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Electronic rearrangement in the dilute vanadium and niobium hydrides
Journal of the Less-Common
Metals, 130 (1987)
ELECTRONIC REARRANGEMENT NIOBIUM HYDRIDES* A. MOKRANIa, C. DEMANGEATa
209 - 217
IN THE DILUTE
209
VANADIUM
AND
and G. MORAITISb
aL.M.S.E.S. (Unite’associe’ au Centre National de la Recherche Scientifique 306) Universite’ Louis Pasteur, 4, rue Blaise Pascal 67070 Strasbourg Cedex (France) bUniversity of Antananarivo,
Department
of Physics, Antananarivo
(Madagascar)
(Received May 5,1986)
Summary The electronic (or chemical) binding energy between one hydrogen atom and another hydrogen atom or a substitutional impurity in vanadium and niobium is estimated in a tight-binding formalism. The hydrogenhydrogen binding energy is found to be considerably repulsive for nearest and next-nearest neighbouring positions. In the case of vanadium-based alloys we have estimated the binding energy between hydrogen and substitutional scandium, titanium, chromium and manganese. For niobium-based alloys we have estimated the binding energy between hydrogen and substitutional yttrium, zirconium and molybdenum. In both vanadium and niobium the binding energy between hydrogen and substitutional impurities located at nearest neighbouring positions is found to be attractive for impurities located to the left of the host metal in the periodic table and changes sign when the impurities are located to the right.
1. Introduction The calculation of the configurational entropy of hydrogen in bodycentred metals has to take into account the large repulsive interaction between hydrogen atoms at nearest and next-nearest neighbouring positions [l]. The environment of the interstitial in the niobium and vanadium hydrides can be obtained by nuclear magnetic resonance (NMR) [2] or Mijssbauer experiments [ 31 if a reasonable choice of the hydrogen-hydrogen binding energy with distance is given. One of the purposes of this paper is therefore to estimate the chemical binding energy of a pair of hydrogen atoms at tetrahedral positions in vanadium and niobium in terms of the distance between them. In our calculation we expect that hydrogen atoms *Paper presented at the International Symposium on the Properties and Applications of Metal Hydrides V, Maubuisson, France, May 25 - 30, 1986. 0022-5088/87/$3.50
0 Elsevier Sequoia/Printed
in The Netherlands
210
are located at tetrahedral positions, a result obtained by recent ion channelling experiments [ 4 - 61. The presence of the substitutional impurities alters the properties of the metal hydrides. These changes have been attributed to hydrogenimpurity attraction (or trapping effect) for impurities located to the left of the host metal in the periodic table and to hydrogen-impurity repulsion (or antitrapping effect) for impurities located to the right. By means of magnetic after-effect measurements, Vargas et al. [7] have found that in a ferromagnetic host the hydrogen is most generally trapped by substitutional impurities located to the left of the host in the periodic table. In addition, Shirley and Hall [8] have discussed the trapping of hydrogen by metallic substitutional impurities in vanadium, niobium and tantalum; their estimations for electronic and elastic contributions appear to be comparable. It should also be noted that measurements of the proton spin-lattice relaxation times lead to an antitrapping effect for manganese impurities in titanium dihydride [9] and to a trapping by vanadium impurities in the niobium hydride [ 21. This paper is organized as follows. Section 2 is devoted to the determination of the electronic structure of a single hydrogen atom in vanadium and niobium. In addition, the description of one substitutional impurity (scandium, titanium, chromium, manganese in vanadium and yttrium, zirconium, molybdenum in niobium) is reported. Section 3 is devoted to results obtained for the electronic binding energies between two hydrogen atoms in terms of the distance between them. Section 4 reports the estimation of the binding energy between a hydrogen atom and a substitutional impurity. These results are compared with recent experimental results and a theoretical result.
2. The electronic structure of one hydrogen atom and one substitutional ‘d’ impurity in vanadium and niobium The band structure of a pure transition metal is described by an spd Slater-Koster (S-K) fit to a first principles calculation [lo]. The metallic s, p and d orbitals are labelled IRmo), where R is the metallic site, m is the orbital symmetry and u is the spin. T is the tetrahedral position occupied by hydrogen [ 61 and 1Tsu) is the corresponding extra s orbital of energy E;‘. Et” is related to the number of electrons N+” in the s orbital by E$’ = Eff + U,SSNS-a(T)
(1)
The position of the bands of pure metal were fixed so as to give the experimental value for the work function of vanadium and niobium [ll]. The value of the parameter U may be taken from ref. 12 but other values [ 131 have been taken into account in order to test this parameter.
