Physica status solidi: Volume 13, Number 2 February 1 [Reprint 2021 ed.] 9783112502068, 9783112502051

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physica status solidi

V O L U M E 13 • N U M B E R 2 • 1966

Contents Original Papers

Page

G . A . SMOLENSKII, N . N . K B A I N L K , N . P . K H D C H U A , V . V . ZHDANOVA, a n d I . E . M Y L N I K O V A

The Curie Temperature of LiNbO,

309

G . B . A B D U L L A E V , S . I . M E K H T I E V A , D . S H . A B D I N O V , G . M . A L I E V , a n d S . G . AT.TF.VA

Thermal Conductivity of Selenium

315

P . PERETTO, D . DAUTREPPE e t P . MOSER

E t u d e e t essai d'interprétation du stade I I I de recuit dans le nickel

325

L . A . B U R a n x a n d A . C. MCLAREN

Transmission Electron Microscope Study of Natural Badiation Damage in Zircon (ZrSiO,) 331 L. I. VAN TORNE

Speoific Electrical Conductivities of Tantalum-Molybdenum

Alloy Single

Crystals

345

B . r . M a H H t e j i H S , A. M. T o i K a i ë B H E . H . BOATOBIII

D. J . D. THOMAS

TenjioBoe pacumpemie KpHCTajuimrecKHX aaoTa, KucJiopoaa H MeTaHa . . . .

351

Growth and Structure of Evaporated Silicon Layers

359

J . F. PÉTROFF e t A. A I ITH 1KB E t u d e topographique des figures de ohoc dans le fluorure de lithium

373

F . T E L L E e t P . LAUGIN I E

Propriétés diélectriques d u sulfate de glycocolle

387

L. I . VAN TORNE

Struoture of Boron-Nitride Fibers

395

J . KLUGE

Kernresonanzuntersuchungen a n diffundierenden Punktdefekten in AgBr. .

.401

A . C. MCLAREN a n d P . P . PHAKEY

Electron Microscope S t u d y of Brazil Twin Boundaries in Amethyst Quartz . . 413

V . SADAGOPAN a n d H . C . GATOS

Superconductivity i n t h e Close-packed Intermediate Phases of t h e V - I r , N b - I r ,

R. GOBRECHT

N b - R h , T a - R h , N b - P t , T a - P t , a n d Other Related Systems

423

Photoelektrische Eigenschaften von Lithiumantimonid

429

H . M . OTTE a n d H . A . L i p s r r r

On t h e Interpretation of Eleotron Diffraction P a t t e r n s f r o m "Amorphous" Boron 439 (Continued on cover three)

Contents — Continued

Page

R . GEVERS, H . BLANK, a n d S . AMELINCKX

Extension of the Howie-Whelan Equations for Electron Diifraotion to NonCentro Symmetrical Crystals 449 J . V A N L A N D U Y T , R . G E V E B S , a n d S . AMELINCKX

L. FBITSCHE

Ordering of Interstitial Impurities in Niobium

467

Representation of a Lattice Electron in a Uniform Electric Field

487

F . BELEZNAY a n d G . PATAKI

Remarks on the Recombination of Electrons and Donors in n-Type Germanium 499 F. POBELL

Isomerieverschiebung der 23,8 keV-y-Linie von '"Sn in verschiedenen Legierungsphasen 509

R . P . GUPTA a n d B . DAYAL

Lattice Dynamics of Zinc

519

M. WILKENS

Modifizierte Bloch-Wellen und ihre Anwendung auf den elektronenmikroskopischen Beugungskontrast von Gitterfehlern 529

E. NEMBACH

Untersuchung der Wechselwirkung magnetischer Flußfäden mit Versetzungen und Oberflächen in Niob (I) 543

A . BOUURET e t D . DAUTKEPPE

Observation des domaines on rubans dans les lames minces de fer pur par microscopie électronique 559 L . E . P O P O V , E . V . KOZLOV, a n d N . S . GOLOSOV

Configuration of Antiphase Boundaries in Ordered AuCua-Type Solid Solutions 569 E . B U D E W S K I , W . BOSTANOFF, T . V I T A N O F F , Z . STOINOFF, A . K O T Z E W A u n d R . K A I S C H E W

Zweidimensionale Keimbildung und Ausbreitung von monoatomaren Schichten an versetzungsfreien (lOO)-Fläohen von Silbereinkristallen 577

Short NoteB (listed on the last pages of the issue)

Pre-printed Titles and Abstracts of papers to be published in this or in the Soviet journal „H3BKa TBepaoro Tejia" (Fizika Tverdogo Tela).

International Conference on Magnetic Resonance and Relaxation — XIYth Colloque AMPERE, Ljubljana, Yugoslavia, September 5—10, 1966 Preliminary Announcement

The "International Conference on Magnetic Resonance and Relaxation — X l V t h Colloque A M P E R E " will be held at the Nuclear Institute "JoZef Stefan", Ljubljana from 5th to 10th September, 1966, under the auspices of the Federal Council for the Coordination of Research, the Federal Nuclear Energy Commission of Yugoslavia, and the Groupement A M P E R E . Sessions will include introductory lectures, round-table discussions and a limited number of short communications on magnetic resonance and relaxation (physical problems and new techniques). A parallel session will be devoted to dielectric relaxation in solids. Additional information m a y be obtained from the Secretary, X l V t h Colloque A M P E R E , Nuclear Institute "Joief Stefan", Ljubljana, Yugoslavia.

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 f i E L , Grenoble, A. P I E K A R A , Poznan, A. S E E G E R , Stuttgart, 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 13 • Number 2 • Pages 307 to 590 and K 53 to K 164 February 1, 1966

A K A D E M I E - V E R L A G .

B E R L I N

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Original

Papers

phys. stat. sol. IB, 309 (1966) Institute of Semiconductors, Academy of Sciences of the USSR, Leningrad

The Curie Temperature of LiNb(V) By G. A . SMOLENSKII,

N. N . KRAINIK, and

I. E .

N . P . KHTJCHUA,

V . V . ZHDANOVA,

MYLNIKOVA

Single crystals of LiNb0 3 are grown by the flux method. Measurements are made in the temperature range 20 to 1170 °C of the dielectric constant e, dielectric loss and relative thermal expansion Al/l along the polar axis (the three-fold axis x3) and perpendicular to the glide plane (xj. A sharp maximum in s corresponding to a decrease in volume is observed in the x3 direction at 1140 °C. This temperature is assumed to be the Curie temperature of L i N b 0 3 which corresponds to the transition from an electrically ordered state (ferroor ferrielectric, rather than antiferroelectric) into a paraelectrical state. The temperature dependences of e and AIfl below the Curie temperature show several similar anomalies. Some of these anomalies may be due to phase transitions. BbipameHbl MOHOKpHCTajMM LiNbOj (jWIIOCOBbIM MeTOflOM. IIpOH3BeAeHM H3MepeHHH HHajieKTpHHeCKOii npOHHUaeMOCTH £ H nOTepb H OTHOCHTejIbHOrO yHJiHHeHHH Al/l B HHTepBajie TeMnepaTyp 20 HO 1 1 7 0 ° C Bnojib nonnpHoii OCH (OCH T p e T b e r o nopHHKa x3) H nepneHUHKyjiHpHO K IIJIOCKOCTH CKOJib>KeHHH (A^). IIpH 1140 °C B HanpaBjieHHH x3 o6Hapy>KeH pe3KHft MaKCHMyM e, KOTopoMy COOTBCTCTByeT yMeHbiueHHe o6beMa. IIpeHnoJiaraeTCH, HTO 3Ta TeMnepaTypa HBJIHCTCH TeMnepaTypoti KiopH LiNb0 3 , KOTopoli cooTBeTCTByeT n e p e x o n H3 BJICKTPHHGCKH ynopHnoieHHoro (cnopee cerHeTO- HJIH cerHeTHBJieKTpHHecKoro, neM aHTHcerHeTosjieKTpHqecKoro COCTOHHHH) B napasJieKTpHqecKoe cocToroHHHe. HH»E TeMnepaTypbi K i o p n Ha K P H B H X 3aBHCHM0CTeil e — f(T) H M/l = f(T) o C H a p y w e H

pan aHOMaJinii, TeMnepaTypbi KOTOPHX ynoBjieTBopHTejibHO corjiacyroTCH Hpyr c flpyroM. HeKOTopbie H3 STHX aHOMajiHH, n0-BH«nM0My, HBJIHIOTCH $a30BbiMH nepexo«aMH.

Matthias and Remsika [1] showed for the first time that L i N b 0 3 is a uniaxial ferroelectric. B y investigating single crystals of L i N b 0 3 and L i T a 0 3 they observed hysteresis loops. But no more details were reported about L i N b 0 3 . In his later works Megaw [2, 3] proposes to classify L i N b 0 3 and L i T a 0 3 as "frozen ferroelectrics". The Curie temperature of L i N b 0 3 has not been reported up to the present. In [4] an increase in the Curie temperature is observed for L i T a 0 3 - L i N b 0 3 solid solutions if the concentration of the second component increased up to 70 mol%. 2 ) Extrapolation of the Curie temperature to pure lithium niobate yields a value of about 1170 °C. This is the temperature of the melting point of polycrystalline LiNbO a [4]. On this basis the authors [4] suggest that LiNbO a may be considered as a pyroelectric or a "frozen ferroelectric". In [6] an X-ray study of L i N b 0 3 is made at high temperatures (20 to 700 °C). At 587 °C a phase transition is observed which was believed to be due to a transition from a ferroelectric to a paraelectric state. Originally an ilmenite structure The term "Curie temperature" used in this work does not mean that the phase transition to the paraelectric state is of the second type. 2 ) The Curie temperature of polycrystalline L i T a 0 3 is about 650 °C [5]. 21*

310

SMOLKNSKII, K R A I N I K , K H U C H U A , ZHDANOVA, a n d

MYLNIKOVA

(FeTi0 3 ) was ascribed t o L i N b 0 3 [7], However, later investigations b y Bailey [3] showed t h a t the structure of L i N b 0 3 is significantly different f r o m t h a t of ilmenite: the space group of L i N b 0 3 is R3C (the ilmenite space group is R3C). Moreover t h e sequence of cations along the t r i a d axis is different in LiNbO a and F e T i 0 3 (in L i N b 0 3 the sequence of cations in this direction is L i - N b - L i - N b ) . The space group R3C of L i N b 0 3 was confirmed b y n e u t r o n diffraction experiments carried out b y Shiosaki and Mitsui [8]. The polarity of L i N b 0 3 was also confirmed b y the existence of a piezo- and pyroeffect at room t e m p e r a t u r e [9]. Megaw [3] derives t h e crystalline structure of lithium niobate from the perovskite structure b y continuous ion displacements a n d assumes t h a t a transition into the perovskite phase m u s t occur at high t e m p e r a t u r e s with ion displacements of the order of 1 A. B u t this assumption is d o u b t f u l since t h e u n i t cell of L i N b 0 3 remains rhombohedral till 700 °C [6] and is quite different f r o m the perovskite cell. Thus, the n a t u r e of the dielectric properties of lithium niobate (a compound whose structure has not been adequately investigated) still remains uncertain. On t h e other hand, much attention is paid t o lithium niobate due t o its large importance as a material for non-linear optics. I t was possible t o use L i N b 0 3 for modulating light of a laser source due t o a strong electrooptic effect. L i N b 0 3 [10, 11] is reported t o have great advantages in comparison with other nonlinear materials for coherent second harmonic generation. This is connected with a large negative birefringence of this material in t h e visible and near infrared and a large nonlinear coefficient which is an order of magnitude higher for L i N b 0 3 t h a n for other known crystals. Moreover, lithium niobate can be used for optical p a r a m e t r i c devices, for electromechanical transducers, and microwave applications. The above-mentioned results indicate t h a t careful investigations of L i N b 0 3 in a wide t e m p e r a t u r e range u p t o t h e melting point are of great interest. I t is also desirable t o carry out measurements on oriented single crystals. I n the present work single crystals of L i N b 0 3 have been grown and t h e temper a t u r e dependence of dielectric permeability, losses, and t h e r m a l expansion have been investigated in t h e t e m p e r a t u r e range 20 t o 1170 °C. I n [9, 12, 13] it has been reported t h a t L i N b 0 3 single crystals were grown by t h e Czochralski technique. I n this work single crystals have been grown b y t h e flux method. Li 2 C0 3 and LiCl " p r o analysi" and Nb 2 O s of 99,5% p u r i t y were used and a mixture consisting of 5 m o l % Li 2 C0 3 , 5 m o l % Nb 2 0 5 , and 90 m o l % LiCl was p u t in a platinum crucible and heated in furnace t o a temperat u r e of 1250 t o 1300 °C. Then the furnace was cooled slowly t o 800 °C at the r a t e of 5 t o 7°/h. I n this way crystals with an area u p t o 1 cm 2 have been obtained. The colour of the crystal varied from pale pink t o brown. After annealing in air at a t e m p e r a t u r e of 1120 t o 1150 °C t h e crystals became colourless. W h e n t h e measurements were repeated at high t e m p e r a t u r e s t h e crystals in some cases h a d at first a light green and t h e n a pink colour. I t m a y be supposed t h a t crystal coloration is caused b y the character of oxygen defects. The melting points of different samples varied in the t e m p e r a t u r e range from 1185 t o 1210 °C which m a y be also connected with t h e n u m b e r and character of oxygen defects. Two single crystals without any twins a n d visible defects were chosen for optical and X - r a y experiments. The crystals were oriented by using X - r a y technique. We chose the usual orientation of a crystal of space group R3C [14], i.e. x3 is t h e three-fold polar axis and x± is t h e direction perpendicular t o the

The Curie Temperature of LiNb0 3

311

glide plane and therefore perpendicular to the three-fold axis. The crystals were cut to an accuracy of ^ 2°. Measurements of e and tg

ALIEVA

F i g . 4. Temperature dependence of X for amorphous Se doped with thallium. 1 - Se (B-5), 2 - S e + 0.05 w t % Tl, 3 - Se + 0.125 w t % Tl, 4 - Se + 1 w t % Tl

»

300

305

310 T(°K1

315

Keyes has shown that B = 0.06 gives the best agreement with the experimental values for semiconductors. We have also used this value for the calculation. Above 350 ° K the heat transferred by photons according to equation (6) was also taken into account. It is seen that there is a satisfactory agreement between the experimental and calculated values at high temperatures. A slight difference at lower temperatures is due to the fact that the studied specimens of selenium contain different defects, which play a significant part at low temperatures and decrease the value of X. Fig. 4 presents the temperature dependence of 1 for amorphous selenium with admixtures of thallium. Curves 2 to 4 refer to 0.05, 0.125, and 1% Tl. It is shown that when Tl is added to pure amorphous selenium (curve 1) the change of X at Tg decreases initially (curve 2) and then disappears completely. This is an indication of the crystallizing action of Tl [43], i.e. the addition of a thallium admixture curves 3 and 4, Fig. 4) and heat treatment (curves 3 and 4, Fig. 2) result in the same changes in the temperature dependence of X. The volume of the crystalline phase of the selenium specimens doped whith 0.05, 0.125, and 1% Tl was calculated according to equation (4) and found to be 25, 64, and 44% respectively. References [1] G. M.

ALIEV

and G. B .

[2] G . B . ABDULLAEV,

[3] [4] [5] [6] [7] [8]

ABDULLAEV,

M. I. ALIEV,

Dokl. Akad. Nauk S S S R 116, 598 (1957).

A . A . BOSHSHALIEV,

and

G. M. ALIEV,

Voprosy

metallurgii i fiziki poluprovodnikov, trudy tretego soveshchaniya po poluprovodnikovym materialam, Izd. Akad. Nauk S S S R , Moskva 1958. G . B . A B D U L L A E V and A . A . B A S H S H A L I E V , Zh. tekh. Fiz. 27, 1971 (1957). D. SH. ABDINOV, G. B. ABDULLAEV, and G. M. ALIEV, Dokl. Akad. Nauk Azerb. S S R 20, 27 (1964). B. D. ALIEV, G. B. ABDULLAEV, and G. M. ALIEV, Trudv Inst. Fiz. Akad. Nauk Azerb. S S R 11, 5 (1963). C H . M . A S K E R O V , G . M . A L I E V , and E . A. A K H U N D O V A , I Z V . Akad. Nauk Azerb. S S R , Ser. fiz.-mat. tekh. Nauk, No. 1, 83 (1964). G. B. ABDULLAEV, D. SH. ABDINOV, and G. M. ALIEV, Dokl. Akad. Nauk Azerb. S S R 21, 18 (1965). G . B . A B D U L L A E V , S . I . M E K H T I E V A , D . S H . A B D I N O V , and G . M . A L I E V , phys. stat. sol. 11, 891 (1965).

T h e r m a l Conductivity of Selenium

323

G . K . W H I T E , S . B . WOODS, a n d T . M . ELFORD, P h y s . R e v . 1 1 2 , 1 ( 1 9 5 8 ) . A . F. I O F F E , Zh. tekh. Fiz. 2 2 , 12 (1952). a n d E . K . M A L Y S H E V , Zh. t e k h . Fiz. 13, 12 ( 1 9 4 3 ) . M. I . V E L I E V a n d G. M. A L I E V , I Z V . Akad. N a u k Azerb. SSR, Ser. fiz.-mat. t e k h . N a u k , No. 2, 97 (1964). H . I . AMIRCHANOV, Izv. A k a d . N a u k Azerb. SSR, No. 4, 3 (1946); No. 4, 39 (1949). E . D . D E V Y A T K O V A a n d I . A . SMIRNOV, Zh. t e k h . Fiz. 2 2 , 1 9 4 4 ( 1 9 5 7 ) . A . EISENBERCJ a n d A . V . T O B O L S K Y , J . Polymer Sei. 61, 4 8 3 ( 1 9 6 2 ) . G . G A T T O W a n d G . H E I N R I C H , Z. anorg. allg. Chem. 331, 275 (1964). A . V. IOFFE

and

A. V . KURTNER

H . ORTMAN a n d K . U E B E R R E I T E R , K o l l o i d - Z . 1 4 7 , 3 ( 1 9 5 6 ) .

H . KREBS a n d W. MORSH, Neorganicheskie polimery, I L , Moskva 1961. R . G. K L E M E N S , Solid S t a t e P h y s . 7 , 1 ( 1 9 5 8 ) . C. K I T T E L , P h y s . R e v . 7 5 , 9 7 2 ( 1 9 4 9 ) . A. F . I O F F E , Fizika poluprovodnikov, Izd. A k a d . N a u k S S S R , Moskva 1957. W . DE SORBO, J . chem. P h y s . 21, 764 (1953); 21, 1144 (1953). S. N. Z H U R K O V a n d B. J . L E V I N , Vestn. Leningr. Univ., No. 3, 45 (1960). M. ELLERSTEIN STITAR, J . P o l y m e r Sei. 1, 4 4 3 ( 1 9 6 3 ) . M . ELLERSTEIN STUAR, J . P o l y m e r Sei. 2, 3 7 9 ( 1 9 6 4 ) .

G.

Stekloobraznoe Sostoyanie, O N T I , Moskva/Leningrad 1935. a n d M . V E L I E V , Uehennye Zapiski Azerb. Gos. U n i v . im. S. M . K i r o v a , No. 1, 59 (1960). G. T A M M A N a n d A. K O H L H A A S , Z. anorg. allg. Chem. 182, 49 (1929). G. TAMMAN, Z. anorg. allg. Chem. 197, 1 (1931). B. B. K U L I E V , S . A . ABASOV a n d K H . M. K H A L I L O V , Fiz. t v e r d . Tela 7, 1860 (1965). K . E I E R M A N N , Z . Kolloide u n d Polymere 180, 163 (1962). A. E U C K E N , Ann. P h y s . (Germany) 3 4 , 1 8 5 ( 1 9 1 1 ) . J . R . D R A B B L E a n d H . J . G O L D S M I D , Thermal Conduction in Semiconductors, IL, Moskva 1963. L. GENZEL, Z. P h y s . 135, 177 (1953). A. B. HYMAN, Proc. P h y s . Soc. B 69, 743 (1956). TAMMAN,

K . KOCHARLI

D . N . NASLEDOV a n d B . V . SOKOLOV, Z h . t e k h . F i z . 2 8 , 7 0 4 (1958).

J . J . DOWD, Proc. P h y s . Soc. B 64, 783 (1951). F . E C K A R T a n d H . R A B E N H O R S T , Ann. P h y s . (Germany) 6 , 1 9 ( 1 9 5 7 ) . D . M. CHIZHIKOV a n d V. P . S H A S T L I V Y , Selen i selenidy, I z d . A k a d . N a u k SSSR, Moskva 1964. A . N . G E R R I T S E N a n d P . VAN D E R S T A R , Physica 9, 5 0 3 ( 1 9 4 2 ) . R . W. POWELL, Proc. I n t e r n a t . Conf. Thermodynamics a n d T r a n s p o r t P r o p e r t i e s Fluids, I n s t . Mech. Engs., London 1958 (p. 182). R. W . K E Y E S , Phys. Rev. 1 1 5 , 5 6 4 (1959). G. KREBS, Poluprovodnikovye materialy, I L , Moskva 1954. (Received

22

physica

October

11,

1965)

325

P. PERETTO et al. : Interprétation du stade I I I de recuit dans le Ni phys. stat. sol. 13, 325 (1966) Centre d'Études Nucléaires de Grenoble

Etude et essai d'interprétation du stade III de recuit dans le nickel Par P. PERETTO,

D. DAUTREFPE

et

P.

MOSER

Une étude par traînage magnétique du Nickel pur ou dopé irradié aux neutrons et aux électrons à très basse température permet un essai d'interprétation du stade I I I . Ce stade serait dû à la destruction d'agglomérats d'interstitiels, dont la structure évoluerait au cours du stade I I . A possible interpretation is given of stage I I I recovery in nickel, based on magnetic aftereffect studies on pure or doped samples irradiated by electrons or neutrons at low temperatures. This stage could be due to the annealing of interstitial aggregates whose structure is evolved during stage I I .

1. Introduction L a résistivité du nickel déformé à froid ou irradié présente un stade de recuit vers 350 °K. A. Seeger [1] et L. Stals [2] attribuent ce stade à l'annihilation des interstitiels dans les lacunes. Ce modèle ne permet pas, à notre avis, une interprétation cohérente des stades I et II. Nous pensons que le stade I I I résulte de la destruction d'amas d'interstitiels disposés en plaquettes alors que, suivant le modèle que nous avons déjà proposé [3, 4] le stade I est associé à l'annihilation des interstitiels dans les lacunes d'abord avec, puis sans, corrélation et le stade I I à la disparition des di-interstitiels. 2. Données experimentales Notre argumentation est étayée par des expériences de traînage magnétique [5, 6] pratiquées sur du nickel de fusion de zone soit pur soit pollué de fer et irradié aux électrons ou aux neutrons à diverses doses. Sur les Fig. 1 à 3, nous présentons les bandes de traînage qui apparaissent au cours d'une montée linéaire de température. On peut suivre sur la Fig. 4 la

Fig. 1. Perméabilité initiale du nickel irradié à 28 °K aux neutrons (1,2 • 1 0 " nvt) au cours d'une montéelinéaire en température (60°/h) après stabilisation de 3 " (traits supérieurs) et 1 1 1 " (traits inférieurs) a) en traits pleins: nickel pur, b) en traits pointillés: nickel + 0 , 3 % fer 22»

\J 25

75

125

175 TCK)

225

275

325

326

P . PERETTO,

D . DAUTREPPE

et

P . MOSER F i g . 2. Perméabilité initiale du nickel irradié à 20 ° K aux électrons de 2,5 MeV (10 1 8 électrons/cm 2 ) au cours d'une montée linéaire de température (60°/h) après stabilisation de 3 " (traits supérieurs) et 1 6 2 " (traits inférieurs) a) en traits pleins: nickel pur, b) en traits pointillés: nickel + 0 , 3 % fer

50

75 T(°K)

Fig. 3. Perméabilité initiale du nickel pur de fusion de zone irradié à 77 ° K a u x neutrons (7 • 1 0 u n v t ) au cours d'une montée linéaire de température (60°/h) après stabilisation de 3 " (traits supérieurs) et 1 6 2 " ( t r a i t inférieur)

50

100

m m )

150 m )

ZOO —

—

250

-

F i g . 4. R e c u i t isochrone d'un nickel pollué par 0 , 3 % de fer, irradié à 28 ° K aux neutrons (1,2 • 10 17 n v t ) ( E n coordonnées obliques les températures de fin de recuit). Chaque dépression représente un défaut. Les t r a i t s pointillés sont tracés au fond des bandes de traînage magnétique. On peut suivre ainsi l a disparition des défauts et le déplacement de la bande I I I

Interprétation du stade III de recuit dans le nickel

327

Fig. 5. a) en traits pleins: courbes de recuit de la résistance, à gauche d'un nickel irradié aux électrons d'après Walker, à droite d'un nickel Johnson Matthey non parfaitement pur irradié aux neutrons (2,5 • 1 0 " nvt) à 28 °K, b) en trait pointillé: annihilation du champ de traînage magnétique de la bande I I I rapporté à 175 °K

modification de ces bandes au cours d'un traitement thermique isochrone. Ces résultats expérimentaux montrent l'existence, dans certaines conditions d'une bande I I I dont la disparition coincide avec le stade I I I de recuit de la ré-istivité (Fig. 5). L'apparition de cette bande I I I dans le nickel pur irradié aux neutrons, nécessite une dose élevée (7 • 1018 nvt) (Fig. 2). P a r contre, si le nickel est pollué par du fer, la bande I I I apparait déjà nettement pour une dose de 1,2 • 1017 n v t (Fig. 1). On ne la trouve pas après irradiation aux électrons, que le nickel soit pur ou pollué (du moins pour des doses inférieures à 1018 électrons de 2,5 MeV). P e n d a n t le recuit, l'énergie d'activation moyenne se déplace fortement (de 0,3 eV à 0,8 eV environ), mais le spectre d'énergie d'activation, bien que croissant légèrement au cours d u recuit, reste relativement étroit ( i 0,05 eV) : les défauts sont donc assez bien définis. Enfin, la bande I I I en se déplaçant vers les hautes températures laisse derrière elle des bandes qu'il est difficile de séparer et qui disparaissent sur place. Cette bande I I I a été également observée par Balthesen et al. [9]. 3. Interprétation des données experimentales Nous présumons que le défaut I I I est de n a t u r e interstitielle puisque nous n'avons pas pu, jusqu'ici, le créer par trempe. Cette hypothèse est en accord avec les résultats d'énergie stockée puisque l'annihilation de ce défaut est accompagnée d'une importante libération d'énergie stockée [7]. Pour expliquer nos résultats expérimentaux, nous sommes conduits à a d m e t t r e que l'origine de la bande I I I est la réorientation d ' a m a s d'interstitiels susceptibles de grossir par un traitement thermique partiel. 3.1 Formation

des

amas

3.1.1 Irradiation de nickel pollué par 0,3°¡0 de fer (Fig. 1) Nous notons i„-Fe le défaut composé de n atomes de nickel interstitiels, piégés par un atome de fer. Les interstitiels de nickel i, dans un premier temps, migrent et se piègent sur les atomes étrangers de fer pour donner naissance à une bande nouvelle i-Fe. Lorsque ce défaut éclate, l'interstitiel libéré peut

328

P . PERETTO, D . D A U T R E P P E e t

P . MOSER

de même être piégé par un défaut i-Fe non encore détruit pour former le défaut ia-Fe, amorce nécessaire de l'amas i n -Fe. Si T1 est le temps de destruction du défaut i-Fe et T2 le temps de migration de l'interstitiel entre chaque défaut de ce type, la concentration en défaut i2-Fe sera de la forme exp (— TJTj). Le temps T2 dépend essentiellement de la concentration initiale de défauts i-Fe. Il existera donc une concentration critique au-dessous de laquelle les amas ne se formeront pas, cette concentration calculée d'après les valeurs expérimentales des énergies de migration de l'interstitiel 0,15 eV) et de guérison de l'interstitiel piégé (•—• 0,20 eV) est de l'ordre de 50 ppm. Pour les doses utilisées dans l'irradiation aux électrons on n'atteindrait pas cette valeur (Fig. 3). Au contraire, la présence de cette même bande dans un échantillon identique irradié aux neutrons à une dose créant une concentration d'interstitiels sensiblement équivalente (Fig. 1) s'explique en supposant une répartition inhomogène de défauts et des zones où la concentration critique est dépassée. 3.1.2 Irradiation

du nickel pur (Fig. 2)

La petitesse de la bande II attribuée aux di-interstitiels peut s'expliquer par l'existence d'un potentiel répulsif entre interstitiels de nickel qui entraverait le processus d'agglomération tout ou moins en son début par l'accroissement considérable qu'il entraine pour le temps T2. Seules de fortes irradiations en créant des germes assez gros permettent la formation de ces amas. Ainsi se comprend l'absence de bande III dans un nickel pur irradié faiblement aux neutrons (1017 nvt) et la présence d'une telle bande pour de fortes irradiations (7 • 1018 nvt). 3.1.3 Ecrouissage

du nickel

La bande III obtenue dans ces conditions résulte de l'agglomération de rangées voire de dipôles d'interstitiels formés lors du croisement des dislocations. 3.2 Structure

et réorientation

des amas

Par analogie avec les amas de Guinier-Preston, parce que la structure d'amas d'interstitiels a été souvent proposée, parce que cette agglomération doit se faire préférentiellement entre les plans denses, nous supposons que les amas sont des plaquettes orientées dans les directions [III] (Fig. 6). Ces plaquettes peuvent ne pas atteindre une taille suffisantes pour être accessibles à la microscopie électronique. Il est possible de trouver dans les plaquettes l'asymétrie nécessaire au couplage magnétique avec l'aimantation spontanée qui est à l'origine de traînage magnétique. La réorientation des plaquettes se ferait dans leurs plans et mettrait en jeu un nombre d'interstitiels bien inférieur à n, car seuls sont intéressés à cette réorientation les interstitiels périphériques c'est-àdire un nombre d'environ |/n qui migrerait suivant le parcours d'énergie minimum (Fig. 6). On conçoit que les énergies de réorientation puissent varier considérablement avec la taille de la plaquette, cette variation étant rapide pour de petites plaquettes (n petit) pour se stabiliser quand la dimension des plaquettes croît (saturation de l'énergie de réorientation par de grands n). On peut ainsi

329

Interprétation du stade I I I de recuit dans le nickel F i g . 6. a) Structure proposée pour l e d é f a u t r e s p o n sable de l a bande I I I : en t r a i t s pleins les plans [ I I I ] , le segment est l a trace de l a p l a q u e t t e d'interstitiels, le carré l ' a t o m e de fer en substitution, b ) E x e m p l e d'une p l a q u e t t e s i 5 «Fe : les t r i a t s pleins représentent l'intersection des plans [100] avec le plan [111], les t r a i t s pointillés sont les projections des directions de facile a i m a n t a t i o n [111] sur le p l a n de la figure, les p e t i t s cercles sont les a t o m e s de nickel en insertion ^à l a cote — .

