Low temperature dynamics of small γ-Fe203 particles

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E. Vincent, J. Hammann, P. Prené, E. Tronc

To cite this version:

E. Vincent, J. Hammann, P. Prené, E. Tronc. Low temperature dynamics of small γ-Fe203 particles.

Journal de Physique I, EDP Sciences, 1994, 4 (2), pp.273-282. �10.1051/jp1:1994137�. �jpa-00246904�

J. Phys. I France 4

(1994)

273-282 FEBRUARY 1994~ PAGE 273

Classification Physics Abstracts

05.40J 75.30G 75.60J

Low temperature dynamics of small ~i-Fe203 partiales

E. Vincent

(~),

J. Hammann

(~),

P. Prené (~) ^and E. Tronc (~)

(~) Service de Physique de l'Etat Condenséj C.E.A. Saclay, Orme des Merisiers, 91191 Gif-sur- Yvette Cedex, France

(~) Laboratoire de Chiinie de la Matière Condenséej C.N-R.S.

(URA

1466)~ Université Pierre et Marie Cune~ 4 place Jussieu, 75252 Paris Cedex 05j ^France

(Received

15 October 1993, accepted 1 December

1993)

Abstract We have studied trie _response of _a system of small (+~⁴

nm)

ferrimagnetic partiales

to _a field reversai at low temperature

(0.6

to 12

K).

The reversai of trie partiale magnetic moments

through trie anisotropy barners _is shown _to be governetl by thermally activated dynamics down to 0.6 K, with _no evidence for quantum effects. We propose a data analysis _m terms of the unique

variable T

Ln())j

which accounts for _a non-constant distribution of anisotropy barriers.

1 Introduction.

Since the

pioneering

^{works of} Louis Néel _on

magnetic grains il,

_2],

superparamagnetism

has been

widely

^studied [3]. ^{The interest in this field has been}

recently

revived

by

theoretical

studies [4, Si of the

magnetization

reversai of small

particles through

their amsotropy

barrier,

which have shown that quantum

tunneling

_processes [6,

ii

could have _a

significant

contribution for small but

macroscopic partiales (a

few

nm)

ai net very low temperatures

(of

order 1

K).

The

challenge

of

evidencing

_{a new}

example (actually

the second _one, after

superconductivity

and related

junction phenomena)

of the

irruption

of quantum ^mechanics in _Dur macroscopic world has motivated _a lot of

experimental

work [8-

iii.

In most of these studies, the

samples

_are

collections of

magnetic abjects

of different sizes and

shapes, supposedly magnetically decoupled.

They

present a wide distribution of

anisotropy

_energy barriers

(except perhaps

in

il Ii ),

at least wide

enough

ta

produce (through

the drastic

exponential dependence

of charactenstic limes

on barrier

heights)

_a

quasi-logarithmic time-dependence

^{of trie} relaxation. Trie

loganthmic slope

of trie relaxation _{curve is} defined _as trie

magnetic viscosity;

ils temperature

dependence

reflects the thermal

(or possibly quantum)

nature of the excitations, ^but is aise

weighted by

trie _energy barrier distribution. In trie present

literature,

_numerous situations

[8-10]

^{in which}

trie

magnetic

viscosity faits ta decrease

regularly

for

decreasing

temperature are ascribed ta

quantum

tunneling

_processes. In this _{paper, we} present an

experimental study

of small

(ca.

4

nm) fernmagnetic +f-Fe203 Particles.

^Down ta 0.6

K,

the results remain

compatible

with

thermally

activated

dynamics;

we show how ibis _cari be checked without _a

priori assumptions conceming

trie distribution of anisotropy barriers.

2 S

ample.

Maghemite +f-Fe203

is _a metastable

phase

of hematite cx-Fe203 It bas trie inverted

spinel

structure of trie

magnetite Fe304,

with vacancies _on trie B-site and _a

ferrimagnetic

arrangement between the A and B sites [12].

X-ray

diffraction measurements [13] ^{bave shown that trie} vacancies do net order _at

long

_range in

partides

less than _ca. 20 _nm in size. Trie Curie temperature is 590 °C.

