Crossover from equilibrium to nonequilibrium dynamics in a reentrant AuFe ferromagnet

HAL Id: jpa-00246702

https://hal.archives-ouvertes.fr/jpa-00246702

Submitted on 1 Jan 1992

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Crossover from equilibrium to nonequilibrium dynamics in a reentrant AuFe ferromagnet

P. Mitchler, R. Roshko, W. Ruan

To cite this version:

P. Mitchler, R. Roshko, W. Ruan. Crossover from equilibrium to nonequilibrium dynamics in a reentrant AuFe ferromagnet. Journal de Physique I, EDP Sciences, 1992, 2 (12), pp.2299-2309.

�10.1051/jp1:1992282�. �jpa-00246702�

Classification

Physics

Abstracts

75.508 75.50L 75.60L

Crossover from equilibrium to nonequilibrium dynamics in

_a

reentrant Aufe ferromagnet

P.

Mitchler,

R. M. Roshko and W. Ruan

Department

of

Physics, University

of Manitoba,

Winnipeg,

Manitoba, Canada R3T 2N2

(Received 26 May 1992, accepted in

final

form Ii August 1992)

Abstract. Measurements of the

decay

of the low field thermoremanent

magnetization,

_over four decades of observation time, 6 _s

w t

w10~

s. as a function of temperature T and wait time t~, ^{have been}

performed

on two Aufe

alloys

located _on either side of the multicritical

point

_{: a} Au~ofejo

spin

glass and _a

strongly

bond-disordered reentrant Au83Fej~ ferromagnet. The

spin glass

relaxation isotherms exhibit the canonical symptoms ^of

nonequilibrium

ordering below

Tso,

the isotherms _are characterized by an inflection

point,

_{on a}

logarithmic

time

perspective,

which is sensitive to the wait time t~,

display

_{a memory} for _a

history

of field reversals, and _are all

compatible

with _an

analytical representation

based _{on a} stretched

exponential

function,

M~(i)=Mo+Miexp[- (i/r)~~"],

with _r =t,~~ and _a weakly temperature and wait time dependent exponent _{n m} 0.7. The reentrant ferromagnet exhibits _a

temperature-driven

_crossover

between _two distinct relaxation

regimes

_{a «}

high

temperature»

regime

coincident with the

ferromagnetic

phase, characterized

by

an extremely

rapid

approach to

equilibrium, negligible

wait time effects, and _a weak power law

decay, M~(t)

=

Mo

₊

Mi

t~~, with _a weakly temperature

dependent

_exponent 0.04 _{w m w} 0.08, consistent with the

predictions

of droplet fluctuation models of random

Ising ferromagnets,

and _{a «} low temperature »

regime

coincident with the reentrant

phase,

with

nonequilibrium, age-dependent

dynamics which _are virtually

indistinguishable

from those in the

« pure » spin glass

phase

and describable by the _same stretched

exponential

relaxation function. This

study

represents ^the first

systematic comparison

of relaxation dynamics in the

reentrant and

ferromagnetic

phases ^of a system ^with

sequential

transitions, and

provides

compelling

evidence for the

equivalence

of the reentrant and

spin

glass phases.

1. Introduction.

Ferromagnets

with substantial

exchange

bond

disorder,

due _{to some} forni of

quenched

structural

randomness, frequently exhibit,

_upon

cooling, apparently sequential magnetic

transitions from paramagnet to

ferromagnet

to a low temperature

phase

with many of the elements

(strong irreversibility, displaced hysteresis loops) commonly

associated with

spin glass freezing.

The Aufe system, ⁱⁿ

spite

of its well documented

metallurgical complexities [I],

is

widely recognized

as the

archetype

of such _{« reentrant »} behaviour and exhibits the

classic

symptoms [2]

: the low

frequency dynamic

_response first increases

rapidly

with

decreasing

temperature, ^then _passes

through

a

typical ferromagnetic Hopkinson

maximum

(if

the

susceptibility

is not limited

by

the

demagnetizing factor)

followed

by

a

weakly

temperature

dependent

_«

plateau

_», and then

abruptly

decreases

(in

_{a manner} reminiscent of a pure

spin glass

below its

freezing

temperature

Tso)

_as the

ferromagnetic phase collapses

into the

« reentrant »

phase.

Infinite

ranged

vector

spin

models for

magnetic

_systems with

competing

exchange

interactions

[3] interpret

the _reentrant transition _as the _onset of _transverse

spin glass freezing

in the presence of

longitudinal ferromagnet ordering

_; while local

probes,

such _as M6ssbauer spectroscopy

[4]

and inelastic neutron

scattering [5],

_appear to support ^this

identification, they

do _not indicate that this

phenomenon

is critical in the _sense of _a

genuine cooperative thermodynamic phase change,

_nor has there been any

systematic attempt

to characterize the relaxation

dynamics

within the _reentrant

phase,

which _are

expected

to exhibit

nonequilibrium

^{behaviour similar} to that which dominates the relaxation response of pure

spin glasses.

According

to a

phenomenological theory [6-8] developed by

Fisher and Huse for

Ising

systems ^with

short-range interactions,

the low

lying

excitations which dominate the

long-

distance and

long-time

correlations in the ordered

phase

of systems ^{with discrete broken} symmetry are

droplets (or domains)

^of

coherently flipped spins

enclosed

by

_a

single

wall.

Droplet

^formation

requires

activation _over energy barriers B which grow with

length

scale L _as B

~L*, leading

to

droplet

lifetimes _r

exp(flL*),

and hence _to

logarithmic droplet growth, L(t)~ (Tlnt)~~~.

