2014 We discuss the local spin susceptibility of substitutional disordered paramagnetic .... disorder comes solely from the randomness of ua .... B 2 (1970) 740.
LE JOURNAL DE PHYSIQUE
TOME
1279
36, DÉCEMBRE 1975,
Classification
Physics Abstracts 8.524
LOCAL SPIN SUSCEPTIBILITY IN DISORDERED ALLOYS
(*)
F. BROUERS Laboratoire de
Physique des Solides (**), Bâtiment 510, Université Paris-Sud, Orsay, France (Reçu le 2 juin 1975, accepté le 24 juillet 1975)
Résumé. Nous discutons la susceptibilité locale de spin des alliages de substitution paramagnétiques désordonnés dans le cadre de la théorie de la diffusion multiple. Nous établissons une expression de la susceptibilité qui présente un intérêt pour l’étude de l’effet des interactions entre amas sur la condition de formation d’amas magnétiques dans l’alliage. 2014
We discuss the local spin susceptibility of substitutional disordered paramagnetic Abstract. alloys in the framework of the multiple scattering expansion of the T-matrix. We derive an expression which is of interest to investigate the effect of cluster-cluster interactions on the magnetic cluster formation condition in alloys. 2014
1. Introduction. Recently a number of papers have been devoted to the description and the study of the influence of local environment on the magnetic properties of disordered binary substitutional -
alloys [1-6]. A possible approach
to that problem is to consider cluster of limited size in the alloy, to calculate its enhanced static susceptibility within a molecular field approximation and then to determine the condition of local instability. The divergence of the susceptibility is correlated to the apparition of a local moment in the cluster. For a given cluster, this condition depends on three factors : a
1. The number of atoms of each constituent in the
neighbourhood of the central atom of the cluster, which appears explicitly in the definition of the local susceptibility. 2. The non-interacting susceptibilities which are functions of the cluster partial densities of states, depending strongly on the cluster configuration. 3. The concentration and local environment dependence of the relative position of the constituent subbands the neglect of which can lead to completely misleading results in some cases. ,
The first of these effects was considered by Roth [1] and then by Dvey-Aharon and Fibich [4] in a more detailed manner. The importance of the two other factors has been emphasized by Brouers et al. [5] and Van der Rest et al. [6]. They cannot be discarded.
(*) Partially supported by ESIS Programme. (**) Laboratoire associé au C.N.R.S.
However the theory is not yet able to provide a good description of the local environment effects close to the ferromagnetic transition. When nearly magnetic clusters are present in the system they give rise to non negligible cluster-cluster interactions and when magnetic clusters are formed, they polarize the non-magnetic clusters. Until now this effect has been neglected. Brouers et al. [5] have considered cluster-cluster interactions indirectly in an approximate way by averaging the medium outside the cluster. This
method however does not contain an important aspect of the cluster-cluster interaction i.e. the statistical nature of local environment fluctuations in the alloy. The purpose of this paper is to show that it is possible to derive an expression for the local susceptibility which allows a statistical approach to local magnetic properties in disordered alloys. This expression can be deduced from a formula established by Dvey-Aharon and Fibich [4] (D. F.). These authors have used a sophisticated diagrammatic method to derive their formula of local susceptibility. We first want to show how the D. F. formula can be derived straightforwardly using the multiple-scattering T-matrix expansion. We shall then transform the expression into a form convenient for investigating the effect of cluster-cluster interactions on magnetic cluster formation in alloys. 2. T-matrix expansion of the susceptibility. The molecular field theory enables a simple expression to be obtained for the local susceptibilities in concentrated alloys. The magnetization XrzfJ of the a-th cell resulting from an external unit field in the fi-th cell
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197500360120127900
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1280
is obtained from the corresponding quantity Y’ by the relation
non-interacting
This equation is the direct consequence of the molecular field approximation : the local magnetic field Hy acting on conduction electrons is equal to
Hy = Ir; + l Uy Xyp .
with
Starting from
H° is the value of the extemal field in the y-th cell (Ir; byp) and the molecular field in the y-th cell Xyp is proportional to the local magnetization Xyp uy p =
effective intraatomic Coulomb interaction (uï Uff/2 J-li). For pure metals, the solution of (1) is easily obtained by Fourier transform but, for concentrated alloys, only approximate solutions can be found because of the lack of translational symmetry. A simple and compact form of X0152P can be derived however. If we isolate the diagonal non-interacting susceptibility, eq. (1) can be rewritten : and to
The summation on the right hand side of (2) can be expressed as an expansion in x° in the following way. We introdt;ce’the T-matrix defined by :
and
separating the diagonal and non-diagonal interacting susceptibilities one can write
non-
an
=
multiplying by X’,
we
Making use of (5), we have
Introducing (7) into
the first r.h.s. term of
(6),
we
obtain
have
which
yields the expression of Xa.¡J in terms of the T-matrix
In the
multiple scattering theory,
the
T-operator can be expanded in terms of the atomic r-matrix
with 1
Let we
consider first the define the quantity us
- U/J Xip
diagonal susceptibility.
where the x° in (13) are non-diagonal, from (10) and using (13), one can write
If
starting
If we form
use
the definition
of ’tex (12),
eq.
