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Electronic structure of hydrogen in nickel and its trapping by impurities
Journal of the Less-Common
Metals, 130 (1987)
275
275 - 283
ELECTRONIC STRUCTURE OF HYDROGEN IN NICKEL AND ITS TRAPPING BY IMPURITIES* Z. BADIRKHANa,
J. KHALIFEHa
aPhysics Department, bLMSES,
and C. DEMANGEATb
Jordan University, Amman (Jordan)
Universite’ Louis Pasteur, 4, rue Blaise Pascal, 67070 Strasbourg Cedey (Fmnce)
summary The electronic structure of a hydrogen atom in nickel is obtained using a generalized Slater-Koster fit to an ab initio band-structure calculation for the transition metal combined with an extra s orbital for the hydrogen. The hamiltonian of the system is mainly characterized by hopping integrals between the interstitial and its nearest neighbours. The physical parameters in the tight-binding hamiltonian are adjusted with the help of the Friedel sum rule. The hydrogen atom induces resonance in the lowest part of the nickel band structure. A decrease of 0.25 cc, of the total magnetic moment per added hydrogen is observed together with an increase of the density of states at the Fermi level. This work is needed as a starting point for further calculations such as the hydrogen-impurity binding energy.
1. Introduction A fundamental understanding of trapping by lattice defects in transition metals remains a challenging problem. For example, diffusion coefficient, hydrogen embrittlement, pressure-composition-temperature phase diagrams and induced electrical resistivity are sensitively influenced by its interaction with trapping centres. A detailed knowledge of the interaction processes of hydrogen with defect structures is therefore required. By means of magnetic after-effect measurements the bindingL energy of hydrogenimpurity complexes in nickel has been investigated [l]. The conclusion of these workers is that the binding energy of diatomic metal-hydrogen complexes in nickel is composed of electronic as well as elastic contributions which are of the same order of magnitude. It is difficult to say whether the strain field produced by the impurity is responsible for the trapping or whether it arises from’ modification of the screening clouds around this impurity. Moreover, one of the crucial *Paper presented at the International of Metal Hydrides V, Maubuisson, France, 0022-5088/87/$3.50
Symposium on the Properties May 25 - 30,1986. 0 Elsevier
Sequoia/Printed
and Applications
in The Netherlands
2’76
questions is whether the strain effect acts in the same direction as the electronic perturbation. Our objective (for the future) is therefore to extend to nickel alloys a model developed previously for the palladium system P, 31. To determine the electronic (‘chemical’) binding energy between one hydrogen atom and a transition metal impurity we have to start with the determination of the electronic structure of the isolated hydrogen atom. The modification of the electronic structure of the substitutional impurity is simply represented in a tight-binding scheme by a localized matrix element of the perturbed potential [4] which satisfies Friedel’s screening rule. The chemical binding energy between hydrogen and impurities therefore shows a change of sign when one goes from impurities located to the right of nickel in the periodic table to impurities located to the left [ 51. For example, for impurities located to the left of nickel and for a small charge difference between the impurity and the host, the binding energy is found to be attractive [6]. To determine the elastic binding energy between one hydrogen atom and a transition metal impurity, it is necessary to know the strain effect resulting from these two point defects. If we use the current asymptotic approximation, this may be expressed in terms of a product of the dipole force tensor of each species. However, this approximation may not be justified when the impurities are at nearest or next-nearest positions. Then, we have to introduce a microscopic description of the forces [7,8] and of the force constants [9] of the alloy. The forces resulting from the presence of one interstitial hydrogen in nickel are deduced from the variation of the total energy of the alloy up to first order in the atomic displacement. This tight-binding calculation is based on a rigidly moving wavefunction basis. Expressions of the distribution of the forces are derived in terms of the variation of the hopping integrals and of the energy levels [lo]. Once this has been done, the elastic binding energy can be obtained if we know the lattice Green function of the alloy [ 91. Section 2 is devoted to the description of the tight-binding model used for the electronic structure of hydrogen in nickel. Section 3 reports the numerical results and the conclusions are reported in Section 4.
