

Itòcancies or interstitiats rSuriace slates Fig. 3. Schematic energy band mode] for metal with oxide layer and surface states, from Ramsey and Garlick [30)

Metal

Aluminum oxide

Vacuum

14

F . R . BKOTZEN

tron emission, Sujak and Bojko [36] noted that electronegative gas particles tended to change the counting rate. Recent experiments in ultra-high vacuum carried out by Scharmann and Seibert [37] shed some light on the role that the oxide layer plays. In the case of photostimulated emission from aluminum surfaces, which were prepared by milling in high vacuum (10~9 Torr), the relatively small emission increased with time. Scharmann and Seibert attributed this to gradual adsorption of gas, which through the creation of a dipole layer lowers the work function of the material. When the specimens were abraded at a pressure of about 10~5 Torr, the emission decayed with time. The ability to emit electrons decreased with time, when the abraded specimens were stored in vacuum; presence or absence of light had virtually no effect on this loss of emitting ability. These authors also carried out experiments in which the specimens were not exposed to light. Gradual heating of aluminum produced emission "glow curves", provided the surfaces were scraped at a pressure of about 10~5 Torr. At pressures of about 10~ 8 Torr, no emission was observed. These epxeriments clearly indicate that the presence of an oxide layer is essential to thermo-stimulated emission. The emission was thought to originate in centers near lattice defects in the oxide. As the oxide layer grows, however, the emission diminishes largely because of the annealing of defects. I t was mentioned earlier that Grunberg and Wright [5] had observed an emission peak in cold-worked and evaporated aluminum at illumination with light of 4700 A. Scharmann and Seibert [37] observed a shift of the 7 5 Ei threshold wavelength to values as / high as 6800 A when the specimens 6 were scraped. They were unable 2 i to find the selective maximum at '2 4700 A reported by Grunberg and cur/ Wright. Strelka [38] was also unable to detect this peak, although his v— '2.1 experimental conditions were simicur lar to those of Grunberg and Wright. ^ ! 1, h0 Bojko, Pirog, and Sujak [39] and J t u ^ hSujak [40] considered the possibility that stray irradiation of low wavelengths transmitted by the filter had been responsible for Grunberg and Wright's results. Conrad and Levy [41] were equally unable to find selective emission maxima in the visible portion of the spectrum from filed and X-radiated metals, although filing and X-radiation clearly raised the treshold wavelength for

\

\

V

iA

T

r

>7 "

Fig. 4.

Schematic representation of distribution of volta potential in an oxide layer, according to Schaaffs [44] a) Volta potential as a periodic function of layer depth, b) sources and sinks for electric fields across a cut in the oxide layer

15

Emission of Exoelectrons from Metallic Materials

photoemission above that of the undisturbed metals. Similarly, Petrescu [42] could not find any spectral selectivity and concluded that exoelectron emission does not come from F'-centers. More recently, Ramsey and Garlick [30] reported that they, too, had been unable to confirm the strong emission peak at 4 700 A. The nature of the oxide, no doubt, affects the emission. Vlasakova [43], for example, had shown that the emission maxima that occurred during heating were located at different temperatures when steel was oxidized under different conditions. B y using a variety of oxidizing temperatures it was possible to form different iron oxides and to correlate these with the peaks of the emission "glow curves". In connection with exoelectron emission from abraded metal surfaces, the work by Schaaffs [44] deserves mention. He attempted to link the volta potential with exoelectron emission and, for this purpose, measured the former in various metals under oxidizing conditions. I t became clear that the volta potential did not vary monotonically with the growth of the oxide layer but displayed various extrema. Schaaffs suggested that abrasion would create cuts in the oxide which would lay bare layers between which volta-potential differences of the order of one tenth of a volt prevail (Fig. 4). In view of the very small dimensions involved, field-emission regions are created, which can cause the observed emission. 2.2 Emission

from freshly

evaporated

metals

Significant work with freshly evaporated metal films was carried out by Wiistenhagen [45] whose results, particularly with aluminum, again revealed the importance of oxygen to the emission process. He observed that no emission took place from a freshly deposited aluminum film at 4 X 10~7 Torr; the emission became strong, however, as soon as the oxygen pressure was raised to 2 x X 10~ 5 Torr. Resembling Lohff's [25] results with abraded aluminum specimens under controlled atmospheres, Wiistenhagen found that evaporated aluminum virtually ceased to emit electrons when the oxygen pressure was lowered during the test. Yet, after the pressure was restored the emission was resumed at a lower level, indicating that the material lost its ability to emit, regardless of the oxygen pressure (Fig. 5). This was interpreted as an indication for the existence of surface sites from which electrons are emitted preferentially and which are gradually eliminated even in the absence of oxygen. Hrbek and Vlasakova [46], who found extraW ordinarily strong photoemission from vapordeposited aluminum films, concluded that the electrons do not emanate just from temporarily JO3 occupied F-centers (cf. [5]). In their opinion, the F-centers in the oxide are ionized by the irradiaI tion of light, and electrons from the metal reV 2 constitute the F-centers. Newly formed alu10 minum oxide, they believed, has a particularly defective structure and therefore a great number of potential emission centers. The latter con10' cept is similar to that advanced by Ramsey

\

Fig. 5. Emission from vapor-deposited aluminum as a func time, from Wiistenhagen [45]. Curve I : no oxygen from ( = 1 min to t = 10 min. Curve I I : continuous oxygen atomosphere

0.1

, i

Y i i

10

Time (min J-

100

16

F . R . BBOTZEN

and Garlick [30]. Also, as the oxide layer grows, diffusion becomes more sluggish so that the emission is slowed down. 2.3 Emission

during

and after plastic

deformation

One of the significant advantages that plastic deformation offers over an abrasion process is the opportunity to control its extent and to measure quantitatively the degree of deformation. Moreover, in contrast to abrasive deformation, a fairly clear picture of the mechanism of plastic deformation has gradually evolved over the last years. Yet, it is conceivable that the emission mechanism from abraded metals differs fundamentally from that of plastically deformed ones. In 1960, Meleka and Barr [47] reported that electron emission in plastically deformed metals originatedfrom the slip bands in the surface. Their evidence was gathered from exposures of a highly photosensitive stripping emulsion which had been applied to zinc single crystals. Deformation of the crystal produced black lines in the emulsion, which corresponded exactly to the slip lines on the crystal surface. Although they were dealing clearly with a manifestation of the Russell effect, they hypothesized that the regions act as emission sites for electrons where dislocations intersect the crystal surface. In a more recent paper, Hempel, Kochendorfer, and Tietze [48] reviewed the effects of mechanical work on exoelectron emission and reported several of their own experiments with aluminum and carbon steel. They found that illumination of the aluminum specimens with light of wavelength shorter than 5000 A was required for the emission. Etching tended to increase the emission rate, while heat-treating lowered it. A tensile strain of at least 1% was required to produce emission. Like several earlier investigators they, too, noted the absence of emission during complete darkness, pointing toward a photoelectric phenomenon. Hempel and his co-workers suggested tentatively that defects in the oxide layer are the sources for the emitted electrons which, as earlier proposed by Nassenstein [6], create discrete states in the forbidden band of the oxide. In two papers to which the present author contributed [49, 50], an attempt was made to correlate quantitatively the mechanically induced changes in polycrystalline aluminum with exoelectron emission. The experimental arrangement consisted of a Geiger-Miiller counter which was installed on the specimen in such a manner that strain, strain rate, stress, and temperature of the specimens could

Timéis!

mo

Fig. 6. Emission rate as a function of time during stepwise straining of aluminum, from von Voss and Brotzen [49], Straining was halted after each step for a b o u t 10000 s. The first step ended a t a strain of a p p r o x i m a t e l y 7 % , subsequent steps are a b o u t 2 % each. Lines represent calculated emission

17

Emission of Exoeleetrons from Metallic Materials

be controlled. In the first series of tests [49] carried out a t room temperature, the specimens were strained a t different rates and, a f t e r a given strain, allowed t o rest until t h e emission h a d returned t o t h e background value. The experiments showed clearly t h a t the emission r a t e during straining a n d t h e subsequent decay during rest depended strongly on the strain history of the specimen (Fig. 6). This was t a k e n as an indication for structure sensitivity of t h e emission during and a f t e r plastic deformation. I n the second series of experiments [50], the aluminum specimens were deformed a t a low t e m p e r a t u r e and emission "glow curves" were obtained during subsequent annealing u p t o room t e m p e r a t u r e . The model adopted for t h e interpretation of the results was based on t h e concept that a) point defects, such as vacancies, are created in the aluminum crystals during plastic deformation (cf. [26]), b) these defects, depending on the temperature, are sufficiently mobile to migrate toward sinks, including the surface, c) some of t h e point defects succeed in locating themselves in t h e oxide layer, where t h e y create discrete levels in the forbidden band, and d) these centers are the origin for the photo or thermionic emission in a yet unspecified manner. I t was assumed t h a t the centers introduced in this m a n n e r into the oxide layer decay as a first-order reaction. Using the known d a t a on vacancy creation by plastic deformation and vacancy annihilation b y sinks, such as dislocations, a diffusion equation for t h e point defects waas solved, yielding t h e r a t e of arrival of defects in t h e surface layer. After superposing a first-order decay the results obtained corresponded closely t o the experimental observations. I n the set of experiments which led t o a "glow c u r v e " [50], the same model was adopted, although t h e mathematics of t h e theoretical interpretation were complicated by t h e t e m p e r a t u r e sensitivity of the diffusion coefficient. I n explaining the observed "glow curve", it was considered t h a t during deformation a t low temp e r a t u r e s vacancies are created b u t remain virtually immobile. As t h e temperat u r e is gradually raised, t h e vacancies migrate toward sinks. Some of the vacancies reach t h e surface layer and create preferential sites for photoelectric emission. - — Ï Ï X ! -80 -100 -120 -140 40 0 -40 Since no f u r t h e r non-equilibrium vacancies are —I 1 1 r • Emission, exp. 50 created during t h e annealing process, the num— Emission.computed ber of centers in the surface decreases as a first order reaction, producing the observed »Resistivity change, exp. 40 m a x i m u m (Fig. 7). This m e t h o d provided a .Resistivity changent computed way of measuring a migration energy of the point defects involved in the process. Inter- ; 30 estingly enough the value of 0.44 eV found in this m a n n e r was also reported in t h e literature [51 t o 53] for annealing of defects introduced in \ 20 aluminum by quenching f r o m elevated temperatures a n d by electron b o m b a r d m e n t . I t should 10 be noted t h a t annealing of defects in aluminum Fig. 7. Comparison of e x p e r i m e n t a l w i t h c o m p u t e d e m i s s i o n rates f r o m a l u m i n u m a n d electrical r e s i s t i v i t i e s for a n n e a l i n g a t a cons t a n t t e m p e r a t u r e rate of 0.27 °C/s after tensile strain of 7 . 5 % a t 133 ° K , from Claytor a n d B r o t z e n [50] 2

physica 22/1

0

0.6

0.8 z - y r —

W

18

F . R . BROTZEN

—Rate of energy release

o

ltQ |

Fig. 8. Comparison of calorimetric data by van den Beukel [54] with emission rates obtained by Claytor and Brotzen [50], after deformation of aluminum at 133 ° K

70

0 r r a

may well involve the migration of divacancies rather than vacancies. Claytor and Brotzen [50] also measured the electrical resistivity at 4.2°K of aluminum treated in a manner identical to the specimens used for emission experiments. A significant drop in the resistivity was observed during continuous heating near 230 °K, while the peak in the emission curve occurred near 220 °K. The emission data also corresponded closely to the calorimetric data for aluminum by van den Beukel [54] (Fig. 8). These experiments tend to create the impression that the emission from the insulating surface layer is engendered by structural changes in the underlying metals. The introduction of the concept of vacancy motion from the bulk metal to the surface by von Voss and Brotzen [49] yields a solution, according to which the emission rises after the deformation is stopped and before the usual decay sets in. This effect, which was actually observed [17, 49], is due to the high initial concentration of defects in the bulk, which continue to diffuse to the surface. Sujak, Gieroszynski, and Mader [55], however, noted such a large increase in the emission rate upon cessation of deformation that they suggested as its cause the deposition of positive ions on the rapidly growing oxide layer, which would enhance the emission. Yet, as more electrons are emitted, some of the ions are neutralized so that eventually an equilibrium situation is created. A series of significant experiments with plastically deformed aluminum was performed at the Institute for Experimental Physics at the University of Wroclaw under the direction of B. Sujak. I t had been shown by this group in 1960 [56] that aluminum plastically deformed by indentation and subsequently filed to remove the indentation responded to photostimulation by emitting electrons for prolonged periods in the area of the former impression. Subsequently Sujak [57] measured the emission from aluminum as a function of deformation by bending. The decay of the emission after the bending had stopped could be described by the same function (cf. equation (1)) that Pimbley and Francis [27] had used for the description of decay after abrasion. During deformation, a rise in the emission rate approximately proportional to the third power of the strain was observed. Sujak was able to justify it on the basis of lattice-vacancy formation and dislocation movement. After working with aluminum coated with oxide layers of varying thickness, Sujak and his co-workers arrived at the conclusion that the rupture of the oxide layer during deformation plays an important role in the emission process. After aluminum-oxide layers of varying thickness were deposited electrolytically (1% oxalic acid) on aluminum, a definite amount of tensile strain, £0, had to be applied before emission could be detected [58]. I t appeared therefore, that the

19

Emission of Exoelectrons from Metallic Materials Fjg. 9. Tensile strain required to initiate emission from aluminum as a function of oxide-layer thickness; from Gieroszyrtski, Mader, and Sujak [58]

