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Atomic-scale structure of alkali halide solid solutions
Solid State Communications, Vo1.63,No.Z, Printed in Great Britain.
ATOMIC-SCALE
STRUCTURE
pp.91-95,
0038-1098187 $3.00 + .OO Pergamon Journals Ltd.
1987.
OF ALKALI
HALIDE
SOLID
SOLUTIONS
M. Peressi Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy A. Baldereschi Dipartimento di Fisica Teorica and GNSM-CNR Universita di Trieste, Italy and Institut de Physique Appliquke, EPF-Lausanne, Switzerland. (Received 10 April 1987 by E. Tosatti)
A simple model is proposed to explain the crystallographic properties of solid solutions of the alkali halides. The random system is simulated with a periodic superlattice whose energy is evaluated with the Born-Mayer model generalized to include ionic polarization. The resulting average lattice parameter and anioncation distances agree with recent EXAFS data on Kl_,Rb,Br, RbBrl_,I,, and KCll-,Br,. The results show that ionic polarization plays a significant role in the determination of both interatomic distances and heats of formation.
sion of the model, i.e. we neglect all second-neighbour and Van der Waals interactions, but we include the effects of ionic polarization which have been found to be important in analogous circumstances of low atomicsite symmetry as for example in the presence of isolated defects or impurities and lattice vibrations 14), and at crystal surfaces ‘e). Ionic polarization plays also an important role in the cohesion of alkali-halide molecules 17) where again the symmetry is lower than that of the solid.
The crystallographic structure of solid solutions A1-,B,C of alkali halides has been investigated with X-ray diffraction and more recently with the EXAFS technique. X-ray data I) show that in most pseudobinary solutions the lattice parameter follows Vegard law ‘), i.e. it varies linearly with composition between the values for the AC and BC end compounds. Accurate EXAFS data s)*4)*5)are available for Kl_,Rb,Br, RbBrl-,I,, and KCll_,Br,. They indicate that the nearest neighbour distances have a bimodal distribution similar to that already observed in semiconducting alloys ‘)*“)**), and their value is intermediate between that of the pure compound and the average Vegard value. In Kl_,R&Br 5), for example, the NN distances r(K-Br] and r[Rb-Br] vary linearly with composition and their total variation is about 40% of the difference between NN distances in the pure compounds, i.e. the variation is more pronounced than that observed in semiconducting alloys.
Although the distribution of A and B ions in the homogeneous Al_,B,C solid solution is random, we simulate the alloy with a periodically repeated supercell. Since the macroscopic symmetry of the real alloy is cubic, we choose a cubic supercell and distribute in it anions and cations according to cubic symmetry. With these criteria a given cell allows only for particular compositions x and symmetry-restricted atomic displacements from the ideal sites of the rocksalt virtual crystal.
The local structure of metallic ‘) and semiconducting 7)*10)-13) pseudobinary alloys has been the subject of several theoretical investigations. These studies however do not apply to ionic solid solutions which are substantially different both in lattice structure and in type of binding. For ionic materials, only the case of infinite dilution, i.e. the isolated impurity limit, has received some attention in the past 13)*14).
Iu the present work we use the smallest cubic supercell which allows for atomic distortions, i.e. the FCC unit cell with 16 ions (Fig. 1). The ions within the supercell can be grouped into shells of symmetryequivalent ions. The sublattice centered at the origin contains three shells with 1,6, and 1 ions, respectively, while the other sublattice contains two shells with 6 and 2 ions, respectively. According to our filling criteria, each shell can contain only one kind of ions, so that the supercell allows only for the compositions z = l/8, z = l/4, and the complementary concentrations z = 718 and z = 3/4. The fillings with composi-
In thii work we intend to verify if the Born-Mayer model, which has successfully explained the cohesive properties of pure alkali halides 15), can be applied to the atomic-scale structure of alkali-halide solid solutions and their heat of mixing. We use the simplest ver91
92
Vol.
