DLELECTRIC RELAXATION AND THE ELECTRICAL CONDUCTIVITY OF ICE CRYSTALS B Y H. GR;I;NICHER c. JACCARD P. SCHERRER AND A. STEINEMANN Physikalisches Institut der Eidg. Techn. Hochschule Zurich Schweiz. Received 5th February 1957 The results of previous and recent extended dielectric investigations on pure hexagonal ice crystals and on mixed-crystals containing hydrofluoric acid are summarized. Measure-ments of the d.c. conductivity were performed using electrodes of different nature. Quantitative electrolysis experiments demonstrate the ionic character of the electrical conductivity. The theory of the dynamic behaviour of dielectrics is applied to ice assuming that polarization changes only occur at lattice imperfections of two types orientational defects (i.e. vacant and doubly occupied hydrogen bonds) and ionized states (i.e.H3O+ and OH-). For both cases the possible proton jumps are computed statistically. The theory gives the correct dispersion behaviour and shows that the static dielectric constant of pure ice (- 100) is due to proton jumps at molecules adjacent to doubly occupied bonds. The theory is extended to mixedcrystals with HF. It can be concluded that at medium high HF concentrations the predominant dispersion mechanism takes place by proton transfer at H3O+ ions whereas at the highest concentrations the vacant bond mechanism prevails. Finally it is shown that the present theory of lattice imperfections in ice is consistent with all known structural and physical facts inchding the zero-point energy. 1. RESULTS OF DIELECTRIC INVESTIGATIONS The dielectric properties of pure ice crystals are of great interest and have been studied repeatedly since 1924.1 The most reliable results were obtained in 1952 on polycrystalline ice2 and with single crystals an anisotropy of the dielectric constant of about 15 % was determined.3 Two features of the dielectric pro-perties are astonishing : (i) Ice shows an ordinary Debye dispersion with an extremely low dispersion frequency (cod = 2 x 104 c/sec at - 10” C) compared with that of the liquid (od - 1011 c/sec) and of other polar substances.The temperature dependence of the relaxation time is given by The values 2.3 B = 5.3 x 10-16 sec and E = 13-2 kcal/mole which corresponds to 0.575 eV have been verified in the present investigation. (ii) The static permittivity of ice is about 100 and is thus even higher than that of liquid water.The measurements of the real and imaginary part of the dielectric constant E’ and E” fit the semicircles of the Cole plot E”(E’) well except for very low frequencies where deviations occur even under the conditions of highest purity. For low temperature measurements the data give the appearance of fitting the Cole plot better because the measurements have not been extended to sufficiently low frequencies. In this case the static permittivity obtained by extrapolation of the Cole plot is too large due to the d.c. conductivity q. (2) The imaginary part E” of the Debye dispersion should vanish at low frequencies since it is proportional to the frequency. Actually the deviations are such that E” diverges as 1/0.This behaviour might be interpreted as due to a frequency-r d = B exp (EJkT). (1) €;(go) = E i true + 2CJ‘O7/EO. 5 1-1. GRANICHER c . JACCAKD P . SCHERRER AND A . STEINEMANN 51 independent d.c. conductivity 00 but this explanation does not account for the observed increase in E' (fig. 1). Tt has been known for a long time that the deviations from the ideal Debye dispersion are favoured by impurities.4.5 A systematic investigation of these effects as a function of frequency temperature impurity content crystal thickness and ageing time has now been completed. The choice of the added impurities was based on the following two considerations.6 (i) Reproducible results are only to be expected if mixed crystals with ice are formed and if the impurity anion is able to replace an oxygen atom in the lattice.This requires that the size of the I o4 I 0' E'(V)-€b, to' 10 I I I I I I 10' I 10 I o2 10' s e ~ ' 10' I o5 frequency V FIG. 1.-Frequency dependence of E' for 5 nearly pure ice crystals at - 10" C. atoms be the same. (ii) As shown below the dielectric properties and the elec-trical conductivity are connected with the occurrence of lattice defects in ice. Therefore the addition of impurities should give rise to lattice defects (i.e. vacant proton sites) in a predictable manner. Hydrofluoric acid proved to be ideally suited for this purpose 6 and was used for most of the experiments in the concentration range from 10-7 to 10-1 M HF. The water used had a conductivity of - 5 x 10-7 L?