Phase transition in ferromagnetism

A thermodynamic approach to ferromagnetism and phase transitions

M. Fabrizio

a

, C. Giorgi

b

, A. Morro

c,*

a

Department of Mathematics, Piazza di Porta S. Donato 5, 40126 Bologna, Italy b

Department of Mathematics, Via Valotti 9, 25133 Brescia, Italy c

University of Genova, DIBE, Via Opera Pia 11a, 16145 Genova, Italy

a r t i c l e

i n f o

Article history:

Received 30 December 2008

Received in revised form 30 March 2009 Accepted 13 May 2009

Available online 31 May 2009 Communicated by K.R. Rajagopal Keywords: Ferromagnetic–paramagnetic transition Saturation effect Continuum thermodynamics Logarithmic potential

a b s t r a c t

The paper provides a modelling of the magnetization curve and of the ferromagnetic–para-magnetic transition within a continuum thermodynamic setting. The general model of the nonlinear, time dependent behaviour of ferromagnetic materials is accomplished by regarding the magnetization vector as an internal variable, namely as a vector field whose time evolution is a constitutive equation subject to the requirements of the second law of thermodynamics. The exchange interaction of the magnetization is modelled through a dependence of the free energy on the magnetization gradient. Consistent with the non-simple character of the material, the second law allows for a non-zero extra-entropy flux. A general three-dimensional scheme is elaborated which seems to be new in the literature. The three-dimensional setting is then established for stationary and homogeneous fields thus finding the collinearity and the corresponding form of the magnetic susceptibility. The whole evolution problem for the temperature and the magnetization is provided so that temperature-induced transition processes are allowed. The model accounts also for the dependence of the saturation magnetization on the temperature. Also for the sake of comparison with the existing literature, the evolution equations for the direction and the intensity of magnetization are derived. Known models, such as those of Landau–Lifshitz and Gilbert, are recovered as particular cases of saturated bodies. Next, the model is made more specific so as to account in detail for the saturation, the residual or spontaneous mag-netization and the coercive field. First, the classical potential, which traces back to Ginz-burg, and the Weiss model are revisited. The corresponding lack of the saturation effect or the description via implicit relations are emphasized. Hence, a new potential, with a log-arithmic dependence on the magnetization, is investigated which provides the residual magnetization and the coercive field in an explicit way and satisfies expected properties of the residual magnetization as a function of the temperature.

Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Ferromagnetic materials exhibit a linear relation between the magnetic field H and the magnetization M. The non-linear relation provides the standard hysteretic phenomena. Nonnon-linearity and hysteresis occur below a characteristic tem-perature called the (magnetic) Curie temtem-perature hc. Above the Curie temperature, the materials are paramagnetic in that

the relation is linear with a coefficient, the magnetic susceptibility, which is inversely proportional to the difference h hcbetween the current temperature h and hc; such a dependence is the content of the Curie–Weiss law. The fact that

so different M  H curves are parameterized by the temperature allows us to cast the passage from one curve to another within the scheme of phase transitions.

0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2009.05.010

*Corresponding author. Tel.: +39 10 3532786; fax: +39 10 3532134. E-mail address:angelo.morro@unige.it(A. Morro).

Contents lists available atScienceDirect

International Journal of Engineering Science

There is a wide variety of approaches to the modelling of magnetization in ferromagnetic bodies. At the bottom, the mod-els deal with the time evolution of the magnetization. We mention that this problem has been addressed in the works of Landau et al.[1,2]which model the time evolution of M in a magnetically-saturated body under the action of a time-depen-dent field H. Next, in the framework of micromagnetics and gyromagnetics, a similar phenomenological approach has been set up by Gilbert[3,4]and then improved by Brown[5]who replaced the field H with an effective field Heff. By selecting an

appropriate form of Heff, Mallinson[6]has derived a description of the switching effect in damped gyromagnetics. Recent

papers exhibit more involved models to account for the evolution of domain walls in ferromagnets (see, e.g.,[7–9]). Such models are well-motivated from micromagnetics but are essentially isothermal in character. Non-isothermal models are provided by Maugin[10]about magnetoelasticity through internal variables and next by Maugin and Fomethe[11]to model phase-transition fronts in deformable ferromagnets.

As a particular case, namely in stationary conditions, the evolution equation is expected to provide the M  H relation of the magnetization curve. Upon the assumption that M and H have a common fixed alignement, the evolution equation was first set up in the form (see, e.g.,[2,12])

_

M ¼

a

0

l

0H 

a

1ðh  hcÞM þ

a

2M3þ

a

3

D

M:

This non-isothermal model accounts for the transition, at the Curie temperature hc, between the paramagnetic and the

fer-romagnetic behaviours. Yet, the corresponding magnetization curve

a

0

l

0H ¼

a

1ðh  hcÞM 

a

2M3;

does not allow for saturation in the ferromagnetic regime (h < hc). In 1907, Weiss developed a theory of ferromagnetic

do-mains structure known as mean field theory. In essence he arranged the Langevin potential in order to describe the non-iso-thermal paramagnetic–ferromagnetic transition and to account for the saturation phenomena [13]. Such a model is motivated by statistical physics but is one-dimensional in character.

The purpose of this paper is threefold. The first fold is to model the nonlinear, time dependent behaviour of ferromagnetic materials within a thermodynamic framework in three-dimensional setting. By analogy with[14], this is accomplished by regarding the magnetization M as an internal variable, namely as a phase field whose time evolution is given by a constitu-tive equation subject to the requirements of the second law of thermodynamics. The exchange interaction of the magneti-zation is modelled through a dependence of the free energy on the gradient of M. Consistent with the non-local character of the material, the second law allows for a non-zero extra entropy flux. A general form of the evolution equation is then de-rived and known models, such as those of Landau–Lifshitz, Gilbert and others, are recovered as particular cases of saturated bodies. The second fold is to improve the model of the M  H relation in ferromagnetism so as to account in detail for the saturation, the residual or spontaneous magnetization and the coercive field. This part is realized by starting with the clas-sical potential which traces back to Ginzburg and showing that a different behaviour (ferromagnetic–paramagnetic) occurs according as h < hcor h > hcbut the saturation effect does not follow. The Weiss model is then examined whence it follows

that transition and saturation are allowed though via implicit relations. A new singular potential, with a logarithmic depen-dence on the magnetization, is investigated which provides the residual magnetization and the coercive field in an explicit way and satisfies expected properties of the residual magnetization as a function of the temperature. The third fold is to pro-vide the whole set of evolution equations, for the temperature and the order parameter, in the three-dimensional frame-work. As a result, our model describes temperature-induced reversible transitions between the paramagnetic and the ferromagnetic regimes. Hence, we can control the phase transition process by acting on the external heat source and the ap-plied magnetic field. Moreover, the scheme is set up in a general way so as to allow also for the dependence of the sponta-neous magnetization, relative to the saturation magnetization, on the temperature. These features seem to be new in the literature.

The advantage of the present approach is the unified, thermodynamically-consistent scheme of constitutive equations and evolution equations which in turn provide the magnetization curve. The pertinent equations prove to be characterized by the free energy as a thermodynamic potential. The scheme is three-dimensional in character and, also for the sake of com-parison with the existing literature, the evolution equations for the direction and the intensity of M are derived. It is of inter-est that all of the schemes appeared in the literature (e.g., time-dependent or stationary, Lagrangian) are recovered as particular cases. In particular, the evolution equation of the direction reduces to that of Landau and Lifshitz if the exchange and anisotropic interactions are neglected. Also, in static conditions, Brown’s equation is obtained with a general form of the effective magnetic field. Next, as an application, the appropriate potential for crystals of iron is determined through the data for the residual magnetization versus h=hc. Moreover, a one-parameter free energy is shown to provide a satisfactory

descrip-tion of both the ferromagnetic and the paramagnetic behaviour according as the temperature is below or above the Curie temperature.