Small

partides

of

+f-Fe203

_can

yield electrostatically-stabilized dispersions

in aqueous medium [14]. ^Trie

acidity

of trie medium contrais trie

density

of electrical

charges

_on trie surface of trie

partides.

Trie surface electrical

charge

promotes

partide-solvent

interactions and

partide-partiale repulsions

which compete with Van der Waals and

magnetostatic

interac-

tions. Via the

pH,

it is thus

possible

to central trie

partide aggregation [15-17]. Aggregation

is minimal _near

pH

₌ 2 and maximal _near

pH

= 7 where flocculation _occurs because of surface

charge canceling.

Colloids of

+f-Fe203

_were

prepared according

to [18]. ^The

precipitate

obtained

by alkalizing

an

Fe~+-Fe~+

_{aqueous mixture with NaOH}

was treated with

HCI04

until the ratio

Fe~+/

Fe~+

_{in trie}

partides

_was

r~ 0.03 On

dispersai

in water, it

yielded

_a sol of

pH

= 2.1 The

dispersion

_was frozen

by adding

_a

stiffening polymer (polyvinylic alcohol).

^Electron

microscopy

observations of the

composite

have shown that the

partides

_are

spheroidal,

their diameter

being

distributed around _a

peak

value of 4.2 nm, with _a fuit width at half maximum

(FWHM)

of

3.0 _nm. The amount of Fe in the

composite

_was measured to be 0.37 _x 10~3 _Fe

mole/g,

and

the composite

density

is

equal

to 1.33

g/cm3.

This

yields

_a

separation

of the

partides,

from center to center, ^{of the} order of 6 times their _mean

diameter, assuming

that the

particles

_are

homogeneously

distributed in the

sample

and have _a

density

of 4.87

g/cm3.

3 Static

properties.

The

magnetic

measurements have been

performed

in _a

pumped

He-3 cryostat. The magneti- zation was measured

by extracting

^the

sample

eut of the detection coil while

integrating

trie induced

voltage.

^The

sensitivity

of the set-up

corresponds

to _a

magnetic

moment of _a few

10~~

_{emu. Ail}

along

this _paper, the

magnetization

values in _emu have been normalized to the

volume of

+f-Fe203 (which

amounts to

r~

0.5% of trie total volume of

resin).

Figure

1 shows trie

magnetization

curve obtained at 4.3 K

dunng

a field

cycle

from H

= 0

to 15 koe and back. The

sample

has been cooled in _zero field from 40 K, but still shows _an initial zero-field

magnetization

^of 32 _emu related to

previous

field

history.

Upon increasing

trie

field,

_no ^real saturation is

found;

^around 5

koe,

trie

magnetization

becomes reversible but continues to increase

roughly linearly

with _a weak

slope dM/dH

_ce

3.2 x 10~3 _emu. This continuous increase, also observed in

[17],

^is

likely

to be due _{to a}

progressive

increase of the net

ferrimagnetic

moments for

increasing field;

the observed

slope

is indeed of the _same order _as the value which _can be

crudely

estimated from the ratio of trie

applied

to

exchange

fields [19].

Linearly extrapolating

^trie

high-field

behavior _{to zero}

field,

_we

con determine _a "saturated

magnetization

at zero field" M~~ = 191 _emu

(see Fig. 1).

After

magnetizing

^the

sample

under _a field of15

koe,

the remarient

magnetization

found at _zero field is 92 _{emu =} 0.48 Ms~. This proportion is very close to the 50% value which is to

