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Lattice dynamics of the exchange charge model of alkali halides
Solid State Communications, Vol. 5, PP. 731- 734, 1967. Pergamon Press Ltd. Printed in Great Britain
LATTICE DYNAMICS OF THE EXCHANGE CHARGE MODEL OF ALKALI HALIDES* R.L. Marstont andB.G. Dick Department of Physics, University of Utah, Salt Lake City, Utah, U. S. A. (Received 3 July 1967 by J.A. Krumhansl)
Certain difficulties associated with the shell model of ionic crystals are not removable by considering the longrange electrical interactions deriving from small positive exchange charges. These exchange charges are located in the regions of overlap between nearest-neighbor Ions and represent the distortion of charge distributions on the shells.
THE SHELL model of ionic crystals has been successful in accounting for their observed phonon frequencies.’ Nevertheless, some difficulties remain. The most notable of these relates to the elastic constants: Since it reduces to the rigid-ion model in elastic theory4, the SM cannot account for the observed failure of the Cauchy relations for alkali-halides. S Also, the most successful versions of the SM require ten parameters to fit the theoretical model to all available experimental data,’ 2 including inelastic neutron-scattering data. Since the neutron data are not available for all alkali-halides, a preferred model would entail at most seven parameters which could be evaluated by use of elastic, dielectric, and lattice constants alone without recourse to the neutron data. In addition, the more elaborate versions of the SM’ 2 lead to an unphysical value for at least one parameter. It is as if the parameters were able by virtue of the flexibility inherent in their multiplicity to represent effects present in the crystal other than those intended.
exclusion of charge from regions of overlap of adjacent ions. The ECM represents these regions by point charges. Here, we use the ECM in conjunction with a simple SM in which the short-range (ZR) interactions are confined to act between nearest-neighbor shells but both ions are taken to be polarizable. Since it involves many-s body forces, the ECM is not in general equivalent to any SM. ~ There is evidence that such a model might correct some of the deficiencies of the SM. Cowley et al.2 have suggested that neglect of longrange (I~3juadrupoleinteractions may be responsible for some of the unappealing features of their most successful SM, mentioned above; and the ECM does treat these effects. Furthermore, Dick6 has shown that the ECM, through its manybody character, can count qualitatively, at least, for the failure of the Cauchy relations in alkalihalide crystals. Like the SM, the ECM lends Itself to analysis in terms of simple, classical physical concepts. The regions of exchange-charge density are replaced by small, point, positive charges which are constrained to lie on the lines between centers of nn shells, as shown in FIg. 1. The
The exchange charge model6 (ECM) SUpplements the dipolar distortion of the SM with ion charge distortions of higher multipole order, These distortions result from the Paull principle ~Th1s work was supported by the National Science Foundation
positions of the exchange charges are given by r~= y r 0~,where a vector locating a un negative-ion shell r0~is in the h-direction with respect to the positive-ion shell. The magnitude of an exchange charge, being related to the Born-Mayer
lPresent address: Wilkes College, Wilkes-Barre, Pennsylvania, U. S. A. 731
732
EXCHANGE CHARGE MODEL OF ALKALI HALIDES
+
•
±
•
Lattice site Exchange charge Shell or core center
Negative—ion shell
Vol. 5, No. 9
can result in models with different characteristics, as we shall see. The theory of lattice dynamics begins with an expression for the total potential energy of the crystal,
shell
=
(3)
~22’~kk’~kk’
core
hell—N ore
where £ and ~‘ run over lattice sites, and k and k’ run over particles (shells and cores) at a lattice In theenergies ECM, Itinto is convenient to given separate thesite. 2-particle a LR
2
\/ ~
•
±
±
electrical contribution, a SR contribution, and a third part which will be explained shortly: k2 22’ tkki
POSITIVE ION NEGATIVE ION FIG. 1
=
22’ ~kk’>LR
22’ + ~kk’~SR
dd, ~~kk’
~
=
____
-
dk,E(~_
~)
Rkk~
~ kEy~
repulsion energy (1)
IrOhI/p)
22’ (tkkl)
‘
(4)
The LR energy is expanded in multipoles:
A simple shell model with exchange charges.
