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Electronic structure of the intermetallic compound Y2Ni17
Journal of AUoys and Compounds, 178 (1992) 379-384 JAL 5038
379
Electronic structure of the intermetallic compound Y2Ni, v V. Cri~an, A. Verues, A. J ~ o s i , V. P o p e s c u a n d D. Kapusi "Babe~-Bolyai" University, Physics Department, Ro-3400 Cluj-Napoca (Romania) (Received July 11, 1991)
Abstract The electronic densities of states in Y2Ni17 are calculated for various occupation numbers of the yttrium and nickel positions using the tight-binding d-band model in conjunction with the recursion method and the virtual crystal approximation. Some yttrium atoms are displaced from the 2b to the 2c positions and some nickel atoms from the 4f and 12j to the 4e positions. The low temperature specific heat is calculated and compared with the experimental value. The density of states for both yttrium and nickel atoms were lowered when the atomic positions deviated from the ideal TheNi17 crystal positions. This might explain the occurrence of metamagnetism in Y2Nil~when some of the yttrium atoms are displaced from 2b into 4f positions.
1. I n t r o d u c t i o n Intermetallic c o m p o u n d s of yttrium and a transition metal M ( M - N i , Co, Fe, Mn) h a v e g a i n e d g r e a t i n t e r e s t o v e r the last d e c a d e o w i n g to t h e i r l a r g e v a r i a t i o n in m a g n e t i c b e h a v i o u r . In the Y - N i s y s t e m , an i n c r e a s e in y t t r i u m c o n c e n t r a t i o n d e c r e a s e s the s p o n t a n e o u s m a g n e t i z a t i o n f r o m p u r e nickel to t h a t o f Y2Ni17 w h i c h is w e a k l y f e r r o m a g n e t i c a n d to t h a t o f YNi~ w h i c h is p a r a m a g n e t i c . H o w e v e r , with a f u r t h e r i n c r e a s e in y t t r i u m c o n c e n t r a t i o n a r e s u r g e n c e o f f e r r o m a g n e t i s m o c c u r s a n d YeNi17 a n d YNi3 are w e a k l y f e r r o m a g n e t i c a g a i n [ 1 ]. It w a s p o i n t e d o u t b y G i g n o u x et al. [1, 2] t h a t t h e s e c h a r a c t e r i s t i c s c a n b e e x p l a i n e d in t h e itinerant e l e c t r o n m o d e l a n d in t e r m s o f t h e s p e c i a l s h a p e o f t h e d e n s i t y o f s t a t e s (DOS) o w i n g to t h e 3d nickel b a n d a n d 4d y t t r i u m b a n d h y b r i d i z a t i o n at the F e r m i level. B a n d s t r u c t u r e c a l c u l a t i o n s f o r R2Ni17 c o m p o u n d s ( R = r a r e earth) w e r e m a d e b y Shimizu et al. [3] b y t h e r e c u r s i o n m e t h o d a n d b y o n e o f t h e a u t h o r s [4] u s i n g t h e A P W m e t h o d .
2. T h e c r y s t a l l i n e s t r u c t u r e a n d t h e m e t h o d o f c o m p u t a t i o n T h e c o m p o u n d Y2Ni17 h a s t h e h e x a g o n a l Th2Ni17-type s t r u c t u r e , s p a c e g r o u p P 6 J m m c . T h e a t o m i c p o s i t i o n p a r a m e t e r s u s e d in t h e c o m p u t a t i o n s
0925-8388/92/$5.00
© 1992- Elsevier Sequoia. All rights reserved
380 were z(4f) = 0 . 1 1 , z( 12k) = 0 . 0 , x ( 1 2 k ) = 1/6 and x(12j) = 0.0 [51. The lattice p a r ameter s were a = 8 . 3 0 7 × 10 -1° m and c = 8 . 0 4 × 10 -1° m [6]. The yttrium atoms are located in the standard ideal crystalline structure in 2b and 2d positions and the nickel atoms in 4f, 6g, 12j and 12k positions. This implies that the nearest-neighbour transition-metal surroundings of the two rare-earth sites are almost identical, whereas the nearest rare-earth neighbours are quite different for the rare-earth sites. Crystallographic examination [7] revealed that for Ho2Co17 and Ho2Fe17 the structures were p a r tly disordered. It was found that some transition metal atoms could o c c u p y the 4e positions at the expens e of the 4f and 12j positions and that some of the rare-earth atoms could o c c u p y the 2c positions at the expense of the 2b positions. We applied these ideas to Y2Nil~ in order to study the effect of this disorder on the DOS. It was found that the structure of Y2Ni16 is constructed by replacing some of the nickel dumbbell atoms of Y2Ni,7 by yttrium atoms. The peak in the DOS for Y2Ni,6 at the Fermi level is lower than that of Y2Ni,v and this fact is related to the m e t a m agnet i s m found in Y2Nil~ [1 ]. In our computations we used the formula y(2d)y1 _x(2b)Yx(2C)Ni9(12k, 6g)si6_y(12J)Ni2_z(4f)Niz+y(4e) with z = 2x, the recursion m e t h o d [8], the Slater tight-binding d-band model and one orbital on each atom. The transfer integrals are given by the two centred integrals dda, ddTr and ddtt [9, 10], and the formula given by Pettifor [ 11 ]. The atomic potentials [3] are E(Y 2 b ) = 0.0 Ryd.