N°2 LOW TEMPERATURE DYNAMICS OF SMALL ~f-Fe203 PARTICLES 275

~~~ Mso

H

(

~ T=4.3 K

~ ^{i oo ~~~}

/~~

^{~~~ ~~}

5 0 5 10 15 20

H

(koe)

Fig. l. Magnetization _{curve, as a} function of field, at T ₌ 4.3 K When decreasing the field tram

H = 15 koe, _a remanent magnetization Mrem ₌ 92

emu is round _at H ₌ 0, and _{a coercive} field HC ₌ 500 Oe _is measured. A "saturated magnet12ation ai _zero field" M~~

= 191 _{emu is} estimated by

a linear extrapolation of the high-field behavior.

be

expected

when ail

magnetic

moments are

oriented, along

their

randomly

distributed _easy axes, in trie direction selected

by

trie

previously applied

field [19]. ^{Trie coercive field which}

suppresses this _remanence is

Hc

= 500 De.

Trie

anisotropy

_energy

KA

of trie

particles

can be calculated from the

magnetization

curve

in

figure

1. It

corresponds

to trie work needed to

magnetize

^the

^sample

up to saturation. We estimate KA

by summing

the work up ^to M~~.

M~o

KA _"

/

_{H dM}

ce 8 _x 10~

erg/cm~. (1)

o

This value is _an average over the

anisotropy

directions of the

partides

with respect to the field. In addition to the bulk

magnetocrystalline effects,

it is

expected

to indude other

important contributions the

shape (departure

from

sphericity)

of the

partides plays

of _course

an

important rote, together

with

possible

^{surface effects and} constraints due to

magnetostriction

[31.

Figure

2 presents ^{the variation of the}

sample magnetization

with temperature ^under _a 300 De field. The

sample

has

initially

been cooled in _zero

apphed

field from 80 K _to 4.3 K. A residual

magnetization

Mo ₌ 94 _emu is found at 4.3 K. This _remanence, which

appeared

after

previous measurements under

higher fields,

could trot be

suppressed;

it arises from much

larger magnetic

entities than those which _con

respond

at low temperatures and low fields.

However,

ail the

forthcoming

measurements

(induding

^the

relaxations,

^Sect.

Si

^have been done under

these _same conditions. In order to

only

show trie _response of trie

sample

to _a 300 De field

applied

at 4.3

K,

we have subtracted

Mo

from the results in

figure

2. The zero-field cooled

(ZFC)

_curve shows _a broad

peak

around 13 K; this temperature

corresponds

to the

blocking

of

abjects

^of a

typical

volume Χ such that

~ÎÎ Î ~Îo

~~~

Choosing

KA

" 8 _x 10~

erg/cm3 jour result),

t

= 1000 _s

(ZFC experimental

lime

scale)

and _To

=

10~~° _s

(see

below in Sect.

4),

T

Ln(t/To)

CÎ 30 and _we obtain _a volume Vb "

6.7 _x 10~~°

cm3, equivalent

to

spheres

of diameter S-o _nm

_

25

~

FC

~ H=300 De

E

fl

_~~

ZFC

à

~

0 10 20 30 40 50

T

(K)

Fig. 2. Zero-field cooled

(ZFC)

and field-cooled

(FC)

magnetization _{as a} function of temperature, for _a field of 300 Oe. The frozen magnetization Mo " 94 _emu round at 4.3 K alter zero-field cooling

and before applying the field has been subtracted tram the data in order to emphasize the eoEect of the field.

The broadness of trie

peak

in

figure

2 indicates that trie distribution of _energy barriers is very wide.

Actually,

trie

peak

of trie ZFC _curve does not

correspond

to trie

peak

of trie

size distribution. For

mcreasing

temperature,

larger

and

larger objects align

with trie

field;

beyond

the

peak

of the size

distribution,

the

objects

of

larger

sizes become less _numerous, but

they

contribute

significantly

to the enhancement of the

magnetic

_response because of their

larger magnetization.

It _can be checked

easily

that the ZFC

magnetization peaks

around temperatures

corresponding

to _{one or} two standard deviations above the _mean value of _a Gaussian distribution.

On the other

hard,

_our size determination _may be underestimated for trie

following

_reasons:

ii) neglecting

the temperature variation of the _remanence Mo _may lower the position of the ZFC

peak, (ii)

the 300 De field somewhat reduces the

anisotropy barriers,

and

(iii)

zero-field

cooling

from

only

80 K leaves out the contribution of the

largest objects

with _a

higher blocking

temperature. We therefore consider that _our size estimate remains

approximate; nevertheless,

ii agrees

fairly

well with trie _mean value of 5.2 _nm deduced from Môssbauer measurements

[20,

21] ^{and with the}

peak

value of 4.2 _nm

(3

nm

FWHM)

observed

by

electron microscopy.

