- Email: [email protected]
Overlap polarization and lattice dynamics of ionic crystals
Solid State Communications,Vol. 16, pp. 1023—1026, 1975.
Pergamon Press.
Printed in Great Britain
OVERLAP POLARIZATION AND LATFICE DYNAMICS OF IONIC CRYSTALS H. Bilz, M. Buchanan, K. Fischer and R. Haberkorn Max-Planck-Institut für Festkorperforschung,7 Stuttgart 1, Heilbronnerstr. 69, W. Germany and U. Schroder Fachbereich Physik, Universität Regensburg 84 Regensburg, W. Germany (Received 3 December 1974 by M Cardona)
The lattice dynamics of cubic ionic crystals is re-investigated. The consideration of overlap polarization justifies the introduction of a positively charged shell at the site of the nearly unpolarizable cation in a simple shell model, by giving it a physical basis as a representation of this polarization. The model gives a satisfactory description of the dispersion curves of many cubic ionic crystals with only six or seven parameters.
IN THE LATTICE dynamics of ionic crystals dipolar models (such as the shell model and its extensions) have been successfully used for the description of dispersion curves.1 From the point of view of the Lowdin—Lundqvist microscopic theory,2 these models are missing the overlap polarization complement of the rigid overlap (Born—Mayer type) potential.3 The problem was mentioned by Dick and Marston4 and by Cowley eta!.5 in their discussion of the shell model for alkali halides. Here, we introduce a model description of the overlap polarization. The argument is illustrated in Fig. 1. We take advantage of the fact that in many diatomic ionic crystals the “centre of gravity” of the overlap region lies close to the midpoint of the positive ion, as the positive ion radius is usually much less than that of the negative ion [Fig. 1(a)] Thus, the effect of the overlap polarization is best described in the simplest model as being centred at the positive ion. This corresponds to the introduction of “pseudo-deformabilities” for the positive ion. In Fig. 1 (b), we show the effect on the overlap regions of a displacement of the positive ion. In addition to the long and short range .
1023
dipoles which are induced at the negative ion sites by the positive ion displacement, due to the free ion polarizability, there exists a redistribution of charge in the overlap regions. Charge is repelled from the regions where the overlap is increased, as indicated in Fig. 1. In this simplest approximation, the redistribution has a dipole character, and corresponds to a flow of positive charge in the direction opposite to the motion of the positive ion. The model proposed here is very similar to the model VI of Cochran, Cowley, Brockhouse and Woods.5 Their discussion assumed, however, that the positive charge was simulating a deformabiity of the negative ion, which led to the development of the breathing shell model.6 A more detailed discussion has been given by Cochran.’ He discusses the implications of model VI (including a discussion of the positive shell charge), the breathing shell model, and charge transfer (rigid overlap model).2 The three-body forces in the charge transfer model, while able to account for the deviation from the Cauchy relation, are not able to give the observed lowering of the LO branch at L.7
OVERLAP POLARIZATION AND LATTICE DYNAMICS OF IONIC CRYSTALS Vol. 16, No.8
1024
8
l~o]
(~~1 Na!
6-
K-
1
—
loot]
V. IN ;4 o z
_____
Ui
a
Ui LL
OVERLAP CHARGES
(a)
I
c—
0
0.5
REDUCED WAVE VECTOR
FIG. 2. Dispersion curves of Nal in the three high
IONIC MOTION
symmetry directions. The experimental points are taken from reference 5. The solid line is given by a 6-parameter overlap shell model and the dashed line by an 8-parameter breathing shell. The values of the parameters are given in Table 1. ELECTRONIC CHARGE FLOW
A and B (non-central), the ionic polarizabiity of the negative ion represented by the shell charge Y_ and the shell—core interaction constant K_, and, in addition, the overlap polarizability, represented by a shell charge Y÷at the positive ion and a shell—core force constant K+; for some crystals, Z must be varied from one (or
FIG. 1. (a) Regions of overlap (charge depletion) shown clustered about the small, almost unpolarizable, positive ions. The parameters describing the polarizability in our model are indicated. (b) Effect of the overlap region during the displacement of a positive ion. There is a net charge shift in the same direction as the displacement, or correspondingly, a shift of positive charge in the opposite direction.