e(Y 2 d ) = 0 . 0 1 5 Ryd.
E(Y 2 c ) = 0.005 Ryd.
e(Si 12j) -- - 0.22 Ryd.
e(Ni 6 g ) = - 0 . 2 4 Ryd.
e(Ni 4e) = 0.205 Ryd.
e(Ni 1 2 k ) = 0 . 2 2 Ryd.
e(Ni 4 f ) = + 0.19 Ryd.
and e(b - c) = (0.5 - x ) e ( Y 2b) + (0.5 + x ) e ( Y 2c) e0fe) = (0.6 - y)e(Ni 12j) + (0.2 - z)e(Ni 4f) + (0.2 + y + z)e(Ni 4e) for the virtual yttrium and nickel atoms (1 Ryd. = 13.57 eV). The n u m b e r of electrons on yttrium and nickel atoms have been assumed to be 2 and 1.4 respectively. It was shown that the Fermi level is situated at a peak in the DOS [1, 3]. In order to c hoos e the m os t suitable orbital we c o m p u t e d the DOS for x y , yz, zx, x ~"_ y 2 and 3z 2 - r 2 orbitals for the standard structure. The results showed that the m axi m um of the DOS at the Fermi level (1700 st at es/ ( R y × f . u . ) ) was obtained for the 3z 2 - r e orbital.
381
DOS
Tofal
I 12k
I
6g I 12j
L -030
EF -Q02 E( R y d ) ~
Fig. 1. The DOSs for the standard c o m p o u n d Y2Nilv.
7O90 ! /! 6000
,, \.
/
,.(12j) , .(/.f) y ( Yl-x 2 dYxJ2c)_.(12~ ) .NI9. ( 2INI6-y b )NI2-z .
,
5000
....... ..........
~ooo
......
(~e}
J~y+z
x=O.O ~z=O.O x = 0 5 ; z ='I.0 x = 1 0 )z= 2.0 x = I . 5 ;z=3.0
3000
\ 2000
p
\,,
1000 ~ - = : = : = -
~5
-._,.~.......... ::.~-~: ......... T:=~:~.~: =..=.=..=..=~-.~..~.. . . . . . ,_ . . . . . . . . . . . . . . . . . . . . . .
~
1's
~
-
~s
~
~s Y
F i g . 2. T h e t o t a l D O S standard posiUons.
at t h e F e r m i
level for different
atomic
positions.
P means
DOS for
382
oosE ] y =0.0
! I
Y
L
J
y =0.5
j
/
y=lO
/
_1
y=1.5
i -
(130
EF
-0.15 E{Ryd)~
Fig. 3. The total DOS as a function of energy for different atomic positions ( x f z = 0 . 0 ) .
3. R e s u l t s The calculated DOSs in Y2Ni17 for the standard c o m p o u n d are shown in Fig. 1. The partial DOSs of yttrium atoms are situated far from the Fermi level. The contributions to the total DOS at the Fermi level are mainly due to the nickel 12k-contributing 1286 s t a t e s / ( R y d . × c e l l ) , and to the nickel 12j contributing 285 states/(Ryd. × c e n ) . The DOSs of the dumbbell atom are situated above the Fermi level. As in ref. 3, t he band width of the nickel electrons is lkrger than that of the yttrium electrons, which disagrees with the explanation given by Gignoux et al. [11 ]. The DOS at the Fermi level is ver y sensitive to the position of the atoms in the unit cell. Figure 2 shows that the variation in the DOS is very important, mainly f o r x = z = 0.0, and that w he n y > 2 the DOS remains almost unchanged.
