- Email: [email protected]
Magnetic properties of hexagonal-close-packed Co and Ni
PHYSICA
Physica B 176 (1992) 227-231 North-Holland
Magnetic properties of hexagonal-close-packed Co and Ni G . D . Maksimovi6 and F . R . Vukajlovi6
Institute for Theoretical Physics (020), Boris Kidri( Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Yugoslavia Received 2 August 1991
The Stoner model of ferromagnetism in combination with the self-consistent linear-muffin-tin paramagnetic calculations are used for the investigation of magnetic properties of Co and Ni in the hexagonal-close-packed (HCP) structure. Very good agreement is obtained with previous more complicated spin-polarized calculations by means of a new fixed-spinmoment method and experiment.
1. Introduction
Molecular beam epitaxy (MBE) has made possible the stabilization of metastable phases of elements not previously found to occur naturally [1, 2]. As a consequence, in the last few years there has been an increasing interest in theoretical investigations of the stability conditions of 3d [3-6] and 4d [7] magnetic cubic materials. Recently, the search for new magnetic materials was extended to 3d transition elements in the HCP phase [8]. We also investigated magnetic properties of HCP Cr in our previous work [9]. In ref. [8] it was investigated which of the 3d HCP metals can be grown by MBE in the ferromagnetic phase and their structural and magnetic properties were predicted in the framework of self-consistent spin-polarized calculations by means of a new fixed-spin-moment method [3] (FSM). Since the fixed-spin-moment method surveys an entire surface in the moment-volume space, it requires much more calculations than standard spin-polarized calculations. In our previous works [10, 11] we investigated magnetic properties of MoPd 3 and FePd 3 compounds using the Stoner model of ferromagnetism [12] and the method suggested by Krasko for FCC Fe [6]. The method of Krasko [6] enables one to identify all magnetic phases of
iron; stable, metastable and unstable. This method is much easier and cheaper than spin-polarized calculations. We obtained very good agreement for the magnetic moment of these compounds with experimental values. Here we use the same method for the investigation of magnetic properties of HCP Co and HCP Ni. In section 2 we present a brief description of the Stoner method. Section 3 gives the magnetic properties of HCP Co and Ni.
2. Method of calculation
According to Stoner [12], material is magnetic if I. N(ef)~> 1, where I is so-called Stoner exchange parameter and N(ee) is the density of states (DOS) at the Fermi level, el. As HCP Ni and HCP Co have both two atoms per unit cell, the Stoner parameters of these materials are obtained on the basis of the generalized expression for compounds [13]:
I = ~ i,[ N'(ef) ] 2 J
'
(1)
where N,(ef) is the nonmagnetic DOS at the Fermi energy ef on site t, N ( e f ) is the total DOS
0921-4526/92/$05.00 © 1992-Elsevier Science Publishers B.V. All rights reserved
228
G.D. Maksimovi(, F.R. Vukajlovi( / Magneticproperties of HCP Co and Ni
at Fermi energy and I, is the Stoner parameter at site t, In the case of HCP Ni and H C P Co the Stoner parameter for unit cell I is given by I = Ia~omic 2
(2)
We calculate the atomic Stoner parameter Iatom,c according to Gunnarsson [14], using quantities found in paramagnetic L M T O calculations [15, 16]. We also noted in our previous works [10, 11] that all quantities needed to calculate magnetic moments per cell m are obtained from paramagnetic L M T O calculations. The condition for the existence of a magnetic phase is: 1 U(m, s) = 7 '
(3)
where N(m, s) is the average DOS at a given Wigner-Seitz radius s, defined over an energy interval around the Fermi level el, containing m electrons ( m / 2 above and m / 2 below el) [10, 11]. Intersection of the curve N ( m , s ) with the horizontal line 1/I gives all stable magnetic solutions if:
ON(m, s) <0. Om
ture. The experimental values for the lattice constant a and for the c/a ratio are 2.51 A and 1.622, respectively [19], and the magnetic moment per atom is 1.6~B [20]. The values of a and c/a correspond to a Wigner-Seitz radius s = 2.604a.u., which we used in our L M T O paramagnetic calculations. The DOS is presented in fig. 1. The DOS at the Fermi level is: N ( e f ) = 4 4 . 4 3 5 states/Ry/spin per cell ( e l = - 0 . 0 6 8 4 Ry). Using Gunnarsson's work [14] and eq. (1) we obtained the Stoner parameter for the unit cell of HCP-Co: 1 = 0,03632 Ry. For the Stoner product we obtained: 1. N(e~) = 1.61 > 1, which implies magnetic order in HCP Co. From fig. 2, as described in section 2, we obtained for H C P Co a stable ferromagnetic state with a magnetic moment per unit cell of m = 3.3~B, i.e. m = 1.65/~ B per atom. This value is in fair agreement with experiment [20], and with previous theoretical work of Podgorny and Goniakowski [8], who applied the FSM procedure, which is HCP-Co 180
(4)
I
I
I
I
I
~60 140
3. Results LLI C3
The calculations were done using the LMTOASA method, without inclusion of the so-called combined correction terms [15, 16]. The irreducible part of the Brillouin zone was sampled at 294 k-points. The exchange-correlation potential used in the calculations was the Von B a r t h Hedin potential [18] . In the self-consistent band calculations reported here, the Dirac equation was solved neglecting spin-orbit coupling but including mass-velocity and Darwin terms. Our calculations have been done with Skriver's computer code.
