GENERAL DISCUSSION Dr. Y. Haven (Philips’ Res. Lab., Eindhoven) said: In order to compare the diffusion theory of Brinkman and Schwarzl with transition-state theory it seems appropriate to compare their eqn. (13) with the corresponding equation in the transition-state theory, which reads as follows, if we confine ourselves to classical statistical mechanics and apply it to the special case under discussion : - 1 v - v Ki = R$ I vX I - a 1 exp (- F ) d q / exp (- q ) d q . (13A) top of the barrier hole i Here IzI = an average velocity, A = integration length at the top of the barrier and R = coefficient, accounting for the deviation from “ equilibrium ” (a kind of reflection coefficient). The factor R is < 1, but according to transi- tion-state theory, by hypothesis, R = 1, in zeroth approximation.Apart from a slight difference introduced by the integration over the barrier, the two eqn. (13) and (13A) are completely equivalent, with the following cor- respondence, where A,ff is an effective width of the barrier. Therefore, with respect to applica- tion, the diffusion theory of Brinkman and Schwarzl and the transition-state theory will yield the same results. The transition-state theory, however, has the advantage over the diffusion theory in as far as in the approximation R = 1, the quantities on the right of eqn. (13A) can be determined from theory, so that this equation is liable to experimental verification. Dr. H. C. Brinkman and Mr. F. Schwarzl (Delft) (cornmunicated): In reply to Dr. Haven we wish to point out that the difference between our eqn.(13) and Haven’s eqn. (1 3A) may be considerable, as our factor, , and Haven’s factor, 1 exp (- Y - v /kT) dq, have a different origin and may give quite different results. Furthermore in our diffusion theory the forward as well as the backward diffusion is taken into account. This is not possible in a logically consistent way in the transition state method. Dr. J. Wood and Dr. A. Suddaby (Sir John Cass College) (cornmunicated) : We should like to make two comments which are relevant to Dr. Brinkman’s paper. First, the Smoluchowski equation follows from the Planck-Fokker equation when terms of the order (l/y)2 and higher are neglected. When the restoration of thermodynamic equilibrium arises by means of intermolecular forces which may be represented by a friction constant, Kirkwood 1 has shown that the Planck- Fokker equation follows as a consequence of the molecular random motion in a liquid.In doing this he has related the friction factor 7 to the intermolecular forces. Secondly, when in these circumstances the Planck-Fokker equation is applied to a molecular rate process we have shown2 that the rate constant is given by exp ( V - v /kT) dq {J ( ) I-’ 0 w kT Q complex y h Q initial k j = - - exp (- AE/kT), when terms of 0 (l/y)2 are neglected, and AE > kT. The symbols apart from 1 Kirkwood, J. Chem. Physics, 1946, 14, 180. 2 Wood and Suddaby, to be published. 72GENERAL DISCUSSION 73 o and y are identical with those employed in the transition state method. y, the friction constant (units sec-1) depends on the intermolecular forces, w (also sec-1) is the curvature at the maximum of the plot of energy against reaction path.The more exact expressions when AE is comparable with kT, and when terms 0 (l/y)2 are included have also been obtained by us. Prof. A. R. Ubbelohde (Imperial College) said: A problem of fundamental importance with respect to dislocations is to determine whether any entropy is to be attributed to them, or whether they introduce purely energy terms in the overall energy of a condensed state. If each of the various kinds of dislocations introduces only energy terms, the ideal crystal when it has fully attained equilibrium will squeeze them out at all temperatures. Kinetically this may require a con- siderable lapse of time, especially at low temperatures. If, on the other hand, an entropy as well as an energy can be attributed to any kind of dislocation, then the familiar considerations about the increase of point defects with rise in temperature in thermal equilibrium might apparently apply also to dislocations which are co- operative defects.Theoretically it is difficult to calculate entropy contributions from dislocations. It may be suggested, however, that one way of proceeding is to calculate the dis- turbances in the vibrational spectrum of the crystal, arising from the presence of dislocations. For the various modes, wherever the vibrational spectrum traverses frequencies of vibration at which the specimen exhibits marked relaxation peaks, strong absorption is found. This contribution aims to draw attention to the con- sequence that the vibrational spectrum of the condensed phase will be traversed by absorption bands at the relaxation frequencies.In a way analogous to the constitution of Bloch-Brillouin zones in metals, these absorption bands of forbidden frequencies distort the vibrational spectrum of the ideal crystal. The consequent modification of the vibrational entropy, c,ibd In T corresponds with an entropy contribution from each dislocation. When this contribution is positive. as may happen for certain types of dislocations, the crystal containing the appropriate proportion of these dislocations will be thermodynamically more stable than the ideal crystal free from dislocations, at temperatures above absolute zero. Studies of the interaction between mechanical relaxation frequencies and the presence of dislocations may help to throw light on the possible thermodynamic stabiliza- tion of dislocations, which is otherwise difficult to establish experimentally.Prof. F. C. Frank (Bristol) (communicated): The basic answer to Prof. Ubbelohde’s enquiry about entropy contributions from dislocations is that there can be no coupling between dislocation strains and vibrations where Hooke’s law is obeyed. Hence, at low temperatures, this coupling which can increase the entropy exists only within a small radius from the dislocation line, and can have only a finite effect, whereas the strain energy density has a long-range dis- tribution giving a logarithmically