Electronic structure of NiAl by the FLAPW-method

PHYSICA

Physica B 182 (1992) 267-277 North-Holland

Electronic structure of NiA1 by the FLAPW-method O.V. F a r b e r o v i c h a n d S.V. V l a s o v Voronezh State University, 1 Universitetskaya Square, 394693 Voronezh, Russian Federation

K . I . P o r t n o i a n d A . Yu. L o z o v o i All-Union Institute of Aircraft Materials, 17 Radio Street, 107005 Moscow, Russian Federation

Received 3 October 1991 Revised 24 February 1992 The band structure calculation of the B2-phase of NiAI by the full-potential linearized augmented-plane-wave method is used for the qualitative description of interatomic interaction in the compound. Calculations of partial contributions to the joint density of states provides us with a more consistent interpretation of the imaginary part of the dielectric function e2(~0). The results of FLAPW calculations are compared with those obtained within the standard "muffin-tin" LAPW approach.

I. Introduction

T h e B2-phase of NiAI belongs to a well studied class of intermetallic electronic compounds with 3 electrons per atom. Intermetallic NiAI, as well as FeA1 and CoAl, has a rather high melting t e m p e r a t u r e ( T m = 1911K) and a high o r d e r - d i s o r d e r transition temperature ( T c > Tm) [1]. This means that interatomic interaction in the c o m p o u n d is sufficiently strong, though it strongly depends on the kind of atoms. Nevertheless, the nature of this interaction is not clear yet. In the time of H u m e - R o t h e r y it was supposed that three valence electrons of aluminum produce a metallic bond. However, this does not explain the high stability of the compound. Jelatt et al, [2] regard the bond between a transition metal and a non-transition element as a result of c o m p e t i t i o n of two physical mechanisms. The Correspondence to: Dr. A. Yu, Lozovoi, All-Union Institute of Aircraft Materials, 17 Radio Street, 107005 Moscow, Russia.

first is related to the weakening of the bonds between the atoms of the transition metal due to the increase of the distance between them during a c c o m m o d a t i o n of the non-transition element. The second one is due to the formation of a covalent bond between the d-states of the transition metal and the p-states of the non-transition element. Besides, there are indications that charge transfer from aluminum to nickel takes place in the c o m p o u n d [3-8]: it has been interpreted as the filling of the d-shell with valence electrons of aluminum. For example, Ellis et al. [9] obtained the self-consistent configuration Ni-°12Al °12 as a result of cluster calculations. The ionic character of the compound in question was also pointed out by Lee et al. [101. A rather original treatment of the charge transfer in NiAI was offered by Lui et al. [11] to explain " a b n o r m a l " core-level shifts (positive for nickel and negative for aluminum). Authors state that the direction of the charge transfer in the c o m p o u n d depends on the orbital symmetry of the states, which leads in the result to the elec-

0921-4526/92/$05.00 © 1992 - Elsevier Science Publishers B,C. All rights reserved

268

O.l'. l:arherorich ('t al.

l:leclr
tron transfer from nickel to aluminum (for cxample, nickel gains d electrons but loses morc sp electrons). The rippled relaxation of the NiAI (I 1 0) surface came to light recently [12, 13[ and this immediately raised a new interest for thc intermetallic. The magnitude of the surface rippling was successfully calculated by the methods of the e m b e d d e d - a t o m [14], mixed basis pseudopotential [15] and full-potential linearizcd plane wave ( F L A P W ) [10]. The main physical mechanism underlying the surface relaxation proposed in these papers consists in the restoration of the electrostatic equilibrium broken by the "'spilling" of charge into the vacuum region. This chargc acts to screen the density discontinuity caused by the formation of the surface and consists of sand p-electrons of aluminum due to their extended nature. The purpose of this work is to investigatc these two problems (high thermodynamic stability and surface relaxation phenomena) from a single point of view. To this purpose we propose a model describing the interatomic interaction in a B2-compound consisting of a transition metal with a nearly full d-shell and a non-transition metal. This model is based on the F L A P W band structure calculations for bulk NiAI. The result of the band structure calculations allow one to analyse the imaginary part of the dielectric function e2(w) whose fine structure and interpretation remain rather indefinite (see section 3.3). The results obtained on the basis of the F L A P W method are compared, where it is possible, with the results of "'muffin-tin" L A P W ( M T - L A P W ) , to estimate the non-MT corrections which a p p e a r in a relatively close-packed compound.

