IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998

3867

Temperature- and Time-Dependent Preisach

Model for a Stoner–Wohlfarth Particle System

Alexandru Stancu,

Member, IEEE,

and Leonard Spinu

Abstract— In this paper, a dynamic (time and temperature

dependent) Preisach model is described. In the Preisach plane
one uses a double well potential distribution calculated with the
Stoner–Wohlfarth model. Both static and dynamic interactions
for a ultrafine particulate system are taken into account in order
to simulate various magnetization processes, such as: field-cooled
(FC), zero field-cooled (ZFC), thermo-remanent magnetization
(TRM), magnetic relaxation in applied field (MR), and the major
hysteresis loop (MHL) as a function of temperature. The results
of the simulations are in good qualitative agreement with ex-
perimental data obtained for ferromagnetic ultrafine particulate
systems.

Index Terms—Magnetization processes, Preisach model, static

and dynamic interactions, Stoner–Wohlfarth model.

I. I

NTRODUCTION

T

HE classical Preisach model (CPM) [1] describes static
magnetization processes, that is, processes in which the

effect of relaxation is neglected. Visintin [2] defined hysteresis
as a rate independent memory effect, so the viscous-type
effects are not included. Mayergoyz [3] considers both static
and dynamic hysteresis. A system is characterized by static
hysteresis if its branches of hysteresis nonlinearities are in-
fluenced only by the past extremes values and not by the
speed of input variations. The wiping-out (A property) and
congruency (B property) properties are the necessary and
sufficient conditions for a system to be correctly described by
a (static) CPM. In order to extend the application of similar
methods to explain magnetization processes in systems which
are not obeying to both A and B properties various Preisach-
type models have been developed [4]–[6]. Most of them are
considering only static magnetization processes.

The most important time effect on magnetization phenom-

ena is the magnetic viscosity (also named magnetic after
effect). It includes all the influence of time without taking
into account [7] the influence of the sample inductance, eddy
currents, irreversible relaxation of chemical or topological
microstructure of the material and relaxation phenomena with
characteristic time less than 10

s. The magnetic after effect

is essentially due to the energy barriers which may be over-
come by thermal energy. One may identify three categories of
magnetic viscosity: 1) reversible aftereffect (associated with
diffusion of impurity atoms), 2) irreversible aftereffect, and 3)

Manuscript received February 9, 1998; revised June 25, 1998.
The authors are with “Alexandru Ioan Cuza” University, Faculty of Physics,

Department of Electricity Iasi, 6600, Romania (e-mail: [e-mail]).

Publisher Item Identifier S 0018-9464(98)07210-0.

quantum tunneling of magnetization [8]; the irreversible after
effect is due au fond to the thermal activation [9]. Another
time effect is the accommodation which appears when the
minor loops are cycled between two fields; this effect, which
is in disagreement with the wiping-out property, was included
in the Preisach model by Della Torre [10] and is explained
as the result of the motion of particles within the Preisach
distribution.

Mayergoyz developed the dynamic Preisach model (DPM)

[11] in which the irreversible after effect is considered in a
purely phenomenological manner. Thermally induced mag-
netic relaxation has also been included in Preisach-type models
in a more physical way by Mayergoyz [12] and by Souletie
following the ideas initially developed by N´eel [13] which will
be referred to as the Preisach–N´eel model (PNM).

The PNM was used recently to explain the effect of the

finite temperature

and observation time

on the Henkel

plots of ac and thermally demagnetized systems [14], [15] and
the maximum in field dependence of the isothermal remanent
magnetization and thermoremanent magnetization for the spin
glasses [16]. A similar approach has been used by Bertotti [17]
to deal with the domain wall dynamics in metals.

In all the approaches of the PNM, until now, only the linear

expression of the energy barrier as a function of the field was
used. In this paper we present a PNM for a Stoner–Wohlfarth
(SW) [18] particle system which introduces a nonliniarity in
the energy barrier expression.