12

0

1

J

4

logC

layer where the emission originates is laid bare by cracks in the oxide layer which are introduced by the tensile deformation. Yet, these cracks must attain a certain width before emission can take place. As thicker layers were deposited, the minimum strain required to cause emission, g0, rose to a maximum at a thickness of about 1 ¡xm (Fig. 9). Since the average crack width in the oxide layer could be measured microscopically, a ratio of this crack width to the thickness of the oxide layer, B/D, was established. This ratio tended to decrease monotonically with increasing oxide thickness. Further experiments by Gieroszynski, Mader, and Sujak [59] carried out with aluminum covered with oxide layers of varying thickness were designed to test the effect of moisture content in the counter atmosphere. The rise in £0 with increasing oxide thickness was less pronounced when the moisture content of the atmosphere rose. This observation was interpreted in terms of a neutralization by the moisture of high electric fields across the small cracks in the oxide (cf. [44]). These fields ordinarily tend to restrain the emission but are reduced by the effects of the moisture. Thus, the additional parameter of high fields across the cracks in the oxide was introduced by these authors. Sujak and his co-workers extended their investigations to measurements in vacuum [60]. Their results, obtained by means of an electron multiplier, confirmed the earlier observations made with a Geiger counter, namely, that increasingly greater strains were necessary to initiate emission when thicker oxide layers were employed. There exists then a critical thickness for which emission occurs only upon fracture of the specimen. This limiting thickness of the oxide layer was the topic of a special investigation by Sujak and Gieroszynski [61]. They were able to show that the limiting oxide thickness increased with the frequency and the intensity of the stimulating light as well as with the magnitude of the applied accelerating field. The value of the strain to initiate emission, e0, was found to decrease exponentially with greater fields applied the specimen and the grid. It was suggested that this applied field distorts the fields across the fissures in the oxide, thus causing the observed effect. Moreover, £0 depended also on the wavelength of the illumination used for photostimulation [62]. Increasing wavelengths required higher degrees of deformation for the initiation of emission, while increasing intensities of the light lowered the value of e0. Sujak, Gieroszynski, and Gajda [63] studied specifically the decay of photostimulated emission from oxide-covered aluminum, which was deformed to fracture. During the initial portion of the decay, the emission decreased exponentially with time. The corresponding decay constant was lowered when the oxide thickness was increased. While the externally applied field seemed to

20

F . R . BROTZEN Fig. 10. Model of an electrically charged crack in the oxide layer, according to GieroszyAski and S u j a k [64], E„ is the vector of electric field strength across the crack and P the transitional metal-oxide layer where photostimulated exoelectron emission originates

have no effect on the decay constant in the case of freshly prepared specimens, the decay was more rapid at higher applied fields in samples annealed at 500 °C for one minute. Increasing light intensity caused a significantly more rapid decay. Gieroszynski and Sujak [64] used these data for the development of a model for the emission process, which incorporated some of the elements mentioned earlier. The mechanism proposed by them (Fig. 10) considered the creation of fissures in the oxide layer as a result of plastic deformation. A strong electric field lies across these fissures and opposes the outflow of electrons from the emitting layer at the bottom of the cracks. This bottom is viewed as a semiconducting, transitional metal-oxide layer which contains emitting centers created possibly by the influx of lattice defects from the underlying metal and stimulated to emission by illumination. Electrons emitted by the transitional layer may be unable to leave the cracks and thereby eet up a space charge that varies with time. This, in turn, affects the kinetics of the emission process. Through the use of this model, Sujak and his co-workers were able to explain the observed dependence of the initial decay rate upon oxide-laver thickness and light intensity: Emission without photostimulation was oberserved in aluminum covered with a relatively heavy oxide layer ( > 5 0 n m ) [65]. With increasing strain (Fig. 11) and at constant strain rate and temperature, the emission in vacuum reached two maxima at relatively small deformation values before decaying to the vanishing point. When the same experiment was repeated in air in the absence of photostimulation, no emission was detected. It should be noted that similar peaks were obtained when heavily coated aluminum specimens were strained in vacuum while being illuminated. The major increase in emission,

Fig. 11. Comparison between the emission curve during deformation — (e) without illumination in vacuum ( ) and in air ( — — — — ). Thickness of oxide layer I ) 70 nm. F r o m Gieroszynski and S u j a k [65]

Emission of Exoelectrons from Metallic Materials

21

both in vacuum and air,took place when the specimen fractured, but was absent when no illumination was applied. The strains at which the emission maxima were observed decreased as the oxide coating became thicker. An increase in the strain rate raised the emission of the second of the observed peaks. I t was especially noteworthy that interruption of the deformation process caused an abrupt drop in the emission without the usual decay in time. Gieroszyriski and S u j a k [65] attributed emission in the dark from heavily coated aluminum to the large fields ( « 10 7 V c m - 1 ) caused by the separation of charges when the cracks in the oxide are formed. The photoelectric nature of exoelectron emission from plastically deformed metals had also been noted by Bernard, Guillaud, and Goutte [66]. They observed t h a t the photo-emission from gold was substantially increased by the application of stress. This increase was most obvious in the regions of maximum stress, at least in the case where the surface had not been cleaned completely. In an interesting set of experiments, Andreev and Palige [67] determined the actual change in the contact potential with strain in molybdenum and tantalum, indicating t h a t the work function of these metals was reduced as a result cf plastic deformation (Fig. 12). In their opinion, the structural changes concurrent with plastic deformation lead to surface defects. As a consequence, the surface dipole moment, which prevails because of the asymmetry of the electronic shells of the surface atoms, is lowered. I n this connection is should be reported that earlier work by Yoshiro [68] and Tanaka [69] revealed that the surface potentials of aluminum, magnesium, and titanium were modified by tensile deformation. After the deformation was halted, the potential gradually returned to its original value in a manner strongly reminiscent of exoelectron emission after deformation (Fig. 13). m

Fig. 13. Decay of the surface potential of aluminum after deformation, from Yoshiro [68]

~0

7 2 3 Time (mini—-

22

F . R . BKOTZEN

The effects of methods of deformation other than simple straining in tension have been investigated. Following Kramer's [1] experiments with copper, Koch [70] had tried in 1954 to determine emission from aluminum plates subjected to cyclic stresses. He noted that the emission began when the deflection reached a value corresponding to only 0.25% strain. When a newly prepared specimen was tested in this manner, emission was generally observed after a certain number of cycles. Later, Grosskreutz and Benson [71] tested the possibility of using exoelectron emission as an indicator for incipient fatigue failure in aluminum. They found that an oxidizing atmosphere was required to produce even a weak intensity of emission. Hempel and his co-workers [48] also tested aluminum in fatigue and determined that a stress differential of 2 kp/mm 2 was required to produce emission. This value, however, could be modified by surface treatments. Langenecker and Ray [72] studied the effect of ultrasonic irradiation at 20 kHz on the emission from previously abraded and filed aluminum (Fig. 14). The emission rate during the normal decay was raised whenever the ultrasound was applied, and the increase depended strongly upon the intensity of the irradiation. Langenecker and Ray believed that the introduction of acoustic and thermal energies causes similar effects on the exoelectron emission. They felt that the acoustic energy is absorbed preferentially by dislocations and transmitted to the terminal points of the dislocation lines on the surface where the oxide layer is highly distorted. Following Meleka and Barr [47], Langenecker and Ray assumed that dislocation end points are the sources of exoelectron emission. It is of special interest here that Bohun [73] had already predicted emission through ultrasonic irradiation, since it was known that this treatment led to luminescence in certain non-metallic crystals.

Time

(mini—

Fig. 14. Effect of ultrasound on exoelectron emission from aluminum, from Langenecker and Kay [72]. The different decay curves refer to different methods of abrasion. Emission peaks occur during ultrasonic irradiation. In part c) the peak is obtained by heating

Emission of Exoelectrons from Metallic Materials

23

Fig. 15. Electron emission decay at room temperature from aluminum quenched from 378 °C, according to Claytor et al. [74]

2.4 Emission

after

quenching

I n a recent paper, Claytor, Gragg, and Brotzen [74] demonstrated t h a t aluminum disks a n d aluminum wire emitted electrons a f t e r rapid quenching f r o m elevated temperatures. The model used by von Voss and Brotzen [49] was applied a n d gave satisfact o r y q u a n t i t a t i v e agreement with the decay curve (Fig. 15). The reason for the choice of this experiment was t h e earlier indication t h a t migration of vacancies f r o m the bulk metal into t h e oxidized surface is responsible for t h e emission f r o m a l u m i n u m ; such vacancies can be introduced also by quenching. The wires were chosen thin enough t o minimize quenching stresses. Interestingly enough, the emission was detected only at quench t e m p e r a t u r e s below 450 °C. The energy of formation of t h e defect responsible for t h e emission was found t o be considerably higher (1.8 eV) t h a n t h a t for vacancies and implied t h a t the n a t u r e of the migrating defect was still uncertain. Scharmann and Seibert [37] had noticed also t h a t t h e threshold wavelength for photostimulated emission from aluminum in ultrahigh v a c u u m was raised not only b y mechanical working but also by quenching of t h e specimen a t 10 s Torr pressure. Aging of t h e specimen caused the threshold value t o shift back to lower wavelengths. These experiments might also be interpreted in t e r m s of defect migration. 2.5 Emission

after

irradiation

I t is difficult to generalize t h e results and conclusions presented b y different investigators on the emission f r o m irradiated metals, because experimental techniques v a r y greatly among different researchers. Much work has been done b y scientists in Vienna in connection with emission f r o m oxidized and X-radiated metals during a heating cycle. The resulting emission curves resembled very much t h e "glow curves" well known f r o m luminescence studies. Hieslmair a n d Müller [75] found t h a t oxidized metals had generally higher emission rates t h a n etched metals. When metals were oxidized a t different temperatures, subjected to X-radiation, and t h e n tested for emission a t a cons t a n t heating rate, the ensuing emission curves for different metals varied in peak intensities but not in t h e temperature a t which the m a x i m a occurred. Characteristic for all metal oxides were maxima at 160 a n d 260 °C. These were also found in metals and alloys which were not intentionally oxidized and in those t h a t had been etched prior to X-radiation (Fig. 16). Hieslmair a n d Müller interpreted these interesting results as a strong indication for t h e origin of exoelectrons in t h e surface layer and not in t h e metal itself and suggested the possibility t h a t the acutal emitting region is an adsorption layer which is virtually identical for all the oxide surfaces. Vogel [76] later confirmed these results b y using electron-bombarded metal oxides a n d b y replacing t h e Geiger-

24

F . R . BROTZEN Fig. 16. Emission "glow c u r v e " for zinc and zinc oxide after X-radiation, from Hieslmair and Müller [75]

counter technique of Hieslmair and Müller by an electron-multiplier technique. He, too, concluded that at least part of the emission originates in a layer which is independent of the nature of the underlying metal. It is important that the peaks were observed well above room temperature and are therefore not related to the low-temperature maxima observed after deformation [50]. Birgfellner [77] studied the depth of the layer which in X-radiated metals is responsible for the exoelectron emission. He found that in copper, zinc, iron, and an iron-nickel alloy the removal by etching of about 20 ¡im was necessary to eliminate the X-ray-induced emission. Birgfellner and Müller [78] checked some of the results obtained by Hieslmair and Müller on copper, zinc, aluminum, and iron which, prior to testing for emission in a heating cycle, had been X-radiated. By varying the heating rate they found activation energies of and

1.40 eV for the peak at 160 °C 2.0 eV for the peak at 260 °C .

Since these values were again the same for all metals tested, they reaffirmed the conclusion that emission takes place from traps that are independent of the base metal. Balarin [79] disagreed with the method of evaluation of these activation energies and, in an attempt to recompute these values from Birgfellner and Müller's data, found activation energies which were only about half as great as those listed above. Furthermore, Balarin pointed out that the frequency factor, which entered the rate equation, was also too high. He considered the similarity that exists between the annealing of radiation damage in metal lattices and exoelectron emission. Considering that accurate knowledge of the frequency factor and of the energy is essential for the understanding of the emission process, Specht and Klein [80] undertook a careful investigation of the emission from aluminum bombarded at 90 ° K with electrons with energies from 0.4 to 1 keV. They found that the emission decay from one kind of trap can be represented by a reaction of the first order (cf. [49]) and that the decay constant is given by the relation

X = A0 e- El kT, where is the decay constant at infinite temperature and E the activation energy. For the two peaks which Specht and Klein observed below room temperature, the energies were near 0.6 eV. Values for ?.0 could not be determined with the same accuracy, since the results were very sensitive to the method of determination. It is noteworthy that the first of the peaks occurs at about the same temperature as that observed by Claytor and Brotzen [50] after deformation of aluminum, while the second peak was not observed after deformation. In some cases, the emission from bombarded and irradiated

25

Emission of Exoelectrons from Metallic Materials Fig. 17. Thermostimulated delayed electron emission from a n aluminum surface.

65

39

132 m

T(%! 1% 222 252 278 305 332

1

2

3

5

a ) After scraping of the surface, b) after [¿-irradiation from a Sr 9 0 source (about 5 x 10 1 1 c m ' 1 ) , c) after X - r a d i a t i o n (29 kV). F r o m Scharmann and Seibert [37]

0

~1 r. The frequency spectrum (49) is shown in Fig. 3. In the angular range (41) CO

(

p - W caused by the Hall conductivity is small compared to the damping, and the spectrum (49) turns to (45). Thus, there are no points below the straight line co = n Q. The formulas for the wave spectrum and the damping have been obtained y0 - i A assuming that the parameter \w\ = 1/2 is small | we find (50) In a magnetic field of H « 3 X 104 Oe and for (p r^ 5° the number of waves is r0 « 10.

Fig. 3 The dispersion of the waves with discrete spectrum near cyclotron frequencies

56

A. YA. BLANK a n d

E. A.

KANER

4. Excitation of Waves with Discrete Spectrum by an External Electromagnetic Field For external electromagnetic waves incident upon t h e metal surface, the natural oscillations (45) or (49) will be excited in the bulk. The excitation of these waves may lead to resonant as well as to non-resonant behaviour — according to whether the external field frequency co coincides with one of the fundamental frequencies a) n , r or not. Due to t h e skin-effect in the metal, electromagnetic oscillations with given frequency and arbitrary values of the wave vector k are excited including the natural wavelength k = R. The interference of these waves inside the metal for non-resonant excitation (a> 4= « M>r ) leads to a spatially periodical splash system as described in Section 2. The surface impedance increase (24) is caused essentially by the non-resonant excitation of the DSW (45). The resonant excitation of a discrete frequency will reveal itself in a fine structure of the CR line if the distance between neighbouring frequencies (I n, — " h i , r ~ 11) is great compared to the damping decrement |Im con, r |. I n the opposite case, the frequency spectrum becomes practically continuous. The condition t h a t the damping decrement I m con , r is small compared to the diis given by stance | a > „ , — (x> , -1| r

r

n r

Im (On,

T —

(On,

r

œ 2

r-1

50 kOe in the millimeter wavelength range. The corresponding slope angles are 1 to 2 degrees.

