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Point defect parameters in β-PbF2 revisited
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Solid State Ionics 179 (2008) 438 – 441 www.elsevier.com/locate/ssi
Point defect parameters in β-PbF2 revisited P. Varotsos ⁎ Solid State Section, Physics Department, University of Athens, Panepisthmiopolis, Zografos 157 84, Athens, Greece Received 27 September 2007; received in revised form 28 February 2008; accepted 28 February 2008
Abstract The defect parameters in β-PbF2 that have been determined to date from the association and extrinsic regions of the isobaric conductivity plot as well as from conductivity measurements under various pressures, are studied. We find that, in the low temperature range where bulk elastic and expansivity data are available, the defect volumes scale linearly with the defect enthalpies with a slope which is governed by bulk qualities. A deviation from linearity is observed in the high temperature range from which the relevant parameters for the anion Frenkel formation process are deduced. © 2008 Elsevier B.V. All rights reserved. Keywords: Frenkel defects; Defect volumes; Fluorites; Conductivity
1. Introduction The major reason for the interest on the fluorite-structured materials during the last three decades has been the discovery that these systems exhibit superionic conductivity at high temperature. Here we focus on one of them, i.e., β-PbF2, which is probably the most prominent fluoride superionic conductor because of its low transition temperature (≈700°K) and high ionic conductivity (σ = 1 Ω− 1 cm− 1 at 800°K) (e.g., see Refs. [1,2]). This has been the subject of a large number of investigations. Among these studies, Figueroa et al. [3] have made low temperature dc-conductivity measurements under various pressures on PbF2 either pure or doped with various alkali metals. At their lower temperatures of these measurements at which the conductivity is due to free vacancies thermally dissociated from the electric dipoles consisting of the impurity and the fluorine vacancy (created for reasons of charge compensation), the activation volume υact is given by: υact ¼ 1=2 υa þ υfm
ð1Þ
where υa, υfm are the volumes that correspond to the association process and the free (fluorine) vacancy motion, respectively. Moreover, Figueroa et al. [3] made dielectric relaxation measurements at various pressures, which give the volume υm,b for ⁎ Tel.: +30 210 7257688; fax: +30 210 9601721. E-mail address: [email protected]. 0167-2738/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2008.02.055
the (re)orientation process of the above dipoles. All these volumes determined for all the alkali ions Li+, Na+, K+ and Rb+ have been found to be positive. In addition, Figueroa et al. [3] noticed that the ratio of the activation volumes to the activation energies (which are, in fact, activation enthalpies designated hereafter with hact), is the same for all dopants. An explanation on this important experimental finding was then proposed by Lazaridou et al. [4], based on an early model termed cBΩ model, that will be summarized later. In accordance with this explanation, the ratio υact/hact should solely be governed by macroscopic properties of the bulk solid. Actually, it was shown [4] that the migration enthalpy (hf,m) and volume (υf,m) for the free ( f ) fluorine vacancy motion, as well as the corresponding quantities hm,i, υm,i for the fluorine interstitial (i) migration, which have been determined from earlier conductivity measurements [5,6] under pressure, exhibit ratios υf,m/hf,m and υm,i/hm,i that are not only equal to those υact/hact found by Figueroa et al. [3] but also equal to the value predicted by the cBΩ model. In the meantime, however, new conductivity measurements in β-PbF2 appeared [7] which lead to values of the ratio υ/h for the defect migration as well as for the anion Frenkel defect formation processes. It is therefore of interest to investigate whether these new data –and hence the totality of data observed to date – result in values consistent with the behavior expected on the basis of the cBΩ model. This constitutes the main objective of the present paper. The variation of the pressure (P) affects the formation Gibbs energy, gf, as well as the migration Gibbs energy, gm, of defects
P. Varotsos / Solid State Ionics 179 (2008) 438–441
in solids. The defect volumes for the formation process (υf) and migration process (υm) are defined as [8]: υf ¼
dgf j dP T
and
υm ¼
dgm j dP T
ð2Þ
where T denotes the temperature. When both processes, i.e., formation (f) and migration (m) are operating, the experimental results are described in terms of an activation Gibbs energy gact, on the basis of which an activation volume is defined: υact ¼
dg act j dP T
ð3Þ