For particulate SW systems in which all the easy axes of the

particles are parallel to the applied field direction, in the CPM,
the particle magnetic moment is associated with a point in
the Preisach plane whose coordinates are the switching fields

[Fig. 1(a)]. In the PNM, one associates to the same

point a double well potential calculated for each particle in
zero applied field [Fig. 1(b)]. For systems with a single particle
volume the Preisach distribution in both models are identical.

II. PNM

FOR A

SW P

ARTICLE

S

YSTEM

For an ensemble of uniaxial single domain particles, having

their easy axes aligned, and with a field making an angle
with this direction, the energy of a particle can be written as
follows:

(1)

where

is the anisotropy constant,

is the particle volume,

is the saturation polarization of the particle,

is the angle

between polarization vector and field direction, and

is the

0018–9464/98$10.00



1998 IEEE

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998

(a)

(b)

Fig. 1.

(a) The rectangular hysteresis loop and (b) the associated Stoner–

Wohlfarth double well potential.

angle between the direction of the polarization vector and easy
axis (EA) direction. In general the angle

is given by the

expression

(2)

where

is the angle made by the

direction with the

(

) plane. The stable equilibrium positions of the polar-

ization vector of the particle can be determined by minimizing
the expression of the energy

, i.e., imposing the

equilibrium conditions. It is easy to prove that the polarization
vector lies in the (

) plane. The values of the magnetic

moment in the two stable equilibrium positions are

(3)

where

are the stable equilibrium angles between the

polarization vector and the easy axis direction. The unstable
equilibrium position corresponds to the maximum of the
energy

. The heights of the energy barriers corre-

sponding to the magnetic moment vector rotation in and out
of the field direction are

(4)

The critical field

at which the polarization switches be-

tween

positions is given by the well-known formula

(5)

In (1)

is the effective field acting on a particle, which in

the case of a noninteracting particles systems is equal to the
external applied field. When the interactions between particles
are significant the effective field is given by

(6)

where

is the applied field and

is an interaction

field due to the neighbor particles. Like in the CPM, we
shall consider that the interaction field is collinear with the
applied field direction. Thus, the interaction field is shifting
the hysteresis loop on the field axis with

, and

the effective field is given by

(7)

The SW model does not take into account the thermal

effects, and therefore in this model the magnetic moment
cannot overcome the energy barriers given by (4). When
relaxation effects are considered, the evolution in time of the
number of particles in the two equilibrium states, in a two
level model, is described by the master equation

(8)

where

are the relative number of particles in the “ ” and

“ ” positions, and

are the relaxation times in and out of

the field direction, respectively.

Usually, for the relaxation times one operate with the

asymptotic expressions given by N´eel [19]

(9)

where

is a microscopic attempt frequency which is about

s

.

Solving the appropriate kinetic Fokker–Planck equation one

obtains for the relaxation time more accurate expressions giv-
ing the field and temperature dependence of the preexponential
factor; quite recently, Coffey et al. [20], [21] studied the effect
of the applied field direction on the N´eel relaxation time of a
S–W ferromagnetic particle.

If the relaxation time does not depend on time, the solution

of the master equation (8) is given by

(10)

where

(11)

is the total relaxation time and

is the initial relative

number (at

) of particles in the “ ” equilibrium state.

The particles for which

are called blocked particles

(

and

; if

the particles are

named superparamagnetic and for them

(12)

and the magnetization is given by

(13)

STANCU AND SPINU: TEMPERATURE- AND TIME-DEPENDENT PREISACH MODEL

3869

The condition

is the equation of a critical curve which

separates in the Preisach plane the blocked and superparamag-
netic particles. Due to the strong dependence of

on the field

and to the fact that if

then

and if

then

, one may use for the critical curves

where

is the experimental duration. The general expressions of the

thermal critical curves as a function of the coercive

and

interaction field,

is given by

(14)

where

.

For a better description of the PNM for a SW particle sys-

tem, in the following sections we shall analyze the particular
cases

and

for which an analytical expression

of the energy barrier could be given. Also, we shall investigate
the general case for any

using the interpolated expression

of the energy barrier given by Pfeiffer [22]

(15)

with

(16)

A. Case

In this case, the heights of the energy barriers corresponding

to the magnetic moment vector rotation out of the field
direction and into the field direction are

(17)

where

and

and

are 0

and

180 , respectively.