58

A . Y A . BLANK

and

E. A. KANER:

Waves with Discrete Spectrum in Metals

References [1] M. YA. AZBEL, Zh. eksper. teor. Fiz. 39, 400 (1960). [2] E. A. KANER, Zh. eksper. teor. Fiz. 44, 1036 (1963). [3] E. A. K A N E R and V. F. G A N T M A K H E R , Zh. eksper. teor. Fiz. 45, 1430 (1963). [ 4 ] C . C . G R I M E S , A . F . K I P , F . SPONG, R . A . STRADLING, and P . P I N C U S , Phys. Rev. Letters 11, 455 (1963). [5] E. A. K A N E R and A. Y A . B L A N K , J . Phys. Chem. Solids (1967) (in the press). [6] E. A. K A N E R and V. G . SKOBOV, Zh. eksper. teor. Fiz. 45, 610 (1963). [ 7 ] E . A . K A N E R a n d V . G . SKOBOV, P h y s i c s [8] C. C. GRIMES

and

A. F . KIP,

Phys. Rev.

2, 165 (1966). 132, 1991 (1963).

[9] I . F . KOCH, R . A . STRADLING, a n d A . F . K I P , P h y s . R e v . 1 3 3 , A 2 4 1 (1964).

[10] D. G. HOWARD, Phys. Rev. 140, A1705 (1965). (Received

March

7,

1967)

59

U. BERTELSEN et al.: Môssbauer Effect in F e F 3 phys. stat. sol. 22, 59 (1967) Subject classification: 6.1; 18.2; 18.4; 22.5

Physics Department I, H. C. Orsted Institute, University of

Copenhagen

Môssbauer Effect in FeF 3 By U . BERTELSEN, J . M. KNUDSEN, a n d H . KHOGH The antiferromagnetic compount F e F 3 is studied by the Môssbauer technique in the temperature range from 300 to 400 °K. A Néel temperature T N of (362.4 + 0.2) ° K is deduced from the data. If the reduced hyperfine field is represented by h(T) = = D (1 - T/Tx)9, it is found that 0 = 0.358 + 0.013 in the region 0.81 < T/T-$ < 0.99. The shapes of the peaks in the Môssbauer spectra in the vicinity of the Néel temperature are discussed. La composition antiferromagnétique F e F 3 a été examinée par effet Môssbauer entre 300 et 400 °K. La température de Néel est déterminée à (362.4 + 0.2) °K. Si le champ hyperfin réduit est représenté par h(T) = D (1 - TjT^f on trouve fi = 0.358 + 0.013 pour 0.81 < T / T x < 0.99. La structure des spectres aux environs de la température Néel est discutée.

1. Introduction The Môssbauer effect is a particularly suitable technique for the investigation of magnetic ordering phenomena in solids. FeF 3 is an antiferromagnetic compound, which shows weak ferromagnetism [1], The crystal structure of F e F 3 has been studied by X - r a y diffraction by Hepworth et al. [2], and it was found that it crystallizes with a bimolecular rhombohedral cell of space group R3C, with two iron atoms at 000, 1/2 1/2 1/2 and six fluorine atoms at + (z, 1/2 — x, 1/4), +(1/2 — x, 1/4, x), +(1/4, x, 1/2 —x), where x has the value - 0 . 1 6 4 . F e F 3 has been investigated by neutron diffraction by Wollan et al. [3], and they found that the iron ions couple antiferromagnetically via the intervening fluorine ion to each of their six nearest neighbours. A Néel temperature of 394 ° K was found by Wollan et al. An earlier Môssbauer study of F e F 3 has been made by Buchanan and Wertheim [4]. 2. Experimental Procedure I n our experiments we have used a drive system of the constant velocity type. I t is a " l a t h e " spectrometer from N.S.E.C. which we have automatized in such a way that the velocity setting is automatically changed at pre-fixed time intervals and the collected number of counts printed out. The advantage of the " l a t h e " spectrometer is the rather low velocity broadening (the halfwidth of potassium ferrocyanide at room temperature has been measured to less than 0.24 mm/s with our spectrometer). The good resolution is especially an advantage when we want to observe the broadening of the resonance lines in the vicinity of the Néel temperature. The disadvantage of the spectrometer is the high stability which is required of the single channel analyzer to avoid drift of the counting efficiency. This problem has been overcome. Another disadvantage is that it is necessary to move the oven with the absorber instead of the source to avoid rather severe geometric corrections. The problem of

U. Bertelsen, J. M. Knudsen, and H. Kbogh

60

moving the oven without introducing additional velocity broadening has been solved. The oven is able to keep the temperature of the absorber constant over several days with an accuracy of + 0 . 3 °K. The maximum temperature gradient across the absorber has been measured to be less than 1.0 °K. The total uncertainty in the quoted temperatures is + 0 . 4 °K, where the contribution from the temperature gradient across the absorber is found by weighting the different areas of the absorber with their temperatures and calculating the second moment of this temperature distribution with respect to its mean value. The temperature difference between the absorber and melting ice was measured by a copper-constantan thermo-electric couple, which was calibrated at the boiling point of water. A 2 mCi source of Co57 electroplated on copper was used. The absorber material F e F s was bought from Alfa Inorganics. The purity of the material was better than 9 9 % . T h e absorbers were produced by pressing polyethylene or polypropylene and FeF 3 crystalline powder into rigid tablets containing around 10 mg natural iron per cm 2 . The tablets were enclosed in aluminium foils. The absorbers were checked by X-ray diffraction both before and after the experiments. The 14.41 keV y-rays were detected by a Reuter-Stockes proportional counter containing a mixture of xenon and methane. The measured velocities were calibrated with the room temperature spectrum of iron [5] and with the spectrum of nitroprusside [6]. 3. Experimental Data Seventeen Mossbauer spectra of the antiferromagnetic compound FeF 3 have been measured in the temperature range from 295 to 380 °K. In Fig. 1 some of these spectra are shown. The splitting between the lines is seen to decrease with increasing temperature and at the Neel temperature the lines coalesce to one single line. The splitting of the lines is proportional to the hyperfine field at the iron nucleus. The hyperfine field is mainly due to the Fermi contact interaction between the nuclear magnetic dipole moment and the spin-polarized s-electrons at the nuclear position. The excited spin 3/2 level in Fe 5 7 is thus split into four levels and the spin 1/2 ground state into two levels. The selection rules Amn = 0, + 1 give the well known six-line spectrum. The polarization of the s-shells is due to exchange interaction with the partly filled 3d shell, and the magnitude and direction of the hyperfine field is thus directly proportional to the average magnitude and direction of the electronic spin of the 3d shell, where the average is taken over a time comparable to the Larmor precession time. Especially for the ferric ion which has the structure 3d 5 , is this proportionality true, because there is no contribution from angular momentum of the half-filled 3d shell. As the magnetic ordering which we want to investigate is ordering of the 3d spins, we see that the magnitude of the hyperfine field directly gives the degree of ordering. The four parameters which determine the Mossbauer spectrum are the isomer shift, the quadrupole interaction term, the magnetic splitting of the ground J

) The impurities being mainly hydrogen fluoride and water.

Môssbauer Effect in FeP 3

61

state, and the magnetic splitting of the excited nuclear state. These four parameters have been found by a least square fit from the positions of the six lines. From the results it was found that the quadrupole splitting is less than 0.03 mm/s corresponding to 1.4 X 10~ 9 eV. I t was further found that the isomer shift within the uncertainty remains constant through the T-295.2 °K transition from the ordered state to the paramagnetic state. Any > possible discontinuity must be less than 0.02 mm/s compared to the discontinuity of —0.011 + + 0.006 mm/s measured in iron T-M2°K at the Curie point [5]. The • ^ isomer shift relative to Co57 in copper was found to be 0.285 ± 0.009 mm/s at room tem- TO' = — 2 are forbidden under the given assumptions by the selection rules [4], [9]. Arnm (a) are the creation and annihilation operators of the states with quantum numbers (rn , m, cr), respectively. rn is the set of all quantum numbers of the w-th molecule except TO and a. 3. Spin-Orbit Interaction The spin-orbit interaction operator of the crystal in the representation of second quantization in hole formalism is given by H

* =

1

I°

V

w

"(r>x

V

I

^

^

+

•

(We assume that the operator of spin-orbit interaction acts only within one molecule, i.e. we neglect spin-orbit interaction between adjacent molecules). vn (r) is the potential energy of the n-th molecule in point r, originating from the nuclei and non-optical electrons. The total Hamiltonian of the crystal (when neglecting the vibrations and rotations of molecules as well as overlapping of the wave functions of the optical electrons of adjacent molecules) in hole formalism is given by [5] where

H = H0 + He + H, , H0 = W0 (R) + 2" % N[Al

(5) Afn )

is the Hamiltonian of the crystal in zeroth approximation, He =

8«

s Bn .(fn, in-; B) N [At A/J n,fn', »',fn'

+

E . Majeenîkovâ

116

is the operator of electrostatic interactions in the lattice. N[. . .] means the normal product of operators in the usual definition. Further we shall consider the operator of spin-orbit interaction as a perturbation, and it will suffice to limit ourselves to the first order of the perturbation theory. As a result of the compensation of spins and of the orthogonality of wave functions, the mean value of the spin-orbit interaction in the ground state is equal to zero. The wave functions of the unperturbed state of the crystal are 0

,

0

=

1

t)

y = l

= _ L exp ( - 4 ]/A7 \ »

t)z ]

I y =

P

=

Km

yfY

1

(v) uy „ exp(i KRp)

,

(6)

H = 1, 2, . . ., a, where a is the number of molecules in the unit cell. (u y/J ) is a

uy v = ¿V,,, satisfying the system of equations an unitary matrix, £ «-*/; "y y=1 Z H % { K ) u

y

l

l

l

=

W

(

ï \ K ) u

y

„ .

(7)

V'

is the Hamiltonian of the crystal expressed in the representation of wave functions W ] i ( v , t). The respective secular equation determines the energies W ^ ( K ) of quasi-continuous cxciton bands (Davydov splitting) of the states (6). The wave functions of localized exciton states v) are given by the expressions (4). The solution of the Schrôdinger equation y

f>W i h —— dt

=

H

W ,

where H is given by the expression (5) we shall assume in form of a linear combination of undisturbed wave functions (6) V i ï l i v ,

t

)

=

Z

( F K \ f ' K )

< p g > ,

t)

,

(8)

while the coefficients (F K \ f K) are the solutions of the equation W ] ( F K \ f K )

+

Z < < P Z \ H s \ < t > % l > ( F K \ f ' K ) = 0 ,

(9)

/>+/

where + e«*")

A(r,)}

)

,

+ - i ¿(r«))|

E0 is the amplitude of the electric field of the light wave, pt = —i h grad4. In the optical range the wavelength of light is great compared with the lattice constant so that (k a) dj a> i—i « a}

S0 S60®t>>® ki S 6£0 ao 6

0

10 20 30 40 50 60 70 80 Bleaching time (min)

Fig. 5. The effect of optical bleaching in the 4490 A band on the two main bands in SrF 2 . O 4490 A band, A 6350 A band

136

H . W. D E N H a b t o g and J. A b e n d s

Bleaching Urne (min)

Bleaching time (mini

Fig. 7. The effect of optical bleaching in the 4490 or the (5350 A band on the E S l l signal intensity at 300 " I . o 4490 A band, A 6350 A band

Fig. 6. The effect of optical bleaching in the 6350 A band on the two main bands in SrF a . O 4490 A band, A 6350 A band

Secondly, additively colored SrF 2 single crystals were irradiated with light corresponding to the 4490 or the 6350 A band inside the microwave cavity. The E S R signal intensity as a function of the bleaching time is given in Fig. 7. 4. Discussion 4.1 Electron

spin

resonance

The crystal structure of SrF 2 is of the fluorite type. Each F~-ion is surrounded by four Sr + + -ions, while each Sr + + -ion has eight F~-ions as nearest neighbors. From the ionic conductivity data obtained by Ure on CaF 2 [11] it seems reasonable to assume that the predominating defects in SrF 2 are of the Frenkel type. In analogy with the experiments of Arends [9] on CaF 2 one might expect that electrons introduced by additive coloration are trapped in F~-vacancies. In this case the F electron is surrounded by four Sr + + -ions and by six F~-ions as next-nearest neighbors. On the basis of this model, the interaction of the trapped electron with the magnetic field H and with the surrounding nuclei can be described by the following Hamiltonian: 3e = g p S - H + i a L l i=1

l S r

- S + ¿ 4 H - S , 3=1

(1)

where ¡3 is the Bohr magneton, S is the electron spin, Is r and are the spins of the Sr and F nuclei of the first and second shell, respectively. a'Sr and aj,< are the isotropic hyperfine constants of the Sr and F nuclei, respectively; H is the static magnetic field. In (1) we have assumed that the anisotropic hyperfine interaction constant is small compared to the isotropic one; only interactions with nearest and nextnearest neighbor nuclei are taken into account. With the selection rules Am s = + 1 , A % „ = 0, we can derive from (1) the E S R transition frequencies hv = gfi H+

£

4

air «4r

+ 2

6

F

a

,

(2)

F-Centers in SrF 2

i P1S n1 o^ or—1— iOI XXX CO lO

137

where

« a

-

EH e.