B exp (-
-
is gtv~nby
£L~~~k Rkk~
1 ~R22”~1~’y kk
(5)
- :XY~~~)R22:dkXkkIY
kk
AIrOhIeXP
=
(-
t~I/~) .
(2)
The quantities B, A, and p are parameters to be determined for a particular alkali-halide by comparing theoretical predictions of measurable quantities with experiment, ~‘,
When charge is excluded from the regions of overlap, it is assumed to redistribute itself evenly over the two shells involved in the overlap. In order to represent this, for each exchange charge -‘-qh a quantity of charge -~qais restored to each shell. Neutrality of the unit cell is thus maintained. ,
Although the presence of these exchange charges is responsible for the repulsive forces between nn Ions, we are here concerned with their LR electrical interactions. It is convenient to associate the exchange charges with the shells by means of multipole expansions In order to simplify analysis. However, there Is arbitrariness In this procedure; and the various ways of doing it
+ *dk~EXy(~ ~
+ kk’
The number of Cartesian subscripts on a moment dk indicates its order (i. e. dkXy represents a quadrupole moment, dk represents a dipole moment). The ex2ansions were carried up to terms of order (Rn, )“ (monopole-octopole and quadrupole-dipole interactions). ~,
The multipole moments are generally functions of the positions of the particles indexed; in fact, where exchange charges are involved, the moments are also functions of the positions of nn particles, as can be seen from the form of equation (2). Using the harmonic approximation the moments and their products in equation (5) were expanded to second order in relative displacements. As mentioned above, different models are obtained, depending on how the exchange charges are associated with the shells. Two ways were considered as representative. For ECM No. 1,
Vol. 5, No. 9
EXCHANGE CHARGE MODEL OF ALKALI HALIDES
the exchange charges were associated with only the positive-ion shells, while for ECM No. 2, each exchange charge was associated half with each shell which shares it. In either case, the multipole moments of the shells and associated exchange charges are not only functions of the exchange-charge magnitudes and positions, but also of the rates of change of the exchange-charge magnitudes as functions of positions of the sharing shells. The difference between the two models arises principally from the fact that the net charge of an ion is a function of position for ECM No. 1, but constant (±e)for ECM No.2. Thus ECM No. 1 involves fluctuation of the ionic-charge magnitudes as well as distortion of the shellcharge distributions as lattice waves traverse the crystal, while ECM No. 2 involves only fluctuating charge distortions. For both ECM’s, the net charge of a unit cell is constant (0). For the SM, the second term (ZR) on the right-hand side of equation (4) is composed of core-shell interaction energies and Born-Mayer repulsion energies. In keeping with the harmonic approximation, the Born-Mayer expression equation (1) Is expanded to second order in relative displacements of the nn shells involved, and only the second-order terms are kept. The last term (XCL) of equation (4) arises from the simultaneous use of Born-Mayer repulsion and exchange charges. Dick and Overhauser showed for a pair of helium atoms that the interactions of exchange charges with nuclei and electrons and each other are responsible for a net repulsion. In fact, the resultant energy of these interactions can be approximated by the Born-Mayer formula [equation (1)]. It is evident that use of both concepts, exchange charges and Born-Mayer repulsion, entails some redundancy. To avoid this, the last term (XCL) of equation (4) represents the exclusion of those exchange-charge interactions which duplicate the intended role of the Born-Mayer formula for the LR interactions, The terms excluded were so chosen that any part of the interaction between nn ions which involves exchange charges or corresponding shell compensations (i. e. any part other than those involving only charges of fixed magnitude) was removed as being redundant with the BornMayer SR interaction. We note that the interactions of exchange charges with part of their own ions (ions with which they are associated mathematically) are automatically excluded by the use of the multipole expansions. Using the ECM potential energy just de-
