Physica status solidi: Volume 23, Number 1 September 1 [Reprint 2021 ed.] 9783112496220, 9783112496213

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plxysica status solidi

V O L U M E 23 • N U M B E R 1

1967

Classification Scherno 1. Structure of Solids 1.1 Alloys. Metallurgy 1.2 Solid-State Phase Transformations 1.3 Surfaces 1.4 Films 2. Non-Crystalline State 3. Crystallography 3.1 Crystal Growth 3.2 Interatomic Forces 4. Microstructure of Solids 5. Perfectly Periodic Structures 6. Lattice Mechanics. Phonons 6.1 Mossbauer Investigations 7. Acoustic Properties of Solids 8. Thermal Properties of Solids 9. Diffusion in Solids 10. Defect Properties of Solids (Irradiation Defects see 11) 10.1 Defect Properties of Metals 10.2 Photochemical Reactions. Colour Centres 11. Irradiation Effects in Solids 12. Mechanical Properties of Solids (Plastic Deformations see 10)see 10.1) 12.1 Mechanical Properties of Metals (Plastic Deformations 13. Electron States in Solids 13.1 Band Structure. Fermi Surfaces 13.2 Excitons 13.3 Surface States 13.4 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 j 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 )

p h y s . s t a t . sol. 2 3 (1967)

Author Index A. Y. N. G. S. J. J. D. L.

R , ADAMS ADDA I . AKIMENKO ALBRECHT AMELINCKX ARENDS ASKILL E . ASPNES AVRAMOV

D . J . BACON G . R . BARSCH B . W . BATTERMAN P . BAUMANN H . BAURICH C . BENOIT A LA GUILLAUME U . BERG I . M . BERNSTEIN A . VAN DEN B E U K E L

K99 223 93 K17 549 137,713 263, K 2 1 K79 K61 527 577 K147 K99 K137 295 K87 539 165

M . BITTER D . A . BLACKBURN B . BLASZCZYK H . BLEHER V . L . BONCH-BRUEVICH D . BORSAN H . BOTTGER A . BOURRET W . BRODKORB A . D . BROTHERS W . BRUCKNER I . S . BUDA P . BYSZEWSKI

K155 177: 2 2 3 419 K13 761 K113 325 207 K35 697 475 229 K53

Z. K. J. N.

P . CHANG P . CHIK H . CRAWFORD J R CROITORU

577 113 301 621, 627

I . V . DAKHOVSKII - J . L . DAVIDSON B . DAYAL P . H . DEDERICHS A. DEVENYI A . DEZOR I . DIMA L . DOBROSAVLJEVIC G . DUESING J . M . DUPOUY € . DUFUIS

229 K139 K93 377 621, 627 K159 K113 509 481 K109 509

R . M . EASSON P . ECKELT

K129 307

V. F. P. R.

D . EGOROV EICHHORN G. ELISEEV ENDERLEIN

703 237 587 127

H . FELLENZER V. A. FINKEL D . FRAITOVÄ

171 K97 501

H . GABRIEL R . R . GALAZKA J . M . GALLIGAN J . A . GALLOWAY" L . GENZEL R . GEVERS W . GISSLER S . CODERSKA P . GÖRLICH R . D . GRETZ R . GRIGOROVICI R . G . J . GRISAR G . D . GUSEINOV W . GÜTH E . Y U . GUTMANAS I ) . HAARER H . W . DEN HARTOG B . HEINRICH H . HEMMERICH 0 . HENKEL H . - P . HENNIG W . HENRION B . HILCZER G . HILDEBRANDT P . HLAWICZKA P . S. H O R . H . HOPKINS W . HÖRSTEL T . HORVÄTH L . HRIVNÄIC M. HULIN

195 K39 K139 97 613 549 K155 419 313 453 621,627 613 461 K13 571 633 713 501 481 K151 599 K67 419 K147 K129 489 697 639 K71 189 563

S . IKONOPISOV 1. P . IPATOVA D . IWANOV

K61 467 663

S. B. V. A. H.

K9 K53 501 K5 313

G . KALASHNIKOV KALIÀSKA KAMBERSKY A . KAMINSKII KARRAS

768

Author Index

R . KEIPER F . R . KESSLER

. . . .

M . KESTIGIAN

. . . .

A . G. KHACHATURYAN . K . KLEINSTÜCK . . . . M. KLEMAN A . A . KLOCHICHIN .

.

.

Z. KNESL J . S . KNOL J . KOLODZIEJCZAK.

.

.

A . I. KOSINSKAYA .

.

.

G. G. KOVALEVSKAYA

.

G . KOTITZ J . KOVÄR A . I . KRASILNIKOV.

.

.

.

.

.

A. A. KUKHARSKII.

.

.

J. VAN LANDUYT

.

.

G . KRETZSCHMAR F . KROUPA

H . LANGE Y . E . LAPSKER J . LECIEJEWICZ

.

127 K25 289 745 . . . . 237,475 207 467 185 K83 . .423, K53, K57 57 313 755 189 587 639 K143 447 549 K67 K5 K123 709 335 709 709 539 721 521 413 K83 147 429 697

. . . . . . . .

S . L E MONTAGNER . D . LEPSKI

.

.

A . L E TRAON F . L E TRAON

J. 0. M. Li J. Loos A . E . LORD J R R . LUDEKE W . VAN DER LUGT . S . LUNGU V . G. LYAPIN D . W . LYNCH

.

.

587 223 K133 K125 353 481 577 Kl 97 K9 703 . 253 K123

M . A . MANKO G . MARTIN A . A . MARYAKHIN . GH. MAXIM P . H . E . MEIJER . D . MEISSNER D . L . MILLER K. D. A. G. H. A.

.

.

.

.

MOORJANI J . MORGAN I . MOROSOV 0 . MÜLLER P . MÜLLER MURASIK

E . M. NADGORNYI . . . E. NAGY . . . . . . . CHR. N A N E V . . . . .

. . . . . .

D . N . NASLEDOV . . . P . NOVAK . . . . . .

. . . . . .

571 K49 663 755 K45

A . OTTO

. . . .

S.