two). These parameters are fitted to neutron data, not to macroscopic entities (three elastic constants, two dielectric constants, and the infrared resonance fre-
Table 1. The values of the parameters used for the curves shown in Fig. 2. The notation is similar of that of reference 5 A BSM OSM
10.52 11.8
B+B”
A’
--1.01 1.47 —1.14 —
B’
Z
—0.025 0.913 1 —
A meaningful physical model ought not to be required to fit neutron measurements to better than a few per cent, as anharmonic effects, for example, are ignored. It should be mentioned, with respect to fitting the LO branches, that these vibrations are often nearly degenerate with their own two phonon decay states, leading to shifts of frequency of a few per cent,8 not given in a harmonic model. For a sensible description of the dispersion curves, then, we find that our model requires only six or seven parameters: two nearest neighbour force constants
Y.
K+
Y
K
G
54.5
—3.36 —4.16
123.6 227.5
81.2
1.45
—
quency). One should be cautious about using the elastic constants for macroscopic fits, as they are usually determined as isothermal of adiabatic at room temperature, and they can deviate by as much as 10—30 per cent from the extrapolated values of the neutron scattering data in some crystals.9’° Since the equations of motion for this model are equivalent to those of a simple shell model with just these parameters, there exist several fairly good calculations in the literature (for NaBr,9 KC1,11 RbBr,12
Vol. 16, No.8
OVERLAP POLARIZATION AND LATTICE DYNAMICS OF IONIC CRYSTALS
and CsCl 13) using, in effect, just this model. The success of these fits, which were often thought to be unphysical because of the positive shell charge, is no longer surprising in the light of the re-interpretation, In addition, many calculations exist using a positive shell charge where the situation is somewhat masked by the use of many parameters. We investigated a variety of systems (KBr, Nal, MgO and CaO) and found good agreement with this overlap shell model using six or seven parameters. As an example, we show in Fig. 2 the result of our re-investigation of Nal. The parameters of the two models illustrated are given in Table 1. An eight parameter breathing shell model (with second neighbour forces) fitted to neutron data (dashed curve), is needed to give about the same fit as a six parameter fit fitted overlap shell model (solid curve).* Several points about the model should be mentioned, First, if the static ionic charge Z is treated as a free parameter, it tends to obtain values close to one in the alkali halides or two in the alkali earth oxides. If the overlap polarization is introduced, it is found that second neighbour forces are less important than is found in an extended shell model or a breathing shell model: compare, for example the size of the second neighbour force constants of model VI of reference S to those in the breathing shell model given in Table 1. By comparison with the breathing shell model, the description of the elastic constants
1025
is much simpler, as second neighbour forces are not required to correct for the influence of the breathing deformabiities, and the use of a positive shell rather than breathing means that the deviation from the Cauchy relation is not related to the lowering of the LO frequency at L. Furthermore, it thus avoids the anomaly in the LO branch near the X point that the breathing shell model exhibits. In order to check the reliability of the eigenvectors given by the overlap shell model, we recalculated the second-order Raman spectra of MgO (given by a single eigenvector-dependent term14). The eigenvectors of a given parameter overlap shell model gave a result which was as good as the previous calculation using the eigenvectors of the eight parameter breathing shell model D of Sangster et al. ~ To summarize, a simple overlap shell model is well able to describe the dispersion curves of alkali halides including those with cesium chloride structure, and the dispersion curves of the alkaline earth oxides. It avoids the problem the breathing shell model has as a result of the introduction of second neighbour forces, and yields good fits to measured dispersion curves without the introduction of a large number of parameters as in the usual extended shell model. *
A five parameter overlap shell model, with the sum of the two polarizabilities given by the Clausius— Mosotti relation, yields nearly the same fit, although the implications of the model are not consistent with this restriction.
REFERENCES I.
For a review refer to COCHRAN W., CriticalRev. Solid State Sci. 2, 1(1971).
