DISLOCATIONS IN IONIC CRYSTALS (Structure, Charge Effects and Interaction with Impurities)

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DISLOCATIONS IN IONIC CRYSTALS (Structure,

Charge Effects and Interaction with Impurities)

R. Smoluchowski

To cite this version:

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DISLOCATIONS IN IONIC CRYSTAL S

(Structure, Charge Effects and Interaction with Impurities)

UniversitC de Paris, Facultk des Sciences, Orsay (Essonne), France

RBsumB.

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Un rtsumk de nos connaissances thkoriques actuelles sur les dislocations et Ieurs crans est suivi d'une discussion des diffkrents aspects des effets de charge dans les cristaux purs ou dopks. On compare les conclusions avec certaines des plus rkentes donnkes expkrimentales. Fina- lement, on rtsume la thkorie et les expkriences sur la tension critique de cisaillement dans ces solides.

Abstract.

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A summary of our present theoretical knowledge of the structure of dislocations and of their jogs is followed by a discussion of the various aspects of charge effects in nominally pure and in doped crystals. The conclusions are then compared with some of the more recent experimental data. Finally a resume is given of the theory and experiments concerning the influence of impurities on the critical shear stress in these solids.

It is well known that a considerable gap exists halides will be treated only if they introduce or illus- between our theoretical and experimental understan- trate essentially new concepts and properties.

ding of individual dislocations and their regular arrays on the one hand and the quantitative aspects of the more random, three-dimensional networks of dislocations in real crystals [l]. While considerable progress is being made [2] to bridge this gap the problem of predicting properties of such crystals is still essentially unsolved. The situation is somewhat better for ionic crystals such as alkali halides than for metals and semiconductors because the former are transparent in samples of reasonable thickness while the latter have to be studied as thin films which, one often suspects, are not necessarily representative of the bulk material. An enormous amount of work

1. Structure of dislocations.

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The fundamental difference between the structure of dislocations in pure metals and in pure ionic crystals stemms from the fact that in the latter one deals with positive and negative ions while in metals charge problems play little role if any. An immediate consequence of the charge balance requirement is the fact that a pure edge dislocation with a (110) slip plane and a12 [lie] Burgers vector can be constructed by inserting two half planes of ions

as illustrated in figure (1) and a pure screw dislocation consists of two helicoidal surfaces one made uplof

positive, the other of negative ions. The fact that bin- has been done and published on dislocations in ionic

crystals and it is not the purpose of this paper to present a complete review of this field. The accent will be placed on certain selected topics such as atomic arrangement, charge effects, critical shear stress, etc. with only illustrative reference to studies of gross mechanical properties. Crystals other than alkali

(*) Permanent address : Solid State and Materials Program, Princeton University, Princeton, N. J. USA. The author wishes

to express his appreciation to the University of Paris for its FIG. 1. -Edge dislocation in NaCl using correct ratio of hospitality, to the National Science Foundation for financial ionic radii and theoretical ionic positions. Configurations I

assistance and to the Fulbright Commission for a travel grant. and I1 differ by b/2. (Whitworth 6.)

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C 3 - 4 R. SMOLUCHOWSKI

ding energy of alkali halides can be satisfactorily calculated using a semi-empirical point lattice model permitted an early calculation [3] of dislocation energies and of the position of ions in the cores. The total energy of a dislocation E is a sum of the energy E, of the elastically deformed region and E, the energy of the plastically deformed inner core. That part of the energy which is due to the elastic deformation around a dislocation is given for NaCl approxi- mately by

E, = ( A

+

B sin2 a) In (Rlr,) (1) where A = 0.393 (0.182) eV and B = 0.122 (0.064) eV per atomic distance along the dislocation line at 80 OK (and at 1 000 OK). Here a is the angle between the dislocation and the Burgers vector, R is the cut-off radius of the elastic region and r, is the radius of the plastically deformed core. The energy E, varies only by about 10 percent as the slip plane rotates, around the fixed Burgers vector, from a (1 10) plane to a (100) plane. The energy of the core is computed by minimizing its energy as a function of position of individual rows of ions parallel to the center of the dislocation and matching it with the elastic solution at larger distances. These calculations are done for the center of the dislocation lying in one or in the other of the two half planes of an edge dislocation and for a screw dislocation coinciding with a row of ions or lying between them. The final result can be expressed as

with typical values of the parameters given in Table I for NaCl at 80 OK :

