Acoustic susceptibility of an insulating spin-glass in an applied magnetic field

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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 415

Classification

Physics

Abstracts

75.50Lk 62.25 ₊ k

Acoustic susceptibility of

_an

insulating spin-glass in

_an

applied magnetic field

P.

Doussineau,

A. Levelut and,W. Sch6n

(*)

Laboratoire

d'Acoustique

et

Optique

de la Matidre Condens6e

(**),

Universitb Pierre _et Marie Curie, Tour 13, 4

place

Jussieu, 75252 Paris Cedex 05, France

(Received

8 November 1990,

accepted16

November

1990)

Abswact. The

propagation

of

longitudinal

acoustic _waves of

frequency

between 30 MHz and 800 MHz has been studied in the

insulating 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 _an

applied

magnetic field _up to 9 Teslas. The results _are the

following.

i) The

velocity

shows _an

anisotropic

behaviour. It

depends

_on the

angle

between the field and the acoustic wavevector. ii) The initial

slope

^{of the}

velocity

_versus the _square of the

magnetic

field which _{measures a} non-linear

magneto-elastic

coefficient presents a very

rapid

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 _a

complicated

behaviour

: a maximum followed

by

a minimum and

finally

_an increase with the field. v) For

high

fields the

velocity

increases with the field, the lower the temperature the steeper ^the

slope,

and _a trend towards saturation is observed at the lowest temperatures for the

highest

fields. In the _same field _range the attenuation decreases with the field. All these data _are

interpreted

^within the framework of the static

Sherrington- Kirkpatrick

model

including

two

spin-strain coupling

mechanisms _a Waller mechanism and _a field induced mechanism. It is shown that, besides terms

coming directly

from the two

coupling

mechanisms, crossed terms not

previously

taken into account _are

important.

The value of the

jump

of the non-linear

magneto-elastic

coefficient, the minimum of the

velocity

below 2~, and the succession of _a maximum and _a minimum above _{T~ for} the velocity are well

explained.

The

dynamical

effects _are

briefly

considered for the

paramagnetic phase

in low field.

I. Introduction.

Because of time-reversal invariance the

coupling

of

spins

with _an elastic strain is

generally

either

quadratic ~van

Vleck

mechanism)

_or bilinear

ovaller mechanism)

in

spin

_operators.

As _a consequence, acoustic

experiments provide specific

information _on

spins

in solids _; for instance

they

allow the measurement of

four-spin

correlation functions. This

interesting

(*)

New address _: GEC-ALSTHOM, Division Transports, Tour

Neptune,

Cedex 20, 92086 Paris La Ddfense.

(**)

Associated with the Centre National de la Recherche

Scientifique (URA 800).

property

applies

to any

magnetic materials, including spin-glasses. However, probably owing

to technical

difficulties,

the acoustic studies of

spin-glasses

_are rather _rare

[1-3], especially

if their number is

compared

to the multitude of

experiments using

other

techniques [4].

The

situation has been

slowly changing

for the last few _years and several measurements of sound

velocity

and attenuation have been

published [5-9],

in

particular

_on

insulating spin-glasses.

Recently

_we have

reported

a detailed acoustic

study

of _a cobalt

fluoro-phosphate spin-glass (COF~)o_~(BaF~)o_~(NaPO~)o~

from both

experimental

and theoretical

points

of

view,

^but limited to the _case where _no

magnetic

field is

applied _[7].

In what follows this

study

is referred to as I. In the

present

^article we

give

_a

_report

of _an extension of the work described in I.

Experiments

_on the _same

compound

have been

performed

in _a

magnetic

field _up to 9 Teslas

(T).

Several _new features have been observed _:

I)

a minimum of

velocity

_{as a} function of the field below the

spin-freezing

temperature T~ and _a maximum followed

by

_a minimum above T~

it)

at

higher

fields _a

strong

increase of

velocity

which saturates at the

highest

fields

iii)

_a

rapid

variation _{near T~} of the initial

slope

of the elastic constant _{as a} function of the _square of the

magnetic

field

iv)

an attenuation decrease when the field increases. The

velocity

data

are

interpreted

in the framework of the

Sherrington-Kirkpatrick

model

[10].

Tliis model considers

only

the static _case.

Unfortunately

the

spin dynamics theory

used in I _cannot be extended when _a finite

magnetic

field is

applied.