Si 1016 a/mg were, on X-ray evidence, glassy (metamict). However, t h e electron diffraction p a t t e r n s and micrographs showed t h a t these crystals were composed of slightly misoriented zircon crystallites a b o u t 100 Ä in size. After annealing, these metamict crystals showed dislocation loops and networks and t h e single crystal character was restored. The dislocation loops appear t o lie on {101} planes a n d have Burgers vectors parallel to . Inspection of t h e crystal structure of zircon suggests t h a t this is a likely slip system for zircon. Both interstitial and vacancy dislocation loops were observed. These results suggest a mechanism for t h e transformation f r o m normal t o metamict zircon and this is discussed. Zirkonproben unterschiedlicher H e r k u n f t zeigen beträchtliche Unterschiede der physikalischen Eigenschaften. Allgemein wird angenommen, daß die Unterschiede die Folge eines Strahlungsdamage sind, das durch radioaktive Verunreinigungen, insbesondere U r a n und Thorium, verursacht wird. An einer Anzahl Zirkonproben wurde mit Durchstrahlungselektronenmikroskopie das Strahlungsdamage in Abhängigkeit von einer a-Strahlendosis (D) untersucht. F ü r Dosen D < 1014 a/mg wurde in den Kristallen keine Strahlungsschädigung beobachtet. I n Kristallen, f ü r die D = 1015 a/mg war, wurden dunkle Flecke beobachtet. Temperungsexperimente und Elektronenspinresonanzuntersuchungen weisen darauf hin, daß die dunklen Flecke durch Ausscheidungen von Sauerstoff auf Zwischengitterplätzen hervorgerufen werden. Kristalle mit D > 1016 a/mg waren glasartig (metamikt), wie Röntgenstrahluntersuchungen zeigten. Elektronenbeugungsbilder und Mikrophotographien zeigten jedoch, daß die Kristalle aus schwach fehlgerichteten Zirkonkristalliten zusammengesetzt sind, deren Abmessungen etwa 100 Ä betragen. Die Kristalle zeigten nach dem Tempern Versetzungslinien und -netzwerke; der Einkristallcharakter war wieder hergestellt. Die Versetzungslinien scheinen in der {101 }-Ebene zu liegen mit Burgersvektoren parallel zu a30Boro npeBpameHHH npn 43,8 ° K o t 15 no 38 nacoB. norpeuiHOCTb H3MepeHHü b HH3K0TeMnepaTypH0H a-HCHëH

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W . F . STREIB, T . H . JORDAN, and W . N . LIPSCOMB, J. chem. P h y s . 37, 2 9 6 2 ( 1 9 6 2 ) . E . M . HÖRL and L . MARTON, A c t a cryst. 14, 11 (1961). K . KLUSIUS, Z. phys. Chem. B3, 45 (1929). A . R . UBBELOHDE, Z. phys. Chem. 37, 183 (1963). C. A . SWENSON, J. chem. P h y s . 23, 1963 (1955). R . H . BEAUMONT, H . CHIHARA, and J. A . MORRISON, P r o c . P h y s . Soc. 78, 1462 (1961).

[8] R. STEVENSON, J. chem. Phys. 27, 673 (1957). [ 9 ] W . F . GIAUQUE and H . L . JOHNSTON, J. A m e r . Chem. Soc. 51, 2300 (1929).

[10] J. W. STEWART, J. Phys. Chem. Solids 12, 122 (1959). [11] B . T. MaHHiejiHH H A . M. T o j i K a i e B , H3. TBepji. Tejia 5, 3413 (1963). [12] A . M . T o J i K a n e B [ 1 3 ] J. H . COLWELL, E . K . [14] n . A . E e 3 y r J i b i f t , 8, 744 (1966). [ 1 5 ] J. H . COLWELL, E . K .

H B . I \ M a H m e j i H i i , O n 3 . TBepH. T e j i a 7, 2125 (1965). GILL, and J. A . MORRISON, J. chem. P h y s . 39, 635 (1963). H . r . B y p M a H P . X . M u H H c j i a e B , O H 3 . TBepn. T e j i a GILL, and J. A . MORRISON, J. chem. P h y s . 42, 3145 (1965).

[16] L. A. K . STAVELEY, J. Phys. Chem. Solids 18, 46 (1961). (Received

November

11,1965)

T).

J.

D . THOMAS

: Growth and Structure of Evaporated Silicon Layers

359

phys. stat. sol. 13, 359 (1966) Standard

Telecommunication

Laboratories Ltd., Harlow,

Essex

Growth and Structure of Evaporated Silicon Layers By D . J. D . THOMAS Silicon, on heating in a vacuum at pressures of less than 10" 6 Torr, reacts with residual organic vapours to form silicon carbide which may grow epitaxially. Growth of evaporated silicon layers commences in etch pits produced during the annealing pretreatment. Deposition continues until the etch pits become filled, when the layer becomes continuous by lateral spreading from the initial islands. Most of the defects observed are due to stacking disorders formed in the initial growth islands. Many of these defects are removed by interaction with one another as the layers become thicker. Im Vakuum unter Drücken von weniger als 10~6 Torr erhitztes Silizium reagiert mit restlichen organischen Dämpfen und bildet Siliziumkarbid, das epitaxial aufwachsen kann. Das Wachstum aufgedampfter Siliziumschichten beginnt an Atzgruben, die während einer vorangegangenen Temperung erzeugt werden. Das Wachstum erfolgt solange bis sich die Ätzgruben gefüllt haben. Danach erfolgt eine flächenhafte Verbreiterung der anfänglich aufgewachsenen Inseln zu einer durchgehenden Schicht. Der größte Teil der beobachteten Defekte rührt von Stapelfehlern her, die in den anfänglich gewachsenen Inseln gebildet wurden. Viele dieser Defekte verschwinden wieder infolge gegenseitiger Wechselwirkung mit zunehmender Dicke der Schichten.

1. Introduction In recent year, there has been considerable interest in the growth and structure of epitaxial silicon layers [1 to 3]. Most of the work has been concerned with layers produced by the reduction of silicon compounds at high temperatures O 900 °C). The layers produced by these means have been almost completely free of growth defects but this was achieved only after considerable efforts had been made to remove contaminating impurities in the apparatus used for making the layers. The most important impurities were found to be oxygen and organic vapours which formed either oxide or carbide on the surface of the substrate and which promoted the growth of defects. Similar effects to those just described have been observed in evaporated silicon layers. Due, however, to the conditions under which the layers were grown, the way in which the growth took place was modified. In particular this was due to the initial heat treatment of the substrate. After the heat treatment the substrate was covered with a layer of silicon carbide and also had a large number of etch pits. It is shown in this report how the structure of these layers is determined by tha condition of the substrate and how growth of the silicon takes place. In addition to silicon substrates, some experiments were carried out using gold/silicon substrates. 2. The Preparation ol the Silicon Layers [4] The silicon layers were produced in a standard CVC 18" evaporator. During the course of the work a liquid nitrogen chevron baffle in the mouth of the diffusion pump and a Meissner trap in the chamber were incorporated in the system in order to reduce the organic vapour concentration. Both the source and the'

360

D . J . D . THOMAS

substrate were heated by electron bombardment and the pressure during growth was less than 10~6 Torr as measured by an ionization gauge [5]. The silicon substrates were chemically polished and growth rates were usually about 0.5 ¡xm/min. 3. The Structure of Evaporated Silicon Films 3.1

The

effect

of

pretreatment

On heating in the vacuum equipment to about 1200 °C the silicon surface became covered by a randomly oriented second phase whose electron diffraction pattern was consistent with that of the cubic form of silicon carbide (Fig. l a ) . The carbide was present in two basic forms. The first was composed of small randomly orientated crystallites whose diameter and thickness was about 1000 A . Many of the crystallites had an internal structure, consisting of a series of parallel lines of spacing about 100 A and which were probably due to stacking disorders. This was confirmed by streaking in the electron diffraction pattern.

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Growth and Structure of Evaporated Silicon Layers

361

When improvements had been made in the evaporator to reduce the concentration of organic vapours a second type of carbide growth was observed. In this case the carbide was present as irregular segmented ribbons. When the organic vapour concentration was still further reduced heating at temperatures below 1200 °C produced both randomly orientated and epitaxial silicon carbide (Fig. lb). In the latter case (111) [110] were parallel to the silicon (111) [110]. The silicon carbide crystals also showed considerable twinning. Further heating experiments were carried out to determine the effect of temperature and gaseous ambient on the silicon surface. Heating at temperatures near the melting point produced clean silicon surfaces. When hydrogen was added to the gas ambient more perfectly orientated silicon carbide was produced but the effect of oxygen was inconclusive. During the annealing treatment the silicon substrate became etched and the surface covered by etch pits whose diameter was a few microns and whose sides were parallel to

3.5 Carte des désorientations

efficaces

s° = 2 sin 6 '

lorsque

le plan réflecteur

est

(010)

L'expression (3) se simplifie alors et devient, p o u r u n e dislocation contenue d a n s le plan (011) et de vecteur de Burgers \ [011] 8(A0) = —

sin 95 (1 + t g 0) .

(4)

Le système de coordonnées ej lié au p l a n réflecteur a u n e orientation simple p a r r a p p o r t au système de boucles (Fig. 11): e[ est perpendiculaire à la surface d u cristal, dirigé vers l'intérieur du cristal, e 2 est parallèle à l'alignement des points d'émergence des dislocations le long d ' u n des côtés d u carré, il est égalem e n t parallèle à la t r a c e des plans réflecteurs sur la surface. L'origine d u système de coordonnées est au centre d u carré. L'expression (4) devient, en fonction des coordonnées x'\, wAm M l + tg 6) ^ . . S ( A 6 )

=

— ï r c —

r/2

-

*-»)» +

'

( 6 )

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n 4H

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(2

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2

1) a]

2

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(6)

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e) x

= •v

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—

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[ 1/2 ( x ^ M - [a/ "- (2 >1 • 1) «P ~~ 1/2 (x' ) + [x'2 - (2 n - 1) a]2 J ' ^ ' L a Fig. 12 m o n t r e la distribution des p o i n t s de la surface d u cristal pour lesquels la variation de l'écart à l'incidence de Bragg est de i 2 secondes d'arc. N o u s avons pris comme longueur d ' o n d e d u r a y o n n e m e n t X 0,71 Â et 2 ¡i.m comme valeur du p a r a m è t r e 2a, ce qui correspond à la valeur effectivement mesurée entre les points d'émergence de d e u x dislocations successives sur la Fig. 7. Nous avons pris comme hypothèse que les régions pour lesquelles la désorientation était inférieure à 2 secondes d ' a r c contribuaient t r è s p e u à la f o r m a t i o n des images directes. C'est en effet l'ordre de g r a n d e u r de la largeur d u profil de réflexion pour le cristal p a r f a i t . Les régions qui contribuent à la f o r m a t i o n de l'image sont celles p o u r laquelle la désorientation a u n e valeur supérieure; ce sont les régions hachurées sur la figure. Nous avons représenté avec des hachures différentes les régions pour lesquelles la désorientation a u n signe différent. N o u s pouvons faire les remarques suivantes sur la figure : 1. D a n s la zone centrale entre les d e u x séries de boucles, les désorientations se compensent et cette région ne contribue pas à la f o r m a t i o n de l'image, ainsi que nous l'avons observé expérimentalement.

Etude topographique des figures de choc dans LiP

385

Fig. 12. Carte des déaorientations autour des familles de boucles de dislocations dont les points d'émergence forment un carré ABCD. Plan de la figure: (001); réflexion: 020. Les dislocations dont les points d'émergence sont sur BC et AD ne contribuent pas aux désorientations car leur vecteur de Burgers est parallèle au plan réflecteur. Hachures simples', régions pour lesquelles les désorientations sont supérieures ou égales à 2 secondes; hachures croisées". régions pour lesquelles les désorientations sont inférieures ou égales à —2 secondes

2. La forme générale de la carte des désorientations correspond bien à celle des images observées. Compte tenu du fait que l'image sur la topographie est due aux désorientations créées tout le long des lignes de dislocation, et non dans un plan seulement, l'accord quantitatif avec la dimension des images est satisfaisant. 3. Les désorientations correspondant aux deux séries de boucles sont de signes opposés. Nous l'avons vérifié expérimentalement en prenant une topographie avec un réglage éloigné du réglage maximum : les régions dont la désorientation est de signe opposé à celle que l'on fait subir au cristal pour l'éloigner du réglage maximum se trouveront en position de réflexion et apparaîtront avec un contraste important, celles dont la désorientation est en sens opposé seront à peine visibles sur la topographie. Nous avons montré ainsi que chacune des deux moitiés des images doubles correspondait à des désorientations de sens opposés dont les signes sont en accord avec les prédictions de notre modèle.

3.6 Carte des désorientations efficaces lorsque le plan réflecteur est (111)

Nous avons calculé la carte des désorientations de la même manière. La Fig. 13 montre les points de la surface du cristal pour lesquels la désorientation est égale à 2 secondes d'arc. Nous retrouvons approximativement la forme de l'image expérimentale (Fig. 5d): une petite tache centrale entourée par un demi-cercle.

Fig. 13. Carte des désorientations à la surface du cristal lorsque le plan réflecteur est (111). L'image obtenue en reliant les points pour lesquels la désorientation est de 2 secondes environ, est semblable à celles observées sur les topographies avec la réflexion 222

• SUB) >2' "SUß)~f5 °6k8ìff\ Ao)°„ •»jA* AO>°H \2 K> D0 ~ N0 \Atu onr) N0 {*»>% +¿mV) aus den DjD0-Werten im Minimum und im temperaturunabhängigen Bereich die temperaturabhängige Verbreiterung (3) und wegen

AcoS-Tr1

(4)

die Spin-Gitter-Relaxationszeit T r (Aco^ ist dabei die Halbwertsbreite im Vergleichskristall, und N und iV0 sind die Anzahl der Spins im Kristall bzw. Vergleichskristall.) Diese Meßmethode der Relaxationszeit ist den Verfahren der direkten Bestimmung unterlegen, da sie so kleine Relaxationszeiten voraussetzt, daß diese sich auf die Linienbreite auswirken. Außerdem bestehen in der indirekten Auswertung möglicherweise Fehlerquellen. In der Arbeit werden nur relative Änderungen der Spin-Gitter-Relaxationszeit betrachtet und T1 daher auch nur in willkürlichen Einheiten angegeben. Trotzdem kann wegen der oben benutzten Addition der Linienbreitenanteile ein systematischer Fehler bei der Tj-Bestimmung wirksam werden. Da im undotierten Kristall bei uns die Linienbreite durch die Inhomogenität des äußeren Feldes bestimmt wird, wäre z. B . auch eine geometrische Addition vertretbar [3]. Auch in diesem Falle ergibt sich aber aus den Meßwerten der gleiche relative Gang, so daß die daraus gezogenen physikalischen Folgerungen ihre Gültigkeit behalten. Die Messungen wurden in der Regel an AgBr-Polykristallen durchgeführt. Probemessungen an Einkristallen ergaben keine abweichenden Ergebnisse. Die Verunreinigungen an zweiwertigen Kationen lagen im Ausgangsmaterial nach Leitfähigkeitsmessungen bei 2 bis 5 • 10 - 6 . 1 ) Der Zusatz wurde der Schmelze unter Vakuum zugegeben und diese dann durch einen Luftstrom schnell abgekühlt. Danach temperten wir zur Homogenisierung mehrere Tage dicht unterKonzentrationsangaben in Molenbrüchen.

Kernresonanzuntersuchungen an Punktdefekten in AgBr

403

halb des Schmelzpunktes und schreckten die schwefeldotierten Kristalle in Wasser ab, um höhere Fremdkonzentrationen einzubauen. Bei den kadmiumund kupferdotierten Kristallen war letzteres nicht notwendig und wegen der hohen Diffusionskonstanten dieser Kationen auch nicht möglich.

3. Meßergebnisse 3.1 Schwefeldotierte

Kristalle

3.1.1 Schwefeldotierte Kristalle bei tiefen

Temperaturen

An schwefeldotierten AgBr-Kristallen konnte das von Seifert [4] beschriebene DID 0 -Minimum bei 150 °K, das von ihm auf die Spin-Gitter-Relaxation unassoziierter Silberlücken zurückgeführt wurde, nicht gefunden werden. Der Versuch zur Reproduktion wurde unternommen, weil es unwahrscheinlich schien, daß die benötigte hohe Silberlückenkonzentration auf so tiefe Temperat u r e n abschreckbar ist [5]. 3.1.2 Bromierte schwefeldotierte

Kristalle

Bei der Bromierung entstehen Defektelektronen im Kristallgitter nach der Bruttoreaktionsgleichung 2 ) y Br 2 (g) + Ag k ^ AgBr (f) +

+ e-.

(5)

Durch Messung der Gewichtszunahme wurde festgestellt [6], d a ß m a n zum völligen Durchbromieren des Kristalles pro S""-Ion zwei Defektelektronen verbraucht. W e n n eine Ausscheidung auszuschließen ist, sind somit nach der Bromierung gitterpositive Schwefelatome Sa und negative Silberlücken nie vorhanden. Nachdem die Assoziate S a unterhalb 200 °K mit Hilfe der Dipolrelaxation vergeblich gesucht wurden [7], lag es nahe zu versuchen, freie Silberlücken durch ein X)/D 0 -Minimum nachzuweisen, wie das bei den Messungen von Reif an CdBr 2 -dotiertem AgBr bei 0 °C gelang. Durch Bromierung bei möglichst tiefen Temperaturen (bis herab zu Zimmertemperatur) wurde erreicht, daß die Kristalle klar durchsichtig blieben. (Eine Trübung weist eindeutig auf Ausscheidung des Schwefels hin.) Zum Teil wurden die Meßproben aus dünnen Kristallscheiben zusammengesetzt, die in 17 Stunden bei Zimmertemperatur durchbromiert werden konnten. E s gelang jedoch in keinem Falle, ein Lückenminimum nachzuweisen. Die Deutung liegt wahrscheinlich darin, daß sich der neutralisierte Schwefel zusammen mit den Silberlücken vorwiegend in größeren, jedoch noch nicht lichtstreuenden Aggregaten ausscheidet. 2

) In dieser Arbeit werden die Fehlstellen wie folgt bezeichnet: Das Hauptzeichen gibt das vorhandene Fremd- oder Eigenion durch sein chemisches Symbol an. Für die Leerstelle steht das Hauptzeichen • . Der obere Index bezeichnet die Überschußladung der Fehlstelle relativ zum Idealgitter (• positiv, ' negativ). Der untere Index gibt die Platzbesetzung an (a Anion, k Kation, O Zwischengitterplatz). Nachgestelltes eingeklammertes ,,g" weist auf die Gasphase und ,,f" auf die Oberfläche hin. Defektelektronen werden mit e- bezeichnet. 27

physica

404

J. K L U G E

3.1.3

Bestrahlte

schwefeldotierte

Kristalle

Seifert [4] berichtete von einer Lagerungszeitabhängigkeit der statischen Verbreiterung abgeschreckter Ag2S-dotierter Kristalle und führte das auf die Einstellung des thermodynamischen Gleichgewichtes zurück, bei der die dissoziierten Schwefelionen S^ und Bromlücken O j verschwinden. Die Zahl dieser Fehlstellen sollte sich durch Bestrahlung ebenfalls vermindern lassen [8], Es wurden jedoch auch nach außerordentlich langer Belichtung bei Zimmertemperatur keine Änderungen gefunden. Anschließende Versuche, den von Seifert beschriebenen Lagerungseffekt zu reproduzieren, schlugen ebenfalls fehl. 3.2 Kationdotierte +

3.2.1 -5

Cu -dotierte

Kristalle Kristalle

-3

Es wurde mit 5 • 10 bis 10 CuBr dotiert. Ab 5 • 10~4 weist eine Trübung der Kristalle auf einen unvollständigen Einbau des Zusatzes bei Zimmertemperatur hin. An allen untersuchten Kristallen fand man ein deutliches D/Z) 0 -Minimum bei etwa 110 °C (Fig. 1). Der Verlauf in der Umgebung des Minimums stimmt im einzelnen nicht mit dem überein, der sich aus den von Reif [1] theoretisch berechneten Werten für 1 (T1 Spin-Gitter-Relaxationszeit) ergibt. Dies ist jedoch in Anbetracht der in die Rechnung und die Auswertung eingehenden Vernachlässigungen auch nicht zu erwarten. Eine Abhängigkeit der statischen Verbreiterung vom Zusatz lag dicht an der Fehlergrenze und kann nicht mit Sicherheit festgestellt werden. Fig. 2 zeigt die reziproke Spin-Gitter-Relaxationszeit in willkürlichen Einheiten in Abhängigkeit von der Stärke der Dotierung. Fig. 1. Temperaturabhängigkeit der Linienbreite (ausgedrückt durch Z>/Z>0) der Brel-Kernresonanzabsorption eines mit 5 • 10- 4 CuBr dotierten AgBr-Polykristalies. Die eingezeichneten Werte wurden durch Mittelung aus mehreren Messungen gewonnen

50

100 Temperatur (°C1-

Fig. 2. Aus der Breite der Br"-Kernresonanzabsorptionslinie bestimmte reziproke SpinGitter-Relaxationszeit in Abhängigkeit von der CuBr-Dotierung

Kernresonanzuntersuchungen an Punktdefekten in AgBr 3.2.2

Cu+-Cd++-dotierte

405

Kristalle

Als Arbeitshypothese wurde angenommen, d a ß das Minimum bei Cu+dotierten K r i s t a l l e n durch Platzwechselvorgänge von K u p f e r i o n e n auf Zwischengitterplätzen erzeugt wird. Dies vorausgesetzt, v e r m i ß t m a n bei den Cu+dotierten Kristallen ein Minimum bei 0 °C durch Silberlücken, die zur Ladungskompensation dieser Kupferionen notwendig sind. Zur K l ä r u n g der Verhältnisse wurden durch zusätzliche CdBr 2 -Dotierung Silberlücken eingebracht und deren E i n f l u ß auf das Cu + -Minimum untersucht. Man fand an solchen K r i s t a l len, wie n i c h t anders zu erwarten, zwei Minima. Die Auswertung wird dadurch erschwert, d a ß die an Cu-Cd-dotierten K r i s t a l l e n bei 110 °C gemessene Verbreiterung nicht allein auf den Kupferzusatz zurückzuführen ist. D e n n wie Vergleichsmessungen an Cd-dotierten Kristallen zeigen, bewirkt der K a d m i u m zusatz noch bei 110 °C eine Verbreiterung. D a h e r ist es nicht r a t s a m , das Verhältnis K a d m i u m zu K u p f e r so groß zu machen, wie es wünschenswert wäre, um einen sehr starken E f f e k t zu erzielen, und zweitens m u ß die Minimumstiefe der Cu-Cd-dotierten Kristalle auf einen kupferfreien K r i s t a l l gleicher K a d m i u m k o n z e n t r a t i o n bezogen werden. N i m m t m a n wieder eine Addition der auf die Cu-Dotierung zurückzuführenden temperaturabhängigen Verbreiterung Acojj 1 und der sonstigen Verbreiterungsursachen an, so gilt analog zu (3) Acug"

(6)

110 °c

Die K o n s t a n t e C ist unabhängig von der K a d m i u m k o n z e n t r a t i o n und deshalb für die beabsichtigten B e t r a c h t u n g e n relativer Änderungen uninteressant. Bezüglich der Addition der Linienbreitenanteile gilt das in A b s c h n i t t 2 Gesagte. Auch hier stellen sich bei geometrischer Addition qualitativ die gleichen E r g e b nisse ein, so d a ß diese Annahme die Folgerungen nicht s t a r k beeinflußt. Die Meßwerte zweier Versuchsreihen sowie die daraus nach (3) und (6) gewonnenen temperaturabhängigen Verbreiterungen i m Silberlücken- und im K u p f e r m i n i m u m (Acog bzw. Aoj^ u ) sind in Tabelle 1 zusammengefaßt. Tabelle 1 Zusatz

I

Cu Cu+Cd Cd

II

Cu Cu+Cd Cd

D/Do -130

0 °C

110 °C

0,92

1,00 0,37

0,87 0,62

0,97

0,32

0,79

0,95 0,97

1,00 0,60 0,61

0,88 0,82 0,95

0

ly — x> 2/3 — z y, x, 2/3 + z x — y, x, z

where w = 0.465, x = 0.415, y = 0.272, z = 0.120, a = 4.913 Â, c = 5.405 À

2. Electron Microscope Observations Thin specimens suitable for transmission electron microscopy were prepared from gem-quality amethyst quartz by the method described previously [3] and examined in JEM-6A electron microscope operating at 100 kV. Typical diffraction contrast fringe patterns due to planar defects (boundaries) parallel to {1011} planes are shown in Fig. 2, 3, 4, and 5. Fig. 2 and 3 are bright-field electron micrographs of the fringe patterns observed with g = {1011}. I t will be seen that across the boundaries the thickness extinction contours are continuous and there is no change of contrast. I t follows that there can be no change of crystal orientation across the boundaries, and this is confirmed by the single crystal nature of the associated selected area diffraction patterns. I t will be seen that the fringe profiles are symmetrical but that the contrast (and hence the sign of a) reverses at each successive boundary. Other features of the fringe patterns which can be seen (particularly in Fig. 3) are: l.The intensity along the fringes varies with the background intensity. 2. On passing a thickness extinction contour, the additional fringe is created by forking of the central fringe. I t was also found that these fringe profiles became asymmetrical in dark field. These features are typical of fringe patterns with » = ¿ 2 Til3 or ± jt/3 for the exact Bragg condition [6, 7, 8]. When g = {1010} the fringe profiles are again symmetrical, but now there is no reversal (and, therefore, no change in the sign of a) at successive boundaries. This is illustrated in Fig. 4. These fringes must, therefore, have a = ^ n. This follows from the fact that exp (i at iß ^ — 00 t- "TS « _ d

_-O _ «H

8 « a

8>

Superconductivity in t h e Close-packed Intermediate Phases .•2 §

ti ^ ^ «8 P. 3> m «-2

8 J a -P ts a ® M ÇL, h «-rt » a cS

« w o

ft

V fi

02

(N®tOCOlOlOlOt't-H!OCO

CI

Ci to CO Tj* M1 1—J 1 I—I CO 1 1 1(5 tco co > ! -=«3

TJI CO ce o (M Tjf CO O H\

kH

1 t - H

\u>-H\

kn

kH

(10)

In the centro-symmetrical case wji = w _ h and therefore tjj — r T h e final result is that apart from introducing the quantity r in the equations (7) one has also to put VH+iu>H V-H + iW-H _ ^ T II kH tu

Fig. 1. Geometry of the crystal foil showing the meaning of some symbols 30

physica

452

R. Gevers, H. Blank, and S. Amelinckx

This system of equations now adequately describes scattering by a centro-symmetrical perfect crystal taking normal and anomalous absorption into account. I t should be noted that Friedel's law is obeyed, i.e. I T , h = fo H Wo H = h,-H Is,H = fHVH

=

(12)

= y>o,-H V>o,-H ,

= V-HV-H-

(13)

We shall now discuss how this theory has to be adapted for non-centro symmetrical crystals. 3.2 Equations for the non-centro symmetrical

case

The origin of the coordinate system can no longer be chosen such that F ( r ) = V(—r) and W{r) = W(—r). The Fourier coefficients for V as well as for W will now in general be complex. We shall put Vh = \vh\

= |i>n|

,

WH = |w H |e'>H,

6h

(14a, b)

,

w__H = \wH\ e-ivH,

(15a, b)

where Qh and i-eIS, H + is, —H • (33) In particular one can write IS,H

— IS, —H

=

1/2[/S,h+/S, -H]

TH • [} O . _ 4. — Sln H T

(34)

h

For sin = 1 and %n pa 10 TJJ this ratio is about 40%. It thus turns out that Friedel's law is still obeyed for the transmitted beam, but NOT for the scattered beam. The violation of Friedel's law in this case is clearly a consequence of the anomalous absorption since the effect disappears if Tu —> oo (i.e. if W(r) = 0). Intuitively one can see that the effect described here is a direct consequence of the dynamical interaction between transmitted and scattered beam. An electron emerging from the crystal as part of the transmitted beam has been reflected on equal number of times from either side of the set of reflecting planes H. Any difference in reflecting power between -\-H and —H would therefore be averaged out in the transmitted beam. This averaging process is reflected mathematically in the fact that the expression for a 2 (25) is symmetrical in qn and q~H and in the fact that TG and H only enter in IT through A. The electrons in the scattered beam on the other hand have been reflected one time more from the -]-H than from the —H side of the set of reflecting planes. Consequently asymmetry subsists. This asymmetry is apparent from formula (24b) which contains qn linearly. We shall now discuss the influence of the magnitude of fin on the rocking curves, for transmitted and scattered beams. From formula (24a) it is clear that IT is asymmetrical in SU, whereas IS is symmetrical in SH (from (24b)). This is also the case for a centro-symmetrical crystal. For the centro-symmetrical case the asymmetry of /T leads to enhanced transmission for su > 0; this is the so-called Borrmann effect. However, in the non-centro symmetrical case the asymmetry of ITT H depends on the value of /3hFor cos

> 0, i.e. for —

< (¡h 0 than for SJJ < 0. However, for cos < 0, i.e. for - < 3 0. I t thus becomes clear that for (T; > 0 easy transmission occurs for sh > 0, whilst for Cj < 0 the wave which is most strongly excited for s g < 0 is most easily transmitted. In the centro-symmetrical case, and for a primitive cubic structure, the maxima of the wave field I coincided with the atomic positions, whereas the maxima of I I were located between the atom planes; this is no longer the case. Let us consider the simplest hypothetical non-centro symmetric structure built on a primitive cubic lattice, with one atom at Q, in an excentric position, as shown in two dimensions in Fig. 2. All atoms are assumed to be the same and have an atomic scattering factor equal to unity. The origin is chosen in one of the atoms. The phase angle Oh is the same as the phase angle of the structure amplitude F„ = 1 + e 2 , l i H e = (2 cos a

TT

I /RV

\

6H = 7T HQ + (0 or 7T)

HQ) e:nH°

,

(0( if C3S 31 H O > 0, n if cos n H Q < 0).

(41) , Aii42 \

( )

Howie-Whelan Equations for Electron Diffraction Fig. 2. Two-dimensional non-centro symmetrical structure. All atoms are considered to be equivalent

O

O X

O

O X

o X

O

457 O X

X

O X

O

O X

X

O X

O X

O

O

O

O

O

X

[100.1 Let us consider the diffraction vector H = — [100]. The expressions (40 a) and (40 b) then become (for s = 0)

where 0// =

xja .