edge 1.81 X 107 1.37 X 107

screw 1.37 X 107 0.39 X 107 A reasonable value for R is (N n)% where N is dislocation density and r, is of the order of the interionic distance a. For R equal 10 p i.e. for well annealed crystals one obtains total energies of about 5.8 eV per atomic plane which is smaller than the 8 eV/molecule energy of the perfect crystal lattice. These theoretical results have been confirmed for LiF by Gilman and Johnston [4] who measured the minimum size of dislocation half-loops which are stable near the surface. The fact that slip in alkali halides occurs on the (1 10) rather than on (100) planes is according to this calculation the result of a much higher core energy in the

latter case. The elastic energy, as mentioned above, is about the same for both slip planes. Buerger [5] pointed out that slip on (100) planes would be prefer- red because they are more densely packed than the (110) planes but that the former lead to electrostatic faulting : For a glide of one half Burgers vector there is no net binding across the (100) slip plane while there is always binding for a fliO] (110) slip system. An important consequence of the presence of strong Coulombic interactions in alkali halides is the fact that a stacking fault lying in the (110) plane would produce an electrostatic repulsion between nearest neighbors across that plane or would require an absence of every other ion among these pairs of neighbors. In both cases the energy would be prohibiti- vely high and thus one expects that in ionic crystals dislocations do not dissociate, there are no stacking faults and no partial dislocations. In this respect the situation is much simpler than in metals.

2. Structure of jogs.

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As is well known jogs can be formed on dislocations by absorption or creation of vacancies or interstitials and also by interaction of two dislocations. An intersection produces in each dislocation a jog equal to the component, normal to its slip plane, of the Burgers vector of the other dislocation. Depending upon the angle between the slip planes and between the Burgers vectors of the two dislocations the jog consists of adjoining steps in both or in one of the two half-planes which form an edge dislocation. In the first case it is neutral (see Fig. 2), very stable and

FIG. 2. - Schematic representation of a vacancy (V), a neutral jog (B) and a charged jog (H) in the edge of the two half-planes of a dislocation. The plane of solid circles is b/2 above the plane of dotted circles. (Whitworth 6).

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provided by the heat dissipated by the dislocation itself. For practical purposes one can consider the charged jog to be glissile too.

Between two neutral intersection jogs there is only strain interaction but there is a Long range Coulombic interaction between charged jogs on the same dislocation. It is thus favorable for jogs of opposite sign on one of the half plane to move towards each other and to combine. It is also favorable for a charged jog on one half plane to combine with an oppositely charged jog on the other half plane so as to produce in effect a neutral intersection jog. The gain of energy in this process is [7] about 0.2 eV, and it follows that in equilibrium there is preference for neutral jogs at lower temperatures and for charged jogs at higher temperatures.

The mobility of all jogs along a dislocation is small if it requires a net dislocation climb. On the other hand, the mobility of a pair of adjacent equally charged but oppositely oriented jogs is large because it corresponds to a diffusion of a vacancy or of an interstitial along the dislocation without requiring any climb.

3. Charge effects in pure crystals.

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The observa- tion and theory of charged dislocations in ionic crystals are fairly :recent developments. The problem even in pure crystals is complicated and thus the discussion given below will be limited to a few simpler considerations.

a) STATIONARY DISLOCATIONS. - It is well known that the charge density of all point defects ave- raged over a suitably chosen small volume has to vanish. If linear and planar defects such as dislocations, surfaces, etc. which are sinks and sources of point defects are present then they have to be included too. The main consequence, as far as dislocations are concerned, is that in an ideally pure ionic crystal the density of positive vacancies has not to be equal to that of negative vacancies because of the additional charge compensation provided by charged dislocation jogs. Since the energy of formation of a positive ion vacancy g+ is expected to be less than the energy of formation of a negative ion vacancy g- one expects more positive than negative jogs, As a result a dislocation in such an ideal equilibrium condition should be positively charged and be surroun- ded by a negatively charged cloud of predominantly positive ion vacancies. Various aspects of this rather idealized situation have been discussed in detail by Eshelby et a1 [S].