As _a consequence the

interpretation

of _our

attenuation data is

only

schematic. Nevertheless _{we can}

explain

most of _our data

qualitatively

and _{even some} of them

quantitatively.

Several

spin-strain coupling

mechanisms have been

considered, including

_a

coupling

induced

by

the field which _was

evidently

absent in I. Short

reports

^{of this work have been}

previously published [I1, _12],

without

taking

into account the field induced

spin-strain coupling.

The

organization

of the article is _as follows. We first present our

experimental

results which include _a detailed

study

in very low fields and studies in medium and

high

fields

(Sect. 2).

After _a short sketch of the

Sherrington-Kirkpatrick

model

(Sect. 3)

we

apply

the method to the calculation of the static elastic _constant for _any

spins

S _m

1/2 (Sect. 4)

and

then,

with _more

details,

for

spins

S

=

1/2 (Sect. 5).

A

dynamical model

valid

only

for S

=

1/2

in low fields is

presented

in section 6.

Finally

_we discuss _our data in the frame of these theories

(Sect. 7)

and

we conclude

(Sect. 8).

As

already

mentioned the present ^{article is} the continuation of another refered to _as I. As far _as

possible

_we have used the _same

symbols

in both. However this article is intended to be

self-contained.

2.

Experiment.

2.I METHOD. Ultrasonic _{waves were}

propagated

in _a

sample

^of the

magnetic insulating

amorphous compound (COF~)o_~ (BaF~)o_~ (NaPO~

_{)o_~. In} this material the

freezing

of the

spins

occurs at T~ =1.8 K

[6],

a critical temperature which is within the

explored temperature

range 1.2 K

< T _< 4.2 K.

Longitudinal

ultrasonic _waves with

frequency varying

from 30 MHz

up ^to 800 MHz _were used. The

sample

_was immersed in _a

liquid

helium bath whose

temperature was

changed by pumping.

Two different set-ups were used. In the first _one,

magnetic

fields _up to 9 T _were

produced by

_a

superconducting

coil. In this _case, the

magnetic

field _was

applied parallel

to the acoustic wavevector. In the second _case,

magnetic

fields _up to 2.I T _were delivered

by

_an

electromagnet.

Then the

magnetic

field _can be rotated in _a

plane containing

the wavevector. In both _cases the data _were collected at fixed

temperature

as a

function of the

magnetic

field

applied

after the

temperature

was reached. A very

good

accuracy was obtained for the relative sound

velocity change

measurements. It _was about 10~^~ at 100

MHz, decreasing

at

higher frequencies

due to

stronger absorption

in the material.

Changes

in the ultrasonic

absorption

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

_we

present

our data _as functions of the _square of H. The full

magnetic

field range studied in the

experiments

will be divided into three

parts

for

clarity.

2.2. I

High jieliig,

This domain _concerns the field

higher

than 2 T

(H~

~ 4

T~).

The

velocity

results _are shown in

figure

I for several fixed

temperatures.

The measurements were done

only

at the lowest

frequency (30 MHz).

The

velocity change

is first

nearly

linear with

H~

_{up _to}

an upper limit which increases with temperature

(it

is

equal

to about

30

T~

_{at I.4 K and it is}

larger

than 80

T~

_{at 4.2}

K),

the lower the

temperature

^the

larger

the

slope

of

AV/Vo

versus

H~.

_{At the}

highest

fields the _curves bend

around, showing

_a trend

towards saturation of the

velocity change.

^This behaviour is

clearly

observed

only

^{for the} two lowest temperatures

(I.4

K and 1.8

K).

It is worth

noting

that the

asymptotic

value at 1.8 K is

larger

than it is at I.4K. In the _same field range the attenuation decreases

roughly

_as

H~,

_as shown in

figure

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 detail

by varying

the

frequency

and the

angle

_q~ between the

magnetic

field and the

ultrasonic 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. Relative

velocity change

for

longitudinal

elastic waves of

frequency

30 MHz

propagating

in the

spin-glass (CoF2)o

5

(BaF2)o.2(NaP03)o.3

_{as a} function of the square of the

magnetic

field for the full field range up ^to 9T. The _wavevector is parallel to the field. A set of data _was taken at the critical

temperature 2~ = 1.8 K

(o).

The other _sets

correspond

to _a temperature lower than _T~

(+)

and two temperatures

larger

than T~

(x)

and

(*).