V(r) ....

/~

V l ( r , l Y/(,~:,,).

rC(_2,,.

V(K)cxp(iKrl

r~

I .

k h

and similarly for p(r). Here the indices .s and t cnumeratc the kinds of atoms and different atoms of the samc kind. index L substitutes the pair (/. m), 1 denotes thc interstitial region, and r,, =- r

b,

,

where b , arc the coordinates of basis atoms in the unit cell. The Hamillonian can be written as

where HMI is a Hamiltonian in the M T - L A P W method: t~/,~ I

caep 4 (/3

I )me

+ ~ V,.,(r,,/Y,,,,0(rC .¢L,I. H~ corresponds to the interstitial region and H,~, to the non-spherical components of the potential insidc MT-spheres: 1211 =

~\~ V ( K ) e x p ( i K r ) O(r~ 1), a.~ ~t

,~, /

V(r,)Y / (t',)O(rC

~(2,)

* (ll II)

The calculations were performed in two steps 117]. First, the self-consistent band structure was determined with the cxact Hamiltonian only for the interstitial region:

H,,M, = FtM, ~ r,',, 2. Calculation

We calculate the band structure of NiAI with the lattice constant of 2 . 8 8 6 A by the F L A P W method [16]. In this method the charge density p ( r ) and the crystal potential V ( r ) are the series of spherical harmonics inside MT-spheres .(2,, and of the plane waves outside those spheres:

and then the electronic structure was recalculated with respect to non-spherical corrections inside the MT-spheres (one more iteration) having lixed the Fourier expansion of thc density and potential within the interstitial rcgion. Thc latter step uses self-consistent eigenfunctions of the Hamiltonian /~/~.,~ as basis functions.

O.V. Farberovich et al. / Electronic structure o f N i A l by the F L A P W - m e t h o d

For the evaluation of the Coulomb part of the potential we use the method developed by Weinert [18]. We pick the optimum value of the parameter n which is responsible for the rapid convergence of the Fourier series with respect to the reciprocal lattice vectors K by calculating the Madelung constant with the same radii of the MT-spheres and the same number of plane waves as in the FLAPW-calculations. Note that the same result can be obtained according to Weinert's recommendations [18]. The exchange and correlation potential is approximated by the form of Vosko et al. [19]. The Fourier components of the exchange-correlation potential in the interstitial region are fitted by the least-squares method with the use of the singular value decomposition procedure. The charge density in the interstitial region is calculated in 2000 random points in the irreducible wedge of the Wigner-Seitz cell thus stabilizing the Fourier components of the exchange-correlation potential with an accuracy of better than 1 mRy. Calculating the band structure we use the basis set of 117 augmented plane waves, while the charge density and crystal potential in the interstitial region are expanded into 925 plane waves. Inside the atomic spheres the density and potential are represented by spherical harmonics with the value of l up to 9. Calculations were carried out within the scalar-relativistic approximation (SRA) using 32 points of general position in the irreducible wedge of the Brillouin zone. MT-radii were determined from intersections of zero-iteration atomic potentials and placed to the nearest point of the radial mesh so that the MT-spheres do not overlap. The criterion of self-consistency is the stabilization of eigenvalues within 0.1 mRy of the total energy, l m R y , and of the potential, 0.1%. After each iteration the electroneutrality condition is checked to be fulfilled, the excess charge (about 10 -4 electrons per unit cell) being distributed uniformly over the unit cell. Having achieved self-consistency the energy band structure was recalculated in the standard mesh of 35 points along symmetric directions in the irreduc-

269

ible domain of the Brillouin zone in order to construct the density of states (DOS) and joint density of states (JDOS). Self-consistency for the MT-LAPW method (e.g. ref. [20]) was looked for with the same parameters.

3. Discussion

3.1. Density of states The energy band structure of NiAI (fig. 1) coincides with the results of previous calculations [1,3, 21-23] to the accuracy of a few millirydbergs, while the final electronic charge density agrees perfectly with the FLAPW-density of the central layer of the NiA1 (1 1 0) film [10]. DOS has been calculated using the hybrid tetrahedron method with the quadratic interpolation of the eigenvalues [24]. Beyond the MTapproximation we find a slight expansion of the valence band mainly due to the s-states (fig. 2). E, R y 1.0

0.9 0.B 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

X

I-

M

X

R

r-

M

Fig. 1. B a n d s t r u c t u r e o f N i A I (E v : 0 . 7 7 0 3 R y ) .