At equilibrium, there cannot be any net flow of vacancies between dislocations and other parts of the crystal. In particular Eshelby et a1 [8] have pointed

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C 3 - 6 R. SMOLUCHOWSKI

detail by Whitworth [6]. In discussing moving dislocations a clear distinction has to be made between old dislocations which may or may not have broken away from the charge cloud and from impurities along their cores and fresh dislocations formed at large plastic deformations which had no time to reach equilibrium with vacancies or impurities. In view of the fairly strong binding energy [l01 of a vacancy to the compression side of a dislocation the first mechanism is quite probable especially at higher temperatures. Since in a pure crystal the concentration of positive and negative vacancies far away from stationary dislocations is equal, the sign of the charge on such a moving fresh dislocation will be determined by the effectiveness with which it picks up vacancies. Bassani and Thomson [l01 have pointed out that a negative ion vacancy is more tightly bound to the compression side of a dislocation than a positive ion vacancy. One concludes thus that this mechanism will lead also to a net positive charge on the dislocation.

4. Charge effects in impure crystals. - All crystals are impure and in particular alkali halides have usually a substantial amount of divalent positive ions, dissolved in them substitutionally, which are compen- sated for by positive ion vacancies. At higher temperatures the number of these extra vacancies is negligible as compared to those required by the usual thermo- dynamic considerations. At very low temperatures the impurities usually precipitate out and the crystal behaves again as if it were pure. At intermediate temperatures on the other hand the concentration of positive ion vacancies is equal to the number of divalent impurities and is thus temperature independent. The actual number of single vacancies in either one of the three ranges depends upon the degree of association between positive vacancies, negative vacancies and impurities.

a) STATIONARY DISLOCATIONS.

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At high temperatures where the number of vacancies is essentially independent of purity of the crystals the charge of the dislocations will be positive as in pure crystals. With lowering of the temperature a point will be reached at which the actual concentration of positive ion vacancies will be equal to the concentration calculated ignoring the requirement of charge neutrality and ignoring the presence of impurities. At this temperature the effective charge on the dislocations vanishes and below it it becomes negative. Finally at still lower temperatures precipitation begins which removes the impurities from the solution and once again the dislocation may become positively charged. The occurence of these two isoelectric temperatures

has been first pointed out by Eshelby et a1 [8] and correlated with experiment which showed that the yield stress of NaCl drops from its high value at very low temperatures reaching a minimum at 3000K, rises to a maximum at 600 OK and then falls again. Parallel observations have been made on AgCl by Slifkin and his collaborators [l11 which indicate that at the isoelectric point in this crystal dislocation damping reaches a maximum.

It is clear that the existence of a binding energy between impurities and dislocations may lead to a direct accumulation of charge of either sign at equilibrium conditions. The charge changes immediately as soon as the dislocation breaks away from the immobile impurities which are most likely situated at or near oppositely charged jogs [7]. In the range of temperatures where the positive ion vacancies form dipoles with the divalent impurities the dislocation may pick up the vacancies producing a cloud of positively charged and rather immobile impurities around it. A net negative charge of the dislocations will be thus expected both in situ and immediately upon breaka- way from the impurities.

b) MOVING DISLOCATIONS. - A fresh dislocation moving through an impure crystal will behave like those in a pure crystal except for the fact that doping may change the sign of the majority of free vacancies and alter the charge picked up by the dislocation on its way. Thus in a crystal in which divalent impurities have introduced an excess of positive ion vacancies the moving dislocation will pick up a negative charge in spite of the fact that, as mentioned above, the negative ion vacancies may have a somewhat higher binding energy to the dislocation core. Conversely by doping a crystal with negative divalent impurities the opposite charge pick-up would be expected. There is, on the whole little chance of picking up charged impurities by a moving dislocation because their binding energy and their mobility are small. Whitworth has pointed out that a stationary divalent impurity may remove a positive ion vacancy from a moving dislocation thus making the dislocation more positive 161.

