Ionic conductivity of fluorite type crystals CaF2, SrF2, BaF2, and SrCl2 at high temperatures

Journal of Physics and Chemistry of Solids 62 (2001) 1349±1358

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Ionic conductivity of ¯uorite type crystals CaF2, SrF2, BaF2, and SrCl2 at high temperatures B.M. Voronin*, S.V. Volkov V. I. Vernadskii Institute of General and Inorganic Chemistry, 32±34 Palladin Avenue, 03680 Kiev 142, Ukraine Received 13 July 2000; accepted 16 January 2001

Abstract The ionic conductivity of ¯uorite-type crystals (CaF2, SrF2, BaF2, and SrCl2) has been studied at high temperatures including the melting points. Represented in detail, the technique of measurements is based on the use of a capillary-type cell made of hexagonal boron nitride. Measurements were carried out on polycrystalline specimens obtained by the solidi®cation of corresponding melts. A common feature of all ¯uorites is that the temperature dependences of conductivity in the superionic region are described by sigmoid-like curves in the Arrhenius plot. The behaviour of the apparent Arrhenius energy and molar conductivity in the superionic region as well as on melting crystals has been characterized. From comparative analysis of these data, it is suggested that the conductivity anomaly is mainly due to a non-uniform course of anion sublattice disordering in crystals up to melting points. The variation of transport properties is well correlated with the relative sizes of cations and anions in the ¯uorite lattice, that assumes a regularity in the degree of sublattice disorder in these solids near the melting points. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Inorganic compounds; D. Defects; D. Electrical conductivity; D. Phase transitions

1. Introduction By the present time, a large number of experimental and theoretical studies on the defect properties of crystals with the ¯uorite-related structure have been carried out. Nominal pure crystals are known to demonstrate a `normal' behavior under ordinary conditions, i.e. to have a rather low ionic conductivity, which is described in terms of a classical approach [1] and is due to the formation and motion of isolated anti-Frenkel defects. At the same time, for a number of ¯uorite-type compounds (SrCl2 [2±5], CaF2 [3,6], SrF2 [4], BaF2 [4,5] et al.) the presence of a broad peak in the heat capacity curve has been observed at a temperature Tc (speci®c heat maximum) essentially below the melting point Tm. According to the results of O'Keeffe and coworkers [7] the electrical conductivity of crystalline salts (CaF2, SrF2, SrCl2) increases continuously with temperature and in the premelting region becomes comparable to that of melts. Precision measurements carried out later for SrCl2 [8,9] and b-PbF2 [8,10] have allowed the peculiarity to be * Corresponding author. Fax: 1380-44-444-3070. E-mail address: [email protected] (B.M. Voronin).

revealed that the ionic conductivity, starting from the intrinsic region, grows not linearly but according to a sigmoidlike curve when plotting in Arrhenius coordinates, so that the apparent activation energy (Arrhenius energy) has a maximum at temperature close to that for heat capacity. It is now widely accepted that at high temperatures ¯uorite compounds undergo a continuous (diffuse) transition to a state of relatively heavy, but far from massive, dynamic disorder of the anion sublattice, and the ion transport occurs via a hopping mechanism ([11±15] and references therein). However, when comparative reviewing, e.g. the data in Refs. [10,14,16±22], it should be admitted that many aspects of the superionic behavior of pure ¯uorites, such as the exact nature and the extent of the disorder, the defect structure and the mechanisms of conduction, have still not been fully clari®ed. In our view, a detailed comparative analysis of thermodynamic and transport properties over a temperature range embracing the superionic and molten states could make a useful contribution to the elucidation of the phenomenon under consideration [9,15]. Such an approach supposes taking as broad a series of structurally related compounds as possible to clarify here the role of ionic parameters. Meanwhile, it is to