R

270

O.V. Farberovich et al. ,' Electronic structure q f N i A I by the F l . A t ' W - m e t h o d

Site- and s y m m e t r y - d e c o m p o s e d densities ol states for NiAI (fig. 3) show the strong hybridization of the p- and the d-states of aluminum and the p-states of nickel with the nickel d-states all over the occupied zone. S-states of aluminum reside mainly at the bottom of the valence band and hence their hybridization with nickel d-states is practically absent. On the other hand, the s-states of nickel are more homogeneously distributed within the valence band mixing with the aluminum s-states as well as with the nickel d-states. There is a fall of the s-state densit~ curve of aluminum at the energy of the main p e a k of the nickel d-states, which indicates that the charge transfer from aluminum to nickel takes place in the compound. The electron density p(r) provides us with more detailed information about the charge redistribution in the crystal caused by the formation of the bonds between atoms.

E, eV lO

8

6

4

2

o

,'T~

280

! [3 ~

C

z/

240 i D i

~,

I/

'~ A

200

/ / / /

E

160

L

J"

1

120

\ v

8Q

40

3.2. Bonding FLAP 0.8

0.6

0.4

E, Ry

Changes in electron density accompanying the appearance of the chemical bonds in a compound are usually small in comparison with the value of the density itself, so the crystal structure is mainly responsible for the shape of p(r). For this reason analyzing the interatomic interaction in NiAl we have constructed a difference of the self-consistent crystal charge density of the c o m p o u n d and a superposition of the self-consistent densities of the isolated atoms placed on their own sites of the lattice:

O

0.2

Fig. 2. Total densities of states (in states per Ry per unit cell): results of M T - L A P W and F L A P W compared with the experimental XPS-spectrum [4] (dashed line). The Fermi level is confirmed with zero energy in all curves.

In their other features the M T - L A P W and F L A P W curves practically coincide and agree well with the experimental data and other calculations (table 13.

Ap(r) =

p(r)

p,,(r)

Table 1 C o m p a r i s o n of energy positions of features of the density of states curves obtained b;' the different m e t h o d s (energy in eV). Method

A

B

(57

D

xPs ups

0.63 o.s

1.s4 2.1

3.2

4.54

04

SXS-ALK~ SXS-NiL m FLAPW MT-LAPW APW

0.7

3.3

5.5 4.8 4.7(I 4.70 4.8

11.7

(1.55 (I.58 [).7

1.8 1.94 1.91 1.9

3.17 3.2(I 3.(I

E

11.33 11.29 11

Ref.

[4] 125] [26] [26] this work this work [4]

271

O.V. Farberovich et al. / Electronic structure of NiAl by the FLAPW-method " - - - - - t - -

1.6

,

1

,

Ni-s

1.2 0.8 0.4

4""

-

2.0

~

~

AI- p

J

1.0

/ 120

Ni-d

I

I

E,I

Ef

Ai-a

80

40

",o,w

0.0

O.2

0.4

0.6

0.8

E~ Ry

0.0

0.2

0.4

0-6

0.8

E, Ry

Fig. 3. Site-and symmetry-decomposeddensitiesof states (in statesper Ry per atomper spin). and Oo(r) = 7 ;

0

,(r - R . - b . , ) ,

R n ,st

where R, are the coordinates of the unit cells. The choice of po(r) as the reference state charge density can be justified as follows [3]. The charge density in the crystal is supposed to be formed in two steps. In the first step neutral atoms are placed on the lattice sites of the ordered compound and their atomic charge densities are superposed. In the second step atoms interact with each other to form a redistributed self-consistent charge density of the real crystal. In this sense po(r) corresponds to the charge density of a hypothetical crystal with non-interacting atoms, and Ap(r) to a charge redistribution caused by the formation of chemical bonds. We construct the charge density of the reference state with the next valence electron configurations of the neutral atoms [3]: 3d94s I for nickel and 3s24p I for aluminum. Isolated atoms in this case have the lowest total energy in SRA.

