On the influence of statistical fluctuations of concentration on the I s vs. T curve of disordered ferromagnetic alloys

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On the influence of statistical fluctuations of concentration on the I s vs. T curve of disordered

ferromagnetic alloys

P. Mazzetti, G. Montalenti, G.P. Soardo

To cite this version:

P. Mazzetti, G. Montalenti, G.P. Soardo. On the influence of statistical fluctuations of concentration

on the I s vs. T curve of disordered ferromagnetic alloys. Journal de Physique, 1972, 33 (1), pp.113-

117. �10.1051/jphys:01972003301011300�. �jpa-00207202�

ON THE INFLUENCE OF STATISTICAL FLUCTUATIONS OF CONCENTRA-

TION ON THE Is

^vs.

T CURVE OF DISORDERED FERROMAGNETIC ALLOYS

P.

MAZZETTI,

^G. MONTALENTI and G. P. SOARDO Istituto Elettrotecnico Nazionale ^« Galileo Ferraris _»,

Gruppo

^{Nazionale Struttura} della Materia del C. N.

R.,

¹⁰¹²⁵

Torino, Italy (Reçu

^{le 21}

juillet 1971)

Résumé. ²⁰¹⁴ Les courbes réduites de l’aimantation à saturation Is ^en fonction de la température ^T

des alliages ferromagnétiques désordonnés montrent des déviations prononcées ^par rapport ^aux memes courbes des métaux ferromagnétiques purs. Il a été proposé que ^ces déviations soient dues a des fluctuations statistiques de la concentration dans l’alliage. Pour vérifier cette théorie on a calculé les courbes Is en fonction de T d’alliages ferromagnétiques désordonnés en tenant compte ^{de la} variation de Is ^{et de la} température ^{de Curie avec} la concentration de l’élément dissous. Les calculs faits _par un ordinateur ont été exécutés ^en supposant ^une distribution de _gauss des concentrations,

dont la variance est liée à un paramètre 03BB, qui représente ^en première approximation la distance de corrélation du champ moléculaire de Weiss. On suppose donc _que les régions du matériel séparées

par ^une distance plus grande que 03BB n’interagissent pas mutuellement. Les courbes théoriques ^de Is

en fonction de T de plusieurs alliages ^{de Fe ou de Ni sont calculées pour} des valeurs différentes de 03BB,

et elles sont comparées ^{aux courbes}

expérimentales,

^{obtenues par des mesures de couple de torsion}

sur des échantillons ^en forme

d’ellipsoides

^{dans une balance} magnétique. ^{Le meilleur accord entre}

courbes calculées et expérimentales ^est obtenu pour ^une distance de corrélation 03BB de l’ordre de dix distances interatomiques, ^{en très bon accord avec} les résultats d’autres méthodes de mesure.

Abstract. ²⁰¹⁴ The curves of saturation magnetization ^{Is vs.} temperature ^{T of} disordered ferro-

magnetic alloys show consistent deviations with respect ^{to the ones of} pure ferromagnetic ^{metals. It}

has been

suggested

that these deviations may be due to statistical fluctuations of concentration in the

alloy. ^{In order to}

quantitatively

^{study the problem, the Is vs.} T curves of disordered ideal alloys ^are

computed

taking ^{into account} the variation of Is and of Curie temperature ^{as a} function of solute concentration. The computer calculations are made assuming ^a gaussian distribution of concentra-

tions, whose variance is related to a parameter 03BB, ^which roughly represents the correlation length ^of

the Weiss molecular field. Regions ^{of material} separated by ^{a distance} larger ^{than 03BB are} thus assumed

as non interacting. ^The theoretical Is vs. T curves of several Fe or Ni base alloys ^are computed ^for

different values of 03BB, ^and compared ^{to the} experimental ^{curves. The best} fitting between theoretical and

experimental

^{curves is obtained in the many cases which were examined} by assuming ^{a correla-}

tion length of the order of ten atomic distances, ⁱⁿ agreement with other types ^of estimates.

