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Acoustic susceptibility of an insulating spin-glass in an applied magnetic field
P. Doussineau, A. Levelut, W. Schön
To cite this version:
P. Doussineau, A. Levelut, W. Schön. Acoustic susceptibility of an insulating spin-glass in an applied magnetic field. Journal de Physique I, EDP Sciences, 1991, 1 (3), pp.415-440. �10.1051/jp1:1991143�.
�jpa-00246341�
J.
Phys.
11 (1991) 415-440 MARS 1991, PAGE 415Classification
Physics
Abstracts75.50Lk 62.25 + k
Acoustic susceptibility of
aninsulating spin-glass in
anapplied magnetic field
P.
Doussineau,
A. Levelut and,W. Sch6n(*)
Laboratoire
d'Acoustique
etOptique
de la Matidre Condens6e(**),
Universitb Pierre et Marie Curie, Tour 13, 4place
Jussieu, 75252 Paris Cedex 05, France(Received
8 November 1990,accepted16
November1990)
Abswact. The
propagation
oflongitudinal
acoustic waves offrequency
between 30 MHz and 800 MHz has been studied in theinsulating spin-glass (CoF2)05(BaF2)02(NaP03)03.
This was achieved in the temperature range 1.2 to 4.2K which includes the critical temperature T~ = 1.8 K, with anapplied
magnetic field up to 9 Teslas. The results are thefollowing.
i) Thevelocity
shows ananisotropic
behaviour. Itdepends
on theangle
between the field and the acoustic wavevector. ii) The initialslope
of thevelocity
versus the square of themagnetic
field which measures a non-linearmagneto-elastic
coefficient presents a veryrapid
variation(a jump)
at 2~. ii@ Below 2~ the velocity presents a minimum, whereas the attenuation has a maximum for H around I T. iv) Above T~ the
velocity
shows acomplicated
behaviour: a maximum followed
by
a minimum and
finally
an increase with the field. v) Forhigh
fields thevelocity
increases with the field, the lower the temperature the steeper theslope,
and a trend towards saturation is observed at the lowest temperatures for thehighest
fields. In the same field range the attenuation decreases with the field. All these data areinterpreted
within the framework of the staticSherrington- Kirkpatrick
modelincluding
twospin-strain coupling
mechanisms a Waller mechanism and a field induced mechanism. It is shown that, besides termscoming directly
from the twocoupling
mechanisms, crossed terms notpreviously
taken into account areimportant.
The value of thejump
of the non-linearmagneto-elastic
coefficient, the minimum of thevelocity
below 2~, and the succession of a maximum and a minimum above T~ for the velocity are wellexplained.
The
dynamical
effects arebriefly
considered for theparamagnetic phase
in low field.I. Introduction.
Because of time-reversal invariance the
coupling
ofspins
with an elastic strain isgenerally
either
quadratic ~van
Vleckmechanism)
or bilinearovaller mechanism)
inspin
operators.As a consequence, acoustic
experiments provide specific
information onspins
in solids ; for instancethey
allow the measurement offour-spin
correlation functions. Thisinteresting
(*)
New address : GEC-ALSTHOM, Division Transports, TourNeptune,
Cedex 20, 92086 Paris La Ddfense.(**)
Associated with the Centre National de la RechercheScientifique (URA 800).
property
applies
to anymagnetic materials, including spin-glasses. However, probably owing
to technical
difficulties,
the acoustic studies ofspin-glasses
are rather rare[1-3], especially
if their number iscompared
to the multitude ofexperiments using
othertechniques [4].
Thesituation has been
slowly changing
for the last few years and several measurements of soundvelocity
and attenuation have beenpublished [5-9],
inparticular
oninsulating spin-glasses.
Recently
we havereported
a detailed acousticstudy
of a cobaltfluoro-phosphate spin-glass (COF~)o_~(BaF~)o_~(NaPO~)o~
from bothexperimental
and theoreticalpoints
ofview,
but limited to the case where nomagnetic
field isapplied [7].
In what follows thisstudy
is referred to as I. In thepresent
article wegive
areport
of an extension of the work described in I.Experiments
on the samecompound
have beenperformed
in amagnetic
field up to 9 Teslas(T).
Several new features have been observed :I)
a minimum ofvelocity
as a function of the field below thespin-freezing
temperature T~ and a maximum followedby
a minimum above T~it)
athigher
fields astrong
increase ofvelocity
which saturates at thehighest
fieldsiii)
arapid
variation near T~ of the initialslope
of the elastic constant as a function of the square of themagnetic
fieldiv)
an attenuation decrease when the field increases. Thevelocity
dataare
interpreted
in the framework of theSherrington-Kirkpatrick
model[10].