-1 cm-1.At the lower limit of concentrations the behaviour became random due to the remaining impurities of the water and to natural lattice defects. No clear homogeneous mixed crystals were obtained for concentrations above 10-1. Fig. 1 shows that the permittivity at very low frequencies increases with the impurity content. This increase turns out to be due to the occurrence of a second dispersion regiun.7 This is evident from the behaviour of the loss factor tan 8 curves of fig. 2 where two maxima can be distinguished. The maximum at high frequencies is due to the Debye dispersion and is unaffected by not too high impurity concentrations. The additional maximum at low frequencies is significant for a new dispersion which is shifted to higher frequencies with increasing im-purity content.With this dispersion is associated an a.c. conductivity UF which is proportional to the square root of the HF concentration. From the behaviour of tan 8 the static limit of E' can be calculated and is found to be proportional to U F ~ though it is not directly measurable. For these nearly pure crystals in the frequency range of about 1 c/sec the increase of E' is steeper (cc OF) and leads to values of the order of 104 (fig. 3). However a critical concentration o 52 RELAXATION OF ICE CRYSTALS 3 x 10-4 M HF (corresponding to OF = 5 x 10-7 Q-1 cm-1) exists above which the permittivity E’ drops abruptly to values of about 25. This is evidence that the polarization mechanism above and below the critical concentration is completely different, 10 -tan b w l -I Olr / / I I //”V I I I 1 lo-’ 10 10’ 10’ 10’ rec-‘ 10’ ’ lo6 frequency V FIG.2.-Frequency dependence of tan 6 for the same crystals as in fig. 1 at - 10” C . I I 1 I ‘ 2 ‘ 3 ’ 4 ‘ 5 ‘ 6 I ! FIG. 3.-DieIectric constant E’ as a function of the a.c. conductivity OF (= const. x N F ~ ) at - 10” C for different frequencies as a parameter. In the low concentration range the Debye dispersion-especially its relaxation time-is not changed by the addition of HF. The Debye dispersion as observed in pure ice and the low frequency dispersion are therefore of different origin. The low frequency dispersion is fairly well described by the theory of Macdonald,s which is an extension to a.c. fields of the theory of Jaffe.9 The main assumptions of this theory are as follows a cd [To face page 5 H .GRANICHER c. JACCARD P . SCHERRER AND A . STEINEMANN 53 The crystal of a high frequency permittivity EZ contains NI neutral centres which are partly ionized by thermal activation. The charge carriers thus formed are free to move in the crystal but are not discharged (blocking electrodes). Con-sequently under the influence of an electric field the charge carriers build up a space-charge which is denser near the electrodes and which gives rise to an addi-tional polarization. Its frequency dependence closely resembles a Debye dis-persion. Among the relations which can be compared with the experiment the proportionality between the static value of the impurity contribution AEF and the thickness of the crystal capacitor is well fulfilled.The a.c. conductivity, however does not depend on the thickness. This gives direct evidence that the low frequency dispersion is caused by space-charge effects. The dependence of c;'.(w) on the HF concentration for nearly pure crystals-as described above-is in agreement with the theory for very low frequencies and for the static limit. Since not all assumptions made by the theory 8 are appropriate for ice other properties investigated are only in qualitative agreement with the theory. For HF concentrations above the critical value only the dispersion region of the Debye type is observable. The relaxation time is much shorter in this region (e.g. o d = 5 x l o 5 c/sec at - 10" C for a crystal with OF = 5 x 10-6 Q-1 cm-1) and its activation energy has dropped from the value of 0.575 eV for pure ice to ETL = 0.23 eV.The static limit of the Debye contribution is now only about 25 After a second minimum it increases to values of about 70 toward the highest accessible HF concentrations. These results strongly suggest that the mechanism leading to the Debye dispersion in pure and nearly pure ice breaks down at the critical concentration. The properties of ice with high M F content are to be explained by a different polarization mechanism. 2. ELECTRICAL CONDUCTMTY MEASUREMENTS The occurrence of the space-charge polarization focuses interest on the electrical conductivity and the nature of the charge carriers in ice. Conductivity measurements are rare and have so far been made only with a.c.fields.lo.11 As shown before the frequency behaviour is very complex. It is therefore difficult to compensate capacitive effects in a.c. conductivity measurements. The determination of the d.c. conductivity is a tricky problem too. Space-charge effects in d.c. fields have been observed.12 They lead to considerable changes in the potential distribution and to a time-dependence of the current in accordance with theoretical work? The potential irregularities were measured with clear ice rods in static fields.13 The space-charge effects were marked