211
The perturbed hamiltonian H for a spin u is given by H = HO + ITsu)E~~(Tsu~+
~Rma)v”“‘“(Rm’uj
x R.lll.Ul’
+ ,c,‘l
Rmu)(3%Tsul
+ c.c.)
(2)
where H, is the pure metal hamiltonian and C.C. means complex conjugate. The hopping integrals pg$ between the hydrogen and the metal are taken from an S-K fit, using an orthonormalized basis, to the VH2 and NbHz band structures self-consistently calculated by the augmented plane wave (APW) method [14]. Concerning the umm”’(R) term on the nearest neighbour R of the hydrogen atom, we restrict ourselves to the most important component s$u [ 151. The generalized phase shift Z;(E) is given by Z;(E)
=-
FrTrace{G(E’)
- G”(E’)} dE’
where the trace is over all metallic and hydrogen sites. The number of electrons in the s orbital NSu(T) is given by [ 131 N=‘(T) = N;“((E,) -
;
rG;;(E)
d.E
(4)
where iV.SP(&,)is the filling of the s bound state on site T, at energy Ei,. We have to solve, in terms of s%u,eqn. (1) and Friedel’s screening rule, i.e. (5) We have done the numerical calculations for different sets of parameters. In fact in ref. 14 the sensitivity of the S-K parameters to the fitting of different APW levels was investigated. There can be substantial differences depending on how many bands above the Fermi level are fitted. In the present calculation we want to obtain most accurately the bands below the Fermi level, because the binding energies between hydrogen and another point defect are related only to occupied states. Thus, we use a slightly different set of parameters than those quoted in ref. 14. This new set of parameters is designed to fit very well the first band which has the strongest hydrogen-metal hybridization. A more detailed discussion will appear elsewhere [16]. Therefore we restrict ourselves to two sets of parameters. The first set is that obtained from an S-K fit to the VH2 and NbHz structures. In fact, the hydrogen-metal distance for hydrogen at the T site in b.c.c. vanadium and niobium metals is a little different from that in the VH2 and NbHz structures. The second set of parameters includes a correction owing to the different bond distance. This was made through a power law
212
described in Harrison [17]. In all cases, a bound state is found below the metal conduction bands [ 181. The electronic structure of a substitutional impurity in vanadium and niobium is described through a single localized potential [19]. Therefore, the phase shift per spin for a substitutional impurity at site 0 is given by Z,(E) = -
g
2 Ig{l - cG;fm(E)}
where Vr is the matrix element of the impurity potential between m orbitals. We have restricted m to s and d orbitals; moreover only spherical terms of Vr have been retained. In this case Vr = V,” for d symmetry (instead of V, (I’,,‘) and V, (r,,)) and I$’ = Vi for s symmetry. We have to solve, in terms of Vi and V$ eqn. (6) and Friedel’s screening rule, i.e. zZ,OtEF)
= 41 -
qM
where qI and qM are the nuclear charge of the impurity and the host respective. Equation (7) has been solved for 0.25 < n < 0.75 (q = V$Vt) and for lqI - qM1 < 2. For this charge difference the localized approximation for Vr appears reasonable because no bound states are extracted from the conduction bands.
3. Hydrogen-hydrogen
binding energy in vanadium and niobium
The purpose of this section is to give an estimation of the electronic binding energy between a hydrogen atom located at tetrahedral position T (see Fig. 1) and another at tetrahedral position Ti (i = 1 - 5), the T-Ti distances are given in Table 1. We use the results obtained recently [20] where it was shown that the binding energy AETTi between one hydrogen at site T and another at site Ti is given by
L\ETTi= -‘JEp{z%i(E)
- Z,(E) - Z,i(E)} dE + AEb,
(8)
The factor two is for the spin, whereas the Z’s are the phase shifts for the pair of hydrogen atoms at T and Ti (2&(E)) and for isolated hydrogens at T and Ti (Z,(E), 2$(E)). The phase shift for isolated impurities is described in Section 2. The superscript (1) in Z!&(E) means that the perturbing potentials appearing in this phase shift a& related to the noninteracting perturbing potentials. Equation (8) is split into a bound state term located below the metal conduction bands and the usual band term
213
Fig. 1. Illustration of the lattice model. The circles indicate metal atoms and the squares indicate the tetrahedral interstitial sites which may or may not be occupied by hydrogen atoms. Five shells of neighbours of a central site T are labelled by Ti to Tg. Coordinates of these sites are reported in Table 1.
