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Pressure dependence of effective ionic charges in alkali halides
J. Phys. Chem. Solids Vol. 52, No. 2. pp. 435436, Printed
in Great
0022-3697/91 $3.00 + 0.00 6 1991 Pergamon Prew plc
1991
Britain.
PRESSURE DEPENDENCE OF EFFECTIVE CHARGES IN ALKALI HALIDES
IONIC
W. KUCHARCZYK Institute of Physics, Technical University, W6lczadska 219, 93-005 L6di, Poland (Received 25 July 1990; accepted 5 September 1990)
Abstract-An estimation of the pressure dependence of the Szigeti effective ionic charge in alkali halides is presented. In the method the empirical connection between electronic and lattice ionic&y proposed by Lawaetz is taken into account. The connection links e: with the ratio of Phillips’ electronegativity difference and the plasma energy of a free-valence-electron gas. Changes in the dielectric electronegavity parameter are related to the photoelastic effect. Keywords: Szigeti effective charge, alkali halides, photoelastic
effect.
INTRODUCTION
METHOD
It is known that the charge of ions in crystals differs from its formal value related to valence. The deviation is often described in terms of the Szigeti effective charge [l]
Lawaetz [3] has pointed out that the Szigeti effective charge can be roughly related to the ratio
(e:)* = %(~o--~m) 4IIN(c,
+ 2)2 &’
(1)
where p is the reduced mass of an ion pair, N is the density of ion pairs, E,, and E, are low- and high-frequency dielectric constants, respectively, and wr is the transverse optic mode frequency. The definition of e,* involves local field effects expressed by the Lorentz factor. The effective ionic charge is of great interest in investigations of the dynamics of crystal lattices, and is connected to the splitting of the triply-degenerate optic mode of a diatomic cubic crystal into longitudinal and transverse type modes. The experimental values of e: in the alkali halides are known with an accuracy of 1% from the measurements of Lowndes and Martin [2]. Since the effective charge is related to many different phenomena in crystals, e,* has been studied within several different theoretical approaches. For example, Lawaetz [3] has found a simple empirical relationship between e: and electronic ionicity in the framework of the Phillips-Van Vechten theory. Ab initio calculations of e,* have been performed with the use of the local-density-approximation by Mahan [4]. In alkali halides an interesting correlation between effective ionic charge and interatomic separation has also been reported [5]. The aim of the present paper is to calculate changes in the Szigeti effective ionic charge due to hydrostatic pressure in alkali halides. 435
e: 2: C/ho,.
(2)
where C is the heteropolar part of the effective energy gap as defined by Phillips [6] and u+, is the effective plasma frequency. The parameter C is related to the ionic binding, and in the alkali halides can be expressed as C = be*(l/r,
-
7/r,)exp(-k,R/2),
(3)
where r, and r, are the cation and anion radii, respectively, k, is the Thomas-Fermi screening wave number, R is the nearest-neighbour distance and b is a coefficient compensating for the deficiency of the free-electron model [6-81. The parameter b is introduced to account for the fact that the linearized Thomas-Fermi screening factor, being a longwavelength approximation, will overestimate the screening on the scale of the bond length. The screening wave number k, can be written as k, =
2(a,)“2(3N/II)“6,
(4)
where a, is the Bohr radius and N is the number of valence electrons per unit volume. Equation (4) shows that the factor k, is proportional to R-U* and wp can be written in the form 0: = Ne*jc,m.
(5)
Here co is the permittivity of free space, e and m are the electron charge and mass, respectively. According
436
W. KUCHARCZYK Table 1. Pressure dependence of the Szigeti effective ionic charge in alkali halides; the two different experimental values reported for LiF and NaF correspond to different input data de:/dP(10e3 Theory Crystal
This work
Ref. 14
-0.28 -0.78 -0.95 -0.47 -1.1 -1.3 -1.9 -0.93 -1.8 -2.1 -2.8 -1.2 -2.1 -2.6 -3.3
-0.60
LiF LiCl LiBr NaF NaCl NaBr NaI KF KC1 KBr KI RbF RbCl RbBr RbI
to eqn (5) up varies as one can obtain de:/dR
= ef[l/ZR
R -312. Employing
Experiment
Ref. 13
Ref. 13
-0.48
-0.52
-0.90
-0.35
-1.16
-0.81
-1.35
-1.28
-3.27 -4.95
-2.23 -2.45
-2.6
-3.4 -4.6 -7.15
-1.87 -1.89
-3.5 -4.9
eqns (1 HS)
-lc,/4 + (l/b)(db/8R)].