If _a

spin glass

^is

subjected

to an ideal

quench

^{from infinite}

temperature to a temperature T _~

Ts~

in _zero

magnetic

field,

then,

after _a time t~ ^has

elapsed,

the system ^{will consist of domains of}

typical

linear dimension

L(t~)

^within which the

spin

configuration

is

locally equivalent

to _one of the _two pure

equilibrium

states

[6]

characteristic of the

temperature T,

and two

temporal

relaxation

regimes

_are

distinguishable [7]

_{: an «}

early epoch

_»

(In

t « In

t~)

where relaxation _{occurs on}

length

scales L _«

L(t~) through

activated

equilibrium droplet fluctuations,

and the correlations

C(t) decay

_as

C(t)~L(t)~Y~

(In t)~Y~*,

and a « late

epoch

_»

(In

t

»Int~)

where the

dynamics

are a consequence of

activated

nonequilibrium

domain

growth

and the

decay

is

govemed by

_a

nonequilibrium

exponent ^A ~y which leads to _{a more}

rapid logarithmic

_power

law, C(t)~L(t)~~

(ln

t

)~

~~*. Thermoremanent relaxation measurements allow both

regimes

to be observed in the

same

experiment [9]:

the

sample

^is cooled

rapidly

in _a field H _{to a} temperature

T~TS~,

and, ^after a time t~ has

elapsed,

the field is

abruptly

reduced

by

an amount

AH which is small

enough

that the

overlap length

between the

equilibrium

states

(T, H)

and

(T,

H

AH)

is much

larger

than the

typical (T, H)-domain

size

L(t~).

On _a

logarithmic

time

perspective,

the relaxation _curves exhibit _an age

(t~ )-dependent

^inflection

point [9],

which

represents

^the crossover between the two relaxation

regimes,

and

which,

in

practice,

is often modelled

by

_a stretched

exponential

function

M(t )

_exp

[- (t/r

_)~ ^~

],

with

r = t,n~mt~. ^The theoretical

justification

for this choice is

provided by

the

mesoscopic

domain model of

Koper

and Hilhorst

[10]

which

postulates

that, ^if a

nonequilibrium spin glass

is located in _a heat bath of constant temperature ^{T and in} a constant

magnetic

field

H

then,

for any

arbitrary

choice of

(Ti, Hi ),

the system ^{will be}

composed

of

(Tj, Hi )-domains

which _are characterized

by

a

specific

set of

(Tj, Hi )-spin correlations,

and which will grow

with time but _never exceed the

overlap length ^I (Tj T, Hi

H

),

which defines the maximum

length

^scale over which the

spin

correlations for the two

thermodynamic equilibrium

states

(Ti, Hi )

and

(T, H)

_are

indistinguishable.

The relaxation response function of _a

spin

in _a domain with _a

time-varying

size

s(t)

is _a

complicated

functional of

s(t)

and, _as in the Fisher and Huse

approach,

this model features _an

interplay

between two

lengths,

a characteristic size I

(t

for _a

growing

domain and the

overlap length

^I

(AT,

AH ). ^In a field

jump experiment

of the

type ^described

above,

where the system ^is

quenched

to

T~TS~

at t =0 in _a field

Ho

which is

subsquently

turned off _{at t}

= t~, ^{the model}

predicts that,

when AH

=

Ho

is

sufficiently

small that

I(t~)~i(0, _AH),

the relaxation _rate exhibits _a maximum _at r'w t t~ m t~,

signalling

_a crossover between _a slow

equilibrium decay

and _{a more}

rapid nonequilibrium

stretched

exponential decay.

In this paper, we present a detailed

comparison

of thermoremanent relaxation in the

spin glass

^and reentrant

ferromagnetic configurations

of

Aufe,

and _we show that the _reentrant

ferromagnet

is characterized

by

two distinct relaxation

regimes corresponding

_to the _two

sequential

ordered

phases

_{: a} low temperature

nonequilibrium regime,

coincident with the reentrant

phase,

in which the relaxation

dynamics

_are

essentially

identical _to those in the pure

spin glass phase,

and _a

high temperature equilibrium regime,

coincident with the

ferromagnetic phase,

with _an

entirely

different functional

representation.

2.

Experimental

details.

Two

alloys

of Aufe, _{one a}

spin glass containing nominally

10 _at. fGFe and the other _{a reentrant}

ferromagnet containing nominally

17at.fGFe _were

prepared by

arc

melting appropriate

amounts of 99. 99 fG pure Au wire and 99.99 9b pure Fe wire _on the water cooled copper hearth

of _a

titanium-gettered

_{argon arc} fumace

using

_a tungsten ^{electrode. The choice of these}

compositions

_was based _on the definitive

dynamic magnetic

characterizations of Sarkissian

II

], and,

_{as our} static measurements

subsequently confirmed,

the

similarity

of the fabrication

and heat _treatment

techniques ultimately

ensured very

comparable magnetic products.

Each

alloy

_was

repeatedly

inverted and remelted in order to achieve _a

homogeneous consistency,

and calculations based _on total

melting

losses indicated that the true concentrations _were

(at worst)

within _± 0.3 at.fG of the nominal

values,

in each _case. Each of the

original ingots

_was

cold rolled into _a

sheet,

and _one

sample

_was

prepared

from each sheet: the reentrant

Au~3Fej~ ferromagnet

_{was a}

thin, needle-shaped sample,

with dimensions 0.08 _{mm x} 0.8 _mm x10.4 _mm, while the

Au~ofejo spin glass

was a

thicker, rod-shaped sample (for

enhanced

sensitivity),

with dimensions 0.9 _mm x0.9 _mm x5.2 _mm. Each

sample

_was

encapsulated

in _a qualtz ^{tube in} an argon

atmosphere,

annealed for 3

days

at

900

°C,

and then

quenched rapidly

into water, ^{in order} to

generate

an

atomically

disordered state, and _an EDAX

analysis

_was consistent with _a

high degree

of

homogeneity

and

negligible clustering.

The static

magnetization

and _remanent relaxation measurements _were

performed

with _a variable temperature, ^variable

frequency SQUID

susceptometer,

operating

in the

magnetometer ^mode

(w

=

0),

as described in detail elsewhere in the literature

[12].

3. Data

analysis

and discussion.

