Magnetization process : model of the potential function in the case of a thermal demagnetization

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Magnetization process : model of the potential function in the case of a thermal demagnetization

P. Molho, J.L. Porteseil

To cite this version:

P. Molho, J.L. Porteseil. Magnetization process : model of the potential function in the case of a thermal demagnetization. Journal de Physique, 1983, 44 (7), pp.871-878.

�10.1051/jphys:01983004407087100�. �jpa-00209671�

Magnetization process : model of the potential ^function

in the case of a thermal demagnetization

P. Molho and J. L. Porteseil

Laboratoire Louis Néel, CNRS-USMG,166X, ^{38042 Grenoble} Cedex, ^France

(Reçu ^le 6 janvier 1983, révisé le 9 mars, accepté ^{le 23 mars} 1983)

Résumé.

Le modèle de la fonction potentiel, ^{conduit aux lois de} Rayleigh pour les cycles d’hystérésis ^en champ

faible d’un corps ferromagnétique désaimanté par ^un champ alternatif. Dans le cas d’une désaimantation ther-

mique, ^la paroi peut occuper n’importe quel minimum de V(x), ^{et la structure en domaines est} initialement très métastable. En étudiant toutes les configurations ^où peut ^{se trouver la} paroi, dans le formalisme développé par

Néel, ^on étudie l’effet du champ magnétique ^{sur une telle structure en domaines.}

On montre que les lois de Rayleigh ^{ne doivent} plus ^être vérifiées, ^et qu’en particulier ^la première ^rémanence

doit avoir une variation, ^{somme d’un terme linéaire et d’un terme} quadratique, ^en fonction de l’amplitude ^du cycle

de champ; que dès le deuxième cycle, ^la structure est stabilisée, mais que les cycles d’hystérésis ^{sont décalés du côté}

où le champ agit ^la première ^fois.

On compare ^ces résultats avec le modèle des grains de Preisach étendu aussi au cas de la désaimantation ther-

mique.

Abstract.

The model of the potential ^function V(x), ^{for the} hysteresis cycles ^{in a weak field of a} ferromagnetic sample, demagnetized by ^an alternating field, ^{leads to} Rayleigh’s laws. In the case of a thermal demagnetization, ^the

Bloch wall may be present in every well of V(x), ^{and the domain structure is} initially metastable.

Considering ^all configurations ^{where we can} find the Bloch wall, ^{in Néel’s} formulation, ^we study ^{the effect of a} magnetic ^{field on such a domain} structure. We show that Rayleigh’s ^{laws should no} longer ^{hold and} that, ^{for ins-} tance, the first remanence should vary with the amplitude ^{of the} cycle ^{as the sum of a} linear and a quadratic ^term.

The structure is stabilized when the first cycle ^is over, but the limit cycle is shifted to the side where the field is

applied initially.

We compare these results with Preisach’s grains model, ^{extended to the case} of thermal demagnetization.

Classification

Physics ^Abstracts

75 . 60E

1. Introduction.

Model of the potential ^function [1, 2] ^{allows to describe} Rayleigh’s laws, observed in the hysteresis cycles ^of ferromagnetic samples, in weak field.

This model has been developed ^for « prepared » samples. ^This preparation ^{is a} demagnetization by

an alternating field whose amplitude slowly decreases,

from a value large ^with respect ^{to the coercive} field, ^to

zero. This kind of demagnetization consists in

« shaking » the domain structure, ^so that it reaches a

stabilized state, ⁱⁿ which the Bloch walls movements are globally reproducible ^{from one} cycle ^{to another.}

It is not the same in the case of a thermal demagne- tization, by cooling ^{from a} temperature ^above Tc.

Cooling ^{creates a} metastable domain structure, ^which will change during ^{the first} cycles. ^We present ^{here the} model of the potential function, developed by ^N6el [1],

extended to the case of a thermal demagnetization.

2. The modeL

This model aims at describing ^the hysteresis loops ^of ferromagnetic materials in fields which are weak with respect ^to the coercive force. In this field _range, the only significant magnetization processes ^are reversible and irreversible displacements ^{of Bloch}

walls. The model deals with the motion of a plane wall, ^of constant area, separating ^two domains of

opposite magnetizations ± Ms, under the effect of a

magnetic ^field parallel ^{to and} having ^{the same direc-}

tion as the magnetization ^{in one} of the domains.