ATOMIC-SCALE STRUCTURE OF ALKALI HALIDE SOLID SOLUTTONS
+ BAC
2 =P I
Fig.1 Wigner-Seitz unit cell of the FCC superlattice considered in the present work. The unit cell contains 16 atoms. Non-equivalent atomic sites are indicated with different symbols.
tions z = l/4 and z = 314 will not be considered any further since in this case the supercell reduces to twice the SC cell with 8 ions, and all atomic displacements from ideal sites are symmetry forbidden. For the compositions z = l/8 and z = 718 only the six ions which are octahedrally coordinated to the central one can have a common radial relaxation 6 so that the crystallographic structure is fully described by the lattice constant a and the internal distortion parameter c = 6/a. The lattice energy at zero temperature and pressure is evaluated with the Born-Mayer model. Neglecting all interactions beyond nearest - neighbours, the Van der Waals interactions, and the zerepoint kinetic energy of the ions, the lattice energy per unit supercell is the sum of three terms
Lp.(%4 + qd.(~,~).
(1)
The first term is the Madelung energy
where o(c is a diitortion-dependent Madelung function (Fig. 2) which for E = 0 has the value a(0) = 55.922 which corresponds to the well known Madelung constant properly scaled to the dimension of our supercell. The second term is the repulsive energy between A-C and B-C nearest-neighbour pairs. For z = l/8 it can be written
63, No. 2
(--a/‘lpAC)
I
-
\
and a similar expression obtained by exchanging the A and B indices holds for z = 718. The parameters BAC,BBC,PAC, and p~c are adjusted to reproduce the lattice constant and bulk modulus of the pure AC and BC compounds. We use the values given in the first column of Table VIII of Ref. 15 for all materials except RbBr where we use the values B = 2.61* 10s8 erg and p = 0.324 A which reproduce the NN distance of 3.427 A asmeasured by EXAFS 51. The last term in equation (1) describes the main effects of the ionic p+ larization induced by the distortion of the lattice and contains charge-dipole interactions only. Considering that only the C ions can polarize, Ed can be written iiS:
Epd.(a,e) =
-9,
(4)
where pc is the polarizability of the C ions, whose value is taken from Ref. 18, and f(c) is a distortion dependent structural sum. Thii term vanishes for an undistorted lattice, i.e. f(0) = 0, but lf the ions move away from the ideal sites, they experience a Madelung field and polarize. Only those ions which do not occupy the ideal sites of the virtual crystal can polarize. The Madelung function Q(C) and the polarization function f(c) have been computed with the Ewald,method r51. It is convenient to introduce the following analytic approximate expressions for both a(s) and f(c) which are valid for Ic( < .005 : Q(E)=
55.922 - 693.08e2,
(5)
f(c) =
1.028 * 10’ c2.
(6)
The equilibrium values of the lattice constant o(z) and of the internal distortion parameter e(z) of Kr_,Rb,Br, RbBrr_,I, and KClr_,Br, have been computed for z = l/8 and z = 718 by minimizing the total lattice energy. In order to understand the releVance of the ionic polarization we have also computed the equilibrium values of a(z) and c(z) by neglecting the polarization contribution. The resulting values of a(z) and c(z) are given in Fig.3 and compared with experiment. The calculated values of the heat of mlxii are given in Fig.4 . From Fig.3 we see that the lattice constant of the supercell follows quite closely Vegard law. In the unit supercell several NN distances are present at equilibrium because of the displacements of the six ions near the center of the cell. After weighted average we obtain A-C and B-C distances which are closer to the ra
Vol. 63, No. 2
ATOMIC-SCALE STRUCTURE OF ALRALI HALIDE SOLID SOLUTIONS
-0.006
l0.003
-0.003
93
~0.008
;
Fig.2 Madehmg constant a(e) as function of the internal distortion parameter. Dots indicate the results of computations while the full line represents the quadratic approximation given in the text.