Figure

I shows the temperature

dependence

of the low field static

magnetization

of the

Au~ofeio spin glass

and the _reentrant

Au83Fej7 ferromagnet,

measured under both field

cooled

(PC)

and _zero field cooled

(ZFC)

conditions. The

spin glass

exhibits the canonical

cusp-like

structure _at

Ts~

=

32K,

indicative of random

spin freezing,

and the

gradual

bifurcation of the

M~c

and

Mz~c

_curves below

Ts~ symptomatic

of

irreversibility

_; the reentrant

ferromagnet

exhibits

multiple

structure

(identified by

the vertical _arrows in

Fig. lb)

consistent with the

dynamic

measurements of

Sarkissian[11],

and thus with the formation of _a

ferromagnetically

ordered state below about T~ m100

K,

which

subsequently collapses

_near

T m 40 K into _a low

temperature

reentrant

phase,

within which the temperature

dependence

of the PC and ZFC

magnetizations

mimic those in the pure

spin glass phase.

In order to

fully appreciate

the

significance

of the relaxation

dynamics

of the Aufe system ⁱⁿ its reentrant

configuration,

it is first essential to characterize the thermoremanent

decay

within the pure

spin glass phase,

and

figure

2 summarizes the essential features of this

decay

at a

typical

measurement temperature

T~

=

22K. Each relaxation isotherm in

figure

2a _was

obtained

by cooling

the

sample

in _an

applied

field

H~

₌ ^{5.0 Oe from} a reference

temperature

la)

Au~fe~~

Ha=500e 16)

Au~fe~~

Ha.05

/~

~OO~OOOOOO~

~

Fcoo°°

'

~

~o j

^{°~ ~}

~i

°~°°~

~

~~

Cl

, j LO _,

f I '.

.°

i

w O 'O ©

~ O_*

2 ~

O_O ~

*

°

' O

° O. ZFC ~~'

.' e

5o o ioo

T(K)

Fig. I. (al The temperature dependence of the field cooled (FC) and _zero field cooled (ZFC) static

magnetization

of the

Au~ofejo spin

glass measured in _an

applied

field H~ ₌ ^5.0 Oe. (b) The temperature

dependence

of the FC and ZFC static magnetization of the reentrant Au83Fe17

ferromagnet

measured in

an

applied

field H~ ₌ 0.5 Oe. The vertical arrows

identify

the structural features which also characterize

the measurements of the

dynamic

_response in reference [11].

(~) T~= 22 K (bl

fl

6O ~

i

^'

I

w Q~

7

9 _.~ 9

~~ .~ _m

~

s

s

2 3 4 2 3 4

log t

Fig. 2. (a) The

decay

of the therrnoremanent

magnetization

of the Au~ofeio spin glass at a typical

measurement temperature

T~

₌ 22 K,

plotted

_{on a}

logarithmic

time scale, for five different wait times

t~. ^{The vertical} arrows mark the locations of the inflection

points

as determined from the maxima in the relaxation rate S(t ), and the solid _curves illustrate

typical

fits _to

equation

(I ) for two extreme wait times.

(b) The

numerically

calculated relaxation rate S(t

w

am~lafn

_t for four of the five relaxation _curves in

figure

2a,

plotted

on a

logarithmic

time scale.

T~

=

60K in the

paramagnetic regime,

where relaxation effects _were

negligible,

to the

measurement temperature

T~

=

22K

(the cooling procedure

was very

reproducible

and

yielded cooling

^times

consistently

close to t~ = 500 s),

waiting

for _a

predetermined

time t~ to

elapse

at constant temperature

T~,

then

abruptly removing

the

applied

field and

recording

the

decay

over four decades of observation time 6s

~ t

w10~s.

_On

a

logarithmic

time

perspective,

the relaxation isotherms all exhibit well defined inflection

points

which coincide, within

experimental

error, with the effective age of the system, so that t,~~ = t~ + t~ ^this age

dependent

behaviour is _a manifestation of the

nonequilibrium

nature of the

spin glass

state, and translates into _a maximum in the relaxation rate

S(t)w %M~/%fn

t at t~ = t~ + t~ ^which propagates ^towards

longer

observation times with

increasing

_{t~, as} ^{shown in}

figure

2b. The existence of

macroscopically long

relaxation times in the

Au~ofeio spin glass ultimately implies

that the

decay

of the thermoremanent

magnetization depends

on the

history

of the

sample

and should thus

display

a memory for past events. This is

vividly

illustrated in

figure 3,

which shows the time

dependence

of the thermoremanent

magnetization

measured _at

T~

₌ 22 K

after

the

sample

_was

subjected

to the sequence of field reversals

depicted

in the upper half of the

figure

_; the oscillations observed in _zero

applied

field _{are a} manifestation of the

principal

of

superposition [13]

which _assumes that the memory of _a field

change

is not

erased

by

a

subsequent reversal,

and which describes the _net relaxation response as the _sum of

a sequence of

essentially independent

_responses.

.

,' P~ i

~ ~ ~~

'~

,

m

°., ~/

~

2 3 4

log

Fig.3.-The

memory effect: the time

dependence

of the therrnoremanent

magnetization

of the

Au~ofeio spin glass

after the

sample

was cooled in _zero field to the measurement temperature T~

= 22 K and then subjected to the sequence of field pulses shown in the upper half of the

figure.

In order to

permit quantitative comparisons

with the reentrant

data,

it _was instructive _to establish _a functional

description

of the relaxation isotherms in

figure

2a. Of the various

empirical

and theoretical

representations _proposed

in the

literature,

the

simplest

formulation which _was

capable

of

consistently

and

accurately replicating

the

principal

structural features of the

experimental

data _over

virtually

the entire observational time

window,

consisted of the

superposition

of _a stretched

exponential

and _a constant term

MR(t)

=

Mo

₊

Mi

_exp

i- (t/r

_)~

~~i (I)

(It

should be mentioned here that double

logarithmic plots

of

M~(t)

as a function of

in

_t, constructed _to

verify

the Fisher and Huse

prediction

^that

M~ (in t)~

_Y, do not

yield

straight

lines _over any extended

temporal interval,

but rather exhibit continuous _curvature with

a

monotonically increasing negative slope, although

the

systematic

trend towards

larger

values of _y with

increasing

observation time is

generally

consistent with the _crossover from slower _to

faster

dynamics expected

^within the

droplet

fluctuation

formalism.)