383 1.2 | ,.~. 1.1 ! / •~' /
,
,
\
,
,
.~,
,,1~
x=O.O ; z=O.O \
12j
.
....
. ...............................
4- ...............
4,
1 0.9' 0.8
0.4
~
~
...... K.,.
03 L
o2
0s
_ _ _ _
~
.~
-,~
l'.s
~'
-,x... ~ ~ - ~ .
~s
12k t
3
- -
I
3s
4
Y Fig. 4. The o r b i t a l e l e c t r o n o c c u p a t i o n
for x=z=0.
The shape of the DOS also changes drastically with x, y and z. For example, for x = z = O when y = 0 . 5 , the DOS of the nickel 12k atoms a p p r o a c h e s the Fermi level but the DOSs of the ot her atoms remain unchanged. For y = 1.0, the nickel 12k DOS moves towards higher energy and the DOS of the nickel atoms at 6j, 4e and 4f moves towards lower energy, the total DOS at the Fermi level being three times smaller than that for y = 0. The changing shape of the DOS is due to the changing location of the atomic DOS with r e s p e c t to the Fermi level. For y > 1.0, the m o v e m e n t of the different DOS peaks p r o c e e d s in the same direction. In Fig. 3 the total DOSs are shown for different y values and for x = z = O. The orbital electron occupat i on numbers (Fig. 4) lead to a m a x i m u m for the nickel 12k and 4f partial DOS and to a minimum for the nickel 12j partial DOS for y = 0 . 2 5 . For y > 0 . 2 5 the DOS for each at om reaches an almost constant value. W h en the yttrium atoms are not fully occupying the 2b, 2d positions displacement of the nickel 12j atoms to the 4e positions leaves the orbital electron o c c u p a t i o n almost unchanged irrespective of what kind of atomic positions we refer to. This suggests that the yttrium atoms when m o v e d from 2b to 2c positions have an effect on the DOS only when x and z are very small. The coefficient of the electronic specific heat com put ed for various DOSs at the Fermi level has smaller values than the experimental value unless x=z=O and y < 0 . 7 5 . The results show that (a) the total DOS is lowered when the yttrium, a n d / o r nickel ato m s are r e m o v e d from their standard positions and that (b) the shape o f the DOS at the Fermi level as a function of energy and the
384 orbital electron occupation positions.
axe v e r y s e n s i t i v e a t s m a l l c h a n g e s
of atomic
Acknowledgments O n e o f t h e a u t h o r s , A. V., w i s h e s t o e x p r e s s h i s g r a t i t u d e t o t h e O r g a n i z i n g C o m m i t t e e f r o m M i i n s t e r a n d t h e O p e n S o c i e t y F o u n d a t i o n ( S O R O S ) o f ClujNapoca for the financial support given during the Mtinster Conference.
References 1 2 3 4 5 6 7 8 9 10 11 12
D. Gignoux, R. Lemaire and P. Molho, J. Magn. Magn. Mater., 21 (1980) 119. D. Gignoux, R. Lemaire, P. Molho and F. Tasset, J. Magn. Magn. Mater., 21 (1980) 307. M. Shimizu, J. Inoue and S. Nagasawa, J. Phys. F, 14 (1984) 2673. V. Cri§an, I. Pop. M. Popescu and V. To§a, Proc. Pth Int. Workshop on Rare-Earth Magnets, Bad Sode~ Germany, 1087, Springer, Berlin, 1987, p. 165. W. B. Pearson, Lattice Spacings and Structures of Metals and Alloys, Vol. 2, Pergamon, Oxford, 1967, p. 344. K. H. J. Buschow, J. Less-Common Met., 11 (1966) 204. A. Christensen and R. G. Hazell, Acta Chem. Scand. A, 34 (1980) 455. R. Haydock, V. Heine and M. J. Kelly, J. Phys. C, 8 (1975) 2591. J. C. Slater and G. F. Koster, Phys. Rev., 04 (1954) 1498. R. K. Sharma, Phys. Ref. B, 12 (1979) 2813. D. G. Pettifor, J. Phys. F, 7 (1977) 613. D. Gignoux, R. Lemaire, P. Molho and F. Tasset, J. Appl. Phys., 52 (1981) 2087.