This shows t h a t the maxima of y>1 are shifted over a;0/2 with respect to the atom rows passing through the origin, whereas the maxima of ipu are displaced over "¡T + IT with respect to the same atom rows. ¿t Z The amplitudes of if 1 and y>1Y depend on the ratio £ H / T J I as well as on /SJJ, i.e. on the relative phase of % with respect to VJI- For small values of QH one can consider all atoms as being concentrated in the vicinity of the planes x fv 0, a, . . . The maxima of I now practically coincide with these planes, suggesting t h a t this may be the reason why for /?// > 0 this wave is the most strongly absorbed. The statement j}H = 0 means in fact t h a t the maxima of the Fourier component H for the "scattering" potential and those for the same Fourier component of the "absorbing" potential coincide in space. If, however, cos < 0, e.g. flu — u, this means t h a t scattering and absorbing potentials have maxima located in alternating planar regions. I t is then not astonishing to find t h a t now the wave field of which the maxima coincide roughly with the atomic position is nevertheless the easily transmitted one, because for this wave field the minima correspond with the regions of large absorption. The fact t h a t "absorbing" and "scattering" potentials have Fourier components which need to be in phase is a consequence of the absence of a centre of symmetry. 5. Consequences of the Theory The most striking prediction of the theory is t h a t Friedel's law is violated in the scattered beam but not in the transmitted beam. I n principle this could be verified by measuring the intensities under exactly the same diffraction

458

R. G e y e r s , H. B l a n k , and S. Amelinckx

conditions, for the two reflections + H and — H. Since intensities in electron diffraction are so sensitive to slight variations in the diffracting conditions, this approach is not practical. However, nature has provided a more practical approach. Certain non-centro symmetrical crystals, such as tetragonal ferroelectric barium titanate, acquire a domain structure whereby neighbouring domains have exactly parallel lattices but oppositely oriented polar fourfold axes. The planar interface is a so-called 180° wall. In zinc sulfide and in BeO a mode of twinning occurs occasionally whereby domains oppositely oriented [111] or [0001] axis are formed within the same crystal. In amethyst quartz fine lamella of alternatively left and right handed crystal parts are sometimes observed [13]. Let us first consider in particular the 180° walls in BaTi0 3 . The present theory predicts two contrast features for such domains. Both result from the fact that in one domain the reflection + His excited whereas in the other domain the reflection — i f or a reflection equivalent to — H is excited simultaneously under exactly the same diffraction conditions. 1. When the interface of the 180° wall is inclined with respect to the foil plane a fringe pattern with a = 2 6h should be observed as a result of the phase difference i 2 Oh between Sh and The phase angles dn have been calculated for a number of reflections in B a T i 0 3 and the results are shown in Table 1. We can conclude that this angle can, for a favourable reflection, become large enough to produce a distinct fringe pattern (22°). Further below we discuss observed examples. 2. The background intensity in the two domains should be the same in the bright-field image, but should be different in the dark-field image for conveniently chosen reflections. Under optimum conditions the intensity difference may be as large as 40% (see (34)). It is further worthwhile to point out that the phenomenon described here contributes also to the difference in background intensity at 90° walls in the dark field. First we shall briefly discuss the contrast at 90° walls perpendicular to the foil surface. The fringe pattern associated with 90° walls inclined with respect Table 1 Extinction distances and phase angles for different reflections in barium titanate

hkl

tH

20Ö 002 400 004 301 103 110 101 220 202

418 417 936 965 1183 1149 537 530 604 609

(Ä)

6a 0 8° 50' 0 17° 20' 8 ° 30' 25° 10' 0 8° 23' 0 8° 30'

teii/tH

459

Howie-Whelan Equations for Electron Diffraction

\

\

®

fi-im] I: nou Mi

c'

a

® ri:[ioi] l mou

c'

/

\

F i g . 3. R e l a t i v e orientation of c r y s t a l a x i s a n d d i f f r a c t i o n v e c t o r s used to d e t e r m i n e the t y p e of coupling in bariumti tanate

to the foil plane have been discussed intensively in previous papers [3]. For 90° walls perpendicular to the foil the difference in background intensity in bright field as well as in the dark field is partly due to the difference in extinction distance for the reflections which are simultaneously excited in both domains. There may also be an «-difference, but this effect is small, except perhaps for foils appreciably inclined with respect to the incident beam. The difference in extinction distance only occurs evidently if crystallographically non-equivalent reflections are excited in both domains. Let us first discuss the consequences of the failure of Friedel's law. We shall call the foil plane (010). The reflecting planes have then indices of the type { AOZ}. Let them be (hOl) in I and (h'Ol') in II. With the choice of axis shown in Fig. 3 the indices of the simultaneously reflecting sets of planes in the two domains satisfy the relations h' — I ,

I' = h

in the case of head to tail coupling, i.e. with (hOl) in part I corresponds (lOh) in part II. (We neglect the small orientation difference of 36'.) With the diffraction vector g1 = [101], parallel to the boundary plane and related to the lattice of part II, corresponds the simultaneously active diffraction vector [101] related to the lattice of part I. We can therefore say that for a diffraction vector like g1 reflections + H a n d — H are simultaneously excited, the first in part I, the second in part II. On the other hand for a diffraction vector like g2, perpendicular to the boundary plane, the indices are the same in both parts, i.e. [101]. Let us now consider the case of head to head (or tail to tail) coupling. The reference system in both parts is chosen as shown in Fig. 3. Corresponding planes are now related by the relations h = I', I — — h'. For a diffraction vector gl parallel to the boundary plane the indices in part I are [101] against

460

R . GEYERS,

H . BLANK,

and

S . AMELINCKX

Table 2 9i parallel to boundary plane Head to tail Head to head (or tail to tail)

Contrast in DF -\-H and —H operate simultaneously No contrast

92 perpendicular to boundary plane No contrast Contrast in DF II and — II operate simultaneously

[101] in part II. The latter indices are equivalent by symmetry to [101], i.e., crystallographically equivalent reflections operate in both parts. On the other hand for a diffraction vector g2 perpendicular to the boundary plane the corresponding indices are [101] in part I and [101] in part I I ; the latter is equivalent to [101]. We have again + H and — H reflections operating simultaneously in part I and II. This discussion suggests a simple procedure to distinguish between the two types of coupling. In Table 2 we have summarized the conclusions. One can formulate the simple rule: If contrast, i.e. a difference in background intensity occurs in the dark-field image for a diffraction vector parallel to the 90° wall and none for a diffraction vector perpendicular to the boundary plane, we have head to tail coupling. If the reverse is true, we have head to head (or tail to tail) coupling. In the bright field no contrast should arise. Let us now briefly discuss the consequences of a difference in extinction distances. Table 1 gives a survey of extinction distances for a number of reflections in B a T i 0 3 . I t is clear that differences of up to three per cent exist for reflections which are simultaneously excited on both sides of a 90° wall perpendicular to the foil. I t can be shown (Appendix) that for a suitable specimen thickness the relative intensity change associated with a relative change in extinction distance can be about eight times larger than the relative change in extinction distance itself. 5 ) As a consequence reasonable contrast can result. For diffraction vectors, other than the g1 and gz (parallel and perpendicular to the 90° wall, i.e. of the type {hOh} or {/¿OA}) considered above, there are in general differences in extinction distance in the two regions and, therefore, complications arise due to the just mentioned effect. 6. Experimental Evidence 6.1 The observations

of Tanaka

and Honjo on

BaTiOa

In their paper on domain configurations in B a T i 0 3 , Tanaka and Honjo [11] reproduce the required photographs to make the determination of the type of coupling. Their Fig. 15 a shows a dark-field image of 90° walls perpendicular to the foil plane which are crossed by 180° walls in the way shown schematically in Fig. 4. A difference in intensity is found for a diffraction vector parallel 5

) See equation (A 2) of the Appendix.

461

Howie-Whelan Equations for Electron Diffraction Fig. 4. Schematic view of configuration of 90° and 180° domain wails as observed by Tanaka and Honjo [11]

to the domain wall. This difference is in the opposite sense after crossing the 180° boundary (Fig. 4). This observation shows in the first place that the contrast is not due to a difference in extinction distance, because in that case the difference in intensity would not invert at the 180°. I t is therefore due to the failure of Friedel's law and it proves that in this case head to tail coupling is present (Fig. 3 a). The image using a diffraction vector g2 perpendicular to the boundary plane, and which would allow to confirm even more strongly our conclusion, was unfortunately not made by these authors. The same authors also reproduce photographs of the contrast at 180° under different diffraction conditions, using the reflections originating from one row of the reciprocal lattice. Among this sequence of photographs (Fig. 20 of [11]) several images are due to the simultaneous operation of more than one reflection. B y far the best contrast is observed for the best two-beam case of the sequence (the second from the top in Fig. 20 [11]). This seems to suggest that the main contrast might not be due to multiple beam effects, as proposed by the authors. On the other hand, this effect indicates that a two-beam theory might be adequate. 6.2 Present

work

We shall now give evidence for the two phenomena predicted by the theory developed here. 6.2.1 Fringe 'patterns at inclined 180° walls in

BaTi03

Fig. 5 shows a weak fringe pattern. The interface is not plane as can be judged from the zig-zag shape of the fringes. This particular shape results from the tendency of the 180° wall to adopt the shape of minimum energy. The orientation of minimum energy for a 180° wall in B a T i 0 3 is the cube plane. In the case shown the wall cannot adopt this orientation because it is attached to a microtwin consisting of two closely spaced 90° walls. The 90° walls are apparently much more bound to { 1 0 1 } type planes (referred to the cubic phase) and the 180° walls are therefore forced to remain roughly in a plane inclined with respect to the foil surface. As a result of this partly head to head (or tail to tail) coupling occurs along such a wall (Fig. 6). The zig-zag shape minimizes the electrostatic energy associated with the head to head or tail to tail coupling. This observation gives direct evidence for the presence of the phase factor 6HThe fringes have a-characteristics. 6.2.2 Failure

of Friedel's

law

Fig. 7 shows the bright-field and the dark-field image of a fission fragment irradiated foil of B a T i 0 3 . The irradiation was used to pin the very mobile 180° walls, so as to allow dark-field images to be made. I t is clearly visible that the 180° wall itself is visible in both bright and dark field. However, an intensity difference within the domains is only observed in the dark-field image.

462

R . GEVERS, H . B L A N K , a n d

S . AMELINCKX

Fig. 5. F r i n g e s a t 180° wall in b a r i u m t i t a n a t e

/(t ? \ -» iV

\8-i--8i-0 \d2 = -8,=d e^-e^d

Fig. 6. Schematic view of 180° wall a t t a ched to a microtwin consisting of two superposed 90° walls

Howie-Whelan Equations for Electron Diffraction

463

F i g . 7. B r i g h t - a n d d a r k - f i e l d i m a g e s of 180° walls in B a T i 0 3 p i n n e d b y fission f r a g m e n t t r a c k s . N o t e t h e d i f f e r ence in b a c k g r o u n d i n t e n s i t y in t h e d a r k - f i e l d i m a g e o n l y

Since the images are not complementary, it is very suggestive to conclude t h a t anomalous absorption is responsible for the phenomenon, as described in the theoretical section. 7. Effects of Non-Centro Symmetry on the Fringe Patterns at Planar Interfaces One might question whether or not symmetry properties of fringe p a t t e r n s at planar interfaces are affected b y the absence of a centre of symmetry in the crystal. We have to consider different types of interfaces: i) A stacking fault in a given domain. The same diffraction vector is t h e n active in the crystal p a r t s on both sides of the fault and the excitation errors are also the same in both parts, i.e.

(18)

r)

and, therefore, form an orthogonal set with norm unity: /

V

r ) 0r',n'(ki,

r ) d3r =

dr,v- dkj_,k'±

8n,n•

(19)

0 for all v, v', and kj_, and hence tinn'

l i m (4 K T-+00

I M ) 2 J [ M t t ? f D* V, v'

M-t-oc =

M~>oo

2

+v—v' 4=Xß—fi'

Mtt>

f

exp

J

M

^

( Wv, n

(Wv,

D ( W , , n -

W^v', i

n

-

W,',

W„',n>

n> - h

h co)

-

t

ft co)

co)

di (25)

+

D{Wßll

-

W,M

»

n

—

h c o ) ,

a

)

494

L.

FRITSCHE

where D(/±W

—

h

sin [(A W — A co) T j K ] (Alf - h(o)jh

ca)

Evidently, the contribution of the second sum on the right hand side of (25) vanishes. The first sum may be simplified by using lim - L 4 |Z)2| T—>oo & 1

= =

2 n h è (ATP - h m)

so that cy M W

=

lim

Y

¡z — i Ml 2-1 ¿vi üf/2-l

TT

¿vi ¡z — i

M

M-+oc"

E V

£ , J ±

f

al, t

M I

H

(2n)

J

W

^

h

k

^

d ( W , ,

n

-

Wu>,

n

. -

h

œ ) .

This formula agrees with Callaway's result (24) except for the quantity M which, in the present treatment, appears in place of N, and has been introduced as being infinite from the very beginning. However, in the further treatment of his expression (24), Callaway tacitly assumes that N —> oo employing a theorem of Argyres [4] for any function f(x) possessing a Fourier transform •> X/2-1 W

L v=-N/2

1

ii/2-1 Z

oo r

CO t ( v - v ' ) =

2 J

(26)

f ( x ) 6 * " « > d z

i — — oo

y'=—NI2

J

— oo

which only holds rigorously if N

oo.

3. The Effect of Interband Tunneling (Zener Effect) Interband transitions also occur in the absence of an optical perturbation and become important if the external field is sufficiently high. This effect was first studied by Zener [11] and will briefly be discussed in the following. As pointed out in Section 1, the modified Houston functions (3') obey equation (7), whereas an exact solution of the problem in question is required to satisfy (je

o = o.

(27)

Since the functions (3') form an orthogonal set, W(r, t) may be represented as W ( r ,

0 = 2 " k0, fe

n

o>fejL.0

• F M~ 1' 2 KXnn,(k) ÔHx,h'± ô(kx -

K),

where Xnn'{k)

= ^ Ju*(k,

r)-^unik,

r) d 3 r

is the generalization of (8), and M denotes the same number as in the preceding Section. Hence, equation (29) assumes exactly the form of (22) (with h m = 0 and N replaced by M oo), and M„n'> is now defined by KI2 / n M (;:.\kL) = / A*n(k) F xnn,(k) A,, nik) dkx . -K/2

Due to the fact that the above derivation requires M oo, the obtained result provides finite tunneling rates at finite fields, as one should expect. At a first sight Argyres' theory [4], based on the assumption of "Kane" states, seems to yield the same result. However, N appears in place of M oo which gives rise to a similar contradiction as discussed in the preceding Section. The transition rate wnn> vanishes identically if F is smaller than a threshold

F0 =

Ann.(0)KI2n(N-l),

496

L.

FRITSCHE

a result which, of course, is physically meaningless since F0 depends, via N, on the arbitrarily chosen length of the specimen in the ^-direction. This inconsistency is eliminated in Argyres' theory by using the theorem (26), which implies N — o o , so t h a t his formula takes exactly the form of our result. Following the lines of his f u r t h e r t r e a t m e n t the transition rate can immediately be presented in the form -F2 a exp ]8 n A2 E 1 » where

AW

m ,

(30)

El 12 = ~ h F !l rt i w n 00

„ x-r fW = 1+ 2 Z r

1 ^

¡2nn . \ ^v^i cos ( _ 4 , ) - 2 £

8 / 2 ¡J, .\2h 2 (K/2f rf ) it/TYg '

o-n

. /2 jtn A \ am A0) ,

(31)

. . 4, = 4..'«>),

and A is defined b y ¿»»'(k l) = 4> + A k \ . The quantity ¡1 denotes the reduced mass of the bands, i.e. p-i = m " 1 + m-} . The two Fourier series in (31) represent a periodic function of A0¡F a with period unity. For a 2 1 these series can easily be summed u p t o yield |

{[(*/«) (2 AJF a -

1) + 1] -

[1 + (W*) 2 ]}

for

0 < AJF a < 1

which represents a saw-tooth function of width 2 jr/a. The result of Kane's t h e o r y [2] agrees with (30) except t h a t f(F) = 1. I n fact, the contribution of t h e two Fourier series is of the order of Ttja which, in general, will be small compared t o unity. Argyres surmises t h a t the effect of electron collisions will drastically diminish the contribution of these terms. Certainly the most important case is when the collisions are sufficiently numerous so t h a t none of the conduction electrons can gain more energy t h a n Eg in the external field, which would be necessary to kick another electron out of the valence band. As previously mentioned, in this field region below the electrical breakdown the electrons do not cross the entire conduction band, and hence the Houston function — representing an electron between subsequent collisions — cannot be expanded in a Fourier series a n y more. As a result, there is no way t o define K a n e functions in this case since t h e y represent the Fourier coefficients of the expansion ( l i b ) . Thus, Argyres' prediction cannot be reasoned within the frame of the " K a n e " state picture. However, as has been shown b y the author [7], a theory which is based on Houston functions and accounts for collisions as discussed above arrives at the result t h a t the effect of the collisions is, in fact, t o cancel out t h e two oscillatory terms in (31).

Representation of a Lattice Electron in a Uniform Electric Field References [ 1 ] W . V . HOUSTON, P h y s . R e v . 5 7 , 184 ( 1 9 4 0 ) .

[2] E. 0 . KANE, J . Phys. Chem. Solids 12, 181 (1959). [3] E. N. ADAMS, J . chem. Phys. 21, 2013 (1953). [4] P. N. ARGYRES, Phys. Rev. 126, 1386 (1962). [ 5 ] J . CALLAWAY, P h y s . R e v . 1 3 0 , 5 4 9 (1963). [6] J . CALLAWAY, P h y s . R e v . 1 8 4 , A 9 9 8 ( 1 9 6 4 ) .

[7] L . FRITSCHE, phys. stat. sol. 11, 381 (1965). [ 8 ] W . FRANZ, Z. N a t u r f . 1 3 , 4 8 4 ( 1 9 5 8 ) .

[9] L. V. KELDYSH, Soviet Phys. - J . exp. theor. Phys. 7, 788 (1958).

[10] W . FRANZ, phys. stat. sol. 1. K 4 (1961).

[ 1 1 ] C. ZENER, P r o c . R o y . S o c . 1 4 5 , 5 2 3 ( 1 9 3 4 ) . (Received

November

19,

1965)

497

F. BELEZNA Y and G. PATAKI: Recombination of Electrons and Donors in n-Ge

499

phys. stat. sol. 13, 499 (1966) Research Institute

for Technical

Physics

of the Hungarian

Academy

of Sciences,

Budapest

Remarks on the Recombination of Electrons and Donors in n-Type Germanium1) By F . BELEZNAY and G.

PATAKI

The cascade theory of the recombination of electrons into ionized donors is critically examined in this paper. I t is shown that inLax's model, and also in the work of Hamann and McWhorter, the neglect of lower lying states leads to an overestimate of the sticking probability. It is also shown, that the model proposed by Ascarelli and Rodriguez — which neglects the higher excited states — does not lead to the experimental value for the total capture cross-section. B HacTOHmeit paSoTe KpHTHHecKH HccnezjoBaHa KacKajjHaa TeopHH peKOM0HnaIJHH OJICHTpOHOB C HOHH3HpOBaHHbIMH HOHOpaMH. Il0Ka3aH0, HTO B MOHeJIH JIaKCa npeHefiperaiomett HH3KO jiemamnMH COCTOHHHHMH a TAKWE B paGoTe TaManHa H MaK B o p T e p a nepeoiienena ,,Bep0HTH0CTb n p m n i n a H H H " („Sticking Probability"). C Hpyroii cTopoHM, na>ne cTporaa pa3pa6oTKa MonejiH AcKapejuiH h P o n p n r e 3 a , B K0T0pbie npeHeSperaiOTCH BHCOKO Jiewcamne BoSywueHHtie COCTOHHHH, He p;aeT c o B n a n a r o m e e c 3KcnepHMeHT0M 3HaqeHHe nojiHoro ce^eHUH 3axBaTa.

1. Introduction Investigating the mechanism of recombination, t h e electron-donor recombination in n-type Ge has a special role. This can mainly be explained by the f a c t t h a t the electron states of shallow donors can be described much better in t h e effective mass approximation t h a n any other t y p e of recombination centre. F r o m the experimental point of view, on the other hand, the donors of the group V of the periodic table are well controllable impurities. The theoretical treatment of this type of recombination is especially interesting since the capture cross section is larger t h a n the geometrical area of electron orbit of the ground state [1], To interpret this " g i a n t " cross section, two different theories were developed. L a x [2] and later H a m a n n and McWhorter [3] supposed t h e capture of electrons b y ionized donors t o be occurring on the highly excited states accompanied b y emission of a phonon. The carriers captured, with f u r t h e r emission and absorption of phonons, will either reach the ground state or t h e y will be ejected into t h e conduction band. The L H W theory is based on the assumption t h a t only highly excited states are responsible for the capture. Ascarelli and Rodriguez [4], on the other hand, proposed a model, where t h e capture t a k e s place through the lower lying states of the centres, namely through t h e states Is, 2s, 3s, 4s. After subsequent transition between s-states of the donor the elect r o n reaches the ground state. The capture as well as the various transitions are accompanied b y one-phonon emission or absorption. I n t h e AR theory the contribution of states with angular momentum differring from zero were neglected since — according to their estimate — the probability A part of this work was presented at the International Symposium on Recombination in Warsaw, 1965. 33

physica

500

F . BELEZNAY

and

G. PATAKI

of capture and transition is greatly reduced with increasing angular momentum. The purpose of the present paper is to investigate the above theories critically and to improve the treatment of Ascarelli and Rodriguez. In Sections 2 and 3 the insufficiencies of the theories in question are considered, while Section 4 contains the improved theory. I n Section 5 the results of calculations as well as discussion of the model can be found. The detailed calculation of the transition probabilities is given in the Appendix. 2. The Role of Highly Excited States in the Recombination I n case of highly excited states the motion of electrons can be described by classical mechanics: The position of the particles may vary continuously, thus, it is suitable to introduce the distribution function f(r, v). The origin of the system of coordinates is chosen to be in the centre of impurity. I t was proved in reference [3] that, if the mean free path for acoustic-phonon interaction is large compared to the greatest bound-state radius of interest, the distribution function depends on the total electron energy only. Since the subsequent emission and/or absorption of phonons are independent events, the wandering of electrons between various energy states, the cascade process can be considered as continuous Markov process in total energy. In the limit u0 - > — oo (u0 is the energy of the ground state), for the sticking probability the following homogeneous integral equation was deduced [3]: P{u) =

o / P(w') k(u, u') du' ,

— OO

(1)

P(-oo) = 1 , where P(u) is the sticking probability, i.e. the probability that the electron of energy u will reach the ground state, and k(u, u') is the normalized transition probability, in other words, k(u, u ) du du' is the probability of transition from energy shell (u, u + dw) to the energy shell («', u + du'). The expression for k(u, u') was given in [3]. The capture rate BT of electrons has the following form: Bt =

0

co

Jdu P{u) J dw' K(u', u) f0(u') , — oo

(2)

0

where K(u', u) is the transition probability per unit time corresponding t o k(u', u). This expression presumes that the lower lying states are irrelevant. I n other words, the used limit w0 - > — oo is only correct if the finite value of u0 does not alter the results significantly. As a rough check of this assumption, the change of BT was investigated in reference [3] by cutting off the integration over the sticking probability — the u integration in (2) — at an energy corresponding to the state 5s. The change was found to be less than 10 per cent. Above check, however, is not sufficient, as the function K(u', u) sharply decreases if (u' — u) is large. Thus, cutting off the integration only means that, within the frame of validity of the classical approximation, the electron capture to a lower lying state is a very unlikely event compared with the capture to the highly excited states. I t seems to be more important that in expression (2) the P(u) itself should depend on the actual value of u0, because the P(u) is the probability that the electron from energy stase u will reach the ground state u0.

Recombination of Electrons and Donors in n-Type Germanium

501

I t is easy to demonstrate how markedly the arbitrary choice of u 0 m a y vary the value of sticking probability P(u). According to paper [3] the sticking probability, at energy corresponding to the state 3s, is equal to 1 for all values of parameter y and temperature. The discrete state 3s, however, does really exist (as proved by spectroscopical experiments), thus, only the quantum mechanical treatment of paper [4] is adequate to determine the sticking probability for this state. The calculations gave a value of 0.08 for 4 °K. The more precise calculations of present paper give a still lower value for sticking probability of state 3s (see Section 5). Since the sticking probability is a monotonous function of energy, the value of P(u) will decrease and thus it results (see equation (2)) a decreasing of the capture cross section, too. i 3. The Role of the Lower Lying Excited States in the Recombination I t has already been mentioned that in the AR model, the recombination takes place through lower lying excited states of the centres. Since only the s-states were taken into account, the different states of donors could be characterized with the single index n. The electron captured on the state % either reaches the ground state or makes a transition to a state nz or will be ionized. Let wHj denote the probability per unit time of transition from state nx to state n2 and the probability of ionization per unit time. The corresponding normalized probabilities of the above transitions (k, hnt and are given as follows 2 ): kn,«, =

1

-f- £

In! = Aw ^

,

(3)

+ S wnin,yi .

(4>

Only t h a t part of captured electrons (e.g. on the state n) will really recombine which, after making transitions of arbitrary number, will reach the ground state; the remaining part getting sooner or later rejected into the conduction band. This can be expressed mathematically as follows: Pn

+

£

I{n

= 1,

(5).

where Pn is the sticking probability and JW is the probability of being ionized after making transitions of number i. Using (3) and (4) equation (5) can be written in the form ii? + 2 K«. = 1 ,

(6)'

n2

so t h a t we can write i)

°°

i= 2

or finally

( )

(i)

n2 =

n2

2 kn^i Pn2 •

(7)

n2

Equation (7) corresponds to the Kolmogorov equation (1), but for discrete case. I n paper [4] it was assumed t h a t the ratio of the probability of ionization and 2 ) For the sake of better comparison with the continuous treatment of Section 2, the notations used are different from that of paper [4] in the following sense:

Wn^-^Wn^, 33*

p«,»,-*

P(n

^n

>

S

n

P

n •

502

P. B e l e z n a y and G. P a t a k j

recombination may be determined by the probabilities of single-step recombination and ionization only. This can be expressed mathematically in the form Kni 1 1 7, 1T* ) , K n1l~r n21

¿V

_ W«,1 ßnt +

(8)

The quantities w , l t l and /?Wi (see Table 1) can be determined by calculating the matrix elements of the phonon induced transition (see Appendix). The numerical values of w„t „2 prove that the approximation (8) is erroneous. In fact, conTable 1 Values of ivn n . and ßn. at T = 4 °K 1

»2

2

3

4

2.65- 107

5.61 •105 1.64- 109

1 nt tiz (S" )

w

»1 2 3 4

1.18 •108 1.17 •107 3.25 •106

2.95 •109 3.26 •10»

—

8.51 •109

-

ßn, 4.90 106 4.52 108 7.25 1010

sidering an electron on the state nx = 3, the probability of making a direct transition into the ground state is less than that of the process through intermediate state nz — 2. Similarly, the processes of reaching the ground state through higher excited states (e.g. through n z = 4) cannot be neglected. These possibilities are taken into account in the system of equations (7). The previous arguments prove that the higher excited states (w2 = 4) would give a significant contribution to the capture of electrons. I n this case, however, concerning paper [4] a further objection arises. In fact, the contribution of states with higher angular momentum, as has been estimated in [4], is proportional to

1

\

\2ft + «

/

where

Eid* (9) ti ca (»I nl) ' q being the phonon wave number, a* the effective Bohr radius, the ionization energy of the donor, c9 the speed of the sound. For n2 = 1, n^ = 3 we obtain 2.nin2 12, but if n v n 2 Sg 3 the value of is less than 0.6 which means that at nx = 4, ng = 3 already, the effect of states with higher angular momentum is not negligible and they will be even more important at larger nx and n2— qa* —

4. The Proposed Model As it has been pointed out in Section 3 the A R treatment is insufficient because of two reasons : First, the number of excited states and transitions involved is inadequate, and secondly, the states with non-zero angular momentum were neglected. In order to take into account all the possible transitions, instead of equation (8) we obtain the following expression : Pnlh = Z Pnj, Kfl] , (10) n2l2 3 where P » ^ is the sticking probability for state (w1( ïx) ) and khj] is the normalized 3 ) Further on the states of electron will be denoted as (n, I), because only the states with quantum number m = 0 are relevant (see Appendix).

Recombination of Electrons and Donors in n-Type Germanium

503

probability of the transition from state ( n v into the state (n2, l2). The equation (10) represents an infinite set of equations since a hydrogen-like i m p u r i t y has an infinite number of states. The highest excited states, however, cannot be realized, because they would be destroyed, e.g. b y the Coulomb forces of t h e neighbouring donors. There is a further reason to use a finite set of equations instead of (10). I n fact, considering the sticking probability on a lower lying state, e.g. the state 4s, not all the possible transitions of the wandering electron are important. The probability of reaching the ground state, say, through state 10s, is very small. I n the actual calculation, therefore, an arbitrary b u t finite value of n should be chosen; let us denote it nh and all the states (n, I) u p to the limit rab must be taken into consideration: Kb Ba-l , Pnih

= Z

Z

Pnknn±

.

(11)

«i = l h = 0 I n order t o obtain the normalized probabilities, however, one has t o take into account all such states (nz > nh) for which the probabilities of single-step transitions are large enough. This procedure, however, gives b u t a few additional states, because the transition probabilities w"' ¡[ become very small with increasing value of (n1 — nz). To have the normalized probabilities kn[i\, the transition probabilities per unit time must be determined. These are given for A„lBl 1 in reference [4], while for the general case they can be found in the Appendix of the present paper. I t can be seen from the equations (3) and (4) t h a t the quantities /Sni are needed, too. These may, however, be neglected based on the d a t a for = l% = 0) given in [4], I n fact, for the case w2 = nt + 1, one obtains W«, « + 1 kn,n +1 •ft. A 1} ' This formula gives just the ratio of probabilities of making a transition t o t h e state wa = «!-)- 1 and being ionized. With numerical values of [4] we obtain ^"•" + 1 > 103 1(1) ~ '

Thus, normalizing the probabilities ¡' the quantities ¡3n will be negligible. The consequence of the previous remarks is t h a t the approximation (8) used in [4] does not hold since not the highest value of w ni „ z was compared with /?Mj. 5. Results and Discussion of the Results I n the calculation n h = 5nvas taken, but for the determination of normalized probabilities all states u p t o n = 7 were t a k e n into account. 4 ) Table 2 shows the transition probabilities k ni i lt calculated on the basis of equation (A9), at 4 °K for all relevant values of % l t and also the d a t a given by Ascarelli and Rodriguez for l 1 == 0. The variation of sticking probability with temperature is given in Table 3. To illustrate the effect of t h e states with non-zero angular momentum, the sticking probabilities at 4 °K were calculated both with and without states I =|= 0. The corresponding values are given in Table 4. where t h e d a t a given b y A R model are also listed. I t is clear f r o m Table 4 t h a t the states 4

) For the sake of easier comparison in the calculation the parameters of paper [4] were used: E1 = 20 eV, ca = 5 • 105 cm s- 1 , q = 5.35 g cm- 3 .

-504

F . BELEZNAY

and

G . PATAKI

with Z =}= 0 are extremely important: Their neglecting leads to an overestimation of the sticking probability. It can also be seen that the application of the formula (8) which does not take into consideration the possibility of indirect transitions leads to erroneous results. In Table 5 all relevant transition probabilities per unit time from states (2,0) and (3,0) are given. The data of this Table show that from point of view of transitions the states with ?=)= 0 are important and the estimation used in [4] based on the value of the parameter X,h „2 (see (9)) is not correct. In Table 6 the contribution of all important states to total capture cross section is given for different values of temperature. To determine the total capture cross section the expression Gnl eff = Gnl Pnl Table 2 Values of Wn'l at T = 4 °K

(»,'-)

1,0 Wn,l s- 1 )

2,0 2,1 3,0 3,1 3,2 4,0 4,1 4,2 4,3 5,0 5,1 5,2 5,3 5,4

1.18 •10» 8.28 •106 1.17 •108 7.45 •104 1.01 •104 3.25 •10« 2.03 •105 3.37 •103 2.18 •101 1.34 •106 8.36- 104 1.49 •103 1.34 •101 5.51 •io- 2

Results of [4] 4.08 • 108 4.09 • 107 1.16 • 107

4.87 • 10«

Table 3 Sticking probability at different temperatures (n, I)

2,0 2,1 3,0 3,1 3,2 4,0 4,1 4,2 4,3 5,0 5,1 5,2 5,3 5,4

T = 1 °K 1.000 1.000

10° 10"

9.563 i o - 1 5.740 i o - 1 7.622 IO"2 2.150 i o - 1 3.676 i o - 2 3.075 i o - 3 2.767 i o - 4 6.890 i o - 3 1.637 IO"4 5.623 i o - 6 3.791 i o - 7 4.021 i o - 8

T = 2 °K

T = 3 °K

T = 4 °K

T =

9.931 i o - 1 6.050 i o - 1 3.166 i o - i 2.863 10-2 1.682 i o - 3 6.609 i o - 3 2.345 i o - 4 9.657 10-« 6.219 i o - 7 1.524 i o - 4 1.382 1 0 " 6 2.253 i o - 8 7.813 1 0 - i o 4.864 i o - 1 1

8.496 • i o - 1 6.338 • IO-2 6.318 • IO-2 2.430 • i o - 3 8.888 • i o - 5 6.534 • i o - 4 1.191 • i o - 5 3.437 • i o - 7 1.796 • i o - 8 1.760 • i o - 5 8.810 • i o - 8 1.014 • i o - 9 2.717 • i o - 1 1 1.363 • i o - 1 2

5.334 • i o - 1 1.423 • IO"2 2.082 • IO-2 6.415 • i o - 4 1.894 • i o - 5 1.738 • IO"4 2.344 • i o - 6 5.884 • i o - 8 2.821 • i o - 9 5.179 • 10-« 1.918 • i o - 8 1.912 • i o - 1 0 4.624 • i o - 1 2 2.110 • i o - 1 3

1.835 2.803 3.609 9.996 2.710 2.623 2.621 5.902 2.698 8.767 2.347 2.057 4.658 1.985

5

°K io-

1

io-

3

10"3 XO-5

10- 6 10- 5 10-' 10- 9 10

-10

io-

7

io-

9

10- 11

XO-13 io-

1 4

10 °K

,

5.263 6.720 5.663 1.451 3.820 3.900 2.938 6.031 2.669 1.469 2.808 2.189 4.731 1.930

Recombination of Electrons and Donors in n-Type Germanium

505

was used. Starting from the results of [4] the value of ani is less by factor Q~21 than an0, where Qn 188¡n 2 , thus, the contribution of states with 1=j= 0 can be neglected. This can be supported also by the fact t h a t the sticking probability itself decreases markedly with increasing value of I, as it is shown on Table 6. Table 4 The numerical values of sticking probability calculated using different approximations (i 1 = 4 °K) (71,0)

Results of paper [4]

States I 4= 0 States I 4= 0 included neglected 5.33 2.08 1.74 5.18

2,0 3,0 4,0 5,0

• • • •

10- 1 10- 2 lO"4 10- 6

9.70 8.58 7.12 5.68

• • • •

10- 1 10" 1 10" 1 10" 1

9.99 8.30 1.60 1.99

- 10" 1 • IO"2 • 10" 3 • IO"4

Table 5 n I Special values of w a t 4 °K 3,0

2,0

V i

Klh 1.169 • 107 2.952 • 109 5.080 • 109

1.177 • 108

1,0 2,0 2,1 3,0 3,1 3,2 4,0 4,1 4,2 4,3 5,0 5,1 5,2 5,3 5,4

—

2.645 • 10' 5.389 • 107 2.176 • 10' 5.612 • 105 0.795 • 106 2.865 •105 4.003 • 104 0.726 • 105 0.883 • 105 2.985 • 10 4 5.107 • 10» 3.967 • 102

1.635 1.341 2.728 2.142 1.493 1.069 2.099 2.082 0.938

• 109 •1010 • 1010 • 1010 • 108 • 109 • 109 • 109 • 109

Table 6 a (cm2) at different temperatures T =

K O )

° K

T =

4

° K

T =

6

° K

T =

10

° K

2,0

5.65-

io-

1 3

2.19

io-

1 4

4.46

10-

1 4

6.05

io-

3,0

3.34-

10"

1 3

7.66

io-

1 4

7.29

IO"

1 5

5.02

10-1«

4,0

1.19 • i o - 1 4 1 0 - 1 5

1.98

io-

1 5

1.58

10-

1 6

9.98

]0-i«

io-

1 6

9.11

5,0

Otot fftot

3

from

[4]

-13

io-

1 3

1.01

1 0

2.27 • io-

1 2

7.36

io-

1 3

io-

IO"17

1 5

1 9

5.11

io-

1 4

6.56 • io-

2.84

io-

1 3

1.00

1 5

IO-13

506

F . BELEZNAY

and

G . PATAKT

According to the data of Table 6, the main contribution to the total capture cross section, excepted very low temperature (1 to 2 °K), is given by the lower lying excited states n = 2,3. The states with non-zero angular momentum, on the other hand, result a considerable diminution of the sticking probability compared to that of A R model. Here we have to mention that the parameter values used in [4] (deformation potential, sound velocity) are too high and also a factor 4 is used unjustified in order to take into account the equivalent minima (see objections of Hamann and McWhorter). Thus, the correct theoretical value of the total cross section is by one order of magnitude less than the experimental one [1]. On the other hand, as pointed out in Section 3 the continuous model also leads t o the overestimation of sticking probability P(u). Analysing the above theories of electron-donor recombination, certain objections could be raised. The definition of the sticking probability seems to be somewhat arbitrary. I n fact, Table 5 shows, if the electron is captured, say, on the state (2,0), the average time of making transitions to the higher excited states is less than the experimentally measured lifetime. Thus, the definition of sticking probability would be: the probability that the electron will reach the state n = 2 (rather than the ground state). This is supported also by the fact that using more realistic parameters of Ge the transition probabilities per unit time become by one order of magnitude less than given in Table 5. On the other hand, the value of sticking probability does not change since determining the normalized probabilities the parameters in question fall out. Finally, it is interesting to remark that in case of deep impurity levels the L a x cascade hypothesis is probably inadequate to determine the large recombination cross section in lack of excited states as was shown by Bonch-Bruevich and Glasko [10]. Present model, however, is still applicable knowing the lower states of impurity atoms. The authors express their thanks and acknowledgement to the follow-workers of the Electronic Computer Centre of the Ministry of Heavy Industries for having made the numerical calculations figuring in present paper in due time. Appendix The perturbation potential, taking into account the electron-acoustic phonon interaction only, is H = Eidivs{r),

(Al)

where is the deformation potential, s(r) is the displacement of the atom occupying the position r\ The equation (Al) gives a non-vanishing matrix element only for longitudinal acoustic phonons. The transition probabilities may be obtained in a way similar to that given in [4]. For n 2

7i,o(â) •

The integration over the angles can be carried out using the " 3 j " symbols: /i,/sin « »{r,,„ yw r „ , _ {'"• + 0 0 where the " 3 j " symbol ^ ^

+ " < " + " } " ' (\0

_

r

0

o !

n

l

l

+ k

( h + h ' +

2

n

[

M

+

h

+

h

+

* + 2 V -

v

hx\k^\

Z

[ K - ^ i - 1)1 ( « , - t , - 1)! ( r h ± h)Hrh + h)!]1/2 _ _ _ 1), ( 2 + kt + 1)! (2 + k2 + 1)! k l _ i),

*

'

^

^

^ l1+h

+ k1 + k2+

( A 9 )

—

where the length is measured in effective Bohr-radii a*, P = — + — is H

a dimensionless quantity, Q = q a* is identical with the parameter [4]. (q is the wave number of phonon accompanying the transition.)

i

n 2

used in

508

F . BELEZNAY and G. PATAKI: Recombination of Electrons and Donors in n-Ge

If Q 1, t h e t e r m w i t h = e x p l i c i t f o r m of P(x) (see [8])

lz =

=

kz — X =

0 will d o m i n a t e .