2. 3.
LOWDIN P.O.,Adv. Phys. 5, 1(1956); LUNDQVIST S.O.,Ark. Fysik. 6,25 (1952); 9,435 (1955); 12, 263 (1957). GLISS B., ZEYHER R. and BILZ H.,Phys. Status Solidi (b) 44, 910 (1970).
4.
DICK B.G.,Phys. Rev. 129, 1583 (1963); MARSTON R.L. and DICK B.G.,Solid State Commun. 5,731(1967).
5.
COWLEY R.A., COCHRAN W., BROCKHOUSE B.N. and WOODS A.DB.,Phys. Rev. 131, 1030 (1963).
6. 7.
SCHRODER U.,Solid State Commun. 4, 347 (1966). ZEYHER R.,Phys. Status Solidi (b) 48, 711(1971).
8. 9.
COWLEY R.A.,Adv. Phys. 12,421(1963). REID J.S., SMITH T. and BUYERS W.T.L.,Phys. Rev. BI, 1833 (1970).
10.
KRESS W., Phys. Status Solidi (b) 62, 403 (1974).
1026
OVERLAP POLARIZATION AND LATTICE DYNAMICS OF IONIC CRYSTALS Vol. 16, No.8
11.
COPLEY J.R.D., MACPHERSON R.W. and TIMUSK T., Phys. Rev. 182,965 (1969).
12.
ROLANDSON S. and RAUNIOG.,J. Phys. C4, 958 (1971).
13.
ROLANDSON S. and RAUNIO G.,Phys. Rev. B4, 4617 (1971).
14.
HABERKORN R., BUCHANAN M. and BILZ H.,Solid State Commun. 12,681(1973).
15.
SANGSTER M.J.L., PECKHAM G. and SAUNDERSON D.H.,J. Phys. C3, 1026 (1970).
Note added in proof We have been informed by Dr. R. Bauer, MPI fuer Metall forschung, Stuttgart, about his recent calculations on overlap forces in simple alkali halides. As a result, second nearest neighbour forces between cations seem to be very weak due to the cancellation of repulsive and attractive parts in the potential.
Pergamon Press.
Printed in Great Britain
OVERLAP POLARIZATION AND LATFICE DYNAMICS OF IONIC CRYSTALS H. Bilz, M. Buchanan, K. Fischer and R. Haberkorn Max-Planck-Institut für Festkorperforschung,7 Stuttgart 1, Heilbronnerstr. 69, W. Germany and U. Schroder Fachbereich Physik, Universität Regensburg 84 Regensburg, W. Germany (Received 3 December 1974 by M Cardona)
The lattice dynamics of cubic ionic crystals is re-investigated. The consideration of overlap polarization justifies the introduction of a positively charged shell at the site of the nearly unpolarizable cation in a simple shell model, by giving it a physical basis as a representation of this polarization. The model gives a satisfactory description of the dispersion curves of many cubic ionic crystals with only six or seven parameters.
IN THE LATTICE dynamics of ionic crystals dipolar models (such as the shell model and its extensions) have been successfully used for the description of dispersion curves.1 From the point of view of the Lowdin—Lundqvist microscopic theory,2 these models are missing the overlap polarization complement of the rigid overlap (Born—Mayer type) potential.3 The problem was mentioned by Dick and Marston4 and by Cowley eta!.5 in their discussion of the shell model for alkali halides. Here, we introduce a model description of the overlap polarization. The argument is illustrated in Fig. 1. We take advantage of the fact that in many diatomic ionic crystals the “centre of gravity” of the overlap region lies close to the midpoint of the positive ion, as the positive ion radius is usually much less than that of the negative ion [Fig. 1(a)] Thus, the effect of the overlap polarization is best described in the simplest model as being centred at the positive ion. This corresponds to the introduction of “pseudo-deformabilities” for the positive ion. In Fig. 1 (b), we show the effect on the overlap regions of a displacement of the positive ion. In addition to the long and short range .