2. Balance equations

An undeformable ferromagnetic material occupies the regionX#R3. The electric field E, the magnetic induction B, the electric displacement D and the magnetic field H satisfy Maxwell’s equations, in the space-time domainX R,

r

 E ¼  _B;

r

 H ¼ _D þ J; ð2:1Þ

r

 B ¼ 0;

r

 D ¼

q

; ð2:2Þ

where J is the current density and

q

is the charge density. The superposed dot denotes the time derivative andris the gra-dient operator. The balance of energy in electromagnetic materials is based on the view that E  H is the vector flux of energy of electromagnetic character. This view follows from Poynting’s theorem which merely shows that



r

 ðE  HÞ ¼ H  _B þ E  _D þ E  J ð2:3Þ

is a consequence of Maxwell’s equations.

Since the body is undeformable, then the balance of energy is taken in the form (see[15]) _e ¼ 

r

 ðE  H þ qÞ þ r;

where e is the internal energy density, q is the heat flux and r is the heat supply, namely energy per unit volume and unit time provided by external sources. By means of the identity(2.3)we have

_e ¼ H  _B þ E  _D þ J  E 

r

 q þ r: ð2:4Þ

The second law of thermodynamics is taken as the statement that the Clausius–Duhem inequality holds for any set of functions which satisfy Maxwell’s equations(2.1) and (2.2)and the energy equation(2.4). Also because of possible nonlocal effects, the entropy flux is likely to be different from q=h, h being the absolute temperature. Hence, letting

g

be the entropy density and k the extra-entropy flux vector, we write the Clausius–Duhem inequality in the form

_

g

P

r

 ðq=hÞ 

r

 k þr

h: ð2:5Þ

The extra-entropy flux k is required to satisfy the boundary condition Z

@X

k  n da ¼ 0 ð2:6Þ

for the whole body (see[16]). This allows(2.5)to provide the standard global statement of the second law, d dt Z X

g

d

v

P Z X r hd

v

 Z @X 1 hq  n da: By(2.4) and (2.5)we have _e  h _

g

 H  _B  E  _D  J  E þ1 hq 

r

h h

r

 k 6 0: For later convenience we consider the free energy density

w¼ e  h

g

:

Hence, the Clausius–Duhem inequality becomes _

wþ

g

_h  H  _B  E  _D  J  E þ1

hq 

r

h h

r

 k 6 0: ð2:7Þ

Having in mind a model for ferromagnetism, we disregard polarization and let

D ¼



0E; B ¼

l

0ðH þ MÞ: ð2:8Þ

This assumption is consistent with the fact that polarization does not contribute to magnetization in bodies at rest (see e.g.

[12], p. 83, and [17]).

Upon substitution of(2.8)in(2.7)we find that the Clausius–Duhem inequality takes the form

_

wþ

g

_h 

l

0H  _H 

l

0H  _M 



0E  _E  J  E þ 1

hq 

r

h h

r

 k 6 0: ð2:9Þ

Restrictions placed by the inequality(2.9)are now evaluated for a rather general set of constitutive equations.

3. Thermodynamic restrictions

Let w;

g

;q; k and _M be given by (constitutive) functions of the set of variables

C

¼ ðh; E; H; M;

r

h;

r

MÞ:

The vector quantities q, k and the time derivative _M are allowed to depend also on the higher-order gradients

Maxwell’s equations read

l

0ð _H þ _MÞ ¼ 

r

 E;

r

 H ¼



0_E þ J; ð3:1Þ

r

 M ¼ 

r

 H;



0

r

 E ¼

q

: ð3:2Þ

Eq.(3.2)hold at any time t, as a consequence of(3.1), provided they hold at an initial time t0and the continuity equation

r

 J þ _

q

¼ 0

holds. Hence we can take the values of _H; _E as arbitrary whereas _M is provided by the pertinent constitutive equation, _

M ¼ ^Mð

C

Þ:

The function ^M may be viewed as

c

times the d’Alembertian inertia couple density,

c

being the gyromagnetic ratio. The space dependence of E and H is required to be appropriate so thatr E;r H satisfy Eq.(3.1)andr _M;r _E satisfy

r

 _M þ

r

 _H ¼ 0;



0

r

 _E þ

r

 J ¼ 0:

The chain rule allows us to write the inequality(2.9)in the form

ðwhþ

g

Þ _h þ wrh

r

_h þ ðwH

l

0HÞ  _H þ ðwM

l

0HÞ  _M þ ðwE



0EÞ  _E þ wrM

r

M  J  E þ_ 1

hq 

r

h h

r

 k 6 0; ð3:3Þ where the indices h;rh;H; M; E;rM denote partial derivatives. The arbitrariness of _h,r_h and _H, _E requires that

g

¼ wh; wrh¼ 0 and wH¼

l

0H; wE¼



0E: As a consequence, w¼1 2

l

0H 2 þ1 2



0E 2 þ

W

ðh; M;

r

MÞ:

Upon some rearrangements, the inequality(3.3)becomes

ð

W

M

l

0H 

r



W

rMÞ  _M þ

r

 ð

W

rMM  hkÞ þ k _

r

h J  E þ 1

hq 

r

h 60: ð3:4Þ

A simple scheme arises by letting

hk ¼

W

rMM_ ð3:5Þ

and hence(2.6)requires that Z @X 1 h

W

rM _ M  nda ¼ 0: ð3:6Þ

Look now at the corresponding conditions which guarantee the validity of(3.4). By(3.5)we have ð

W

M

l

0H 

r



W

rMÞ  _M þ k 

r

h¼ ½hð ^

W

M

r

 ^

W

rMÞ 

l

0H  _M; where ^

W

¼

W

h; _ M ¼ ^Mð

C

Þ: Hence(3.4)reduces to ½hð ^

W

M

r

 ^

W

rMÞ 

l

0H  ^M  J  E þ 1 hq 

r

h 60: ð3:7Þ

The inequality(3.7)holds if

J  E P 0; q 

r

h 60; ð3:8Þ

and

The inequalities(3.8)are satisfied by Ohm’s and Fourier’s laws, namely J ¼

r

E; q ¼ 

j

r

h

with positive-valued functions

r

and

j

ofC. The inequality(3.9)is a restriction on the constitutive equation for the time derivative _M and hence on the time evolution of the magnetic polarization M. To save writing, we can express the inequality

(3.9)as

N  ^M 6 0; ð3:10Þ

where

N :¼ dM

W

^

l

0H; dM

W

^ :¼ hð ^

W

M

r

 ^

W

rMÞ: ð3:11Þ The time dependence of the magnetization M and the M  H relation are the subject of the next sections. The inequalities

(3.9) and (3.10)are investigated to determine possible forms of the d’Alembertian inertia couple. Remark 1. Henceforth, we let

ð

W

rMMÞ  n ¼ 0;_ on @

X

; ð3:12Þ

so that(3.6)is satisfied. Incidentally, there are approaches to magnetic modelling where the extra-entropy flux does not oc-cur (see, e.g.,[10]). Yet the boundary condition is placed just in the form(3.12).

Remark 2. In terms of the Gibbs free energy eW¼W

l

0H  M we have

N ¼ dM e

W

h : Hence(3.9)becomes dM e

W

h  _M 6 0:

4. Restrictions on the evolution of M

We now have to find a function ^MðCÞ compatible with(3.10). Letting

w ¼ ^M ð4:1Þ

we can write the inequality(3.10)as

N  w 6 0; ð4:2Þ

where N and w are parameterized by M. Let

v

¼ N þ



M  w;



2 R: The inequality(4.2)is equivalent to

v

 w 6 0 ð4:3Þ

and(4.3)holds if

w ¼ K

v

þ bM 

v

þ

m

M  ðM 

v

Þ;

where

m

2 Rþ_{, b 2 R and K 2 Sym}þ_{, Sym}þ_{being the space of positive semidefinite tensors. This is so because, upon}

substi-tution, we have

v

 w ¼ 

v

 K

v



m

jM 

v

j2:

Accordingly we can write the following statement. Proposition 1. The inequality(4.2)holds if

w ¼ KðN þ



M  wÞ þ bM  ðN þ



M  wÞ þ

m

M  ½M  ðN þ



M  wÞ; for every

m

2 Rþ_{, every b;}

_

_{2 R, and every K 2 Sym}þ_.

