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Defect interactions in intrinsic silver chloride
I. Phys. Chem. Solids. 1975, Vol. 36. pp. 12334235.
Pergmon
Press.
Printed in Great Britain
DEFECT INTERACTIONS IN INTRINSIC SILVER CHLORIDE A. R. ALLNATTt Department of Chemistry, University of Western Ontario, London, Canada N6A 5B7
J. CORISH Department of Chemistry, University College, Dublin, Ireland and P. w. M. JAC0BSt.t Physics and Chemistry of Solids, Cavendish Laboratory, Cambridge, England (Received 29 October 1974;in reoisedform 13January 1975) Abstract-Approximate formulae are developed for the activity coefficients of vacancies and interstitials in intrinsic silver chloride. The numerical values are compared with the DebyeHiickel approximation and the useful range of the formulae is discussed.
The purpose of this article is to develop a simple but useful approximation for the activity coefficient (y) of interstitial ions and vacancies in intrinsic silver chloride. Practical calculations of defect interactions in ionic crystals are almost always performed in the manner first suggested by Teltow[l] and by Lidiard[2]. This method consists of allowing for short range interactions by the formation of pairs and treating long range interactions in the Debye-Hiickel approximation developed originally for ionic interactions in dilute solutions. Recent detailed analyses of the ionic conductivity of AgCl[3-61 have indicated that the Debye-Hiickel approximation might not be an adequate representation of defect interactions. A general method of calculating defect interactions has been developed by Allnatt and Cohen [7] and this has been applied to extrinsic NaCl by Allnatt, Loftus and Rowley[8] and to intrinsic AgCl by Sevenich and Kliewer[9]. However, the latter calculation leads to a complicated result not readily amenable to the routine numerical computation necessary for the iterative calculations which must be performed in fitting conductivity data. Sevenich and Kliewer (SK) included contributions from cycle sums and from those terms from the modified virial expansion of the remainder which contains pairs and triplets of defects (pair and triplet diagrams of the Mayer expansion). The Debye-Hiickel result is an approximation to the cycle sum[8]. SK simplified the evaluation of the integrals over the Brillouin Zone (BZ)-formulae Al0 and All of Ref. [9]-by using a sphericalized BZ and a continuum approximation for the Fourier transform l(t) of (4~(Rl)-‘, where R is the vector joining two defect sites, namely Member, Chemical Physics Centre, University of Western Ontario. SPermanent address: Department of Chemistry, University of Western Ontario, London, Canada N6A 5B7.
l(t) = A-?-*
(1)
where A is a volume of the primitive unit cell in the direct lattice, equal to 2a3 where a denotes the nn anion-cation distance. Integration over the whole of reciprocal space, regarding the BZ as infinite, gives the Debye-Hiickel limiting law
where K2=4~b(Cv+2C,)/A b = e*/4moe,kT
(3)
E, is the relative permittivity, CV is the site fraction of vacancies, and CI is the site fraction of interstitials. (CI is the number of occupied interstitial sites divided by the number of available interstitial sites so that in intrinsic AgCl Cv = 2C,). Our purpose is to find a more accurate expression for the contribution to y from the cycle diagrams. Three approximations will be used. (i) Only terms to order O(K*) will be retained in the SK formulae, eqn AlO, All. (ii) The integrations will be carried out over a sphericalized BZ. (iii) The Allnatt and Cohen approximation for the Fourier transform AC(t) = t-*- b,,a*+O(f*)
(4)
where i, j = I, 2, 3 and 1 denotes a vacancy site and 2, 3 the two types of interstitial site in the unit cell, will be used. The bt, coefficients have been evaluated by SK. These approximations (i)-(iii) have been tested thoroughly for extrinsic and intrinsic NaC1[8, to]. The
1233
1234
A. R. ALLNAIT,J. COREH and P. W. M. JACOBS