M

®

—

i i a

Here m s and mIlx are t h e magnetic q u a n t u m n u m b e r s belonging to S and respectively; gIoi is t h e nuclear gr-value of nucleus a ; [>n is t h e nuclear m a g n e t o n a n d 0 F ( k , a) is t h e value of t h e t r a p p e d electron wave function a t t h e &-th nucleus of element a (Sr of F). Strontium is present in the form of four isotopes Sr 84 , 86 Sr , Sr 88 (all with I = 0), and Sr 87 with a nuclear spin 9/2. The abundance of Sr 87 is 7.02% [12]. The statistical chance for a t r a p p e d electron to be surrounded b y four non-magnetic Sr nuclei is 74.5%, while t h e probability t h a t one Sr 87 and three non-magnetic Sr nuclei are nearest neighbors is 22.5%. W e shall consider only those F-centers which are surrounded by four Sr nuclei without a magnetic m o m e n t , i.e. only interactions with six F 1 9 nuclei are t a k e n into account. Because t h e F 1 9 nuclei have a nuclear spin 7 = 1 / 2 we expect seven hyperfine lines with relative intensities 1 : 6 : 1 5 : 2 0 : 1 5 : 6 :1. The experimentally observed E S R spectrum is in good agreement with this expectation. The intensity ratios agree within 5 % with those predicted. We have not been able to observe a n E S R spectrum due t o F-centers surrounded b y one magnetic strontium nucleus. Although t h e probability for such centres to occur is a b o u t 22.5%, the relative intensity will be small because this spect r u m will consist of t h e n groups of seven lines each. For t h e F-centers considered here t h e next-nearest neighbor F~-ions give rise t o t h e seven-line spect r u m , while all other nuclei contribute t o the line width of t h e individual lines. If we assume t h a t only t h e second shell of twelve F 1 9 nuclei contribute to t h e observed line width, t h e second moment of the line is given b y = f Z

o oo pq

(3)

where e is the electronic charge, r is the vector from the initial position of vacancy to the maximum of potential barrier, k is the Boltzmann constant, and T is the absolute temperature. Let 2 x and 2 y respectively denote the number of vacancies in (a) and (b) positions. For the rate of change of vacancies in (a) and (b) positions we have da: /aeE\ ( aeE\ - = - 2 x C o 1 e x p ( — ) + 22/Wl e x p ^ - — j , di dy „ (aeE\ „ / ae E\ - = 2* exp ( _ j - 2 y co, exp ( - — ) . Here a is the anion-cation separation. Since

a e

(1)

is aa very very ssmall quantity for is

usual electric fields, we retain only the first-order terms erms in "JTJL~ and get 2 lcT (

dx d

V

¥

10Jl

=

(

, aa ee EE\\ , aeE\ aeE\

I l^

, o n

1

/(1

(

a eaeE\ E

\

a eaeE\ E\

¥kr)

(2)

'

to be periodic in time of the form We assume the electric field E to E = E0e~imt

.

(3)

a

b fl

a

Fig. 1. Interstitial impurity ion "I" with four nearest neighbour positions consisting of two (a) positions and two (b) positions

Dielectric Properties of Impurity Cations in Alkali Halides

143

Here co is the angular frequency of electric field. Substituting in equation (2) the value of E from equation (3) the following solutions are obtained for equation (2) : Nj

a e iVj

2

kT

4 m1 - i co

Ni a e N-, w = — H y 2 kT

0

co, _ En 4 c«! — » to

(4)

. ,

e~'mt.

0

Here N t is the concentration of impurity ions. The total polarization will be P = ,

(2 y -

2 X) =

E0 e~iwt

,

(5)

where ¡x is the component of dipole moment along the field direction. From equation (5) we derive . W

2 Jt a2 e2 Ni ekT

=

4 co cox co2 + 16 cof'

... (6)

where e is the static dielectric constant of the crystal. This is a familiar Debye type equation with maximum value of tan > ' E% - E \m /

he

H , ¿7Jn

\ h /

\h 8F /

,

(2.11a)

x ,

4 n- x 2

x

2

2

y/2

I 2fi

co

By J (ma -

ie

yeH

\m j mv)

vH

he +

(n'

—

n)

• eH M

è

_

4 nz

x2

n0 e h2

to

I

2 ijl \i/2 / e \ 2

\h 6V )

\m )

vHc

H, /e \he]

Jn'

n(a)

(

Ai

^n"

[k

+

n |

" *

* ~

~T~

~

g

y)]Vu

x (3.9)

V H]

where Eg is the direct band gap. When the electric and magnetic fields are parallel, the Airy function is independent of qy and the integrations over qx and qy as well as the summation over n" can be performed [2] yielding the following for a : 2 2 _ S 1 2 - l l X \" {(Pc Y • i) (2TOC)1/2 (2TO2 V)1/2 /e H\ a — 32 n* \m J n0 c h* co (EV - » co) (ftc) 00 x

i n' n

J

r

Ai 2 (z)

dz

+ \

f Of

1/2 "

h /

(3.10)

Optical Absorption in Electric and Magnetic Fields

191

In (3.10), we have replaced qz as a variable of integration by h2 +7—(qz '2 M h 0F

An'n+hQ z =•

-

ij).

The expression of a in (3.10) is evaluated for low fields in the Appendix. From terms n', n so t h a t An> n + h Q > 0, we obtain the result _

C2 / 2 \ 1/2 ( » ) "

(PCV Î)2

'

C ¿3

(2W(E% - h t o )

2

X

^j^fe+Tfi) exp [— T( hTp ) J' (eHy^l

h8 F

\»/*

while from terms n', n so that An>n

I"

4 /zIM'„ + A i2\3/2-|

h Q 0 .

-{i;

The steps which usually occur in the optical absorption for phonon-assisted transitions in a magnetic field are smeared out by the tunneling induced by the electric field. The case for phonon absorption can be obtained by replacing An.n + h Q by An-n - h Q in (3.11 to 12). 4. Discussion Our calculations show that in the presence of a component of electric field along the direction of the magnetic field, the absorption peaks and edges in the case of direct transitions and the steps in the absorption in the case of indirect transitions tend to be broadened. The reason for this broadening is t h a t this component of electric field induces electron tunneling between the conduction and valence bands. This differs from the effect of the component of electric field perpendicular to the magnetic field which alters the motion of the electrons without inducing tunneling, at least within the limits of the effective mass approximation. The parallel component of electric field tends to shift the magneto-absorption peaks etc. to longer wavelengths by an amount 0 F . However, when the shift resulting from this Franz-Keldysh effect is larger t h a n the separation between adjacent peaks, edges or steps, much of the structure which appears in the magneto-absorption will be washed out. This will occur for electric fields such t h a t 0 F > cof. c , m cv . Using the values of the effective mass appropriate to Ge [4], this imposes the condition that s z < 1.5 Xl0~ 3 B 3 ! 2 for the peaks etc. not to be washed out by tunneling where e2 is in V/cm and B is in G. For magnetic fields of order 104 G, this condition requires t h a t the component of electric field along the magnetic field be less than 1500 V/cm for the structure in the magneto-absorption to be observable. The s component of electric fields acts to broaden the peaks etc. in the same sort of way as collisions do.

(311)

H. N. Spector

192

Appendix To evaluate the integral appearing in (3.10), we follow t h e same procedure used by Fritsche [10]. The asymptotic expansions for the Airy functions when l^n'nI > h Sir, i-e. in t h e low field limit, are

Ai(

2>1,

*)=2^expHz3/2);

(A la)

-

The integral we wish t o evaluate is oo p

where we have put

1, we can introduce a new variable A, i.e. z — /J = 4-/3" 1 ' 2 A2. Then

When /9

¿i

( 2

3 2

A \ /

and

ZV2 w

3

piU

and the integral I 0 can readily be evaluated.

n

j

Zl/2

(z -

fill*

~

OO e x p ( - l ^ /

2

j |

d A

e X

p ( - A

2

) =

| l / | -

o

fS~ ^ e x p

i - / ? ^ ) .

'

(A.3)

When — 1 we can use the asymptotic expansion (A.la) for t h e integral in the range 0 to oo and t h e asymptotic expansion (A.lb) in the range fi to 0. This approximation neglects t h e fact t h a t the asymptotic expansions (A.l) are not valid in the vicinity of z tm 0. We now have ...

/ „ = -

J = T

„

(-2)3/2 + . sin2 dz( - « ) 1/2 (2 - 0)1/2 ^ 8

W-iVn) 0

°° I d

4 COS — «8/2

(~ bz'2)

exp

^1 f dZ

in J 0

sl/2(2

_ 0)1/2

M 1

%i/2(|0|-,)i/2 + i ^ / 0

/"

ex P

/

,1/2

4 \ - y « * 0)1// •

(A-4)

Optical Absorption in Electric and Magnetic Fields

193

I n addition t o the first term which gives the result w h e n e z = 0, we h a v e the second term that gives rise t o oscillations as a function of e 2 . References [1] H. N. SPECTOR, Physica 30, 1917 (1964). [2] H. N. SPECTOR, Physica 32, 1551 (1966). [3] G. CIOBANU, R e v . R o u m . P h y s . 10, 109 (1965).

[4] Q. H. P. VREHEN and B. LAX, Phys. Rev. Letters 12, 471 (1964). Q. H. F. VREHEN, Phys. Rev. Letters 14, 558 (1965). Q. H. F. VREHEN, Phys. Rev. 145, 675 (1966). [5] M. REINE, Q. H. F. VREHEN, and B. LAX, Phys. Rev. Letters 17, 582 (1966). [6] A. G. ARONOV, Soviet Phys. - Solid State Phys. 5, 402 (1963). A. G. ARONOV and G. E. PIKUS, J . exp. theor. Phys. 49, 1904 (1965). [ 7 ] J . M . LTJTTINGER a n d W . KOHN, P h y s . R e v . 9 7 , 8 6 9 ( 1 9 5 5 ) .

[8] J. ZAK and W. ZAWADSKI, Phys. Rev. 145, 536 (1966). [9] M . CHESTER a n d L . FRITSCHE, P h y s . R e v . 1 3 8 , A 9 2 4 ( 1 9 6 5 ) .

[10] L. FRITSCHE, p h y s . s t a t . sol. 11, 381 (1965). (Received

13 physi;a 22/1

April

11,

1967)

K . R . EVANS and W . F . FLANAGAN: Analysis of the Cottrell-Stokes L a w

195

phys. stat. sol. 22, 195 (1967) Subject classification: 10.1; 21.1 Materials Technology Unit, Boeing Company, /Seattle, Washington (a) and Research Laboratories, General Motors Corporation, Warren, Michigan

(b)

An Analysis of the Cottrell-Stokes Law By K . R . EVANS ( a ) * ) a n d W . F . FLANAGAN ( b ) The retarding force profiles available for copper and its solid-solutions are shown to explain reported deviations from the Cottrell-Stokes law. I t is shown t h a t the CottrellStokes law is not valid and only approximated when a large athermal force must be overcome by an applied stress attempting to translate a dislocation segment past a rate-controlling obstacle to dislocation motion. E s wird gezeigt, daß die Verläufe der Verzögerungskräfte, die für Kupfer und seine festen Lösungen verfügbar sind, die berichteten Abweichungen vom Cottrell-Stokesschen Gesetz erklären können. Das Cottrell-Stokessche Gesetz ist nicht mehr anwendbar und nur näherungsweise gültig, wenn eine große athermische Kraft durch eine angelegte Spannung überwunden wird, die einen Versetzungsabschnitt über einen geschwindigkeitsgesteuerten Widerstand in eine Versetzungsbewegung überführt.

1. Introduction I t is a common and experimentally justifiable practice to separate the applied stress, r , imposed upon a material into two components as R =

T* +

xG ,

(1)

where r * is the thermal and strain-rate sensitive component of the stress and x Q is the athermal component of the stress. The Cottrell-Stokes law states that the ratio of the flow stresses measured at two temperatures, TTJXT,, is independent of the state of deformation. Agreement with the law indicates that the same obstacle is responsible for the existence of both x* and r G , since in this case they would increase proportionately with one another as the specimen work-hardens. Confirmation of the Cottrell-Stokes law has been reported for the stage I I deformation of single crystals of copper [1, 2, 3], aluminum [3, 4, 5], silver [3], and nickel [6], and for polycrystalline aluminum [3] and copper [7]. Deviations from the Cottrell-Stokes law have been reported for pure f.c.c. single crystals deformed in easy glide [7, 8, 9, 10] but after a small pre-strain the law is generally obeyed. Significant deviations from the law occur when impurities are present. For example, the deviation is greater for 99.4% Ni than 9 9 . 9 8 % Ni [6], while controlled additions of impurities to copper result in systematic deviations from the law during easy glide [11, 12]. I t has been shown for coppersilicon solid solutions that the law is not obeyed until deformation has proceeded well into stage I I , and the state of deformation where this occurs is dependent upon the solute content [13]. Typical results are shown in Fig. 1 where the flow stress ratios, x n °k/ti94 are compared for copper-silicon and 99.999% copper single crystals with increasing states of deformation. !) Now a t Shell Development Company, Emeryville, California. 13«

196

K . R . E V A N S a n d W . F . FLANAGAN

1.6

Fig. 1. The Cottrell-Stokes ratio for copper and coppersilicon single crystals as a function of increasing state of deformation

1.1 99.999 %CU 0

2

$

6

8

10

-

12

Considerable significance has been attached to the validity of the CottrellStokes law, in that it has served as a primary source of evidence for the accuracy of short-range stress theories of work hardening [3, 14, 15, 16, 17]. If agreement with the Cottrell-Stokes law has physical significance, the systematic deviations reported should be in accord with the proposed physical models. It is, in fact, difficult to make such a correlation. I t is the purpose of this paper to show, by considering the nature of obstacles to dislocation motion in f .c.c. metals, that the Cottrell-Stokes ratios and their variation with solute content and state of deformation have a physical basis. 2. Discussion The nature of the retarding force profile of an obstacle to dislocation motion is of fundamental importance in analyzing thermally activated deformation mechanisms. A schematic diagram of such a profile is shown in Fig. 2 where L is the mean distance between rate-controlling obstacles, b is the Burgers vector, Fs is the retarding force near 0 °K which is characteristic of the rate-controlling obstacle, X is the distance over which the thermally activated event occurs, H is the activation enthalpy, and V is the activation volume. Mitra and Dorn [10] have calculated the retarding force profile of rate-controlling obstacles in oxygen free, high conductivity (OFHC) copper single crystals. Their results showed that the shape of the profile was independent of the state of deformation with F = 2 . 5 x l 0 ~ 6 d y n while the athermal force, F q = Tg Lb, increased linearly with 1 ¡L. Evans and Flanagan [13] performed similar calculations for copper-silicon single crystals and found F% = 2.3 X X 10~6 dyn, independent of the state of deformation and solute content. I n contrast, a significant difference was found in the values for F ( } , which, besides being dependent upon the state of deformation, varying linearly with 1 ¡L, were also strongly dependent upon solute content. The latter feature is shown in Fig. 3 where the retarding force-distance profiles are plotted for single crystals of OFHC copper from the results of Mitra and Dorn [10] and copper-silicon solid solutions [13] at various stages of deformation. States (1) and (a) correspond to the onset of stage I I in each case.