(6)
From eqn (6) it follows that de:/dP
kbar-‘)
= e,*[k,R/4 - l/2 - (R/b)(~%/dR)]/3&
(7)
where B is the isothermal bulk modulus. Apart from (l/~)(~~/~R), the parameters occurring in eqn (7) are known independently. Previously, we have proposed a method of evaluating ionic radii from the observed optical susceptibilities [9, lo]. The approach has been extended to study photoelastic phenomena in alkali halides [l I]. Taking into account the experimental values of the hydrostatic photoelastic constants p,r + 2pn we have found that, to a first approximation, the factor (l~~)(~~~~R) can be treated as a constant within the aikali halides, and that (i/b)(ab/dR) 2:0.701 [ll].
etical values of de,*/dP are also included in Table 1. The second column gives de:/dP obtained from {~/e~)(~e~~~~) calculated in 1141on the basis of a Born-Mayer potential. The third column gives de:/dP evaluated in [13] employing the Born-Mayer potential. The fourth column presents de,*/dP estimated in [13] on the assumption that (eZ)2/V is constant with respect to pressure. In conclusion, the empirical relationship given by eqn (2), despite its simplicity, seems capable of describing the pressure dependence of ef and leads to reasonable agreement with experiment. Acknowledgement-This
work was carried Research Project CPBP 01.06,6.04.
REFERENCES 1. Szigeti B., Trmrs Faraday Sot. 45, 155 (1949). 2. Lowndes R. P. and Martin D. H., Proc. R. Sac. A308,
473 (1969). 3. Lawaetz P., Phys. Rev. Len. 26, 697 (1971). 4. Mahan G. D., Phys. Rev. B34, 4235 (1986). 5. Niwas R., Goyal S. C. and Shanker J., J. Phys. Chem.
RESULTS The values of de:/dP derived for alkali halides using the factor (l/b)(~%jaR) = 0.701 and values of B obtained using data from [12] are listed in the first column of Table 1. Our results predict that def/dP decreases with pressure, in agreement with experiment [13]. The experimental values have been evaluated from the pressure dependence of the longwavelength optic phonon frequencies and the highand low-frequency dielectric constants. The two different experimental values reported for LiF and NaF correspond to different input data. Other theor-
out under
6. 7. 8. 9. 10.
Solids 36, 219 (1977). Phillips J. C., Rev. mod. Phys. 42, 317 (1970). Van Vechten J. A., Phys. Rev. 182, 891 (1969). Levine B. F., J. them. Phys. 59, 1463 (1973). Kucharczyk W., Acta crystailogr. B43, 454 (1987). Kucharczyk W., J. Phys. Chem. Solids So, 233 (1989).
Il. Kucharczyk W., J. Phys. Chem. Solids 50, 709 (1989). Numerical Data and Functional 12. Landolt-Bornstein, Relationships in Science and Technology, New
Series,
Group III, Vol. 11. Springer, Berlin (1979). 13. Mitra S. S. and Namjoshi K. V., J. them. Phys. 55, 1817 (1971). 14. Lowndes R. P. and Martin D. H., Pruc. R. Sot. A316, 3.51 (1970).
in Great
0022-3697/91 $3.00 + 0.00 6 1991 Pergamon Prew plc
1991
Britain.
PRESSURE DEPENDENCE OF EFFECTIVE CHARGES IN ALKALI HALIDES
IONIC
W. KUCHARCZYK Institute of Physics, Technical University, W6lczadska 219, 93-005 L6di, Poland (Received 25 July 1990; accepted 5 September 1990)
Abstract-An estimation of the pressure dependence of the Szigeti effective ionic charge in alkali halides is presented. In the method the empirical connection between electronic and lattice ionic&y proposed by Lawaetz is taken into account. The connection links e: with the ratio of Phillips’ electronegativity difference and the plasma energy of a free-valence-electron gas. Changes in the dielectric electronegavity parameter are related to the photoelastic effect. Keywords: Szigeti effective charge, alkali halides, photoelastic
effect.
INTRODUCTION
METHOD
It is known that the charge of ions in crystals differs from its formal value related to valence. The deviation is often described in terms of the Szigeti effective charge [l]
Lawaetz [3] has pointed out that the Szigeti effective charge can be roughly related to the ratio
(e:)* = %(~o--~m) 4IIN(c,
+ 2)2 &’
(1)
where p is the reduced mass of an ion pair, N is the density of ion pairs, E,, and E, are low- and high-frequency dielectric constants, respectively, and wr is the transverse optic mode frequency. The definition of e,* involves local field effects expressed by the Lorentz factor. The effective ionic charge is of great interest in investigations of the dynamics of crystal lattices, and is connected to the splitting of the triply-degenerate optic mode of a diatomic cubic crystal into longitudinal and transverse type modes. The experimental values of e: in the alkali halides are known with an accuracy of 1% from the measurements of Lowndes and Martin [2]. Since the effective charge is related to many different phenomena in crystals, e,* has been studied within several different theoretical approaches. For example, Lawaetz [3] has found a simple empirical relationship between e: and electronic ionicity in the framework of the Phillips-Van Vechten theory. Ab initio calculations of e,* have been performed with the use of the local-density-approximation by Mahan [4]. In alkali halides an interesting correlation between effective ionic charge and interatomic separation has also been reported [5]. The aim of the present paper is to calculate changes in the Szigeti effective ionic charge due to hydrostatic pressure in alkali halides. 435
e: 2: C/ho,.
