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TRANSPORT ATOMIQUE ET
THERMODYNAMIQUE DES DÉFAUTSLATTICE DEFECTS IN IONIC CRYSTALS
A. Lidiard
To cite this version:
A. Lidiard. TRANSPORT ATOMIQUE ET THERMODYNAMIQUE DES DÉFAUTSLATTICE DEFECTS IN IONIC CRYSTALS. Journal de Physique Colloques, 1973, 34 (C9), pp.C9-1-C9-10.
�10.1051/jphyscol:1973901�. �jpa-00215376�
TRANSPORT ATOMIQUE ET THERMODYNAMIQUE DES DEFAUTS
LATTICE DEFECTS IN IONIC CRYSTALS
A. B. L I D I A R D
Theoretical Physics Division, A E R E , Harwell, Didcot, Berks, U. K.
Rbsumb. - Cet article de revue se propose de definir le cadre de I'ensemble de la confkrence.
I1 traite de nornbreux domaines dans lesquels d'importants progrks ont CtC recemrnent accornplis, ainsi que des aspects fondamentaux. L'article englobe ( i ) etats, structure et energies des defauts, (ii) defauts ponctuels a ou trks proches de I'equilibre thermodynamique et (iii) la production de dkfauts par irradiation. L'accent est porte plus particulikrement sur les nouvelles techniques experirnen- tales et thkoriques et leurs implications (par exemple, microscopie Clectronique a basse temp6 rature et irradiations impulsionnelles d'une nanoseconde en ce qui concerne i'aspect experimental, rnodkles vibroniques des centres colores et simulations des defauts sur ordinateur en ce qui concerne I'aspect thkorique). L'amelioration de la connaissance de la structure et la production des defauts apportke par ces techniques et celles plus anciennes doivent permettre des progrks rapides dans la comprehension de processus plus complexes (par exemple durcissement par irradiation, diffusion dans les solides a forte concentration de defauts, etc ...).
Abstract. -This review article aims to set the scene for the entire conference. It describes a number of areas where important progress has recently been made, as well as commenting on more established understanding. The review covers (i) defect states, structures and energies, (ii) point defects in or near thermodynamic equilibrium and (iii) the production of defects by irradiation and radiation damage phenomena. Attention is particularly drawn to new experimental and theo- retical techniques and their implications (for example, low temperature electron microscopy and nanosecond pulse irradiations on the experimental side, vibronic nlodels of colour centres and computer simulations of defects on the theoretical side). The growing knowledge of defect struc- ture and defect production which these, and more established, techniques give is likely rapidly to advance the understanding of more complex processes (e. g. irradiation hardening, diffusion in more highly defective solids, etc.).
1. Introduction. - T h e object of this review is t o set the scene for the conference a s a whole, although it must be recognised a t the outset that any brief review of such a wide field must be based on a rather personal selection of illustrations. We d o not, therefore, claim completeness but only t o highlight sonie of the important areas. T h e other invited papers provide additional guidance t o the subject. As starting points we may take the International Conference on Colour Centres in Ionic Crystals (Reading, England, 1971 ; unpublished) a n d the Europliysics Conference on Atomic Transport in Solids and Liquids (Marstrand, Sweden, 1970 ; [I]), although only a part of this second conference was concerned with ionic crystals.
There appear t o have been no comparable meetings o n dislocations in ionic crystals ; thus it was one of the objects of this conference to highlight those aspects of the subject common t o dislocations, colour centres and transport a n d t o empliasise the interplay between concepts in these different but related areas ( I ) .
One may divide the interests a n d material of the present conference into the following areas
(ii) Point Defects in o r near Thermodynamic Equilibrium,
and (iiil Production of Defects by Irradiation and Radiation Damage Phenomena.
Under (i) we include theory, spectroscopy a n d new techniques, especially electron microscopy a n d the study of defect clusters. Under (ii) we include atomic and ionic transport, the solution a n d precipitation of impurities and space charges a t dislocations and interfaces. Lastly, under (iii) we comment upon the production of defects by radiation, the observation of radiation damage and its effect upon mechanical proper-ties. Of course, a n y such div~sion of the field is somewhat arbitrary and one could find themes running across these sub-divisions. F o r example, one sees currently a wide extension of interest a n d understanding t o new materials (e. g. the fluorites, CdF,, KMgF,, etc.) which only a few years ago were very imperfectly understood. We shall, however, use the above division of the subject a s being the most fundamental division convenient f o r this meeting.