spective distances in the pure AC and BC compounds than to the virtual crystal NN distance, in accordance with EXAFS data. The agreement is very good for Kr_,Rb,Br, whereas for RbBrr_,I, and KClr_,Br, the calculated internal distortions are smaller than those measured experimentally. We stress that the diitortion of the C sublattice is induced by the difference between the repulsive forces of A-C and B-C pairs, but the equilibrium A-C and EC bond lengths are determined by all forces acting on the ions. Theoretical values of the heats of formation (see Fig. 4) can only be compared with experiment for KClr_,Br, and in thii case good agreement is obtained. The calculations performed neglecting the chargedipole interaction energy in es.(l), show that polarization effects are not negligible. As can be seen in Fig.3, these effects are not very important for the lattice parameter, but they increase significantly the internal distortion and bring the calculated anion-cation distances in closer agreement with experiment. Polarization effects are specially important in cationic alloys such as Kr_,Rb,Br where the displaced ions are the highly polarizable Br- anions. The agreement with EXAFS data is very good for this system. Anionic alloys polarize less and the inclusion of charge-dipole interactions is not sufficient to reproduce the experimental data. Ionic polarization also reduces the values of the heats of formation as can be seen for KCli-,Br, in Fig.4 .
For curiosity we have also minimized the lattice energy neglecting both polarization effects and inter-nal distortions. The resulting theoretical predictions are very poor. The bimodal distribution of NN distances is of course absent, and a(z) deviates considerably from both Vegard and Retgers laws (additivity of distances and of volumes, respectively). The heats of formation are much larger than experimental data (see Fig. 4). These simple computations however allow one to understand the importance of anharmonic forces in alkali-halide solid solutions. Separating the lattice energy E(o, B = 0) into harmonic and anharmonic terms we have evaluated how much each one contributes to the small deviations of e(z) from Vegard law. Harmonic terms alone give the wrong sign for the deviations, (i.e. a sub-linear behaviour), and only with the inclusion of the anharmonic terms one obtains the experimentally observed positive bowing. Including anharmonic terms, positive deviations from Vegard law are predicted for all alkali-halide solid solutions. The supercell used in this work is the smallest cubic cell which can have internal diitortionz. Its A-B sublattice however cannot distort and only a fraction of C ions are allowed by symmetry to move offsite. Nevertheless with this simple supercell and a simple version of the Born-Mayer model we have obtained satisfactory agreement with experimental data. The encouraging results obtained here suggest that the local structure of alkali-halide solid solutions can be fully explained by using more refined versions of the BomMayer model (i.e. including second-neighbour repul-
Vol. 63, No. 2
ATOMIC-SCALE STRUCTURE OF &KALE HALIDE SOLID SOLUTTONS
9c
I'I',',',',',',','m
K,.x Rb,Rr
-
1 I I , I , I , I , , , , , , , , , 0.0
0.2
0.4
0.6
1.0
0.8
X
8 L
2
200
100
0 0.0
0.2
3.3
I
I
0.4
’
I
’
I
0.5
X
’
I
0.8
r
I
’
I
*
1
’
I
1
1 1 0.2
1
1
1 1 0.4
1
02
0.6
0.4
’
’ ’ 0.6
’
’
’ ’ 0.8
1
’
1 1.0
X
Fig.3 Anion-cation nearest-neighbour distances (in A) calculated in thii work for Ki_,Rb,Br, RbBrr_,I,, and KCli_,Br, at z = l/8 and z = 7/8. Distances calculated with/without polarization effects are represented by dots/crosses. EXAFS data (open circles) are also indicated for comparison. Data for Kr_,Rb,Br and RbBrt_,I, are from Ref. 5 while those for KCli_,Br, are from Ref. 4.
Fig.4 Predicted heats of solution (in cal./mole of ion pairs) as a function of composition for the systems Kr_xRbxBr, RbBri_,I,, and KCli_,Br,. The points indicate the experimental values for KCli_,Br, quoted in Ref. 19 (solid circles), Ref. 20 (open squares), and Ref. 21 (open triangles). For the latter system the predictions obtained by neglecting simultaneously polarization effects and internal distortions (dotted curve) and neglecting only polarization effects (broken curve) are also represented.
sions and Van der Waals interactions) and using bigger supercells which should better simulate the real disordered systems. In particular the study of bigger supercells will allow one to determine the statistical distribution of the AC and BC bond-length values around their average and in turn to analyse and interpret the line shapes of EXAFS spectra.