3.1. HCP-Co The native form of Co is a hexagonal struc-
120
~-" 100 LLJ
.<
80 60
(Z) E~
20 0 -0.80 -0.60 -0.40 -0.20 0.00
0.20
0.40
ENERGY [ R y ]
Fig. l. DOS for s = 2.604 a.u. for HCP Co versus energy. Fermi energy is: ef = - 0 . 0 6 8 4 R y .
G.D. Maksimovi6, F.R. Vukajlovi6 / Magnetic properties of HCP Co and Ni
more computationally intensive. In their work, the contour plot of magnetic field H and pressure p indicates the ferromagnetic ground state with s = 2.568 a.u. and m = 1.55/~ B. We want to stress here that we obtained excellent agreement with the experimental value of magnetic m o m e n t , using the experimental value of Wigner-Seitz radius s = 2.604 a.u. in our calculations.
Co-HOP
56
48
40
3.2. HCP-Ni
32
t/Z
W e o b t a i n e d t h e S t o n e r p a r a m e t e r for u n i t c e l l
of H C P Ni: I = 0.038452 Ry. As already noted by Papaconstantopoulos et al. [21] and in ref. [8], Ni has an extremely large D O S at the Fermi level. We obtained for this value: N(ef) = 76.325 s t a t e s / R y per atom (fig. 3). This suggests a case of strong ferromagnetism (I. N(ef) = 2 . 9 4 ) . From fig. 4 as described in section 2, we obtained for H C P Ni a stable ferromagnetic state with magnetic m o m e n t per atom: m = 0.66/z B. In ref. [8] the contour plot indicates the global equilibrium point at m = 0 . 5 9 / z B and s = 2 . 5 6 7 a . u . , which is the value of Wigner-Seitz radius that we used in our L M T O calculations
24 Z
16
0
0,0
229
1.0
2.0
3.0
4.0
5.0
m (P3)
Fig. 2. Average DOS N(rn, s) for HCP Co for s = 2.604 a.u. versus magnetic moment per cell. The ferromagnetic phase corresponds to the intersection of 1/1 with N(rn, s).
175.00
150.00
hcp
-
Ni
1250O © o 10000
©
S D
7500
5000
25 00 -~ / O 00
q
0.00
~
'-
J
'
I
020
'
'
'
I
~
040
i
J
r
0 60
i
j
080
i ~ 7 ~ ,
r
1 OO
,
P
l
,
120
ENERGY (Rx) Fig. 3. DOS for s = 2.567 a.u. for HCP Ni versus energy. Fermi energy is: ef
=
0.728 Ry.
G.D. Maksimovi(, F.R. Vukajlovi6 / Magnetic properties of HCP Co and Ni
230
Ni-HCP
I
i
I
L
64
i I
55
48
V ~- 4o ~
32
%
B 0 ".0
2.0
3 2.
4.0
5.0
Fig. 4. Average DOS N(rn, s) for HCP Ni for s = 2.567 a.u. versus magnetic moment per cell. The ferromagnetic phase corresponds to the intersection of 1/1 with N(m, s).