0022-3697/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(01)00036-1

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be stated that along with investigations based on the modern experimental and computer facilities, no appropriate attention was given to the study of ionic conduction proper with reference to high-temperature region. Fairly correct results were only obtained for SrCl2 [8,9] and PbF2 [8,10]. As to the data for alkaline-earth ¯uorides [7], the authors themselves pointed out that `the emphasis is on the order of magnitude of the conductivity over wide temperature ranges rather than on precision measurements'. The present paper deals with the high-temperature electrical conductivity of alkaline-earth ¯uorides, CaF2, SrF2 and BaF2. The accent is made on the tracing of the behavior of the compounds in a comparison with each other during superionic transition and subsequent melting. To have a fuller comparative picture, also included here are the results published earlier for SrCl2 [9], which were obtained under identical conditions. 2. Experimental procedure We used high-purity calcium and strontium ¯uorides and chemically pure barium ¯uoride, which had been predried at 1208C and stored in a dry atmosphere. The starting material SrCl2´6H2O of high purity was ®rst recrystallized and then dehydrated by slow heating and melting in a stream of dry hydrogen chloride and then argon. Unhydrated SrCl2 thus obtained was stored in sealed ampoules. To carry out experiments with samples in both the solid and molten states, a capillary-type cell was chosen as the most suitable one for obtaining correct results [23]. It was necessary, however, to overcome dif®culties connected with the high melting temperatures of alkaline earth ¯uorides and corrosivity of their melts towards conventional materials of capillary cells. This problem has been solved by using our original cell, which had exhibited its serviceability in the determination of the electrical conductivity of ¯uoridecontaining melts up to 1600±16508C [24]. Such a cell is shown schematically in Fig. 1. It included a molybdenum crucible as a sample container and a shaped cylindrical element made of sintered boron nitride (BN) of hexagonal modi®cation, which proved to be a good electrical insulating and chemically inert material. In the BN element there were two blind holes for thermocouples and a through hole ending in a capillary, where the `internal' electrode (tungsten wire wound at the end into a dense cylindrical spiral) was placed. The second electrode was the molybdenum crucible itself together with the `external' electrode (tungsten wire wound at the end into a ¯at spiral), the latter being placed at the bottom of the crucible around the BN element. The cell assembly was mounted directly in an electric furnace ®tted with a temperature control system; puri®ed argon was fed into the furnace during operation. The heating element of the furnace was a vertical hollow cylinder made of graphite, where the inner space was the working zone.

Fig. 1. High-temperature capillary-type cell for conductivity measurements of solid and molten ¯uorides: (1) tungsten electrodes, (2) tungsten wires for suspension, (3) thermocouples, (4) cylindrical BN element, (5) molybdenum crucible, (6) sample under investigation (molten or solidi®ed), (7) capillary.

The crucible with a weighed amount of salt was placed on an insulating stand (BN) in the working zone. All the rest of the cell assembly was ®rst ®xed in a lifted position and then sunk into the crucible after melting the sample, the melt ®lling the capillary and the electrode space inside the BN element. The top of the furnace was covered with a heat shield (graphite) having channels for the passage of thermocouples, suspension wires and electrodes; electrical insulation was attained by means of corundum or porcelain tubes inserted into the channels. The thermocouples were encased in corundum jackets. The thermocouple PtRh (6%)±PtRh (30%) was the main one; to check and correct its readings (below 13008C), the insertion of a commonly used PtPtRh(10%) thermocouple into the cell was envisaged. Measurements were made during stepwise temperature variation (thus the system was allowed to reach thermal equilibrium), going in this way from the molten to crystalline phase and back. An individual BN element was designed for each of the salts. The cell constant was determined by the ordinary calibration method based on the measured resistances of the cell with molten sample (approximately 10 points with 108C step) and on using the corresponding data on the speci®c conductivity of molten salts under consideration studied before under the same conditions [24,25]. A constant value of ca. 40 cm 21 was preset by the chosen geometric dimensions of the capillary, which were about 2 mm in diameter and 12 mm in height. An R5083 autobalance admittance bridge operated at 0.1± 100 kHz was employed as a measuring instrument. Under our experimental conditions, only a slight dispersion of conductance (2±3%) was observed, which fell mostly within a frequency range below 1 kHz. In the vicinity of 10 kHz the phase angle was negligible, therefore, all measurements were