divergent integral. Hence, at low temperatures, the crystal with a low concentration is certainly unstable with respect to the crystal having none.Prof. Ubbelohde’s statement of the situation is of course slightly erroneous. In order that dislocations shall exist in the equilibrium state, it is not sufficient that their entropy contribution SD be positive, but that TSo shall be greater than their enthalpy contribution HD. “ An appropriate proportion of dislocations ” in equilibrium is not to be expected : rather, we should expect none, or very many. Above some critical temperature the latter condition could be the stable one. This formally justifies an identification of the liquid with the very highly dislocated crystal : but this description is not a very useful one, because dislocation defini- tions become non-unique at excessively high dislocation densities.Prof. F. C. Frank (Bristol) (communicated): I think there is a basic miscon- ception in Dr. Dryden’s comment. The low concentrations of lattice defects con- templated should be without any eEect on the steady-state polarization ultimately J:74 GENERAL DISCUSSION established, which is an equilibrium quantity: they provide only the mechan- ism by which it is established and their concentration determines the rate of establishment. This simple statement assumes that one of the two quantities I have called a~ and ag greatly predominates over the other. There are two conceivable polarization states according to which predominates. The situation will be more complex if they are nearly balanced, but in no case should we expect polarization proportional to exp (- EDlkT).Prof. J. S. Koehler (1ZIimi.s) said: With regard to the paper by Seeger, Douth and Pfaff : (i) Is 7; related to the yield stress in pure metals? (ii) In what respects does this theory differ from that of Mason? (iii) Will the presence of partial dislocations influence matters ? Dr. A. Seeger (Stuttgart) (communicated) : The answers to Prof. Koehler's questions are as follows: 7; is certainly nut related to the yield stress of pure metals at sufficiently high temperatures, since thermal energy helps the dislocations to overcome the Peierls stress. At high enough temperatures it will bring down the " effective Peierls stress " to virtually zero. Provided we are justified in neglect- ing quantum effects 7; will be equal or closely related to the critical shear stress at 0" K if in the undeformed crystals all dislocation lines lie along close-packed direction or if the dislocation arrangement is such that during the operation of the Frank-Reed sources the dislocation rings can be " trapped " along the close- packed directions.If, however, the critical shear stress (at all temperatures) of pure metal crystals is determined nut by the length of the Frank-Reed sources but by the elastic stress fields of the dislocation and by the so-called dislocation point, 7; should not be related to the critical shear stress. The reason for this is that under these conditions the majority of the dislocation lines will run along other directions than the close-packed ones, and will therefore not " feel " the Peierls stress.There is considerable experimental evidence that the critical shear stress of pure metal crystal is due to the stress contributions mentioned above ; 1 therefore there should be nu connection between 7; and the critical shear stress of pure metal crystals. This conclusion is in agreement with the fact that the numerical values of 7; deduced in our paper are by an order of magnitude larger than the critical shear stress of copper 2 and aluminium 3 9 4 single crystals extrapolated to 0" K. Mason's theory and the theory presented in our paper have this in common that both of them connect the Bordoni relaxation peak with the Peierls stress. The details, however, are quite different as may be seen from the fact that the numerical values of 7; deduced in our paper are two orders of magnitude larger than those derived by Mason.The main difference between the two theories is that accord- ing to Mason's mechanism the activation energy of the relaxation process should depend linearly on the length I between pinning points due to impurities or between dislocation nodes. (This is at variance with a large number of observations ac- cording to which the temperature of the Bordoni peaks is remarkably insensitive to impurity content, on pre-strain, and neutron bombardment.) In the present theory however the partition of the relaxation peak is to a very good approxima- tion independent of the state of perfection of the crystals. As was also pointed out in a discussion between Weertman5 and Mason the mechanism of thermally activated kink formation, discussed in our paper, is thought to make the process considered by Mason impossible.1 Seeger, 2. Naturforschg., 1954, 9a, 758, 870. 2 Blewitt, Coltman and Redman, Rep. Con$ Defects in Solids (London, 1955), p . 369. 3 Sash and Koehler, Physic. Rev., 1956, 101, 972. 4 Noggle and Koehler, J. Appl. Physics, 1957, 28, 53. 5 Weertman, Physic. Rev.,GENERAL DISCUSSION 75 To a first approximation the fact that the dislocations in metals like copper and silver comist of a pair of partial dislocations should not influence the dis- cussion presented in our paper. If eqn. (4.1) holds for each of the partial dis- locations it will also hold for the extended dislocation formed by a pair of partials.The reason for this is that the sine terms on the right-hand side of eqn. (4.1) due to the partial dislocations, add to another sine term for the extended dislocation. This holds even if the individual sine terms are “ out of phase ”, i.e. if the separa- tion between the partial dislocations is not a multiple of a. If the separation between the partial dislocations depends on temperature, the parameter ri (in eqn. (4.1) for the complete dislocation) will be temperature-dependent (even at temperatures low enough to render Dietze’s mechanism 1 for the temperature dependence of 7; ineffective). Some caution is therefore necessary if the theory is to be applied to data which have been obtained over a wide range of