By replacing w via(4.1)and taking advantage of the identity

M  ½M; ðM  _MÞ ¼ jMj2M  _M;

_

M ¼ 



ðK þ

m

jMj21ÞM  _M þ



bM  ðM  _MÞ  KN þ bM  N þ

m

M  ðM  N Þ; ð4:4Þ where K 2 Symþand



;b;

m

are functions ofCsubject only to

m

P0. At this stage the rescaled free energy ^Wis any function of h;M;rM. Hence, Eq.(4.4)is the most general three-dimensional evolution equation for M compatible with thermodynamics. We are now in a position to show that the evolution equation (4.4) generalizes both Landau–Lifshitz and Gilbert equations. Indeed, as we show inProposition 2,(4.4)provides two equations which are more general. They reduce exactly to Landau–Lifshitz and Gilbert equations in saturation conditions.

Denote by m the unit vector of M, m ¼ M=jMj, and let

K¼

a

1m  m þ

a

2ð1  m  mÞ;

a

1;

a

2>0: ð4:5Þ

Of course, if

a

1¼

a

2then K reduces to a scalar times the unit tensor 1. Indeed, for any vector

v

the application of K as in

(4.5)provides

K

v

¼

a

1

v

kþ

a

2

v

?;

where

v

kand

v

?are the parallel and the perpendicular parts of v, relative to the pertinent unit vector m.

Since

ðm  mÞM  _M ¼ 0; M  ðKuÞ ¼

a

2M  u; for any vector u, then

KðM  _MÞ ¼

a

2M  _M; M  KN ¼

a

2M  N :

Proposition 2. Let K be as in(4.5). The three-dimensional evolution equation(4.4)becomes

_

M ¼ KN þ

s

1M  M  N 

s

2M  _M; ð4:6Þ

or _

M ¼ 

a

ðN  MÞM þ 

c

M  N þ kM  ðM  N Þ; ð4:7Þ

where

s

1;

s

2; 

a

; 

c

and 

c

are parameterized by the constants in(4.4)and by

a

1;

a

2and jMj2.

Proof. Application of M to(4.4)provides the expression for M  ðM  _MÞ. Substitution in(4.4)and some rearrangements allow us to write(4.4)in the more compact form(4.6), where

s

1¼

m

þ b2

a

2þ

m

jMj2 ;

s

2¼



ð

a

2þ

m

jMj2Þ þ b 1 þ



bjMj2

a

2þ

m

jMj2 : By(4.5)we can write(4.6)as _ M ¼ 

a

1Nk

a

2N?þ

s

1M  ðM  N?Þ 

s

2M  _M whence _ M ¼ 

a

1Nk ð

a

2þ

s

1jMj 2 ÞN?

s

2M  _M: ð4:8Þ

Vector multiplication of(4.8)by M provides the expression for M  _M. Substitution in(4.6)and some rearrangements yield ð1 þ

s

2

2jMj 2

Þ _M ¼ 

a

1Nk ð

a

2þ

s

1jMj2ÞN?þ

s

2ð

a

2þ

s

1jMj2ÞM  N?þ

s

22ðM  _MÞM: ð4:9Þ Inner multiplication by M provides M  _M. Hence(4.9)simplifies to

_ M ¼ 

a

1Nk

j

1N?þ

j

2M  N?; ð4:10Þ where

j

1¼

a

2þ

s

1jMj2 1 þ

s

2 2jMj 2 >0;

j

2¼

s

2

j

1: Now, because Nk¼ 1 jMj2ðN  MÞM; N?¼  1 jMj2M  ðM  N Þ; M  N?¼ M  N ;

we can write(4.10)formally in terms of N only as in(4.7), where 

a

; k >0 and 

c

2 R are given by



a

¼

a

1 jMj2; 

c

¼

s

2

a

2þ

s

1jMj 2 1 þ

s

2 2jMj 2; k¼

a

2þ

s

1jMj 2 jMj2ð1 þ

s

2 2jMj 2 Þ: 

It is worth looking at the simplest case of(4.4)which follows by letting b;

m

;



¼ 0, or, equivalently, by letting

s

1;

s

2¼ 0 in

(4.6), namely _

M ¼ KN ¼ 

a

ðN  MÞM þ k0M  ðM  N Þ; ð4:11Þ

where k0¼

a

2=jMj2. The same relation follows from(4.7)by letting 

c

¼ 0 and k¼ k0. As we see in Section6, the form of K is

crucial to the splitting of(4.4)into two separate evolution equations, one governing the evolution of jMj, the other the direc-tion of M. Henceforth we examine the role played by ^Wthrough N .

Eq.(4.6)is now examined in the stationary regime to prove the following statement. Proposition 3. For any given H, the stationary states for M are solutions to the equation

hð ^

W

M

r

 ^

W

rMÞ ¼

l

0H;

W

^ ¼ 1

h

W

ðh; M;

r

MÞ; ð4:12Þ

subject to the constraint

r

 M ¼ 

r

 H and the boundary condition

^

W

rM n ¼ 0 on @

X

:

Proof. By(4.6), the stationary states solve the equation 0 ¼ KN þ

s

1M  ðM  N Þ;

which can be rewritten as

0 ¼

a

1Nkþ ð

a

2þ

s

1jMj2ÞN?: ð4:13Þ

Since

a

1;

a

2;

s

1>0 then(4.13)implies that N?; Nk¼ 0 and hence N ¼ 0. The equilibrium condition(4.12)follows from

both(4.4) and (4.11). As a consequence, existence of one (or more) solutions depends on the convexity (or non-convexity) ofWwith respect to M. 

Lemma 1. If f is a function which depends onrM throughr M then

r

 frM¼ 

r

 frM:

Proof. This identity follows by the observation that, in indicial notation, @ @Mq;p ¼ @ @ð

r

 MÞj @ð

r

 MÞj @Mq;p ¼



pqj @ @ð

r

 MÞj ; and hence ½ðfrMÞpq;p¼



pqj @f @ð

r

 MÞj " # ;p ¼ ð

r

 frMÞq;

where ; p denotes partial differentiation relative to the p-th coordinate. 

Here we assume thatWdepend onrM only throughr M so that, with an abuse of notation, the additive free energyW takes the form

W

¼

W

ðh; M;

r

 MÞ:

As a consequence, owing toLemma 1, we have

dM

W

^ :¼ hð ^

W

Mþ

r

 ^

W

rMÞ; ð4:14Þ

while the boundary condition(3.6)holds if

ð _M  ^

W

rMÞ  n ¼ _M  ð ^

W

rM nÞ ¼ 0 on @

X

: ð4:15Þ Remark 3. By virtue ofLemma 1, when ^Wdepends onrM only throughr M the stationary equation becomes

hð ^

W

Mþ

r

 ^

W

rMÞ ¼

l

0H

subject to the boundary condition ^

4.1. An example of free-energy function

Quite a general (rescaled) free-energy function ^W¼W=his given by ^

W

¼ Fðh; jMjÞ þ1

2½c1M 

r

 M þ c2j

r

 Mj 2

þ c3jM  ej2;

where F is a non-convex function of jMj, c1and c2are constants, c3is parameterized by h and e is a possibly-privileged unit

direction. Hence we have dM

W

^ ¼ h

1

jMjFjMjM þ c1

r

 M þ c2

r



r

 M þ c3ðhÞðM  eÞe

 

: ð4:16Þ

If the material is isotropic then c3¼ 0. The term M r M in the free energy is considered for the sake of generality but does

not seem to be motivated on physical grounds.

Landau et al.[2]regardWas though c1;c3¼ 0. Their position amounts to assuming that K ¼

a

1 and that

Fðh; jMjÞ ¼1 2a1ðh  hcÞM 2 þ1 4a2M 4_; where a1and a2are allowed to depend on h.

We now exhibit a three-dimensional setting and letWdepend on M, through a logarithmic function of M2. This depen-dence is motivated by an investigation of one-dimensional potentials which is shown in Section8.