unless the water had been outgassed carefully before freezing. In our experiments water of conductivity of 10-6 to 10-7 Q-1 cm-1 was used and further purified after solidification by a zone-refining process.The electrodes were thin metal foils of Au Pt or Ag frozen on the crystal surface. Even under the conditions of highest purity and with the greatest care the current always decreased with time. The decay was proportional to 1 - (t/to)* at the beginning and became exponential later. Completely reproducible results could not be obtained. However if Au electrodes were applied by evaporation in a high vacuum while the crystals were kept at liquid air temperatures time-independent currents were observed in static fields. The conductivity was 10-8 to 10-9 Q-1 cm-1 at - 10" C with an activation energy of about 0-6 eV. Further experiments in particular to lower temperatures are in progress. By analogy with a useful method in semiconductor technique so-called " sand-wich '' electrodes were tried.The surface of a pure ice crystal was covered by a thin layer of ice containing HF in high concentration. Pt or Au electrodes were then frozen on the contaminated layers. Such " sandwich " crystals always showed ohmic contact and a time-independent conductivity at all temperatures. The potential distribution was measured using 2 potential probes and found t 54 RELAXATION OF ICE CRYSTALS be linear. Thus one is sure to measure the true conductivity. With an applied tension of about 100 V the conductivity between 250" and 125" K is represented by (11, with E = (0.325 f 0405) eV. C varies for different crystals from 3.6 to 5.0 Q-1 cm-1. The corresponding values of o(T) are 1-0 to 1-4 x 10-6 Q-1 cm-1 at 250" K. The diffusion of fluorine atoms into the bulk of the crystal was found to be negligible.The nature of the charge carriers might be electronic or ionic. Though the intrinsic absorption lies at 1670 A which corresponds to a forbidden energy gap of 7-42 eV extrinsic conduction seems possible. The results of thermolumin-escence 14.15 and paramagnetic resonance experiments 16 on irradiated ice specimens show the occurrence of electronic processes. The non-linear behaviour of the conductivity of halide-contaminated ice was interpreted as electronic but later reported to have been ionic in all cases.17 Using pure ice crystals with " sandwich " electrodes quantitative electrolysis experiments at - 10" C were performed. Measuring the volume of hydrogen formed at the cathode of the crystal one finds within experimental errors (1-2 %) that the conduction is entirely ionic by a proton transfer mechanism.Further evidence for ionic conduction is obtained from the current/voltage characteristic. With tensions below about 1 V which is the dissociation potential of water. practically no current is observed. o(T) = Cexp (- E&T) 3. CRYSTAL STRUCTURE OF ICE Under ordinary conditions water crystallizes in a hexagonal structure of space-group D& - P63/mm~. Each oxygen atom is surrounded by four oxygen atoms at a distance of 2-76 A. The tetrahedra thus formed are practically ideal. Bernal and Fowler 18 suggested the following three rules for the hydrogen positions : (i) the H atoms lie on the lines connecting neighbouring 0 atoms; (ii) there is only one H atom on each such linkage; (iii) each 0 atom has two H atoms at a short distance (0.99& and hence water molecules are preserved.The special type of bond -OH . . . 0- is called the " hydrogen bond ". Pauling 19 calculated the number of possible configurations which are com-patible with these three rules and found (3/2)" Assuming that these configur-ations have practically the same energy and thus equal probability (Pauling hypothesis) he successfully explained the observed zero-point entropy of ice SO = 0.82 cal/mole deg. = R In (3/2) as a result of the disorder in the H arrange-ment. The time and space average of a crystal with such a disorder is described by half-hydrogens lying on the two possible sites of each bond. Very accurate neutron diffraction studies with D20 single crystals 20 proved that this " half-hydrogen " model is essentially the correct long-range description of the ice structure.Completely polar structures (i.e. of the Rundle type 21 and the inter-pretation of the microstructure as antiparallel twins of a polar structure 22) cannot be maintained. If the three rules must be strictly fulfilled configurational changes by a trans-lational motion of protons along the hydrogen bonds or by rotations of molecules would require a simultaneous process on a chain through the crystal or on a closed ring of 6 or more molecules. Such processes are very unlikely in view of the high activation necessary. In addition movements on closed rings give no change in polarization and therefore cannot contribute to the permittivity.An ice crystal obeying the Bernal-Fowler rules and having no imperfections such as those described below and no dislocations stacking faults etc. is henceforth called ideal H . G R ~ N I C H E R c . JACCARD P . SCHERRER AND A . STEINEMANN 55 because it would show no electrical conductivity and a static dielectric constant of only 3.2. 