The interactions between the wall and the lattice

are described by ^a conservative potential energy

V(x) ^{which is a} random function of the abscissa of the wall. This is because of local fluctuations of the

magnetocrystalline ^and exchange energies.

The effect of a magnetic ^{field H on} the wall is equi-

valent to a hydrostatic pressure U

2 Ms ^{H, and}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004407087100

872

Fig. ^1.

^Schematic representation ^{of the} potential ^func-

tion V (x). ^{For the} meaning ^of symbols A, B, C, ^{see text.}

the equilibrium of the wall is governed by ^the equa- tion U ⁼ d V/dx. It is stable if d2 V/dx2 > 0.

To give ^a convenient picture ^{of the} complicated irregularities ^{of the material,} the x-axis is divided into

equal segments ^of length ^{2 1. The curve} V(x) ^{is then}

simulated by ^a succession of arcs of parabolas, ^each

one being ^defined by ^the slopes ^{of its} tangents ^{at the} ends of a segment (Fig. 1). The values of successive

slopes ^are random and without correlation. A slope Po, characteristic of the magnetic hardness of the

material, ^{allows us to} define the following ^dimension-

less quantities : ^reduced slope p ⁼ PIPO, and reduced field h ⁼ U/Po. ^{The reduced} displacement ^{of the} wall, j ⁼ x/2 1, represents ^the magnetization. Only

reduced variables will be used in the following. ^The

wall is assumed to move under the effect of a field h,

whose modulus remains smaller than a positive quantity ^{8 which in turn} is much smaller than _one,

i.e. I h I e 1.

The study of the wall motion on the various para- bolic arcs leads us to define three different types ^of slope : slopes larger ^{than e} (type A), bracketed between

-

8 and + e (type B), and smaller than - 8 (type C).

As the modulus of h remains smaller than _s, the wall

can neither get ^{over the} slopes ^A when the field

increases, ^nor get ^{over the} slopes C when the field decreases.

Let w(p) dp ^{be the} probability ^of finding ^a slope between p and p ⁺ d ; ^{co(p) is assumed to be Gaussian,}

namely w(p) ⁼ I/.vn exp( - p2). ^The probability ^of finding ^a slope ^{of a} given type ^is

for type B, ^and 1/2 - q/2 ^for types ^{A or C. Since}

8 is much smaller than 1, ’1 ⁼ 2 8/J; ⁺ 0(83).

The potential ^function V(x) ^{is now} completely

defined by Po ^{and I,} and the motion of the wall on a

succession of arcs of parabola ^can be studied.

2.1 MOTION OF THE WALL UNDER AN INCREASING POSITIVE FIELD h. - Whichever method was used to

demagnetize ^the specimen, the wall is initially ^located

at a minimum of V(x), i.e. between a negative slope ^on

the left-hand side, ^{and a} positive slope ^{on the} right-

hand side. Hence, ^it belongs ^{to a} region ^of V(x) beginning by ^{one of the} following sequences of slopes : CB+, CA, ^{B _} B+ ^{or B-} A, ^where B+ ^{and B _ stand}

for the slopes ^{between 0 and} 8, and - 8 and 0, respec-

tively. When the field increases, the wall will move to the right, ^and stop ^{at the first} slope ^{A it} encounters. If the calculations are to be performed up ^to the second order in 8 and since the probability ^{of a} slope ^{B is of}

order _8, only ^the sequences including ^{no more than}

two slopes ^{B have to} be taken into account.

Hence, the _sequences of arcs to be considered are the

following : ^{CA, B_ A,} CB+ ^Cm A, ^B_ B+ ^{Cm A and} CB+(cm B) A where the integer m > ⁰ stands for the number of slopes C, ^{and the} parenthesis repre- sents all the possible permutations ^{of one} slope ^B

with m slopes ^C.

An increasing field drives the wall in the direction of positive ^{x. If the} slope ^of V(x) ^is steadily increasing,

the wall will move in a reversible fashion. If, ^{on the}

other hand, ^the slope passes through ^a maximum, ^then

decreases (inflexion point), the obstacle is no longer

able to hinder the displacement ^{of the wall, which will}

start to move irreversibly, ^{until it meets a} slope slightly higher than the inflexion slope. ^{We will be} only concerned here with the irreversible displace-

ments of the wall.