TABLE 1 Stability of the system corresponding a/4(0, 2, 1) and Ti
j=
to two hydrogen atoms at lattice positions T =
ofT
Lattice coordinates of Ta
1
Ti = a/4(1, 2, 0)
2
T2 = a/4(0,
2, 1)
q(4)“2
3
T3 = a/4(2, 1, 0)
;(6)l12
4
T4 = a/4( 2, 0,
5
T5 = a/4(0, 1, 2)
neighbour
Distance T-Ti
i) ;(10)“2
Binding energy (meV) V
Nb
370
41
25
147
-65
-147
-105
-166
-23
aThe metal atom at site A in Figs. 1 and 2 is taken as the origin of the coordinates.
4
214 EF
AETTi = 2(E&, + E&
-ET” - E,bi) - 2s
V$
i (E) - ZT (E)
Ebottom -‘nil
a
(9)
E$ = EbTi are the bound states of localized hydrogen atoms at tetrahedral positions and EiTi and E& are the values of the new bound states (owing to hydrogen-hydrogen interactions). The quantities appearing in eqn. (9) are obtained in terms of matrix elements of the Green operator G described in an adspace of N + 2 particles (N metal atoms and two hydrogen atoms) where
G = (E-H)-’
(10)
where H is given by H = Ho + ITsu)EF(Tsu] + lTisu)E$(Tisul
+
C
(1Rmu)u~“(RmuI
R.mEd
+ IRmu)u~“(Rmul)
+ C (lRma)~~+(Tsul
R,m
+ IRmd~$CTisUI
+ C.C.)(11)
where pg;(/lg$i) are the hopping integrals between I Rmu) and I Tsd (TisU>)* Direct interactions p”+ri between hydrogen atoms have been neglected. Z&?(E) is given by the relation i Z&(E)
=-
Im E RJTrace(G
- G”) dE’
(12)
The hydrogen-hydrogen binding energies are reported in Table 1 for a set of hopping integrals corresponding to the VH2 and NbHz structures [14]. The change with distance of these parameters does not significantly alter the results [ 161.
4. Hydrogen-impurity
binding energies in vanadium and niobium
The purpose of this section is to estimate the binding energies between hydrogen and scandium, titanium, chromium and manganese in vanadium metal and between hydrogen and yttrium, zirconium, molybdenum in niobium. To obtain an estimation of the electronic binding energy between a hydrogen atom at tetrahedral position T and a substitutional impurity at nearest neighbouring position Ri or next-nearest neighbouring position Rz (see Fig. 2) we need to know the electronic structure of both interstitial and substitutional impurities. This has been done in Section 2. The binding energy Al& between a hydrogen atom at site T and a substitutional impurity at site R = 0 is given by
215
Fig. 2. Illustration of the hydrogen-substitutional impurity geometry. The circles indicate metal atoms and the squares indicate the tetrahedral interstitial site T occupied by a hydrogen atom. The filled circles indicate first (RI) and second (Rz) nearest neighbouring metal atoms of T.
AE,, =
-2pFIZg)YE) -Z,(E) -Z,(E)}
dE + AEbs
(13)
Equation (13) is split into a bound state term located below the metal conduction bands and the usual band term A&J = 2(E& - E,b) - ZjF{Z$?(E)
- Z,(E) - Z,(E)} dE
(14)
where E& is the position of the interstitial bound state in the presence of the substitutional impurity. The quantities E&, and Z&?&E) are obtained in terms of matrix elements of the Green operator G defined in the adspace of N + 1 particles (N metallic atoms and one hydrogen atom) and relative to the hamiltonian H defined by I Rmo)u!i?U(Rmol + C )Omu) V,““(Omul H = Ho + ITsu)E~~CI’SU~ + z m R,mEd
+ aC,( IRmu)P~XI’sul + c.c.)