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Fig. 2. Arrhenius plot of the conductivity of SrCl2. The straight lines (a) and (b) approximate the experimental data assigned to the extrinsic and intrinsic (classical) regions respectively. Previous data are represented by a long dash curve [27], short dash curve [7] and triangles [8]. Here and in other ®gures, the arrows show special temperature points of the superionic region (see text).

made at this frequency. The precision on the reported results of the speci®c conductivity has been estimated to be ca. ^3%. This evaluation includes both the effect of compression of already solidi®ed samples (within the inner capillary space) on cooling and the error of the data for ¯uoride melts [24], which were employed in the calibration procedure and now recommended as reference data [26]. 3. Results The speci®c electrical conductivity of SrCl2, CaF2, SrF2, and BaF2 in the crystalline and molten states is shown in Figs. 2±5 in the generally accepted form of Arrhenius plots of ln(s T ) vs. T ±1. As was expected, the ionic conductivity of all salts under consideration exhibits a continuous increase with temperature, this increase being so substantial that only a slight jump is observed upon melting. (The jump value was found by slight extrapolation of experimental data for crystal to the melting point since in close vicinity to this temperature pre-melting effects distorted the measurement results; therefore, these results were rejected.) Another feature typical of them is that the Arrhenius relation for intrinsic conductivity is not observed

any more in the high-temperature region. As can be seen from Figs. 2±5, the experimental points are described by wellde®ned sigmoid-like curves showing ®rst a positive and then a negative deviation from the Arrhenius dependence with rising temperature. Previous papers on the high-temperature solid conductivity of SrCl2 [7,8], CaF2 and SrF2 [7] do not contain suitable details for a quantitative comparison, with the data obtained having been plotted only. To make visual comparisons and some approximate evaluations, we have tried to reproduce carefully their experimental plots in the present work. Note that precision measurements on the SrCl2 single crystal [8] were carried out in a limited temperature range of 915± 1118 K, while in other work [7], when studying the solidi®ed ¯uorite samples, the techniques employed were not designed for high-precision data. 3.1. SrCl2 For this crystalline phase (see Fig. 2) our results are very close to the data of Carr et al. [8] relating to the hightemperature region, but with decreasing temperature they become closer to the data of Beniere et al. [27] obtained

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B.M. Voronin, S.V. Volkov / Journal of Physics and Chemistry of Solids 62 (2001) 1349±1358

Fig. 3. Arrhenius plot of the conductivity of CaF2. The data from previous works are designated as 1 [28], 2 [29], 3 [30] and 4 [7].

at lower temperatures. Besides, two relatively linear portions observed in the `low-temperature' ranges 690± 770 K and 830±905 K may be regarded as characterizing the conductivity behavior in the extrinsic and intrinsic (in the classical sense) regions respectively. A straight line ®tting of the portion assigned to the intrinsic conductivity leads to an Arrhenius energy value of ca. 200 kJ/mol, which is somewhat higher compared with the value of 176 kJ/mol from Ref. [27]. When extrapolating the data of Carr et al. [8] to the melting point 1148 K, a s value of 1.09 S/cm is obtained, which is comparable with our value of 1.31 S/ cm. The corresponding curve from the work of Derrington et al. [7], being sigmoid-like too, is displaced markedly to higher temperatures and conductivities, resulting in s < 1.75 S/cm at Tm < 1160 K. On the other hand, at the intermediate temperature of 915 K one can compare the conductivity values of 0.0058 (present work), 0.011 [7], 0.0108 [8] and 0.0069 S/cm [27]. Thus, we believe that our and previous [8,27] results give together a reliable description of the speci®c conductivity behavior of

crystalline SrCl2 in a wide temperature range, including the melting point. 3.2. CaF2 The main difference between previous [7] and our results presented in Fig. 3 is that the former do not show a sigmoidlike conductivity behavior of this compound in the hightemperature region, although near the melting point 1690 K they give comparable values of s (3.05 against 4.00 S/cm, see also Table 1). As the temperature decreases, our data approach closely the `low-temperature single crystal' data of Barsis and Taylor [28] (for example, the discrepancy does not exceed 10% at 1083 K). 3.3. SrF2 Here, in contrast with the CaF2 case (cf. Figs. 3 and 4), both our data and that of Derrington et al. [7] demonstrate a similar (sigmoid-like) behavior, being very close together

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Fig. 4. Arrhenius plot of the conductivity of SrF2. Previous data are designated as 1 [28], 2 [31] and 3 [7].

