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Crystal structure and anion disorder in β-PbF2
Solid State lonics 9 & 10 (1983) 521-524 North-Holland PublishingCompany
521
CRYSTAL STRUCTUREAND ANION DISORDER IN ~-PbF2 Rolf Bachmann and Heinz Schulz Max-Planck-lnstitut fur Festk~rperforschung Heisenbergstr. 1, 7000 Stuttgart 80, F.R.G. The f l u o r i n e d i s t r i b u t i o n in ~-PbF2 was obtained by analysing the e la s t ic neutron d i f f r a c t i o n data of Dickens et al. ( i ) . The f l u o r i n e d i s t r i b u t i o n at high temperatures shows that the conduction path connects neighbouring regular positions of the f l u o r i t e structure. The path is not a straight line but bent towards the octahedral void at ( I / 2 , 1 / 2 , 1 / 2 ) , which however is not a point of the conduction path. Interpreting the f l u o r i n e d i s t r i b u t i o n in terms of an effective one partical potential i t is shown that the defect concentration of the f l u o r i n e ions is less than 5%. A potential barrier of 0.26(5)e~ for ionic motion is obtained, which agrees with the activation energy of cond u c t i v i t y at elevated temperatures.
1.
INTRODUCTION
A number of compounds with the f l u o r i t e structure are good ionic conductors at elevated temperatures. They e x h i b i t a diffuse transition (2, 3) to a highly conducting phase with conductivities comparable to the melt (4,5) and low a c ti vation energies. For the study of the highly conducting B-PbF2 is best suited because i t has the highest conductiyity and lowest t r a n s i t i o n temperature of a l l f l u o r i t e s . For this reason ~-PbF2 has been studied many times in recent years. The results of d i f f e r e n t experiments however s t i l l disagree. Structure investigations using neutron d i f f r a c t i o n (1) gave f l u o r i n e concentrations at i n t e r s t i t i a l sites of more than 35%. A X-ray study (6) showed a very broad d i s t r i b u t i o n of the f l u o r i n e ions with no clear i n t e r s t i t i a l position. Measurement of conductivity, NMR (7), specific heat (8) and molecular dynamics simulations (9) gave defect concentrations of only a few per cent. Two explanations were offered for this discrepancy: Part of the ions at i n t e r s t i t i a l positions do not represent true defects but ions "relaxed" to d i f f e r e n t positions because of the interaction with true Frenkel defects. The other explanation is that large anharmonic thermal v i brations lead to large displacements of the fl u o rine ions. To decide between the two hypotheses the neutron d i f f r a c t i o n data of Dickens et al. were reanalysed. 2.
REFINEMENTS
Two d i f f e r e n t models for the f l u o r i n e d i s t r i b u tion were refined. One model with a l l f l u o r i n e ions at t h e i r regular positions and anharmonic temperature factors only and another model with one i n t e r s t i t i a l position at (x,x,.5) ( x ~ . 1 7 ) and anharmonic temperature factors f o r the regular f l u o r i n e site. These models are compared with the models of Dickens et al. who used the 0 167-2738/83/0000-0000/$ 03.00 © 1983 North-Holland
regular f l u o r i n e site and two i n t e r s t i t i a l sites at (x,x,x) (x~.35) and (x,x,.5) (x~.17) to refine the high temperature data. For temperatures up to 673K, i . e . below the diffuse t r a n s i t i o n , anharmonic temperature factors alone were enough to obtain a good f i t between measured and calculated i n t e n s i t i e s . For temperatures above the transition the purely anharmonic model was s i g n i f i c a n t l y worse than the models with i n t e r s i t i a l sites, the model with one i n t e r s t i t i a l site and anharmonic temperature factors being s l i g h t l y better than the model with two i n t e r s t i t i a l sites. The refinement alone hQweyer does not allow to distinguish between thermal motion and i n t e r s t i t i a l sites. Additional positions in the refinement can be a purely mathematical aid. 3.
THE DISTRIBUTION OF THE FLUORINE IONS
From the refined data, the j o i n t p r o b a b i l i t y density function (PDF) of the f l u o r i n e ions was calculated using the following r e l a t i o n (10): PDF(x) = ~'j wj pdfj(x) w.: ~ pdf~
occupancy of site j p r o b a b i l i t y density function of site j , which is the Fourier transform of the refined temperature factors of site j . The sum is taken over a l l refined f l u o r i n e positions.