approximations for the cycle sums may be inadequate at very high defect concentrations but the omission of all but cycle sums would in any case be incorrect in this range. From Fig. 8 of SK we see that the cycle sum approximation would be adequate up to 650 K. If we regard -In 7~ t In ~DHLand -In x + In ~DHLas regular functions of K that can be Taylor expanded? then on retaining terms to O(K’) the SK formulae AlO, All
on using the values for the bi, coefficients from SK. The temperature-dependence of E, was allowed for [4] in evaluating A and K. These formulae are compared with the Debye-Hiickel approximation to the cycle sums
yield
in Fig. 1 by plotting -In (r/rDw) against KR, where y is the mean defect activity coefficient y = (y~yvb)“’and R, is the distance of closest approach, taken to be V!&I/~ in intrinsic AgCI. Also shown are plots of -In (y(‘)/y~~) and -In (y’*)/yDn)for extrinsic NaCl (R, = %%I), where y”’ refers to the equivalent set of approximations (i)-(iii) used here and y(2) is the interpolation formula derived from Allnatt, Loftus and Rowley’s results[l] of a numerical integration over the BZ. The corresponding vacancy concentrations are shown in the upper scale as these values give one a better “feel” for the useful range of the approximations than do perhaps values of KR~. This plot of -In (y/y~~) actually exaggerates the differences between y”’ and y’*’which agree to better than 4% for 0 < KR~ 5 0.35, which can be considered the limit of the userul range of the approximations for NaCI. Approximations (i)-(iii) are not likely to be any less satisfactory for AgCl than they are for NaCl. The deviations of the Debye-Hiickel approximation from the accurate cycle sum are seen to be smaller for intrinsic AgCl than for extrinsic NaCl and are unlikely to a major source of error in fitting conductivity data [5].
(3 where A = 4nb, AI = t-*, I,,, I,* are given by (4), and OEZ means the integration is carried out outside the BZ, and
(6) On substituting from (4) for II, (neglecting terms of 0 (2 t”) in the Fourier transform) in eqns (5) and (6) and carrying out the integrations analytically we find -In
yv = (AK/~v)(]
-In “/I = (AK/&TH~
-0*61904~a}
(7)
-0.54244~~)
(8)
-lnym=;Kb/(l+K&)
tThis is not strictly correct but the poles are far enough away for the procedure to be legitimate when K is in the range in which the cycle sum approximation will be valid. 0.349
0.15r
I 0.199 I 603
0.523 I 643
1.40 I 1.22 I 663
(9)
Acknowledgements-P. W. M. Jacobs thanks the Master and Fellows of Churchill College, Cambridge, for the award of an Overseas Fellowship. The calculations were performed on the IBM 360 computer at University College Dublin and .I. Corish thanks the Computer Centre there for the use of their facilities. 3.14 I 2.56 3.60 I I 723 743
5.59 I ,^,
CV
x
CV
x
IO4
NaCl
KRq Fig. 1. Plots,of -In(y/ypn) against K& for extrinsic NaCl (at 773K) and for intrinsic AgCI. yvr is the mean defect activity coefficient for AgCl calculated from the cycle sums with the aid of approximations (i)-(iii). y”’ is the activity coefficient for vacancies in NaCl evaluated using the equivalent approximations. y (*)is the exact cycle sum result for vacancies’in NaCl from Ref. [S]. The numbers between the lines for -In (#‘)/yDH) and -In ($“/yDH) show the percentage difference between the accurate and approximate cycle sum activity coefficients.
Defect interactions in intrinsic silver chforide REFERENCES
I. Teltow J., Ann. Phys. 5, 63, 71 (1949). 2. Lidiard A. B., in Hundbuch der Physik (Edited by S. Fhtgge) Vol. XX, Part 2, pp. 246-349.Springer Verlag, Berlin (1957). 3. Mttller P., Phys. Status Solidi 12, 775 (1965). 4. Co&h J. and Jacobs P. W. M., J. Phys. Chem. Solids 33,1799 (1972). 5. Corish J. and Jacobs P. W. M., Phys. Status Solidi, (bf67,263 (1975).
1235
6. Friauf R. J. and Aboagye J., Phys. Rev. B 11, 1654(1975). 7. Alinatt A. R. and Cohen M. H., J. Chem. Phys. 40,1860,1871 (1964). 8. Allnatt A. R., Loftus E. and Rowley L. A., Crystal Lattice Defects 3, 77 (1972). 9. Sevenich R. A. and Kliewer K. L., J. Chem. Phys. 48, 3045 (1968). 10. Yuen P. S. and Ailnatt A. R., J: Phys. C: Solid State Phys. in press.