Z,(E) and Z,(E) are given in Section 2. The third phase shift 2$(E) expressed in terms of diagonal matrix elements of the Green operator G
(15)
is
216
Z&;(E) = -
$j%race(G - G") dE’
(16)
Details can be found in refs. 15 and 16. The hydrogen-substitutional impurity binding energies are reported in Table 2 for nearest and next-nearest neighbouring positions. It can be shown that they are in relative agreement with recent experimental results [2, 7, 91 and in disagreement with the sole existing theoretical calculation [ 211. TABLE 2 Stability of the system corresponding to one hydrogen atom at site T = a/4(0, 2, i) and one substitutional impurity at sites Rr and Rz Substitutional
impurity
in V
(meV) Nearest neighbour of T(Rr) Next-nearest neighbour of T(R2)
Binding
Substitutional
impurity
in Nb
(meV) Nearest neighbour of T(Rr) Next-nearest neighbour of T(R2)
Binding
SC
Ti
Cr
Mn
-84 -20
-53 -5
77 -20
86 -74
Y
Zr
MO
Tc
-53 -25
-40 -11
energy
energy
81 -4
97 -84
The results correspond to ?j = V$V,” = 0.5 (no qualitative change is found when 77 is varied from 0.25 to 0.75). The metal atom at site A in Figs. 1 and 2 is taken as the origin of the coordinates.
5. Conclusion This report was initially devoted to a calculation of the binding energy between one hydrogen atom and another point defect. Usually the binding energies are split into a chemical and an elastic term. We have not succeeded yet in calculating the elastic term. Interesting results are however reported for hydrogen-hydrogen and hydrogen-substitutional impurity chemical binding energies. Hydrogen-hydrogen repulsion appears to be related to an electronic origin and can be compared with a blocking model. For the hydrogen-substitutional impurity binding energy our present estimation has to be completed by a calculation of the elastic binding energy.
References 1 G. Boureau, Z. Phys. Chem., N.F., 143 (1985) 89. 2 L. Lichty, J. Shinar, R. G. Barnes, D. R. Torgeson and D. T. Peterson, Phys. Lett., 55 (1985) 2895. 3 R. Wordel, F. J. Litterst and F. E. Wagner, J. Phys. F, I5 (1985) 2525.
Rev.
217 4 E. Yagi, S. Nakamura, T. Kobayashi, K. Watanabe and Y. Fukai, J. Phys. SOC. Jap., 54 (1985) 1855. 5 E. Yagi, T. Kobayashi, S. Nakamura, Y. Fukai and K. Watanabe, Phys. Rev. B, 31 (1985) 1640. 6 E. Ligeon, R. Danielou, J. Fontanille and R. Eymery, J. Appl. Phys., 59 (1986) 108. 7 P. Vargas, H. Kronmiiller and M. C. Bohm, Z. Phys. Chem., N.F., 143 (1985) 229. 8 A. I. Shirley and C. K. Hall, Acta Metall., 32 (1984) 49. 9 M. Belhoul, G. A. Styles, E. F. W. Seymour, T. T. Phua, R. G. Barnes, D. R. Torgeson, R. J. Schoenberger and D. T. Peterson, J. Phys. F, 15 (1985) 1045. 10 D. A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids, Plenum, New York, 1986. 11 H. B. Michaelson, J. Appl. Phys., 48 (1977) 4729. 12 C. D. Gelatt, H. Ehrenreich and J. A. Weiss, Phys. Rev. B, 17 (1978) 1940. 13 J. Khalifeh and C. Demangeat, Philos. Mug. B, 47 (1983) 191. 14 D. A. Papaconstantopoulos and P. M. Laufer, J. Less-Common Met., 130 (1987) 229. 15 A. Mokrani, Thesis, Strasbourg, 1987. 16 A. Mokrani, C. Demangeat, D. A. Papaconstantopoulos and A. C. Switendick, to be published. 17 W. A. Harrison, Electronic Structure and the Properties of Solids, Freeman, San Francisco, 1980; p. 504. 18 A. Mokrani, C. Demangeat, D. A. Papaconstantopoulos and A. C. Switendick , Meet. French Physical Society, Nice, 1985, Bull. Sot. Fr. Phys. (Paris) Suppl., 57 (1985) 65. 19 C. Demangeat,J. Phys. F, l(l971) 755. 20 G. Moraitis and C. Demangeat, Phys. Lett. A, 83 (1981) 460. 21 P. Jena, R. M. Nieminen, M. J. Puska and M. Manninen, Phys. Rev. B, 31 (1985) 7612.