The _sequences CA, ^{B _} A, CB + A, B _ B + ^{A and} CB+ B+* A ^with BI > B+ give purely ^reversible

contributions in an increasing ^field. Only ^the following

sequences result in irreversible motion :

m being ^an integer > ^1.

The calculation presented ^hereafter consists in

summing up the contributions of all the possible types of sequences. Every contribution is equal ^to

the length of the irreversible path, multiplied by ^the probability of the sequence, and by ^the probability

that the wall lies initially ^{in a} given potential ^{well of}

the _sequence. As previously indicated, ^{we limit}

ourselves to second order in E.

2.2 PROBABILITY OF A GIVEN SEQUENCE.

As ^the

slopes ^{are assumed to} be without any correlation,

the probability ^{of a} sequence is simply equal ^{to the} product ^{of the} probabilities of the various slopes

encountered. The probability ^{of a} slope ^of type A ^{or C}

is 1/2 - 8/.J1c, ^{that of a slope B is 2} and ^{that of}

a B+ ^{or B_ is} 8/.J1c. ^{For a} given slope B+ ^{with a}

value p, ^the slopes B *, ^where B * B+, all lie between

0 and _p. Hence, ^the probability ^of B* ^is P/.J1c. ^{As p}

is uniformly distributed between 0 and E, the overall

probability ^of finding Bt ^{is :}

Hence the probability ^of finding ^{a succession} B+ B* ^with B* B+ ^is E2 /2 ^n.

Since the _sequences CB+ ^{B_ A,} CB+ B* A, B _ B + ^{Cm A and} CB+(cm B) A ^{contain two} slopes

of type B, ^their probabilities ^{are of order 82. Accor-} dingly, it will be sufficient to calculate the irreversible

path of the wall in these _sequences to the zeroth order in E. On the contrary, ^the sequence CB+ ^{Cm A has a} probability of order 8 : the corresponding path ^will

have to be expanded up ^to the first order in 8.

2.3 IRREVERSIBLE PATHS OF THE WALL IN A SEQUENCE : AN EXAMPLE.

The wall is initially located in a well of V(x) between the two first slopes ^on the left-hand side of the sequence.

Let us consider for example ^a sequence of type CB+ ^{C2 B _ A} (Fig. 1). ^{As the field} increases, ^{the wall} first moves reversibly in the well CB+, until the field reaches the value of the slope B +. ^{At that} point, ^the

wall starts to move irreversibly until it meets, between B _ and A, ^a slope higher ^than B +. Then its motion becomes reversible again, ^{and it} stops ^{at the} point

where the slope ^{is h. The} average abscissa of the point

of slope B+, between B_ and A, calculated for all the

possible ^{values ofB_ and A,} differs from the abscissa of the point ^of slope ^B_ by ^a quantity ^{of order 8} (Fig. 2). ^{As there are two} slopes ^{B in the} sequence, the contribution due to that small difference is of third order in 8 and can be neglected. The irreversible path

is equal ^to 3 in units of x/2 ^1.

In the same way, if, ^after starting irreversibly ^{at a} point ^of slope B+, ^{the wall meets a} slope ^B > B+,

it will have again ^a reversible motion until the field becomes equal ^{to B.} This additional reversible dis-

placement ^is proportional ^to the difference B - B+,

and hence of order E. Such a situation can occur only

Fig. ^2.

Example of variation of the slope ^{of V} (x), ^function

of the abscissa. i : is the starting position of irreversible motion, f : is the final position of the irreversible path.

in _sequences of type CB, (cl B) A, ^whose proba- bility is of order E2, ^{and the} corresponding ^correction

can be neglected.