Using (7) of the effective field

, for the thermal critical

curves we obtained the following:

(18)

For an aligned particle system the anisotropy field is equal

to the coercive field

; passing to the (

) coordinate

system, rotated with 45 with respect to the (

) system

[see Fig. 2(a) and (b)], (18) is becoming

(18 )

where

. At

and/or

(for

) in the (

) coordinate system,

the thermal critical curves (18 ) are two lines [

and

in Fig. 2(a)] which separates the Preisach plane in 3 regions,
labeled “ ,” “ ,” and “memory region.” For the particles from
“ ” and “ ” regions there is only one possible orientation of
the magnetic moment of the particle, parallel, and antiparallel,
respectively, with the direction of the applied field; for these
particles there is no energy barrier to surmount in order to
achieve the equilibrium position of the magnetic moment, the
energy having only one stable equilibrium position. For the
particles from the “memory region,” there are two possible
orientations for the magnetic moment of each particle. The

(a)

(b)

Fig. 2.

The Preisach plane in a positive applied field

H

a

, (a) for

h

t

= 0 (no

relaxation process—like in the static CPM) and (b) for

h

t

6= 0 in the PNM

for a Stoner–Wohlfarth particle system with

= 0



.

state is determined in this case by the “history” of the
temperatures and applied fields.

For

and/or

, (

) the thermal critical

curves (18 ) are becoming two parabolas [Fig. 2(b)] which
intersect each other in two points (

,

) and

(

,

). The branches represented with dotted

lines in Fig. 2(b) are not physical because these branches
correspond to “ ” and “ ” regions in the Preisach plane where
there is no energy barrier and the condition

is not fulfilled. Thus, the “memory region” is separated in
four regions where the corresponding particles have different
behavior: 1) for the particles in the region 1, the thermal
energy corresponding to the experimental temperature

and

observation time

is not sufficient to change their state and

they remain blocked in their previous state; 2) particles in the
regions 2, 3, and 4 are superparamagnetic, experimental time

being sufficient for achieving of the thermal equilibrium.

For a certain magnetization process the configuration of

the regions in the Preisach plane mentioned before will be
modified, due to changing of the thermal critical curves with
the applied field

, temperature

, and time . For example,

as the temperature

or time

increases,

increases, and

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998

the thermal critical curves are moving diminishing the region
corresponding to the blocked particles

[region

1 in Fig. 2(b)] and increasing the region corresponding to
superparamagnetic ones

[regions 2, 3, and 4 in

Fig. 2(b)]. The magnetic moment of the sample as a function
of time and temperature is given by

(19)

where

is the Preisach distribution,

and

are weight functions for superparam-

agnetic and blocked states, respectively. For a system with
aligned axes on the field direction the expression of the
magnetization corresponding to the thermal equilibrium (13)
becomes

(20)

The weight function for the blocked particles may be better

written as

, history of

, and

due to the fact that

the magnetization of these particles depends on the history
of the system temperature and applied fields. To be more
precise, one imagines a magnetization process for which
(temperature) decreases at

const. The critical curves

are moving in the Preisach plane determining a decreasing of
the region corresponding to the superparamagnetic region. In
their moving, the critical curves determine an increase of the
region in the Preisach plane for which the inequality

is

not fulfilled. Thus, all the particles in the zones swept by the
critical curves will be blocked in the state corresponding to the
critical curve “passing through” the point in the Preisach plane
where the particle’s moment is associated. In this case, the
weight function

history of

and

can be written

as

history of

and

(21)

where

is the field in which the particle is blocked and

is the temperature at which the particle is blocked.

B. Case

In this case the analytical expressions for the energy barriers

and

are

(22)

Fig. 3.

The configuration of the Preisach plane and the critical curves when

a linear dependence of the energy barrier as a function of the magnetic applied
field is taken into account (like in [14]–[16]). The dotted lines represent the
critical curves in the PNM for a S–W particle system for

= 45



.