temperatures. The preceding discussion shows that it is presumably justified to apply the fornzalfrarnework of the theory to the existing data on f.c.c.metals. With regard to the rmgrzitzide of T; there might be an influence of the extension into partial dislocations. Other things being equal, 7; is predicted to be smaller for partial dislocations than for complete unextended dislocations.29 3 Although it is true that aluminium should not in all respects be compared to the noble metals, it is tempting to associate the empirical result of T ~ / G being larger for aluminium than for copper or silver with the fact that in aluminium the dislocations are practically unextended whereas in the noble metals they are well separated into partial dis- locations. Dr. J. Volger (Eindhoven) said: As one of the authors of the paper on the dielectric measurements on smoky quartz referred to at the end of the paper by Seeger et ai., I would make the following comment.The relaxation time of the dielectric loss mechanism in smoky quartz is characterized by a very small activation energy and a somewhat unusual pre-exponential factor, of the order of 10-8sec. The same orders of magnitude are found with the low-temperature losses in Fez03 with electrons trapped at donor centres. A connection with dis- location damping may suggest itself. However, unless the colour centres in quartz (which are without doubt chemical impurities) and the donor centres in the Fe203 investigated (excess Fe) are lined up along dislocations, it is most likely that the dielectric relaxation peaks found are due to point defects.This point of view is also taken in my paper and seems to be supported by recent considerations of Frohlich.4 Dr. J. S. Dryden (Sydney) (communicated) : Seeger, in this discussion, suggests that the mechanisms of the dielectric absorptions discussed by Volger may be similar to the mechanical relaxation in deformed crystals as discussed by Seeger, Donth and Pfaff. Without wishing to comment as to whether this is correct or not I want to point out that one of Seeger’s arguments is not valid. This argu- ment is that a frequency factor of 109 sec-1 does not occur in other examples of dielectric absorption. However, to take one example, in a barium magnesium titanate which we have investigated the frequency factor associated with a di- electric relaxation involving the movement of ions is 4 x 109 sec-1. Dr.A. Seeger (Stuttgart) (communicated) : In reply to Dr. Volger and Dr. Dryden I should like to thank these authors for their comments on the low-temperature dielectric relaxation in quartz. I agree with their remarks that frequency factors of the order 109 sec--1 have been found in dielectric relaxation phenomena not connected with dislocations. As indicated in our paper the mechanical losses occurring in the same temperature region are unlikely to be of the Bordoni type. 1 Dietze, 2. Physik, 1952, 132, 107. 2 Dietze, 2. Physik, 1952, 131, 156. 3 Seeger, Handbuch der Physik, VII/l, Ziff. 72. 4 Frohlich, Discussions, Colloqiie A.M.P.E.R.E., St. Malo (France), 1957.76 GENERAL DISCUSSION Since so far no alternative explanation for the mechanical losses has been advanced, and since frequency factors of the order lO9sec-1 are very rare in mechanical relaxation phenomena in crystals an experimental investigation into a possible interrelation between the mechanical and the dielectric losses is still thought to be interesting, In order to indicate just one possibility we mention that the following inter- pretation seems to be compatible with the existing data.Both the mechanical and the dielectric losses are due to the same process, or the same group of processes. The mechanical losses show up in both unirradiated and irradiated crystals. The dielectric losses, however, occur only in irradiated crystals, due to the introduction of an electric dipole moment upon irradiation.Prof. J. H. de Boer (Geleen) said: Dr. Cole, in the presentation of his con- tribution, referred to solid solutions of rare gases and hydrogen halides. Could he, possibly, give some more information? It does not look possible that a non- interrupted row of mixed crystals of rare gases and the low-temperature forms (ordered forms) of the hydrogen halides could exist. Dr. H. Granicher (Zurich) said: The dielectric constants of solid hydrogen halides show in the low temperature phase a dispersion at low frequencies similar to the behaviour of ice crystals. Further common features are the rather large dipole moments of the molecules and structural evidence that the crystals are built up by a relatively rigid system of hydrogen bonds. It is therefore suggested that the dielectric properties of the hydrogen halides might be explained by a lattice imperfection mechanism similar to that of ice described in our paper.Dr. J. S. Dryden (Sydney) said: By the same method as used in our paper we have calculated the energy barrier for the movement of a positive ion vacancy in AgCl to be 0.31 eV. This agrees to within 30 % with the experimental value of 0.36 eV obtained from d.c. conductivity.1 I have learnt recently from Dr. Morrison (National Research Council, Ottawa) that the energy barrier for the movement of C1 ions in NaCl is nearer 1.3 eV than 1.7 eV, the value quoted in table 3 of our paper. This makes the agreement with the calculated value poorer for NaCl but improves the consistency of the ratio expt./calc. between NaCl and NaBr.Dr. Y. Haven (Philips’ Res. Lab., Eindhoven) said: Energies of activation are often obtained by drawing two intersecting straight lines through the points of measurements in a logarithmic plot of ionic conductivity against 1/T. For the ionic conductivity of alkali halides containing divalent ions, however, it is possible to account for the interaction between impurity defects and thermally produced defects (as described in the paper of Compaan and Haven in this Discussion). When doing so, one obtains lower energies of activation for jumping and higher energies of activation for producing defects, than is ordinarily found in literature. This may account for the difference between