Tliis model considersonly
the static case.Unfortunately
thespin dynamics theory
used in I cannot be extended when a finitemagnetic
field isapplied.
As a consequence theinterpretation
of ourattenuation data is
only
schematic. Nevertheless we canexplain
most of our dataqualitatively
and even some of them
quantitatively.
Severalspin-strain coupling
mechanisms have beenconsidered, including
acoupling
inducedby
the field which wasevidently
absent in I. Shortreports
of this work have beenpreviously published [I1, 12],
withouttaking
into account the field inducedspin-strain coupling.
The
organization
of the article is as follows. We first present ourexperimental
results which include a detailedstudy
in very low fields and studies in medium andhigh
fields(Sect. 2).
After a short sketch of the
Sherrington-Kirkpatrick
model(Sect. 3)
weapply
the method to the calculation of the static elastic constant for anyspins
S m1/2 (Sect. 4)
andthen,
with moredetails,
forspins
S=
1/2 (Sect. 5).
Adynamical model
validonly
for S=
1/2
in low fields ispresented
in section 6.Finally
we discuss our data in the frame of these theories(Sect. 7)
andwe conclude
(Sect. 8).
As
already
mentioned the present article is the continuation of another refered to as I. As far aspossible
we have used the samesymbols
in both. However this article is intended to beself-contained.
2.
Experiment.
2.I METHOD. Ultrasonic waves were
propagated
in asample
of themagnetic insulating
amorphous compound (COF~)o_~ (BaF~)o_~ (NaPO~
)o_~. In this material thefreezing
of thespins
occurs at T~ =1.8 K
[6],
a critical temperature which is within theexplored temperature
range 1.2 K
< T < 4.2 K.
Longitudinal
ultrasonic waves withfrequency varying
from 30 MHzup to 800 MHz were used. The
sample
was immersed in aliquid
helium bath whosetemperature was
changed by pumping.
Two different set-ups were used. In the first one,magnetic
fields up to 9 T wereproduced by
asuperconducting
coil. In this case, themagnetic
field was
applied parallel
to the acoustic wavevector. In the second case,magnetic
fields up to 2.I T were deliveredby
anelectromagnet.
Then themagnetic
field can be rotated in aplane containing
the wavevector. In both cases the data were collected at fixedtemperature
as afunction of the
magnetic
fieldapplied
after thetemperature
was reached. A verygood
accuracy was obtained for the relative sound
velocity change
measurements. It was about 10~~ at 100MHz, decreasing
athigher frequencies
due tostronger absorption
in the material.Changes
in the ultrasonicabsorption
down to 0.02 dB cm~ can be detected.bt 3 ACOUSTICS OF A SPIN-GLASS IN A MAGNETIC FIELD 417
2.2 RESULTS. The elastic constant is an even function of the
magnetic
field H.Consequently
wepresent
our data as functions of the square of H. The fullmagnetic
field range studied in theexperiments
will be divided into threeparts
forclarity.
2.2. I
High jieliig,
This domain concerns the fieldhigher
than 2 T(H~
~ 4
T~).
Thevelocity
results are shown infigure
I for several fixedtemperatures.
The measurements were doneonly
at the lowestfrequency (30 MHz).
Thevelocity change
is firstnearly
linear withH~
up _toan upper limit which increases with temperature
(it
isequal
to about30
T~
at I.4 K and it islarger
than 80T~
at 4.2K),
the lower thetemperature
thelarger
theslope
ofAV/Vo
versusH~.
At thehighest
fields the curves bendaround, showing
a trendtowards saturation of the
velocity change.
This behaviour isclearly
observedonly
for the two lowest temperatures(I.4
K and 1.8K).
It is worthnoting
that theasymptotic
value at 1.8 K islarger
than it is at I.4K. In the same field range the attenuation decreasesroughly
asH~,
as shown infigure
2 for 86 MHz ultrasonic waves.2.2.2 Intermediate
fieliig.
This domain stretches out from about 0.3 T up to 2 T. It was studied in detailby varying
thefrequency
and theangle
q~ between themagnetic
field and theultrasonic wavevector.
v.30 MHz
~o ~~~ ~~~
~o°~
,3 K
oo ++ ~"~
~o# ~
#
DP I
°
"
~o° z °
+~ < °
4
++~ m fl
a
~
~ x
>
~
°
x <
~
a x m
~
_
+g XX~
~
fl
I 4 +~ i B x ~'
o xX
$
°E
+ x~
«
~ uo
+ °
x a-B
,
+ ° x~ ~/
a $
+ ° x~ ~
° i
+ x~ "
°
°
x ~
~
x ~* M
+~ x *
x
_/
°>~~° ~~
w*** 4 2
~
°
+o x w*"
~~ ~~
~ ~
*"
o 20 40 60 0 2000 4000 6000
H~ (T~1 H~ lT~l
Fig.
