DIELECTRIC RELAXATION PROCESSES IN LITHIUM, SODIUM AND POTASSIUM HALIDES BY J. S. DRYDEN AND R. J. MEAKINS Division of Electrotechnology, C.S.I.R.O., Sydney, Australia Received 14th January, 1957 The dielectric properties of the lithium, sodium and potassium halides containing divalent cation impurity have been studied at frequencies and temperatures where the relaxation absorption curves are clearly separate from the background loss due to d.c. conductivity. In several cases two distinct absorptions were observed, the higher fre- quency one being present only at higher concentrations of impurity. Activation energies have been obtained for the lower frequency absorption which is due to orientation of the dipole formed by association of a divalent cation and metal ion vacancy. These energies agree with those obtained from d.c.conductivity measurements for the movement of cations provided these are correctly interpreted, but are lower than the values generally accepted. A method is suggested for calculating these energy barriers using elastic constants which gives excellent agreement with experiment. A mechanism is suggested for the second absorption. It has been demonstrated that there is dielectric absorption present in alkali halides containing divalent cation impurities 1 9 2 and that this absorption can be obtained quite distinct from the loss due to d.c. conductivity.3 Dielectric ab- sorption in alkali halides is important in presenting a simple case of a dielectric relaxation process and the one in which it appeared most likely that some theor- etical calculations could be made of the activation energy. I t was decided, there- fore, to extend the investigation to a number of alkali halides.The activation energies for the movement of cations in alkali halide crystals have been measured previously from d.c. conductivity and diffusion experiments. However, the measurement of a frequency of maximum dielectric absorption appeared to have advantages over these other methods, particularly after pre- liminary experiments demonstrated that it was not necessary to have large single crystals from which to obtain samples. EXPERIMENTAL MATERIALS.-The criterion of purity required of the alkali halides used in this work was that they should show no dielectric absorption in the region of the absorption peak which occurred after the addition of divalent impurity.With NaC1, NaBr, KCl, KBr and KI the commercial A.R. or reagent-grade materials were found to be suitable, provided that the crystals for measurement were chosen from the clear region of the mass solidified from the melt. LiCl was prepared from A.R. Li2SO4. H2O and A.R. BaC12.2H20. For the bromide and iodide, the sulphate was first reacted with A.R. K2CO3 to give Li2CO3 and the latter purified by conversion to the chloride and reprecipitation. LiBr was then prepared by heating the Li2CO3 with A.R. NHdBr, and LiI was obtained by bubbling HI into a suspension of Li2CO3 in water. NaI was prepared similarly from A.R. Na2C03. For NaF, the A.R. Na2C03 was heated with a slight excess of A.R. NH4F. The divalent impurities were the best commercial grades available and, where con- venient, were added as the halides,P_and in other cases as the carbonates in the reaction mixture. PREPARATION OF SPECIMENS FOR MEASUREMENT.-with LicI, NaCI, NaBr, KCl and KBr the crystals were prepared simply by melting the material (20-40 g, according to density) in a platinum crucible in a furnace and allowing it to cool slowly. For NaF 3940 RELAXATION IN HALIDES and KF, the air in the furnace was replaced by H2, and for LiBr and KT, atmospheres of H2 and HBr and H2 and HI, respectively, were used.NaI and LiI were melted in silica ware. The powdered material was introduced into the bottom of a silica tube 2.5 cm int. diam. sealed at the bottom and having a con- striction about 7-8cm from the bottom.It was then heated under vacuum for about 4 h to remove moisture and the portion of the tube below the constriction sealed off under vacuum and detached. The material was then melted and solidified in the furnace and the silica vessel transferred to a dry box to be broken. In each case on solidification, the mass separated into a clear, glassy region at the outer edge and a granular, poorly-crystalline region at the centre. The former gave small cubic crystals when broken up with a probe and these were split with a sharp blade to give flakes about 1 mm in thickness and a few mm in length and breadth. The number of splits required for a sample varied from about 10 to 35 according to size and, as the thickness had to be determined at this stage by estimate, there were appreciable variations throughout the sample.To reduce the amount of water absorbed by the freshly split surfaces, the breaking-up of the mass was begun when its temperature had decreased to about 130" C, and this operation and the splitting were performed on a hot plate. As an additional precaution with LiBr, NaI and LiI the hot plate was housed in a dry box. DIELECTRIC MEASUREMENTS.