the several energies of activation cited in table 1 of the paper of Dryden and Meakins.This same correction makes the high-temperature energy of activation for diffusion of Na in pure NaCl larger, and the energy of activation for diffusion of C1 in NaCl smaller. Therefore, the energy of activation for the movement of C1- and Br- ions in NaCl and NaBr, respectively, will undoubtedly be much smaller than that cited in table 3 of Dryden and Meakins’ paper. For C1- in NaCl, this value should be smaller than 1 eV. Prof. F. G. Fumi (Universita di Palerrnu) said : Dryden and Meakins schematize the logarithmic plot of the ionic conductivity of a doped alkali halide crystal against 1/T in the impurity-sensitive range with two straight lines, of which the one referring to the lower temperature region has a steeper slope. They assume that the difference in slope of these straight lines is related directly to the binding energy of the complex between the divalent impurity cation present in the crystal and the positive-ion vacancy.In effect one should expect several distinct regions 1 Compton and Maurer, J. Physic. Chem. Solids, 1956, 1, 191.GENERAL DISCUSSION 77 in the impurity-sensitive range, and the straight line of the authors for the lower temperature region (1.4 < lO3/T < 1.6) corresponds most likely to the impurity precipitation region of the Gottingen school.1 Thus the difference between the values of AE given in columns 4 and 6 of table 1 of their paper does not give the binding energy of the impurity complex. The elastic method used by Dryden and Meakins to estimate the energy barrier for the movement of ions in an alkali halide crystal has naturally an heuristic character as the elastic method used by Buffington and Cohen2 for metals; for one thing, the choice of the strained volume TJO is rather arbitrary.I think one should be careful in assuming that an estimate of this sort really allows one to say which is the path of movement of ions in an ionic solid : in particular, electrostatic terms such as the coulombic interaction between the moving ion and the two neighbouring vacancies, which are neglected in an elastic calculation, will be important in determining the actual path of movement, and they favour the “straight” path AA’ (fig. 6a of Dryden and Meakins’ paper) against the “ interstitial ” path AXZX’A’ by something like a couple of electron volts.Dr. Y. Haven (Philips’ Rex. Lab., Eindhoven) said: When energies of activation from conductivity data in the form K = KO exp (- U/kT) are compared with energies of activation from dielectric loss data, it is convenient to compare U + kT++ E (dielectric loss) because of the relation K m nyezdzjkT, where y is the jump frequency. When doing so one obtains LiF LiCl LiBr LiI E (diel. loss) (Dryden, Meakins) 0.65 0.42 0.40 0.35 U + kT (Haven) 0.73 0-47 0.45 0.43 Thus the energy of activation for jumping is smaller if a vacancy is associated with a divalent ion, than when the vacancy is free. Previously 3 this was seen as an indication that for instance in NaCl, a Naf does not jump straight through between two C1 ions from one equilibrium site to another, but makes a detour passing more near the tetrahedral hole.In that case the repulsion of the Naf ion by the divalent ion will be smaller at the barrier than at the neighbouring lattice site, thus lowering the energy barrier. Dr. A. B. Lidiard (Harwell) said: In connection with the paper of Dryden and Meakins I would like to make two minor criticisms and one general comment. First, eqn. (6) is correct as it stands and no factor 4 should be inserted : the angular frequency at which maximum absorption occurs is 2(wl + w2), where w2 is defined as the frequency of impurity-vacancy exchange and w1 is the jump frequency for an associated vacancy going to a particular but arbitrary one of the four neighbour- ing positions common to both the vacancy and the impurity (fig.4 of their paper). This was shown explicitly a year or two ag0.4 When w2 < w1 we then obtain A = f 0 i . r and not 4fo/n. The comparison of A withf& in table 1 thus indicates the occurrence of entropies of activation for the w1 jumps of about 216. My second criticism is that eqn. (7) needs a factor 4rr on the right-hand side to be ~ o r r e c t . ~ At 300” K, I then obtain 0.055 for the maximum loss in NaCl containing 0-075 mole % CaC12, which is four times larger than the estimated experimental value of 0.013. Although, as the authors point out, this estimate may not be accurate owing to the use of many small crystals, the result is consistent with the 1 see, e.g., Zuckler, TIzesis (Gottingen University, 1949). 2Buffington and Cohen, Acta Met., 1954, 2, 660; see also Brooks, Zmpurities and 3 Haven, Defects in Crystalline Solids, Rristol, 1955, p 261.4 Lidiard, Conference on Defects in Crystalline Solids (Bristol, 1954)) Physic. SOC., Imperfections (Amer. SOC. Met., 1955). 1955, p. 283.78 GENERAL DISCUSSION previous results of Haven,l also on NaCl + CaC12, and of Teltow and Wilke 2 on AgBr + CdBr2. In both these cases the magnitude of the loss coming from the impurity-vacancy complexes is only about one-third of that predicted by the theory given in 0 2 of ref. (1). This may well indicate that the electric dipole moment of the complex is considerably smaller than the d%a assumed in that treatment; a strong attraction of the electrons on the anions neighbouring the complex to the divalent cation may well reduce the dipole moment below the value 6 e a . Some confirmation of this view is obtained from the results of Busse and Teltow 3 on dielectric loss in AgBr + AgzSe : here the complexes are Se2- ions and interstitial Ag+ ions.The maximum loss in this case was found to be only one-twentyfifth of the value expected from the simple ionic model : transfer of electrons from the Se2- ion to the interstitial Ag+ will very probably occur and will reduce the dipole moment accordingly. Dr. R. A. Sack (Cambridge) said: Dr. Dryden and Dr. Meakins are led to compare the frequency factor A and the expression 4foi.r by using eqn. (6) for the transition probability w1. Yet the Debye relation T = 1/2w for jumps between stable dipole orientations applies only if the dipole moment is completely reversed ; in the present case, if the field is along one of the crystal axes (see fig.4), two of the four possible jumps do not alter the relevant dipole component at all; the other two take it only half-way to a reversal. The product of the number of possible transitions and their mean efficiency is thus 4 x + = 1. Hence the factor A should be compared with f&; in table 1 this would improve the agree- ment for LiF and KI, but make it worse in the other cases. Prof. F. C. Frank (Buisiol) said: It happens that last weck P was at the Nottingham conference of the Physical Society, at which Schneider presented a study by paramagnetic resonance of sodium chloride containing 10-5 or 10-4 parts of manganous chloride. This gave a very clear indication of the various situations of manganous ions in the crystal, which may help in the interpretation of Dryden's observations.The spectra observed were, in order of increasing temperature, from room temperature to about 500" C , I think : (A) a broad un- resolved hump giving place to (B) and (C), a complex orientation dependent spectrum of many lines, out of which emerged (D), an orientation-independent spectrum of 6 lines. (B) and (C) can be resolved into two parts, (B) having symmetry about (110) axes, and (C) having symmetry about (100) axes about 1/10 as strong as (B). These are interpreted as : (D), manganous ions in cubic environment, normal substitutional cations ; (B) manganous ions with cation vacancies in nearest neighbour positions ; (C), manganous ions with cation vacancies in second nearest neighbour positions ; and (A), manganous ions close to other manganous ions.As no precipitate is visible in the ultramicroscope, 1 would surmise that the latter are very likely coherently segregated in monatomic layers, like Preston-Guinier zones. Without referring to these, since vacancies are trapped both in nearest and in second nearest neighbour positions to a divalent cation, there should be two different relaxation components, for jumps between two nearest neighbour positions, and for jumps between nearest and second nearest neighbour positions. Dr. J. S . Dryden (Sydney) said: The model used by Granicher, Jaccard, Scherrer and Steinemann and by Bjerrum to explain the dielectric properties of ice appears to me to contain a feature which is inconsistent with experiment.It is claimed that lattice defects must be present in the crystal before polarization, other than atomic and electronic, can occur. The number of these lattice im- perfections is a function of temperature and consequently the magnitude of the 1 Haven, Conference on Defects in Crystalline Solids (Bristol, 1954), Physic. Soc., 2 Teltow and Wilke, Naturwiss, 1954, 41, 423. 3 Busse and Teltow, Noturwiss., 1957, 44, 11 1. 1955, p. 261.GENERAL DISCUSSION 79 polarization should vary in proportion to exp (- E&T) whereas in fact (€0 - E ~ ) varies with 1/T. The same authors make use of a quantity which they call the " ax. conduc- tivity ". The measurements of tan 8 at any frequency is the same as measuring an a.c. conductance at that frequency ; for example, each point in fig.2 of the paper of Granicher et al. can be related to an a.c. conductance but beyond this the term has no meaning. However, Granicher et al. use the term having some other property in mind and I would be interested to know what this is. The dielectric absorptions discussed by Volger are very small, parts in 105 in some instances, and I would like to know what errors are associated with the curves in his figures. Dr. A. Steinemann (Zurich) said: In reply to J. S. Dryden, the conductivity correction (eqn. (1.2)) can be derived from the Debye equations by adding a d.c. term a&~o to E". It can be shown that, for the Cole plot, semicircles still exist, but have an increased radius. In the alkali halides polarization processes only occur at lattice defects and hence (6; - EL) is proportional to the defect con- centration.In ice crystals, however, in principle every water molecule is able to re-orient and to contribute to the polarization. The fraction of polarized molecules in static fields depends on the ratio eSF/kT of the electric to the thermal energy. The number of defects only affects the rate of reorientation processes, but not the equilibrium polarization. The d.c. conductivity was measured by the time-independent current in a static applied field. The a.c. conductivity results from the bridge measurements, if the complex permittivity of the crystal is interpreted as a parallel circuit of a real permittivity and a loss resistance. In $ 8 of our paper and elsewhere d.c.and a.c. conductivities are compared. With the term a.c. conductivity we then mean the low-frequency limit of the a.c. conductivity designed by OF. Prof. R. M. Barrer (Imperial College) said: R. S. Bradley has recently measured the conductivity of ice from 0 to - 25" C, and obtains CT = 23.4 exp (- 12,30O/RT) ohm cm-1. As far as the energy barrier is concerned, this agrees only moderately with Granicher's value 0.6 eV = 13.8 kcal. What accuracy can be achieved with such measurements? Is there likely to be anything like grain boundary con- ductivity in polycrystalline ice? I would also like to draw attention to work done at Basle by Kuhn and Thurkauf on self-diffusion in ice, on which a brief report appears in the 10th Solvay Corzgr. Clzern.1 They found that DzO and H2018 diffused at the same rate in ice, with D - 10-10 cm2 sec-1 at - 1 O C. This means a molecule diffusion mechanism, except in the unlikely event that a coupled diffusion of OH- and H 3 0 f occurred.Prof. Kuhn is, I think rightly, convinced that a place interchange between neighbour- ing pairs of water molecules is the controlling mechanism, and not a diffusion of holes or defects. The time needed for adjacent molecules to interchange was - 5 x 10-6 sec at - 1" C, i.e. such changes occurred with a given water mole- cule 2 x 105 times per sec. Since such events occur so rapidly I would expect them to have an influence on the dielectric