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A new model for lattice dynamics (“breathing shell model”)
Solid State Communications Vol. 4, pp.34?-348, 1966. Pergamon Press Ltd. Printed In Great Britain.
A NEW MODEL FOR LATTICE DYNAMICS (“BREATHING SHELL MODEL”) U. Schröder~ Physikalisches Institut der Universität Frethurg. (Received 23 May 1966 by G. Lelbfried)
If we introduce the isotropic deformation of ions as a new degree of freedom, we can show that the normal modes can be calculated from a few macroscopic parameters and the results agree with the neutron scattering measurements. The calculations are carried out for KBr and NaL
LATTICE dynamics of non-metallic crystals have improved remarkably during the lastfew years taking into account the po]arizabillties of the lattice ions. ~ Essentially this reduces the number of force constants necessary for the description of experimental dispersion curves as measured by neutron spectrometry. ~ In the alkali halides, for example, the main discrepancies between measured curves and the theoretical ones, calculated by using the Kellermann model5, are removed by its extension to the so-called simple-shell modeL6
Na! 100°K
—~“~
•
7
:::~~,~~1L
[OO~]
...
[~~OJ
:[
~
6 ~
•..~ •..•
•
......
-~
—
V
r
~.
•
•
5
~
..
-
3
Nevertheless, there remain regions in the Brillouin- zone where the improvement is rather poor. An instructive example Is the longitudinal optical mode at q = (1/2, 1/2,1/2) in Nal and ICBr (Figs. 1 and 2). At this poInt the strongly polarized anions are at rest while the nearly rigid catlons move In a symmetrical way. Since the electron “shell” in this model generally can only be displaced against the core but not be deformed, the result for this mode gives no Improvement over the rigid-ion-model.
3
I
2
~ /
.......
..
0
~
•
~
2
....... •
Q VECTOR COORDIfrS4TE.
~
FIG. I The dispersion curves for the lattice vibrations of sodium Iodide. Cornparison of the experimental curves (dots) with calculated ones based on the different models.
To overcome these difficulties one can introduce formal parameters (pseudo-polarizablllties of the cations additional springs to second neighbours etc.). These lead to excellent fits for alkali halides using about 9 parameters2 and for homopolar crystals (Ge, SI, Diamant) using about 10 12 parameters. ~ -
The fitting of formal parameters to measured dispersion curves seems to be unsattsfactory from a physical point of view. In addItion, it restricts the calculation of precise
____________________
*Present address: Iñstltut für theor. Physik der Universitãt Frankfurt. 347
A NEW MODEL FOR LATTICE DYNAMICS
348
KBr
9Q0
K 7
RIGID ICI~14COEt
[oo~]
Cc~]
.L
,1.
6 ~
•:.•~
....
‘1 —
I
I
••.•~~T
FIG. 2
6
The dispersion curves for the lattice vibrations of potassium bromide. Comparison of the experimental curves (dots) with the calculated ones based on the different models.
•..••~_~ .
•....
— ..
1p~0,0)
Vol. 4, No. 7
..•
~~0.~7.0) ~ ~Q~] ~ REDUCED WAVE VECTOR COORDINATE.
c
dispersion curves, densities etc. to the few crystals which are measured bytoinelastic 8 has proposed considerneutron the scattering. Lax effect of quadrupole interactions. It seems plausible that there is some influence of these and higher multipoles to the lattice modes. We feel, however, that the essential point might be to take into consideration the compressibility of the ions. From a quantum-mechanical point of view the displacement of the electron shell against the core corresponds to an intermixing of the s-type ground-state-function with the p-type excited wave function (demonstrating the dipole-character of the apprc~cirnation). In the same way higher multipoles are connected with d-, f-, etc. type wave functions. But, it seems likely that low-energy s-type wave functions play an important role. They lead to a spherical extension or compression of the ions corresponding to a new Internal degree of freedom [V in eq. (1)1 * Introducing this concept into the equations of motion given by Cowley ~~j• 2 [eqs. (1) and (2) for the shell model we obtain the following set of equations: 2u = (R + ZCZ)u + (T + ZCY) w + Qv (1) rnw 0=(T~+YCZ)u+ (S+ YCY)w+Qv ]
0 = Q+ (u + w) + H V (1) All quantities of eqs.(1) agree with those used by Cowley et al. The additional quantities Q and H desciffi~the coupling between the change of v&ume of the shell and the shift of shells and cores. if only one ion Is polarizable, they have the following form:
Q =(~), —
sin (q~r
2 =
3 H
0) sin(q1r0) sin (q ,r ~) k
-i A ~
= +~
2
(A
(2)
+
Q and ~ degenerating into vectors and H into a scalar since v is a scalar. Within the approximation of a refined simple shell model (additional springs A’, B’ and B” In the notation of Woods et al. (4 Appendix 2)t corresponding to model ~~TCowley etal. 2 but putting cx, = 0) all parameters are uniquely determined by the three elastic constants, the transverse optic mode w0 and results the obtained dielectricinconstants this way for c0 and Nal c. and The KBr are shownin Figs. land2. The agreement of the calculated and the measured dispersion curves is very satisfactory.
