B. PRASAD and R. S. SRIVASTATA: Pseudopotential and Lattice Vibrations 327

phys. stat. sol. (b) 56, 327 (1973)

Subject classification: 6; 4; 21.3; 31.3

Department of Physics, Bunaras Hindu University, Vurunasi

Pseudopotential and Lattice Vibrations of Simple Metals BY

B. PRASAD and R. S. SRIVASTAVA

The phonon dispersion curves and binding energies of Rb, Cs, and Ba have been calculated using Harrison’s modified point ion pseudopotential with two adjustable parameters. The exchange and correlation effects in the Hartree dielectric function have been adeqiiately considered. The theoretical results are found to be in good qualitative agreement with the experimental results.

Es werden die Phononen-Dispersionskurven und Bindungsenergien von Rb, Cs und Ba mit dem modifizierten Punktionen-Pseudopotential von Harrison mit zwei anpaBbaren Parametern berechnet. Die Austausch- und Korrelstionseffekte in der Hartreeschen dielek- trischen Funktion werden adPquat berucksichtigt. Es wird gefunden, daB die theoretischen Ergebnisse in guter qualitativer Ubereinstimmung mit den experimentellen Werten sind.

1. Introduction The model potential method for studying the electronic structure and pro-

perties of simple metals was introduced by Heine and Abarenkov [l, 21. The method they proposed is essentially a fusion of the fundamental ideas in two older methods. As in the pseudopotential method [3 to 51, the fundamental idea is to replace the deep core potential a t each ion site by a weak model potential which is constructed to preserve the eigenvalues of the metal’s Schrodinger equation. The parameters of their model potential are determined in the spirit of the “quantum defect method” [6 ] . Several calculations of phonon dispersion curves in simple metals have been performed in recent years, using a variety of model potentials and pseudopotentials. For instance pseudopotential method has been used by Harrison [7, S] to compute dispersion relations for A1 and Pb. Vosko et al. [9] used an orthogonal plane wave approach for a non-local pseudo- potential to calculate dispersion curves in Na, Al, and Pb. Shaw and Pynn [lo] used Shaw’s [ll] optimized non-local model potential in a calculation for Mg. Ho [12] and Schneider and Stoll[13] used model potential technique. for studying the several properties of simple metals. A number of properties of alkali metals have been calculated by Brovman et, al. [14] using the two parameter model potential. Wallace [ 15 to 171 has incorporated Born-Mayer exchange repulsion between the first nearest neighbour ion cores and calculated the phonon fre- quencies, binding energies, and Gruneisen parameters in the case of Na, K, and Li using Harrison’s modified point ion pseudopotential with two adjustable param- cters. These calculations are in good qualitative agreement with the experi- mental results. The same method has been used here to study the lattice vi- brations of b.c.c. rubidium, cesium, and barium. The short range Born-Mayer exchange repulsion between ion cores is assumed to be negligible here. The two pseudopotential parameters were chosen so that the crystal binding energies were within 9% of the experimental one’s which are given in Table 1.

328 B. PRASAD and R. 8. SRIVASTAVA

2. Theoretical Details

We follow exactly the same theory as given by Wallace 1151 for Na and K . The bare pseudopotential form factor W , is represented by Harrison's modified point-ion model [ 5 ] , which contains two parameters P and p :

where q = IQ + kl,l with Q and It,, the phonon wavevector and reciprocal lattice vector, respectively; /3 and e are the model parameters whose values have been adjusted so as to obtain reasonable agreement with the experimental values of the binding energies. Z is the valence of the metal and Qo is the volume of the atomic polyhedra. I n order to introduce the exchange and correlation effects in the electron-electron interaction, Hubbard [18], Sham [19], Kleinman [20] and Geldart and Vosko [21] suggested various modifications to the Hartree di- electric function. I n the scheme of Hubbard and Sham the effective dielectric function can be written as

where T = q/2 k,, 1 = (n kE')-l and k, is the Fermi wave vector. The function f (q ) corrects for exchange and correlation effects among the conduction electrons. I n the Hartree approximation f ( q ) = 0. Following Hubbard and Sham a reason- able form for f ( q ) is

(3) where 6 is the exchange correlation parameter. In some recent calculations [9, 221, was computed from the compressibility of electron gas. Following Pines-Nozieres [23] interpolation approximation the parameter 5 can be written as

(4)

where r, is the radius of the sphere whose volume is average conduction-electron volume, which is given by

( 5 )

f ( q ) = 472 (!I2 + E kE*) >

6 = 0.910/(0.458 + 0.012 r,) ,

4 7c r: /3 = Q,/Z .