(we also used a c/a ratio equal to 1.622). The self-consistent A P W calculations carried by Papaconstantopoulos et al. [21] for s = 2.59 a.u. gave a huge m o m e n t of m = 0.76/x B per atom. According to ref. [8] this larger value of the magnetic moment (in ref. [21]) is caused by a small number of k-points used in A P W calculations a n d / o r by the tight-binding interpolation scheme. Our value for the magnetic moment of H C P Ni lies between the values of refs. [8] and [21]. Indeed, it is closer to value of ref. [8]. We also confirm that the HCP phase of Ni, if epitaxially grown, will probably be strongly ferromagnetic. T o sum up, we applied the Stoner model of ferromagnetism in order to investigate magnetic properties of HCP Co and HCP Ni. Using the Stoner criterion for compounds we found that magnetic order in these materials exists and we
calculated the magnetic moment per atom of H C P Co and HCP Ni. Here, the magnetic moments were found to agree well with both earlier band calculations and with experiment. In previous theoretical work of Podgorny and Goniakowski [8] the spin-polarized calculations have been carried out using the FSM scheme. They obtained information about the magnetic moment of HCP Co and HCP Ni using the combined electronic pressure p(m, V) and the magnetic field H(m, V) contour plots. The H = 0 and p = 0 contours have a special significance, the former being equivalent to the re(V) curve. An intersection of the zero-pressure and zerofield contours gives the ground state of the system. It is obvious that such an analysis is much more complicated and more time consuming, than our calculations of magnetic moment, where we used one N(m, s) curve for a given Wigner-Seitz radius s. As noted above, all the quantities needed to calculate the magnetic moment are obtained from paramagnetic L M T O calculations, which are easier and cheaper than spin-polarized calculations. Bearing in mind that metastable phases of HCP 3d transition metals could possibly be obtained by technique of molecular beam-epitaxy, we hope that our work may be a helpful guide for experimental investigations of these materials.
Acknowledgements This work was supported by Yugoslavia's SFRJ G o v e r n m e n t support program number P52, US Department of Energy under contract number JF 817-71 and Serbian Scientific Fund.
References [1] G.A. Prinz, Phys. Rev. Lett. 54 (1985) 1051. [2] M. Maurer, J.C. Ousset, M. Piecuch and M.F. Ravet, Europhys. Lett. 9 (1989) 803. [3] V.L. Moruzzi, P.M. Marcus, K. Schwarz and P. Mohn, Phys. Rev. B 34 (1986) 1784. [4] V.L. Moruzzi, P.M. Marcus and P.C. Pattnaik, Phys. Rev. B 37 (1988) 8003.
G.D. Maksimovi6, F.R. Vukajlovi( / Magnetic properties of HCP Co and Ni [5] V.L. Moruzzi and P.M. Marcus, Phys. Rev. B 38 (1988) 1613. [6] G.L. Krasko, Phys. Rev. B 36 (1987) 8565. [7] V.L. Moruzzi and P.M. Marcus, Phys. Rev. B 39 (1989) 471. [8] M. Podgorny and J. Goniakowski, Phys. Rev. B 42 (1990) 6683. [9] G.D. Maksimovi6, Z.S. Popovi6 and F.R. Vukajlovi6, Solid State Commun. 79 (1991) 631. [10l G.D. Maksimovi~, Z.S. Popovi6, F.R. Vukajlovi6 and K. Vuleti6, Phys. Stat. Sol. (b) 160 (1990) 635. Ill] G.D. Maksimovi6, Z.S. Popovi6 and F.R. Vukajlovi6, J. Phys. Condens. Mat. 2 (1990) 4907. [12] E.C. Stoner, Proc. Roy. Soc. (London) A169 (1939) 339.
[13] [14] [15] [16] [17] [18] [19] [20] [21]
231
S.S. Jaswal, Solid State Commun. 52 (1984) 127. O. Gunnarsson, J. Phys. F 6 (1976) 587. O.K. Andersen, Phys. Rev. B 12 (1975) 3060. H.L. Skriver, The LMTO Method: Muffin-Tin Orbitals and Electronic Structure (Springer, New York, 1984). P.M. Marcus and V.L. Moruzzi, Phys. Rev. B 38 (1988) 6949. U. yon Barth and L. Hedin, J. Phys. C 5 (1972) 1629. N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976) p. 662. AIP Handbook, 2nd Ed. (McGraw-Hill, New York, 1963). D.A. Papaconstantopoulos, J.L. Fry and N.E. Brener, Phys. Rev. B 39 (1989) 2526.