Using the

T h e t r a n s i t i o n probabilities of this special case c a n be w r i t t e n in t h e f o r m

which is equal t o f o r m u l a ( 2 3 ) of reference [4] e x a c t l y . References [1] S . H . KOENIG, R . D . BROWN, a n d W . SCHILLINGER, P h y s . R e v . 1 2 8 , 1 6 6 8 ( 1 9 6 2 ) .

[2] M. LAX, Phys. Rev. 119, 1502 (1960). [ 3 ] D . R . HAMANN and A . L . M C W H O R T E R , Phys. Rev. 1 8 4 , A 2 5 0 ( 1 9 6 4 ) .

[4] G. ASCARELLI and S. RODRIGUEZ, P h y s . R e v . 124, 1321 ( 1 9 6 1 ) ; 1 2 7 , 167 (1962).

[5] B . V. GNEDENKO, Theory of Probability, G I T L , Moscow 1954 (p. 111). [6] A. MESSIAH, Quantum Mechanics I , North-Holland Puhl. Comp., Amsterdam 1961 (p. 427). [7] A. MESSIAH, Quantum Mechanics I I , North-Holland Publ. Comp., Amsterdam 1962 (p. 1 0 5 9 ) .

[8] Y . L . LUKE, Integrals of Bessel Functions, McGraw-Hill, Inc., New York 1962 (p. 246). [9] A. ERDELYI,

W . MAGNUS,

F . OBERHETTINGER,

and

F . G . TRICOMI,

Higher

cendental Functions I , Chap. 3, McGraw-Hill, New York 1953. [ 1 0 ] V . L . BONCH-BRUEVICH and V . B . GLASKO, Fiz. tverd. Tela 4 , 5 1 0 ( 1 9 6 2 ) . (Received November

20,1965)

Trans-

F . POBELL: Isomerieverschiebung der 23,8 keV-y-Linie von

119

509

Sn

phys. stat. sol. 13, 509 (1966) Kommission für Tieftemperaturforschung der Bayerischen Akademie der Nebenstelle Reaktor station Garching

Wissenschaften,

Isomerieverschiebung der 23,8 keV-y-Linie von in verschiedenen Legierungsphasen

119

Sn

Von F.

POBELL

Es wird über Messungen der Isomerieverschiebung der 23,8 keV-y-Linie von 119 Sn in Legierungen von Zinn mit sechs seiner Nachbarn im Periodensystem der Elemente berichtet. Die Messungen wurden über den gesamten Konzentrationsbereich der Legierungen durchgeführt. Innerhalb einer einzelnen Phase ist die Isomerieverschiebung konstant. Der Übergang zu einer neuen Phase ergibt eine starke Änderung ders-Elektronendichteim 1 1 9 Sn-Kern. Die Ergebnisse werden diskutiert und mit Messungen der Knight-Verschiebung verglichen. Measurements are made of the isomeric shift of the 23.8 keV-y-line of U 9 Sn in alloys of tin with six of its neighbours in the periodic system. The measurements are made over the complete range of concentration of the alloys. The isomeric shift ist constant within a single phase. The transition to a new phase results in a strong change in the s-electron density within the 119 Sn nucleus. The results are discussed and compared with knight-shift measurements.

1. Einleitung Die Verschiebung der Resonanzlinie bei Messung der „rückstoßfreien" Kernresonanz [1] erlaubt es, die s-Elektronendichten am Kernort in Verbindungen oder Legierungen zu bestimmen. In der vorliegenden Arbeit wird über Messungen der Isomerieverschiebung [2] der 23,8 keV-y-Linie von 1 1 9 Sn in Legierungen verschiedener Konzentration von Zinn mit einigen seiner Nachbarn im periodischen System der Elemente berichtet. Bisher sind Werte der Isomerieverschiebung über den gesamten Konzentrationsbereich von Legierungen nur für 119 Sn in Pd-Ag, in Au-Ag und in Sn-Ag [3], für 197 Au in einigen Goldlegierungen [4], sowie für 5 7 Fe in Cu-Ni [5] und in F e - P d [6] angegeben worden. Von den Messungen über einen bestimmten Konzentrationsbereich seien die Untersuchungen von Shirane et al. an F e - R h erwähnt [7]. Die Verschiebung der Resonanzenergie wird bestimmt durch die elektrostatische Wechselwirkung der Protonen mit den Elektronen, die eine endliche Aufenthaltswahrscheinlichkeit im Atomkern haben, den s-Elektronen. Unterschiedliche Umgebung des resonanten Kerns in Quelle und Absorber führt zu einer relativen Verschiebung der Energieniveaus der Kerne. Man erhält das Ergebnis als Produkt einer kernphysikalischen und einer Festkörpergröße. E s gilt für die Isomerieverschiebung [8] A ( < r » >

a

- g ) (|y S;A (0)|» - |y S;Q (0)| 2 ).

(1)

Dabei sind Z Kernladung, A(0), y>s, Q(0) s-Elektronenwellenfunktionen am Kern der Absorber- bzw. Quellenatome, a, g mittlerer quadratischer

510

F . POBELL

Ladungsradius des angeregten- bzw. des Grundzustandes; dessen Definition lautet /e(f)r«dT

dr Der resultierende Energieunterschied ist proportional der Differenz der sElektronendichten am Kernort in Absorber und Quelle; er wird mit Hilfe des Dopplereffekts durch Bewegen einer der Proben relativ zur anderen kompensiert. Dabei wird A E positiv gezählt, wenn sich die Quelle auf den Absorber hin bewegt, d. h. wenn die Energiedifferenz zwischen angeregtem- und Grundzustand im Absorber größer als in der Quelle ist. ^

7

fe(r)

2. Yersuchsdurchführung Als Legierungspartner von Zinn wurden seine Nachbarn im periodischen System der Elemente und die entsprechenden Metalle der nächsten, 6. Reihe des Systems gewählt. Für Legierungen, die weniger als 3 % Sn enthalten, wurde auf 8 9 , 8 % 1 1 9 Sn (normale Häufigkeit 8 , 6 % 1 1 9 Sn bei allen anderen Legierungen) angereichertes Zinnmetall verwendet. Die zinnärmsten Legierungen enthalten 1 A t % Sn. Die Legierungen wurden in einem Hochfrequenz-Induktionsofen meist unter Vakuum (p < 10~ 5 Torr), zum Teil auch unter Wasserstoffatmosphäre (p = 1 at) zusammengeschmolzen. Die Substanzen wurden mindestens einige Minuten flüssig gehalten und dann innerhalb von 1 bis 2 h auf Raumtemperatur abgekühlt. Die für die Verwendung als Absorber benötigten dünnen Scheiben (20 bis 100 ¡im) wurden zum größten Teil durch Auswalzen der Legierungen gewonnen. Legierungen mit mehr als 25 A t % Sb und mehr als 50 A t % B i sind sehr spröde; aus ihnen wurden Absorber durch Zerstoßen der Substanzen und Binden des Pulvers mit Araldit erhalten. Die Quelle wurde durch Neutronenbestrahlung von metallischem Zinn, das u. a. 9 7 , 1 5 % 1 1 8 Sn und 0 , 6 4 % 1 1 9 Sn enthielt, hergestellt. Die Quelle wurde durch einen elektromagnetischen Antrieb relativ zum Absorber mit sinusförmiger Geschwindigkeit bewegt. Gemessen wurde die Transmission der 23 keV-y-Linie von 1 1 9 Sn durch den Resonanzabsorber als Funktion der Geschwindigkeit der Quelle. Die Spektren wurden mit einem Vielkanalanalysator nach der Methode des zeitrichtigen Einschreibens aufgenommen [9]. Als Detektor diente ein NaJ(Tl)-Kristall ( 2 " x l / 8 " ) . Als Selektivabsorber für die Zinn-Röntgenstrahlung wurde 72 mg/cm 2 Palladium verwendet. Jede Messung wurde zweimal durchgeführt; das erste Mal innerhalb der ersten Tage nach dem Herstellen der Absorber. Die bei dieser Messung durch die Temperaturdifferenz zwischen Quelle (20°K) und Absorber (110 °K) hervorgerufene Josephson-Verschiebung [10] wurde korrigiert. Vor der zweiten Messung wurden die Legierungen ca. ein J a h r bei Zimmertemperatur gelagert; bei dieser Messung hatten Quelle und Absorber eine Temperatur von 77 °K. 3. Meßergebnisse Bei keiner der verwendeten Legierungen wurde eine auflösbare Quadrupolaufspaltung der Kernniveaus beobachtet. Einige Transmissionslinien zeigten jedoch eine kleine Unsymmetrie, die darauf schließen läßt, daß in diesen Legierungen nicht alle Zinnatome gleiche Umgebung hatten. Ungleiche Umgebung kann zu unterschiedlicher Isomerieverschiebung und unterschiedlicher, hier nicht auf-

Isomerieverschiebung der 23,8 keV-y-Linie von

U9

Sn

511

gelöster Quadrupolwechselwirkung führen. Unsymmetrische Spektren wurden nur bei Legierungen beobachtet, die aus Kristallgemengen bestehen. Wahrscheinlich setzen sich diese Spektren aus den Linien der beiden unterschiedlichen Kristalle der Gemenge zusammen. Auch die Halbwertsbreiten der Linien, die ihre Maxima im Gebiet der Kristallgemenge und ihre kleinsten Werte für die reinen Phasen hatten, stützen diese Annahme. Es wurde nicht der Versuch unternommen, die gemessenen Linien in ihre möglichen Einzelkomponenten zu entfalten. Als Lage der Linie wird ihr Schwerpunkt angegeben. Diese Lage der Transmissionslinien ist auf die metallische Zinnquelle bezogen. Die Resultate der beiden Meßreihen zeigen keine den Meßfehler übersteigende systematische Differenz. Als Ergebnis wird der Mittelwert der Resultate der beiden Meßreihen angegeben. Dieser Mittelwert, die Isomerieverschiebung für 77 °K ^ T 110 °K, ist als Funktion der Zusammensetzung der Legierungen in den unteren Hälften der Fig. 1 und 2 aufgetragen. I m oberen Teil der Figuren sind die Phasendiagramme der jeweiligen Legierung nach [11, 12] dargestellt. Der angegebene Fehler ( + 0,015 mm/s) wurde bestimmt aus den Unsicherheiten in der Eichung des Antriebs (1,5%), der Messung der Maximalgeschwindigkeit (0,5%) und aus der Unsicherheit für die Bestimmung der Linienschwerpunkte aus den Spektren (1,6%). Zur Kontrolle dieses Fehlers wurde die mittlere Abweichung zwischen den Werten der ersten und zweiten Meßreihe bestimmt; sie beträgt + 0,018 mm/s. Bei den Systemen Bi-Sn und Cd-Sn wurden mit den zinnärmsten Legierungen (1 A t % Sn) möglicherweise nicht die bisher nicht genau bekannten Löslichkeitsgrenzen von Zinn in diesen Metallen unterschritten [11]. Es können deshalb die auf 100% des Wirtsgitters extrapolierten Werte bei diesen beiden Metallen nicht unbedingt als Werte bei unendlicher Verdünnung betrachtet werden. 4. Diskussion Die Übereinstimmung der hier gemessenen Isomerieverschiebungen mit den von Bryukhanov et al. [13] bestimmten Werten für 1 bis 3 A t % Sn in In, T1 und Pb ist nur für P b (+0,51 mm/s in [13]) gut, während für I n (+0,27 mm/s) und T1 (+0,32 mm/s) keine Übereinstimmung innerhalb der Fehlergrenzen besteht. Auch die in weiteren Arbeiten der genannten Gruppe [3, 14] gemessenen Werte für I n (+0,32 mm/s) und SbSn (+0,11 mm/s) liegen außerhalb des hier bestimmten Bereichs. Übereinstimmung herrscht jedoch mit einigen von Cohen [15] bestimmten Werten von Antimon-Zinn-Legierungen (Sb0)45Sn0)55: +0,20 m m / s ; SbSn: + 0 , 1 5 mm/s; Sb0l55Sn0,45: +0,26 mm/s; Fehler jeweils ± 0 , 0 3 mm/s). Bei den von Roberts et al. untersuchten Goldlegierungen (Au-Ni, A u - P d , Au-Cu, Au-Ag, A u o ^ i - C u z - o ^ - N i i - z ) war die Isomerieverschiebung über den gesamten Bereich eine lineare Funktion der Konzentration. Die erwähnten Legierungen von Gold sind ausgezeichnet durch vollständige Löslichkeit der beiden Partner ineinander. Es treten nur magnetische Übergänge auf, die offensichtlich keinen meßbaren Einfluß auf die Isomerieverschiebung haben [4], Ebenso fanden die bereits genannten russischen Autoren [3], daß die Isomerieverschiebung der 23,8 keV-y-Linie von 119Sn als Verunreinigung in dem vollständig löslichen Ag-Au-System eine lineare Funktion der Konzentration ist. Dagegen wurde für 119 Sn in Ag-Pd erhalten, daß die Verschiebung nach einem steilen Anstieg im palladiumarmen Gebiet sich nur noch schwach mit der Kon-

512

F . POBELL

(s/wui]w

S rH*

IlS

X C\4 0 und kleiner als in reinem Blei für AZ < 0. Für eine bestimmte Legierung zeigte die Knight-Verschiebung innerhalb eines einphasigen Gebietes eine schwache, lineare Abhängigkeit von der Konzentration. Das Verhalten der Isomerieverschiebung in Zinnlegierungen zeigt große Ähnlichkeit mit den von Drain [23] an Ag-Cd-Legierungen und von Bloembergen und Rowland [24] an Hg-Tl-Legierungen gemessenen Knight-Verschiebungen. Auch für die Knight-Verschiebung in den untersuchten Legierungen gilt, daß sich die Resonanzenergie innerhalb einer Phase gar nicht oder aber linear mit der Konzentration ändert; beim Übergang von einer Phase zu einer anderen werden auch für die Knight-Verschiebung sprunghafte Änderungen beobachtet. Für die Knight-Verschiebung gilt ~ =

N(EF) F ,

wobei v Resonanzfrequenz, ß Bohrsches Magneton, N^E-p) Zustandsdichte an der Fermigrenze bedeuten. In den Diskussionen führen die Autoren die beobachteten Änderungen der Knight-Verschiebung im wesentlichen auf die Änderungen der Zustandsdichte N(Ef) zurück [23, 24]. Es wurden von Drain vereinfachende Annahmen für die Elektronendichte am Kernort |y>s(0)j2 gemacht und dann aus der Knight-Verschiebung der Verlauf der Zustandsdichte berechnet. Die Isomerieverschiebung wird dagegen nur durch die Elektronendichte, nicht durch die Zustandsdichte N(EV) bestimmt. Die vorliegenden Messungen der Isomerieverschiebung, die innerhalb eines Legierungssystems ein völlig analoges Verhalten wie die KnightVerschiebungen zeigen, lassen darauf schließen, daß vor allem die Änderung von |y>s(0)|a für den beobachteten Verlauf der Knight-Verschiebung verantwortlichist. Zur weiteren Klärung dieser Frage wären Messungen der KnightVerschiebung und der Isomerieverschiebung an den gleichen Legierungen notwendig. Bisher ist die elektronische Struktur der Legierungen zu wenig bekannt, um eine quantitative Interpretation der vorliegenden Messungen zu ermöglichen.

Isomerieverschiebung der 23,8 keV-y-Linie v o n

119

Sn

517

Herrn Prof. H. Maier-Leibnitz danke ich für die Förderung der Experimente. Herrn Prof. F. X. Eder, Herrn Prof. P. Kienle und Herrn Dr. W. Wiedemann möchte ich für das Interesse, das sie den Messungen entgegenbrachten, und für Diskussionen über die Ergebnisse danken. Literatur [1] R . L . [2] O. C. L. R. [3] V . A .

MÖSSBAUER, Z . P h y s . 1 5 1 , 124 (1958). KISTNER u n d A . W . SUNYAR, P h y s . R e v . L e t t e r s 4 , 4 1 2 ( 1 9 6 0 ) . WALKER, G . K . WERTHEIM u n d V . JACCARINO, P h y s . R e v . L e t t e r s 6 , 9 8 ( 1 9 6 1 ) . BRYTJKHANOV, N . N . DELYAGIN u n d V . S . SHPINEL, S o v i e t P h y s . — J . e x p .

theor. P h y s . 20, 1400 (1965). [4] L . D . ROBERTS,

R . L . BECKER, F . E . OBENSHAIN u n d

J . O . THOMSON, P h y s .

Rev.

137, A 895 (1965). [5] G . K . WERTHEIM u n d J . H . WERNICK, P h y s . R e v . 1 2 3 , 7 5 5 ( 1 9 6 1 ) .

[6] G. BEMSKI u n d X . A. DA SILVA, J . appl. P h y s . 35, 1081 (1964). [7] G . SHIRANE, C. W . CHEN, P . A . FLINN u n d R . NATHANS, P h y s . R e v . 1 3 1 , 1 8 3 ( 1 9 6 3 ) .

[8] A. J . F . BOYLE u n d H . E . HALL, R e p . Progr. P h y s . 25, 441 (1962). [9] E . KANKELEIT, Z . P h y s . 1 6 4 , 4 4 2 (1961). [ 1 0 ] J . D . JOSEPHSON, P h y s . R e v . L e t t e r s 4, 3 4 1 ( 1 9 6 0 ) .

[11] M. HANSEN u n d K . ANDERKO, Constitution of B i n a r y Alloys, 2 n d E d . , McGraw-Hill Book C o m p a n y Inc., N e w Y o r k 1958. [12] T . HERMANN u n d O . ALPAUT, J . less c o m m o n M e t a l s 6 , 1 0 8 ( 1 9 6 4 ) . [13] Y . A . BRYTJKHANOV, N . N . DELYAGIN u n d Y . S . SHPINEL, S o v i e t P h y s . — J . e x p . t h e o r . P h y s . 20, 55 (1965). [14] V . A . BRYTJKHANOV, N . N . DELYAGIN, R . N . KUZMIN u n d Y . S . SHPINEL, S o v i e t P h y s .

- J . exp. t h e o r . P h y s . 19, 1344 (1964). [15] R . L . COHEN, p r i v a t e Mitteilung. [16] J . L . BEEBY, P h y s . R e v . 135, A 130 (1964). [17] V . I . GOLDANSKII u n d E . F . MAKAROV, P h y s . L e t t e r s ( N e t h e r l a n d s ) 1 4 , 1 1 1 ( 1 9 6 5 ) . I . B . BERSTJKER, V . I . GOLDANSKII u n d E . F . MAKAROV, P r e p r i n t .

[18] V. A. BELYAKOV, P h y s . L e t t e r s (Netherlands) 16, 279 (1965). [19] W . MEISSNER, H a n d b u c h der experimentellen P h y s i k , Vol. X I V , 1935. [ 2 0 ] W . MEISSNER, H . FRANZ u n d H . WESTERHOFF, A n n . P h y s . ( G e r m a n y ) 1 3 , 5 0 5 ( 1 9 3 2 ) .

[21] S. VALENTINER, Z. P h y s . 115, 11 (1940). [22] R . J . SNODGRASS u n d L . H . BENNETT, P h y s . R e v . 1 3 4 , A 1294 ( 1 9 6 4 ) .

[23] L . E . DRAIN, Phil. Mag. 4, 484 (1959). [24] N . BLOEMBERGEN u n d T . J . ROWLAND, A c t a m e t a l l . 1, 7 3 1 ( 1 9 5 3 ) .

(Received November 29,

34*

1965)

R. P.

GUPTA

and B. D A Y A L : Lattice Dynamics of Zinc

519

phys. stat. sol. 13, 519 (1966) Department

of Physics,

Banaras Hindu

University,

Varanasi

Lattice Dynamics of Zinc By R . P . GUPTA a n d B .

DAYAL

A tensor force model including up to sixth-neighbour interactions is developed for zinc. The model involves a total of twenty-three force constants of which eighteen are determined from certain room temperature experimental elastic constants. The condition for crystal equilibrium which interrelates the force constants is also satisfied, so that the model is elastically consistent. The other five constants are not determined. The theoretical dispersion curves involving only the eighteen force constants determined are computed and compared with the experimental neutron scattering results of Borgonovi et al. Good agreement is found. The theoretical results of D e Wames et al. are also discussed. Ein Tensormodell der Kräfte, das die Wechselwirkung bis zu den sechsten Nachbarn einschließt, wird für Zink entwickelt. Das Modell enthält insgesamt 23 Kraftkonstanten, von denen 18 aus den experimentellen elastischen Konstanten bei Zimmertemperatur bestimmt werden. Die Bedingung für das Kristallgleichgewicht, die die Kraftkonstanten miteinander verknüpft, ist erfüllt, so daß das Modell elastisch konsistent ist. Die anderen fünf Konstanten werden nicht bestimmt. Die theoretischen Dispersionskurven, die nur die 18 bestimmten Kraftkonstanten enthalten, werden berechnet und mit den experimentellen Ergebnissen von Borgonovi et al. für Neutronenstreuung verglichen. Es wird gute Übereinstimmung gefunden. Zusätzlich werden theoretische Folgerungen von D e Wames et al. diskutiert.

1. Introduction Among all the hexagonal close-packed metals studied so far zinc has led to several theoretical problems in the field of lattice dynamics. I t is strongly anisotropic and has an axial ratio (c/a) 1.855 which is much higher than the ratio 1.633 for an ideal close-packed lattice. Several workers have made extensive investigations on this metal both from theoretical and experimental points of view. The neutron scattering measurements to study the phonon dispersion curves were carried out by Maliszewski [1], Borgonovi et al. [2], and Maliszewski et al. [3, 4], the most extensive being those of Borgonovi et al. All these workers also made attempts to fit their dispersion curves by suitable theoretical models. Borgonovi et al. [2] could show a good agreement between the experimental and the theoretical dispersion curves by the use of a four neighbour tensor force model. But their model suffers from some serious drawbacks. These authors used entirely the neutron scattering data for a determination of their force constants. As shown by De Wames et al. [5] the elastic constants calculated from these force constants are not in good agreement with the experimental ones. Further, their force constants show an internal inconsistency and do not satisfy an internal relationship between the force constants which must be satisfied for crystal equilibrium. The non-fulfilment of this condition gives rise to two different values of c44 when the secular equation is solved in the [0001] and [0110] symmetry directions for long wavelength limit. The greatest drawback of their model lies in the fact t h a t the atoms which have been considered by

520

R . P . GUPTA

and

B . DAYAL

them to be fourth neighbours are in reality sixth neighbours. This amounts to considering interactions of the first three and the sixth neighbours and neglecting those of the fourth and the fifth without assigning any reason. Maliszewski et al. [4] kept in view the high axial ratio of this metal and derived expressions for a four neighbour tensor force model. They, however, could not be successful in fitting the model to the experimental dispersion curves. Comparison of the theoretical and the experimental curves was also done for two and three neighbour central and two neighbour tensor force models, but a very poor agreement was obtained with the experimental dispersion curves. Recently De Wames et al. [5] have been able to show a good agreement between the theoretical and the experimental dispersion curves by employing a six neighbour modified axially symmetric model involving eighteen force constants. But as these authors have themselves shown the elastic constants calculated from their force constants do not show a good agreement with the experimental ones, the discrepancies being of the order of 18 and 27 per cent in c44 and c13, respectively. In an earlier paper [6] the present authors had given an electron gas model for hexagonal metals and reported good agreements for beryllium [6, 7] and magnesium [8]. The effect of conduction electrons in the equation of motion was also considered and the number of force constants kept very small. Attempts were made to fit this model for zinc also, but a reasonable agreement between the experimental and the theoretical results could not be obtained. This is, however, not surprising in view of the strong anisotropy and high axial ratio exhibited by this metal. In an attempt to find a model which could give a reasonable agreement between the theoretical and the experimental results the authors have extended the fourth neighbour tensor force model to include the fifth and the sixth neighbours interactions. The force constants were determined using all the experimental elastic constants except c13 and some of the neutron scattering frequencies. The condition for crystal equilibrium was also incorporated. A good agreement has been obtained between the experimental and the theoretical dispersion curves. 2. Results The distribution of the neighbours in the case of zinc is slightly different from that found in the cases of magnesium and beryllium on account of its high axial ratio. The first and the fourth neighbours are each six in number and lie in the basal plane at distances a and a |/3, respectively. The second, third, and the fifth neighbours are located on the c/2 planes at distances /c 2 /4 + a 2 /3, )/c2/4 + 4 a 2 /3 , and /c 2 /4 -f- 7 a 2 /3 , respectively, their numbers being six, six, and twelve, respectively. The sixth neighbours are two in number and lie on the c planes at a distance c. The coordinates of the representative atoms of these neighbours are given in Appendix I. The cartesian system of axes has been followed and is so oriented that the x- and the z-axes lie along the positive directions of ax and a3 , respectively. av o a , and a3 are the primitive lattice vectors and l^l = |a 2 | = a = 2.6649 A and \a3\ = c = 4.9468 A. Proceeding in the usual way the secular equation for the determination of the frequencies may be written as |D(g) - 4n 2mv*I\

= 0 ,

(1)

521

Lattice Dynamics of Zinc

where m is the mass of an atom and I is a unit matrix. Since the hexagonal close-packed structure consists of two interpenetrating simple hexagonal lattices and contains two atoms in the unit cell, the dynamical matrix D(g) is a 6 x 6 matrix and can be written in the form D (q)

=

B B*

(2)

A

A and B are each 3 x 3 submatrices and B* is the complex conjugate of B. I n the tensor force model the elements A a n d Bij of the submatrices can be expressed as Aj = U D f j exp {-iqrp) , p (3) Bij = 2 m exp {—iqrp>). p refers to atoms in the lattice containing the reference atom and p to those in the other interpenetrating lattice. rp is the distance of the pth atom from the reference atom and D f j is the tensor force constant. The force constant matrices are derived from symmetry considerations of the lattice discussed by Begbie [9]. These are given in Appendix I for the representative atoms. The elements of the dynamical matrix are given in Appendix I I . The elastic constants expressed in terms of the force constants are given in Appendix I I I . The crystal equilibrium condition leading to an internal relationship between the force constants has also been discussed in this Appendix. Begbie [9], Collins [10], and Iyengar et al. [11] have also discussed this condition. The secular determinant gets easily factorised for the [0001] and [0110] symmetry directions. The dispersion relations in these directions are, therefore, obtained in the form of very simple expressions. The total number of force constants occurring in the model is twenty-three, but the dispersion relations need only eighteen of them. These have been determined using all the room temperature experimental elastic constants except c13 and some neutron scattering frequencies. The recent measurements on the elastic constants of zinc are those of Alers and Neighbours [12] and Garland and Dalven [13]. The former measurements are employed in this work. They are given in Table 1. Table 1 Elastic constants of zinc in units of 1011 dyn/cm 2 Constant

Experimental (values used here)

De Wames et al. (calculated values) 15.39 6.78 3.15 3.62 3.85

16.368 6.347 3.879 3.64 5.30

The internal relationship between the force constants due to the crystal equilibrium condition is [3 {a3 + 3Ca) + & + 4 y 3 + 14 e3] =

¿

2

[3 (ft + A, +

yx

+ y a + 2 £ l + 2 e2) + 8 d j .