relaxation properties of ice, and would like to hear the views of Granicher et al. about this. Dr.H. Griinicher (Zurich) said: In reply to Prof. Cole, we are quite aware of the problem connected with the electric field F which appears in our eqn. (5.3) to (5.6). The term e6Frepresents the change of the electrostatic energy of a proton displacement. Since actually work is delivered for such a movement, the field in question must be the externally applied field. Prof. A. R. Ubbelohde (Imperial College) said: With reference to the paper by Granicher, Jaccard and Steinemann, we have recently studied proton conductance in solid organic acids and other crystals containing hydrogen bonds.2 So far 1 1956, pp. 77 et seq. 2 6 . Pollock and Ubbelohde, Trans. Furuduy SOC., 1956, 52, 1112.80 GENERAL DISCUSSION as the preliminary findings extend, it seems likely that systems of co-operative hydrogen bonds in crystals arranged as networks, sheets or spirals, permit proton migration mechanisms that require considerably lower activation energies than when hydrogen bonds are present as isolated pairs in the crystal.This lends considerable support to a co-operative Grotthus type of conductance, in which the proton co-operative systems polarize in the external field and relax from these specific positions by one or more micro-mechanisms. The net consequence of these two operations is to lead to proton migration. Crystals such as organic acids offer certain advantages over ice in the study of co-operative hydrogen bonds because the limits of the co-operative systems are more clearly defined. The existence of polymorphs of practically the same free energy as ordinary ice makes it likely that boundaries between one crystallite and the next involve small regions where the ideal arrangement is modified ; this may confuse estimates of residual entropy based on the assumption of an ideal lattice of infinite extent. Concerning the paper by de Boer and the remarks by Dr.Mannil, we are en- gaged in various studies of crystal transformations in which a single crystal is subjected to temperature cycles, traversing the transformation temperature T,, and in which the persistence of crystal axes is followed by X-ray methods. These bring out the great importance of traces of impurities, such as water, for deter- mining the extent to which the single crystal traverses the same path in detail. It may be added that one of the best ways of removing these traces of water is to subject the crystals to very high vacua and take them through a cycle of temperatures including T,.Co-operative defects that arise in the neighbourhood of T‘, because of the coexistence of domains of closely similar structure, greatly facilitate the removal of impurities that migrate rapidly around the transformation temperature. Recent work has shown that this ‘‘ thermal shock ” cyclic treatment is effective, for example, for nitrates and for sulphocyanides. Dr. Y. Haven (Philips’ Res. Lab., Eindhoven) said: In order to discuss the di- electric properties of ice, it is appropriate to note that one can discern “ individual ” and “ collective ” properties. The relaxation of a vacancy-divalent ion-pair in alkali halides is an individual property and independent of the concentration of lattice defects.The relaxation in ice, however, is a collective property as all molecules contribute to the equilibrium. If the average jump frequency of the molecules is raised because the concentration of lattice defects is raised, the re- laxation frequency will be raised, as has been observed for the larger concentra- tions of fluoride ions. This type of relaxation is closely connected with the electrical conductivity of ice. There might be found, however, another relaxation process when fluoride ions are present. If the corresponding hydrogen vacancies associate to the fluoride ions, one obtains a dipole of the type as in alkali halides containing divalent ions with “ individual ” properties.A dielectric relaxation corresponding with the orientation of these dipoles may be expected at much higher frequencies than used by Granicher et al. (say 108 sec-1) or at much lower temperatures. Dr. M. Eigen (Giittingen) said: In general, it is not possible to determine the ionic concentration and mobilities separately by conductivity measurements, as the specific conductivity contains these quantities in form of the product c(u+ + u-). Earlier considerations of some authors resulted in the conclusion that these three quantities may be nearly the same as in water, because the conductivity of ice shows the same values as in water. Bjerrum was the first to point out that these conclusions would lead to difficulties, as the electrostatic interaction of the fast moving ions in ice cannot be described by the static dielectric constant as for water.Using a dynamical value he calculated a heat of neutralization which is appreciably higher than that of water. The results of Dr. Granicher and co-workers support these assumptions by reporting a value of ionic con- centration between 3 x 10-9 and 6 x 10-11 mole/l. (The respective value ofGENERAL DISCUSSION 81 water would be 2 x 10-8 molell.) Assuming equal conductivities of both media, the lower concentration has to correspond to higher mobility. We came to very similar conclusions by a rather different approach. We studied fast protolytic reactions in solution using relaxation methods (e.g. sound absorption, dispersion of the dissociation field-effect). In these investigations we found for the rate constant of the neutralization reaction Hf + OH- -+ H20 the very high value of 1.3 x 1011 I./mole sec which suggests that protons and hydroxyl ions recombine spontaneously if they approach to a distance of 2 to 3 H bonds.1 This is only possible if the mobility of these ions in H-bonds is higher than normal in water.In order to determine the mobility of protons in H-bonds (e.g. ice crystals) directly, we developed a special technique, which has recently been described 2 (the stationary field-method). Preliminary results show that in ice the mobilities of the proton (and also of the hydroxyl