Table I presents a

comprehensive

list of the best fit values of the

parameters Mo,

n, and _r,

along

with _a number of other relevant

experimental

parameters, ^for a sequence of relaxation isotherms

(including

those in

Fig. 2)

with measurement temperatures

T~ spanning

much of the

spin glass phase,

and the solid _curves in

figures

2a illustrate

typical

^{fits for} two extreme wait times. For wait times t~ w 240 _s,

equation (I) provides

_a

complete description

of the measured isotherms _over the entire

experimental

time interval

(6

s _~ t _w

10~ s),

and the characteristic time _r agrees, within

experimental

_error, with the location of the inflection

point

_t~n~ deduced from the maxima in the relaxation _rate S

(t ).

This

correspondence

continues _to hold for

longer

wait times t~ _~ ^{l 000} s,

although

the range of

validity

of the stretched

exponential

fit becomes

progressively

_narrower,

with

systematic

deviations apparent at short observation times

(solid

_curve for t~ =

3 600 _s in

Fig. 2a).

Such behaviour is consistent with

previous investigations [14],

^which suggest ^{that the}

applicability

of the stretched

exponential

is restricted to

experimental

time intervals in the immediate

vicinity

of the effective

experimental

_age of the system, where the relaxation _rate is dominated

by

^the influence of the

aging

_process, but that it is

incompatible

with the short time

(t

«

t~) equilibrium

_response, which is

expected

to be _a weak power law

decay [14].

The

magnitude

of the exponent n is

typical

of

spin glasses,

and exhibits _a

slight

increase with

measurement temperature ^for

T~/Ts~

~

0.8,

_as well _{as a} weak inverse

dependence

_on the wait time t~, ^which is

just resolvable, experimentally.

The introduction of the constant term

Mo

is difficult _to

justify

on

physical grounds,

and is

probably symptomatic

of _a further limitation inherent in the stretched

exponential representation,

which vanishes _too

rapidly

to

provide

a proper

description

of the actual

long

term

decay

to zero

magnetization [9]

_;

nevertheless,

such _{a term} is _an essential

ingredient

of other

empirical representations [9, 15]

of

spin glass relaxation,

_even those

[15]

which do not

explicitly

invoke _a stretched

exponential

component, ^{and exhibits similar} temperature

dependent systematics [9].

Table I.

Au~ofejo spin glass

parameters.

Tel) ~(s) ~(8) _n (s)

13 60 640 600+200 730+10 0.68+0.01 1.63+0.01 6-10'

20 60 610 600+200 630+8 0.71+0.01 1.16+0.01 6-10'

22 60 610 360+100 440+6 0.67+0.01 0.96+0.01 6-10'

22 240 720 700+100 690+8 0.67+0.01 0.93+0.01 6-10'

22 1200 1680 2000+600 2050+60 0.65+0.01 0.88+0.02 10-10~

22 1800 2280 3000+600 2060+70 0.62+0.01 0.98+0.02 20-10'

22 3600 4050 4000+1000 4006+240 0.69+0.01 0.98+0.04 160'-10'

24 60 570 250+100 290+3 0.70+0.01 0.70+0.01 6-10'

26 60 780 350+100 330+4 0.72+0.01 0A7+0.01 6-10'

28 60 630 250+100 185+3 0.73+0.01 0.30+0.01 6-10'

The characterization of the thermoremanent relaxation in the reentrant

Au~~fei~ ferromagnet proceeded by adapting

the

experimental

^and

^analytical techniques developed

^for the

Au~ofejo spin glass.

^The

tendency

for residual relaxation effects _to

persist

well above the

peak

in the ZFC

magnetization (see Fig. lb)

necessitated the choice of _a

comparatively high

reference

temperature T~

₌ ^{140 K for}

establishing

the _zero level of

magnetization,

_so that the

reentrant

study

was

severely

constrained

by

^the

anomalously large cooling

^intervals

T~-T~ (~100K

for the reentrant

phase)

and

correspondingly long cooling

times t~

(typically

l 000

s),

which allowed the

cooling

_process to assume a dominant role in

defining

the age of the system.

Figure

4 summarizes the

thermorenianent _decay

_for

a sequence of

representative

measurement temperatures

T~ spanning virtually

the entire ordered

phase (15

Km

T~w100K),

with all isotherms

corresponding

to an identical

cooling

field

H~

₌ ² Oe and wait time t~ =

60 _s. While the structure of these isotherms

(particularly

at the lower

temperatures)

is much broader and

considerably

more subtle than that associated with

« pure »

spin glass

^relaxation

(undoubtedly

a consequence of the

comparatively

slow

cooling rates),

there is nevertheless _an

unmistakable, progressive change

ⁱⁿ curvature

throughout figure 4,

^from concave down to _concave up with

increasing

measurement temperature

T~. Furthermore, quantitative analysis

of the isotherms reveals that there _{are nvo} distinct types

of relaxation

behaviour, corresponding

to two distinct

temperature regimes

each of which coincides with _one of the _two ordered

phases

which characterize the reentrant sequence, and the isotherms in

figure

4 have been

grouped accordingly

in order to reflect this

dichotomy

_:

(a)

a low temperature (« reentrant

») regime

in

figure

4a with

T~

_~ 35

K,

where the isotherms all possess inflection

points

and exhibit _a wait time

dependence

^indicative of

nonequilibrium

processes and describable

analytically by

the

superposition

of _a stretched

exponential

and _a

constant term as in

equation (I),

and

(b)

_a

high

temperature («

ferromagnetic ») regime

in

figure

4b with

T~~35K,

where

aging

effects _are

negligible

and where the

decay

is

incompatible

with _a stretched

exponential representation,

but is consistent with _a weak power law

dependence

of the form

M~(t)

=

Mo

₊

Mj t-m, (2)

(a) (b)

~~ Tm(K)

, so

~ ~~~ ~~

f

_~

70

80

uo

io ioo 1

tog t

Fig.