2. Formalism The electronic structure of the dilute alloy of hydrogen in nickel is obtained using a Mater-Koster (S-K) fit to an ab initio calculation for the pure metal combined with an extra orbital for hydrogen. The metallic s, p and d orbitals of the host are labelled by IRmo), where R is the metallic site, m is the orbital symmetry with m = (s,~,y,z, xy,yz,zx, X* - y2,3z2 r*) and u is the spin direction. If X is the impurity interstitial site at the octahedral position [ll] and 1Xso) is the corresponding extra s orbital of energy Ef’, then the perturbed hamiltonian per spin u is given by
277
H = Hi + C IRdo)vg(R)(Rdal a.d with
+ 2 (IRrnu>/3~~~hsal + c.c.) a.m
Hi = Ho + Ihsu)E~“(XsuI
(I)
(2)
where Ho is the pure metal hamiltonian per spin u. The second term in eqn. (l), u:(R), represents the diagonal disorder in the 3d orbitals centred at the crystalline sites R and represents the variation of the energy levels owing to alloying. 0% is the hopping integral, between the host and the interstitial orbitals and C.C. indicates complex conjugate. Neglecting the overlap integrals, the local density of states (LDOS), per spin u, at the hydrogen site X is given by nr(E) = - Im{n-‘G,$‘(E)}
(3)
where Gr{(E) is the matrix element of the Green function at site h [12,13]. The variation of the local density of states at a metallic nearest neighbour site R is found to be 6@(E)
= - Im n-‘~{G~~“(E) I m
1
- Gk,“,“O(E)}
(4)
where GEE” is the matrix element of the Green function at site R. To solve our problem, we have to satisfy Friedel’s sum rule, i.e. the number of external electrons brought by the hydrogk impurity (one electron) is equal to the total number of displaced states up to the Fermi level EF
I
=
1
(5)
(7
Here, P(E) P(E)
=-
is the number of displaced states up to energy E defined as F
jTrace{G’-‘(E -m
‘) - G’O(E’)} dE’
(6)
Now, the total number of s electrons per spin u in the hydrogen orbital at site X is given by NSU(X)= sEK’,“(E) dE --m
(7)
The variation of the number of electrons at site R, owing to the presence of hydrogen is
6W(R)
= &(E) -0a
dE
(8)
278
As usual, the effective level of hydrogen in the host Ef” is related to P’(X) by the equation [ 141 Er
= Eit + ax + UTW-“(X)
(9)
where Eft is the atomic level of the hydrogen atom in its ground state, (Y! is the crystal field, i.e. the variation of the impurity energy level owing to its existence in the metallic crystal and Ur is the Coulomb correlation term on the hydrogen site. For pure ferromagnetic metal with unequally populated bands of different spins [15], there appears an energy difference AE between them, defined as AE = Jddrp
(10)
where /.I is the magnetic moment of pure metal defined by the difference in the local density of states rig(E)) for opposite spins /..t=
c
EF{n;(E) - n;-‘(E)}
s m --m
dE
(11)
and Jda’ is the exchange correlation term. The relation between the exchange integral and the total variation of the LDOS on a metallic site R, owing to alloying, along with the diagonal disorder term u:(R) is u:(R) + J&W’(R)
= u;“(R) + J&W”(R)
(12)
with [16] J dd' =
udd + udd 5
(13)
3. Numerical results and discussion The aim of this section is to point out the details of the method by which the physical parameters, Ei” and u:(R), existing in the tight-binding model hamiltonian have been obtained. The problem of one hydrogen impurity in nickel is mainly characterized by an effective energy level at the octahedral position and hopping integrals coupling the interstitial with its first nearest neighbours. These hopping values between hydrogen and metallic orbitals are of the types (ssu, spu, sdo) and are obtained from Switendick’s work [17]. In contrast, for pure nickel we use an “spd” S-K fit to the band structure obtained by Wang and Callaway [ 181. This is not entirely satisfactory since the S-K fit of Switendick consists of a nonorthogonal basis set of atomic orbitals, whereas that of Demangeat et al. [19] for pure nickel uses an orthogonal basis set. A more elaborate calcula-
