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Linear potential and phonons in calcium and strontium
0038-1098/81/231083-03502.00/0
Solid State Communications, Vol. 38, pp. 1083-1085. Pergamon Press Ltd. 1981. Printed in Great Britain.
LINEAR POTENTIAL AND PHONONS IN CALCIUM AND STRONTIUM K.S. Sharma and C.M. Kachhava Physics Department, University of Rajasthan, Jaipur 302 004, India
(Received 16January 1981 byE. Burstein) The success recently achieved by the linear form of electron-ion pseudopotential in the case of alkalies, A1 and Pb has led to further apply it to the bivalent metals, Ca and Sr, for the calculation of phonon propagation along the three symmetry directions. The results obtained compare well with the value based on the more sophisticated formulations of Animalu and Moriarty. The reliability of these results is confirmed through an excellent reproduction of the experimental binding energy and compressibility. A successful prediction of elastic constants further highlights the usefulness of the present simple approach. THE PHONON SPECTRA in calcium and strontium have been theoretically investigated by Animalu [1 ] through the use of Heine-Abarenkov psuedopotential [2], and by Moriarty [3, 4] on the basis of generalized s-d hybridization pseudopotential theory [5, 6]. The present motivation has its roots in two grounds. First, the experimental data for phonon dispersion in these metals are not available; secondly, even the inclusion of s-d hybridization in the most sophisticated theory of Moriarty [3] has led to the conclusion that the hybridization is found to make important contribution neither to the binding energy nor to the phonon frequencies. Thus, it would appear that the complexities like the inclusion of s-d hybridization and nonlocality in any pseudopotential formalism are redundant for the cases under consideration. We, therefore, attempt to predict certain physical properties like phonon dispersion relations, binding energy, compressibility and elastic constants of Ca and Sr through a simple one parameter form of the local pseudopotential, the linear potential, which has been so successfully used by the present authors for the prediction of these properties in Sodium [7] and other alkali metals [8], A1 [9] and Pb [I0]. The proposed potential, V(r), is an attractive form of the electron-ion pseudo-potential, which varies linearly with distance in the core region (with effective core radius RM) and acquires the Coulombian character outside the core. Its matrix elements for the electronic scattering from the state k to k + q are given, in Rydbergs, by
_
~2oRMq 16rrz 3 [sinqR M 1aRM (1 _ c o S q R M ) ] ,
(1)
where z is the valence, and ~2o represents the atomic volume. The self-consistent type of the dielectric screening, e(q). due to Singwi et at [11 ] appropriately explains the behaviour of the electron gas in these metals, which form is therefore used to obtain the relevant screened form factors. Defining the energy wavenumber characteristics by [12]
F(q)
=
aoq2 -
v(q)
1-
16---7
1 ,
(2)
the phonon frequencies v (q, ~) for wave-vector q and polarization branch p are obtained as eigenvalues of the Dynamical matrix D~0 (q) appearing in the secular determinantal equation:
[Dc~#(q)--My (q, p)II
V(q) = (k + qlVlk)
Present address: Department of Physics, M.S.J. College, Bharatpur-321001, India.
= 0,
(3)
where I is the unit matrix and M is the ionic mass. The core-overlap contribution to the dynamical matrix is neglected in view of the smallness of the core size compared to the interatomic separation in these metals, whereas the direct core-core Coulombian contribution is estimated in the well-known and standard technique of Kellerman [13]. The electronic contribution refers to the electron-ion pseudopotential. As far as other physical properties are concerned, the small wave-number behaviour of the calculated phonon dispersion curves along the [100] longitudinal, [100] transverse, and the [110] lower transverse branches can be easily exploited to predict c11, c12 and c44. On the other hand, the calculations of the binding energy and the compressibility of these metals could be made on the lines of Kachhava and Sharma [14].