Metals, 130 (1987)
ELECTRONIC REARRANGEMENT NIOBIUM HYDRIDES* A. MOKRANIa, C. DEMANGEATa
209 - 217
IN THE DILUTE
209
VANADIUM
AND
and G. MORAITISb
aL.M.S.E.S. (Unite’associe’ au Centre National de la Recherche Scientifique 306) Universite’ Louis Pasteur, 4, rue Blaise Pascal 67070 Strasbourg Cedex (France) bUniversity of Antananarivo,
Department
of Physics, Antananarivo
(Madagascar)
(Received May 5,1986)
Summary The electronic (or chemical) binding energy between one hydrogen atom and another hydrogen atom or a substitutional impurity in vanadium and niobium is estimated in a tight-binding formalism. The hydrogenhydrogen binding energy is found to be considerably repulsive for nearest and next-nearest neighbouring positions. In the case of vanadium-based alloys we have estimated the binding energy between hydrogen and substitutional scandium, titanium, chromium and manganese. For niobium-based alloys we have estimated the binding energy between hydrogen and substitutional yttrium, zirconium and molybdenum. In both vanadium and niobium the binding energy between hydrogen and substitutional impurities located at nearest neighbouring positions is found to be attractive for impurities located to the left of the host metal in the periodic table and changes sign when the impurities are located to the right.
1. Introduction The calculation of the configurational entropy of hydrogen in bodycentred metals has to take into account the large repulsive interaction between hydrogen atoms at nearest and next-nearest neighbouring positions [l]. The environment of the interstitial in the niobium and vanadium hydrides can be obtained by nuclear magnetic resonance (NMR) [2] or Mijssbauer experiments [ 31 if a reasonable choice of the hydrogen-hydrogen binding energy with distance is given. One of the purposes of this paper is therefore to estimate the chemical binding energy of a pair of hydrogen atoms at tetrahedral positions in vanadium and niobium in terms of the distance between them. In our calculation we expect that hydrogen atoms *Paper presented at the International Symposium on the Properties and Applications of Metal Hydrides V, Maubuisson, France, May 25 - 30, 1986. 0022-5088/87/$3.50
0 Elsevier Sequoia/Printed
in The Netherlands
210
are located at tetrahedral positions, a result obtained by recent ion channelling experiments [ 4 - 61. The presence of the substitutional impurities alters the properties of the metal hydrides. These changes have been attributed to hydrogenimpurity attraction (or trapping effect) for impurities located to the left of the host metal in the periodic table and to hydrogen-impurity repulsion (or antitrapping effect) for impurities located to the right. By means of magnetic after-effect measurements, Vargas et al. [7] have found that in a ferromagnetic host the hydrogen is most generally trapped by substitutional impurities located to the left of the host in the periodic table. In addition, Shirley and Hall [8] have discussed the trapping of hydrogen by metallic substitutional impurities in vanadium, niobium and tantalum; their estimations for electronic and elastic contributions appear to be comparable. It should also be noted that measurements of the proton spin-lattice relaxation times lead to an antitrapping effect for manganese impurities in titanium dihydride [9] and to a trapping by vanadium impurities in the niobium hydride [ 21. This paper is organized as follows. Section 2 is devoted to the determination of the electronic structure of a single hydrogen atom in vanadium and niobium. In addition, the description of one substitutional impurity (scandium, titanium, chromium, manganese in vanadium and yttrium, zirconium, molybdenum in niobium) is reported. Section 3 is devoted to results obtained for the electronic binding energies between two hydrogen atoms in terms of the distance between them. Section 4 reports the estimation of the binding energy between a hydrogen atom and a substitutional impurity. These results are compared with recent experimental results and a theoretical result.