2.4 AVERAGE START AND STOP POSITIONS.

a) ^{In the}

situations which lead to an irreversible motion, ^the wall starts at the first slope B+, except ^{when that} slope ^is immediately ^followed by ^a slope B* > B +,

in which case the wall starts only ^at the second slope B*. ^These special situations are the sequences

CB+ B+* Cm A(B*+ > B+) ^which belong ^{to the class} CB+(Cm B) ^A.

b) ^{Two cases have to} be considered for the stop position. If it lies between two slopes ^{of finite} moduli,

C and A, ^{it will} coincide on average with the middle of segment CA, except ^{for a} correction of order 8. On the other hand, if the wall stops ^{between a} slope ^{B and a} slope A, ^its average position ^will coincide with the

point ^of slope B, ^{with a} correction which is again ^of

order e. Accordingly, in the class CB+(Cm B) A, ^the

sequences which are terminated by ^{BA have to be}

considered separately ^from those which are terminated

by ^CA.

c) Corrections of order 8 have to be taken into account for sequences CB+ ^C"’ A, ^whose proba- bility ^is proportional ^{to c. The wall starts irrever-} sibly ^{from the} slope B+ ^and stops between the last

slope’ ^{C and the} slope ^A (Fig. 2). ^{As first} approxima- tion, ^{the final} position ^{lies on average at the middle}

of the segment, CA, ^and hence the path ^is equal

to m + 1/2 in units of x/2 ^{l. In} fact, the wall stops between C and A at the point ^{where the} slope ^h’

is equal ^to B+. ^The expectation value for all the pos- sible slopes B+ smaller than h is h/2. ^{If we take the} origin of the x-axis at the middle of, CA, the reduced

abscissa j ^{of the} point between C and A where the

slope ^is equal ^to h/2 ^{is :}

The expectation value of this abscissa for all the allowed values of p and q ^{is :}

where E ⁼ Euler constant ⁼ 0.577 22.

That correction has to be added _up to the main contribution m + 1/2.

2. 5 PROBABILITY OF OCCUPATION OF THE VARIOUS CONFIGURATIONS.

For the sake of simplicity, ^we

consider here the case where all the wells of the curve

V(x) ^{have the same} probability ^of being initially

occupied by the wall. This corresponds ^{to a} demagne-

874

tization of the specimen by cooling ^{it in a zero field}

from a temperature higher than the Curie point, ^when

the quenching ^{can be} regarded ^as infinitely ^{fast. This}

simple assumption will allow us to calculate the theo- retical magnetization ^{curves of an} ideally quenched

domain structure. A more complete derivation would have to take into account, ^a relaxation leading ^{to a} weighted average ^on the various types ^{of wells. Never-} theless, ^{the essential} physical ^{results derived hereafter}

would not be basically ^modified.

The restriction to two slopes ^{B at most in} every sequence leads to situations with 1, ^{2 or 3} potential

wells :

-

The CB+ B* ^A (Bf B+) ^contain only ^one well; ^the probability ^of occupation ^{is 1.}

-

The CB+ ^{B_ A,} CB+ ^{C"’ A, B_} B+ ^{Cm A,} CB+ Cm B+ A, CB+ B+ Cm A ^and CB,(CM B-’) A

contain two potentials ^{wells; the} probability ^{of occu-} pation ^is 1/2 for each of them. The second well is of type ^{CA or B_} A ; ^when occupied, ^its contribution ⁱⁿ

increasing ^{field is} purely reversible.

-

The CB + C(B+ Cm - 2 ) ^CA contain three poten- tial wells, ^the probability ^of occupation of each well is

1/3. The situations where the second well is occupied

were already taken into account with the CB + ^Cm A ; the situations where the third well is occupied ^lead only ^to reversible contributions.

Table I summarizes the various configurations,

their probabilities, ^the corresponding irreversible

paths ^{of the wall, the} probabilities ^of occupation, ^and

the contributions to the irreversible magnetization

after summation over all the possible values of ^m.

The _sequences belonging ^{to the class} CB , (C I B) A

are split into five subclasses which result in different

paths ^{of the wall, the} probabilities ^of occupation, ^and

the position and value of the second slope ^B.

The contributions ^to the magnetization ^{are no} longer expressed ^{in terms of the} arbitrarily ^fixed quantity ^{s, which was used to} define the three types of slopes, ^{but in terms of the} physical quantity h (reduced field), which is of the same order.

Finally, the irreversible magnetization acquired

under a field h is :

Table I.

Field increasing from ^{0 to h.}

Table II.