If experiments refer to migration (m) of a bound (b) defect, e.g., the case of the (re)orientation of electric dipoles consisting of an aliovalent impurity and a neighboring (bound) vacancy or interstitial, the results are described in terms of an activation energy gact,b which is associated with an activation volume υm,b ≡(dgm,b/dP)T for the (re)orientation process. In the vast majority of the experiments studying the pressure variation of either the dc ionic conductivity or the diffusion coefficients in various materials υact was found to be positive. The same holds for the υm,b values obtained from electrical relaxation measurements under various pressures. However, some noticeable exceptions have been reported, in which negative activation volumes have been found [9–12]. This case is of particular importance since it provides the basis for the explanation [8,13,14] of the generation of low frequency electric signals that are observed before earthquakes [8,15–18]. The present paper is organized as follows: In Section 2 we briefly summarize the cBΩ model, whereas its application to the data of β-PbF2 is discussed in Section 3. Finally, in Section 4, we present our main conclusion. 2. The model that interconnects the defect Gibbs energy with the bulk properties The defect Gibbs energy gi is interconnected with the bulk properties of the solid through the relation [8,19–21]: gi ¼ ci BX
ð4Þ
where the superscript i refers here to the defect process under consideration, i.e., i =f and m for the formation and migration, respectively. Concerning the symbols, B is the isothermal bulk modulus, Ω the mean volume per atom and ci is dimensionless which can be considered – to the first approximation – independent of temperature and pressure. By differentiating Eq. (4) with respect to pressure, we find that the volume υi = (dgi/dP)T is given by: dB jT 1 ð5aÞ υ i ¼ ci X dP or, equivalently
υi ¼
g i dB jT 1 B dP
ð5bÞ
439
Similarly, by differentiating Eq. (4) with respect to temperature we find the entropy si = −(dgi/dT)T. We then insert this result as well as Eq. (4) into the relation hi = gi + Tsi which finally gives for the enthalpy hi: dB i i j h ¼ c X B T bB T ð6Þ dT Ρ Thus, taking the ratio of Eqs. (5a), (5b) and (6), we get: υi dB dB j j ¼ 1 = B T bB T ð7Þ dP T dT Ρ hi Whenever the temperature is small and Tsi bbhi, Eq. (7) can be approximately written as: υi 1 dB j 1 c hi B dP T which is reminiscent of an equation obtained in Ref. [22] that was derived, however, on a different basis [8], i.e., the dynamical theory of defect migration in solids. The validity of Eqs. (4) and (7) has been checked for various processes in a variety of solids [8,20,21]. The same procedure can be also applied to mixed ionic solids since their B-values used in Eq. (4) can be estimated from the corresponding B-values of the pure constituents [23]. Furthermore, we note that Eq. (4) has been recently found of value for the behavior in the formation of Schottky defects in high Tc-superconductors [24] as well as for the temperature dependence of the viscosity of glass forming liquids as we approach the glass transition [25,26]. 3. Investigation of the compatibility of the parameters from conductivity measurements under pressure with the cBΩ model Eq. (7) indicates that for a given host crystal the ratio υi/hi should be the same irrespective of the process and the kind of the dopant and that its value is solely determined by the bulk properties. In Fig. 1, we plot the experimental values of υi versus hi from various sources as follows: First, four points (solid dots) correspond to the association parameters from the conductivity studies of Figueroa et al. [3] in β-PbF2 doped with Li, Na, K and Rb (see their Tables III and IV), whereas two other points obtained by the same authors in two undoped samples P1 and P2. Second, three points (open, full and inverted triangle) correspond to the (free) fluorine vacancy migration parameters, as they have been reported by Oberschmidt and Lazarus [5], Samara [6] and Murin et al. [7], respectively. Third, two points (solid and open square) show the (free) fluorine interstitial migration parameters reported by Samara [6] and Murin et al. [7], respectively. Finally, the two remaining points (full and open diamond) correspond to the values deduced for the anion Frenkel formation process by Oberschmidt and Lazarus [5] and Murin et al. [7], respectively. An inspection of this figure reveals that all the points, except of the two ones that correspond to the formation process, lie more or less on a straight line, as they should in accordance with the cBΩ model. In the same figure, for the sake of comparison,
440
P. Varotsos / Solid State Ionics 179 (2008) 438–441
Fig. 1. Experimental values of the defect volume versus the defect enthalpy for various processes in β-PbF2: ♢ and ♦ refer to anion Frenkel formation parameters reported by Murin et al. [7] and Oberschmidt and Lazarus [5], respectively; Δ,and ▿ — fluorine vacancy migration by Oberschmidt and Lazarus [5], Samara [6] and Murin et al. [7], respectively; □ and ■ — fluorine interstitial migration by Murin et al. [7] and Samara [6]; ⁎ association parameters for two samples P1 and P2 of pure β-PbF2 measured by Figueroa et al. [3]; ● — association parameters for β-PbF2 doped with K, Rb, Na and Li, respectively, by Figueroa et al. [3].