-A three-electrode assembly was used, consisting of two brass electrodes 2-5cm in diam., enclosed in a brass cell. Silica gel was placed in the bottom of the cell and the whole assembly heated for an hour or more at 130" C. The crystal splits were then taken from the hot plate and placed between the electrodes while they were still hot, the dry box being again used for the very deliquescent materials. After cooling a little, the joints in the assembly were completely sealed with Apiezon sealing compound.The temperature of the sample during measurement was controlled by immersing the electrode cell in a liquid bath contained in a dewar. The bridge used for the measurements has been described recently by Thompson.4 The capacitance between the electrodes was usually about 10pF of which about two-thirds was contributed by the alkali halide dielectric. ANALYSIs.-The concentrations of calcium impurity in KCl and in NaCl were estimated spectrophotometrically using a method described by Watson and Scott.5 RESULTS AND DISCUSSION No dielectric absorption other than that arising from the presence of d.c. conductivity was detected in any of these compounds in the absence of divalent impurity.This confirms the conclusions of Haven 1 and of Jacobs 2 for NaCl and of Burstein et al.6 for KCl. In the presence of a divalent impurity with an ionic radius approximately the same as that of the cation of the alkali halide, two absorptions may be detected. That occurring at the higher frequency is present only when the impurity con- centration is above a certain level. It was considered desirable, in most cases, to eliminate the higher absorption or reduce it to a low level. If the first sample prepared was found to contain an appreciable amount of the upper absorption, a portion of it was diluted with more alkali halide and recrystallized. This process was repeated until only the main absorption remained.MAIN ABSORPTION.-The main absorption will be discussed first. In all cases the dielectric loss factor E" could be described, as a function of frequency, by a Debye curve where T is the relaxation time and f the frequency. This signifies that the ab- sorption is characterized by a single relaxation time. As examples, the variation of dielectric loss factor with frequency is given in fig. 1 and 2 for NaBr and for LiCl. The latter is chosen as an example of compounds with a low activation energy. In this case, comparatively low temperatures had to be used before the absorption peak was clear of the background.J . S . DRYDEN AND R . J . MEAKINS 41 Tn all cases the frequency fmaX at which maximum absorption occurs obeys the following expression as a function of temperature (2) The values of A and of AE obtained for the different compounds are listed in table 1 (columns 2 and 4).fmaX = A exp (- AE/kT). 3.6- g 3 . 4 - 53 2 - 0 Y 3.0- lo9 frequency (c/s) FIG. 1 .-Dielectric absorption at various temperatures for sodium bromide with calcium impurity. t lo9 frequency ( c / s ) FIG. 2.-DieIectric:absorption at variousztemperatures focdithium chloride with magnesium impurity. In most of the alkali halides only one divalent impurity was added but in KC1 three impurities, Sr, Ca and Ba, were used in the measurements (see fig. 3). Some experiments were made with NaCl containing Cd and Mn. No absorption was observed with Cd and with Mn a complex absorption curve was obtained. It is possible, however, that both these impurities could have been added under different conditions to give similar dielectric results to those obtained for other divalent impurities.42 RELAXATION 1N HALIDES Activation energies from other dielectric absorption measurements are listed in column 5.Those of Breckenridge7 are not included because his AE values were determined from measurements at one temperature only and in many cases his absorption curves are not well defined. '015- c .- c . - c 3 ZI L 2 -!? ,010 v 0 I J 0 L - I 1 I I I I f 0 I 2 3 4 5 b loq frequency ( c / s ) FIG. 3.