Field decreasing from h ^{to 0.}

2.6 FIELD DECREASING FROM h TO 0.

The increasing

field h applied previously ^{drives the wall} against ^the slopes A ^{located at the} right-hand ends of the sequences. Among ^the configurations considered _up to now, only ^the CB+ B* ^A (B* B+) ^will give ^an

irreversible contribution when the field decreases down to 0. The other situations will not contribute since the wall will be stopped as soon as it meets a

negative slope ^on the left of the slope ^{A, and has} only ^a reversible movement up ^to this point : sequen-

ces CB+ B_ A, CB+ ^C"’ A, ^B- B+ ^{C"’ A and} CB+

(C"’ B) ^A. On the other hand, ^{no new} configuration leading ^to irreversibility ^{was created,} because the wall which had a reversible motion as the field increas- ed remained in the same potential ^{well. The situation}

is thus summarized in table II.

2. 7 FIELD DECREASING FROM 0 TO - h.

Among ^all

the configurations considered previously, only ^the CB+ ^{B_ A will} play a role. Their probability ^{of occu-} pation ^is equal ^to 1, ^{since the} increasing ^field previously applied, emptied all the wells CB+. ^The CB+ Bf A (Bf B+) ^have already contributed to the irreversible magnetization while the field was

decreasing ^{from h to 0. In} all the other sequences, the wall is stopped by ^{the first} slope ^{C on its} left; ^{from now}

on, it is trapped ^{in a well of} type ^{CA or} CBA, ⁱⁿ which its motion remains purely reversible. All these situations have been destroyed ^as regards ^irrever- sibility.

On the other hand, ^new configurations ^{which have}

not contributed in a positive field have to be taken into account when the field decreases from 0 to - h.

Table III.

Field decreasing from ^{0 to - h.}

876

These are the following sequences, terminated ^on the

right-hand ^side by B - A or B - B + A :

with m > ^{1. These} configurations resulted in reversible motion in positive ^{fields. In a} negative ^field, they ^will

lead to irreversible contributions; the calculation is

analogous ^{to that} presented ^{for h} > 0. Table III summarizes the results for all the configurations ^which

have to be considered.

The overall irreversible change ^of magnetization

between 0 and - h is :

The irreversible contributions between 0 and h on

the one hand, and 0 and - h on the other, ^are symme-

trical, except ^for two terms :

-

the term arising ^{from the}

during ^{the first} magnetization : since the two wells

CB+ ^{and B_ A have the same} probability of occupa-

tion, only ^one half of these configurations ^contribute

to irreversibility. ^{On the} contrary, when the field decreases from 0 to - h, only the wells B- A initially

contain the wall, ^{and the} CB+ ^{B_ A} contribution is

1 h2

- 1 h2 T hat is the difference between virgin curve, and the subsequent ^ones.

Among ^{the B_} B+ ^Cm A, ^which play ^a role when the field increases, ^{the CB_} B+ ^{C’" A have} symmetrical configurations, ^which play a ^{role in} decreasing ^fields.

These are the CA"’ B_ B+ ^{A. On the} contrary, ^the AB_ B+ ^{C’" A, which are the most} metastable confi-

gurations, ^{have no} symmetrical ^{ones in} decreasing

fields and would give irreversible contributions for both positive ^and negative initial fields. Since they ^are destroyed ^{between 0 and + h,} they ^{will no} longer

contribute between 0 and - h. This asymmetry ^results in a positive shift of the hysteresis loop, ^of magnitude

5 h2 ²

16 7 r

2. 8 SUBSEQUENT VARIATIONS OF THE FIELD. - After the field has decreased down to - h, ^{all the} configura-

tions apart ^{from the CB 2 A have been} destroyed ^as regards irreversibility. ^{For all the} subsequent cycles,

the only contributions arise from :

-

the CB+ ^B_ A, ⁱⁿ positive ^and increasing ^or negative ^and decreasing field ;

-

the CB+ B* A (B+* B+) ⁱⁿ positive field,

whether increasing ^or decreasing;

-

the CB* B- A (B* > B_) ⁱⁿ negative ^field,

whether increasing ^or decreasing.