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. 361 . 413 . 263 . 335 . K75 . K49 . K105 . K125 . 353 . 563 . 587 . 365 . K9

. . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. 461 . K109 . 613 . 613 . K117 . 277 . Kl . 365 . 113 . 489

. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

PAL

W . PAUL R . F . PEART B . PEGEL H . PEISL G. PETÖ P . PETRESCU V . PETRESCU J . R . PEVERLEY M . PICARD I . Z. PINSKER M . PORSCH V . V . PROKLOV A . M . RAMAZANZADE P . REGNIER K . P . REINERS K . F. RENK N . REZLESCU H . J . ROHDE J . - Y . RONCIN J . RÖSELER M . RÜHLE A . L . RUOFF A . G . SAMOILOVICH W . SCHILLING D . SCHMID W.SCHÜLKE G . E . R . SCHULZE B . SCZANIECKI P . K . SHARMA W . A . SLBLEY H . P . SINGH D . SLPPEL S . V . SLOBODCHIKOV Y . N . SMIRNOV

. . . .

B . P . SOBOLEV E.SONDER H . SPALT I . SPINULESCU-CARNARU T . SPRINGER M. D . STAFLEU J . STANKOWSKI P . M. STARYK A . STELLA H . STOLZ M . STUART J . STUKE A . V . SUBASHIEV V . K . SUBASHIEV P . SÜPTITZ E . SUTTER

.

.

171

. 229 . 481 . 633 . K87 . 475 . Kl59 . 361 . 301 . K93 . 237 . 755 . K97 . K5 . 301 . K75 . 157 . K155 675,683 . . K159 . . 93 . . 697 . . 729 . . 595 . . K99 . . 467 . . 447 . . 9 . . K25

Author Index I . V . SVECHKAREV J . B . SWAN

K133 171

M . L . SWANSON

649

W . SZYMAÄSKA

69

J . F . VERWEY F . VOIGT M . VOOS V . V . VOROBJOV P . I . VORONYUK

T . L . TANSLEY

241

R . VOSZKA

H . TEICHLER J . TELTOW

341 9

A . R . DE VROOMEN

P . THOMA

253

W . WAIDELICH

K . B . TOLPYGO

429

F . F . Y . WANG

IC. D . TOVSTYUK R . TROC

75,93 K123

P . ULLIIANN S. I . URITSKY

313 57

M . P . USIKOV

745

S . WANG S . WAPLAK

G . VAUBEL F . VÀVEA S . VELEVA

105 253 K121 K61

137 607 295 K97 75 K71 C75,683 K75 289 387,401 K159

A . WATTERICH

K71

G. WEBER H.H.WEBER

K17 703

M. WILKENS L . WOJTCZAK

K . VACEK

769

H.C.WOLF T . ZAKRZEWSKI A . ZYGMUNT

113 K163 633 K39 K123

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. GÖRLICH, 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. SEITZ, Urbana, 0 . S T A S I W , Berlin, M. S T E E N B E C K , Jena, F. STÖCKMANN, Karlsruhe, G. SZIGETI, Budapest, J . TAUC, Praha Editor-in-Chief P. GÖRLICH 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. COCHRAN, Edinburgh, R. COELHO, Fontenay-aux-Roses, H.-D. DIETZE, 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. KUBO, Tokyo, M. M A T Y A S , Praha, H. D. MEGAW, Cambridge, T. S. MOSS, Camberley, E. NAGY, Budapest, E. A. N I E K I S C H , Jülich, L. P A L , Budapest, M. RODOT, 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 23 • Number 1 • Pages 1 to 438, K1 to K104, and A l to A32 September 1, 1967

AKADEMIE-VERLAG•BERLIN

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Contents Page

Review Article P . SÜPTITZ a n d J . TELTOW

Transport of Matter in Simple Ionic Crystals (Cubic H a l i d e s ) . . . .

9

Original Papers A . I . KOSINSKAYA a n d S. I . URITSKY

W.

SZYMAI&SKA

On the Classical Theory of the Optical Properties of Free Carriers in Semiconductors Conduction Band Structure of HgTe from Thermomagnetic Measurements

57 69

K . D . TOVSTYUK a n d P . I . VORONYUK

Galvanomagnetic Effects in P b T e

75

N . I . AKIMENKO, P . M . STARYK, a n d K . D . TOVSTYUK

Magnetic Properties of P b T e

93

D . J . MORGAN a n d J . A . G A L L O W A Y K . VACEK

Sum Rules for Effective Masses in Energy B a n d Theory Electroluminescence of AgCl Crystals Excited b y Short Field Pulses.

97 105

K . P . CHIK, M . WILKENS u n d M . RÜHLE

Die Interpretation des elektronenmikroskopischen Beugungskontrastes von Einschlüssen nahe einer kräftefreien Folienoberfläche. .

113

R . ENDERLEIN a n d R . KEIPER

On t h e Dielectric Function of a Semiconductor in an External Electric Field

127

J . ARENDS a n d J . F . VERWEY

S.

LUNGU

E S R on UV Irradiated Lead Halides a t 80 °K Point Defects in Neutron-Demaged Silica Glass

137 147

I . SPINULESCU-CARNARU

The Orientation of Crystallites in ZnTe Thin Films

157

A . VAN D E N B E U K E L

On t h e Relation between Self-Diffusion, Melting, and V a c a n c y - I m purity Binding in Dilute Alloys

165

J . B . SWAN, A . OTTO, a n d H . F E L L E N Z E R

Observed Retardation Effects on the Energy of the co + -Surface Plasmons in Thin Aluminium Foils

171

D . A . BLACKBURN

Z. KNESL

Non-Equilibrium Vacancy Concentrations in Metals Subject to Thermal Gradients and Electric Fields The Equilibrium Position of an Orientation J u n c t i o n of an Edge Dislocation Dipole

177 185

L . HRIVNXK a n d J . KOVÄR

H.

GABRIEL

Absorption of Ultrasound in Indium Antimonide Mössbauer Spectra in the Presence of Electron Spin Relaxation (I) .