733
fined the equations of motion were obtained and solved in the usual manner, following Born and Huang. ~‘ The long-wave approximation for ionic crystals~ (found to be applicable where ECM’s ~re concerned) was used to relate the predictions of the models to the measured elastic constants. Details of these lengthy calculations are available in a thesis by Marston. The results of these investigations are that it is not possible to find a set of parameters A, B, p, and y which can predict the three elastic constants of KBr using ECM No. 1. Furthermore, it is not possible to solve the elastictheory equations consistently for the parameters of ECM No. 2 for either Na! or KBr. A non-linear least square procedure was used In an attempt to fit theoretical phonón dispersion curves to the inelastic neutron-scattering data of Woods, et al.’ ; that is, to the phonon frequencies as i1iii~ctionof wave vector in the three symmetry directions of KBr. It appears impossible to improve upon the SSM predictions with either ECM. The effects on theoretically predicted phonon frequencies, especially those of the LO mode for k in the (111) direction, were also Investigated by varying individually the parameters of ECM No. 1. The effects on the SR terms, introduced by means of the condition for static equilibrium, completely countered the weaker effects on the LR terms. Furthermore, varlations which tended to improve the fit of the (111) LO modes adversely affected some predicted frequencies of other modes and directions. These results strongly suggest that a modification of the SR interactions, as opposed to the LR interactions, is needed to improve upon the predictions of the SSM. Specificially, the need Is demonstrated for a modification of the SSM which wuld have the effect of reducing the repulsion for the LO mode in the (111) direction, but would have little effect on the predictions of other frequencies. Second-mi interactions between positive Ions (van der Waals interactions) were considered in this regard. ~ It was found that the van der Waals effect at the point k = (~,~, k)/2r0 can cause a 2~%change in frequency of the LO mode, whereas a 10% change is needed to bring the SSM prediction into agreement with experiment. The conclusion is drawn that the failure of the exchange-charge concept to improve upon the predictions of the SSM implies the unimpor-
734
EXCHANGE CHARGE MODEL OF ALKALI HALIDES
tance of long-range effects other than those inherent in the SSM. This suggests that improvements on the SSM lie in the direction of careful
Vol. 5, No. 9
treatments of short-range interactions which do not increase the number of model9 parameters. is a useful Schrbder’s step in this breathing direction. shell model -
References 1.
WOODS A.B.D., BROCKHOUSE B.N., COWLEY R.A. and COCHRAN W., Phys. Rev. 131, 1025 (1963).
2.
COWLEY R.A., COCHRAN W., BROCKHOUSE B.N. and WOODS A.B.D., Phys. Rev. 131, 1030 (1963).
3.
DICK B. G., in Lattice Dynamics (ed. by WALLIS R. F.) p. 159. Pergamon Press, London (1964).
4.
WOODS A.B.D., COCHRAN W. and BROCKHOUSE B.N., Phys. Rev. 119,
5.
DICK B.G.,
6.
DICK B.G. and OVERHAUSER A.W.,
7.
BORN M. and HUANG L., Dynamical Theory of Crystal Lattices, Oxford University Press, New York (1954).
8.
MARSTON R. L., unpublished thesis, University of Utah (1967).
9.
SCHRODER U., Solid State Comm. 4, 347 (1966).
980 (1960).
Phys. Rev. 129, 1583. (1963). Phys. Rev. 121, 90 (1958).
Gewisse Schwterlgkeiten in zusammenhang mit dem Schalenmodell des Ionenkrlstalls können nicht eliminiert werden, wenn man die weitreichenden Wechselwirkungen von kleinen positiven Austauschladungen berücksichttgt. Diese Austauschiadungen befinden sich in den Uberlappungsgebieten zwischen benachbarten lonen und representieren die Verzerrungen in den Ladungsvertellungen der Schalen.
