Physica status solidi: Volume 19, Number 1 January 1 [Reprint 2021 ed.] 9783112498286, 9783112498279

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plrysica status solidi

V O L U M E 19 • N U M B E R l . 1967

Classification Scheme 1. S t r u c t u r e of Solids 1.1 Alloys. Metallurgy 1.2 Solid-State P h a s e T r a n s f o r m a t i o n s 1.3 Surfaces 1.4 Films 2. Non-Crystalline S t a t e 3. Crystallography 3.1 Crystal G r o w t h 3.2 I n t e r a t o m i c Forces 4. Microstructure of Solids 5. Perfectly Periodic S t r u c t u r e s 6. L a t t i c e Mechanics. P h o n o n s 6.1 Mossbauer Investigations 7. Acoustic Properties of Solids 8. T h e r m a l Properties of Solids S). Diffusion in Solids 10. Defect Properties of Solids (Irradiation Defects see 11) 10.1 Defect Properties of Metals 10.2 P h o t o c h e m i c a l Reactions. Colour Centres 11. I r r a d i a t i o n E f f e c t s in Solids 12. Mechanical Properties of Solids (Plastic Deform a t i o n s see 10)see 10.1) 12.1 Mechanical Properties of Metals (Plastic Deformations 13. Electron S t a t e s in Solids 13.1 B a n d S t r u c t u r e . F e r m i Surfaces 13.2 E x c i t o n s 13.3 Surface S t a t e s 13.4 I m p u r i t y a n d Defect S t a t e s 11. Electrical Properties of Solids. T r a n s p o r t P h e n o m e n a 14.1 Metals. Conductors 14.2 S u p e r c o n d u c t i v i t y . Superconducting Materials a n d Devices 14.3 Semiconductors 14.3.1 Semiconducting Films 14.3.2 Semiconducting Devices. J u n c t i o n s (Contact Problems see 14.4.1) 14.4 Dielectrics 14.4.1 H i g h Field P h e n o m e n a , Space Charge Effects, Inhomogeneities, I n j e c t e d Carriers (Electroluminescence see 20.3; J u n c t i o n s see 14.3.2) 14.4.2 Ferroelectric Materials a n d P h e n o m e n a 15. Thermoelectric a n d T h e r m o m a g n e t i c P r o p e r t i e s of Solids 16. P h o t o c o n d u c t i v i t y . P h o t o v o l t a i c E f f e c t s 17. Emission of Electrons a n d I o n s f r o m Solids 18. Magnetic Properties of Solids 18.1 P a r a m a g n e t i c Properties 18.2 F e r r o m a g n e t i c Properties 18.3 F e r r i m a g n e t i c Properties. F e r r i t e s 18.4 A n t i f e r r o m a g n e t i c Properties (Continued

on cover three)

phys. stat, sol. 19 (1967)

Author Index E . D . ALUKER S. AMELINCKX A . T . AMOS C. ANGHEL V . V . ANTONOV-ROMANOVSKII . . . . M . ARNDT A . AXMANN H . BACHERT R.G.BARNES G . R . BARSCH W . BARTH R . D E BATIST A . R . BEATTIE V . Z . BENGUS P . BENNEMA A . VAN DEN BEUKEL G . BJÖRKMAN J . BLOK K . W . BÖER I R . BOESONO 0 . V . BOGDANKEVICII B . I . BOLTAKS F . BORSA Y U . S. BOYARSICAYA H . P . VAN DE BRAAK R . H . BRAGG V . L . BROUDE P . BROUERS W . J . CASPERS Z . P . CHANG M . S . R . CHARI P . CHARSLEY G . A . CHASE C. CHERKI J . M . CHRISTIE R . COELHO P . C . J . COREMANS J. B. P. H. G. T.

G . DASH DAYAL DELAVIGNETTE P . DIBBS DÖHLER D . DZHAEAROV

35 683 587 613 417 K79 721 K59 359 129,139 515 77 577 533 211 177 863 K107 203 107 K5 705 359 441 385 K99 395 867 385 139 169 K63 645 K91 631 K91 177

K31 729, 751 683 KLL 555 705

R . ENDERLEIN G . J . ERNST L.ERNST I . N . EVDOKIMOV G . J. O . EYOKU

673 107 89 407 ILL

0 . E . FACEY A . S . FILIPCHENKO

565 435

R.FISCHER W . FRANK W.FRANZ B . FRITZ

757, K 1 0 3 239 597 515

E . I . GAVRILITSA A . GEMPERLE A.M.GEORGE R . GEVERS R . N . GHOSHTAGORE W . GISSLER A . GLODEANU P . GÖRLICH R . M . GRETCHISHKIN D . T . GRIGGS G . GRIMWALL 0 . GRÜTER F . GURU G . J . VAN GURP G . D . GUSEINOV E . GUTSCHE P . GUYOT

609 333 117 77 123 721 K43 K23 K1 631 863 217 339 173 K7 823 K95

M. R. A. M. M.

K99 353 185 833 543

L . HAMMOND P . HARRISON K . HEAD HENZLER HOLZMANN

E . IGRAS 1. V . IOFFE M . Z . ISMAILOV

K67 51 K7

P . W . M . JACOBS L.JAHN E . JAHNE W . A . JESSER B . JIMÉNEZ G.A.JONE S

565 K75 823 95 805 811

M . D . KARKHANAVALA H . KARRAS A. KEIPER R . KEIPER C. A . K E N N E D Y E . M . KERIMOVA R.KERN J.D.KEYS G . A . KHOLODAR E . KIERZEK-PECOLD 623, J . KOCÍK J . KOLODZIEJCZAK 2 3 1 , 3 7 3 , 6 2 3 , K 5 1 , V . G . KOLOSKOVA S. N . KOMNIK

117 K23 K59 673 203 K7 211 KLL 41 K55 333 K55 441 533

876

Author Index

E . KÖSTER A . I . KRASILNIKOV D. KUHLMANN-WILSDORF R . LABUSCH P . T . LANDSBERG M . J . VAN LANGEN M . C. LEMMENS R . D . LEVINE P . F . LINDQUIST H . A . LIPSITT Y U . A . LOGATCHOV W.LUDWIG W . VAN DER LUGT P . LUKÄC M . LUPULESCU E . D . LYAK

153, 6 5 5 K5 95 715 777 107 107 587 K99 K99 K35 313 327 K47 613 533

E . MACHERAUCH M . J . MALACHOWSKI J . MAN R . MANAILA E . S . MASHKOVA E . MAURER A . C. MCLAREN E . MEISEL J . MENDIOLA I . P . MEZINA M . P . MIKHAILOVA P . MIOSGA D. D . MISHIN V . A . MOLCHANOV S . B . VAN DER MOLEN H . MOTHES H . MUGHRABJ L . E . MURR

793 K27 543 K19 407, 425 805 631 K23 805 35 429 597 KL 407, 425 327 K23 251 7

D. R. M. R.

429,435 K87 KILL K31

N. E. A. H.

NASLEDOV NEWNHAM NIZAMETDINOVA NUSSBAUM

D . D . ODINTSOV H . M . OTTE

407 K99

V . B . PAKIISKII P . PAUSESCU A . N . PECHENOV K . - H . PFEFFER H . PICK A. POLICEC

525 K19 K5 735 313 613

S . RAAB S . I . RADAUTSAN

K59 609

C. V . RAMAIAH C. RAMASASTRY E . I . RASHBA A . I . RASULOV R . C. RAU R . A . REESE J . A . RETCHFORD A . G. REVESZ B . F . ROTHENSTEIN A . F . RUDOLPH M . RÜHLE

K15 K15 395 K7 645 359 631 193 613 K79 263, 279

J . A . M . SALTER R . P . SANTORO S . SCHÄFER CH. SCHWINK A . SEEGER E . F . SHEKA M . M . SHUKLA J . SILCOX B . S I M ON S . V . SLOBODCHIKOV G. SMITH V . SOSHKA B . SPRUSIL J . SPYRIDELIS H . H . STILLER R . STOCKMEYER A . M . STONEHAM R . STRUIKMANS

K63 K87 297 217 251 395 729 57, 6 3 211 429 577 425 K83 683 781 781 787 K107

L . J . VAN TORNE J . TREUSCH Y . P . VARSHNI B . YELICKY M . P . VERMA M . YERSCHUEREN J . VILLAIN V . L . VINETSKII B . YLACH O. VÖHRINGER P . VOSTRY G. DE VRIES M . M . A . VRIJHOEF R . H . WADE A . K . WALTON T . WARMIKTSKI J . H . P . VAN WEEREN K . WETZIG R . P . ZHITABU S . ¿UKOTYNSKI M . M . ZVEREV

855 603 459 K39 751 77 767 41 543 793 K83 107 177 57, 63, 8 4 7 111 K67 K107 K71 441 K51 K5

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, 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 , Aachen, J. D. E S H E L B Y , Cambridge, 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, M. MATYAS, 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 19 • Number 1 • Pages 1 to 452, K 1 to K 1966, and A 1 to A 30 January 1, 1967

AKADEMIE-YERLAG

• BERLIN

Subscriptions a n d Orders for single copies should be addrcssed to A K A D E M I E - V E R L A G G m b H , 108 Berlin, Leipziger Straße 3 - 4 or to Buchhandlung K U N S T U N D W I S S E N , Erich Bieber, 7 S t u t t g a r t 1, Wilhelmstr. 4 - 6 or t o Deutsche B u c h - E x p o r t u n d - I m p o r t G m b H , 701 Leipzig, Postschließfach 160

Editorial Note: "physica status solidi" undertakes t h a t an original paper accepted for publication before the 8 t h of any m o n t h will be published within 50 days of this date unless t h e author requests a postponement. I n special cases there m a y be some delay between receipt and acceptance of a paper due to t h e review a n d , if necessary, revision of the paper.

S c h r i f t l e i t e r u n d v e r a n t w o r t l i c h f ü r d e n I n h a l t : Professor D r . D r . h . c. P . G ö r l i c h , 102 B e r l i n , N e u e S c h ö n h a u s e r S t r . 20 b z w . 69 J e n a , H u m b o l d t s t r . 26. R e d a k t i o n s k o l l e g i u m : D r . S. O b e r l ä n d e r , D r . £ . G u t s c h e , D r . W . B o r c h a r d t . A n s c h r i f t d e r S c h r i f t l e i t u n g : 102 B e r l i n , N e u e S c h ö n h a u s e r S t r . 20. F e r n r u f : 42 6 7 8 8 . Verlag: Akademie-Verlag G m b H , 108 B e r l i n , L e i p z i g e r S t r . 3 — 4 , F e r n r u f : 2 2 0 4 4 1 , T e l e x - N r . 011773, P o s t s c h e c k k o n t o : B e r l i n 3 5 0 2 1 . — D i e Z e i t s c h r i f t „ p h y s i c a s t a t u s s o l i d i " e r s c h e i n t jeweils a m 1. d e s M o n a t s . B e z u g s p r e i s e i n e s B a n d e s M D N 72,— ( S o n d e r p r e i s f ü r die D D R M D N 60,—). B e s t e l l n u m m e r dieses B a n d e s 1068/19. J e d e r B a n d e n t h ä l t zwei H e f t e . G e s a m t h e r s t e l l u n g : V E B D r u c k e r e i „ T h o m a s M ü n t z e r " B a d L a n g e n s a l z a . — V e r ö f f e n t l i c h t u n t e r d e r L i z e n z n u m m e r 1310 d e s P r e s s e a m t e s b e i m V o r s i t z e n d e n des M i n i s t e r r a t e s d e r D e u t s c h e n D e m o k r a t i s c h e n R e p u b l i k .

Contents Review Article L. E. MURR

page Calibration and Use of an Electron Microscope for Precision Micromeasurements in Thin Film Materials

7

Original Papers E . D . ALUKER a n d I . P . MEZINA

The Temperature Dependence of Radioluminescence Yield in Alkali Halide Crystal Phosphors

35

V . L . VINETSKII a n d G . A . KHOLODAR

I. V. IOFFE

On the Electric Conductivity of Semiconductors Caused by the Ionization of Thermal Lattice Defects

41

On the Theory of the Lattice Oscillations of Dielectric Crystals in an External Electric Field

51

R . H . W A D E a n d J . SILCOX

Small Angle Electron Scattering from Vacuum Condensed Metallic Films (I)

57

R . H . W A D E a n d J . SILCOX

Small Angle Electron Scattering from Vacuum Condensed Metallic Films (II)

63

R . D E BATIST, R . GEVERS, a n d M . VERSCHUEREN

L. ERN ST

Magnon Contribution to the Low-Temperature Specific Heat of UO»

77

Field-Emission Microscopy of Germanium

89

W . A. JESSER a n d D . KUHLMANN-WILSDORF

On the Theory of Interfacial Energy and Elastic Strain of Epitaxial Overgrowths in Parallel Alignment on Single Crystal Substrates . .

95

I R . BOESONO, G . J . E R N S T , M . C . L E M M E N S , M . J . VAN L A N G E N , a n d G . DE V R I E S

A Zener-Damping Peak in Fe-4 W t % Si

107

A . K . WALTON a n d G . J . O . EYOKU

Piezo-Hall Effect in n-Type Germanium

Ill

A . M . GEORGE a n d M . D . KARKHANAVALA

Studies of the Electrical Properties of Uranium Oxides (III) . .