I.Fig.
2.Fig.
I. Relativevelocity change
forlongitudinal
elastic waves offrequency
30 MHzpropagating
in thespin-glass (CoF2)o
5
(BaF2)o.2(NaP03)o.3
as a function of the square of themagnetic
field for the full field range up to 9T. The wavevector is parallel to the field. A set of data was taken at the criticaltemperature 2~ = 1.8 K
(o).
The other setscorrespond
to a temperature lower than T~(+)
and two temperatureslarger
than T~(x)
and(*).
All the data arearbitrarily
put at zero for zero field.Fig.
2. Attenuation oflongitudinal
elastic waves offrequency
86 MHz propagating in thespin-glass (CoF2)05(BaF2)02(NaP03)03
as a function of the fourth power of the magnetic field for the full field range at T=
2.3 K.
The effect of the
angle
is shown infigure 3,
where thevelocity change
of a 148 MHz ultrasonic wave at IA K at various fixed values of themagnetic
field isplotted
as a function of theangle.
The main results are thefollowing.
Thevelocity change
can beanalysed
asbV/Vo
=A+Bsin~
q~ where coefficients A and B
depend
on themagnetic
field(their
variations are shown in
Fig. 4).
The effect is moreanisotropic
forhigh
fields. It isstronger
when the field isapplied perpendicular
and lower when the field isparallel
to the ultrasonicwavevector. Because the behaviours are
qualitatively
similar whatever the direction of thefield,
in thefollowing
the results will bepresented only
for amagnetic
fieldapplied
perpendicular
to the ultrasonic wavevector.o
T.I.4 w
w
v.14B MHz
o
~
x
+ + fl
w a
~ ~ > o
+ + /+ + ~ m
~
~/~
~
$ o
~
l
$ + + X X X X ~ ~
< a b4
x w fl
~
g
x x ,- o ° ~
x x x rJ W
x o b4 °
~ o a >
"
a w
o o w ~
x w m
, J >
, J z Do
, J m
°, Jo m ~ M
" ~,
_~~
J' x T-I 4 K ~
~ «
o
~ v.14B MHz
~ l'
~~
90 0 I 2
ANGLE (degreesl H~ lT~)
Fig.
3.Fig.
4.Fig. 3. Relative velocity
change
oflongitudinal
elastic waves offrequency
148 MHzpropagating
in thespin-glass (COF~)o~(BaF2)o.2(NaPO~)o~
as a function of the angle q~ between theapplied
magnetic field and the ultrasonic wavevector at the fixed temperature T= IA K. Theorigin
of the velocity change is taken at its value withoutmagnetic
field. The different linescorrespond
to3V/Vo
=
A + B sin~ ~, where A and B are free parameters
depending
on themagnetic
field. The varioussymbols correspond
to different values of themagnetic
field(+)
H= 0.4I T;(x)
H=0.56T;(o)
H
=
0.83 T ;
(*)
H=
1.18 T.
Fig.
4. Parameters A(o)
and B (*) describing theanisotropy
of the effect of amagnetic
field on thevelocity
of elastic waves in thespin-glass (COF~)05(BaF2)02(NaPO~)03
as a function of the square of the field.Figure
5 shows severalexamples
ofvelocity changes
at 148 MHzplotted
as a function of the square of themagnetic
field. For T< T~ a remarkable feature is observed the
velocity
firstdecreases,
then passesthrough
a minimum andfinally
increases asexpected
from thehigh-
field results shown in
figure
I. At T~ T~ the behaviour iscompletely
different and morecomplicated
: thevelocity
firstincreases,
then goesthrough
a maximum which is followedby
aminimum and
finally
increasesagain.
The minimum is not observed belowH~
=
4
T~
for thebt 3 ACOUSTICS OF A SPIN-GLASS IN A MAGNETIC FIELD 419
1
1
v .14B MHz
0 1 gi
~ x
~
)
X
X $
x
x ~
x b4
~
~
I
1 x14B
I
x ~
i.3 ~
~
j
~ i X
fi
x
~ ~ x ~
#
~ ~ x
x
j
~
~ <
x
_
~
fl
-I
x x x
o ~
a
o
»
o «
° fl
~
(
o o o 0
°269
c~
~ o
°2.3
~
jf
Do o a a
«
~ w
w z + ~
m
>
*
~ 506 Q
~ ~
~ ~
Z
+ *~
~
~ w
w w ~
~ w* * fl
* j
I
G
+
+
~
+ +
+ + +
4.2 +
+
o
~2 j~2j ~2 j~2j
Fig.