4. CHARACTERISTIC LATTICE IMPERFECTIONS IN ICE Bjerrum 23 pointed out that the electric properties of ice cannot be explained unless two types of lattice imperfections are assumed : (i) Orientational defects are generated by the rotation of a H20 molecule around one of its four bonds preferentially around one of the two bonds with a close-lying proton. The two normal bonds (B) -OH . . . 0- thereby give one doubly occupied bond (0) -OH.. HO- and one vacant bond (L) -0. . . . 0-. The reaction equation for this process can be written 2B+ D + L. (1) The mass action law then gives for equal numbers of D- and L-defects * JvbM=lN~2 = (ND/NB>~ = a2 exp (- Eo/kT) = ‘W02. (2) The number NB of normal bonds is twice the number N of 0 atoms per unit volume. The constant a as yet undetermined takes into account the lattice disturbance in the surroundings of a lattice defect. W D = WL is the probability of finding a doubly or a vacant bond on any linkage. By rotation of a molecule adjacent to a D- or an L-defect the defect moves to a neighbouring linkage and is thus able to diffuse in the crystal. It is important to note that with orientational defects the molecules only change the direction of their dipole moments.(ii) Ionized states. If a proton moves along a hydrogen bond to the neigh-bouring molecule an H3O+ and an OH- ion are formed. By subsequent trans-lational motions of protons both sorts of ion states can move from one molecule to another This dissociation is formulated as a reversible reaction as it is usual for liquid water 2H20 + H3O+ + OH-. (3) In this case the mass action law gives for thermal equilibrium where EI is the activation energy for pair formation and b is a constant similar to a. As the lattice imperfections diffuse in the crystal their concentration will be homogeneous in the bulk of the crystal and imperfections of the same type re-combine according to the reaction equations. But it must be emphasized that no recombination is possible between orientational defects and ionized states.However one feature is common to both types when a lattice defect has moved past a certain molecule-either due to ordinary diffusion or due to a field-induced motion-the molecule is left behind in an orientation depending on the path and the nature of the defect. This means that a change in polarization has occurred. Since in all processes only one proton per molecule actually is shifted it is much more convenient to consider the polarization change brought about by the in-dividual shifts rather than to compute the dipole moments of all the molecules. 5. DYNAMIC THEORY OF DIELECTRJCS The general theory 24 of the dynamic behaviour of a dielectric containing par-ticles of charge e with two equilibrium positions separated by a distance 6 and a * Upright and cursive capitals are used for temperature-independent and temperature-dependent quantities respectively 56 RELAXATION OF ICE CRYSTALS potential barrier of height E between them leads to the differential equation d dt - (Jv; - 4) =- (W12 + W21) (Jy; - Jl) + (W12 - W21)(M2 + m.(1) ./v; and M2 are the number of particles in position 1 and 2 respectively. Wg are the probabilities for a transition from position i to j . In the absence of an ex-ternal field F the probabilities are all the same namely, (2) but become different under the action of the field. Since e6F< kT one obtains Wo = V r exp (- E/kT), W12 = Wo[l f (e6F/ZkT)] 21 and W12 f W21 = 2W0, and Thus eqn.(1) can be written W12 - W21 = Wo eGF/kT. and can be solved for a periodic field F = FoIexp iwt by (3) (4) ( 5 ) Comparing this equation with the formula for a Debye dispersion and considering c* = E' - i~" one obtains and Td = 1/2w0, (.A5 + XI) being the total number of particles. 6. APPLICATION TO HEXAGONAL ICE The foregoing theory holds only if interactions among the charged particles are negligible. This assumption is fulfilled for ice. One easily calculates 25 that only every 107th molecule has to reorient under the action of a field of 1 V/cm in order to produce the polarization corresponding to a static permittivity of 100. In the following only the case of the polarization mechanism by motion of doubly occupied bonds is outlined; the calculations for vacant bonds are completely analogous.If a doubly occupied bond is present and with sufficient thermal activation one of its two protons is able to jump to an adjacent normal bond. By this process the D-defect has moved to a previously normal bond and a polarization change arises. Since every molecule has four outgoing bonds pointing in different direc-tions the probability of finding a D-defect on a given bond e.g. parallel to the z-direction is W~14. With the value of WD from eqn. (4.2) the probability which must be introduced in eqn. (5.10) is in this case (1) and thus the relaxation time is (2) W6 = (1/4)W~ vr exp (- E:,D'/kT) = (1/4) av exp -(+Eo + E$:?)