5. Evolution in the three-dimensional space

The whole evolution of the system is described by the equation for _h (balance of energy) in addition to that for _M. We now set up the corresponding scheme without any restriction on the amplitude and the direction of M. Later on, we investigate the case when the system is saturated (jMj ¼ constant) or the direction of M is fixed in space (M=jMj ¼ constant). However, to avoid formal difficulties, we restrict attention to isotropic materials so that the free energy depends onrM through jrMj2. For definiteness, the pertinent coefficient is taken to be proportional to the temperature h. Though with more in-volved formulae, the anisotropic case might be considered by following along the same lines.

Let the free energyWbe given by

W

¼ Gðh; MÞ þ1 2

j

hj

r

Mj 2 ; ð5:1Þ where Gðh; MÞ ¼ gðhÞ  bðuðhÞ þ 1Þ lnð1  M2=M2sðhÞÞ  bM 2 =M2sðhÞ ð5:2Þ

and, for simplicity, uðhÞ takes the classical form u ¼ ðh  hcÞ=hc.

The function MsðhÞ models the dependence of Mson h as is the case for the relation

MsðhÞ ¼ Msð0Þð1  BhbÞ ð5:3Þ

which is often referred to as Bloch’s law (see[18,19]). Here B is the Bloch constant and is obviously dependent on the mate-rial. The Bloch exponent b equals 3=2 for bulk materials but equals roughly 1=2 for some nanoparticle specimens[20].

We let K ¼

a

1 so that the evolution equation(4.6)applies with

a

1¼

a

2¼

a

. Moreover, let

s

1;

s

2¼ 0, namely



;b;

m

¼ 0.

Eq.(4.6)then reduces to _ M ¼ 

a

ðdM

W

^

l

0HÞ: Hence, by(5.1)we have _ M ¼

al

₀H  2

a

b ðu þ M 2 =M2sÞ Msð1  M2=M2sÞ M Ms 

aj

h

D

M: ð5:4Þ

As a check on the validity of the evolution equation(5.4)we restrict attention to stationary ( _M ¼ 0) and uniform (DM ¼ 0) conditions. Eq.(5.4)gives

l

0H ¼ 2b u þ M2_=M2 s Msð1  M2=M2sÞ M Ms : ð5:5Þ

As a consequence M and H are collinear. Letting M; H be the components in the common direction we find that the differ-ential susceptibility

v

¼dM dH

is a function of M=Ms¼ n namely

v

¼

l

0M 2 s 2b ð1  n2_Þ2 u þ ðu þ 3Þn2_{ n}4; n2 ½0; 1: ð5:6Þ

If h > hcthen u > 0. Hence,

v

>0, in that 3n2>n4, and the material is paramagnetic. Also, for small values of n we have

v

’

l

0M 2 s 2b hc h hc ; ð5:7Þ

whence the differential susceptibility

v

varies with h as ðh  hcÞ1, which is the content of the Curie–Weiss law (see[2]).

If h < hcthen, by(5.7),

v

<0 for small values of n whereas, by(5.6),

v

>0 as n2approaches 1 or the body is almost

sat-urated. This is the typical behaviour of ferromagnetic materials. Again, the differential susceptibility varies with h as ðh  hcÞ1for small values of n. If h ! 0 then u ! 1 and(5.4)reduces to

_ M ¼

al

0H þ 2

a

b M2 s M:

Similar, though more involved, models and conclusions are obtained by letting u be given, e.g., as in Section8.2. It is of interest to look at the function HðMÞ, or HðnÞ, as h > hcor h < hc. By(5.5)we have

l

0Ms 2b H ¼

ðu þ n2Þn

1  n2 : ð5:8Þ

Fig. 1shows the right-hand side of(5.8)as h ¼ 1:2hcand h ¼ :8hc.

The evolution of the material is completed by accounting also for the balance of energy. Now, by(5.1) and (5.2)we have

g

¼ g0_þb hc lnð1  M2_=M2 sÞ þ 2 bM2 M3 s u þ M2=M2s 1  M2_=M2 s M0 s 1 2

j

j

r

Mj 2 ; e ¼ w þ h

g

¼1 2ð

l

0H 2 þ



0E2Þ þ g  hg0þ lðh; M; Ms;M0sÞ;

where a prime stands for the derivative with respect to the temperature h and lðh; M; Ms;M0sÞ ¼ b M2 M2 s þ 2bhM 2 M3 s ½h  hcð1  M2=M2sÞM 2 hcM3sð1  M 2_=M2 sÞ M0s: The balance of energy in the form(2.4)then yields

½hg00_{þ l} hþ lMsM 0 sþ lM0 sM 00 s _h þ ðlM

l

0HÞ  _M  J  E þ

r

 q  r ¼ 0; ð5:9Þ where J and q are to be viewed as given by the pertinent constitutive functions and r is a possible given function (heat sup-ply). Eqs.(5.4) and (5.9)constitute the system of evolution equations for the two fields hðx; tÞ; Mðx; tÞ.

Remark 4. Ferrimagnetic materials, like ferromagnets, hold a spontaneous magnetization below the Curie temperature hc and are paramagnetic above hc. However, the amount of spontaneous magnetization in ferrimagnetic materials, such as ferrites and magnetic garnets, is smaller than in ferromagnets. This is so because a ferrimagnetic material consists of

different sublattices with opposed but unequal magnetic moment, whereas in antiferromagnetic materials the magnetic moments of the two sublattices are equal and opposed. For instance, in ferrite the sublattices are given by two families of ions, Fe2þand Fe3þ. In addition, in magnetic garnets there is a temperature below hc, called magnetic compensation point, at which the spontaneous magnetization vanishes. They also exhibit a third critical temperature corresponding to the angular momentum compensation point[21]. The modelling of such materials requires the occurrence of two distinct vector phase variables, say M1and M2, each obeying an evolution law like(4.11).

6. Evolution of direction and amplitude

Also for a more direct comparison with some models of magnetization in matter, we now examine two representations of the field M. Let Ms>0 be the saturation value of magnetization. We can write

Mðx; tÞ ¼ Ms/ðx; tÞmðx; tÞ; /2 ½0; 1; jmj ¼ 1: ð6:1Þ

Hence, m is the unit vector of M. The saturation magnetization Msis here regarded as a (temperature-independent) constant.

As shown by measurements on nanoparticles (see[20]), the approximation is reasonable in many circumstances. In saturation conditions,

Mðx; tÞ ¼ Msmðx; tÞ; /¼ 1:

If the magnetization has a constant direction we can represent M as Mðx; tÞ ¼ Msnðx; tÞe; n2 ½1; 1; jej ¼ 1;

where e is the fixed unit vector. More generally, for a variable direction we can write

Mðx; tÞ ¼ Msnðx; tÞrðx; tÞ; n2 ½1; 1; r 2 U; ð6:2Þ

where U is a solid cone,

U ¼ fr : jrj ¼ 1; r 2 U ) r R Ug:

If U is the set of possible directions then the two representations are equivalent by letting /¼ jnj; r ¼ ðsgn nÞm:

We now proceed by representing M as Msjnj m, regarding n and m as independent quantities and looking for the separate

evolution equations _

m ¼ ^mð

C

Þ; _n ¼ ^nð

C

Þ: By the representation(6.2)we have

_ M ¼ Msð _nr þ jnj _mÞ;

r

M ¼ Msð

r

n r þ jnj

r

mÞ; jnj _m ¼ n_r; jnj

r

m ¼ n

r

r; and hence ^

W

n¼ Msð ^

W

M r þ ^

W

rM rÞ;

W

^rn¼ Ms

W

^rMr; ^

W

m¼ Msð ^

W

Mjnj þ ^

W

rM

r

jnjÞ;