189 195

A . BOURRET e t M . KLEMAN

R u b a n s d'aimantation dans les lames minces de fer p u r

207

G . MARTIN, D . A . BLACKBURN e t Y . ADDA

Autodiffusion au joint de grains de bicristaux d'argent soumis ä une pression hydrostatique

223

A . G . SAMOILOVICH, I . S . B U D A , a n d I . V . D A K H O V S K I I

Anisotropy of Thermomagnetic Effects in n-Ge I-

229

Contents

4

Pago F . EICHHORN, D . S I P P E L , a n d K .

T. L. TANSLEY

KLEINSTÜCK

Influence of Oxygen Segregations in Silicon Single Crystals on the Halfwidth of the Double-Crystal Rocking Curve of Thermal Neutrons

237

Spectral Response of p-n Heteroj unctions

241

H . P . M Ü L L E R , P . THOMA, a n d G . V A U B E L

The Phosphorescence of Anthracene Single Crystals and its Spectrum

R . F . PEART and J .

H. J . ROHDE

ASKILL

253

The Mechanism of Diffusion in B.C.C. Transition Metals

263

Double Injection into CdS Single Crystals

277

F . F . Y . WANG a n d M. KESTIGIAN

Magnetic Susceptibilities of Sodium Neodymium Tungstate and Sodium Terbium Tungstate

289

and M. Voos Optical Measurement of Compensation in Highly Doped Silicon. . .

295

C. B E N O I T Ä LA G U I L L A U M E

J . H . CRAWFORD J R . , W . A . S I B L E Y , a n d E . SONDER

P.

ECKELT

The Influence of Electron-Trapping Impurity on Defect Creation and Bleaching in KCl Irradiated at 78 °K

301

Energy Band Structures of Cubic ZnS, ZnSe, ZnTe, and CdTe (Korringa-Kohn-Rostoker Method)

307

P . GÖRLICH, H . KARRAS, G . KOTITZ, a n d P . ULLMANN

Thermoluminescence Investigations on Doped Alkaline Earth Fluorides

313

Untersuchung des Jahn-Teller-Effektes an Ionen mit Spin

325

H . BÖTTGER B . PEGEL and D.

H.

TEICHLER

1 . . . .

LEPSKI

On the Isopte Effect in Interstitial Diffusion (III)

335

Berechnung des Peierls-Potentials im Diamantgitter mit Hilfe der Pseudopotential methode

341

J . R . PEVERLEY and P. H. E.

MEIJER

Entropy and Susceptibility of a Pure Dipole-Dipole Substance. . .

S . P A L a n d P . K . SHARMA

On Lindemann's Melting Criterion

M . PORSCH a n d J .

RÖSELER

Recoil Effects in the Polaron Problem

P. H.

DEDERICHS

353 361 365

Dynamical Scattering Theory for Crystals with Point D e f e c t s . . . .

377

S . WANG

Optical Absorption of an Interstitial Alkali-Earth Cation in an Alkali-Halide Crystal

387

S.WANG

Electronic States of the M A -Centers in KCl

401

R. LUDEKE and W .

PAUL

Growth and Optical Properties of Wurtzite and Sphalerite CdSe Epitaxial Thin Films

413

Contents

5 Page

B . HILCZER, B . BLASZCZYK, a n d S . GODERSKA

Effect of Storage Temperature on the Time Dependence of Charge Density in Carnauba Wax and Methyl Polymethacrylate Electrets

419

J . KOLODZIEJCZAK

Effect of Strong Electric Fields on Magneto-Resistivity and Hall Coefficient in Semiconductors

423

V . G . L Y A P I N a n d K . B . TOLPYGO

Partial Account of Interelectron Correlation in the Theory of Band Structure of Crystals with Sphalerite and Diamond L a t t i c e s . . . .

429

Short Notes J . - Y . RONCIX a n d K . MOORJANI

On the Absorption Spectrum of Gaseous and Solid Xenon

K1

A . A . K A M I N S K I I , Y . E . L A P S K E R , a n d B . P . SOBOLEV

Induced Emission of N a C a C e F 6 : N d 3 + a t Room Temperature.

. . .

K5

Some Data of Acousto-Electrical Domains in Photoconductive Cadmium Sulphide

K9

S . G . K A L A S H N I K O V , A . I . MOROSOV, a n d V . V . PROKLOV

W . GÜTH u n d H . BLEHER

Lawinendurchbruch in Germanium bei optischer Band-Band-Anregung

KU

G . ALBRECHT a n d G . W E B E R

J . ASKILL

Hyperfine Fields Induced by Spin-Orbit-Coupling in Binary Intermetallic Compounds

K17

Tracer Diffusion of Tungsten in Molybdenum

K2L

F . R . KESSLER a n d E . SUTTER

Absorption and Photoconductivity of a Series of Selenium-Tellurium Mixed Crystals K25 W . BRODKORB

Green's Function Method in the Theory of Antiferromagnetic Thin Films with Body Centred Cubic Crystal Structure K35

R . R . GAL4ZKA a n d T . ZAKRZEWSKI

Heavy Hole Effective Mass of Cd 0 ^Hg,, 9 Te P . NOVAK

K39

On the Inversion Splitting in Octahedrally Coordinated E^ Electronic States K45

E. NAGY a n d G. PETÖ

Ordering in the Alloy Cu 3 Au-V

K49

P . BYSZEWSKI, B . KALINSKA, a n d J . KOLODZIEJCZAK

Interband Faraday Rotation and Ellipticity in Germanium and Gallium Arsenide K53

6

Contents Pago

J.

KOLODZIEJCZAK

Free Carrier Effect of the Generation of Higher Harmonics in Semiconductors K57

S . I K O N O P I S O V , L . AVRAMOV, a n d S . V E L E V A

On the Electroluminescence of Indium Oxide Layers in Contact with an Electrolyte K61

H . LANGE a n d W .

HENRION

Thermoreflectance of CdS and Se Single Crystals

K67

R . VOSZKA, T . HORVÄTH, a n d A . W A T T E E I C H

A Possible Model for V-Centres Stable at Room Temperature in XIrradiated KCl(Ca) Crystals K71

H . PEISL, H . SPALT, a n d W .

D.

E.

ASPNES

WAIDELICH

X - R a y Diffuse Scattering in y-irradiated LiF

K75

Note on Collision Broadening of Franz-Keldysh Effect

K.79

W . YAK DER LUGT a n d J . S . K N O L

The Knight Shift in Potassium from 23 to 123 °C

W . SCHULKE a n d U .