More particularly, ^the irreversible contribution which is linear in h, is cancelled as soon as the field reaches the value - h, and all the subsequent ^irre-

versible contributions are quadratic ⁱⁿ h, ^{as shown}

on table IV.

Figure ^3a gives ^{a schematic} picture ^{of the} hysteresis loops predicted by ^{the model, in the case} where all the

potential wells have the same probability ^of having trapped ^{a wall.}

3. Comparison ^{with an} alternating demagnetization.

In the case considered by ^Neel [1], ^the specimen ^has previously undergone ^a demagnetization by ^{an alter-} nating field whose amplitude ^has steadily ^decreased

down to zero from a value higher than the coercive field. After such a process has been completed ^{the wall}

is constrained to lie in regions ^{or «} domains » of type CA; ^{CBA and} CB 2 A (if the calculations are to be

. performed up ^to second order in e). All the metastable

configurations ^{have been} destroyed by ^the alternating

field.

Table IV.

Subsequent ^variation of field between - h and h.

Fig. ^3.

^Schematic hysteresis cycle ^{when the} sample ^is demagnetized by cooling ^{it in a zero} field from a temperature above Tc : 1) ^{in the} potential ^function model, 2) ^{in the} Preisach’s grains model. These cycles ^are drawn for h

0.5, while the model is defined for h 1. First of all, this value is in correspondence ^{with the} experimental ^{values we will} present. Then, ^{in the case of a} demagnetization by ^{an alter-} nating ^field Rayleigh’s laws may be observed _{up to h}

0.8, while also established for h ^1.

In the double well configuration CB+ ^B_ A, ^{the wall} initially lies in the well which is limited by ^the slope ^B having ^the higher ^{modulus. Hence,} when the field starts increasing ^{from 0,} only ^the CB+ ^{B_ A with} B+ > I B- I give ^an irreversible contribution 1 h . h2 When the field decreases from h, only ^the CB+ B* ^A

1 h2 with B+* ⁺ B+ play a ^{+p Y role,} and j(h) - j(0) ^{J() J()} = 1 h . _{8 03C0}

Between 0 and - h, ^the CB + B- A and the CB-* B- A 3 2 with B * > B_ contribute, and/(0) -./(- h ^{J( ) J( 7r} = 3/8 h2 ,

One thus finds a loop equivalent ^to the limit loop

reached at the second cycle ^{in the case of thermal} demagnetization, except for the aforementioned shift

arising ^{from the more} metastable configurations.

The situation considered by N6el leads to the well-known Rayleigh ^{laws, i.e.} quadratic changes ^of

magnetization in fields smaller than H c. ^{In contrast,}

it can be remarked that, ^{for a thermal} demagnetiza-

tion :

-

The Rayleigh ^{laws are no} longer obeyed. ^The

irreversible magnetization ^{on the} virgin ^{curve is the}

sum of a linear and a quadratic ^term.

-

Particularly, ^{the remanent} magnetization ^after

the field has decreased down to 0 for the first time is also the ^sum of a linear and a quadratic ^{term. This}

linear term will be destroyed by ^the subsequent

variation of the field 0 -+ - h.

-

The metastable configurations ^are destroyed ^as

soon as the first cycle ^{has been} completed, ^{and the} Rayleigh ^{laws are} again valid from the beginning ^of

the second cycle.

It may be noticed that the linear irreversible

magnetization arises from the _sequences CB+ ^{C "’ A}

and CA m B _ A. The path of the wall in such sequences is quite ^{similar to the} path ^{shown on} figure ^{2 for}

another type ^of sequence. The linear contribution to the magnetization ^{is due to} the first-order probability

of finding sequences with only ^one slope ^B.

The above calculations were performed by assuming equal probabilities for the initial occupation ^of every well. In a real experiment, ^some relaxation will

obviously ^take place ^{as the} specimen is cooled down

through T c ^{at a finite rate. The most} metastable wells AB_ B+ C, in which the wall is bracketed by ^two

weak slopes, may be assumed to undergo ^the strongest relaxation; but this effect will also be present ^{for the} wells of types CB+ ^{C and AB_ A.} Hence, the contribu- tions listed in tables I and III have to be weighted by

different occupation probabilities. ^{However, the lea-} ding ^{term in the} irreversible magnetization ^{will remain} proportional ^{to h. In fact, the} comparison ^{of the}

measured and calculated linear terms will afford a means, among others, ^of estimating ^{the relaxation}

which takes place during ^the quenching process. In ^a

forthcoming paper, ^we will _compare the predictions

of the model with experimental ^{results obtained on} polycrystalline gadolinium.