279
tion should avoid this inconsistency by using an orthogonal basis set for the metal hydride as already done for some CaFz structures [ 201. To determine the position of hydrogen effective levels for both spin directions at the octahedral position in nickel, the values of Eft, ai and Uy, as well as the position of nickel bands relative to Eit are required. Therefore, it is convenient to choose a vacuum as the zero energy. The position of the host bands is fixed to reproduce the experimental value (-0.19 au.) for the work function of nickel [21]. As usual, in such kinds of calculation, the crystal field c~yoh is neglected. Since the value of the Coulomb integral Urh is not calculated in this project, values already available in the literature have been used [ 121. The variation of the energy levels Ei” because of alloying for both spin directions corresponding to the hydrogen orbital and the variation of the d energy levels on nearest neighbours u:(R), remain to be determined. This work has the advantage of avoiding the local neutrality condition, i.e. the total number of electrons at the hydrogen site is equal to one (&N”“(X) = l), as was discussed by Demangeat et al. [ 191. Eqns. (5), (9) and (10) are satisfied self-consistently for a correlation value Ur = 0.26 a.u. A charge transfer from metal to metalloid is present (see Table 1). The impurity is overscreened by electrons from the host d bands. An estimation of the number of electrons in the first nearest neighbours region indicates that the perturbation probably exceeds our prediction. A virtual bound state appears in the bottom of the d bands, mainly because of the minority spin direction (see Fig. 1).
TABLE I Parameter
Z?EFY I'm %Z'(EF)~
Nsf(X)c NS+h)C C,,N=‘(?I)~ 6~ t(~)e GN&(R)~ X;,&N~(R)~
Calculated value 0.378
0.624 1.002 0.715 0.904 1.620 -0.050 -0.011 -0.061
aNumber of displaced states per spin u up to the Fermi level Z”(E~) (U = f, 4). bThe total number of displaced states up to EF for two spin directions X,Z’-‘(EF). CIntegral local density of states per spin u up to EF for the hydrogen site N”(x). dThe total number of electrons at the interstitial positions &,NSO(h). eThe variation of the LDOS per spin u integrated up to EF at a nearest neighbour metallic site &NO(R). fThe total variation of the number of electrons at a nearest neighbour site &$N”(R).
I 30.0
1
- 25.0
I!
- 20.0
.II. - 15.0
- 10.0
- 5.0
0.0
Fig. 1. Local density of states per atomic unit at the hydrogen site: -majority spin (t) direction; - . -. -, for the minority spin (4) direction; represented by the full line.
-, for the the total is
- 5.0
-4.0
- 3.0
- 2.0
-1.0
1JF “‘q.1 *-
fl’_._._
- 0.0
.d’,
-1.0
- 2.0
Fig. 2. The variation of the local density of states per atomic unit in the d metallic band: --, for the spin (t) direction; - . -. -, for the spin (4) direction,
281
The potential perturbes, essentially, the majority spin band causing a downward shift in the density of states on nickel atoms (see Fig. 2). In contast, no appreciable variation in the LDOS of the minority spin band is noticed. The number of displaced states per spin u is sketched as a function of energy (Z’(E)) in Fig. 3. A positive variation of the total density of states at the Fermi level d Z”(E)/dE is obtained.
- 0.5
0.0
Fig. 3. Number of displaced states as a function of energy: - - -, for the spin (t) direcfor the spin (4) direction; the full line represents the total number of tion; -*--.-, displaced states &,Z”(E).
The change in the total magnetic moment per added impurity (dp/dc), the variation in the magnetic moment at the nickel site R, (&a) and the local magnetic moment at the impurity site (piocal) are calculated (see Table 2).
TABLE 2 Calculated values for the total magnetic moment per added impurity (dp/dc) the variation in the magnetic moment at the nickel site R (s/&) and the local magnetic moment at the impurity site X (/Jre& Wldc
bR
(PB)
(PC(B)
PlOCd (C(B)
(a.u.)