/kT, TD = 1/2W6 = (2/vp) exp (+Eo + c t i ) / k T H. GRANICHER c. JACCARD P. SCHERRER AND A .STEINEMANN 57 Assuming e.g. that the electric field is in the z-direction the following possibilities for proton jumps (by rotation) exist if the doubly occupied bond is on one of the bonds parallel to z the defect is transferred to one of the two neighbouring bonds which had no close-lying proton before. Similarly a D-defect on one of the linkages oblique to the z direction is able to move over to a bond parallel to z. The shift component in the z direction is for all jumps z = (413)ro~ = (4/3) 0.99 A. These favourable jumps are only a fraction of the total of 24 possible jumps which can take place at the 4 different molecules of the elementary cell. Therefore the quan-tity 82 in eqn. (5.9) has to be replaced by a statistical average 62. This average is defined by the fraction of favourable jumps times the sum of the squares of the shift components in field direction of the individual jumps.The fraction of favourable jumps in the case under consideration being 1/4 one obtains The calculation of €I - EL with eqn. (5.9) is more complicated. The same result is found for fields in the x- or y-directions thus no anisotropy appears. However for fields perpendicular to z it seems plausible to exclude jumps of protons from the z direction which would be at a high angle to the field direction if another proton is in position to jump in the xy-plane. By this restric-tion 7/48 rn 1/6 of the jumps are rejected. Defining the anisotropy as a frequency and temperature independent quotient A one finds 1 / ''IlC - E.T ' c = 14.6 %.A = I Esllc - ECO TABLE 1 A O/O It c I C theor. 106.4 91-3 14.6 theor. 124.0 106.0 14.6 expt. e i at 0" C expt. 105 91 13 + 4 E at - 40" C 129 104 19 It 4 dc6 theor. 0*44/deg. 0.38/deg. -dT expt. 0.6 5 0.1 0.3 5 0.1 - _ In table 1 the theoretical results for + E& = E its temperature coefficient de;/dT and the anisotropy A are compared with the experimental values.3 The quantities introduced in eqn. (5.9) are + ( N 2 +Nl) =N=3-08 X 1022 molecules per cm3 computed from the X-ray density E&= 3.2 and the 0-H distance t - 0 ~ = 0-99 A. The theoretical results are estimated to be accurate to f 1/2 % the uncertainty arising mainly from the 'OH value which is accurately known for D20 20 but not equally well for H2O. The ab-solute accuracy of the experimental values is t 4 % but the relative accuracy for crystals of the same orientation is 5 1 to 2 %.With these values the limits of error given in table 1 for dc/dT and A are obtained. The theoretical values for the polarization mechanism by doubly occupied bonds are in good agreement with the experiment.3 A systematic deviation only occurs for eiC at low temperatures where the omission of a conductivity correc-tion (1.2) leads to too high ci values. The computation for a vacant bond mechan-ism gives exactly the same results. A difference between the two mechanisms arises only in the relaxation time since it is to be expected that the activation energies Erot are different in general. Both mechanisms are present and hence the measured relaxation time depends on the sum of the dispersion frequencies.As is shown in 3 7 there is evidence that W D > OL. The observed dielectric properties of pure ice crystals must be ascribed to the doubly-occupied bond mechanism 58 RELAXATION OF ICE CRYSTALS Since the lattice dimensions and the bond energies are only slightly affected by the replacement of H by D the dielectric properties of D202 are very nearly the same as for H20 except for a change of d2 in the pre-exponential factor B (eqn. (1.1)). This is to be expected by this theory because a 42-change occurs in the oscillation frequency v,. Similar statistical calculations were made for the translational mechanism of H3O+ and OH- ionized states. One finds and thus the contribution of the translational mechanism Ae;rms to the static permittivity is 23 only.In principle since the orientational and the translational mechanisms are independent the true static permittivity should be 125. The fact that down to frequencies of 1 cJsec only the rotational contribution is observed shows that the dispersion frequency of the translational mechanism is very low. 7. EXTENSION OF THE THEORY TO CRYSTALS CONTAINING HF As mentioned in $ 1 HF forms mixed crystals with ice and one has good reasons to assume that each HF molecule replaces one H20 molecule in the lattice. As for each such replacement one H atom is missing a number of vacant bonds equal to the number NF of fluorine atoms is present. In eqn. (4.2) Jvz must now