Classification Physics Abstracts :

17.64

1. Introduction. ^- It is known that the ^curves of the reduced

magnetization IS(T)/IS(o)

^{versus reduc-}

ed

temperature TITe

^of disordered

ferromagnetic

ideal

alloys

^lie

always

^{below the ones of the corres-}

ponding

pure

ferromagnetic

metal. This

typical

behaviour is

reported

^{for some Fe or Ni base}

^alloys

in

figures

^{1 and} 2. It has been

proposed

^{that this ano-}

malous behaviour is related ^{to the} fluctuations of concentration of the solute element ^{in the}

alloy [ 1 ].

^In

fact,

^{to a first}

approximation,

^the

^sample

^{can be}

thought

^as

being

divided into many small

regions

ⁱⁿ

which the

spins

^are

strongly correlated,

^{but which}

behave

independently

^{of one another for what}

concerns the

magnitude

and the direction of their average

magnetization.

^Each

^region

^{is thus defined so}

as the local Weiss field is

essentially

determined

by

the

magnetic

^{moments of the atoms} present within it. On the other

hand,

^{because of}

thermodynamical

reasons, any ideal disordered

alloy

^is characterized

by

statistical fluctuations of concentration of the solute element : therefore the concentration of each

region

^{may in}

general

differ from the

alloy

average one,

by

^{an amount}

statistically given by

^{a convenient}

distribution

function,

^which

depends

^on the average volume of the

regions.

^{Since the} saturation _magne- tization and the Curie temperature

depend

^on

concentration,

^{the curve}

/s

^{vs. T of} the whole

sample

is

given,

^{in this}

approximation, by

^{the sum} of many ideal Brillouin curves, ^one for each

region,

^charac-

terized

by

^different

Is(O)

^and

Tc

values.

This

theory

was never verified so

far,

^{but for} a

single

^{case due to} Kranz and

Bodewig [2],

^whose

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01972003301011300

114

FIG. 1. ^- Reduced curves Is(T)IIs(O) ^vs. T/ Tc ^{of some Fe base} alloys. The reduced curve of Fe _pure metal is shown for compa-

FIG. 2. ^- Reduced curves Is(T)IIs(O) ^vs. T/Te ^{of some Ni base} alloys. ^{The reduced curve of} Ni pure metal is shown for _compa-

rison.

treatment,

however,

^{made use of several}

approxi-

mations and limited the

fitting

^between

experimental

and

computed

^{curves to one}

single point.

In this _paper on the basis of the concentration fluctuation model the theoretical reduced curves are

computed

for several Fe or Ni base

alloys,

^{and the}

corresponding experimental

curves are obtained

by

a torque ^{balance method. In the}

computations

^the

average width of the upper defined

regions

^{is taken}

as a parameter. ^The

comparison

^of

experimental

and theoretical curves

permits

^{to evaluate the value}

of  for which the best

fitting

is obtained. For all examined

alloys

^this parameter ^{is found to be of the} order of ten atomic

distances,

ⁱⁿ

good agreement

with other estimates obtained

by

different methods.

2.

Computations

^and

experiments.

^- As outlined in the

preceding section,

^the

computations

^{of the}

reduced curves for different

alloys

^were

performed according

^{to the}

following assumptions :

a)

^the

sample

^is

ideally

^{divided into non}

interacting regions

whose linear dimension is considered as a parameter ;

b)

the fluctuation of the solute element concen-

tration of each

region

^{is described}

by

^a

gaussian

distribution function centred on the average ^concen- tration of the

alloy.

^The variance J can be related tao À

through

^the

equation :

where

CA

^and

CB

^{are the} concentrations of the

alloy

constituent elements

[2] ;

c)

^{the reduced curves}

IS(T)/IS(0)

^vs.

T/T

^{of each}

region

^is coincident with the one of _{the pure} metal

(Fe

^or

Ni). However, 1,(0)

^and

7.

^{are variable for}

each

region according

^{to the}

gaussian

concentration distribution. Their values are taken to

depend

^{on the}

local concentration

according

^{to the} literature data

[3].

The

I,

^{vs. T curve for the whole}

sample

^{is then}

computed

^{as the}

weighted

^{sum of the curves relative}

to the various

regions

^into which the material is

thought

^{to be}

divided,

^the statistical

weight being given by

^the

gaussian

distribution.

Computations

were made for différent values of the variance _6, that

is, through

eq.