Physica B 176 (1992) 227-231 North-Holland
Magnetic properties of hexagonal-close-packed Co and Ni G . D . Maksimovi6 and F . R . Vukajlovi6
Institute for Theoretical Physics (020), Boris Kidri( Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Yugoslavia Received 2 August 1991
The Stoner model of ferromagnetism in combination with the self-consistent linear-muffin-tin paramagnetic calculations are used for the investigation of magnetic properties of Co and Ni in the hexagonal-close-packed (HCP) structure. Very good agreement is obtained with previous more complicated spin-polarized calculations by means of a new fixed-spinmoment method and experiment.
1. Introduction
Molecular beam epitaxy (MBE) has made possible the stabilization of metastable phases of elements not previously found to occur naturally [1, 2]. As a consequence, in the last few years there has been an increasing interest in theoretical investigations of the stability conditions of 3d [3-6] and 4d [7] magnetic cubic materials. Recently, the search for new magnetic materials was extended to 3d transition elements in the HCP phase [8]. We also investigated magnetic properties of HCP Cr in our previous work [9]. In ref. [8] it was investigated which of the 3d HCP metals can be grown by MBE in the ferromagnetic phase and their structural and magnetic properties were predicted in the framework of self-consistent spin-polarized calculations by means of a new fixed-spin-moment method [3] (FSM). Since the fixed-spin-moment method surveys an entire surface in the moment-volume space, it requires much more calculations than standard spin-polarized calculations. In our previous works [10, 11] we investigated magnetic properties of MoPd 3 and FePd 3 compounds using the Stoner model of ferromagnetism [12] and the method suggested by Krasko for FCC Fe [6]. The method of Krasko [6] enables one to identify all magnetic phases of
iron; stable, metastable and unstable. This method is much easier and cheaper than spin-polarized calculations. We obtained very good agreement for the magnetic moment of these compounds with experimental values. Here we use the same method for the investigation of magnetic properties of HCP Co and HCP Ni. In section 2 we present a brief description of the Stoner method. Section 3 gives the magnetic properties of HCP Co and Ni.
2. Method of calculation
According to Stoner [12], material is magnetic if I. N(ef)~> 1, where I is so-called Stoner exchange parameter and N(ee) is the density of states (DOS) at the Fermi level, el. As HCP Ni and HCP Co have both two atoms per unit cell, the Stoner parameters of these materials are obtained on the basis of the generalized expression for compounds [13]:
I = ~ i,[ N'(ef) ] 2 J
'
(1)
where N,(ef) is the nonmagnetic DOS at the Fermi energy ef on site t, N ( e f ) is the total DOS
0921-4526/92/$05.00 © 1992-Elsevier Science Publishers B.V. All rights reserved
228
G.D. Maksimovi(, F.R. Vukajlovi( / Magneticproperties of HCP Co and Ni
at Fermi energy and I, is the Stoner parameter at site t, In the case of HCP Ni and H C P Co the Stoner parameter for unit cell I is given by I = Ia~omic 2
(2)
We calculate the atomic Stoner parameter Iatom,c according to Gunnarsson [14], using quantities found in paramagnetic L M T O calculations [15, 16]. We also noted in our previous works [10, 11] that all quantities needed to calculate magnetic moments per cell m are obtained from paramagnetic L M T O calculations. The condition for the existence of a magnetic phase is: 1 U(m, s) = 7 '
(3)
where N(m, s) is the average DOS at a given Wigner-Seitz radius s, defined over an energy interval around the Fermi level el, containing m electrons ( m / 2 above and m / 2 below el) [10, 11]. Intersection of the curve N ( m , s ) with the horizontal line 1/I gives all stable magnetic solutions if:
ON(m, s) <0. Om
ture. The experimental values for the lattice constant a and for the c/a ratio are 2.51 A and 1.622, respectively [19], and the magnetic moment per atom is 1.6~B [20]. The values of a and c/a correspond to a Wigner-Seitz radius s = 2.604a.u., which we used in our L M T O paramagnetic calculations. The DOS is presented in fig. 1. The DOS at the Fermi level is: N ( e f ) = 4 4 . 4 3 5 states/Ry/spin per cell ( e l = - 0 . 0 6 8 4 Ry). Using Gunnarsson's work [14] and eq. (1) we obtained the Stoner parameter for the unit cell of HCP-Co: 1 = 0,03632 Ry. For the Stoner product we obtained: 1. N(e~) = 1.61 > 1, which implies magnetic order in HCP Co. From fig. 2, as described in section 2, we obtained for H C P Co a stable ferromagnetic state with a magnetic moment per unit cell of m = 3.3~B, i.e. m = 1.65/~ B per atom. This value is in fair agreement with experiment [20], and with previous theoretical work of Podgorny and Goniakowski [8], who applied the FSM procedure, which is HCP-Co 180
(4)
I
I
I
I
I
~60 140
3. Results LLI C3
The calculations were done using the LMTOASA method, without inclusion of the so-called combined correction terms [15, 16]. The irreducible part of the Brillouin zone was sampled at 294 k-points. The exchange-correlation potential used in the calculations was the Von B a r t h Hedin potential [18] . In the self-consistent band calculations reported here, the Dirac equation was solved neglecting spin-orbit coupling but including mass-velocity and Darwin terms. Our calculations have been done with Skriver's computer code.