522

R . P . GUPTA a n d

Fig. 1. Dispersion curves in [0001] direction. The curves are the results of the present calculations. The experimental data have been shown by O , X , -f for longitudinal optical, longitudinal acoustical, transverse optical, and transverse acoustical branches, respectively

B . DAYAL

Fig. 2. Dispersion curves in [0110] direction. The curves are the results of the present calculations. The experimental data have been shown by O, x , + , A , A for longitudinal optical, longitudinal acoustical, transverse optical X» transverse acoustical X> transverse optical j| and transverse acoustical || branches, respectively

This relationship was also used while determining the force constants. Either side of this equation is found to have a value 16.6180 in units of 103 dyn/cm. The calculated force constants in units of 103 dyn/cm are given below. 25.9012

Ci = - 0 . 3 0 4 7

1.0438

«2 = - 1 . 9 6 5 4

0.7430

\ r ) W\ r ) ; 1=1

= £ W ) e2"i{Si(r,+(9"' r - R ) ) . n=0

(2.6)

Der Unterschied zu [5] besteht darin, daß jetzt auch die Amplitudenverhältnisse cln =cln{r) ortsabhängig sein können und daß die in den Ortskoordinaten lineare Phase K([, r) durch eine allgemeine Eikonalfunktion Sl0 = 8l0(r) ersetzt wird. Für die Amplitudenverhältnisse gilt als Nebenbedingung die Normierung 2 Kl2 = 1 • n

(2-7)

Durch die Querstriche über den Funktionssymbolen Wl,1 und cln wird angedeutet, daß sie sich bei analoger Bedeutung wie im Ansatz (2.5) jetzt auf modifizierte Bloch-Wellen beziehen. Das gleiche gilt für weiter unten noch einzuführende Funktionen, die eine jeweils analoge Bedeutung in [5] haben. Die Anwendung des A-Operators auf den noch sehr allgemein gehaltenen Ansatz (2.6) ergibt eine größere Anzahl von Einzelgliedern, die in folgender Weise zusammengefaßt werden: A f = AJ p + A ä f + A 3 ^ . 35

physica

(2.8)

M. WlLKENS

532 Dabei bedeutet

A ^ = - 4 »« 2 Z ( V ^ ) 2 ¥ cln e 2 n i S " ,

(2.9a)

Aaf = 2 n i Z Z { ( W c l n , ^7Sln)+f c^Si}e2niSn ,

(2.9b)

l

l

A mit

3

{ < i n

n

n

A9 l + 2 ( W , V 4 ) + V1 A 4 } *niSi S^EEE^ + (gn,r-R).

(2.9c) (2.9d)

I n allen praktisch vorkommenden Fällen bleiben [grad q>l\ und |grad cln\ klein gegenüber 2 n [grad Sln\ m 2 n k0. Demnach gilt A30, n folgt. 0n = 0o, „ + A© n bezeichnet den Glanzwinkel, den die eingestrahlte Elektronenwelle mit der durch gn gekennzeichneten Gitterebenenschar des ungestörten Bezugsgitters bildet. sn ist der Anregungsfehler im ungestörten Bezugsgitter. s n kann man deshalb formal als lokalen Anregungsfehler im gestörten Gitter bezeichnen. Setzt man jetzt (2.9a) mit (3.5)ff. und mit der Säulennäherung (3.1) in die erste Näherung (2.11) ein, so erhält man nach Abspalten der Phasenfaktoren ein System von N Gleichungen für die N Amplitudenverhältnisse c n . Der Index l möge vorerst außer Betracht bleiben. 2 k0 [8Q/Sz -sn]cn+ZUn-n'Cn>= (Bezüglich Un_n> ist (2.3) zu beachten.) 35*

0 ;

n = 0 . . . N -

1.

(3.9)

534

M. WlLKENS

Nicht-triviale determinante

für die

erfordern das

cn

eQ

U -

2

dz

1

n

k0

(3.10) SQ.

U + n

2

k0

Sn

'dz

"

'[

Die Randbedingungen für Q erfordern, daß Q auf der Eintrittsgrenzfläche verschwindet. Dazu müssen die in (3.10) unbestimmten Komponenten dQ/dx und 6 Q / Q y herangezogen werden. Beschränkt man sich auf den symmetrischen Laue-Fall, was in der Säulennäherung keine wesentliche Einschränkung bedeutet, so liegt die Grenzfläche senkrecht zur Strahlrichtung, d. h. senkrecht zur z-Achse, etwa bei z = 0. Die oQ/Qx und dQ/dy können dann = 0 gesetzt werden. Der Einfachheit halber ersetzen wir hinfort 8/öz durch d/dz und fassen die x- bzw. ^-Abhängigkeit des Yerschiebungsfeldes R als Parameterabhängigkeit auf. Ist das Potential reell, was vorerst vorausgesetzt werden soll, so ist die Matrix von (3.10) hermitisch. Die N rellen Eigenwerte AQ'jdz ergeben sich aus einem Polynom iV-ten Grades, und für die Amplitudenverhältnisse gelten die Orthogonalitäts- und Normierungsgleichungen Zc n

l

n

(c™)*

=

d,,„

(3.11)

.

(* bedeutet konjugiert komplex.) Aus den Eigenwerten dQ'jdz folgen durch Integration die gesuchten Funktionen Ql(z)

o Damit ist die erste Näherung erfüllt. Die zweite Näherung (2.12) zerfällt wegen der mit der Gitterperiodizität oszillierenden Faktoren exp (2 n i(g„, r — R)) in ein System von N Differentialgleichungen + ^.^JJeS^ = 0 ;

n =

0 . . . N - l .

(3.13)

Dabei wurden in (2.12) bzw. (2.9b) nur solche Glieder berücksichtigt, die klein in erster Ordnung sind. Die Glieder U¥lldz auflösen. Multipliziert man (3.13) mit (c™)* und summiert über n, so folgt mit (3.11)

, 1 rrtU

= -

dz

mit

-a m ' 1 -e 2 * i sind gleich Null oder rein imaginär. Ist am• =4= 0, kann man das Glied mit ,

Wg = { 1 1 1 } B CBepxCTpyKType L 1 2 . r i 0 K a 3 a H 0 , HTO B6JIH3h aHTHa3Hoii r p a m m b i HMeeT MecTO pejiaKcauHH aioMHoro nopH«Ka h 3HaiHTejibHoe OTKJioHeHHe cocTaBa OT CTexwoMeTpHHecKoro. H a i i i ; c n o , HTO CBepxnHCJioKamra B ycJioBHHX TepMOHHHaMHHecKoro paBHOBecHH HOBOJibHO CHJibHO 3aKpenjieHbi. TeMnepaTypHan 3aBHcnMocrb HanpHJKCHHH CTapTa CBepxanc-noKauHM oSHapywHBaeT MancHMyM npH «a 0 , 8 Tk.

Recently it was shown [1 to 4] that the long-range order parameter and the concentrations of the components in the ordered solid solutions under conditions of thermodynamic equilibrium are different at antiphase boundaries (APB) and in the volume of domains. For a b.c.c. superstructure of CsCl type the equilibrium values of the long-range parameter and of the concentrations of the components at an A P B were calculated in the approximation of Gorski, Bragg, and Williams (for a near-stoichiometric substance AB) [1, 4]. This work gives the corresponding calculation for a f.c.c. L l 2 superstructure. An A P B of ^

(4)

+ -¡rmvi + ij >

pVb = ¿ ( 1 2 - 24 C l + 12 ci Vab

571

V{j

,

(5)

- 1 2 c 5 - i ^ .

C l

There is t h e following obvious relation between the concentrations c0, a n d cv c2, . . ., cn of t h e A-component: : N = c0 (N -

2 n Nq Q) + 2 £ NQ Q a ,

(6)

where c is the mean concentration of the alloy. Making use of (2) t o (6) and t a k i n g into account t h a t for a f.c.c. lattice z = 12, z1 = 6, and z 11 = 3, we find E = [6 N -

(12n

+ 3)NQ n

+ 6 NqQ

/

Q] (CJ _ i

^

«+

1 1 \ — — rfi + Ci ^ -]. x — jg rji rji + 1J v +

+ 3 Nq Q | c j + ^

V{jv

- 6 N [2 c (« A b - w BB ) + ^bb] ,

(7)

where v = 2 vAB — vAA — v B B • The entropy of the configuration (S) is defined f r o m t h e n u m b e r of distinguishable rearrangements of A a n d B atoms on the lattice sites for the given concentrations of components and degrees of order -(N W

=

— 2n NQ Q) N(£ !

x

n

i=1

!

- (N

—2n

NQQ)

N?o)

.(10)

)

( c o - ~ V o ) ( 1 - c0 - — Vo

Vl)

(L "

(ci + ~

Cl +

T

(l - Ci +

(11)

V^j (12)

ln

1 - Ci — — rji

Antiphase Boundaries in Ordered AuCu3-Type Solid Solutions 2 6N - (12 n + 3) NqQ N — 2 n NqQ

1 12

kT v

( Ci + x

2NqQ Cm — N — 2n NqQ

c0 + Ci_i + 2 ^ + c i + i

I ">) (c° ~

(c° +

( Ci ~ Ï

~ c° ~~ T

573

I

VoJ 11

\8 T"4)

t

Vo )

(* ~ c° +

(13)

T

Here i = 1, 2, 3, . . ., n and rji+i = r)0. The equation (12) is not valid for the values i — 1. I t should be replaced for this case by equation (11). In the absence of the A P B (Q = 0) equations (10) and (12) coincide with that of the GorskiBragg-Williams theory. The system (10) to (13) may be solved numerically. For the case of a solid solution of stoichiometric composition (c0 = 0.25) with a small density of A P B (NQ Q/N < 1) the equations (10) to (13) can be written as follows: kT 4

% =

5

i r

kT

rn-x + 2 V i +

Vi+1

l n

(Cl

(1 + 3 Vo) (3 + Vo) ( i - ^ o - s ^ '

+

T'?1)(1

kT (Ci = - - In J

+

~Ci

= 1

k

+

T Vt) (l~Ci j-J-J

\l

Ci_! + 2 Ci + ci+1

(14)

T

V l

+

)

T m)

— ,

(16)

~Ci-TVij

— 4 c0 = T

12 v

3 In (1 + 3 ij„) (1 — Vo)V (l - ^ - 43 4- Mijj) (l - c, +4 -L * i 3 \/ 1 \ Ict + - - raj Iet - — tjij ( 3 - 3 Vo) (3 + >,„)»

'

(17) *

'

Equations (14) to (17) were solved numerically using the electronic computer "Minsk". For every temperature a number of planes n was chosen so that the difference between r)n calculated from (16) and r]0 from (14) was less than 0.001. The results of the calculations are given in Fig. 1 and 2 where the dependence of rji and Ci on temperature are plotted for different i. I t is seen that the longrange order parameter ryt is much smaller on the A P B than in the matrix. As the temperature rises the A P B is widened into a great number of planes. Near Tk the degree of order is essentially different from rj0 in already 10 atomic planes adjoining APB. For the superstructure considered an additional effect is found: the equilibrium concentrations of the components on the A P B differ from the mean values even in alloys of stoichiometric composition.

574

L . E . POPOV,

E . V . KOZLOV,

and

N . S . GOLOSOV Fig. 1. Temperature dependence of the longrange order parameters in the v i c i n i t y of a { 1 1 1 } antiphase boundary in L l 2 superstructure. The figures a t the curves denote the numbers of the atomic planes (counted from the antiphase boundary)

Let us consider one of the consequences of the above effects. In [1] it is shown that in cases when the A P B is connected with superdislocations the processes which make the long-range order parameter on the A P B approach its equilibrium value rj1 < rj0 lead to the pinning of superdislocations. The stress r k necessary to make superdislocations move from an A P B being at thermodynamic equilibrium is, according to [1], given by the equation VO -

=

VI

2 B

(18)

"

where and y0 are the surface energies of an A P B at equilibrium and during the slip of the dislocation, respectively, and 2 & is the Burgers vector of the superdislocation. Using equations (2) to (5) we find rt =

NQ

Z11

V BFO

-

NL)

•

(19)

From the temperature dependence of r k given in Fig. 3 (plotted in units of NQ Z11 V/24 b) one sees that r k maintains a high value up to T k .

Fig. 2. Temperature dependence of the equilibrium concentrations of the components near the a n t i phase boundary

Fig. 3. Temperature dependence of the " s t a r t " stress of the superdislocations under equilibrium conditions

Antiphase Boundaries in Ordered AuCu3-Type Solid Solutions

575

Let us estimate r k for Cu3Au at T — 0.8 Tk. Taking k Tk = 0.82 v (according to the Gorski-Bragg-Williams approximation) we find r k = 4 kp/mm 2 . The real value must obviously be much higher (about 2 times), because the Gorski-BraggWilliams approximation underestimates the values of v and rja. Thus, in the Ll 2 type superstructure the slipping dislocations are pinned rather strongly at equilibrium APB in a wide range of temperatures. I t should be mentioned in conclusion t h a t the decrease of the APB energy which occurs as the degree of order on the APB approaches the value rjx also leads to an increase of the equilibrium width of the superdislocations. References [1] N . BROWN, P h i l . M a g . 4 , 6 9 3 ( 1 9 5 9 ) . [2] H . KIKUCHI a n d W . CAHN, J . P h y s . C h e m . S o l i d s 2 8 , 137 ( 1 9 6 2 ) . [3] L . E . POPOV, E . Y . KOZLOV, a n d N . V . KODZEMYAKIN, D o k l . A k a d . N a u k S S S R 1 5 7 ,

1442 (1964). L . E . POPOV, E . V . KOZLOV, a n d N . Y . KODZEMYAKIN, IZV. V U Z S S S R , F i z . , N o . 1, 1 2 9

(1965). [4] E . V . KOZLOV a n d L . E . POPOV, F i z . M e t a l l o v i M e t a l l o v e d e n i e 1 8 , 9 3 9 ( 1 9 6 4 ) . [5] M. A . KRIVOGLAZ a n d A . A . SMIRNOV, T e o r i y a u p o r y a d o c h i v a y u s h c h i k h s y a s p l a v o v ,

Fizmatgiz, Moskva 1958. (Received August

19,1965)

E. BUDEWSKI et al.: Zweidimensionale Keimbildung an (lOO)-Flächen von Ag

577

phys. stat. sol. 13, 577 (1966) Institut

für

'physikalische

Chemie, Bulgarische

Akademie

der

Wissenschaften

Zweidimensionale Keimbildung und Ausbreitung von monoatomaren Schichten an yersetzungsfreien (100)-Flächen von Silbereinkristallen Von E . B U D E W S K I , W . B O S T A N O F F , T . V I T A N O F F , Z . STOINOFF, A . K O T Z E W A u n d R . KAISCHEW

Elektrochemische Erscheinungen, die das Wachstum von isolierten versetzungsfreien Flächen von Silbereinkristallen begleiten, deuten auf ein Wachstum durch zweidimensionale Keimbildung hin. Die Bildungsgeschwindigkeit von Keimen neuer Netzebenen sowie die Ausbreitungskinetik monoatomarer Schichten über versetzungsfreie Flächen werden näher untersucht. The electrochemical phenomena connected with the electrolytic growth of dislocationfree planes of single crystals of silver indicate that the mechanism of growth is associated with two-dimensional nucleation. The kinetics of the formation of new lattice nets, and the propagation of monoatomic layers are closely investigated. 1. Einleitung

Nach der klassischen Kristallwachstumstheorie ist das Aufwachsen einer neuen Netzebene auf einer intakten Kristallfläche mit der Überwindung der Energieschwelle für die Bildung eines zweidimensionalen Keimes verbunden. Kleine, auf der Fläche abgeschiedene Inseln der neuen Netzebene sind nämlich instabil und zeigen die Tendenz, sich wiederum in die Mutterphase aufzulösen. Nur wenn ein solches Gebilde infolge von thermodynamischen Schwankungen eine bestimmte kritische Größe überschreitet, kann es weiter als „zweidimensionaler Keim" für die Ausbreitung der Netzebene über die ganze Kristallfläche wirken. Die Keimgröße hängt von der Übersättigung ab, indem sie mit Zunahme derselben kleiner wird. Dasselbe gilt auch bei der Elektrokristallisation, nur macht sich in diesem Falle die Übersättigung durch das Auftreten einer Überspannung bemerkbar. Die Aktivierungsenergie für die Entstehung einer neuen Netzebene auf der wachsenden Fläche ist gleich der Arbeit zur isotherm-reversiblen Bildung eines zweidimensionalen Keimes. Diese Arbeit ergibt sich als umgekehrt proportional der Überspannung, so daß sich die Wahrscheinlichkeit für die Bildung eines solchen Keimes als proportional exp (— k'Jrj) erweist [1], Man müßte also einen linearen Zusammenhang zwischen dem Logarithmus der Keimbildungszeit x und 1 ¡r] erwarten [2]: log t = ^ - log

.

(1)

Die Konstante k2 enthält die Keimbildungsarbeit Ak, die aus dieser experimentell bestimmbaren Konstante nach (2) Ak = 2A±Ik2 berechnet werden kann.

578

B U D E W S K I , BOSTANOFF, VITANOFF, STOINOFF, K O T Z E W A u n d

KAISCHEW

Aus der Keimbildungsarbeit Ak kann die spezifische Randenergie x oder die Zahl n, die den Keim bei der gegebenen Überspannung bilden, nach Vj'2

=

^2,3 k T e„ k^l*

(3)

und 1 o

7 2e

(4)

berechnet werden, wo e0 die elektrische Elementarladung und / die Fläche pro Atom in der entsprechenden Ebene bedeuten. Dieser klassische Wachstumsmechanismus einer intakten Kristallfläche findet in zwei nacheinander folgenden Schritten statt: Entstehung eines zweidimensionalen Keimes, welcher sozusagen einatomare Wachstumsstufen erzeugt, die dann durch weitere Anlagerung von Atomen die Ausbreitung der neuen Netzebene über die ganze Fläche ermöglichen. Nach Ausbreitung der Netzebene über die ganze Fläche verschwinden auch die Wachstumsstufen, und der weitere Wachstumsvorgang muß durch erneute Keimbildung eingeleitet werden. Die Notwendigkeit einer Keimbildung bei diesem Wachstumsmechanismus bringt mit sich auch die Notwendigkeit einer verhältnismäßig großen Übersättigung zur Verwirklichung des Wachstumsvorganges. Beim Wachstum aus Gasphase, Lösung oder Schmelze ist die Geschwindigkeit des zweiten Schrittes — die Ausbreitung der Netzebene über die Fläche — viel größer, so daß die Wachstumsgeschwindigkeit der Fläche in Richtung der Flächennormale (die sogenannte lineare Kristallisationsgeschwindigkeit) durch die Bildungsgeschwindigkeit der zweidimensionalen Keime limitiert wird. Wie wir sehen werden, liegen die Verhältnisse bei der Elektrokristallisation offensichtlich umgekehrt. Die lineare Kristallisationsgeschwindigkeit wird durch die Ausbreitungsgeschwindigkeit der Wachstumsstufe über die wachsende Fläche limitiert. Aus diesem Grunde ist auch die Stromstärke bei einem konstanten Potential proportional der Randlänge der sich ausbreitenden neuen Netzebene und unterliegt infolgedessen unregelmäßigen Schwankungen. Wie Frank [3] gezeigt hat, ist auch ein anderer Wachstumsmechanismus möglich, der keine Injizierung von Wachstumsstufen erfordert. Das ist der Fall, wenn die wachsende Fläche durch eine oder mehrere Schraubenversetzungen durchstoßen wird. Die dadurch gebildeten monoatomaren Stufen machen die Bildung von zweidimensionalen Keimen überflüssig, da diese Stufen beim Wachstum nicht verschwinden und dem Wachstumsvorgang dauernd Wachstumsstellen bieten. Wie weiter von Frank gezeigt wurde, windet sich dabei die Wachstumsstufe in eine Spirale ein, wobei der Abstand zwischen den Windungen sich proportional dem Radius des zweidimensionalen Keimes ergibt, welcher der vorhandenen Übersättigung entspricht. Ein solcher Wachstumsmechanismus ermöglicht das Flächenwachstum auch bei sehr kleinen Übersättigungen und wurde auch bei der Elektrokristallisation beobachtet [4], Allerdings können auch bei diesem Wachstumsmechanismus Keimbildungserscheinungen auftreten, wenn zwei Schraubenversetzungen entgegengesetzter Windungsrichtung gemeinsam wirken und der Abstand zwischen denselben größer ist als der Durchmesser des entsprechenden zweidimensionalen Keimes.

Zweidimensionale Keimbildung an versetzungsfreien (lOO)-Flächen von Ag

579

2. Experimenteller Teil Eine Weiterentwicklung der sogenannten Kapillarmethode zum elektrolytiKchen Wachstum von Einkristallen [5] ergab die Möglichkeit, Einkristallelektroden herzustellen, die nur eine einzige kristallographische Fläche der Elektrolyse darbieten [6], Die Methode besteht darin, daß man einen passend orientierten Silbereinkristall elektrolytisch in eine Glaskapillare als Einkristallfaden hineinwächst. Die dafür verwendete elektrolytische Zelle ist in Fig. 1 dargestellt, a ist die Kathode, d. h. der Keimkristall, c die Silberanode und b ein Glasröhrchen, das unten in eine Kapillare endet, wie das noch einmal seitlich in der Figur veranschaulicht ist. Die Zelle ist von unten mit einem planparallelen Glasfenster verschlossen, so daß eine direkte mikroskopische Beobachtung der Front des in der Kapillare wachsenden Einkristallfadens möglich ist. Durch Anwendung eines mit Wechselstrom modulierten Gleichstromes kann man erreichen, daß die wachsende Frontfläche, welche je nach der Orientierung des Keimkristalls eine Würfel-, Oktaeder- oder Rhombendodekaederfläche sein kann, die ganze lichte Weite der Kapillare ausfüllt. Auf diese Weise erhält man einflächige Elektroden, welche sehr geeignet sind für das Studium des Mechanismus der Elektrokristallisationsvorgänge an einzelnen Kristallflächen. Besonders günstig ist dabei die Verwendung von elektrischen Impulsmethoden, welche die elektrolytisch abgeschiedene Metallmenge genau zu dosieren gestatten. Gleichzeitige mikroskopische Beobachtung mit Hilfe einer Interferenzkontrasteinrichtung erlaubt es weiter, die elektrischen Parameter des Wachstumsvorganges mitmorphologischen Fragen zu verbinden.

F i g . 1. D i e e l e k t r o l y t i s c h e Zelle, a K e i m k r i s t a l l , b G l a s k a p i l l a r e , c S i l b e r b l e c h a n o d e , d T e m p e r i e r z w i s c h e n s t ü c k 38

physica

580

B U D E W S K I , BOSTANOFF, VITANOCF, STOINOFF, KOTZEWA u n d

Fig. 4

KAISCHEW

Fig. 5

Fig. 2 bis 5. Verringerung der Oberflächendichte der Schraubenversetzungszentren a n einer kubischen Fläche eines Silbereinkristalls beim W a c h s t u m . Vergrößerung 300fach

Sehr wichtig f ü r das Folgende ist die Tatsache, d a ß m a n unter bestimmten Bedingungen vollkommen schraubenversetzungsfreie Flächen erhalten kann, was durch Kontrolle der Versetzungsdichte mittels kathodischer Wachstumsimpulse bewiesen werden k a n n [6]. Fig. 2 bis 5 illustrieren den Vorgang der allmählichen Verminderung der Versetzungsdichte im Laufe der Herstellung einer solchen versetzungsfreien Würfelfläche sehr deutlich. Versetzungsfreie Flächen besitzen ein ganz spezifisches Wachstumsverhalten, welches auf zweidimensionale Keimbildung hinweist. 3. Wachstum unter galvanostatischen Bedingungen Beim Einschalten eines k o n s t a n t e n W a c h s t u m s s t r o m e s von etwa 0,5 mA/cm 2 steigt die Überspannung auf einen W e r t von etwa 7 bis 12 mV u n d beginnt periodisch zu schwanken (Fig. 6). Diese Schwankungen sind nur f ü r versetzungs-

Zweidimensionale Keimbildung an versetzungsfreien (100)-Flächen von Ag

581

freie Flächenelektroden charakteristisch und verschwinden sofort, wenn aus irgendwelchen Gründen eine mikroskopisch sichtbare Stufe erscheint. Offenbar sind die erwähnten Schwankungen durch Keimbildungserscheinungen bedingt: Beim Einschalten des Wachstumsstromes steigt die Überspannung schnell und erreicht den kritischen Wert, bei welchem die zweidimensionale Keimbildung ausgelöst wird. Die nachfolgende allmähliche Ausbreitung der neuen Netzebene über die wachsenden Fläche verursacht ein Fallen der Überspannung, und zwar in dem Maße, wie die Länge der wachsenden monoatomaren Stufe größer wird. Erreicht schließlich die wachsende Netzebene die Flächenbegrenzungen, so ist die Überspannung wiederum bis zum kritischen Wert angewachsen, und der Vorgang beginnt von neuem.

iiiiniiia

Fig. 6. Schwankungen der Überspannung beim Wachstum einer versetzungsfreien kubischen Fläche unter konstantem Strom, i = 1,9 • 10 4 A / c m 2 ; Abszisse 0,5 s/cm, Ordinate 5,2 mV/cm

Fig. 7. Schwankungen der Überspannung beim Wachstum einer versetzungsfreien kubischen Fläche unter konstantem Strom, i = 3,8 • 10 4 A / c m 2 ; Abszisse 0,5 s / c m , Ordinate 5,2 m V / c m

582

B U D E W S K I , BOSTANOKF, V J T A N O F F , STOINOFF, K O T Z E W A u n d K A I S C H E W

Aus der Periode dieses Schwankungsvorganges und der entsprechenden Stromdichte läßt sich die Dicke der auf der Fläche wachsenden Schicht berechnen. Die experimentellen Daten ergeben eine monoatomare Schicht. Die Steigerung der Stromdichte bedingt eine Verminderung der Schwankungesperiode im umgekehrten Verhältnis (Fig. 7 und 8), was beweist, daß das Flächenwachstum, unabhängig von der Stromdichte, über monoatomare Schichten erfolgt. Es zeigte sich weiter, daß die kritische Überspannung ebenfalls von der Stromdichte unabhängig ist, und die Übersättigung, berechnet aus dieser Überspannung, ergab sich zu etwa 50%. Dieser Wert fällt mit dem allgemein angenommenen Wert der kritischen Übersättigung bei zweidimensionaler Keimbildung gut zusammen. Manchmal, insbesondere bei größeren Stromdichten, breiten sich über die wachsende Fläche sichtbare Stufen aus, welche das oben entworfene Bild gut makroskopisch veranschaulichen. Dies sieht man in Fig. 9, welche zwei Stadien

m PI n ü

Hü

mI i m • mp

-

f

Fig. 8. Schwankungen der Überspannung beim Wachstum einer versetzungsfreien kubischen Fläche unter konstantem Strom, i = 5,8 • 10" 4 A/cm 2 ; Abszisse 0,2 s/cm, Ordinate 5,2 mV/cm

Fig. 9. Dickere Wachstumsstufe über eine versetzungsfreie kubische Fläche in zwei Wachstumsstadien. Vergrößerung 300fach

Zweidimensionale Keimbildung an versetzungsfreien (lOO)-Flächen von Ag

583

Fig. 10. Schwankungen der Überspannung beim Wachstum einer dickeren Wachstumsstufe unter konstantem Strom, i = 5,8 • 10" 4 A/cm 2 ; Abszisse 0,5 s/cm, Ordinate 2,6 mV/cm

der Ausbreitung einer solchen Stufe wiedergibt. Dabei beobachtet man dieselben Stromschwankungen wie die oben beschriebenen (Fig. 10), die jetzt sichtbar mit der Entstehung und Ausbreitung der dicken Schicht über die Fläche verbunden sind. 4. Wachstum unter potentiostatisclien Bedingungen Bei einem konstanten Potential verhalten sich die versetzungsfreien Würfelflächenelektroden anders. Beim Anlegen einer kleinen Klemmenspannung bis zu etwa 7 mV an der Zelle bleibt diese elektrisch gesperrt. Durch die Zelle fließt kein Strom, abgesehen von dem kleinen kapazitiven Strom im Moment des Einschaltens der Spannung. Der Sperrwiderstand der Elektrode ist dabei von der Größenordnung 10 MQ. In diesem Potentialbereich verhält sie sich also wie eine vollkommen polarisier bare Elektrode. Erst wenn die Klemmenspannung einen bestimmten Wert, der im Mittel zwischen 8 und 12 mV liegt, überschreitet, beginnt durch die Zelle ein mit der Zeit schwankender Strom zu fließen. Man sieht sofort, daß dieses Bild vollkommen demjenigen der Bildung zweidimensionaler Keime entspricht. Die Messung der Abhängigkeit der Stärke dieses Stromes von der Zeit wurde in der Weise untersucht, daß man zunächst die Zelle mit einer Spannung von 5 mV sperrte und dann einen Spannungsimpuls überlagerte. Wurde dabei die Dauer des Impulses so gewählt, daß die Keimbildungszeit bei der gegebenen Impulshöhe überschritten wurde, so floß nach dem Ablauf des Impulses so lange ein Strom, bis sich über die ganze Fläche eine neue Netzebene ausbreitete. Dies wurde durch graphische Integration der Strom-Zeit-Kurven bewiesen, die in Fig. 11 wiedergegeben sind. Man sieht, daß diese Kurven bei den einzelnen Versuchen eine ganz verschiedene Form haben. B ? i allen besitzt jedoch das Integral der Stromstärke über die Zeit denselben Wert, und dieser Wert entspricht der Ausbreitung einer vollen Netzebene über die wachsende Würfelfläche. Unter der Annahme einer bestimmten Form des Keimes (bzw. der sich über die Fläche ausbreitenden Netzebene) und einer bestimmten Entfernung seines

584

B U D E W S K I , BOSTANOFF, VITANOFF, STOINOFF, KOTZEWA u n d

KAISCHEW

Fig. 11. S t r o m - Z e i t - K u r v e n an einer versetzungsfreien kubischen F l ä c h e bei einer Sperrspannung von 5 mV. Die m i t einem Pfeil markierten Zeiten entsprechen der Impulsvorgabe. Impulsamplitude 12 m V , Impulsdauer 4 ms

Bildungsortes vom Zentrum der kreisförmigen Würfelfläche läßt sich die Form der Strom-ZeitKurven auf rein geometrischer Grundlage berechnen. Dabei muß allerdings noch die naheliegende Annahme gemacht werden, daß bei einem konstanten Potential die Stromstärke i der Peripherielänge L der sich über die Kristallfläche ausbreitenden neuen Schicht proportional ist: i = kr] L , (5) wobei wiederum r] die Überspannung bedeutet. Die Konstante k kann als eine spezifische Leitfähigkeit (D- 1 cm _ 1 ) des Schichtrandes aufgefaßt werden. Bei der Annahme einer bestimmten, zeitlich konstanten symmetrischen Form der wachsenden Schicht läßt sich die Peripherielänge L als Funktion des Radius r des eingeschriebenen Kreises, L = L(r) , darstellen. Da jedoch die Stromstärke pro Einheit der Peripherielänge konstant ist, ist auch die Ausbreitungsgeschwindigkeit konstant, und somit ist r eine lineare Funktion der Zeit: ze„

(6)

wo / die Fläche, die auf ein Bauelement in der betrachteten Ebene entfällt, bedeutet. Da einerseits die Stromstärke der Peripherielänge der wachsenden Schicht und andererseits auch die Zeit t dem eingeschriebenen Radius r proportional ist, ist die L(r)-Funktion der «(i)-Funktion homomorph und ergibt eine Vorstellung für den Verlauf der Strom-Zeit-Kurve bei der Ausbreitung der neuen Netzebene über die Fläche. Die ¿(r)-Funktion kann nun auf rein geometrischem Wege berechnet werden. Für größere Werte von r, bei denen einer der Ränder der neuen Netzebene die Flächenbegrenzungen zu berühren beginnt, enthält diese Funktion den inneren Radius R der Glaskapillare, die ja in unserem Falle die Kristallfläche begrenzt, sowie die Koordinaten des Entstehungsortes des Keimes. Aus diesem Grunde ist es zweckmäßig, die dimensionslosen Parameter Q = rjR und X=L\R einzuführen. Die /(0 in the whole interval for the first sample, whereas for the diffusion layer this dependence is approximately linear up to £ 0.1. Calculation of the slopes of the straight lines from the data of the table and equations (1) and (2) as well as from Fig. 5 of referenoe (5), gives large differences, namely, formulae (1) and (2) yield k » 0.4, 0.035, and 0.166 and from Fig. 5 we obtain k • 16, 4, and 5.