Table 1 Conductivity properties of ¯uorites in superionic and molten states Compound

T (K)

Crystal

s (S/cm) SrCl2 CaF2 SrF2 BaF2

To ˆ 900 Tc ˆ 985 Tm ˆ 1148 To ˆ 1100 Tc ˆ 1370 Tm ˆ 1690 To ˆ 1150 Tc ˆ 1380 Tm ˆ 1740 To ˆ 1050 Tc ˆ 1245 Tm ˆ 1620

3.74 £ 10 23 0.164 1.31 1.44 £ 10 23 0.325 4.00 5.04 £ 10 23 0.364 4.25 6.77 £ 10 23 0.462 3.92

L cryst/L melt (%)

Melt

L (S´cm 2/mol) 9.14 73.11 9.51 117.0 12.58 146.8 18.74 159.0

EA (kJ/mol) 227 427 40.1 197 322 71.2 214 330 53.4 205 266 46.3

L (S´cm 2/mol)

EA (kJ/mol)

118.9

21.7

61

189.2

17.4

62

201.3

19.0

73

203.3

19.8

78

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Fig. 5. Arrhenius plot of the conductivity of BaF2. The numbered straight lines are the results of previous `low-temperature' studies on single crystal samples: 1 [32], 2 [33], 3 [34], 4 [35].

at the highest and lowest temperatures of measurements and to `low-temperature single crystal' data of Bollmann et al. [31]. 3.4. BaF2 For this compound we have not found high-temperature conductivity data published previously. However, there are a number of works on the `low-temperature single crystal' conductivity [32±35], which show a marked scatter. Our results (see Fig. 5) seem to be closer to the data of Bollmann [33]. 4. Discussion. 4.1. Conductivity behavior with temperature Taking into account the non-Arrhenius high-temperature conductivity behavior of crystals under consideration, it seemed useful to consider the behavior of apparent Arrhenius energy, EA, which in this particular case is given by the equation: E A ˆ 2 R‰d…lnsT†=d…T 21 †Š where R is the gas constant. The experimental plots of

Arrhenius energies of crystals as functions of temperature are shown in Fig. 6 in comparison with those for the `lowtemperature' region, where they are constants. These plots, being of asymmetrical shape, pass through a maximum with rising temperature and then slope down gradually to a level much lower than that observed for classical intrinsic conduction. Such a behavior of Arrhenius energies explains, in particular, why the constant EA values given in previous works for single crystals (cf., e.g. Refs. [28,29] and [28,31]) tend to a higher level as the conductivity measurements are extended to higher temperatures. SrCl2 is exceptional among the group of salts in the presence of a much narrower and higher peak in the EA(T ) curve. As can be noticed, the behavior of Arrhenius energies resembles greatly that of crystal heat capacities in the same temperature regions [3±5]. It is therefore clear that the anomalies observed in both properties are of common origin related with diffuse transition to superionic state [3,7], which is speci®c to ¯uorite structure. These hightemperature (superionic) regions, where transition occurs, may be conveniently distinguished by their `special' temperature points, which correspond to (i) the beginning of the observed deviations at T0, (ii) the position of maximum in the property (EA or Cp) curve at Tc and (iii) crystal melting proper at Tm. The numerical data collected in Table

B.M. Voronin, S.V. Volkov / Journal of Physics and Chemistry of Solids 62 (2001) 1349±1358

Fig. 6. The behavior of apparent Arrhenius energies at high temperatures in (a) SrCl2, (b) CaF2, (c) SrF2, and (d) BaF2. The dashed horizontal straight lines denote the upper and lower levels for EA values, which correspond to the ªlow-temperatureº intrinsic conductivity according to previous works (see references in Figs. 2±5).

