Physica status solidi: Volume 1, Number 7 1961 [Reprint 2021 ed.] 9783112500200, 9783112500194

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plrysiea status solidi

BAND 1 • HEFT 7 • 1961

Contents Page

1. Review Article A.

SEEGEB

Recent Advances in the Theory of Defects in Crystals

669

2. Original Papers M . G . A . B E R N A R D a n d G . DURAPFOURG

Laser Conditions in Semiconductors H.

PUFF

Zur Theorie der Photoelektronenemission von Metallen, Teil I I , Die Verteilungen der Photoelektronen

E.

GUTSCHE

699

704

Über den Einfluß hohen Druckes auf die Grundgitterabsorptionskante von Kadmiumsulfid-Einkristallen

716

Z . GYULAI, E . HABTMAKN u n d B . JESZENSKY

Zerreißfestigkeitsmessungen an NaCl-Nadelkristallen (Whiskern)

726

K . W . BÖER u n d U . KÜMMEL

Vorprozesse des Wärmedurchschlages, Teil I I I , Charakteristische Inhomogenitäten und ihre Kinetik in elektrisch hoch belasteten CdS-Einkristallen H . BEBGER

730

Über das Ausheilen von Gitterfehlern frisch aufgedampfter CdSSchichten, Teil I

739

P . BRAUER u n d M . KOLB

Über das „Anreiben" von Alkali- und Erdalkali-Halogenid-Thallium-Leuchtstoffen

758

W . H A U B E N B E I S S E R , D . L I N Z E N u n d E . GLAUCHE

Über die Richtungsabhängigkeit der kritischen Mikrowellenfeldstärke bei Nebenresonanz in Mangan /Magnesium /Cobalt/FerritEinkristallen 8. Short Notes (listed on the last page of the issue) 4. Pre-printed Titles and Abstracts ol Original Papers to be published in this or in the Soviet journal ,,H3HKa TßepHoro TeJin" (Fizika Tverdogo Tela).

764

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, K. W. B Ö E R , New York, W. D E K E Y S E R , Gent, ~W. F R A N Z , Hamburg, P. G ÖR 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. NÉ EL, Grenoble, A. P I E K A R A, Poznan, N. R I E H L , München, 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 NN, Karlsruhe, G. S Z I G E T I , Budapest, J . T A Ü C , Praha Editor-in-Chief K. W. BÖER, New York Advisory Board M. B A L K A N S K I , Paris, P. C . B A N B U R Y , Reading, R . B E R N A R D , Paris, W. B R A U E R , Berlin, W . C O C H R A N , Cambridge, 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 , Birmingham, H. K. H E N I S C H , Reading, 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 , Cambridge Mass., M.MATYA'S, Praha, H. D.MEGAW, Cambridge, T. S. M O S S , Camberley, E. N A G Y , Budapest, E. A. N I E K I S C H , Erlangen,

L. PAL, Budapest, M. RODOT, Bellevue/Seine, B. V. R O L L I N , Oxford, H . M . R O S E N B E R G , Oxford, K . M . V A N V L I E T , Minneapolis, R . Y A U T I E R , Bellevue/Seine

Volume 1 • Number 7 • Pages 667 to 774 and K 147 to K 182 1961

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

•

B E R L I N

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Review

Article

Max-Planch-Institut fiir Metallforschung, Stuttgart, und Institut fur theoretische und angewandte Physik der Technischen Hochschule Stuttgart

Recent Advances in the Theory of Defects in Crystals By ALFRED SEEGER

List of Contents 1. Introduction I. General 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Continuum

Theory

of Lattice

Defects

Continuum approach versus atomistic approach The basic equations of the linear theory of internal stresses The geometrical fundamental equation (linear theory) Examples for the incompatibility tensor and the dislocation density tensor The calculations of internal stresses (linear theory) The analogy to magnetostatics The double force tensor and the elastic dipole Extension to non-linear theory — Survey Riemann-Cartan dislocation geometry Non-linear elastic theory Propagation of elastic waves in a self-stressed medium with non-linear elastic properties II. Application

to Special

Dislocation

Problems

13. Application t o scattering problems 14. The calculation of self-energies and interaction energies 15. Application to extended dislocation a) Cross slip of extended screw dislocations in f. c. c. crystals b) The effect of jog lines on the energy of jogs in extended dislocations c) Cutting of an extended dislocation through another dislocation d) Comparison with experiments and experimental determination of stacking fault energies 16. Approach t o ferromagnetic saturation 17. Stored energy of dislocation arrangements 18. The macroscopic crystal density and the second-order elastic strains 19. Scattering of phonons (lattice vibrations) and low temperature heat conductivity III. Point Defects; 20. 21. 22. 23. 24.

Calculations

Based on Electron

Theory

and on Atomistic

Models

General discussion Electrical resistivity due to dislocations The charge condition The energy of vacancies in monovalent metals Application t o various defects a) Binding energy of a vacancy pair in noble metals b) Energy of high-angle grain boundaries in metals c) Line energy of jog lines 25. A general method for treating the effect of strains in atomic models with application t o interstitial atoms 26. Remarks on the electrical resistivity of point defects 45*

670

A.SEEGER 1.