In the (

) coordinate system the critical curves are

represented by the implicit equations

(23)

where

is given by

(for 45

).

The weight functions for the superparamagnetic region

and for the blocked region

,

history of

, and

are calculated with the general

expression (13) where

and

are given by (22)

and

and

by the expressions

(24)

In Fig. 3 we show the thermal critical curves for

;

the critical curves intersect each other only in one point with
the coordinates (

).

C. Case: General

For a general value for

, it is not possible to find an

analytical expression for the energy barriers and therefore for
the thermal critical curves. Using the interpolated expressions
for the energy barriers given by Pfeiffer in [22], the equations
of the thermal critical curves for any

are given by

(25)

In Fig. 4(a) we represent the critical curves for three values
for

, (0, 15, and 45 ). One observes the same behavior of

the critical curves like in the

case: two points of

intersection, one with the coordinates (

)

and one for

, and for a

value depending on the

(26)

In Fig. 4(b) one shows the critical curves in the case

, calculated with the exact formula (23) (represented with

STANCU AND SPINU: TEMPERATURE- AND TIME-DEPENDENT PREISACH MODEL

3871

(a)

(b)

Fig. 4.

(a) The Preisach plane in a positive applied field

H

a

for

h

t

6= 0 in

the PNM for a Stoner–Wohlfarth particle system with

= 0



(solid line for

the greatest

h

c

value),

= 15



(dashed line), and

= 45



(solid line for the

smallest

h

c

value) and (b) comparison between critical curves for

= 45



,

calculated with the exact formula (23) (represented with simple solid lines)
and calculated with the approximated relations (25) (represented with solid
lines and circle symbol).

circles) and calculated with the approximated relations (25)
(represented with full lines); we remark that the approximated
critical curves reproduce very well the exact ones for the
values in the region where these curves have physical sig-
nificance. The weight functions used in (19) in the general
case are calculated using the general formula (13).

III. S

TATIC AND

D

YNAMIC

I

NTERACTIONS IN THE

PNM

When one applies an external field to a particulate medium

each particle is feeling not only this field but also the field
created by the neighbor particles. In classical recording media
the relaxation effects may be neglected at the room temperature
due to the fact that the energy barriers are sufficiently high
in comparison to the thermal energy. In the CPM which
was intensively used to model these media, the field created
by the neighbor particles were taken as constant in time.
The main assumption in this model is that the interaction
fields are not the same for all the particles; these fields are
distributed. The interaction field distribution is considered as
statistically independent with the coercive field distribution.

The interaction field distribution is usually taken as Gaussian,
centered on the

axis (zero mean interaction

field). These interactions will be referred to as static statistical
interactions.

A system with only static statistical interactions have to

fulfill both the wiping-out and congruency properties, as we
mentioned before. In many experiments made on classical
recording media the congruency property is not satisfied. This
was also referred to as a violation of the “statistical stability”
of the Preisach distribution [23], [24]. Della Torre explained
this violation as an effect of the static mean field interactions.
In his moving Preisach model (MPM) the effective field acting
on each particle is the sum between the applied field, the
static statistical interaction field, and the interaction mean
field which may be computed in an iterative process [25],
[26]. The MPM is giving results in good agreement with the
experimental data measured on media where the after effect
may be neglected.

When the relaxation is important in the system, the statistical

interaction field is fluctuating at a high rate. This was proven
by Dormann [27], [28] to be equivalent to an increase of
the particle anisotropy. These interactions are qualitatively
different from the static ones; they will be referred to as the
dynamic interactions. In the terms of the Preisach model, that
means that the dynamic interactions will influence the coercive
field distribution. To be more precise, in the PNM presented
in this paper, the following interaction effects are taken into
account:

• static statistical interactions;
• dynamic interactions.

The effect of the mean field interactions in a PNM will be

the subject of a future paper.

