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Phonon dispersion relations in beryllium using the pseudo potential approach
PHYSICS
Volume 23, number 11
PHONON
DISPERSION PSEUDO
LETTERS
RELATIONS POTENTIAL
12 December 1966
IN BERYLLIUM APPROACH
USING
THE
V. C. SAHNI, G. VENKATARAMAN and A. P. ROY Nuclear Physics Division, Atomic Energy Establishment Trombay, Bombay
74, India
Received 23 November 1966
Phonon dispersion relations in beryllium are calculated using the pseudo potential method and compared with the experimental data of Schmunk et al.
We report in this letter some results for phonon dispersion relations in beryllium obtained from first principles using the pseudo potential approach of Sham [l]. Be has the h.c.p. structure, and in this there are two atoms per unit cell. The dynamical matrix will consequently be six-dimensional, and the eigenvalues of this matrix will be proportional to the squares of the six frequencies associated with the phonon wave vector q. In the psueudo-potential approach, there will be three contributions to the elements of the dynamical matrix arising respectively from (i) Coulomb interactions between the ions and that between the ions and the uniform compensating background (ii) core-core repulsion and (iii) electronic effects. The contribution from (i) was computed using the methods previously used by Kellermann [2]. The contribution from (ii) was ignored since the size of the core in Be is small compared to the nearest neighbour distance [3]. The electronic contribution was computed using the following expressions derived by Roy and Venkataraman [4] in the pseudo potential formalism for lattices with a basis (notation for the dynamical matrix is the same as in the book by Born and Huang [5]).
- LA-‘/
0.2
9 /Qmax
0.4
0.6
0.8
1.0
9/9max
Fig. 1. Phonon dispersion relations in beryllium. The dashed lines denote theoretical calculations while the solid lines denote the experimental results of Schmunk et al. [7]. 633
Volume 23. number 11
PHYSICS
=+Cl&
D$
(&j
=?&~[(T+Q)~
LETTERS
12 December
(7+9Eb+q)
1966
- (7)(y (T)@E(T)F(k)
(7+g)gexp[-i7.r(k’k)]E(T+q)~
(1) .
In the above equations n is the number of atoms per unit cell, Z the valency and M the mass of the ion. Further T is a reciprocal lattice vector, r(k’k) is the vector distance of the k’ th ion from the k th ion in the unit cell and F(k) = c cos [T . r(k 'k)] . The function E(q) in the local pseudo potential approximation is k’ given by E(q)
= -$[Wq)]2
[x(q)
/dq)]
(3)
,
where U(q) is the Fourier transform of the local pseudo potential, E(q) is the wave vector dependent dielectric function of the electron gas and x(q) is as defined by Sham. We have calculated U(q) using the recent band structure results of Louks and Cutler [6] and the averaging procedure of Sham. The electronic contribution to the dynamical matrix was computed using this U(q). For c(q) we used Sham’s expression which takes some account of exchange effects. The results for the dispersion relations along [OOOl] and [OliO] directions are summarised in fig. 1 where the dashed lines show the theoretical curves and the solid lines the experimental results of Schmunk et al. [7]. The agreement between theory and experiment for the transverse branches along [OOOl] is fairly good. This is not surprising since these branches are not much affected by the conduction electrons [4]. The real test of our calculations lies in the results for the longitudinal branches for which the screening effects are very large. Here the agreement leaves room for improvement in spite of the fact we have used a potential based on a careful band structure calculation. It seems a better treatment of exchange and screening is necessary before improved quantitative agreement can be obtained. Refmences 1. L.J.Sham. Proc.Roy.Soc. (London). A283 (1965) 33. 2. E.W.Kellermann, Phil.Trans.Roy.Soc.A238 (1940) 513. 3. C.Herring and A.G.Hill, Phys.Rev.58 (1940) 132. 4. A. P.Roy and G.Venkataraman, to be published. 5. M. Born and K. Huang. Dynamical theory of crystal lattices (Oxford Univ. Press, 1954). 6. T.A.Louks and P.H.Cutler, Phys.Rev.133 (1964) A819. and K.A.Strong. Phys.Rev.128 (1962) 562. 7. R. E .Schmunk. R. M. Brugger, P.D.Randolph *****
THE
LOW
TEMPERATURE
SPECIFIC
B. C . PASSENHEIM, University
HEAT
OF
D. C . McCOLLUM Jr. of California, Riverside.
Received
21 November
EuS
IN A MAGNET
FIELD*
and J. CALLAWAY California
1966
The heat capacity of ferromagnetic EuS has been measured in a magnetic field at liquid helium temperatures. The experimental results compare favorably with spin wave calculations and indicate first and second neighbor exchange interactions of Jl/kg = 0.20°K and of J2/kB = -0.06’K respectively.
The heat capacity of EuS, a ferromagnetic insulator with the NaCl structure, has been measured at liquid helium temperatures for internal fields of 0 [I], 5 820 and 10 800 Oe. These measurements have been compared with the predictions 634
of spin wave theory for the Heisenberg net as given by Holstein and Primakoff
ferromag[2]. Terms
* Supported by the U.S.Air Force Office of Scientific Research.
