POINT DEFECTS PRODUCED BY IRRADIATION AND THEIR ANNEALING IN IONIC AND COVALENT CRYSTALS BY F. P. CLARKE Atomic Energy Research Establishment, Harwell, Berks. Received 8th February, 1957 Atoms are displaced by interaction with energetic particles and point defects are pro- duced. The diffusion of these defects to normal lattice positions may be described mathematically, using as basis a simple physical model, and the description obtained in this way accords with certain experimental data. On this treatment the activation energy for movement of any type of defect is considered to be a discrete quantity. Another method of describing these annealing phenomena is based upon the as- sumption that the rate processes follow a type of equation familiar in gaseous kinetics. It appears that if this description is to accord satisfactorily with certain experiments a distribution of activation energies has to be assumed.These different treatments are contrasted. MODELS FOR THE DISORDER PRODUCED BY HEAVY PARTICLE MOVEMENT THROUGH A LATTICE A heavy charged particle penetrating a solid may lose its energy (i) through inelastic collisions in which there is excitation of the electronic system of the atoms or ions, or (ii) by elastic collisions in which there is momentum transfer to the nuclear mass of the stationary atoms. For high particle energies the losses will be overwhelmingly by excitation ; for lower energies elastic collisions will predominate, but there is not necessarily a sharp cut-off separating these processes, which actually overlap.For a carbon atom moving in diamond for example, ionization losses may be ignored for energies of the moving atom below about 104 eV. The figure will be higher the higher the atomic number of the moving atom, and for potassium in potassium chloride it is about 105 eV. To move an atom from its position in the lattice some minimum energy will be required. Determinations of the minimum energy have not been made for covalent or ionic solids, but the figure is of the order 25 eV. This is based on an estimate by Seitz which has worked out fairly well for metal systems. Hence energy transfer to the nuclear mass which exceeds this minimum value may result in an atomic displacement within the lattice. For uncharged particles such as neutrons, elastic collisions with the nucleus may be made, and at these a considerable proportion of the energy of the particle may be given up.If this energy is greater than -25 eV a displaced atom may result. If the energy is greater than twice this, the knocked-on atom may itself displace other atoms and continue to do so until its energy is used up. With the range of neutron energies met with in a typical pile experiment, the knock-on energy for atoms with atomic numbers greater than 10 is well below the range at which ionization losses might occur, and so only elastic collisions need be con- sidered in interactions between the primary displaced atom and the atoms remaining in the crystal. The maximum energy which a knock-on can lose by non-relativistic elastic collision is given by EM = 4M1M2 2E, where E is the energy of the incident (M1+ M2) atom and hfl its mass.M2 is the mass of the initially-stationary atom. For low energy encounters it is assumed that the collisions are equivalent to those 141142 DEFECTS PRODUCED BY IRRADIATION between hard spheres of a suitably chosen radius.1 In this case collision will impart any energy between Em and zero with equal probability. Hence a two- dimensional representations of the way in which a primary knock-on might dissipate its energy is shown in fig. 1. The incident charged particle makes a collision at A, and a lattice atom is knocked-on to make a further collision at B. Here the atom gives up some random amount of its energy to a second atom, and both proceed to make further collisions. At some collisions, such as C, only sufficient energy is imparted to cause one displacement and an interstitial atom results.In general it is to be expected that the ratio of interstitials to vacancies will be fractional near the point of primary knock-on and greater than unity at the outermost parts of the region of disorder. FIG. 1. interstitial - track of interstitial 0 vacancy - - - track of incident particle. It will be noticed that in the figure the path of a knocked-on atom is shown longer near the point of primary impact. This is because the probability of collision varies inversely with the particle energy, and at around 100 eV or so a particle will make a collision about once every atomic spacing. As the number of incident particles, and hence the number of disordered regions, increases, a point will eventually be reached when the regions of disorder begin to overlap appreciably. When considering the annealing kinetics both of these possibilities have to be considered separately.An alternative model for the distribution of displaced atoms is based on the following arguments. It has been stated that the energy required to force an atom from its place in the lattice to an interstitial position is probably