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Relation between the ion-ion potential and phonon spectra in alkali metals
Solid State Coninunications, Vol. 23, pp. 443
—
445, 1977. Pergamon Press. Printed in Great Britain.
RELATION BETWEEN THE ION-ION POTENTIAL AND PHONON SPECTRA IN ALKALI METALS
*
R. Day, F. Sun Department of Physics The Pennsylvania State University University Park, Pennsylvania 16802 USA and P. H. Cutler ** E.S.I.S., Universit~de Li~ge Institut de Physique B-4000 Sart Tilman/Li~ge, BELGIUZ4 (Received 4 April 1977 by R. H. Silsbee) Pair potentials and phonon curves for the alkalis were computed using an exact treatment of the conduction electron potential in a nonlocal a priori pseudopotential. It was found that pseudopotentials which yield quite similar dispersion curves can produce different ion—ion potentials.
We have recently done a priori nonlocal pseudopotential calculations which studied the significance of an exact versus an approximate treatment of 3 It the was conduction found that electron the exact ~,otentreatment tial) of the conduction electron potential (first formulated by Harrison4) had an appreciable effept upon the. phonon spectra of simple metals such as the alkalis, and excellent results were obtained .for these fully nonlocal calculations. This letter reports calculations, of the effec— tive ion—ion potential U Cr) for the alkali metals, that were based upon an a priori pseudo— potential which treated the conduction electron potential (i.e., the orthogonalization hole density and v 0~) exactly. Figure 1 shows two normalized energy wavenumber character (F~(q)) curves for Na. The F~(q) curves were calculated from two a priori nonlocalthat except pseudopotentials the core electron that wave werefunctions identical were chosen to be slightly different in the two calculations. The pair potential U(r) was calculated from FN (q) using equation (1), where *2
U(r)
=
z re
2
*2 + ~
¶
~ ~ sin qrqr FN (q)dq
However, the U(r) given by the solid curve does not exhibit any oscillations about zero which is a characteristic of the ion—ion potential in metals. The dashed curve does contain oscillations at largeU (r) r as it should. it is the evident that pseudopotentials which yield similar phonon spectra can give pair potentials that are quite different in their qualitative aspects (i.e., the oscillations at large r). This reaffirms Torrens • ~ observation that dis tinctly different pair potentials may reproduce very well the experimental phonon dispersion curves, and that, therefore, it is not suffi— cient to specify the pair potential by requiring it to predict good phonon spectra. The pseudopotentials used to calculate the curves in Figures 1, 2 and 3 used Singwi et al. .~6 dielectric function to describe the we found electron gas screening. However, 78 contrary to the assertions that the Hartreeofdielectric some researchers function yields a pair potential with a first minimum close to the nearest neighbor distance. The discrepancy may be due to the use of model potentials in the calculations by Shyu and Gaspari,7 and Duesbury and Taylor.8 Finally in an attempt to theoretically verify cochran’s9 method for extracting pair potential from phonon spectra, we computed phonon dispersion curves using a truncated F~(q). The truncation is a consequence of Cochran’s method which results in an F~(q) that is zero beyond the first reciprocal lattice vector. Hence, for Cochran’s method to be valid the truncation
.
The quantity z~ is the effective valence from pseudopotential theory and e is the electronic charge. Figure 2 shows the pair potentials U(r) corresponding to the F~(q) and Figure 3 shows the phonon spectra. The phonon spectra are both in excellent agreement with experiment,
*
This research was performed in part for the Applied Research Laboratory, The Pennsylvania State University under contract with Naval Sea Systems Command and in part under the auspices of the joint ESIS project (Electronic Structure in Solids) of the University of Antwerp and ~~the Universit~ de Li~ge. Permanent address: Department of Physics, 104 Davey Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802 USA. 443
ION—ION POTENTIAL AND PHONON SPECTRA IN ALKALI METALS
444
.0
Vol. 23, No. 7
.02
.9 .8 .7
.6 5 L
/
\\
I I I
.4
\
/
.3-
3~O
2.0
40
q/k,
1.
Normalized energy wavenumber character (F~(q)) curves for Na caicu— lated from a nonlocal a priori pseudopotential.
Slightly different
core electron wave functions were used to calculate the solid and dashed curves representing the energy wavenumber characteristic.
I
-.003-
I
I
I
I
I
I
I
I
I
\•._._/
I
5.0 2.
6.0
7.0
8.0
9.0 10.0 11.0 r IN BOHR UNITS
I
12.0
13.0
Na pair potentials calculated from the F~(q) curves shown in Figure 1.