Introduction

The present report is an extension and up-to date version of lectures and mimeographed notes presented at the summer school on "Defects in Crystalline Solids" held in 1959 at the University of Cambridge, England. The director of the summer school, Professor N. F. MOTT, had asked the author to emphasize in t h a t course the possibility of carrying out quantitative calculations on defects and of obtaining reliable numerical results on activation and interaction energies etc. This feature has been preserved in the present paper. The paper is not intended to be a report on the literature of the field, but it a t t e m p t s rather to present in a coherent and unified way some recent developments with which the author and his collaborators have been associated. References will frequently been given to summaries or to recent papers which reflect the present situation of the field. References to earlier work can easily be obtained from these. For the general background on the physical properties of defects in crystals reference may be made to "Imperfections in Crystals" by H . G. VAN BUEREN [1], The present subject divides itself in a natural way into two main p a r t s : Theoretical approaches which can be applied to all crystals, and methods which are designed specifically for particular classes of crystals, e. g. metals, ionic crystals, homopolar crystals etc. The first p a r t almost coincides with the continuum theory of lattice defects, in which the individual properties of the crystals and their chemical and crystallographic nature enter essentially through the elastic constants. The second p a r t comprises the methods based on the atomistic pictures of the various types of crystals. I n order to yield quantitative results, the models used in such calculations must be sufficiently detailed. They will therefore only apply to certain classes of crystals, e. g. metals, ionic crystals, valence crystals. More often t h a n not the models are valid for still narrower classes only, e. g. monovalent metals, noble metals, or alkali halides. For metals the appropriate discipline to base an atomistic theory of defects on is the electron theory of metals. For ionic defects we have similarly a starting point in the socalled Born theory of ionic crystals. Due to the author's interests, and in order to preserve a certain unity of presentation, the atomistic part of the present paper will only deal with metals. A summary of the older applications of Born's theory to lattice defects in ionic crystals has been given elsewhere [2], Recent years have brought a number of detailed calculations on ionic crystals (by F . G. FUMI and his school as well as others), without changes in the foundations of the subject, however. Very little work has so far been done on semi-conductors. A promising approach to a theory of point defects in germanium and silicon has recently been published by SWALIN [3]. The organization of the paper reflects the subdivision just discussed. I n p a r t I we shall treat the general theory of the continuum approach. P a r t I I will give the application of the continuum theory to special problems. P a r t I I I reports on the calculations based on electron theory and on models for point defects. I. General Continuum Theory of Lattice Defects 2. Continuum

approach

versus

atomistic

approach

A physicist usually associates with the concept of a particular defect in a crystal, e. g. a dislocation, a vacancy, or an interstitial, a model involving a certain arrangement of atoms or ions. However, due to the complicated force laws between atoms, the quantitative treatment of any realistic model of the atomic