about 25 eV. This energy is required to overcome the potential saddle separating the normal lattice atom and the interstitial. Once over this saddle the atom will drop into a potential trough and an amount of energy, depending on the trough depth, will be released. Alternatively, if an amount of energy less than that required for displacement be given to an atom, this will be given up to the lattice as vibrational energy.In either event energy transfer by elastic collisions may cause a localized rise in temperature. This will operate over a spherical region surrounding the atom of origin, or if a number of atoms in a line are excited in this way a cylindrical hot region will result. These regions have been called spherical and cylindrical thermal spikes.2 Their duration is about 10-12sec and the magnitude of the temperature rise is not considered sufficient to allow general recombination of any vacancies or interstitials in this time. Brinkman 3 has suggested that during the last part of its movement through the lattice, the primary knock-on must be regarded as having a diffusive-typeF .P. CLARKE 143 movement. Thus below some critical energy the remaining kinetic energy of the moving atom will be dissipated collectively over a small volume of the crystal. This may cause a rise above the melting point. The duration of this displacement spike, as Brinkman has called it, is estimated at about 10-11 sec. Any interstitials or vacancies within the melted region must conceptually be replaced by high and low local degrees of atomic disorder. Within the duration of the spike the local strains due to these abnormal concentrations would not persist and so in resolidifi- cation most of the vacancies and interstitials would have disappeared within this region. The size of the region is estimated as under 50 diam. in a tyEical solid. During the initial part of the movement of the knocked-on atom, however, vacancy pairs can still be formed together with thermal spikes.A diagrammatic representation of these ideas (after Brinkman) is shown in fig. 2. On this figure, FIG. 2. Reproduced, with acknowledgment, from J. Appl. Physics, 1954, 25, 962. suggested features of the displacement spike are also shown: (i) the formation of dislocation loops within the displacement spike during the resolidification, and (ii) the formation of small microcrystals of new orientation. Brinkman considers that the main product of the displacement spike should be the dislocation loops, though more rarely an array of such loops could form the boundary of a microcrystal. Lomer and Cottrell4 consider that these loops would disappear rapidly after resolidification.These models have been suggested initially for the metallic state, where a clear-cut means of distinguishing between them has not been attained. As far as covalent and ionic materials are concerned most of the thinking so far has been in terms of point defects or small combinations of point defects. The important point about these models is that on all of them we would expect some vacancy- interstitial pairs to be formed, though in different amounts on each model. The most fruitful approach as far as annealing is concerned is to look for physical property changes which will allow changes in the numbers of these point defects to be followed as functions of temperatures and time. In some ionic solids, notably the alkali halides, point defects may be produced by X-, or y-irradiation, at energies where direct momentum transfer is ruled 0~t.5.2 In such cases there is no question of displacement spikes occurring.Electrons144 DEFECTS PRODUCED B Y IRRADIATlON will produce point defects distributed randomly, provided they are energetic enough to penetrate the crystal. To sum up, with any of the types of radiation considered, point defects will be produced. These may be distributed randomly in the lattice (electron irradi- ation) or in groups (neutron irradiation) as well. In considering models for the annealing processes it will be assumed that any point defects or groups of point defects will not be present in sufficient numbers to interact with each other. For neutron irradiation, if the displacement spike model is valid, we can ignore the groups as far as point defect annealing is concerned, since the spike will itself anneal them.Hence it may be approximated that the distribution of defects is random. (i) DISCRETE ACTIVATION ENERGY tribution of the defects may be divided into ANNEALING PROCESSES On the simple damage models which have so far been considered the dis- (i) those pairs within each others perturbing influence, (ii) those pairs outside this region. Now, interstitials and vacancies will cause the surrounding lattice to become distorted. Fig. 3 shows a two-dimensional representation of the way in which FIG. 3.