In the system one has both blocked and superparamagnetic

particles. The blocked particles are creating the static statis-
tical interaction field and the superparamagnetic particles the
dynamic interaction field. The mean interaction field is given
by all the particles.

In the time and temperature dependent magnetization pro-

cesses the weight of the blocked or superparamagnetic parti-
cles is changing. The effect of this modification is taken into
account in the PNM as a variation of the standard deviation of
the interaction field distribution. The variation of the dynamic
interactions intensity is inducing a variable displacement of the
Preisach distribution in the

direction. One may mention that

similar displacement, but in the opposite direction and with
other physical motivation, was considered in [16] to explain
the appearance of a maximum in the field dependence of the
isothermal remanent magnetization and the thermoremanent
magnetization in spin glasses.

IV. M

AGNETIZATION

P

ROCESSES IN THE

PNM

A. Experimental Magnetization Processes for
Fine and Ultra-Fine Particle Systems

The magnetic characterization of fine particle systems is

usually obtained through a set of experimental curves. The
magnetization processes in commercial recording media which

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998

are frequently used are the remanent curves: IRM (isother-
mal remanent magnetization), DCD (dc demagnetization), and
ARM (anhysteretic remanent magnetization) processes [29],
[30]. With these data one calculates the well-known Henkel
plot [31], deltaM plot [32] to evaluate the interaction field
distribution. One may also use other remanent magnetization
processes [33] and the generalized deltaM plot [34]. The rema-
nent magnetic moment is usually considered time independent
(relaxation effects are neglected). The magnetization processes
measured in the presence of the applied field, like the first
magnetization (FM) curve and major hysteresis loop (MHL)
are used to evaluate the weight of the reversible magnetization
part in the total magnetization.

In the ultra-fine particulate systems, as we pointed before,

the relaxation effects are the most influential in the measured
value of the sample magnetic moment. For these particle
systems the commonly used experiments are those time and
temperature dependent (when the variable parameter is time
or/and temperature). The zero field-cooled (ZFC) and field-
cooled (FC) processes are the most important experimental
curves used in the characterization of these systems. In the
ZFC process the magnetization is measured in a constant
applied field

when the temperature

is increased. The

initial state is obtained after a thermal demagnetization. The
FC curve is obtained by measuring the magnetic moment of
the system during its cooling (

is decreasing) in the presence

of a constant applied field. The initial state of the system in
the FC process is at a sufficiently high temperature to assure
a thermal equilibrium state. The FC and ZFC curves are often
measured for the same applied field.

The FC and ZFC curves have been measured for many

particulate systems in order to evidence the effect of the
applied field and that of the packing ratio for the particles in
the system which is directly related to the interaction strength
between the particles [28], [35], [36]. It was observed that the
temperature where the ZFC curve has a maximum

moves

with both the applied field and packing ratio. The experiments
made on nanometric iron oxides particle systems shows that
if the applied field is increasing

is decreasing and if the

packing ratio is increasing the

is increasing [35], [36].

These effects were explained in activation energy models as
an increase of the energy barrier height in the system with the
increase of the interactions barriers [28], [35], [36] (effect due
to dynamic interactions). The SW expression for the energy
barrier was able to explain the influence of the applied field
[36]. In all these models the static statistical interactions were
not taken into account.

Other magnetization processes usually used for the char-

acterization of the fine and ultrafine particle systems are the
thermoremanent magnetization (TRM) and the relaxation in
an applied field (MR). The TRM process is achieved by
cooling down the system in applied magnetic field from a
temperature high enough to assure the thermal equilibrium to
the measuring temperature at which the field is removed. The
MR process is obtained cooling down the sample in zero field
from a temperature where the irreversibility had vanished, to
the measuring temperature where a magnetic field is applied,
and the magnetization is recorded as a function of time.

B. Preisach Distribution: Algorithm

In this section we shall describe the PNM algorithm we

designed to explain some typical magnetization processes for
ultra-fine particulate systems.