(i) Defect States, Structures and Energies,
2. Defect states, structures and energies. -
(1) A number of good reviews on point defects in ionic 2 . 1 TI~IEOKY. - There are two parts to the theoretical crystals are gathered together in volume 1 of Point Dcfccts problem in generai :
in Solids [2]. Other recent reviews on point defects in ~.clation
to transp01.t be found i n volumes I and 2 of physics of (1) the calculation of the electronic states for a
Electrolytes [3]. given defect structure a n d lattice configuration,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973901
C9-2 A. B. LIDIARD (ii) the evaluation of the response of the lattice to
forces exerted on it by the defect, which may be calculated in (i).
For the first of these we need to have a good idea of the principal features of the defect structure in order to limit the calculations to a practicable number of structures, while to evaluate the lattice response we need good physical models of the perfect solid itself. Both calculations are generally quite difficult although in any given situation one may be more so than the other. Thus for the F-centre the most sophisticated part of the effort has up till now largely gone into calculating the electronic states since, with these often rather compact wave-functions, it is a relatively simple matter to handle the lattice relaxation and polarisation. By contrast, in calcu- lations of Schottky or Frenkel formation energies, part (i) is answered by the assumptions of the ionic model so that part (ii) has attracted the most mathe- matical effort. Although, in general, problenis do not divide as sharply as these two examples, it is convenient to order our remarks against these two parts of the problem. For the electronic part we can take Stoneham's review at the Reading Conference as our starting point [4], while for the lattice relaxation calculations the review by Lidiard and Norgett [ 5 ] dating from the same time will be found perti- nent.
Possibly the most basic question engaging colour centre theorists at present is the description of the low-lying excited states of F-centres and the way these depend upon and couple to lattice relaxation.
The states of major interest are (i) the unrelaxed excited state obtained by F-light absorption from the cubic Is-like ground state and (ii) the relaxed excited state obtained from this by subsequent lattice relaxation, and from which luminescent emission eventually occurs. Experimentally it appears that the 2s-like state lies above the 2p-like state in the unrelaxed configuration while in the relaxed excited state the order is reversed, the 2s-like level lying lower. The problem is to identify the nature of these states more closely and to explain the manner of this cross-over. One significant step taken since the Reading Conference is Ham's demonstration [6] that one can solve exactly a vibronic model suggested by the experimental work on Stark effects in F-centre lumi- nescence, namely (non-degenerate) 2s and 2p elec- tronic states interacting via a coupling to a triply- degenerate T , , vibrational mode. Explicit results have so far been given only for the strong-coupling limit, but these do show how a crossing of s- and p-like excited states can occur, while other features of the model are also in qualitative agreement with experimental results on the effects of stress and other perturbations on F-centre luminescence.
Calculations on the thermal activation energies for F-centre motion by Brown and Vail [7] broke
new ground and have recently been extended to the discussion of the motion of FA-centres and to the stability of the tr saddle-point )) co~ifiguration of the type I1 FA-centres in their relaxed excited state 181.
Although the agreement with experiment is quali- tative rather than quantitative these calculations must be regarded as a successful first attack on a group of rather complex colour centre problems.
It is to be hoped that they will encourage others to tackle other complicated centres containing foreign ions, e. g. Z-centres.
Trapped hole centres were reviewed by Itoh at the Reading Conference in both their experimental and theoretical aspects [9]. On the theoretical side, the description of these centres (V,, H, V,, etc.) as halogen molecule-ions X i embedded in a crystal lattice appears to work well on the whole, although a rigorous description of the remaining small covalent interactions with the ions of the lattice has yet to be provided.
As already noted, the evaluation of the lattice relaxation caused by the presence of a centre is often rather simple, e. g. for the ground states of F-centres and thcir aggregates. But this is also often far from being a good or allowable approximation, as in the case of defects bearing a net electric charge or having large elastic misfit (e. g. interstitials). For these defects there are various methods in use [4], [5], most notably (i) the Kanzaki Fourier transformation method ( 2 ) , (ii) the Green's function method of Tewary and (iii) generalised Mott-Littleton methods. All three are in principle general methods but all detailed appli- cations of the first two have so far involved particular assumptions about the symmetry of the defect, although the formulation by Stoneham and Bartram [59]
allows the description of symmetry lowering via the Jahn-Teller effect. An important development of the last year or two is the production by Norgett of a general computer program tr package )) based on the third method (tr HADES D) which is able to handle a wide variety of defects and models. It has so far been used notably for (i) rare gas impurities [5], [lo], (ii) Schottky and Frenkel defects [5], [ I l l , (iii) defect clusters [I21 and (iv) V,-centres [13].