References
1) The 2)
1.0
I
K Clr_xBrx 1
0.0
X
I
3.1 0.0
1.0
subject has recently been reviewed by V. Hari Babu and U. V. Subba Rao, Prog. Crystal Growth and Charact. 8, 189 (1984). L. Vegard, 2. Physik 5, 1’7 (1921).
3) J. B. Boyce and J. C. Mikkelsen, in EXAFS and Near Edge Structure III, edited by K. 0. Hodgson, B. Hedman and J. E. Penner-Hahn (SpringerVerlag, Berlin, 1984), p.426.
Vol. 63, No. 2
ATOMIC-SCALE STRUCTURE OF ALKALIHALIDE SOLID SOLUTIONS
4) T. Murata, in EXAFS and Near Edge Structure III, edited by K. 0. Hodgzon, B. Hedman and J. E. Penner-Hahn (Springer-Verlag, Berlin, 1984), p.432. 5) J. B. Boyce and J. C. Mikkelsen, Phys. Rev. B31, 6903 (1985). 6) J. C. Mikkelzen and J. B. Boyce, Proc. 17th Int. Conf. on tie Physics ofSemiconductors (SpringerVerlag, New York, 1985), p.933. 7) A. Balzarotti, M. Czyzyk, A. Kisiel, N. Motta, M. Podgomy, and M. Zimnal-Stamawska, Phys. Rev. B30, 2295 (1984); Phys. Rev. B31, 7526 (1985); Proc. 17th Int. Coti on the Physics of Semiconductors (Springer-Verlag, New York, 1985), p.807. 8) N. Motta, A. Balzarotti, P. Letardi, A. KisieI, M. J. Czyzyk, M. Zimnal-Stamawska, and M. Podgomy, Solid State Commun. 53,509 (1985); Solid State Commtm. 55,413 (1985). 9) W. B. Pearson, The Crystal Chemistry and Physics of MetaJs and Alloys, (Wiley, Interscience), ch.lV and cited references. 10) T. Fukuy, Jpn. J. Appl. Phys. 23, L208 (1984); J. Appl. Phys. 57,5188 (1985). 11) P. Boguslawski and A. Baldereschi, Proc. 17th hL. Conf. on the Physics of Semiconductors (SpringerVerlag, New York, 1985), p.939.
12) G.
P. Srivzstava, J. L. Martins, and A. Zunger, Phys. Rev. B31, 2561 (1985) and references therein. 13) C. K. Shih, W. E. Spicer, W. A. Harrison, and A. Sher, Phys. Rev. B31, 1139 (1985). 14) J. R. Hardy and A. M. Karo, The Lattice Dynamics and Statics of Alkali-Halide Crystals (Plenum, New York, 1979). 15) M. P. Tosi in Solid State Physics, edited by F.Seitz and D . Tumbull (Academic Press, New York, 1984)) 10, p.1 . 16) F. W. de Wette, W. Kress, and U. Schroeder, Phys. Rev. B32,4143 (1985) and cited references. 17) M. P. Tosi and M. Doyama, Phys. Rev. 160, 716 (1967) and cited references. 18) J. R. Tessman, A. K. Kahn, and W. Shockley, Phys. Rev. 92, 890, (1953). 19) D. L. Fancher and G. R. Barsch, J. Phys. Chem. Solids 30, 2503 (1969) and cited references. 20) W. E. Wallace, J. Chem. Phys. 17,109s (1949). 21) H. H. Landolt and R. Boemstein, Numerical Data and Functional Relationships in Science and Te& nology, edited by K. H. Hellewege (Springer-Verlag, Berlin-Heidelberg-New York 1973), group II& 7a.
95
ATOMIC-SCALE
STRUCTURE
pp.91-95,
0038-1098187 $3.00 + .OO Pergamon Journals Ltd.
1987.