4. Relaxation

dynamics: general

remarks.

We have

probed

the

slowing

down of the

dynamics

of the

particles

with

decreasing

temperature

by measuring

^the

magnetization

relaxation in response to _a field reversai ai low temperature.

In the fictitious _case of _a sample constituted of

strictly

identical

partiales,

ail oriented in the

same direction, ^trie

magnetization

reversai

implies

_crossmg barners of _a

unique

value

U;

this

gives

_use to an

exponential decay:

M(tj

₌

M~~j-Hj

₊

AM~~

exp

(-))

,

(3j

where AM~C

=

M~C(+H) M~C(-H)

_is the total

magnetization

_excursion between the field-cooled values

corresponding

to the opposite values of the

field,

and

T(U)

= To exP

~)) ₍₄₎

B

N°2 LOW TEMPERATURE DYNAMICS OF SMALL ~f-Fe203 PARTICLES 277

is the characteristic lime for _crossing,

by thermally

activated

dynamics,

_a borner U ai tem-

perature T with _an attempt lime _To

In the realistic _case of _a distribution

P(U)

of

anisotropy

barriers

U, equation (3)

is modified to

Mit)

₌

M~~i-H)

₊

~M~~ £°°

^exP

1- Al _PIU)dU

15)

where the summation _{runs over} the whole barrier distribution. Due _to the

exponential depen-

dence of _{T on}

U,

the function _exp

(-))

_{in the}

_integrand

_increases

very

abruptly

from 0 to 1 around the value of U such that

T(U)

₌ t.

Namely,

for _{e-g- t}

= 100 _s, trie function is

equal

to 0 for U < 26kBT and

equal

to 1 for U _> 33kBT within _a 0.5% _accuracy.

Equation (5)

is thus _very well

approximated by

Mit)

"

M~~(-H)

₊

AM~~ / _{PIU)dU (6)}

where Ut

"

UIT

=

t)

=

kBT.Ln()). Equation (6)

shows that the main

dependence

of the _mag-

netization relaxation _on lime and temperature occurs

through

the variable

Ut "

kBT Ln());

in other

words,

most of the

magnetization change

at temperature T and

lime t is due to

crossing

of barriers of order

Ut

= kBT

Lu( ().

This famous property ^has

already

been

widely

used to

explain

_some aspects ^{of the}

dynamics

of spin

glasses [22,

23] ^and of

randomly

disordered

ferromagnets

[24,

25].

In order to describe magnetic

viscosity

measurements, let _{us now} consider the

logarithmic

derivative of the

magnetization:

ÎÎÎ _~~~~

%~ T(Î)~~~ T(Î) ^~~~~~~

^~~~

~~~~ Î~ro Î~~~ Î~

^~

^~~~~

~~~~~~°~~ ~~~ ^~~~ ~ ~~~

Within

exactly

the _same

approximation

_as above

(from Eq. (5)

to

Eq. (6)),

the function

)

_exp

(-))