-0.246
-0.040
-0.189
0.26
ur
(14)
6p, = (u~{GN+(R) - &V’(R)}
(15)
C(1oca1=
(16)
PBWW
The value obtained
-
NJ
@)I
for dl/dc
is in agreement
with experiments
[ 22,231.
4. Conclusion The local electronic environment of a hydrogen interstitial is studied for a realistic value of the Coulomb correlation term using a set of selfconsistent coupled equations. Our semiquantitative calculation shows that a small amount of charge is transferred from the neighbouring metallic sites to the impurity. The total number of electrons in the region of the first nearest neighbours is found to be different from the number of electrons brought by the impurity. This suggests a need to extend the calculation to farther nearest neighbours, and to introduce the intersite Green function matrix elements. This work is required as a starting point for further calculations such as the chemical binding energy of hydrogen pairs, the determination of the dipole force tensor and the elastic binding energy between two interstitials.
References 1 H. Kronmiiller and P. Vargas, Philos. Mug. A, 51 (1985) 59. 2 C. Demangeat, M. A. Khan, G. Moraitis and J. C. Parlebas, J. Phys., 41 (1980) 1001. 3 J. Khalifeh, G. Moraitis and C. Demangeat, Phys. Lett. A, 93 (1983) 235. 4 C. Demangeat, J. Phys. F: Met. Phys., 1 (1971) 755. 5 C. Demangeat, F. Gautier and J. C. Parlebas, J. Phys. F: Met. Phys., 8 (1978) 1879. 6 Z. Badirkhan, J. Khalifeh and C. Demangeat, to be published. 7 J. Khalifeh, G. Moraitis and C. Demangeat, Egypt. J. Sol., 7 (1985) 157. 8 J. Khalifeh, G. Moraitis and C. Demangeat, in P. Ziesche (ed.), 15th Annu. Symp. Electronic Structure of Metals and Alloys, Gaussg, 1985, p. 116. 9 G. Moraitis, J. Khalifeh and C. Demangeat, J. Less-Common Met., 101 (1984) 203. 10 G. Moraitia, J. Khalifeh and C. Demangeat, Philos. Mag. B, 49 (1984) 533. 11 B. Lengeler, Phys. Rev. Lett., 53 (1984) 74. 12 J. Khalifeh and C. Demangeat, Philos. Mag. B, 47 (1983) 191. 13 Z. Badirkhan, M.Sc. Thesis, University of Jordan, Amman, 1986. 14 D. M. Newns, Phys. Rev., 178 (1966) 423. 15 N. F. Mott and H. Jones, Metals and Alloys, Dover Publications, New York, 1936. 16 C. Demangeat, J. Phys. F: Met. Phys., 5 (1975) 1637. 17 A. C. Switendick, Ber. Bunsenges. Phip. Chem., 76 (1972) 535. 18 C. S. Wang and J. Callaway, Phys. Rev. B, 15 (1977) 298. 19 C. Demangeat, M. A. Khan and J. C. Parlebas, J. Magn. Magn. Mater., 15 - 18 (1980) 1275.
283 20 21 22 23
D. H. H. W. 54
A. Papaconstantopoulos and P. M. Laufer, J. Less-Common B. Michaelson, J. Appl. Phys., 48 (1977) 4729. J. Bauer and E. Schmidbauer, 2. Phys., 164 (1961) 367. E. Martin, H. J. Bauer, S. M. Filipek and B. Beranovski, - 57 (1986) 1505.
Met., 130 (1987)
229.
J. Magn. Magn. Mater.,
Metals, 130 (1987)
275
275 - 283
ELECTRONIC STRUCTURE OF HYDROGEN IN NICKEL AND ITS TRAPPING BY IMPURITIES* Z. BADIRKHANa,
J. KHALIFEHa
aPhysics Department, bLMSES,
and C. DEMANGEATb
Jordan University, Amman (Jordan)
Universite’ Louis Pasteur, 4, rue Blaise Pascal, 67070 Strasbourg Cedey (Fmnce)
summary The electronic structure of a hydrogen atom in nickel is obtained using a generalized Slater-Koster fit to an ab initio band-structure calculation for the transition metal combined with an extra s orbital for the hydrogen. The hamiltonian of the system is mainly characterized by hopping integrals between the interstitial and its nearest neighbours. The physical parameters in the tight-binding hamiltonian are adjusted with the help of the Friedel sum rule. The hydrogen atom induces resonance in the lowest part of the nickel band structure. A decrease of 0.25 cc, of the total magnetic moment per added hydrogen is observed together with an increase of the density of states at the Fermi level. This work is needed as a starting point for further calculations such as the hydrogen-impurity binding energy.