14.0
ION—ION POTENTIAL AND PHONON SPECTRA IN ALKALI METALS
Vol. 23, No. 7 I
4.0
I
-
I
I
I
I
I
I
(001)
I
I
445 I
I
(III)
I
(II0~~
~ REDUCED WAVEVECTOR q/(2i,-/a) 3.
Na phonon dispersion curves calculated from the F~(q) of Figure 1. Experimental points are from Woods at al. [10]
procedure should have no significant effect upon the predicted pair potential and phonon spectra. It was found that setting F~(q) to zero beyond the first reciprocal lattice vector measurably affected our phonon dispersion results for Li and somewhat less for K but had no significant effect for Na. The U(r) of Li, Na, and K showed a similar dependence upon the truncation of FN (q). That is to say that for Na the truncation of F~(q) hardly affected U(r) while for K,U(r) was changed, though by only a few percent for most values of r. The trunca—
tion of F~(q)for Li produced changes in U(r) which were quite significant in comparison to Na and K. Hence though Cochran’s method is valid for Na and K it should be used with care when applied to less free electron—like metals such as Li where the truncation procedure of F~(q) may not be as acceptable. Acknowledgment - We wish to express our appreciation to Professor M. T. Pigott of the Applied Research Laboratory of The Pennsylvania State University for his support and encouragement.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
P. H. Cutler, R. Day and W. F. King III, J. Phys. F: Metal Phys. 5, 1801 (1975). R. S. Day, F. Sun and P. H. Cutler, J. Phys. F: Metal Phys. 6, L297 (1976) F. Sun, R. Day, P. H. Cutler and W. F. King III, J. Phys. F: Metal Phys. 6, Ll (1976). W. A. Harrison, Phys. Rev. 181, 1036 (1969). I. A. Torrens, Interatomic Potentials (Academic Press, 1972), P. 140. K. S. Singwi, A. Sj~lander, M. P. Tosi and R. H. Land, Phys. Rev. Dl, 1044 (1970). W. M. Shyu and G. D. Gaspari, Phys. Lett. 30A, 53 (1969). M. S. Duesbury and R. Taylor, Phys. Lett. 30A, 496 (1969). R. F. S. Cochran, Proc. Roy. Soc. 1276, 308 (1963). A. D. B. Woods, B. N. Brockhouse, R. H. March and A. T. Stewart, Phys. Rev. 128, 1112 (1962).
—
445, 1977. Pergamon Press. Printed in Great Britain.
RELATION BETWEEN THE ION-ION POTENTIAL AND PHONON SPECTRA IN ALKALI METALS
*
R. Day, F. Sun Department of Physics The Pennsylvania State University University Park, Pennsylvania 16802 USA and P. H. Cutler ** E.S.I.S., Universit~de Li~ge Institut de Physique B-4000 Sart Tilman/Li~ge, BELGIUZ4 (Received 4 April 1977 by R. H. Silsbee) Pair potentials and phonon curves for the alkalis were computed using an exact treatment of the conduction electron potential in a nonlocal a priori pseudopotential. It was found that pseudopotentials which yield quite similar dispersion curves can produce different ion—ion potentials.
We have recently done a priori nonlocal pseudopotential calculations which studied the significance of an exact versus an approximate treatment of 3 It the was conduction found that electron the exact ~,otentreatment tial) of the conduction electron potential (first formulated by Harrison4) had an appreciable effept upon the. phonon spectra of simple metals such as the alkalis, and excellent results were obtained .for these fully nonlocal calculations. This letter reports calculations, of the effec— tive ion—ion potential U Cr) for the alkali metals, that were based upon an a priori pseudo— potential which treated the conduction electron potential (i.e., the orthogonalization hole density and v 0~) exactly. Figure 1 shows two normalized energy wavenumber character (F~(q)) curves for Na. The F~(q) curves were calculated from two a priori nonlocalthat except pseudopotentials the core electron that wave werefunctions identical were chosen to be slightly different in the two calculations. The pair potential U(r) was calculated from FN (q) using equation (1), where *2
U(r)
=
z re
2
*2 + ~
¶
~ ~ sin qrqr FN (q)dq
However, the U(r) given by the solid curve does not exhibit any oscillations about zero which is a characteristic of the ion—ion potential in metals. The dashed curve does contain oscillations at largeU (r) r as it should. it is the evident that pseudopotentials which yield similar phonon spectra can give pair potentials that are quite different in their qualitative aspects (i.e., the oscillations at large r). This reaffirms Torrens • ~ observation that dis tinctly different pair potentials may reproduce very well the experimental phonon dispersion curves, and that, therefore, it is not suffi— cient to specify the pair potential by requiring it to predict good phonon spectra. The pseudopotentials used to calculate the curves in Figures 1, 2 and 3 used Singwi et al. .~6 dielectric function to describe the we found electron gas screening. However, 78 contrary to the assertions that the Hartreeofdielectric some researchers function yields a pair potential with a first minimum close to the nearest neighbor distance. The discrepancy may be due to the use of model potentials in the calculations by Shyu and Gaspari,7 and Duesbury and Taylor.8 Finally in an attempt to theoretically verify cochran’s9 method for extracting pair potential from phonon spectra, we computed phonon dispersion curves using a truncated F~(q). The truncation is a consequence of Cochran’s method which results in an F~(q) that is zero beyond the first reciprocal lattice vector. Hence, for Cochran’s method to be valid the truncation
.