Advances in the Theory of Defects in Crystals

671

a r r a n g e m e n t of such a defect is necessarily r a t h e r involved. H e a v y numerical computations, o f t e n applicable t o one specific m e t a l or ionic c r y s t a l only, are f r e q u e n t l y r e q u i r e d t o o b t a i n reliable n u m e r i c a l r e s u l t s . I t is t h e r e f o r e d i f f i c u l t t o arrive in t h i s w a y a t q u a n t i t a t i v e r e s u l t s of general validity. T h e c o n t i n u u m a p p r o a c h neglects entirely t h e a t o m i s t i c s t r u c t u r e of t h e defects. I n i t s simplest f o r m it considers t h e defects as point, line or s u r f a c e imperfections e m b e d d e d in a n elastic m e d i u m . T h e elastic s t r e s s e s caused b y t h e m are n o t d u e t o e x t e r n a l loads; t h e y are t h e r e f o r e called internal stresses. I n t e r n a l stresses a n d s t r a i n s h a v e been considered for a long t i m e in connection with t e m p e r a t u r e stresses a n d similar problems of self-strained elastic bodies. T h e y h a v e come into prominence only recently, however, in connection w i t h t h e t h e o r y of defects in crystals. T h e continuum t h e o r y t o be p r e s e n t e d in t h i s p a p e r will be a l m o s t entirely a t h e o r y of i n t e r n a l stresses a n d strains. T h e special f e a t u r e of t h e absence of applied loads h a s lead to c o m p u t a t i o n a l m e t h o d s specifically a d o p t e d t o i n t e r n a l stresses. The role of t h e loads as t h e " s o u r c e s " of t h e stresses is t a k e n over b y t h e "incompatibilities" of t h e s t r a i n fields, which a r e t h e source f u n c t i o n s for t h e i n t e r n a l stresses. T h e i n t r o d u c t i o n of t h e incompatibility enables u s t o e x t e n d t h e concept of a n imperfection as a singularity in a n elastic m e d i u m t o continuously distributed defects. If we consider a single defect, e. g. a dislocation line, it is in general n o t possible t o f i n d t h e continuous distribution of incompatibilities associated w i t h it w i t h o u t r e s o r t t o an atomistic t r e a t m e n t of t h e dislocation core. E v e n so, t h e t h e o r y of continuously d i s t r i b u t e d defects is m o s t h e l p f u l in a c t u a l calculations, since it helps t o avoid or t o h a n d l e t h e divergencies which a r e associated w i t h t h e singularities of t h e stress fields. A p a r t f r o m i t s i n a d e q u a c y in t h e core of t h e defects, t h e r e a r e a n u m b e r of o t h e r shortcomings of t h e c o n t i n u u m approach. F o r p u r p o s e s of illustration, we consider t h e p a r t i c u l a r topic of t h e interaction of lattice v i b r a t i o n s w i t h defects. Acoustic v i b r a t i o n s w i t h wave-lengths long c o m p a r e d t o t h e lattice p a r a m e t e r of t h e crystal a r e v e r y well described b y t h e c o n t i n u u m t h e o r y . Processes peculiar to s h o r t wavelengths, in p a r t i c u l a r U m k l a p p - p r o c e s s e s , c a n n o t be t r e a t in t h i s way, since Umklapp-processes a r e n o t possible in a c o n t i n u u m . (The c o n t i n u u m m a y be looked a t as a crystal of vanishing lattice c o n s t a n t ; t h e r e f o r e no reciprocal lattice exists.) I n c r y s t a l s with m o r e t h a n one a t o m in t h e s m a l l e s t u n i t cell we c a n n o t t r e a t t h e optical vibrational m o d e s b y t h e t h e o r y of elasticity, even if t h e y h a v e v e r y long wave-lengths. A t f i r s t sight it m i g h t a p p e a r t h a t a serious l i m i t a t i o n t o t h e use of elasticity t h e o r y in t h e t h e o r y of crystal defects is t h e f a c t t h a t m o s t c r y s t a l s follow t h e laws of elasticity only u p t o applied stresses which a r e several o r d e r s of m a g n i t u d e smaller t h a n t h e theoretical shear s t r e n g t h , which is of t h e o r d e r one t e n t h t o one t h i r t i e s t of t h e shear m o d u l u s 0 . Long before t h i s t h e o r e t i c a l s t r e n g t h is reached, t h e crystals either d e f o r m plastically, i. e. p e r m a n e n t l y , or f r a c t u r e . H o w e v e r , b o t h t h e plastic b e h a v i o u r a n d t h e f r a c t u r e of crystals are d u e t o t h e s t r e s s concent r a t i o n s a t a n d t h e motion of imperfections in t h e c r y s t a l s or a t t h e c r y s t a l surfaces. T h e imperfections themselves, however, a r e e m b e d d e d in a p e r f e c t m a t r i x which t o t h e b e s t of our knowledge behaves elastically u p t o t h e stresses arid s t r a i n s corresponding t o t h e theoretical s t r e n g t h of t h e m a t e r i a l . T h e r e f o r e t h e t h e o r y of elasticity is well suited t o be applied t o t h e s t u d y of d e f e c t s in crystals. Most of t h e c o n t i n u u m calculations on defects a r e b a s e d on t h e linear t h e o r y of elasticity. T h e reason for t h i s is m a t h e m a t i c a l simplicity, resulting f r o m t h e

672

A.SEEQER

validity of the superposition principle. The linear theory of elasticity is the appropriate and often employed tool to calculate the interaction between defects in crystals which are not too close together. As a matter of fact, it is not rarely used even under conditions of large strains or small distances between defects where one would not a priori expect the linear theory of elasticity to be applicable, with results that at least do not contradict other evidence. There are a number of problems, however, in which the linear elasticity theory is completely insufficient. As a matter of course, to these problems belong the questions as to the detailed atomic arrangement in the core of the defects, which can only be answered by atomistic treatments. In other problems the failure of the linear elasticity is not due to the use of the concept of the continuum but due to the linearity of the theory and the validity of the superposition principle. Fortunately, in the majority of these cases it suffices to go one step further, i. e. to use the second-order theory of elasticity. Among the second order problems that have been solved by the use of the quadratic theory of elasticity are the effect of dislocations on the macroscopic density of crystals and the scattering of lattice vibrations of long wavelengths by dislocations. 3. The basic equations

of the linear theory

of internal

stresses

[4,5]

Sections 3 to 8 will be based on the linear theory of elasticity. We assume that the relation between the stress tensor a and the strain tensor € is given by Hooke's law a — C • • e, (1) where C is the rank four tensor of the elastic constants. Furthermore, throughout sections 3 to 8 the strains are considered to be infinitesimal. The condition of static equilibrium is D i v o = V - o = —/, (2) where / is the density of the external volume forces. The classical way of solving problems with non-vanishing / is to introduce a displacement field s, related to the strain tensor e by1) e = Def s = y ( \ / s + s V ) .

(3)

to insert (3) and (1) into (2), and to obtain the equilibrium condition in terms of displacements. In static problems associated with defects, we are mainly interested in problems without external forces, i. e. / = 0. In regions containing dislocation lines a univalued and continuous elastic displacement field cannot be defined. Therefore, c cannot in general be defined by (3). This statement is equivalent to saying that in the presence of dislocations the de St. Venant conditions of compatibility V x £ X V = 0 are violated. It is natural to introduce the incompatibility tensor

(4)

r) = Ink E = V x e X V as a measure of the degree of violation of the compatibility conditions (4).

(5)

We use the Gibbs notation of vector analysis in section 2 to 7 («V means the transpose of V s ) and the tensor calculus with the Einstein summation convention in section 10 to 12.