be replaced by the total number N L tot = c/y;. therm + NF-This leads to The thermal defect concentrations are thereby very much reduced and for high HF content the concentration MDtot = c/Ybthem of doubly-occupied bonds tends toward zero whereas the number of vacant bonds becomes equal to NF.A further consequence of the presence of HF is an increase of the H3O+ con-centration according to the dissociation equation (3) The mass action law gives (4) and hence the number of H3O+ ions is proportional to N$. (5) Only the case of low dissociation is considered so that NF can be regarded as con-stant (NF > N:). The H3O+ ions formed by dissociation of HF influence the dissociation equilibrium of the water molecule (eqn. (4.3)). One has to combine eqn. (4.4) with (7.4) and to introduce the total number of H3O+ states The result-similar to- the case of orientational defects treated above-is that the OH- concentration diminishes with increasing NF and Ntot starting from the value JV+ rises with NI; for low dissociation up to the limiting value given by eqn.(5). The expression (5.8) for the dielectric constant has now to be written in the following more generalized form : HF -1- H20 +- H3O+ + F-. NJ&/(NNF) = c2 exp (- EF/kT), M$ = c(NNF)~ exp (- EF/%T). in both equations .N;’,t = .N+ + A’$ (6) where the dispersion frequencies ar I f . GRXNICHER C . JACCARD P . SCHERRER AND A . S T E l N E M h N N 59 wrot = COD -+- COL 1= v,.("/CrD ,,,/ZNB) exp (- E$s'/kT) td2NB) exp (- E%i/kT) (8) and Wtrans = V t ( 4 W N ) exp (- E & d W - (9) * Since permittivity measurements are limited for experimental reasons to fre-quencies above 0.1 c/sec mechanisms of polarization with a dispersion frequency o d below 1 c/sec due to a small number of the respective defects (Jvb NL .N+ or N-) cannot be observed.In all measurements of ice with or without added im-purities the space-charge contribution is always present. Therefore only the I I I I I I I I I I - - - + + + \+ + ' \+ ++ x x' E,(pure) - 0 5 7 5 eV + \ c 0 -I - 3 O C v v I I I I I I I I I I 16' 2 9-' cm-' Id' (NF =number o f f ~ u O f l n $ a r O n l s ) ' 2 - . ~ F (-IOoc) FIG. 4. Debye dispersion with the highest can be observed. Its relaxation time Td is plotted as a function of the a.c. conductivity OF in fig. 4. Mechanisms with values of wd 5 Wspaw&arge are always masked by the space-charge polarization.In very high concentration crystals the conductivity leads to tan 8 ;t 103 and prevents even the measurements of the space-charge polarization at lower frequencies. One is now in position to discuss the expressions (7)-(9) as a function of in-creasing HF content. The observed static permittivity E w 100 corresponds to the value A& as computed in 5 6. Hence one concludes that wtrans << 1 c/sec. In nearly pure crystals Jvb decreases whereas Jyt tot increases with rising HF content. The fact that at the critical concentration the static limit of the Debye contribution drops from - 100 to - 25 shows (i) that in pure ice wg > WL or the predominant,polarization mechanism is (ii) that in concentrations above the critical value the translational mechanism In this concentration range-between the two minima of E; (fig.3)-A'& is prac-tically equal to J$ according to eqn. (5). The proportionality between wtraw the one involving doubly-occupied bonds and E!,q' < E$&) ; by H3Of states becomes effective. * A second term with can be omitted since it is never important 60 RELAXATION OF ICE CRYSTALS and NF* is actually fulfilled (fig. 4). NL tot however increases as NF and one has to expect that at still higher concentrations the rotational mechanism by vacant bonds will overtake the translational mechanism. This explains the new increase of ei at the highest HF concentrations above the second minimum and the change of the slope of the T~(Q) curve. Theoretically one can predict that E would rise to - 125 if crystals of sufficiently high HF content could be produced.8. INTERPRETATION OF THE CONDUCTIVITY AND DISCUSSION OF THE ACTIVATION ENERGIES It is quite clear that the d.c. conductivity in pure ice crystals requires proton transfers along the hydrogen bonds since the ionic character of the conduction has been ascertained. The conductivity will therefore depend on the activation energy Ez for ion pair formation and on EZans and EGans for translational motion. If a crystal had in the mean no polarization before a d.c. field is applied, a H3O+ state can move by proton transfer on a chain of molecules through the crystal. The molecules on the chain are left in an orientation which corresponds to a polarization opposite to the field.The existence of a time-independent current requires that after a certain time the same chain is again able to conduct another ion-state. This is possible only if the molecules have changed their polarization from opposite the field to a polarization parallel to the