1083
1084
LINEAR POTENTIAL AND PHONONS IN CALCIUM AND STRONTIUM
Table 1. Potential parameter RM (a.u.), binding energy Uo (Ryd/atom), compressibility f2o K (Ryd/atom), and elasn'c constants (x 10 -3 Ryd a.u. -3) of Ca and Sr Metal
RM Uo
~oK
Ca
Present Modarty Exptl. Present Exptl.
CI1 C12 C44
Sr
2.70
3.00
-- 1.488 -- 1.378 -- 1.457
-- 1.342 -- 1.206 -- 1.355
0.307 0.303
0.378 0.300
1.791 1.236 1.428
1.359 1.302 1.169
The values of the potential parameter RM, determined for a best fit of the phonon spectra of Ca to the results of Moriarty, and of Sr to the results of Animalu and Moriarty are included in Table 1. The calculated phonon frequencies for Ca have been displayed in Fig. 1, where the best available results of Moriarty [4] have also been depicted by O, A for the sake of comparison. It is gratifying to note that the phonon frequencies from the present calculation, based on a simple potential, exhibit good accord with the values of Moriarty [4]. The average absolute percentage deviations [AAPD] for the [100] longitudinal and transverse branches are respectively 4.9 and 2.8, whereas for [111 ] longitudinal and transverse branches these values are respectively 6.2 and 5.2. The matching between the
two sets of frequencies is excellent for [110] longitudinal and the lower and upper transverse branches, for which the AAPD are 3.2, 3.7 and 1.8 respectively. The results of Animula, being higher by more than 10%, cannot provide a meaningful comparison, and hence these have not been included in Fig. 1. The phonon dispersion curves for Sr, obtained from the present simple approach, have been compared with those of Animula [1] and Moriarty [3] in Fig. 2. It may be observed that the phonon frequencies from the present work show an excellent agreement with the values of Animula for longitudinal branches, and an AAPD of the order of 7.5 for the transverse branches; whereas the AAPD is nearly 4 for both the longitudinal and transverse branches obtained by Moriarty. It is gratifying to note that these deviations are of the order of differences between the results of Animula and Moriarty. A further test for the applicability of the simplemetal pseudopotential theory to alkaline earth metals is provided through the successful reproduction of the other metallic properties. The computed values of binding energy, compressibility and elastic constants are reported in Table 1, which also contains the experimental data of binding energy [3] and compressibility [ 15 ], while the experimental data for elastic constants of these metals have not been so far made available in the literature. For the sake of comparison, we include the best set of binding energy values also from the Moriarty's work [3]. It is gratifying to note that the present values of binding energy show an exquisite
4.0
[roo]
[J"]/
[~,o]
L 3.0
o
L
"7
2.0
1.0
I 0.5 2~"
Fig. I. Phonons in Ca.
I 1.0
Voi. 38, No. 11
0.5
4"2
0
0.5
Vol. 38, No. 11
LINEAR POTENTIAL AND PHONONS IN CALCIUMAND STRONTIUM 40
Re~.~ts of oresenl" work Results of Moriort'y Res4JIts of Antmolu
3O
[,oo]
1085
[,,o]
'f
'an e2_
o_ ~_ 2 , 0
OIO
I 05
I 1.0
0.5
0
0.5
C2---~ ~) Fig. 2. Phonons in St. accord with the experimental data, and are in better agreement with these data as compared to the Moriarty's best set of results. Also a very good agreement obtained. between the presently calculated $20K values, where K represents the Bulk modulus, and the experimental data justify the philosophy envisaged in the present investigation, and also confirms the validity of the calculated phonon dispersion curves. The usefulness of the present simple approach is further highlighted through the successful prediction of elastic constants, displayed in Table 1. Though the absence of experimental data for these constants have refrained us from making a useful comparison, yet the overall success met by the theory justifies the applicability of the second order perturbation theory to Ca and Sr. In conclusion, it may be observed that the theoretical results for phonon frequencies and elastic constants are useful and instructive for other applications. We may term these results as reliable in view of the close harmony between the experimental and theoretical values of binding energy and compressibility.
ties. One of the authors (K.S.S.) thanks U.G.C., New Dehli for providing him a teacher fellowship.