OF ALKALI
HALIDE
SOLID
SOLUTIONS
M. Peressi Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy A. Baldereschi Dipartimento di Fisica Teorica and GNSM-CNR Universita di Trieste, Italy and Institut de Physique Appliquke, EPF-Lausanne, Switzerland. (Received 10 April 1987 by E. Tosatti)
A simple model is proposed to explain the crystallographic properties of solid solutions of the alkali halides. The random system is simulated with a periodic superlattice whose energy is evaluated with the Born-Mayer model generalized to include ionic polarization. The resulting average lattice parameter and anioncation distances agree with recent EXAFS data on Kl_,Rb,Br, RbBrl_,I,, and KCll-,Br,. The results show that ionic polarization plays a significant role in the determination of both interatomic distances and heats of formation.
sion of the model, i.e. we neglect all second-neighbour and Van der Waals interactions, but we include the effects of ionic polarization which have been found to be important in analogous circumstances of low atomicsite symmetry as for example in the presence of isolated defects or impurities and lattice vibrations 14), and at crystal surfaces ‘e). Ionic polarization plays also an important role in the cohesion of alkali-halide molecules 17) where again the symmetry is lower than that of the solid.
The crystallographic structure of solid solutions A1-,B,C of alkali halides has been investigated with X-ray diffraction and more recently with the EXAFS technique. X-ray data I) show that in most pseudobinary solutions the lattice parameter follows Vegard law ‘), i.e. it varies linearly with composition between the values for the AC and BC end compounds. Accurate EXAFS data s)*4)*5)are available for Kl_,Rb,Br, RbBrl-,I,, and KCll_,Br,. They indicate that the nearest neighbour distances have a bimodal distribution similar to that already observed in semiconducting alloys ‘)*“)**), and their value is intermediate between that of the pure compound and the average Vegard value. In Kl_,R&Br 5), for example, the NN distances r(K-Br] and r[Rb-Br] vary linearly with composition and their total variation is about 40% of the difference between NN distances in the pure compounds, i.e. the variation is more pronounced than that observed in semiconducting alloys.
Although the distribution of A and B ions in the homogeneous Al_,B,C solid solution is random, we simulate the alloy with a periodically repeated supercell. Since the macroscopic symmetry of the real alloy is cubic, we choose a cubic supercell and distribute in it anions and cations according to cubic symmetry. With these criteria a given cell allows only for particular compositions x and symmetry-restricted atomic displacements from the ideal sites of the rocksalt virtual crystal.
The local structure of metallic ‘) and semiconducting 7)*10)-13) pseudobinary alloys has been the subject of several theoretical investigations. These studies however do not apply to ionic solid solutions which are substantially different both in lattice structure and in type of binding. For ionic materials, only the case of infinite dilution, i.e. the isolated impurity limit, has received some attention in the past 13)*14).
Iu the present work we use the smallest cubic supercell which allows for atomic distortions, i.e. the FCC unit cell with 16 ions (Fig. 1). The ions within the supercell can be grouped into shells of symmetryequivalent ions. The sublattice centered at the origin contains three shells with 1,6, and 1 ions, respectively, while the other sublattice contains two shells with 6 and 2 ions, respectively. According to our filling criteria, each shell can contain only one kind of ions, so that the supercell allows only for the compositions z = l/8, z = l/4, and the complementary concentrations z = 718 and z = 3/4. The fillings with composi-
In thii work we intend to verify if the Born-Mayer model, which has successfully explained the cohesive properties of pure alkali halides 15), can be applied to the atomic-scale structure of alkali-halide solid solutions and their heat of mixing. We use the simplest ver91
92
Vol.
ATOMIC-SCALE STRUCTURE OF ALKALI HALIDE SOLID SOLUTTONS
+ BAC
2 =P I
Fig.1 Wigner-Seitz unit cell of the FCC superlattice considered in the present work. The unit cell contains 16 atoms. Non-equivalent atomic sites are indicated with different symbols.
tions z = l/4 and z = 314 will not be considered any further since in this case the supercell reduces to twice the SC cell with 8 ions, and all atomic displacements from ideal sites are symmetry forbidden. For the compositions z = l/8 and z = 718 only the six ions which are octahedrally coordinated to the central one can have a common radial relaxation 6 so that the crystallographic structure is fully described by the lattice constant a and the internal distortion parameter c = 6/a. The lattice energy at zero temperature and pressure is evaluated with the Born-Mayer model. Neglecting all interactions beyond nearest - neighbours, the Van der Waals interactions, and the zerepoint kinetic energy of the ions, the lattice energy per unit supercell is the sum of three terms
Lp.(%4 + qd.(~,~).