Short Notes F i g . 2 . Dependence of Rg upon magnetio f i e l d f o r b o t h samples. (E 831 e { H«o iuivalent H a l l c o n s t a n t which i s c a l c u l a t e d from a H a l l voltage at a d e f i n i t e f i e l d s t r e n g t h , Rneq i s the equivalent Hall ct oo nBs t a=n t0 ) e x t r a p o l a t e d

K63

B(H6)

-

Prom curve b i n P i g . 2 as w e l l as from t h e o r y ( 5 ) i t f o l l o w s t h a t t h e H a l l c o n s t a n t becomes s a t u r a t e d i n l a r g e •i f i e l d s and i s e q u a l t o ——, where n i s t h e mean v a l u e of t h e c o n c e n t r a t i o n . Prom P i g . 4 of ( 1 ) i t may be seen t h a t t h e c u r v e s 3-39 and 5 - 1 - 0 s a t u r a t e l i k e w i s e w i t h o u t changing. t h e i r s i g n . This i s expected i f R_ would change i n agreement w i t h e q u a t i o n ( 1 ) of r e f e r e n c e ( 1 ) ( s e e a l s o ( 6 ) ) . Prom t h e s t a n d p o i n t of t h e e q u a t i o n s ( 1 ) and ( 2 ) i n paper ( 1 ) i t i s s u r p r i s i n g t h a t , a l t h o u g h t h e samples 5-2 and 28-1 have a p p r o x i m a t i v e l y t h e same h o l e c o n c e n t r a t i o n 1S —3 of n^ = 3x10 cm , t h e i r Eg dependences on magnetio f i e l d a r e very d i f f e r e n t . The dependences of t h e Hall c o n s t a n t , g i v e n i n P i g . 3 of ( 2 ) , a r e v e r y s i m i l a r t o t h e c u r v e s of P i g . 2 . E s p e c i a l l y t h e Eg dependence of sample 14-7 r e s e m b l e s t o curve b and t h e o t h e r c u r v e s t o curve a of P i g . 2 . A s i m i l a r v e r y s t r o n g dependence of t h e H a l l c o n s t a n t on magnetic f i e l d s t r e n g t h has been l i k e w i s e observed a t 4 . 2 and 77 °K f o r ^ o . o s ^ o . g s 1 ® ' a 1 * 1 "" 1 « 11 » a c c o r d i n g t o t h e d a t a of t h i s work, ¿ E < 0 and has an a b s o l u t e magnitude of about 0.02 eV. Hence, i t f o l l o w s t h a t t h e c o n c l u s i o n s of ( 1 ) do n o t f u l l y agree w i t h t h e e x p e r i m e n t a l d a t a and t h a t , on t h e

K64

physica status solidi 13

contrary, it may be suggested to interpret them as being due to a special inhomogeneity of the material. This suggestion can be easily verified by measuring the galvanomagnetic phenomena in higher magnetic fields and grinding the surface layers or by measuring the galvanomagnetic coefficients in magnetic fields oriented in various directions towards the mentioned gradients. References (1) V.I. IVANOV-OMSKII, B.T. KOLOMBETS, A.A. MALKOVA, V.K. OGORODMIKOV, and K.P. SMEKALOVA, Izv. Akad. Nauk SSSR, Ser. Fiz. 28, 1057 (1964). (2) V.I. IVANOV-OMSKII, B.T. KOLOMBETS, A.A. MALKOVA, V.K. 0G0R0DBIK0V, and K.P. SMEKALOVA, phys. stat. sol. 9, 613 (1965). (3) ff. GIRJAT, Regional Symposium on "Survey of works about the development of semiconductor electronics", Jablonna, September 28 to October 1, 1964. (4) V.K. OGORODNIKOV, private communication. (5) I. HLASBTK, Solid-State Eleotronics 8, 461 (1965). (6) C. HILSTJM and A.C. ROSE-IBBES, Semiconducting III-V Compounds, Pergamon Press, Oxford/London/New York/Paris 1961 (p. 154). (Received October 23, 1965)

Short Notes

K65

phys. stat. sol. 1J., K65 (1966) Institute of Semiconductors, Academy of Sciences of the USSR, Leningrad The Distribution of Spin Density in Paramagnetic Perovskite Crystals By V.S. LVOV and M.P. PETEOV The distribution of spin density in crystals of the ABP^ type with the perovskite structure (A = Na + , Rb + , Tl + ; B = Mn

o.

, Ni

p i

, Co

p,

...) is recently intensively investigated

with the M E method (1,2,3). Our results of spin density f_ 19 (per one bond with a magnetic ion) on the nuclei P and ^Na in MaNiPj can be compared with the f s ~values in TlMnP^ and EbMnP^ (Table 1). The f s ~value on the nucleus A depends essentially on the type of the paramagnetic ions B. The present paper deals with the theoretical explanation of this fact. It shall be noted that with decreasing atomio distances J J the spin density on the nucleus in the row TlMnP--NaNiPo 19 P continuously increases. Besides, the f g -value on the nucleus A in EbMnP^ is greater than in TIMnPj. Therefore one could expect that the f s ~value on the nuoleus A in NaNiP^ would be greater than in other crystals. However, no f_-value po S has been found for Na. The method of MO LCAO (4,5) may be Table 1 The values of spin densities per one bond and the lattice parameters of the primitive cells TN

(°K)

Parameters çf the cells (A)

C*s>P

w

TlffinFj (1)

83

a = 4.25

0.51

- 0.030

EbMnP 3 (2,3)

93

a = 4.14

0.52

- 0.052

149

\ = 3.85

0.56

0 + 0.025

NaJ¥iF3

K66

physica status solidi 13

used to analyse these experimental facts. Ve consider the complex ABgF.,2 which has cubic symmetry. Fig. 1 shows the coordinate axes for eaoh atom. The presence of a spin density on the nuoleus of the ion A depends entirely on the polarization of s-orbitals as this ion has a cubic surrounding. Therefore we shall consider only 1U0 whioh are invariants of the cubic group. We shall take into account 3d-orbitals of paramagnetic ions (dz2, dx2_y2, d x z , d x y , d y z ) and 2porbitals of F (p. Using group theoretical methods x py. p„). z (6) we find the possible invariant combinations of the previously mentioned AO. In our case only three exist:

V . ? •

^-TfrZZrl; ul y~(i ¥ d

j«1

J

*dJ),

where the atoms F and B of a complex are denoted by the indices i and respectively. It is important that s does not belong to MO including d z 2 and d z 2_ y 2 AO of the atom B.

Short notes

K67

Analysing the cubic complex BAgFg we get the same results. Aooording to (1) we find three HO

-1/2 r

$b

. N3,/2[OO9a

We assume that ot,

T

+

^

+

cpF ] .

and Silc « 1. The spin density is due

to the unpaired electron and the polarization of the filled orbitals by this electron. The d_,, d.,„, d„„ orbitals of o. xy yz Hi -ions wioh have an octahedral surrounding are filled* It is natural to believe that the MO ^ b which consists 2+ mainly of this Ni AO is also filled; therefore , in the first approximation no spin density exists upon the HO $ unb In BbHnP where one•.electron is present per eaoh d*y , d , d. ys„ orbital of Mn *, there is one electron on §uno. and a spin density exists on the nucleus Rb. The f s-value may be expressed as a function of the overlap Integrals S ^ and the parameters of oovelenoy a, p, ¡f, the latter depend on the direction of eleotron spin. In this oase we have

This formula indioates that the spin density may be negative as well, which agrees with experiment (Table 1). In this way the experimental results may be Interpreted in the following manner: the nuoleus A interacts with the d^y, d xz' d yz or * )i ' tals o f the i o n B * unpaired electrons are present on these orbitals, a spin density exists on the nucleus A. Otherwise, there is no spin density on the nuoleus A. We are grateful to Prof. G.A. Smolenskii for his interest in this paper and many helpful discussions.

K68 ( 1 ) M.P. 2156 (2) R.E. 3306

physica status s o l i d i

13

References PETROV and G.A. SMOLENSKII, F i z . t v e r d . T e l a J j (1965). PAYEE, R . E . FORMAN, and A. KAHN, J . ohem. P h y s . 42> (1965).

( 3 ) M.P. PETROV, G.A. SMOLENSKII, and P-.P. SIRNIKOV, P i z . t v e r d . T e l a Z> 3699 ( 1 9 6 5 ) . ( 4 ) C. ROOTHAN, R e v . mod. P h y s . 2 3 , 1 ( 1 9 5 1 ) . ( 5 ) R . G . SHULMAN and S. SUGANO, P h y s . R e v . 1 3 0 , 507 ( 1 9 6 3 ) . ( 6 ) L . D . LANDAU and I . M . LIFSHITS, Quantum M e c h a n i c s , Moscow 1963. ( R e c e i v e d December 1 , 1 9 6 5 )

Short Notes

K69

phys. stat. sol. IJj K69 (1966) Advanced Materials Research and Development Laboratory, Pratt & Whitney Aircraft, North Haven, Connecticut On the Temperature and Strain-Rate Dependence of the Plow-Stress of a Solution-Hardened Niobium Alloy By B. HARRIS and D.E. PEACOCK Attempts have been made to describe the deformation of b.c.c. metals in terms of a single deformation meohanism,and it has been shown that the mechanical behaviour of many of these metals appears to fit the general rate equation (1) e « NAbv exp(-H(r,T)/kT), where the symbols have their usual meanings (1). Prom this relationship the stress and temperature dependence of the activation enthalpy H have been deduced with the aid of assumptions of the relative constancy of such variables as the density of mobile dislocations 5 and the vibration frequenoy v> which are inoluded-in the pre-exponential term. Per the metals Nb, Ta, W, Mo, V, and fc-Fe the relationship H ~ 28 kT (2) is fairly well applicable below a characteristic temperature TQ. At T • T Q , H » H Q is the height of the potential barrier to dislocation motion at 0 Conrad (1), Dorn and Rajnak (2), Christian and Masters (3), and others show that the values of H Q for the different metals are consistent with the idea of a single deformation meohanlsm. An analysis such as that of Conrad and Wiedersich (4) implies that a linear relationship exists between the temperature dependence of the flow stress (d«r/9T)£ and the strain-rate dependence of the flow stress T~1 (3 W, and 5 % Mo the temperature dependence of Young's modulus dlnE/dT between 80 and 400 °K is

p h y s i c a s t a t u s s o l i d i 13

K70

T - W K 3.0

•

' ° i 'ft

1.0 55

1 }« •0 30

60

T'2S6°K

65

70

° *

•

75

trl2'10~*s'

P i g . 1 . Change i n f l o w s t r e s s A(r f o r a 1 0 - f o l d change i n s t r a i n r a t e . P u l l symbols denote i n c r e a s e s and open symbols decreases in s t r a i n r a t e

é,=6"10~ss~'

about 7x10 (°K) ^ UUIUJ compared - 55 (°K> ,0^-1 w i t h a v a l u e of 8x10~ 70 75 80 85 f o r pure Nb. In a d d i t i o n , M T-2tt'K t h e product Efc ( a b e i n g t h e ¿ t r U ' l O V 3.0 c o e f f i c i e n t of l i n e a r e x 2M p a n s i o n ) was found t o be 30 35 True stress a, {kg mm') c o n s t a n t over t h e same temp e r a t u r e range ( 6 ) and s i n c e no e l e c t r i c a l o r magnetic anomal i e s could be d e t e c t e d , i t was suggested t h a t t h e s e e f f e c t s a r e simply a consequence of t h e l a t t i o e dynamics of t h e a l l o y . The a c t u a l t e m p e r a t u r e dependence of t h e f l o w s t r e s s of t h e a l l o y does not d i f f e r g r e a t l y from t h a t of pure Nb, but i t can be shown t h a t t h e v a r i a t i o n w i t h t e m p e r a t u r e of t h e f u n c t i o n (o"/E) f o r t h e a l l o y i s almost n e g l i g i b l e above 150 °Z i n comparison w i t h t h a t of Nb ( 5 ) . Unless, t h e r e f o r e , t h e f u n o t i o n (8o-/dln6)j v a r i e s l i n e a r l y with temperature, equation ( 2 ) cannot be s a t i s f i e d . i

• •

•

o

1—

°

*

» »

.

9

,

.

±

» .

« A U •

i

Using t h e m a t e r i a l and methods d e s c r i b e d p r e v i o u s l y ( 5 ) , s t r a i n - r a t e change and temperature-change experiments were c a r r i e d out on t h i s a l l o y , t h e u s u a l p r e c a u t i o n s being t a k e n t o aohieve e q u i l i b r i u m a f t e r t e m p e r a t u r e changes and t o avoid machine e f f e o t s a f t e r s t r a i n - r a t e changes. The r e s u l t s are shown i n P i g s . 1 and 2 . I t appears from P i g . 1 t h a t t h e change i n f l o w s t r e s s Ac f o r a 1 0 - f o l d change i n s t r a i n r a t e i s a f u n c t i o n of s t r e s s o n l y . The mean v a l u e of t h e r a t i o «¿/«r, =* « 1.037, t h e s t a n d a r d d e v i a t i o n f o r 52 measurements a t a l l t e m p e r a t u r e s and s t r a i n r a t e s being 0.004 ( f f = A d ) . The cons t a n c y of (Tg/ff-j i s e q u i v a l e n t t o constancy of t h e r a t i o Aff/ff^, i . e . t h e change i n f l o w s t r e s s i s p r o p o r t i o n a l t o t h e a p p l i e d

K71

Short Notes F i g . 2 . Change i n f l o w s t r e s s w i t h changes i n t e m p e r a t u r e A v a s f u n c t i o n of s t r a i n

0.25 / A T

T-=WK

Û.20 •

s t r e s s , which i s an e x p r e s s i o n of t h e C o t t r e l l - S t o k e s law ( 7 ) . The r e l a t i o n s h i p

% S

0.15

Ao/c = c o n s t i s independent of t e m p e r a t u r e , which i s not t r u e of pure b . o . c . T=360°K m e t a l s . I t can be seen from F i g . 3 ° Temperature raised t h a t t h e v a l u e s of A a / A l n i from 0.05 • Temperature lowered t h e s e e x p e r i m e n t s agree w e l l w i t h T,*l 2 v a l u e s o b t a i n e d from c o n s t a n t s t r a i n - r a t e t e s t s in reference 60 70 ( 5 ) . The ohanges i n f l o w s t r e s s True stress Eg + Ep may be expressed by 1' 06*. _A_(hv» - EQ - E ho V 9 P

n

) 2 2kT /

O)

(Eg energy gap, E^ phonon energy, T absolute temperature). The experimental data of ( ot •hv)1/2 vs. hv for different temperatures are plotted in Pig. 1a. Over a certain energy interval the experimental points form a straight line to a good approximation. The point of intersection of this line with the energy axis allows to obtain the value of the energy gap E^ . E ^ is plotted in Pig. 2 as a function of temperature. At low temperatures QE^/flT is in good agreement with earlier data (1). However, 1) It is assumed that the matrix elements for emission and absorption of phonons are equal.

physioa status solidi 13

K92

/

0 o o 0 . °/ 0/ o/ o/ y

s 60

o o

^/

oc, o 0/

J S

/I

o

o / ooo///

/è-

20

o/

0

s

oo v

° /

o

80

J

W

2.0

22

o

o / o/

MJÌ

A / y

Af

7

y

o/ 2À

2.6 hv(eV)-

2.8

Pig. 1. Absorption edge of GaP at various temperatures the value of E ^ at 0 °K in our measurements is equal to (2.31 + 0.005) eV, which is somewhat lower than in (1). At higher temperatures the energy gap varies linearly with the coefficient 9^/aT = (5 + 0.25)x10 - 4 eV, in accordance with Folberth's data (5). One may expect a deviation from linearity at hv-Eg^O.1 eV due to the beginning of transitions from the third valence band which is split off by spin-orbital interaction. However, in our experiment such a deviation was hot observed. It is seen (Pig. 1a) that at hV - E g i 2 0.3 eV the experimental points begin to deviate from linearity. Supposing that this deviation is caused by transitions to the higher branches of the conduction band and that the bands remain parabolic, the

K93

Short Notes P i g . 2 . D i r e c t and i n d i r e c t energy gap as a f u n c t i o n of temperature

absorption c o e f f i c i e n t f o r t h e new t r a n s i t i o n s may be o b t a i n e d by means of s u b t r a c t ing (tt- a , ) where corresponds t o the s t r a i g h t l i n e in P i g . 1a. as a f u n c t i o n of hv i s shown i n P i g . 1b. The a b s o r p t i o n edge of t h e second t r a n s i t i o n s ( E ^ ) was d e t e r m i n ed by e x t r a p o l a t i n g t h e l i n e a r p a r t t o t h e energy a x i s . ^ The v a l u e s of E„ a t d i f f e r e n t temp e r a t u r e s are shown i n P i g . 2. 3 . At a c e r t a i n photon energy a s h a r p r i s e of a b s o r p t i o n due t o d i r e c t t r a n s i t i o n s was observed ( P i g . 3 ) . At low t e m p e r a t u r e s (T à 233 °K) t h e a b s o r p t i o n c o e f f i c i e n t i n c r e a s e s t o a peak, f a l l s s l i g h t l y , and t h e n c o n t i n u e s t o r i s e more slowly a g a i n . This peak i s a broadened l i n e due t o d i r e c t t r a n s i t i o n s i n t o an e x c i t o n s t a t e . For allowed d i r e o t t r a n s i t i o n s t h e a b s o r p t i o n c o e f f i c i e n t i n t h e r e g i o n of c o n t i n u o u s a b s o r p t i o n i s g i v e n ( 6 ) by

rv^cnto where

Z

.

( - i r ^ - ) 3 ' 2 1IPt v l I 2 \ * 1

I E«

^

s,MhZ

»

htf6 E0 ,

(2)

V « E q ). When [ln(1-AO)]~"2 is plotted vs. hv equation (3) yields a series of straight lines with slopes depending only on B g x . The exciton binding energy has been found by plotting the experimental values in such a diagram. The minimum absorption in the "valley" is taken to be & 0 . The exciton binding energy is equal to (0.006 + 0.001) eV at T » 77.3 °K and decreases by about 30 £ as the temperature inoreases to T » 233 °K. 3) A more accurate analysis of the experimental data (3) leads to values of the exciton binding energy at T • 153 to 233 whioh are somewhat lower than those deduoed in (3).

K95

Short Notes Pig. 4. Absorption coefficient of GaP up to 3 eV at T = 77.3

^

Ï 3-10*

The direct energy gap EQ was determined from the relationship EQ = " tem P e " ratures above 233 °K the exciton absorption peak is not observed because of the strong broadening of the exciton line. Therefore it is not possible to determine E q with sufficient accuracy. In Pig. 2 B Q is plotted as a function of temperature. At low temperatures E q is desoribed (3) by

4 210

r-m*

23

3.0 hv(eV)

•

E q = [2.885 + 0.005 - (1.25 + 0.05)xT 2 x10" 6 J eV. At higher temperatures E Q varies linearly with the coefficient ôE0/ôTB (6.5 + 0.55)x10" 4 Comparison of 8E^/3T and 3E0|9T shows that the conduction band minimum at the centre of the Brillouin zone moves faster with temperature than the absolute minimum X^ . This conclusion is opposite to that of Zallen and Paul (2). Pig. 4 shows the spectral dependence of the absorption coefficient for direct transitions at 77.3 °K over a wide energy range. The absorption edge due to transitions from the third valence band (split off by spin-orbital coupling) to the oonduotion band is clearly visible in the region of 2.93 eV. By inspection of Pig. 4 one can estimate the spin-orbital splitting to be A 0 = (0.09 + 0.01) eV. References (1) M. GEBSHENZON, G.S. THOMAS, and R.E. DIETZ, Proc. Intern. Conf. Semicond. Phys., Exeter, July 1962 (p. 752).

£96

physica status solid! 13

(2) R. ZALLEN and W. PAUL, Phys. Rev. 134, A1628 (1964). (3) V.K. SDBASHIEV and G.A. CHALIKYAN, Fiz. tverd. Tela Jj 1237 (1965). (4) V.K. SUBASHIEV and S.A. ABAGYAN, Proc. Intern. Conf. Semicond. Phys., Paris, July 1964 (p. 225). (5) O.G. POLBERTH, Z. Naturf. 10a, 502 (1955). (6) R.J. ELLIOTT, Phys. Rev. 108, 1384 (1957). (Received December 28, 1965)

Short Notes

K97

phys. s t a t . s o l . 12, K97 (1966) Institute of Physics, Bulgarian Academy of Sciences, Sofia Optioal Transitions in Crystalline and Vitreous AsoS^ By G r . GETOV, B. KANMLAROV, P. SIMIDTCHIEVA, and R. AMDREYTCHIN Interesting results as regards the dependence of both the band structure and the type of electron transitions in solids on their crystalline or vitreous state, i . e . on the existence of long-range or short-range order, may be expected from a comparison of the optical absorption in the absorption edge region of crystalline and vitreous AS2S3. Recently Gorban and Dashkovskaya ( 1 ) established the presence of indirect transitions in AsgS-j c r y s t a l s . For the energy of phonons they obtained the value of 0.04 eV ( 8 = 465 °K), without, however, specifying whether these are optical or acoustical phonons. Moreover, these authors note that the values f o r the absorption coefficient obtained by them have been v i r t u a l l y increased because of the presence of structural defects and mechanical damages in the crystals which they used. The optical absorption in the absorption edge region f o r vitreous ASgS^ at room temperature has been studied by Zorina ( 2 ) . Referring to a theoretical work of Almazov ( 3 ) , she considers that certain analogies between the crystalline and amorphous states give s u f f i c i e n t ground to apply the concepts and c r i t e r i a f o r electron transitions in crystals to vitreous ASgSj. Almazov ( 3 ) , however, e x p l i c i t l y stresses that the f o r mulas deduced by him apply only to the calculation of the thermodynamic properties of a fermion system, and that i t is an approximation valid only on the assumption that a number of limiting conditions have been f u l f i l l e d ( f o r instance, parabolic type of dependence of energy on quasi-impulse). Moreover, he insists that the kinetic and two-band problems in his work are discussed only qualitatively and tentatively. But even i f we agree with Zorina that the problem of the role of phonons in electron transitions and of the presence of

K93

p h y s i o a s t a t u s s o l i d i 13

i n d i r e c t t r a n s i t i o n s i n t h e oase of amorphous s o l i d s can be solved i n t h e same way a s i n t h a t of c r y s t a l s , a d i s c u s s i o n of t h e r o l e of phonons would presuppose measurements of t h e a b s o r p t i o n c u r v e s i n t h e v i c i n i t y of t h e end of t h e fundament a l a b s o r p t i o n edge, not only a t room but a l s o a t low temperatures. The purpose of t h e p r e s e n t paper i s t o check t h e p r e s e n ce of i n d i r e c t t r a n s i t i o n s i n v i t r e o u s AS2S3, and t h u s i n d i r e c t l y t o check t h e analogy a p p l i e d i n ( 2 ) . F u r t h e r m o r e , f o r t h e sake of comparison and i n view of c e r t a i n r e s e r v a t i o n s i n t h e work of Gorban and Dashkovskaya ( 1 ) on t h e i n f l u e n c e of d e f e c t s i n t h e c r y s t a l s t r u c t u r e , we a l s o r e p e a t e d t h e i r i n v e s t i g a t i o n s of As 2 S^ c r y s t a l s . We made use of n o n - p o l a r i z e d l i g h t , a s t h e r e i s no major d i c h r o i s m of t h e band-gap energy

(1).

The o p t i c a l a b s o r p t i o n i n t h e v i c i n i t y of t h e a b s o r p t i o n edge of v i t r e o u s ASgS^ was measured at room t e m p e r a t u r e and a t l i q u i d n i t r o g e n t e m p e r a t u r e . From t h e square of the a b s o r p t i o n c o e f f i c i e n t we o b t a i n e d a band-gap energy w i t h d i r e c t t r a n s i t i o n s of 2.21 eV a t room t e m p e r a t u r e and of 2.35 eV a t liquid nitrogen temperature. 1 /P The dependence of K ' on t h e energy of f a l l i n g photons for vitreous i s shown i n F i g . 1 . In our measurements t h e a b s o r p t i o n c o e f f i c i e n t ohanged from 150 t o 1 c m - 1 , being l i m i t e d only by t h e t h i c k n e s s of t h e specimen and t h e c o n d i t i o n s of measurement. There i s no t e m p e r a t u r e dependence of t h e shape of t h e c u r v e s , a l l t h e s t e p s a r e s t i l l p r e s e r v e d , not a s i n g l e of them being ' f r o z e n ' a t l i q u i d n i t r o g e n temp e r a t u r e . Hence, i f we assume t h a t i n t h e case of amorphous s o l i d s t h e q u e s t i o n of t h e n a t u r e of t h e e l e c t r o n t r a n s i t i o n s ( i n t h e sense of whether t h e y a r e d i r e c t or i n d i r e c t ) and of t h e p a r t i c i p a t i o n of phonons i s solved in t h e same manner a s i n t h e c a s e of c r y s t a l s , we do not have s u f f i c i e n t ground t o a c c e p t a s a u t h e n t i c Z o r i n a ' s ( 2 ) a s s e r t i o n about t h e presence of i n d i r e c t t r a n s i t i o n s w i t h 0 . 1 3 eV phonons i n v i t r e o u s AS2S3 The t e m p e r a t u r e dependence of t h e a b s o r p t i o n curve i n

Short Notes

2 J 0 0

2 3 0

1 2 0

2 3 0

Eie!/)

Pig. 1.

I W

2 3 0

K99

2 A 0

1 5 0

•

2 £ 0

2.70

EleV!

2 8 0

•

Fig. 2

Pig. 1. Vitreous ASgS^. 1 - T 293 °K; 2 - T = 90 °K Pig. 2. Crystalline As 2 S 3 . 1 - T = 293 UK; 2 - T = 90 UK the absorption edge region for vitreous As2S^ i s expressed by i t s displacement towards shorter wavelengths with a decrease in temperature. The mean value for the temperature coefficient of the energy band gap in the examined temperature range i s - 6.7x10"^ eV/deg. This value for vitreous AS2SJ found by us i s close to that for c r y s t a l l i n e ASgS^ ( - 7x10"4 eV/deg according to our measurements and - 10x10~4 eV/deg a f t e r ( 1 ) ) . Measurements were also made on the absorption in the absorption edge region for ASgS^ single c r y s t a l s . As can be seen from Pig. 2, the value obtained by us for the phonon energy i s in good agreement with the r e s u l t s of Gorban and Dashkovskaya CO ( - 0.04 eV). Acknowledgements are due to Prof. I . Kostov of Sofia University for the c r y s t a l s he kindly placed at our disposal, as well as to Mrs. M. Nikiforova for synthesizing the vitreous material.

K100

physica status solidi 13

References (1) I.S. GORBAN and R.A. DASHKOVSKAYA, Piz. tverd. Tela 6, 2339 (1964). (2) B.A. ZORINA, Piz. tverd. Tela 7, 332 (1965). (3) A.B. ALMAZOV, Piz. tverd. Tela 1320 (1963). (Received December 28, 1965)

Short Notes

K101

p h y s . s t a t . s o l . 13^ K101 (1966) Department of Solid S t a t e P h y s i c s , F a c u l t y of Mathematics and P h y s i c s , Charles U n i v e r s i t y , Prague High-Temperature E l e c t r i c a l C o n d u c t i v i t y of Cadmium T e l l u r i d e By P. HOSCHL Some s i n g l e c r y s t a l s of cadmium t e l l u r i d e p r e p a r e d by d i v e r s e methods and v a r i o u s a u t h o r s have a very h i g h p - t y p e Ft ft r e s i s t i v i t y of 10 t o 10 Q, cm a t room t e m p e r a t u r e . This shows t h a t cadmium t e l l u r i d e oan be c l a s s i f i e d as a " s e m i - i n s u l a t o r " . Allen ( 1 ) and Gooch, Hilsum, and Holeman ( 2 ) e x p l a i n e d s i m i l a r p r o p e r t i e s of g a l l i u m a r s e n i d e a s a compensation of s h a l low donor l e v e l s by deep a c c e p t o r l e v e l s . Our measurements of t h e c o n d u c t i v i t y of cadmium t e l l u r i d e samples were made i n t h e t e m p e r a t u r e range 300 t o 1350 °Z, and t h e a c t i v a t i o n energy of t h e a c c e p t o r l e v e l s and t h e r m a l band gap were d e termined. The h i g h - t e m p e r a t u r e measurements were made on samples cut o f f from t h e ends of i n g o t s p r e p a r e d e i t h e r by Bridgeman's v e r t i c a l method ( 3 ) or by zone m e l t i n g ( 4 ) . Cadmium and t e l lurium w i t h a n a l y s i s c e r t i f i c a t e s of 99.999 % p u r i t y were used f o r t h e c r y s t a l p r e p a r a t i o n a f t e r d i s t i l l a t i o n i n h y d r o gen t o remove s u r f a c e o x i d e s . Measurements were c a r r i e d out in the high-pressure vessel described in (5), at a pressure of 100 atm N2 t o p r e v e n t r a p i d s u b l i m a t i o n of t h e sample a t h i g h t e m p e r a t u r e s . The d . c . p o t e n t i a l drop was measured by means of an e l e c t r o m e t e r and g r a p h i t e e l e c t r o d e s a t t a c h e d 3 to t h e sample. The dimensions of t h e sample were 12x5x5 mm . Ohm's law was v a l i d f o r the g r a p h i t e e l e c t r o d e s over t h e whole t e m p e r a t u r e range 300 t o 1350 °K. The t e m p e r a t u r e was measured by means of a Pt—PtRh thermocouple w i t h an accuracy of + 3 °C. Measurements on samples prepared by t h e Bridgeman t e c h n i q u e (B) and by zone m e l t i n g (Z) are shown i n F i g . 1 . The i n d i c e s 1 and 2 r e f e r t o subsequent measurements on sample B. The d i f f e r e n c e between t h e r e s u l t s on t h e same sample i s caused by changes of s u r f a c e c o n d u c t i v i t y , as was l a t e r shown

physio*. status solidi 13 Pig. 1. Temperature dependence of the resistivity of p-type cadmium telluride in the temperature range 300 to 1350 °K by measurements made after grinding off the surface layer. Similar curves have been measured on oadmium telluride samples by Appel (6) and Appel and Lautz (7). The scattering of carriers in oadmium telluride is caused by the interaction of carriers with longitudinal optical phonons and ionized impurities (8,9). In the temperature range studied where T =» 6 ( 0 ® 247 °K (10) Debye temperature) the scattering is caused only by longitudinal optioal phonons. The temperature dependence of the hole mobility is only weak and may be expressed by ,Up « T V 2 [e8/T-1] . (The Howarth-Sondheimer function Gie~S/k°T (11) is nearly equal to unity.) Assuming that the semiconductor is not degenerate and that p N a - lijj and p ffj), the carrier oonoentration p in the temperature range 300 to 600 K may be expressed by equation (1) or (3), depending on whether the semiconductor is internally compensated or not: J A ^ W

3

'

N

vexP

[-EA/*OT].

( 1 )

Short Notes

P- ( J

a

n

ANV)

1 / 2

exp E - E A / 2 ! c o T ] .

K103

(3)

i s the f a c t o r of spin degeneracy of the acceptor l e v e l . In order to decide whether the semiconductor i s compensated or not an estimate of the pre-exponential f a c t o r in relationships ( 1 ) and ( 3 ) was made, assuming that saturation ooours in the temperature region of about 600 °K and that ¿Up(T)«^lp(300 °K) 10 2 cm2/Vs and = 0.35 mQ. I t was shown that the semiconductor i s compensated and has an acceptor A C

concentration NjjWlO 15 cm""3. Assuming that the perature dependence of OL may be expressed by 0E « T2

O

om"J and a donor concentration semiconductor i s compensated} the temthe e x t r i n s i c e l e c t r i c a l conductivity the following r e l a t i o n s h i p :

[ e 0 / T - L ] exp [ - E A / V 0 T ] .

(4)

Calculation of the acceptor a c t i v a t i o n energy gives E^ => 0.66 eV. By comparing the r e s u l t s of our measurements with those of other authors ( 6 , 7 ) we i n f e r that cadmium t e l l u r i d e e x h i b i t s i n t r i n s i c conductivity in the temperature range 820 to 1350 °K. The temperature dependence of the i n t r i n s i c conductivity 0t i s expressed by the r e l a t i o n 0 t ~ T 2 ¡ > 0 / T - l ] exp [ - E g / 2 J c 0 T ] .

(5)

Prom here we obtain E^ = 1.59 eV, in good agreement with values computed from measurements of the o p t i c a l absorption edge and of photoconduotive e f f e c t s . Considering that samples prepared by various methods and d i f f e r e n t workers, using elements of various o r i g i n , give very similar r e s u l t s , we expect that deep acceptors are

K104

physica status solidi 13

self-compensated by shallow donors, produced by electrically active intrinsic point defects or more complicated associates of such defects. The author is very much indebted to Prof. E. Klier and Dr. R. Kuzel C.Sc. for their valuable advice and helpful criticism. References (1) J.W. ALLEN, Nature 187. 403 (1960). (2) C.H. G00CH, C. HILSUM.and B.R. HOLEMAN, J. appl. Phys., Suppl. ¿2, 2069 (1961). (3) P. HOSCHL, Thesis, Charles University, Prague 1961. (4) E. SUBERTOVA, Thesis, Charles University, Prague 1964. (5) (6) (7) (8)

P. HOSCHL and C. KONAK, Czech. J. Phys. B13, 437 (1963). J. APPEL, Z. Naturf. 9a, 265 (1954). J. APPEL and G..LAUTZ, Physica 20, 1110 (1954). N. SEGA1L, M.S. LORENZ,and R.E. HALSTED, Phys. Rev. 129, 2471 (1963).

