Virtual phonon exchange in glasses

HAL Id: jpa-00208320

https://hal.archives-ouvertes.fr/jpa-00208320

Submitted on 1 Jan 1975

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.

Virtual phonon exchange in glasses

J. Joffrin, A. Levelut

To cite this version:

J. Joffrin, A. Levelut. Virtual phonon exchange in glasses. Journal de Physique, 1975, 36 (9), pp.811-

822. �10.1051/jphys:01975003609081100�. �jpa-00208320�

VIRTUAL PHONON EXCHANGE IN GLASSES

J. JOFFRIN and A. LEVELUT

Laboratoire d’Ultrasons

(*),

Université

Pierre-et-Marie-Curie,

Tour

13, 4, place Jussieu,

75230 Paris Cedex

05,

^France

(Reçu

^le

5 février 1975, accepté

^{le 9 avril}

1975)

Résumé. ²⁰¹⁴ Cet article montre

l’importance

de l’interaction indirecte produite par l’échange ^de

phonons

^{entre les défauts} des matériaux

amorphes.

^Le

couplage

^entre les défauts et les phonons

étant fort, ^{il en résulte un}

couplage

^{indirect intense; en}

particulier,

^le temps ^de relaxation transversale

produit

par ^cette interaction est très court (T2 ~ ^10-9 s). ^{Les mesures} d’atténuation ultrasonore

qui ^{avaient été} analysées antérieurement sous

l’hypothèse

² T1 ⁼ T2 ^sont réexaminées avec ce nouveau point ^{de vue.}

Abstract. ²⁰¹⁴ In this article we demonstrate the

importance

of the indirect interaction via the

phonon field between

pairs

^of defects in an amorphous ^material. Using only ^the

approximate

^value

of the lattice

coupling

^constant it is shown that the exchange interaction is large. ^{The most}

important

conclusion that can be drawn from this calculation is that the order of magnitude ^{of the} T2 ^relaxation

time of a spin

packet

^is quite ^small

(10-9

s). It is therefore not

possible

^to

interpret

^the

experimental

data under the

hypothesis

² T1 ⁼ T2 ^{hitherto used; we} re-analyse the available results from this

new

point

^{of view.}

Classification

Physics ^Abstracts

7.142 - 7.270

1. Introduction. ^- The

phonon

^{field has}

long

^been

recognized

^{to be an efficient}

coupling

^of

particles

or excitations in

crystals.

^{The most famous}

example

is the case of

superconductivity

^{where a}

pair

^of

electrons is bound

by

^virtual

exchange

^of

phonons.

The same mechanism has also been invoked in the

case of

paramagnetic impurities

ⁱⁿ

^{crystals [1] ;}

however because of the small value of the

coupling

constant of each

spin

^{with the}

^lattice,

^{even for non-}

Kramers

ions,

this mechanism

yields

^a

negligible

contribution to the linewidth measured in E.P.R.

[2].

On the contrary, ^{in the case of helium 3 or 4}

impurities

in helium 4 or 3

single crystals,

^the

coupling

^between

the

impurities

^{via the}

phonon

^{field seems to be rather}

strong

and probably

contributes to the anomalous value of the diffusion coefficient of

impurities,

^as

observed in N.M.R.

[3].

^One

might

also ask whether the anomalous value at low

temperature

^{of the} dielectric constant of KCI

crystals containing

^mole-

cular

impurities (OH-)

^{is not due to the} appearance of a collective effect ordered

by

^the

phonon

^field

[4].

In view of these

examples,

^the

following question

can be raised : how does the

phonon

^field govern certain

properties

^of

amorphous

^materials

and,

ⁱⁿ

particular, following

the model of Anderson et al.

[5]

and

Philipps [6],

the value of the

T2

relaxation time of the

spin population ?

It is

generally

admitted that an

amorphous

^material

may be

thought

^{of as an}

assembly

of defects with

randomly

distributed characteristics. Each of these defects has two conformation states

separated by

^a

potential barrier,

^crossed

by tunneling. ^Thus,

^each

defect _may be

represented by

^{an effective}

spin -1

ⁱⁿ

a local

magnetic

field whose

intensity

and orientation vary from site to site. The _energy

separation

^{2 E of}

two states of the

assembly

^{extends over a}

large

range of energy ; it has been established

by

^{thermal measu-}

rements at low temperature that the distribution

n(E)

^of

spins

per unit _energy varies _very

smoothly

with E until a maximum value of the order of 1

eV, giving

^{a linear T} variation for the

specific

^heat

[7, 8].