Its considerable value comes from its ability to handle accurately and ecoilomically very low symmetry defects and it can do this because it employs very efficient energy ~ninimisation procedures (rr variable metric)) methods). An example illustrating this comes from the use made of it by Catlow [I21 who shows how it allows us to understand the rather unusual interstitial structures seen in neutron diffrac- tion studies of fluorite compounds. One broad conclu- sion coming out of this work is the readiness of anions in this structure to relax in
<
1 1 1>
direc- tions towards the centres of the empty anion cubes ;( 2 ) Sometimes called the method of lattice statics though
this title has really a wider meaning.
LATTICE DEFECTS I N IONIC CRYSTALS C9-3
this can easily lower the symmetry of defects below one's immediate naive expectation.
Just as these point defect calculations have given considerable insight into structures and processes, so we should expect similar calculations for disloca- tions to be equally valuable, e. g. in answering some of the questions raised in Whitworth's review [14]
(core structure, jog energies, dislocation-impurity interaction, etc.). Here the recent work of Granzer et al. 1151 should be particularly mentioned, while the development of a flexible program package for dislocations by Norgett et al. [I61 should also be noted. Since the models used by the two groups are different, it will be interesting t o compare the sensiti- vity of the predictions of dislocation properties to variations in the assumed physical model. Point defects work has confirmed what has been known for some time in lattice dynamics, namely that pola- risable point-ion models are bad and that it is necessary t o include explicitly the dependence of the Born- Mayer overlap forces upon the state of polarisation of the ions and vice versa, as is done in shell models.
Recent work [lo], [ l l ] also shows how one can (and sometimes must) get away from purely empirical analyses of interatomic forces (such as those of Fumi and Tosi [17] by including information from quantum mechanical calculations of greater o r lesser sophis- tication [IS]. I n general, now that the accurate predic- tion of the response of any given lattice model is feasible, greater attention will inevitably shift towards the determination of better physical models.
In concluding this section on the theory of defect structures, attention should be drawn t o related calculations on planar defects [19], [20]. The work by Stoneham and Durham [20] is a discussion of the interesting phenomenon of the ordering of shear planes which occurs in many non-stoichiometric oxides and which has been of considerable interest to solid state chemists. I t brings out clearly that, because the ordering results from elastic interactions, this phenomenon is analogous to some other macro- scopic ordering phenomena, e. g. void lattice formation and the formation of interstitial solute superlattices in certain bcc metals [21]. Figure 1 shows an example of the void lattice seen in irradiated Mo. Although void formation is currently of niajor interest only in metals, there is no fundamental reason why voids should not also be seen in other materials, including ionic crystals, wherever a mechanism exists for pro- ducing a steady state excess of vacancy over intersti- tial defects [22].
9 . 2 EXPERIMENT. - The greater part of the expel-i- mental material bearing directly on defect states and structure presented at this conference is spectroscopic in nature and exploits the well-tried techniques of' optical and n~icrowave spectroscopy.
We have already mentioned in section 2 . 1 the interest in the nature of the relaxed excited state of
FIG. 1 . - An electron micrograph of zone-refined M o after neutron irradiation at 450 "C showing the bcc void lattice so produced. The upper picture is a bright field image showing the void lattice, projected along a
<
100>
direction. The lower picture shows an electron diffraction pattern from a part of the same specimen, showing the region in the vicinity of the (000) reflection : the f o ~ ~ r extra spots visible are from extra reflections res~~lting from the void array, in this case along the<
01 l>
directions. (After Sass and Eyre [54].)the F-centre. At the Reading Conference Mollenauer and Baldacchini presented data on the ENDOR spectrum of this state in KI [23], although unfortuna- tely there are n o theoretical predictions with which to compare their results for this material. At the present conference they present similar results on KBr, giving clear proof of the diffuse nature of the relaxed excited state in this material too, in agreement with some theoretical predictions. These experiments d o not in themselves tell us whether the state is mainly 2s-like o r 2p-like.
