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Relativistic spin-polarized single site scattering theory
Journal
of Magnetism and Maguctic Materials 140-144 (1995) 37-38 M M --
ELSEVIER
A
Relativisticspin-polarizedsinglesite scatteringtheory AC Jenkins *, P. Strange Physics Department, Keele Unicersiry, Keele, Staflordshire, ST5 5BG, UK
Abstract
We present a fully relativistic single site scattering theory for magnetic systems. The radial Dirac equations for an electrcn in a potential with a magnetic field component are solved and scattering amplitudes found. New coupling is observed due to the removal of previous approximations. The impbcations for observables is discussed.
Most present-day work on the electronic structure of condensed matter is bajed on density functional theory [1,9-l. In the late 1970s this was generalized to treat relativistic systems /3,4]. McDonald and Vosko (31 developed the theory for a many electron system in the presence of a ‘spin-only’ magnetic field, i.e. ignoring diamagnetic effects. This is a suitable ‘Jask for describing electrons in condensed matter, In the mid 1980s Strange et al. I51and Feder et al. [6] tacklcd this problem. They described a relativistic electron scattr .;ng from a single effective poteniial with a magnetic component. The parameters from this theory were then used in a multiple scattering theory [7] to perform relativistic spin-poiarized KKR calculations. From this work many observables have been calculated [S]. The radial Kohn-Sham-Dirac equation for an electron experiencing a potential with ;i magnetic component becomes two infinite sets of coupled partial differential equations, for each value of the ltf, quantum number. By neglecting coupling between states 1 and I f 2 this can be reduced to sets of four coupled differential equations. These have been solved [s] and incorpotated into a single site scattering theory. Recently Jenkins and Strange [9] have reintroduced the coupling and made the alternative approximation that the wavefunctions for 1~ 6 are set equal to zero. This brings the number of coupled differential equations to a number of urder ten, depending upon mj. These were solved and the authors found that for some applications the previously ignored coupling may be significant. Here we show explicitiy that the error introduced by the neglect of this coupling is of the same order as the neglect of higher order terms diagonal in 1. From the Lippman-Schwinger equation it has been
* Corresponding author. Rx -1-44-782-711093.
shown that the Kohn-Sham-Dirac equation can be separated into radial and angular parts, the radial part is a+f;T+)) ar
=~cf~~(r)-(E-V~~~(r))rg~;(r) +Beff(r}~G(d’, KS
K’, mj)
rg?,Jr), (la)
a( W%(r))
= +B’“jr)~G(-K”.-
K’, m,)ct$$(r).
(lb)
2 K is the usual spin-angular quantum number [IO]. For the derivation of these equations see Strange et al. [5,9]. The full wavefunction inside the region of the potential is then
(2)
If it is asztmed that both the potential, Vcff(r), and the magnetic field. Bee(r), are zero outside the muffin-tin radius, r,,,, then the radial solutions outside the muffin-tin sphere have the form of a radi.31 spherical wave 191 and these two solutions can be matched at rm to calculate scattering amplitudes. In principle this means matching two infinite column vectors for each value of K (= KI, . . .,K”, . . . ) - but, due to the cutoff in I and the fact that mj remains a good quantum number, this reduces to, at
03048853/95/$09.50 8 19% Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00846-u
A.C. Jenkins, P. Strange/Journal
of Magnetism and Magnetic Materials 140-144 hW5~ 37-38
worst, the matching of two 12 component vectors. For details of the matching procedure see Ref. f9f. In Fig. 1 we show the scattering amplitudes for a single Pa atom in a spin-only magnetic field as a function of
WY
(Ryd)
energy. The quantum numbers for this particular scattering amplitude are K = 2, K’ = - 1, mi = l/2, i.e. this scatter-
ing amplitude represents coupling between the I = 2 and are drawn, for various magnetic fields, and the more pronounced curves are for greater fields. In Fig. 2 the scattering amplitude for K = -4, K’ = -4, m, = l/2 (i.e. I- I’ = 3) is drawn for the same magnetic fields. All the different field curves lie virtually on top of one another, however, we see that in the energy range of interest in condensed matter between zero and one Rydbergs, these scattering amplitudes are of the same order of magnitude. From this we can conclude that the error in omitting coupling between 1 and 1 f 2 is of the same order as that caused by omitting I values above the highest 1 value in the outer electronic shells. Hence it makes no sense to include this coupling without expanding the size of the spherical harmonic basis set. Conversely, it makes no sense to increase the size of the basis set and hence the MUGmatrix if this coupling is ignored.