1023
dipoles which are induced at the negative ion sites by the positive ion displacement, due to the free ion polarizability, there exists a redistribution of charge in the overlap regions. Charge is repelled from the regions where the overlap is increased, as indicated in Fig. 1. In this simplest approximation, the redistribution has a dipole character, and corresponds to a flow of positive charge in the direction opposite to the motion of the positive ion. The model proposed here is very similar to the model VI of Cochran, Cowley, Brockhouse and Woods.5 Their discussion assumed, however, that the positive charge was simulating a deformabiity of the negative ion, which led to the development of the breathing shell model.6 A more detailed discussion has been given by Cochran.’ He discusses the implications of model VI (including a discussion of the positive shell charge), the breathing shell model, and charge transfer (rigid overlap model).2 The three-body forces in the charge transfer model, while able to account for the deviation from the Cauchy relation, are not able to give the observed lowering of the LO branch at L.7
OVERLAP POLARIZATION AND LATTICE DYNAMICS OF IONIC CRYSTALS Vol. 16, No.8
1024
8
l~o]
(~~1 Na!
6-
K-
1
—
loot]
V. IN ;4 o z
_____
Ui
a
Ui LL
OVERLAP CHARGES
(a)
I
c—
0
0.5
REDUCED WAVE VECTOR
FIG. 2. Dispersion curves of Nal in the three high
IONIC MOTION
symmetry directions. The experimental points are taken from reference 5. The solid line is given by a 6-parameter overlap shell model and the dashed line by an 8-parameter breathing shell. The values of the parameters are given in Table 1. ELECTRONIC CHARGE FLOW
A and B (non-central), the ionic polarizabiity of the negative ion represented by the shell charge Y_ and the shell—core interaction constant K_, and, in addition, the overlap polarizability, represented by a shell charge Y÷at the positive ion and a shell—core force constant K+; for some crystals, Z must be varied from one (or
FIG. 1. (a) Regions of overlap (charge depletion) shown clustered about the small, almost unpolarizable, positive ions. The parameters describing the polarizability in our model are indicated. (b) Effect of the overlap region during the displacement of a positive ion. There is a net charge shift in the same direction as the displacement, or correspondingly, a shift of positive charge in the opposite direction.
two). These parameters are fitted to neutron data, not to macroscopic entities (three elastic constants, two dielectric constants, and the infrared resonance fre-
Table 1. The values of the parameters used for the curves shown in Fig. 2. The notation is similar of that of reference 5 A BSM OSM
10.52 11.8
B+B”
A’
--1.01 1.47 —1.14 —
B’
Z
—0.025 0.913 1 —
A meaningful physical model ought not to be required to fit neutron measurements to better than a few per cent, as anharmonic effects, for example, are ignored. It should be mentioned, with respect to fitting the LO branches, that these vibrations are often nearly degenerate with their own two phonon decay states, leading to shifts of frequency of a few per cent,8 not given in a harmonic model. For a sensible description of the dispersion curves, then, we find that our model requires only six or seven parameters: two nearest neighbour force constants
Y.