BERG

H . P . SINGH a n d B .

DAYAL

K83

Measurement of the Compton Band on Beryllium Single Crystals . . K87 X-Ray Determination of the Thermal Expansion of Zinc Selenide . . K93

V . A . F I N K E L , J . N . SMIRNOW u n d V . V . WOROBJOV

Zum Problem der thermischen Ausdehnung von Gadolinium

A. R . ADAMS, F . BAUMANN, a n d J .

. . . K97

STUKE

Thermal Conductivity of Selenium and Tellurium Single Crystals and Phonon Drag of Tellurium K99

Pre-printed Titles and Abstracts of papers to be published in this or in the Soviet journal ,,iI>H3HKa Teepjioro T e j i a " (Fizika Tverdogo Tela)

A1

Contents

7

Systematic List Subject classification:

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

1.2 1.4

K49 157, 413

3.1

413

4

113, 137, 157, 237

5

K93

6

335, 361, 365, 423

6.1

195

7

189,

K9

8

189, 353, K93, K97,

9

9, 165, 223, 263, 335,

K99 K21

10

9, 147, 185, 263, 341, 377,

10.1

165, 177, 223

10.2

301, 387, 401, K71,

11

147, 301

13

171, 195, 253, K 1 3 , K 5 3 ,

13.1

57, 69, 75, 93, 97, 127, 307, 365, 387, 413, 423, 429, K 2 5 , K57, K67, K79,

13.2

K75

K75 K57

K87

387, K 1

13.3

171

13.4

277, 295, 301, 313, 325, 387, 401,

14

9

14.3

75, 93, 423, K 3 9 , K 5 7

14.3.2

241

14.4

9

14.4.1

127, 241, 277, 419, 423, K9, K 1 3 , K 7 9

15

9, 69, 229, K 3 9 , K 9 9

16

241, K 1 3 , K25,

18

57, 93, 353,

18.1

289, K 1 7

18.2

207

18.4

289,

K49

K53

K35

19

137, 325, K17,

20

57, 127, 171, K53,

20.1

241, 313, 387, 401, 413, Kl,

20.2 20.3

K45

K83 K79

K5 105, 253, 295, 313, K 5 ,

K61

K 5 , K25,

K67

K39,

8

Contents

21

165, 171, 263, K21

21.1

263, K17,K49

21.1.1

207, 263

21.2

K83

21.3

K.87

21.4

K97

21.6

223

22

75, 93, 423, 429

22.1.1

57, 97, 229, 263, 341, K13, K53

22.1.2

237, 295, 341

22.1. 3

K25, K67, K99

22.2.1

K53

22.2.3

189, 423

22.4.1

277, 307, K9, K67

22.4.2

307, 413, K93

22.4.3

69, 157, 307

22.5

137

22.5. 1

9, 105

22.5.2

9, 301, 387, 401, K71, K75

22.5. 3

9, 313

22.6

K61

22.8

147, 241, 289, K5, K39

22.9

253

23

K1

Contents of Volume 23 Continued on Page 441

Review Article phys. stat. sol. 23, 9 (1967) Subject classification: 10; 9; 12; 14; 14.4; 15; 22.5.1; 22.5.2; 22.5.3 Institut fur Kristallphysik

der Deutschen AJcademie der Wissenschaften

zu

Berlin

Transport of Matter in Simple Ionic Crystals (Cubic Halides) By P . SUPTITZ a n d J .

TELTOW

Contents List of principal' 1.

symbols

Introduction

2. Theoretical

background

and their

migration

Charge and elementary statistics of point defects Defect interactions Mobility of defects Remark on surface effects

3.1 3.2 3.3 3.4 3.5

Diffusion Ionic conductivity Dielectric relaxation Anelastic relaxation Residual methods

3. Experimental

4.

— Point defects

2.1 2.2 2.3 2.4

techniques

Results

4.1 Effects involving free migration of defects 4.2 Effects involving reorientation of diatomic defects 5.

Conclusion

References

List of Principal Symbols Chemical symbols: M, X Mf, X f • a, c, O •, '

Cations and anions of the host lattice (MX) Impurity cations and anions Vacancy Incorporation on anion, cation, and interstitial site (subscripts) Single positive and negative excess charge (superscripts)

Thermodynamic

functions:

G, H, S, V Increment of Gibbs free energy, enthalpy, entropy, and volume, respectively (due to defect formation, if without subscript) a Subscript relating to formation of associates

10 d m ma s

P . SUPTITZ a n d J. T E L T O W

Subscript Subscript Subscript Subscript

relating relating relating relating

to to to to

(reorientation of) defect dipoles migration of single defects migration of associates segregation of (unassociated) defects

Other symbols: a D Da E I ITC J k K nn nnn P Q tM, tx u v Fm iv W x

y (5 Ac, A m x'1 ¡x v vDcb a r T0 O)

Lattice parameter (large cube edge in Fig. 1) Diffusion coefficient Pre-exponential factor for diffusion Electric field strength Current Ionic thermo-current, depolarization current upon linear heating Current density Boltzmann constant Constant of the law of mass action Nearest neighbour Next-nearest neighbour Polarization Heat of transport Transport numbers for cations and anions Electric mobility (for subscripts see x) Volume per lattice molecule M X Molar volume Jump probability (see Sections 2.3.2 c, 2.3.4) Experimental activation enthalpy for diffusion Molar fraction of incorporated defects (subscripts 1, 2, i meaning species 1, 2, i; a associates, d dipoles, f foreign ions, • vacancies, • a anion vacancies, • • divacancies, Q interstitials) Statistical weight factor entering the law of mass action Loss angle Electrical and mechanical relaxation strength Debye length, effective radius of charge cloud Effective dipole moment Frequency factor (see Section 2.3.1) Debye limiting frequency Conductivity Relaxation time Pre-exponential factor of relaxation time Field frequency times 2n 1. Introduction

For many years, matter transport in crystals has found widespread interest as the most direct experimental approach to point defects and their migration under the influence of forces. Table 1 lists the various forces and the observed effects, implying the restriction on atomic (not purely electronic) defects and on migration over at least one shortest lattice separation. Valuable information on point defects obtainable in special cases from other observations (e.g. lattice dilatation, electronic and thermal conductivity, optical absorption, ESR, broadening of N M R lines) will not be considered in this report.