(1)
The first term is the Madelung energy
where o(c is a diitortion-dependent Madelung function (Fig. 2) which for E = 0 has the value a(0) = 55.922 which corresponds to the well known Madelung constant properly scaled to the dimension of our supercell. The second term is the repulsive energy between A-C and B-C nearest-neighbour pairs. For z = l/8 it can be written
63, No. 2
(--a/‘lpAC)
I
-
\
and a similar expression obtained by exchanging the A and B indices holds for z = 718. The parameters BAC,BBC,PAC, and p~c are adjusted to reproduce the lattice constant and bulk modulus of the pure AC and BC compounds. We use the values given in the first column of Table VIII of Ref. 15 for all materials except RbBr where we use the values B = 2.61* 10s8 erg and p = 0.324 A which reproduce the NN distance of 3.427 A asmeasured by EXAFS 51. The last term in equation (1) describes the main effects of the ionic p+ larization induced by the distortion of the lattice and contains charge-dipole interactions only. Considering that only the C ions can polarize, Ed can be written iiS:
Epd.(a,e) =
-9,
(4)
where pc is the polarizability of the C ions, whose value is taken from Ref. 18, and f(c) is a distortion dependent structural sum. Thii term vanishes for an undistorted lattice, i.e. f(0) = 0, but lf the ions move away from the ideal sites, they experience a Madelung field and polarize. Only those ions which do not occupy the ideal sites of the virtual crystal can polarize. The Madelung function Q(C) and the polarization function f(c) have been computed with the Ewald,method r51. It is convenient to introduce the following analytic approximate expressions for both a(s) and f(c) which are valid for Ic( < .005 : Q(E)=
55.922 - 693.08e2,
(5)
f(c) =
1.028 * 10’ c2.
(6)
The equilibrium values of the lattice constant o(z) and of the internal distortion parameter e(z) of Kr_,Rb,Br, RbBrr_,I, and KClr_,Br, have been computed for z = l/8 and z = 718 by minimizing the total lattice energy. In order to understand the releVance of the ionic polarization we have also computed the equilibrium values of a(z) and c(z) by neglecting the polarization contribution. The resulting values of a(z) and c(z) are given in Fig.3 and compared with experiment. The calculated values of the heat of mlxii are given in Fig.4 . From Fig.3 we see that the lattice constant of the supercell follows quite closely Vegard law. In the unit supercell several NN distances are present at equilibrium because of the displacements of the six ions near the center of the cell. After weighted average we obtain A-C and B-C distances which are closer to the ra
Vol. 63, No. 2
ATOMIC-SCALE STRUCTURE OF ALRALI HALIDE SOLID SOLUTIONS
-0.006
l0.003
-0.003
93
~0.008
;
Fig.2 Madehmg constant a(e) as function of the internal distortion parameter. Dots indicate the results of computations while the full line represents the quadratic approximation given in the text.
spective distances in the pure AC and BC compounds than to the virtual crystal NN distance, in accordance with EXAFS data. The agreement is very good for Kr_,Rb,Br, whereas for RbBrr_,I, and KClr_,Br, the calculated internal distortions are smaller than those measured experimentally. We stress that the diitortion of the C sublattice is induced by the difference between the repulsive forces of A-C and B-C pairs, but the equilibrium A-C and EC bond lengths are determined by all forces acting on the ions. Theoretical values of the heats of formation (see Fig. 4) can only be compared with experiment for KClr_,Br, and in thii case good agreement is obtained. The calculations performed neglecting the chargedipole interaction energy in es.(l), show that polarization effects are not negligible. As can be seen in Fig.3, these effects are not very important for the lattice parameter, but they increase significantly the internal distortion and bring the calculated anion-cation distances in closer agreement with experiment. Polarization effects are specially important in cationic alloys such as Kr_,Rb,Br where the displaced ions are the highly polarizable Br- anions. The agreement with EXAFS data is very good for this system. Anionic alloys polarize less and the inclusion of charge-dipole interactions is not sufficient to reproduce the experimental data. Ionic polarization also reduces the values of the heats of formation as can be seen for KCli-,Br, in Fig.4 .