_is

_equal

_to

zero

everywhere

except around t ₌ T, 1e

$

= 30 + 3 The

magnetic viscosity

therefore reads: B

~~~~

=

AM~C(T)

_P

kBT

Ln(~))

^kBT.

⁽⁹⁾

n t _To

In the _case of _spm

glasses

[26], ^{it has been taken}

advantage

of this property to estimate the spectrum ^of response times from the measured

magnetization

relaxation.

In _an

experiment performed

at _a fixed temperature

T,

the _range of barriers which _can be

probed during expenmental

times t

(such

_as,

typically,

^10° < t _s < 10~ _is limited to

$

r~ Ln

r~ 25 35.

(10)

B To

If

P(U)

_can be considered constant within this _range, equation

(loi

shows that _a linear Lnt

dependence

of the relaxation will be obtained at each temperature. This is the _case if the number of

partiales

with volume _{spanning a} +15% _range does not vary ^too

abruptly (assuming

that the

anisotropy

^barrier is

proportional

to the

volume). However, concerning

the temper-

ature

dependence

of trie measured

viscosity,

it may become _more dillicult _to consider that

P

(kBT Ln()))

does not vary

significantly

_over T-variations of _a factor 10 to 20.

Any depar-

0

ture from _a strict

proportionality

to temperature ^{of the}

viscosity

[9, 10] can as well be ascribed to _a P

(kBT Ln()))

variation _as to

possible

non-thermal

dynamics,

as was also

emphasized

o

in [8, 24] ^We argue that _a

scaling

of trie results _{as a} function of trie unique variable T

Ln())

substantially helps

to _remove this

difficulty.

^°

5. Relaxation

dynamics:

measurements.

The relaxation measurements have been

performed along

the

following procedure.

Trie

sample

is first heated to 50

K,

and _a +300 De field is

applied;

then the

sample

is field-cooled down

to the measurement temperature To

(between

0.6 and 12

K).

We have checked that the field-

cooled

magnetization

at TO does Dot show _any relaxation _{m our range} of time, _{nor any}

cooling

rate

dependence. ^Then,

the field is reversed from +300 to -300

De,

and the

magnetization

relaxation at To towards the _new

equilibrium

_state under -300 De is recorded from _a few seconds to _a few hours.

Figure

3

displays

_a few

typical

relaxation _curves. At temperatures ^below 5

K,

_we have measured the

magnetization by extracting

the

sample

_{on a} twice shorter _range thon at

higher

temperatures, ^{in order} to minimize the weak

heating

related _to the _movement. This results in

slightly

more scattered

data,

_{as can} be _seen from

figure

3 where both

procedures

_are

compared

at 4.3 K As _a first

approximation,

each of the _{curves can} mdeed be considered linear in Ln t,

although

_a ^careful examination reveals _a

slight upwards (resp. downwards)

curvature for

higher (resp. lower)

temperatures, ^{which will be}

mterpreted

below.

) Îi

_1.10

fl

E

4.3 K

é

~

(

_{+300 -b -300 Oe 11.1 K}

°

loi 102 103 104 105

t

(sec)

Fig. 3. -Typical relaxation _curves ai TO " 11.1, 4.3 and 1.10 K, obtained alter field-cooling the

sample tram 50 K to T0 under +300 Oe, and reversing the field to -300 Oe ai TO- The _curves have been arbitrarfly shifted along the vertical scale. At low temperatures, the

magnetization

has been

measured with

a shorter extraction _range, yielding _{more noisy} data; bath procedures _are compared ai 4.3 K.

For each of the _{curves, we} bave determined _{an average}

slope dM/dLn

t

by fitting

_a linear

Ln t behavior in the 5 _to 6000 _s range. We have divided

(according

_to

Eq. (9))

the

slope by

AM~C(T);

the latter

quantity

shows _a 7% _variation in the 0.6 _to 12 K _range

explored.

This

N°2 LOW TEMPERATURE DYNAMICS OF SMALL ~f-Fe203 PARTICLES 279

"corrected

viscosity"

is

displayed

in

figure

4 _{as a} function of temperature. Between 0.6 and 7

K,

a nice

proportionality

to temperature is

observed,

which within the

experimental

_accuracy

can

extrapolate

to _{a zero} value at _zero temperature. ^As far _as P