*It should be mentioned that the effect of coinpressibility of the ions cannot be described by changing the spring-constants to nearest neighbours. The first discussion of the importance of the deformation of ions is by Szigeti. ~
There is no additional parameter to this model. t
Vol. 4, No. 7
A NEW MODEL FOR LATTICE DYNAMICS
An even better agreement can be obtained by assuming both ions polarizable and the ionic charge less than one.
349
theory and the above mentioned refinements will be discussed in a forthcoming paper.
The proposed model seems to enable one to calculate precise dispersion without knowledge from neutron scattering. The details of the
The author wishes to thank Prof. Dr. H. Bilz and his group and Prof. Dr. S. Filigge for many stimulating discussions and helpful criticism.
References 1. COCHRAN W., ~p. Progr. Phys. Earlier references are given here.
~, 1 (1963).
2. COWLEY R. A., COCHRAN W., BROCKHOUSE B. N. and WOODS A. D. B., Phys. Rev. 131, 1030 (1963). —
3. KARO A. M. and HARDY J. R., Phys. Rev. 129 (1963). 4. WOODS A. D. B., BROCKHOUSE B. N. and COWLEY R. A., Phys. Rev. For experiments on C, Si and Ge see Ref. 7.
i~i,
1025 (1963).
—
5. KELLERMANN E. W., Phil. Trans. Roy. Soc. London, A238, 513 (1940). 6. WOODS A. D. B., COCHRAN W. and BROCKHOUSE B. N., Phvs. Rev. 119, 980 (1960). 7. DOLLING G. and COWLEY R. A. (preprint). 8. LAX M., Lattice dynamics (Edited By Wallis R. F.), p. 179. Pergamon Press, London (1965). 9. SZIGETI B., Proc. Roy. Soc. A204, 51(1950).
Am Beispeil von KBr und Nal wlrd gezeigt, da~durch die Einfflhrung der isotropen Deformation als neuen Freiheltagrad die Normalschwingungen mit wenigen makroskopischen Parametern serh befriedigend beschrieben werden kOnnen, ohne da~elne Anpassung an Neutronenbeugimgsmessungen erforderlich ist.
A NEW MODEL FOR LATTICE DYNAMICS (“BREATHING SHELL MODEL”) U. Schröder~ Physikalisches Institut der Universität Frethurg. (Received 23 May 1966 by G. Lelbfried)
If we introduce the isotropic deformation of ions as a new degree of freedom, we can show that the normal modes can be calculated from a few macroscopic parameters and the results agree with the neutron scattering measurements. The calculations are carried out for KBr and NaL
LATTICE dynamics of non-metallic crystals have improved remarkably during the lastfew years taking into account the po]arizabillties of the lattice ions. ~ Essentially this reduces the number of force constants necessary for the description of experimental dispersion curves as measured by neutron spectrometry. ~ In the alkali halides, for example, the main discrepancies between measured curves and the theoretical ones, calculated by using the Kellermann model5, are removed by its extension to the so-called simple-shell modeL6
Na! 100°K
—~“~
•
7
:::~~,~~1L
[OO~]
...
[~~OJ
:[
~
6 ~
•..~ •..•
•
......
-~
—
V
r
~.
•
•
5
~
..
-
3
Nevertheless, there remain regions in the Brillouin- zone where the improvement is rather poor. An instructive example Is the longitudinal optical mode at q = (1/2, 1/2,1/2) in Nal and ICBr (Figs. 1 and 2). At this poInt the strongly polarized anions are at rest while the nearly rigid catlons move In a symmetrical way. Since the electron “shell” in this model generally can only be displaced against the core but not be deformed, the result for this mode gives no Improvement over the rigid-ion-model.
3
I
2
~ /
.......
..
0
~
•
~
2
....... •
Q VECTOR COORDIfrS4TE.
~
FIG. I The dispersion curves for the lattice vibrations of sodium Iodide. Cornparison of the experimental curves (dots) with calculated ones based on the different models.
To overcome these difficulties one can introduce formal parameters (pseudo-polarizablllties of the cations additional springs to second neighbours etc.). These lead to excellent fits for alkali halides using about 9 parameters2 and for homopolar crystals (Ge, SI, Diamant) using about 10 12 parameters. ~ -
The fitting of formal parameters to measured dispersion curves seems to be unsattsfactory from a physical point of view. In addItion, it restricts the calculation of precise
____________________
*Present address: Iñstltut für theor. Physik der Universitãt Frankfurt. 347
A NEW MODEL FOR LATTICE DYNAMICS
348
KBr
9Q0
K 7
RIGID ICI~14COEt
[oo~]
Cc~]
.L
,1.