1= 0. Several. curves
for a
0.10
Fig. 2. The K = -4, K’ = -4, m, = l/2 scattering amplitude for a single Pt atom potent:al for several different magnetic fields. In summary
then, we have shown explicitly
that two of
the approximations which go into spin-polarized reiativistic electronic structure calculations are of the same order and hence that there is nothing to be gained by removing one of these approximations without the other. References [l] P. Hohenberg and W. Kohn, Phys. Rev. 5 136 (1964) 864. [2] W. Kohn and L.J. Sham, Phys. Rev. A 140 (1965) 1133. [3] A.H. MacDonald and S. Vosko, J. Phys. C: Solid State Phys. 12 (1979) 2977. [4] M.V. Ramana and A.K. Rajagopal, J. Phys. C: Solid Stale Phys. 12 (1979) I-845. 151P. Strange, LB. Staunton rrd B.L. Gyorffy, J. Phys. C: Solid State Phys. 17 (1984) 3355. [6] R. Feder, F. Rosicky and B. Ackccmann, 2. Phys. B 52 (1983) 31. [7] P. Strange, H. Ehert, J.B. Staunton and B.L. Gyorffy, 1. Phys. Condens. Matter 1 (1989) 2959. 181 S. Wilson, I.P. Grant and B.L. Gyorffy, eds., The Effects of Relativity in Atoms Molecules and the Solid 9tate (Plenum Press, New York, 1991).
[9] A.C. Jenkins and P. Strange, J. Phys. Condens. Matter 6 (1994) 3499. [lo] M.E. Rose, Relativistic Electron Theory (WiIey, New Yoix, 1961).
of Magnetism and Maguctic Materials 140-144 (1995) 37-38 M M --
ELSEVIER
A
Relativisticspin-polarizedsinglesite scatteringtheory AC Jenkins *, P. Strange Physics Department, Keele Unicersiry, Keele, Staflordshire, ST5 5BG, UK
Abstract
We present a fully relativistic single site scattering theory for magnetic systems. The radial Dirac equations for an electrcn in a potential with a magnetic field component are solved and scattering amplitudes found. New coupling is observed due to the removal of previous approximations. The impbcations for observables is discussed.
Most present-day work on the electronic structure of condensed matter is bajed on density functional theory [1,9-l. In the late 1970s this was generalized to treat relativistic systems /3,4]. McDonald and Vosko (31 developed the theory for a many electron system in the presence of a ‘spin-only’ magnetic field, i.e. ignoring diamagnetic effects. This is a suitable ‘Jask for describing electrons in condensed matter, In the mid 1980s Strange et al. I51and Feder et al. [6] tacklcd this problem. They described a relativistic electron scattr .;ng from a single effective poteniial with a magnetic component. The parameters from this theory were then used in a multiple scattering theory [7] to perform relativistic spin-poiarized KKR calculations. From this work many observables have been calculated [S]. The radial Kohn-Sham-Dirac equation for an electron experiencing a potential with ;i magnetic component becomes two infinite sets of coupled partial differential equations, for each value of the ltf, quantum number. By neglecting coupling between states 1 and I f 2 this can be reduced to sets of four coupled differential equations. These have been solved [s] and incorpotated into a single site scattering theory. Recently Jenkins and Strange [9] have reintroduced the coupling and made the alternative approximation that the wavefunctions for 1~ 6 are set equal to zero. This brings the number of coupled differential equations to a number of urder ten, depending upon mj. These were solved and the authors found that for some applications the previously ignored coupling may be significant. Here we show explicitiy that the error introduced by the neglect of this coupling is of the same order as the neglect of higher order terms diagonal in 1. From the Lippman-Schwinger equation it has been
* Corresponding author. Rx -1-44-782-711093.
shown that the Kohn-Sham-Dirac equation can be separated into radial and angular parts, the radial part is a+f;T+)) ar
=~cf~~(r)-(E-V~~~(r))rg~;(r) +Beff(r}~G(d’, KS
K’, mj)
rg?,Jr), (la)
a( W%(r))
= +B’“jr)~G(-K”.-
K’, m,)ct$$(r).