The quantity z~ is the effective valence from pseudopotential theory and e is the electronic charge. Figure 2 shows the pair potentials U(r) corresponding to the F~(q) and Figure 3 shows the phonon spectra. The phonon spectra are both in excellent agreement with experiment,
*
This research was performed in part for the Applied Research Laboratory, The Pennsylvania State University under contract with Naval Sea Systems Command and in part under the auspices of the joint ESIS project (Electronic Structure in Solids) of the University of Antwerp and ~~the Universit~ de Li~ge. Permanent address: Department of Physics, 104 Davey Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802 USA. 443
ION—ION POTENTIAL AND PHONON SPECTRA IN ALKALI METALS
444
.0
Vol. 23, No. 7
.02
.9 .8 .7
.6 5 L
/
\\
I I I
.4
\
/
.3-
3~O
2.0
40
q/k,
1.
Normalized energy wavenumber character (F~(q)) curves for Na caicu— lated from a nonlocal a priori pseudopotential.
Slightly different
core electron wave functions were used to calculate the solid and dashed curves representing the energy wavenumber characteristic.
I
-.003-
I
I
I
I
I
I
I
I
I
\•._._/
I
5.0 2.
6.0
7.0
8.0
9.0 10.0 11.0 r IN BOHR UNITS
I
12.0
13.0
Na pair potentials calculated from the F~(q) curves shown in Figure 1.
14.0
ION—ION POTENTIAL AND PHONON SPECTRA IN ALKALI METALS
Vol. 23, No. 7 I
4.0
I
-
I
I
I
I
I
I
(001)
I
I
445 I
I
(III)
I
(II0~~
~ REDUCED WAVEVECTOR q/(2i,-/a) 3.
Na phonon dispersion curves calculated from the F~(q) of Figure 1. Experimental points are from Woods at al. [10]
procedure should have no significant effect upon the predicted pair potential and phonon spectra. It was found that setting F~(q) to zero beyond the first reciprocal lattice vector measurably affected our phonon dispersion results for Li and somewhat less for K but had no significant effect for Na. The U(r) of Li, Na, and K showed a similar dependence upon the truncation of FN (q). That is to say that for Na the truncation of F~(q) hardly affected U(r) while for K,U(r) was changed, though by only a few percent for most values of r. The trunca—
tion of F~(q)for Li produced changes in U(r) which were quite significant in comparison to Na and K. Hence though Cochran’s method is valid for Na and K it should be used with care when applied to less free electron—like metals such as Li where the truncation procedure of F~(q) may not be as acceptable. Acknowledgment - We wish to express our appreciation to Professor M. T. Pigott of the Applied Research Laboratory of The Pennsylvania State University for his support and encouragement.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
P. H. Cutler, R. Day and W. F. King III, J. Phys. F: Metal Phys. 5, 1801 (1975). R. S. Day, F. Sun and P. H. Cutler, J. Phys. F: Metal Phys. 6, L297 (1976) F. Sun, R. Day, P. H. Cutler and W. F. King III, J. Phys. F: Metal Phys. 6, Ll (1976). W. A. Harrison, Phys. Rev. 181, 1036 (1969). I. A. Torrens, Interatomic Potentials (Academic Press, 1972), P. 140. K. S. Singwi, A. Sj~lander, M. P. Tosi and R. H. Land, Phys. Rev. Dl, 1044 (1970). W. M. Shyu and G. D. Gaspari, Phys. Lett. 30A, 53 (1969). M. S. Duesbury and R. Taylor, Phys. Lett. 30A, 496 (1969). R. F. S. Cochran, Proc. Roy. Soc. 1276, 308 (1963). A. D. B. Woods, B. N. Brockhouse, R. H. March and A. T. Stewart, Phys. Rev. 128, 1112 (1962).