4.-(a)

Decay

of the thermoremanent

magnetization

at several

representative

temperatures

T~

within the reentrant

phase

of the

Aug~fej~ ferromagnet,

for _a wait time t~ = 60 _s,

plotted

_{on a}

logarithmic

time scale. The vertical _arrows mark the locations of the inflection

points.

(b) Decay of the thermoremanent

magnetization

for t~ ₌ 60s _at Several

representative

temperatures T~ ^{within the}

ferromagnetic phase

of

Aug~Fej7.

symptomatic

of relaxation fron~ _a

fully equilibrated

field cooled _state.

Figure

5

provides

a

typical

illustration of the

quality

of th± fits for selected isotherms in each of the two relaxation

regimes.

and tables II and III list all the relevant

experimental

parameters,

including

the best fit values of the _constants

Mo,

_{n, r,} and _m in

equations (I

and

(2).

^Based on a

comparison

with the relaxation

dynamics

of _a pure

spin glass,

as defined

by figure

2 and table I, the

magnitude

and variation of the stretched

exponential

_{parameters n} and _r in table II _are

completely

consistent with

orientationally

random

spin freezing

within the low

temperature

reentrant

regime. However,

the constraints

imposed

_on the

experimental cooling

rates

by

the

necessity

to cool the system over

relatively large

temperature ^intervals

T~ T~

are

responsible

fur _some apparent

systematic discrepancies

_; in

particular,

while the

experimental cooling technique

is unable to

distinguish

bet~N.een

cooling

intervals

differing by typically

A

(TR T~ )/T~

^0. I, so

that the nominal

experimental

_{age t~+t~} of the system ^is

essentially independent

of

measurement temperature

T~

within the _reentrant

phase,

the location of the inflection

point

t~n~ = r

(see

vertical _arrows in

Fig. 4a),

and hence the _true age of the system, ^exhibits an

obvious inverse correlation _to

T~, shifting

from t,n~ _~ t~ + t~ ^to t,n~ _~ t~ + t~ with

increasing

measurement temperature. ^{This behaviour}

is,

however,

entirely

consistent with the absence of

aging

observed

throughout

the

ferromagnetic phase,

and

merely

confirms that the system ^is

continuously

in

equilibrium throughout

the initial

portion

of the

cooling interval,

and that

aging

processes do _{not commence} until the system has been cooled

through

_some effective reference temperature

Ti

^{which lies} well below

T~

but _{close to} the

ferromagnetic-reentrant phase boundary (although

fine

tendency

^for t~n~ to exceed all of the obvious

experimental

time

(a) (b)

5emu/g

M~

05emu/g

70K

90K

~2 3 4 2 3 4

log t

Fig.

5. -(a)

Typical

fits (solid curves) of

Eq.

(I) to some of the _reentrant relaAtion isotherms of

Aug~fej~

for _two different wait times t~ ₌ 60 s(.) and t~ ₌ 7 200 _s (A). The vertical _arrows mark the locations of the inflect:on points, and the _zero magnetization levels of the relaxation _curves have been shifted by arbitrary amounts for clarity of

presentation,

_so that only, the scale of the relaxation has been indicated in the lower left of the figure. (b)

Typical

fits (solid curves) of

equation

(2) to some of the

ferromagnetic

relaxation isotherms of

Aug~fej~.

As in

figure

_a,

only

the scale of the relaxation has been indicated in the upper

right

of the

figure.

Table II.

Au~~Fei7

reentrant

phase.

t~(S) t~+t~(s) tan(s) ~s) _n (s)

16 60 900 3500+1000 2860+l10 0.77+0.01 2.98+0.01 6-10'

17.5 60 900 3000+1000 2330+l10 0.83+0.01 2.92+0.01 5~10'

20 60 960 2500+500 2560+140 0.88+0.01 2.37+0.01 5-10'

20 7200 8220 _> 10' (1.9+0.9)x10~ 0.87+0.01 2.75+0.03 6~10'

21.6 60 1200 1600+600 1800+180 0.91+0.01 2.82a0.01 6~10'

22.5 60 1260 1200+300 1000+70 0.92+0.01 2.80a0.02 5-10'

22.5 7200 8260 4000+1000 4800+300 0.89+0.01 2.7l+0.02 5~10'

23 60 l160 1200+500 1760+50 0.93+0.01 2.78+0.01 5~2.7x10'

23 7200 8220 _> 10' (5.4+0.3)x10' 0.89+0.01 2.77a0.03 5~l.5x10'

23.5 60 1200 _< 10 7.0a2.0 0.96+0.01 2.68+0.03 5-10'

Table III.

Au83Fe17 ferromagnetic phase.

T~O~) ~(s) _m

40 60 0.06+0.01 1.69+0.01

45 60 0.06+0.01 1,64+0.01

60 60 0.04+0.01 lA7+0.02

65 60 0.04+0.01 lA6+0.01

60 60 0.04+0.01 1.38+0.01

60 7200 0.04+0.01 1.38+0.01

65 60 0.06+0.01 1.37+0.01

70 60 0.06+0.01 1.30+0.01

70 7200 0.04+0.01 1.30+0.01

75 60 0.08+0.01 1.27+0.01

80 60 0.07+0.01 1.18+0.01

80 7200 0.06+0.01 1.18+0.01

86 60 0.06+0.01 1.12+0.01

90 60 0.08+0.01 1.05+0.01

100 60 0.08+0.01 0.85±0.01

constants at the lower measurement

temperatures

^{is further evidence that slow}

cooling

rates

distort the

anticipated correspondence

between t~ + t~ and tin~,

and,

in

fact,

accelerate the

aging process).