-The way in which a lattice distortion will assist defect movement. A and B are forced towards V, thus favouring movement of I to normal lattice site B.this distortion might take place in an ionic lattice. The defects perturb the poten- tial barrier to diffusion, so as to favour the mutual annihilation of the defects. In interstitial movement the saddle of the potential barrier is lowered to, say, E, so that the frequency of movement towards the defect under thermal excitation at temperature To K will now be given by v = vo exp (- E/kT), where vo is the lattice frequency. In vacancy movement, the potential barrier for an atom adjoining the vacancy and between it and the interstitial will be lowered, and the frequency of move- ment of this sort of atom into the vacancy will also be increased. This is equiv- alent to a movement of the vacancy towards the interstitial with the same jump frequency. Thus, suppose that the effect of the perturbation is to lower the barrier for vacancy movement from 0-5 eV to 0.4 eV at room temperature, then on the average the vacancy will jump towards the interstitial one hundred times for every jump that it would otherwise have made.Furthermore the nearer theF. P. CLARKE 145 interstitial the vacancy gets the higher will its jump frequency become. Thus it may be assumed (Fletcher and Brown) that the total jump time for a vacancy to become annihilated, if it starts off within the region of distortion, will approximate to the jump time for the initial movement. Stage 1 will comprise those defect pairs within a certain distance of each other corresponding to (i) above, each of which will annihilate on movement.Outside this distorted region the defects will move according to a random walk process. Some will wander back to the distorted regions and recombine. This will comprise stage 2 of the annealing. Others will wander away and may become annihilated at defects other than the ones of origin, or they may become trapped at the surface or at dislocations. These possibilities will comprise stage 3 annealing. We can classify these processes as being unimolecular or bimolecular, or inter- mediary between these two. Stage 1 annealing rate is dependent on the total number of defects per cm3 within the distorted regions and hence we can write dN/dt = - Nv (a unimolecular reaction), where N is the number of close pairs per cm3. In stage 2, dN/dt will be propor- tional to (no.of interstitials per cm-i) x (no. of vacancies one jump away from region of strain around an interstitial) K (density of interstitials) x (density of vacancies x vol. of spherical region containing the first shell of atoms around the region of disorder) Hence the annealing process may be divided into stages. K NI x NV OC N v ~ (a bimolecular reaction), where N, and NV are the densities of interstitials and vacancies remaining after stage 1 process has finished. In stage 3 there will be a steady reduction in the density of vacancies due to capture at dislocations, etc., as well as due to recombination. Hence the process will appear to be intermediate between unimolecular and bimolecular. Mathematical analyses of these stages have been made by Fletcher and Brown 6 for vacancy diffusion, assuming the interstitials to be stationary.Their first analysis assumes an isotropic medium as well as the other physical approximations outlined above. The second analysis deals with a discrete lattice and develops a set of differential equations on a more rigorous basis. The solutions to the equa- tions developed on the analyses may be referred to as the " isotropic lattice " and " discrete lattice " solutions respectively. The physical ideas behind the forming of these differential equations may be summarized. Vacancies in sites which have the same symmetry with respect to the interstitial, have the same probability of escaping or recombining with the interstitial. Adjacent to any site are several others to which the vacancy might jump.The number of these sites and the individual probability for jumping to each is taken into account. The computations involved in this approach are very laborious unless further approximations are made, and for stage 1 annealing there would seem to be some difficulty in determining the relative jump frequencies in the various possible directions. However, by correctly choosing values for the radius of the distorted region around an interstitial a good correspondence can be obtained between the " isotropic lattice " and " discrete lattice " solutions. The isotropic lattice solutions are shown in table 1 for the various annealing stages. One important aspect common to all these solutions is the fact that the annealing dependence is a dependence in the ratio t/T, where Y = 1 / ~ i s already defined.A detailed analysis of annealing data based on this analysis has been made by Brown et aZ.7 for electron bombardment of germanium, using electrical con- ductivity as a measure of the number of defects. The damage here is mainly the production of isolated vacancy interstitial pairs. Fig. 4 and 5 show how this application has been made and how successful it has been.146 DEFECTS PRODUCED BY IRRADIATION TABLE 1 NFO STAGE 3 : NF = 1 + 4nrC2(b>N~o - (3. Total number of vacancies present at time, t = total left due to each annealing stage N = NM+ NL + NF. Ni = number of vacancies originally in ith type of site, T = the appropriate jump time, ri and r, = radii of initial vacancy position, and that of deformed region, measured <b) is the magnitude of the change in radius ri made at each jump in the appropriate stage, from interstitial.averaged over all possible types of jump. '0 0 0 (I, P C 0 +. > U 3 'p c 0 u rc - c .