The Preisach distribution was considered as the product be-

tween two statistically independent distributions (as a function
of coercive and interaction/shift fields)

(27)

In the simulations Gaussian distributions were used

(28)

(29)

where

and

are the standard deviations and

is the

most probable coercive field. In order to take into account the
increase of the energy barriers height as an effect of dynamic
interactions,

is a function of saturation superparamagnetic

moment of the sample

(30)

where

is the most probable value of the

distribution in the absence of the dynamic interactions and

is a positive constant proportional with the strength of

dynamic interactions.

We also considered the dependence of the static statistical

interactions on the weight of the superparamagnetic part of
magnetization. When all the particles are blocked (at very
low temperatures) these interactions are the most important.
As the temperature increases the number of blocked particles
diminishes and in the same time the static interaction distribu-
tion standard deviation is decreasing; the dynamic interactions
intensity is increasing, so the

is increasing (see Fig. 5).

When all the particles are superparamagnetic the static statistic
interactions are negligible (zero standard deviation—singular
Preisach distribution). Taking that into account we used for

the expression

(31)

where

is the standard deviation of the

dis-

tribution in the absence of the dynamic interactions,
is a positive constant which value is chosen so that for the
maximum value of the

is very close to zero.

C. Simulations

The PNM presented in the previous sections was used

to simulate the MHL, ZFC, FC, MR, and TRM processes
for a S–W fine particle system characterized by a Preisach
distribution with

Oe,

Oe, and

Oe, and a direction of the applied field with

regard to easy axis

. The MHL for three values of the

temperature

and

K, and the same value of the

Oe parameter are shown in Fig. 6.

The effect of the dynamic interactions on the variation

of the ZFC and FC versus temperature is shown in Fig. 7.

STANCU AND SPINU: TEMPERATURE- AND TIME-DEPENDENT PREISACH MODEL

3873

Fig. 5.

The Preisach distribution for two different values of the experimental

temperature and/or experimental time (

h

t

): 1) at low temperature the static

statistical interactions are important and 2) at high temperature the dynamic
interactions produce the displacement of the Preisach distribution and the
diminishing of the effect of the static statistical ones.

Fig. 6.

The MHL for the following values of the sample temperature 10,

50, and 80 K. The parameter characteristic of the dynamic interactions is

1h

c0

= 1000 Oe.

The strength of the dynamic interactions in the sample was
given by the different values for the

parameter, 100,

1000, 1500 Oe. As the dynamic interactions increases the
maximum corresponding of the ZFC curve shifts toward
higher temperatures and becomes more broad, as in numerous
experimental data on oxide ultrafine particles (for example,
[28, p. 345] which concerns maghemit nanoparticles dispersed
in a polymer matrix with different degrees of dispersion).
The FC curve changes too with the dynamic interactions,
diminishing the value of the magnetization corresponding to
the lowest value of the experimental temperature. It must be
noted that in a real fine particle system as the packing fraction
ratio increases and the interparticle distances between particles
diminishes the static statistic interactions also increase. The
increasing of the static statistic interactions can be taken into
account by increasing the standard deviation

which

cannot explain the shift of the ZFC maximum but causes a
flattening of the ZFC and FC curves. In experiments, one
obtains a global effect of the interactions (both static and
dynamic interactions).

Fig. 8 shows the temperature dependence of the ZFC and

FC for three different values of the applied field 30, 60, and
120 Oe. The position of the ZFC peak shifts toward lower

Fig. 7.

The dependence of the field-cooled magnetization and the zero field

cooled magnetization on the temperature for several values of the parameter
characteristic of the dynamic interactions

1h

c0

= 100; 1000;and 1500 Oe.

Fig. 8.

The dependence of the field-cooled magnetization and the zero field

cooled magnetization on the temperature for several values of the applied
field

H

a

: 30, 60, and 120 Oe. The parameter characteristic of the dynamic

interactions is

1h

c0

= 1000 Oe.

Fig. 9.

The dependence of the ZFC curve on the angle

for an applied

field

H

a

= 30 Oe. The parameter characteristic of the dynamic interactions

is

1h

c0

= 1000 Oe.

temperatures as the field is increasing, as we mentioned that
one observes experimentally [36].