The Pdf combines the contributions of d i f f e r e n t positions to the overall f l u o r i n e d i s t r i b u t i o n and gives the p r o b a b i l i t y of finding a f l u o r i n e ion in a volume element of the crystal. The PDFs of a l l models are very similar. Below the transition they show that the f l u o r i n e distribution is concentrated near the regular pos i t i o n at (1/4,1/4,1/4) with the largest displacements along (1,1,1). Above the phase tran-
R. Bachrnann, H. Schulz / Crystal structure and anion disorder in ~-PbF2
522
s i t i o n the PDF is continuously smeared out and connects neighbouring regular fluorine positions, as can be seen in f i g . 1. The PDF also shows the conduction path, which is not a straight l i n e but bent towards the octahedral void. No maximum in the PDF was found outside the regular f l u o r i n e positions at a l l temperatures.
In the case of disorder the above formula gives a pseudo potential which is strongly temperature dependent and an apparent potential barrier is obtained which is too low (10).
V[meV]
t ÷
-Q54
x[A]
240
//
220
!//
200
/J/
0.54
Figure 2: Effective one p a r t i c l e potential of the f l u o r i n e ions. The minimum is at the regular fluorine position, i) 298K, 2) 473, 573 and 6231<, 3) 673K, 4) 723-973K
01 --,-
Figure 1: Joint p r o b a b i l i t y density function of the fluorine ions in ~-PbF2 at 773K.
4.
THE EFFECTIYE ONE PARTICLE POTENTIAL OF THE FLUORINE IONS
The effectiye one p a r t i c l e potential V(x) is given by: V(x) = ~o " kT In(PDF(x)) k: Boltzman constant T: temperature Yo: arbitrary constant The temperature dependence of this potential allows to distinguish between thermal motion and i n t e r s t i t i a l sites (disorder). I f no disorder is present, the real physical potential is obtained in which the ions are moying. Except for temperatures near a phase transition this potential is almost independent of temperature (thermal expansion leads to small changes) and the potential barrier is equal to the activation energy of mob i l i t y for mobile ions.
Fig. 2 shows the effective one p a r t i c l e potential near the regular f l u o r i n e position for d i f f e r e n t temperatures. Between room temperature and 623K the potential slowly becomes softer which can be explained by thermal expansion. This softening becomes rapid at the transition. Above the trans i t i o n no further change in the potential is observed. This is true for the potential along the whole conduction path as can be seen in f i g . 3. (The y a r i a t i o n of the potential near the barrier is smaller than the error in determining the potential at this point. At the potential barrier the PDF is very low which leads to the high error). The average potential barrier for ionic motion is 0.26(5)eV. This agrees with the activation energy of conductivity above the transition of O.2(2)eY (11,12). Both the very weak temperature dependence of the potential and the agreement between the potent i a l barrier and the activation energy of cond u c t i v i t y show that the broad d i s t r i b u t i o n of the fluorine ions in B-PbF2 is due to anharmonic thermal motion. The defect concentration is too small to be detected and only an upper l i m i t of 5% can be given. Thus the neutron data also lead to low defect concentrations and the disagreement between the d i f f e r e n t experimental techniques is removed.
R. Bachmann, H. Schulz
20(
fi > 100 L.J
50 F
/Crystal structure and anion disorder in ~-PbF2
5C [4]
We would like to thank Prof. Hutchings for sending us the manuscript before publication. One of us (R.B.) g r a t e f u l l y acknowledges the support of a Krupp doctoral fellowship.
[5]
REFERENCES
[8]
[2] [3]
Dickens, M.H., Hayes, W., Hutchings, M.T. and Smith, C., J. Phys. C 15 (1982) 40434060. Derrington, C.E., Navrotsky, A. and O'Keeffe, M., Sol. State Com. 18 (1976) 47-49. Dworking, A.S. and Bredig, M.A., J. Phys. Chem. 72 {1968) 1277-1281.
Figure 3: Effective one particle potential of the fluorine ions along the conduction path. Solid l i n e : 973K, broken l i n e : 773K, dotted l i n e : 723K
F
ACKNOWLEDGEMENT
[1]
523
[6] [7]
[9] [10] [11] [12]
Derrington, C.E., Lindner, A. and O'Keeffe, M.,J. Sol. State Chem. 15 (1975) 171-174. Benz, R., Z. Phys. Chem. N.F. 95 (1975) 25-32. Koto, K., Schulz, H. and Huggins, R.A., Solid State Ionics 1 (1980) 355-365. Bonne, R.W. and Schoonman, J., J. Electrochem. Soc. 124 (1977) 28-35. Schr6ter, W. and N61ting, J., J. de Phys. C6 (1980) C20-C23. Walker, A.B., Gillan, M.J. and Dixon, M., J. Phys. C 15 (1982) 4061-4073. Bachmann, R., Ph.D. Thesis, Universit~t Karlsruhe (Feb. 1983). Carr, V.M., Chadwick, A.V. and Shaghafian, R., J. Phys. C 11 (1978) L637-641. Gordon, R.E. and Strange, J.H., J. Phys. C 11 (1978) 3213-3223.