3.1. HCP-Co The native form of Co is a hexagonal struc-
120
~-" 100 LLJ
.<
80 60
(Z) E~
20 0 -0.80 -0.60 -0.40 -0.20 0.00
0.20
0.40
ENERGY [ R y ]
Fig. l. DOS for s = 2.604 a.u. for HCP Co versus energy. Fermi energy is: ef = - 0 . 0 6 8 4 R y .
G.D. Maksimovi6, F.R. Vukajlovi6 / Magnetic properties of HCP Co and Ni
more computationally intensive. In their work, the contour plot of magnetic field H and pressure p indicates the ferromagnetic ground state with s = 2.568 a.u. and m = 1.55/~ B. We want to stress here that we obtained excellent agreement with the experimental value of magnetic m o m e n t , using the experimental value of Wigner-Seitz radius s = 2.604 a.u. in our calculations.
Co-HOP
56
48
40
3.2. HCP-Ni
32
t/Z
W e o b t a i n e d t h e S t o n e r p a r a m e t e r for u n i t c e l l
of H C P Ni: I = 0.038452 Ry. As already noted by Papaconstantopoulos et al. [21] and in ref. [8], Ni has an extremely large D O S at the Fermi level. We obtained for this value: N(ef) = 76.325 s t a t e s / R y per atom (fig. 3). This suggests a case of strong ferromagnetism (I. N(ef) = 2 . 9 4 ) . From fig. 4 as described in section 2, we obtained for H C P Ni a stable ferromagnetic state with magnetic m o m e n t per atom: m = 0.66/z B. In ref. [8] the contour plot indicates the global equilibrium point at m = 0 . 5 9 / z B and s = 2 . 5 6 7 a . u . , which is the value of Wigner-Seitz radius that we used in our L M T O calculations
24 Z
16
0
0,0
229
1.0
2.0
3.0
4.0
5.0
m (P3)
Fig. 2. Average DOS N(rn, s) for HCP Co for s = 2.604 a.u. versus magnetic moment per cell. The ferromagnetic phase corresponds to the intersection of 1/1 with N(rn, s).
175.00
150.00
hcp
-
Ni
1250O © o 10000
©
S D
7500
5000
25 00 -~ / O 00
q
0.00
~
'-
J
'
I
020
'
'
'
I
~
040
i
J
r
0 60
i
j
080
i ~ 7 ~ ,
r
1 OO
,
P
l
,
120
ENERGY (Rx) Fig. 3. DOS for s = 2.567 a.u. for HCP Ni versus energy. Fermi energy is: ef
=
0.728 Ry.
G.D. Maksimovi(, F.R. Vukajlovi6 / Magnetic properties of HCP Co and Ni
230
Ni-HCP
I
i
I
L
64
i I
55
48
V ~- 4o ~
32
%
B 0 ".0
2.0
3 2.
4.0
5.0
Fig. 4. Average DOS N(rn, s) for HCP Ni for s = 2.567 a.u. versus magnetic moment per cell. The ferromagnetic phase corresponds to the intersection of 1/1 with N(m, s).