ion, or defect proton) are higher by 1 to 2 orders of magnitude, and the ionic concentration is lower by 2 to 3 orders of magnitude than in water (in agreement with Dr.Granicher’s results). The proton mobility in H-bonds differs from electron mobility in metals only by 1 to 2 orders of magnitude, approximately corresponding to the square root of their mass ratio. This result strongly suggests a quantum-mechanical mechanism (tunnel effect) for the proton transfer within the H-bond. The high value of the proton mobility mentioned should be determinable by Hall-effect measurements in very high electric and magnetic fields. Such measure- ments are in progress. On the other hand, the above-mentioned stationary field method is generally applicable to semiconductors with very low concentrations of charge carriers, and permits one to determine the mobilities in cases where Hall-effect measurements become difficult.As the properties of protons and defect-protons correspond to those of electrons and holes, interesting analogies between H-bond systems and electronic semi- conductors arise. So it was possible to test ap,n rectifier consisting of two ice crystals, one contaminated with HF (proton donor) the other one contaminated with LiOH (proton acceptor). Instead of normal metal electrodes, one has to use a special electrode system containing acid and base solutions or exchangers and unpolarizable electrodes. As H-bond systems are present in biological membranes, the above-mentioned effects may be important for biological processes. Prof. F. C. Frank (Bristol) said : I have a bad conscience about the Bjerrum theory of dielectric polarization in ice, which Granicher and his colleagues appear to have satisfactorily established.When I thought of this theory just about ten years ago, I was quite excited with it, and went and told Onsager about it, who happened to be in England at the time. He told me he had discussed the same theory in lectures two or three years before. This depressed me somewhat. Neither of us ever published it, so it is Bjerrum’s theory. One of my reasons for not publishing was that I did not know how to make it unique or exact. Let me use the letters P and N to denote hydroxonium ions and hydroxyl ions, and call these the intrinsic defects of class (A), the ionic defects, and then call D and L the orientaticnal defects with two protons or none between a pair of oxygen atoms respectively, class (B).Then if either class were absent or immobile, the crystal would behave as a polarizing dielectric. A positive current carried along any particular path by a defect of one of the classes can only be followed along the same path by a negative current carried by the same class, or a positive current carried by the other class. This is the well-known situation with regard to the Grotthus conduction mechanism-that ion-transfer and reorientation processes must both occur for steady-state conduction : if either occurs alone, there is only a polarization. Given non-polarizing electrodes, the lesser of the two quantities, OA = ( I I P ~ P +- IINPN) and UB (HDPD + HLPL), 1 Eigen and De Maeyer, 2. Elektrochenr., 1955, 59, 986. 2 Eigen and De Maeyer, 2. Elektrockeni., 1956, 60, 1037.82 GENERAL DISCUSSION where n and p signify concentration and mobility respectively, gives the steady state conductivity, while the difference between these two gives the polarizing current which determines the dielectric constant.It is easy to see with regard to the charges, by considering the effect of transferring two defects along the same path, that e p + eD = ep - eL = -eN - eL = -eN + eD = e , and I estimated approximately, eD = - eL -+e, e p =- eN - t e . However, just because the dielectric constant of ice is so large, it appeared to me that it could be interpreted by predominance of either a~ or OB, so far as the order of magnitude is concerned. An approximate calculation of the consequences of this mechanism is not difficult, but for an exact calculation there are some complex correlation problems to be solved.These are intimately connected with Pauling’s problem concerning the entropy of ice. Pauling’s problem is to estimate the number of configurations of ice without restriction as to the degree of polarization -the formal problem relating to the dielectric constant is to estimate the number of configurations corresponding to various degrees of polarization. I think it is still the case that neither problem has been solved exactly. Pauling’s two deriva- tions of R In (3/2) for the residual entropy of ice are certainly not rigorous. It is possibly in these correlation corrections that one should seek the explanation of the anisotropy in the dielectric constant of ice, rather than in the unsatisfactory explanation offered by Granicher (“ it seems plausible to exclude jumps .. . at a high angle to the field direction ”). The direction of the applied field cannot discontinuously affect the jumping frequencies : the field causes only a very slight bias in the frequencies of jumps which occur spontaneously all the time. Dr. A. Steinemann (Zurich) said: Dr. Bradley’s value 3 for the activation energy of the electric conduction in pure ice of 0.5 eV is considerably smaller than our present value (0.6). This might be due to impurities and the too small temperature range of measurements. Since impurities give rise to lattice imperfections the energy term corresponding to the formation of these imperfections does not appear in the total activation energy.Hence only the highest measured activation energy is significant for pure ice. The diffusion coefficient of fluorine in our ice crystals was about 5 x 10-11 cm2 sec-1 at - 10” C in agreement with Kuhn’s results for the self-diffusion of H2018. These diffusion processes are only possible by vacant molecular sites and (or) interstitial molecules. If interstitial molecules are considered as rather free to rotate, a dielectric constant higher than 100 would be expected theoretically. The ionic conductivity of ice, however, cannot be explained unless orientational defects and ionized states are present simultaneously