6 ~
•:.•~
....
‘1 —
I
I
••.•~~T
FIG. 2
6
The dispersion curves for the lattice vibrations of potassium bromide. Comparison of the experimental curves (dots) with the calculated ones based on the different models.
•..••~_~ .
•....
— ..
1p~0,0)
Vol. 4, No. 7
..•
~~0.~7.0) ~ ~Q~] ~ REDUCED WAVE VECTOR COORDINATE.
c
dispersion curves, densities etc. to the few crystals which are measured bytoinelastic 8 has proposed considerneutron the scattering. Lax effect of quadrupole interactions. It seems plausible that there is some influence of these and higher multipoles to the lattice modes. We feel, however, that the essential point might be to take into consideration the compressibility of the ions. From a quantum-mechanical point of view the displacement of the electron shell against the core corresponds to an intermixing of the s-type ground-state-function with the p-type excited wave function (demonstrating the dipole-character of the apprc~cirnation). In the same way higher multipoles are connected with d-, f-, etc. type wave functions. But, it seems likely that low-energy s-type wave functions play an important role. They lead to a spherical extension or compression of the ions corresponding to a new Internal degree of freedom [V in eq. (1)1 * Introducing this concept into the equations of motion given by Cowley ~~j• 2 [eqs. (1) and (2) for the shell model we obtain the following set of equations: 2u = (R + ZCZ)u + (T + ZCY) w + Qv (1) rnw 0=(T~+YCZ)u+ (S+ YCY)w+Qv ]
0 = Q+ (u + w) + H V (1) All quantities of eqs.(1) agree with those used by Cowley et al. The additional quantities Q and H desciffi~the coupling between the change of v&ume of the shell and the shift of shells and cores. if only one ion Is polarizable, they have the following form:
Q =(~), —
sin (q~r
2 =
3 H
0) sin(q1r0) sin (q ,r ~) k
-i A ~
= +~
2
(A
(2)
+
Q and ~ degenerating into vectors and H into a scalar since v is a scalar. Within the approximation of a refined simple shell model (additional springs A’, B’ and B” In the notation of Woods et al. (4 Appendix 2)t corresponding to model ~~TCowley etal. 2 but putting cx, = 0) all parameters are uniquely determined by the three elastic constants, the transverse optic mode w0 and results the obtained dielectricinconstants this way for c0 and Nal c. and The KBr are shownin Figs. land2. The agreement of the calculated and the measured dispersion curves is very satisfactory.
*It should be mentioned that the effect of coinpressibility of the ions cannot be described by changing the spring-constants to nearest neighbours. The first discussion of the importance of the deformation of ions is by Szigeti. ~
There is no additional parameter to this model. t
Vol. 4, No. 7
A NEW MODEL FOR LATTICE DYNAMICS
An even better agreement can be obtained by assuming both ions polarizable and the ionic charge less than one.
349
theory and the above mentioned refinements will be discussed in a forthcoming paper.
The proposed model seems to enable one to calculate precise dispersion without knowledge from neutron scattering. The details of the
The author wishes to thank Prof. Dr. H. Bilz and his group and Prof. Dr. S. Filigge for many stimulating discussions and helpful criticism.
References 1. COCHRAN W., ~p. Progr. Phys. Earlier references are given here.
~, 1 (1963).
2. COWLEY R. A., COCHRAN W., BROCKHOUSE B. N. and WOODS A. D. B., Phys. Rev. 131, 1030 (1963). —
3. KARO A. M. and HARDY J. R., Phys. Rev. 129 (1963). 4. WOODS A. D. B., BROCKHOUSE B. N. and COWLEY R. A., Phys. Rev. For experiments on C, Si and Ge see Ref. 7.
i~i,
1025 (1963).
—
5. KELLERMANN E. W., Phil. Trans. Roy. Soc. London, A238, 513 (1940). 6. WOODS A. D. B., COCHRAN W. and BROCKHOUSE B. N., Phvs. Rev. 119, 980 (1960). 7. DOLLING G. and COWLEY R. A. (preprint). 8. LAX M., Lattice dynamics (Edited By Wallis R. F.), p. 179. Pergamon Press, London (1965). 9. SZIGETI B., Proc. Roy. Soc. A204, 51(1950).
Am Beispeil von KBr und Nal wlrd gezeigt, da~durch die Einfflhrung der isotropen Deformation als neuen Freiheltagrad die Normalschwingungen mit wenigen makroskopischen Parametern serh befriedigend beschrieben werden kOnnen, ohne da~elne Anpassung an Neutronenbeugimgsmessungen erforderlich ist.