-Change in frequency of maximum absorption with radius of divalent cation; KCl containing Ba2+, Sr2+ and Ca2+ ; temperature 41" C. A sec-1 compound and impurity LiF (Mg) 7 x 1012 ref. (3) LiCl (Mg) 4 x 1012 LiBr (Mg) LiI (Mg) NaF (Ca) NaCl (Ca) 1.6 x 1013 NaBr (Ca) 8 x 1012 NaI (Ca) 6 x 1012 KF (Sr) 8 x 1012 KCl (Ca) 8 x 1012 KCl (Sr) 8 x 1012 KCI (Ba) 8 x 1012 KBr (Sr) 7 x 1012 KBr @a) 7 x 1012 KI (Sr) 9 x 1011 TABLE 1 AE from dielectric absorption eV this liter- paper ature 1.2 x 1013 0.653 0.42 0.40 0.35 8 x 1012 0-87 6 x 1012 0.68 0.7 1 4.5 X 1012 0.62 3 x 1012 0.56 068 4.5 x 1012 0.64 4.5 X 1012 0.67 0.7 6 4.5 x 1012 0.70 3 x 1012 065 3 x 1012 0.68 3 X 1012 0-58 AE from d.c.conductivity eV AE from AE from nuclear diffusion magnetic recalc. eV resonance liter- from eV ature resuits in literature 0658 - 0.7 9 0.5910 0-418 0.5610 0-398 0-388 0.45 - 0.8 9 0.7915 0.8615 0.7 15 0.7816 0.9415 0.7 1543 J . S. DRYDEN A N D R . J . M E A K I N S MECHANISM OF ABSORPTION For every divalent cation present in the crystal there will be an alkali metal ion absent.Some of these divalent cations, which have a net positive charge and metal ion vacancies, which have an effective negative charge, will be associ- ated. The absorption in these compounds arises from the orientation of the dipoles so formed. The vacancy has twelve possible equivalent positions around the impurity ion (fig. 4) and the orientation of the dipole can change either by @ Metal ion @ H a l i d e ion 0 Divalent impurity 0 Metal ion vacancy FIG. 4.-A sodium chloride lattice containing a divalent cation impurity and a metal ion vacancy on adjacent sites. (i) an alkali metal ion in one of the positions marked 1, 2, 3, 4, moving to the vacant site, or (ii) by an exchange of places between the vacancy and the divalent ion.The relaxation time, determined from these experiments, of dipole orienta- tion is equal to 1/2(wl + waj, where w1 and w2 are the probabilities per second of transitions (i) and (iij occurring. For these transitions the ions must pass over energy barriers El and E2 respectively, and w1 and ~ ' 2 will be proportional to exp (- AEl/kT) and to exp (- k * / k T ) . If the two energy barriers differ by more than a few per cent, one of the transition probabilities will be so much lower than the other that the activation energy determined experimentally will be the smaller of the two. It is not possible to say which of these energy barriers will be the smaller. For the lithium and sodium salts and of Ba in potassium salts, where the radii of the impurity ion and of the alkali metal ion are close to the same size the energy barriers will differ only in so far as the extra charge on the impurity affects it.For Sr and Ca in KC1 the divalent cations are 15 % and 25 % smaller respectively in radii than potassium and it might be thought that the barrier to movement of the impurity in these cases will be considerably lower than that of the potassium.44 RELAXATION I N HALIDES However, this may not be so since the net positive charge on the impurity ion will result in the nearest negative ion neighbours being attracted closer to the impurity and there will be a contraction in this part of the crystal. This is supported by the experimental results which show only a slight decrease in energy barrier with ionic radius (see table 1).COMPARISON OF AE WITH THAT ESTIMATED BY OTHER METHODS There are two established methods for studying the movement of vacancies in ionic compounds, viz., d.c. conductivity and diffusion. More recently, nuclear magnetic resonance studies have been used.17 The activation energies, where they have been determined by these methods, are included in table 1. The energy barriers for movement of a cation vacancy are equal to (in KF) or smaller '4 '11 FIG. 5.-D.c. conductivity in KCl (Sr) and KC1 @a) as a function of 117'. The full line is taken from curves given by Kelting and Witt.15 than those for an anion vacancy in all of these compounds, and hence the activa- tion energies obtained from conductivity measurements should be those of the cation.Since the fundamental process in both the conductivity and the dielectric mechanisms is the jumping of a cation from one position to an adjacent vacant site the activation energies should be approximately the same. The only differ- ence should arise from the presence of a divalent ion on a next nearest lattice site in the dielectric case. However, it can be seen that with the exception of those of Havens on LiF, LiCl and LiBr the activation energies from d.c. conductivity are higher than those measured from dielectric absorption to the extent of about 0.2 eV. An important difference between Haven and other workers is that Haven calculated his activation energies at higher temperatures. When some of the other workers' curves for the variation of conductivity with 1/T are re-examined, activation energies close to those from dielectric absorption are obtained.An example of this is shown in fig. 5 in which the conductivity curves 15 for KC1 containing Sr and Ba are redrawn. These curves consist of three parts: (a) a linear