All the data _are

arbitrarily

_put at zero for _zero field.

Fig.

2. Attenuation of

longitudinal

elastic _waves of

frequency

86 MHz propagating ⁱⁿ the

spin-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 in

figure 3,

where the

velocity change

of _a 148 MHz ultrasonic _wave at IA K at various fixed values of the

magnetic

field is

plotted

as a function of the

angle.

The main results _are the

following.

The

velocity change

_can be

analysed

_as

bV/Vo

=A

+Bsin~

q~ where coefficients A and B

depend

_on the

magnetic

field

(their

variations _are shown in

Fig. 4).

The effect is _more

anisotropic

for

high

fields. It is

stronger

when the field is

applied perpendicular

and lower when the field is

parallel

to the ultrasonic

wavevector. Because the behaviours _are

qualitatively

similar whatever the direction of the

field,

in the

following

the results will be

presented only

for _a

magnetic

field

applied

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

of

longitudinal

elastic _waves of

frequency

148 MHz

propagating

in the

spin-glass (COF~)o~(BaF2)o.2(NaPO~)o~

_{as a} function of the angle _q~ between the

applied

magnetic field and the ultrasonic wavevector at the fixed temperature ^{T= IA K. The}

origin

of the velocity change is taken at its value without

magnetic

field. The different lines

correspond

to

3V/Vo

=

A ₊ B sin~ _~, where A and B _are free parameters

depending

_on the

magnetic

field. The various

symbols correspond

to different values of the

magnetic

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 the

anisotropy

of the effect of _a

magnetic

field _on the

velocity

of elastic waves in the

spin-glass (COF~)05(BaF2)02(NaPO~)03

as a function of the square of the field.

Figure

5 shows several

examples

of

velocity changes

at 148 MHz

plotted

_{as a} function of the square of the

magnetic

field. For T

< T~ a remarkable feature is observed the

velocity

first

decreases,

then passes

through

_a minimum and

finally

increases _as

expected

from the

high-

field results shown in

figure

I. At T~ T~ ^{the behaviour is}

completely

different and _more

complicated

_: the

velocity

first

increases,

then goes

through

_a maximum which is followed

by

_a

minimum and

finally

increases

again.

The minimum is not observed below

H~

=

4

T~

_{for the}

bt 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. Relative

velocity change

of

longitudinal

elastic _waves of

frequency

148 MHz

propagating

in the

spin-glass (COF~)o~(BaF~)o~(NaPO~)o~

_{as a} function of the square of the

magnetic

field,

applied perpendicular

to the ultrasonic wavevector for different temperatures. The vertical shift for the various sets of data is

arbitrary.

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 ofvarious

frequencies propagating

in the spin-glass

(COF~)o~(BaF~)o~(NaPO~)o~

_{as a} function ofthe _square ofthe magnetic field measured _at the

same temperature ^T ₌ ^{1.3 K. The}

velocity

is measured with respect to its value in _zero field which is in fact

frequency dependent.

The number _near each set of data

gives

the

frequency

in MHz.

highest temperatures.

The

frequency dependence

of the

velocity change

is shown in

figure

6 at 1.3 K. The minimum is shifted to

higher

temperatures ^{when the}

frequency

increases. The

variation at low fields is

independent

of the

frequency.

In the _same field range the ultrasonic attenuation _a shows also _some

interesting

features.

Figure

7 presents some results transformed into the

imaginary _part

of the elastic constant C

(3m~~ _Co

⁼

^~a~~~~

^{as a function of the square of the}

^magnetic

^{field. For}

w 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 to

H~

= 4

T~.

_The

frequency dependence

of the

imaginary

part ^{of the elastic} constant is shown at T

= 1.3 K in

figure

8.

2.2.3 Low

fieliig.

For low

applied fields,

the

velocity changes

have been studied in detail.

Some _curves at a

frequency

of 269 MHz _are shown in

figure

9. In this field _range the

behaviour is

frequency independent

between 90 MHz and 800

MHz,

as can be _seen in

figure

6. This is in contrast with what _we have observed in _zero field

experiments

in the _same

frequency

_range. In this

figure

all the _curves at different

temperatures

have

arbitrarily

the

same value for _zero field. This _means that the _zero field temperature

dependence

which has

I 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 attenuation

by

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 the

magnetic

field. The data

were obtained with

longitudinal

_waves of

frequency

148 MHz.

They

_are shown for two temperatures

given

^{in Kelvins}

by

the number _near each _set of data. The _zero of the vertical scale is

arbitrary

and

corresponds

to the value in _zero field which is in fact temperature dependent.

