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A simple tight-binding estimate of the dipole force tensor in α-palladium hydrides
J. Khalifeh, G. Moraitis, C. Demangeat
To cite this version:
J. Khalifeh, G. Moraitis, C. Demangeat. A simple tight-binding estimate of the dipole force tensor in α-palladium hydrides. Journal de Physique, 1982, 43 (1), pp.165-171.
�10.1051/jphys:01982004301016500�. �jpa-00209374�
A simple tight-binding estimate of the dipole force tensor
in 03B1-palladium hydrides
J. Khalifeh, G. Moraitis and C. Demangeat
Laboratoire de Magnétisme et de Structure Electronique des Solides (*), Université Louis-Pasteur, 4, rue Blaise-Pascal, 67070 Strasbourg cedex, France (Recu le 2 avril 1981, révisé le 20 ao0t, accepté le 23 septembre 1981 )
Résumé. 2014 Le tenseur dipolaire des forces P résultant de la présence de l’hydrogène dans le palladium est déduit
à partir de la structure électronique de l’alliage. La structure électronique de l’impureté est décrite en utilisant le formalisme de la fonction de Green dans le cadre de l’approximation des liaisons fortes. Une compensation subtile
entre les termes attractifs (termes de bandes) et les termes répulsifs (électron-électron et ion-ion) permet une des- cription en pseudopotentiel. Le calcul liaisons fortes est basé sur un déplacement rigide des orbitales et permet d’obtenir les forces agissant sur les voisins de l’hydrogène en termes de dérivée premières 2014 par rapport au dépla-
cement 2014 de la variation totale de l’énergie. Une estimation numérique de P dans le cas où l’hydrogène est en position octaédrique dans une matrice de palladium a été faite. Si l’on suppose une décroissance exponentielle des intégrales de saut en fonction de la distance, on peut obtenir un accord satisfaisant avec les résultats expérimentaux.
Abstract 2014 A model for the dipole force tensor P arising from the inclusion of interstitial hydrogen in 03B1-palladium hydrides is given in terms of the electronic structure of the alloy. The electronic structure of the impurity is described
in the tight-binding approximation of the Green function formalism. A subtle compensation between attractive
(band terms) and repulsive (electron-electron and ion-ion terms) leads to a small perturbing pseudopotential.
This calculation based on a rigidly moving wave function basis gives the forces acting on the nearest neighbour
metallic atoms to the hydrogen impurity in terms of the first derivative of the total energy relative to the atomic
displacement. A numerical estimate of P is performed in the case of interstitial hydrogen at octahedral sites in a
palladium host. A reasonable account of the experiments is obtained when an exponential fall-off of the hopping integrals with distance is assumed.
Classification -
Physics Abstracts 62.20 - 71.55
1. Introduction. - Johnston and Sholl [1] ] have recently estimated the lattice distortions and relaxation
energies due to hydrogen impurities in fcc and bcc transition metals using experimental results for the
change in volume of the crystal due to the interstitial atom. Also Dietrich and Wagner [2] have estimated the elastic interaction between two hydrogen atoms in
the incoherent phase of palladium hydrides using the experimental value of the dipole force tensor P. Up
to now, only phenomenological pair potentials have
been used for a theoretical estimation of P in fcc transition metals [3]. On the other hand, it has been shown recently that the tight-binding method could
be a useful tool for a semi-quantitative electronic
structure of .sp impurities in transition metals [4].
Moreover, it has been shown that this method can
give a reasonable insight into the electronic structure
of hydrogen in a-palladium hydrides [5]. The tight- binding method has also been used for an estimation of the dipole force tensor in dilute alloys of transition
metals [6, 7].
Our objective is to extend this previous model [6, 7]
to interstitial hydrogen in a-palladium hydrides by using a purely tight-binding Hamiltonian for the electronic structure. The band structure of the pure transition metal is described by a spd Slater-Koster fit to first-principle calculations [8] whereas the
impurity is described by an extra-orbital s. In the
case of hydrogen at an octahedral position in a fcc
transition metal, the hopping integral between the
hydrogen s and nearest metallic spd orbitals can be
defined in terms of three different two centre integrals (ss a, spa and sd(1). A more complete description can
be found in [5].