Volume 23, number 11
PHONON
DISPERSION PSEUDO
LETTERS
RELATIONS POTENTIAL
12 December 1966
IN BERYLLIUM APPROACH
USING
THE
V. C. SAHNI, G. VENKATARAMAN and A. P. ROY Nuclear Physics Division, Atomic Energy Establishment Trombay, Bombay
74, India
Received 23 November 1966
Phonon dispersion relations in beryllium are calculated using the pseudo potential method and compared with the experimental data of Schmunk et al.
We report in this letter some results for phonon dispersion relations in beryllium obtained from first principles using the pseudo potential approach of Sham [l]. Be has the h.c.p. structure, and in this there are two atoms per unit cell. The dynamical matrix will consequently be six-dimensional, and the eigenvalues of this matrix will be proportional to the squares of the six frequencies associated with the phonon wave vector q. In the psueudo-potential approach, there will be three contributions to the elements of the dynamical matrix arising respectively from (i) Coulomb interactions between the ions and that between the ions and the uniform compensating background (ii) core-core repulsion and (iii) electronic effects. The contribution from (i) was computed using the methods previously used by Kellermann [2]. The contribution from (ii) was ignored since the size of the core in Be is small compared to the nearest neighbour distance [3]. The electronic contribution was computed using the following expressions derived by Roy and Venkataraman [4] in the pseudo potential formalism for lattices with a basis (notation for the dynamical matrix is the same as in the book by Born and Huang [5]).
- LA-‘/
0.2
9 /Qmax
0.4
0.6
0.8
1.0
9/9max
Fig. 1. Phonon dispersion relations in beryllium. The dashed lines denote theoretical calculations while the solid lines denote the experimental results of Schmunk et al. [7]. 633
Volume 23. number 11
PHYSICS
=+Cl&
D$
(&j
=?&~[(T+Q)~
LETTERS
12 December
(7+9Eb+q)
1966
- (7)(y (T)@E(T)F(k)
(7+g)gexp[-i7.r(k’k)]E(T+q)~
(1) .
In the above equations n is the number of atoms per unit cell, Z the valency and M the mass of the ion. Further T is a reciprocal lattice vector, r(k’k) is the vector distance of the k’ th ion from the k th ion in the unit cell and F(k) = c cos [T . r(k 'k)] . The function E(q) in the local pseudo potential approximation is k’ given by E(q)
= -$[Wq)]2
[x(q)
/dq)]
(3)
,
where U(q) is the Fourier transform of the local pseudo potential, E(q) is the wave vector dependent dielectric function of the electron gas and x(q) is as defined by Sham. We have calculated U(q) using the recent band structure results of Louks and Cutler [6] and the averaging procedure of Sham. The electronic contribution to the dynamical matrix was computed using this U(q). For c(q) we used Sham’s expression which takes some account of exchange effects. The results for the dispersion relations along [OOOl] and [OliO] directions are summarised in fig. 1 where the dashed lines show the theoretical curves and the solid lines the experimental results of Schmunk et al. [7]. The agreement between theory and experiment for the transverse branches along [OOOl] is fairly good. This is not surprising since these branches are not much affected by the conduction electrons [4]. The real test of our calculations lies in the results for the longitudinal branches for which the screening effects are very large. Here the agreement leaves room for improvement in spite of the fact we have used a potential based on a careful band structure calculation. It seems a better treatment of exchange and screening is necessary before improved quantitative agreement can be obtained. Refmences 1. L.J.Sham. Proc.Roy.Soc. (London). A283 (1965) 33. 2. E.W.Kellermann, Phil.Trans.Roy.Soc.A238 (1940) 513. 3. C.Herring and A.G.Hill, Phys.Rev.58 (1940) 132. 4. A. P.Roy and G.Venkataraman, to be published. 5. M. Born and K. Huang. Dynamical theory of crystal lattices (Oxford Univ. Press, 1954). 6. T.A.Louks and P.H.Cutler, Phys.Rev.133 (1964) A819. and K.A.Strong. Phys.Rev.128 (1962) 562. 7. R. E .Schmunk. R. M. Brugger, P.D.Randolph *****
THE
LOW
TEMPERATURE
SPECIFIC
B. C . PASSENHEIM, University
HEAT
OF
D. C . McCOLLUM Jr. of California, Riverside.
Received
21 November
EuS
IN A MAGNET
FIELD*
and J. CALLAWAY California
1966
The heat capacity of ferromagnetic EuS has been measured in a magnetic field at liquid helium temperatures. The experimental results compare favorably with spin wave calculations and indicate first and second neighbor exchange interactions of Jl/kg = 0.20°K and of J2/kB = -0.06’K respectively.
The heat capacity of EuS, a ferromagnetic insulator with the NaCl structure, has been measured at liquid helium temperatures for internal fields of 0 [I], 5 820 and 10 800 Oe. These measurements have been compared with the predictions 634
of spin wave theory for the Heisenberg net as given by Holstein and Primakoff
ferromag[2]. Terms
* Supported by the U.S.Air Force Office of Scientific Research.