The other

types

^of

experiments

which have been

performed

^on

glasses

^are

transport experiments :

thermal

conductivity [7, 8]

and acoustic _propaga- tion

[9].

^These

experiments give

information on the

dynamical

behaviour of the

spin

system

(relaxation

of the

population,

^{mean free}

path

^{of the}

phonons etc...)

^{and all of them are related to the}

change

^of

properties

^{of each}

spin

^as the strain associated with the

phonon

^varies.

In

fact,

any strain of the environment of the defect modifies the

parameters

^{of the double}

well ;

^conse-

quently,

^{an elastic wave} may induce resonant transi- tions between the

spin

^levels

provided

^{that the fre-}

quency of the

phonons

^{is on}

speaking

^{terms with}

the energy 2 E of the

spin.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01975003609081100

Following

^this

simple model,

^the

dynamics

^{of the}

spin

system ^is

conveniently

^described

by

^{the same}

concepts

^{used in a}

paramagnetic

system :

spin

popu-

lation, longitudinal (Tl)

^and transverse

(T2)

^relaxa-

tion

times,

^line

saturation,

self-induced transparency for the

propagation

^{of an acoustic wave in a resonant}

medium,

^etc...

However, ^{until now the} results of

experiments

ⁱⁿ

amorphous

materials have been

interpreted

^{with the}

use of the a

priori

^{relation 2}

Tl

⁼

T2 [9, 10] ;

^such

an

approach

is attractive because

magnetic dipolar

interactions or

exchange

interactions do not occur

here for fictitious

spins.

However,

^we would like to

investigate

the follow-

»ing problem :

^{due to the}

large spin-phonon coupling

observed in most of the

glasses,

what is the influence of the

phonon

^{field on} the static

properties

^{of the}

spins ?

^In

particular,

what is the indirect

coupling

induced between two

spins

^{and what are its conse-}

quences ?

In

fact,

^{as we show}

below,

^the indirect interaction is

easily represented by

^an

exchange

^{constant between}

pairs

^of

spins;

^we shall first calculate the order of

magnitude

of this interaction.

Once it has been

estimated,

^we shall calculate more

carefully

^the différent relaxation

times;

^a

technique developed by

^{Van Vleck}

thirty

years ago for a mixture of

spins [11]

will be used. Because

T2

appears ^to be smaller than

Tl,

^{we are led to} recalculate the beha- viour of an acoustical ^wave in such a resonant medium.

Finally, currently

available results whose

interpreta-

tion

depended

^{until now on the}

hypothesis Tl T2

will be

re-interpreted

within the framework of our

theory.

^{In this} way, ^some

apparently contradictory

data can be reconciled.

A few elements of the

theory developped

^{here are}

not

entirely

new, but

they

^are included in order to

present

^a self-contained paper, from the

starting equations

^{to the} numerical estimation of the para- meters introduced in the model.

The difficult

question

of the anomalous _propaga- tion of a wave in a resonant medium when the sta-

tionnary

^{state is not} reached is

beyond

^the scope of this

article ;

this is the

specific problem

^of self-induced

transparency.

2.

Spin-phonon coupling.

^- The first

step

^{is an} exhaustive

description

^{of the}

coupling

^{between one}

spin

^{and the}

phonons.

^{We examine}

successively

^the

microscopic

mechanism of the

coupling

^{and the}

tensorial

aspect

^{of the}

coupling

^constant.

Any

^{strain of} the environment of a defect results in a modification of the different

parameters

^{of the} double well. In the notation of

Phillips [12] they

^{are :}

2 J,

^the

asymmetry

of the double

well ; 1,

the distance between the two local

minima ; Vo,

the barrier

height (Fig. 1)

^and

Q,

the oscillation

frequency

^{in an indi-}

FIG. 1. ^- Double-well potential ^with asymmetry ² J, ^barrier height Vo and distance 1 between the two minima.

vidual well. The last three

parameters

^{combine to}

give

^a

single parameter A,,

^the

coupling

energy :

where

When there is no strain the relative

energies

^of

the two

spin

^{states are :}

Thus,

^{in the} basis which

diagonalizes

^{the static}

hamiltonian

Ko,

^the

coupling

Hamiltonian is

simply

8 ^{is a}

component

of the strain tensor.