673

Advances in the Theory of Defects in Crystals

Y) is the source function for the internal stresses that are present in dislocation problems in spite of / = 0. The »¡-approach to dislocation stresses is indispensable in problems involving continuous dislocation distributions. 4. The geometrical fundamental equation (linearized theory) [6, 7, 8]

In oder to replace (3) by a more general expression, we define the (second rank) distorsion tensor (3 by ds = dr • (3 ,

(6)

where ds and dr denote the differences in displacement and position of two neighbouring points, ds is not necessarily a perfect differential. The strain tensor € is defined as2)

+

(?)

which reduces to (3) if (4) is satisfied. The elementary definition of the Burgers vector b is b = 6 ds = $dr-p e e

.

where E refers to the Burgers circuit. Applying Stokes theorem to

6 = J dF • a .

(8) we obtain (9)

The (second rank) dislocation density tensor a is defined as follows 3 ): db = dF • a .

(10)

Here dF is a vector characterizing an infinitesimal surface element in the crystal. db is the resulting Burgers vector of the dislocation lines (of infinitesimal strength) threading the surface element. By comparison of (8), (9) and (10) we obtain the geometrical fundamental equation a = V x P = Curl (3 ,

(11)

which is the counterpart to and just as important as (2). From (11) we have [9]

V • « = Div a = 0

(12)

as the continuum formulation of the fact that dislocation lines cannot end in the interior of the crystal. Comparing (5) with (11), we obtain the following relation between the dislocation density tensor and the incompatibility tensor: YJ = Sym { a X V } = — Sym{Curlot} .

(13)

rj is a symmetrical tensor of rank two, which has, on account of (12), three independent components. From (12) it follows that V • »1 = 0 .

(13a)

2 ) (J is in general an asymmetric tensor, p is the transpose of (3. In some sections (16,17) w e shall use ~ to denote Fouriertransforms. W e hope that the context helps to avoid misunderstandings. 3 ) The tensor of the dislocation density was first introduced by J. F. NYE [9]. The original definition differs somewhat from the one used here.

674

A.SEEGER

5. Examples

for the incompatibility

tensor

r), and the dislocation

density

tensor

a

A symmetrical interstitial atom is often represented by a "center of dilatation", i. e. a finite amount A V of volume is inserted at a point r0. From (7) we have (/ = second order unit tensor, d(r) = three dimensional ¿-function) r) = AV ( V X / X V) defined by o — Ink X = V X X X V (18) X is determined by the fourth order partial differential equation that is obtained by inserting (18) via (1) into (5), and that contains t] as the source function. For an isotropic homogeneous medium (Poisson's ratio v, shear modulus G, modulus of compression K ) this equation can be written as with the subsidiary condition where

2G

X

V 4 X ' = >1

(19)

Div x ' = 0 ,

(20)

'=

X

-^--

2 X i

I

(21)

(Xr = first scalar invariant of x)- (19) can be replaced by V 4 X = V = 2G(y, +

T

-^

i ? i

l).

(19a)

In an infinite medium, the solution of (19) satisfying (20) is X'M = — ^

f f f *](*•') \r — r'\ dxr,.

(22)

The Airy stress function x used in the elementary treatment of straight edge dislocations parallel to the z-direction is proportional to and therefore but a special case of the present approach.

Advances in t h e Theory of Defects in Crystals

675

For a singular dislocation (22) gives (£jki = Levi-Civita tensor) X'a

=

S o 71

y

m

fe"

b

i

$ \r — r'\dLt. L

(22a)

The integral is to be extended over the dislocation line. The preceding theory has been extended to anisotropic media [12]. However, it has not yet been possible to give a general expression, valid for all crystal classes, for the differential operators occuring in the anisotropic theory. Special considerations have to be carried out for each particular symmetry. The necessary formulae are given explicitly for the important case of cubic symmetry, however [12]. 7. The Analogy

to

magnetostatics

K R O N E R [13] has developed a most useful analogy between the theory of internal stresses and strains as described in sections 2 to 6 and the theory of the magnetic field of distributions of stationary electric currents. Table 1 contains a list of the corresponding physical quantities, differential operators, and equations. We hope that this table is understandable without any further comments (see also the review

a r t i c l e b y DE WIT [ 1 0 ] ) . Table 1 Correspondences in elasticity a n d magnetism Elasticity vector quantity tensor r a n k two tensor r a n k four Div Ink Div I n k = 0 Def I n k Def = 0 Burgers vector b incompatibility tensor r) strain tensor € stress tensor a stress function tensor yj elastic constants C (or Q, K) displacement s equation (3) equation (5) equation (17) equation (18) equations (19), (19a) equation (20) equation (22)

Magnetism scalar quantity vector tensor r a n k two div curl div curl = 0 grad curl grad = 0 current I current density J magnetic intensity H magnetic induction B vector potential A permeability /t scalar potential if H = grad yi curl H = J div B = 0 B = curl A V2 A = —fiJ div A = 0

4 ^71

Jf Jf Jf lr — r ' \

The analogy between elastostatics and magnetostatics may be extended further by introducing the concepts of parelasticity and dielasticity and of elastic polarizability [14, 15, 16]. We shall illustrate these concepts by giving examples.