(14) takes the
In the approximation considered by Dvey-Aharon and Fibich for NiCu the diagonal and non-diagonal
1281
non-interacting susceptibilities are site independent and equal respectively to To and F,. The effect of the disorder comes solely from the randomness of ua which is equal to zero if a is occupied by Cu and u if a is occupied by Ni, eq. (15) with LatJ. given by (13) reduces to formula (3.11) of D. F.
where
=
1 if a is
otherwise and
occupied by a
= 11- UFO ur 0 . R°‘°‘R2)
Ni atom and
zero
We shall show that the multiple scattering expression can yield again this result in an elegant and much more direct way. A more compact expression for l0153(J can be derived which has a form such that the results of the theory of localization of electrons can be used to discuss the effect of cluster-cluster interactions on the moment formation in disordered alloys. If we define Ë as the susceptibility of a medium with interatomic electron-electron interaction on each site except on site oc, one can write
and therefore
is the number of
0
paths of 1 nearest-neighbour steps between Ni atoms which start and end at site a and T,,(,’) is similar to R (’) but excludes all paths which cross the site a at any intermediate step. Let us now consider the non-diagonal interacting susceptibility in the D. F. approximation, one obtains
The
interacting susceptibilities can be expressed in non-interacting susceptibilities using the
term of the
T-matrix.
immediately where T" is given excluding the site
by a.
an
expression similar to (11) (23), one can write :
but
From
with
where R,,(,’) is the number of paths of 1 nearest-neighbour steps between Ni atoms which start and end at site fil.
where Sââ is the number of paths of 1 nearest-neighbour steps between Ni atoms that start at site a and end at site fi excluding only those paths which cross site fi at an intermediate step.
Substituting (24) into (22)
we
obtain the compact
form
XlXfJ = 1 In the D. F.
We obtain
model,
0 + 1 (lx) x +
- IXfJ {,BIO + IXfJ -Y2013xx(or»
one
aa
has
(26)
+
1
If we use this approximation in the diagonal susceptibility X. given by (26), one gets an expression which Collecting (17), (18) and (19), yields the final. is a compact form identical to (16) and (20). The instaexpression (3.16) of D. F. As shown by D. F., the bility condition of the susceptibility reads expression for the enhanced diagonal susceptibility Xa.a (16) can be expressed only in term of the paths which avoid a, the central atom of the cluster. They This condition is quite general. In the D. F. NiCu demonstrate and use the relation model, where the local moment effect comes solely
T,(,’)
has the same where the interatomic distance.
meaning
as
T(’)
and d is
from the number N of Ni atoms and the various of arranging them on the first shell,
possibilities
1282
E:) is expanded for an isolated cluster by considering only interactions on the shell of first neighbours. If
one
and
solving eq. (33), (34)
limits the summation to first order, the
divergence condition reads
This is the simplest approximation, the Nm;n model. There is an instability of the local susceptibility if N is larger than a critical number of Ni neighbours. If one stops after four steps the instability condition reads
K is the number of nearest-neighbour pairs of Ni atoms in the first shell. 2 Lo is the number of self-
four step paths staying in the first shell around the cluster central atom a. If one expands the instability condition
avoiding
one can
eq.
check that up to third order in
T we
obtain
(30).
3. Conclusions. Using the multiple scattering in a simple and straightderived we have formalism, forward manner the expressions used by DveyAharon and Fibich for the investigation of the formation of magnetic clusters in paramagnetic NiCu alloys. Although it has been shown by Brouers et al. [5, 6] that one cannot ignore the variation with local environment of the non-interacting diagonal and non-diagonal susceptibilities and therefore that the D. F. model is probably too crude to correctly describe the cluster properties of NiCu, the discussion in the present paper has been rewarding. -
K is the number of
of Ni 2 L is the number of four shell, first which in the shen around a. stay steps paths For a given concentration the probability for a Ni atom to have an environment such that the condition (30) holds is given by
nearest-neighbour pairs
atoms in the first
’
where P12(Nl x) is the binomial distribution factor and WNKL is the probability of having N nearestneighbours arranged in K pairs and N triplets. The expression (31) is the starting point of the D. F. analysis. Here again the multiple scattering theory can be used to derive a closed expression for C. The multiple scattering expression for the T-matrix corresponding to scattering on the first shell reads [5] :
3.1 We have derived a more general expression for the instability condition within the D. F. model ; 3.2 the method we have developed in this paper provides a natural starting point to go beyond the first shell approximation, this should be done for NiCu where second neighbours are thought to play a non negligible role [6] ;
of the expressions we have derived are quite general and could be most useful to investigate the effect of cluster-cluster interactions due to fluctuations of local environment in the medium. In particular the expression 3.3
where the coordinates R, R’, R" correspond to atoms on the first shell. In the D. F. model, the diagonal Td and non diagonal Tnd can be written as :
From the definition
(24) of âa, we have
some
for the local susceptibility is of interest. The correction E Il (13) corresponds to paths on the lattice excluding the central site and its structure is analogous to the site-diagonal Green’s function self-energy used to investigate the localization of electrons in disordered systems. This analogy will be discussed in a forthcoming paper [7]. ,
Acknowledgment. Ducastelle for
some
-
We
are
grateful
useful discussions.
to Dr. F.
1283
References
[1] ROTH, L., Phys. Rev. B 2 (1970) 740. [2] BENNEMANN, K. H. and GARLAND, J. W., J. Physique Colloq. 32 (1971) C1-750. [3] GAUTIER, F., BROUERS, F. and VAN DER REST, J., J. Physique Colloq. 35 (1974) C4-207. [4] DVEY-AHARON, H. and FIBICH, M., Phys. Rev. B 10 (1974) 287.
[5] BROUERS, F., GAUTIER, F. and VAN DER REST, J., J. Phys. F. 5 (1975) 975. [6] VAN DER REST, J., GAUTIER, F. and BROUERS, F., J. Phys. F. 5 (1975) 995. [7] BROUERS, F., KUMAR, N. and LITT, C., submitted to J. Physique.