Using

^the

spin

operators

Sx, Sy, Sz

^{for a}

spin -1

formula

(3)

may be written :

Also

The formulas

(4)

^{must not be}

misinterpreted :

time reversal invariance for

example

^is

required

^for

4

and therefore also for the

operators Sx... Sz

which are not true

spin operators ;

^{on the other}

hand,

X, Y,

^{Z refer to the axes for the}

spin

space, whereas

oc,

B

^{= x, y, z refer to}

geometrical

^{axes of the}

material.

The

latter,

for the sake of

consistency,

^{must be}

kept

similar

throughout

^the

spin assembly,

^{while the}

former are local

spin

^{axes ; there is no}

simple

^relation

between the ^two systems ^{of axes.}

Later _on,

however,

^it will be easier to

keep

^the

same systems ^{of axes from}

spin

^to

spin ;

^the

price

that must be

paid

for this is a

change

in orientation of the tensor of the

coupling

^{constants from site i}

to site

j.

^{As a}

result,

^{for each}

spin

^{there is a different}

spin-phonon

^tensor

G,m,

^{with no}

^simple

^relation

among these tensors,

except

^that

they originate

ⁱⁿ

the same

GafJ,m ^given

ⁱⁿ

⁽⁴⁾

^so

^long

^{as the symmetry}

of the defects is the ^same.

Finally,

^{we must}

emphasize

^{that the}

spin popula-

tion defined

by

^the energy difference 2 E is

composed

of

spins

^with different characteristics : A and

Jo

are related

only by (3). Consequently,

^the

^G

^{i tensors}

are different not

only

because local axes are tilted but also because A and

Jo

^{are not the same ; in this}

sense the

spin packet

^of energy 2 E is not

homogeneous

even if it is so within the E.P.R. standards

(inhomo-

geneous

coupling).

3.

Spin-spin coupling.

^{- It has}

already

^{been known}

for some time that the indirect interaction between

spins i and j

^{leads to a non}

vanishing coupling

^term

JCi’j nt only

^{if the}

elementary

process

corresponds

^to

creation and annihilation of a

phonon

^{of the same}

mode

[2]; otherwise,

interferences reduce the pro-

babîlity

^{of the} process ^{to zero.}

As a consequence, the _energy reduction of one

spin

^{must be}

compensated

^for

by

^a

corresponding

increase in _energy of the other

spin.

This rule is

quite simple :

^{it means that the}

only

^{resultant terms}

in

3CO

^are the secular tenns, those which commute with

x- 0

⁺

icio.

^{The first one is}

where i

and j

^{refer to like}

spins (Ei

⁼

Ej)

^{and also to}

unlike

spins (Ei * Ej).

The second one is

Here, i and j

^are

necessarily

^like

spins.

^{The cross}

coefficient

Ji,

^{is zero.}

We shall write

At this stage ^the

qualitative

behaviour of

Jij

is

strongly dependent

^on the model that we use for

3C,,

even if we consider the material to be

isotropic

^from

the elastic

point

^{of view}

(pure compressional

or , shear modes without

dispersion).

a)

^The

simplest

^{case to} be considered

corresponds

to

isotropic

individual

coupling :

A rather

straightforward

calculation

gives

R

= Rij

is the distance

between,

^the

spins i

^and

j.

qT is the modulus of the wave vector of the

phonons

with

polarization i(i

⁼

L, Ti

^or

T2)

^and energy hco ⁼ 2

E ;

^{with this}

type

^of

coupling,

^{i is}

necessarily

L. Under the same

hypothesis,

III

The essential feature of

[8]

^is

^{the R -1} dependence

of

JXX

as R tends to zero. At the

opposite ^limit,

^the

interaction has an

oscillatory decreasing behaviour ;

q,i’ is a

characteristic

length

^{for the two}

spins

ⁱ

and j

which serves as a scare for their

distance ;

is the wave

length

^{of the}

phonons

^on

speaking

^terms

with each

pair

^of

spins.

In the different cases that we shall

consider, ^ÂS

is

large

ⁱⁿ

comparison

^{with the mean} distance between

spins;

^{hence we shall not} be concerned with the

oscillatory

^{behaviour of}

(8) (R large).

b)

^{In the second} case, it is

supposed

^{that the}

coupling

^{has no}

symmetry

^{at all. As can be}

easily shown,

^{the smallest}

anisotropy

^{of the}

coupling

introduces a new R

dependence

in the interaction.

Even if the numerical coefficient obtained in the model is

only

^a

rough approximation,

^{this is not}

too severe a limitation for our model : the R

depen-

dence of

Xi.

^{is the most}

important

^feature.