5.Fig.
6.Fig.
5. Relativevelocity change
oflongitudinal
elastic waves offrequency
148 MHzpropagating
in thespin-glass (COF~)o~(BaF~)o~(NaPO~)o~
as a function of the square of themagnetic
field,applied perpendicular
to the ultrasonic wavevector for different temperatures. The vertical shift for the various sets of data isarbitrary.
The number near each set of data indicates the temperature in Kelvins. The critical temperature is T~ = 1.8 K.Fig. 6. Relative
velocity change oflongitudinal
elastic waves ofvariousfrequencies propagating
in the spin-glass(COF~)o~(BaF~)o~(NaPO~)o~
as a function ofthe square ofthe magnetic field measured at thesame temperature T = 1.3 K. The
velocity
is measured with respect to its value in zero field which is in factfrequency dependent.
The number near each set of datagives
thefrequency
in MHz.highest temperatures.
Thefrequency dependence
of thevelocity change
is shown infigure
6 at 1.3 K. The minimum is shifted tohigher
temperatures when thefrequency
increases. Thevariation at low fields is
independent
of thefrequency.
In the same field range the ultrasonic attenuation a shows also some
interesting
features.Figure
7 presents some results transformed into theimaginary part
of the elastic constant C(3m~~ Co
=~a~~~~
as a function of the square of themagnetic
field. Forw 10
T< T~ the
imaginary part
of the elastic constant has a maximum before to decrease. For T ~ T~ it increases with the field up toH~
= 4
T~.
Thefrequency dependence
of theimaginary
part of the elastic constant is shown at T= 1.3 K in
figure
8.2.2.3 Low
fieliig.
For lowapplied fields,
thevelocity changes
have been studied in detail.Some curves at a
frequency
of 269 MHz are shown infigure
9. In this field range thebehaviour is
frequency independent
between 90 MHz and 800MHz,
as can be seen infigure
6. This is in contrast with what we have observed in zero fieldexperiments
in the samefrequency
range. In thisfigure
all the curves at differenttemperatures
havearbitrarily
thesame value for zero field. This means that the zero field temperature
dependence
which hasI x
A~
(
(i i$
g i x
~a~
~i
o + + ~
i i~
a a >
a
~ b4
x a ° o 3.6 f~ ~
x ~
~
o
ji
~9i
fl
Do fl
# x
1 ~
~
nD >
~ i z
x >
+
~ ~
~
o
o + 1 ~
x
q
x14B
+ ~x
~ ~
o m
4
m
x r
p
+x >
x
~_
a fl
i'~ fl
Bo2+x 3
b4 rJ
+ rJ
x x o
a x z
+ fl
x 269 ~
° ~
T.I.3K °
v.14B MHz x x
x Q
i~
"
o 5 ° 5
H~ lT~l H~ lT~l
Fig.
7.Fig.
8.Fig.
7.Imaginary
part of the elastic constant(it
is related to the attenuationby
3m~ ~
=
~'
a
~~ ~~
C w lo
of the
spin-glass (CoF2)o
5
(BaF2)o.2(NaP03
)o.3 as a function of the square of themagnetic
field. The datawere obtained with
longitudinal
waves offrequency
148 MHz.They
are shown for two temperaturesgiven
in Kelvinsby
the number near each set of data. The zero of the vertical scale isarbitrary
andcorresponds
to the value in zero field which is in fact temperature dependent.Fig.
8.Imaginary
part of the elastic constant for thespin-glass (COF~)o_~(BaF~)o_~(NaPO~)o~
as afunction of the square of the
magnetic
field. The data were taken at 1.3 K for variousfrequencies
of thelongitudinal
elastic waves. The number near each set of datagives
thefrequency
in MHz. The zero of the vertical scale isarbitrary.
Itcorresponds
to the value without magnetic field which is in factfrequency dependent.
been studied
previously
is discarded. This initialslope
ofAV/Vo
as a function ofH~
iscompletely
different below and above T~. It isnegative
for T< T~,positive
for T ~ T~ and zerojust
at 7~.3. The
Sherrington-llirkpahick
model and itsgeneralization.
Many
features observed inspin-glasses
are accounted for(at
least as a firstapproximation) by
a model
proposed by Sherrington
andICirkpatrick (SK) [10].