521
CRYSTAL STRUCTUREAND ANION DISORDER IN ~-PbF2 Rolf Bachmann and Heinz Schulz Max-Planck-lnstitut fur Festk~rperforschung Heisenbergstr. 1, 7000 Stuttgart 80, F.R.G. The f l u o r i n e d i s t r i b u t i o n in ~-PbF2 was obtained by analysing the e la s t ic neutron d i f f r a c t i o n data of Dickens et al. ( i ) . The f l u o r i n e d i s t r i b u t i o n at high temperatures shows that the conduction path connects neighbouring regular positions of the f l u o r i t e structure. The path is not a straight line but bent towards the octahedral void at ( I / 2 , 1 / 2 , 1 / 2 ) , which however is not a point of the conduction path. Interpreting the f l u o r i n e d i s t r i b u t i o n in terms of an effective one partical potential i t is shown that the defect concentration of the f l u o r i n e ions is less than 5%. A potential barrier of 0.26(5)e~ for ionic motion is obtained, which agrees with the activation energy of cond u c t i v i t y at elevated temperatures.
1.
INTRODUCTION
A number of compounds with the f l u o r i t e structure are good ionic conductors at elevated temperatures. They e x h i b i t a diffuse transition (2, 3) to a highly conducting phase with conductivities comparable to the melt (4,5) and low a c ti vation energies. For the study of the highly conducting B-PbF2 is best suited because i t has the highest conductiyity and lowest t r a n s i t i o n temperature of a l l f l u o r i t e s . For this reason ~-PbF2 has been studied many times in recent years. The results of d i f f e r e n t experiments however s t i l l disagree. Structure investigations using neutron d i f f r a c t i o n (1) gave f l u o r i n e concentrations at i n t e r s t i t i a l sites of more than 35%. A X-ray study (6) showed a very broad d i s t r i b u t i o n of the f l u o r i n e ions with no clear i n t e r s t i t i a l position. Measurement of conductivity, NMR (7), specific heat (8) and molecular dynamics simulations (9) gave defect concentrations of only a few per cent. Two explanations were offered for this discrepancy: Part of the ions at i n t e r s t i t i a l positions do not represent true defects but ions "relaxed" to d i f f e r e n t positions because of the interaction with true Frenkel defects. The other explanation is that large anharmonic thermal v i brations lead to large displacements of the fl u o rine ions. To decide between the two hypotheses the neutron d i f f r a c t i o n data of Dickens et al. were reanalysed. 2.
REFINEMENTS
Two d i f f e r e n t models for the f l u o r i n e d i s t r i b u tion were refined. One model with a l l f l u o r i n e ions at t h e i r regular positions and anharmonic temperature factors only and another model with one i n t e r s t i t i a l position at (x,x,.5) ( x ~ . 1 7 ) and anharmonic temperature factors f o r the regular f l u o r i n e site. These models are compared with the models of Dickens et al. who used the 0 167-2738/83/0000-0000/$ 03.00 © 1983 North-Holland
regular f l u o r i n e site and two i n t e r s t i t i a l sites at (x,x,x) (x~.35) and (x,x,.5) (x~.17) to refine the high temperature data. For temperatures up to 673K, i . e . below the diffuse t r a n s i t i o n , anharmonic temperature factors alone were enough to obtain a good f i t between measured and calculated i n t e n s i t i e s . For temperatures above the transition the purely anharmonic model was s i g n i f i c a n t l y worse than the models with i n t e r s i t i a l sites, the model with one i n t e r s t i t i a l site and anharmonic temperature factors being s l i g h t l y better than the model with two i n t e r s t i t i a l sites. The refinement alone hQweyer does not allow to distinguish between thermal motion and i n t e r s t i t i a l sites. Additional positions in the refinement can be a purely mathematical aid. 3.
THE DISTRIBUTION OF THE FLUORINE IONS
From the refined data, the j o i n t p r o b a b i l i t y density function (PDF) of the f l u o r i n e ions was calculated using the following r e l a t i o n (10): PDF(x) = ~'j wj pdfj(x) w.: ~ pdf~
occupancy of site j p r o b a b i l i t y density function of site j , which is the Fourier transform of the refined temperature factors of site j . The sum is taken over a l l refined f l u o r i n e positions.