As a first

example,

^we calculate the

following

^{term :}

and consider

only

the contribution of the shear modes

polarized perpendicularly

^{to the vector}

R(Tl

^modes

of

Fig. 2).

^{We obtain :}

JZZ

is obtained

from (12) by letting

^{E - 0}

(or

qT -->

0)

Thus in the limit

qT R --> 0, JXX

^{as well as}

JZz

increase

as R - 3 ;

this is the same

dependence

^{as that}

of the

dipolar magnetic

interaction.

Another

example

^is

given by

^{the term :}

FIG. 2. ^- Diagram showing ^{the three} polarizations L, Tl and T2

relative to phonon modes with wave vector q. The vector R joining

the two spins ^is parallel ^{to Oz.}

We

obtain,

for the contribution of

longitudinal

modes

only :

and : ^{o o}

Here too, in the short distance range, the

leading

tenn varies as

R - 3.

In the

general

^case, the contribution to

JXX

^{of terms}

with

coupling

^constant

Gll§ x Gi xl 16 ^npv$

^{is either}

zero or a linear combination of

only

three functions

of qt R :

these appear in eq.

(8), (12),

^and

(12bis).

Finally,

^we may conclude that the total contribu- tion to

JXX

varies in the short _range limit as

R - 3 ;

in the

long

range

limit,

^the

leading

^{term is cos}

(qt R )/R.

For

JZZ,

^we obtain either zero or

R - 3,

^{for all R.}

4. Calculation of the relaxation times

Tl

^and

T2.

^-

The

similarity

between the formal treatment of the static

properties

of the materials and those of ^an

assembly

^{of true}

spins

is clear. We would like ^now to tum to the

dynamical properties ^and,

in the first

place,

estimate the

longitudinal

relaxation time

Ti

and the transverse relaxation time

T2 using

^our

formalism.

4.1 LONGITUDINAL RELAXATION TIME

Ti.

^-

Ti

is the relaxation time of the difference of

population

between two levels of a

spin packet separated by

^an

energy 2 E. We calculate

Tl resulting

from the direct process

only,

which is the most effective one at low temperatures.

The

starting point

is formula

(4)

^{where we}

keep only

^{the first term in}

:Iec.

^The

probability

per unit of time for a

spin i

^{to fall}

from S.’

⁼

+Itosz,= -lis

In this formula _{q, I} refer to the wave vector q and

polarization i

of the thermal

phonons,

^{V is the volume}

of the

crystal, N(co.)

the thermal excitation of a

mode of

frequency

_roq, ^and

F(q,,r)

^and

q

^{are unit}

vectors

parallel

^{to the}

displacement

^{and wave vector.}

The total contribution of the different strains 80 is :

We

separate

^the

angular integration

^on the direction of the wave vector and the summation over a,

fi,

y, ô.

For the

integration,

it is tedious but not difficult to calculate all the terms. A number of them

give

^a

null

contribution;

the others

give typically

^{a coefli-}

cient

1/10

that is almost

independant

^{of the}

polariza-

tion

[13].

Now, taking

^{into account the} coefficient

1/10,

the number of

contributing

^{terms and our}

ignorance

of the exact form of the G tensor, ^{we set :}

The summation over i has to be

applied only

^to

v«r5;

^for VL ~ 1.5 _vT, this

gives :

On the other hand :

With these

partial results,

^{one has :}

A standard

procedure

^then

provides

the relaxa-

tion

time Tl

^{of the}

population

^of

spins

^of energy difference 2 E

1 1 / 17 B

with

In

(16), (17)

^we have written a double mean value

«

(GX)2 >

instead of

(Gi)2,

^{because even} if all the

spins

^{i have the same} energy 2

E,

^the

population

^is

made _up of

spins

^with different d

and Jo,

and because these

quantities

^are correlated with the deformation

potential ;

each of the

averagings

^is

designated by

^a

pair

of brackets.

To

give

^a numerical value to

Ti

^{we retum first to}

formula

(4).

^{The first}

operation

^{is to} describe the correlation between

84 f88, d, DA01be

^and

do.

^Follow-

ing Phillips [12]

^{in his} paper ^on

dilatation,

^{we shall} take all the functions of L1 and

A.

^{to be}

independent,

and then take the correlations to be :

where b is of the order of a deformation

potential (b - 84/88)

^{and a is a} number between 0 and 1.

a ⁼ 1 is the extreme case of

complete

correlation.

Phillips

^{uses a} value of a close to

10-’ [12].

On the other

hand,

where

Du/De

^is of the order

unity [12].

