Local spin susceptibility in disordered alloys

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Local spin susceptibility in disordered alloys

F. Brouers

To cite this version:

F. Brouers. Local spin susceptibility in disordered alloys. Journal de Physique, 1975, 36 (12), pp.1279-

1283. �10.1051/jphys:0197500360120127900�. �jpa-00208375�

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. ²⁰¹⁴ Nous discutons la susceptibilité ^{locale de}

spin

^des alliages de substitution parama-

gnétiques 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.

Abstract. ²⁰¹⁴ We discuss the local spin susceptibility of substitutional disordered paramagnetic 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.

Classification

Physics ^Abstracts

8.524

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}

a cluster of limited size in the

alloy,

^to calculate its enhanced static

susceptibility

^{within a molecular}

field

approximation

^{and then to} determine the condi- tion of local

instability.

^The

divergence

^{of the sus-}

ceptibility

^is correlated to the

apparition

^{of a local}

moment in the cluster. For a

given cluster,

^this

condition

depends

^{on three factors :}

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

depen-

dence 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.

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 sus-}

ceptibility

which allows a statistical

approach

^{to local}

magnetic properties

ⁱⁿ disordered

alloys.

^This expres- sion 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 concen-}

trated

alloys.

^The

magnetization

_XrzfJ ^{of the a-th cell}

resulting

^{from an} external unit field in the

fi-th

^cell

(*) Partially supported by ^ESIS Programme.

(**) Laboratoire associé au C.N.R.S.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197500360120127900

1280

is obtained from the

corresponding non-interacting quantity Y’ ^by

^{the relation}

This

equation

^{is the direct} consequence of the mole- cular field

approximation :

^{the local}

magnetic

^field

Hy acting

^on conduction electrons is

equal

^to

Hy = Ir; + l Uy Xyp .

H°

is the value of the extemal field in the

y-th

^cell

(Ir;

⁼

byp)

and the molecular field in the

y-th

^cell

uy

^{Xyp is}

proportional

^{to the local}

magnetization

_Xyp

p

and to an 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

suscep-

tibility,

eq.

(1)

^can be rewritten :

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 :

with

Starting from

and

separating

^the

diagonal

^and

non-diagonal

^non-

interacting susceptibilities

^{one can write}

Making

^{use of}

(5),

^{we have}

Introducing (7)

^{into the} first r.h.s. term of

(6),

^we

obtain

multiplying by X’,

^{we 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 - U/J Xip

Let us consider first the

diagonal susceptibility.

^If

we define the

quantity

where the

x°

ⁱⁿ

⁽¹³⁾

^are

non-diagonal, starting

from

(10)

^and

using (13),

^{one can write}

If we use the definition

of ’tex (12),

eq.

(14)

^{takes the}

form

In the

approximation

considered

by Dvey-Aharon

and Fibich for NiCu the

diagonal

^and

non-diagonal

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}

occupied by

^{a Ni atom and zero}

otherwise and

= 1 - ur 0 . ^R2)

is the number of

1 UFO

0

R°‘°‘

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}

immediately

where

R,,(,’)

^{is the number of}

^paths

^{of 1}

nearest-neigh-

bour

steps

^{between Ni atoms which start and end at} site

fil.

where

Sââ

^is the number of

paths

^{of 1}

nearest-neigh-

bour

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.

We obtain

where

T,(,’)

^{has the same}

^meaning

^as

^T(’)

^{and d is}

the interatomic distance.

Collecting (17), (18)

^and

(19), yields

^{the final.}

expression ^(3.16)

^{of D. F. As shown}

by

^D.

F.,

^the

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

demonstrate and use the relation

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

The

interacting susceptibilities

^{can be}

expressed

ⁱⁿ

term of the

non-interacting susceptibilities using

^the

T-matrix.

where T" is

given by

^an

expression

^{similar to}

(11)

^but

excluding

^{the site a. From}

(23),

^{one can write :}

with

Substituting (24)

^into

(22)

^we obtain the

compact

form

0

+ 1 (lx)

x ⁺

XlXfJ = ₁

- IXfJ {,BIO + IXfJ

^{aa +}

^{-Y (or» 2013xx (26)}

In the D. F.

model,

^{one has} 1

If we use this

approximation

^{in the}

diagonal

suscep-

tibility

X.

given by (26),

^one gets ^an

expression

^which

is a

compact

^{form identical to}

(16)

^and

(20).

^{The insta-}

bility

condition of the

susceptibility

^reads

This condition is

quite general.

^{In the D. F. NiCu}

model,

^{where the local moment effect comes}

solely

from the number N of Ni atoms and the various

possibilities

^of

arranging

^{them on the first}

shell,

1282

E:)

^is

expanded

^{for an} isolated cluster

by considering only

interactions on the shell of first

neighbours.

If one 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 L} is the number of four steps

paths

^which stay ^{in the} first shen around a.

For a

given

concentration the

probability

^{for a Ni}

atom to have an environment such that the condi- tion

(30)

^{holds is}

given by

where

P12(Nl x)

is the binomial distribution factor

’ and

WNKL

^{is the}

probability

^of

having

^{N nearest-}

neighbours 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] :

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}

and

solving

eq.

(33), (34)

K is the number of

nearest-neighbour pairs

^{of Ni}

atoms in the first shell. 2

Lo

^{is the} number of self-

avoiding

^four

step paths staying

in the first shell around the cluster central atom a. If one

expands

the

instability

^condition

one can check that _up to third order in ^T we obtain eq.

(30).

3. Conclusions. ^-

Using

^the

multiple scattering formalism,

^we have derived in a

simple

^and

straight-

forward manner the

expressions

^used

by Dvey-

Aharon and Fibich for the

investigation

of the for- mation 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 discus- sion in the

present

paper has been

rewarding.

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] ;

3.3 some of the

expressions

^we have derived are

quite general

and could be most useful to investi-

gate

^the effect of cluster-cluster interactions due to fluctuations of local environment in the medium. In

particular

^the

expression

for the local

susceptibility

^is of interest. The cor-

rection

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 disor- dered

systems.

^This

analogy

will be discussed in a

forthcoming

paper

[7].

^,

Acknowledgment.

^{- We are}

grateful

^{to Dr. F.}

Ducastelle ^{for some} useful discussions.

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.