117

R . N . GHOSHTAGORE

G. R.

BARSCH

Diffusion from a Thin Film into a Dissociating or Evaporating Solid

123

Adiabatic, Isothermal, and Intermediate Pressure Derivatives of the Elastic Constants for Cubic Symmetry (I)

129

G . R . B A R S C H a n d Z . P . CHANG

Adiabatic, Isothermal, and Intermediate Pressure Derivatives of the Elastic Constants for Cubic Symmetry (II)

139

E . KÖSTER

Verformungs- und Temperaturabhängigkeit der Magnetisierungsprozesse in Nickeleinkristallen (I)

153

M. S. R.

The Electrical and Thermal Resistivities of Iron

169

CHARI

4

Contents Page

G. J . VAN GURP Temperature Dependence of the Critical Field in Superconducting Vanadium

173

A . VAN D E N B E U K E L , P . C . J . COREJIANS, a n d M . M . A . V R I J H O E F

On the Kinetics of Short-Range Ordering in AuAg (50,50) and CuAl (85,15)

177

A. K. HEAD

Unstable Dislocations in Anisotropic Crystals

185

A. G. REVESZ

Thermal Oxidation of Silicon: Growth Mechanism and Interface Properties

193

K . W . BOER a n d C. A . K E N N E D Y

Vacuum H e a t Treatment of CdS Single Crystals

203

P . B E N N E M A , R . K E R N , a n d B . SIMON

Interpretation of Deviating Points in the Relation between the R a t e of Crystal Growth and the Relative Supersaturation

211

CH. SCHWINK u n d 0 . GRÜTER

Untersuchungen der Bereichstruktur von Nickeleinkristallen (II)

217

J . KOLODZIEJCZAK

W. FRANK

On the Scattering Processes in Semiconductors

231

Die kritische Schubspannung kubischer Kristalle mit Fehlstellen tetragonaler Symmetrie (IV)

239

H . MUGHRABI a n d A . SEEGER

The Study of Defects in Quenched Nickel

251

M. RÜHLE

Elektronenmikroskopie kleiner Fehlstellenagglomerate in bestrahlten Metallen (I)

263

M. RÜHLE

Elektronenmikroskopie kleiner Fehlstellenagglomerate in bestrahlten Metallen (II)

279

S. SCHÄFER

Messung von Versetzungsgeschwindigkeiten in Germanium . . . .

297

W . LUDWIG a n d H . PICK

Scattering of Phonons a t Defect Planes in the Simple Cubic Lattice W . VAN DER L U G T a n d S . B . VAN D E R M O L E N

Nuclear Magnetic Resonance in Liquid Gallium Alloys

313 327

A. GEMPERLE a n d J . KOCIK

Transmission Electron Microscopic Observation of Lattice Deformation in Antiphase Boundaries of Ordered Pe-Si Alloys . . . .

333

F. Guiu

Temperature and Strain R a t e Dependence of the Flow Stress in Molybdenum

339

R . P . HARRISON

The Effect of Cadmium Additions on the Fracture Strength of Sodium Chloride Single Crystals

353

F . BORSA, R . G . B A R N E S , a n d R . A . R E E S E

Nuclear Magnetic Resonance and Mössbauer Effect Study of Sn 119 in Rare Earth-Tin Intermetallic Compounds

359

J . KOLODZIEJCZAK

On t h e Theory of Galvanomagnetic Phenomena in Strong Electric Fields

373

H . P . VAN D E B R A A K a n d W . J . C A S P E R S

Disorder Resistivity and Spin Depolarization in Rare-Earth Alloys

385

V . L . BROUDE, E . I. RASHBA, a n d E . F . SHEKA

A N e w Approach to the Vibronic Spectra of Molecular Crystals

395

Contents

5 Page

I . N . EVDOKIMOV,

E . S . MASHKOVA,

V . A . MOLCHANOV, a n d D . D . O D I N T S O V

Dependence of the Ion-Electron Emission Coefficient on the Angle of Incidence

407

V . V . ANTONOV-ROMANOVSKII

On the Kinetics of the Crystal Phosphor Luminescence E . S . MASHKOVA,

417

V . A . MOLCHANOV, a n d V . SOSHKA

Influence of the Thermal Vibrations of the Crystal Lattice upon the Energy Distribution of Scattered Ions M . P . MIKHAILOVA,

425

D . N . N A S L E D O V , a n d S . V . SLOTBODCHIKOV

The Effect of a Magnetic Field upon Illuminated InAs p - n Junctions

429

A . S . F I L I P C H E N K O a n d D . N . NASLEDOV

On the Mixed Mechanism of Electron Scattering in InSb Crystals Y u . S . BOYARSKAYA,

435

R . P . Z H I T A R U , a n d V . G . KOLOSKOVA

Effect of Different Lattice Defects on the Mobility of Dislocations in Alkali Halide Crystals

441

Short Notes D . D . M I S H I N a n d R . M . GRETCHISHKIN

Electron Microscopic Platinum Alloys O. V . BOGDANKEVICH,

M . M . ZVEREV,

Examination

of

High-Coercive

Cobalt-

A . I . RASULOV,

1

K

5

K

7

A . I . KRASILNIKOV, a n d A . N . PECHENOV

Laser Emission in Electron-Beam Excited ZnSe G . D . GUSEINOV,

K

E . M . KERIMOVA, a n d M . Z .

ISMAILOV

On Heat Conductivity of A m B V I - T y p e Semiconductors

J . D. K E Y S a n d H . P . DIBBS

On the Bonding in Bismuth Telluride

KLL

C. RAMASASTRY a n d C. V . RAMAIAH

Dielectric Constant of Sodium Chlorate Crystals at Microwave Frequencies K15 R . MANAILA a n d P . PAUSESCU

The Tetragonal to Cubic Transition in the System Cd J Mgi_ z Mn 2 0 4 K 19 P . GORLICH,

H . KARRAS,

E . MEISEL, a n d H . MOTHES

I R Absorption of Uncoloured and Additively Coloured Synthetic Crystals and CaF 2 /Na Crystals K 23 M . J . MALACHOWSKI

Current-Voltage-Temperature Junctions

Dependence of Symmetric Tunnel K 27

R . H . NUSSBAUM a n d J . G . DASH

Relaxation Narrowing and Thermal Shift of Mossbauer Lines . . . K 31 Y u . A . LOGATCHOV

Model Potential Approximation for Zj-Centres in Ionic Crystals . . K 35 B.

VELICKY

Polaron Effect on the Optical Absorption Edge of Semiconductors K 39

A.

GLODEANU

Helium-Like Impurities in Semiconductors

K 43

6 P. LTJKAC

Contents Spannungsrelaxation in Cd-Einkristallen

K 47

S . ZUKOTYNSKI a n d J . KOLODZIEJCZAK

On the Determination of the Non-Parabolicity of Energy Bands

. K 51

J . KOLODZIEJCZAK a n d E . KIERZEK-PECOLD

Free Carrier Electro-Optical Kerr Effect in Semiconductors . . . . K 55 S . RAAB, H . BACHERT, a n d A . K E I P E R

Spectral Investigation of Emission Inhomogeneities in GaAs Injection Lasers K 59 P . CHARSLEY AND J . A . M . SALTER

Further Comments on the Analysis of Measurements of the Lattice Thermal Conductivity K 63 PRE-PRINTED TITLES AND ABSTRACTS

of papers to be published in this or in the Soviet journal ,,®H3Hi;a TBepaoro Tejia" (Fizika Tverdogo Tela) A SYSTEMATIC LIST

Principal subject Classification: 1.1 1.2 1.3 1.4 3.1 3.2 5 6 6.1 8 9 10 10.1 10.2 12 12.1 13.1 13.4 14.1 14.2 14.3 14.3.2 14.4 15 17 18.2 19 20 20.1 20.2 20.3

Corresponding papers begin on the following pages:

177 K 19 89, 1 9 3 7, 57, 6 3 211 K 11 333 313, 395, 425 359, K 31 77, K 7, K 6 3 123 95, 297, 353, 441 185, 2 3 9 , 2 5 1 , 2 6 3 , 2 7 9 , 3 3 9 , K 4 7 41, K 2 3 129, 139 107 K 51 203, K 35. K 4 3 169, 3 8 5 173 41,111,231,435 429, K 27 K 15 117, 3 7 3 407 153, 2 1 7 , K 1 327 K 55 K 39 K 5, K 5 9 35, 417

Contents of Volume 19 Continued on Page 455

1

Review Article phys. stat. sol. 19, 7 (1967) Subject classification: 1.4; 4; 10 Materials Research Laboratory of the Pennsylvania State University Park, Pennsylvania

University,

Calibration and Use of an Electron Microscope for Precision Micromeasurements in Thin Film Materials By L. E. M u r b Contents 1. Introduction 2. Image magnification calibration 3. Diffraction pattern calibration 4. Examples of special electron diffraction

4.1 Double and multiple diffraction 4.2 E x t r a diffraction spots 4.3 Streaking in the diffraction pattern 5. Image-pattern rotation calibration 6. Some aspects of diffraction contrast

6.1 Bright-field images 6.2 Dark-field images 7. Structural 8. Closure References

geometry

in thin metal

foils

1. Introduction In recent years, the electron microscope has become virtually a common place research instrument in most universities and many industries, as well as hospitals and clinics throughout the United States and Europe. Its use is essentially indispensable in many areas of chemistry, physics, biology, medicine, metallurgy, and numerous other related areas concerned with atomic and molecular details, and micromeasurements in organic and inorganic materials. Although considerable information can be obtained by simple image observation in certain substances, qualitative interpretation and precision micromeasurements of fine scale detail cannot be attained without a fairly rigorous calibration of microscope constants. These constants relate to direct magnification calibration, image rotation characteristics, camera constant information for the interpretation of electron diffraction data, and image-diffraction pattern correlations. I t is the purpose of this paper to present some of the fundamental techniques for electron microscope calibration, and to illustrate various uses to which the calibrated microscope can be put in the particular case of thin metallic specimens of engineering interest.

L . E . MURE

2. Image Magnification Calibration In order to determine the relative dimensions of image detail, the magnification characteristics of the microscope must be determined. Magnification, in a physical sense, is controlled by the intermediate lens strength of an electron microscope. In fact, the lens strength can be related directly with the image magnification. Thus, a direct and effective means of magnification calibration is to relate image detail with the intermediate lens current. At lower magnification (1000 to 10000 times) the most effective means of calibration is the use of a suitable diffraction grating. As an example, a 28800 lines per inch grating (in the form of a standard 1/8 inch diameter replica grid) can be used, and images taken at various intermediate lens current settings on a particular low magnification range. From this information, a magnification plot can be constructed by computing magnification directly from

where M is the magnification, Nx is the total number of lines per unit length in the calibrated grid, and n is the number of lines per unit length as measured in the image. At higher magnifications, suitable fine scale detail may then be calibrated on the high end of the lower magnification scale, and the image detail compared in successive image enlargements. The ultimate check on very high magnification would then involve the direct imaging of atomic plane spacings or molecules of known dimensions. Many electron microscopes today are designed for what might be called multirange operation. That is, a range selector places a particular portion of the intermediate lens coil in operation, and magnification is therefore limited by the effective coil arrangement, and the peak current flow. It is then necessary to calibrate each range independently, since the magnification-current response will differ from one to the other. I t is also necessary to calibrate each range at the particular operating voltage desired, since magnification at a particular current setting will increase with increasing operating potential. 3. Diffraction Pattern Calibration Selected area electron diffraction patterns of an image area are often indispensible for the complete identification of a material's properties. Diffraction patterns can also supply information inexcessible in the image detail of an electron micrograph. It is therefore necessary, to calibrate the dimensional properties of a diffraction pattern in much the same way that image magnification is obtained. The principles of electron diffraction are somewhat more involved than direct beam transmission since the image detail in the former is structure sensitive, and corresponds to lattice periodicity. The basis for the interpretation of electron diffraction patterns thus stems from the Bragg relationship 2dsin6

= nX ,

(2)

where d is the spacing between atomic planes (h k I) in the lattice, d is the angle of diffraction from the plane (h k I), n is the order of diffraction, and X is the wavelength of the diffracted beam.

Use of an Electron Microscope for Precision Micromeasurements

9

We observe from equation (2) t h a t as the electron beam strikes a plane surface, the diffraction image will be controlled by the accelerating potential since 12 3 A= 7 - —^(A), yv (i + io-® V)

(3)

where V is the operating voltage in volts; and by the angle of diffraction 6, considering only first-order reflections (n = 1 ) . F o r a totally r a n d o m structure of the material, the resulting p a t t e r n will be a succession of rings about the main beam spot. The spacing and intensity of the rings will be governed chiefly b y d and the structural characteristics of the material. F o r a cubic structure (simple cubic, f.c.c., or b.c.c.) d =

|/A2 + F + i 2

,

(4)

where a is the lattice constant (mean interatomic spacing), a n d h, k, I are t h e Miller indices of the reflecting plane. The angle 6 or t a n 2 6 corresponds to t h e spacing of the rings in the diffraction image. And, because of the large distance from the specimen plane to the final photographic image plane (L), the ring spacing or radius f r o m the center spot can be represented b y R = 26 L , (5) where t a n 2 6 2 d for the conditions specified. Substituting equation (5) for sin (7 d, and (4) for d into (2) with n = 1 results in

or we could rewrite

]jh2 +k2 + l2

(£)-*•

|H' + t' + l'

.J *.

(7)

We observe f r o m equation (7) t h a t if the left-hand t e r m is known for e a c h successive ring, and the radius is accurately measured, t h e n knowing t h e product A L will allow the interatomic spacing (a) to be determined. On the other hand, if a is known for a particular material, the resulting rings can be related to h, k, I indices. Ring p a t t e r n s result from polycrystalline materials where the grain size is much smaller t h a n the aperture area and of random orientation. I n the case of crystalline areas greater t h a n the diffraction (field limiting) aperture area, the resulting electron diffraction p a t t e r n consists of symmetrical arrays of spots, each spaced according to (7) along a radius vector (i?) from t h e center or main beam spot. Each spot therefore relates to a reflecting plane (h k l ) or a direction \ h k l \ . If the surface plane is then continuous in the area observed in diffraction, the beam direction normal to this surface will be given by [H K L], The arrangement of spots h, k, I about this pole axis along radius vectors corresponding to planes h, k, I, will t h e n be governed by Hh + Kk + Ll = 0 .

(8)

We can therefore denote (H K L) as the crystallographic surface orientation. We observe, therefore, t h a t measuring the radius vector of each spot in such a diffraction p a t t e r n will allow the plane (H K L) to be identified. I t also allows the directions \h k I] t o be determined by identifying each spot as it relates to the indices (h kl).