W

^rm¼ Ms

W

^rMjnj; and _ M  dM

W

^ ¼ Msðr  dM

W

^Þ _n þ MsðjnjdM

W

^Þ  _m: Now, Msr  dM

W

^ ¼ Mshð ^

W

M

r

 ^

W

rMÞ  r ¼ h½ ^

W

n

r

 ðMs

W

^rMrÞ ¼ hð ^

W

n

r

 ^

W

rnÞ ¼: dn

W

^; MsjnjdM

W

^ ¼ Mshð ^

W

M

r

 ^

W

rMÞjnj ¼ h½ ^

W

m

r

 ðMs

W

^rMjnjÞ ¼ hð ^

W

m

r

 ^

W

rmÞ ¼: dm

W

^: Hence, _ M  dM

W

^ ¼ dn

W

^^nþ dm

W

^ ^m and(3.9)becomes ðdn

W

^

l

0MsH  rÞ^n þ ðdm

W

^ 

l

0MsjnjHÞ  ^m 6 0: ð6:3Þ Since nr ¼ jnjm then ndn

W

^ ¼ m  dm

W

^:

In view ofProposition 1, a sufficient condition for the validity of(6.3)is that the evolution functions ^nand ^m take the form ^

n¼ 

x

ðdn

W

^

l

0MsH  rÞ; ð6:4Þ

^

m ¼ 

a

1Nk 

a

2N?þ bm  N þ 

m

m  ðm  NÞ; ð6:5Þ where the decomposition Nk, N?is relative to m and

N ¼ M þ 



m  _m; M ¼ dm

W

^ 

l

0MsjnjH: ð6:6Þ

Also,

x

; 

a

1; 

a

2; 

m

are non-negative valued functions ofCand b; 



are real valued. Because jmj ¼ 1, ^m satisfies the constraint

_

m  m ¼ ^m  m ¼ 0

and hence it follows from(6.5)that 0 ¼ 

a

1Nk m:

This condition holds by merely requiring that 

a

1¼ 0. Hence, letting 

a

2¼ 

a

we can write(6.5)in the form

^

m ¼ 

a

½ð1  m  mÞM þ 



m  _m þ m  ðbN þ 

m

m  NÞ: ð6:7Þ By means of(6.7)we can now prove the following statement.

Proposition 4. If n – 0 then the evolution equation for m can be expressed in terms of M in the form _ m ¼ 

c

m  M þ km  ðm  M Þ; ð6:8Þ where 

c

¼  bþ 



½b2_{þ ð}

_a

_{þ }

_m

_Þ2  ð1 þ 



_bÞ2 þ 



2_ð

_a

_{þ }

_m

_Þ2;  k¼ 

a

þ 

m

ð1 þ 



_bÞ2 þ 



2_ð

_a

_{þ }

_m

_Þ2: Proof. Since jmj ¼ 1 then m  _m ¼ 0, j _mj ¼ jm  _mj, and

m  ðm  _mÞ ¼  _m: Also,

ð1  m  mÞM ¼ m  ðm  M Þ:

As a consequence, by(6.7)the evolution equation becomes

ð1 þ 



_{bÞ _m ¼ }

_

_ð

_a

_{þ }

_m

_{Þm  _m þ bm  M þ ð}

_a

_{þ }

_m

_{Þm  ðm  M Þ: ð6:9Þ} Apply m to(6.9), derive the expression of m  _m and then replace it in(6.9). Upon some algebraic rearrangements we ob-tain(6.8). 

As expected, the evolution equation(4.4)follows from(6.4) and (6.7)once we make appropriate identifications. They are given by

x

¼

a

1 M2s ;

a

¼

a

2 M2 sjnj 2; b¼ b Msjnj ; 

m

¼

m

; 

_

_¼

_

_M3 sjnj 3

while K has the form(4.5). At saturation (jnj ¼ 1), the constancy of

a

1;

a

2;b;

m

;



is equivalent to that of

x

; 

a

; b; 

m

; 



.

As we shall see in a moment, the term dmW^, which occurs in M , accounts for exchange and anisotropic interactions. If

such effects are neglected then(6.8)reduces to _

m ¼ 

c

0m  H  k0m  ðm  HÞ; ð6:10Þ

where

c

0¼

l

0Msjnj

c

; k0¼

l

0Msjnjk:

Two particular cases are of interest, namely the magnetic saturation, which occurs if jnj ¼ 1, and the fixed alignment of M, which means that the magnetic direction r ¼ sgn n m is constant. They are now examined by allowing for non-isothermal conditions and assuming that the temperature is below the critical temperature hc.

6.1. Magnetic saturation

In magnetically-saturated bodies, jnj ¼ 1, and hence M ¼ dm

W

^

l

0MsH:

Also, dmW^¼ MsdMW^ so that M ¼ MsðdM

W

^ 

l

0HÞ ¼: 

l

0MsHeff; which ascribes to Heff¼ H  1

l

0 dM

W

^ ð6:11Þ

the role of effective magnetic field.

Let h < hcbe constant in space and time. We look for equilibrium configurations by applying the evolution equation(6.8).

If _M ¼ 0, that is _m ¼ 0, then we have 

c

m  M þ km  ðm  M Þ ¼ 0: Hence the equilibrium condition becomes

l

0Msm  Heff¼ 0; ð6:12Þ

where Heff is the effective field given by(6.11). Eq.(6.12)traces back to Brown[5]and ascribes to- dMW^=

l

0the meaning of

the field arising from exchange and anistropic interactions. Indeed, the standard form of Heff ([5, p. 48])

Heff¼ H  1

l

0Ms fmþ 2

l

0Ms

r

 ðA

r

mÞ; follows from(3.11) and (6.11)by letting

W

¼ f ðMÞ þ

r

M  A

r

M;

where A is a symmetric fourth-order tensor. Let f depend on M in the anisotropic form

f ðMÞ ¼1

2

l

0M  QM where Q 2 Symþ_{. Hence we have}

Heff¼ H  MsQm þ 2

l

0

Ms

r

 ðA

r

mÞ: ð6:13Þ

The last term, ð2=

l

0ÞMsr ðArmÞ, accounts for exchange interactions and penalizes magnetization inhomogeneities. If the

exchange interaction is isotropic in space (for instance such is the case for a cubic cell, see[2]) then A ¼ ð

l

0A=2Þ1, where A is

the so-called exchange constant which depends on the lattice geometry (body-centred, face-centred cubic crystals). Denote by Dx;Dy;Dz and ex;ey;ezthe eigenvalues and the eigenvectors of Q. Hence we can express f as a function of

Mx¼ Msmx;My¼ Msmy;Mz¼ Msmz, the components of M relative to the eigenvector basis, in the form

f ðMÞ ¼1 2

l

0M

2

sðDxm2xþ Dym2yþ Dzm2zÞ;

where the coefficients Dx;Dy;Dz>0 characterize the anisotropy of the body. Of course, m2xþ m2yþ m2z¼ 1. If Dx;Dy;Dzare

dis-tinct values then the energy term f accounts for biaxial anisotropy, f ðMÞ ¼1

2

l

0M 2

s½Dzþ ðDx DzÞmx2þ ðDy DzÞm2y:

If, instead, Dx¼ Dy¼ D?and Dz–D?then f accounts for uniaxial anisotropy with easy direction ez,

f ðMÞ ¼1 2

l

0M

2

s½Dzþ ðD? DzÞð1  m2zÞ:

In such a case the body is transversely isotropic (relative to the z-axis) and the effective field takes the form Heff¼ H  MsðDzmzezþ D?m?Þ;

where mz¼ m  ezand m?¼ m  mzez. In particular, for strongly anisotropic materials D?is negligible with respect to Dzand

hence

Heff’ H  DzðM  ezÞez: ð6:14Þ

In transversely-isotropic materials the evolution equation(6.8)accounts also for the damping switching[6], namely the switching of a component, say mz, generated by the z-component of the magnetic field opposite to the initial value mzð0Þ.

Finally, in isotropic bodies Dx¼ Dy¼ Dz¼ D > 0 and A ¼ ð

l

0A=2Þ1 so that f simplifies to the constant

f ðMÞ ¼1 2

l

0M

2 sD

Heff¼ H þ MsA

D

m ¼ H þ A

D

M:

Accordingly, when the body is magnetically isotropic and subject to uniform fields (rm ¼ 0) the effective field reduces merely to the applied field H.

Section7is devoted to the relation of these particular cases with known models appeared in the literature.