These authorsrepresent
a SG asan
assembly
of NIsing spins coupled together by
random Gaussianexchange
constants/,
scaledby
a standard deviation$ $
anda mean value
(IN.
The model may be extended to the case ofspins S~1/2
and the effect of amagnetic
field H and aligand
field D may beincorporated [13].
Then the Hamiltonian of thespin
system is3C=-~z/~S;)-g~pBH£S;-DjjS) (I,j=1,2,..,N ). (I)
~,,J
, ;
bt 3 ACOUSTICS OF A SPIN-GLASS IN A MAGNETIC FIELD 421
w
*
* * 3.0
*
*
*
*
~ 4.2
*I
.
~ ~
+ + +
a o
fi
a r
~ o o o
x rJ
~
i-B ° $
x
x
x I-a
i~
x o
x
v .269 MHz
o 04 OS
H~ lT~l
Fig.
9. Relative velocitychange
oflongitudinal
elastic waves offrequency
269 MHzpropagating
in thespin~glass (COF~)o
~
(BaF~)o ~(NaPO~)o
~ as a function of the square of the
magnetic
field in a reduced field rangeshowing
the temperaturedependence
of the initialslope.
The number near each set of datagives
the temperature in Kelvins. The critical temperature is 2~ = 1.8 K.As
usual,
g~ is the Land6 factor and pB is the Bohr magneton.Using
the famous «replica
trick » the
averaged
free energy is calculated and it is found to be :fl
=
NkT( p$m~+ (pjj~ ~p~- q~) E(In try e~~~ ~~~)) (2)
with
0~
=p $
m +
p lv~
t +
pgL
MB H q~~=
(pjj~ ~p
q +
pD.
2 The Gaussian
expectation
value is defined as :~"~
"
fi
~~~~
~~~~~~The values of the three parameters m, p and q have to be
self~consistently
determined for every value of thetemperature T,
themagnetic
field H and theligand
field Dby
:m =
E( (S~~)
=£ (S;)
; q=
E
((S)()
=
£ ($)~
P#E(lRj~)=(z(().
The
symbol ( )~
denotes a thermal average over aone~spin
space while the upper bar indicates an average over the disorder. The effective operatorp3C~j
=
0~
S + q~~S~
whichoccurs in the definition of the average
( )
is used for any otherhigher
orderparameters.
For the calculation of an elastic constant in a
magnetic
field we needthree~spin
correlation functions(in
this casethey
are notequal
to0)
andfour~spin
correlation functions :~
"E(ls~lj)
=
z@
h
=
E( ls~l
i
lsl j)
=( z (S~) IS,)
~ ~
~( j ~ 3) ~ ~3
fi~
,
r =
E( ls~l i)
=
( z (Sf)
s =
E(iS~>1)
=
z@~
,
~
j~( j~2j j~2) ~ j~2j j~ j2
j fi~ I ,
,
~ ~
E(lsll)
"
i lsl~
,
w
=E(ls~lj lslj) =~zfi)
It is useful to put :
f
=
3 v + s 4 u. A
higher~order
correlation functions is of interestx =
E( (S~()
=
jj ($)~
We recall that the
freezing temperature
T~ of thespins (spin~glass transition)
and theirordering temperature
To are related to thespin-spin
interaction constantsland $
respectively.
For S=
1/2
the relationssimply
read :1=4kT~ $=4kTo.
When we write some
quantities (such
as/ $,
g~,D,
T~ and To) without upperindex,
thismeans that an elastic strain may be present. On the contrary the upper index 0
specifies
thatno strain is
applied.
4. The static elastic constant in the SK model for any
spins
SmI/2.
The
magnetic
contribution to the elastic constant C is :3C
=
~~~ l(3)
il dE
e =o
~lJ is the volume of the
sample
while E is a static elastic strain.It is assumed that the two
parameters
which characterize thespin~spin
interactions becomeI= j°(I
+yE)
andI= $(I
+yE)
when the strainE is
applied.
In the same manner theLandb factor and the
ligand
field tum out to be g~ =g((I
+fE)
and D=
D°+
GEbt 3 ACOUSTICS OF A SPIN~GLASS IN A MAGNETIC FIELD 423
respectively.
y andf
are dimensionless coefficients G has the dimensions of an energy.Then the
strain-dependent
free energy reads :~~~~ ~~~~ ~~~~~~
~~'~~ ~~~ ~~~~~
~~ ~~'~~~ ~~ ~~~
E
(In trj
e~~~~~~ ~ ~~~~~~))
(4)
with
0H(E)
=P4(1+ y~)m+ pJ°(i
+y~) It
+
pp~g[(i
+f~)H
wD(E)
=
(PJ°)~ (i
+y~)2 ~p-q)
+
p(D°+ G~).