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the fact that in normal laboratory conditions most dislocations carry a negative charge in accord with estimate of the upper isoelectric point which lies higher than a few hundred degrees C. At higher temperatures the charge becomes positive as expected. The study of the lower isoelectric point is difficult because of our general ignorance of the details of precipitation phenomena in alkali halides. It should be pointed out however that a few experimenters found consistently positively charged dislocations at room temperature. One of the explanation of this difference may be a strong impurity effect, another could be an incomplete knowledge of the actual direction of motion of dislocations. It is important to note in this connection that the observed strain depends on the motion of all, edge and screw, dislocations while the observed charge effect is due only to the motion of the excess edge dislocation of a particular orientation. Thus a proper evaluation of the experimental data is very difficult and often only qualitative. Usually the charge of the dislocations is deduced from the observations of an electric potential caused by a unidirectional or oscillatory deformation of a crystal. Conver- sely a mechanical deformation can be observed as a result of an applied field as it has been done by Sproull [l21 and others. Unidirectional movement is usually produced by making an indentation on a free surface of the crystal. Oscillatory deformation is usually produced by bending and a typical path swept by dislocation is of the order of a few hundred microns. Probably the most direct evidence for the transport of charges by dislocations and for the sign of this charge was obtained by Dupuy et a1 [13]. By spraying the surface of a deformed crystal with a mixture of powdered sulfur and red lead oxide they could determine the sign of the charge carried by the emerging dislocations and even the position of the dislocation source on the slip plane (Fig. 3). They confirm a negative charge on dislocations in LiF at room temperature. Strurnane et a1 [l41 have shown that a prolonged oscillation of a crystal leads to a decreased electric signal. Such an effect may be interpreted as an evidence either for a redistribution of the compensating charge cloud or for a gradual loss of charge by the dislocation because of interactions with stationary impurities and a gradual approach to truer equilibrium. Reasonable assump- tions permit obtaining about 0.4 eV for the binding energies between positive ion vacancies and dislocations.

Davidge [l 51 has made many studies using push-pull deformation which indicate that, as expected, screw dislocations do not carry a charge and that negative

b

0

FIG. 3. -a) Distribution of charges on a slip plane as revea-

led by powder deposition technique, 6 ) Dotted line delineates the birefringent zone around the slip plane, arrow indicates direction of slip. (Dupuy 13.)

ion doping produces a positive charge on edge dislocations. This is in agreement with the results of Rueda and Dekeyser [l61 obtained by an indentation technique assuming that dislocations move away from the indenter. The importance of the knowledge of the active slip system is best illustrated by the bending experiments of Caffyn and Goodfellow [17]. Assuming that more dislocations are moving away from the neutral axis of a bent specimen than towards it they obtained a positive charge of the dislocation. The same data interpreted in term of dislocations generated at the surfaces leads to exactly opposite conclusions [15]. A very careful experiment in which only one slip system was opera- tive (Fig. 4) was made by Frohlich and Suisky [IS]

FIG. 4. -Compression in the vertical direction produces initially slip only on the slip plane G1 leading to charge deposition on faces F1 and F2. At higher deformations slip occurs also on other planes such as G z . (FrBhlich and Suisky 18.)

who confirmed the negative charge of dislocations in pure and in Ca-doped NaCl and a positive sign in OH-doped NaCl.

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C 3 - 8 R. SMQLUCHOWSKI charge was investigated in NaCl by Davidge [l51 who

was able to deduce from it the free energies of formation of positive and negative ion vacancies :

g + = 0.95 eV and g- = 1.17 eV at 0 OK. The charge reversal occurred between 402 and 543 OC as the excess impurity concentration was increased from 5.5 X 10b7 to 7.2 X 1 0 - ~ . Similar temperature variation of the charge was observed by Strumane and De Batist [l41 who found that in NaCl up to about 250 OC and above 550-600 OC the dislocations are positive while in the intermediate range they are negative in qualitative agreement with the theory [7]. The transition temperature of 250 OC agrees well with the onset of associa- tion between vacancies and divalent impurities [19]. The same authors confirm also the energy of formation of vacancies quoted by Davidge. The effect of doping on charge of moving dislocations was studied among others by Rueda and Dekeyser 1161 who found that 5 X 1 0 - ~ of Na202 in NaCl was enough to reverse the sign of the charge from negative to positive. They found also that dislocations in LiF behaved very similarly to those in NaCl while in MgO they had at room temperature an opposite sign. Caffyn et a1 [l71 obtained a sign reversal in NaCl on doping with 3 X 10-" of NaOH. Rueda [20] has shown also that the well known Gyulai-Hartley effect can be accounted for by negative charge transported by dislocations, by a change in the space-charge polarization and by creep. Thus the increase of conductivity upon deformation is not necessarily an evidence for the formation of charge carriers such as positive ion vacancies or free electrons.

There were several estimates [6,14] made of the net charge carried by moving dislocations. Typical values are in the range of 10-3 to 1OP4 esu per cm or one electronic charge per 50 or so atomic spacings along the dislocation. Etch pits count is essential in this type of measurements.