1. Introduction A fundamental understanding of trapping by lattice defects in transition metals remains a challenging problem. For example, diffusion coefficient, hydrogen embrittlement, pressure-composition-temperature phase diagrams and induced electrical resistivity are sensitively influenced by its interaction with trapping centres. A detailed knowledge of the interaction processes of hydrogen with defect structures is therefore required. By means of magnetic after-effect measurements the bindingL energy of hydrogenimpurity complexes in nickel has been investigated [l]. The conclusion of these workers is that the binding energy of diatomic metal-hydrogen complexes in nickel is composed of electronic as well as elastic contributions which are of the same order of magnitude. It is difficult to say whether the strain field produced by the impurity is responsible for the trapping or whether it arises from’ modification of the screening clouds around this impurity. Moreover, one of the crucial *Paper presented at the International of Metal Hydrides V, Maubuisson, France, 0022-5088/87/$3.50
Symposium on the Properties May 25 - 30,1986. 0 Elsevier
Sequoia/Printed
and Applications
in The Netherlands
2’76
questions is whether the strain effect acts in the same direction as the electronic perturbation. Our objective (for the future) is therefore to extend to nickel alloys a model developed previously for the palladium system P, 31. To determine the electronic (‘chemical’) binding energy between one hydrogen atom and a transition metal impurity we have to start with the determination of the electronic structure of the isolated hydrogen atom. The modification of the electronic structure of the substitutional impurity is simply represented in a tight-binding scheme by a localized matrix element of the perturbed potential [4] which satisfies Friedel’s screening rule. The chemical binding energy between hydrogen and impurities therefore shows a change of sign when one goes from impurities located to the right of nickel in the periodic table to impurities located to the left [ 51. For example, for impurities located to the left of nickel and for a small charge difference between the impurity and the host, the binding energy is found to be attractive [6]. To determine the elastic binding energy between one hydrogen atom and a transition metal impurity, it is necessary to know the strain effect resulting from these two point defects. If we use the current asymptotic approximation, this may be expressed in terms of a product of the dipole force tensor of each species. However, this approximation may not be justified when the impurities are at nearest or next-nearest positions. Then, we have to introduce a microscopic description of the forces [7,8] and of the force constants [9] of the alloy. The forces resulting from the presence of one interstitial hydrogen in nickel are deduced from the variation of the total energy of the alloy up to first order in the atomic displacement. This tight-binding calculation is based on a rigidly moving wavefunction basis. Expressions of the distribution of the forces are derived in terms of the variation of the hopping integrals and of the energy levels [lo]. Once this has been done, the elastic binding energy can be obtained if we know the lattice Green function of the alloy [ 91. Section 2 is devoted to the description of the tight-binding model used for the electronic structure of hydrogen in nickel. Section 3 reports the numerical results and the conclusions are reported in Section 4.