The Pdf combines the contributions of d i f f e r e n t positions to the overall f l u o r i n e d i s t r i b u t i o n and gives the p r o b a b i l i t y of finding a f l u o r i n e ion in a volume element of the crystal. The PDFs of a l l models are very similar. Below the transition they show that the f l u o r i n e distribution is concentrated near the regular pos i t i o n at (1/4,1/4,1/4) with the largest displacements along (1,1,1). Above the phase tran-
R. Bachrnann, H. Schulz / Crystal structure and anion disorder in ~-PbF2
522
s i t i o n the PDF is continuously smeared out and connects neighbouring regular fluorine positions, as can be seen in f i g . 1. The PDF also shows the conduction path, which is not a straight l i n e but bent towards the octahedral void. No maximum in the PDF was found outside the regular f l u o r i n e positions at a l l temperatures.
In the case of disorder the above formula gives a pseudo potential which is strongly temperature dependent and an apparent potential barrier is obtained which is too low (10).
V[meV]
t ÷
-Q54
x[A]
240
//
220
!//
200
/J/
0.54
Figure 2: Effective one p a r t i c l e potential of the f l u o r i n e ions. The minimum is at the regular fluorine position, i) 298K, 2) 473, 573 and 6231<, 3) 673K, 4) 723-973K
01 --,-
Figure 1: Joint p r o b a b i l i t y density function of the fluorine ions in ~-PbF2 at 773K.
4.
THE EFFECTIYE ONE PARTICLE POTENTIAL OF THE FLUORINE IONS
The effectiye one p a r t i c l e potential V(x) is given by: V(x) = ~o " kT In(PDF(x)) k: Boltzman constant T: temperature Yo: arbitrary constant The temperature dependence of this potential allows to distinguish between thermal motion and i n t e r s t i t i a l sites (disorder). I f no disorder is present, the real physical potential is obtained in which the ions are moying. Except for temperatures near a phase transition this potential is almost independent of temperature (thermal expansion leads to small changes) and the potential barrier is equal to the activation energy of mob i l i t y for mobile ions.
Fig. 2 shows the effective one p a r t i c l e potential near the regular f l u o r i n e position for d i f f e r e n t temperatures. Between room temperature and 623K the potential slowly becomes softer which can be explained by thermal expansion. This softening becomes rapid at the transition. Above the trans i t i o n no further change in the potential is observed. This is true for the potential along the whole conduction path as can be seen in f i g . 3. (The y a r i a t i o n of the potential near the barrier is smaller than the error in determining the potential at this point. At the potential barrier the PDF is very low which leads to the high error). The average potential barrier for ionic motion is 0.26(5)eV. This agrees with the activation energy of conductivity above the transition of O.2(2)eY (11,12). Both the very weak temperature dependence of the potential and the agreement between the potent i a l barrier and the activation energy of cond u c t i v i t y show that the broad d i s t r i b u t i o n of the fluorine ions in B-PbF2 is due to anharmonic thermal motion. The defect concentration is too small to be detected and only an upper l i m i t of 5% can be given. Thus the neutron data also lead to low defect concentrations and the disagreement between the d i f f e r e n t experimental techniques is removed.
R. Bachmann, H. Schulz
20(
fi > 100 L.J
50 F
/Crystal structure and anion disorder in ~-PbF2
5C [4]
We would like to thank Prof. Hutchings for sending us the manuscript before publication. One of us (R.B.) g r a t e f u l l y acknowledges the support of a Krupp doctoral fellowship.
[5]
REFERENCES
[8]
[2] [3]
Dickens, M.H., Hayes, W., Hutchings, M.T. and Smith, C., J. Phys. C 15 (1982) 40434060. Derrington, C.E., Navrotsky, A. and O'Keeffe, M., Sol. State Com. 18 (1976) 47-49. Dworking, A.S. and Bredig, M.A., J. Phys. Chem. 72 {1968) 1277-1281.
Figure 3: Effective one particle potential of the fluorine ions along the conduction path. Solid l i n e : 973K, broken l i n e : 773K, dotted l i n e : 723K
F
ACKNOWLEDGEMENT
[1]
523
[6] [7]
[9] [10] [11] [12]
Derrington, C.E., Lindner, A. and O'Keeffe, M.,J. Sol. State Chem. 15 (1975) 171-174. Benz, R., Z. Phys. Chem. N.F. 95 (1975) 25-32. Koto, K., Schulz, H. and Huggins, R.A., Solid State Ionics 1 (1980) 355-365. Bonne, R.W. and Schoonman, J., J. Electrochem. Soc. 124 (1977) 28-35. Schr6ter, W. and N61ting, J., J. de Phys. C6 (1980) C20-C23. Walker, A.B., Gillan, M.J. and Dixon, M., J. Phys. C 15 (1982) 4061-4073. Bachmann, R., Ph.D. Thesis, Universit~t Karlsruhe (Feb. 1983). Carr, V.M., Chadwick, A.V. and Shaghafian, R., J. Phys. C 11 (1978) L637-641. Gordon, R.E. and Strange, J.H., J. Phys. C 11 (1978) 3213-3223.