Fig.

8.

Imaginary

part ^{of the elastic} constant for the

spin-glass (COF~)o_~(BaF~)o_~(NaPO~)o~

_as a

function of the square of the

magnetic

field. The data _were taken at 1.3 K for various

frequencies

of the

longitudinal

elastic _waves. The number _near each set of data

gives

the

frequency

in MHz. The _zero of the vertical scale is

arbitrary.

It

corresponds

to the value without magnetic field which is in fact

frequency dependent.

been studied

previously

is discarded. This initial

slope

of

AV/Vo

as a function of

H~

_is

completely

different below and above T~. ^It is

negative

for T< T~,

positive

for T ~ T~ ^and zero

just

at 7~.

3. The

Sherrington-llirkpahick

model and its

generalization.

Many

^{features observed} in

spin-glasses

are accounted for

(at

least _{as a} first

approximation) by

a model

proposed by Sherrington

and

ICirkpatrick (SK) [10].

These authors

represent

a SG _as

an

assembly

of N

Ising spins coupled together by

random Gaussian

exchange

constants

/,

scaled

by

_a standard deviation

$ $

_and

a mean value

(IN.

_{The model may be} extended to the _case of

spins S~1/2

and the effect of a

magnetic

field H and _a

ligand

field D may be

incorporated [13].

Then the Hamiltonian of the

spin

system ^is

3C=-~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 velocity

change

of

longitudinal

elastic waves of

frequency

269 MHz

propagating

in the

spin~glass (COF~)o

~

(BaF~)o ~(NaPO~)o

~ as a function of the square of the

magnetic

field in _a reduced field _range

showing

the temperature

dependence

of the initial

slope.

The number _near each set of data

gives

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 the

temperature T,

the

magnetic

field H and the

ligand

field D

by

:

m =

E( (S~~)

=

£ (S;)

_{; q}

=

E

((S)()

=

£ ($)~

P#E(lRj~)=(z(().

The

symbol ( _)~

denotes _a thermal average over a

one~spin

_space while the upper bar indicates _an average over the disorder. The effective operator

p3C~j

=

0~

^S + q~~

S~

_which

occurs in the definition of the _average

( )

is used for any other

higher

order

parameters.

For the calculation of _an elastic constant in _a

magnetic

field _we need

three~spin

correlation functions

(in

this _case

they

_are not

equal

to

0)

and

four~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 interest

x =

E( (S~()

=

jj ($)~

We recall that the

freezing temperature

_T~ ^of the

spins (spin~glass transition)

and their

ordering temperature

_{To are} related to the

spin-spin

interaction constants

land $

respectively.

For S

=

1/2

the relations

simply

read _:

1=4kT~ $=4kTo.

When _we write _some

quantities (such

_as

/ $,

_g~,

_D,

_T~ and To) without _upper

index,

this

means that _an elastic strain _may be present. On the contrary ^the upper index 0

specifies

that

no strain is

applied.

4. The static elastic constant in the SK model for any

spins

Sm

I/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 the

spin~spin

interactions become

I= j°(I

₊

_yE)

_and

I= $(I

₊

_yE)

_{when the strain}

E is

applied.

In the _{same manner} the

Landb factor and the

ligand

field tum out to be g~ =

g((I

₊

_fE)

_{and D}

=

D°+

_GE

bt 3 ACOUSTICS OF A SPIN~GLASS IN A MAGNETIC FIELD 423

respectively.

_y and

f

_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[

_MB

_HI ~

+

Y(PJ°)~ (P )