portion from which the activation energy is usually calculated, (c) the region in which the curve is much steeper and the slope of which is taken to be the sum of the energy to create a defect and the activation energy for the movement of a defect and (6) a region between these which may be present and which is linear over a small temperature range only. The energies listed in column 7 of table 1 have been obtained from region (b) and it can be seen that there is good agreement with the energy barriers from dielectric relaxation (columns 4 and 5).It can be concluded therefore that the energy barrier to movement of a cation is not in- fluenced to any significant extent by the presence of a divalent impurity at a next nearest lattice site. The difference between the energies calculated from the slope of part (a) and those determined from dielectric measurements is presumably the energy of association of the divalent cation-cation vacancy complex and the temperature at which the curve changes shape ((a) to (b)) that at which dissociation is complete.J . S. DRYDEN AND R . J . MEAKINS 45 When the divalent impurity and the metal ion are nearly equal in radii the associ- ation energy is therefore 0.15-0.20 eV but can be lower when the divalent impurity is smaller ; for example, in KCl containing Ca it is approximately 0.09 eV only.There have been several attempts to calculate the association energies in alkali halides 18 but the calculations yield values of 0.3-0.4 eV, i.e. higher than experi- mental by a factor of two. ESTIMATION OF THE ENERGY BARRIERS FROM ELASTIC PROPERTIES * Examination of the lattice structure suggests that an alkali ion moving from a position A to a vacant site A1 faces the smallest energy barrier if it moves via the interstitial position Z (fig. 6(a)). In passing along such a route AA1 it passes P P FIG. 6.--Illustrating path of a cation in moving from one lattice site to an adjacent vacant site. (b) and (c) show the distortion necessary to allow an ion to be in positions X and Z respectively.twice through a position in which it is coplanar with three halogen ions. Such a position is shown as X at the centre of an equilateral triangle LMN (fig. 6(b)), using NaCl as the example. The distortion necessary for a cation to be in the interstitial position Z is illustrated for NaCl in fig. 6(c). The energy is greater when the cation is at X than at Z . Therefore a calculation of the energy difference between positions X and A will give a value of the energy barrier which a cation has to overcome in moving from A to A1. A rough method can be devised to calculate this energy difference. If a volume vo of a substance whose compressibility is /3 is subjected to a uni- form volumetric compressive (or dilational) strain A then the stored energy is given by E = (1 /2/3)A2~0.(3) * This section is a result of collaboration between the authors and Dr. P. G. Harper and Dr. J. J. O'Dwyer of this laboratory.46 RELAXATION I N HALIDES Referring to fig. 6(b) we can write a linear strain associated with the position of the alkali ion at X as and calculate the corresponding volumetric strain as ( 5 ) Using vo = Z03, the values of /3 selected by Hojendahl19 and ionic radii of _ - vo 10 Zachariasen 20 the calculated activation energies are given in table 2. TABLE 2 pound LiF c1 Br I NaF c1 Br I KF c1 Br I KC1 Br dijt. A 2.01 2-57 2.75 3.02 2.3 1 2.8 1 2.98 3.23 2.67 3.14 3.29 3.53 energy difference 2 A V / V ~ ref. (19) calc. from A cmzdyne-1 eqn. (3) B nearest corn- neighbour S:Fzf eV 2.01 1.64 0.37 0.83 1 .5 0 ~ 10-12 0.61 2.49 2.10 0.39 0.66 3-34 0.38 2.64 2.25 0.39 0.64 4.23 0.3 5 2.87 2.46 0.41 0.59 5-89 0.28 2.31 1.89 0.42 0.81 2.07 0-66 2.79 2.30 0.49 0-77 4-18 0.54 2.94 2-44 0.50 0.75 4-98 0.51 3.17 2-64 0-53 0.72 6.94 0-43 2-66 2.18 0.48 0.81 3.25 0.65 3.14 2-56 0.58 0-83 5.53 0.64 3.29 2.68 0.61 0.83 6.56 0.63 3-52 2.88 0.64 0.82 8.37 0.53 0.64 0.63 e;;t. 0.65 1.05 0.42 1.10 0.40 1-15 0.35 1.25 0-87 1.33 0.68 1.25 0.62 1-2 0.56 1.3 0.65 1.05 Sr 0.68 1.05 0.67 1.05 0.58 1.1 calc. I Two main objections can be raised to this calculation. First, the formula holds only within the elastic limit while the volumetric strains in this model are of order 0.5. However, the strain is spread over a much larger volume than that used in the model and throughout most of this volume the elastic limit may not be exceeded.If then A+) dr equals A%, for some suitable volume integration the formula would be correct. Secondly, the strain is more two-dimensional than three-dimensional at its source but again this should be rapidly transmitted into a volume effect. For these reasons the excellent absolute agreement between the calculated and experimental values