(U

^{= T}

Ln()))

con 0

be considered constant, ^the

T-proportionality

is characteristic of

purely thermally

activated processes; but since the

explored

temperatures vary from 0.6 to 12

K,

the concemed _energy barriers vary

by

the _same factor

20,

and there is _no guarantee that the distribution

P(U)

of the

partiales

remains constant over such _a range. A limit _to this

approximation

_con be _seen in the

viscosity

data above 8

K,

which show _a saturation: the most

likely explanation

is that such

larger partiales

_are

significantly

less _numerous than the smaller _ones.

This,

in tum, prevents _us from _a

simphstic interpretation

of the

low-temperature

data: _a

departure

from _pure thermal

activation, due e-g- ^to the influence of quantum

tunneling

_processes, could be hidden

by

_a decrease of the

partiale

distribution for smaller sizes.

_

6

p

~Î

4 O °

~ ~ ~ ~

à

O

= ^o

- O

] _~o° 7-Fe203

~ W

fl

&Ù

~ S°

~'0

2 4 6 8 10 12 14

T

iKi

Fig. 4. Viscosity data _{as a} function of temperature. ^The mean logarithmic slope

dmjdLn

^t

^(t

ⁱⁿ

^si

of the relaxation _curves has been normalized to the total magnetization excursion AM

C(T)

between the two field-cooled values measured with +300 and -300 Oe.

A natural

presentation

of the data

[22,

23] ^{follows from}

equation (6):

_{one can} try to

plot

ail relaxation _curves at ail temperatures as a function of the

single

variable T

Ln()),

which is the

magnitude

^{of the} _energy barrier which

strongly

dominates the

dynamics

ai

tÉme

_{t and} temperature T. As far _as the

dynamics

is

thermally activated,

ail data _can be

expected

to fait onto a

single

continuous _curve. Trie _more drastic

approximation

^of _a constant distribution of barriers would

imply

that this _curve be _a

straight fine;

this approximation is non needed any more, since the distribution

P(U)

is _now

simply proportional

to the

slope

of the _curve.

The check of

continuity

_m the T

Ln()) plot

necessitates _a reliable

knowledge

of the total

magnetization (a

measurement of the

oÀy

drift with time of the

magnetization

would not be

sufficient).

We have indeed measured this total value for ail data points;

but,

_as noticed

above,

the

sample

indudes _a frozen

magnetization

which

slightly depends

_on the

history.

When

plotting

_{our raw} relaxation data _{as a} function of

T.Ln( )

,

we have observed

that, followmg

the

chronological

order of data

taking,

_our results _were

assembling

onto three

parallel

_curves, shifted from each other

by

_a few _emu, and

corresponding

to three series of measurements between which trie

sample

had been

kept

several

days

at room temperature. We bave therefore attributed

these _two small shifts to a variation at _room temperature ^{of trie} remanent

magnetization

(amounting

to 7 and 5% of trie

remanence),

and corrected for them. Trie result is

presented

in

figure

5: ail relaxation _{curves at} ail temperatures ^{assemble in} a

unique

_curve. Trie best

continuity

of this _curve is obtained

by choosing

_{To =}

^10~~°

_s; this is _a rather sensitive _test for To

,

since the continuity is

already

somewhat

downgraded lover

the whole temperature

range)

for choices of10~~ _or 10~~~

s. This

value,

_a little bit

higher

than trie

typical

attempt times of

paramagnetism,

is indeed of trie correct order of

magnitude

when

compared

with trie dassical models of

superparamagnetism

_[3]. Trie inset of

figure

5 shows _{a zoom} of trie

low-temperature region. Indeed,

_{a very}

slight tendency

to _a

departure

from thermal activation ai trie lowest

temperatures 0.6-0.8 K _may be

suspected,

^which

appeared

also in trie

viscosity plot

of

figure

4;

complementary

data should be of interest, but _were

beyond

trie

possibilities

of trie present

expenment.

oo

90

~°

"~~~N~,

~

m

'§~

~ _%

~ _~~

20 40 60 80

~

~o=1 o-1°sec