q

) ₍₆₎

The

following

calculations _can be put into _a compact ^{form with the}

help

of _a

(3

_x

3) symmetric

matrix

£,

whose elements _are functions of the

spin

_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

_~IIP

_{L13 ~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

+ 2

f (fl~i

_@

B

H) fl GL12

+

(fl

~)~

L22)

~~~

JV

=

N/~lJ

is the number of

spins

per unit volume and

Co

is the elastic constant in the same

material without

spins.

The matrix elements L,~ ^are

given

in

Appendix

A.

A

physical meaning

_can be ascribed _to the different terms of

3C/Co.

The first term

(proportional

to y is the contribution of the Waller mechanism

(modulation

of the

spin~spin

interaction

by

the

strain)

its Hamiltonian reads

[14]

Jcw=-~yszfs,j.

~

,.j

The last _term

(proportional

to

G~

_{is due to the} Van Vleck mechanism

(modulation

of the

ligand field)

_; the

corresponding

Hamiltonian is

[15]

_:

3Cvv

=

GE

jj ^$I.

This

operator

becomes _a scalar

(equal

_to

-NGS/4)

for

spins

S

=1/2.

The fourth term

~proportional

to

f~

_comes from the modulation of the Landd factor

by

the strain. It vanishes when _no

magnetic

field is

applied

this is _a field induced mechanism. The

corresponding

Hamiltonian reads

[16]

3CFI ~

fg[

_MB HE

I $

,

The other terms

originate

in cross~effects. Some of them

disappear

in absence of

magnetic

field.

We recall that the

magnetic susceptibility

_x is defined _{as :}

x ₌

$ ₍₈₎

and it reads

X

~~(~~@B)~flLll ₍₉₎

This

explains

the

similarity

of the

magnetic susceptibility

_x and the field~induced contribution to the elastic constant

(3C/Co)~i.

We

emphasize

that the

spin~averages appearing

in the matrix

£ depend

_on the

magnetic

field H.

Finally

_we underline that the

expression

of

3C/Co

is valid

only

if the

replica symmetric

solution of the SK model

is,

I-e- above the

generalized

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)

the

equations

obtained in the

previous

section become

simpler

because several elements

L;j

_are then

equal

to _zero.

However the

equations

remain awkward. This is due to _numerous coefficients

(1/2)~.

These coefficients _are eliminated if

spin operators

_s, with

eigenvalues

_± I _are used. From here _we

adopt

this rule _{as we} did in I for the

spin dynamics.

^Therefore all the _averages have their value between 0 and I instead of 0 and

(1/2)~

for _a

n-spin

_average. We

distinguish

the _new averages

by

_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 elements

I;y

are related _to the old _ones

by

_:

~°11 "

4

Lll>

_~°13

~ 81~13> ~°33 "

161~33

>

while the other three elements _are

equal

to _zero. More

explicitly

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

₂

g(d e)2/oj ^IA

£i~

=

2(~ii e)/A

_{; A}

=

(1 fle2) _ii

_g

_{(1 4)/oi}

_{+ 2}

_g(~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 the

change

of the elastic constant reads _:

~~

=

~~~~~

(y~[(1 ^#~)

+ 2

g~ iii

_{~ii~ 4}

gij~ _di#/@

+ 2

i~~ f~/0~]

₊

Co

2

Co

°

+ 4 _y

fh [g0ijj

_Jii

ij~ @]/0

₊ 2

f~ h~ijj) ₍₁₀₎

h is _a reduced

(dimensionless) magnetic field,

defined

by

h=g[MBH/2k7~.

As

expected

the Van Vleck

mechanism,

_as well _as the cross-effects where G

appeared,

have vanished. To _our

knowledge

the cross-effect

proportional

to

yf

has _never been taken into account

previously.

It is

important

in _our acoustic

experiments

because

they

were

performed

on a system ^which combines two conditions _{: a}

high spin density sample

and _an

applied magnetic

field.

It has been shown in I that 3

C/Co

must be

negative

for any temperature. ^This property ^still holds if _a

magnetic

field is

applied.

Indeed the

magnetic

contribution for _an

arbitrary

value of the

magnetic

field _can be obtained

only by

the _means of _a

computer. However,

for _very low and for very

high fields, analytic expressions

_can be calculated.

They

_are

reported

in the

following

two sections.

5.2 LOW FIELD EXPANSIONS. Here the _case

S=1/2

is

considered,

with

$#0

and

p

_{g~ p}

B H _« I. Since the elastic constant is _{an even} function of the

magnetic field, incre~sing-

power

expansions

of 3C

only

contain terms H~ where _n is _{an even}

integer.

In fact _we limit

ourselves to terms

proportional

to

H°, H~

_and

H~.

Indeed the term

independent

of

H has

already

been calculated in I. We have calculated the terms

proportional

to

Hi.

_{The results}

are so much

complicated

that _we do not

give

them in the present ^{article. We}

only

mention that _some of them

diverge

when

approaching

the critical temperature both from

above and from below.