F L A P W allows to plot Ap(r) in the whole unit cell because this method does not make any shape approximations for the charge density. The difference of densities Ap(r) in NiA1 looks as a superposition of almost spherical regions of excess charge which surround the nickel atoms and reside near the boundaries of the nickel MT-spheres, shifting along the (1 1 1) direction to the origin (see fig. 4). High density that appears due to the overlapping of these spherical regions near the Ni-Ni (1 0 0) directions may be interpreted as the formation of covalent-type bonds between the neighboring nickel atoms (fig. 4(a),(d)). The density profile along the Ni-A1 (1 1 1) direction refers to the bonds of the ionic-covalent type (fig. 4(e)). Both types of bonds are formed by the p- and d-electrons of nickel and aluminum and by the nickel s-electrons mixing with them. The remaining part of the nickel s-electrons with the energies lying in the lower half of the valence band as well as the aluminum s-electrons take part in the metallic-type bond.

O.V. lVarberoYiHl ¢,f al.

272

t~/l¢clrot*lc.Wrtic[Htx. ¢~/ ,\"t.41 I~ Itt(' l'l..,lf'VJ-megtlr~J T

•

•

....

,

r

• .

bl

1.5

.

.

.

.

I

1.0

1.0

05

0.5 0,0 0.5

!,,3~

1.0 1.5 i 1.5 ~'.0 3.5

,).b

~

.

4.'.,

.1

.

.

\

O.O (}.5 10 15

C'

L 10

212,

30

40

50

0

]5

t). !5 L(1) I5LJ 0.5

1(

i.~

.-'( :

P/

3.O

~,~

Fig. 4. Profiles of the difference of the crystal s¢lf-connistcnl chaFgc density and the s u p c r p o s i l i o n el a t o m i c ch:u eL' dcllsilic'. Ap(r) p(r) p~,(r) (m 10 c a u ' ) b e t w e e n n c i g h b o r i n g a t o m s : (a) Ni Ni. (1 ()(I): (h) Ni Ni. ~1 l ii' : (c) AI AJ. l ( l ( I ) : (d} AI AI. ( 1 1 0}: (e) AI- Ni, { I 1 1 }. H o r i z o n t a l axis is i n t c r a t o m i c distancc in au. zero in (c) c o r r e s p o n d s t~ AI atom. I ) a s h c d Hue i n d i c a t e s the regions w h e r e the curve c a n n o t bc d r a w n cxactlx due t¢~ d i s c o n l i n u i t \ a! lhc M'l'-sphcrc hotlll(l;.H; {;lt~oul 1() + C/~ttl~).

The first two bond types provide thc high thermodynamic stability of the compound. The statement of Jelatt et al. [2] concerning the weakening of the bonds between the transition metal atoms in a compound may appear in the present case to be not quite exact. The bonds between the nickel atoms are strengthened by the contribution of aluminum p- and d-electrons that comes from the hybridization, so the total effect can prove to be positive, This situation resembles partly one in AILi and AI~Li compounds where Li donates its valence electrons to strengthen the A I - A I directional bonds [27]. It is not yet clear why the N i - A I bond proves to be polarized. Probably, this is so because of the fact that electrons fill mostly bonding states which have larger amplitude on nickel atoms having more localized orbitals [2]. Such a complexity of bonding in NiAI is con-

lirmed to a ccrtain degree by mechanical propertics of the single crystal. The metallic part oI bonding imparts some ductility to the intermetallic [28], while the presence of the exccptional direction of the "'hard" deformation {l()(I), other directions being "'soft", givcs us ground to suppose that the Ni-Ni (1()()) bond has not the least important rolc in thc con> pound. Note that there is no definite indication of the existence of the direct Ni-Ni bond in the litcrature. Fuggle ct al. [29] suggest that there exists an interaction between Ni atoms via the intermediate aluminum atoms. Sarma et al. [301 conclude from their cluster calculations that the direct N i d - N i d interaction contributes to the bonding energy in the next way: d - d coupling causes a gcneral redistribution of states by "'pushing out" bonding-anti-bonding d-sp states,