Transport of Matter in Simple Ionic Crystals

11

Table 1 Matter transport in crystals Effective forces Kind

Generation

Observation

Atomic process

chemical *)

concentration gradient

matter current levelling the concentration (diffusion)

migration of defects

electrical

electric field

ionic current, ionic conductivity

migration of charged defects

dielectric loss

relaxation or migration of charged defects

stress field

anelastic deformation, diffusional creep

migration of defects

vibrations, sound field

attenuation, sound absorption

relaxation or migration of defects

temperature gradient

concentration gradient (Ludwig-Soret effect)

migration of defects

mechanical

thermal

migration of charged ionic e.m.f. (Seebeck effect) defects *) Meaning a force due to a gradient of a chemical potential.

Generally the observed macroscopic transport effect depends on a number of atomic variables, such as concentrations and mobilities of the different kinds of migrating defects. These in turn are functions of external parameters given by the experimental conditions. Hence a successful quantitative analysis requires the large-scale variation of these conditions (temperature, pressure, frequency, doping species, concentration, etc.) and the combination of various investigation methods (Table 1) on the same material which should have a simple crystal structure enabling a simple geometry of atomic jumps. Only ionic crystals allow the investigation of matter transport by sensitive electrical methods. These considerations excuse the restriction on the transport properties of the alkali and alkaline-earth halides and of cubic heavy-metal halides, as far as the data allow the evaluation in terms of defect parameters. For other ionic crystals we refer to more general reviews and compilations [1, 2, 3, 4], This article centres on the progress made within the last decade. Thus it aims at supplementing former reports [5, 6, 7, 8] on (part of) the same subject. In contrast to some of these, a short account is given of the experimental technique which recently has been enriched by important new methods. I n cases where new measurements and the improved crystal quality have led to more precise values of the defect parameters, earlier work will not be mentioned. For brevity, omission of a number of closely related topics was indispensable: effect of plastic deformation and of irradiation, transport phenomena governed by the presence of dislocations and/or colour centres, surface and boundary effects, polarization capacity, reorientation of substituted molecular ions like O H - , and lattice calculations of defect parameters. Even so, it is regretted t h a t some of the remaining problems can only be touched, within the given frame.

12

P . SUPTITZ a n d J . T E L T O W

2. Theoretical Background — Point Defects and their Migration 2.1 Charge and elementary

statistics of point

defects

The basic concepts on point defects in ionic crystals, due to Frenkel, Schottky, Wagner, and others, have been treated in many articles and textbooks. For a survey also of the historical aspects as well as for a bibliography up to 1956, see the famous encyclopedia article by Lidiard [5]. The following short outline is given only for recollection and for the purpose of introducing some notation useful for the interpretation of the experimental results. The simplest possible types of atomic defects in ionic crystals are vacancies, lattice ions on interstitial sites, and foreign ions on lattice or interstitial sites. Assuming that the repulsion by the neighbouring lattice ions outweighs by far the other terms in the formation enthalpy of an interstitial ion, the resulting well-defined interstitial sites, as shown in Fig. 1, have high site symmetry and are separated from each other by potential walls of the order 1 eV. The incorporation of an ion on a certain site is not only a question of its size, but depends on its electronic structure, too. In some cases relatively small substitutents leave the highly symmetrical central position, forming a quasimolecule with one of their neighbours (Li in KC1). Neither these nor the dumbbell model for interstitial ions (two displaced ions on both sides of a lattice site) are considered here, being unimportant for transport. Exceptional crystal structures offering a wealth of possible sites to the small cation (a-AgJ, a-CuJ) are excluded, too. Electric neutrality generally requires the incorporation of at least two kinds of defects with opposite charge. The situation is described conveniently in terms of excess (virtual, effective) charge, i.e. the difference between the local charge states after and before the formation or incorporation of the defect. In what follows, the superscripts x , •, and ' mean zero, single positive and negative excess charges, respectively. Introducing the symbol • for a vacancy 1 ) and denoting the site by the subscripts " c " (cation), " a " (anion), and O (interstitial) 2 ), respectively, the charge state of the possible inherent defects in a monovalent ionic crystal M X (consisting of ions Mcx and X a x ) is seen from the symFig. 1. Cubic unit cell with basis of cation, anion, and interstitial sites for simple ionic crystal structures. The occupation of the sites a, b, c, d in the different structures is the following: Structure

Cations

Anions

NaCl CaF 2 ZnS

a (4c 4d, 6b, 12a) a (4c 4d, 6b, 12a) a (4c 4d, 6b, 12a)

b (4c 4d, 6a, 12b) c d (4a 4b, 6d/c, 12e/d) c (4a 4b, 6d, 12c)

CsCl

a t ( 6 b j , 12 a 2 , 8 b,)

b j (6a 2 , 12b 2 , 8 a , )

Interstitial sites c d (4a 4b, 6d/c, 12c/d) b (4c 4d, 6a, 12b) b (4c 4d, 6a, 12b) d (4a 4b, 6c, 12d) a 2 ( 2 b ! 4b 2 , 4a, 8a 2 , 8b 2 ) b2 (2a x 4a 2 , 4 b t 8b 2 , 8a 2 )

Number and kind of the nearest neighbours in the 1st, 2nd, and 3rd shell are given in brackets, " / " meaning " o r " . The subscripts 1 and 2 refer to the CsCl structure only. N o t e that in the ZnS and CsCl structures there are t w o kinds of interstitial sites with different environments

2)

A symbol is preferred which cannot be confounded with that of a chemical element. These facilitate the comparison of chemically identical defects in various crystals.