2. Formalism The electronic structure of the dilute alloy of hydrogen in nickel is obtained using a Mater-Koster (S-K) fit to an ab initio calculation for the pure metal combined with an extra orbital for hydrogen. The metallic s, p and d orbitals of the host are labelled by IRmo), where R is the metallic site, m is the orbital symmetry with m = (s,~,y,z, xy,yz,zx, X* - y2,3z2 r*) and u is the spin direction. If X is the impurity interstitial site at the octahedral position [ll] and 1Xso) is the corresponding extra s orbital of energy Ef’, then the perturbed hamiltonian per spin u is given by
277
H = Hi + C IRdo)vg(R)(Rdal a.d with
+ 2 (IRrnu>/3~~~hsal + c.c.) a.m
Hi = Ho + Ihsu)E~“(XsuI
(I)
(2)
where Ho is the pure metal hamiltonian per spin u. The second term in eqn. (l), u:(R), represents the diagonal disorder in the 3d orbitals centred at the crystalline sites R and represents the variation of the energy levels owing to alloying. 0% is the hopping integral, between the host and the interstitial orbitals and C.C. indicates complex conjugate. Neglecting the overlap integrals, the local density of states (LDOS), per spin u, at the hydrogen site X is given by nr(E) = - Im{n-‘G,$‘(E)}
(3)
where Gr{(E) is the matrix element of the Green function at site h [12,13]. The variation of the local density of states at a metallic nearest neighbour site R is found to be 6@(E)
= - Im n-‘~{G~~“(E) I m
1
- Gk,“,“O(E)}
(4)
where GEE” is the matrix element of the Green function at site R. To solve our problem, we have to satisfy Friedel’s sum rule, i.e. the number of external electrons brought by the hydrogk impurity (one electron) is equal to the total number of displaced states up to the Fermi level EF
I
=
1
(5)
(7
Here, P(E) P(E)
=-
is the number of displaced states up to energy E defined as F
jTrace{G’-‘(E -m
‘) - G’O(E’)} dE’
(6)
Now, the total number of s electrons per spin u in the hydrogen orbital at site X is given by NSU(X)= sEK’,“(E) dE --m
(7)
The variation of the number of electrons at site R, owing to the presence of hydrogen is
6W(R)
= &(E) -0a
dE
(8)
278
As usual, the effective level of hydrogen in the host Ef” is related to P’(X) by the equation [ 141 Er
= Eit + ax + UTW-“(X)
(9)
where Eft is the atomic level of the hydrogen atom in its ground state, (Y! is the crystal field, i.e. the variation of the impurity energy level owing to its existence in the metallic crystal and Ur is the Coulomb correlation term on the hydrogen site. For pure ferromagnetic metal with unequally populated bands of different spins [15], there appears an energy difference AE between them, defined as AE = Jddrp
(10)
where /.I is the magnetic moment of pure metal defined by the difference in the local density of states rig(E)) for opposite spins /..t=
c
EF{n;(E) - n;-‘(E)}
s m --m
dE
(11)
and Jda’ is the exchange correlation term. The relation between the exchange integral and the total variation of the LDOS on a metallic site R, owing to alloying, along with the diagonal disorder term u:(R) is u:(R) + J&W’(R)
= u;“(R) + J&W”(R)
(12)
with [16] J dd' =
udd + udd 5
(13)
3. Numerical results and discussion The aim of this section is to point out the details of the method by which the physical parameters, Ei” and u:(R), existing in the tight-binding model hamiltonian have been obtained. The problem of one hydrogen impurity in nickel is mainly characterized by an effective energy level at the octahedral position and hopping integrals coupling the interstitial with its first nearest neighbours. These hopping values between hydrogen and metallic orbitals are of the types (ssu, spu, sdo) and are obtained from Switendick’s work [17]. In contrast, for pure nickel we use an “spd” S-K fit to the band structure obtained by Wang and Callaway [ 181. This is not entirely satisfactory since the S-K fit of Switendick consists of a nonorthogonal basis set of atomic orbitals, whereas that of Demangeat et al. [19] for pure nickel uses an orthogonal basis set. A more elaborate calcula-
279
tion should avoid this inconsistency by using an orthogonal basis set for the metal hydride as already done for some CaFz structures [ 201. To determine the position of hydrogen effective levels for both spin directions at the octahedral position in nickel, the values of Eft, ai and Uy, as well as the position of nickel bands relative to Eit are required. Therefore, it is convenient to choose a vacuum as the zero energy. The position of the host bands is fixed to reproduce the experimental value (-0.19 au.) for the work function of nickel [21]. As usual, in such kinds of calculation, the crystal field c~yoh is neglected. Since the value of the Coulomb integral Urh is not calculated in this project, values already available in the literature have been used [ 121. The variation of the energy levels Ei” because of alloying for both spin directions corresponding to the hydrogen orbital and the variation of the d energy levels on nearest neighbours u:(R), remain to be determined. This work has the advantage of avoiding the local neutrality condition, i.e. the total number of electrons at the hydrogen site is equal to one (&N”“(X) = l), as was discussed by Demangeat et al. [ 191. Eqns. (5), (9) and (10) are satisfied self-consistently for a correlation value Ur = 0.26 a.u. A charge transfer from metal to metalloid is present (see Table 1). The impurity is overscreened by electrons from the host d bands. An estimation of the number of electrons in the first nearest neighbours region indicates that the perturbation probably exceeds our prediction. A virtual bound state appears in the bottom of the d bands, mainly because of the minority spin direction (see Fig. 1).