Now the three averages m, p and q
depend
on H and E.It is easy to calculate the derivatives of the free energy with respect to the strain :
$
=
NkT( ) jp $m2+ (pJ°)2 (1+ y~)~p2- q2)j
+fpg[
p~ Hm +pGp (5)
((
=
NkTj y2(pJ0)2 ~p2 q2)
+pG )
+
+
IYPJ°Yn +fPg[
MBHI ~
+
Y(PJ°)~ (P )
q
) (6)
The
following
calculations can be put into a compact form with thehelp
of a(3
x3) symmetric
matrix£,
whose elements are functions of thespin
averages(see Appendix A)
calculated for E
=
0. The relative
change 3C/Co
of the elastic constant C=
Co
+ 3C turns out to be :+
2(fl~)~ (fl~)(L12
~IIPL13 ~ll~)
+(fl~)~ (L22P~
2~23P~
+L33 ~~)l
+ 2
Yf(fl~~@B lf)[(fl~)Lll
~ll +(fl~)~ (L12P L13~)1
+ 2
Yfl
G((fl~) L12
~ll +
(fl~)~ (L22P L23 ~)l
+
f~(fl ~~
@ B
H)~ Ll1
+ 2f (fl~i
@B
H) fl GL12
+(fl
~)~L22)
~~~JV
=
N/~lJ
is the number ofspins
per unit volume andCo
is the elastic constant in the samematerial without
spins.
The matrix elements L,~ aregiven
inAppendix
A.A
physical meaning
can be ascribed to the different terms of3C/Co.
The first term(proportional
to y is the contribution of the Waller mechanism(modulation
of thespin~spin
interaction
by
thestrain)
its Hamiltonian reads[14]
Jcw=-~yszfs,j.
~
,.j
The last term
(proportional
toG~
is due to the Van Vleck mechanism(modulation
of theligand field)
; thecorresponding
Hamiltonian is[15]
:3Cvv
=GE
jj $I.
This
operator
becomes a scalar(equal
to-NGS/4)
forspins
S=1/2.
The fourth term~proportional
tof~
comes from the modulation of the Landd factorby
the strain. It vanishes when nomagnetic
field isapplied
this is a field induced mechanism. Thecorresponding
Hamiltonian reads
[16]
3CFI ~
fg[
MB HEI $
,
The other terms
originate
in cross~effects. Some of themdisappear
in absence ofmagnetic
field.
We recall that the
magnetic susceptibility
x is defined as :x =
$ (8)
and it reads
X
~~(~~@B)~flLll (9)
This
explains
thesimilarity
of themagnetic susceptibility
x and the field~induced contribution to the elastic constant(3C/Co)~i.
We
emphasize
that thespin~averages appearing
in the matrix£ depend
on themagnetic
field H.
Finally
we underline that theexpression
of3C/Co
is validonly
if thereplica symmetric
solution of the SK modelis,
I-e- above thegeneralized
de Almeida~Thouless line[17, 18].
5. The static elastic constant in the SK model for
spins
S=1/2.
5,I GENERAL RESULTS. For
spins
S=
1/2 (with eigenvalues
±1/2)
theequations
obtained in theprevious
section becomesimpler
because several elementsL;j
are thenequal
to zero.However the
equations
remain awkward. This is due to numerous coefficients(1/2)~.
These coefficients are eliminated ifspin operators
s, witheigenvalues
± I are used. From here weadopt
this rule as we did in I for thespin dynamics.
Therefore all the averages have their value between 0 and I instead of 0 and(1/2)~
for an-spin
average. Wedistinguish
the new averagesby
a circumflex and we have :Jii=2m,@=4q, fl=I,d=8a=Jii,$=8b=Jii,d=8c,f=I,§=I,
Q=16u=
@,fit=16w= @,b=16v,I=3@-4f+1,I=64xandj=256y.
Similarly
the new matrix elementsI;y
are related to the old onesby
:~°11 "
4
Lll>
~°13~ 81~13> ~°33 "
161~33
>
while the other three elements are
equal
to zero. Moreexplicitly
the new set of elements reads :L12
=°1 L23
"°1 L~
= 0
iii
"
((1 4)(1 t/0~) 2(Jii d)~/0~) IA
i~~
=
2
jii g(i q)/oi I
2g(d e)2/oj IA
£i~
=
2(~ii e)/A
; A=
(1 fle2) ii
g(1 4)/oi
+ 2g(~ii e)2/03
bt 3 ACOUSTICS OF A SPIN~GLASS IN A MAGNETIC FIELD 425
where
=
T/7$
and g=
T~/l~° (-
co < g <).