we draw the straight line having a slope equal to that predicted from Eq. (7) when considering the following bulk parameters at the room temperature: β = 5.7 × 10− 5 K− 1 [27], B = 63.0 GPa [27], (dB/dP)T = 7.13 [28]. The latter two values come from the adiabatic elastic data of Ref. [28]. However, their difference from the isothermal ones is small. As for the (dB/dT)P, we used the value − 0.057 GPa K− 1 deduced from the elastic constants by Manaresh and Pederson [29] from 300 to 850 K (The measured adiabatic values of Bs were converted to the isothermal values, B, using standard thermodynamic equations [8]). That the aforementioned straight line in Fig. 1 – drawn on the basis of cBΩ model – passes almost through the vast majority of the experimental points is remarkable, especially when one considers the following: if one uses Zener's model [30] –which states that gi is proportional to the shear modulus – we find [31] that the calculated values of the ratios υm,υ/gm,υ, and υm,i/gm,I differ greatly from the experimental results, i.e., by a factor of about 4. We now comment on the fact that the two points in Fig. 1 that correspond to the anion Frenkel formation process seem to deviate from the straight line that passes through all the other points. More precisely the values of the ratio υf/hf deduced from the experimental results of Oberschmidt and Lazarus [5] and Murin et al. [7] are smaller by ~ 20% and 30%, respectively, from the one predicted by the cBΩ model. We clarify that these two points of Fig. 1 come from conductivity measurements in the high temperature range (i.e., in the so called intrinsic region), while all the others from appreciably lower temperatures. In particular, the so called intrinsic region II, which extends from 410 to 490 K (see Table 1 of Ref. [32]), the Arrhenius energy deduced from the slope of the conductivity plot lnσT vs 1/T (P = const) is equal to hf,m + hf, F/2 since in this region the anion
(free) vacancies are the more mobile species, while in the intrinsic region III (extending from 530 to 615 K) – where the fluorine interstitials are the more mobile species – the Arrhenius energy is hf,i + hf, F/2. We now recall that the straight line in Fig. 1 has been drawn upon considering the elastic data and expansivity data of the right hand side of Eq. (7) measured at room temperature. Unfortunately, measurements of (dB/dP)T at higher temperatures (and in particular in the intrinsic regions mentioned above) are not yet available, thus we are not able to draw the corresponding straight line of the intrinsic range(s). In other words, the deviation of the aforementioned two points – which corresponds to the formation process – from the straight line drawn in Fig. 1, should not be misinterpreted as showing that the cBΩ model exhibits a departure from the real situation. Such a deviation has not been found [33] in α-PbF2, the high pressure behavior of which has been studied by angular-dispersive synchrotron X-ray powder diffraction techniques [34]. The latter, through the fit of third order Birch–Murnaghan equation of state gave the values of B and (dB/dt)T, which enable us to show [33] that the cBΩ model leads to values of the Frenkel defect formation volume and the migration volumes from the fluorine vacancy and fluorine interstitial that are comparable to those obtained from conductivity measurements under pressure. 