An Analysis of the Cottrell-Stokes Law

197

10 20 30

W 50 60 70 80 90 Distance H0'scm) -

0 Fig. 2. Schematic force-distance profile characteristic of a rate-controlling obstacle to dislocation motion

Fig. 3. Calculated force-distance curves for OFHC copper [10] and copper-5 a/o Si single crystals [13]

The results are of particular interest for they can account for the observed agreement with the Cottrell-Stokes law in certain cases, and the obvious disagreement observed for other cases. We may express the force acting on a dislocation segment during activation as, F = rLb

= Fs + rQLb,

(2)

where F s is the retarding force of the rate-controlling obstacle at the test temperature T and imposed strain rate, s. From equation (2), the Cottrell-Stokes ratio may be expressed in general terms as T(r„e,e) r{Tvi,e)

=

Fs(T2,i) + FQ(e) FgHTvi) + FQ(e) '

M

From equation (3) the Cottrell-Stokes law will appear to be obeyed only if [Fs(T2,

e) -

Fa(Tlt

¿)]


Fa(T„

e),

the

ratio approaches unity and becomes relatively insensitive to an increase in state of deformation and FG. The effect of solute content on values of Fg for the retarding force profiles at the onset of stage I I are given in Table 1 along with corresponding variations in L. From these data it is seen that the CottrellTable 1 The retarding force parameters Fg and FQ for single crystals of copper and copper-silicon Material

FqX 10® (dyn)

i g x l O 6 (dyn)

Cu 99.999% [13] Cu OFHC [10] Cu-5 a/o Si [13] Cu-7 a/o Si [13]

15.0 8.8 3.0 2.2

2.5 2.3 2.3

_

Lx 106 (cm) 20.0 24.6 0.48 0.32

198

K. R.

EVANS

and W.

F . FLANAGAN:

Analysis of the Cottrell-Stokes Law

Stokes ratio for 99.999% copper would be very low, and the associated increase of FQ with increasing state of deformation would result in a small decrease in TTJTT^ Similarly, the Cottrell-Stokes ratio is expected to be large for t h e solidsolution alloys at low states of deformation due to t h e small values of F G , with t h e decrease in t h e ratio with increasing s t a t e of deformation being due to the increase in FQ with 1 \L. I n this manner the experimental observations of Fig. 1 may be accounted for. The implication of t h e results is significant for the Cottrell-Stokes law is shown to be a limiting approximation having no physical significance for which short-range theories of hardening m a y be based upon. Advocates of short-range hardening theories have dismissed deviations from t h e Cottrell-Stokes law as being due to undefined impurity interactions while maintaining absolute validity of the law for a basis of hardening theories [15, 16]. I n fact, short-range hardening theories are not compatible with t h e experimental observations. Variation of t h e Cottrell-Stokes ratio with state of deformation and solute content are related to t h e independent n a t u r e of t h e rate-controlling obstacles [13] with respect to the strongly dependent athermal force, FG. The results do indicate t h a t attention must be focused on t h e manner t h a t FG is related to t h e structure of materials as produced by alloying and/or prior deformation history. References [1] M. A . ADAMS a n d A. H . COTTRELL, P h i l . Mag. 4 6 , 1187 (1955).

[2] M. J. MAKIN, Phil. Mag. 3, 309 (1958). S. B A S I N S K I , Phil. Mag. 4 , 3 9 3 ( 1 9 5 9 ) .

[ 3 ] Z.

[4] A. H . COTTRELL a n d R . J . STOKES, P r o c . R o y . Soc. A 233, 17 (1955). [ 5 ] S . K . MITRA, P . W . OSBORNE, a n d J . E . D O R N , T r a n s . M e t . S o c . A I M E 2 2 1 , [6]

(1961). P. H A A S E N , Phil. Mag.

1206

3, 384 (1958).

[ 7 ] P . R . THORNTON, T . E . MITCHELL, a n d P . B . HIRSCH, P h i l . M a g . 7 , 3 3 7 ( 1 9 6 2 ) .

[8] J. D I E H L and R. B E R N E R , Z. Metallk. 51, 522 (1960). [9] R. BERNER, Z. Naturf. A 15, 689 (1960). [ 1 0 ] S. K . MITRA and J . E. D O R N , Trans. Met. Soc. AIME [11] P. C. J. GALLAGHER, Phil. Mag. 11, 355 (1966).

224, 1062 (1962).

[ 1 2 ] F . R . N . NABARRO, Z . S . B A S I N S K I , a n d D . B . HOLT, A d v . P h y s . 1 3 , 1 9 3 ( 1 9 6 4 ) . [ 1 3 ] K . R . EVANS a n d W . F . FLANAGAN, t o b e p u b l i s h e d .

[14] N . F . MOTT, Trans. Met. Soc. A I M E 2 1 8 , 9 6 2 (1960).

[15] Z. S. BASINSKI, Disc. Faraday Soc. 38, 172 (1964). [16] D. KUHLMANN-WILSDORF, Trans. Met. Soc. AIME 224, 1047 (1962). [17] G. SAADA, Electron Microscopy and Strength of Crystals, Interscience, New York 1 9 6 3 (p. 6 5 1 ) . (Received

April

5,

1967)

E. J. KRAMER and CH. L. BAUER: Variations of Young's Modulus in Nb

199

phys. stat. sol. 22, 199 (1967) Subject classification: 10.1; 1.2; 14.2; 18; 21 Department of Metallurgy, University of Oxford (a) and Department of Metallurgy and Materials Science, Carnegie Institute of Technology, Pittsburgh, Pennsylvania

(b)

Variations of Young's Modulus throughout the Mixed State in Niobium By E . J . KRAMER (a) a n d CH. L . BAUER

(b)

Variations of Young's modulus throughout the mixed state in niobium have been measured as a function of external magnetic field. The variations then are converted to magnetization values utilizing the two-phase model of the mixed state. Results are in satisfactory agreement with the measured magnetization curve, and thus support the applicability of the two-phase model. In Niobium wird die Änderung des Young'schen Moduls durch gemischte Zustände als Funktion des äußeren Magnetfeldes gemessen. Die Änderungen werden in Magnetisierungswerte umgewandelt, wobei das Zwei-Phasenmodell der gemischten Zustände benutzt wird. Die Ergebnisse befinden sich in befriedigender Übereinstimmung mit der gemessenen Magnetisierungskurve und unterstützen die Anwendbarkeit des Zwei-Phasenmodells.

1. Introduction The magnetization of a type I I superconductor varies from t h a t of a perfect diamagnet at a lower critical field H eA to zero at an upper critical field H c 2 . These critical fields designate the boundaries of t h e mixed state - below H c i the flux-excluding Meissner state obtains, and above H c 2 t h e normal state obtains. Flux penetration in t h e mixed state is implemented by t h e creation of quantized vortices of supercurrent, termed fluxoids. Pinning of fluxoids by crystalline imperfections, such as dislocations, permits lossless transport of current in the mixed state. However, t h e specific interactions responsible for t h e pinning are unresolved. One possible interaction has been proposed by Webb [1] who hypothesized t h a t t h e core of a fluxoid is characterized by the elastic moduli of the normal state, rather t h a n by those of t h e surrounding superconducting matrix. Thus, t h e fluxoid behaves like an elastic inhomogeneity and interacts with dislocations. If this hypothesis is correct, Young's modulus, as measured as a function of external magnetic field in the mixed state, should bear a specific relation to the magnetization curve of the material. I t is the purpose of this investigation to demonstrate t h a t such a relationship does exist. 2. Experimental Procedure Variations of Young's modulus in a niobium single crystal cylinder, characterized by a length t o diameter ratio of 5:1, were measured as a function of the magnetic field H, applied parallel to the longitudinal axis of the cylinder. The crystal was purchased from t h e Materials Research Corporation, Orangeburg, New York, and subsequently deformed plastically three percent in tension. I m p u r i t y content is estimated from the measured residual resistivity ratio of 70 to be about 300 ppm.

200

E . J . KRAMER a n d CH. L . BAUER

Young's modulus Y is determined with the aid of a composite oscillator similar to that utilized by Cooke [2], consisting of a niobium specimen bonded to a quartz piezoelectric transducer. Fractional modulus variations of the specimen A YjY are related to fractional resonant frequency variations of the composite oscillator A/// by the expression

where m a and m c denote the specimen and composite oscillator mass, respectively. The composite oscillator, driven in longitudinal resonance at approximately 80 kHz, functions as one arm of an ac bridge in which A/// can be determined within 0.1 ppm. Magnetization values M(H) are determined independently from the throw of a fluxmeter when the specimen is translated in a uniform magnetic field from one coil to another wound in opposition. Further details concerning specimen preparation and experimental apparatus are presented elsewhere [3]. 3. Results and Discussion Variation of A Y j Y at 4.2 ° K , relative to Y when H = 0, is depicted in Fig. 1 as H is cycled from zero to above Hc2 and back to zero. The curve for ascending fields is virtually independent of H below the lower critical field Hc i (1.2 kOe) and above H c 2 (3.0 kOe), whereas the mixed state is characterized by a rapid rise during the first 0.4 kOe and a linear variation thereafter. The curve for descending fields varies linearly from Hc2. to 1.0 kOe, and then becomes nearly independent of H. The remanent A Y/ Y of 23 ppm can be obliterated by warming the specimen above its transition temperature. In contrast, a linear variation of the stiffness moduli between He i and Hc 2 is observed by Alers and Karbon [4] for a series of lead-thallium alloys. It is shown subsequently that a two-phase model of the mixed state, first proposed by Rosenblum and Cardona [5] and later theoretically justified by Caroli, DeGennes, and Matricon [6], can account for all aforementioned observations. This model is implicit in the hypothesis of Webb also. The quantity A F/ Y can be related to the total superconducting to normal Young's modulus variation (A Y/Y) s n if it is assumed that the normal and superconducting phases contribute to the modulus in proportion to their volume fractions; viz. (2)

- 0

0.5

10

1.5

2.0

25

35 30 fflkOe)-

i.0

Fig. 1. Fractional variation of Young's modulus AY IY a.t 4.2 ° K , relative to that when II -= 0, with i n c r e a s i n g ( # ) and decreasing external magnetic field H for a deformed niobium crystal driven in longitudinal resonance at approximately 80 kHz

Variations of Young's Modulus throughout the Mixed State in N b

201

where f(H) is the volume fraction of normal phase material. Furthermore, if the volume of normal phase associated with each fluxoid is independent of the fluxoid spacing, f(H) merely denotes the product of the density of fluxoids threading unit area and the volume per unit length of the normal core of a fluxoid F n . Since the density of fluxoids is equal to the magnetic induction B divided by the fluxoid quantum o

=

B

(3)

The second equality in (3) is due to the constraint f(Hc2) = 1. Therefore, M(H) can be estimated by combining (2) and (3) to yield 4 Ti M{H)

= B -

H

=

A Y/Y (a

- 1 Hc J

H

w'SO

(4)

Values of M(H), derived from (4) and the data of Fig. 1, are compared with the measured magnetization curve in Fig. 2. The derived values of M(H) are in excellent agreement with the measured curve for ascending fields and after H is reduced to zero. The discrepancy between derived and measured values during demagnetization is attributed to surface deterioration by handling, and aging in the interval between Young's modulus and magnetization measurements (nearly one year). Nembach [7] has shown t h a t the ascending magnetization curve and the remanent magnetization of similar, sharp-edged, niobium cylinders are unaffected by surface condition, but t h a t the demagnetization curve may be altered significantly. Thus, derived and measured values of M(H) are deemed to be in satisfactory agreement, and indicative of the applicability of the two-phase model. The linear variation of the stiffness moduli of lead-thallium alloys with H between i / c i and H c 2 is consistent with the two-phase model also. The fractional stiffness variation ACjC is given by an expression analogous to (4); viz., AC ( C ) s n [ # c 2 ~ ^ci

Hc2 ^

(5)

^

where (AC/C)SJl denotes the superconducting to normal stiffness variation and = (#c2 - H)I(HC° — Hcl) + 4 nM(H)\HcX. The function g(H) is a measure of the deviation of the magnetization curve from linearity between Hc i and Hc2, being zero for a linear curve. However, since the ratio i / c l / / / c 2 is approxig(H)

Vig. 2. Magnetization —4 n M , derived from (4) and the data displayed in Fig. 1, plotted versus increasing ( # ) and decreasing external magnetic field H. The solid line denotes the measured magnetization curve

2.0 25 HikOel •

202

E.

J . KRAMER

and

CH. L . B A U E R :

Variations of Young's Modulus in Nb Fig. 3. a) Measured magnetization curve of Bon Mardion, Goodman, and Lacaze [8J for a lead-25 a t % thallium alloy. b) Fractional stiffness variation A C / C , generated f r o m t h e magnetization curve of Fig. 3 a via (5). Within experimental error (0.2 ppm), A C / C varies linearly with H

2ß 23 HlkOe)—

mately equal to x~2, where x is the Ginsburg-Landau parameter, g(H) cannot cause an appreciable nonlinear modulus variation for superconductors characterized by a sufficiently large x. The stiffness variation, generated from t h e measured magnetization curve of Bon Mardion, Goodman, and Lacaze [8] for a lead-25 a t % thallium alloy, is displayed in Fig. 3. The quantity A C j C , calculated from (5), does not reflect the due to the large value of x for this alloy

nonlinear variation of M(H) near 7 / c l (x = 5.2). Detection of nonlinear variations of A Y / Y with H in niobium is assisted by t h e small value of x (x = 1.3) and t h e large value of (A Y/ Y) sn . Since (A Yj F) 3n for lead-thallium alloys is about one-twentieth of t h a t for niobium while t h e limit of resolution is about the same (0.2 ppm), nonlinearities in the lead-thallium moduli are much less pronounced. The appreciable demagnetization factor of the plate-like samples utilized by Alers and K a r b o n also will obscure nonlinearities by reducing the field at which flux penetration first occurs. In principle, a discontinuity in Young's modulus should accompany t h e second order phase transformation at ATC 2- However, Hc 2 probably varies along t h e length of t h e niobium crystal due to inhomogeneous distribution of impurities produced during crystal growth. Indeed, t h e magnetization measurements indicate t h a t , although H c 2 is 2.9 kOe at the center, one end does not become fully normal until 3.1 kOe. Therefore, it is not possible t o resolve a modulus discontinuity less t h a n the equivalent of + 0 . 1 kOe on Fig. 1, which amounts to about 5 p p m . Acknowledgements

The authors express their appreciation to t h e United States Atomic Energy Commission for support of the present research, to J . Good of the Clarendon Laboratory, Oxford, for measurement of the magnetization curve, and t o G. A. Alers for communication of his results prior to publication. One of t h e authors (E. J . Kramer) also received partial financial support during t h e course of this investigation in the form of a NATO Postdoctoral Fellowship. References [1] [2] [3] [4]

W. W. WEBB, Phys. Rev. Letters 11, 191 (1963). W. T. COOKE, Phys. Rev. 50, 1158 (1936). E. J. KRAMER, Ph.D. Thesis, Carnegie Institute of Technology, Pittsburgh 1966. G. A. ALERS and J. A. KARBON, Bull. Amer. Phys. Soc. 10, 347 (1965).