The expression for band structure energy Ebs may be written as

Ebs = .z ' fJ*(q) &Y) F ( q ) 3

P

where S(q) is the structure factor defined as

and F(q) the energy wave-number characteristic which is written as

The secular determinant for calculating phonon frequencies is obtained in the usual way as

iD,,(q) ~ wz I w21 = 0 , (9)

Pseudopotential and Lattice Vibrations of Simple Metals 329

where na is the mass of the atom, I the unitary matrix, and co the circular fre- quency. The dynamical matrix consists of two major contributions, viz., D:B(q) due to electrostatic energy between the ions; and DX(q) due to band structure energy, The contribution due to the ion-ion exchange repulsion is expected to be small and negligible here. Thus

Dndd = %(g) + Z / d P ) . (10)

The contributions for D:p(q) in the case of b.c.c. metals are taken directly from the paper of Srivastava and Srivastava [24]. The total band structure contri- butions to the 0: /? element of the dynamical matrix for wave vector q =+ 0 is given by

( ' 1 ) 2

e p ( q ) = 2 2 F ( 4 + h), (Y + h)B- 3 2' h, h, . h h

3. Computational Results

The model parameters and some relevant data used in the calculation are listed in the Table 1, which gives also the experimental and theoretical values of crystal binding energies for Rb, Cs, and Ba. The observed crystal binding ener-

T a b l e 1 Model parameters, some relevant data (atomic units), and the calculated and observed

crystal binding energies of the metals studied

metals 1 z

! Rb I 1

Ba ' 2 cs 1 1

~-

QO

587.0 744.5 421.7

B

85 105 131

e

0.88 1.07 1.30

- ~~

-zb (Ryd) ~~

calculated 1 observed ~~~~~

~ ~ 5 1 I ~ 7 1

0.392 ~ 0.3674 1 0.3696 0.376 0.3462 0.3468 1.336 ~ 1.296 I 1.311

gies for Rb, Cs, and Ba have been taken from the references [25] and 1271. The secular determinant for 47 points, which is equivalent to finding the vibrational frequencies for the 1000 points of the Brillouin-zone, was solved. Calculated phonon dispersion curves in the three symmetry directions are compared in Fig. 1

Fig. 1. Calculated phonon disper- sion curves (solid lines) compared with the 120 OK measurements of

Copley e t a]. [26] for Rb

330 B. PRASAD and R. S. SRIVASTAVA

9 4

5 9 4

with the experimental results of Copley et al. [26] for Rb. As is evident from the Fig. 1 a reasonably good agreement with the experimental results is obtained. The calculated phonon dispersion curves for Cs and Ba have been shown in Fig. 2 and 3, respectively. No comparison has been made for these elements for lark of experimental data on it.

4. Summary and Discussion

('alculations of the phonon frequencies and binding energies for Rb, C's, and Ba have been made using a theory based on long range Coulomb interaction among the ions, short range Born-Mayer repulsion between ion-cores (which is assumed to be negligible here) and a local pseudopotential interaction between ions and conduction electrons. Fig. 1 compares the calculated phonon frequencies for b C.C. Rb with the 180 "I( measurements of Copley et al. [26]. The agreement between theory and experiment is qualitatively good. However, a t boundary in LOOl]-direction, the theoretically calculated frequency is slightly lower than the experimental one. Such discrepancies reflect the limitations of local pseudo- potential and the approximate treatment of the exchange and correlation effects amongst the conduction electrons. We might conclude by considering the steps that could be taken to bring the theory into better agreement with experiment. For this, first we recall, the well known remarks of Harrison [5] that the pseudo- potential perturbation theory is by nature approximate and should not be ex- pected to reproduce all the experimental results. One can expect to improve the agreement by including higher order pseudopotential term in the dynamical matrix. Secondly, the accuracy of the Hubbard and Sham approximation in- cluded in this work is quite unknown at present. Lastly the neglect of the short range Born-Mayer exchange repulsion term between ion cores may have effected the results slightly.

Pseudopotential and Lattice Vibrations of Simple Metals 331

A cknowledgem,ents

The authors are very grateful t o Prof. D. C. Wallace for encouraging ub to undertake this calculation and for giving helpful advice while the work was in progress. We would like to thank Prof. 3. Dayal for his keen interest in the work and Dr. R. N. Srivastava, Shri J. A. Mathew, B. N. Lal, and G. K. Srivastava for the help in calculation. One of us (B. P.) thanks Dr. 0. P. Mathur for his help in many wags.

References

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(Receiwed December 11, 1972)