In this spectroscopic field there are also a number of interesting contributions on Z-centres (i. e. those colour centres in alkali halides involving divalent cation impurities), particularly the so-called 2, and Z 2 centres. The interest in these goes back to the early days of colour centre physics, but although the 2 , - centre has been well established [24] as a n associate of an F-centre with an impurity-cation vacancy pair the structure of the Z,-centre remains uncertain.
This centre can be produced by purely thermal conver- sion in the dark and appears quite definitely to be a two-electron centre [25], despite earlier work [26]
on the thermal partition between F- and Z2-centres which appeared to show that it could only involve one F-centre (Fig. 2). Perhaps this work should be repeated in order to establish the consistency ( o r otherwise) with thc two electron models.
Another area in which divalent cations arc impor- tant is in the tlierniol~~minescence of LiF : MgZi and especially dosirneter grade LiF : M g 2
' .
Thc0 - 4 A. B. LIDIARD
FIG. 2. - The absorption coefficient of additively coloured KC1 containing 2.5 x 10-4 mole fraction SrClz showing the thermal equilibrium between the F and the Zz bands at several temperatures : 80 OC (curve 1) ; 120 OC (curve 2) ; 160 0C (curve 3). (All absorption measurements were made at liquid nitrogen temperature following a quench from the anneal temperature.) These results are consistent with a simple parti- tion [F]/[Z2] = 0.8 x 1 0 4 e ~ p ( ~ 0 . 2 8 eV/kT) which would indicate that only one F-centre is contained in the Z r complex.
(After Camagni ct of. [26].)
present conference is no exception in showing an interest in the mechanisms of thermoluminescence in these materials. However, in interpreting thermo- luminescent behaviour in terms of known-defects, e. g. impurity-vacancy pairs and larger aggregates, it may be well to recall the results of Lengweiler et a/. [27] presented at the Reading Conference. These workers analysed the thermoluminescent emission by both wavelength and temperature and so showed that any given glow peak may have more than one wavelength band associated with it ; this could obviously spoil any attempts to find straightforward correlations between total thermoluminescent intensity and other measures of the defects present (e. g.
dielectric relaxation).
Important non-spectroscopic but still rather direct techniques for obtaining information about point defect symmetry and structure are those methods which yield the force tensor P i j of the defects, notably the diffuse X-ray and neutron scattering methods [28], [29]. This force tensor is
where P m is the force exerted on atom m of the lattice located a t position Rm ( i and j Cartesian components).
For defects with cubic symmetry P i j is a scalar
while tetragonal and orthorhombic defects have respectively two and three independent components.
For all defects the macroscopic volume change caused by a random distribution of molar fraction c in a cubic crystal is
where K is the compressibility of the crystal and Vc is the cell volume. Techniques for measuring P i j have been developed particularly at several centres in Germany and at the present Conference Balzer et a/. [30] show how P i j for H- and V,-centres are obtainable from macroscopic dimension changes when the defects are aligned optically. These and other results bearing on P i j should be of direct inte- rest t o theoreticians in view of their close relation to the forces on the atoms around the defect and thus to the basic potential models used (eq. (1)).
One can point to current conflicts between theory and experiment here. First, the inference that the inter- stitial anion is a tetragonal centre [31] is difficult to understand even though explicit calculations have only been made for cubic and trigonal configura- tions 1321. Secondly, the inferred net outwards relaxa- tion around anion vacancies [31] appears to conflict with the long-range nature of the electrostatic forces exerted by such a defect on the ionic lattice [33].