Transport of Matter in Simple Ionic Crystals

13

bols M 0 , and X ¿ . Also, the symbol Mf^ for a substituting divalent cation Mf 2 + is self-explanatory, as is Xf t i for the foreign anion Xf 2 ~. There are three possible ways of charge compensation in pure ionic crystals: a) Two material defects of opposite charge, giving the well-known four disorder types. The formation reactions in crystals MX are Mc

M o + Oc

Xa

X ¿ + Da

Mc + X a MX (d)

cationic Frenkel disorder,

(la)

anionic Frenkel disorder (sometimes called anti-Frenkel disorder) , (lb) Dc + Dk + MX (d) Schottky disorder, and (lc) M5 + anti-Schottky disorder, (Id)

MX(d) meaning a MX molecule at a dislocation or at a free surface. The superscript x is omitted. 3 ) b) Charge compensation by surplus electrons or holes. This case, though important and widespread, will not be considered here, being not essential for matter transport in the crystals selected above. c) Near exterior or interior surfaces (e.g. dislocations), local deviations of the neutrality are compensated by free charges on the surfaces. This case will be discussed briefly in Section 2.4. Specializing on case a), statistical thermodynamics shows that, owing to the high configurational entropy gain per defect involved in low defect concentrations, every crystal must have a certain degree of thermal disorder in equilibrium. Let xlt x2 be the molar fractions 4 ) of the two defects present, then, in the limit of small concentrations, - 1 - = K(T,

p) = — eGlkT YiYz

=

_ L e-Slk YiVi

e

H/kT _

(2)

G, H, and S are the Gibbs free energy, enthalpy, and entropy of formation per defect pair, and y1 y2 are the numbers of possible sites per lattice molecule MX for the defect in consideration, e.g. 2 for interstitials in the NaCl structure (Fig. 1). Note that in this article all symbols for thermodynamic functions mean difference values relating to the defect-free crystal. Equation (2) applies also to the case x1 =(= x2 (more than two kinds of inherent defects, impure crystals with aliovalent additions). Owing to the strong dependence on T, at low temperatures the difference determined by the inevitable aliovalent impurities ("extrinsic" range). Only in sufficiently pure crystals there exists an intrinsic high-temperature range where On the other hand, doping with aliovalent ions in definite amounts is an important tool to vary the concentrations of the inherent defects or even to "salt out" one type of them. Equation (2) is written in the form of a mass action law for the reactions ( l a ) to (Id), i.e. for "dissociation" of a lattice ion (Frenkel), removing two ions from the lattice (Schottky) or incorporating them on interstitial sites (anti-Schottky),

3 ) Note that the conservation of charge, mass, and lattice site occupation is clearly seen in the reaction equations by virtue of the chosen defect symbols. 4 ) In this report, all concentrations (x) are given in molar fractions relative to a lattice molecule (MX). The corresponding number of particles per unit volume is x/v, where v is the volume per MX.

14

P . SUPTITZ a n d J .

TELTOW

respectively. In either case the numerator 1 on the left-hand side corresponds to the reaction partners in the undisturbed lattice. Thus equation (2) is the prototype of all mass-action law equations for reactions between point defects. In a first approximation valid for very low concentrations, they obey the same laws as do atoms in gases or ions in dilute solutions. Together with the neutrality condition these equations are indispensable for calculating the mutually dependent defect concentrations. In complicated cases dealing with many kinds of defects, an approximate graphical method introduced by Dutch workers [3] is of value if the reaction constants K(T) are known. Sufficient mobility of the defects involved is an important condition for these calculations. There will be no equilibrium unless all micro-states of the defect distribution over the lattice sites are run through successsively within the duration of experiment. This condition, excluding the low-temperature range, is more restrictive for the Schottky types of disorder, defect migration over many shortest ion distances (at least to the nearest dislocation) being involved. If in equation (2) K(T, p) is known from experiments, then according to thermodynamics H is given by (0(ln K)ld(llkT))p or, assuming H and S to be independent of T, by the slope in a plot of In K vs. 1 /T. Likewise the entropy S may be obtained as — k(d(T In y1 y2 K)jdT)v or directly from equation (2) under the same condition. In the harmonic approximation, S is related to the variation of the vibration frequencies of the crystal due to the incorporation of the defects. Additional pre-exponential contributions have been discussed by various authors (see [3], p. 427). — The free volume V of formation, as defined by H = U + p V ([/internal energy of formation), is given by ¿T(8(ln K)ldp)T. V is important for estimating the volume distortion of the lattice around the defect. Concerning atomistic calculations of H, S, V, etc., for special defects in given crystals, we refer to reports [5] and [6] and to some more recent papers (cf. [9] (S for LiF, NaCl, KC1, KBr), [10] (H, Hm for NaCl, KC1, RbCl), [11] (H, Hm for LiF, NaCl, NaBr, K halides, RbCl), [12] (H for CaF 2 ), and [13] (H for all alkali (except Cs) halides). 2.2 Defect

interactions

The elementary statistics of defects may be improved by taking their interaction into consideration. In the case of defects with excess charge, the interaction forces [14] are mainly electrostatic and may be approximated by Coulomb's law. The resulting pairs (aggregates, associates) of oppositely charged non-recombining defects are analogous to molecules in gases or to complexes in dilute solutions. Three examples are given, written again as dissociation reactions, juxtaposition of symbols meaning spatial neighbourhood: • c Da ^ Dc + da Mfc • 0 Mf;, -(- •Sa, and free volume F a of pair formation may be evaluated as above. I t follows that the degree of association generally increases with increasing impurity concentration x{, but even in the limit xr ->• 0 it does not vanish entirely. A straightforward extension of the theory accounts for "excited" states of the pair where its partners are not in nearest neighbour (nn) positions, but occupy next-nearest neighbour (nnn) or even more remote sites. Of course this rough approach neglects all wide-range interactions between unassociated defects, as well as interactions between them and pairs, or between pairs alone. The latter are responsible for the pre-stages of segregation (e.g. [5, 15]). To account for the remaining wide-range Coulomb interaction of the unassociated defects at least, the Debye-Hiickel theory of strong electrolytes has been used for better approximation. Also in crystals, every charged defect will be surrounded by a statistically diffuse charge cloud consisting of other mobile charged defects attracted and repelled according to their charge sign. The effective radius of the cloud, the so-called Debye length, is for single charged defects