A.SEEGER

676

Interstitial carbon atoms in a-iron are well known to give rise to the Snoek effect. Such a carbon atom occupies one of three different kinds of sites, located on the x-, y-, or z-cube edges in the body-centered cubic crystal. It is surrounded by strains of tetragonal symmetry which are larger in the direction of the cubeedge on which the atom is located (from now on called the direction of the interstitial atom) than in the direction perpendicular to it. In an otherwise stress-free crystal the interstitial atoms are randomly distributed over the x-, y-, and z-sites. An external stress will in general favour energetically one of the types of sites over the others by increasing the separation of the two iron atoms neighbouring the, say, «-sites of the carbon atoms. If the temperature is high enough to enable the carbon atoms to jump by thermal activation from one site to the next, we get a larger population of the x-sites than of the y- or z-sites. The redistribution and reorientation of the carbon atoms gives rise to an additional "anelastic" strain which adds to the "elastic" strain that was obtained as the reaction to the applied stress with fixed directions of the carbon atoms. These anelastic relaxation effects, and the internal friction and magnetic effects associated with them, are known as the Snoek effect of carbon-atoms in a-iron. Since the reorientation of the carbon atoms increases the elastic strains, a crystal containing such defects is called "parelastic" in analogy to paramagnetism and parelectricity. The carbon atoms are called elastic dipoles; their preferred direction is called the direction of the dipole. However, as we shall see in section 8 in line with the analogies listed in table 1, an elastic dipole is not described by a vector but by a second-order tensor. Elastic dipoles have been discussed earlier, although under different names [17, 18, 19]. A recently discovered example is that of interstitial atoms in f. c. c. crystals, e. g. Ni interstitials in a nickel crystal [20, 21, 22], Examples for dielastic materials are crystals which contain inclusions or inhomogeneities with elastic constants that are different from those of the matrix. An applied stress will induce elastic dipoles at the interface between the matrix and the inclusion. The strains associated with these induced dipoles will, generally speaking, enhance the total strain under a given stress, if the inclusionis softer than the matrix (e.g. a void or a lattice vacancy); they will reduce the total strain if they are harder, e. g. they have larger elastic constants than the surrounding matrix. A full discussion of the inclusion problem has been given by ESHELBY [23], Both elastic dipoles and induced dipoles at inclusions can be described quantitatively by the concept of the elastic polarizability [14, 4]. This concept has been used to calculate the relaxation strength of the relaxation effect due to the reorientation of the interstitial dipoles in f. c. c. crystals [24], 8. The double force

tensor

and the elastic

dipole

In sections 2 and 3 we have mentioned that imperfections in crystals give rise to internal stresses, i. e. stresses without resulting external forces. However, a very useful concept is that of a double force, which goes back to BOTJSSINESQ and is discussed in some detail by LOVE [17]. Fig. 1 shows that double forces without moment can be generated by combining two or four forces of equal magnitude but different directions in such a way that neither a resulting force nor a resulting torque ensues. If we let the magnitude of the forces tend to infinity and at the same time the distance between the points of application of the forces shrink to zero in such a way that the product of the two quantities remains finite, we obtain a second rank tensorial quantity which is called the double force tensor. For a

677

Advances in the Theory of Defects in Crystals

fixed cartesian system of coordinates the double forces of the type shown in Fig. l a correspond to the diagonal components Pi{ of the double force tensor, whereas the off-diagonal components Pq = P 3 i (i 4= j) correspond to arrangements of the type of Fig. l b . The double force tensor P has been successfully used by various authors as the continuum analog of defects of the type of the carbon interstitial in «-iron as discussed in section 7. P may be split in an invariant way into two tensors, the 3

deviator P' with zero trace, and the trace 2J P a times the unit tensor I. The B d r r = f f f

y)b • X A

dtr .

(43)

In analogy to magnetostatics we write EAB

=

bA MAB

bB

(44)

where M A B is the (in general asymmetric) dislocation inductance tensor. Equations (43), (44), and (22) show, that M A B can be written as a double line integral over A and B. K R O N E R finds

Mf* =

s s O 71 i j k n p i X

2-—

§§ (Vj Vpf) AB

dL„ dLi + dL% dLi + dLi dLf dnk

(45)

A variety of other forms of this equation are possible. (45) has been evaluated for a number of special forms of the dislocation lines. A general formula can be obtained, if one of the dislocations, say A, is straight. In connection with problems occuring in the theory of cross-slip of extended dislocations (see section 15), this formula has been integrated in the special case that B is composed of arcs of hyperbolas [40]. The preceding equations can be extended to an arbitrary number of interacting dislocation lines by summing over the superscripts A and B. To avoid divergenphysica

684

A.SEEGER

cies in the calculation of self inductances, in principle one has to use a continuous Yj-distribution in the dislocation core. To a good approximation, however, we may put MAA = y MAB,

where A and B are two dislocation lines separated by a

small cut-off distance r0, which may depend on the character (orientation) of the dislocation line. 15. Application

to extended

dislocations

[40,

41]

In the theory of plastic deformation it is important to study the thermally activated processes connected with intersecting of dislocation lines, jog formation, and cross-slip. In most crystals a theoretical treatment of these processes is not feasible without a detailed consideration of the atomic arrangement in the dislocation cores. I f we deal with extended dislocations, however, the main contributions to the formation and activation energies can be calculated from the elastic interactions of the partial dislocations, using the method outlined in section 14. For small separations of the partial dislocations, this method has to be supplemented by considerations based on Peierls' model (for a detailed description of Peierls' model, see e. g. [2]). a) Cross slip of extended screw dislocations in f. c. c. crystals [40, 42] The activation energy Uq for cross slip has been calculated as a function of the stacking fault energy y, the number n of piled- up dislocations, and the applied < 20 • I0~3 the interpolated stress r. For 20 < n < 102 and 4 • 10~3 $'|

h ca (h a>)3 cos # 0

Eç—W 4 jg^

— 0)