(2)
where C is the heteropolar part of the effective energy gap as defined by Phillips [6] and u+, is the effective plasma frequency. The parameter C is related to the ionic binding, and in the alkali halides can be expressed as C = be*(l/r,
-
7/r,)exp(-k,R/2),
(3)
where r, and r, are the cation and anion radii, respectively, k, is the Thomas-Fermi screening wave number, R is the nearest-neighbour distance and b is a coefficient compensating for the deficiency of the free-electron model [6-81. The parameter b is introduced to account for the fact that the linearized Thomas-Fermi screening factor, being a longwavelength approximation, will overestimate the screening on the scale of the bond length. The screening wave number k, can be written as k, =
2(a,)“2(3N/II)“6,
(4)
where a, is the Bohr radius and N is the number of valence electrons per unit volume. Equation (4) shows that the factor k, is proportional to R-U* and wp can be written in the form 0: = Ne*jc,m.
(5)
Here co is the permittivity of free space, e and m are the electron charge and mass, respectively. According
436
W. KUCHARCZYK Table 1. Pressure dependence of the Szigeti effective ionic charge in alkali halides; the two different experimental values reported for LiF and NaF correspond to different input data de:/dP(10e3 Theory Crystal
This work
Ref. 14
-0.28 -0.78 -0.95 -0.47 -1.1 -1.3 -1.9 -0.93 -1.8 -2.1 -2.8 -1.2 -2.1 -2.6 -3.3
-0.60
LiF LiCl LiBr NaF NaCl NaBr NaI KF KC1 KBr KI RbF RbCl RbBr RbI
to eqn (5) up varies as one can obtain de:/dR
= ef[l/ZR
R -312. Employing
Experiment
Ref. 13
Ref. 13
-0.48
-0.52
-0.90
-0.35
-1.16
-0.81
-1.35
-1.28
-3.27 -4.95
-2.23 -2.45
-2.6
-3.4 -4.6 -7.15
-1.87 -1.89
-3.5 -4.9
eqns (1 HS)
-lc,/4 + (l/b)(db/8R)].
(6)
From eqn (6) it follows that de:/dP
kbar-‘)
= e,*[k,R/4 - l/2 - (R/b)(~%/dR)]/3&
(7)
where B is the isothermal bulk modulus. Apart from (l/~)(~~/~R), the parameters occurring in eqn (7) are known independently. Previously, we have proposed a method of evaluating ionic radii from the observed optical susceptibilities [9, lo]. The approach has been extended to study photoelastic phenomena in alkali halides [l I]. Taking into account the experimental values of the hydrostatic photoelastic constants p,r + 2pn we have found that, to a first approximation, the factor (l~~)(~~~~R) can be treated as a constant within the aikali halides, and that (i/b)(ab/dR) 2:0.701 [ll].
etical values of de,*/dP are also included in Table 1. The second column gives de:/dP obtained from {~/e~)(~e~~~~) calculated in 1141on the basis of a Born-Mayer potential. The third column gives de:/dP evaluated in [13] employing the Born-Mayer potential. The fourth column presents de,*/dP estimated in [13] on the assumption that (eZ)2/V is constant with respect to pressure. In conclusion, the empirical relationship given by eqn (2), despite its simplicity, seems capable of describing the pressure dependence of ef and leads to reasonable agreement with experiment. Acknowledgement-This
work was carried Research Project CPBP 01.06,6.04.
REFERENCES 1. Szigeti B., Trmrs Faraday Sot. 45, 155 (1949). 2. Lowndes R. P. and Martin D. H., Proc. R. Sac. A308,
473 (1969). 3. Lawaetz P., Phys. Rev. Len. 26, 697 (1971). 4. Mahan G. D., Phys. Rev. B34, 4235 (1986). 5. Niwas R., Goyal S. C. and Shanker J., J. Phys. Chem.
RESULTS The values of de:/dP derived for alkali halides using the factor (l/b)(~%jaR) = 0.701 and values of B obtained using data from [12] are listed in the first column of Table 1. Our results predict that def/dP decreases with pressure, in agreement with experiment [13]. The experimental values have been evaluated from the pressure dependence of the longwavelength optic phonon frequencies and the highand low-frequency dielectric constants. The two different experimental values reported for LiF and NaF correspond to different input data. Other theor-
out under
6. 7. 8. 9. 10.
Solids 36, 219 (1977). Phillips J. C., Rev. mod. Phys. 42, 317 (1970). Van Vechten J. A., Phys. Rev. 182, 891 (1969). Levine B. F., J. them. Phys. 59, 1463 (1973). Kucharczyk W., Acta crystailogr. B43, 454 (1987). Kucharczyk W., J. Phys. Chem. Solids So, 233 (1989).
Il. Kucharczyk W., J. Phys. Chem. Solids 50, 709 (1989). Numerical Data and Functional 12. Landolt-Bornstein, Relationships in Science and Technology, New
Series,
Group III, Vol. 11. Springer, Berlin (1979). 13. Mitra S. S. and Namjoshi K. V., J. them. Phys. 55, 1817 (1971). 14. Lowndes R. P. and Martin D. H., Pruc. R. Sot. A316, 3.51 (1970).