To obtain an expression for the dipole force tensor
P we have taken into account the effect arising from
the rigid displacement of d orbitals only. Let Bm’m be
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004301016500
166
a transfer integral between two metallic atoms at
sites A and p given by :
where gm > is a d metallic orbital, H the Hamiltonian of the unrelaxed alloy, and Ho the Hamiltonian of the unrelaxed pure metal :
In the two centre approximation and for the chemical
potential AV discussed in [5] it can be shown that :
which means that the hopping integrals between the
metallic sites are unaffected by the chemical potential
of the impurity. If we consider now the effect of the relaxation on one site A it will affect the distance Rfp
between the two sites by a quantity ul given by :
The resulting effect on the transfer integral is a varia-
tion bfl"" defined by :
A similar expression can be obtained for the variation of the transfer integral, fl[§/ between the s orbital of
hydrogen at site I and the nearest metallic neighbours li
at an unrelaxed distance Ri) :
where the radial part of fl[§/(Rj) is given in terms of
sd a only.
In section 2, we derive a general formulation for the field of forces induced on the metallic atoms by the
interstitial hydrogen. Section 3 is devoted to a nume-
rical estimate of the forces on nearest, next nearest and third nearest palladium atoms to the hydrogen impurity in terms of two parameters q IND and q OND
which characterize the typical exponential fall-off
of the sd and dd Hamiltonian matrix elements res-
pectively. qIND and q IND are different but we will not
differentiate between them here. Finally, section 4 pre- sents a possible extension of this calculation for a
lattice distortion.
2. Forces resulting from the presence of a hydrogen
interstitial atom at an octahedral site in a fcc para-
magnetic transition metal. - The aim of this section is to relate the hydrogen-metal forces F(£) on palladium
sites A to the electronic structure of a-palladium hydrides. Once this has been done, the dipole force
tensor P can be obtained and compared with expe- rimental results.
The force Fo.), on site A, induced by the defect is
given by (a = x, y, z) :
where bE denotes the variation of the total energy due to the relaxation of the lattice whereas the subscript
zero means that the derivative is evaluated in the unrelaxed configuration; u,,,(A) is the atomic displa-
cement. In the framework of the Hartree-Fock appro-
ximation, the change in the energy induced by the
defect is given by :
Ebs is the band structure term and is defined as the
sum of one electron energies; E, results from the electron-electron interactions which are counted twice in Ebs, and from the ion-ion interactions. The one
electron Hamiltonian in the tight-binding scheme is given by :
where 03BCm > (m = s, p, d) represents the orbital of symmetry m centred on site p, either metallic or
interstitial, Bmm" is the transfer integral and Em is the
energy level. The effect of the relaxation is to replace
the Hamiltonian H by :
and the orbital I À > by :
as discussed in [6]. Using the fact that the relaxation will not modify the total number of electrons, bEbs
is obtained from the following expression :
where EF is the Fermi energy of the pure metal and
5N(JE’) is the number of states displaced by the relaxa- tion up to energy E ; it can be written as :
For the calculation of F(A), from (2.1), it is enough to
calculate the variation of the total energy of the system up to first order with respect to the displacement of
one site only :
G(E) denotes the Green function of the alloy when
4A) = 0 while bw is a pseudopotential which describes
the electron scattering due to the distortion around the defect; this pseudopotential takes into account
the effect of the displacement of the atomic orbitals as
well as the change in the lattice potential. The matrix
elements of bw in the basis of the undisplaced atomic
orbitals are given by [6, 7, 9] :
The energy shifts and the change in the transfer inte-
grals result from the modification of the crystal poten- tial 6V and the basis function ðt/JT(r). Keeping only
the terms proportional to the displacement, we obtain formally :
6’Eb. is the variation of the band structure energy
resulting from the potential modification, 6V :
with
and
ðCfJ Ebs expresses the contribution of 6g/2(r) ; it is written
as :
The variation of the total energy, 6E, given by (2.2)
contains a repulsive term 6E, given by :
The electron-electron term has been approximated by :
and the ion-ion term hy : -.-
n
It can be shown [6] that 6’Eb., combined with 6E,,
leads to the following expression :
where A V(r) is the perturbation introduced by the
interstitial when u(À) == 0 while bn ion (r) denotes the modification of the ionic density in the alloy due to u( À). Finally, for 6E, we obtain :
6"Ebr can be decomposed into two terms :
with
wherd AN denotes the variation due to the interstitial impurity when 4A) = 0. The prime in the summation
means that the sum is restricted to lattice sites only.