2. The electronic structure of one hydrogen atom and one substitutional ‘d’ impurity in vanadium and niobium The band structure of a pure transition metal is described by an spd Slater-Koster (S-K) fit to a first principles calculation [lo]. The metallic s, p and d orbitals are labelled IRmo), where R is the metallic site, m is the orbital symmetry and u is the spin. T is the tetrahedral position occupied by hydrogen [ 61 and 1Tsu) is the corresponding extra s orbital of energy E;‘. Et” is related to the number of electrons N+” in the s orbital by E$’ = Eff + U,SSNS-a(T)
(1)
The position of the bands of pure metal were fixed so as to give the experimental value for the work function of vanadium and niobium [ll]. The value of the parameter U may be taken from ref. 12 but other values [ 131 have been taken into account in order to test this parameter.
211
The perturbed hamiltonian H for a spin u is given by H = HO + ITsu)E~~(Tsu~+
~Rma)v”“‘“(Rm’uj
x R.lll.Ul’
+ ,c,‘l
Rmu)(3%Tsul
+ c.c.)
(2)
where H, is the pure metal hamiltonian and C.C. means complex conjugate. The hopping integrals pg$ between the hydrogen and the metal are taken from an S-K fit, using an orthonormalized basis, to the VH2 and NbHz band structures self-consistently calculated by the augmented plane wave (APW) method [14]. Concerning the umm”’(R) term on the nearest neighbour R of the hydrogen atom, we restrict ourselves to the most important component s$u [ 151. The generalized phase shift Z;(E) is given by Z;(E)
=-
FrTrace{G(E’)
- G”(E’)} dE’
where the trace is over all metallic and hydrogen sites. The number of electrons in the s orbital NSu(T) is given by [ 131 N=‘(T) = N;“((E,) -
;
rG;;(E)
d.E
(4)
where iV.SP(&,)is the filling of the s bound state on site T, at energy Ei,. We have to solve, in terms of s%u,eqn. (1) and Friedel’s screening rule, i.e. (5) We have done the numerical calculations for different sets of parameters. In fact in ref. 14 the sensitivity of the S-K parameters to the fitting of different APW levels was investigated. There can be substantial differences depending on how many bands above the Fermi level are fitted. In the present calculation we want to obtain most accurately the bands below the Fermi level, because the binding energies between hydrogen and another point defect are related only to occupied states. Thus, we use a slightly different set of parameters than those quoted in ref. 14. This new set of parameters is designed to fit very well the first band which has the strongest hydrogen-metal hybridization. A more detailed discussion will appear elsewhere [16]. Therefore we restrict ourselves to two sets of parameters. The first set is that obtained from an S-K fit to the VH2 and NbHz structures. In fact, the hydrogen-metal distance for hydrogen at the T site in b.c.c. vanadium and niobium metals is a little different from that in the VH2 and NbHz structures. The second set of parameters includes a correction owing to the different bond distance. This was made through a power law
212
described in Harrison [17]. In all cases, a bound state is found below the metal conduction bands [ 181. The electronic structure of a substitutional impurity in vanadium and niobium is described through a single localized potential [19]. Therefore, the phase shift per spin for a substitutional impurity at site 0 is given by Z,(E) = -
g
2 Ig{l - cG;fm(E)}
where Vr is the matrix element of the impurity potential between m orbitals. We have restricted m to s and d orbitals; moreover only spherical terms of Vr have been retained. In this case Vr = V,” for d symmetry (instead of V, (I’,,‘) and V, (r,,)) and I$’ = Vi for s symmetry. We have to solve, in terms of Vi and V$ eqn. (6) and Friedel’s screening rule, i.e. zZ,OtEF)
= 41 -
qM
where qI and qM are the nuclear charge of the impurity and the host respective. Equation (7) has been solved for 0.25 < n < 0.75 (q = V$Vt) and for lqI - qM1 < 2. For this charge difference the localized approximation for Vr appears reasonable because no bound states are extracted from the conduction bands.
3. Hydrogen-hydrogen
binding energy in vanadium and niobium
The purpose of this section is to give an estimation of the electronic binding energy between a hydrogen atom located at tetrahedral position T (see Fig. 1) and another at tetrahedral position Ti (i = 1 - 5), the T-Ti distances are given in Table 1. We use the results obtained recently [20] where it was shown that the binding energy AETTi between one hydrogen at site T and another at site Ti is given by
L\ETTi= -‘JEp{z%i(E)
- Z,(E) - Z,i(E)} dE + AEb,
(8)
The factor two is for the spin, whereas the Z’s are the phase shifts for the pair of hydrogen atoms at T and Ti (2&(E)) and for isolated hydrogens at T and Ti (Z,(E), 2$(E)). The phase shift for isolated impurities is described in Section 2. The superscript (1) in Z!&(E) means that the perturbing potentials appearing in this phase shift a& related to the noninteracting perturbing potentials. Equation (8) is split into a bound state term located below the metal conduction bands and the usual band term
213
Fig. 1. Illustration of the lattice model. The circles indicate metal atoms and the squares indicate the tetrahedral interstitial sites which may or may not be occupied by hydrogen atoms. Five shells of neighbours of a central site T are labelled by Ti to Tg. Coordinates of these sites are reported in Table 1.