1 illustrate the transport properties of the salts with reference to these `special' temperature points. For SrCl2, the transition begins approximately at T0 < 900 K or somewhat lower (see Fig. 6(a)), as is corroborated by heat capacity [5], thermal expansion [36] and neutron diffraction [37] data as well.

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As to ¯uorides, we have estimated the T0 values proceeding from the high-temperature heat capacity data [4,6]. An interesting fact is, in particular, that the maxima in the Arrhenius energy curves for all compounds fall at somewhat lower temperatures Tc as compared with the peak positions for Cp (cf. with 1000 (SrCl2), 1420 (CaF2), 1450 (SrF2), and 1265 K (BaF2) according to our revision of the data in Refs. [2,4,6]). One may assume in according to observation of Schoonman [38] that the positive deviation relates to a change from intrinsic conductivity by anion vacancies to intrinsic conductivity by anion interstitials having a higher migration energy [13]. When using available data [13,27±34], our estimates result in the maximum increments in EA to be 74, 45, 48 and 22 kJ/mol for SrCl2, CaF2, SrF2, and BaF2, respectively. These are rather small values compared with increments in Arrhenius energy between T0 and Tc observed experimentally (see Fig. 6 and Table 1). Therefore, a change to the interstitialcy mechanism can indeed contribute somewhere close to the onset of positive deviations in the conductivity plots only. The ionic conductivity involves the product of the defect concentrations and their mobilities. So, as pointed out by Chadwick [13], either of these terms could show an anomalous increase as Tc is approached, although, in the light of the available information [11±19,21,22], it seems logical to assume the defect concentrations are being perturbed. Moreover, we suggest that the anion sublattice disordering lasts as high as the melting point of crystals. Indeed, if one adopts that the conductivity above Tc is alone due to an anion interstitial mobility and that the mobility behavior with temperature is de®ned by the same manner as at lower temperatures [13,27±34,38], then the conductivities of salts would increase from Tc to Tm no more than by the factors of 5.7 (SrCl2), 4.4 (CaF2), 5.5 (SrF2) and 5.5 (BaF2). (Meanwhile, such evaluation for mobility seems to be overestimated [38]. The details concerning the defect mobilities as well as predominant types of mobile defects in the superionic region are still the subject of controversy. At any rate, we should keep in mind that the effective ion mobility is related with the defect concentration.) Actually, the conductivities are increasing by the factors of 8.0, 12.3, 11.7 and 8.5 ,respectively (see Table 1). These comparisons, in accord with most previous works (for example, neutron scattering and diffraction [16,17,21], molecular dynamics simulation [18±20,22] and thermodynamic models [5,15,39,40]), lead to conclusion that the conductivity anomaly between T0 and Tm are mainly controlled by the generation of mobile defects. A rapid rise in s is in toto attributed to the generation of an excess of mobile defects due to their attractive interactions decreasing effectively the energy of defect formation [12,14,20,22,39±41]. In the particular case of neutron scattering and diffraction investigations of the nominally pure ¯uorite phases, the results have been interpreted in terms of a model of short-lived (,10 ±12 s) clusters comprising anion Frenkel

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Fig. 8. Correlation between the r1/r2 ratio and molar conductivity at the melting point for a number of ¯uorite-type crystals.

Fig. 7. The molar conductivity of (1) SrCl2, (2) CaF2, (3) SrF2, and (4) BaF2 plotted on a reduced reciprocal temperature scale, Tm/T.

interstitials, anion vacancies and surrounding relaxed anions [16,17,21]. The following negative deviation in the conductivity plots above Tc can be due to the retardation of the disordering process by defect±defect repulsions [12,13,21] or by saturation in defect clusters concentration [14] or by assuming the defect mobilities have reached a critical value [38], although Hainovsky and Maier reported [40] that their `cube root' thermodynamic model describes the conductivity anomaly in b-PbF2 without such assumptions. 4.2. Comparison of molar conductivities The molar conductivity, L , is known to be preferable when comparing the behavior of several salts. In our case of capillary-type measurements, the L values in the solid state are de®ned in the best way by the relation