4. Conclusions In the low temperature range in β-PbF2, where all the necessary elastic and expansivity data are available, we do find that the eleven points (hi, υi) in Fig. 1 – that include the association parameters as well as those for the migration processes of the fluorine vacancy and the fluorine interstitial experimentally determined to date – do lie on a straight line having a slope comparable to that predicted from the cBΩ model. In the high temperature range, from which the parameters for the anion Frenkel formation process are deduced, a deviation (ranging from 20% to 35%) is found when comparing the value of υf/hf calculated from the cBΩ model with the one obtained from the experimental results. This deviation is attributed to the lack of elastic (and expansivity) data under pressure in the latter temperature range. References [1] J. Schoonman, Solid State Ion. 1 (1980) 121. [2] J. Schoonman, D.J. Dirksen, G. Blasse, J. Solid State Chem. 7 (1973) 245. [3] D.R. Figueroa, J.J. Fontanella, M.C. Wintersgill, A.V. Chadwick, C.G. Andeen, J. Phys., C 17 (1984) 4399. [4] M. Lazaridou, K. Alexopoulos, P. Varotsos, Phys. Rev., B 31 (1985) 8273. [5] J. Oberschmidt, D. Lazarus, Phys. Rev., B 21 (1980) 2952. [6] G.A. Samara, J. Phys. Chem. Solids 40 (1979) 509. [7] I.V. Murin, O.V. Glumov, W. Gunsser, M. Karus, Radiat. Eff. Defects Solids 137 (1995) 251. [8] P. Varotsos, K. Alexopoulos, Thermodynamics of Point Defects and their Relation with the Bulk Properties, North Holland, Amsterdam, 1986. [9] P.C. Allen, D. Lazarus, Phys. Rev., B 17 (1978) 1913. [10] P. Varotsos, K. Alexopoulos, Phys. Rev., B 21 (1980) 4898. [11] J.J. Fontanella, C.A. Edmondson, M.C. Wintersgill, Macromolecules 29 (1996) 4944. [12] J.J. Fontanella, M.C. Wintersgill, D.R. Figueroa, A.V. Chadwick, C.G. Andeen, Phys. Rev. Lett. 51 (1983) 1892.
P. Varotsos / Solid State Ionics 179 (2008) 438–441 [13] P. Varotsos, K. Alexopoulos, K. Nomicos, Phys. Status Solidi, B 111 (1982) 581 see also; P. Varotsos, Physics of Seismic Electric Signals, TerraPub, Tokyo, 2005. [14] P. Varotsos, N. Sarlis, M. Lazaridou, P. Kapiris, J. Appl. Phys. 83 (1998) 60. [15] P. Varotsos, K. Alexopoulos, K. Nomicos, M. Lazaridou, Nature 322 (1986) 120 see also; P. Varotsos, K. Alexopoulos, Tectonophysics, 110 (1984) 73 ibid110 (1984) 99. [16] P. Varotsos, K. Alexopoulos, K. Nomicos, M. Lazaridou, Tectonophysics 152 (1988) 193 see also; P. Varotsos, M. Lazaridou, Tectonophysics 188 (1991) 321; P. Varotsos, K. Alexopoulos, M. Lazaridou, Tectonophysics 224 (1993) 237. [17] N. Sarlis, M. Lazaridou, P. Kapiris, P. Varotsos, Geophys. Res. Lett. 26 (1999) 3245. [18] P. Varotsos, N. Sarlis, M. Lazaridou, Phys. Rev., B 59 (1999) 24. [19] P. Varotsos, K. Alexopoulos, Phys. Rev., B 15 (1977) 4111 ibid 15 (1977) 2348; J. Phys. Chem. Solids 38 (1977) 997; J. Physique Lettr. 38 (1977) L455; phys. stat. sol. (a) 47 (1978) K133; Phys. Rev. B 18 (1978) 2683. [20] P. Varotsos, K. Alexopoulos, Phys. Rev., B 22 (1980) 3130 see also;
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M. Lazaridou, C. Varotsos, K. Alexopoulos, P. Varotsos, J. Phys. C: Solid State 18 (1985) 3891. [21] P. Varotsos, K. Alexopoulos, Phys. Rev., B 24 (1981) 904. [22] C.P. Flynn, Point Defects and Diffusion, Clarendon, Oxford, 1972. [23] P. Varotsos, K. Alexopoulos, J. Phys. Chem. Solids 41 (1980) 1291. [24] H. Su, D.O. Welch, W. Wong-Ng, Phys. Rev., B 70 (2004) 054517. [25] J.C. Dyre, Rev. Mod. Phys. 78 (2006) 953. [26] J.C. Dyre, T. Christensen, N.B. Olsen, J. Non-Cryst. Solids 532 (2006) 4635. [27] R.B. Roberts, C. White, J. Phys. C 19 (1986) 7167. [28] D.S. Rimai, R.J. Sladek, Phys. Rev., B 21 (1980) 843. [29] M.O. Manaresh, D.O. Pederson, Phys. Rev., B 30 (1984) 3482. [30] C. Zener, J. Appl. Phys. 22 (1951) 372. [31] P. Varotsos, J. Appl. Phys. 101 (2007) 123503. [32] A. Azimi, V.M. Carr, A.V. Chadwick, F.G. Kirkwood, R. Saghafian, J. Phys. Chem. Solids 45 (1984) 23. [33] P. Varotsos, Phys. Rev., B 76 (2007) 092106. [34] L. Ehm, K. Knorr, F. Maedler, H. Voigtlaender, F. Buseto, A. Casseta, A. Lausi, B. Winkler, J. Phys. Chem. Solids 64 (2003) 919.