field. These molecular turns which are necessary to reactivate the chains for conduction are possible only with the aid of orientational defects-doubly occupied or vacant bonds. Hence the concentration of these defects and the activation energies for rotation Ei$) and E:;? have to be considered too. Because of the many processes involved it is not yet possible to decide which processes are rate-determining. But it is hoped that the conductivity experiments with H20 and D20 crystals which are in progress will give a clearer insight, The a.c.conductivity of HF-contaminated crystals is much easier to interpret because with high HF concentrations only H3O+ states and vacant bonds occur. As shown in 6 7 the number NL, is then practically equal to NF whereas the H3O+ concentration is given by eqn. (7.5). Hence a~ = const. N F ~ exp - ( ~ E F + Eiki + E&,,)/kT. The activation energy of UF determined by dielectric measurements is 0-32 eV. For the d.c. conductivity with " sandwich " electrodes the same mechanism prevails. In the electrode layer the H3Of concentration is much higher than in the bulk of the crystal and the diffusion of the ions would soon be counteracted by a diffusion potential since a negative space-charge of the immobile F- ions remains in the surface layer.However the surface also contains a high number of L-defects which have a tendency to diffuse into the bulk. The molecules which permitted this diffusion by molecular rotations are left behind in a state of polar-ization pointing toward the surface. Thus the combined diffusion of H3Of ions and vacant bonds does not give rise to a resultant diffusion potential. One there-fore has to assume that the bulk of" sandwich " crystals is homogeneously crowded by defects which were generated by the HF content of the electrode layers. This explains the much higher conductivity of " sandwich " crystals compared with crystals with evaporated electrodes and the occurrence of the same activation energy (0.325 eV) as in the a.c. conductivity of HF contaminated crystals.The activation energies are now summarized and compared with the experi-mental values. For the dielectric relaxation time Td in pure ice we found (eqn. (6.2)) ETPuTe = *Eo + = 0.575 eV. (2) In the fluorine contaminated crystals where the translational mechanism prevails, combining eqn. (7.9) with (7.5) gives GF = +EF + E&ans = 0.23 eV (3 H . G R ~ N I C I - I E R c . JACCARD P . SCHERRER AND A . STEINEMANN 61 For the a.c. conductivity of these crystals and the d.c. conductivity of " sandwich " crystals we found (eqn. (1)) (4) The difference between eqn. (3) and (4) leads to I?:; = 0.095 eV. It was shown in Q 7 that > E!,q>. Therefore the energy for formation of orientational defects must be within the limits 0.96 < EO < 1 .1 5 eV. The theoretical values 23 are 0.9 < EO < 1 . 0 eV. The interpretation that the L-mechanism is predominant only for HF concentrations above the second minimum of E permits the estimate < 0.02 eV. A further separation of the activation energies is not yet possibIe. With the limits for E D and the empirical value of the pre-exponential factor B (eqn. (1.1)) the number of orientational defects in pure ice turns out to lie between 2 x 1015 and 4 x 1013 per cm3 at - 10" C. The ratio between normal and defect bonds is 107 to 109. Thus the vital assumption of the dynamic theory of dielectrics that the interaction among the jumping particles can be neglected is well fulfilled. It may be worthwhile to note that the degrees of dissociation of H20 and HF according to the reactions (4.3) and (7.3) are considerably smaller in ice than in liquid water.The conclusion drawn in $ 7 that Wtrans < 1 means that the ion concentration in pure ice is smaller than the values given above for orientational defects by at least a factor 103. This is to be expected since the permittivity of the bulk effective for dissociation is 3.2 in ice but is 81 in water. It must be emphasized that it is immaterial for the present theory whether the molecular rotations or the translational motions of protons along the hydrogen bonds are classical activation processes or whether they are of quantum-mechanical nature (tunnelling). The distinction will be made by comparison of the behaviour of H20- and D2O-ice. EUF = +EF + ELans + EiL; = 0.325 eV.9. NEW CONCEPT OF THE LATTICE DISORDER IN ICE In order to explain the observed zero-point entropy of ice Pauling 19 postulated that the (3/2)N possible arrangements for the H atoms in the hexagonal ice lattice are of practically equal energy and equal probability. In the past few years this assumption was criticized,23 * because electrostatic energy calculations showed that different configurations may have energy differences comparable with kT at the melting point. Recent computations 26 taking into account higher order interactions than considered previously 23 led to considerably smaller energy differences. However from the present point of view these configurational energies are not essential for the explanation of the zero-point entropy. The experimental facts-in particular the zero-point entropy the neutron diffraction data,20 the results of nuclear magnetic resonance 27-28 and the electric properties as interpreted by the present theory of lattice imperfections in ice-are consistent with the concept summarized in the following statements.