L ˆ s´V m:p: ; where Vm.p. is the molar volume of the solidi®ed salt just below the melting point; such a formula is following from the measurement technique used. To calculate the Vm.p. values, we used the data on the density of melts [26] and volume change on melting salts [42±44]. The data on the volume change on melting BaF2 were not found, therefore, we have taken in the given case a value of 5% because for other similar compounds (CaF2, SrF2, SrCl2) these values are within 4±6%. In Fig. 7 the L curves are plotted as a function of reciprocal reduced temperatures Tm/T. This makes it possible to conveniently illustrate the idea that sublattice disordering in ¯uorites, being continuous and limited in principle, seems to be more and more extended in the series of crystals SrCl2 , CaF2 , SrF2 , BaF2, when the melting

points are approached. We already pointed out a different degree of disorder in SrCl2 and K2S obtained from the thermodynamic model calculations [15]. Indeed, in the ¯uorite structure the size of interstitial voids is known to correspond to cation one. Then the cation radius-to-anion radius ratio r1/r2 can be regarded as the valid parameter that characterizes the ease for anions to move through, or occupy, these sites [3] and hence also determines the degree of disorder that is reached at the melting point. The representative in this respect is the comparison of molar conductivities, L m.p., at the melting points for a number of ¯uorite-type crystalline phases (Fig. 8). To get a fuller picture, also included here are the data on the conductivity of b-PbF2, SrBr2 and BaCl2. (The L m.p. values have been evaluated proceeding from the data on thermal expansivity [21] and speci®c conductivity [10] of crystalline b-PbF2 and on the density of melts [26], volume change of melting [44] and speci®c conductivity of crystals [45] in the case of SrBr2 and BaCl2). In the latter two salts, the hightemperature superionic ¯uorite modi®cation is formed by a ®rst-order phase transition at 918 and 1193 K followed by the melting points at 930 and 1233 K, respectively [45]. One can see a distinct correlation between L m.p. and r1/r2 values, though the differences in the melting temperatures of salts reach over 800 K. Such a behaviour implies that an increase in the relative size of interstitial sites leads to a greater disorder near the melting point and, as a consequence, to an increase in conductivity. These assumptions are strongly supported by our results on the thermodynamics of ionic disorder in high-temperature ¯uorite-type ¯uorides, which have been obtained following the approach in Ref. [15] and will be discussed elsewhere. Finally, the data in Table 1 allow to compare the transport properties on `crystal-melt' transition. As is seen, the ratio of the molar conductivities of crystal and melt (L cryst/L melt) at the melting point (i.e. the fraction of solid-phase conductivity) increases in the order SrCl2, CaF2, SrF2, BaF2. Simultaneously, the EA values for crystal and melt tend, in

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general, to approach with decreasing relative size of mobile ions, although, there is a feeling that other ionic parameters (polarizability, mass) should be taken into consideration. These tendencies are again assumed to re¯ect the different levels of the crystal sublattice disorder and support the view that the positional disorder of ions on melting is the less pronounced, the higher the degree of its `preparedness' in the precedent crystal phase.

[18] [19] [20]