=

{4 E9 /S»(0 0 , (p)

•

(12)

8 ) Die relativen Verteilungen des normalen Photoeffektes stellen im Hinblick auf die experimentelle Unkenntnis der optischen Konstanten die sichersten Aussagen der Theorie dar. 9 ) Es sei daran erinnert, daß die Unstetigkeit der berechneten Energieverteilung an der oberen Grenze Folge der Beschränkung auf T = 0° K ist; bei endlichen Temperaturen fallen die Kurven stetig auf Null ab. 10 ) Daß der Anregungsprozeß eine untere Grenze der Energieverteilung zur Folge hat, die über der Photonenenergie liegt, ist bereits von M A Y E R und THOMAS [7] bemerkt worden. Das dementsprechend? Pehlen langsamster Elektronen wird von METHFESSEL [9] als typisch für den Volumenphotoeffekt dünner Schichten angesehen, jedoch scheinen uns die dort angeführten Messungen diese These nicht überzeugend zu untermauern. 11 ) Die Berücksichtigung indirekter Übergänge führt nicht zur Anregung von Elektronen mit Energien unterhalb der Photonenenergie. 12 ) An dieser Stelle ist ein interessanter, von D I C K E Y [8] zitierter Deutungsvorschlag zu erwähnen, demzufolge an der Entstehung des Überschusses langsamer Elektronen Prozesse beteiligt sein könnten, bei denen ein Photon simultan von zwei Elektronen absorbiert wird; derartige Prozesse lassen sich offenbar auch bei freien Elektronen keineswegs allein mit Hilfe der Erhaltungssätze ausschließen. Wir können über ihre Bedeutung hier nichts aussagen; ihre Berücksichtigung würde den Verzicht auf die Ein-Elektronen-Näherung im Anregungsprozeß erfordern und damit unter anderem auch in die Theorie der optischen Konstanten wesentliche Mehrteilchen-Aspekte bringen.

712

H . PUFF

Für den Grenzwert dieser Größe für E - > 0 ergibt sich wegen ** 2 f*XK

= n . (14) Entsprechend ist die Verteilung in der Ebene senkrecht zur Einfallsebene bebestimmt durch üy = 0

bzw.

9> = y >

9» =

3

y-

(lö)

Aus (4) folgt sofort, daß die Energie-Winkel-Verteilungen des normalen Effektes nicht nur in diesen beiden, sondern sogar in jeder beliebigen Ebene symmetrisch sind (sie sind aber natürlich nicht axialsymmetrisch in bezug auf die OberflächenNormale). Dagegen ist die Verteilung des selektiven Effektes nur in der Ebene

^) E> X I

,

E+W)

für a) Ef + h co — W > h (o) =

T

< h c o < ( h wYX,

(2(/iF—[/^.i

l j + F{E,

E+W

T

(2 fW —

0, Eq— W ,

< h m < (h tu)5Sn,

für E$ — W > E > 0 , / S

(2 fW -

< h co n = y H ,

(7a)

mM = y M ,

(7b)

mex

He,

(7c)

yHD,

(7d)

= y

a>D = (os = -

m = z

(7e)

y h M

M'

m+ = mx +

Miel = m ) f c i ;

l

J

r-

i Ttiy — (m~)*

=

£

i

m+ m" =

= x'y>z

i — a

fur

a

k(l)

(7f)

ßikx

k mz = / l — m+rnr

==1—

1—

ak.(t)

ap^k(t)

« a*k

=

™kx + i m-ky

;

= y

™kx

+

*] für

a L k = mkx — i m k y ;

k ^

(7h)

0,

m k y = — [ak — aL*] (7i)

und erhalten mit der magnetodynamischen Gleichung (1) unter Vernachlässigung des Landau-Lifshitz-Dämpfungsterms f ü r die zeitliche Ableitung der relativen transversalen Magnetisierungskomponente m+ die Gleichung: ak eikt

=

~

W M

% & kt

= i £

E ¥

[I

( a k

+

l - - £ a k , a p kk' -

-

comZ

^

a

a,—k'"

2 Jfc"

k"

ak'

(10)

Jfc'"2 '

+k"'—k'

(.fc*0) wenn Terme höherer Ordnung in ak entsprechend der Suhlschen Theorie vernachlässigt werden 7 ). Dabei gilt: 40 B

=

0

%

= ° f

+

° f

(Nx

+

Ak"

=

(Nx

+

+

N?

+

Nl> -

N f

N

+

N

-

N?