For curiosity we have also minimized the lattice energy neglecting both polarization effects and inter-nal distortions. The resulting theoretical predictions are very poor. The bimodal distribution of NN distances is of course absent, and a(z) deviates considerably from both Vegard and Retgers laws (additivity of distances and of volumes, respectively). The heats of formation are much larger than experimental data (see Fig. 4). These simple computations however allow one to understand the importance of anharmonic forces in alkali-halide solid solutions. Separating the lattice energy E(o, B = 0) into harmonic and anharmonic terms we have evaluated how much each one contributes to the small deviations of e(z) from Vegard law. Harmonic terms alone give the wrong sign for the deviations, (i.e. a sub-linear behaviour), and only with the inclusion of the anharmonic terms one obtains the experimentally observed positive bowing. Including anharmonic terms, positive deviations from Vegard law are predicted for all alkali-halide solid solutions. The supercell used in this work is the smallest cubic cell which can have internal diitortionz. Its A-B sublattice however cannot distort and only a fraction of C ions are allowed by symmetry to move offsite. Nevertheless with this simple supercell and a simple version of the Born-Mayer model we have obtained satisfactory agreement with experimental data. The encouraging results obtained here suggest that the local structure of alkali-halide solid solutions can be fully explained by using more refined versions of the BomMayer model (i.e. including second-neighbour repul-
Vol. 63, No. 2
ATOMIC-SCALE STRUCTURE OF &KALE HALIDE SOLID SOLUTTONS
9c
I'I',',',',',',','m
K,.x Rb,Rr
-
1 I I , I , I , I , , , , , , , , , 0.0
0.2
0.4
0.6
1.0
0.8
X
8 L
2
200
100
0 0.0
0.2
3.3
I
I
0.4
’
I
’
I
0.5
X
’
I
0.8
r
I
’
I
*
1
’
I
1
1 1 0.2
1
1
1 1 0.4
1
02
0.6
0.4
’
’ ’ 0.6
’
’
’ ’ 0.8
1
’
1 1.0
X
Fig.3 Anion-cation nearest-neighbour distances (in A) calculated in thii work for Ki_,Rb,Br, RbBrr_,I,, and KCli_,Br, at z = l/8 and z = 7/8. Distances calculated with/without polarization effects are represented by dots/crosses. EXAFS data (open circles) are also indicated for comparison. Data for Kr_,Rb,Br and RbBrt_,I, are from Ref. 5 while those for KCli_,Br, are from Ref. 4.
Fig.4 Predicted heats of solution (in cal./mole of ion pairs) as a function of composition for the systems Kr_xRbxBr, RbBri_,I,, and KCli_,Br,. The points indicate the experimental values for KCli_,Br, quoted in Ref. 19 (solid circles), Ref. 20 (open squares), and Ref. 21 (open triangles). For the latter system the predictions obtained by neglecting simultaneously polarization effects and internal distortions (dotted curve) and neglecting only polarization effects (broken curve) are also represented.
sions and Van der Waals interactions) and using bigger supercells which should better simulate the real disordered systems. In particular the study of bigger supercells will allow one to determine the statistical distribution of the AC and BC bond-length values around their average and in turn to analyse and interpret the line shapes of EXAFS spectra.
References
1) The 2)
1.0
I
K Clr_xBrx 1
0.0
X
I
3.1 0.0
1.0
subject has recently been reviewed by V. Hari Babu and U. V. Subba Rao, Prog. Crystal Growth and Charact. 8, 189 (1984). L. Vegard, 2. Physik 5, 1’7 (1921).
3) J. B. Boyce and J. C. Mikkelsen, in EXAFS and Near Edge Structure III, edited by K. 0. Hodgson, B. Hedman and J. E. Penner-Hahn (SpringerVerlag, Berlin, 1984), p.426.
Vol. 63, No. 2
ATOMIC-SCALE STRUCTURE OF ALKALIHALIDE SOLID SOLUTIONS
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