REFERENCES
I. 2. 3. 4. 5. 6. 7. 8.
A.O.E. Animula,Phyg Rev. 161,445 (1967). V. Heine & I. Abarenkov, Phil. Mag. 9, 451 (1964). J.A. Moriarty, Phyg Rev. B6, d445 (1972). J.A. Moriarty, Phys. Rev. 138, 1338 (1973). W.A.Harrison, Phys. Rev. 181, 1036 (1969). J.A. Moriarty, Phys. Rev. B5, 2066 (1972). K.S. Sharma & C.M. Kachhava, Solid State Commurt 30,749 (1979). K.S. Sharma & C.M. Kachhava, Czech. J. Phys.
S30, 531 (1980). 9. I 0. 11. 12.
13. Acknowledgements - The authors acknowledge gratefully the computer facilities extended for this work by 14. the Director, C.E.E.R.I., Pilani, and to Shri R.P. Agrawal for useful discussions. They are also thankful to Dr M.P. 15. Saksena for providing the necessary departmental facili-
K.S. Sharma & C.M. Kachhava, Indian J, Pure AppL Phy~ 18,536 (1980). K.S. Sharma & C.M. Kachhava, Indian J. Phys. (at press). K.S. Singwi, A. Sj61ander, M.P. Tosi & R.H. Land, Phys. Rev. B1, 1044 (1970). W.A.Harrison, Pseudopotentials in the Theory of Metals. Benjamin, New York (1966). E.W.Kellerman, Phil Tran~ Roy Soc. (London) A238, 513 (1940). C.M.Kaehhava & K.S. Sharma, Phys. Status Solidi (b) 97,601 (1980). C. Kittel, Introduction to Solid State Physics, V edn., p. 85. Wiley, New York (1976).
Solid State Communications, Vol. 38, pp. 1083-1085. Pergamon Press Ltd. 1981. Printed in Great Britain.
LINEAR POTENTIAL AND PHONONS IN CALCIUM AND STRONTIUM K.S. Sharma and C.M. Kachhava Physics Department, University of Rajasthan, Jaipur 302 004, India
(Received 16January 1981 byE. Burstein) The success recently achieved by the linear form of electron-ion pseudopotential in the case of alkalies, A1 and Pb has led to further apply it to the bivalent metals, Ca and Sr, for the calculation of phonon propagation along the three symmetry directions. The results obtained compare well with the value based on the more sophisticated formulations of Animalu and Moriarty. The reliability of these results is confirmed through an excellent reproduction of the experimental binding energy and compressibility. A successful prediction of elastic constants further highlights the usefulness of the present simple approach. THE PHONON SPECTRA in calcium and strontium have been theoretically investigated by Animalu [1 ] through the use of Heine-Abarenkov psuedopotential [2], and by Moriarty [3, 4] on the basis of generalized s-d hybridization pseudopotential theory [5, 6]. The present motivation has its roots in two grounds. First, the experimental data for phonon dispersion in these metals are not available; secondly, even the inclusion of s-d hybridization in the most sophisticated theory of Moriarty [3] has led to the conclusion that the hybridization is found to make important contribution neither to the binding energy nor to the phonon frequencies. Thus, it would appear that the complexities like the inclusion of s-d hybridization and nonlocality in any pseudopotential formalism are redundant for the cases under consideration. We, therefore, attempt to predict certain physical properties like phonon dispersion relations, binding energy, compressibility and elastic constants of Ca and Sr through a simple one parameter form of the local pseudopotential, the linear potential, which has been so successfully used by the present authors for the prediction of these properties in Sodium [7] and other alkali metals [8], A1 [9] and Pb [I0]. The proposed potential, V(r), is an attractive form of the electron-ion pseudo-potential, which varies linearly with distance in the core region (with effective core radius RM) and acquires the Coulombian character outside the core. Its matrix elements for the electronic scattering from the state k to k + q are given, in Rydbergs, by
_
~2oRMq 16rrz 3 [sinqR M 1aRM (1 _ c o S q R M ) ] ,
(1)
where z is the valence, and ~2o represents the atomic volume. The self-consistent type of the dielectric screening, e(q). due to Singwi et at [11 ] appropriately explains the behaviour of the electron gas in these metals, which form is therefore used to obtain the relevant screened form factors. Defining the energy wavenumber characteristics by [12]
F(q)
=
aoq2 -
v(q)
1-
16---7
1 ,
(2)
the phonon frequencies v (q, ~) for wave-vector q and polarization branch p are obtained as eigenvalues of the Dynamical matrix D~0 (q) appearing in the secular determinantal equation:
[Dc~#(q)--My (q, p)II
V(q) = (k + qlVlk)
Present address: Department of Physics, M.S.J. College, Bharatpur-321001, India.