1 _ / «

v

\4 n rt 2

y/2

(4)

where £ xi is the sum of the molar fractions of the charged defects, disregarding their charge sign, v is the volume per molecule of the pure salt, and rt is the "thermal radius", i.e., the distance between two defects where their Coulomb energy equals the thermal energy, in international units e2/4 n e e0 kT. Therefore increasing defect concentrations mean shrinking charge clouds, increasing their energy. At the same time the activity coefficients

will decrease more and more below their limit 1. Here rx is the nearest possible distance of approach for two defects before getting associated [5] or, according to the original treatment by Bjerrum, rx = rtj2 [14], with negligible influence of this choice on the results as long a s x r ^ l . Now the /,• enter all mass action equations mentioned above as corrective factors with the respective concentrations of the free charged defects. These in turn must increase in order that the equation remains valid after "switching on" the interaction. This produces an anomalous increase of the corresponding transport effect, especially at high temperatures where xt is large even in undoped crystals. In the contrary, association is mainly a medium- or low-temperature phenomenon. This primitive approach to the underlying many-body problem, though unsatisfactory from a theoretical point of view, is useful in practice, allowing to extend the validity range of the statistical calculations to values of f{ » 0.5 [14], i.e. up to total defect concentrations of 10" 3 for s T = 8500 °K, or 10~5 for e T = 1800 °K (v = 0.05 nm 3 ; denominator 1 + x rx in equation (5) ne-

16

P . SUPTITZ a n d J .

TELTOW

glected). On t h e other hand t h e value = 0.99 defines t h e concentration u p t o which t h e Debye interaction m a y be neglected. These limits are 1/50 of those given above. A t t e m p t s t o account for t h e defect interactions in a systematical way b y a cluster expansion of t h e partition function [16] are promising b u t suffer f r o m convergence difficulties. F r o m these calculations it was inferred t h a t t h e above simple interaction t h e o r y is legitimate at least for s T > 5000 °K and xt < 10" 4 , giving ft « 0.6. This paper also contains a detailed discussion of t h e problems inherent in the theory of strong electrolytes and in its application to ionic crystals. 2.3 Mobility

2.3.1

of

defects

General

The motion of a defect from one equilibrium position in t h e lattice t o t h e next one is f u n d a m e n t a l for all t r a n s p o r t processes. I n fact, this migration of defects under t h e influence of external forces violates t h e conditions of t h e r m o d y n a m i c equilibrium. Therefore, a n elaborate t r e a t m e n t of these processes should use nonequilibrium thermodynamics (Howard and Lidiard [6]) which indeed provides a n elegant a n d unified description of defect migration especially in complicated cases where t h e cross-terms in t h e set of equations between fluxes a n d forces are i m p o r t a n t (mutual drift effects, two a n d more fields acting on t h e migrating defects, etc.). Such t r e a t m e n t is out of scope here, t h e more so because it cannot give additional insight in t h e underlying atomic processes, being essentially phenomenological. This criticism does not apply t o physical conclusions from t h e Onsager relations. B u t their derivation for crystals f r o m first principles requires t h e fulfilment of certain conditions [6], For an atomistic description, we have t o resort t o t h e commonly used simple model of thermally activated jumps, though it lacks a sound theoretical foundation. According t o this model, a n y defect will sometimes be able t o cross t h e lowest potential saddle of its surroundings, owing t o t h e r m a l fluctuations. The result is a mean j u m p frequency w t o a particular neighbour site, W

= v e x p ^ = , e x p ^ e x p ^ - ,

(6)

where Gm is t h e Gibbs free energy of activation a n d v a vibration frequency of t h e order of t h e Debye limiting frequency (1012 s - 1 ) . For a critical discussion of recent refinements in t h e derivation of equation (6), we refer to [220] (foundation), [17] (one-dimensional computer simulation), and [212] (allowance for anharmonic terms in the crystal potential). At low t e m p e r a t u r e s quantum-mechanical deviations are possible [18]. F r o m t h e pressure dependence of w, a free activation volume F m of migration m a y be derived as above, accounting for t h e dependence of v on p. F m is a measure for t h e expansion or t h e lattice due to t h e presence of t h e jumping a t o m in the saddle point configuration. Now t h e most probable j u m p mechanisms, easily t o be exemplified in Fig. 1, are t h e following: as a) J u m p of M c into a d j a c e n t Q c o r °f X a into a d j a c e n t n a > a s similar jumps of Mf c of X f a (vacancy mechanism).

Transport of Matter in Simple Ionic Crystals

17

b) J u m p of or X ¿ on an adjacent interstitial site, similar for Mf 5 or X f ¿ (direct interstitial migration). c) M q pushes an adjacent Mc on another adjacent interstitial site in order to occupy itself t h e vacated lattice site (interstitialcy mechanism). d) J u m p s of ions from lattice sites t o adjacent interstitial sites and vice versa (Frenkel disorder, distribution of foreign ions on both lattice and interstitial sites). Mechanism a) may proceed in steps d), if energetically favourable. I t is assumed t h a t all these different jump mechanisms are duly described by equation (6). I n what follows, t h e connection between w and t h e quantities measured by means of t h e various transport effects is outlined. 2.3.2

Diffusion

Diffusion of defects results from the statistical succession of a large number of jumps described by equation (6) (random walk), leading to a net current of defects down a concentration gradient (grad x). Any class i of parallel jump paths of length r, forming an angle i)i with the direction of t h e gradient (0 < u Q assuming u Q > u a taken from the text, whereas in Table I I I of [134] the values are interchanged n ) [207] a 15 ) [217] a 16 ) [203] Q, k'1 S = 10.3

T a b l e 3c Vacancy pairs Q c D a Crystal KCl KBr NaCl AgCl AgBr *) ) 3 ) 4 ) 5 ) 6 ) ') 8 ) 2

H -Ä» (eV)

Há (eV)

10« r 0

H ni;l (eV)

1.341) 1.413) 1.303) 0.826) «)

1.041) 1.193) 1.253)

2.981) 0.158 3) 0.1523)

1.502)

[152] A e [82] D, calculated with H = 2.31 eV [150] Ae [55] D, calculated with H = 2.12 eV [183] a assuming H = 1.25 eV may apply to n a [110] D

M

1.314) 1.0 5 ) 7 ) < 1.658)