Thus

It is clear that the most

important

^term is the first

one since b L-- 1 eV is much

larger

^{than J.}

The second ^mean value is taken on different sites : for a

given

^{value of} E, ^{J and}

Jo may be

^{different and}

their

probability

distribution is

[6]

Using (20)

^and

(19)

^{we can} calculate

(GX)2 >

which is defined as

The final result is

Thus the factor

[Log

²

El Amin] -1

^{acts as a reduction}

factor for the

coupling

between the

phonons

^{and the}

spins.

^This

explicit analytical

^form

depends

^{of course}

on the model used for

p(A, do) ;

²

E/dm;n is probably

a

large

^number.

Nevertheless,

^{due to the}

logarithm,

this factor is not very sensitive to the detailed des-

cription

^{of the}

properties

^{of the}

spin assembly. Phillips

takes a value of 50 for

Log AmaxI Amin

^where

Using

^the

following

^values

we obtain

for a

spin population corresponding

^{to an} energy difference of 700 MHz.

The calculation based on this model calls for two final comments. For true

spins,

^{which are} all identical

(same

^level

splitting

^{and same}

spin-lattice constant),

it is

rigorously

^{correct that the}

population

^relaxes

exponentially;

the relaxation time is

unambiguously

defined. For

amorphous materials,

^the

spin population

with _energy

splitting

^{2 E is not}

homogeneous

^{in the}

sense that the

phonon coupling

^constants

depend

on A and

A,

^at least. The relaxation of the

population

of a

spin packet

^is

probably

^not

exponential.

^We

have avoided the

difficulty by taking

^{a mean value}

for the

coupling,

but this is

only

^an

approximation.

On the other

hand,

^{with so}

large

^a value for the

spin-phonon coupling,

^one may ask whether the Raman processes may ^not be

competitive

^{with the}

direct _process. A detailed calculation shows that

they

are

equally

^{efficient at}

temperatures

of about 5 K

(lower

^{than for}

paramagnetic salts).

^{Since most}

of the

experiments

^{have been}

performed

^at

tempera-

tur.es less than 5

K,

^{it suffices to consider}

only

^the

direct _process.

4.2 TRANSVERSE RELAXATION TIME

T2.

^{- The calcu-}

lation of

T2

^{is a} well known

problem

ⁱⁿ

paramagnetic

resonance. Two concepts ^{are useful.}

First,

^{we distin-}

guish

between like

spins

and unlike

spins :

^{in a}

given spin assembly,

^{we call like}

spins

those whose _energy

are the _same;

they

constitute a

spin packet.

^Their

mutual interaction does not

change

^the energy of a

pair

^{as a}

whole,

^{even if we consider}

flip-flop

processes

induced by terms such as (S’ ^S’

⁺

^Si S+). Secondly,

in the method of moments

developped

^at

length

in textbooks

[14],

the notion of thé secular part

3e! int

of the

coupling

Hamiltonian

je;nt

^is

used ; by

^defini-

tion, JC;nt

^{commutes with}

Ko,

the total Zeeman

Hamiltonian ;

^{the different moments of the}

spin

^dis-

tribution ^are related to

JCt

^and

Jeo by

^formulas

given

^below.

In the case of

amorphous materials, Jint is simply

For our

problem,

^a

good approximation

^is to , consider that the contribution to the different moments comes

essentially

from unlike

spins

^since

they

^are

more numerous than the like

spins.

The second moment is

given by

Its value must be calculated in some detail.

Van Vleck has

given

^an

expression [13]

^for

M2

for a

spin

^S

= 1 :

where N is the number of

spins

^of

type

ⁱ

belonging

^to

the same

packet

^of energy

2 E,

^and

JzÙ"z

^is

given by (13)

and

(4).

Using

^{the same}

approximations

^{as for}

Ti (with

In

(25), « (Gz)’ »

^{is a mean value for}

Gz

^over

all sites and over the total

spin population,

^and

«

(G Iz)’ » is

^{the mean value of}

G.

^over all sites for the

spin population

^with energy 2 E

(like spins only).

Further,

^{we shall use} the result :

valid for a

simple

cubic lattice

[14] (a

is the minimum distance between two

spins).