379
Electronic structure of the intermetallic compound Y2Ni, v V. Cri~an, A. Verues, A. J ~ o s i , V. P o p e s c u a n d D. Kapusi "Babe~-Bolyai" University, Physics Department, Ro-3400 Cluj-Napoca (Romania) (Received July 11, 1991)
Abstract The electronic densities of states in Y2Ni17 are calculated for various occupation numbers of the yttrium and nickel positions using the tight-binding d-band model in conjunction with the recursion method and the virtual crystal approximation. Some yttrium atoms are displaced from the 2b to the 2c positions and some nickel atoms from the 4f and 12j to the 4e positions. The low temperature specific heat is calculated and compared with the experimental value. The density of states for both yttrium and nickel atoms were lowered when the atomic positions deviated from the ideal TheNi17 crystal positions. This might explain the occurrence of metamagnetism in Y2Nil~when some of the yttrium atoms are displaced from 2b into 4f positions.
1. I n t r o d u c t i o n Intermetallic c o m p o u n d s of yttrium and a transition metal M ( M - N i , Co, Fe, Mn) h a v e g a i n e d g r e a t i n t e r e s t o v e r the last d e c a d e o w i n g to t h e i r l a r g e v a r i a t i o n in m a g n e t i c b e h a v i o u r . In the Y - N i s y s t e m , an i n c r e a s e in y t t r i u m c o n c e n t r a t i o n d e c r e a s e s the s p o n t a n e o u s m a g n e t i z a t i o n f r o m p u r e nickel to t h a t o f Y2Ni17 w h i c h is w e a k l y f e r r o m a g n e t i c a n d to t h a t o f YNi~ w h i c h is p a r a m a g n e t i c . H o w e v e r , with a f u r t h e r i n c r e a s e in y t t r i u m c o n c e n t r a t i o n a r e s u r g e n c e o f f e r r o m a g n e t i s m o c c u r s a n d YeNi17 a n d YNi3 are w e a k l y f e r r o m a g n e t i c a g a i n [ 1 ]. It w a s p o i n t e d o u t b y G i g n o u x et al. [1, 2] t h a t t h e s e c h a r a c t e r i s t i c s c a n b e e x p l a i n e d in t h e itinerant e l e c t r o n m o d e l a n d in t e r m s o f t h e s p e c i a l s h a p e o f t h e d e n s i t y o f s t a t e s (DOS) o w i n g to t h e 3d nickel b a n d a n d 4d y t t r i u m b a n d h y b r i d i z a t i o n at the F e r m i level. B a n d s t r u c t u r e c a l c u l a t i o n s f o r R2Ni17 c o m p o u n d s ( R = r a r e earth) w e r e m a d e b y Shimizu et al. [3] b y t h e r e c u r s i o n m e t h o d a n d b y o n e o f t h e a u t h o r s [4] u s i n g t h e A P W m e t h o d .
2. T h e c r y s t a l l i n e s t r u c t u r e a n d t h e m e t h o d o f c o m p u t a t i o n T h e c o m p o u n d Y2Ni17 h a s t h e h e x a g o n a l Th2Ni17-type s t r u c t u r e , s p a c e g r o u p P 6 J m m c . T h e a t o m i c p o s i t i o n p a r a m e t e r s u s e d in t h e c o m p u t a t i o n s
0925-8388/92/$5.00
© 1992- Elsevier Sequoia. All rights reserved
380 were z(4f) = 0 . 1 1 , z( 12k) = 0 . 0 , x ( 1 2 k ) = 1/6 and x(12j) = 0.0 [51. The lattice p a r ameter s were a = 8 . 3 0 7 × 10 -1° m and c = 8 . 0 4 × 10 -1° m [6]. The yttrium atoms are located in the standard ideal crystalline structure in 2b and 2d positions and the nickel atoms in 4f, 6g, 12j and 12k positions. This implies that the nearest-neighbour transition-metal surroundings of the two rare-earth sites are almost identical, whereas the nearest rare-earth neighbours are quite different for the rare-earth sites. Crystallographic examination [7] revealed that for Ho2Co17 and Ho2Fe17 the structures were p a r tly disordered. It was found that some transition metal atoms could o c c u p y the 4e positions at the expens e of the 4f and 12j positions and that some of the rare-earth atoms could o c c u p y the 2c positions at the expense of the 2b positions. We applied these ideas to Y2Nil~ in order to study the effect of this disorder on the DOS. It was found that the structure of Y2Ni16 is constructed by replacing some of the nickel dumbbell atoms of Y2Ni,7 by yttrium atoms. The peak in the DOS for Y2Ni,6 at the Fermi level is lower than that of Y2Ni,v and this fact is related to the m e t a m agnet i s m found in Y2Nil~ [1 ]. In our computations we used the formula y(2d)y1 _x(2b)Yx(2C)Ni9(12k, 6g)si6_y(12J)Ni2_z(4f)Niz+y(4e) with z = 2x, the recursion m e t h o d [8], the Slater tight-binding d-band model and one orbital on each atom. The transfer integrals are given by the two centred integrals dda, ddTr and ddtt [9, 10], and the formula given by Pettifor [ 11 ]. The atomic potentials [3] are E(Y 2 b ) = 0.0 Ryd.