10

L. E. Murr

002 r

,î,

*

? 000,0

1

b-

i TI 002 ( MO)

Fig. 1. Selected area electron diffraction patterns from a JEM 6A electron microscope fitted with a tilt stage. Upper pattern is face-centered cubic nickel, lower pattern is body-centered cubic iron. Both the nickel and iron samples were prepared by vapor deposition onto NaCl crystal surfaces at 10~4 Torr vacuum. Ring indices are indicated on each pattern (100 kV operation)

Use of an Electron Microscope for Precision Micromeasurements

11

I t should be observed that the left-hand term of equation (7) is merely structure sensitive and depends only on allowed values of h, k, I. Since a is also known for many materials (particularly metals), it is possible to compute the product (A L), which is usually referred to as the camera constant, for some known set of diffraction rings; where R for each ring (h k I) may be measured. Fig. 1 illustrates typical ring patterns for face-centered cubic (f.c.c.) nickel, and body-centered cubic (b.c.c.) iron prepared by vapor deposition of the pure metal onto rock salt substrates at low vacuum (10~ 4 Torr). Direct measurement of the ring radii corresponding to some (h k I) in each case allows two straight lines to be constructed. The camera constant can then be computed from the slope of each curve. As it turns out, (A L) is constant for any particular operating potential (V) and for any standard specimen stage arrangement. However, when V is changed or the stage characteristics altered, (A L) will also vary correspondingly. An efficient means of identifying ring indices or spot indices (h k I) for a set of constant conditions is to computerize the data for any desired metal or other material with a known lattice constant and structural characteristic (b.c.c., f.c.c., etc.). 1 ) Typical examples of the spot patterns obtained from single-crystal sections of metal films are shown in Fig. 2. In addition to the diffraction spots, Fig. 2 also illustrates the existence of symmetrical electron scattering by traversing the relatively thick metal sections. Such scattering is usually referred to as Kikuchi electron diffraction, and the resulting dark and bright lines designated as Kikuchi lines. Each set of lines (bright and dark) corresponds to a particular index (h k I) as do the spots. Thus, the spacing between sets of Kikuchi lines (h k I) corresponds to the spot radii (hkl). For more detailed information concerning Kikuchi diffraction, the reader is referred to references [1], [2], and [3]. Fig. 2 illustrates schematically the symmetry of spots for the surface orientations specified, i.e., (100), (110), (112). Similar patterns for direct comparison with other orientations can also be sketched utilizing equation (8) as a basis for constructing the schematic. The proof of a particular surface orientation is then obtained by comparing the distance of the spot indices from the center spot, or more accurately the ratio of the directions, i.e.

"(f)-

Mr)-

H t ) -

-

The solution of the crystallographic surface plane from an unknown pattern thus involves a trial and error manipulation of indices (spot direction magnitudes) and/or ratios of the spot distances from the center spot and comparison with the calibrated radii for a particular known material. In some cases, more rigorous analysis is necessary involving stereographic projections of the crystallographic poles. 4. Examples of Special Electron Diffraction 4.1 Double

and multiple

diffraction

I t might seem logical to assume that if an electron beam passes through two types or layers of materials simultaneously, diffraction will result from both if the total layer thickness is not so large that the beam energy is scattered and A Fortran computer program for the simultaneous computation of diffraction radii for any f.c.c. or b.c.c. metal and for any particular camera constant (XL) is available on request from the author. Requests should be addressed to 14 Holly Circle, State College, Pennsylvania, U. S. A.

L . E . MURR

Fig. 2. Single crystal (spot) diffraction patterns form large grains in annealed 304 stainless steel. Graphical representation of patterns shows symmetry of the more common patterns observed in metals, i.e. (100), (110), (112) surfaces nearly normal to the beam direction. The appearance of Kikuchi lines in the patterns indicates the diffracting areas are generally thick (1500 to 2500 A) and maintain a high degree of angular perfection

Use of an Electron Microscope for Precision Micromeasurements

13

dissipated as heat. Thus, for a polycrystalline layer superposed on a single crystalline layer, one might expect a set of rings from the polycrystalline diffraction, and a superposed set of symmetrical spots resulting from diffraction in the single crystal layer. Ideally, we therefore observe double diffraction from the composite section.

F i g . 3. E l e c t r o n diffraction p a t t e r n s from a X a C l - A l double f i l m layer e x h i b i t i n g double diffraction f r o m multiple p r i m a r y sources. B o t h f i l m s are a p p r o x i m a t e l y 5 0 0 À t h i c k , with t h e X a C l the f i r s t or p r i m a r y diffracting l a y e r , a) NaCl ( 1 0 0 ) surface, b) NaCl ( 1 1 1 ) surface, c) NaCl ( 1 1 0 ) surface, d) NaCl ( 1 1 0 ) surface. A l u m i n u m sub-layer is polycrystalline-fine grain

14

L . E . MURK

Frequently, t h e double-diffracted beam produces strong secondary centers which in t u r n produce a complete diffraction p a t t e r n in the layer in which t h e y operate. We t h e n have w h a t might be t e r m e d multiple diffraction. Several examples of multiple electron diffraction have been previously reported for double layers such as a single crystal metal film with a superposed oxide layer (Heidenreich [1]). This phenomena is becoming increasingly common in electron diffraction analysis, and especially in the examination of double metal composites, such as electroplated t h i n foils. Multiple diffraction f r o m such double layer composites is enhanced b y a difference in t h e crystal sizes, the effect becoming increasingly pronounced as the difference becomes greater. I n the present work, t h e double layer consisted of a recrystallized and/or adhered cleavage steps of single crystal NaCl on a vapor deposited polycrystalline substrate of high p u r i t y aluminum. The NaCl was observed in several orientations including (100), (110), and (111). Electron diffraction microscopy was performed with a J E M 6A electron microscope operated a t 100 kV using double condenser illumination. Observations were made with a n d without a specimen tilting stage. Typical results of multiple electron diffraction from a NaCl-Al double layer are shown in Fig. 3. I n Fig. 4, prominent features of the multiple diffraction p a t t e r n s shown in Fig. 3 are sketched with reference to the Bragg conditions producing the p a t t e r n s observed. The single crystal spot p a t t e r n s in the examples shown originate in t h e NaCl, (100)

m

a

b m

0

(no)

d

"Fig. 4. Schematic diagrams of strong XaCl diffraction centers and doubly diffracted A1 rings shown in the p a t t e r n s of Fig. 3

Use of an Electron Microscope for Precision Micromeasurements

15

while the characteristic ring pattern originates in the aluminum substrate. In cases where strongly diffracting spots occur, this condition acts as an incident beam for diffracting again from the polycrystalline aluminum. I t will be observed also that only prominent reflections in the polycrystalline ring pattern are visible in the case of such secondary incident diffraction, which can occur at a large number of strong symmetrically diffracting indices (Fig. 3a), or only a relatively few such as in Fig. 3 c. In the case of Fig. 3 a, as shown in Fig. 4a, all A

= cos

1

j c o s tan

+ cos coA sin tan - cos

11

tan

1

1

/ WT I — - tan V^o

\jc.cos t a n " 1

• cos ft)B sin tan Qc.

cos cos tan

1

- cos - 1 \ cos t a n cos a) c sin t a n - 1

r

j

1

sin t a n - 1 sin

tan tan

|,

/w-i

m cos r / wt \ t a n 0 T | cos tan 1 ( ^ ^ t a n S m ) V^a /J L /. IWT Vn \ r / Ti7_ \ ^^r- tan 0 T 1 sin t a n - ^ t a n ^ ) ] } , ( w

P

t a n •T)

( f * tan

0T)

cos t a n sin t a n - 1

(17)

(^tanflT)](^tanflT)]}.

We observe from equation (17) that true object detail is obtained directly from the image by simply measuring the projection widths of the boundary planes, and the projected two-dimensional intersection angles. The accuracy, however, and in fact the initiation of such an analysis rests entirely on the precision with which the electron microscope has been calibrated. I t is not sufficient in many instances to rely on calibration precision as a means of obtaining true object analyses. T h a t is, while crystallographic orientations can be determined from selected area electron diffraction patterns as outlined in Section 3, the subsequent analysis of geometrical features in the image based on this information alone could be in considerable error unless the exact, or nearly exact orientations can be assured. Several methods can be employed in an attempt to exactly determine the crystallographic surface orientations; and these usually involve a Kikuchi analysis [2, 18], or a direct indexing and consideration of diffraction spots and directions in the selected area diffraction pattern [2], A simpler method recently demonstrated by Laird et al. [19] considers the symmetry of diffraction spot intensities as a criterion for nearly exact orientations. I n this respect, the patterns shown in Fig. 2 are nearly exact orientations, especially the (100) orientation. r i g . 15. Schematic representation of image of Fig. 14. [w v w] denotes crystallographic directional nature of twin planes {111}. W i : WA, WB, and Wc define the projection of the boundary planes in the image, to A, ">B: and coc define the angles between grain boundary planes measured directly from the image (Fig. 14). Tb denotes the twin crystal of grain B 6 ) I t should be observed that sign convention following is dictated by the convention employed in measuring the boundary tilt angles as shown in Fig. 13.

Use of an Electron Microscope for Precision Micromeasurements

3-dimensional foil section

top surface

slip plane (Wfcc

bottom surface

2-dimensionat electron diffraction projection of act/Ve slip plane

Fig. 16. Measurement of the radii of curvature of dislocations in Inconel 600 alloy (76% Ni, 16% Cr, 7 % Fe). Upper schematic of crystal section shows dislocations in an active slip plane projected on a twodimensional frame (photographic emulsion in the electron microscope). Lower portion shows the true twodimensional electron microscope image. In the true case shown, the grain surface is close to (110), and 6 is approximately 35°. The angle 0 positions the radius vector and can be measured by an extension of the bisector of the angle (BOT) as indicated (100 kV operation)

31

L. E. Mukk

32

For still other analytical considerations in thin foils, it is necessary to insure that the surface of the specimen is coincidental with the crystallographic orientation and perpendicular to the beam normal. This is particularly true for certain geometrical features such as internal surface junctions (e.g. Fig. 12), dislocation lines, and other defect structures. Fig. 16 illustrates a typical analytical procedure for the determination of the true radius of curvature of a dislocation line. In such an analysis, the true radius of curvature of the dislocation line relative to the object system of the foil is given, with reference to the schematic representation of Fig. 16, by R

0

=

R

" ( c o s 2

J. J J a

NaCL-

•

KCL-

a

J

100200300W500m

NaBrTl

w-l

J K

IsCl\Tt

a

V

V

KBr-

J Af

\ nV CsBr-

J

NaJ71

i

T("K) -

KJ\

CsJTL

Fig. 1. Temperature dependence of luminescence of alkali halide crystals activated with thallium. a) Activator luminescence at X-ray excitation; b) thermoluminescence after X-irradiation at liquid nitrogen temperature

37

Radioluminescence Yield in Alkali Halide Crystal Phosphors

KJ-Tl 350tf\

//\\

"7 a V_

300'^ 285W

550°K m

I Inm)-

Fig. 2. Radioluminescence spectra of a K J - T l crystal (3 mol% T1J in the melt)

1

L

300 W

1 1 500 600 M 500 600 M X (nm) —»-

Fig. 3. Radioluminescence spectra of KC1-T1 and K C l - I n crystals at various temperatures

temperature decrease a thermoluminescence peak (high-temperature peak) is observed which is fainter than the low-temperature peak. If the crystals are Xirradiated at temperatures much higher than liquid nitrogen temperature, for example at room temperature for KBr-Tl and KJ-Tl, the intensity of the high-temperature peak considerably increases [4]. I t is interesting to note that the low-temperature decrease shifts continuously towards higher temperatures if for a given halogen the alkali metal is interchanged in the order Na-K-Cs. (An exception presents CsJ-Tl.) An analogous character of I(T) is found in indium-activated phosphors [8]. The investigation of emission spectra of X-ray luminescence shows that at sufficiently high activator concentrations in thallium-activated iodides and indium-activated phosphors activator luminescence prevails at all temperatures (Fig. 2 and 3) [8, 9]. In chlorides and bromides activated with thallium the low-temperature decrease of the activator luminescence is accompanied by an increase of the visible band (Fig. 3) [5, 9]. In the same spectral region luminescence of non-activated crystals is to be observed, for example a-bands [10]; however, at high concentrations the contribution of this luminescence to the total intensity of the visible band does not

Fig. 4. Radioluminescence spectra of ICBr-Tl crystals at 105 °K. (1) Photoluminescence at 1 S 0 — 3 P i excitation; (2), radioluminescence of a crystal with a concentration of 0.5 mol% TlBr in the melt; (3) radioluminescence of a crystal with a concentration of 3.0 mol% TlBr in the melt; a) radioluminescence of a non-activated crystal, b) radioluminescence of a crystal w i t h a concentration of 0.5 mol% TlBr in the melt

Xlnml-

38

E . D . ALUKER a n d I . P . MEZINA

exceed a small percentage (Fig. 4). I t is also necessary to note t h a t at sufficiently high thallium concentrations changes in the activator concentration do not result in a redistribution of the intensity between the visible and activator bands (Fig. 4). These facts are in favour of the assumption that the visible band is also connected with a recombination via activators. 4. Discussion Fig. 5 shows schematically the general character of the temperature dependence of the activator luminescence yield at X-ray excitation. In our previous works we suggested t h a t the second high-temperature decrease is connected with a quenching within the centres [4], In the same papers we assumed t h a t the first high-temperature decrease is due to external quenching what is well known for zinc-sulphide phosphors [11]. In the present paper we consider the region of the low-temperature decrease. The position of the thermoluminescence peaks corresponding to a destruction of the atomic activator centres Tl° and In° is known for a number of phosphors. Table 1

No.