(1),

^{of the} parameter ^Â.

The

following alloys

^were examined : FeAl 22

% (atomic) ;

^{FeSi 12}

% ;

^{NiAl 10}

% ;

^{NiCr 5}

% ;

^NiCu

20

% ;

^{NiV 5}

%.

^The

corresponding computed

^curves

are

reported

^{in reduced form in}

figures

³

through 8,

in which the values of  and Q are shown for each

curve.

On the same

figures

^the

experimental

^reduced

curves are also

reported.

^These curves were obtained

by

^{means of a} torque magnetometer ^which

permitted

to

perform

measurements between - 200°C and 800 OC on

ellyptical polycrystalline specimens

^under

a

saturating magnetic

^{field. Since the residual} crys- talline

anisotropy

^{and the induced}

magnetic

^aniso-

tropy ^{were made} very small in

comparison

^with

the

shape anisotropy,

^{the measured} torque ^was pro-

portional

^to

I/.

^The

paramagnetic

^{effect which is} present ^near

T,

^{was corrected}

by taking

measurements

of IS

^{vs. T} under different

magnetic fields, high enough

to saturate the

sample,

^and

extrapolating

^{to zero}

field.

This method

permitted

^to

perform

very ^accurate measurements of

I,

^{vs. T at}

high

temperature, ^while

some lack of

precision

^{at low} temperature may result from a non

perfect compensation

^{of the residual}

crystalline anisotropy.

Since the concentration fluc- tuation effect is

particularly large

^at

high

temperature, this measurement

technique

^seems

quite

convenient.

FIG. 3. ^- Computed (full lines) ^and experimental (dotted line)

curves Is(T)IIs(O) ^vs. T/Td ^{of FeAl 22} % (atomic). ^{For each} computed ^{curve the value of} the parameter ^Â (linear ^dimension of non interacting regions) ^{and the} corresponding ^{value of a} (variance ^{of the} gaussian distribution function of concentrations)

for which the computation ^{was made are shown.}

FiG. 4. ^- Same as figure ^{3 for FeSi 12} % (atomic) alloy.

3. Discussion. ^-

According

^{to the made} assump-

tions,

^{one can} expect

that,

^{for a}

given

average

concentration,

^the

larger

the variation of

Is(O)

^and

T,,

with the solute element

concentration,

^the

larger

^the

deviation of the

experimental

^{reduced curve from the}

one of the _pure metal. This behaviour is confirmed

by

^the

comparison

^of

figure

^{1 and 2}

(Is(O)

^and

^Te

being

^faster

decreasing

functions of concentration in Ni than in Fe base

alloys [3]),

^{and thus}

^provides

a first

qualitative

^{check as to the}

validity

^{of the}

concentration fluctuation

theory.

^{For the same} reason,

FIG. 5. ^- Same as figure 3 for NiAl 10 % (atomic) alloy.

FiG. 6. ^- Same as figure ^{3 for NiCr} 5 % (atomic) alloy.

FIG. 7. ^- Same as figure ³ for NiCu 20 % (atomic) alloy.

116

FIG. 8. ^- Same as figure ³ for NiV 5 % (atomic) alloy.

the deviation of the

computed

^{curves with} respect ^to the one of the _pure metal is mush

larger

^{for a}

given

in the Ni base

alloys

than in the Fe base ones

(Fig.

³

through 8).

It should be

noted, however,

^{that the} agreement between

computed

^and

experimental

^{curves is not}

very

good

^over the whole reduced temperature range whatever the value of Â.

Experimental

^curves syste-

matically

^{show an initial}

slope larger

^{than the corres-}

ponding

^{best fit}

computed

curves, almost as if À were an

increasing

^{function of} temperature. ^{This fact} does not find _any

physical explanation,

^{and it seems}

more

plausible

^{that at low} temperature ^other

pheno-

mena contribute to the modification of the

7g

^{vs. T}

curve of disordered

ferromagnetic alloys.