Our calculation is

performed

ⁱⁿ two steps : the

paramagnetic phase

first and then the

spin~

glass phase.

We

give

them in terms of the dimensionless reduced parameters 0 and h.

5.2.I

Paramagnetic phase.

The

expansions

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)

^{' ~~}

(@~-

l

The _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

to

h~

_is

always negative

since 0 g _~ 0 and the other factor is _a

squared quantity.

5.2.2

Spin~glass phase.

The _averages Jii, _@, d and have the

following expansions

~*

~

lo

_g

Ii _40)1~ ^~

^~

^~°

^~

io

_g

(i <o)12 (

o²

to)

^~

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)

e

to e2(e2

₊ 2

to)

~~~ ~

l~

_g

(1 40)1 lo

_g

(1 40)l~ (0~- to)

^~

2

lo

0

Lj~

₌ ^h

lo

_g

(1 40)l~ (02 lo)

2

i~

o²

o210

_g

(1 40)1 to

_{+ g}

_(o

²

_i~)

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}

^{~°~~ ~°~ ~°}

_o²

02- 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~)

2

i~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)

near

T~

(I.e.

for 0

m

I)

we put ⁰ = 1+ 3

(then (3 «1).

We get in the

paramagnetic 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 ~}

~°

_FI

Co

^l g I g)2

~

(13b)

~~ ^~

=

~~~2

y

fg

^{2 3}

^~~

~°

_C

Co _(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 two

phases

II ^~

-

llll~ ^Y~

+

~~°' ii~~ Ii ^{~~~~ ~l} ₍₁₅₎

~

^~~

~~ _°'~

~

~

~~~'

g

~~

_~'^~

~

~

~~

These results deserve several comments.

I) ^The term

proportional

to

h~

_{exhibits a finite}

discontinuity

at T~:

~ _)~- ~ ^~~

=

l~~~ ^{"~~~~ ~~}

^{" ~}

~)~ ^{~~~~ h~ (17)}

o o o

3(1- g)

it)

The finiteness of the

discontinuity

is not a trivial result.

Indeed,

in the calculation several terms have _a

divergence

but

they fortunately

cancel.

iii)

The

discontinuity

vanishes if the Waller coefficient _y is

equal

to zero. On the contrary if the coefficient

f

of the field-induced mechanism is

equal

to zero, a

discontinuity

remains.

iv)

The term

proportional

to

h~

_{in the elastic constant}

comes from _a contribution

proportional

to

s~h~

_{in the free energy}

expansion.

Therefore the coefficient _may be

interpreted

_{as a} non-linear

magnetoelastic

coefficient.

v) This coefficient is

always negative

in the

paramagnetic phase

since _g

< I. On the

contrary

^it may have any

sign

in the

spin-glass phase.

Exactly

at T~ the three contributions read

~~

=

~~~ 2[~ (h(

~°

_W ~ f~0

/(1

g

)

3C

A~k7~

~

~2

~

PI

Co ~

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 _are

singular

and

they

must be calculated

directly.

5.2A

Comparison

between _X and

3C/Co.

From the definitions

given by (3)

and

(8),

_a

relationship

between 3C and _x is

easily

deduced

d~(3

C I

d~fl d~x

dH~

~lT

dH~ 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 kT

W

⁺

^~f t

^{~ ~}

~

~~~ ~~~~

For H

=

0, iii _simply

_reads

:

40

~~~

l

g(1 40)/°

When _an elastic strain _s is

applied,

the two characteristic temperatures ^become

7~= 7~(1+ _ys)

_and

_To

=

7j~(1+ ys).

In the _{same manner} the reduced temperature

becomes

0/(1+ ys)

while the ratio g remains

unchanged. 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~

^{° o}

^{w 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 _a

specific

interaction mechanism since

they

_are

proportional

to

y~,

_y

f

and

f~ respectively.

Moreover _we have _now

some

physical

_argument for the

discontinuity

of

d~(

3 C

)/dH~

at

7~.

Indeed in the

paramagnetic phase

@o =

0 for any T

~ T~ ^and

consequently d@o/d0

₌

0, d~@o/d0~

₌ 0. On the contrary, ⁱⁿ the

spin-glass phase,

in the

vicinity

of T~ we have _:

~

dqo d2qo

₂

40" (1-°)+j(1-°)

+.

~"-l _~"j.