In crystals other than those of the alkali halide group data are available only for AgCl and AgBr. Sonoike [21] has shown that in both these materials dislocations acquire a negative charge on doping with Cd+2 and positive charge on doping with S-'. Slifkin and co-workers [l11 found that at room temperature dislocations in AgCl are negatively charged as expected from the condensation of excess vacancies introduced by metallic impurities. Furthermore there is reversal of charge at the isoelectric point which increases from 20 OC for 0.30 ppm to about 80 OC for 20 ppm impurity content. The energy of formation of a positive ion

vacancy at 0 OK appears to be about 0.6 eV and the formation entropy about 8 OK. The charge density is about one electronic charge per several hundred atomic spacings along the dislocation in a pure (1 ppm impurity) and twice as high in an impure (20 ppm of divalent cationic impurity) crystal. They estimate that in the latter case there is a jog or an impurity every ten, a negative jog every few hundred and a positive jog every 104 atom lengths.

An interesting effect of X-ray irradiation upon charge of dislocations in NaCl was observed by Davidge [22]. He concludes that at room temperature dislocations in irradiated and bleached crystals are originally negative but acquire temporarily a positive charge during vibration. One can interpret these results as a preferential pick-up of negative ion vacancies (bleached F-centers) by the moving dislocation rather than of interstitial halogens which at these temperatures are highly mobile and tend to form rapidly stable complexes. At rest the excess negative ion vacancies escape from the dislocations and the charge returns to its normal negative value. Similar results on LiF were obtained by Dupuy [l31 who concluded that there were about 4 positive charges per 1 000 spacings along dislocation.

6. Charged jogs as electron traps.

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The existence of charges localized on dislocations may lead to other observable effects in alkali halides besides charge transport. Kawamura and Okura [23] have suggested that the influence of deformation on photoconductivi- ty may be explained by photoelectrons falling into 0.2 eV deep traps at dislocations. It would be interesting to see whether this effect disappears when the dislocations are negatively charged. Adair and Squire [24] have recently reported the existence of a magnetic torque associated with lattice defects in X-ray irradiated LiF. They suggest tentatively that the anisotropy of this effect may be related to a non-isotropic distribution of charged dislocations and of the electrons trapped at them. A careful study of the Kawamura and Okura traps and of dislocation orientation and doping may lead to a better understanding of this new phenomenon.

The problem of calculating the structure and the energy of an electron trap at a positively charged jog is very complicated. It is possible however to make very rough estimates. For instance the strength of binding of a hydrogen-like orbit is inversely proportional to the square of the dielectric constant E and thus it is clear

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lation of Mayburg's data [25] suggests that there E may be up to a factor of two lower than in the ideal crystal. The wave function will have probably a rather elongated shape oriented parallel to the dislocation and extending some 10 to 15

k

away from the charged jog (Fig. 5) which permits neglecting the

FIG. 5. - Schematic representation of the probable charge distribution of an electron trapped at a positive jog.

structure of the jog itself. The observed depth of the trap, its cross-section of about 10-13 cm2 and the strong coupling to the dislocation, as required by the magnetic data, seem reasonable on this model.

7. Interaction cf impurities with dislocations. - One of the most widely investigated properties of ionic crystals is their mechanical behaviour in the plastic region. Certain observations can be interpreted in terms of the usual interaction between dislocations and the lattice, or between the dislocations themselves, others require a careful consideration of impurities. The first group of phenomena in general confirms to the usual concepts of the behavior of dislocations in all crystals and is not particularly characteristic of ionic crystals. The turning point in this area was the work of Johnston and Gilman [26] who have shown

that in LiF the yield point is primarily a function of the velocity and of the number of edge and screw dislocations. Since then a great progress in this field has been made but care has to be taken in generalizing results obtained on one alkali halide to other ionic crystals and especially in drawing analogies between them and metals. This has been particularly discussed by Haa- sen and his co-workers [27]. These authors have also studied the dependence of flow stress z on dislocation density N confirming Taylor's relation z

-

N I f 2 .

Creep has been investigated among others by Christy [28] and by Ilschner and Reppich [29] but the phenomenon seems to be complicated and without a gene- rally valid quantitative understanding. The mechanism of strain hardening in rocksalt ceramics has been described in considerable detail by Stokes [30].