- .- &. 0 0 c .- c u 5 u, Annealing time (min) (4 Annealing curves at digerent temperatures for a group of samples having the same initial conductivity and given the same bombardment. The sample numbers appear above the curves, the annealing temperatures at the right end. FIG. 4. Reproduced, with acknowledgment, from Physic. Rev., 1953, 92, 592. From fig. 4(a) an equivalent annealing curve at 220" C is constructed. This is done by superimposing the annealing curves at different temperatures, after adjusting the time-scale proportionality. This can be done since the annealing kinetics are a function of t/r.The composite curve is shown in fig. 4(6). The fitting of various processes is shown in fig. S(a), (b), (c) and (d). It would seem that (d) is the best fit. However, the lattice frequency constant determinedF. P . CLARKE 147 SO 20 I 0 0 I n C 0 u i 9, v W P 0 c4 r c W 4- a - U L L 0 C Y u b 0 c 0 .- c U 0 L t& Equivalent annealing time a t 2 2 0 ' ~ (rain) (b) Composite annealing curve obtained by adjusting the time scales for superposition of the curves of (a) with respect to the curve at 220" C. The arrows above the curves indicate the ten minute annealing points for the original curves. FIG. 4.-(After Fletcher and Brown.) Reproduced, with acknowledgment, from Physic. Rev., 1953,92, 592. 8 0 r 7 0. Monomolccu I ar bo 5 0 .3 0 - Equivalent annoo~ing t i m e a t 2 2 0 ' ( m i n ) (4 Theoretical fit of the composite experimental annealing curve of fig. 4(6) with ii bimolecular or single monomolecular recombination process. FIG. 5. Kt.pvoiliicetl, ,r.itli cickno wlerig,neut, from Physic. Rev., 1953, 92, 594.148 DEFECTS PRODUCED BY I R R A D I A T I O N I h c e 9, V Q, a W W 9, 0 Y t c 0 L w Y L c r 9, V + 0 c 0 .- c U 9, LL Equivalent annealing time a t 2 2 0 ' ~ ( m i n ) (4 Theoretical fit of the experimental annealing curve with two monomolecular recombination processes. FIG. 5.-(After Fletcher and Brown.) Reproduced, with acknowledgment, from Physic. Rev., 1953, 92, 595. n c 01 V a# Q V m 0 W E c 0 0) , I L Y - w L L t 0, U .c 0 c Equivalent annealing t i m e a t 2 1 0 ~ ~ (min) (c) Theoretical fit of the experimental annealing curve with two monomolecular recombination processes, plus the recombination during liberation.FIG. 5. Reproduced, with acknowledgment, from Physic. Rev., 1953, 92, 595.I CI & c a u Y c E 0 E q u i v a l e n t annealing t i m e a t 2 2 0 " ~ ( r n i n ) (4 Theoretical fit of the experimental annealing curve with the recombination during liberation and bimolecular recombination. FIG. 5.-(After Fletcher and Brown.) Reproduced, with acknowledgment, from Physic. Rev., 1952, 92, 596. from the time constant T is - 1017 sec-1, and the capture radius is about 20 A. The first of these values is highly improbable and the second would seem to be rather large. Curve (c) however, gives a lattice frequency of 1013 sec-1 and a capture radius of 0.9a, where a is the cube edge of the unit cell.For this and other detailed reasons it is considered that this represents the best overall description of the processes. (ii) VARIABLE ACTIVATION ENERGY In order to explain the kinetics of resistance changes in metal films, Vand 8 suggested that the defects responsible for the changes might be distributed in activation energy. Following this concept, Primak 9 has made a detailed mathematical analysis of the kinetics of annealing processes having a spectrum of activation energies. The approach is similar in principle to that used in the exhaustion theory of transient creep.10 It is assumed that the annealing process obeys a differential equation of the kind - (dN/dt) = kqr, (1) where N is the concentration of kinetic processes, t the time and k a frequency factor related to the rate of the process.y is called the order of reaction by analogy with gaseous kinetics. Then if some property is related to N by the relation p = fN and the rate constant may be written as A exp (- E/kT), the equation becomes150 DEFECTS PRODUCED B Y I R R A D l A T l O N For a process having a discrete activation energy this may be integrated to give the propertyp at some time t, in terms of the property at time t = 0, 1 - p = po[l - (1 - y)Bt exp (- E/kT)]' - = Po#,, where 6,, is termed the characteristic annealing function. Thus a plot of log t against 1/T obtained at a fixed p , should be linear. Very often, however, this is not so. If the processes are distributed in activation energy, the measured value of the property P(t) will be the integral of all the discrete processes occurring at the FIQ.