must be regarded to some extent as fortuitous. Of greater significance is the uniformity in the ratio between these energy barriers. The ratios for the potassium halides while they agree to within 5 % amongst them- selves are lower than the ratios for the lithiurn and sodium halides. This is partly because the divalent impurity, strontium, is smaller than potassium as has been discussed earlier.J FREQUENCY FACTORS So far, the discussion has been concerned with energy barriers. The other quantity which is obtained from these experiments is the frequency factor A . These frequency factors are listed in table 1 , column 2. The frequency at whichJ . S . DRYDEN A N D R . J . MEAKINS 47 maximum absorption occurs, is equal to w~/T, where w1 is the probability of a cation jumping into a vacant site in unit time. On the simple two position model, w1 =(w0/2n)exp (- AE/kT), provided there is no entropy of activation.21 However, in the case being con- sidered here, each cation vacancy has four positions into which it can move so that an extra factor of 4 must be included in eqn. (6). Therefore, provided that no entropy differences have to be taken into consideration, and that the lower energy barrier is for univalent cation movement rather than divalent cation movement, the frequency factors in the alkali halides should be equal to 4fol.r.This quantity is listed in column 3 of table 1 for those compounds in which fo is known? It can be seen that the agreement between 4fo/n and A is good to within a factor of three or better. This is almost within experimental error. MAGNITUDE OF THE ABSORPTION The disadvantage of using a large number of small crystals in these measure- ments instead of one larger section of a single crystal is that the magnitude of the absorption cannot be obtained accurately. However, it is possible to estimate the magnitudes from the area and thicknesses of the different slabs.For a sample of NaCl containing 0.075 mole % Ca2+ the maximum loss factor has been estimated to be 0.013. On the assumption that all the divalent impurities and cation vacancies are associated the value calculated for this concentration by means of the following equation 23 is 0.025 : where n = the number of divalent cations per cm3, a = half the lattice constant and e is the electronic charge. In view of the approximate nature of the cal- culation of ezrn from the experimental result no significance can be attached to these two values except that they are of the same order. SECOND ABSORPTION As mentioned earlier, a second absorption is detected in many of the alkali halides when the concentration of divalent cation is greater than a certain limit. This absorption was studied more thoroughly in NaCl than in the other com- pounds.In fig. 7 the dielectric loss factors are shown in three samples of NaCl containing different amounts of Ca2f, the scales have been adjusted so that the main absorption is the same magnitude in each. It can be seen that for a con- centration of 0.075 mole % Ca the loss factor curve is close to a Debye curve and there is oiily a trace of the second absorption present. As the concentration increases beyond this value the magnitude of the second peak increases until at 0.95 mole % it is comparable with the main absorption. The upper absorption region, unlike the main absorption, is always wider than a Debye curve. In NaCl the activation energy for the two regions is the same to within 4 %. The difference in the frequencies of maximum absorption arises from the difference in frequency factors, the value being 1 x 1015 sec-1 for the second absorption compared with 1.3 x 1013 sec-1 for the main absorption.The activation energy was not measured in any other compound but from the fact that the two absorptions moved with temperature at a similar rate it was con- cluded that the activation energies were nearly the same. An exception to this occurred in NaF containing Ca in which an absorption with a lower activation energy was sometimes detected. Since the activation energies are the same or nearly the same it is concluded that the second absorption is also associated with the movement of a cation vacancy. In considering possible mechanisms of this absorption the X-ray studies of Miyake and Suzuki24 on NaCl + CaC12 solid solutions are relevant.48 RELAXATION I N HALIDES These authors report that when crystah were grown from melts containing more than 0.5 mole % of Ca2+, extra diffuse reflections were obtained on X-ray photo- graphs. These authors did not analyse their crystals but by analogy with Kelting and Witt's experiments on the solubility of divalent impurities in KCl 15 it can be assumed that the concentration of Ca in the solid is about one-tenth that in the melt.This means that the extra reflections appear when the concentration of Ca in the crystals is about 0-05 mole %, which is in reasonable agreement with the concentration at which the second dielectric absorption appears, viz. 0.075 mole %. loq frequency ( c / s ) FIG.7.