Solid State Ionics 179 (2008) 438 – 441 www.elsevier.com/locate/ssi
Point defect parameters in β-PbF2 revisited P. Varotsos ⁎ Solid State Section, Physics Department, University of Athens, Panepisthmiopolis, Zografos 157 84, Athens, Greece Received 27 September 2007; received in revised form 28 February 2008; accepted 28 February 2008
Abstract The defect parameters in β-PbF2 that have been determined to date from the association and extrinsic regions of the isobaric conductivity plot as well as from conductivity measurements under various pressures, are studied. We find that, in the low temperature range where bulk elastic and expansivity data are available, the defect volumes scale linearly with the defect enthalpies with a slope which is governed by bulk qualities. A deviation from linearity is observed in the high temperature range from which the relevant parameters for the anion Frenkel formation process are deduced. © 2008 Elsevier B.V. All rights reserved. Keywords: Frenkel defects; Defect volumes; Fluorites; Conductivity
1. Introduction The major reason for the interest on the fluorite-structured materials during the last three decades has been the discovery that these systems exhibit superionic conductivity at high temperature. Here we focus on one of them, i.e., β-PbF2, which is probably the most prominent fluoride superionic conductor because of its low transition temperature (≈700°K) and high ionic conductivity (σ = 1 Ω− 1 cm− 1 at 800°K) (e.g., see Refs. [1,2]). This has been the subject of a large number of investigations. Among these studies, Figueroa et al. [3] have made low temperature dc-conductivity measurements under various pressures on PbF2 either pure or doped with various alkali metals. At their lower temperatures of these measurements at which the conductivity is due to free vacancies thermally dissociated from the electric dipoles consisting of the impurity and the fluorine vacancy (created for reasons of charge compensation), the activation volume υact is given by: υact ¼ 1=2 υa þ υfm
ð1Þ
where υa, υfm are the volumes that correspond to the association process and the free (fluorine) vacancy motion, respectively. Moreover, Figueroa et al. [3] made dielectric relaxation measurements at various pressures, which give the volume υm,b for ⁎ Tel.: +30 210 7257688; fax: +30 210 9601721. E-mail address: [email protected]. 0167-2738/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2008.02.055
the (re)orientation process of the above dipoles. All these volumes determined for all the alkali ions Li+, Na+, K+ and Rb+ have been found to be positive. In addition, Figueroa et al. [3] noticed that the ratio of the activation volumes to the activation energies (which are, in fact, activation enthalpies designated hereafter with hact), is the same for all dopants. An explanation on this important experimental finding was then proposed by Lazaridou et al. [4], based on an early model termed cBΩ model, that will be summarized later. In accordance with