&-The course of an anneal. The characteristic annealing function By sweep- ing across the initial activa- tion energy spectrum. temperature of anneal at any instant, within a narrow activation energy range dE Thus the process of anneal may be represented by a function O,,, sweeping across the po against E curve as shown in fig. 6. The point of inflection in the O,, function is given by EO = kT In (Bt) and so as time proceeds, and at any given tempera- ture, the progress of across the initial activation energy spmtrum becomes slower. For low values of y, O,, is steep over the major portion as shown in the figure, and may be approximated by a step function such that P(t) = po(E)dE.1:: Hence it may be shown that 1 dP kT d(ln t)' PO(E0) = - - Hence a plot of po(E0) against EO may be obtained by plotting -!- dP against kT In (Bt). kT d(ln t) An interesting point revealed by this analysis is that very small departures from linearity in the annealing relation between property changes and In t, are most important in determining the activation energy spectrum. Primak and Szymanski's 11 results for vitreous silica illustrate this and are shown in fig. 7. With a discrete activation energy the order of reaction of - 10 would have to be chosen to explain the annealing curves shown. With a distribution in activation energies, however, an order of reaction around one can be chosen to explain them.Of course it is possible that the initial assumption with regard to the rate of the process may be invalid. For example, an annealing process discussed by Kinchin and Pease 12 may be explained with an order of 6. It is not necessary, however, to assume a distribution in activation energy to make this order lower. Kinchin and Pease have shown, from physical considerations similar to those already outlined, that the problem (in a layer lattice) may be treated by applying diffusion equations to the random migration of interstitials to circular sinks assum- ing a discrete activation energy. Marx et al.,l3 Overhauser,l4 and Dienes 15 have analysed isothermal annealing curves on the basis of an activation energy varying due to the presence of other defects.Dienes has emphasized the effect of this on the apparent order of reaction.F. P . CLARKE 151 l o g t ( t i n seconds) (4 Annealing of the radiation-induced changes in the density of an irradiated vitreous silica H, 1000" C. sample. A , 300" C, B, 450" C , C, 550" C, D, 650" C, E, 725" C, F, 800" C , G, 900" C , FIG. 7.-(After Primak.) Reproduced, with acknowledgment, froin Physic. Rev., 1956, 101, 1269. Act,i'vation encr 9 7 (b) The initial distribution of density changes over activation energy from the annealing data above. FIG. 7-(After Primak.) Reproduced, with acknowledgment, from Physic. Rev., 1956, 101, 1271.152 DEFECTS PRODUCED B Y IRRADIATION The suggestion is that an activation energy EO will be modified by the presence of N defects such that it becomes E = EO - uN where, for simplicity, K is a constant, The rate eqn.(1) may then be written dN - = ANyexp(- Eo/kT)exp (orN/kT). dt For a fixed N , the solution of this may be written t = k' exp (Eo/kT) exp (- orN/kT). Hence, although the plot of 1/T against log t should still be linear, the effective activation energy will vary with N. The effect of this on the annealing curve is that a small change of activation energy with N, leads to a large change in the FIG. 8a.-Annealing curves at different temperatures ("C) for a group of samples having the same initial optical absorption at the wavelength of measurement and given the same bombardment. time-scale for any given isothermal anneal. An annealing curve of low order and variable E is very similar to one described by a high y and discrete E.This suggestion was introduced to correct the rate equation dN/dt = ANY exp (-Eo/kT) which assumes an isotropic medium for the whole of the annealing process, The physical concept underlying this suggestion, that the lattice strain near the defect will change the activation energy for diffusion of other defects, is of course the basis of part of Fletcher and Brown's analysis. If it be assumed that a defect will cause a lattice relaxation appreciable over a sphere of radius 5 atomic spacings, then one would expect a general perturbation of all activation energies to exist for defect concentrations in excess of about 1019-1020 per cm3. It is interesting to make an analysis of annealing curves in such a way that some of the alternative approaches can be compared.In the annealing of neutron- caused damage in magnesium oxide, the decay of a particular centre may be fol- lowed by means of the optical absorption arising from an electron trapped at the centre. It is estimated that the absorption in question is caused by about 5 x 1016 defects, which are identified16 as magnesium ion vacancies. The an- nealing curves at various temperatures, and for different specimens, are shown in fig. 8a. Assuming that the annealing can be represented by the type of expression in which t/r is the ultimate variable, then an equivalent annealing curve for 500" C can be composed as shown in fig. 8(6). The first part of this anneal can be ap-F . P . CLARKE 153 proximated to the combination of two unimolecular processes for low annealing times.However, attempts to fit the rest of the curve by a combination of the physical processes described mathematically by Fletcher and Brown were not successful. V - 0 C c 0 0 0 0 V E - 0 0 U 0 LI I - 5 - 0 I 3 4 5 6 7 8 1 0 9 t i m e ( i e c ) (b) FIG. 8b.