TABLE 1 Stability of the system corresponding a/4(0, 2, 1) and Ti
j=
to two hydrogen atoms at lattice positions T =
ofT
Lattice coordinates of Ta
1
Ti = a/4(1, 2, 0)
2
T2 = a/4(0,
2, 1)
q(4)“2
3
T3 = a/4(2, 1, 0)
;(6)l12
4
T4 = a/4( 2, 0,
5
T5 = a/4(0, 1, 2)
neighbour
Distance T-Ti
i) ;(10)“2
Binding energy (meV) V
Nb
370
41
25
147
-65
-147
-105
-166
-23
aThe metal atom at site A in Figs. 1 and 2 is taken as the origin of the coordinates.
4
214 EF
AETTi = 2(E&, + E&
-ET” - E,bi) - 2s
V$
i (E) - ZT (E)
Ebottom -‘nil
a
(9)
E$ = EbTi are the bound states of localized hydrogen atoms at tetrahedral positions and EiTi and E& are the values of the new bound states (owing to hydrogen-hydrogen interactions). The quantities appearing in eqn. (9) are obtained in terms of matrix elements of the Green operator G described in an adspace of N + 2 particles (N metal atoms and two hydrogen atoms) where
G = (E-H)-’
(10)
where H is given by H = Ho + ITsu)EF(Tsu] + lTisu)E$(Tisul
+
C
(1Rmu)u~“(RmuI
R.mEd
+ IRmu)u~“(Rmul)
+ C (lRma)~~+(Tsul
R,m
+ IRmd~$CTisUI
+ C.C.)(11)
where pg;(/lg$i) are the hopping integrals between I Rmu) and I Tsd (TisU>)* Direct interactions p”+ri between hydrogen atoms have been neglected. Z&?(E) is given by the relation i Z&(E)
=-
Im E RJTrace(G
- G”) dE’
(12)
The hydrogen-hydrogen binding energies are reported in Table 1 for a set of hopping integrals corresponding to the VH2 and NbHz structures [14]. The change with distance of these parameters does not significantly alter the results [ 161.
4. Hydrogen-impurity
binding energies in vanadium and niobium
The purpose of this section is to estimate the binding energies between hydrogen and scandium, titanium, chromium and manganese in vanadium metal and between hydrogen and yttrium, zirconium, molybdenum in niobium. To obtain an estimation of the electronic binding energy between a hydrogen atom at tetrahedral position T and a substitutional impurity at nearest neighbouring position Ri or next-nearest neighbouring position Rz (see Fig. 2) we need to know the electronic structure of both interstitial and substitutional impurities. This has been done in Section 2. The binding energy Al& between a hydrogen atom at site T and a substitutional impurity at site R = 0 is given by
215
Fig. 2. Illustration of the hydrogen-substitutional impurity geometry. The circles indicate metal atoms and the squares indicate the tetrahedral interstitial site T occupied by a hydrogen atom. The filled circles indicate first (RI) and second (Rz) nearest neighbouring metal atoms of T.
AE,, =
-2pFIZg)YE) -Z,(E) -Z,(E)}
dE + AEbs
(13)
Equation (13) is split into a bound state term located below the metal conduction bands and the usual band term A&J = 2(E& - E,b) - ZjF{Z$?(E)
- Z,(E) - Z,(E)} dE
(14)
where E& is the position of the interstitial bound state in the presence of the substitutional impurity. The quantities E&, and Z&?&E) are obtained in terms of matrix elements of the Green operator G defined in the adspace of N + 1 particles (N metallic atoms and one hydrogen atom) and relative to the hamiltonian H defined by I Rmo)u!i?U(Rmol + C )Omu) V,““(Omul H = Ho + ITsu)E~~CI’SU~ + z m R,mEd
+ aC,( IRmu)P~XI’sul + c.c.)
