-

m

a

t

.

m

t

r

l

-

s

c

i

]

1

2

J

u

n

2

0

0

1

Magnetic

Field

scaling

of

Relaxation

curves

in

Small

Particle

Systems

`

Oscar

Iglesias

and

Am´

ılcar

Labarta

Department

de

F´

ısica

Fonamental,

Facultat

de

F´

ısica,

Universitat

de

Barcelona,

Diagonal

647,

08028

Barcelona,

Spain

(Printed

on:February

1,

2008,

Last

version:

28/05/2001)

We

study

the

effects

of

the

magnetic

field

on

the

relaxation

of

the

magnetization

of

small

mon-

odomain

non-interacting

particles

with

random

orientations

and

distribution

of

anisotropy

constants.

Starting

from

a

master

equation,

we

build

up

an

expression

for

the

time

dependence

of

the

magne-

tization

which

takes

into

account

thermal

activation

only

over

barriers

separating

energy

minima,

which,

in

our

model,

can

be

computed

exactly

from

analytical

expressions.

Numerical

calculations

of

the

relaxation

curves

for

different

distribution

widths,

and

under

different

magnetic

fields

H

and

temperatures

T,

have

been

performed.

We

show

how

a

T

ln(

t/τ

0

)

scaling

of

the

curves,

at

different

T

and

for

a

given

H,

can

be

carried

out

after

proper

normalization

of

the

data

to

the

equilibrium

magnetization.

The

resulting

master

curves

are

shown

to

be

closely

related

to

what

we

call

effective

energy

barrier

distributions,

which,

in

our

model,

can

be

computed

exactly

from

analytical

expres-

sions.

The

concept

of

effective

distribution

serves

us

as

a

basis

for

finding

a

scaling

variable

to

scale

relaxation

curves

at

different

H

and

a

given

T,

thus

showing

that

the

field

dependence

of

energy

barriers

can

be

also

extracted

from

relaxation

measurements.

PACS

Numbers:

75.10.Hk,75.40.Mg,75.50.Tt,75.60.Lr.

I.

INTRODUCTION

Time

dependent

phenomena

in

small-particle

systems

have

been

the

subject

of

an

increasing

number

of

exper-

iments

because

of

their

interest

as

non-equilibrium

phe-

nomena

in

spin

systems,

1

for

magnetic

recording

materi-

als

technology

2

and

even

as

a

possible

way

to

prove

ex-

perimentally

the

existence

of

macroscopic

quantum

tun-

neling

phenomena

in

magnetic

materials.

3

,

4

Whereas

the

basis

of

a

theory

of

the

magnetic

after-effect

dates

back

from

old

studies

on

rock

magnetism,

5

–

7

the

interpreta-

tion

of

several

experimental

results

is

still

waiting

for

suitable

theoretical

models

that

capture

the

relevant

fac-

tors

and

parameters

that

can

play

a

role

in

the

explana-

tion

of

these

phenomena.

One

of

the

points

that

has

not

been

completely

clarified

is

the

influence

of

a

magnetic

field

in

the

relaxation

of

small-particle

systems.

Relaxation

in

zero

field

is

usually

analyzed

in

terms

of

parameters

such

as

the

so-called

magnetic

viscos-

ity

S

,

8

fluctuation

field

9

–

11

and

activation

volume,

12

,

13

which

are

susceptible

to

misinterpretations.

In

the

last

years,

several

authors

14

–

20

have

proposed

an

alternative

method

to

analyze

relaxation

curves

based

on

a

T

ln(

t/τ

0

)

scaling

of

the

relaxation

data

at

different

temperatures

that

avoids

the

above

mentioned

problems

and

gains

in-

sight

on

the

microscopic

details

of

the

energy

barrier

dis-

tribution

f

(

E

)

producing

the

relaxation.

16

,

17

In

this

con-

text,

the

purpose

of

this

article

is

to

extend

this

kind

of

analysis

to

the

case

of

relaxation

in

the

presence

of

a

magnetic

field.

We

want

to

account

for

the

experimen-

tal

studies

on

the

relaxation

of

small-particle

systems,

which

essentially

measure

the

acquisition

of

magnetiza-

tion

of

an

initially

demagnetized

sample

under

the

ap-

plication

of

a

magnetic

field.

19

,

21

–

24

In

this

kind

of

ex-

periments,

the

field

modifies

the

energy

barriers

of

the

system

that

are

responsible

for

the

time

variation

of

the

magnetization,

as

well

as

the

final

state

of

equilibrium

towards

which

the

system

relaxes.

The

fact

that

usu-

ally

the

magnetic

properties

of

the

particles

(anisotropy

constants,

easy-axis

directions

and

volumes)

are

not

uni-

form

in

real

samples,

adds

some

difficulties

to

this

anal-

ysis

because

the

effect

of

the

magnetic

field

depends

on

them

in

a

complicated

fashion.

In

a

previous

study,

14

,

17

we

started

to

address

some

of

these

peculiarities,

show-

ing

how

experimental

relaxation

data

must

be

treated

in

order

to

compare

relaxation

curves

at

different

tem-

peratures

and

fields

making

simple

assumptions

about

the

sample

composition.

Here,

we

will

present

the

theo-

retical

background

that

supports

this

phenomenological

approach,

as

well

as

detailed

numerical

calculations

of

the

time

dependence

of

the

magnetization

of

a

system

of

non-interacting

randomly

oriented

small

monodomain

particles

with

uniaxial

anisotropy

and

with

a

distribution

of

anisotropy

constants.

In

a

first

approximation,

we

will

neglect

inter-particle

interactions

leaving

for

a

future

in-

vestigation

the

effects

of

long-ranged

dipolar

interactions

between

the

particles.

The

paper

is

organized

as

follows.

In

Sec.

II,

we

present

the

basic

features

of

model

show

how

the

dis-

tribution

of

energy

barriers

of

the

system

is

influenced

b

the

application

of

a

magnetic

field

with

the

help

of

the

concept

of

effective

energy

barrier

distribution.

In

Sec.

III

we

introduce

the

Two-State

Approximation

(TSA)

for

the

calculation

of

the

thermal

dependence

of

the

equilib-

rium

magnetization

In

Sec.

IV,

we

derive

the

equation

governing

the

time

dependence

of

the

magnetization

from

a

master

rate

equation

in

the

TSA.

The

results

of

numer-

ical

calculations

based

on

the

above

mentioned

equation

1