The property which appears to be characteristic of dislocations in ionic crystals is their interaction with charged impurities. The best understood is the role of divalent cationic impurities which has been elucidated in a quantitative manner first by Fleischer [31]. He has pointed out that interaction of dislocations with substitutional impurities or vacancies is too weak to account for the experimental facts and that the basic phenomenon is a short range interaction between dislocations and impurity-vacancy dipoles. The latter form large tetragonal lattice distortions which couple strongly to the eIastic fields of the dislocations if the glide planes are one lattice spacing away. On the basis of this model it is possible to obtain good agreement with the observed dependence of the critical shear stress z on temperature, which helps to overcome the coupling between the two strain fields, and on concentration c, which determines the mean distance d

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C-'/' between impurity dipo-

les in the slip plane. In this connection the importance of knowing the state of dispersion of impurities has been pointed out by Johnston [33]. Figure 6 shows

FIG. 6. - Influence of temperature and purity on the critical shear stress in quenched NaCl containing 2 ppm (dotted line)

and 18 ppm (full line) of impurity. See text for definition of

G. (Hess 32 and Frank 38.)

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C 3 - 1 0 R. SMOLUCHOWSKI elastic interactions between dislocations. The activa-

tion energy thus obtained is about 0.6 eV for both concentrations. Fleischer's model is applicable to deformations below room temperature where the defect mobility is negligible. Near room temperature defects begin to move and in particular the dipoles in the proximity of the dislocation can alter their orientation [34] so as to minimize the energy in analogy with the Snoek atmospheres in Fe-C alloys [35]. The critical shear stress is then determined by the force necessary to lift the dislocation from the strain valley. As pointed out by Pratt and al. [36] the depth and half-width of this valley are both inversely proportional to temperature and thus the maximum slope is temperature independent. It follows that the force holding the dislocation is temperature independent and is proportional to impurity concentration in accord with experiment. With increasing temperature two phenomena take place, first the Snoek atmosphere of dipoles can be dragged along by the dislocation and also the impurity-vacancy dipoles become dissociated. Both lead to a rapid drop in the critical stress in accord with observations.

In slowly cooled crystals the situation is entirely different because there the dipoles form aggregates and it is the interaction of these aggregates, rather than of individual dipoles, with the dislocation which determines the mechanical property. This model, worked out in detail by Frank [37], is analogous to Fleischer's model for individual dipoles discussed above. Knowing the degree of aggregation from Lidiard's theory [38] and from the work of Cook and Dryden [39] and assuming that the aggregates form platelets one obtains a correct dependence on temperature and on concentration of aggregates.

i

of clusters

l

FIG. 7. - Influence of temperature on the critical shear stress in slowly cooled LiF containing Mg. See text for definition of X . (Johnston 33 arid Frank 38.)

Figure 7 shows the relatively small influence of temperature up to a point at which the aggregates break up into individual dipoles in Mg-doped Li [33]. Here

X

= z,(Cz,113

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1) where C is a constant. The dissociation of aggregates occurs of course at different temperatures for different materials. The interaction energy between a dislocation and an aggregate turns out to be about 2.1 eV. It is not possible to say whether this is the energy of the elastic interaction between a dislocation and an aggregate or the energy necessary for the dislocation to cut the aggregate. Frank is inclined to the second alternative because of the expected large number of dipoles (50 to 80) in each aggregate. For NaCl containing Ca, after air cooling from 300 OC, it is on the other hand necessary to assume that the most important clusters are those containing only two dipoles. Finally figure 8 illustrates the

I I

400 500 600 Tabs

FIG. 8. - Stress at 0.1 % deformation as a function of temperature in NaCl containing Ca. Curves represent theoretical behavior for edge and screw dislocations assuming mobile Snoek atmospheres and dipole break up. (Pratt 36 and Frank 38.) behavior of stress at 0 . 1

%

deformation, which is closely related to the critical shear stress, in the high temperature region. It is clear that the determining factor is the coupling of the dipoles to screw rather than to the edge dislocations. This is in accord with the fact that, as shown by Johnston and Gilman [26], for the same velocity of motion, screw dislocations require higher stresses than edge dislocations.

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t o a satisfactory model for strain ageing. Both in NaCl [40] a n d in AgCl [41] containing divalent impurities the strain ageing can be accounted for by the diffusion of impurities, rather than of vacancies alone, t o the freshly displaced dislocations.