One of the most important recent experimental developments I believe is the elegant work of Hobbs et a/. [34] on the application of electron microscopy to the study of radiation damage in alkali halides, made possible by their development of the liquid helium stage. Although electron microscopy has played a major role in the study of lattice defects and radiation damage in metals and other materials, its application to the alkali halides has been effectively prevented by the very highly damaging effect of the electron beam on these materials arising, of course, from their sensitivity to ionisation. The liquid helium stage not only significantly reduces damage rates in some halides but, more importantly, it so inhibits interstitial mobility that the point defects produced do not aggregate in the times needed to make good electron microscope observations. These studies of the structure and formation of dislocation loops in irradiated alkali halides are likely to have a substan- tial impact on many parts of the subject (e. g. inter- stitial aggregation, effect of irradiation on mechanical properties, etc.). One of the surprising results is the observation that the dislocation loops formed by defect aggregation at both room temperature and very low temperatures are perfect interstitial loops (Fig. 3), even though all the defects produced by ionisation are produced in the anion sublattice. It simply does not seem possible to account for the observed absence of stacking faults in these loops on the basis of anion interstitials alone. The authors have thus proposed a mechanism of loop growth in which cations also join the extra half-plane in such a way that the resultant effect is the growth of the loop
LATTICE DEFECTS IN IONIC CRYSTALS CO-5
< ~ I o > The suggestion therefore seems both necessary to
t
/<I 00) explain the electron microscope observations and tobe energetically feasible. It should be of considerable interest to both theory and experiment as providing another (( molecule-like )> colour centre, the identifica-
3. Point defects in or near thermodynamic equili- ther- (i) atomic and ionic transport, e. g. ionic conduc-
(a) tivity, self- and impurity-diffusion, motional effects in nuclear magnetic resonance, etc.,
<!lo> (ii) relaxation phenomena associated with ther-
<TOO> mally activated reorientation of complex defects,
e. g . dielectric and inelastic relaxation of impurity-
FIG. 3. - Schematic diagram showing (a) the arrangement of ions in a perfect interstitial loop with Burgers vector 4
<
110 >a n d ( b ) the sort of configuration which would result from the mechanism proposed, i. e. a perfect loop of Burgers vector
3
<
110>
with a number of halogen nlolecules embeddedin vacancy pairs in the vicinity. (After Hobbs c,t a/. [34].)
and the formation of anion vacancy-cation vacancy pairs each containing a halogen molecule X, formed by the association of two H-centres. In other words, the resultant effect is that each pair of H-centres ((digs its own hole )), the dug-out material (anion plus cation) joining the loop. Whatever the details of the mechanism by which this is accomplished, the process appears certainly t o be exothermic. Thus, in the case of KCI, existing calculations of the energies of H-centres and of vacancy pairs show that 3-4 eV will be liberated per molecule added to the loop (").
( 3 ) This figure can be estimated froni the energies needed
for the following steps :
(i) form a vacancy pair adding the anion and cation to the edge of the dislocation loop (requires 2 1.3 eV ; [35]),
(ii) remove 2 H-centres froni the crystal giving two neutral CI atoms in a state of rest a t infinity (gains 2.8 to 3.1 eV : [MI),
(iii) combine the two CI atoms to give a Clz molecule in a state of rest a t infinity (gains 2.5 eV).
(iv) insert the C12 molecule into the empty vacancy pair (this term has not been calculated yet hut o n the basis of cspc- rience with rare gas atoms trapped into vac;tncics and vacitnc!
pairs, we would not expect the energy required to be more thlin about 0.5 eV).
defect pairs,
-
(iii) the solution and dissolution of impurities, (iv) space charges a t dislocations and interfaces, important for charged dislocations, the photographic process, ionic thermopower, etc.
All these aspects are represented at the present conference. This area of study has made considerable progress over the years on the basis of rather simple ideas of defect structure, often only the basic geome- trical features of co-ordination and symmetry being required. As we turn to more complex materials, however, these simple models may need modification, especially when paired o r clustered defects are involved, e. g. in UO,,,, Y3+-doped alkaline earth fluorides and in Fe, -,O.
3 . 1 TRANSPORT. - Basic transport properties such as ionic conductivity, self- and impurity-diffusion depend on products of defect mobility with defect concentration or, more generally, with some function of defect concentration. Defect mobilities and ju11ip rates are largely intrinsic properties of the defect, rather little influenced by interactions at long range and sometimes even surprisingly little affected by the presence of another defect close by (4). Mostly, defect jump rates are represented by an Arrhenius forniula
where Ag is the free energy barrier between the neigh- bouring sites available to the defect. Such a formula follows from a generalised rate theory based on classical statistical mechanics; it would thus be expected to hold at high temperatures (T 9 (I,,) where the distributions of atomic positions and velocities conform to classical statistical mechanics.
A number of examples exist where H' has been observed
( 4 ) Conipai-c c. g. the j u m p fi-equencies of free cation vacan-
cies in alkali halides with thosc of cation \lacarlcics moving around divalent cation impi~rities in thc first co-ordiniition sliell [35].