As mentioned

earlier,

the _zero field

equilibrium (t~

₌

co)

relaxation in

spin glasses

is

expected,

on the basis of both Monte Carlo simulations

[16]

_as well _as critical fractal-cluster model calculations

[17],

to be characterized

by

a slow

power-law decay

below

Ts~,

and this

expectation

is

supported by

both static and

dynamic

measurements of the relaxation _rate S in various canonical

spin glass _systems [17].

In

ferromagnets,

_remanent relaxation has

traditionally

been attributed _to domain rotation and wall

motion,

and models

[18]

which invoke activation processes over a smooth distribution of barrier

heights predict logarithmic

time

dependences. However,

_more recent theoretical

investigations

of

Ising ferromagnets, specifi- cally

those with

quenched

^random

exchange disorder,

_suggest

[8]

that power law

decay

is also

a feature of the

equilibrium dynamics

of random

ferromagnets. Apparently,

the low

frequency, long length

scale fluctuations in the ordered

phase

of such systems ^with

spontaneously

broken discrete symmetry are

droplets

of

coherently

reversed

spins

surrounded

by

domain walls which

are

pinned

to the disorder, so that

droplet

annihilation

requires

thermal activation _over

large free-energy

barriers which grow with

length

scale. In

fact,

the

long-time

behaviour of the

spatially averaged temporal spin

autocorrelation function C

(t)

w

(S,(0) Sj(t)) (S,)~)

,

where

(.. )

denotes _an infinite-time average and

(.. )

~

a

configuration

_average, is dominated

by droplet

excitations with

anomalously long

relaxation times

which,

in random

ferromagnets (through

_a combination of activated

droplet

lifetimes and Boltzmann

droplet

excitation

probabilities [8]),

are

exponentially

_i-are in

in

_t,

yielding

a power law

decay

:

c

(t j

~- kiln ti

~ ~- tiT)

(3j

for random

exchange

where y = I.

(This

result appears not to be

specific

_to

Ising spins,

since

recent models

[15]

for activated relaxation of

dispersive

excitations _{on a}

percolation

distribution of fixed finite-sized

domains,

which _are based _on

quite general geometric

considerations

independent

of such

specifics

as

spin dimensionality,

and which have been

applied successfully

to pure

Heisenberg ferromagnets [19], yield

the _same

asymptotic behaviour.)

Moreover, ^Fisher and Huse have

pointed

out

[8]

that the

region

^of

validity

of the

predicted asymptotic equilibrium decay (3) expands

to include

progressively

shorter obser- vation times _as the bond disorder becomes _more extreme, so that

nearly

tricritical

ferromagnets

should represent ^{the best}

experimental

candidates for

assessing

the

applicability

of

equation (3).

Since these _are

precisely

the conditions which favour _reentrant

ferromagnetism,

it follows that the behaviour of the relaxation isotherms within the

ferromagnetic phase

of the

reentrant

Aug~fei~ alloy (viz.,

a weak power law

decay

with _an exponent m 0.04

0.08,

and

negligible

wait time

effects)

_may be

interpreted

_as direct confirmation of the theoretical

predictions.

In summary, measurements of the low field thermoremanent relaxation _on either side of the tricritical

point

in the Aufe system ^{have revealed the existence of} a

temperature-driven

crossover between two distinct relaxation

regimes

in

ferromagnets

where the

exchange

bond

disorder is sufficient _to generate

sequential (or

_« reentrant

»)

behaviour _one of the

regimes

is coincident with the _«

high

temperature »

ferromagnetic phase,

and, within the

slow-cooling

constraints of the current

investigation,

is characterized

by

an

extremely rapid approach

to

equilibrium

and

by

a weak power law

temporal decay

of the thermoremanent

magnetization

consistent with the

predictions

of

droplet

fluctuation models

[8]

of random

Ising ferromagnets,

while the other

regime

coincides with the

« low

temperature

» reentrant

phase,

and exhibits

nonequilibrium, age-dependent

relaxation

dynamics

which _are

virtually indistinguishable

from those in the _« pure »

spin glass phase,

and which _are describable

by

a common relaxation

function which has been linked _{to a}

mesoscopic picture [10]

of

growth

and

fragmentation

of

spin glass

domains.

Acknowledgements.

This work has been

supported

in part

by

_{a grant} from the Natural Sciences and

Engineering

Research Council of Canada.

References

[I] CRANE S. and CLAUS H., Solid State Commun. 35 (1980) 461.

[2] COLES B. R., SARKISSIAN B. V. B. and TAYLOR R. H., Philos. Mag. B 37 (1978) 489.

[3] GABAY M. and TouLousE G.,

Phys.

Rev. Lett. 47 (1981) 201.

[4] LANGE S., ABD-ELMEGUID M. M. and MICKLJTz H.,

Phys.

Rev. B 41 (1990) 6907.

[5] MIREBEAU I., HENNION M. and LEQUIEN S., J.

Appl.

Phys. 63 (1988) 4077.

[6] FISHER D. S. and HusE D. A.,

Phys.

Rev. B 38 (1988) 386.

[7] FISHER D. S. and HusE D. A., Phys. Rev. B 38 (1988) 373.

[8] HusE D. A. and FISHER D. S.,

Phys.

Rev. B 35 (1987) 6841.

[9] NORDBLAD P., SVEDLINDH P., LUNDGREN L. and SANDLUND L., Phys. Rev. B 33 (1986) 645.

[10] KOPER G. J. M. and HILHORST H. J., J.

Phys.

France 49 (1988) 429.

[I I] SARKISSIAN B. V. B., J.

Phys.

F _: Metal

Phys.

ii (1981) 2191.

[12] KUNKEL H., RosHKo R. M., RUAN W. and WILLIAMS G., Philos. Mag. B 63 (1991) 1213.