(i) The Pauling (half-hydrogen) model is an essentially correct description of the average (long-range) structure of ice. Local ordering of protons along a certain axis leading to a spontaneous polarization may be possible to an amount not exceeding 20 x.20 One easily verifies that ordered (polarized) regions of opposite orientation or ordered with disordered regions can be joined with each other without violating the Bernal-Fowler rules for the hydrogen positions. It is not possible to speak of twin boundaries between such regions in the usual sense.(ii) An ice crystal obeying strictly the Bernal-Fowler rules is called ideal, though it has a statistically disordered hydrogen arrangement. In such a crystal changes among the (3/2)N possible configurations do not occur. Such recon-structive transitions would require to break a considerable number of bonds and * for further references see ref. (22) 62 RELAXATION OF 1CE CRYSTALS hence need a high activation energy. They are therefore very unhkely even near the melting point. Hence the ideal crystal remains in the same incidental con-figuration established during freezing and therefore possesses zero-point entropy. (iii) The following types of lattice imperfections account for the electric pro-perties of ice : orientational defects i.e.doubly occupied bonds and vacant bonds : ionized states i.e. H3O+ and OH- ions. The diffusion of these lattice imperfections causes configurational changes among the ( 3 / 2 ) N possibilities. These changes are however only of local (short-range) character the number of lattice imperfections being so small. In addition, molecules on the diffusion path of an imperfection are left behind in a certain orientation (5 4) which is not necessarily the one of least configurational energy. Since the formation and diffusion of these lattice defects require thermal activa-tion their concentrations decrease exponentially with lowering temperature or if the crystal is quenched the imperfections freeze in and are no longer able to diffuse.Therefore at very low temperatures no configurational changes occur at all. (iv) The final conclusion is that the temperature where thermal equilibrium among the ( 3 / 2 ) N hydrogen arrangements is established lies well above the melting point,22 if the values for the configurational energy of 23 and others are assumed to be correct. Recent calculations26 lead to smaller energy differences which are compatible with freezing-in in the range of liquid-air temperatures. The observation of a zero-point entropy of the value So = R In ( 3 / 2 ) shows that the ideal as well as the real crystal is disordered with respect to the hydrogen arrange-ment and that the configurations are frozen-in at temperatures where the spread in configurational energy is still smaller than kT.Detailed papers on the present investigation will appear in Helu. phys. Acta. 1 Errera J. Physique Rad. 1924 5 304. 2 Auty and Cole J. Chem. Physics 1952 20 1309. 3 Humbel Jona and Scherrer Helw. phys. Acta 1953 26 17. 4 Granier Compt. rend. 1924 179 1314. 5 Smyth and Hitchcock J. Amer. Chem. SOC. 1932,54,4631. 6 Granicher Scherrer and Steinemann Helv. phys. Acta 1954 27,217. 7 Griinicher Jaccard Scherrer and Steinemann Helv. phys. Acta 1955 28 300. 8 Macdonald Physic. Rev. 1953 92,4. 9 Jaffe Ann. Physik 1933 16 217 ; J. Chem. Physics 1952 20 1071. 10 Johnstone Proc. Trans. Nova Scotian Inst. 1912 13 126. 11 Murphy Physic. Rev. 1950 79 396. 12 Joffe The physics of crystals (McGraw Hill,.New York 1928) chap. 7. 13 Oplatka Helv. phys. Acta 1933 6 198. 14 Grossweiner and Matheson J. Chem. Physics 1954 22 1514. 15 Ghormley J. Chem. Physics 1956 24 1111. 16 Matheson and Smaller J. Chem. Physics 1955,23 521. 17 Workman Truby and Drost-Hansen Physic. Rev. 1954,94 1073. 18 Bernal and Fowler J. Chem. Physics 1933 1 515. 19 Pauling Nature of the Chemical Bond (Cornell University Press Ithaca N.Y., 20 Peterson and Levy Acta Cryst. 1957 10 70. 21 Rundle J. Physic. Chem. 1955 59 680. 22 Granicher Helv. phys. Acta 1956,29 213. 23 B j e m Dan. Mat. Fys. Medd. 1951 27 nr. 1. 24 Frohlich Theory of Dielectrics (Clarendon Press Oxford 1949) chap. 3. 25 Debye Polar Molecules ('The ChemicaI Catalog Co. New York 1929) p. 102. 26 Piker and Polissar J. Physic. Chem. 1956 60 1140. 27 Bloembergen Purcell and Pound Physic. Rev. 1948,73 679. 28 Pake and Gutowsky Physic. Rev. 1948,74,919. 1948) p. 301