Pergmon
Press.
Printed in Great Britain
DEFECT INTERACTIONS IN INTRINSIC SILVER CHLORIDE A. R. ALLNATTt Department of Chemistry, University of Western Ontario, London, Canada N6A 5B7
J. CORISH Department of Chemistry, University College, Dublin, Ireland and P. w. M. JAC0BSt.t Physics and Chemistry of Solids, Cavendish Laboratory, Cambridge, England (Received 29 October 1974;in reoisedform 13January 1975) Abstract-Approximate formulae are developed for the activity coefficients of vacancies and interstitials in intrinsic silver chloride. The numerical values are compared with the DebyeHiickel approximation and the useful range of the formulae is discussed.
The purpose of this article is to develop a simple but useful approximation for the activity coefficient (y) of interstitial ions and vacancies in intrinsic silver chloride. Practical calculations of defect interactions in ionic crystals are almost always performed in the manner first suggested by Teltow[l] and by Lidiard[2]. This method consists of allowing for short range interactions by the formation of pairs and treating long range interactions in the Debye-Hiickel approximation developed originally for ionic interactions in dilute solutions. Recent detailed analyses of the ionic conductivity of AgCl[3-61 have indicated that the Debye-Hiickel approximation might not be an adequate representation of defect interactions. A general method of calculating defect interactions has been developed by Allnatt and Cohen [7] and this has been applied to extrinsic NaCl by Allnatt, Loftus and Rowley[8] and to intrinsic AgCl by Sevenich and Kliewer[9]. However, the latter calculation leads to a complicated result not readily amenable to the routine numerical computation necessary for the iterative calculations which must be performed in fitting conductivity data. Sevenich and Kliewer (SK) included contributions from cycle sums and from those terms from the modified virial expansion of the remainder which contains pairs and triplets of defects (pair and triplet diagrams of the Mayer expansion). The Debye-Hiickel result is an approximation to the cycle sum[8]. SK simplified the evaluation of the integrals over the Brillouin Zone (BZ)-formulae Al0 and All of Ref. [9]-by using a sphericalized BZ and a continuum approximation for the Fourier transform l(t) of (4~(Rl)-‘, where R is the vector joining two defect sites, namely Member, Chemical Physics Centre, University of Western Ontario. SPermanent address: Department of Chemistry, University of Western Ontario, London, Canada N6A 5B7.
l(t) = A-?-*
(1)
where A is a volume of the primitive unit cell in the direct lattice, equal to 2a3 where a denotes the nn anion-cation distance. Integration over the whole of reciprocal space, regarding the BZ as infinite, gives the Debye-Hiickel limiting law
where K2=4~b(Cv+2C,)/A b = e*/4moe,kT
(3)
E, is the relative permittivity, CV is the site fraction of vacancies, and CI is the site fraction of interstitials. (CI is the number of occupied interstitial sites divided by the number of available interstitial sites so that in intrinsic AgCl Cv = 2C,). Our purpose is to find a more accurate expression for the contribution to y from the cycle diagrams. Three approximations will be used. (i) Only terms to order O(K*) will be retained in the SK formulae, eqn AlO, All. (ii) The integrations will be carried out over a sphericalized BZ. (iii) The Allnatt and Cohen approximation for the Fourier transform AC(t) = t-*- b,,a*+O(f*)
(4)
where i, j = I, 2, 3 and 1 denotes a vacancy site and 2, 3 the two types of interstitial site in the unit cell, will be used. The bt, coefficients have been evaluated by SK. These approximations (i)-(iii) have been tested thoroughly for extrinsic and intrinsic NaC1[8, to]. The
1233
1234
A. R. ALLNAIT,J. COREH and P. W. M. JACOBS