LATTICE DYNAMICS OF THE EXCHANGE CHARGE MODEL OF ALKALI HALIDES* R.L. Marstont andB.G. Dick Department of Physics, University of Utah, Salt Lake City, Utah, U. S. A. (Received 3 July 1967 by J.A. Krumhansl)
Certain difficulties associated with the shell model of ionic crystals are not removable by considering the longrange electrical interactions deriving from small positive exchange charges. These exchange charges are located in the regions of overlap between nearest-neighbor Ions and represent the distortion of charge distributions on the shells.
THE SHELL model of ionic crystals has been successful in accounting for their observed phonon frequencies.’ Nevertheless, some difficulties remain. The most notable of these relates to the elastic constants: Since it reduces to the rigid-ion model in elastic theory4, the SM cannot account for the observed failure of the Cauchy relations for alkali-halides. S Also, the most successful versions of the SM require ten parameters to fit the theoretical model to all available experimental data,’ 2 including inelastic neutron-scattering data. Since the neutron data are not available for all alkali-halides, a preferred model would entail at most seven parameters which could be evaluated by use of elastic, dielectric, and lattice constants alone without recourse to the neutron data. In addition, the more elaborate versions of the SM’ 2 lead to an unphysical value for at least one parameter. It is as if the parameters were able by virtue of the flexibility inherent in their multiplicity to represent effects present in the crystal other than those intended.
exclusion of charge from regions of overlap of adjacent ions. The ECM represents these regions by point charges. Here, we use the ECM in conjunction with a simple SM in which the short-range (ZR) interactions are confined to act between nearest-neighbor shells but both ions are taken to be polarizable. Since it involves many-s body forces, the ECM is not in general equivalent to any SM. ~ There is evidence that such a model might correct some of the deficiencies of the SM. Cowley et al.2 have suggested that neglect of longrange (I~3juadrupoleinteractions may be responsible for some of the unappealing features of their most successful SM, mentioned above; and the ECM does treat these effects. Furthermore, Dick6 has shown that the ECM, through its manybody character, can count qualitatively, at least, for the failure of the Cauchy relations in alkalihalide crystals. Like the SM, the ECM lends Itself to analysis in terms of simple, classical physical concepts. The regions of exchange-charge density are replaced by small, point, positive charges which are constrained to lie on the lines between centers of nn shells, as shown in FIg. 1. The
The exchange charge model6 (ECM) SUpplements the dipolar distortion of the SM with ion charge distortions of higher multipole order, These distortions result from the Paull principle ~Th1s work was supported by the National Science Foundation
positions of the exchange charges are given by r~= y r 0~,where a vector locating a un negative-ion shell r0~is in the h-direction with respect to the positive-ion shell. The magnitude of an exchange charge, being related to the Born-Mayer
lPresent address: Wilkes College, Wilkes-Barre, Pennsylvania, U. S. A. 731
732
EXCHANGE CHARGE MODEL OF ALKALI HALIDES
+
•
±
•
Lattice site Exchange charge Shell or core center
Negative—ion shell
Vol. 5, No. 9
can result in models with different characteristics, as we shall see. The theory of lattice dynamics begins with an expression for the total potential energy of the crystal,
shell
=
(3)
~22’~kk’~kk’
core
hell—N ore
where £ and ~‘ run over lattice sites, and k and k’ run over particles (shells and cores) at a lattice In theenergies ECM, Itinto is convenient to given separate thesite. 2-particle a LR
2
\/ ~
•
±
±
electrical contribution, a SR contribution, and a third part which will be explained shortly: k2 22’ tkki
POSITIVE ION NEGATIVE ION FIG. 1
=
22’ ~kk’>LR
22’ + ~kk’~SR
dd, ~~kk’
~
=
____
-
dk,E(~_
~)
Rkk~
~ kEy~
repulsion energy (1)
IrOhI/p)
22’ (tkkl)
‘
(4)
The LR energy is expanded in multipoles:
A simple shell model with exchange charges.