(lb)
2 K is the usual spin-angular quantum number [IO]. For the derivation of these equations see Strange et al. [5,9]. The full wavefunction inside the region of the potential is then
(2)
If it is asztmed that both the potential, Vcff(r), and the magnetic field. Bee(r), are zero outside the muffin-tin radius, r,,,, then the radial solutions outside the muffin-tin sphere have the form of a radi.31 spherical wave 191 and these two solutions can be matched at rm to calculate scattering amplitudes. In principle this means matching two infinite column vectors for each value of K (= KI, . . .,K”, . . . ) - but, due to the cutoff in I and the fact that mj remains a good quantum number, this reduces to, at
03048853/95/$09.50 8 19% Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00846-u
A.C. Jenkins, P. Strange/Journal
of Magnetism and Magnetic Materials 140-144 hW5~ 37-38
worst, the matching of two 12 component vectors. For details of the matching procedure see Ref. f9f. In Fig. 1 we show the scattering amplitudes for a single Pa atom in a spin-only magnetic field as a function of
WY
(Ryd)
energy. The quantum numbers for this particular scattering amplitude are K = 2, K’ = - 1, mi = l/2, i.e. this scatter-
ing amplitude represents coupling between the I = 2 and are drawn, for various magnetic fields, and the more pronounced curves are for greater fields. In Fig. 2 the scattering amplitude for K = -4, K’ = -4, m, = l/2 (i.e. I- I’ = 3) is drawn for the same magnetic fields. All the different field curves lie virtually on top of one another, however, we see that in the energy range of interest in condensed matter between zero and one Rydbergs, these scattering amplitudes are of the same order of magnitude. From this we can conclude that the error in omitting coupling between 1 and 1 f 2 is of the same order as that caused by omitting I values above the highest 1 value in the outer electronic shells. Hence it makes no sense to include this coupling without expanding the size of the spherical harmonic basis set. Conversely, it makes no sense to increase the size of the basis set and hence the MUGmatrix if this coupling is ignored.
1= 0. Several. curves
for a
0.10
Fig. 2. The K = -4, K’ = -4, m, = l/2 scattering amplitude for a single Pt atom potent:al for several different magnetic fields. In summary
then, we have shown explicitly
that two of
the approximations which go into spin-polarized reiativistic electronic structure calculations are of the same order and hence that there is nothing to be gained by removing one of these approximations without the other. References [l] P. Hohenberg and W. Kohn, Phys. Rev. 5 136 (1964) 864. [2] W. Kohn and L.J. Sham, Phys. Rev. A 140 (1965) 1133. [3] A.H. MacDonald and S. Vosko, J. Phys. C: Solid State Phys. 12 (1979) 2977. [4] M.V. Ramana and A.K. Rajagopal, J. Phys. C: Solid Stale Phys. 12 (1979) I-845. 151P. Strange, LB. Staunton rrd B.L. Gyorffy, J. Phys. C: Solid State Phys. 17 (1984) 3355. [6] R. Feder, F. Rosicky and B. Ackccmann, 2. Phys. B 52 (1983) 31. [7] P. Strange, H. Ehert, J.B. Staunton and B.L. Gyorffy, 1. Phys. Condens. Matter 1 (1989) 2959. 181 S. Wilson, I.P. Grant and B.L. Gyorffy, eds., The Effects of Relativity in Atoms Molecules and the Solid 9tate (Plenum Press, New York, 1991).
[9] A.C. Jenkins and P. Strange, J. Phys. Condens. Matter 6 (1994) 3499. [lo] M.E. Rose, Relativistic Electron Theory (WiIey, New Yoix, 1961).