K+
Y
K
G
54.5
—3.36 —4.16
123.6 227.5
81.2
1.45
—
quency). One should be cautious about using the elastic constants for macroscopic fits, as they are usually determined as isothermal of adiabatic at room temperature, and they can deviate by as much as 10—30 per cent from the extrapolated values of the neutron scattering data in some crystals.9’° Since the equations of motion for this model are equivalent to those of a simple shell model with just these parameters, there exist several fairly good calculations in the literature (for NaBr,9 KC1,11 RbBr,12
Vol. 16, No.8
OVERLAP POLARIZATION AND LATTICE DYNAMICS OF IONIC CRYSTALS
and CsCl 13) using, in effect, just this model. The success of these fits, which were often thought to be unphysical because of the positive shell charge, is no longer surprising in the light of the re-interpretation, In addition, many calculations exist using a positive shell charge where the situation is somewhat masked by the use of many parameters. We investigated a variety of systems (KBr, Nal, MgO and CaO) and found good agreement with this overlap shell model using six or seven parameters. As an example, we show in Fig. 2 the result of our re-investigation of Nal. The parameters of the two models illustrated are given in Table 1. An eight parameter breathing shell model (with second neighbour forces) fitted to neutron data (dashed curve), is needed to give about the same fit as a six parameter fit fitted overlap shell model (solid curve).* Several points about the model should be mentioned, First, if the static ionic charge Z is treated as a free parameter, it tends to obtain values close to one in the alkali halides or two in the alkali earth oxides. If the overlap polarization is introduced, it is found that second neighbour forces are less important than is found in an extended shell model or a breathing shell model: compare, for example the size of the second neighbour force constants of model VI of reference S to those in the breathing shell model given in Table 1. By comparison with the breathing shell model, the description of the elastic constants
1025
is much simpler, as second neighbour forces are not required to correct for the influence of the breathing deformabiities, and the use of a positive shell rather than breathing means that the deviation from the Cauchy relation is not related to the lowering of the LO frequency at L. Furthermore, it thus avoids the anomaly in the LO branch near the X point that the breathing shell model exhibits. In order to check the reliability of the eigenvectors given by the overlap shell model, we recalculated the second-order Raman spectra of MgO (given by a single eigenvector-dependent term14). The eigenvectors of a given parameter overlap shell model gave a result which was as good as the previous calculation using the eigenvectors of the eight parameter breathing shell model D of Sangster et al. ~ To summarize, a simple overlap shell model is well able to describe the dispersion curves of alkali halides including those with cesium chloride structure, and the dispersion curves of the alkaline earth oxides. It avoids the problem the breathing shell model has as a result of the introduction of second neighbour forces, and yields good fits to measured dispersion curves without the introduction of a large number of parameters as in the usual extended shell model. *
A five parameter overlap shell model, with the sum of the two polarizabilities given by the Clausius— Mosotti relation, yields nearly the same fit, although the implications of the model are not consistent with this restriction.
REFERENCES I.
For a review refer to COCHRAN W., CriticalRev. Solid State Sci. 2, 1(1971).
2. 3.
LOWDIN P.O.,Adv. Phys. 5, 1(1956); LUNDQVIST S.O.,Ark. Fysik. 6,25 (1952); 9,435 (1955); 12, 263 (1957). GLISS B., ZEYHER R. and BILZ H.,Phys. Status Solidi (b) 44, 910 (1970).
4.
DICK B.G.,Phys. Rev. 129, 1583 (1963); MARSTON R.L. and DICK B.G.,Solid State Commun. 5,731(1967).
5.
COWLEY R.A., COCHRAN W., BROCKHOUSE B.N. and WOODS A.DB.,Phys. Rev. 131, 1030 (1963).
6. 7.
SCHRODER U.,Solid State Commun. 4, 347 (1966). ZEYHER R.,Phys. Status Solidi (b) 48, 711(1971).
8. 9.
COWLEY R.A.,Adv. Phys. 12,421(1963). REID J.S., SMITH T. and BUYERS W.T.L.,Phys. Rev. BI, 1833 (1970).
10.
KRESS W., Phys. Status Solidi (b) 62, 403 (1974).
1026
OVERLAP POLARIZATION AND LATTICE DYNAMICS OF IONIC CRYSTALS Vol. 16, No.8
11.
COPLEY J.R.D., MACPHERSON R.W. and TIMUSK T., Phys. Rev. 182,965 (1969).
12.
ROLANDSON S. and RAUNIOG.,J. Phys. C4, 958 (1971).
13.
ROLANDSON S. and RAUNIO G.,Phys. Rev. B4, 4617 (1971).
14.
HABERKORN R., BUCHANAN M. and BILZ H.,Solid State Commun. 12,681(1973).
15.
SANGSTER M.J.L., PECKHAM G. and SAUNDERSON D.H.,J. Phys. C3, 1026 (1970).
Note added in proof We have been informed by Dr. R. Bauer, MPI fuer Metall forschung, Stuttgart, about his recent calculations on overlap forces in simple alkali halides. As a result, second nearest neighbour forces between cations seem to be very weak due to the cancellation of repulsive and attractive parts in the potential.