as underlined by Prof. Frank in his discussion remark. Our theory is able to interpret the dielectric as well as the conductive properties of pure and impure crystals.This fact together with the evidence given by Eigen in his contribution makes it appear that a con- nection between the dielectric properties and the mechanism of self-diffusion is rather improbable. Grain boundary conductivity has been observed and inter- preted by a special surface diffusion mechanism (bipedal random walk) by Murphy.4 Dr. J. L. Meijering (Philips’ Res. Lab., Eindhaven) said: I agree with Prof. Frank that Pauling’s value R In (3/2) for the theoretical residual entropy of disordered ice is not quite correct. A better approximation yields a 5 % higher value : 0.85 cal/mole deg. (In the corresponding quadratic lattice the entropy can be computed rather accurately and one finds 63 % more than R In (3/2).) As the experimental values for the residual entropy appear to be about 0.80 cal/mole deg.(0.82 for H20 and 0-77 for D2O) one can now say that there are indications of partial order of the dipoles in ice. 3 Trans. Faraday Soc., in press. 4 J. Chem. Physics, 1953, 21, 1831.GENERAL DISCUSSION 83 Dr. H. Granicher (Zurich) said: In reply to Prof. Frank and Dr. Meijering, we agree that the computation of the number of possible hydrogen configurations in an ice crystal as given originally by Pauling is crude. A full proof of the value ( 3 / 2 ) N has been given by Bjerrum (our ref. (23)). This derivation is based on the three-dimensional ice structure model and is considered to be correct for an infinitely large crystal for which contributions of the surface can be neglected. Deviations from the value (3/2)N were not known to us before, but apparently do not exceed the inaccuracy of the experimental values.The experimental accuracy (& 0.15 cal/mole deg.) is not such as to justify Dr. Meijering's con- clusion that ice should be partially ordered. It is very probable that stacking faults occur in ice crystals. The sequence of the close-packed o-layers in hexagonal ice along the c-axis may be interrupted by one or more layers which have the orientation of cubic "diamond" ice. (Stacking faults of this type are j'ery common in synthetic crystals of ZnS.) In contrast to the suggestion of Prof. Ubbelohde, such stacking faults in ice do not necessitate any disturbance in the H-arrangement. Since " diamond " ice has the same number of possible H-configurations, such imperfections do not affect the interpretation of the measured zero-point entropy.Dr. J. L. Meijering (Philips Res. Lab., Eindhoven) (communicated): The cal- culations of the residual entropies of ice and of the quadratic pseudo-ice lattice will bc published in Philips' Research Reports. The reason why Pauling's result is too low may be sketched as follows. Consider a group of dipoles which has a plane of symmetry traversed by IZ bonds. In each bond the hydrogen has two possi bfc positions. If the number of configurations of each isolated half-group is P, then P2j'is that number for the whole group, where f stands for the probability that all IZ interconnecting bonds " fit ". There are 2" possible combinations of the H positions in these bonds.Let pi be the chance that a half-group of dipoles shows one of those coinbinations (Zpi = 1) ; then, because of symmetry, f = Epi2. This quadratic expression is a minimum (f = (+)") if all pi are equal, which is as- sumed in Pauling's treatment. This random distribution of the hydrogens is, however, excluded by the requirement that each oxygen must have two " near " and two " far " hydrogen neighbours. Inspection of small groups of dipoles shows indeed that the pi are unequal, which makes f > (t)". And in building an infinite lattice by repeatedly doubling in size the group of dipoles this excess of con- figurations cannot be compensated, becausef can never be smaller than (+)". Dr. H. Griinicher (Zurich) said: Prof. Munster's statement that a knowledge of the temperature at which configurational changes freeze-in is not essential for the explanation of the zero-point entropy itself, is certainly correct. But it is emphasized that the existence of configurational changes is of great importance for the interpretation of most of the physical properties of ice, in particular for the dielectric behaviour. Dr. A. B. Lidiard (Harwell) said: The use of eqn. (4) and (5) in Volger's paper implies the assumption of a Lorentz internal field. I do not wish to discuss this question in general, but I think it is worth pointing out that there are situations connected with lattice defects where the use of this field seems to be incorrect. I would refer in particular to the type of dipole studied in the alkali halides by Dryden and Meakins. Such a dipole changes its orientation by a jump of the vacancy which is associated with the impurity from one neighbouring position to another. In both positions it has the same cubic environment of field-induced dipoles (except for a small change caused by the difference in polarizability between an impurity ion and a normal ion) and thus its energy of interaction with this environment is unchanged. Hence the Lorentz internal field should not be in- cluded in this case, since no work is done by it when the dipolar orientation changes; is then given simply by 4n-Np2/3kT. If the situation described by Dr. Volger in quartz should be similar then somewhat larger and more " normal " moments would be inferred.84 GENERAL DISCUSSION Dr. J. Meinnel (Rennes) said: Many results found in Prof. Freymann's labor- atory (Rennes) are very similar to those of Dr. Volger. The study of ZnO (pure or doped) have shown no less than five different mechanisms with different activa- tion energies, giving rise to dielectric losses. In the study of Sic losses were found with very low activation energies (0.01 eV for gray Sic, 0.03 eV for green Sic) ; these low temperature losses (T below 50" K) may be interpreted as due to trapped elcc- trons. Dielectric studies of selenium have given valuable information on the homogeneity and on the impurity centres in this material.