1 2 3 4 5

Phosphor

KC1-T1 NaBr-Tl KJ-T1 CsJ-Tl KJ-In

Thermoluminescence peak corresponding to a destruction of atomic centres (data presented in the references) 300 113 175 128 140

[12] [13] [14] [13] [13]

Low-temperature peak according to our data 295 115 175 125 135

A comparison of the cited data in Table 1 on thermoluminescence peaks with our data on low-temperature peaks shows t h a t the low-temperature peak of thermoluminescence is connected with a destruction of atomic activator centres. The coincidence of the position of the low-temperature peak with the region of the low-temperature decrease gives evidence of the connection of the lowtemperature decrease with the formation of Tl° centres. A further evidence on the connection of the low-temperature decrease with the formation of Tl° centres is the continuous displacement of the position of the low-temperature decrease with the change of the alkali metal. Indeed, it is natural to believe t h a t there is a correlation between the stability of Tl° centres and the differences of the ionization potential of free atoms of thallium and the alkali metals (the ionization potential of free thallium atoms is 6.12 eV, of sodium 5.14 eV, of potassium 4.34 eV, and of cesium 3.89 eV); that is to say, the stability of Tl° centres should increase and the low-temperature decrease

Fig. 5. General character of the temperature dependence of the activator luminescence in alkali halide crystals at X-ray excitation a) low-temperature decrease, b) first high-temperature decrease, c) second high-temperature decrease, d) low-temperature peak of thermoluminescence (Tl°), e) high-temperature peak of thermoluminescence (T1++)

Radioluminescence Yield in Alkali Halide Crystal Phosphors

39

should be displaced towards higher temperatures while changing the alkali metal from sodium to cesium. As already noted this is observed in the experiment (Fig. 1). The temperature dependence of photoconductivity under F-band excitation which was recently published in [15] is an impressive corroboration of the consideration stated above about the connection of the low-temperature decrease with the formation of Tl° centres. In the region of the low-temperature peak there is a sudden decrease of photoconductivity in KC1-T1 and KBr-Tl crystals which is connected with the capture of electrons by thallium centres. In pure crystals a decrease of photoconductivity is not observed in this temperature range. We will discuss the possible mechanism of the low-temperature decrease in greater detail, i.e. we will try to understand how the formation of Tl° centres causes the decrease of the activator luminescence yield. To simplify the question we will consider an "ideal" phosphor with an activator as the sole defect. In such a phosphor only two recombination mechanisms of electrons and holes are possible which compete with each other. 1 ) 1. Recombination of an electron with a hole localized on an activator — upper recombination. 2. Recombination of a hole with an electron localized on an activator — lower recombination. In the region of the plateau (Fig. 5) Tl° centres are not stable, that is, the electron which becomes localized on the activator stays there for a very short time and is again thermally excited in the conduction band. Obviously, the probability is small that in such a short time a hole can reach the electron and recombine with it. On the contrary, T1++ centres are stable in the region of the plateau, that is, the hole which becomes localized on the activator stays there as long as an electron reaches the hole and recombines with it. Therefore in the region of the plateau the probability of the upper recombination considerably exceeds the probability of the lower recombination, and on the whole proceeds an electronic recombination luminescence. However, if the temperature is lowered, the life time of Tl° centres increases which leads to an increase of the probability of the lower recombination. Therefore, at sufficiently low temperatures, the lower recombination can compete with the upper recombination. Depending on the correlation between the emission spectra and luminescence yield at the upper and lower recombination there are various cases possible. 1. If the emission spectra at the upper and lower recombination differ from each other, the decrease of the activator luminescence yield (electronic recombination luminescence) is accompanied by an increase of the luminescence intensity of an other spectral composition (recombination luminescence of holes). This case seems to be realized in chlorides and bromides activated with thallium (Fig. 3 and 4). Indeed, in the papers [10, 16] it is shown that the activator luminescence in such a phosphor becomes excited only in the electronic peaks of thermoluminescence. ') The probability of direct band-band recombination is negligibly small in samples with a large band gap.

40

E. D. ALUKER and I. P. MEZINA: Radioluminescence Yield in Alkali Halides

2. If the emission spectra at the upper and lower recombination coincide, several cases are possible. a) The electronic recombination luminescence yield is larger than the recombination luminescence yield of holes. 2 ) In such a case we observe a low-temperature decrease of the activator luminescence without increase of the intensity of bands of other spectral composition. This case is realized in iodides activated with thallium (Fig. 2). b) The electronic recombination luminescence yield coincides with the recombination luminescence yield of holes. In such a case the low-temperature decrease of yield cannot be observed. In the papers [10, 16] it was shown t h a t the activator luminescence in phosphors activated with indium becomes excited in the electronic and hole peak of thermoluminescence. Therefore one can expect that such a case can be realized in phosphors activated with indium. Indeed, at sufficiently high activator concentrations when the model of the ideal phosphor is true, the low-temperature decrease in K J - I n is absent [8]. In that way, the spectral features observed in the experiment agree well with the proposed mechanism of the low-temperature decrease of radioluminescence yield. Acknowledgement

The authors are very much obliged to K. K. Shvarts for the suggestion of the theme and the guidance of the work. References [1] R. K. SWANK, Annual Rev. nucl. Sci. 4, 111 (1954). [2] CH. B . LUSHZIK, E . R . ILMAS, G. G. LIIDIA, T . A . SOVIK, T. I . EKSINA, a n d I. V . JAEK,

Dokl. na IV. koordinatsionnom soveshchanii po sintezu, proizvodstvu i primeneniyu stsintiliatorov, Kharkov, 1965. [3] K. K. SHVARTS, G. K. VALE, and B. Y. ZUNDE, Trudy Inst. Phys. i Astr. Akad. Nauk ESSR 12, 77 (1960). [ 4 ] K . K . SHVARTS, E . D . ALUKER, I . P . MEZINA, a n d M. M. GRUBE, R a d i a t s i o n n a y a

Fiz.

(Riga) 1, 73 (1963). [ 5 ] E . D . ALUKER, I . P . MEZINA, a n d K . K . SHVARTS, R a d i a t s i o n n a y a (1965).

F i z . ( R i g a ) 3, 4 5

[6] I. K. PLAVINS, Optika i Spektroskopiya 7, 71 (1959). [7] E. D. ALUKER, Phys. Letters (Netherlands) 14, 17 (1965). [ 8 ] E . D . ALUKER, G. F . DOBRZANSKII, a n d I . P . MEZINA, R a d i a t s i o n n a y a F i z . ( R i g a ) 4 ,

99 (1966). [9] E. I). ALUKER and I. P. MEZINA, Izv. Akad. Nauk Latv. SSR, Ser. fiz. tekh. Nauk 4, 17 ( 1 9 6 4 ) .

[10] I. V. JAEK and M. F. OKK, Trudy Inst. Phys. i Astr. Akad. Nauk ESSR 23, 155 (1963). [ 1 1 ] M. SCHON, Z. P h y s . 1 1 9 , 4 6 3 ( 1 9 4 2 ) .

[12] CH. B. LUSHZIK, Trudy Inst. Phys. i Astr. Akad. Nauk ESSR 3, 3 (1955). [13] R. A. KINK and G. G. LIIDIA, Dokl. na IV. koordinatsionnom soveshchanii po sintezu, proizvodstvu i primeneniyu stsintiliatorov, Kharkov, 1965. [14] N. H. HERSH, J. chem. Phys. 31, 909 (1959). [15] V. E. ZIRAPS and I. K. VITOLS, Dokl. na XIV. soveshchanii po luminestsentsii, Riga, 1965.

[ 16] I. V. JAEK and M. F. OKK, Trudy Inst. Phys. i Astr. Akad. Nauk ESSR 21, 96 (1962). [17] V. S. VAVILOV, Deistvie izluchenii na poluprovodniki, Moscow 1963. (Received 2

September

13,

1966)

) An inverse ratio of yield was not observed in our experiments.

V. L. VINETSKII and G. A. KHOLODAR: Conductivity of Semiconductors

41

phys. stat. sol. 19, 41 (1967) Subject classification: 14.3; 10; 13.4; 22.6

Institute of Physics of the Ukrainian Academy of Sciences (a) and Physics Department of the Kiev Shevchenko State University (b ), Kiev

On the Electric Conductivity of Semiconductors Caused by the Ionization of Thermal Lattice Defects By V . L . V I N E T S K I I (a) and G. A . KHOLODAB (b)

Theoretical calculations are made of the temperature dependences of the equilibrium carrier concentrations and intrinsic lattice defects in a semiconductor with self-activated conductivity. Deviations from stoichiometry and electrical activity of both defect components as well as intrinsic conductivity are taken into account. The high-temperature equilibrium conductivity of cuprous oxide crystals is found experimentally. Comparison between theory and experiment suggests that for cuprous oxide the conductivity is self-activated for temperatures above 300 °C. For crystals with a low concentration of excess oxygen N0, good agreement between the theory and experiment is obtained if it is assumed that only one component of the thermal defects is electrically active. The formation energy W oi & nonionized intrinsic defect is found to be 2.6 eV, the ionization energy Ed of this defect being 0.64 eV, and the effective atomic concentration in the lattice sites 1024 to 1025 cm - 3 . For crystals with a high concentration N0 the mechanism of self-activated conductivity is more complex. B TeopeTHHecKOft n a c r a paSoTi.i BbmiicjieHbi TeMnepaTypHhie 3aBHCHM0CTH paBHOBeCHMX KOHIjeHTpaUHH HOCHTejieil TOKa H C0ÔCTB6HHLIX ï[e({)eKTOB pemeTKH B nojiynpoBOflHHKe c co6cTBeHHoaeeKTHbiM xapaKTepoM TeMHOBOft npoBojjHMOCTH, C yneTOM OTKJIOHeHHH OT CTeXHOMeTpHH, SJieKTpHHeCKOÏt aKTHBHOCTH o6enx K0MN0HEHT ne 300 ° C . XJJIH KpncTajuiOB c Majioft KOHijeHTpauHeH H36biT0HH0r0 KHCJiopoaa N0 n p e n n o j i o w e H H e o TOM, HTO jiHinb OHHa K0Mn0H6HTa TenjioBoro ne •—< K [3]. I t is shown, t h a t the oscillatory external field can lead to the parametric excitation of the coupled magnetoacoustical waves. The change of the optical frequency is considered in Section 2. Within an order of magnitude

I. V. IOFFE

52

where u is the displacement, e(0) and e(a>) is the dielectric susceptibility at the frequencies zero and a>, respectively, and p is the density. The nature of the change of the optical frequency is the same as in the case of the acoustical waves. The value Aco0 can reach observable values in ferroelectric crystals. The effects considered in Section 3 are significant in ferroelectric, ferro- and antiferromagnetic crystals.

2. The Optical Mode in an External Electric Field The part of the energy of the dielectric crystal caused b y the external electric field can be written in the form [4, 5] W=~JdVDiEi,

(1)

where D is the induction. D has the static part D 0 , caused by E0, and the oscillatory part Dm, caused by the lattice oscillations: D0 = e(0) E0,

Dm = e(co) Ea .

Taking into account t h a t 0 , v-i T n s, n , s, n s, n~\ £ik = Sik + L \-eikl Ui + S iklrn m J s, n

(*)

(n is the cell index, s the atom index, and £,• j = elk if ut = u{m = 0 , ¿lu = — s, n > 8Q ui k -> 0) we find t h a t the strength caused by the external field and acting on the atom with indexes s and n is equal to F

r

=

^

) E o k

j d V e h l E

k

(

m

) .

(3)

r0 (V 0 is the cell volume, we take the case in which e% ^ S 8 , n Fr- i Uk , 2j m — n\s- s \n,n' 82y

dul'n Sufn'- '

(5)

53

Theory of the Lattice Oscillations of Dielectric Crystals

m, is the atom mass,

•

.

3. The Ferromagnetic and Antiferromagnetic Dielectric Crystals In this section we shall consider the interaction of the spin and acoustical waves. In the presence of the external electric field the system describing such waves takes the usual form ([2] (3, 1)), but the longitudinal and transverse sound velocities must be replaced by the values sjf and (1 to 4). If cosp = k .s, the ferroacoustical resonance takes place. In the presence of the external electric Ak field the resonance takes place for smaller k, if k \ \ u \ \ E0; — is nearly

(? '

As \ 2

— - 1 and, as was shown in [1], can be of the order of unity. But the . . . /As\2 spin and acoustical waves coupling parameter decreases in the ratio 1 + I — I . Let us consider now the case k \ \ m and k E0. The system of the equations, describing the displacement u and the small part m of the magnetic moment M 0 , is — i m m + Q \n m l — id — [n u] Mo = 0 , m0 c 2 (CO 2 -

f

« +

2

m0c \

Q

+

r

16

71 Q X

= 0 ,

M0J

(n is the unit vector in the direction of the easiest magnetization, x fa 10 to 30; a, ô are the exchange integrals, is the anisotropic constant, rn0 is the electron mass).

I. V. IOFFE

54 The dispersion equation is

(a> — Q) (co — k s_|_) (a) e M0 moc

•Mn {Q + ks\_)

£ k4 Asj_

2m«c

(D + ks°i) (Q+ ksE±)

e°E6 16 n Q x

=

C= r

Mo

Q is the density, y the magnetostriction constant. There are two resonance frequencies because the transverse acoustical wave splites up in two modes. The first of them does not change in the presence of the electric field, and the second increases its velocity with increasing electric field. If Q ->• k « x (the first case), (a> — k sj_) (cy — Q) and if Q -> k 4

eM„ m»c

e M0 C k2

f k2 Q + ksl

2{sl + sf)(Q+

ksE1)m0c

(the second case),

(to — k 4 ) (co

1 T

eMgCk2

— Q)

m0c(Q + k s j_)

Aij_ i 5_L + S_L.

The frequencies of the coupled waves change in comparison with the case E0 = 0. In the expression for w the terms with e Mn m0 c (k s"

Q)

'

must be replaced by the terms with e Ma £ k2

+

1

(A. j.)

2(4+4).

in the first case, or by the terms with eM n

£k2

2(4 + 4).

in the second case. Let us do a remark. We do not take into account the piezoelectric properties of the crystal. I t was shown in [1] that the piezoelectric properties lead to the difference of the sound velocity in the cases k E0 and E0. This leads to the displacement of the point of the ferroacoustical resonance in these both cases. Let us consider the case E0 — E° sin cost, when As = As(i). We restrict ourselves to the case k \\m \ E0 and Q = k 4 . In this case we have for the displacement u the equation 82u ~d(2

L

1

XV /J

8t

~r

ks

-T

m

j

55

Theory of the Lattice Oscillations of Dielectric Crystals

I t can be found from [1, 14] that if As/s is greater then the relative decrement of the ferroacoustical wave and we = Q, the parametric excitation of the coupled wave can take place. Let us consider now the antiferromagnetic crystal. There is [15] a branch of oscillation of the magnetic moment with co = k v. I t was shown in [15], that the x-esonance of the wave of this branch and the sound can take place because v is v{H). The velocity of the coupled waves is equal to = j ( » 2 + *o)

vU

+

Ci=y

4

;

ft]/qs2

v = n M0 sin 0 [(a -

a') [2 (5 - £ - /3']1'2 .