With these new definitions thechange
of the elastic constant reads :~~
=
~~~~~
(y~[(1 #~)
+ 2g~ iii
~ii~ 4gij~ di#/@
+ 2
i~~ f~/0~]
+Co
2Co
°+ 4 y
fh [g0ijj
Jiiij~ @]/0
+ 2f~ h~ijj) (10)
h is a reduced
(dimensionless) magnetic field,
definedby
h=g[MBH/2k7~.
As
expected
the Van Vleckmechanism,
as well as the cross-effects where Gappeared,
have vanished. To ourknowledge
the cross-effectproportional
toyf
has never been taken into accountpreviously.
It isimportant
in our acousticexperiments
becausethey
wereperformed
on a system which combines two conditions : a
high spin density sample
and anapplied magnetic
field.It has been shown in I that 3
C/Co
must benegative
for any temperature. This property still holds if amagnetic
field isapplied.
Indeed themagnetic
contribution for anarbitrary
value of themagnetic
field can be obtainedonly by
the means of acomputer. However,
for very low and for veryhigh fields, analytic expressions
can be calculated.They
arereported
in thefollowing
two sections.5.2 LOW FIELD EXPANSIONS. Here the case
S=1/2
isconsidered,
with$#0
andp
g~ pB H « I. Since the elastic constant is an even function of the
magnetic field, incre~sing-
power
expansions
of 3Conly
contain terms H~ where n is an eveninteger.
In fact we limitourselves to terms
proportional
toH°, H~
andH~.
Indeed the termindependent
ofH has
already
been calculated in I. We have calculated the termsproportional
toHi.
The resultsare so much
complicated
that we do notgive
them in the present article. Weonly
mention that some of themdiverge
whenapproaching
the critical temperature both fromabove and from below.
Our calculation is
performed
in two steps : theparamagnetic phase
first and then thespin~
glass phase.
Wegive
them in terms of the dimensionless reduced parameters 0 and h.5.2.I
Paramagnetic phase.
Theexpansions
of the averages ~ii,#,
d and are as follows :~_
0~ ~2
'~~(@-g)
' ~(@-g)~(@~-l)
~2
~~~
~4~
~~2
~~
~
(@
-g)~ (@~- l)~
'~
(@
-g)~(@~- l)~~
The matrix elements
I;y
limited to the lowest order are :~
0~ 20~
~_
~ 20~
~~
(° gl'
~~ (@g)~ (@~- l)
' ~~(@~-
lThe sum of the three contributions to the relative
change
of the elastic constant is :~ ~ ~~~ ~~'~~
(o
~
g~31°'~+f(~ ~g)l~h~+. (ll)
We notice that the contribution
proportional
toh~
isalways negative
since 0 g ~ 0 and the other factor is asquared quantity.
5.2.2
Spin~glass phase.
The averages Jii, @, d and have thefollowing expansions
~*
~
lo
gIi 40)1~ ~
~~°
~io
g(i <o)12 (
o2to)
~d
=
~° ~°
h
=
to
+~~~ ~° ~°~
~~h~
1@
g(1 40)1
~je
g(1 q0)l~ (@~~ 10)
with
ko=2-17@o+30@o-153~.
The lower index 0 means that the averages are calculated for h
= 0. The matrix elements
I;~
are :(1 qo)
eto e2(e2
+ 2to)
~~~ ~
l~
g(1 40)1 lo
g(1 40)l~ (0~- to)
~2
lo
0Lj~
= hlo
g(1 40)l~ (02 lo)
2
i~
o2o210
g(1 40)1 to
+ g(o
2i~)
11~ ~
L33 = + 4 0 h
(°~ ~o) lo
g(I
@o)l~(0~- io)~
The total contribution due to the three processes is :
3c sG
JekT) i 0~
+ 3
lo ~
(
f~0 ° ~ ~'o 2-
i~
~~~
+
y2(g2(1 q~)3
o 4 g~°~~ ~°~ ~°
o202- to
~ ~
4i kol° g(1 40)1
~4
~40 li(° g)
~~
(02- to)~ (02- to)2
4010
g(1 40)1to ~~li
02 i~ io g(i q~)13
g(1- 40)2 (02- i~)
2i~q~
o~ ~
(i q~)
o~~
~~~ ~
~'~
1°
g(1 40)l~ (°~ lo)
~~
° ~(~ 4°)
~'5.2.3 Behaviour around T~. In order to
study
the behaviour of(3C/Co)
nearT~
(I.e.
for 0m
I)
we put 0 = 1+ 3(then (3 «1).