6.2. Fixed alignment

If the direction e ¼ ðsgn nÞm of M is constant, say r ¼ e, then M ¼ Msne and _M ¼ Ms_ne. Accordingly, m is constant except

when sgn n changes. Such is the case usually assumed for an isotropic magnet in a magnetic field H with constant direction. Indeed, because dmW^¼ ðsgn nÞdrW^, by(6.8)we have

0 ¼ 

c

r  ðdr

W

^

l

0MsnHÞ þ ksgn n r  ½r  ðdr

W

^

l

0MsnHÞ whence

r  ðdr

W

^

l

0MsnHÞ ¼ 0:

As a consequence, when anisotropic and exchange interactions are neglected (drW^ ¼ 0) we have r  H ¼ 0, which means that

H and M are parallel (one-dimensional setting). The same result holds also for magnets with strong uniaxial anisotropy pro-vided that the direction of the applied field H coincides with the easy direction of the body. If such is the case then it follows from(6.14)that Heff is approximately parallel to r. In addition, by(6.4)we have

_n ¼ 

x

ðdn

W

^

l

0HÞ;

x

>0; ð6:15Þ

where H ¼ MsH  r, which governs the evolution of the (magnetic intensity) component n. If, as a special case,

^

W

¼1 2aðh  hcÞjnj 2 þ1 4bjnj 4 then(6.15)becomes _n ¼ 

x

½aðh  hcÞn þ bn3

l

0H

which is in fact the non-isothermal model by Landau et al.[2].

In Section8known models of free energy are shown to follow as particular cases and, morevorer, a new free energy is established.

7. Relation to the Gilbert–Landau–Lifshitz model

It is natural to contrast(6.8)with known evolution equations of micromagnetics which apply to magnetically-saturated bodies. Landau and Lifshitz[1]started with the evolution equation

_l¼ 

c

l H

for a magnetic spin momentum l of an electron in a magnetic field H,

c

>0 being the absolute value of the gyromagnetic ratio. In the continuum limit they wrote

_

M ¼ 

c

M  H

where H is the magnetic field in matter. Dissipation was modelled by adding a torque which pushes M toward the field H. The torque was taken in the form KM  ðM  HÞ, whereK>0, so that

_

M ¼ 

c

M  H 

K

M  ðM  HÞ ð7:1Þ

Later on, Gilbert[3,4]wrote the evolution equation with a dissipative term TM  _M, T > 0, in the form _

M ¼ 

c

M  H þ TM  _M: ð7:2Þ

In both cases, taking the inner product with M gives

M  _M ¼ 0; ð7:3Þ

which means that jMj is constant in time. Hence(7.1)and(7.2)apply to magnetically-saturated bodies. Since jMj is constant, we can replace M with Msm in(7.1), (7.2)to get

_

m ¼ 

c

m  H  km  ðm  HÞ; ð7:4Þ

_

m ¼ ~

c

m  H þ

s

m  _m; ð7:5Þ

where k ¼KMsand

s

¼ TMs,

s

being the dimensionless Gilbert damping constant. Eqs.(7.4) and (7.5)are formally

_ m ¼ 

c

_{m  H  k}_{m  ðm  HÞ; ð7:6Þ} where

c

_¼ ~

c

1 þ

s

2; k  ¼

s

~

c

1 þ

s

2:

Incidentally, comparison of(7.5) and (7.6)shows that

s

¼ k=

c

and ~

c

¼ ð

c

2_{þ k}2

Þ=

c

. This in turn is consistent with the fact that

s

and k are dissipation coefficients. If k ! 0 then

s

! 0 and ~

c

!

c

so that(7.4) and (7.5)reduce to _

m ¼ 

c

m  H:

Remark 5. Let H be constant. By(7.4)we have d

dtðm  HÞ ¼ kjm  Hj 2

P0:

The component of m along H increases until m and H are collinear. Hence the Landau–Lifshitz equation(7.1), in the approx-imation of a constant magnetic field, models a magnetization M, with constant jMj, that tends to orient itself parallel to H. In addition, letting eHbe the unit vector of H, so that H ¼ jHjeH, we have

d

dtðm  eHÞ ¼ kjm  eHj 2

jHj;

which shows that the rate dðm  eHÞ=dt is proportional to jHj.

Recently more involved evolution equations have been considered by replacing H, in(7.5), with an effective magnetic field Heff. A generalization of the evolution equation(7.5)is performed in[8](see also[9]) by replacing H with

Heff¼ f

D

m þ

g

ðe  mÞe þ H; ð7:7Þ

where e is the unit vector of the easy axis of magnetization, H is the sum of the stray field and of the external field (see also

[7,11]). The model(7.7)is a particular case of(6.13)as it follows by letting e ¼ ez,

g

¼ MsDz, MsA ¼1₂

l

0f1. In such a case the

(Gilbert) evolution equation takes the form _

m 

s

m  _m ¼ 

c

m  ½f

D

m þ

g

ðm  eÞe þ H; ð7:8Þ

which reduces to(6.12)in stationary conditions. As a check we see that m  _m ¼ 0. Hence we can write(7.8)also in the form _

m ¼

c

_{fm  ½f}

_D

_{m þ}

_g

_{ðm  eÞe þ H þ}

_s

_{m  ½m  ðf}

_D

_{m þ}

_g

_{ðm  eÞe þ HÞg: ð7:9Þ}

The evolution equations(7.4), (7.5) and (7.8)are particular cases of(6.8) and (6.10)and hence are compatible with ther-modynamics. Eq.(7.4)coincides with(6.10)once the identifications k ¼ k0and

c

¼

c

0are made.

The same identifications hold for the Gilbert equation(7.6)with

c

_{and k}_{in place of}

_c

0and k0. The generalized Gilbert

equation, in the form(7.9), follows from(6.8)by letting the temperature h be constant, the rescaled free energy take the form

^

W

¼

g

2

l

0 jM  ej2 f 2

l

0 j

r

 Mj2 and

c

_¼

_l

0Msjnj

c

;

s

¼ k=

c

:

8. Potentials for one-dimensional models

Henceforth we look for appropriate functionsWto model ferromagnetism and the associated phase transition. For sim-plicity we restrict attention to the one-dimensional case and apply(4.4)by letting b;

c

¼ 0. Denote by M; H 2 R the signifi-cant components of M; H. Eq.(4.4)for M, likewise(6.15)for n ¼ M=Ms, then becomes

_n ¼ 

a

½hð ^

W

n

r

 ^

W

rnÞ  h; ð8:1Þ

where h ¼

l

0H. In stationary conditions, n ¼ constant, we have

h ¼ hð ^

W

n

r

 ^

W

rnÞ: ð8:2Þ Also we let

W

ðh; n;

r

nÞ ¼1 2chj

r

nj 2 þ Vðn; uÞ;

where c is a positive constant and u is a suitable increasing function, of the temperature h, which vanishes at the Curie point hc. The function V is often referred to as the potential. Hence(8.1) and (8.2)become

_n ¼ 

a

ðVn h  ch

D

nÞ; ð8:3Þ

h ¼ Vn ch

D

n: ð8:4Þ

This setting allows us to obtain an immediate connection with other approaches. For instance, a standard free energy ap-plied in the literature (see, e.g.,[12]) corresponds to letting u ¼ ðh  hcÞ=hcand

V ¼ n2

½aðh  hcÞ þ bn2; ð8:5Þ

where a; b are positive constants. In such a case, Eqs.(8.3) and (8.4)become

_n ¼ 2

a

½aðh  hcÞ þ 2bn2n þ

a

h þ 1 2

a

ch

D

n; and

h ¼ ch

D

nþ ½2aðh  hcÞ þ 4bn2n: ð8:6Þ

The potential(8.5)traces back to Ginzburg[2]. The function e

U

¼ Vðn; uÞ  hn

is considered in[22], the potential eUbeing identified with the Lagrangian density.

Equilibrium conditions are sought for homogeneous configurations,rn¼ 0. Hence the equilibrium conditions are the sta-tionary points of the potential. The potential(8.5), as well as similar ones in the literature, does not provide a reasonable set of equilibrium values and a satisfactory scheme for phase transition which occurs when u ¼ 0. Indeed, it is a well-known drawback of the potential(8.5)that it does not allow for saturation. By(8.6), for large values of h in homogeneous config-urations we have

n’ ð4bÞ1=3h2=3h;

as though the material had a permittivity proportional to h2=3_{. This motivates the search for schemes where the saturation is}

allowed.