[13] LUNDGREN L., NORDBLAD P. and SANDLUND L., Europhys. Lett. 1 (1986) 529.

[14J NORDBLAD P., LUNDGREN L., SVEDLINDH P., SANDLUND L. and GRANBERG P.,

Phys.

Rev. B 35

(1987) 7181.

[15] CHAMBERLIN R. V. and HAINES D. N.,

Phys.

Rev. Lett. 65 (1990) 2197.

[16] OGIELSKI A. T.,

Phys.

Rev. B 32 (1985) 7384.

[17] LUNDGREN L., NORDBLAD P. and SVEDLINDH P.,

Phys.

Rev. B 34 (1986) 8164.

[18] STONER E. C. and WOHLFARTH E. P., Philos. Trans. Roy. Soc. 240 (1948) 599.

[19] CHAMBERLIN R. V, and HOLTzBERG F.,

Phys.

Rev. Lent. 67 (1991) 1606.

JOURNAL DE PHYSIQUE I T 2, N' 12, DECEMBER J992 83

regus

au

The

properties

of nuclei G. A. JONES

Sdrie _:

O~fiord Physics

Series12

(Oxford University Press, 1987)

^ISBN :

0-19~851869~2,

193 p., £20.

Monographie

d'introduction h la

physique

du noyau;

7chapitres: propridt£s gdndrales,

forces nud6aires, moddles, d6croissance

alpha

et bdta, transitions

dlectro-magndtiques,

rdactions nuddaires, fission _et fusion,

appendices,

index.

Aspects

of nuclear science

(in

honour of Prof. A. C.

Pappas)

E.

HAGEBO,

B. SALBU Eds.

(Norwegian University Press,

distribu6 par Oxford

University Press, 1987)

ISBN _: 82-00~

18343-2,

177 p., £ 20.

Comptes

rendus d'un

symposium

tenu h Oslo (10/1985), 14 contributions.

Nuclear

chemistry

A.

V#RTES,

I. KISS

Sdrie _:

Topics

in

Inorganic

and General

Chemistry

22

(Elsevier, 1987)

ISBN _:

0-444-99508~0,

619 p., Dfl.

275, $

122.25.

Monographie spdcialis6e

12

chapitres

de la thdorie _aux

applications,

index.

Frontiers of

Heavy

Ion

Physics

N.

CINDRO,

W.

GREINER,

R. CAPLAR EdS.

(World Scientific, 1987)

ISBN _:

9971-50~392-1, 541p.,

£ 62.55.

Comptes

rendus d'une confdrence tenue h Dubrovnik, 15-19/06/87.

TeV

Physics

and

Beyond

R.

DELBOURGO,

J. R. Fox Eds.

(World Scientific, 1987)

ISBN _:

9971-50-301~8,

309 _p., £ 66.85.

Comptes rendus d'une confdrence tenue h Launceston, Australie, 2-7/02/87.

Particle accelerators _;

applications

in

technology

and research

W. SCHARF

(Wiley, 1989)

ISBN :

0-471-92206-4,

663 p., £ 77.50.

Monographie sp6cialis6e,

traduite du

polonais

_; 11

chapitres,

index.

Elementary Particles,

^{3rd Edition} I. S. HUGHES

(Cambridge University

Press,

1991)

ISBN _:

0-521-40739-7,

431 p., £

50/16.95,

$

80/29.95 (relid/brochd).

Monographie

d'introduction 16

chapitres

niveau second

cycle

_;

premibre

Edition _en 1972.

Neutrino

Physics

K. WINTER Ed.

Sdrie _:

Cambridge Monographs

_on Particle

Physics,

Nuclear

Physics,

and

Cosmology,

Vol. I

(Cambridge University Press, 1991)

ISBN _:

0~521-36452-3,

670 _p., £

75,

125.

Monographie sp£cialis6e; 6chapitres:

histoire,

propridt£s intrinsdques

de neutrinos, interaction neutrinos-matidre, Etude

expdrimentale

de l'interaction faible, Etude de la structure des nucldons h l'aide des neutrinos, neutrinos _en

astrophysique

et en

cosmologie.

Elementary

Particles and the

Universe, Essays

in Honor of

Murray

Gell-Mann

J. H. SCHWARTz Ed.

(Cambridge University

Press,

1991)

ISBN _:

0-521-41253-6, 212p.,

£

24.95, S

49.95.

Collection de 18 courts articles; compte ^rendu

partiel

d'un

symposium

tenu au CalTech 27- 27/01/1989.

The

interacting

Boson-Fermion Model F.

IACCHELLO,

P. VAN ISACKER

Sdrie _:

Cambridge Monograph

_on Mathematical

Physics (Cambridge University Press, 1991)

ISBN _:

0-521-38092-8, 312p.,

£

40, $

75.

Monographie

spdcialis6e ⁴ parties, ¹²

chapitres,

index.

Electromagnetism (2e Edition)

I. S. GRANT, W. R. PHILLIPS

Sdrie _: Manchester

Physics

Series

(Wiley, 1990)

ISBN _:

0-471-92712-0,

525 p., £ 14.95.

Livre de cours, contenu

classique,

I* ddition 1975.

Electromagnetics

B. B. LAUD

(Wiley-Halsted Press, 1987)

ISBN _:

0-470-20746-9,

335 p., £ 24.20.

Livre de _cours, niveau

2ndcycle.

An introduction to

applied electromagnetism

C. CHRISTOPOULOS

Sdrie _:

Wiley

Student Series in Electronic and Electrical

Engineer (Wiley, 1990)

ISBN _: 0-

47l-92761-9,

183 p., £ 14.95.

Livre de _cours, niveau

lercycle.

Differential

equations,

their solutions

using symmetries

H. STEPHANI

Edited

by

M. MACCALLUM

(Cambridge University Press, 1990)

ISBN _:

0-521-36689-5,

260 p.,

£12.95, S

19.95.