[5] B . ROSENBLTJM and M. CARDONA, P h y s . R e v . Letters 12, 657 (1964).

P. G . D E G E N N E S , and J. MATRICON, Phys. Letters (Netherlands) [7] E. NEMBACH, phys. stat. sol. 13, 543 (1966). [ 6 ] C . CAROLI,

9 , 3 0 7 (1964).

[8] G. BON MARDION, B . B . GOODMAN, a n d A. LACAZE, P h y s . L e t t e r s ( N e t h e r l a n d s ) 2, 321 (1962).

(Received April 7, 1967)

J . L. ROUTBORT a n d H . S. SACK: LOW T e m p e r a t u r e I n t e r n a l Friction of Al

203

phys. s t a t . sol. 22, 203 (1967) S u b j e c t classification: 10.1; 21 Department

of Engineering

Physics,

Cornell University,

Ithaca, New

York

Low Temperature Internal Friction of Aluminum at 1 Hz1) By J . L . ROUTBOBT 2 ) a n d H . S . SACK

T h e internal friction a n d shear modulus of single a n d polycrystals of Al were m e a s u r e d of 1 H z f r o m 4.2 t o 300 ° K as a f u n c t i o n of t h e a m o u n t of d e f o r m a t i o n , t e m p e r a t u r e a t deformation, a n d g a m m a irradiation. Two Bordoni p e a k s were f o u n d with a c t i v a t i o n energies of (0.16 ± 0.01) eV a n d (0.26 ± 0.03) eV, a n d a t t e m p t frequencies of 5 x 10( 12 "to.6) s - i a n d 3 x 10 kZ< k oM * f i m /

a+ A

p

e

- m : + = (\zna> z t t t Y v /

*

2k < k cM * < *

c

+

* c* b '

(15)

b.

p

One can see immediately that if we suppose that the matrix element V{p) does not vary with p, which is correct in first approximation for a transformation into a L level, the terms E.M._ and E.M.'_, E.M. + and E.M.'+ cancel each other exactly. Physically one can understand this compensation. When a valence electron falls into a deep level the plasmon eigenfields of the electron and the ion are cancelling each other as the particles have opposite charges. If one excludes the small variation of V(p) with p, this destructive interference phenomenon would be total and there would be no secondary emission if the two particles were at rest. In fact we know this is not the case. We have to take the motion of the electron into account. The electrons have individual random motions giving rise to the coupling between the plasmon field and the individual quasi-electrons. This coupling is described in the random phase approximation by

=(§y - s) ^ - ^ • ^

This term can be eliminated by a further canonical transformation which is the second transformation of the original Bohm and Pines theory [16]. The effects of this transformation upon the electronic part of the Hamiltonian are well known: the electron mass is slightly renormalized; the plasmon frequency now depends on k, and this gives rise to a dispersion relation which can be written for lc i) •

(6)

The wave numbers q are defined by q = (2 n/N d) n, with integer n, and are restricted to the first Brillouin zone —njd < q njd. The lattice spectrum has the form / 46 d\ i/2 /o\i/2 and for the coupling parameters jq(s) we find the following Fourier representation : r! U(s) = ) l n ~ (2 Nm Z e"7^"^' • , (8) a r k where K = (2 n/d) n are the "vectors" of the reciprocal lattice. I n the present context we are interested in the case that the particle M is moving in vibrational states around a stable equilibrium position, say s = d/2. W e obtain the corresponding Hamiltonian from (6) by expanding it in a series for the assumed small particle displacement ds = s — dj2. Introducing action and angle variables (J, (p) also for the particle M the Hamiltonian takes the form H^QJ+ZwqJz

+ Z yjTTq cos

is strongly influenced by the lattice spectrum and t h a t there should be no principal difficulties in obtaining also smaller isotope effects than the r a t e theory predicts. A refined model including longitudinal lattice modes would probably yield a non-monotonous behaviour of g(fi), thereby allowing smaller isotope effects t h a n expected from the rate theory. Acknowledgement

The author is indebted to Prof. H.-G. Schopf for reading the manuscript and to Prof. E. Rexer for supporting this work. References [1] C. A . WERT a n d C. ZENER, P h y s . R e v . 76, 1169 (1949).

[2] C. A. WERT, Phys. Rev. 79, 601 (1950). [3] G. H. VINEYARD, J. Phys. Chem. Solids 3, 121 (1957). [4] G. CANNELLI a n d L. VERDINI, R i c e r e a Sci. 3 6 , 2 4 6 (1966). [5] A . D . LE CLAIRE, Phil. Mag. 14, 1271 (1966).

[6] I. PRIGOGINE and T. A. BAK, J. chem. Phys. 81, 1368 (1959). [7] [8] [9] [10]

I. PRIGOGINE a n d R . BALESCU, P h y s i c a 25, 281 (1959). I. PRIGOGINE a n d F . HENIN, J . m a t h . P h y s . 1, 3 4 9 (1960). F . HENIN, P . RESIBOIS, a n d F . ANDREWS, J . m a t h . P h y s . 2, 68 (1961). R . BROUT a n d I. PRIGOGINE, P h y s i c a 2 2 , 621 (1956).

[11] W. JOST and A. WIDMANN, Z. phys. Chem. B 45, 285 (1940). [12] R. C. FRANK, W. L. LEE, and R. L. WILLIAMS, J. appi. Phys. 29, 898 (1958). [ 1 3 ] TH. HEUMANN u n d D . PRIMAS, Z. N a t u r i . 2 1 a , 2 6 0 (1966). (Received

March

15,

1967)

E.

GUTSCHE

and

H . LANGE

: Electroreflectance of CdS and CdSe Crystals

229

phys. stat. sol. 22, 229 (1967) Subject classification: 20.1; 13.1; 13.2; 14.4.1; 22.4.1; 22.4.2 Physikalisch-Technisches Institut der Deutschen Akademie der Wissenschaften zu Berlin, Bereich elektrischer Durchschlag and IV. Physikalisches Institut der Humboldt-Universität zu Berlin

Electroreflectance of CdS and CdSe Single Crystals at the Fundamental Absorption Edge By E . GUTSCHE a n d H .

LANGE

Measurements are presented of the electroreflectance spectra of CdS and CdSe single crystals. Comparison with the photon energies of the interband edges and exciton peaks demonstrates that the low-energy peak of the spectra is dominated by exciton quenching. At higher energies field-influenced interband transitions are also important at room temperature. Es werden Ergebnisse vonMessungen der Elektroreflexion an CdS- und CdSe-Einkristallen vorgelegt. Ein Vergleich der gemessenen Spektren mit den energetischen Lagen von Interbandkanten und Exzitonenlinien zeigt, daß der langwellige Teil der Elektroreflexionsspektren durch den Einfluß des Feldes auf Exzitonen bestimmt ist. Im kurzwelligen Teil der Spektren sind auch feldmodifizierte Interbandübergänge von Bedeutung.

1. Introduction Measurements of electroabsorption and electroreflectance have been carried out during the past years on a variety of substances. I t was the purpose of these investigations to obtain detailed information about t h e positions of interband edges (see, for instance, [1, 2]) and, at t h e f u n d a m e n t a l edge, reduced effective masses [3 to 5]. The results were discussed in terms of field-induced changes of t h e absorption coefficient [6 to 9] and refractive index [10 to 12] in the vicinity of critical points. As to t h e shape of t h e electroreflectance and electroabsorption spectra, qualitative agreement between t h e o r y and experiment was found. The theory failed, however, to explain quantitatively the position of peaks, the temperature dependence etc.. The main discrepancy is t h a t t h e prominent low-energy peak in electroreflectance (usually negative) occurs always at somewhat lower energies t h a n t h a t expected from field-influenced interband transitions. Recent investigations of electroabsorption carried out a t the f u n d a m e n t a l edge of Ge by Handler et al. [13] indicated t h a t the spectra cannot be explained in terms of interband transitions alone, but t h a t "exciton quenching" plays a dominant role. I t is well known t h a t the optical spectra of CdS and CdSe exhibit a pronounced excitonic structure near the fundamental edge, the ionization energy being about six times larger t h a n t h a t in Ge. Therefore, these substances seem to be very suitable for studying the role of excitons in electroreflectance and electroabsorption. I n this paper we present and discuss electroreflectance investigations for these materials.

230

E . GUTSCHE a n d H . LANGE

2. Experimental Electroreflectance measurements were carried out at room temperature using the electrolyte technique introduced in electro-optical experiments by Williams [14] and further developed by Cardona et al. [15]. The lower part of the crystals was immersed in a dilute solution of KC1 in water. Ohmic contacts were made at the upper part, using an In-Ga-alloy. A high d.c. field was produced in a surface layer at the electrolyte-semiconductor interface by applying a voltage of the order 10 V in the blocking direction between this electrode and a platinum electrode also immersed in the electrolyte. An a.c. voltage (5 10 V) of frequency 1 kHz was superimposed. The current was well below 10~ 6 A in all measurements. The samples were irradiated with monochromatic light at near-normal incidence. The spectral resolution was 7 meV. The reflected light intensity was recorded by a photomultiplier. The modulation of the reflectivity was measured by means of a selective microvoltmeter and a homodyne rectifier. The samples investigated were thin platelets (thickness 50 ¡xm) grown by a modified Frerichs technique as well as massive crystals (thickness » 0.5 cm) obtained from a method similar to that introduced by Piper and Polich [16]. In the latter case surfaces suitable for reflectivity measurements were prepared by cleaving. In all experiments the electric field was directed perpendicular to the c-axis. Exactly the same values for zeros and extrema were obtained for A - R / J R curves taken from various cleavage planes of the same crystal.

Electroreflectance of CdS and CdSe Single Crystals

231

3. Results Typical electroreflectance spectra for CdS and CdSe are shown in Figs. 1 and 2 for polarization of light parallel and perpendicular to the c-axis. The same typical behaviour is exhibited by plate-shaped and by massive crystals. The energetic position of the extrema depends, in general, on the d.c. field. This dependence is weak for peak I, whereas a considerable shift to shorter wavelengths with increasing field is observed for the peaks at higher energies. Further, all peaks are broadened and their height is reduced with increasing field. By varying the a.c. voltage, an approximately linear increase of all extrema with increasing voltage is observed. Some crystals show a rather broad electroreflectance spectrum even a t low d.c. voltages. This is presumably due to an unusual high field in the depletion layer caused by a peculiar space charge distribution at or near t h e surface. The peak denoted by I ' in Figs. 2 a and 2 b could be observed for CdSe as well as for CdS, especially on very thin samples. I n contrast to the other peaks this one increases with increasing d.c. voltage. I t s height was in some cases up to one order of magnitude larger t h a n t h a t of peak I. 1 ) A careful study of the behaviour of this peak revealed t h a t it has a trivial origin. Contrary t o the other peaks its height is drastically reduced with increasing sample thickness. Simultaneously t h e position is shifted towards longer wavelength. The peak

A pronounced peak on the low-energy side showing a behaviour different from that of the other peaks was also reported by Cardona et al. [17] and by Seraphin [18] for GaAs and was ascribed to impurities.

232

E . GUTSCHE a n d H . LANGE

is completely absent in samples of sufficient thickness (at not too high d.c. voltages). From these facts we conclude that this peak is simply due to a penetrating beam which is reflected at the backward surface and is thus subjected to electroabsorption in both depletion layers. 4. Discussion From an extensive study of the temperature dependence of the energetic position, shape, and width of exciton lines in CdS over a wide temperature range carried out in this laboratory [19] we have very accurate knowledge of the positions of the n = 1 lines of the excitons A, B , and C for CdS at room temperature. They are indicated in Figs, l a and l b . B y adding the exciton ionization energies the energetic positions of the respective interband edges at room temperature (refering to the top of the valence band) are obtained. They are also indicated in Figs, l a and l b by A', B ' , and C'. If the presented electroreflectance spectra were due to field-influenced interband transitions, one would have to expect a coincidence of the position of the first negative peak (I) with the position of the respective interband edge when damping is neglected. I t can be seen from Figs, l a and l b , however, that peak I is displaced by about 50 meV to lower energies with respect to the corresponding interband edges for both polarisations of the incident light. This discrepancy is far beyond possible experimental uncertainties including the slight variation of the position of peak I by variation of the d.c. field and cannot be accounted for by including damping in interband transitions [12, 20]. A similar argument applies to CdSe. Therefore it must be concluded that the observed electroreflectance spectra are dominated by the influence of the applied high electric field on excitons rather than on simple interband transitions in the one-electron approximation. 2 ) I t might seem surprising that for both polarizations of incident light the same shape of the spectra is observed although for E _]_ c an influence of additional transitions from the r 9 valence band as well as the corresponding exciton transitions must be expected. However, the lines A, n = 1 and B , n = 1 overlap considerably at room temperature and therefore must be accounted for by an effective line which is centered at about 2.48 eV in CdS. 3 ) Thus the lowenergy part of the electroreflectance spectra is governed by field broadening of this composed line for c and of the line B , n = 1 for E \ \c. It is obvious, on the other hand, that the whole shape of the spectra of Figs. 1 and 2 cannot be explained in terms of a single exciton line. In principle, the spectra could be understood by assuming a cooperation of the field influence on two exciton lines, the second one being C, n = 1 for both polarizations. This explanation is suggested by the shape observed for CdSe in Fig. 2, for which material the situation is somewhat simplified by the comparatively large separation of the bands B and C due to the larger spin-orbit interaction. However, the actual energy dependence of the absorption coefficient in the energy region under discussion at room temperature favours an explanation based on the cooperation of field broadening of the exciton line discussed above in the lowenergy part, and a superposition of the electric field influence on the exciton ) See note added in proof. ) The contributions of lines belonging to n > for the present purpose. 2

3

1 can be neglected at room temperature

233

Electroreflectance of CdS and CdSe Single Crystals

line C, n = 1 as well as on the interband edges of the valence bands B and C for E || c and A, B, and C for E _|_ c. Obviously, measurements of electroabsorption in the whole range of t h e fundamental edge at low temperatures (where the electrolyte technique can no longer be applied) are neccessary in order to obtain a reliable decomposition into the various contributions. Such measurements are now in progress. I t should be noted t h a t , from the present point of view, the interpretation of the earlier experimental work on the field-induced shift of an exponential edge (for instance our own work [4] and others) has to be reconsidered. This shift seems to be due to a shift of the low-energy tail of an exciton line broadened by interaction with phonons rather t h a n to the shift of a superposition of "ideal" interband edges with exponentially decreasing probability as postulated in the work of Franz [6]. Therefore one has to be cautious in deducing effective masses from such experiments. The authors are much indebted to Dr. J . Voigt for valuable discussions. References [ 1 ] B . O. SERAPHIN, P h y s . R e v . 1 4 0 , A 1 7 1 6 ( 1 9 6 5 ) .