The above discussion of the effect of divalent cationic impurities o n the critical shear stress of ionic crystals assumes that charge effects d o not influence significantly this property and that only neutral impurity-vacancy dipoles need t o be considered. The reason for this is the fact that, as pointed out by Pratt [42], charge interactions alone are much too weak t o play a role here and also most of the mechanical properties of alkali halides are determined by the behavior of screw dislocations which have been shown to carry essentially n o charge. Koehler et al. [43] o n the other hand tried t o explain much of the mechanical behavior of alkali halides in terms of charge effects using a charge density of 0.12 esu per cm which they obtain from theory. This value is, however, orders of magni- tude higher than direct measurements suggest.

As this survey of some of the more recent developments concerning dislocations in ionic crystals indicates, considerable progress in this area has been achieved. While we are still usually unable t o predict the actual behavior of a particular crystal, neverthe- less the early ambiguities and even contradictions between various models are gradually disappearing and a self-consistent theoretical pattern is emerging.

[l] See for instance LANG (A. R.), Disc. Farad. Soc., 1964, No. 38 and AMELINCKX (S.), NO. 2 suppl. Nuovo Cimento 1958, vol. 7, 569.

[2] AKULOV (N. S.), Phil. Mag., 1964, 9, 767.

[3] HUNTING~ON (H. B.), DICKEY (J. E.) and THOMSON (R.), Phys. Rev., 1955, 100, 1117.

[4] GILMAN (J. J.) and JOHNSTON (W. G.), Tvans. Amev. Inst. Metall. Eng., 1957, 209, 449.

[5] BUERGER (M. J.), Amer. Mineral., 1930, 15, 21, 35. [6] WHITWORTH (R. W.), Phil. Mag., 1964, 10, 801 ;

1965, 11, 83, and private communication. [7] FRIEDEL (J.), (( Dislocations )) Pergamon Press,

Oxford, 1964.

[8] ESHELBY (J. D.), NEWEY (C. W. A.), PRATT (P. L.) and LIDIARD (A. B.), Phil. Mug., 1958, 3, 75. [9] BROWN (L. M.), Phys. Stat. Sol., 1961, 1, 585. [l01 BASSANI (F.) and THOMSON (R.), Phys. Rev., 1956,

102, 1264.

[l l ] SLIFKIN (L.), CHILDS (C.), FUKAI (A.), MCGOWAN (W.) and MILLER (M. G.), to be published.

[l21 SPROULL (R. L.), Phil. Mag., 1960, 5, 815.

[l31 DUPUY (C. H. S.), SCHAEFFER (B.) and SAUCIER (H.), C. R. Acad. Sc. (Paris), 1965, 260, 4481 and Bull. Soc. Franc. Min. et Crist., 1965, 88, 550.

1141 STRUMANE (R.), DE BATIST (R.) and AMELINCKX (S.),

Phys. Stat. Sol., 1963, 3, 1379, 1387 ; 1964, 6,

817 and Sol. State Comm., 1963, 1, 1. [l51 DAVIDGE (R. W.), Phil. Mag., 1963, 8, 1369 and

Phys. Stat. Sol., 1963, 3, 1851.

[l61 RUEDA (F.) and DEKEYSER (W.), Phil. Mag., 1961,

6, 63 ; J. Appl, Physics, 1961, 32, 1799 and Acta Met., 1963, 11, 35.

[l71 CAFFYN (J. E.) and GOODFELLOW (T. L.), Proc. Phys. Soc., 1962, 79, 1285 ; J . AppE. Physics, 1962,

33, 2567 and Phil. Mag., 1962, 7, 1257. [l81 FROHLICH (F.) and SUISKY (D.), Phys. Stat. Sol.

1964, 4, 151.

[l91 DREYFUS (R. W.) and NOWICK (A. S.), Phys. Rev., 1962, 126, 1367.

[20] RUEDA (F.), Phil. Mag., 1963, 8, 29.

[21] SONOIKE (S.), J. Phys. Soc. (Japan), 1962, 17, 575. [22] DAVIDGE (R. W.), J. Phys. Chem. Solids, 1964, 25,

907.

[23] KAWAMURA (H.) and OKURA (H.), J. Phys. Chem. Solids, 1959 8, 161.

[24] ADAIR I11 (T. W.) and SQUIRE (C. F.), Phys. Rev., 1965, 137, A 949 and J. Chem. Physics, in print.

[25] MAYBURG (S.), Phys. Rev., 1950, 79, 375.

[26] JOHNSTON (W. G . ) and GILMAN (J. J.), J. Appl.

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