this explanation, the ratio υact/hact should solely be governed by macroscopic properties of the bulk solid. Actually, it was shown [4] that the migration enthalpy (hf,m) and volume (υf,m) for the free ( f ) fluorine vacancy motion, as well as the corresponding quantities hm,i, υm,i for the fluorine interstitial (i) migration, which have been determined from earlier conductivity measurements [5,6] under pressure, exhibit ratios υf,m/hf,m and υm,i/hm,i that are not only equal to those υact/hact found by Figueroa et al. [3] but also equal to the value predicted by the cBΩ model. In the meantime, however, new conductivity measurements in β-PbF2 appeared [7] which lead to values of the ratio υ/h for the defect migration as well as for the anion Frenkel defect formation processes. It is therefore of interest to investigate whether these new data –and hence the totality of data observed to date – result in values consistent with the behavior expected on the basis of the cBΩ model. This constitutes the main objective of the present paper. The variation of the pressure (P) affects the formation Gibbs energy, gf, as well as the migration Gibbs energy, gm, of defects
P. Varotsos / Solid State Ionics 179 (2008) 438–441
in solids. The defect volumes for the formation process (υf) and migration process (υm) are defined as [8]: υf ¼
dgf j dP T
and
υm ¼
dgm j dP T
ð2Þ
where T denotes the temperature. When both processes, i.e., formation (f) and migration (m) are operating, the experimental results are described in terms of an activation Gibbs energy gact, on the basis of which an activation volume is defined: υact ¼
dg act j dP T
ð3Þ
If experiments refer to migration (m) of a bound (b) defect, e.g., the case of the (re)orientation of electric dipoles consisting of an aliovalent impurity and a neighboring (bound) vacancy or interstitial, the results are described in terms of an activation energy gact,b which is associated with an activation volume υm,b ≡(dgm,b/dP)T for the (re)orientation process. In the vast majority of the experiments studying the pressure variation of either the dc ionic conductivity or the diffusion coefficients in various materials υact was found to be positive. The same holds for the υm,b values obtained from electrical relaxation measurements under various pressures. However, some noticeable exceptions have been reported, in which negative activation volumes have been found [9–12]. This case is of particular importance since it provides the basis for the explanation [8,13,14] of the generation of low frequency electric signals that are observed before earthquakes [8,15–18]. The present paper is organized as follows: In Section 2 we briefly summarize the cBΩ model, whereas its application to the data of β-PbF2 is discussed in Section 3. Finally, in Section 4, we present our main conclusion. 2. The model that interconnects the defect Gibbs energy with the bulk properties The defect Gibbs energy gi is interconnected with the bulk properties of the solid through the relation [8,19–21]: gi ¼ ci BX
ð4Þ
where the superscript i refers here to the defect process under consideration, i.e., i =f and m for the formation and migration, respectively. Concerning the symbols, B is the isothermal bulk modulus, Ω the mean volume per atom and ci is dimensionless which can be considered – to the first approximation – independent of temperature and pressure. By differentiating Eq. (4) with respect to pressure, we find that the volume υi = (dgi/dP)T is given by: dB jT 1 ð5aÞ υ i ¼ ci X dP or, equivalently