(we also used a c/a ratio equal to 1.622). The self-consistent A P W calculations carried by Papaconstantopoulos et al. [21] for s = 2.59 a.u. gave a huge m o m e n t of m = 0.76/x B per atom. According to ref. [8] this larger value of the magnetic moment (in ref. [21]) is caused by a small number of k-points used in A P W calculations a n d / o r by the tight-binding interpolation scheme. Our value for the magnetic moment of H C P Ni lies between the values of refs. [8] and [21]. Indeed, it is closer to value of ref. [8]. We also confirm that the HCP phase of Ni, if epitaxially grown, will probably be strongly ferromagnetic. T o sum up, we applied the Stoner model of ferromagnetism in order to investigate magnetic properties of HCP Co and HCP Ni. Using the Stoner criterion for compounds we found that magnetic order in these materials exists and we
calculated the magnetic moment per atom of H C P Co and HCP Ni. Here, the magnetic moments were found to agree well with both earlier band calculations and with experiment. In previous theoretical work of Podgorny and Goniakowski [8] the spin-polarized calculations have been carried out using the FSM scheme. They obtained information about the magnetic moment of HCP Co and HCP Ni using the combined electronic pressure p(m, V) and the magnetic field H(m, V) contour plots. The H = 0 and p = 0 contours have a special significance, the former being equivalent to the re(V) curve. An intersection of the zero-pressure and zerofield contours gives the ground state of the system. It is obvious that such an analysis is much more complicated and more time consuming, than our calculations of magnetic moment, where we used one N(m, s) curve for a given Wigner-Seitz radius s. As noted above, all the quantities needed to calculate the magnetic moment are obtained from paramagnetic L M T O calculations, which are easier and cheaper than spin-polarized calculations. Bearing in mind that metastable phases of HCP 3d transition metals could possibly be obtained by technique of molecular beam-epitaxy, we hope that our work may be a helpful guide for experimental investigations of these materials.
Acknowledgements This work was supported by Yugoslavia's SFRJ G o v e r n m e n t support program number P52, US Department of Energy under contract number JF 817-71 and Serbian Scientific Fund.
References [1] G.A. Prinz, Phys. Rev. Lett. 54 (1985) 1051. [2] M. Maurer, J.C. Ousset, M. Piecuch and M.F. Ravet, Europhys. Lett. 9 (1989) 803. [3] V.L. Moruzzi, P.M. Marcus, K. Schwarz and P. Mohn, Phys. Rev. B 34 (1986) 1784. [4] V.L. Moruzzi, P.M. Marcus and P.C. Pattnaik, Phys. Rev. B 37 (1988) 8003.
G.D. Maksimovi6, F.R. Vukajlovi( / Magnetic properties of HCP Co and Ni [5] V.L. Moruzzi and P.M. Marcus, Phys. Rev. B 38 (1988) 1613. [6] G.L. Krasko, Phys. Rev. B 36 (1987) 8565. [7] V.L. Moruzzi and P.M. Marcus, Phys. Rev. B 39 (1989) 471. [8] M. Podgorny and J. Goniakowski, Phys. Rev. B 42 (1990) 6683. [9] G.D. Maksimovi6, Z.S. Popovi6 and F.R. Vukajlovi6, Solid State Commun. 79 (1991) 631. [10l G.D. Maksimovi~, Z.S. Popovi6, F.R. Vukajlovi6 and K. Vuleti6, Phys. Stat. Sol. (b) 160 (1990) 635. Ill] G.D. Maksimovi6, Z.S. Popovi6 and F.R. Vukajlovi6, J. Phys. Condens. Mat. 2 (1990) 4907. [12] E.C. Stoner, Proc. Roy. Soc. (London) A169 (1939) 339.
[13] [14] [15] [16] [17] [18] [19] [20] [21]
231
S.S. Jaswal, Solid State Commun. 52 (1984) 127. O. Gunnarsson, J. Phys. F 6 (1976) 587. O.K. Andersen, Phys. Rev. B 12 (1975) 3060. H.L. Skriver, The LMTO Method: Muffin-Tin Orbitals and Electronic Structure (Springer, New York, 1984). P.M. Marcus and V.L. Moruzzi, Phys. Rev. B 38 (1988) 6949. U. yon Barth and L. Hedin, J. Phys. C 5 (1972) 1629. N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976) p. 662. AIP Handbook, 2nd Ed. (McGraw-Hill, New York, 1963). D.A. Papaconstantopoulos, J.L. Fry and N.E. Brener, Phys. Rev. B 39 (1989) 2526.