approximations for the cycle sums may be inadequate at very high defect concentrations but the omission of all but cycle sums would in any case be incorrect in this range. From Fig. 8 of SK we see that the cycle sum approximation would be adequate up to 650 K. If we regard -In 7~ t In ~DHLand -In x + In ~DHLas regular functions of K that can be Taylor expanded? then on retaining terms to O(K’) the SK formulae AlO, All
on using the values for the bi, coefficients from SK. The temperature-dependence of E, was allowed for [4] in evaluating A and K. These formulae are compared with the Debye-Hiickel approximation to the cycle sums
yield
in Fig. 1 by plotting -In (r/rDw) against KR, where y is the mean defect activity coefficient y = (y~yvb)“’and R, is the distance of closest approach, taken to be V!&I/~ in intrinsic AgCI. Also shown are plots of -In (y(‘)/y~~) and -In (y’*)/yDn)for extrinsic NaCl (R, = %%I), where y”’ refers to the equivalent set of approximations (i)-(iii) used here and y(2) is the interpolation formula derived from Allnatt, Loftus and Rowley’s results[l] of a numerical integration over the BZ. The corresponding vacancy concentrations are shown in the upper scale as these values give one a better “feel” for the useful range of the approximations than do perhaps values of KR~. This plot of -In (y/y~~) actually exaggerates the differences between y”’ and y’*’which agree to better than 4% for 0 < KR~ 5 0.35, which can be considered the limit of the userul range of the approximations for NaCI. Approximations (i)-(iii) are not likely to be any less satisfactory for AgCl than they are for NaCl. The deviations of the Debye-Hiickel approximation from the accurate cycle sum are seen to be smaller for intrinsic AgCl than for extrinsic NaCl and are unlikely to a major source of error in fitting conductivity data [5].
(3 where A = 4nb, AI = t-*, I,,, I,* are given by (4), and OEZ means the integration is carried out outside the BZ, and
(6) On substituting from (4) for II, (neglecting terms of 0 (2 t”) in the Fourier transform) in eqns (5) and (6) and carrying out the integrations analytically we find -In
yv = (AK/~v)(]
-In “/I = (AK/&TH~
-0*61904~a}
(7)
-0.54244~~)
(8)
-lnym=;Kb/(l+K&)
tThis is not strictly correct but the poles are far enough away for the procedure to be legitimate when K is in the range in which the cycle sum approximation will be valid. 0.349
0.15r
I 0.199 I 603
0.523 I 643
1.40 I 1.22 I 663
(9)
Acknowledgements-P. W. M. Jacobs thanks the Master and Fellows of Churchill College, Cambridge, for the award of an Overseas Fellowship. The calculations were performed on the IBM 360 computer at University College Dublin and .I. Corish thanks the Computer Centre there for the use of their facilities. 3.14 I 2.56 3.60 I I 723 743
5.59 I ,^,
CV
x
CV
x
IO4
NaCl
KRq Fig. 1. Plots,of -In(y/ypn) against K& for extrinsic NaCl (at 773K) and for intrinsic AgCI. yvr is the mean defect activity coefficient for AgCl calculated from the cycle sums with the aid of approximations (i)-(iii). y”’ is the activity coefficient for vacancies in NaCl evaluated using the equivalent approximations. y (*)is the exact cycle sum result for vacancies’in NaCl from Ref. [S]. The numbers between the lines for -In (#‘)/yDH) and -In ($“/yDH) show the percentage difference between the accurate and approximate cycle sum activity coefficients.
Defect interactions in intrinsic silver chforide REFERENCES
I. Teltow J., Ann. Phys. 5, 63, 71 (1949). 2. Lidiard A. B., in Hundbuch der Physik (Edited by S. Fhtgge) Vol. XX, Part 2, pp. 246-349.Springer Verlag, Berlin (1957). 3. Mttller P., Phys. Status Solidi 12, 775 (1965). 4. Co&h J. and Jacobs P. W. M., J. Phys. Chem. Solids 33,1799 (1972). 5. Corish J. and Jacobs P. W. M., Phys. Status Solidi, (bf67,263 (1975).
1235
6. Friauf R. J. and Aboagye J., Phys. Rev. B 11, 1654(1975). 7. Alinatt A. R. and Cohen M. H., J. Chem. Phys. 40,1860,1871 (1964). 8. Allnatt A. R., Loftus E. and Rowley L. A., Crystal Lattice Defects 3, 77 (1972). 9. Sevenich R. A. and Kliewer K. L., J. Chem. Phys. 48, 3045 (1968). 10. Yuen P. S. and Ailnatt A. R., J: Phys. C: Solid State Phys. in press.