e(Y 2 d ) = 0 . 0 1 5 Ryd.
E(Y 2 c ) = 0.005 Ryd.
e(Si 12j) -- - 0.22 Ryd.
e(Ni 6 g ) = - 0 . 2 4 Ryd.
e(Ni 4e) = 0.205 Ryd.
e(Ni 1 2 k ) = 0 . 2 2 Ryd.
e(Ni 4 f ) = + 0.19 Ryd.
and e(b - c) = (0.5 - x ) e ( Y 2b) + (0.5 + x ) e ( Y 2c) e0fe) = (0.6 - y)e(Ni 12j) + (0.2 - z)e(Ni 4f) + (0.2 + y + z)e(Ni 4e) for the virtual yttrium and nickel atoms (1 Ryd. = 13.57 eV). The n u m b e r of electrons on yttrium and nickel atoms have been assumed to be 2 and 1.4 respectively. It was shown that the Fermi level is situated at a peak in the DOS [1, 3]. In order to c hoos e the m os t suitable orbital we c o m p u t e d the DOS for x y , yz, zx, x ~"_ y 2 and 3z 2 - r 2 orbitals for the standard structure. The results showed that the m axi m um of the DOS at the Fermi level (1700 st at es/ ( R y × f . u . ) ) was obtained for the 3z 2 - r e orbital.
381
DOS
Tofal
I 12k
I
6g I 12j
L -030
EF -Q02 E( R y d ) ~
Fig. 1. The DOSs for the standard c o m p o u n d Y2Nilv.
7O90 ! /! 6000
,, \.
/
,.(12j) , .(/.f) y ( Yl-x 2 dYxJ2c)_.(12~ ) .NI9. ( 2INI6-y b )NI2-z .
,
5000
....... ..........
~ooo
......
(~e}
J~y+z
x=O.O ~z=O.O x = 0 5 ; z ='I.0 x = 1 0 )z= 2.0 x = I . 5 ;z=3.0
3000
\ 2000
p
\,,
1000 ~ - = : = : = -
~5
-._,.~.......... ::.~-~: ......... T:=~:~.~: =..=.=..=..=~-.~..~.. . . . . . ,_ . . . . . . . . . . . . . . . . . . . . . .
~
1's
~
-
~s
~
~s Y
F i g . 2. T h e t o t a l D O S standard posiUons.
at t h e F e r m i
level for different
atomic
positions.
P means
DOS for
382
oosE ] y =0.0
! I
Y
L
J
y =0.5
j
/
y=lO
/
_1
y=1.5
i -
(130
EF
-0.15 E{Ryd)~
Fig. 3. The total DOS as a function of energy for different atomic positions ( x f z = 0 . 0 ) .
3. R e s u l t s The calculated DOSs in Y2Ni17 for the standard c o m p o u n d are shown in Fig. 1. The partial DOSs of yttrium atoms are situated far from the Fermi level. The contributions to the total DOS at the Fermi level are mainly due to the nickel 12k-contributing 1286 s t a t e s / ( R y d . × c e l l ) , and to the nickel 12j contributing 285 states/(Ryd. × c e n ) . The DOSs of the dumbbell atom are situated above the Fermi level. As in ref. 3, t he band width of the nickel electrons is lkrger than that of the yttrium electrons, which disagrees with the explanation given by Gignoux et al. [11 ]. The DOS at the Fermi level is ver y sensitive to the position of the atoms in the unit cell. Figure 2 shows that the variation in the DOS is very important, mainly f o r x = z = 0.0, and that w he n y > 2 the DOS remains almost unchanged.