Z,(E) and Z,(E) are given in Section 2. The third phase shift 2$(E) expressed in terms of diagonal matrix elements of the Green operator G
(15)
is
216
Z&;(E) = -
$j%race(G - G") dE’
(16)
Details can be found in refs. 15 and 16. The hydrogen-substitutional impurity binding energies are reported in Table 2 for nearest and next-nearest neighbouring positions. It can be shown that they are in relative agreement with recent experimental results [2, 7, 91 and in disagreement with the sole existing theoretical calculation [ 211. TABLE 2 Stability of the system corresponding to one hydrogen atom at site T = a/4(0, 2, i) and one substitutional impurity at sites Rr and Rz Substitutional
impurity
in V
(meV) Nearest neighbour of T(Rr) Next-nearest neighbour of T(R2)
Binding
Substitutional
impurity
in Nb
(meV) Nearest neighbour of T(Rr) Next-nearest neighbour of T(R2)
Binding
SC
Ti
Cr
Mn
-84 -20
-53 -5
77 -20
86 -74
Y
Zr
MO
Tc
-53 -25
-40 -11
energy
energy
81 -4
97 -84
The results correspond to ?j = V$V,” = 0.5 (no qualitative change is found when 77 is varied from 0.25 to 0.75). The metal atom at site A in Figs. 1 and 2 is taken as the origin of the coordinates.
5. Conclusion This report was initially devoted to a calculation of the binding energy between one hydrogen atom and another point defect. Usually the binding energies are split into a chemical and an elastic term. We have not succeeded yet in calculating the elastic term. Interesting results are however reported for hydrogen-hydrogen and hydrogen-substitutional impurity chemical binding energies. Hydrogen-hydrogen repulsion appears to be related to an electronic origin and can be compared with a blocking model. For the hydrogen-substitutional impurity binding energy our present estimation has to be completed by a calculation of the elastic binding energy.
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Rev.
217 4 E. Yagi, S. Nakamura, T. Kobayashi, K. Watanabe and Y. Fukai, J. Phys. SOC. Jap., 54 (1985) 1855. 5 E. Yagi, T. Kobayashi, S. Nakamura, Y. Fukai and K. Watanabe, Phys. Rev. B, 31 (1985) 1640. 6 E. Ligeon, R. Danielou, J. Fontanille and R. Eymery, J. Appl. Phys., 59 (1986) 108. 7 P. Vargas, H. Kronmiiller and M. C. Bohm, Z. Phys. Chem., N.F., 143 (1985) 229. 8 A. I. Shirley and C. K. Hall, Acta Metall., 32 (1984) 49. 9 M. Belhoul, G. A. Styles, E. F. W. Seymour, T. T. Phua, R. G. Barnes, D. R. Torgeson, R. J. Schoenberger and D. T. Peterson, J. Phys. F, 15 (1985) 1045. 10 D. A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids, Plenum, New York, 1986. 11 H. B. Michaelson, J. Appl. Phys., 48 (1977) 4729. 12 C. D. Gelatt, H. Ehrenreich and J. A. Weiss, Phys. Rev. B, 17 (1978) 1940. 13 J. Khalifeh and C. Demangeat, Philos. Mug. B, 47 (1983) 191. 14 D. A. Papaconstantopoulos and P. M. Laufer, J. Less-Common Met., 130 (1987) 229. 15 A. Mokrani, Thesis, Strasbourg, 1987. 16 A. Mokrani, C. Demangeat, D. A. Papaconstantopoulos and A. C. Switendick, to be published. 17 W. A. Harrison, Electronic Structure and the Properties of Solids, Freeman, San Francisco, 1980; p. 504. 18 A. Mokrani, C. Demangeat, D. A. Papaconstantopoulos and A. C. Switendick , Meet. French Physical Society, Nice, 1985, Bull. Sot. Fr. Phys. (Paris) Suppl., 57 (1985) 65. 19 C. Demangeat,J. Phys. F, l(l971) 755. 20 G. Moraitis and C. Demangeat, Phys. Lett. A, 83 (1981) 460. 21 P. Jena, R. M. Nieminen, M. J. Puska and M. Manninen, Phys. Rev. B, 31 (1985) 7612.