We get in theparamagnetic phase
(3~0):
bt 3 ACOUSTICS OF A SPIN-GLASS IN A MAGNETIC FIELD 427
~j)~~-~)y~(1-3+.
+~~~ 6g~
~~j~~
° ~ o
(I-g)~ (i-g)4
+.(13a)
~~ ~
=
'~~~~
f2
3 ~~°
FICo
l g I g)2~
(13b)
~~ ~
=
~~~2
y
fg
2 3~~
~°
CCo (i
g)2 (1
g)3(13c)
and in the
spin-glass phase (3
<
0)
:~ )~~=-~~~y2~l+3+.
+(~(2-3g)
20 ~ ~°'~
~~° 3(1-g)~ 3(1-g)~
~' ~~' ~~~~~~ ~ ~~~~~ ~
~
~~~~~) ~~+' (14b)
) ~~
=
~~2 yf
+
~
~2
0 C 0 g
3(j
g)2
(14c)
If T tends towards T~ the sums of the three contributions take the
following
values in the twophases
II ~
-
llll~ Y~
+~~°' ii~~ Ii ~~~~ ~l (15)
~
~~~~ °'~
~
~
~~~'
g
~~
~'~~
~
~~
These results deserve several comments.
I) The term
proportional
toh~
exhibits a finitediscontinuity
at T~:~ )~- ~ ~~
=
l~~~ "~~~~ ~~
" ~~)~ ~~~~ h~ (17)
o o o
3(1- g)
it)
The finiteness of thediscontinuity
is not a trivial result.Indeed,
in the calculation several terms have adivergence
butthey fortunately
cancel.iii)
Thediscontinuity
vanishes if the Waller coefficient y isequal
to zero. On the contrary if the coefficientf
of the field-induced mechanism isequal
to zero, adiscontinuity
remains.iv)
The termproportional
toh~
in the elastic constantcomes from a contribution
proportional
tos~h~
in the free energyexpansion.
Therefore the coefficient may beinterpreted
as a non-linearmagnetoelastic
coefficient.v) This coefficient is
always negative
in theparamagnetic phase
since g< I. On the
contrary
it may have anysign
in thespin-glass phase.
Exactly
at T~ the three contributions read~~
=
~~~ 2[~ (h(
~°
W ~ f~0/(1
g
)
3C
A~k7~
~
~2
~
PICo ~
fi
3C
A~k7~ 2g_j
~
~~° ~~(l-g)~~~
The field-induced contribution can be obtained
by continuity
from(13b)
or(14h).
On the contrary the other two contributions aresingular
andthey
must be calculateddirectly.
5.2A
Comparison
between X and3C/Co.
From the definitionsgiven by (3)
and(8),
arelationship
between 3C and x iseasily
deducedd~(3
C Id~fl d~x
dH~
~lTdH~ ds~ ds~
It is valid for any H and any s, but we use it in the case H
= 0 and s
=
0.
Starting
from(9)
we get :d2x A'(gl
pB)~
d2£jj d£jj
~
ds~
4 kTW
+~f t
~ ~~
~~~ ~~~~For H
=
0, iii simply
reads:
40
~~~
l
g(1 40)/°
When an elastic strain s is
applied,
the two characteristic temperatures become7~= 7~(1+ ys)
andTo
=
7j~(1+ ys).
In the same manner the reduced temperaturebecomes
0/(1+ ys)
while the ratio g remainsunchanged. Consequently
the parameter@o which is a function of the reduced
temperature only
is transfornled into@o(0/(1
+ ye)).
Then we start from :
~
i-40(o/(i+ ye))
~~ ~ ~
i
gii 40(o/(i
+ys))i(i
+ys)/o
and we find out:dill ye d40
~
l
#o j
~
e =o
II g(1 40)/°l~
d° ~°
~~~l
=
Y~°
~
g2 40
3~ ~ ~
i
40 d#o
~
d40 dE~
e o i g
(1 40)/oi~
° ow w
+d@o 2 l
@o d~@o
+ 2 g d° 0 g ° doj
We first remark that each of the three terms in
(18) corresponds
to aspecific
interaction mechanism sincethey
areproportional
toy~,
yf
andf~ respectively.
Moreover we have nowsome
physical
argument for thediscontinuity
ofd~(
3 C)/dH~
at7~.
Indeed in theparamagnetic phase
@o =
0 for any T
~ T~ and
consequently d@o/d0
=0, d~@o/d0~
= 0. On the contrary, in thespin-glass phase,
in thevicinity
of T~ we have :~