These f o ~ ~ r terms thus add up to a gain of hetween i a n d 4 cV.
to conform t o (4) over a wide range of temperature.
The present conference heard again, however, of the inconsistency of the high- and low-temperature data on anion vacancy migration rates with (4), a problem first raised by the measurements of Hartel and Liity 1371 several years ago (Fig. 4). It is perhaps still
FIG. 4. - A n Arrhenius plot of the jump frequency, lo (s-1).
of anion vacancies in KC], illustrating the contrast between those results inferred from high temperature experiments (diffusion and ionic conductivity) and those inferred from colour centre conversion experiments a t low temperatures [37]. The activation energy - a(ln ul)/%(l/kT) is only 0.6 eV at low tem- peratures but 0.9 eV or higher at high temperatures. For sinipli- city we show only the original results of Fuller [55] in the high temperature region ; these were obtained by analysis of the C1 self-diffusion coefficient in pure and doped KC1 into single vacancy and vacancy-pair contributions. There have been several later careful analyses of these diffusion data and of ionic conductivity data leading t o similar or somewhat higher activation energies than those characterising Fuller's results
shown here [56].
not absolutely clear whether the circumstances of the anion vacancy migration observed in these low- temperature colour centre conversion experiments are such that the inferred mobilities should corres- pond exactly with those obtained by analysis of the high temperature transport measurements. But if not, a quantitative explanation which preserves the agree- ment with (4) has still to be provided.
In this connection, we should recall that we do not expect (4) necessarily to hold in the region of the Debye temperature and below ; and very many defect jump processes do of course take place in this low- temperature region. Despite some initial attempts, a satisfactory quantum theory of barrier-limited motion is still lacking. A satisfactory quantum theory of small polaron hopping does, however, exist and this formalism has been applied to V,-centre motion and to the motion of light interstitial impurities in bcc
metals [38] ; in these cases the hindrance to motion is the energy of self-localisation of the particle (electron hole, interstitial hydrogen, etc.) and not the existence of a large potential energy barrier between initial and final sites. Figure 5 shows the dependence of jump rate upon temperature for hydrogen in bcc Ta ;
this is one well substantiated case of quantum diffusion.
FIG. 5. - An Arrhenius plot of the diffusion coefficient of H in Ta showing the curvature which results from quantum effects
as formulated by Flynn and Stoneham [38].
- 5
- 6 -7- .-. u, -8- a
.
$ -9
V -10 n 0 -11
l7 -
0 -12-
-
-13- -14
So far we have said nothing about defect concen- trations except to note the dependence of macro- scopic transport coefficients upon them. This func- tional dependence is mostly direct as, for example, in ionic conductivity, self-diffusion and diffusion of substitutional impurities by means of vacancies : the greater the appropriate defect concentration the larger the macroscopic coefficient. In the case of interstitial solutes, however, vacancies can act as traps, so that the relation of diffusion coefficient to vacancy concen- tration is inverse in this case ; the most important systems of recent interest showing this behaviour are rare gas impurities in ionic solids [39]. In all cases, however, the concentrations of defects in thermody- namic equilibrium are strongly dependent on purity through (i) the existence of Schottky and Frenkel product relations between the concentrations of conjugate defects and ( i i ) the electroneutrality require- ment imposed everywhere except in the immediate vicinity of sources and sinks of defects by the long- range and strength of Coulomb forces between defects carrying net charges. This, predictable, depen- dence on purity allows one to separate out the diffe- rent contributions to diffusion, conductivity, etc., by deliberate doping ; elegant exalnples of the utility of this technique i n the present conference will be found in the contributions by Bknitre [40], by Ftlix [39]
and a nuriiber of others.
-
H IN TANTALUM
-
- -
-
-
3 . 2 STATISTICAL MECHANICS. - TO proceed from the basic knowledge of defect and atomic mobilities and concentrations to observable lnacroscopic quan- tities requires a kinetic theory. This has been reviewed
0 . 5 10 I O ~ I T
LATTICE DEFECTS IN IONIC CRYSTALS C9-7
in detail in recent years 1411-[43]. It may be noted, however, that the correspondence of these treat- ments with the generalised phenomenological flux equations of the type used in the thermodynamics of irreversible processes has not yet been fully worked out, despite the useful insights which this can bring.