Tj. is the Neel temperature, p x is Bohr's magneton; a, a', 8' are exchange integrals ; (j, fi' are the magnetic anisotropic constants; M0 is the magnetic moment of the unit volume, and 0 is the angle between the choosen axis and M 0 . In the presence of the external electric field we have 2

«1

v . If •s0 < v we have two results: The first is that we can obtain the resonance by increasing the external electric field. The second is that the magnetic field necessary for the resonance decreases in the presence of the electric field. I t was shown in [15] that if H» = M0 [ 2 # - p -

?]

i

-

4-

the static deformation changes its sign. If E0 =|= 0, the value H° changes: s%lv2

H°(E0 * 0)

H» (E0 = 0)

1 — sHv*

Let us consider now the 2z-component of the magnetic susceptibility tensor. (The other components do not change in the electric field.) I t can be obtained from the expression [3]: %zz

—

vk* j

M

S

2

4 Ci

VS

2 v\ — Vi v\ —

V

V S2 — l\

V'2 V2 k2 v\ — CO2

v2

1

i v\ — v\ k2 v\ — c o 2 J

v

that, if s^>v, Xzz decreases with increasing E0, and, if s < v, yiz increases with increasing E 0 . Acknowledgement

I am very obliged to B . L. Gelmont, A. A. Klochikhin, I . Y a . Kornblit, and B . I . Shklovskii for helpful discussions.

56

I . V. IOFFE: Theory of t h e L a t t i c e Oscillations of Dielectric Crystals

References [1] I . V. IOFFE, Fiz. t v e r d . Tela 8, 3274 (1966). [ 2 ] A . I . A K H I E Z E R , N . G . B A B J A C H T A R , a n d M . I . KAGANOV, U s p e k h i fiz. N a u k 7 1 , 5 9 3 (1960). [ 3 ] A . I . A K H I E Z E R , N . G . BARJACHTAR, a n d M . I . KAGANOV, Z h . e x p e r . t e o r . F i z . 8 5 , 4 7

(1958). [4] L . D. L A N D A U a n d E . M. L I F S C H I T S , Electrodynamics of Continuous Media, Fizmatgiz, Moscow 1958. [ 5 ] S. I . P E K A R , Zh. exper. teor. Fiz. 4 9 , 6 2 1 ( 1 9 6 5 ) . [6] A. I . ANSELM, I n t r o d u c t i o n t o Semiconductor Theory, Fizmatgiz, Moscow 1962. [ 7 ] F . I O N A a n d G . S H I R A N E , Ferroelectric Crystals, P e r g a m o n Press, New York 1 9 6 4 [8] G. I . SKANAVI, T h e Physics of Dielectrics, Vol. I , G.T.T.I., Moscow 1949. [9] B . M . WUL, E l e k t r i c h e s t v o , N o . 3, 12 (1946).

[10] [11] [12] [13] [14]

A. W . RZHANOV, Zh. exper. teor. Fiz. 19, 502 (1949). M. S. KOSMAN, Zh. exper. t e o r . Fiz. 21, 724 (1951). R . A. COWLEY, P h y s . R e v . 125, 1915 (1962); 125, 1921 (1962); 184, A 981 (1964). R . A. COWLEY, P h y s . R e v . L e t t e r s 9, 159 (1962). L . D . L A N D A U a n d E . M. L I F S C H I T S , Mechanics, Fizmatgiz, Moscow 1965.

[ 1 5 ] N . G . B A R J A C H T A R , M . A . SAVCHENKO, a n d W . W . T A R A S E N K O , Z h . e x p e r . t e o r .

49, 944 (1965). (Received

July 4, 1966)

Fiz.

R. H.

WADE

and J. SILCOX: Small Angle Electron Scattering

(1)

57

phys. stat. sol. 19, 57 (1967) Subject classification: 1.4; 4 ; 21 Department

of Engineering

Physics,

Cornell University,

Ithaca, New

York

Small Angle Electron Scattering from Vacuum Condensed Metallic Films I. Theory By R . H . W A D E 1 ) a n d J . SILCOX We present the basis of a kinematic theory to account for electron diffraction effects from discontinuous vacuum condensed metallic films in which the separate particles act as scattering centres. The diffraction image depends on the variation in the interparticle spacings and is repeated in several layers of reciprocal space. The treatment is valid for any paracrystalline object. Nous présentons une théorie cinématique qui explique la diffraction électronique par des couches minces discontinues, condensées sous vide, dans lesquelles chaque particule isolée joue le rôle de centre de diffraction. L'image de diffraction dépend de la régularité de l'espacement entre les particules et se répète sur plusieurs plans dans l'espace réciproque. Cette théorie peut être appliquée à toute étude concernant les objets de nature paracristalline.

1. Introduction In many investigations on the electrical [1], magnetic [2], tensile [3], and optical [4] properties of thin metallic layers, vacuum condensation is used as the most convenient technique for the preparation of films with known and uniform thickness. Under the usual preparation conditions, metallic films condensed onto amorphous substrates are polycrystalline and have properties which depend both on their thickness and microstructure [5]. The transmission electron microscope studies of Sennett and Scott [6] on a number of metals reveal that, in the initial stages of condensation, polycrystalline films consist of isolated islands which coalesce to form a continuous layer as the film thickness increases. Rather similar results are obtained by Pashley [7] in studies on the growth of epitaxial layers of gold. The discontinuities between the islands may persist to rather large layer thicknesses. Thus, absorption experiments show some vacuum condensed films to have surface areas proportional to the film thickness [8], whilst measured film densities are sometimes found to be well below the bulk values [9]. Furthermore, although polycrystalline ferromagnetic films are usually assumed to behave magnetically as continuous layers [10], some investigations show films to behave as closely interacting but separate particles [11, 12, 13]. Mahl and Weitsch [14] demonstrated t h a t very thin discontinuous films produce small angle diffraction effects in a transmitted electron beam. We have observed small angle electron diffraction from films up to several hundred angstroms thick and with improved angular resolution in the diffraction image, 1

) Now at Centre d'Études Nucléaires, Grenoble, Isère.

58

R . H . W A D E a n d J . SILCOX

have been able to investigate in some detail the reciprocal space in the small angle region around the central spot of the diffraction image. I t was recently pointed out [15] that the small angle electron diffraction technique provides information about film structure which complements that obtained solely from transmission electron microscope images. Simple modifications in the operation of a conventional electron microscope enable it to be used as a high resolution diffraction camera as well as a transmission microscope. Our techniques appear similar to, but not quite identical with, those used by Bassett and Keller [16] in extensive studies of polymers. In agreement with Mahl and Weitsch, we find the small angle diffraction observed in the neighbourhood of the central beam to be particularly sensitive to the state of continuity in a film. We have been investigating the structure of vacuum condensed films of gold, nickel, palladium, and permalloy (80% Ni, 2 0 % Fc) by the following transmission electron microscope techniques: high resolution microscopy, selected area diffraction, small angle scattering in the neighbourhood of the central beam, and low magnification off-focus imaging of magnetic structure. The magnetic and electrical properties of some of the films have also been investigated. Our present purpose is to describe in detail only the small angle scattering results and their relation to the direct images obtained by transmission microscopy. We discuss here a kinematic interpretation of small angle diffraction from metallic films in which the diffraction image is regarded as the Fourier transform of an assembly of particles. In part I I our experimental observations are described in the perspective provided by this preliminary analysis. 2. The Discontinuous Film as a Diffraction Grating First, consider a continuous film composed of uniform diameter columnar crystallites extending through the film thickness and remaining in contact at the grain boundaries. Fig. 1 schematically illustrates the scattering of electrons from a small region of such a film and demonstrates the distinction between the small angle scattering around the central beam and that around the Bragg diffracted beams. Since there is no interference between the Bragg scattered beams, the diffraction spot broadening is due to the individual crystallite sizes and shows the usual intensity profile of the shape transform. The

I \ \ 7

/

2

\

n-2 n-1 n

J

/ \

N

1 I

1

\

i

\

I

i

\

(Io-tn)

Fig. 1. An electron beam of intensity I 0 incident on a continuous polycrystalline film is Bragg scattered within the individual crystallites (1, 2, . . . , ( » — 1), w) with intensity I n • The undeflected transmitted electron beam has a mean intensity / 0 — ( being the average scattered intensity). The spatial dependence of the local variations (/„ — In) about the average value is shown in the lower part of the diagram

Small Angle Electron Scattering from Metallic Films (I)

59

governing dimension in the central beam is the size of the film or, more realistically, the lateral coherence length of the electron beam. The central beam can be regarded as composed of a mean intensity 7 0 — «?• -

1

1 ii

jKk? . -v i/z^ii

;

4

* .V % t |f

Ss * y Ki i "S ' 'JiiL4.

:

•

-

• '

-,

'

"

.

"

:

-

If PÈfélì. *

'p

;

ffci

®SS1

Oj/jm

Fig. 3. Through focus series from a nickel film 80 A thick condensed under a vacuum of 2 x 10~6 Torr onto a room temperature carbon substrate at an average rate of 1/2 A/s. Note the marked fringes around the crystallites which change contract above and below focus and the absence of fringes in the in-focus micrograph (b)

68

R . H . W A D E a n d J . SILCOX

% -

f

*

p «lfr.ii

' Mjujij" *

m

SP

"It*

**

-

*

fe»

IMP

«

»v »

#

j y

I P

*•

"

w

ii • S l *

* 0

* -

•

9

-f . * ***

IrP fit

H

•

#

• Jr.

•

1

i t Bk A *

u

—

"'^•¿¡¡sHm'' ' ¡¡it ~: >

Fig. 4. A permalloy f i l m a b o u t 300 A thick condensed under a v a c u u m of 8 x 10~ 8 To ronto a room t e m p e r a t u r e carbon s u b s t r a t e at a r a t e of a b o u t 3 A/s. The ring corresponds to a spacing in t h e image of « 40 A. a) Out of focus micrograph a n d small angle diffraction p a t t e r n with t h e electron beam normal to the f i l m

tinuities between crystallites discussed in P a r t I, Section 2 a n d clearly illust r a t e d in Fig. 2 for gold. For t h e present group of metals t h e gaps are not resolved in t h e direct in focus image suggesting t h a t t h e y are either too narrow t o be visible or t h a t there is insufficient amplitude difference between t h e beam passing t h r o u g h the gaps a n d t h r o u g h the crystallites t o produce a noticeable intensity variation in the image. We f i n d very good agreement, according t o t h e accuracy of our measurements, between t h e measured crystallite sizes a n d t h e values o b t a i n e d f r o m small angle diffraction. I n comparison with t h e results for gold we find a r e m a r k able regularity in crystal size which u n d o u b t e d l y accounts for t h e sharpness of t h e small angle diffraction ring f r o m t h e metals in this group. The i n t e r p r e t a t i o n of t h e observed small angle diffraction as due t o an imperfect two-dimensional grating is supported by experiments in which specimens are tilted within the electron microscope t h r o u g h angles u p to 60°. This permits t h e observation of a large portion of t h e reciprocal space surrounding t h e central diffraction spot. For these observations, t h e long camera length method was used allowing almost a complete grid square t o contribute to t h e diffraction

Small Angle Electron Scattering from Metallic Films ( I I )

69

QJjM

F i g . 4. b ) The same area of f i l m tilted about 60° to the electron beam. N o t e the t w o spots in the diffraction pattern laying in the direction normal to the linear structure in the micrograph which is again slightly o f f focus

pattern thereby averaging local variations of the diffracted intensity within the grid square which arise from buckling of the foil. Fig. 4 shows the pattern from a permalloy film tilted through an angle of about 60°. Two side spots parallel to the axis of tilt replace the diffuse ring since tilting the columnar structure can be regarded as changing a film into a one-dimensional grating. When this experiment is carried out with the highest possible resolution, side fringes [6] running parallel to the axis of tilt are observed in the diffraction pattern. 3.2.2 Fast rate of condensation Condensing films at rates of more than 5 A/s under conditions otherwise identical to the slow rate films produces apparently continuous films over the thickness range 100 to 700 A. Micrographs of such films show larger crystallites than the slow rate films and show much less strongly, if at all, the out-of-focus fringes around individual crystallites. Small angle scattering is weak and is limited to a diffuse continuous decrease in intensity around the central spot. Fig. 5 illustrates a through-focus series showing Fresnel fringes and is to be compared with Fig. 3.

R . H . W A D E a n d J . SILCOX

i * ^

- J ®

% I*

4

V ^ r * LT"

•

* •

A** -

/

* s *'j

I

- I

, v.;

-G

f »

1*

'

m - K *

,

-

- i t * H f i f ! >, „

3*

- s?*

»•

. Y

»


- w

'

* * 3f*

L ^ F F

*

£

J

. \

9

'

,

>

.

•*»»•

V

*

© '"JT'

.

«Off

: g* *

%?,

,

•

; . - wti., if - -•% i

„

.

' I

«

>

»>,„. j, • - * S f e V

v

I F S

V %

m.