Monographie sp6cialis6e.

Introduction to

Group

Characters

W. LEDERMANN

(Cambridge University

Press,

1987)

ISBN _:

0-521-33781-X,

227 p., £

8.95, S

14.95.

Monographie spdcialis£e, math6matiques grin£rales.

I* Edition _en 1977.

Les mdthodes tensorielles de la

Physique

: Calcul tensoriel dans _un continuum

structurd,

Vol. 2

J. WINOGRADZKI

(Masson, 1987)

ISBN _:

2-225-80575-X,

270 p., 235 F.

Math6matiques gdndrales, monographie spdcialisde,

analyse tensorielle, th60rie des espaces h connection affine et espaces de Riemann.

Contrblabilitd

Exacte,

Perturbations et Stabilisation de

Systkmes

distribuds Tome I _: Contrblabilitd Exacte

Tome 2 _: Perturbations J. L. LioNs

Sdrie _: Recherche en

Mathdmatiques Appliqudes

RMA 8-9

(Masson, 1988)

ISBN _: 2-225-

8l477-5 et

81474-0,

540 ₊ 270 _p., 450 F les deux _tomes.

Monographie. Equations

_aux ddrivdes

partielles

de la

m6canique

et de la

physique.

On numerical

approximation

in bifurcation

theory

M.

CROUZIEX,

J. RAPPAZ

Sdrie Recherches _en

Mathdmatiques Appliqudes

RMA 13

(Masson/Springer, 1989)

ISBN _: 2-

225-81794-4,

163 p., 160 F.

Monographie spdcialis£e

: bifurcation des solutions des

£quations

_aux ddriv£es

partielles, approxima-

tion et mdthode des 616ments finis.

Cyclic Cohomology,

within the differential

envelope (an

introduction to Alain Connes' _non- commutative differential

geometry)

D. KASTLER

Sdrie _: Travaux _{en cours} 30

(Hermann, 1988)

ISBN _:

2-7056-6044-5,

183 p., 180 F.

Monographie spdcialisde.

Feuilletages riemanniens, quantification gdomdtrique

et

mdcanique

P.

DAZOR,

N.

DESOLNEUX-MOULIS,

J.-M. MORVAN

Sdrie Sdminaire sud-rhodanien de

gdomdtrie

^VII

(Hermann, 1988)

^ISBN :

2-7056-6076-3,

184 _p., 160 F.

Monographie spdcialisde.

TD et TD de

Topologie gdnkrale

A. FAISANT

(Hermann, 1988)

ISBN _:

2-7056-5745-2,

304 p., 98 F.

Livre d'exercices.

Chapitres topologie

fonctions continues espaces

sdpards,

compacts, connexes

espaces

mdtriques

_espaces nomads, _espaces fonctionnels.

Chaque chapitre

comporte un rappel de _cours, des dnoncds d'exercices et de

probldmes puis

leurs solutions. Niveau 2nd

cycle.

Almost

Complex Homogeneous Spaces

and their Submanifolds

K. YANG

(World Scientific,

distributed

by Wiley, 1987)

^ISBN :

9971-50-377-8,

l12 p., £ 22.35.

Monographie spdcialisde.

Matrix

Logic

_;

theory

and

applications

A. STERN

(North-Holland, 1988)

ISBN _:

0-444-70432-9,

218 p.,

Dfl.130,

68.50.

Monographie spdcialisde. Chapitres

_: Calcul matriciel

logique

conventionnelle formulation matri- cielle application ^{h la}

physique

fondamentale, au calcul.

Differential

Equations

C. M.

DAFERMOS,

G.

LADAS,

G. PAPANICOLAOU EdS.

Sdrie _: Lecture Notes in Pure and

Applied

Mathematics Series, V. II 8

(Marcel Dekker, 1989)

ISBN _:

0-8247-8077-9, 787p., $150.

Comptes

rendus de la confdrence

EQUADII~F

1987.

Ondelettes et

opkrateurs

I _: Ondelettes

Y. MEYER

Sdrie _: Actualitds

Mathdmatiques (Hermann, 1990)

ISBN _:

2-7056-6125-0,

220 p., 186 F.

Monographie spdcialisde.

Perturbation Methods in

Optimal Control

A. BENSOUSSAN

Sdrie _: Series in Modern

Applied

Mathematics

(Gauthier-Villars, 1988)

ISBN _: 2-04-016469-

3,

573 p.

Monographie spdcialis6e.

Mathematical

physics

P. K. CHATTOPADHYAY

(Wiley, 1990)

ISBN _:

0-470-21719-7,

352 p., £ 28.30.

Cours de 2nd

cycle. Chapitres:

Variables

complexes; dquations

diffdrentielles, du 2ndordre,

probmmes

aux limites _; fonctions

spdciales

fonctions de Green _; espaces vectoriels et matrices thdorie des groupes.

The

nonequilibrium

statistical mechanics of _open and closed systems K.

LINDENBERG,

B. J. WEST

(VCH, 1990)

ISBN _:

3-527-26638-0,

448 _p., DM

88,

£34.50.

Monographie

_{portant sur}

quelques sujets

choisis _en

mdcanique statistique classique

et

quantique

des

systdmes

ferrnds _{et ouverts.} Huit

chapitres

en deux

parties.

Horizons of

Physics

A. W.

JOSHI,

Ed.

(Wiley Eastem, 1989)

ISBN _:

0-470-21329-9,

383 p., £ 32.15.

Collection de

quinze

articles d'introduction h la

physique contemporaine

rassemblds h l'intention de l'lndian Association of

Physics

Teachers.

Principaux sujets

abord6s dans _ce volume (let d'une s6rie) _: relativit6,

m6canique quantique, philosophie

de la

physique,

_rayons X, effet Mbssbauer, semiconduc- teurs

amorphes

et

liquides, physique

nuc16aire,

particules

£I£mentaires,

quarks

et

gluons,

structure des hadrons,

astrophysique.