-Composite annealing curve obtained by adjusting the time scales for super- position of the curves of (a) with respect to the curve at 500°C. The arrows indicate the ten minute annealing points for the original curves. The theoretical fit of this curve 0.1 (1 - e-d2) + 0.6( 1 - e-rjloo) with two monomoIecular recombination processes is shown by the dotted curve. Fig. 9 shows the activation energy spectrum as obtained from the type of analysis developed by Primak.The two unimolecular processes of the former analysis are now replaced by two definite bands. It must be borne in mind that the effect of the interaction of the characteristic annealing function with the initial activation energy spectrum is to make the spectrum which is produced by analysis of the data, broader than it should be. This broadening might be as FIG. 9.-The initial distribution of defects in activation energy, from the annealing data of fig. 8. much as several times kT for steeply rising parts of the spectrum. Hence the width of the first peak might be due to this and other complicating effects, and the band could be interpreted as representing a discrete activation energy. The second band, however, seems too broad to allow of this interpretation.I54 DEFECTS PRODUCED BY IRRADIATION An interpretation to describe the processes by means of an activation energy varying with the number of defects was not made, since it was considered that the number of defects was insufficient to make this approach physically justified, CONCLUSION When certain radiations penetrate solids, atomic displacements may result.These displacements may be distributed randomly or in groups. Once the dis- tribution is known it is possible in principle to set up diffusion equations to describe their subsequent behaviour as the temperature of the solid is raised. Attempts to do this (Fletcher and Brown, Kinchin and Pease) have been successful for particular cases. The solution of these diffusion equations is complicated for any but the simplest cases.Hence approximations have usually to be made which make the final mathematical description of the processes less accurate. The defect concentration is estimated by measuring physical properties (optical, electrical, density, etc.) which vary with this concentration. The exact way in which this variation takes place is usually uncertain and so before the diffusion equation can be set up, approximations have to be made which again may weaken the final mathematical description. Another important factor which may be obtained from physical measurements is the activation energy for diffusion of defects. If one is sure of the type of defect concerned, this activation energy may be estimated theoretically or by separate diffusion experiments. A common procedure, however, has been to assume that the kinetic behaviour will follow a certain type of equation (e.g.dN/dt = ANY exp (- E/kT)) and to deduce the activation energy from this. Largely as a result of using this procedure the concepts of a continuous distribution in activation energy, and an activation energy varying with the number of defects present has arisen. On the first of these concepts it has been demonstrated that, with an assumption about the way a defect with a discrete activation energy would behave, the continuous distribution may be obtained from experimental data. The final picture emerging in this case would seem to depend for its validity on the correctness of assuming a simple equation for the discrete annealing process. The approaches made by Fletcher and Brown, Kinchin and Pease, Primak, Marx et al., Overhauser and Dienes all emphasize that the concept of an order of reaction is meaningless unless it is based on an underlying physical under- standing of the processes involved. The author would like to thank Dr. A. H. Cottrell, F.R.S., and Dr. J. H. 0. Varley for informative discussions and criticisms on the subject matter of this paper. 1 Bohr, K. Danske Vidensk. Selsk. Mat. Fys. Medd., 1948, 18, 8. 2 Seitz and Koehler, Solid-State Physics (Academic Press, New York, 1956), vol. 2, 3 Brinkman, J. Appl. Physics, 1954, 25, 961. 4 Lomer and Cottrell, Phil. Mag., 1955,46, 711. 5 Varley, J. Nuclear Energy, 1954, 1, 130. 6 Fletcher and Brown, Physic. Rev., 1953, 92, 585. 7 Brown, Fletcher and Wright, Physic. Rew., 1953, 92, 591. 8 Vand, Proc. Physic. Soc., 1943, 55, 222. 9 Primak, Physic. Rev., 1955,100, 1677. 10 Cottrell, J. Mech. Physics Solids, 1952, 1, 53. 11 Primak and Szymanski, Physic. Rev., 1956,101, 1268. 12 Kinchin and Pease, Reports Prog. Physics, 1955, 18, 1. 13 Mam, Cooper and Henderson, Physic. Rew., 1952,88, 106. 14 Overhauser, Physic. Rev., 1954, 90, 393 ; 1954, 94, 1551. 15 Dienes, Physic. Rev., 1953, 91, 1283. 16 Clarke, Phil. Mag., 1957 in press. p. 305 et seg.