This approach can be especially useful when diffusion involves the interplay of solute and defects and the defect population is separately affected ; examples are provided by Fredericks' work on double doping [44]
and by the effect of surfaces on defect population and consequently upon solute behaviour [45], [46].
One of the important tests of these kinetic theories provided by ionic crystals is through the ratio of ion mobility t o diffusion coefficient, ),ID, which in general is not given by the simple Nernst-Einstein value q / k T (where q is the ion charge). The departure of AID from q/kT depends on the mechanism of transport and has been calculated for the commonly studied lattices and defect mechanisms [43]. A parti- cularly satisfying comparison of this sort obtained from the work of Corish and Jacobs [47] is shown in figure 6 . This shows Ag self-diffusion coefficients
FIG. 6. - An Arrhenius plot of the self-diffusion coefficient of Ag in AgCI. The solid circles ( 0 ) are experimental points of Weber and Friauf [57] and the open circles (0) are those of Reade and Martin [58]. The solid line is a theoretical line cal- culated using parameters obtained by analysis of the ionic conductivity of a single specimen of AgCI. (After Corish and
Jacobs [47].)
in AgCl (i) calculated fro111 an analysis of the ionic conductivity of AgCl (into vacancy and collinear and non-collinear interstitialcy jumps) and (ii) as measured experimentally. The close agreement of the two emphasises two points. First, the suf5ciency of simple, largely geometric models of the defects involv- ed, e. g. Ag+ interstitials occupying body-centred sites and moving by replacement Jumps (Fig. 7).
Second, the need to include long-range interactions in the statistical therrnody~laniic expressions used in analysing the ionic conductivity. I n this work and elsewhere these interactions are generally anal!scd
FIG. 7. - A section of the AgCl lattice showing possible modes of interstitial displacement. The interstitial ion shown in the botton left-hand ( f , f , f ) position moves to the lattice posi- tion ( 4 , 4, 4) and the ion which was at that position moves to one of the three interstitial positions shown. For Agf interstitials in both AgCl and AgBr displacement of both ions collinearly
(model 1 ) requires the lowest energy of activation.
by an application of the Bjerrum-Fuoss theory of liquid electrolyte solutions, i. e. one separates out the neutral defect pairs and then treats the long-range Coulomb interactions among the defects bearing net charges by Debye-Hiickel theory. This appears to work well in practice but it is intuitive in formulation.
Allnatt et al. [48] have recently developed more funda- mental analyses based on cluster expansion methods : this shows when one may expect the simpler analysis to be accurate and when one should expect to find departures from it.
F r o m the point of view of statistical thermodyna- mics, defect complexes are merely part of the distri- bution function. But they are generally strongly bound and remain together for many jump-times of the component defects so that they have a separate physical identity as we see in many phenomena, e. g. Z,-centre formation, impurity diffusion, dielectric relaxation, etc. Knowledge of impurity complexes built up in this way provides the basis for understand- ing the kinetics of impurity aggregation and precipi- tation.
3 . 3 SURI-ACES A N D DISLOCATIONS. - We have already remarked that the requirement of overall electroneutrality imposed on the concentrations of defects in the bulk does not hold near to sources and sinks of defects [49]. In these regions a space charge exists as a result of the generally different energies of formation of defects of opposite sign. The thickness of these regions, the Debye screening length, is very simply related to the defect concentrations : it is
\vhere t i i is the numbcl- concentration of dcf'cct i bearing net charge (1, a n d i:,, in thc static diclcctl-ic
constant. Some typical results are shown in figure 8.
These ideas have been used to provide theories of space charges a t surfaces a n d a t dislocations, generally assuming that these can act a s infinite sources and
THICKNESS OF DOUBLE LAYER IN
x
vs. TEMPERATURE IN OK200 NaCL +CONCENTRATIONS OF IMPURITY C.
FIG. 8. - Calculated temperature dependence of the thickness,
AD, of the space charge layer at surfaces and dislocations in NaCl containing mole fractions of divalent cations as indicated.