On the other

hand,

^{two facts must} be stressed which

permit

^{to confirm the}

assumptions

^{on which}

the concentration fluctuation

theory

is based :

a)

^{from Table I} one can see that the value of  for which the best

fitting

^between

computed

^and

experimental

^{curves is} obtained is

always

^{of the same}

order of

magnitude

^{for all examined}

^alloys,

^either

Fe and Ni

base, although

^the deviations of the reduced

experimental

^{curves with} respect ^to the pure metal

ones are

quite

^{different for différent}

alloys ;

b)

^the actual best fit values of  are

always

^of

about 8-10 atomic distances : these values seem very

reasonable,

^since

they

^{are also in}

good

agreement with other estimates due to différent methods.

In fact neutron

scattering

measurements in

Fe,

near the Curie temperature, show evidence of the existence of clusters of

spins

whose linear dimension is between 10 and 30

A [4].

Since these clusters behave

as

independent

^or

weakly interacting regions,

^their

linear dimension can be considered as the interaction distance of the Weiss molecular

field,

^{and thus be} identified with Â.

A further check is

provided by

measurements of

7g

and

Tc

^{on thin}

magnetic

^{films as a function of film}

thickness. When this is reduced to about 20

A, changes begin

^{to be observed in both}

IS

^and

Tc

^with respect

to the bulk values. This critical dimension is then

again

ⁱⁿ very

good agreement

^{with the}

presently

found value of À

[5], [6].

TABLE 1

4. Conclusions. ^- It has been shown that the anomalous behaviour of the reduced

Is(T)/Is(O)

^vs.

Z/Te

^curves of disordered

ferromagnetic alloys

^with respect ^{to the} pure Fe or Ni base metal can be

explain-

ed

by

^{means of a}

single theory,

^{based on the} assump- tion that the fluctuations of concentration of the solute element are the

principal

^{cause of the observed ano-}

malies. The

theory

admits the

sample

^{to be divided}

into small

weakly interacting regions,

whose linear dimension is taken as a parameter, ^{since it controls} the width of the

gaussian

distribution function of concentrations. The

magnetization

^{of the whole}

sample

^{at a}

given

temperature ^thus results from the

sum of the

magnetizations

^{of these}

regions.

^{The com-}

parison

^{of the}

experimental

^and

computed

^curves

shows that the best

fitting

is obtained for all examined

alloys by assuming

^the linear dimension of these

regions

^{to be of} the order of 20-30

A.

This

length,

which can be

thought

^{as the} interaction distance of the

exchange field,

^{is thus about the same in all cases.}

Furthermore its value is in

good

agreement ^with

neutron

scattering

measurements of

spin clusters,

and with

experimental

^{data on} the critical thickness of thin

magnetic

^films.

This seems to prove the

validity

^{of the} concentration fluctuation

theory

ⁱⁿ

explaining

^the anomalous beha- viour of the reduced

Tg

^{vs. T curves of disordered}

ferromagnetic alloys,

^{at least in a} temperature

region

close to the Curie temperature. ^On the other

hand,

the

experimental

^{curves show at low} temperature ^a decrease of

Is(T)IIs(O)

^for

increasing TITe

systema-

tically

faster than the one of the

computed

^curves.

This fact cannot be

explained

^{on the basis of this}

theory,

^without

assuming

^{that the} interaction distance of the

exchange

field decreases with

decreasing

tempe-

rature. This

hypothesis

seems,

however,

^{difficult to}

justify,

and therefore one should not

disregard

^the

possibility

^{that more}

complex phenomena

^take

place

in the low temperature

region.

References

[1] ^WENT (J. J.), Physica, 1951, 17, 596.

[2] ^KRANZ (J.) and BODEWIG (C.), ^Z. Physik, 1955, 142,

396.

[3] ^KNELLER (E.), « Ferromagnetismus » (Springer ^Ver- lag, Berlin, 1962), 149, ^161.

[4] ^AVERBACH (B. L.), ⁱⁿ « Magnetic Properties ^{of Metals}

and alloys », ^BOZORTH (R. M.) ^{et al.} (American Society ^for Metals, Cleveland, Ohio, 1959), ^285.

[5] ^DRIGO (A.), ^Nuovo Cimento, 1951, 8, 498.

[6] NEUGEBAUER (C. A.), « Structure and Properties ^of thin Films » (J. Wiley ^and Sons, ^New York, 1959) ^358.