Following [10] closely, it can be shown that bEcan be reasonably approximated by :
where bw(") is expressed in terms of neutral quantities (6#(’), ðB(n»), so that partial cancellations between bEbS
and 6E, allow us to avoid the self-consistent calculation of the charge transfer introduced by the atomic displa-
cements. In what follows we will suppress the exponent (n) so that dE can be written explicitly in the following
form : ,
168
where I is the interstitial site. Let us define the following quantities :
where t is the usual t matrix relative to the point defect, E; is the interstitial hydrogen level at site I and fls"’ is
the hopping integral between the hydrogen s orbital at site I and metallic orbital of symmetry m at site À. In terms of the definitions appearing in equations (2 . 25), (2 . 26) and (2 . 27) we obtain :
where G ° is the Green function relative to the pure, unrelaxed host. The matrix elements t""’ are calculated for each m’ and m" up to first nearest neighbours. It is shown that, due to high anisotropy of the hopping integrals f3’:t, a considerable number of these matrix elements is equal to zero.
Finally, we obtain :
The component « of the force acting on the matrix atom sitting at the À site is then given by :
where
with the notations
Once the forces on sites A have been obtained a
component Paa of the dipole force tensor is given by :
where P a is relative to F°a (03BB) and P1(À) to Fa(03BB).
3. Numerical estimation of the forces and of the
dipole force tensor in a -palladium hydrides. -
Fo A) and F3£), defined by equations (2.33) and (2.34) are the sums of a diagonal term (containing 6£fi and a non-diagonal part (containing 03B4bmm) :
In order to estimate Ffl(£) we replace B )"’ by f3f;t’m as
discussed in the introduction and B; by Eom [11] :
where the superscript 0 means that these quantities
are relative to the unrelaxed pure metal. Ffl(h) will
now be expressed in terms of known quantities
which are the two centre integrals relative to the
Slater-Koster fit of the band structure of pure palla-
dium to a first-principle band structure calculation [8]
and the hopping integrals between the hydrogen s
orbital and the metallic d level of Pd [5,11 ].
Moreover, it can be shown that the dipole force
tensor relative to Fo’(A) and defined by :
is zero. The fact that Ho, the Hamiltonian of pure Pd will be unchanged by a lattice translation implies the following expression :
Table I. - Values in eV of a dipole force tensor
component Paa ( for a = x, y or z) as a function of the
parameter q.
The number of external electrons brought by the
interstitial (Z = 1) is equal to the variation of the
displaced states :
As discussed in [5], AN" = 1 so that (3.4) is found equal to zero. We are left with F"’(A) which is given
in terms of the electronic structure of the unrelaxed
alloy and bfl"" with m and m’ relative to d orbitals
only. It is easily shown that [12] :
# 0"""’ is the product of a radial part expressed in
terms of two centre integrals dd(1, ddn, ddb and an angular part given in terms of directions cosines [13] :
Let us assume [14] the variation with distance of ddi (i = a, 1t, 6) : :
which relates the values of ddi at distance R to the values at distance RO.pOND is then given by the sum of
two terms :
where the first term is independent of q OND and the
second depends on q OND . After lengthy manipulations
an analytic expression has been found for P."’; in particular it has been shown that the forces F ao(À) are
non zero only up to third nearest neighbours of the hydrogen impurity (Appendix A).