8.1. Weiss model of ferromagnetism

We now review the Weiss model of ferromagnetism (see, e.g.,[23]and refs therein) and show how it allows for a poten-tial VWðh; nÞ.

Let L be the Langevin function, on R, defined by LðxÞ ¼ coth x 1

x:

The function L is strictly increasing and odd and moreover LðxÞ ! 1 as x ! 1. As a consequence the inverse L1_maps

ð1; 1Þ into R. Weiss model relates the magnetization M and the magnetic field H in the form H þ bM

l

h ¼ L 1_ðM=M

sÞ ð8:7Þ

where Msis the maximum value of the magnetization, that is M 2 ðMs;MsÞ, b is the molecular-field parameter, and

l

¼ k=m

is the ratio of Boltzmann’s constant over the magnitude of the (atomic) magnetic moment. By(8.7)we have

H ¼

l

hL1ðM=MsÞ  bM: ð8:8Þ

Letting n ¼ M=Mswe can write H ¼ HðnÞ, where H is defined by

HðnÞ ¼

l

hL1ðnÞ  bMsn; n2 ð1; 1Þ: If h ! 0þthen

H ¼ bM;

which is the classical law of diamagnetism or paramagnetism according as b > 0 or b < 0. Now, ferromagnetism indicates that there is a temperature hcsuch that

H0ð0Þ > 0 as h > hc; H0ð0Þ 2 ðbMs;0Þ as h 2 ð0; hcÞ: Consistently, it is expected that H0_{ð0Þ ¼ 0 at h ¼ h}

c, namely

The requirement(8.9)induces a relation between b;

l

, and hc. In this connection, letting z ¼H þ bM

l

h we have ðL1_Þ0 ðnÞ ¼ 1 L0ðzðnÞÞ:

Since L1_{ð0Þ ¼ 0 then zð0Þ ¼ 0. Moreover, L0ð0Þ ¼ 1=3. As a consequence}

ðL1_Þ0 ð0Þ ¼ 1 L0ð0Þ¼ 3: Hence(8.9)provides hc¼ bMs 3

l

:

This means that the parameter

l

is related to the transition temperature hcby

l

¼bMs 3hc :

Upon substitution we can write Eq.(8.8)in the form 1

bMs

H ¼u þ 1 3 L

1_{ðnÞ  n ð8:10Þ}

where u ¼ ðh  hcÞ=hc. Eq.(8.10)provides the value of the magnetic field H in terms of the magnetization M, in that n ¼ M=Ms.

The equilibrium(8.10)may be viewed as the stationary condition for the potential VW such that

V0 WðnÞ ¼ bMs u þ 1 3 L 1 ðnÞ  bMsn H: The integration gives

VWðnÞ ¼ bMs u þ 1 3

K

ðnÞ  1 2bMsn 2  Hn; ð8:11Þ

whereKis the integral of L1_.

To evaluate the residual magnetization Mrwe can go back to(8.10)and require that the right-hand side vanishes at Mr.

We find that nr¼ Mr=Mssatisfies

nr¼ LðdnrÞ; d¼ 3hc

h : ð8:12Þ

To obtain the coercive field Hc, by(8.10)we look for the maximum of the right-hand side. We find that the coercive field Hc,

at nc¼ Mc=Ms, is given by L0ðL1_ðn cÞÞ ¼ 1 d; Hc¼ bMs½ 1 dL 1_ðn cÞ  nc: ð8:13Þ

In essence, the advantage of the Weiss description is that the saturation effect is allowed and the condition nr! 1, as

h! 0, holds. However, both the residual magnetization Mrand the coercive field Hcare provided in an implicit way, through

(8.12) and (8.13).

8.2. A logarithmic potential

Still we let Ms be the saturation value of the magnetization, M 2 ðMs;MsÞ, and n ¼ M=Ms2 ð1; 1Þ while h ¼

l

0H. We

denote by nrthe residual or remnant (relative) magnetization and by hcthe coercive field. By definition, nris the (positive)

value of n at h ¼ 0 whereas hcis the local maximum of h.

We now examine the properties of the magnetization curve associated with the Lagrangian density e

U

¼ bðu þ 1Þ lnð1  n2Þ  bn2 hn; ð8:14Þ

where b is a positive constant and u is a function of h such that uðhÞ 2 ð1; 0Þ as h 2 ð0; hcÞ; uð0Þ ¼ 1; uðhcÞ ¼ 0:

The potential

VlnðnÞ ¼ e

U

ðnÞ þ hn ¼ bðu þ 1Þ lnð1  n2Þ  bn2

is similar to the Weiss potential VWin(8.11). In this sense bðu þ 1Þ lnð1  n2Þ replaces bMsKðu þ 1Þ=3.

For definiteness we can take uðhÞ in the form uðhÞ ¼  1  h hc  q  p ½1 þ a h hc  q p; a 2 ð0; 1 ð8:15Þ and p; q 2 Q. Since u0_{¼ pqh}q1_hq c ½1  a þ 2aðh=hcÞ q f½1  ðh=hcÞ q ½1 þ aðh=hcÞ q gp1

it is apparent that u is a monotonic, increasing function of h as h 2 ½0; hc. Furthermore, it is increasing as h 2 Rþif p ¼ 1. The

standard form of u, namely u ¼ ðh  hcÞ=hc, is a particular case of(8.15)that corresponds to a ¼ 0, p ¼ q ¼ 1 or to a ¼ p ¼ 1,

q ¼ 1=2.

In stationary conditions ð _n ¼ 0Þ Eq.(8.1)amounts to the vanishing of eUnwhence

hðnÞ ¼ 2bðu þ 1Þ n

1  n2 2bn: ð8:16Þ

Let h 2 ð0; hcÞ and hence u 2 ð1; 0Þ. As h ¼ 0, Eq.(8.16)provides the three solutions

n¼ 0; n¼ pffiffiffiffiffiffijuj; n¼pffiffiffiffiffiffijuj

and hence the residual magnetization nris given by

nr¼ ffiffiffiffiffiffi juj p

: ð8:17Þ

Look at the maximum and the minimum of hðnÞ. Observe that 1 2bh 0 ¼u þ ðu þ 3Þn 2  n4 ð1  n2_Þ2 : Since u 2 ð1; 0Þ then h0 vanishes if

n4 ð3  jujÞn2þ juj ¼ 0:

There are then four solutions for n. Two of them, ni;ni, such that

n2i ¼ 1 2ð3  juj  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3  jujÞ2 4juj q Þ;

are found to belong to ð1; 1Þ. The remaining two solutions, no, no, such that

n2o¼ 1 2ð3  juj þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3  jujÞ2 4juj q Þ; are found to be outside ½1; 1. Hence we have

hc¼ 2bni 1  n2i

ðn2i  jujÞ: ð8:18Þ

Borrowing e.g. from[24], we say that the function nrðhÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffi juðhÞj p

; h2 ð0; hcÞ;

is concave, monotonic decreasing and subject to nrð0Þ ¼ 1, nrðhcÞ ¼ 0. Now, by(8.15)we see that nrð0Þ ¼ 1, nrðhcÞ ¼ 0 and,

moreover, a 6 1 makes sgn u constant. We now ascertain monotonicity and concavity of nrðhÞ. Letting x ¼ h=hcwe have

ffiffiffiffiffiffi juj p

¼ sðxÞ :¼ ½ð1  xq_{Þð1 þ ax}q_Þp=2 :

Upon evaluation of s0_{and s}00_{we conclude that monotonicity holds for p; q P 0 and that concavity holds if and only if a ¼ 1 }

_

and q > 1; p 6 2, for small values of



, or a ¼ 1 and q > 1=2; p 6 2. Remark 6. If

p ¼ 2; q ¼3 4; we have

Since q ¼ 3=4 is allowed only if a ¼ 1 then we have

nr¼ 1  ðh=hcÞ3=2; ð8:19Þ

which is exactly Bloch’s law (see (71.7) of[25]or[18]) for the dependence of the spontaneous magnetization on the tem-perature. The standard choice

u ¼ ðh  hcÞ=hc ð8:20Þ

corresponds to a ¼ 0; p ¼ q ¼ 1 or to a ¼ p ¼ 1; q ¼ 1=2, which are limit cases of(8.15).