The effect of the direction of the applied field with regard to

the easy axis direction on the zero-field cooled magnetization
process is presented in Fig. 9. One observes that as the angle

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998

Fig. 10.

The dependence of the MR on the experimental time for an applied

field

H

a

= 30 Oe and three values of the measuring temperature T : 40,

50, and 80 K. The parameter characteristic of the dynamic interactions is

1h

c0

= 1000 Oe.

Fig. 11.

The dependence of the TRM on the experimental time for an applied

field

H

a

= 30 Oe and three values of the measuring temperature T : 40,

50, and 80 K. The parameter characteristic of the dynamic interactions is

1h

c0

= 1000 Oe.

increases the maximum of the ZFC curve is moving to higher
temperature and its value is diminishing.

Figs. 10 and 11 represent the MR and TRM experiments

simulated with the PNM for an applied field

Oe

and three values of the measuring temperature

, 10, 50, and

80 K. These curves display the same characteristics to those
observed experimentally in the magnetic fine particle systems
[28], [37], [38].

V. C

ONCLUSIONS

A new PNM for an aligned single volume S-W fine particle

system, with an angle

between easy axis direction and

field direction, has been developed. Both static statistical and
dynamic interactions have been taken into account to simulate
various temperature, time, and field dependent magnetization
processes.

To our best knowledge it is the first time that the ZFC

and FC processes are simulated using a PNM. The fact that
it was considered a single volume particle system and not a
size particle distribution it is not very restrictive; in the energy

activation models the role of the particle size distribution is to
determine a distribution of the energy barriers of the system
which can be taken into account in the PNM via the Preisach
distribution.

In a future paper we shall further develop the model

to include the static reversible effects (particle easy axes
orientation) with a distribution of the easy axis directions and
mean field interactions. We shall also analyze the waiting time
effects observed in the MR and TRM curves.

Note Added in Proof: Recently, a paper [39] where new

experimental results attest a nonmonotonic field dependence
of the ZFC peak in some systems of magnetic nanoparticles,
has appeared. In Fig. 8 we have presented the field depen-
dence of the ZFC and FC curves, where a decreasing of the
temperature of the ZFC peak is observed for high applied
fields. We mention that in the PNM we have presented we can
explain the nonmonotonic dependence of the ZFC; a detailed
analysis of this problem with a discussion of the influence of
different factors which interplay in the dependence of the ZFC
peak on the applied field (Preisach distribution parameters,
interactions) will make the subject of a future paper.

A

CKNOWLEDGMENT

The authors are grateful to Prof. J. L. Dormann and Prof.

R. W. Chantrell for helpful discussions.

R

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Alexandru Stancu (M’96) was born in Iasi, Romania, in 1957. He received
the M.Sc. and the D.Sc. degrees in physics from the “Alexandru Ioan Cuza”
University, Romania, in 1981 and 1995, respectively.

He is currently an Associate Professor at the Faculty of Physics of

“Alexandru Ioan Cuza” University, and in 1997, he was an Invited Professor at
Universite de Versailles-Saint-Quentin, France. He has performed research on
the numerical modeling of magnetization processes. In addition to numerous
conference presentations, he is the author of more than 50 refereed publica-
tions. He has also co-authored two books (in Romanian): Special Topics in
Magnetism (Iasi, Romania: "Alexandru Ioan Cuza" Univ. Iasi Pub., 1996) and
Laboratory of Electricity and Magnetism (Iasi, Romania: BIT, 1996).

Leonard Spinu was born in Iasi, Romania, in December 1967. He received
the B.Sc. degree in physics from “Alexandru Ioan Cuza” University, Iasi,
Romania, in 1992 and the M.Sc. degree in solid-state physics from the
University Paris XI, Orsay, France. He is currently working toward the Ph.D.
degree conjointly in Laboratoire de Magnetisme et d’Optique de l’Universite
de Versailles, France, and “Alexandru Ioan Cuza” University, Iasi.

His current research interest is in the area of experimental study and

modeling of the magnetization processes in fine particle systems.