υi ¼
g i dB jT 1 B dP
ð5bÞ
439
Similarly, by differentiating Eq. (4) with respect to temperature we find the entropy si = −(dgi/dT)T. We then insert this result as well as Eq. (4) into the relation hi = gi + Tsi which finally gives for the enthalpy hi: dB i i j h ¼ c X B T bB T ð6Þ dT Ρ Thus, taking the ratio of Eqs. (5a), (5b) and (6), we get: υi dB dB j j ¼ 1 = B T bB T ð7Þ dP T dT Ρ hi Whenever the temperature is small and Tsi bbhi, Eq. (7) can be approximately written as: υi 1 dB j 1 c hi B dP T which is reminiscent of an equation obtained in Ref. [22] that was derived, however, on a different basis [8], i.e., the dynamical theory of defect migration in solids. The validity of Eqs. (4) and (7) has been checked for various processes in a variety of solids [8,20,21]. The same procedure can be also applied to mixed ionic solids since their B-values used in Eq. (4) can be estimated from the corresponding B-values of the pure constituents [23]. Furthermore, we note that Eq. (4) has been recently found of value for the behavior in the formation of Schottky defects in high Tc-superconductors [24] as well as for the temperature dependence of the viscosity of glass forming liquids as we approach the glass transition [25,26]. 3. Investigation of the compatibility of the parameters from conductivity measurements under pressure with the cBΩ model Eq. (7) indicates that for a given host crystal the ratio υi/hi should be the same irrespective of the process and the kind of the dopant and that its value is solely determined by the bulk properties. In Fig. 1, we plot the experimental values of υi versus hi from various sources as follows: First, four points (solid dots) correspond to the association parameters from the conductivity studies of Figueroa et al. [3] in β-PbF2 doped with Li, Na, K and Rb (see their Tables III and IV), whereas two other points obtained by the same authors in two undoped samples P1 and P2. Second, three points (open, full and inverted triangle) correspond to the (free) fluorine vacancy migration parameters, as they have been reported by Oberschmidt and Lazarus [5], Samara [6] and Murin et al. [7], respectively. Third, two points (solid and open square) show the (free) fluorine interstitial migration parameters reported by Samara [6] and Murin et al. [7], respectively. Finally, the two remaining points (full and open diamond) correspond to the values deduced for the anion Frenkel formation process by Oberschmidt and Lazarus [5] and Murin et al. [7], respectively. An inspection of this figure reveals that all the points, except of the two ones that correspond to the formation process, lie more or less on a straight line, as they should in accordance with the cBΩ model. In the same figure, for the sake of comparison,
440
P. Varotsos / Solid State Ionics 179 (2008) 438–441
Fig. 1. Experimental values of the defect volume versus the defect enthalpy for various processes in β-PbF2: ♢ and ♦ refer to anion Frenkel formation parameters reported by Murin et al. [7] and Oberschmidt and Lazarus [5], respectively; Δ,and ▿ — fluorine vacancy migration by Oberschmidt and Lazarus [5], Samara [6] and Murin et al. [7], respectively; □ and ■ — fluorine interstitial migration by Murin et al. [7] and Samara [6]; ⁎ association parameters for two samples P1 and P2 of pure β-PbF2 measured by Figueroa et al. [3]; ● — association parameters for β-PbF2 doped with K, Rb, Na and Li, respectively, by Figueroa et al. [3].