[2] M. CARDONA, P. H. POLLAK, and K. L. SHAKLEE, Proc. Int. Conf. Phys. Semicond., Kyoto 1966 (p. 89). [3] T. S. Moss, J. appl. Phys. 32, 2136 (1961). [4] E. GUTSCHE and H. LANGE, Proo. Int. Conf. Phys. Semicond., Paris 1964 (p. 129). [ 5 ] J . STUKE a n d G. WEISER, p h y s . s t a t . s o l . 1 7 , 3 4 3 ( 1 9 6 6 ) . [ 6 ] W . FRANZ, Z. N a t u r f . 1 3 a , 4 8 4 ( 1 9 5 8 ) .

[7] L. V. KELDYSH, Zh. eksper. teor. Fiz. 34, 1138 (1958). [8] J. C. PHILLIPS and B. 0 . SERAPHIN, Phys. Rev. Letters 15, 107 (1965). [9] R. ENDERLEIN and R. KEIPER, phys. stat. sol. 19, 673 (1967). [ 1 0 ] B . O . SERAPHIN a n d N . BOTTKA, P h y s . R e v . 1 3 9 , A 5 6 0 ( 1 9 6 5 ) .

[11] D. E. ASPNES, Phys. Rev. 153, 972 (1967). [ 1 2 ] R . ENDERLEIN a n d R . KEIPER, t o b e p u b l i s h e d .

[13] Y. HAMAKAWA, F. GERMANO, and P. HANDLER, Proc. Int. Conf. Phys. Semicond., Kyoto 1966 (p. 111). [ 1 4 ] R . WILLIAMS, P h y s . R e v . 1 1 7 , 1 4 8 7 ( 1 9 6 0 ) . [ 1 5 ] K . L . SHAKLEE, F . H . POLLAK, a n d M. CARDONA, P h y s . R e v . L e t t e r s , 1 5 , 8 8 3 ( 1 9 6 5 ) .

[16] W. W. PIPER and S. J. POLICH, J. appl. Phys. 32, 1278 (1961). [ 1 7 ] A . G . THOMPSON,

M. CARDONA,

K . L . SHAKLEE,

and

J . C. WOOLLEY,

Phys.

Rev.

146, 601 (1966). [18] B. 0 . SERAPHIN, J. appl. Phys. 37, 721 (1966). [ 1 9 ] J . VOIGT e t a l . , t o b e p u b l i s h e d . [ 2 0 ] B . O . SERAPHIN a n d N . BOTTKA, P h y s . R e v . 1 4 5 , 6 2 8 ( 1 9 6 6 ) . (Received

May

3,

1967)

Note added in proof :

Curves similar to those in Figs. 1 and 2 are also included in a recent survey by M. C A R D O N A , K. L. S H A K L E E , and F. H. P O L L A K , Phys. Rev. 154, 696 (1967). They have been discussed inadequately, however, in terms of interband transitions neglecting exciton effects.

H. J . FISCHBECK: Weakly Bound Bloch Electrons in a Magnetic Field (I)

235

phys. stat. sol. 22, 2 3 5 (1967) Subject classification : 13.1; 18 Pkysikalisch-Technisches

Institut der Deutschen Akademie der Abteilung Theoretische Physik

Wissenschaften

zu

Berlin,

Theory of Weakly Bound Bloch Electrons in a Magnetic Field 1. Nearly-Free Approximation By H . J . FISCHBECK The energy spectrum and the wave functions of the Bloch electron in a magnetic field are calculated and discussed in the nearly-free approximation. The calculation employs a new formulation of the symmetry properties of this problem. Auf der Grundlage einer neuen Formulierung der Symmetrieeigenschaften des Problems werden Energiespektrum und Wellenfunktionen des Bloch-Elektrons im Magnetfeld in quasifreier Approximation berechnet und diskutiert.

I. Introduction The symmetry properties of the Bloch electron in a magnetic field are explained by Brown [1], Zak [2], and the author [3, 4]. The most important results are conclusions about the degeneracy of the energy spectrum and general properties of the wave functions of the Bloch electron in a magnetic field. Apart from Zak's perturbation-theoretical treatment [5], these results have not found concrete applications until now. The present paper is based on [4]. Section 2 gives a short recapitulation of the formalism developed in [4], Section 3 contains the definition of the magnetic translation group presented originally by Zak [2] in a modified and simplified form. Outgoing from the irreducible representations of this group the symmetry-dependent structure of the wave function is derived in a new form, which is improved in comparison with [4]. This is the starting point for the calculation of the nearly-free approximation in Section 4. The discussion of its validity range yields a general survey of the energy spectrum of weakly bound Bloch electrons in a magnetic field, which clearly shows the appearance of the magnetic breakthrough. 2. Canonical Transformation I t is convenient for describing the Bloch electron in a homogeneous magnetic field B to use the Landau radius (h c/|e| B)112 with B = |B| as an adequate unit of length and to introduce as dimensionless quantities the position operator x, the momentum operator g, the lattice vectors S and the reciprocal lattice vectors T by

>-mw* where r, p, R, K the commutation magnetic field B xB Ar with the

« - w *

H ^ r -

have their usual meaning. The components of x and g obey relation xk] = —i Òik. If we represent the homogeneous = rot A by a linear gauged vector potential A(r) = (e/|e|)x constant gauge matrix A, the Hamiltonian of the Bloch elee-

236

H . J . FISCHBECK

fcron in the magnetic field B and the periodic potential V(x) = according to (1) H

V(x);

V (x + S) reads

=

(2)

This Hamiltonian commutes with the so-called magnetic translation operators [3] =

E2)

3)

Indices enclosed in brackets do not refer to vector components. For the proof see [6], p. 86.

(9)

237

Theory of Weakly Bound Bloch Electrons in a Magnetic Field (I)

which holds if [A, B] commutes with both A and B. (8) shows that the set of the operators does not form a group. As it is shown by Zak [2], this set can be extended to a group by addition of suitable phase factors. In order to carry out this, we define the Hilbert space by the periodic boundary condition Ans y> = y> for all f 6 £)" , (10) where N is a (large) integer and S an arbitrary lattice vector. can serve as a representation space for the A $ only then, if both ip and ANS, y> for any S' are elements of ip". This postulate leads in a well known way [L, 2] to the condition < , = v : ~

s

.

a.)

of the "rational magnetic field", q and p are integers without common factor, and p is a divisor of N. S3 is the shortest lattice vector in any fixed direction of the lattice. 0o = (|e| Bjh c) 3 ' 2 Q0 is the dimensionless volume of the elementary cell of the S lattice, whereas Q 0 is the volume of the elementary cell of the R lattice. With e 3 = e Bj\e\ B and S3 = (\e\ Bjh c) 1 ' 2 R3 (11) takes the usual form he

=

(11a);

i20p

v

This rationality condition has been derived in [3, 4] from another point of view in another form. In addition to S3 one always can find two further lattice vec0o=(81xSa,8.) (12) and S j , S2, S3 form a basis of the lattice. With respect to this basis it may be S = j Sj + k S2 + I S3 with integral j, k, I. With this we define instead of the Ag the new operators tOTSSOtbat

¿ 0 ( S ) = e-^JKlP

As-,

S = jS1 + kS2 + lS3,

j, k, I = 0, . . ., N -

1 . (13)

Using (8), (11), (12) and setting S' = j' St + k' S2 + V S3, we obtain for these operators the multiplication formula 4 , ( S ' ) A0(S) = e"»'*»/* A0 (S + S') .

(14)

The product of two operators (13) is again such an operator multiplied by a p-th unit root. Therefore, it is clear that the operators iI(S) = ei2»"?i)(S);

m = 0, . . . , p - l ,

(15)

form a finite group of p N3 unitary transformations of the Hilbert space which is defined for a "rational magnetic field" (11) by the periodic boundary condition (10). In this so-called magnetic translation group we have because of (14) and (15) the multiplication rule Am'(S') -4 m (S) = Am+m> S = jS1 + tSa

+ lS3,

+

f k (S +

S'=j'S1

S'); + k' S2 + I' s3

j

1161

in which one must take the indices of the group elements modulo p. The representation problem of this soluble group is easy to solve. Among the irreducible representations of the group there are only Nzjp2 p-dimensional representations giving rise to non-vanishing projection operators in the Hilbert space. If we write the lattice vectors in the form S = j Sx + (k p + n) S2 + I S3 = S + n S2

(17)

238

H. J.

with j , I = 0, . . read as follows : L$\Am(S))

FISCHBECK

N - 1, k = 0, .

= e'i«i/fe f(«+ |W»

2V/p — 1, and n = 0 , . . •, p — 1, they s

)(i

,

l i

+

where s, t = 0, . . ., p — 1 , (18)

5 i + n a T,) ; (19) iV n2 = 0, . . and W3 = 0, . . ., N — 1 . i> The basis T t , T 2 , T 3 of the reciprocal lattice is defined by (S{, Tk) = 2 n òik. The index t + n of the Kronecker symbol in (18) must be taken modulo p. With the diagonal elements of the representation matrices (18) being different from zero for n = 0 and, consequently, because of (17) for S = S, we set up the projection operators q = i

¿r

P

iy

K

T, +

v-1 I | D\r(^m(S)) m = 0 iS

Am(S)

= ^

s e" i( q+ l(-i!v) S

t„s)

A o { S )

.

(20)

which define the partners belonging to the i-th row of the irreducible representations (18). In order to construct the partners explicitly, we wish to apply these projection operators on suitable basis vectors of the Hilbert space. To do this, we need some further definitions. As we see from (13) and (7), the operators P

0

§ (q'

—

q) òt,(

q

+

f

e-(.72).«-'MSo,T„)

^ T,

e-ik k'~ (S' e " ' 1 ' ! ' e i M « + t < ? / p ) * V t - » > . s . ) x

+ t J-

Tr+¿T0)

(26)

in which p i , t = 0, . . ., p — 1, T r = IT1 + m T3 w i t h 1 = 0,.. ., q — 1 a n d q is t h e p r o j e c t i o n of q on t h e 1,3-plane. These f u n c t i o n s f o r m a c o m p l e t e o r t h o n o r m a l set of s i m u l t a n e o u s eigenfunctions of t h e m u t u a l l y c o m m u t i n g T ,s operators belonging t o t h e eigenvalues c' 1 >' ). I n t h i s c a p a c i t y t h e y a l r e a d y are derived in [4], b u t u n f o r t u n a t e l y in a n i n a p p r o p r i a t e f o r m . F o r fixed q, T r we h a v e p f u n c t i o n s t h a t s p a n t h e p-dimensionl irreducible subspaces of T h e y g e n e r a t e in each of t h e s e i n v a r i a n t subspaces one of t h e irreducible r e p r e s e n t a t i o n s (18) which, however, do n o t d e p e n d on T r . H e n c e , we c a n choose t h e eigenfunctions of t h e H a m i l t o n i a n (6) as t h e linear c o m b i n a t i o n s («(„, s

q,t)

= S C«,,,,(«(i>; T r ) (« I q, t, T r )

(27)

Tr

of t h e f u n c t i o n s (26) w i t h respect t o T r for fixed q a n d t. H e r e a symbolizes t h e q u a n t u m n u m b e r s being n o t conditioned b y t h e s y m m e t r y . T h e coefficients of t h e s e linear c o m b i n a t i o n s m u s t d e p e n d in accordance w i t h t h e p r o d u c t formation § = X ¿p" of t h e H i l b e r t space on t h e variable u ( 1 ) playing n o p a r t in t h e s y m m e t r y considerations. T h e y a r e d e t e r m i n e d b y t h e Schrodinger equation H(uw,

s \ a, q, t) = E.{q)

(um,

s | a, q, t)

a n d t h e n o r m a l i z a t i o n condition -f- oo 2 f Cf B , g ,,(«{„, T r )c;..,,«(«(!), T r ) d w ( 1 ) =

(28)

(29)

T r — oo

T h e energy eigenvalues Ea(q) are p-fold d e g e n e r a t e d .

do n o t d e p e n d on t h e q u a n t u m n u m b e r t.

They

4. Nearly-Free Approximation T h e H a m i l t o n i a n H0 = (h J2) («(,) + vfi) -f- vfo) of t h e f r e e electron in t h e magnetic field is completely s e p a r a t e d . I t is diagonalized b y t h e basis («,1)1») (51?) = (»! f c ) - ^ Dn(^2 um) of t h e H i l b e r t space

=

(2 TT)-1 e * « ' » *)

(30)

I t s s p e c t r u m en(t') = h coc {n + -i- + — 1 t h e solution of t h e ^-dimensional algebraic eigenvalue 4

) Dn(z) denotes parabolic cylinder functions.

Theory of Weakly Bound Bloch Electrons in a Magnetic Field (I)

241

problem of the matrix of the potential generated by these functions would be necessary for the complete adaptation of the functions (26) to the potential. Therefore, we suppose in the following q = 1. It seems to be at first sight a considerable loss of generality because the difference A(1 ¡B) = e QJ2 n h c |it3| of two allowed 1/fi-values (see (11a)) for short R3 vectors can reach the order of the de Haasvan Alphen period. But it is not so. For each direction characterized by a short R3 can be approximated to any degree of accuracy by directions belonging to very long R3 vectors. For this reason the rationality condition (11) even with the restriction q = 1 can be adjusted to any given situation. Thus we start from the unperturbed states (U(i)\ri) (s\q, t, m T3) with the unperturbed energies £nm(q) = h WC n + y

(q3 + m T ^ J

+ y

(31)

and calculate the matrix elements of the perturbing potential by making use of the Fourier expansion V = (•n';q',

Z

T

VT e » '

{ T

-=

e* ("(2) 2 '.- l '(2)

2 " F t e''("(i) R >-»T>)| _ T _ X (q', t', m' T3 |ei(M(2)T>_!'(2) r.+«

X

X (M(i)|n') {s\q, t, m' T„) ,

where

(34)

„ | \ 1/2 + 00

(- r )

2

•!

n

x

i

^

\

and

0

=

(q

+ y

(m

V

o,i=—°°

+

r

T

^

K2

/

m')

Tt,

e~W .