TABLE I Parameter
Z?EFY I'm %Z'(EF)~
Nsf(X)c NS+h)C C,,N=‘(?I)~ 6~ t(~)e GN&(R)~ X;,&N~(R)~
Calculated value 0.378
0.624 1.002 0.715 0.904 1.620 -0.050 -0.011 -0.061
aNumber of displaced states per spin u up to the Fermi level Z”(E~) (U = f, 4). bThe total number of displaced states up to EF for two spin directions X,Z’-‘(EF). CIntegral local density of states per spin u up to EF for the hydrogen site N”(x). dThe total number of electrons at the interstitial positions &,NSO(h). eThe variation of the LDOS per spin u integrated up to EF at a nearest neighbour metallic site &NO(R). fThe total variation of the number of electrons at a nearest neighbour site &$N”(R).
I 30.0
1
- 25.0
I!
- 20.0
.II. - 15.0
- 10.0
- 5.0
0.0
Fig. 1. Local density of states per atomic unit at the hydrogen site: -majority spin (t) direction; - . -. -, for the minority spin (4) direction; represented by the full line.
-, for the the total is
- 5.0
-4.0
- 3.0
- 2.0
-1.0
1JF “‘q.1 *-
fl’_._._
- 0.0
.d’,
-1.0
- 2.0
Fig. 2. The variation of the local density of states per atomic unit in the d metallic band: --, for the spin (t) direction; - . -. -, for the spin (4) direction,
281
The potential perturbes, essentially, the majority spin band causing a downward shift in the density of states on nickel atoms (see Fig. 2). In contast, no appreciable variation in the LDOS of the minority spin band is noticed. The number of displaced states per spin u is sketched as a function of energy (Z’(E)) in Fig. 3. A positive variation of the total density of states at the Fermi level d Z”(E)/dE is obtained.
- 0.5
0.0
Fig. 3. Number of displaced states as a function of energy: - - -, for the spin (t) direcfor the spin (4) direction; the full line represents the total number of tion; -*--.-, displaced states &,Z”(E).
The change in the total magnetic moment per added impurity (dp/dc), the variation in the magnetic moment at the nickel site R, (&a) and the local magnetic moment at the impurity site (piocal) are calculated (see Table 2).
TABLE 2 Calculated values for the total magnetic moment per added impurity (dp/dc) the variation in the magnetic moment at the nickel site R (s/&) and the local magnetic moment at the impurity site X (/Jre& Wldc
bR
(PB)
(PC(B)
PlOCd (C(B)
(a.u.)
-0.246
-0.040
-0.189
0.26
ur
(14)
6p, = (u~{GN+(R) - &V’(R)}
(15)
C(1oca1=
(16)
PBWW
The value obtained
-
NJ
@)I
for dl/dc
is in agreement
with experiments
[ 22,231.
4. Conclusion The local electronic environment of a hydrogen interstitial is studied for a realistic value of the Coulomb correlation term using a set of selfconsistent coupled equations. Our semiquantitative calculation shows that a small amount of charge is transferred from the neighbouring metallic sites to the impurity. The total number of electrons in the region of the first nearest neighbours is found to be different from the number of electrons brought by the impurity. This suggests a need to extend the calculation to farther nearest neighbours, and to introduce the intersite Green function matrix elements. This work is required as a starting point for further calculations such as the chemical binding energy of hydrogen pairs, the determination of the dipole force tensor and the elastic binding energy between two interstitials.
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