References [1] A.B. Lidiard, Crystals with the Fluorite Structure, in: W. Hayes (Ed.), Clarendon Press, Oxford, 1974, p. 101. [2] A.S. Dworkin, M.A. Bredig, Heat content and entropy of strontium chloride from 298 to 1200 K, J. Chem. and Engng Data 8 (1963) 416±417. [3] M.A. Bredig, The order-disorder (lambda) transition in uranium dioxide and other solids of the ¯uorite type of structure, Colloq. Int. CNRS 205 (1972) 180±191. [4] R.I. Yefremova, E.V. Matizen, Heat content of BaF2, SrCl2, and SrF2 at high temperatures, Heat content of BaF2, SrCl2, and SrF2 at high temperatures, Izvest. Sibirsk. Otdel. Akad. Nauk USSR 2 (1) (1970) 3±12 (in Russian). [5] W. SchroÈter, J. NoÈlting, Speci®c heat of crystals with the ¯uorite structure, J. Phys. (Paris), Colloq. C6 41 (1980) 20± 23. [6] B.F. Naylor, Heat contents at high temperatures of magnesium and calcium ¯uorides, J. Amer. Chem. Soc. 67 (1945) 150± 152. [7] C.E. Derrington, A. Lindner, M. O'Keeffe, Ionic conductivity of some alkaline earth halides, J. Solid State Chem. 15 (1975) 171±174. [8] V.M. Carr, A.V. Chadwick, R. Sagha®an, The electrical conductivity of PbF2 and SrCl2 crystals at high temperatures, J. Phys. C: Solid State Phys. 11 (1978) L637±L641. [9] B.M. Voronin, V.D. Prisyazhnyi, Conductivity and thermodynamic characterization of superionic transition in strontium chloride, Elektrokhimiya 25 (1989) 218±224 (in Russian). [10] A. Azimi, V.M. Carr, A.V. Chadwick, F.G. Kirkwood, R. Sagha®an, Point defect parameters for b-PbF2 from a computer analysis of measurements of ionic conductivity, J. Phys. Chem. Solids 45 (1984) 23±31. [11] M.B. Salamon (Ed.), Physics of Superionic Conductors. Springer-Verlag, Berlin/ Heidelberg/New York, 1979. [12] C.R.A. Catlow, Structure and transport in superionic ¯uorites, Comments Solid State Phys. 9 (1980) 157±167. [13] A.V. Chadwick, High-temperature transport in ¯uorites, Solid State Ionics 8 (1983) 209±220. [14] A.R. Allnatt, A.V. Chadwick, P.W.M. Jacobs, A model for the onset of fast-ion conduction in ¯uorites, Proc. Roy. Soc. (London) A410 (1987) 385±408. [15] B.M. Voronin, Some simple thermodynamic approaches to superionic disorder in ¯uorite-type crystals: application to SrCl2 and K2S, J. Phys. Chem. Solids 56 (1995) 839±847. [16] M.T. Hutchings, Anion disorder in the fast-ion phase of ¯uorites. Proceedings of the Symposium on Neutron Scattering, Argonne National Laboratory (AIP Conference Proceedings), 1981. pp. 209±220. [17] M.T. Hutchings, K. Clausen, M.H. Dickens, W. Hayes, J.K.

[21]

[22]

[23] [24]

[25]

[26] [27] [28] [29] [30]

[31] [32] [33] [34] [35] [36]