-Dielectric absorption in sodium chloride containing various proportions of calcium impurity. In a later paper, Suzuki25 proposes that the extra X-ray reflections are due to the existence of platelets having a crystal structure resembling that of CaC12 but retaining coherency with the structure of the NaCl crystal. At the surfaces of the platelets, where there is a change from one structure to the other, divalent cations and cation vacancies are on adjacent lattice sites, thus forming dipoles, and it is possible that the dielectric absorption under discussion is associated with the movement of these dipoles. The environment in which these dipoles move will be different from those in the bulk of the crystal, thus accounting for differences in the entropies of activation for the two cases.VARIATIONS IN RELATIVE MAGNITUDE OF THE TWO ABSORPTIONS WITH HEAT TREATMENT It was found that when samples were heated for some hours at 130" C or higher the magnitude of the second absorption in NaCl containing calcium was decreased and that of the main absorption increased. No experiments were conducted to determine if this process was reversed after storing at room temperature. Miyake and Suzuki24 report changes in their X-ray patterns after heat treatment but their experiments were carried out at rather higher temperatures (400" C). The authors wish to acknowledge the assistance of Mr. J. S . Cook in the preparation of some of the compounds and in the analysis of the NaCl and KC1 samples ; also Mr. P. Buss and Mr.Cook for the construction of a furnace suitable for the crystallization of the various compounds in different atmospheres. APPENDIX In view of the success in calculating energy barriers for the movement of cations from eqn. (3)-(5) it seemed desirable to extend the calculations to the anions. Unlike the cations the path which presents the lowest energy barrier to the transition of an anionJ. S. DRYDEN AND R . J . MEAKINS 49 from an occupied to a vacant site is not the same throughout this group of compounds. In KF where the two ions are of the same size the energy barrier for the transition of an F- ion is the same as that already worked out for Kf but in all the other compounds the negative ion is the larger. The strain at position X (fig. 6) is the same whether the larger or the smaller ion is moving, but it is no longer true that the interstitial position Z is lower in energy than X.In addition to the path shown in fig. 6 the ion can go via the shortest path AAI passing midway between ions M and N. Calculations show that for NaF and for KC1 the path via the interstitial position presents the lower energy barrier whilst in the other cases the lower barrier is via AAI. The values are set out in table 3 to- gether with the experimental results which are available. compound Li F c1 Br I NaF c1 Br I KF c1 Br I TABLE 3 E in eV Calc. expt. 1-84 1-28 1-26 1.10 1.55 1 *80 1 -67 26 1-70 1.18 13 1 a 5 0 0.65 1-54 1-97 1.91 1 Haven, J. Chem. Physics, 1953, 21, 171. 2 Jacobs, Naturwiss., 1955, 21, 575. 3 Dryden and Rao, J. Chem. Physics, 1956,25, 222. 4 Thompson, P.I.E.E., 1956 (in press). 5 Watson and Scott, J. Chem. Physics, 1956, 24, 619. 6 Burstein, Davisson and Sclar, Physic. Rev., 1954, 96, 819 (abstract only). 7 Breckenridge, J. Chem. Physics, 1948, 16, 959 ; 1950, 18, 913. 8 Haven, Rec. trav. chim., 1950, 69, 1471. 9 Lehfeldt, Z. Physik, 1933, 85, 717. 10 Ginnings and Phipps, J. Amer. Chem. SOC., 1930, 52, 1340. 11 Etzel and Maurer, J. Chem. Physics, 1950, 18, 1003. 12 Mapother, Crooks and Maurer, J. Chem. Physics, 1950, 18, 1231. 13 Schamp and Katz, Physic. Rev., 1954, 94, 828. 14 Phipps, Lansing and Cooke, J. Amer. Chem. SOC., 1926,48, 112. 15 Kelting and Witt, Z. Physik, 1949, 126, 697. 16 Wagner and Hantelmann, J. Chem. Physics, 1950, 18, 72. 17 Reif, Physic. Rev., 1955, 100, 1597. 18 Bassani and Fumi, Nuovo Cimento, 1954, 11, 274. 19 Hojendahl, K. danske vidensk Selksab, 1938, 16, no. 2. 20 Kittel, Introduction to Solid State Physics (Wiley, New York, 1953), 1st ed., p. 40. 21 Frohlich, Theory of Dielectrics (O.U.P., London, 1949), 1st ed., p. 68. 22 Barnes, 2. Physik, 1932, 75, 723. 23 Lidiard, Report on Coi:ference of Defects in Crystalline Solids (Bristol, 1954), p. 283. 24 Miyake and Suzuki, J. Physic. SOC., Japan, 1954, 9, 702. 25 Suzuki, J. Physic. SOC., Japan, 1955, 9, 794. 26 Patterson, Rose and Morrison, Phil. Mag., 1956, 1 (8), 393.