~~~0 _{100 200 300 400}

T. Ln

(t /~o)

Fig. 5. AU relaxation _curves ai ail temperatures _are plotted here _{as a} function of T

Ln(t/T0)

_122, 23]

(in

Kelvin), which is the typical _energy barrier crossed by thermal activation at temperature T

alter _a time t The mset is _{a zoom} of the low-temperature region.

The _{master curve} in

figure

5 shows _a

slight

curvature,

varying

from

negative

to

positive

^for

increasing

values of the main barrier

height

T

Ln())

This is in agreement ^with a weak but

systematic

trend

already

observed _on trie relaxation _{curves at} different temperatures

(see Fig. 3).

From this _{we cari} confirm that trie distribution of barriers is trot

flat,

but bas _a

maximum

(inflection

point ^{of trie} master

curve)

around T

Lu( ))

= 150

K,

which

corresponds

to a

blocking

temperature ^{of around} 5 K.

Recalling that,

for

tÉe

reasons discussed in section

3, our size determinations _may be

slightly underestimated,

_we condude from the

magnetic

measurements that the distribution of

partide

sizes is

peaked

around _a

sphere

diameter of 3.7

nm

(inflection

point ^of trie T Lut

curve),

and starts to decrease

significantly

in trie

region

around 4.3 _nm

(viscosity

saturation above 8

K)

to 5 _nm

(ZFC

broad

peak

around 13

K).

This is indeed in

good

agreement ^{with trie data from electron}

microscopy (Sect. 2).

6. Conclusion

We bave studied trie

low-temperature dynamics

of _a system ^{of small} (r~ ⁴

nm) ferrimagnetic partiales

of

+f-Fe203

in the 0.6 to 12 K range.

Beyond

the usual

plot

of the

magnetic viscosity

versus temperature, which

implicitly

_assumes that the distribution of

particle

sizes _is fiai _over

a range of volumes _as

large

_as that of the

explored

temperatures, we have

analyzed

the data

N°2 LOW TEMPERATURE DYNAMICS OF SMALL ~f-Fe203 PARTICLES 281

with the

help

of _a

scaling plot inspired

from the

spin-glass

literature

[22, 23].

^As a function of the natural variable T

Ln()),

which is _a measurement of the main energy barrier crossed

by thermally

^activated

processls

_at temperature T after _a lime t, ail relaxation _curves at ail temperatures ^form a

unique

master curve, whose

slope

is

simply proportional

to the barrier distribution. Whatever the exact

shape

of this

distribution,

the

continuity

of the _{curve ensures} that the observed

dynamics

is

thermally

activated. In such _a

plot, crossing

of barriers

by quasi- macroscopic

quantum

tunneling

would _{appear as a}

split

of the left-hand part

(low-temperature)

of the _curve in small segments, ^{which would have} a

larger slope

thon

imposed by continuity.

Within _our

experimental

_accuracy, no such eflect _con be evidenced.

Indeed, according

to [4], we'can estimate the _cross-over temperature Tc at which the rate of quantum

tunneling

events should be of the _same order of

magnitude

_as

thermally

activated _processes:

Tc =

~~~@, ₍₁₁₎

where

+f is the

gyromagnetic

ratio and KA CÎ 8 10~

erg/cm3

is the

anisotropy

_energy which inhibits the reversai of the

partiale magnetization ("longitudinal anisotropy").

_KA> is the

"transverse" anisotropy; on account of trie cubic symmetry ^{of Fe}

compounds,

it _con be

expected

to be of trie _same order of

magnitude

_as

KA

This

yields

Tc

r~ o-1

K,

1-e- at _or

beyond

trie limit of trie lowest temperatures

explored

here.

In real

macroscopic samples,

_even

carefully prepared,

_a wide distribution of

auisotropy

barriers _seems to be unavoidable, due _{to many} different

dispersion

factors such _as size, _easy-

axis orientation,

shape

etc...

Correctly accounting

^for this unknown distribution is _one of trie

important

dilliculties encountered in the

experiments

which atm to evidence for quantum

tunneling

effects in trie

magnetization

^reversai of small

partiales.

Trie

scaling plot presented

here

helps

to banale finis

dilliculty,

and in

addition gives

_some information _on trie distribution.

It should be

interesting

to confirm with this kind of data

analysis

trie present evidences of quantum

tunneling phenomena

from

viscosity

measurements of small

magnetic partiales.

Acknowledgements.

We thank Laurent

Leylekian

for very

helpful

and

friendly exchanges.

We _are very

grateful

to

Eugene Chudnovsky

and Jean-Louis Dormann for _numerous

enlightening

discussions and crucial remarks.

Note added in

prooE

After this work _was

completed,

_we learnt that another paper

(A.

Labarta et

al., Phys.

Rev.

848

(1993) 10240)

also

emphasizes

trie _use of _a T Lut

plot

^for this kind of expenments.

We recall that the method which _{we propose} here allows the discrimination between thermal and quantum

dynamics

without _any assumption

conceming

the barrer distribution. Provided that reliable and systematic measurements of the total

magnetization

have been done at ail

temperatures, the crucial test

simply

consists m

checking

the continuity of the T Lnt _curve.

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