-

P

k"2

y

y

+

K

+

NP)

~

(13)

mit K = ßof (y»0/2),

¡10 = ©in (y 0 /2),

A0 = o)Q ßof y>0,

Ba ~ co0 ©in y>0

0

(14)

ergibt sich nämlich als spezielle Lösung für das linearisierte inhomogene Gleichungssystem unter Berücksichtigung des Landau-Lifshitz-Dämpfungsterms der Ausdruck: wo

«„((*)

— ^ 60 -j- irj0 Rechnung getragen, wo rj0 = a a> = y AH0 gilt, und AH0 die Halbwertsbreite der 110-Mode bedeutet. Entsprechend der Suhlschen Theorie gehen wir nun von der Lösung (15) des linearisierten inhomogenen Gleichungssystems (9) aus und lösen das homogene Gleichungssystem (10) f ü r die k" =(= O-Spinwellenamplituden ak>r der Nebenabsorption cok" = ~ k" ©in (y)k") e 2 i s V ' ,

tg®k"

fk",

=

2g fk"

o»t// = [AI = /(coH —

(20a)

(20b)

= \ Bk,,\!Ak>,,

\Bk„\*fV Wjf + coex l2 k"2) (toE — N, a>M + wex l2 k"2 + coM sin 2 ©*"), sin

= \ k'i\!\k"\

(20c)

das folgende Gleichungssystem: b(%,

= 6-k"

ai+)' bfl

(21)

e—

2®—Ä;"]'

für die Spinwellenamplituden bk), der Nebenabsorption. Dieses Gleichungssystem geht mit den Spinwellenamplitudenansätzen: i [cu/2— cojc//]t — i[a>/2 — 0)_j.//]6 bf), = Uk„(t) e (22) b%' = ULk„(t) e in die äquivalente Differentialgleichung 2. Ordnung d2 Ii?

= 0

(23)

für die Spinwellenamplitude Ukn über, aus der nach der Suhlschen Theorie die kritische Mikrowellenfeldstärke ermittelt wird. Dabei gelten mit (15) und (20) die Beziehungen (24) a0(t) = « eial + a(0-> e—iat, mit den Abkürzungen ßo (25a) «o+) = 0Js (co0 —co) + irj0 (co„ + co) — i r]0 4 und !eH2 = 2 co*/.

)

= Kn s

1 (cu—co0 ) + irio

i com fa" (n" k"*—Xk" k"

sin@t"cos0j;"

1 (/cm mit Hilfe einer Hochdruckapparatur und einer Versuchsanordnung durchgeführt, die in (1) genauer beschrieben worden ist. Das Ergebnis .einer solchen Messung ist in Fig. 1 dargestellt, in der der Phötometerstrom als Funktion der Wellenlänge aufgetragen ist. Scharparameter ist der Druck. Man sieht, daß sich die Auslöschungsmaxima in gleicher Weise mit steigendem Druck zu kürzeren Wellen verschieben wie die Absorptionskante. Die Auswertung ergab eine innerhalb der Meßgenauigkeit von der Ordnung der Streifen unabhängige Verschiebung zu 1) Von GOBRiCHT und BARTSCHAT (2) wurde diese Methode für eine genaue Untersuchung der Anisotropie des Brechungsindex von OdS benutzt.

Stört Notes

K151

kürzeren Wellenlängen um -(8,4- + 0,3) ' 10"3

cm" 2 .

Dieser Wert liegt zwischen den in (1) ermittelten Werten für die Verschiebung der Absorptionskante für parallel bzw. senk2") recht zur hexagonalen Achse polarisiertes Licht J . Hieraus

\[mu]

Fig. 1: Einfluß hydrostatischen Druckes auf die durch Anisotropie des Brechungsindex von OdS hervorgerufenen Interferenzstreifen im Absorptionsspektrum kann man, ohne auf weitere Einzelheiten wie Ermittlung der Ordnung der einzelnen Streifen usw. näher einzugehen den Schluß ziehen, daß der Druckeffekt des Brechungsindex im untersuchten Spektralbereich innerhalb der oben angegebenen Meßgenauigkeit lediglich in einer entsprechenden Druckverschiebung der Resonanzstelle (der Absorptionskante) besteht. Den Druckeffekt der Absolutwerte der Brechungsindizes für den ordentlichen bzw. den außerordentlichen Strahl kann man daher offenbar in guter Näherung dadurch ermitteln, daß man von bekannten Werten der Brechungsindizes von OdS (2, 3) ausgeht und diese Meßkurven gemäß den in (1) bestimmten Werten für die 2) Der Einfluß der mit der Druckerhöhung verknüpften Dickenänderung der Probe auf diesen Wert kann vernachlässigt werden.

K152

physica status solidi I

Verschiebung der Absorptionskaxite mit dem Druck parallel zu kürzeren Wellenlängen verschiebt. Literatur (1) E. GUTSCHE, phys. stat. sol. 716 (1961). (2) H. GOBRECHT und A. BARTSCHAT, Z. Fhys. 156. 131 (1959). (3) S.J. CZYZAK, W.U. BAKER, R.C. CRANE und J.B. HOWE, J.opt. Soc.Am. 42, 3, 240 (1957). (Received November 12, 1961)