1

«-

. •

f*

I F '

? jM

- 1¿ a L

" /

U

•j

••

I *

•

M

¡Nil

C

i '" :

''*

'MC

5 I Fig. 5- A through-focus1 series of micrographs from a nickel film condensed at a rate of 2 /2 A/s onto a _s carbon substrate held at room temperature under a vacuum of 5 x 10 Torr. Compare the off-focub micrographs a) and c) with Fig. 3 a and c

Small Angle E l e c t r o n S c a t t e r i n g from Metallic F i l m s ( I I )

3.2.3 Oblique incidence

71

condensation

We discuss here only oblique incidence films with 6 ^ 70°. The detailed nature of the observed small angle diffraction patterns is sensitive to the exact conditions of condensation and to the film thickness, but again we can make a clear distinction between the patterns from films prepared under slow and fast condensation rates. Two such patterns are shown in Fig. 6a and b. The pattern from the slow rate film can be directly compared with that from the tilted sample of the slow rate normal incidence film. The vapour beam direction is perpendicular to the direction of the length of the fringes. The regularity in the size and packing of the crystallites in this perpendicular direction is sufficient to give first order diffraction spots. Along the vapour beam direction the length of the crystallites is constant enough to produce discrete shape transform fringes up to the fifth order. In the example shown the spot separation corresponds to a spacing of 63 A, whilst the separation of the fringes runs close to the ratio 1 : 3 : 5 : 7 with the first fringe being equivalent to a separation of about 550 A. Direct micrographs of this film show little contrast in focus,

Fig. 6. Small angle diffraction patterns from permalloy and palladium films condensed onto room temperature carbon films under a vacuum of 6 x 10~6 Torr at oblique angles, a) A palladium film with a thickness of 30 A and the condensation rate 10 A/min, 0 = 85°. The electron beam is normal to the film, b) A permalloy film condensed at a fast rate with 0 = 85°. c) Same specimen as a) with the film tilted so that its normal lies about 15° off the electron beam direction, d) Palladium film about 100 A thick, 6 = 70°, and the film normal lies in the direction of the electron beam, e) The same film as d) with the film normal tilted about 60° with respect to the electron beam, a), b), and c) were recorded with the same camera length. The spots in a) correspond to a spacing in the image of » 50 A. The spots in d) and ring in e) correspond to a spacing in the image of « 70 A

72

R . H . W A D E a n d J . SILCOX

whereas out of focus micrographs show fringes running parallel to the vapour beam direction with a separation of 50 to 70 A. That we have been unable to observe any structure which might be clearly related to the 550 A spacing is consistent with the interpretation that the diffraction fringes are due to thickness effects. A fast rate film yields a small angle diffraction pattern showing diffuse scattering over a wide region around the central spot whilst, perpendicular to the vapour beam direction, an equatorial band showing no scattering extends outwards from the central spot. Direct electron microscope images show a certain amount of structure along the equatorial direction without any characteristic spacing being apparent. Experiments in which the specimen is tilted with respect to the electron beam again enable us to explore more of the reciprocal space surrounding the ccntral spot. For our present purposes we only consider experiments in which the specimen is tilted about the equatorial axis. A large tilt about this axis yields a diffraction pattern with circular symmetry. Fig. 6d and e show respectively the diffraction pattern before and after tilting through about 60° an oblique incidence palladium film for which d = 70°. Tilting a 6 = 85° palladium specimen through small angles in a double tilting stage altered both the spacing and number of side fringes but did not change the general nature of the pattern. We have no accurate calibration of the tilt but can assume a maximum value of 20°. At normal incidence five orders of fringes were visible on the original plate (Fig. 6a) the separation of the first order fringes being 0.12 cm with the camera length used. Keeping the same camera length and tilting the film about the equatorial axis of the pattern to the maximum excursion of the stage which permitted the specimen to be retained in view the number of visible fringes was reduced to three as shown in Fig. 6c and the separation of the first order fringes increased to 0.18 cm. 3.3 Effect

of radiant

heat from

the evaporation

source

In Fig. 7 we show the small angle scattering patterns obtained from two series of nickel films condensed over a range of substrate-vapour beam angles at a slow rate. In series I the heat shield was present during condensation whilst in series I I it was absent. I t can be seen that for increasing 6 the transition in series I is from a ring at normal incidence which breaks up into two arcs at 6 = 20° and eventually to parallel fringes with side spots on the equator, in the 6 = 85° film. The spacing between the side spots remains that of the normal incidence ring. In series I I the normal incidence film shows some diffuse continuous small angle scattering, whilst for 0 = 85° there are fringes parallel to the equatorial direction. Also there is no scattering along the equator which in all cases is found to be perpendicular to the incidence direction of the vapour beam onto the substrate. This is easily determined by measuring the angle between a shadow cast by a particle of dirt and the equatorial axis of the small angle diffraction pattern observed by the long camera length method. In these condensations the substrates were placed at different angles to the vapour beam but at approximately the same distance from the source so that when the normal incidence film has thickness t the oblique incidence films have thickness t cos d. If t = 250 A, then the 6 = 85° film has a nominal thickness of only about 20 A.

Small Angle Electron Scattering f r o m Metallic Films (II)

Fig. 7. Ni films condensed 9 onto room temperature carbon substrates under a vacuum of 2 x 10" Torr at various incidence angles. In I the heat shield was present between the source and substrate during condensation whilst in I I the substrate was unshielded. Both series i and I I were condensed at the same rates. The ring in Fig. 7a corresponds to a spacing in the image of « 50 A

73

74

R .

H .

W A D E

a n d

3.4 Effect of

J.

SILCOX

thickness

Oblique incidence films w i t h a r a n g e of n o m i n a l thickness b e t w e e n 10 a n d 260 A were m a d e b y condensation o n t o s u b s t r a t e s placed a t d i f f e r e n t distances f r o m t h e source. This m e t h o d has t h e d i s a d v a n t a g e t h a t t h e c o n d e n s a t i o n r a t e is d i f f e r e n t for each s u b s t r a t e position. T h e d i f f r a c t i o n p a t t e r n s f r o m such

l-'ig. 8. Showing the change in s e p a r a t i o n of the d i f f r a c t i o n p a t t e r n " w i n e s " with f i l m t h i c k n e s s for p a l l a d i u m films condensed o n t o room t e m p e r a t u r e c a r b o n s u b s t r a t e s u n d e r a v a c u u m of 4 x l 0 ~ * T o r r . T h e n o m i n a l f i l m thicknesses are a) 140 A, b) 35 A, c) 16 A, d) 8 A. The p e r p e n d i c u l a r distance b e t w e e n t h e f i r s t f r i n g e a n d t h e centre s p o t in Fig. 8 b corresponds to a spacing in the image of 200 A. All t h e images were recorded a t t h e s a m e camera length

Small Angle Electron Scattering from Metallic Films (II)

75

a series (Fig. 8) show t h a t there is a contraction of the fringe systems into the equatorial region as the film thickness is increased. 4. Discussion of Results In normal incidence films, we have observed two distinct types of film struct u r e depending mainly on the condensation r a t e but also on the radiant heat received by the substrate from the vapour source. Small angle scattering from slow r a t e films shows a clear diffraction ring indicating t h a t t h e films are discontinuous. Only diffuse small angle scattering is seen f r o m fast r a t e films (or slow rate films exposed to radiant heat from the vapour source) which we conclude to be continuous. I n the cases where small angle diffraction rings are observed, the average interparticle spacing determined from t h e Bragg law is found to agree with the direct measurements of the interparticle separations, centre to centre. When films are tilted with respect to the incident beam t h e small angle ring degenerates into two spots along the tilt axis, and structure appears along the perpendicular direction when the film is sufficiently tilted [6]. These results are in essential agreement with the predictions of P a r t I, Section 3, when the particle shape is taken into account, and show t h a t under appropriate condensation conditions, films up to at least the limit of our observations at thicknesses of several hundred angstroms can consist of isolated columnar crystallites. I n addition, there can be a considerable regularity in the distribution of t h e crystallites within the film plane. If the vapour beam is incident at a glancing angle to a substrate we may expect the crystallites t o grow towards the direction of the vapour beam. Then t h e film consists of an aggregate of tilted crystallites. An electron beam normal to a film should yield a small angle diffraction image similar to those obtained by tilting normal incidence films with respect t o the electron beam. For slow condensation rates this expectation is fulfilled. This model is supported by the experiments in which the specimen is tilted about the equatorial axis until the diffraction p a t t e r n shows circular symmetry. In this position, the electron beam is incident along t h e long axis of the crystals. At fast condensation rates, the diffraction p a t t e r n is similar but lacks the side spots perpendicular to t h e vapour condensation direction which indicate discontinuities in t h a t direction. This form of diffraction image is produced in a discontinuous film by irregular chains of crystallites aligned perpendicular to t h e incident vapour beam direction and in a continuous film by surface corrugations along the same direction. We have observed m a n y patterns intermediate between the two extremes presented here, and detailed discussion of these is left for subsequent publication. Strict comparison of our observations with the models discussed here leads t o one discrepancy which is believed minor. When thickness fringes are observed experimentally in t h e diffraction patterns, t h e y extend parallel to the plane of the diffraction ring into the region of reciprocal space above the gap between the ring and the central spot. I n the model discussed in P a r t I, Section 3, t h e ring structure is repeated with diminished intensity in the reciprocal space above and below t h e plane of the ring. The breakdown of the interference may be due either to slight variations in the orientation of the axis of the columnar crystals, i.e. grain boundaries and the oxide or vacuum interfaces do not come exactly perpendicular to the foil surface or to variations in position of the centres of the crystallites. The fringes with spacing 3 : 5 : 7 will then arise from normal thickness fringe effects, b u t there should be a wide fringe at t h e origin r a t h e r

76

R. H. WADE and J. SILCOX: Small Angle Electron Scattering (II)

t h a n t w o f r i n g e s a t spacing + 1 . I n t e r f e r e n c e in t h e p l a n e of t h e film which b r e a k s d o w n o u t of t h e p l a n e of t h e film also a c c o u n t s f o r s p l i t t i n g t h e wide zero fringe i n t o t w o fringes a t spacing + 1 . I n conclusion, we n o t e t h a t in t h i s a p p l i c a t i o n small angle s c a t t e r i n g is a v a l u a b l e s u p p l e m e n t t o t r a n s m i s s i o n electron microscopy in d e t e r m i n i n g easily t h e s t a t e of c o n t i n u i t y of a condensed film. E x t r e m e l y careful t r a n s m i s s i o n electron microscopy has t o be d o n e t o d e d u c e t h e same i n f o r m a t i o n . T h e t e c h n i q u e looks promising as a m e a n s of d e t e r m i n i n g t h e s t a t i s t i c s of p a r t i c l e s e p a r a t i o n s . Combining t h i s w i t h a knowledge of t h e t o t a l q u a n t i t y of m a t e r i a l deposited will give t h e m e a n p a r t i c l e size. Clearly, t h e m i c r o s t r u c t u r e s we h a v e f o u n d can h a v e a p r o f o u n d influence on t h e p r o p e r t i e s of a film. I n t h e p r e s e n t application, we h a v e n o t f u l l y exploited t h e resolution of t h e t e c h n i q u e which is a b o u t a f a c t o r of t e n b e t t e r t h a n t h e t h r e e lens m e t h o d of Mahl a n d W e i t s c h . T h e m o s t obvious w a y of f u r t h e r i m p r o v i n g t h e resolution is t o use p o i n t e d f i l a m e n t s as a m e a n s of reducing t h e size of t h e c a t h o d e crossover region. Finally, we c o m m e n t t h a t t h e m e t h o d is clearly applicable t o t h e s t u d y of m a n y biological systems a n d some solid s t a t e t r a n s i t i o n s involving large r e p e a t distances in a d d i t i o n t o p o l y m e r s [4] w i t h t h e i n h e r e n t a d v a n t a g e over X - r a y d i f f r a c t i o n t h a t a direct image of a specimen can also be o b t a i n e d a n d t h a t s h o r t exposure t i m e s a r e required t o r e c o r d t h e d i f f r a c t i o n images on p h o t o g r a p h i c plates. A g a i n s t t h i s we m u s t weigh t h e d i s a d v a n t a g e of requiring t h i n specimens. T h e p r e s e n t application a p p e a r s t o b e p a r t i c u l a r l y suited t o s t u d y b y electron d i f f r a c t i o n since t h e specimens a r e t h i n films c o n t a i n i n g insufficient m a t e r i a l t o yield significant X - r a y s c a t t e r i n g . Acknowledgements

T h e a u t h o r s are g r a t e f u l t o D r . N . K i t a m u r a for m a n y h e l p f u l c o n v e r s a t i o n s d u r i n g t h e course of t h i s work, f o r t h e o p p o r t u n i t y of discussing some of t h e results w i t h D r . D . W. P a s h l e y , a n d for t h e helpfulness of D r . R . W . F e r r i e r a n d Mr. R . T . M u r r a y in supplying a r e p o r t of t h e i r work p r i o r t o p u b l i c a t i o n . T h e w o r k was s u p p o r t e d b y t h e U.S. A t o m i c E n e r g y Commission. U s e of t h e facilities p r o v i d e d b y t h e A d v a n c e d R e s e a r c h P r o j e c t s Agency is also acknowledged. References [1] H . MAHL a n d W . WEITSCH, Z. N a t u r f . 1 5 , 1 0 5 1 ( 1 9 6 1 ) .

[2] R. P. FERRIER and R. T. MURRAY, J. Roy. Microscop. Soc. 85, 323 (1966). [3] V . DRAHOS a n d A . BELONG, N a t u r e 2 0 9 , 8 0 1 ( 1 9 6 6 ) .

[4] G. A. BASSETT and A. KELLER, Phil. Mag. 9, 817 (1964). [5] S. TOLANSKY, Phil. Mag. 85, 120 (1944). [6] R. H. WADE and J. SILCOX, Appl. Phys. Letters 8, 7 (1966). (Received

October

10,

1966)

R . D E B A T I S T et al.: Magnon Contribution to the Specific Heat of U 0 „

77

phys. stat. sol. 19, 77 (1967) Subject classification: 8; 18.4; 22.6 Solid State Physics Department, S.C.K.