4. Comparison with Preisach’s grain ^model.

It is well known that the model of V(x) ^{in the case}

considered by ^N6el (alternating demagnetization) ^is equivalent ^to Preisach’s grain ^model [3]. ^The only configurations ^which play a role in this case are the CB2 A, represented by points ^{on a} plane, the abscissa

(x) and ordinates (y) ^{of which are} the values of the left and right slopes ^B, respectively. Every ^« grain »

has the magnetization ⁺ 1/2 ^{or -} 1/2, depending ^on

the location of the wall in the right-hand ^{or’ the left-}

hand well. These values will be represented by ^{+ or -}

on figure ^{4. The} density ^of grains is assumed constant near the origin. Taking ^{into account the} probabilities

of the slopes ^{C and A which bracket} the sequence B 2,

leads to multiplying the contribution of _every « grain »

by 1/4 (terms ^{of order 83 are} neglected).

878

Fig. ^4.

Variations of magnetization j, in the Preisach model. 1) ^{Case of} demagnetization by ^an alternating field;

2) ^{Case of a thermal} demagnetization. ⁺ represents grains

with + 1/2 magnetization, - represents those with - 1/2 magnetization, ^{and *} represents those with zero averaged magnetization. ^{a :} is the initial situation, ^zero field, ^{b and} f : are situations with value h of field, ^{d : is a} situation with - h value of field, ^{c : is a} situation with 0 value of field,

reached decreasingly ^{and e : is a} situation with 0 value of field, ^reached increasingly.

Figure ^4a shows how the virgin ^{curve and} hysteresis loops ^{can be} predicted by ^{this model, in the case of an} alternating demagnetization. ^In the framework of this model, ^{it is} possible ^to represent ^{some metastable}

configurations ^created by quenching the material from above T c; ^{but the} only configurations ^taken

into account are the CB2 A, ^the CB+ ^CA (increasing field) ^and the CAB- A (decreasing field). ^The CB+ ^cm BA, ^for instance, ^{cannot be} represented ⁱⁿ

Preisach’s plane, ^because they ^would require m ^{+ 2}

coordinates in order to describe the _sequence B+ ^{C’" B.}

After a thermal demagnetization, the initial situa- tion is given by ^{the table V.}

A difficulty ^arises because, in the Preisach model under its usual form (alternating demagnetization), only ^{a limited} region ^such that x I ^and I y I ^{E is}

taken into account. On the contrary, in the thermal

case, I x -I and I y I ^{can be} arbitrarily large (slopes A

and C). So, ^{one has to introduce some cutoff for the} possible ^values of x and I y 1. ^The simplest ^solution

consists in defining ^{a distance} 10 by requiring ^{that the}

constant probability w(0) ^{assumed near the} origin ^be normalized, namely m(0) x lo ^{= 1. If a} Gaussian distri- bution w(p) (P) exp(_ 2 ^is assumed, l0 ^is equal

r ^{p( p ) o q}

to J1c.

Figure 4b shows the evolution of the Preisach

diagram ^{after a thermal} demagnetization. ^{It can be}

seen that the Preisach and V(x) ^models give ^diffe-

rent linear and quadratic contributions to the irrever- sible magnetization. ^{Moreover, the} Preisach model does not lead to any shift of the loop ^with respect ^to the origin. ^{After one} cycle, ^the loop has settled down to the loop obtained after an alternating demagneti-

zation (Fig. 3b).

As indicated earlier, this difference arises from the metastable configurations ^{B _} B+ Cm A which cannot be described by the Preisach model.

Table V.

Initial situation in Preisach’s model.

References

[1] NÉEL, L., ^Cahiers Phys. 12 (1942) ^1.

[2] KRONMÜLLER, H., ^Z. Angew. Phys. ³⁰ (1967) ^9.

[3] ^PREISACH, F., Z.f. Physik ⁹⁴ (1935) ^277.