= 0,
(3)
where I is the unit matrix and M is the ionic mass. The core-overlap contribution to the dynamical matrix is neglected in view of the smallness of the core size compared to the interatomic separation in these metals, whereas the direct core-core Coulombian contribution is estimated in the well-known and standard technique of Kellerman [13]. The electronic contribution refers to the electron-ion pseudopotential. As far as other physical properties are concerned, the small wave-number behaviour of the calculated phonon dispersion curves along the [100] longitudinal, [100] transverse, and the [110] lower transverse branches can be easily exploited to predict c11, c12 and c44. On the other hand, the calculations of the binding energy and the compressibility of these metals could be made on the lines of Kachhava and Sharma [14].
1083
1084
LINEAR POTENTIAL AND PHONONS IN CALCIUM AND STRONTIUM
Table 1. Potential parameter RM (a.u.), binding energy Uo (Ryd/atom), compressibility f2o K (Ryd/atom), and elasn'c constants (x 10 -3 Ryd a.u. -3) of Ca and Sr Metal
RM Uo
~oK
Ca
Present Modarty Exptl. Present Exptl.
CI1 C12 C44
Sr
2.70
3.00
-- 1.488 -- 1.378 -- 1.457
-- 1.342 -- 1.206 -- 1.355
0.307 0.303
0.378 0.300
1.791 1.236 1.428
1.359 1.302 1.169
The values of the potential parameter RM, determined for a best fit of the phonon spectra of Ca to the results of Moriarty, and of Sr to the results of Animalu and Moriarty are included in Table 1. The calculated phonon frequencies for Ca have been displayed in Fig. 1, where the best available results of Moriarty [4] have also been depicted by O, A for the sake of comparison. It is gratifying to note that the phonon frequencies from the present calculation, based on a simple potential, exhibit good accord with the values of Moriarty [4]. The average absolute percentage deviations [AAPD] for the [100] longitudinal and transverse branches are respectively 4.9 and 2.8, whereas for [111 ] longitudinal and transverse branches these values are respectively 6.2 and 5.2. The matching between the
two sets of frequencies is excellent for [110] longitudinal and the lower and upper transverse branches, for which the AAPD are 3.2, 3.7 and 1.8 respectively. The results of Animula, being higher by more than 10%, cannot provide a meaningful comparison, and hence these have not been included in Fig. 1. The phonon dispersion curves for Sr, obtained from the present simple approach, have been compared with those of Animula [1] and Moriarty [3] in Fig. 2. It may be observed that the phonon frequencies from the present work show an excellent agreement with the values of Animula for longitudinal branches, and an AAPD of the order of 7.5 for the transverse branches; whereas the AAPD is nearly 4 for both the longitudinal and transverse branches obtained by Moriarty. It is gratifying to note that these deviations are of the order of differences between the results of Animula and Moriarty. A further test for the applicability of the simplemetal pseudopotential theory to alkaline earth metals is provided through the successful reproduction of the other metallic properties. The computed values of binding energy, compressibility and elastic constants are reported in Table 1, which also contains the experimental data of binding energy [3] and compressibility [ 15 ], while the experimental data for elastic constants of these metals have not been so far made available in the literature. For the sake of comparison, we include the best set of binding energy values also from the Moriarty's work [3]. It is gratifying to note that the present values of binding energy show an exquisite
4.0
[roo]
[J"]/
[~,o]
L 3.0
o
L
"7
2.0
1.0
I 0.5 2~"
Fig. I. Phonons in Ca.