— C.E.N.,

Mol

Magnon Contribution to the Low-Temperature Specific Heat of U 0 2 By R . D E BATIST,

R . GEVERS, a n d M. VEKSCHUEREN1)

The contribution of acoustical phonons to the low-temperature specific heat of U 0 2 , as calculated from the single crystal elastic constants, is much too small to account for the observed heat capacity. I t is shown here that this data can however be explained in terms of antiferromagnetic magnons. An expression for the heat capacity due to spin waves is derived, firstly for a simple antiferromagnet having a spherically symmetric dispersion relation, and secondly for U 0 2 , where the dispersion relation is cylindrically symmetric. For U 0 2 it is found that the next nearest neighbour interactions must be included in order to obtain a meaningful dispersion relation. A l'aide des constantes élastiques de l ' U 0 2 monocristallin, on peut calculer la contribution des phonons acoustiques à la chaleur spécifique à basse température; cette contribution est nettement inférieure aux valeurs expérimentales. On a démontré que celles-ci peuvent être interprétées par la théorie des magnons antiferromagnétiques. On donne une expression pour la chaleur spécifique due aux ondes de spin; d'abord pour un corps à structure antiferromagnétique simple, ayant une relation de dispersion à symétrie sphérique; ensuite pour l ' U 0 2 , où la relation de dispersion a une symétrie cylindrique. Il s'est avéré nécessaire d'inclure les interactions entre seconds voisins afin d'obtenir une relation de dispersion réaliste; celle-ci a alors une symétrie cylindrique. 1. Introduction U r a n i u m dioxide crystallizes in t h e fluorite s t r u c t u r e ( C a F 2 ) a n d is antiferrom a g n e t i c below T-g = 3 0 . 5 ° K [1, 2], Specific h e a t m e a s u r e m e n t s a t low t e m p e r a t u r e show t h e c h a r a c t e r i s t i c m a g n e t i c t r a n s i t i o n a n o m a l y in t h e v i c i n i t y of Tjf [2]. B e l o w a p p r o x i m a t e l y 15 ° K , t h e specific h e a t follows v e r y r o u g h l y a T3 law. One could t h e r e f o r e consider t h e a c o u s t i c a l l a t t i c e v i b r a t i o n s t o be responsible for t h e l o w - t e m p e r a t u r e specific h e a t , a n d one c a n c a l c u l a t e a c h a r a c t e r i s t i c D e b y e t e m p e r a t u r e (9D. F i g . 1 r e p r o d u c e s t h e e x p e r i m e n t a l results t a k e n f r o m [2], t o g e t h e r with 60 c a l c u l a t e d for e a c h point (for T < 15 ° K ) . T h e w e a k t e m p e r a t u r e d e p e n d e n c e of dD is n o t a t all unreasonable, a n d Willis has c o r r e l a t e d this 0 D w i t h t h e c h a r a c t e r i s t i c t e m p e r a t u r e for line broadening derived f r o m his n e u t r o n diffraction e x p e r i m e n t s on U 0 2 [3]. A c t u a l l y , this d a t a yield t w o c h a r a c t e r i s t i c t e m p e r a t u r e s , one for t h e o x y g e n l a t t i c e ( 5 4 0 ° K ) a n d one for t h e u r a n i u m l a t t i c e ( 1 8 0 ° K ) . H e identifies t h e higher t e m p e r a t u r e with a n E i n s t e i n t e m p e r a t u r e t o describe t h e o p t i c a l p h o n o n c o n t r i b u t i o n t o t h e specific h e a t , a n d t h e lower one with a D e b y e t e m p e r a t u r e for t h e a c o u s t i c a l b r a n c h e s . I n this w a y he obtains a v e r y r e a s o n a b l e fit t o t h e e x p e r i m e n t a l p o i n t s ; e x c e p t , of course, in t h e region of t h e m a g n e t i c a n o m a l y (see F i g . 1 a n d reference [2]). Present address: Department of Physics, University of British Columbia, Vancouver Canada).

78

R . D E BATIST, R . GBVEES, a n d M . VEKSCHUEREN Fig. 1. Specific lieat CV (taken from [2]) and Debye temperature SD for UOa

T(°K)

I t is well known that the Debye temperature of a solid can be calculated from the velocity of sound waves. Using the single crystal elastic constants of Wachtman et al. [4], averaged as suggested by Anderson [5], 0 D for U 0 2 is found to be 270 °K. Presumably this is a lower limit, since the room temperature elastic constants have been used rather than the values extrapolated to absolute zero. This means that the lattice specific heat is several times smaller than the observed heat capacity. I t is the aim of this paper to propose an alternative interpretation of the low-temperature heat capacity of U 0 2 in terms of antiferromagnetic magnons. First, the magnon contribution to the specific heat will be calculated for a simple magnetic structure, where the dispersion relation is spherically symmetric. This procedure is then applied to the antiferromagnetic structure of UO a ; it is found that the same functional temperature dependence is obtained as in the simple case, even though the dispersion relation is not spherically symmetric.

2. The Heat Capacity of a Simple Antiferromagnet [6] The dynamical behaviour of a system of spins coupled by exchange interactions can be described at sufficiently low temperatures by means of spin waves. In a quantum mechanical treatment, the elementary excitations of such a system are called magnons. The creation and annihilation operators for magnons satisfy the boson commutation relations. The dispersion relation for magnons can be obtained by diagonalizing the Hamiltonian. In this section, we will consider a simple, cubic antiferromagnet, constructed with atoms distributed over two sublattices " a " and " b " , in such a way that all nearest neighbours of an atom on sublattice " a " lie on sublattice " b " , and vice versa. The Hamiltonian is restricted to nearest neighbour exchange and Zeeman contributions: H

=

J

Z S j*

r

S

j + i

- 2 ( i

0

H

A

Z 3

Sij

+

2/z0

H

a

£ i

Szbj

.

(1)

Here, J is the (positive) nearest neighbour exchange integral; Sj is the spin angular momentum operator of atom j; d is the vector connecting atom j with its nearest neighbours; fi0 = -i- g /nB is the magnetic moment of an atom {¡iB is ¿i

Bohr's magneton, g the gyromagnetic ratio); HA is a positive fictitious magnetic field, tending to align the spins along the «-direction; Sz is the «-component of S.

Magnon Contribution to the Low-Temperature Specific Heat of U0 2

79

This H a m i l t o n i a n yields t h e following dispersion r e l a t i o n : (ft cok)2

=(ha>,

+

h

coA)2 -

(h

coe)2

yl

(2)

if = 2

hcoe h m

x

=

7k

=

J z 8

(z

is t h e n u m b e r of n e a r e s t neighbours),

2/j,0Hx,

Z

—

(3)

A

(one assumes a c e n t r e of s y m m e t r y , so t h a t y_k = yk). F o r a simple a n t i f e r r o m a g n e t , a n d for small values of k a, [1 —^fc] 1 ' 2 is p r o p o r t i o n a l t o k, a n d t h e dispersion r e l a t i o n t a k e s t h e spherically s y m m e t r i c form h iok = [h 2 ml + a 2 A:2]1'2 (4) with «0 = KA + 2 coA t»e]V2 . a is a p a r a m e t e r d e t e r m i n e d b y g e o m e t r y , J, a n d S. If a. k h io0, t h e n t h e dispersion relation (4) becomes of t h e same f o r m as in t h e D e b y e m o d e l for acoustical p h o n o n s : h o)k = k d e v i a t e s f r o m t h e value given b y (5), a n d a p p r o a c h e s a c o n s t a n t v a l u e coQ. This i n t r o d u c e s a n Einstein-like c o n t r i b u t i o n t o t h e specific h e a t , which falls off e x p o n e n t i a l l y in a t low t e m p e r a t u r e s . A t sufficiently low values of T , t h i s will be t h e leading contribution. A t i n t e r m e d i a t e t e m p e r a t u r e s such t h a t a k a n d h o:>a a r e c o m p a r a b l e , t h e complete f o r m (4) of t h e dispersion relation should be used. T h e t e m p e r a t u r e d e p e n d e n c e of t h e h e a t capacity for m a g n o n s obeying t h i s dispersion r e l a t i o n can be calculated e x a c t l y (see A p p e n d i x I). T h e result is

( F is t h e molar v o l u m e a n d x0 = h oj0/kBT, x u = h o>^jknT, if « M is t h e m a x i m a l excited f r e q u e n c y ) . A t low t e m p e r a t u r e s , t h e f u n c t i o n F(x0, xit) can be

80

R . D E BATIST, R . GEVERS, a n d M . VERSCIIUEREN

replaced b y F(x0, oo), which is given b y F(x0, oo) = /j(a;0)

E

/lW

(8)

(the error a m o u n t s t o a p p r o x i m a t e l y 5 % ) . T h e f u n c t i o n fi(x0) is calculated in A p p e n d i x I a n d is p l o t t e d vs. x^1 in Fig. 2. F o r small values of x0, i.e. f o r t e m p e r a t u r e s a b o v e 60 = h

2

>3) •

(

n

)

For the UO a lattice, z1 = 4, z2 = 8, and z3 = 6, whereas in the limit of small values of k a

» 8 - B ' l i - i a ' i i - i f l ' i i ,

yf yf

« 6 - a2

k% -

a2

a2

k2y -

k\ .

Hence, the dispersion relation (10) can be written as (h ojhY

=

(h

Wo)2

+

p* k% +

y2

(k;

+

+

[jl0 Ha)]

(12)

kj)

with the parameters h w0, ¡3, y defined by (;h 0J „) 2 = 4 fi, Ha P

=

Ha + 16 8 Jx) ,

8 a 2 [ 8 S2 J\ -

SJ2{%SJl

-4a?[lxl>HASJ1

+

2SJ2(8

,

S ^ + f r H j j ] .

The calculation of the specific heat using the dispersion relation (12) is given in Appendix I I I and yields the following result: = 6

physica

T

'

F{X

> 1. One has to worry about A only at temperatures such that Consequently CO A (V f x4 e~* da; = (4t + 4 x\ + 12 x'h + 24 a;M + 24) . Xii

One can easily verify that A is negligibly small unless xM < 10, or, using equation (7), unless T > 0.1 0 M .

86

R . D E BATIST, R . GEVERS, a n d M . VERSCHUEREX

Appendix II Dispersion

relation for antiferromagnetic

magnons

in UO2

Consider the following Hamiltonian: H = —2 /10 Ha £ Stj + 2 fx0 Ha £ S^j + i J +

Jl H 3

[ »1

+ J2 2J h

(®aj • Saj + i, + S\,j • Shj '

+

+ tl)

+ £

(®aj '

+

+ ^bj ' Say +

+ Sbj • &'b; +

( 2 3 )

\

6, ^ In [2 ft, (1

—

1

(*

and b

x

x

a

2

\

n

+

v

G") ^

^

+

2

i n [ 2 f t ; ( i

-

2

^

+

11) ^

^

+

j

( y d

2

- 2 /Ç] +

e

2

i

'

v

y

,

(25)

.

(26)

2 n h z

6 y8/ q 2 ln[2|??;(l + +

e

] -

(1

x

71 h l

2 71

a^)

y

h

v

x

e

2

2

These expressions are not explicit solutions for the equlibrium elastic strains but yield a first approximation to the strains when the unstrained values or are substituted for the same strained parameters. For cubic crystals, where %ax = yax and xa2 = ya% so that xtx = ye1 and xe2 = ye2, the solutions simplify to _ e

l

~

b ~

.

a i

I n [2

f

l

'

2

7 T T

(I'*]

( 1 + ^ 1 (a2 + a 1 ) 2 j t h 1 (1 + v )

^ ' )

and Gy

/; i a r &

2

2

»h 2 „ a

e1 i

•

W

Interfacial Energy and Elastic Strain of Epitaxial Overgrowths

101

If the substrate is assumed to be infinitely thick compared to the overgrowth thickness, then e2 = 0; and, further, for misfit confined to one dimension by setting equal to zero while leaving ye,x finite and e2 = 0 we obtain 11

_ ~~

6 a, In [ 2 ^ ( 1 + ^ ) 1 / 2 - 2 ^ ] 1

G~\

11 + —i J (a„ + aj) 2 71 \

3.2 Discontinuous

'

^

'

overgrowth

When a deposit initially forms, it commonly consists of small nuclei, in which case the elastic strains in the overgrowth and substrate are inhomogeneous in that they are largest in the plane of the interface and decrease with distance along the normal to the interface. The elastic strain energy for a given strain at the interface is thus much smaller for nuclei overgrowths than for continuous overgrowths. In accordance with the model and calculations of Cabrera [3], we assume that the overgrowth consists of hemispherical nuclei of radius R and that the inhomogeneous elastic strains are radial strains. The total elastic strain energy associated with establishing such a radial elastic strain at the interface was previously given as [3] _ 2 G1 We* (30) ^ - 3(1-,) for the nucleus, and

-

3(1-,)

(31)

for the infinitely thick substrate. The dislocation spacing is still given by equations (20) and (21), and the energy per unit area of the dislocation grid by equation (15), so that the total energy per nucleus is equal to the sum of (30) and (31) and the interfacial energy as given by (3a) multiplied by the contact area of the nucleus. This total energy then is E = n R* (2 E& + C N) +

+ 2Es .

(32)

In an identical fashion to that given in the preceding section, the equilibrium elastic strains are found to be 1 em =

3ba

1

ln[2p'(l+r) r

7

11 + -L j (a, +

for the overgrowth, and 2 m

_ Oj a2 xem

~ T

~

x l 2

--2r] _ „ « el

a2)2R in

h2(l + v)

-R

Snh^l+v)

(33)

W

for the substrate. Here a first approximation may be made, similar to that Cabrera [3] used, by substituting /? for /5". 3.3 Graphical

analysis

The discussion in this section applies in general to the three previously considered cases of 1. finite, continuous overgrowths on infinite substrates, 2. finite, continuous overgrowths on finite substrates, and 3. finite, discontinuous overgrowths on infinite substrates. In all the above cases, the interfacial energy will be discussed as a function of the elastic strain in the overgrowth which

102

W. A. J e s s e k and D.

Kuhlmann-Wilsdorf

Fig. 4. The clastic strain energy per unit area, A*s, the energy of the cross-grid of interfaciai dislocations per unit area, 2 Ed, and the interfaciai energy per unit area, Ei = 2 Ea -i- A's (assuming C = 0), as a function of elastic strain in the overgrowth. The curves are qualitatively the same in all cases considered. The particular curves drawn refer to an infinitely thick substrate of gold, and continuous platinum overgrowths of thicknesses 46 and 250 A as indicated

was seen to have a fixed ratio to that in the substrate, except for case 1. where e2 = 0. Accordingly, et will be used as the independent variable. In order to visualize the minimization of the total energy by introduction of an elastic strain in the interface, the elastic strain energy per unit area of interface and the interfaciai energy per unit area are plotted against e1 in Fig. 4 where C is taken as zero. The value of the strain which eliminates all of the interfaciai dislocations by reducing the misfit to zero is designated as e max . It is equal to the misfit F where F is defined as F =

(35)

°2

(a2

~

ai)

1

\(G1+Gi)G(l-v){a1 + a2)\ 2:ia2F(l +2V)(G1 + G2) '

{

>

where the additional approximation that /S has been made. Similar expressions may be written for finite substrates and nuclei overgrowths. Equation (36) may also be used to solve for the critical misfit F +

NT 4 A\:

1

4 Ny

^r-^r-

'

(6)

where < ) indicates averaging over the thermal energies of the carriers. In order to estimate the influence of the changes of relaxation time let us put r t = r h + At and omit the thermal averaging procedure. A t is of course a function of NtI4 Nv. Considering small stresses for which iVT 4 Nv and Ar rh, one obtains 1 + -Rho

/ At \2rh 1 +

NJ. \ 3 K 4 NyJ K + S

I Ar Ar 4 r hh \4r

+

4

Nr 4~Ny

_Nt \ 3 K Ny) 2 K

2 (K - 1) 2Vt = 1 + (if + 2) (2 # + 1)

(!)

where E = 1+

V PB2 Bs T Cp

.