(9) J. GINTER, phys. s+at. sol. 6j 863 (1964). (10) R.E. HALSTED, M.R. LORENZ, and B. SEGALL, J. Phys. Chem. Solids 22, 109 (1961). (11) D.J. HOWARTH and E.H. SONDHEIMER, Proc. Roy. Soc. A 219, 53 (1953). (Received December 30, 1965)

Short Notes

K105

p h y s . s t a t . s o l . 1^, K105 (1966) S i b e r i a n P h y s i c o - T e c h n i o a l I n s t i t u t e , Tomsk I n v e s t i g a t i o n of Work-Hardening of t h e Ordered Alloy Ni^Mn By L.E. POPOV, E.V. KOZLOV, and N.A. A1EKSAMR0V A long e x t e n s i o n of t h e l i n e a r work-hardening s t a g e ( s t a g e I I ) i s c h a r a c t e r i s t i c of t h e s t r e s s - s t r a i n c u r v e s of ordered Cu^Au s i n g l e c r y s t a l s ( 1 , 2 , 3 ) . This f a c t was f i r s t noted by- Seeger ( 4 ) . The t r u e s t r e s s - c o n v e n t i o n a l s t r a i n c u r ves of p o l y c r y s t a l s of t h i s a l l o y show a good developed l i n e a r p o r t i o n which e x t e n d s from t h e y i e l d p o i n t t o h i g h s t r a i n v a l u e s (about 15 p e r c e n t ) ( 5 , 6 ) . I t seems r e a s o n a b l e t o a s s o c i a t e t h e h i g h s t r e s s a t which s t a g e I I I b e g i n s i n t h e o r dered a l l o y w i t h t h e d i f f i c u l t y of double c r o s s - s l i p i n t h i s case (such a s u g g e s t i o n was put forward r e c e n t l y ( 7 ) f o r PeCo a l l o y s w i t h B2 s u p e r l a t t i c e ) . A f t e r t h e c r o s s - s l i p of one of t h e s u p e r p a r t i a l d i s l o c a t i o n s t h e f u r t h e r motion of b o t h of them r e q u i r e s t h e a d d i t i o n a l shear s t r e s s y / b n e c e s s a r y t o c r e a t e a n t i p h a s e b o u n d a r i e s . The p a r t i c i p a t i o n of s u p e r p a r t i a l s i n t h e d e f o r m a t i o n p r o c e s s must be d i s p l a y e d i n t h e p l a s t i c behaviour of t h e a l l o y as w e l l as t h e b e g i n n i n g of double c r o s s - s l i p i n pure f a c e - c e n t r e d cubic m e t a l s (change i n t h e work-hardening r a t e (9 t o 1 1 ) , appearance of c o a r s e s l i p bands ( 4 , 8 , 1 0 ) , e t c . ) . In a d d i t i o n , t h e c r e a t i o n of a n t i p h a s e b o u n d a r i e s by t h e motion of t h e s u p e r p a r t i a l s must lead t o a s i g n i f i c a n t i n c r e a s e of t h e e l e c t r i c a l r e s i s t a n c e of t h e a l l o y . In t h e p r e s e n t paper t h e work-hardening of p o l y c r y s t a l s of t h e ordered a l l o y Ni^Mn ( w i t h a s u p e r l a t t i c e of Cu^Au t y p e ) o i s i n v e s t i g a t e d i n t h e t e m p e r a t u r e range from - 190 C t o t h e Curie p o i n t f o r atomic o r d e r (505 ° C ) . The g r a i n s i z e was between 0.12 and 0.017 mm. The ordered s t a t e ( w i t h a degree of o r d e r about 0 . 9 ) was o b t a i n e d by slowly c o o l i n g t h e specimens from 525 t o 300 °C d u r i n g 45 d a y s . The e x p e r i m e n t a l methods were d e s c r i b e d i n d e t a i l in (12).

K106

p h y s i c a s t a t u s s o l i d i 13 F i g . 1 . Dependence of some p r o p e r t i e s of t h e a l l o y Ni^Mn on c o n v e n t i o n a l s t r a i n (complete o r d e r ) : 1 - t r u e flow s t r e s s , 2 - work-hardening r a t e , 3 - r e l a t i v e change of electrical resistance, 4 - mean l e n g t h of s l i p lines

20

30 e/Xi-

I t was found t h a t t h e t r u e s t r e s s - c o n v e n t i o n a l s t r a i n c u r v e s can be d i v i d e d , a t a l l t e m p e r a t u r e s , i n t o t h e f o l l o w i n g stages: 1. yielding (stage I ) , 2 . t h e s t a g e w i t h a n e a r l y c o n s t a n t work-hardening r a t e •3>=-||- e x t e n d i n g t o a s t r a i n of 25 t o 30 p e r c e n t ( s t a g e I I ) , 3 . t h e s t a g e w i t h d e c r e a s i n g •& ( s t a g e I I I ) . Typical c u r v e s 0 = f ( e ) and & = f ( e ) a r e r e p r e s e n t e d i n P i g . 1 ( c u r v e s 1 and 2 ) . The e l e c t r i c a l r e s i s t a n c e i n s t a g e I I I i n c r e a s e s w i t h i n c r e a s i n g s t r a i n much f a s t e r t h a n i n t h e s t a g e of l i n e a r work-hardening ( P i g . 1, curve 3 ) . This i s i n good agreement w i t h t h e above assumption a c c o r d i n g t o which t h e s u p e r p a r t i a l s play a s i g n i f i c a n t part in p l a s t i c flow in stage I I I , while i n s t a g e I I p l a s t i c d e f o r m a t i o n i s due t o t h e motion of s u p e r d i s l o c a t i o n s which produce a small amount of new a n tiphase boundaries. Since t h e energy of t h e a n t i p h a s e boundary i s proport i o n a l t o t h e square of t h e l o n g - r a n g e o r d e r p a r a m e t e r , one would expect t h a t t h e s t r e s s a t which s t a g e I I I b e g i n s ( O J J J ) d e c r e a s e s w i t h d e c r e a s i n g degree of l o n g - r a n g e o r d e r .

Short Notes

K107

1.2

Izff 0.8

0£

OU 02 0

iOO TtX)

600 •

Fig. 2. Temperature dependence of some properties of the alloy Ni-jMn 1 - equilibrium resistivity, 2 - long-range order parameter (aocording to (13))» 3 - stress

vs. quenching temperature (the measu-

rements were carried out at room temperature), 4 - stress

G J J J

vs. temperature of measurement

Indeed, as is seen from Fig. 2,

OJJJ

at room temperature

does not depend on quenching temperature as long as the degree of order remains unchanged, and quickly decreases in the temperature range where the long-range order decreases. Surface slip patterns after various strains were observed by light microscope on flat specimens having degrees of order 0.9 and 0.7. After various amounts of strain the specimens were repolished and, in addition, deformed by 3 to 5 per cent. At all strains in stage'II fine uniformly distributed slip lines were observed. As the end of stage II approaches separate coarse slip bands appear in regions of stress concentration. In stage III coarse slip bands lying far from each other and traces of cross-slip are observed in all the grains. In stage III the length of the slip lines decreases with increasing

K108

physica status solidi 13

strain more slowly than in stage II (Pig. 1). All observed changes of the slip patterns are similar to those taking place in pure face-centred cubic metals at the stress at which stage III begins (8). Thus the end of the stage with high linear work-hardening is associated with the beginning of double cross-slip. The critical stress for this mechanism in the case considered is the stress required for the motion of superpartials with the creation of antiphase boundaries, but not the stress of the " .nstriction" of superdislocations, as was suggested by Seeger (4). In conclusion we wish to note that the identical feature of the plastic behaviour of the alloy FeCo with the superlattice B2 which, when disordered, is body-centred cubic were explained by Marcinkowski and Chessin in a similar way (7). References (1) G. SACHS and J. WEERTS, Z. Phys. 6JJ 507 (1931). (2) B.H. KEAR, Acta metall. 12, 555 (1964). (3) E.G. DAVIES and N.S. STOLOFF, Phil. Mag. 9, 349 (1964). (4) A. SEEGER, Hdb. Phys., Vol. VII/12, Springer-Verlag 1958 (p. 188). (5) L.I. VASILEV, CHEM HUN-I, IN DAO-LO, and EE YUE-KUAN, Izv. VUZ SSSR, Fiz.,No. 6, 48 (1959). (6) V.I. SYUTKINA and E.S. YAXOVLEVA, Fiz. tverd. Tela 4,

2902 (1962). (7) M.J. MARCINKOWSKI and H. CHESSIN, Phil. Mag. 10, 837 (1964). (8) A. SEEGER, Dislocations and Mechanioal Properties of Crystals, Wiley 1957. (9) E. MACHERAUCH and 0. V0HRINGER, phys. stat. sol. 6, 491 (1964). (10) G. ZANEL, Z. Naturf. 13a, 795 (1963). (11) CH. SCHWINK, phys. stat. sol. 8, 457 (1965). (12) I.E. POPOV, E.V. KOZLOV, NoA. AIEKSANDROV, and S.T. STEIN, Fiz. Metallov i Metallovedenie, in press. (13) M.J. MARCINKOWSKI and N.J. BROWN, J. appl. Phys. ¿2, 375 (1961). (Received January 4, 1966)

Short Notes

K109

phys. stat. sol. 13, K109 (1966) Institute of Physics, Academy of Sciences of the Azerbaidshan SSB, Baku Effect of Admixtures on Some Physical Properties of Selenium By G.B. ABDULIAEV, G.M. A1IEV, S.I. MEXHTIEVA, and D.SH. ABDINOV With the addition of different admixtures of selenium a minimum or a maximum is observed in the concentration dependence of electroconductivity (0), thermal conductivity (A.), density (ç), and miorçhardness (H) (1 to 3). It has been shown (3) that this fact is related to the formation of admixture complexes with oxygen. Thus, it seems to be interesting to reveal whether there is some correlation between the variations of (J, A., Q , and H for selenium with different admixtures and their affinity for oxygen. Pig. 1 shows variations of "k with the addition of T1 (curve 1), Bi (curve 2), Sb (curve 3), Cd (curve 4), and S (curve 5). The ionization potentials U of these admixtures are 6.11, 7.30, 8.64, 8.99, and 10.36 eV, respectively. Prom Fig. 1 it is clear that with the growth of U the admixture concentrations at which A. reaches its minimum shift towards higher values. This behaviour is also observed with Na, In, and Mn admixtures in selenium. The shift of the minimum towards higher admixture concentrations with the growth of their U values occurs also for ç , H, and 0 . In Pig. 2 the ratios and ^ p ^ are shown as a function of the ionization potential U for thé admixture introduced. 6 is the electroconductivity of pure selenium, and e ec ®min ^ ^ ® max a r e ^ "';roconductivities corresponding to the minimum and maximum in the concentration dependence of 6 (a similar relationship will result if instead of U the electronegativity is 'taken for the admixture introduced). It is shown by Pig. 2 that Tl, Na, Sb, Mn, and other admixtures with an ionization potential less than that of selenium ^Se^ decrease the value of G which passes a minimum at some concentration.

mo

physica status solidi 13

Pig. 1. Thermal conductivity for various admixtures to selenium. 1 - Se + Tl, 2 - Se + Bi, 3 - Se + Sb, 4 - Se + Cd, 5 - Se + S The effect of admixtures on 0 decreases with decreasing difference between Ug e and Ua(j, and simultaneously the minimum in the concentration dependence of Cf shifts to higher concentrations. In the case of U ^ > Ug e (CI, Br, J, 0, etc.) the value of 0 for selenium increases with the growth of the admixture concentration and passes a maximum. The effect of admixture on 6 intensifies with the rise in the difference U ^ - Ug e , i.e. the

ratio increases.

These results can be explained in the following way: Acceptor levels are created by oxygen in selenium, and these determine mainly the electrical properties of selenium. When admixtures are introduced with

Ug e electrically

neutral complexes with oxygen are formed, such as CdO, Sb 2 0^, MnO, and so on. Due to the neutralization of oxygen the concentration of holes, and hence

the value of 0 decreases. Al-

most all admixture atoms with smaller values of U , form

Short Notes

K111

Fig. 2. Dependence of 0 on ionization potential of the introduced admixtures likely complexes with oxygen. At equal concentrations of oxygen and introduced admixture atoms, oxygen is completely compensated and the concentration of the admixture current carriers will approach zero, i.e. the value of G will reach its minimum. For T1 and Na with ionization potentials of 6.11 and 5.14 eV, respectively, fl passes its minimum at admixture 17 -3 concentrations of » 10 cm . This value is in good agreement with the hole concentration in selenium, and as the holes are created mainly by oxygen acceptors, one can suppose that the concentration of ionized oxygen atoms in the selenium lattice is ~ 10 1 7 c m - 3 . It should be noted that the formation of admixture complexes with oxygen in selenium was also confirmed by the use of the electroconductivity method in the case of Mn (5). With decreasing difference U g e - U a d the probability for the formation of complexes of all introduced admixtures remains free. To achieve a oomplete oxygen compensation in this case it is necessary to introduce an admixture concentration

K112

physica status solidi 13 17 «-3 exceeding 10 cm . Then with complete oxygen compensation the concentration of the admixture current carriers in selenium will differ from zero, and hence the value of will be somewhat higher than that in the case of admixtures with low ionization potentials. Thus, the appearance of a minimum in 0 at various concentrations of different admixtures and the unequal decrease of 0 with complete oxygen compensation by different admixtures are related to the probability of admixture-oxygen complex formation in selenium. In addition, the above-mentioned assumption accounts for the appearance of a minimum in the concentration dependence of A,, p, and H, as well as for the shift depending on the admixture values of U (3). For admixtures with- U a d > Uge the probability of oxygen complex formation is ruled out. By creating local levels these admixtures increase the concentration of current carriers and hence the value of 0 . It is easy to imagine that such admixture effect will increase with the difference U ad ~ USe* T h e correlation between lg or lg ^ p and the value of can be presented by the following equation: lg ^min^max)

= 1 .05 (U ad - U g e ).

(1 )

If the values of Ua(j and 0 are known one can thus predict from this equation an increase or decrease in the value of 0 with the addition of any admixture and calculate the value of Cfmin . v(or d^ max The value of 0 . , calculated from equation (1) for a Mn m i -6 1 1 admixture, amounts to 1.83x10 SI cm" , while the experimen—6 —1 —1 tal value of G m i n (jjn) 1*3x10 £L cm The values of lg ^MO. for the admixtures K, Li, Al, Ge, and Si are also shown in Pig. 2, but their effect has not yet been studied experimentally. It should be noted that for admixtures of Ga, S, and Hg a departure from this rule is observed which is likely to be related to their state in selenium. So, for instance, depending

Short Notes

K113

on temperature, sulphur may appear in the form of Sg, Sg, S^, and Sg molecules. These have a value of U equal t o « 3 . 5 eV which is less than the value of Ug e , whereas for atomic sulphur U g > U S e

(U s = 10.A eV).

Depending on the state of sulphur in selenium, 0 may increase or decrease. This is apparently also the case for Ga and Hg. ffe are grateful to Prof. A.E. Eegel for his valuable advices during the discussion of our work. References (1) I.Z. KOZLOVSKII and D.N. NASLEDOV, Zh. tekh. Fiz. 13, 627 (1943). (2) A.E. EEGEL and B.G. TAGIEV, Fiz. tverd. Tela ¡j 2914 (1963); 6, 1001 (1964); 6, 1429 (1964). (3) G.B. ABDULLAEV, S.I. MEKHTIEVA, D.SH. ABDINOV, and G.M. A1IEV, phys. stat. sol. 11_, 3 9 1

(1965); Dokl. Akad. Nauk

Azerb. SSE 20, No. 2, 27 (1964); 21., No. 3, 18 (1965). (4) P. ECKAET, Ann. Phys. (Germany) 14, 233 (1954). (5) G.B. ABDULLAEV, N.I. IBBAGIMOV, SH. V. MAMEDOV, and T.CH. ZNTJTAEII, Dokl. Akad. Nauk Azerb. SSE 21_, No. 4, 13 (1965). (Eeceived January 4, 1966)

Short Notes

K115

phys. stat. sol. 12, K115 C1966) Ivan Franko State University) Lvov Excitation of Wannier-Mott Bxcitons by Fast Electrons A.E. GIAUBERMAN, M.A. RUVINSKII, and A.7. PUWDYK The study of energy loss speotra of fast electrons in crystals is an important souroe of information on elementary excitations in solids; the role of these investigations in the study of plasmons is well known (1). A recent achievement in this field is the experimental investigation of the generation of excitons by fast electrons in alkali halides (2). In this connection we attempt to make a theoretical analysis of the generation of Wannier-Mott non-localized excitons by fast electrons. ^ The production of exoitons is oaused by the Coulomb interaction between the passing electron and the eleotrons of the crystal« This interaction is generally regarded as a perturbation leading to the transition under consideration. The valenoe and oonduotion bands are assumed to be simple, spherical, and non-degenerate. The calculation is made by means of multieleotron wave functions. The wave function of the initial state of the orystal, with its valenoe band filled and the conduotion band empty, is taken as the Slater determinant of orthogonal one-eleotron Wannier functions. The wave function of the final state in which one electron makes a transition from the valence to the oonduotion band is chosen as a linear combination of respective determinants* The coefficients of this expansion, in the continuous approximation, are determined by the effective two-particle Sohrtfdinger equation for large-radius exoitons of the Wannier-Mott type (4,5). The wave functions of both the inoident and scattered fast electron are taken in the form of plane waves. The exohange effect between the passing eleotron and the electrons of the crystal is considered. The laws of conservation of energy and 1) The oase of Frenkel exoitons is treated in (3).

K116

physica status solidi 13

momentum are where E(^), E Hi'), and E e x ( o for the corresponding hole concentrations mentioned above. Thus the hole conductivity effective mass is increased when the hole concentration increases, but the value of the

Short Notes

K121

mass i s much s m a l l e r t h a n t h a t of t h e d e n s l t y - o f - s t a t e mass mentioned above. I t i s p o s s i b l e t h a t t h e s e b o t h r e s u l t s a r e due t o t h e complexity of t h e v a l e n c e band s t r u c t u r e of I I I - V compounds ( 5 , 6 ) . References ( 1 ) O.V. EMELYANENKO, P . P . KESAMANLY, D.N. NASLEDOV, V.G. SIDOROV, and G.N. TALALAKIN, p h y s . s t a t . s o l . 8, K159 (1965). ( 2 ) P . P . KESAMANLY, E . E . KLOTYNSH, D.N. NASLEDOV, and G.N. TALALAKIN, I z v . Akad. Nauk L a t v . SSR ( i n p r e s s ) . ( 3 ) H. LYDEN, Phys. Rev. 124, A1106 (1964)* ( 4 ) W.G. SPITZER and H.Y. PAN, Phys. Rev. 106. 382 ( 1 9 5 7 ) . ( 5 ) R. BRAUNSTEIN and E.O. KANE, J . Phys. Chem. S o l i d s 22, 1423 ( 1 9 6 2 ) . ( 6 ) J . KOLODZIEJCZAK, P r o c . I n t e r n . Conf. Semicond. P h y s . , P a r i s 1964 ( p . 1 1 4 7 ) . (Received January 6, 1966)

Short Notes

K123

phys. stat. sol. 12, K123 (1966) Scientific Laboratory, Ford Motor Company, Dearborn, Michigan (a) and Department of Mining and Metallurgy, University of Illinois, Urbana (b) The Line Tension of a Dislocation By A.D. BRAILSPORD (a) and P. DE CHATEL (b) In two recent papers (1,2) ^ ^ the authors have investigated the problem of the effeotive line tension of a dislocation using apparently quite different approaohes. The purpose of this note is to point out the agreement between both the mathematical results and the underlying physioal concepts involved in the separate treatments. In paper I the equilibrium of a straight dislocation section bowed out under the effeot of external stress is considered. The smooth curve representing the dislocation is approximated by a stepped sequence in the sense of the methods of caloulus. The energy of the dislocation is calculated by adding together the interaction energy (3) between all the infinitesimal kinks and then taking the limit of the kinkheight going to zero. The resultant energy is then minimized with respeot to the line tension, the displacement appropriate to an extensible string being taken as a trial function. In paper II the equilibrium of a olosed dislocation loop is considered. The energy is calculated from the Peach-Koehler formula (4) and the equilibrium is shown to be maintained by the external and self-stresses (5). The effect of the selfstresses is then replaoed by that of the line tension and this yields an integral formula for the latter. In both papers it is stressed that the general expression for the energy is not equivalent to that derived from the assumption of an energy per unit length of dislooation. The first point we would make is that the two approaches of calculating the energy are equivalent, since both the kink-kink interaction energy (3) and the Peach-Koehler for1) Hereafter referred to as I and II.

K124

physica status solidi 13

mula (4) are derived from the general analysis of Burgers (6). The equivalence of the procedures is best shown by the agreement of the results: applying the energy formula (16) of II for the special geometry described in I leads directly to equation (4) of I. (The detailed algebraic steps follow closely the development given in the Appendix of the latter work.) After having formulated the general expression for the energy of a dislocation, the further developments in I and II differ in detail. Specifically, in II, the later work is concerned with investigating the validity of the following relation for the line tension S (7): S = U + d2U/de2,

(1)

where U is an effective energy per unit length of the straight dislocation and 6 is the angle between the Burgers vector and the line element. It is there emphasized that the approximate relation (1) is only correct when the dislocation segment is very large compared with an interatomic distance. In I this point was not a matter of concern. However, it is established easily from equations (11) and (12) therein by noting that if the Sg;f:f of I is used in (1) above to define an energy U, the value of U so derived is given by U

=

Sifr^vT 0 - v c o s 2 6 ) j l n

- ^-j .

(2)

Thus, as long as ( L / b ) » 1 , the value of U is equivalent to the well-known expression for the energy per unit length of a straight dislocation (8). For short dislocation segments, on the other hand, equation (1) becomes increasingly inaccurate. In summary we would emphasize that l.the kink approach used in I is equivalent to the strain energy calculation given in II; 2.both approaches substantiate equation (1) in the limiting case of small displacements of long dislocation segments from a straight line (but hot otherwise); 3.the derivation of (1) as given in I is free from the objections relating to the concept of an energy per unit length made in II.

Short Notes (1) (2) (3) (4) (5) (6) (7) (8)

K125

References A.D. BRAILSFORD, Phys. Rev. 129, A1813 (1965). P. DE CHATEL and I. KOVACS, phys. stat. sol. 10, 213 (1965). A.D. BRAILSFORD, Phys. Rev. 128. 1033 (1962). M.O. PEACH and J.S. KOEH1ER, Phys. Rev. 80, 436 (1950). L.M. BROWN, Phil. Mag. 10, 445 (1964). J.M. BURGERS, Proc. Acad. Sci. Amsterdam 42, 293 (1939). G. DE WIT and J.S. K0EH1ER, Phys. Rev. 116. 1113 (1959). A.H. COTTRELL, Dislocations and Plastic Plow in Crystals, Clarendon Press, Oxford 1961 (p. 38). (Received January 10, 1966)

Short Notes

K127

phys. s t a t . s o l . 12, K127 (1966) I n s t i t u t e de F i s i c a , Centro Atomico Bariloohe, and I n s t i t u t fllr t h e o r e t i s c h e und angewandte Physik der Technisohen Hochschule S t u t t g a r t Nucleation of Deformation Twins i n P.C.C. Metals By

G. SCHOECK A number of t v i n n i n g mechanisms have been proposed f o r deformation twinning i n f . c . c . metals (1 t o 6 ) . I t i s gener a l l y assumed t h a t only a pole meohanism i s able t o e x p l a i n t h e consecutive shear of a l a r g e number of {111}-planes each by a vector ^ < 2 1 1 ) t h a t i s r e q u i r e d t o produce a deformation t w i n . However, a l l the pole mechanisms proposed so f a r encount e r some t o p o l o g i c a l d i f f i c u l t i e s . As a consequence, complicated assumptions have t o be made t o e x p l a i n how a twinning source can o p e r a t e . The purpose of t h i s note i s t o show how by a p p r o p r i a t e d i s l o c a t i o n r e a c t i o n s t h e s e d i f f i c u l t i e s oan be avoided and how a twinning source can operate i n a simple way. In order t o f a c i l i t a t e the d i s c u s s i o n we use ( F i g . 1) Thompson's t e t r a h e d r o n (7) and assume t h a t deformation frrinning t a k e s place on the plane ( a ) , which may be e i t h e r a p r i mary or a conjugate g l i d e p l a n e . If CD i s the Burgers v e c t o r of t h e g l i d e system with the highest resolved shear s t r e s s , then the twinning d i s l o c a t i o n i s found t o be Ca. (For a r e cent review of t h e experimental o b s e r v a t i o n s see r e f e r e n c e s

F i g . 1.•Thompson's o r i e n t a t i o n tetrahedron f o r f . o . c . structures

physica status solidi 13

K128

Pig. 2. Operation of a twinning source by a pole mechanism

/

u

(8) and (9)«) The conditions which a pole configuration has to satisfy in order to produce twinning are the following: 1. There must be a stable pole dislocation not lying in the plane (a). Its Burgers vector must have the component ccA perpendicular to (a). 2. There must be a twinning dislocation Cci which can rotate freely around the pole dislocation. Several proposals for twinning mechanisms (2,3) have been made in which the twinning dislocation fca is generated at one place and forms a pole after moving through the crystal. However, it has not been shown how a stable pole configuration can be formed in this way and especially how the twinning dislocation can rotate freely around the pole. The alternative possibility, in whioh the twinning dislocation splits off the pole dislocation, has been proposed originally by Cottrell and Bilby (1) and is shown in Pig. 2. The pole dislocation CA in plane (b) has been transferred (for instance by intersection) over the distance BS into the twinning plane (a) where it dissociates acoording to CA = Ctt + aA, The sessile Prank partial ocA remains along BS, whereas the twinning dislocation Cot (a Shookley partial) can bow out between the poles B and S. A detailed study of the

Short Notes

K129

geometry involved shows, however, that under the acting stress the twinning dislocation Cot rotates around the pole dislocation CA into a region into whioh the pole dislocation does not extend (i.e. at R upwards and at S downwards). After one turn the two dislocations Ca along XY are on planes one atomic distance apart and cannot pass each other unless a shear stress of about n / 2 0 (where ¿1 shear modulus) acts on it. In order to avoid this difficulty, Venables (5) has modified the above model and proposed that by successive splitting off of unit jogs from the pole dislocation CA at R or S, successive single rotations of the twinning dislocation can occur. After sufficient jogs have been produced the separation of the dislocations along XY can become large enough to allow them to pass each other at the stress present. Although this mechanism is geometrically possible it involves the successive operation of quite a number of complicated steps and it is not at all clear whether it will be realized. Other geometrical configurations have been discussed (10), but always the problem remains that after the first rotation the two branches of the twinning dislocation stop each other. All these difficulties could be avoided if the two dislocations along XY could pass each other, as already pointed out by Ookawa (2). It has so far been overlooked that such a passing can occur if a segment of the twinning dislocation C« reacts with other dislocations of the dislocation forest. The situation is illustrated by dashed lines in Fig. 2. One of the dislocations Ca may encounter a "tree" with Burgers vector AC and can combine with it over a certain length UV according to the reaction AC + Ca = Aa, as discussed by Saada (11). Such reactions have been proposed (5) to act as frictional stress on the expanding twinning dislocations. Since, however, the dislocation between UV has the Burgers vector A a the other twin dislocation C a can pass it one atomic plane below because the interaction between these two dislocations with perpendicular Burgers vectors is weak and a similar passing is already accomplished along RS. Once the

K130

physioa status solid! 13

twinning dislooation has passed UV it can move on freely and by successive rotations around R and S and repeated passizur through UV a twin of large size can be formed* Thus the proposed mechanism is able to explain the nucleation of a deformation twin without any other complicated assumptions. The mechanism is by no means restricted to twinning in f.o.c. structures but may - with proper modifications - be active in other structures whenever two dislocations of opposite Burgers vectors have to pass eaoh other at a close distance. References A.H. COTTRELL and B.A. BILBY, Phil. Mag. 42, 573 (1951). A. OOXAWA, J. Phys. Soo. Japan 12, 925 (1957). H. SUZUII and C.S. BARRETT, Acta metall. 6, 156 (1958). P. HAASEN and A. KING, Z. Metallk. '¡V, 722 (1960). J.A. VENABLES, Phil. Mag. 6, 379 (1961). J.B. COHEN and J. WEERTMAN, Aota metall. H , 996 (1963). N. THOMPSON, Proc. Phys. Soc. (London) B 66. 481 (1953). J.A. VBNABLES, Metallurg. Soc. Conf. 2£, Deformation Twinning, AIME 1965 (p. 77). (9) E. PEISSKER, Z. Metallk. ¿6, 155 (1965). (10) J.P. HIRTH, Metallurg. Soo. Conf. 2^, Deformation Twinning, AIME 1965 (p. 112). (11) G. SAADA, Acta metall. 8, 841 (1960). (Received January 10, 1966) (1) (2) (3) (4) (5) (6) (7) (8)

Short Notes

K131

phys. stat. sol. 12, K131 (1966) Physikalisoh-Technisches Institut der Deutsohen Akademie der Wissenschaften zu Berlin, Bereich elektrischer Durchschlag, und IV. Physikalisches Institut der Humboldt-Universität zu Berlin Temperature Dependence of the PieId-Induced Shift of the Absorption Edge in CdS and CdSe Single Crystals By £. GUTSCHE and H. LANGE AcoOrding to the theory of Franz (1) the field-induced shift of an exponential absorption edge is given by AL

P 2 r t 2 e 2 j-2 24 (kT^m*

CO

F means the electrio field strength, m* the reduced effective mass of electrons and holes in the field direction; 0 is a constant characterizing the steepness of the exponential absorption edge (P = 2.17 for CdS and CdSe (2,3)). The remaining quantities have the usual meaning. In previous papers (4,5) quantitative measurements have been reported of the field-induced shift of the absorption edge in CdS and CdSe single crystals at room temperature. It has been found that all crystals exhibit a quadratic dependence of the shift on field strength. % analyzing the results on the basis of equation (1) reduced effeotive masses have been determined for the conduction-valence band combinations T T g and and the directions parallel and perpendicular to the c-axis. The question was left whether the field-induced shift A

has the T~ dependence predicted by equation (1). The temperature dependence of the shift is studied in the present paper. The measurements were made on plate-shaped single crystals which had been provided with ohmic contacts. Ac voltage was applied, and the field-induced shift was determined from the modulated transmitted light intensity. The field strength was about 10^ V/cm. The measurements were made at an absorption

K132

Pig. 1.

p h y s i c a s t a t u s s o l i d i 13

2 AE/F as a f u n c t i o n of t e m p e r a t u r e . ^ a p p l i e d f i e l d , £ e l e c t r i c v e c t o r of l i g h t wave. The dashed l i n e i n d i c a t e s t h e slope - 2 A

c o e f f i c i e n t of about 200 cm . D e t a i l s a r e d e s c r i b e d i n ( 4 ) and ( 5 ) . The c r y s t a l s were kept i n vacuum on a c r y s t a l h o l d e r whose t e m p e r a t u r e could be v a r i e d between 150 and 400 °K. The measurements a t 77 °K were c a r r i e d out w i t h t h e c r y s t a l s d i r e c t l y immersed i n l i q u i d n i t r o g e n . p The t e m p e r a t u r e dependence of AE/F was measured on s e v e r a l CdS and CdSe c r y s t a l s f o r d i f f e r e n t d i r e c t i o n s of t h e f i e l d and l i g h t v e c t o r s w i t h r e s p e c t t o t h e c - a x i s . In t h e t e m p e r a t u r e range 150 t o 400 t h e exponent of T was always n e a r - 2. Typical c u r v e s a r e shown i n F i g . 1. The mean v a l u e

Short Notes

K133

of the exponent was - 1.95. At lower temperatures this relationship is apparently no longer valid. The value at 77 °K was in all cases considerably smaller than to be expected from QC the T * dependence found at higher temperatures. The above results confirm that the measurement of the field-induced shift of the absorption edge is in fact a suitable method for the determination of effective masses _2in a wide temperature range. The slight deviation from the T dependence is beyond the experimental error. It seems unlikely that this deviation is completely due to the temperature dependence of the effective masses, for this would mean an increase in m * of 15 % between 150 and 400 °Ko A slight temperature dependence of (J seems to be more probable, although 0 should be temperature independent in this region according to earlier measurements of Dutton (see Fig. 3 in (2))„ For temperatures below 100 °Z it has been estimated that correctional terms omitted in equation (1) (see equations (31) in (1)) can no longer be neglected. However, at present it cannot be decided whether the serious deviation at 77 °K is really due to this neglect. References (1) W. FRANZ, Z. Naturf. 13a, 484 (1959). (2) D. DUTTON, Phys. Rev. 112., 785 (1958). (3) R.B. PARSONS, W. WARDZYNSKI, and A.D. YOFFE, Proc. Roy. Soc. A262, 120 (1961). (4) E. GUTSCHE and H. LANGE, phys. stat. sol. 4, K21 (1964). (5) E. GUTSCHE and H. LANGE, Proc. Intern. Conf. Semicond. Phys., Paris 1964 (p. 129). (Received January 11, 1966)

Short Notes

K135

phys. stat. sol. 13., K135 (1966) Institut für Kristallphysik der Deutschen Akademie der Wissenschaften zu Berlin Zum Wanderungsmechanismus von Cu+-Ionen in AgBr Von P. SÜPTITZ Neue Kernresonanzmessungen an Cu^-dotierten AgBr-Kristallen (1) lassen es als wahrscheinlich erscheinen, daß auch bei kleinen Dotierungskonzentrationen neben dem durch Diffusionsmessungen (2,3) festgestellten direkten Zwischengitterplatzwechsel der Cu+-Ionen der in (3) diskutierte Wanderungsmechanismus 2 auftritt. Dabei wird von den Cu+-Ionen vorübergehend ein Zwisohengitterplatz besetzt, und der EUcksprung in die Silberlücke erfolgt zu einem anderen als dem ursprünglichen Nachbarplatz.- Aus den Diffusionsmessungen ergibt sich, daß bei Temperaturen oberhalb von 150 °C und bei kleinen Dotierungskonzentrationen die Wanderung vorwiegend durch Zwischengitterplatzwechsel vor sich geht. Die Ergebnisse schliessen jedoch die Beteiligung anderer Mechanismen nicht völlig aus, so daß für den Fall eines nur geringen Massentransportes nach dem Mechanismus 2 die Ergebnisse nicht im Widerspruch zu (1) stehen. Durch die Polgerungen aus den Kernresonanzmessungen ergeben sich für die Bestimmung besonders der Fehlordnungskonstanten aus den Diffusionsmessungen neue Gesichtspunkte. In A

\

(3) J war die Auswertung unter der Annahme erfolgt, daß ausschließlich Zwischengitterwanderung auftritt. Im Falle der Wirksamkeit mehrerer Mechanismen sind die Gleichungen (8) und (10) aus (3) durch Zusatzglieder zu ergänzen, die sich leicht aus der allgemeinen Gleichung (5) ableiten lassen. Dabei treten weitere Teilchendiffusionskoeffizienten, im vorliegenden Fall Dg, mit in die Rechnung. Sofern das Verhältnis Dg/Dj nicht zu klein ist (bei 250 °C z.B. 1/10), ergeben sich mit 1) In (3), Fig. 3, ist der untere Teil des Ordinatenmaßstabes dahingehend zu berichtigen, daß der Anfangspunkt der 150 °C -Kurve bei D • 2,1x10"? cm2/s liegt.- Im Zusammenhang mit Gleichung (2c) geht es um 6 nächste Zwischengitterplätze, die Zahl 8 ist durch 6 zu ersetzen.