B exp (-
-
is gtv~nby
£L~~~k Rkk~
1 ~R22”~1~’y kk
(5)
- :XY~~~)R22:dkXkkIY
kk
AIrOhIeXP
=
(-
t~I/~) .
(2)
The quantities B, A, and p are parameters to be determined for a particular alkali-halide by comparing theoretical predictions of measurable quantities with experiment, ~‘,
When charge is excluded from the regions of overlap, it is assumed to redistribute itself evenly over the two shells involved in the overlap. In order to represent this, for each exchange charge -‘-qh a quantity of charge -~qais restored to each shell. Neutrality of the unit cell is thus maintained. ,
Although the presence of these exchange charges is responsible for the repulsive forces between nn Ions, we are here concerned with their LR electrical interactions. It is convenient to associate the exchange charges with the shells by means of multipole expansions In order to simplify analysis. However, there Is arbitrariness In this procedure; and the various ways of doing it
+ *dk~EXy(~ ~
+ kk’
The number of Cartesian subscripts on a moment dk indicates its order (i. e. dkXy represents a quadrupole moment, dk represents a dipole moment). The ex2ansions were carried up to terms of order (Rn, )“ (monopole-octopole and quadrupole-dipole interactions). ~,
The multipole moments are generally functions of the positions of the particles indexed; in fact, where exchange charges are involved, the moments are also functions of the positions of nn particles, as can be seen from the form of equation (2). Using the harmonic approximation the moments and their products in equation (5) were expanded to second order in relative displacements. As mentioned above, different models are obtained, depending on how the exchange charges are associated with the shells. Two ways were considered as representative. For ECM No. 1,
Vol. 5, No. 9
EXCHANGE CHARGE MODEL OF ALKALI HALIDES
the exchange charges were associated with only the positive-ion shells, while for ECM No. 2, each exchange charge was associated half with each shell which shares it. In either case, the multipole moments of the shells and associated exchange charges are not only functions of the exchange-charge magnitudes and positions, but also of the rates of change of the exchange-charge magnitudes as functions of positions of the sharing shells. The difference between the two models arises principally from the fact that the net charge of an ion is a function of position for ECM No. 1, but constant (±e)for ECM No.2. Thus ECM No. 1 involves fluctuation of the ionic-charge magnitudes as well as distortion of the shellcharge distributions as lattice waves traverse the crystal, while ECM No. 2 involves only fluctuating charge distortions. For both ECM’s, the net charge of a unit cell is constant (0). For the SM, the second term (ZR) on the right-hand side of equation (4) is composed of core-shell interaction energies and Born-Mayer repulsion energies. In keeping with the harmonic approximation, the Born-Mayer expression equation (1) Is expanded to second order in relative displacements of the nn shells involved, and only the second-order terms are kept. The last term (XCL) of equation (4) arises from the simultaneous use of Born-Mayer repulsion and exchange charges. Dick and Overhauser showed for a pair of helium atoms that the interactions of exchange charges with nuclei and electrons and each other are responsible for a net repulsion. In fact, the resultant energy of these interactions can be approximated by the Born-Mayer formula [equation (1)]. It is evident that use of both concepts, exchange charges and Born-Mayer repulsion, entails some redundancy. To avoid this, the last term (XCL) of equation (4) represents the exclusion of those exchange-charge interactions which duplicate the intended role of the Born-Mayer formula for the LR interactions, The terms excluded were so chosen that any part of the interaction between nn ions which involves exchange charges or corresponding shell compensations (i. e. any part other than those involving only charges of fixed magnitude) was removed as being redundant with the BornMayer SR interaction. We note that the interactions of exchange charges with part of their own ions (ions with which they are associated mathematically) are automatically excluded by the use of the multipole expansions. Using the ECM potential energy just de-