As concerned with PI, after trivial manipulations
we have found the following expression :
where ao is the lattice parameter and Fa1(03BB1) is the sum
of two terms :
As a first approximation we will neglect F"(A’l)
which contains 03B4es [ ; this is the same type of approxima-
tion as in [12] ; there remains F"(Al) which can be
calculated following the same procedure used for
the determination of F"’(A). Replacing 6#0 by
bfl" (3.7) and (3.8) can be straightforwardly used
and (3.9) has to be replaced by :
where G ’(F 12), G °(4) and G °(5) are respectively
intrasite and intersite matrix elements of the Green function G° (see [11] for detail) and qIND is the para- meter entering in the exponential variation of sd(1 with distance. Let us remark that the derivative of the
angular part of Pi: is zero.
We are still left with two parameters q OND (in the
definition of P,,,O,,,N) and qIND (in the definition of
Paa IND. For convenience we will make the following approximation :
However, recent calculations in the case of Ni-H bonds have shown [15J that qIND is much bigger than qOND this probably remains true in the case of Pd-H
170
and should be taken into account in an improved
model. Once the value of q is obtained (Table 1) by comparison with the experimental value [16], the
forces F(A) on palladium atom A are known and report- ed in table II. For nearest and next nearest neighbours they are in qualitative agreement with the calcula- tion of Johnston and Sholl [I].
Table II. - Forces (in 10-4 dyn.) from an interstitial
hydrogen at an octahedral site in palladium The a
component (a = x, y, z) of the forces F J. À.J are report- ed up to third nearest neighbour À.3 to the hydrogen impurity. a is the lattice parameter.
4. Conclusion. - Using a tight-binding method we
have deduced, for the first time, the dipole force tensor
in terms of the electronic structure of hydrogen at an
octahedral position in a palladium host. Our calcula-
tion relies only on one parameter q (once the electronic
structure is known) and comparison with experi-
mental result gives a reasonable value for q. During
the calculation we have expressed the forces on first,
second and third palladium neighbours to hydrogen
in terms of q. Once q is obtained, from comparison of
P with experimental result, the forces F(A) on palla-
dium atom A are known. It is clear, from lattice statics [17] that the atomic displacements 4A) can be
deduced from F(A) through the relation (a, p = x, y, z) :
where À, J1 are lattices sites and G L is the static Green function which is given in terms of the force constants of the alloy. G L is formally given in terms of G °L, the
host lattice Green function and perturbation due to
the hydrogen defect [18]. GL can be related to G°L if we introduce an auxiliary function GIL. GIL is
the lattice Green function of the system hydrogen-
metal with zero coupling between the two components.
Concerning GOL it should be obtained from the electronic structure of the host. It can be obtained
through the long wavelength limit of the phonon
Green function [19J or more directly by an estimation
of the elastic constants [20, 21 J. This remain even now
a rather formidable task and will be discussed in more
detail in a future publication.
Appendix A. - Some elements about the determi- nation of the forces acting on nearest palladium atoms
to the hydrogen impurity. - As discussed in the text, we are left with the non-diagonal part of the forces
F"’(A) due to the fact that the dipole force tensor
with respect to the diagonal part is zero :
As shown by equations (3.7-3.8) the derivative D.,, splits the expression of the forces into two terms, one depending on the parameter, q OND . entering in the exponential, the other depending on the derivative of the angular part of fl. This calculation is trivial but
lengthy and leads to cumbersome expressions for the
forces [22]. They depend, in our approximation, on
the hopping integral sd(1 between the hydrogen impu- rity and its nearest neighbour palladium atoms, on the dd hopping integrals dd(1, ddn, ddb and on the
matrix elements G °(T, 2), GO(4), GO(5), G °(6) of the
Green function Go = [E - Ho]-l between d orbitals.
The definition of these matrix elements of G ° can be found in [11].
Let I = (0, 0, 0) be the location of the interstitial
hydrogen at an octahedral site. The force acting on
the nearest Pd atom Â.1
= a (o,
0, 1) has the followingcomponents : 2
The components of the force acting on the next
nearest. neighbour palladium atoms are found equal
whereas the component of the force on third nearest
neighbour A3
= a
2(0, 2, 1) are :No relation has been found between the y and the z component of the force. The force acting on À,4
= 2 (3,
1, 1) is zero and results from the fact thatwe have retained in our calculations only matrix
elements of H up to first nearest neighbours.
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