273

O.V. Farberovich et al. / Electronic structure of NiAl by the FLAPW-method

Looking at the character of A p ( r ) between neighboring nickel and aluminum atoms (fig. 4(e)) we can suppose that electron charge in the compound flows from the MT-sphere of aluminum to the nickel sphere. The existence of the regions with negative A p ( r ) near aluminum on sections in other directions (fig. 4(b),(d)) confirms this conclusion. On the other hand, the existence of the regions with charge deficit near the nickel atom in the { 1 0 0 ) and { 1 1 0 ) directions indicates that part of the electrons residing in the MT-sphere of nickel moves toward the boundary of the sphere forming charge excess there and getting out partly in the interstitials. The latter is demonstrated clearly by fig. 4(a). Thus the aluminum atom in the compound plays the role of an electron donor indeed. Part of its electrons passes to the nickel atoms participating in the N i - A I {1 1 1) bond and the other part is redistributed over the interstitial region, being involved into other bonds. However, it is difficult to compare this particular result directly with that by Lui et al. [11] who used a reference state different from ours. Note also that the neighboring aluminum atoms prove to be practically non-bonding to each other (fig. 4(c)). Obviously the picture of bonding in NiAI described here appears to be a strongly simplified model of the real situation. Nevertheless let us try to demonstrate its usefulness on the following example. We discuss now the atomic relaxation on the (1 0 0 ) surface of NiAI. According to the simple theory of Finnis and Heine [31] the relaxation is a response to electrostatic forces produced by a smoothing of the surface electron density. Electrons of the subsurface layer are noticeably involved in a redistribution of the charge in the case of NiAi [15]. This fact is proved in particular by the multilayer character of the relaxation [13, 14]. Nickel (aluminum) atoms in the truncated bulk surface interact with four nearest atoms in the second layer, namely with two nickel and two aluminum atoms. Since the Ni-Ni bond is much stronger than that of AI-AI an additional inward force acts on the top-layer Ni atoms. In an ideal

I

E0113

NLAI

(22!3

Fig. 5. Difference of the crystal charge density and the superposition of atomic charge densities on the (2 1 1) plane. Aluminum atoms are situated in the corners, nickel at the center of the [1 1 1] side. A density plot is cut at the value lAp(r)[ - 0.02 e/au 3. infinite crystal the contraction of d - d bonds is blocked by the increase of the kinetic energy of the interstitial electrons. Interstitiais lying under the surface nickel sites are not occupied by any localized bond (it is an A1-AI bridge region, fig. 4(c)), hence the electrons from these regions readily move toward the vacuum region and thus make possible the inward relaxation of Ni atoms. The interstitial sites beneath AI are occupied by the d - d bond of the subsurface neighboring nickel atoms (figs. 4(a) and 5). Besides that, there are electrons localized in the vicinity of these interstitials which take part in Ni-AI ( 1 1 1 ) bonds between aluminum atoms of the surface layer and nickel atoms of the subsurface layer (figs. 4(e) and 5). For this reason the electrons just below the outermost aluminum atoms cannot spread into the open space and an aluminum atom has no opportunity to move into the bulk. On the contrary, this atom that has lost its electrons and has transformed into a positive ion should experience outward relaxation under the surface dipole field. 3.3. e2 (oa)

The imaginary part of the complex dielectric function if only the direct (or k-conserved) interband transitions are taken into account,

O.V. Farberovieh et al.

274

¢

e4~o) -- (2~re/mw) -~~ -

,,,,.

/ _

Electronic s'tructure of NiAI hv the l"L,4PW-method

2d"k (2v)

~

BZ

× le&.<(k)lea(&.(k)

E,.0,)-h
is proportional in the approximation of constant electric dipole matrix elements P,.,,(k) to the joint density of states: e~(w) ~

J(w)/w

~- .

where

2d:~k

f ~8(E,(k)

J(~o)=,~

E,.(k)-h~).

BZ

B Z denotes the Brillouin zone and the indices u and c e n u m e r a t e the states of the valence and conduction bands, respectively. Note that the approximation of constant transition m o m e n t s in the case of NiAI does not affect practically the peak position, it only changes slightly their relative intensities [32], while if indirect transitions are taken into account a certain broadening of the maxima takes place [23]. For J D O S calculations we have used just as for D O S the hybrid tetrahedron method with quadratic interpolation. The imaginary part of the complex dielectric function of B2-NiA1 was studied both theoretically and experimentally many times [21-23, 3236]. Nevertheless, the main maximum ~~ at 2.5 eV is not unambiguously interpreted till now. Some authors [21,34] related this maximum to transitions into states near the Fermi level, while others [22, 32, 36] explained it to be mainly connected with the transitions into occupied states above Basing on the latter statement Sasovskaya et al. [36] related the evolution of the e~ curve when the nickel concentration in the alloy increases to a shift of the A-peak of the density of states to the high-energy region. As a consequence the martensitic transformation B2---~ L1 in NiAI was related to an instability of the J a h n - T e l l e r type that arises when the peak terminates at the Fermi level. In fact, this peak