The usual changeover from intrinsic to extrinsic defect popula- tions is reflected in the behaviour of Ln too.
sinks, i. e. that there are effectively n o limitations associated with jog formation (on dislocations) o r step formation (on surfaces). Surface space charges contribute t o the thermoelectric power of ionic crystals [50] and there appear to be n o limitations on the infinite sink model here. An important prediction of the infinite sink models is the existence of an isoelectric temperature, Ti,,, where the space charge vanishes ; this depends directly on the presence of aliovalent impurities and the sign of the charge cloud is different above and below Ti,,. This is an important prediction of the model and one confirmed in a num- ber of experiments. However, a n infinite sink model is certainly a simplification and the consequences of making more realistic assumptions are described notably in Whitworth's review [14]. There is a need t o extend the precision of experiments which can give information about isoelectric points. On the theoretical side, we need to know more about the interactions of point defects and dislocations in these materials. Little has been done here apart from the early calculations of Bassani and Thompson [51].
Developments such as those now being made by Granzer, Norgett and co-workers should, in due course, be capable of making the necessary pre- dictions.
4. Production of defects by irradiation. - S o far we have discussed defects in alkali halides and other ionic crystals together, since the principles governing their properties are the same. When we come to their production by irradiation, then the singular nature
of the alkali halides sets them apart, however. This sensitivity of the alkali halides to ionisation to give halogen interstitials and vacancies as the primary defects, is still incompletely understood. The basic idea of the Pooley-Hersh mechanism provides a phenomenological explanation of a number of important facts. This mechanism postulates the pro- duction of the primary defects as the result of a
<
110>
replacement collision sequence along a row of anions following upon the radiationless recombination of a n electron with a V,-centre (self- trapped hole). It provides a model showing how atomic displacements arise in the anion sublattice as a result of ionisation (electron and hole (V,) production) and explains the different temperature dependence of damage production in the different materials and the effect of electron trapping impurities (e. g. Pb) in reducing damage rates. Also, at the present confe- rence, Townsend provided a n explanation of the so- called Rabin-Klick diagram for the efficiency of F - centre production on the basis of this mechanism.However, there are some important difficulties. Firstly, actual calculations of the minimum energy necessary for the propagation of a
<
110>
replacement colli- sion sequence in the anion sub-lattice gave values several times higher than the maximum available from electron-\/, recombination. Secondly, the very beau- tiful fast pulse ( - nanosecond) irradiation experi- ments show that the primary products are not anion vacancies and anion interstitials, as in the Pooley model, but instead are principally F-centres and H- centres [51]. At the present conference, Itoh and Saidoh [53] suggest a means by which the replacement sequence can carry the hole along with it, so that the end product is an H-centre with a n F-centre remaining at the site of the original V,-centre. In this proposal, actual electron-V, recombination does not occur.It will be very interesting to see if this proposal can be made quantitative.
Turning now to secondary processes, we have already commented on the importance of the direct observa- tion of clusters of interstitial defects by Hobbs r t
al. [34]. The consequences for the understanding of the effect of radiation on mechanical properties are further described at the present conference by Hobbs and Howett. In particular, it is very interesting t o see the explanation of previously reported dependence of flow stress on F-centre density in terms of the observed morphology of the loops. This dependence, seen as [F]li2 in some materials and [F]'I4 in others, differs according to whether the loops are round and circular as in K I o r whether they are elongated (i. e.
effectively linear defects) as in some other alkali halides.
5. Conclusion. - I n this brief introductory review, we have touched on a number of important topics - though by no means all - in the study of lattice defects in ionic crystals. The interaction between
LATTICE DEFECTS I N IONIC CRYSTALS C9-9
the different parts of this study is real and fruitful low temperature electron microscopy) show that this and will undoubtedly become more so as fundamental study remains an exciting area of fundamental solid understanding of defect structure, production, mobili- state physics.
ties and mutual interactions extends. The stimulus
provided by new theoretical techniques (e. g. the Acknowledgments. - I a m grateful t o my collea- computer simulation of defect structure) and new gues at Harwell, particularly Drs. A. M. Stoneham experimental techniques (e. g. fast-pulse irradiations, and M. J. Norgett, for their comments on this review.
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DISCUSSION
R. V. HESKETH. - Would you regard a curved Ta was simply to show the extent of the curvature Arrhenius plot for self diffusion as a symptom of which does arise when quantum effects are domi- quantum effects, quite generally ? nant. I would emphasize however that this is not a A. B. LIDIARD. - NO, not generally since there system where a large classical potential energy barrier are many well understood effects which can lead to separates neighbouring equivalent defect positions.
greater or lesser curvature in Arrhenius plots. My A quantum theory of such systems has yet to be object in showing the results for H diffusion in bcc.