I 1.0
Voi. 38, No. 11
0.5
4"2
0
0.5
Vol. 38, No. 11
LINEAR POTENTIAL AND PHONONS IN CALCIUMAND STRONTIUM 40
Re~.~ts of oresenl" work Results of Moriort'y Res4JIts of Antmolu
3O
[,oo]
1085
[,,o]
'f
'an e2_
o_ ~_ 2 , 0
OIO
I 05
I 1.0
0.5
0
0.5
C2---~ ~) Fig. 2. Phonons in St. accord with the experimental data, and are in better agreement with these data as compared to the Moriarty's best set of results. Also a very good agreement obtained. between the presently calculated $20K values, where K represents the Bulk modulus, and the experimental data justify the philosophy envisaged in the present investigation, and also confirms the validity of the calculated phonon dispersion curves. The usefulness of the present simple approach is further highlighted through the successful prediction of elastic constants, displayed in Table 1. Though the absence of experimental data for these constants have refrained us from making a useful comparison, yet the overall success met by the theory justifies the applicability of the second order perturbation theory to Ca and Sr. In conclusion, it may be observed that the theoretical results for phonon frequencies and elastic constants are useful and instructive for other applications. We may term these results as reliable in view of the close harmony between the experimental and theoretical values of binding energy and compressibility.
ties. One of the authors (K.S.S.) thanks U.G.C., New Dehli for providing him a teacher fellowship.
REFERENCES
I. 2. 3. 4. 5. 6. 7. 8.
A.O.E. Animula,Phyg Rev. 161,445 (1967). V. Heine & I. Abarenkov, Phil. Mag. 9, 451 (1964). J.A. Moriarty, Phyg Rev. B6, d445 (1972). J.A. Moriarty, Phys. Rev. 138, 1338 (1973). W.A.Harrison, Phys. Rev. 181, 1036 (1969). J.A. Moriarty, Phys. Rev. B5, 2066 (1972). K.S. Sharma & C.M. Kachhava, Solid State Commurt 30,749 (1979). K.S. Sharma & C.M. Kachhava, Czech. J. Phys.
S30, 531 (1980). 9. I 0. 11. 12.
13. Acknowledgements - The authors acknowledge gratefully the computer facilities extended for this work by 14. the Director, C.E.E.R.I., Pilani, and to Shri R.P. Agrawal for useful discussions. They are also thankful to Dr M.P. 15. Saksena for providing the necessary departmental facili-
K.S. Sharma & C.M. Kachhava, Indian J, Pure AppL Phy~ 18,536 (1980). K.S. Sharma & C.M. Kachhava, Indian J. Phys. (at press). K.S. Singwi, A. Sj61ander, M.P. Tosi & R.H. Land, Phys. Rev. B1, 1044 (1970). W.A.Harrison, Pseudopotentials in the Theory of Metals. Benjamin, New York (1966). E.W.Kellerman, Phil Tran~ Roy Soc. (London) A238, 513 (1940). C.M.Kaehhava & K.S. Sharma, Phys. Status Solidi (b) 97,601 (1980). C. Kittel, Introduction to Solid State Physics, V edn., p. 85. Wiley, New York (1976).