383 1.2 | ,.~. 1.1 ! / •~' /
,
,
\
,
,
.~,
,,1~
x=O.O ; z=O.O \
12j
.
....
. ...............................
4- ...............
4,
1 0.9' 0.8
0.4
~
~
...... K.,.
03 L
o2
0s
_ _ _ _
~
.~
-,~
l'.s
~'
-,x... ~ ~ - ~ .
~s
12k t
3
- -
I
3s
4
Y Fig. 4. The o r b i t a l e l e c t r o n o c c u p a t i o n
for x=z=0.
The shape of the DOS also changes drastically with x, y and z. For example, for x = z = O when y = 0 . 5 , the DOS of the nickel 12k atoms a p p r o a c h e s the Fermi level but the DOSs of the ot her atoms remain unchanged. For y = 1.0, the nickel 12k DOS moves towards higher energy and the DOS of the nickel atoms at 6j, 4e and 4f moves towards lower energy, the total DOS at the Fermi level being three times smaller than that for y = 0. The changing shape of the DOS is due to the changing location of the atomic DOS with r e s p e c t to the Fermi level. For y > 1.0, the m o v e m e n t of the different DOS peaks p r o c e e d s in the same direction. In Fig. 3 the total DOSs are shown for different y values and for x = z = O. The orbital electron occupat i on numbers (Fig. 4) lead to a m a x i m u m for the nickel 12k and 4f partial DOS and to a minimum for the nickel 12j partial DOS for y = 0 . 2 5 . For y > 0 . 2 5 the DOS for each at om reaches an almost constant value. W h en the yttrium atoms are not fully occupying the 2b, 2d positions displacement of the nickel 12j atoms to the 4e positions leaves the orbital electron o c c u p a t i o n almost unchanged irrespective of what kind of atomic positions we refer to. This suggests that the yttrium atoms when m o v e d from 2b to 2c positions have an effect on the DOS only when x and z are very small. The coefficient of the electronic specific heat com put ed for various DOSs at the Fermi level has smaller values than the experimental value unless x=z=O and y < 0 . 7 5 . The results show that (a) the total DOS is lowered when the yttrium, a n d / o r nickel ato m s are r e m o v e d from their standard positions and that (b) the shape o f the DOS at the Fermi level as a function of energy and the
384 orbital electron occupation positions.
axe v e r y s e n s i t i v e a t s m a l l c h a n g e s
of atomic
Acknowledgments O n e o f t h e a u t h o r s , A. V., w i s h e s t o e x p r e s s h i s g r a t i t u d e t o t h e O r g a n i z i n g C o m m i t t e e f r o m M i i n s t e r a n d t h e O p e n S o c i e t y F o u n d a t i o n ( S O R O S ) o f ClujNapoca for the financial support given during the Mtinster Conference.
References 1 2 3 4 5 6 7 8 9 10 11 12
D. Gignoux, R. Lemaire and P. Molho, J. Magn. Magn. Mater., 21 (1980) 119. D. Gignoux, R. Lemaire, P. Molho and F. Tasset, J. Magn. Magn. Mater., 21 (1980) 307. M. Shimizu, J. Inoue and S. Nagasawa, J. Phys. F, 14 (1984) 2673. V. Cri§an, I. Pop. M. Popescu and V. To§a, Proc. Pth Int. Workshop on Rare-Earth Magnets, Bad Sode~ Germany, 1087, Springer, Berlin, 1987, p. 165. W. B. Pearson, Lattice Spacings and Structures of Metals and Alloys, Vol. 2, Pergamon, Oxford, 1967, p. 344. K. H. J. Buschow, J. Less-Common Met., 11 (1966) 204. A. Christensen and R. G. Hazell, Acta Chem. Scand. A, 34 (1980) 455. R. Haydock, V. Heine and M. J. Kelly, J. Phys. C, 8 (1975) 2591. J. C. Slater and G. F. Koster, Phys. Rev., 04 (1954) 1498. R. K. Sharma, Phys. Ref. B, 12 (1979) 2813. D. G. Pettifor, J. Phys. F, 7 (1977) 613. D. Gignoux, R. Lemaire, P. Molho and F. Tasset, J. Appl. Phys., 52 (1981) 2087.