Transport of Matter in Simple Ionic Crystals

33

observed to follow a law (15) (W activation enthalpy for diffusion). For self-diffusion this is expected from equations (2), (6), and (7), if only one kind of defect enables the diffusion. Then W=Hj2 + Hm. Contributions of several defects generally cause a curvature in the log D vs. 1 jT plot; the accuracy of experimental points sometimes is not sufficient to distinguish it from a straight line, hence a constant slope does not prove the operation of only one defect. The contributions of various defects to the total self-diffusion can be separated by comparison of measurements on crystals with altered defect concentration (by suitable doping). From the diffusion coefficients for single anion and single cation diffusion (Dm and D x , respectively) the transport numbers tu and tx follow, being defined as

and tx + i M = 1. In this way the transport numbers can be obtained more accurately than by the Tubandt method; for an example see Section 4.1.3.1. Former work on transport numbers is compiled in [1, 2, 5]. Impurity diffusion has been generally considered with regard to the transport mechanism [5, 25] or to the size of the diffusing ions [11, 10], it is of practical importance as well. Values of D0 and W are listed in Table 2. If the diffusing impurities influence the defect concentration of the host crystal, D depends on concentration. In the case of impurities occupying interstitial sites as well as lattice sites, D decreases with rising concentration [25]. — Aliovalent impurities forming associates with vacancies of excess charge diffuse much faster than the host ions, which only have chance encounters with vacancies, and D increases with increasing concentration (Fig. 3). This is exemplified with various systems (see Sections 4.1.3.1 and 4.1.3.2). At sufficiently high impurity concentrations xf (dependent on temperature) the degree of association approaches unity, i.e. D "saturates". In this concentration range the temperature dependence of D yields approximately the activation enthalpy Hm& for migration of the dipole, if the rate-determining step in the motion of the impurity-vacancy pair is the interchange jump (w2) between the divalent ion and its attached vacancy. (?a and the degree of association are obtained from fitting the equations for the association model [5] to the curves D vs. xf. — For cases kTjO& < < 0.1 the concentration where association becomes overwhelming is small

o.s

05 0.4 03 Fig. 3. Diffusion of Zn 2 + in NaCl as a function of Zn concentration at 703 °C. Near 300 ppm D has reached its maximum value. The line drawn through the experimental points is calculated with Ga = = 0.48 eV and a thermal vacancy concentration of 1.5 x i o - 5 . — After Kothman et al. [86] 3

physica 23/1

0.2

0.1 0

50

100

150 200 250 300 Zn concentration (ppmj

34

P . SÜPTITZ a n d J . T E L T O W

Table 4a Impurity defects, alkali halides System

Defect

H & or (? (eV) 0.5 1 ) 0.34 1 ) 0.30 1 ) +0.31 3 ) 0.67 4 )

NaCl: Mf 2 + Mg 2+ Ca2+

MfcDc MgcDc Cacnc

Sr 2 + Ba2+ Zn 2 + Cd 2+

SrcDc BacDc ZncDc CdeDe

Pb2+ Mn 2 +

PbcDc MncDc

+0.47 to 0.41 1 3 ) 0.33 1 ) 0.7 1 4 )

Ni2+ Sm 2 + K+

Nicnc SmcDc KcKc

0.32 18 )

KCl:Mf 2 +

MfcDc

Ca 2 +

CacQc

+0.43 21 ) 0.22 22 ) 0.52 2 3 )

Sr 2 + Ba 2 + Cd 2+ Pb2+ Ce 2(3) +

SrcDc BacDc CdeDc PbcDc

KBr:Ca 2 +

Cacmc

Sr 2 + Ba2+ Cd 2+ cor

SrcDc BacDc CdcDc

LiF:Mg2+

Mgcnc

Mn 2 + Ti 2 ( 3 )+

0.76 1 ) +0.48 10 ) 0.40!)

*) calculated from [62] a, assuming 0.80 eV H„ 2 ) [31] Ae (dc) 3 ) [96] a, 250 to 400 °C

^ma (eV)

0.66 2 ) 0.68 5 ) 0.70 6 ) 0.67') 0.7 8 ) 0.71 2 ) n

)

+0.42 21 ) +0.51 26 ) +0.58 to+0.41 2 7 ) 0.42 2 9 ) 0.46 ^

0.7 31 ) +0.8 3 1 )

MncDc Ti c Do37)

Bi (eV)

0.92») 0.96 4 )

0.66 2 ) 0.69 2 ) 0.68 7 )

0.51 1 0 ) 1 2 ) 0.98 1 3 ) 0.71 1 4 ) 0.78 1 6 ) 0.83 17 )

0.68 2 ) 0.63 1 5 ) 0.7 s )

) [190] D, k~1 S a = 5.7 5 ) [156] Ae 6 ) [179] A e ') [77] (ITC)

0.94 1 ) 0.94 1 ) 0.79 1 )

0.78 1 ) 0.99 1 ) 0.98 10 ) 0.93 1 ) 0.78 1 )

0.67 1 9 ) 0.9...l.l 2 0 ) 0.67")

0.73 1 7 )

0.63 7 ) 0.62 24 ) 0.66 7 ) 0.70 1 7 )

0.66 2 4 )

,0.62 2 2 )

0.73 17 ) 0.74 1 7 ) 0.54 26 ) 1.18 27 ) 1.03 28 ) 12 ) 0.72 2 9 ) 1.14 29 ) 1.88 29 ) 1.28 29 ) 2.12 29 )

0.65 3 0 ) 0.69 7 )

0.75 32 )

0.79 5 ) 0.75 35 )

0.65 33 ) 0.64 34 )

0.8 2 5 )

0.48 3 8 ) 4

B. (eV)

) ) 10 ) u) 12 ) 8

9

[35] [191] Ae [86] D, a 700 °C from diffusion in heavily doped samples

35

Transport of Matter in Simple Ionic Crystals Table 4 a (continued) System

Defect

KJ: Mg2+ MgcDc Ca2+ Ca c D c Sr2+ Sr c D c 13 ) [205] I), 321 to 528 °C 14 ) [167] D, k- 1 S a = 2.2 15 ) [76] Ae, ESR 16) [158] Ae, ESR 17) [146] Ae 18) [67] o19) [147] a, A (dc) 20) [164] A m

ffa

or Ga(+)

(eV)

0.483S) 0.3836) 0.42 ^ 21) [68] a 22 ,\ [71]