goes down and near the Fermi level a fluctuation zone of the nickel states appears [23], the hitter fact explaining the sharp increase of the density of states at the Fermi level as the nickel concentration comes to 59 atC~ [37 I. Peterman et al. [21] revealed some features in the spectrum which were not seen before, either in experimental works or in theoretical calculalions. A weak peak at about 1.4eV, fine structure of the 2.5 eV peak, a shoulder near 2.1 eV and two peaks at 2.4 and 2.6eV were o b t a i n e d Besides, the m a x i m u m at about 4.1 eV was shown to have an asymmetric shoulder ai the low-energy side. The imaginary part of the complex dielectric function calculated in the range up to 8eV is c o m p a r e d with that obtained by Peterman et al.

1400

/ '\ , \

1200

~ '..

\

/ ¢ ~

\ <~ , "-

,

J

,

\

/

"

a

lOOO

800

6oo

4O0

Ev.

.

0

2

.

.

3

.

4

5 ~

6

7

eV

Fig. ~3. hnaginary part of the dielectric constant ~:(¢oJ in arbitrary units: results of M T - L A P W and F L A P W (the latter curve is shifted up by 500 units). Dashed line reprcscnb, an experimental curve [211

O.V, Farberovich et aL / Electronic structure of NiAI by the FLAPW-rnethod

[21] in fig. 6, the decomposition of the resulting curve into a sum of partial contributions being given in fig. 7. The double structure of the main peak at 2.5eV is not observed. The distance between almost equal maxima related to transitions from the 4th, 5th and 6th bands on the Fermi level which could result in such a spectrum is too small, so the transitions from those bands merge in one broad maximum at about 2.0-2.6eV. In addition a sharp peak superimposes on the maximum. This peak is originated by the transitions from the seventh band into the non-occupied states above E v at the points of the general type and in the direction of M - T - R near R.

275

Hence, both types of electron transitions are responsible for the fine structure of the main maximum. Such a conclusion, which we have come to previously on the basis of LAPW-calculations [38], is confirmed by the recent L M T O calculations of Knab and Koenig [39]. All other features of the experimental e2-function [21] are seen in the calculated one. We also find a maximum at 5.8eV which is related to transitions from the second and third bands. The curve %(w) that we have plotted using the results of the M T - L A P W calculation differs very slightly from that of F L A P W (see fig. 6): the shifts of peaks do not exceed 0.1 eV, this being within the experimental errors.

Band 1 20

10 •

°

,

~

•

r

w

160 200

120

100

80 40

300

Band 6

20O 80

40

300

(a)

a

A

B nd 7

Band 4

200

100

B

2OO

\

tO0

(b/,, 0

, 1

t

2

3

Fig. 7. Contributions of transitions from each band in arbitrary units

4 5 W~ eV

6

7

276

O.V. Farberovich et al.

:

Electronic structure o f N i A I hv lhe t:1-4 l ' W - m e t h o d

4. Conclusion The main results of the present study may be formulated as follows: (i) Interatomic bonding in NiAI can be described as a combination of the bonds of three types: A I - N i bonds along the (1 I 1) direction, N i - N i bonds along the (1 0 0 ) direction and metallic-type bonds produced by s-electrons of aluminum and by the part of nickel s-electrons. The statement about the weakening of the N i - N i 1 0 0 ) bond as compared with that of the pure metal may appear to be not quite exact in relation to NiAI, since this bond is accompanied by strong hybridization with p- and d-electrons of aluminum. A l u m i n u m atoms prove to be practically non-bonding to .each other. (ii) The main maximum in the C,(w) curve near 2.5 eV is a sharp peak (originated by the electron transitions from the states of the scventh band into the conduction-band states) on the background of a broad maximum related to transitions from the valence band states to the states near the Fermi surface. (iii) The density of states and the joint density of states of NiA1 does not depend significantly on whether they are calculated with or without the MT-approximation.

Acknowledgements We are grateful to Professor A . A . Katznelson and Dr. V.S. Stepanuk, M o s c o w State University, for their interest and critical c o m m e n t s on the manuscript. One of the authors (A. Lozowfi) wishes to thank Drs. A.V. Ruban and Ju. M. Mishin, All-Union Institute of Aircraft Materials, for useful and informative discussions in the process of this work.

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