733
fined the equations of motion were obtained and solved in the usual manner, following Born and Huang. ~‘ The long-wave approximation for ionic crystals~ (found to be applicable where ECM’s ~re concerned) was used to relate the predictions of the models to the measured elastic constants. Details of these lengthy calculations are available in a thesis by Marston. The results of these investigations are that it is not possible to find a set of parameters A, B, p, and y which can predict the three elastic constants of KBr using ECM No. 1. Furthermore, it is not possible to solve the elastictheory equations consistently for the parameters of ECM No. 2 for either Na! or KBr. A non-linear least square procedure was used In an attempt to fit theoretical phonón dispersion curves to the inelastic neutron-scattering data of Woods, et al.’ ; that is, to the phonon frequencies as i1iii~ctionof wave vector in the three symmetry directions of KBr. It appears impossible to improve upon the SSM predictions with either ECM. The effects on theoretically predicted phonon frequencies, especially those of the LO mode for k in the (111) direction, were also Investigated by varying individually the parameters of ECM No. 1. The effects on the SR terms, introduced by means of the condition for static equilibrium, completely countered the weaker effects on the LR terms. Furthermore, varlations which tended to improve the fit of the (111) LO modes adversely affected some predicted frequencies of other modes and directions. These results strongly suggest that a modification of the SR interactions, as opposed to the LR interactions, is needed to improve upon the predictions of the SSM. Specificially, the need Is demonstrated for a modification of the SSM which wuld have the effect of reducing the repulsion for the LO mode in the (111) direction, but would have little effect on the predictions of other frequencies. Second-mi interactions between positive Ions (van der Waals interactions) were considered in this regard. ~ It was found that the van der Waals effect at the point k = (~,~, k)/2r0 can cause a 2~%change in frequency of the LO mode, whereas a 10% change is needed to bring the SSM prediction into agreement with experiment. The conclusion is drawn that the failure of the exchange-charge concept to improve upon the predictions of the SSM implies the unimpor-
734
EXCHANGE CHARGE MODEL OF ALKALI HALIDES
tance of long-range effects other than those inherent in the SSM. This suggests that improvements on the SSM lie in the direction of careful
Vol. 5, No. 9
treatments of short-range interactions which do not increase the number of model9 parameters. is a useful Schrbder’s step in this breathing direction. shell model -
References 1.
WOODS A.B.D., BROCKHOUSE B.N., COWLEY R.A. and COCHRAN W., Phys. Rev. 131, 1025 (1963).
2.
COWLEY R.A., COCHRAN W., BROCKHOUSE B.N. and WOODS A.B.D., Phys. Rev. 131, 1030 (1963).
3.
DICK B. G., in Lattice Dynamics (ed. by WALLIS R. F.) p. 159. Pergamon Press, London (1964).
4.
WOODS A.B.D., COCHRAN W. and BROCKHOUSE B.N., Phys. Rev. 119,
5.
DICK B.G.,
6.
DICK B.G. and OVERHAUSER A.W.,
7.
BORN M. and HUANG L., Dynamical Theory of Crystal Lattices, Oxford University Press, New York (1954).
8.
MARSTON R. L., unpublished thesis, University of Utah (1967).
9.
SCHRODER U., Solid State Comm. 4, 347 (1966).
980 (1960).
Phys. Rev. 129, 1583. (1963). Phys. Rev. 121, 90 (1958).
Gewisse Schwterlgkeiten in zusammenhang mit dem Schalenmodell des Ionenkrlstalls können nicht eliminiert werden, wenn man die weitreichenden Wechselwirkungen von kleinen positiven Austauschladungen berücksichttgt. Diese Austauschiadungen befinden sich in den Uberlappungsgebieten zwischen benachbarten lonen und representieren die Verzerrungen in den Ladungsvertellungen der Schalen.