we draw the straight line having a slope equal to that predicted from Eq. (7) when considering the following bulk parameters at the room temperature: β = 5.7 × 10− 5 K− 1 [27], B = 63.0 GPa [27], (dB/dP)T = 7.13 [28]. The latter two values come from the adiabatic elastic data of Ref. [28]. However, their difference from the isothermal ones is small. As for the (dB/dT)P, we used the value − 0.057 GPa K− 1 deduced from the elastic constants by Manaresh and Pederson [29] from 300 to 850 K (The measured adiabatic values of Bs were converted to the isothermal values, B, using standard thermodynamic equations [8]). That the aforementioned straight line in Fig. 1 – drawn on the basis of cBΩ model – passes almost through the vast majority of the experimental points is remarkable, especially when one considers the following: if one uses Zener's model [30] –which states that gi is proportional to the shear modulus – we find [31] that the calculated values of the ratios υm,υ/gm,υ, and υm,i/gm,I differ greatly from the experimental results, i.e., by a factor of about 4. We now comment on the fact that the two points in Fig. 1 that correspond to the anion Frenkel formation process seem to deviate from the straight line that passes through all the other points. More precisely the values of the ratio υf/hf deduced from the experimental results of Oberschmidt and Lazarus [5] and Murin et al. [7] are smaller by ~ 20% and 30%, respectively, from the one predicted by the cBΩ model. We clarify that these two points of Fig. 1 come from conductivity measurements in the high temperature range (i.e., in the so called intrinsic region), while all the others from appreciably lower temperatures. In particular, the so called intrinsic region II, which extends from 410 to 490 K (see Table 1 of Ref. [32]), the Arrhenius energy deduced from the slope of the conductivity plot lnσT vs 1/T (P = const) is equal to hf,m + hf, F/2 since in this region the anion
(free) vacancies are the more mobile species, while in the intrinsic region III (extending from 530 to 615 K) – where the fluorine interstitials are the more mobile species – the Arrhenius energy is hf,i + hf, F/2. We now recall that the straight line in Fig. 1 has been drawn upon considering the elastic data and expansivity data of the right hand side of Eq. (7) measured at room temperature. Unfortunately, measurements of (dB/dP)T at higher temperatures (and in particular in the intrinsic regions mentioned above) are not yet available, thus we are not able to draw the corresponding straight line of the intrinsic range(s). In other words, the deviation of the aforementioned two points – which corresponds to the formation process – from the straight line drawn in Fig. 1, should not be misinterpreted as showing that the cBΩ model exhibits a departure from the real situation. Such a deviation has not been found [33] in α-PbF2, the high pressure behavior of which has been studied by angular-dispersive synchrotron X-ray powder diffraction techniques [34]. The latter, through the fit of third order Birch–Murnaghan equation of state gave the values of B and (dB/dt)T, which enable us to show [33] that the cBΩ model leads to values of the Frenkel defect formation volume and the migration volumes from the fluorine vacancy and fluorine interstitial that are comparable to those obtained from conductivity measurements under pressure. 4. Conclusions In the low temperature range in β-PbF2, where all the necessary elastic and expansivity data are available, we do find that the eleven points (hi, υi) in Fig. 1 – that include the association parameters as well as those for the migration processes of the fluorine vacancy and the fluorine interstitial experimentally determined to date – do lie on a straight line having a slope comparable to that predicted from the cBΩ model. In the high temperature range, from which the parameters for the anion Frenkel formation process are deduced, a deviation (ranging from 20% to 35%) is found when comparing the value of υf/hf calculated from the cBΩ model with the one obtained from the experimental results. This deviation is attributed to the lack of elastic (and expansivity) data under pressure in the latter temperature range. References [1] J. Schoonman, Solid State Ion. 1 (1980) 121. [2] J. Schoonman, D.J. Dirksen, G. Blasse, J. Solid State Chem. 7 (1973) 245. [3] D.R. Figueroa, J.J. Fontanella, M.C. Wintersgill, A.V. Chadwick, C.G. Andeen, J. Phys., C 17 (1984) 4399. [4] M. Lazaridou, K. Alexopoulos, P. Varotsos, Phys. Rev., B 31 (1985) 8273. [5] J. Oberschmidt, D. Lazarus, Phys. Rev., B 21 (1980) 2952. [6] G.A. Samara, J. Phys. Chem. Solids 40 (1979) 509. [7] I.V. Murin, O.V. Glumov, W. Gunsser, M. Karus, Radiat. Eff. Defects Solids 137 (1995) 251. [8] P. Varotsos, K. Alexopoulos, Thermodynamics of Point Defects and their Relation with the Bulk Properties, North Holland, Amsterdam, 1986. [9] P.C. Allen, D. Lazarus, Phys. Rev., B 17 (1978) 1913. [10] P. Varotsos, K. Alexopoulos, Phys. Rev., B 21 (1980) 4898. [11] J.J. Fontanella, C.A. Edmondson, M.C. Wintersgill, Macromolecules 29 (1996) 4944. [12] J.J. Fontanella, M.C. Wintersgill, D.R. Figueroa, A.V. Chadwick, C.G. Andeen, Phys. Rev. Lett. 51 (1983) 1892.
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