"

e - i ( » + m ( « + » ' ) T „ T x e , ) e-ITX!'/4

.

H \

2

i J\T=jT,+l.T2

T s X e3| + t(S2, Ts).

x (35)

+

(m'-m)Ta

= (Tv Tt, 0) is the

component of T perpendicular to the magnetic field, is a Laguerre polynomial, and (p = arctg (TJT^. The notation means that the term n' ,m' = = n, m has to be omitted. '"' The perturbed energy spectrum (33) shows the influence of the lattice potential on the Landau spectrum (31). It should be noted that the matrix elements Vnm (q) given by (35) depend only on the component = (ql, q2, 0) of q perpendicular to the magnetic field. On the one hand, the lattice potential removes the degeneracy of the unperturbed spectrum (31), which only de16

physica 22/1

242

H . J . FISCHBECK

pends on q3, b y causing a dependence of the spectrum on qj_ arising to first order from the diagonal element j,l=-oo

\

I

/|T=Tj_=jTl+lT1

(36) On the other hand, it produces gaps in the ^-dependence of the perturbed spectrum. This can be seen as follows: The expressions (33) and (34) fail near such values of q3 for which enm(q3) = en-m'(qa). I n the neighbourhood of these values we obtain the perturbed energies and wave functions in a well known way by adaptation of the two degenerated functions. This leads t o a splitting of the energy bands by twice the absolute value of t h e corresponding matrix element (35) as it is shown in Fig. 2. The perturbation t r e a t m e n t is only available if t h e splitting of t h e Landau levels is smaller t h a n their distance h o>0. The level splitting according to (36) is mainly determined by t h e F o i r i e r components FT of t h e potential and t h e behaviour of t h e Laguerre polynomials Ln. We suppose that the Fourier components FT decrease rapidly if T increases. The Laguerre polynomials fulfil the inequality e~xl2 Ln(x) sS 1 for x Si 0 [7], Therefore, the perturbation theory is always applicable if A « e > | Ft J (37) Here and in the following FTJ is t h e greatest Fourier component for one of '

tm ( f )

t>CJr-

\M

Fig. 3. Validity regions of the nearly-free approximation

k 2

Fig. 2. The q3 dependence of the p e r t u r b e d energy spectrum

243

Theory of Weakly Bound Blooh Electrons in a Magnetic Field (I)

the shortest reciprocal lattice vectors T | in the plane perpendicular to the magnetic field. For large n the functions e~*'2 Ln(x) oscillate in the range 0 < # < j > = 4 n + 2 a s the asymptotic expression [7] (2 & - sin 2 #) + - J

e"*/2 Ln(x) ^ 2 ( - 1 ) " (n v sin 2 tf)"1'2 sin

(38) x = v cos2 •& , v= 4n+ 2, whereas they fall off exponentially to zero for x > v. Hence, the level splitting (36) is small if |Tj_|2/2 > 4 n + 2 or, because equation (1), =

(39)

The functions e - ^ 2 Ln{x) tend to zero according to (38) for large n also in the oscillatory region. Therefore, from the inequality h coc> | Vt j_| 2 [2 jt v (x/v) 112 X X (1 - xjv) 1' 2]-^ 2 with x = |Tj_|2/2 = 4 EJh a>c we get the further condition E1^E0

(40)

(2 n h o)c

for the applicability of the perturbation theory. Fig. 3 shows the validity range of the perturbation theory given by (37), (39), and (40). h a>c measures the field strength. E ± is the energy part of the motion in the plane perpendicular to the magnetic field. In the range I the energy E is smaller than the energy E0 that for free electrons corresponds to the boundaries of the first Brillouin zone. This corresponds to unperturbed circular orbits lying entirely in the interior of the first Brillouin zone of the fc-space. Their energy spectrum is weakly perturbed by the lattice potential. In the range II, however, the unperturbed circular orbits determined by E j_ > E0 intersect boundaries of Brillouin zones. Now, Bragg reflexion is possible, and the orbits are altered considerably also by a weak potential so that their energy spectrum no longer can be described as a weakly perturbed Landau spectrum. Here the perturbation treatment breaks down. Nevertheless, if the field strength is high enough, the electrons with E > E0 in the range III have again a weakly perturbed Landau spectrum. It is the consequence of magnetic breakthrough. Therefore, (40) is no more than a somewhat modified version of Blount's breakthrough condition. The solution of the Schrodinger equation (28) in the range II, which is especially interesting for the experiment, is the subject of a subsequent paper. Appendix

Recalling that (um\n) = (»! ^)- J / 2 D B (/2 Mn>) = ( and using (9), we obtain

n

\

H

e

„

(|/2 w(1))

+ 00 _ _ e~ iT- T>l* J e iT°-»m £)„-(/2 w(1)) D„(|/2 (w 0 ) - TJ) dw ()) = — oo + oo _ 1!2 iTiT = (2nn\n' \ )~ e*!' 2e~ Tu 2 f e^+ and F + (Sr++ V K ) relative to Zf>, respectively. As we have seen earlier, we may replace to a good approximation the fraction of associated impurity ions by the total impurity concentration at moderate temperatures 3 ): [(Sr++ V K )] « c . (13) Thus, studying the dependence of the ratio [F]/[Z2] on the impurity concentration enables us to decide between the models Z® and ZJf', as long as the color center concentration is much less than the impurity concentration. 4. Activation Energy of the F ^ Z 2 Conversion The activation energy of a reaction may provide useful information about the reaction mechanism. For reversible reactions it can be determined by a method described by Franck and Sizmann [15]. Starting from a non-equilibrium state we record the variation of the concentration of a reactant or some proportional quantity x with time at constant temperature (reaction isotherm). 3)

Here we have presumed that the solubility of the impurity complexes is not exceeded.

Kinetics of the Thermal Mutual Conversion of P and Z 2 Centers in KC1 :Sr

In points marked by the same value x = determine the reaction rates

249

on two different isotherms we

(x, and »>1, JZi the ratio of which is a. If T = is the temperature, where corresponds to the equilibrium state, we can give the following expression for the activation energies q{ and qT of the forward and reverse reaction:

a exp ' Z3L T= k

= a exp

~ T* I T, T=

-?r T=

T, T._

qf and qr can be calculated from two such equations. Application of the above method requires that the reaction isotherms either start at or are reducible to the same initial state. 5. Experimental 5.1 Crystal

samples

The KC1: Sr crystals were grown from the melt under nitrogen atmosphere from reagent grade material 4 ) by theKyropoulos method.The crystals remained clear up to molar S r + + fractions of 2 X 10~3 in the melt. At higher impurity concentrations the crystals became opaque indicating an inhomogenous distribution of the S r + + ions. The samples for the optical measurements were colored in potassium vapour by the van D o o m method [16] at 560 °C and 12 Torr. The crystals were slowly cooled down, reheated to 400 °C and quenched to room temperature in benzene. The samples prepared in this way exhibited only an F band with low S r + + concentrations and an additional slight Z 2 band absorption with higher S r + + concentrations. The S r + + ion fraction of the samples was determined after the absorption measurements with a spectrophotometer by comparing the intensity of the Sr line A = 4077 A with that of samples of known composition. The relative accuracy of the concentration measurements was about 20%. 5.2 F ^ Zo center

conversion

The color center concentrations were determined by measuring the absorption constants. The samples were mounted in a simple cryostat allowing us either to promote the F ^ Z 2 conversion in the dark at elevated temperatures or to measure the optical absorption at liquid nitrogen temperature. In the temperature range from 80 to 170 °C only the F and Z 2 bands were present. For determining the Arrhenius dependence of the F Z 2 equilibrium the crystals were annealed at constant temperature until the absorption spectrum exhibited no change with further annealing. The absorption constants kv and kz2 in the maximum of the mutual bands were then determined. In Fig. 4 the logarithm of the ratio kF/kz„ is plotted versus 1/T for samples with varying impurity fractions. The S r + + concentrations of the crystals are listed together with other data in Table 2. 4 ) KC1 (p.a.) was provided by Riedel-de Haen, Hannover, SrCl 2 dehydrated (EL) by Merck, Darmstadt.

250

G. WOLFRAM a n d V .

WITT F i g . 4. T h e r a t i o o f t h e equilibrium c o n c e n t r a t i o n s o f 1 model. This model is known as the Seitz Z2 center model, which consists of an F center bound to an impurity-vacancy complex. From the results, however, no conclusions can be drawn about the spatial configuration of the impurity-vacancy complex and the relative position of the F center. Furthermore, our findings leave unresolved the problem of the missing electron spin resonance. Experiments are in progress to elucidate the Z2 center structure. Acknowledgements

The authors are indebted to Professor Dr. E. Lüscher for his interest and encouragement. They wish also to thank G. Gehrer for useful discussions. References [1] J . H. S C H U I . M A N and W . D. COMPTON, Color Centers in Solids, Pergamon Press 1963 (p. 169). [2] P. CAMAGNI, Nuovo Cimento Suppl. 9, 372 (1958). G. CHIAROTTI, F . FUMI, a n d L . GIULOTTO, D e f e c t s i n C r y s t a l l i n e Solids, T h e P h y s i c a l

Society, London 1955 (p. 317). [3] P. C A M A G N I , S . C E R E S A R A , and G . C H I A R O T T I , Phys. Rev. 118, 1126 (1960). [ 4 ] F . B A S S A N I and F . G . F U M I Nuovo Cimento 1 1 , 2 7 4 ( 1 9 5 4 ) . [5] M. P. Tosi and G. AIROLDI, Nuovo Cimento 8, 584 (1958). [6] G. D. WATKINS, Phys. Rev. 113, 79 (1959). [7] H . KELTIG a n d H . WITT, Z. P h y s . 1 2 6 , 697 (1949).

[8] [9] [10] [11] [12] [13] [14]

A. B. LIDIARD, Ionic Conductivity, Hdb. Phys. Bd. 20, Springer Verlag 1957 (p. 299). H. PICK, Z. Phys. 114, 127 (1939); Ann. Phys. 35, 73 (1939). F. SEITZ, Phys. Rev. 83, 134 (1951). H. OKHURA, Phys. Rev. 136, A 446 (1964). K. KOJIMA, J . Phys. Soc. J a p a n 19, 868 (1964). A. B. LIDIARD, J . appl. Phys. Suppl. 33, 414 (1962). F. A. KRÖGER, The Chemistry of Imperfect Crystals, N o r t h Holland Publishing Company, Amsterdam 1964 (p. 661).

[15] A . FRANCK a n d R . SIZMANN, C h e m i k e r - Z t g . 8 6 , 6 7 1 (1962).

[16] Z. VAN DOORN, Rev. sei. Instrum. 32, 755 (1961). [17] F. LÜTY, Halbleiterprobleme VI, Vieweg & Sohn, Braunschweig 1961 (p. 246). [ 1 8 ] J . H . S C H U L M A N a n d W . D . COMPTON, l o c . c i t . ( p . 5 8 ) .

[ 1 9 ] J . KNOBLOCH, N . RIEHL, a n d R . SIZMANN, Z . P h y s . 1 7 1 , 5 0 5 (1964).

[20] M. CHEMLA, Thesis, University of Paris 1954. [21] E. E. SCHNEIDER, Photographic Sensitivity, Butterworth Scientific Publications, London 1951. [22] A. SMAKULA, Nachr. Ges. Wiss. Göttingen, Math.-phys. Kl. (NF) 1, 85 (1934). [23] H. RÖGENER, Ann. Phys. (Germany) 29, 386 (1937). [24] J . C . G R A V I T T , G . E. GROOS, D. K. B E N S O N , and A . B . SCOTT, J . ehem. Phys. 37, 2783 (1962). (Received

May

2,

1967)

255

YA. A. IOSILEVSKII: Dynamics of Two-Component Crystals phys. stat. sol. 22, 255 (1967) Subject classification: 5; 6; 10 Institute of Solid State Physics, Academy of Sciences of the USSR,

Chernogolovka

Some Model Problems on the Dynamics of Two-Component Crystals By Y A . A . IOSILEVSKII A study is made of the dynamic problem of superstructures such as A B r _ i , with r atoms per a unit cell built up on the basis of the nearest neighbour, central, and non-central model of a simple rhombic, tetragonal, and cubic Bravais lattice. Es wird das dynamische Problem von Superstrukturen wie A B r _ i , mit r Atomen pro Einheitszelle, auf der Basis eines zentralen und nichtzentralen Modells nächster Nachbarn für ein einfaches rhombisches, tetragonales und kubisches Bravaisgitter untersucht.

1. Introduction In the present paper we consider, according to the method described in [1], the dynamic problem for two-component superstructures AB,.-! with an arbitrary number of atoms r per unit cell (r-atomic) built on the basis of a simple rhombic Bravais lattice r o in which only nearest neighbour interactions (along the edges of a unit rectangular parallelepiped) of central and non-central type are taken into account. Such a lattice includes tetragonal r q and simple cubic jPc Bravais lattices as special cases. The indicated models, in spite of their artificial character, prove to be highly useful as they allow to obtain a wide variety of results in a closed and highly simple form. The terminology and the main notation of [1] are maintained. 2. "Reference" Crystal. Singularities 2.1 General

remarks

In the case under consideration the sites xn of the lattice X are given by translations of three mutually orthogonal basic vectors. All non-zero elements of the force constant matrix * ( » „ ) correspond, by condition, to n = (0,0,0), ( ± 1 , 0 , 0 ) , (0, + 1 , 0), and (0,0, + 1 ) (i.e.toac,, = 0 and xn = + ap; p = 1, 2, 3) and they are diagonalized simultaneously in the crystallographic coordinate system with its axes along the vectors ap: A"(ap)

= A**(-ap)

= -

yip dik , ¿ (I = 1) the behaviour of g^(z) can be investigated by means of an other known expansion of the elliptic integral ([6], p. 919), and this, in absolute accordance with van Hove's theorem [5], leads to the expressions

(2.15)

i ft f

2

2

Here gflM{z) contains a weaker singularity (such as z — If zf> and zf> are close (or equal) to each other we have

In \z — z |).

Comparing the expressions (2.5), (2.6), and (2.7) with each other we see that the spectral function g{s)(z) may be expressed by either of the following equiThe values of

3) obtained by various combinations of the two values