(2)

Here is the volume thermal expansion coefficient, T the absolute temperature, c P the specific heat for constant pressure per unit mass, and V the specific volume. By differentiating these relations with respect to pressure one obtains 2 3

) See the following paper in this issue. ) To be published.

131

P r e s s u r e D e r i v a t i v e s of the E l a s t i c C o n s t a n t s for Cubic S y m m e t r y (I)

for the difference between the isothermal pressure derivatives of the two bulk moduli

jy

-R — 1 [ 2 / 9 In B T\ R 1p \ ST )p

I g - 1 (0B*\ J ^ i? 2 U i j r

(ig-l)2f -B 2

1 /9 In ft \ P \ eT )p

'

(4)

This relation was derived by Overton [2] who obtained the temperature coefficient of the isothermal bulk modulus graphically from the plot of inversus T that was obtained from the empirical data of B s according to (1). Since the data given in the literature for the temperature coefficient refer in general to the adiabatic bulk modulus it is more advantageous to calculate the temperature coefficient of the isothermal bulk modulus from the formula

-i(£),+^ ^ -m r +«r-sa+

which can be obtained by straightforward differentiation of (1). All equations given above are valid for arbitrary crystal symmetry. For cubic symmetry the following relations hold between the adiabatic and isothermal second-order elastic constants [6] : cli. -

c sn =

c[2 -

cf2 =

B

T

- B

,

S

(6a)

C44 = C44 = C44 .

(6 b)

By differentiating these relations with respect to pressure and observing (3) Bartels and Schuele [3] arrived at the relations _/*£) \ 8p

IT

\ Sp

/*£\ JT

\8p

_(54\ }T

LT

(7a)

\ Vp IT

(ecfA \ dp

= D >

_ \ dp IT

\Sp IT

In deriving equations (7) it was assumed that equations (6) which are normally used for zero pressure also hold for non-zero pressure. Since this does not at all seem to be obvious a rigorous proof of (7) will be given in the Appendix. The purely adiabatic pressure derivatives of the elastic constants are obtained from the thermodynamic relation/—) = I — I + I — I I-L1 T71. One obtains /

8 r

with4) —

\

W/T

VBT

= —

\8i> Is

Cp

(fl-(f)^T(tl

(

ÖB \ S

~ ) s

_

s

/8B \

_

\1J}T

~

Vß T

l&B \ s

\ W ) ,

T

W js

\0T/P

«>• (8b)

4 ) Following the notation of [6] the second- a n d third-order elastic constants shall be either denoted in tensor notation b y C j j j z . . . (i, j , . . . = 1 , 2 , 3), or in the contracted notation of Voigt b y ... (fi, v, . . . = 1, 2, . . ., 6). The conversion between these two notations is m a d e b y replacing every index pair i j = 11, 22, 33, 23, 13, 12 b y a single index ¡.i =

= 1,2, . . . , 6 .

132

G. R . BABSCH

I t should be pointed out t h a t , while t h e two isothermal and adiabatic shear moduli — (c n — c12) and c44 are equal, and while their isothermal-adiabatic and isothermal-isothermal pressure derivatives are also equal according to (7), the adiabatic-adiabatic pressure derivatives differ from the other derivatives according t o (8). 3. Expression of the Different Pressure Derivatives in Terms of T.O.E. Constants Expressions for t h e effective elastic constants of a cubic crystal under hydrostatic pressure in terms of T.O.E. constants were first given b y Birch [4]. Since no distinction is made in t h a t paper between isothermal and adiabatic deformations these equations must be specified and supplemented so as to allow for t h e various thermodynamic conditions. For the a d i a b a t i c pressure derivatives of t h e a d i a b a t i c constants and for t h e isothermal pressure derivatives of the isothermal elastic constants this is obviously done by simply adding t h e proper subscripts S or T to t h e second- and third-order elastic constants as shown in t h e first and third row of Table 1, respectively. The linear combinations of the T.O.E. constants entering are the partial contractions with respect t o the last index pair. One has, e.g., for the adiabatic case, if Einstein's summation convention is used here (as through the rest of the paper), (9a) (9 b) (9c) Here (10)

are the adiabatic T.O.E. constants in tensor notation. They are defined as t h e third derivatives of the internal energy U with respect to the Lagrangian strain components rjij [8]. Equations of identical t y p e hold for t h e isothermal T.O.E. constants and their linear combinations with t h e free energy F occurring instead of the internal energy V. The isothermal pressure derivatives of t h e adiabatic elastic constants can be expressed in terms of t h e adiabatic pressure derivatives of t h e adiabatic elastic constants according t o t h e equations (8). The T.O.E. constants occurring in this expression are t h e partial contractions csa, cf, cf of the adiabatic elastic constants. Since t h e ultrasonic wave propagation is determined, however, b y the " m i x e d " adiabatic-isothermal T.O.E. constants, (H) and since t h e equations for t h e ultrasonic wave velocities are therefore usually expressed in terms of these mixed quantities [9] it is necessary to relate t h e mixed pressure derivatives also to t h e mixed T.O.E. constants. The mixed T.O.E. constants shall be characterized by t h e omission of a superscript. To this end it is convenient to s t a r t from t h e expression for t h e adiabatic elastic constants of a hydrostatically compressed cubic crystal [6]: C?j*i(p)=y(

0 and /6cs\

—£

\oT

\6T'P

)p

/6cs \

< 0, except for fused Si02, for which a > 0 and ^ /d(cs1

—

cT)\

\eT

lp

> 0. The sign

of the difference | — — 1 is according to (7) of I determined by competing terms in (4) of I. This difference is negative for all materials except for K , NaCl at 523 °K, NaBr, KJ, and fused Si02. Therefore it is for most substances ( ^ r ) < (l^f) < ' e x c e P t f o r K> N a C 1 a t 5 2 3 ° K ' N a B r > a n d K J ' w h e r e the second inequality is reversed, and for fused Si02, where both inequalities are reversed. 11

In case of the pressure derivatives of c12 the intermediate values

j fall

closer to the adiabatic than to the isothermal values for about half of all materials. The difference /8 ^ \

~ 6P

is smaner

IT

than 1 % for Si, Ge, LiF, NaCl at 523 °K,

NaBr, and fused Si0 2 ; it is between 1 and 5% for most materials; and it is

144

G . R . BARSCH a n d Z . P . CHANG

Table 2

m

Pressure derivatives of second-order elastic constants and linear

Material

1 2 3 4 5 6 7 8 9 10 11 12 13a 13b 13c 14 15 16 17 18 19 20 21 22 23 24 25

Al Cu Ag Au Li Na K Fe Si Ge LiF NaF NaCl NaCl NaCl NaBr KC1 KBr KJ RbBr RbJ MgO CsCl CsBr CsJ CuZn Si0 2 (f)

/8C 12 \

[dp ) /8cf2\

/8cf2\

\ 8P Is

\ 8 p )T

/8efi\ I dp )s

/8c?! \ \ 8 p )T

/Bc?i\ \ 8P }T

7.00 6.20 6.98 6.82

7.38 6.37 7.18 7.00

7.71 6.42 7.20 7.09

3.92 5.13 5.74 5.99

4.14 5.21 5.84 6.13

4.47 5.26 5.86 6.22

3.68 4.04 7.39 4.31 5.02 9.40

10.95 10.93 9.37 10.64 11.95 12.66 13.61 12.63

3.91 4.20 7.51 4.32 5.06 9.92 11.59 11.71 11.67 10.67 11.51 12.77 13.47 14.56 13.53

3.98 4.12 7.54 4.34 5.09 10.03 11.68 11.82 11.78 10.27 11.05 12.89 13.54 14.50 13.56

3.29 3.67 5.13 4.07 4.31 2.73 2.03 2.10 2.10 6.88 8.71 1.70 1.71 2.64 3.17

3.46 3.79 5.19 4.08 4.33 2.72 1.99 2.06 2.06 6.85 8.68 1.61 1.61 2.45 3.05

3.52 3.70 5.22 4.09 4.37 2.83 2.08 2.18 2.17 6.44 8.22 1.73 1.68 2.39 3.08

8.75 6.31 5.89 6.02 2.87 -10.60

8.93 6.82 6.30 6.46 3.08 -10.61

8.94 7.07 6.63 6.79 3.52 -10.62

1.78 4.76 4.54 4.55 3.09 -3.84

1.76 5.05 4.93 4.78 3.33 -3.84

1.78 5.30 5.26 5.11 3.77 -3.85

11.01

\ Sp )T

between 5 and 9% for Al, KBr, KJ, the three cesium halides, and CuZn. The difference ( — \ _ is smaller than 1 % for Ag, Fe, Si, Ge, and fused Si0 2 ; \ 8P Is \ SP It it falls between 1 and 6% for most materials, lies between 6 and 8% for Al, KC1, CsBr, and CsJ, and amounts to 13% for CuZn. Because the thermal expansion coefficient is positive for all materials con— ( — i s according to (8) of I negative if (—¿f < 0 . &p /s \ ep /r \si jp This is the case for most materials considered except for LiF, NaF, NaCl, NaBr, KC1, KBr, KJ, RbBr, RbJ, MgO, and fused Si0 2 , for which the temperature coefficient of c12 and therefore the above difference is positive, (cs — — cT)\) is according to (7) of I equal to the corresponding The difference /a—— 6p s \

't

18c

\

/0cf2\

difference for the elastic constants c„. Therefore the inequality (— — 1 P It

-OS

-ca

2.20 2.24 2.24 1.72 0.802 1.20 1.19 2.63 7.97 1.38 1.25 0.149 0.306 0.304 0.588 0.390 -0.428 -0.335 -0.237 -0.390 -0.604 1.16 3.22 3.34 3.41 1.73 -3.38

19.4 31.2 26.0 42.4

19.5 31.1 25.7 41.9

1.001 0.540 44.3 17.2 14.8 22.9 18.3 9.36 9.34 7.43 7.61 7.46 6.43 5.36 5.96

2.31 2.36 2.37 1.78 1.025 1.64 1.60 2.66 8.01 1.41 1.38 0.205 0.372 0.368 0.729 0.440 -0.392 -0.296 -0.204 -0.361 -0.579 1.20 3.56 3.68 3.74 1.89 -3.38

50.5 4.36 3.57 2.90 14.8 -9.79

—

~CTa

Cj2i2a

^112211

-cf

-ob

20.2 31.3 25.7 42.3

7.29 18.2 15.7 27.5

7.51 18.0 15.4 27.2

8.20 18.2 15.4 27.6

0.981 0.520 44.2 17.3 14.9 22.9 18.4 9.41 9.41 7.50 7.62 7.58 6.49 5.41 6.05

0.988 0.510 44.3 17.3 14.9 23.1 18.5 9.48 9.48 7.23 7.34 7.63 6.51 5.38 6.05

0.516 0.298 22.0 9.65 7.94 4.08 1.73 0.943 0.940 4.10 4.88 0.451 0.367 0.611 0.947

0.516 0.290 22.0 9.66 7.96 3.91 1.61 0.870 0.870 3.68 4.54 0.390 0.310 0.520 0.860

0.523 0.280 22.1 9.70 8.03 4.08 1.71 0.941 0.937 3.41 4.26 0.445 0.333 0.493 0.865

50.6 4.31 3.48 2.90 15.0 -9.80

50.7 4.42 3.61 3.01 16.5 -9.81

4.75 2.15 1.76 1.41 8.37 -5.17

4.60 2.13 1.79 1.41 8.92 -5.17

-cf

4.65 2.24 1.92 1.52 10.4 -5.18

-Ce = -C*

7.62 7.55 14.1 14.1 10.5 10.5 14.5 14.3 0.721 0.774 0.477 0.529 0.237 0.260 19.2 19.3 6.07 6.08 6.04 6.07 5.34 5.37 1.94 1.95 1.10 1.09 1.09 1.09 1.19 1.17 0.959 0.930 0.374 0.380 0.348 0.350 0.303 0.300 0.291 0.290 0.159 0.160 12.10 12.11 2.39 2.38 2.13 2.11 1.75 1.73 10.4 10.5 -2.31 -2.31

inequalities is reversed for the 11 materials mentioned in the preceding paragraph, whereas the second inequality is reversed for K , NaCl at 523 °K, NaBr, K J , and fused Si0 2 . For the elastic constant c 44 only two different pressure derivatives occur because of (7b) of I. Since for all substances considered the temperature coefficient of c 4 4 is negative and the thermal expansion coefficient is positive, except for fused Si0 2 , the inequality fused SiO«. The difference

IsI — \I — i> >T a n d ^ " 1 1 - --^'-j , which are increased f r o m l t o 1 4 a n d f r o m l t o 4 % , respectively.

4. Comparison of the Equation of State Coefficients from Ultrasonic and from High-Pressure Compression Data I n t h e literature on high-pressure physics t h e volume change of a solid due t o an applied hydrostatic pressure p is usually written in t h e form 0

(1)

147

Pressure Derivatives of the Elastic Constants for Cubic Symmetry (II)

Iu

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