K136

physica status solidi 13

den gemessenen Diffusionswerten flir die zu lösende kubische Gleichung zwei weitere reelle Wurzeln, von denen nur eine physikalisch sinnvoll ist. Diese führt dann zu größeren Asso.ziationskonstanten (im obigen Beispiel zu H = 3,9, d.h. 20#ige Dissoziation bei dominierender Eigenfehlordnung) als ohne Berücksichtigung des zweiten Mechanismus. Aus Diffusionsmessungen allein kann das Verhältnis D 2 /D^ nicht bestimmt werden, und es kann gegebenenfalls auch nicht entschieden werden, welche der beiden Lösungen die richtige ist. Der Vergleich mit neuen Leitfähigkeitsmessungen (4) legt jedoch nahe, eine kleinere Dissoziation anzunehmen, als sich bei alleiniger Berücksichtigung des Zwischengitterplatzwechsels ergibt. Dies ist wiederum eine Stütze für den durch die Kernresonanzmessungen wahrscheinlich gemachten zweiten Wanderungsmechanismus» (1) (2) (3) (4)

J. P. P. P.

Literatur KLUGE, phys. stat. sol. 12, 401 (1966). SÜPTITZ, phys. stat. sol. 7, 653 (1964). SÜPTITZ, phys. stat. sol. Z> 667 (1964). MÜLLEE, phys. stat. sol. 12, 775 (1965). (Received January 11, 1966)

Pre-printed

Titles and

Abstracts

Papers to be published in "physica status solidi" Vol. 14, No. 1

Review Article J. MERTSCHING, Physikalisch-Technisches Institut der Deutschen Akademie der Wissenschaften zu Berlin,

Theorie

elektromagnetischer Wellen in Metallen und ihrer Wechselwirkung mit Ultraschallwellen (I) Original Papers M. ASCHE, Physikalisch-Technisches Institut der Deutschen Akademie der Wissenschaften zu Berlin und W.M. BONDAR,

Insti-

tut fUr Physik der Ukrainischen Akademie der Wissenschaften, Kiev, Umverteilung heißer Elektronen auf die Täler des L e i tungsbandes von Silizium

Es wird eine Möglichkeit zur Ab-

schätzung der Elektronenumverteilung auf die Täler in Vieltalhalbleitern bei elektrischen Feldern gezeigt. Sie ist auf der Änderung der Besetzungszahlen der Täler bei Druck begründet. Experimente wurden an 20 ß c m n-Si bei 77 °K durchgeführt und entsprechend ausgewertet. Die Resultate stimmen qualitativ mit den Daten aus Anisotropiemessungen der Leitfähigkeit Uberein. W. PLATIKANOWA and J. MALINOWSKI, Institute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia, Trapping and Capture of Holes in Silver Bromide Single Crystals

The r e a c -

tion between a deposit of silver and bromine formed by holes migrating to the surface of a silver bromide crystal has b e e n applied further to investigate the concentration decay of holes in the interior of the crystal. Effective lifetime of photoexcited holes in a thir. surface layer has been shown to be at least an order of magnitude larger than lifetime in the interior« Thermal annealing of the samples increases

substan-

tially the trapping and recombination rate in the interior, without practically changing it for the surface. The buildingup of a layer of adsorbed bromine on the surface of the ex-

K138

Titles and Abstracts, phys. stat. sol.

posed crystal has been supposed in order to explain observations described in the paper. It is shown that the existence of an effective bromine acceptor is necessary for the building of a stable surface latent image. No relation has been found between the lifetime of photoexcited electrons and holes, suggesting that these carriers do not recombine with each other. The nature and concentration of trapping centres as well as the effect of annealing of the orystals has been discussed. R. GROTH, Philips Zentrallaboratorium GmbH., Laboratorium Aachen, Untersuchungen an halbleitenden Indiumoxydschlchten Nach dem "Sprühverfahren" wurden n-leitende Indiumoxydschichten mit Schichtdicken zwischen 0,1 und 0,5 between the direction of the molecular beam and the normal of the film, on the condensation coefficient a and the density of the beryllium films.

K146

Titles and Abstracts, PTT

It is found that for Ts > 300 to 400 °C the condensation coeffioient considerably decreases (sometimes to oc »*0.5). The density of the vacuum oondensates of Be increases with increasing T s and OJ^ and with decreasing^?. A.B. ROITSIN, Kiev, Theory of the Spin-Lattloe Relaxation in a Hon-Ideal Crystal The probability for a change in the spin state due to lattice vibrations is calculated for a one-dimensional non-ideal crystal. S.V. STARODUBTSEV and E.V. PESHIKOV, Moscow, RadiatlonInduoed.Changes in the Properties of Perroelectrics due to an Internal Displacement Pield The radiation-induced changes due to the formation of an internal displacement field in crystals of seignette-salt and of triglycine sulphate which are irradiated with Co^-y-rays are investigated by special methods. It is shown that the formation of an internal displacement field is equivalent to the action of an external displacing field and can be considered as secondary radiation effects. A description is given of the pulse repolarization oocurring in irradiated triglyoine sulphate crystals. V.P. KA1ASHHTK0V, Sverdlovsk, Polarization of Huclear Spins in. a Semiconductor by Direot Current The polarization of nuclear spins is oaloulated for the interaction with hot electrons in strong electric and magnetio fields. Various extreme oases are considered. N.A. PENIff, B.G. ZHUBKIN, and B.A. VOLKOY, Moscow, Effect of the Conoentrations of Donors and Aooeptors on the Electric Conductivity of Heavily Doped n-Si The influenoe of the phosphorus concentration and the degree of the compensation by boron on the eleotrio conductivity of heavily doped n-Si is investigated in the temperature range 4.2 to 78 °Z. A oonoentration dependence is found of the aotivation energy 61 of impurity oonduotion. The dependence of e^ on the phosphorus oonoentration begins at much higher ooncentrations than the dependence of 6 1 on the boron concentration. It is shown that the activation energy of the hopping conductivity (£3) first increases with increasing donor concentration but

T i t l e s and A b s t r a c t s , PTT

K147 17

decreases when t h e donor c o n c e n t r a t i o n exceeds 6x10 cm For a low degree of compensation t h e temperature dependence of c o n d u c t i v i t y i s determined by constant values of e^ and £3. For s t r o n g compensation in samples with high donor concentrations the aotivation energies and e^ seem to be temperature-dependent. S.K. SAVVTNYKH, A. A. EAR TUSH IN, and B.S. ELYACHKO, Novos i b i r s k , I n t e r a c t i o n of E l a s t i c Surface Waves with a SemiF i n i t e Plasma In a t h e o r e t i o a l study i t i s shown t h a t an e l a s t i c wave moving along the boundary between a p i e z o e l e c t r i c and a semiconductor w i l l be damped by t h e i n t e r a c t i o n of the p i e z o e l e c t r i c f i e l d with t h e f r e e o a r r i e r s of the semiconduct o r . The frequenoy dependence of the damping c o e f f i c i e n t i s strongly influenced by the c h a r a c t e r of the s c a t t e r i n g of charge c a r r i e r s at t h e boundary of the media. L.N. DOBBETSOV, Leningrad, P i e l d Emission from a Metal Covered with an Atomic Layer The d a t a on t h e f i e l d emission from tungsten covered with a germanium l a y e r obtained i n (1 t o 3) are discussed from the s t a n d p o i n t of the one-dimehs i o n a l model of t h e adsorption l a y e r ( 4 , 5 ) . (1) I . L . SOKOLSKAYA and N.V. MILESHKINA, P i z . t v e r d . Tela 2> 3389 (1961). (2) I . L . SOKOLSKAYA and N.V. MILBSHKINA, P i z . t v e r d . Tela_£, 2501 (1963). ( 3 ) I . L . SOKOLSKAYA and N.V. MILESHKINA, P i z . t v e r d . Tela 6, 1786 (1964). (4) J . GUBNEY, Phys. Bev. 4 J j 479 (1935). ( 5 ) L.N. DOBBBTSOV, Elektronnaya i ionnaya emissiya, GITTL 1952 ( p . 137 t o 139, 254 t o 255). B.G. ZHUBKIN and N.A. PENIN, Mosoow, E f f e o t of the Compensation on t h e Bxohange I n t e r a c t i o n of Donors i n Heavily Doped n - S i The i n f l u e n c e of t h e compensation on t h e shape of the EPR s p e c t r a i s i n v e s t i g a t e d i n h e a v i l y doped n - S i with phosphorus oonoentration N^ • 10 17 t o 10• 1 flom—"i. These measurements are made with weakly and s t r o n g l y compensated samples. Compensation i s achieved by i n t r o d u c i n g boron i n t o

K148

T i t l e s and A b s t r a c t s , PTT

t h e melt d u r i n g t h e c r y s t a l growth performed by t h e C z o c h r a l s k i method. I t i s found t h a t t h e i n t e n s i t i e s of t h e h y p e r f i n e l i nes i n c r e a s e as N^ i n c r e a s e s t o 6 x 1 0 ^ cm~^. Moreover, a b r o a d ening of t h e EPE l i n e s i s observed in a sample w i t h N-p, = A q o = 1x10 cm~J and s t r o n g compensation. This i s i n t e r p r e t e d a s weakening of t h e exchange i n t e r a c t i o n of t h e phosphorus atoms by t h e e l e c t r i c f i e l d s of t h e n e g a t i v e l y charged a c o e p t o r s . A.A. AUTIPIN, I . N . KUEKIN, L.Z. P0TV0E0VA and L.YA. SHEKUN, Kazan, EPE of Sm^* i n CaW0A Single C r y s t a l s Measurements a r e made of t h e EPE of Snr i o n s i n CaW0A s i n g l e c r y s 3+ t a l s . The Sm ions a r e m a g n e t i c a l l y e q u i v a l e n t . The paramet e r s of t h e s p i n - H a m i l t o n i a n i n d i c a t e a s t r o n g i n f l u e n c e of t h e mixing of t h e ^Hs s t a t e w i t h e x c i t e d s t a t e s (due t o t h e electric crystal f i e l d ; . N.V. VOLKEKSHTEIN and G.V. PEDOEOV, Sverdlovsk, H a l l E f f e c t i n Neodymium and Samarium The H a l l e f f e c t i s measured i n neodymium and in samarium w i t h a p u r i t y of about 99.9 # i n t h e t e m p e r a t u r e range 2.4 t o 350 °Z. The r e s u l t s a r e compared w i t h t h é f i e l d and t e m p e r a t u r e dependences of t h e H a l l e f f e c t i n heavy r a r e - e a r t h m e t a l s . V.L. GUEEVICH and B.D. LAIKHTMAN, Leningrad, Theory of Sound Generation i n P i e z o e l e o t r i c Semiconductors A theory of t h e g e n e r a t i o n of s t a t i o n a r y a c o u s t i c waves of s m a l l amp l i t u d e i s developed f o r p i e z o e l e c t r i c semiconductors i n a constant e l e c t r i c f i e l d . E . I . MISHCHENKO, Sverdlovsk, E l e c t r o n Density D i s t r i b u t i o n and X-Bay S c a t t e r i n g I n t e n s i t y f o r L a t t i c e Atoms and Ions i n a S t a t i s t i c a l Model The i n f l u e n c e of packing ( i . e . t h e compression of t h e atom or t h e ion i n t h e l a t t i c e ) on t h e e l e c t r o n d e n s i t y d i s t r i b u t i o n and X-ray s c a t t e r i n g i n t e n s i t y i s s t u d i e d t h e o r e t i c a l l y on t h e b a s i s of t h e Thomas-PermiDirac model. The l i m i t s f o r t h e a p p l i c a t i o n of t h e s t a t i s t i c a l model a r e g i v e n f o r d i f f e r e n t d e g r e e s of i o n i z a t i o n . An e x p r e s s i o n i s obtained f o r t h e i n t e n s i t y of i n c o h e r e n t X-ray s c a t t e r i n g by compressed i o n s . Tables are given f o r m e t a l l i c ions and atoms. For p o s i t i v e ions t h e degree of packing con-

T i t l e s and A b s t r a c t s , FTT

K149

siderably a f f e c t s the electron density d i s t r i b u t i o n , the form f a c t o r , and t h e i n t e n s i t y of t h e i n c o h e r e n t s c a t t e r i n g . IT.E. EASE, L . S . KORKTENKO, and A.O. RYBALTOVSKII, Moscow, Rhomblo EPR S p e c t r a of and Hd^ + i n CaJPo Measurements 3+ 3+ a r e made of t h e EPE of Dy and Nd i o n s i n CaP 2 c r y s t a l s grown i n t h e p r e s e n c e of oxygen. For b o t h i o n s rhombic s p e c t r a a r e observed f o r which one of t h e magnetic axes c o i n c i d e s w i t h t h e [110] d i r e c t i o n and t h e o t h e r two l i e i n t h e (110) plane w i t h d i f f e r e n t a n g l e s r e l a t i v e t o t h e [001] and [110] d i r e c t i o n s . As a r e s u l t of y - i r r a d i a t i o n new EPR s p e o t r a a r e f o u n d : a t r i g o n a l Dy^ + spectrum, a spectrum w i t h an i n i t i a l 1 s p l i t t i n g A = 0.26 + 0 . 0 3 cm" , and a new rhombic spect r u m . The t e m p e r a t u r e dependence of t h e r e l a x a t i o n t i m e i s measured f o r one of t h e rhombic Nd^ + s p e c t r a and t h e rhombic Dy^+ spectrum. The p o s s i b l e s t r u c t u r e s of t h e c r y s t a l l i n e environment of t h e paramagnetic i o n s i n CaFg a r e d i s c u s s e d . G.F. KHOLUYAUOV, Leningrad, p-n E l e c t r o l u m i n e s c e n c e i n SIC Doped w i t h B> N, and G-a I n v e s t i g a t i o n s are made of t h e i n f l u e n c e of boron and g a l l i u m i n t r o d u o e d by d i f f u s i o n on t h e e l e c t r o l u m i n e s c e n c e (EL) of n-SiC (6H) a d d i t i o n a l l y doped w i t h n i t r o g e n , and of t h e i n f l u e n c e of t h e n i t r o g e n c o n c e n t r a t i o n on t h e EL and photoluminescence (PL) i n t e n s i t y of boron doped s a m p l e s . a s w e l l as on the EL s p e c t r a of p-n j u n c t i o n s . I t i s shown t h a t g a l l i u m causes two t y p e s of EL w i t h peaks a t 2.6 and 2 . 5 3 eV (20 ° C ) . These EL s p e c t r a a r e s t u d i e d i n a wide t e m p e r a t u r e r a n g e . The EL of t h e f i r s t and second type i s c o n s i d e r e d t o be due t o r a d i a t i v e t r a n s i t i o n s of e l e c t r o n s from t h e c o n d u c t i o n band t o t h e g a l l i u m a c c e p t o r l e v e l , and t o t r a n s i t i o n s between t h e l e v e l s of a d o n o r - a o c e p t o r p a i r . The e x p e r i m e n t a l r e s u l t s f o r t h e EL and PL of boron doped samples can a l s o be e x p l a i n e d by t h i s model. The p o s s i b i l i t y of n i t r o g e n being t h e c o - a c t i v a t o r of boron i s d i s c u s s e d . M.E. MARINCHUK, Moscow, E x t e r n a l P h o t o e f f e o t from Heavily Doped Semiconductors; I n t e r e l e c t r o n i o I n t e r a c t i o n A theoretic a l s t u d y i s made of t h e e x t e r n a l s u r f a c e p h o t o e f f e c t f o r t h e case of non-soreened Coulomb i n t e r a c t i o n between t h e e l e c -

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trons. An expression is given for the photocurrent. The influence of the considered interaction on the long-wavelength limit of the photoeffect is discussed. I.V. PROSTOSERDOVA, E.YA. PUMPER, and N.V. TRONEVA, Moscow, Mechanism of Anomalous Diffusion of Zn in InSb The form of the concentration distribution is investigated for the diffusion of Zn in InSb for different boundary conditions assuming that the diffusion takes place with a constant concentration on the surface and with a reserve of atoms in the diffusion layer. It is found that two diffusion currents of Zn exist with comparable concentrations but with diffusion coefficients which differ by about three orders of magnitude. .These currents can exist independently. The interaction of these currents causes the anomalous character of the diffusion of Zn in InSb. I.L. DRICHKO and I.V. MOCHAN, Leningrad, Effect of Microinhomogeneities on the Nernst Effect in InSb In accordance with theoretical results of Kudinov and Moizhes (1) it is shown experimentally that for sufficiently strong magnetic fields the Nernst effect in n-InSb is completely determined by microinhomogeneities. The thermal e.m.f. is weakly influenced by microinhomogeneities. These results are confirmed by the theory which shows that the concentration fluctuations determining completely the Nernst effect lead to a correction of the thermal e.m.f. whioh is independent of the magnetic field strength and lies within the experimental error. (1) V.A. KUDINOV and B.YA. MOIZHES, Fiz. tverd. Tela 7, 2309 (1965). S.V. SEMENOVSKAYA and A.G. KHACHATURYAN, Moscow, Bffects of Statistical Distortions. Short-Range Order, and Thermal Vibrations of Atoms on' the X-Ray Diffuse Scattering in Polycrystalline Substitutional Solid Solutions It is shown theoretically that the short-range order parameters can be determined from X-ray diffuse scattering in polycrystals of binary substitutional solid solutions, in oases when the difference in the geometric dimensions of the atoms (dimension effect)

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and the thermal vibrations of the atoms make contributions to the diffuse scattering. B.S. TSUKERBLAT and YU.E. PEELIN, Kishinev, Theory of Multi-Phonon Non-Radiative Transitions in Paramagnetic Centres An expression is deduced for the probability of a multi-phonon non-radiative transition between levels of different spin multiplicity of a d -type ion in an octahedral crystal field. The calculations are made in the adiabatic approximation including Jahn-Teller splitting. The spin-orbital interaction of the d-electrons is treated as a perturbation. A numerical estimate is given for the transition ^ T „ in the systems s s 3+ 3+ A^O^iCr and MgO:Cr . The largest contribution to the transition probability comes from processes with simultaneous participation of longitudinal and transverse phonons. N.G. BASOV, O.V. BOGDANKEVICH, and YU.M. POPOV, Moscow, Generation of Short-Wavelength Radiation, and Lifetime with Respect to the Spontaneous Emission in Semiconductors It is proposed to use large band-gap semiconductors and dielectrics excitated by an electron beam as generators of coherent radiation. The carrier lifetime with respect to spontaneous emission is calculated for direct transitions, and the minimum pumping strength and optimum length of the exciting pulse are estimated. B.T. GEILIXMAN and V.Z. KRESIN, Moscow, Critical Temperatures in Normal and Anomalous Superconductors The critical temperature T„c in a superconductor is calculated using the model of Fröhlich. An expression is derived which gives a connection between the energy gap at T = 0 °K and T . The rec suits describe the properties of anomalous superconductors. V.l. SIDLYARENXO, P.N. ZAITOV, and YÜ.L. LUKANTSEVER, Omsk, Effect of Crystal Perfection on the Thermal Stability of Colour Centres in Alkali Halide Crystal Phosphors The influence of the nature and the concentration of impurity ions, plastic deformation, the origin and prehistory of the crystals on the thermal stability of P-centres is studied for NaCl crystals. Some characteristics of the thermal dissociation

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Titles and Abstracts, FTT

of F-centres are determined experimentally. Assuming an ionio mechanism for this prooess a discussion is given of a theoretical relation previously derived between the dissociation rate of a definite type of F-centres and parameters which are structure sensitive. The results of the calculations are compared with the experimental data. A.P. KOMAR and N.N. SYUTKIN, Leningrad, Field Emission Microscopy of Ni-Be Alloys Field emission mioroscopic pictures are obtained of a Ni-Be alloy after annealing at various temperatures and quenching. Conclusions are drawn as to the topography of the specific field emission, the geometry and kinetics of the decomposition of the alloy, migration and coagulation of the precipitating phase. A.I. PILSHCHIKOV and E.V. LEBEDEVA, Moscow, Parametric Excitation of Spin Waves in Polycrystalline MgMn Ferrites An experimental study is made of the influence of the structure and magnetic characteristics of polycrystalline ferrites on the instability of spin waves. The samples used are MgMn ferrites in which the grain size, magnetization,and the value A H of the isotropic precession change within wide ranges. It is found for near-stoichiometric samples in constant fields which are sufficient to saturate magnetization, that the threshold field strengths and the losses of the spin waves are independent of the structure of the polycrystal. Measurements in a field range below saturation show that the presence of a domain structure in samples with large magnetization leads to a strong decrease in the threshold field strengths. L.S. MILEVSKII and V.D. KHVOSTIKOVA, Moscow, Dislocation Structure of Diamond-Type Crystals Grown in the [1003-Direction Edge dislocations oriented in the [100] direction are observed in silicon using a polarization microscope. It is found that reaction between these dislocations leads to the formatipn of large edge dislocations with the Burgers vector a[010]. The precipitation of .copper at different types of edge dislocations is; also investigated. V.V. ZHDANOVA and T.A. K0NT0R0VA, Leniijgrad, Thermal Ex-

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pansion of Doped Germanium Measurements a r e made of t h e t h e r m a l expansion c o e f f i c i e n t of n-Ge doped w i t h P, As, Sb, and Si i n t h e t e m p e r a t u r e range 77 t o 350 °K. An i n c r e a s e of t h e t h e r m a l expansion c o e f f i c i e n t i s observed a t l a r g e c o n c e n t r a t i o n s of e l e c t r i c a l l y a c t i v e i m p u r i t i e s . In Si-doped Ge t h e t h e r m a l expansion c o e f f i c i e n t i s p r a c t i c a l l y unchanged. The i n c r e a s e of t h e t h e r m a l expansion c o e f f i c i e n t of Ge doped with e l e c t r i c a l l y active impurities i s a t t r i b u t e d to the i n f l u e n c e of f r e e c a r r i e r s on t h e s p e c i f i c h e a t of t h e c r y s t a l l a t t i c e and on t h e Grtlneisen p a r a m e t e r . A.A. ROGACHEV and S.M. RYVKIN, L e n i n g r a d , Long-Wave l e n g t h Recombination R a d i a t i o n i n Germanium i n t h e Presence of I n t e r a c t i o n between Current C a r r i e r s Experimental i n v e s t i g a t i o n s a r e made of t h e r e c o m b i n a t i o n r a d i a t i o n i n germanium due t o i n d i r e c t o p t i c a l t r a n s i t i o n s i n v o l v i n g an i n t e r a c t i o n between c u r r e n t c a r r i e r s . I t i s shown t h a t a t a h i g h i n j e c t i o n l e v e l a new type of r e o o m b i n a t i o n r a d i a t i o n a p p e a r s on t h e l o n g wavelength s i d e of t h e i n t r i n s i c r a d i a t i o n . G.G. LE0NID0VA and T.R. VOLK, Moscow, Phase T r a n s i t i o n i n BaTiO^ a t High H y d r o s t a t i c P r e s s u r e The c h a r a o t e r of t h e phase t r a n s i t i o n i n BaTiO^ s i n g l e c r y s t a l s i s s t u d i e d a t p r e s s u r e s up t o 8 . 5 k b a r . A n o n l i n e a r dependence of t h e t r a n s i t i o n t e m p e r a t u r e on p r e s s u r e i s o b s e r v e d . The c o n s t a n t s A and B of t h e Devonshire e q u a t i o n ( 1 ) can be determined from measurements under e q u i l i b r i u m c o n d i t i o n s . The d e c r e a s e of B w i t h p r e s s u r e i n d i c a t e s t h e appearance of a second t y p e of phase t r a n s i t i o n i n BaTiO^ a t a d e f i n i t e c r i t i c a l p r e s s u r e , i n agreement w i t h t h e t h e o r y of Landau ( 2 ) and Ginzburg ( 3 ) . ( 1 ) A.P. DEVONSHIRE, P h i l . Mag. AO, 1040 ( 1 9 4 9 ) . ( 2 ) L.D. LANDAU, Zh. e k s p e r . t e o r . P i z . 7, 19 ( 1 9 3 7 ) . ( 3 ) V.L. GINZBURG, Uspekhi f i z . Nauk 77, 621 ( 1 9 6 2 ) . S.Z. BOKSHTEIN and I . L . SVETLOV, P l a s t i c Deformation of Cu and Co Whiskers Complete d e f o r m a t i o n diagrams of p l a s t i c Cu and Co w h i s k e r s a r e o b t a i n e d by a u t o m a t i c r e c o r d i n g d u r i n g t h e t e n s i l e t e s t . Three s t a g e s of d e f o r m a t i o n a r e f o u n d : an e l a s t i c range i n form of a s h a r p peak i n f l o w , t h e r a n g e

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of "slight" slip, and the range of solidification followed by the destruction of the sample. In the stage of "slight" slip the plastic deformation shows transient response which is caused by the frontal propagation of the LUders lines in primary planes. The peaks of stress correspond to the beginning of the LUders range, whereas the minima can be explained by dislocations leaving the crystal due to the slip lines. In the solidification range slip tracks are observed metallographically in secondary planes. The final destruction of the sample takes place at stresses which agree with the strength of ordinary single crystals. It is concluded that in the flow peak at the highest stress mobile dislocations are formed in the whisker. The motion of dislocations occurs in the "slight" slip stage. V.V. VASKIN, V.A. USKOV, and M.YA. SHIROBOKOV, Gorki, Influence of an Internal Electric Field on Impurity Diffusion in Semiconductors The influence of the internal electric field due to ionized impurity atoms and current carriers on the diffusion of impurities in semiconductors is treated by giving an approximate solution of the diffusion equation combined with the Poisson equation. The calculated concentration profile is compared with experimental data obtained for the diffusion of antimony in germanium. It is shown that for sufficiently high surface impurity concentrations or comparable carrier concentrations, for which the influence of the electric field becomes significant, the distribution of the impurity atoms differs from the well-known expression n(x,t) = is » n Q erfc (^VDT^ obtained for the diffusion from a constant source neglecting the influence of the field (n(x,t), n Q concentrations at the coordinate x and time t or t = o, respectively, D diffusion constant). V.P. TEUBITSYTT, Moscow, Equation of State of Solid Hydrogen Calculations are made of the internal energy E(v,t) and pressure P(v,T) in a hydrogen crystal as functions of volume v and temperature T in the pressure range from 0 to 10^ atm. The calculated curve P(v) for T = 0 is compared with experimental data for 0 « P « 2 x 1 0 4 atm.

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V.E. ADAMYAN, A.V. GOLUBKOV, G.M. LOGINOV, and V.N. FEDOROV, Leningrad, Magnetic Susceptibilities of Neodymium Chalcogenides Measurements are made of the magnetic susceptibility x of NdS, NdS* c, NdSe, and NdTe. The dependence of 1/x on temperature obeys the Van Vleok law (1) for Nd . A description is given of an experimental arrangement for a the measurement of x ( T ) wide temperature range. (1) J.H. VAN VXECK, Theory of Electric and Magnetic Susceptibilities, Oxford 1932. N.N. GKIGOBEV, I.II. DYKMAN, and P.M. TOMCHUK, Kiev, Temperature and Mobility of Hot Eleotrons in Polar Semiconductors The distribution function of electrons in doped polar semiconductors in arbitrary electric fields is calculated from the kinetic equations considering the interaction of the electrons with optical phonons and impurity ions and the intereleotronic interaction. For the electron-phonon and inter electronic interaction the relaxation time is not significant. Two branches of solutions are obtained whioh determine the temperature T of the electron gas. The first (stable) branch shows a monotonic dependence of T on the electric field strength F up to a critical value F* at which At this „ aF field F* the mobility ¿a shows a step-like change. Above a definite electron concentration n the value F * increases sharply with increasing n. The dependence of p on F is determined by the competing influences of the Coulomb and lattice scattering meohanisms. For small electron concentrations the mobility decreases with field strength. At sufficiently high electron concentrations the mobility may increase at weak fields. The theory is oompared with experimental results ob13 talned for n-InSb with electron concentrations from 10 J to IK 10 cm . The dependence of the mobility on electric field strength, lattice temperature, and electron concentration is measured. The dependence of the critical field F * on n is determined. A calculation is made of the coefficient p which characterizes the deviation of the field dependence of mobility from the ohmic behaviour.

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Titles and Abstracts, FTT

V.I. VEKSLER and B.A. TSIPINYUK, Tashkent, Seoondary Emission of Exolted Cs Atoms during the Bombardment of Mo and Ta with Past Gs Ions The secondary emission of excited Cs atoms due to the bombardment of plane polycrystalline Ho and Ta targets with fast (800 to 2000 eV) positive Cs ions is studied systematically at various angles of incidence (x). A method is used previously reported in (1). The angular distribution of the number of the secondarily excited atoms n(cp) and that of the root-mean-square value of velooity are obtained for several energies of the primary ions at normal incidence C X " 0 °). The n(