1357

Kjems, P.G. Schnabel, C. Smith, Investigation of thermally induced anion disorder in ¯uorites using neutron scattering techniques, J. Phys. C: Solid State Phys. 17 (1984) 3903± 3940. M.J. Gillan, Collective dynamics in superionic CaF2: II Defect interpretation, J. Phys. C: Solid State Phys. 19 (1986) 3517± 3533. A.M. Brass, Molecular dynamics study of the defect behaviour in ¯uorite structure crystals close to the superionic transition, Phil. Mag. A 59 (1989) 843±859. D. Bingham, A.N. Cormack, C.R.A. Catlow, A molecular dynamic simulation of gadolinium-doped SrF2, J. Phys.: Condens. Matter 1 (1989) 1213±1222. J.P. Goff, W. Hayes, S. Hull, M.T. Hutchings, Neutron powder diffraction study of the fast-ion transition and speci®c heat anomaly in b-lead ¯uoride, J. Phys.: Condens. Matter 3 (1991) 3677±3687. F. Zimmer, P. Ballone, M. Parrinello, J. Maier, The conductivity anomaly in PbF2: a numerical investigation by classical MD and MC simulations, Solid State Ionics 127 (2000) 277± 284. A. Lunden, Ionic conduction in sulphates, NATO ASI Series E 250 (1993) 181±201. B.M. Voronin, V.D. Prisyazhnyi, K.K. Khizhnyak, Ya.Yu. Kompan, Determination of the electrical conductivity of molten magnesium, calcium, strontium, and barium ¯uorides, Soviet Progr. Chem. 46 (1980) 229±233. B.M. Voronin, V.D. Prisyazhnyi, K.K. Khizhnyak, V.N. Zamkov, Yu.K. Novikov, The speci®c electrical conductivity of molten mixtures CaF2-CaCl2, SrF2-SrCl2, and BaF2-BaCl2, Soviet Progr. Chem. 53 (1987) 603±607. G.J. Janz (Ed.), Properties of Molten Salts, Nat. Inst. Stand. Technol. (USA), Standard Reference Database, version 2.0, 1992. M. Beniere, M. Chemla, F. Beniere, Anion diffusion mechanism in strontium chloride single crystals, J. Phys. Chem. Solids 40 (1979) 729±737. E. Barsis, A. Taylor, Lattice disorder in some CaF2-type crystals, J. Chem. Phys. 45 (1966) 1154±1162. W. Bollmann, H. Henniger, Concentration and mobility of ¯uorine ion vacancies in CaF2, Phys. Stat. Sol. (a) 11 (1972) 367±373. P.W.M. Jacobs, S.H. Ong, Point defect parameters for calcium ¯uoride from ionic conductivity measurements at low temperatures, J. Phys. (Paris), Colloq. C7 37 (1976) 331±335. W. Bollmann, P. GoÈrlich, W. Hauk, H. Mothes, Ionic conduction of pure and doped CaF2 and SrF2 crystals, Phys. Stat. Sol. (a) 2 (1970) 157±170. P.W.M. Jacobs, S.H. Ong, Point defect parameters for BaF2 from a computer analysis of ionic conductivity measurements, Cryst. Lattice Defects 8 (1980) 177±184. W. Bollmann, Ionic conductivity of pure and doped BaF2 crystals, Phys. Stat. Sol. (a) 18 (1973) 313±321. E. Barsis, A. Taylor, F 2 conductivity in BaF2 crystals. J. Chem. Phys. 48 (1968) 4357±4361; Interstitial conduction in BaF2 crystals. J. Chem. Phys. 48 (1968) 4362±4367. D.R. Figueroa, A.V. Chadwick, NMR relaxation, ionic conductivity and the self-diffusion process in barium ¯uoride, J. Phys. C: Solid State Phys. 11 (1978) 55±73. G. Daniel, J. NoÈlting, RoÈntgenographische und dilatometrische

1358

[37] [38] [39]

[40]

B.M. Voronin, S.V. Volkov / Journal of Physics and Chemistry of Solids 62 (2001) 1349±1358 untersuchungen zur fehlordnung im strontiumchlorid, Ber. Bunsenges. Phys. Chem. 87 (1983) 562±566. A. Sadoc, Y. Allain, A neutron diffraction study of disorder in SrCl2 at high temperature, Solid State Communs 25 (1978) 739±741. J. Schoonman, Retarded ionic motion in ¯uorites, Solid State Ionics 1 (1980) 121±131. J. Oberschmidt, Simple thermodynamic model for the speci®c-heat anomaly and several other properties of crystals with the ¯uorite structure, Phys. Rev. B: Condens. Matter. 23 (1981) 5038±5047. N. Hainovsky, J. Maier, Simple phenomenological approach to premelting and sublattice melting in Frenkel disordered ionic crystals, Phys. Rev. B: Condens. Matter. 51 (1995) 15789±15797.

[41] D.O. Welch, G.J. Dienes, Phenomenological and microscopic models of sublattice disorder in ionic crystals ÐI. Phenomenological models, J. Phys. Chem. Solids 38 (1977) 311±317. [42] P.W. Mirwald, G.C. Kennedy, The phase relations of calcium ¯uoride (¯uorite) to 60 kbars and 18008C, J. Phys. Chem. Solids 39 (1978) 859±861. [43] P.W. Mirwald, G.C. Kennedy, Phase relations for SrF2 to 50 kbars and 19008C and its compression to 40 kbars at 258C, J. Phys. Chem. Solids 41 (1980) 1157±1160. È ber die volumenaÈnderung beim [44] H. Schinke, F. Sauerwald, U schmelzen und den schmelzprozeû bei anorganischen salzen, Z. anorg. und allg. Chem. 304 (1960) 25±36. [45] C.E. Derrington, M. O'Keeffe, The solid electrolyte behavior of barium chloride and strontium bromide, Solid State Communs 15 (1974) 1175±1177.