Remark 7. At the absolute zero, u ¼ 1 and the function hðnÞ shows a crucial behaviour. For, as u ¼ 1 we have n2i ¼ n

2 o¼ 1

and hence the magnetization curve hðnÞ degenerates into three straight lines, n¼ 1; n¼ 1; h ¼ 2an:

As h > hcwe have u > 0 and hence, as expected, h vanishes only if n ¼ 0.

In conclusion, the logarithmic potential Vlnprovides a good approximation of VWaround n ¼ 0, allows for the saturation

effect, yields the residual magnetization and the coercive field in an explicit way and provides a concave function nrðhÞ.

8.3. The logarithmic potential for crystals of iron

For definiteness we look for the particular function u of(8.15)for crystals of iron. Although iron may be viewed as fer-rimagnetic, we apply the ferromagnetic model since no compensation point occurs.

Our purpose is to determine the parameters a; p; q by means of the experimental data for iron (see[24]) namely the curve of the residual magnetization nrversus the absolute temperature h.

Let p; q > 0. First we observe that by s0_{ðxÞ ¼}1 2pq½1 þ ða  1Þx q þ ax2qðp2Þ=2xq1_{ða  1  2ax}q Þ we have s0_{ðxÞ ! 1 or s}0_{ðxÞ ! 0;}

as x ! 1, according as p < 2 or p > 2. Because the experimental data show that s0ð1Þ ¼ 1 we find a further reason for the

condition p < 2.

Assume p; q are fixed, p 2 ð0; 2Þ. Since s2=p

ðxÞ ¼ 1  xqþ aðxq x2qÞ; letting

y ¼ s2=p_{; zðxÞ ¼ 1  x}q_;

_v

_{ðxÞ ¼ x}q_{ x}2q

we can write

yðxÞ ¼ zðxÞ þ a

v

ðxÞ:

The linear dependence of y on a allows us to find a as the least squares solution.

Otherwise, once a set of points ðxi;siÞ are given, the joint derivation of the parameters a; p; q may be performed e.g. by the

program PLOT (MacOSX).Fig. 2shows the dots extracted from Bozorth[24]. The corresponding parameters turn out to be p ¼ 0:6838; q ¼ 1:7298; a ¼ 0:5777: The curve gives the corresponding function sðxÞ.

As a check of consistency, if we let p ¼ 0:6838; q ¼ 1:7298 and evaluate the corresponding optimal value of a we find that a ¼ 0:568. If, instead, we take q ¼ 1:7298; a ¼ 0:5777 then we find that p ¼ 0:688. In both cases the values (of a or p) are very close to those given by the joint derivation.

9. Conclusions

The thermodynamic analysis of the modelling of magnetic materials is based on h; E, H, M,rh;rM;rrM as the set of independent variables and on the statement of the second law through an inequality involving an extra entropy flux k. We have found that the second law inequality is satisfied if hk ¼WrMM and __ M is subject to(3.10).

The evolution equation is governed by the rescaled free energyWðh; M;rMÞ=h. We find that the logarithmic function

(8.14), with a < b, has two equilibrium solutions. The corresponding values of the order parameter is (1) times the residual magnetization. The choice(5.1)forWwhere G has the logarithmic form(5.2)provides the model for the paramagnetic–fer-romagnetic transition. Also, the Curie–Weiss law is obtained in the approximation of small values of M.

Evolution equations arisen within micromagnetics have been considered, namely the Landau–Lifshitz equation(7.1), the Gilbert equation(7.2)and recent improvements(7.8). The generality of the present thermodynamic approach allows us to frame also the micromagnetic equations as evolution equations for M, generated by an appropriate form of the free energy potential, as a particular form of(4.4).

It is a positive feature of the present approach that the same logarithmic potential(5.2)provides a satisfactory description of both the ferromagnetic (h < hc) and the paramagnetic (h > hc) behaviour and fits the curve of residual magnetization, as a

function of the absolute temperature, for crystals of iron. Acknowledgement

The research leading to this paper has been supported by the Italian MIUR through the Project PRIN 2005 ‘‘Mathematical models and methods in continuum physics”.

References

[1] L.D. Landau, E.M. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjet. 8 (1935) 153–169. [2] L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii, Electrodynamics of Continuous Media, Pergamon, Oxford, 1984. Section 39.

[3] T.L. Gilbert, A lagrangian formulation of the gyromagnetic equation of the magnetization fields, Phys. Rev. 100 (1956) 1243 [Abstract only: full report, Armor Research Foundation Project No. A059, Supplementary Report, May 1, 1956.

[4] T.L. Gilbert, A phenomenological theory of damping in ferromagnetic materials, IEEE Trans. Magn. 40 (2004) 3443–3449. [5] W.F. Brown Jr., Micromagnetics, Interscience Publishers, Berlin, 1963.

[6] J.C. Mallinson, Damped gyromagnetic switching, IEEE Trans. Magn. 36 (2000) 1976–1981.

[7] M. Bertsch, P. Podio Guidugli, V. Valente, On the dynamics of deformable ferromagnets. I. Global weak solutions for soft ferromagnets at rest, Ann. Mater. Pura Appl. 174 (2001) 331–360.

[8] P. Podio Guidugli, G. Tomassetti, On the evolution of domain walls in hard ferromagnets, SIAM J. Appl. Math. 64 (2004) 1887–1906. [9] C. Melcher, Domain wall motion in ferromagnetic layers, Physica D 192 (2004) 249–264.

[10] G.A. Maugin, Vectorial internal variables in magnetoelasticity, J. Mécanique 18 (1979) 541–563.

[11] G.A. Maugin, A. Fomethe, Phase-transition fronts in deformable ferromagnets, Meccanica 32 (1997) 347–362. [12] A. Kovetz, Electromagnetic Theory, Oxford University Press, 2000.

[13] C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1996.

[14] M. Fabrizio, C. Giorgi, A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics, Physica D 214 (2006) 144– 156.

[15] B.D. Coleman, E.H. Dill, Thermodynamic restrictions on the constitutive equations of electromagnetic theory, ZAMP 22 (1971) 691–702.

[16] M. Fabrizio, C. Giorgi, A. Morro, A continuum theory for first-order phase transitions based on the balance of structure order, Math. Methods Appl. Sci. 31 (2008) 627–653.

[17] M. Fabrizio, A. Morro, Electromagnetism of Continuous Media, Oxford University Press, 2003. p. 51. [18] B.D. Cullity, Introduction to Magnetic Materials, Addison-Wesley, New York, 1972.

[19] Z.Q. Yang, C.Y. You, L.L. He, Microstructures and magnetic properties of Co–Cu nanoparticles prepared by arc-discharge, J. Alloys Comp. 423 (2006) 128–131.

[20] M. Wu, Y.D. Zhang, S. Hui, T.D. Xiao, S. Ge, W.A. Hines, J.I. Budnick, Temperature dependence of magnetic properties of SiO2-coated Co nanoparticles, J. Magn. Magn. Mater. 268 (2004) 20–23.

[21] C.D. Stanciu, A.V. Kimel, F. Hansteen, A. Tsukamoto, A. Itoh, A. Kirilyuk, Th. Rasing, Ultrafast spin dynamics across compensation points in ferrimagnetic GdFeCo: the role of angular momentum compensation, Phys. Rev. B 73 (2006) 220402(4).

[22] N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, New York, 1992. p. 177. [23] R.V. Iyer, P.S. Krishnaprasad, On a low-dimensional model of ferromagnetism, Nonlinear Anal. 61 (2005) 1447–1482. [24] R.M. Bozorth, Ferromagnetism, Van Nostrand, Toronto, 1955. Page 431.