Institut für Magnetische Werkstoffe der Deutschen Akademie der Wissenschaften zu Berlin, Jena, Forschungsgemeinschaft Widerstandsanomalien von Magnetit verschiedenen Oxydationsgrades oberhalb des Umwandlungspunktes Von H. SCHRÖDER Obwohl über die elektrischen Eigenschaften von Magnetit am "Umwandlungspuhkt" bei -155 °C eine ganze Anzahl von Untersuchungen vorliegen, sind Temperaturmessungen des elektrischen Widerstandes bis zur Curietemperatur bisher nur von MILES und Mitarbeiten^ 1) veröffentlicht worden. Diese Autoren fanden bei 60 °C ein Minimum und bei 530 °C ein Maximum des elektrischen Widerstandes. Abweichend hierzu ist Messungen von DOMENICALI (2) ein Widerstandsminimum bei 90 °C zu entnehmen, während sich bei den von SAMOCHWALOW und FAKTDOW (3) bis 100 °C durchgeführten Messungen noch keine Anzeichen für einen Minimalwert des elektrischen Widerstandes bemerkbar machen. Nach DOMENICALI (2) wird das Widerstandsminimum, analog den Untersuchungen von BUSCH und LABHART (4) an Siliciumcarbid, auf eine beginnende Entartung des Elektronengases infolge der hohen "Störstellenkonzentration" zurückgeführt. HAUBENREISSER (5) deutet die gleiche Erscheinung auf der Grundlage des ZENERAustausches als Elektron-Schallquanten-Kopplungseffekt. Im Rahmen eigener Untersuchungen über das Halbleiterverhalten von Ferriten bei verschiedenem Oxydationsgrad wurden auch an polykristallinem Magnetit einige Messungen durchgeführt. Es sollte geprüft werden, in welcher Weise der Oxydationsgrad

Short Notes

K153

bzw. der Fe 2 + -Gehalt die Temperaturlage des Widerstandsminimums und -maximums beeinflußt. Da die Probenwiderstände im allgemeinen sehr gering sind und die relativen Widerstandsänderungen - H ^ ^ ) / ! ^ ^ nur wenige Prozent betragen, wurden alle Anlaßbehandlungen, die zu Probenzuständen mit unterschiedlichem Oxydationsgrad führten, in der Meßanordnung vorgenommen, so daß während sämtlicher Versuche die elektrischen Kontakte unberührt blieben. Dadurch war eine exakte, relative Erfassung dei? verschiedenen Probenzustände gewährleistet. Allerdings mußte dabei in Kauf genommen werden, daß die bei etwa 600 °C an Luft durchgeführten Oxydationen nicht als thermodynamische Gleichgewichtsreaktionen abliefen und demzufolge die Proben (theoretisch) nicht vollständig homogen sein konnten* Nach Beurteilung von Bruchgefügen dürfte es sich jedoch nur tun geringe Inhomogenitäten handeln. Die Messungen erfolgten nach dem Spannungsabgriffverfahren mit einem Präzisionsgleichstromkompensator. Während der Versuche wurde ein Vakuum von 10 J bis 10 Torr aufrechterhalten; die Reproduzierbarkeit bei steigender und fallender Temperatur sowie bei Wiederholung war gewährleistet» Die erhaltenen Meßdaten sind in Fig. 1 wiedergegeben. Probenzustand I stellt den analytisch ermittelten .Ausgangszustand "Fe-jO^" mit 24,0 (+ 0,1) Gew-% F e 2 + dar, der durch "partielles" Abschrecken von 1350 °G bei 10""'' Torr erhalten wurde. Die Zustände II, III und IV besitzen in dieser Reihenfolge jeweils geringere Fe 2+ -Konzentrationen. Eine Abschätzung des Zustandes IV ergab etwa 16 Gew-% F e 2 + . Bekanntlich bleibt die Spinellstruktur vom stöchiometrischen Magnetit mit 24,1 Gew-% F e 2 + bis zu 16,6 Gew-% F e 2 + unter Bildung von Kationenleerstellen erhalten. In Fig. 1 ist außerdem die auf ballistischem Wege aufgenommene Temperaturabhängigkeit der Magnetisierung bei einer Aussteuerung von 60 A/cm für den Ausgangszustand I eingetragen (Kurve V), Die Versuche zeigen, daß mit steigendem Oxydationsgrad das Minimum nach höheren und das Maximum nach tieferen Temperaturen verschoben wird. Auffällig ist, daß sich die beiden Ertremwertfolgen symmetrisch zueinander verschieben und mit

physica status solidi I

K154-

den Daten von MILES (1) verträglich sind. Die Verschiebung der Minima nach höheren Temperaturen steht in Übereinstimmung mit einer gleichsinnigen Verschiebung der "Entartungstemperatur" bei Erniedrigung der Trägerkonzentration bzw. mit einer relativen Vergrößerung der "Sprunglängen" im'Dif fusionsmodell"

o

200

m

s-frj

m

Fig. 1: Kurven I bis IVs Temperaturabhängigkeit des spezifischen Widerstandes $ für polykristalline Magnetitprobe in vier Oxydationszuständen. Kurve I 24,0 Gew-% Fe^ + , Kurve IV ca. 16 Gew-% Fe^ + . (Die Zahlenwerte an den Meßkurven bedeuten Absolutwerte des spezifischen Widerstandes in Ohm-cm.) Kurve V: Temperaturabhängigkeit der Magnetisierung M in willkürlichen Einheiten bei 60 A/cm Aussteuerung Die mit Annäherung an den Curiepunkt verursachte Abnahme des (positiven) Temperaturkoeffizienten des elektrischen Widerstandes mit nachfolgendem Vorzeichenwechsel kann man durch Verringerung und Aufhebung einer Austauschwechselwirkung zwischen 3