^{We add the}

following

correlation functions to those

given by (18)

and obtain

Using (27)

and the distribution

probability (20),

^we

calculate « (Gz)’ »

^{for a}

^given

value of the _energy 2 E. The most

important

^{term in}

(27)

^is

clearly

^the

first _one, since b > ^{2 E} >

40

^and

Du/De ~

^{1. We}

obtain :

y is

kept

^{constant in our} model and formula

(28)

no

longer depends

^{on E.}

We then have

If we introduce these different results in

(25)

^we

finally

^obtain

At this stage it is difficult to deduce the linewidth of the

spin-packet

^{if we have no} information on the

. lineshape ;

^{at the} very

least,

the fourth moment

M4

must be estimated in order to

distinguish

^between

Gaussian or Lorentzian

shapes.

The

procedure

^for

M4

follows the same lines as

for

M2.

^The estimation is based on the standard for- mula of Van Vleck and on the

following

observations.

There are two

important

^{terms in}

M4 :

The first one is the sum

rhe second one is

The other terms result from a summation on

i

j k, i j,

ⁱ

j, p

^or

i,

p q with the restric- tion

Ep

⁼

Eq ;

^the conditions

E,

⁼

Ej

⁼

Ele

^and

Ep = Eq ^greatly

reduce the number of

spins entering

into the sum. If

they

^are distributed at

random,

their contribution is small. We discard the

possibility

that

they

^{are clustered.}

The two first contributions to

M4

^are

nearly equal

to

.__.

The result

(33)

shows that the line

shape

^is

typically

Gaussian ; consequently, T2

is related in a

simple

way ^to

M2 by

Its order of

magnitude

^{can now} be estimated.

Using

y ~

10-2 [12], a

⁼

3 A,

^b

= 10-12

^erg,

p ⁼ 2

g/cm’,

^and VL ⁼ 5 x

105

cm

s-1;

; ^we have

This estimation calls for a few remarks.

As a result of the Gaussian line

shape,

the relaxa- tion of the transverse

magnetization

^{is not} exponen- tial.

The numerical value

(35)

^can be considered

good

within an order of

magnitude only ;

^{but this}

shows,

in any case, that the

equality

²

Tl

⁼

T2

^{cannot be}

used for the

interpretation

^{of the}

experimental

^results.

The result

(34)

^{is almost}

independent

^{of E.}

However,

it is clear that when

2 ET21h

is smaller than _one, the method of moments is no

longer

convenient : its becomes

impossible

^to

distinguish

between secular and non-secular terms with

respect

^to

Ko.

^{In the}

following,

^we will consider that we are

beyond

^that

limit,

ⁱⁿ

particular,

^when

considering

^the

propagation

of an acoustic _wave, we shall take its

frequency

^to

be

sufficiently large

^{so that}

wT2

> 1 : this

implies

^a

frequency larger

than 150 MHz.

In the calculation of the moments, there is ^a pro- blem with the

long

range behaviour of the interaction between like

spins

which varies as

q2 ^{R -1}

^cos

(qR)

for

qR »

^{1. For}

example,

^{it is not} evident that

M2

is convergent.

Two facts

support

^the idea that this contribution is

negligible.

i)

The number of like

spins

^{is a} small fraction of the total number : the

spins

^are

uniformly

distributed

over about 1 eV or

10-12

_erg; the

spin packet

^under

consideration

occupies

^{a width}

1iT21 ’" ^10-18

^erg.

The ratio is theri

10-6.

The same reason was

already

invoked in

discarding

^{some terms in}

M4.

ii)

^The

phonon attenuation,

^{which occurs when} R >

À,

^{has to} be taken into account.

Simply adding

an

imaginary part

^{to the} wavevector is mathemati-

cally

^incorrect

[2],

^{but it} introduces a useful ^conver- gence factor and even for a weak attenuation the contribution to

T2

from like

spins

^becomes very small.

For our final

remark,

^{we return to the}

expression

for

MZ ;

^formula

(24)

^{is the}

high temperature

appro- ximation to

M2,

^{and is}

justified

^as

long

^{as 2 E}

^kT,

which is the case for our numerical values. Since most of the

experimental

^{results have} been obtained in the

temperature

range 0.3 K-1

K,

^the

approxima-

tion remains valid for 2 E

10-16

_erg. But even if it were necessary ^to extend the calculation to much lower temperatures

(or

^{to much}

higher energies),

this could be done

nearly exactly

because the

spin

^°

is t (1).

5. Ultrasonic atténuation. ^- We now tum to the

problem

^{of the}

propagation

^{of an elastic wave in}

amorphous materials,

^{and in}

particular

^{to the calcula-}

tion of the attenuation of the wave due to

coupling

with the

pseudo-spin

^{system. The}

problem

^{will be}

limited to the case of a

longitudinal

ultrasonic wave

in which x is the direction of

propagation

^{and vibra-}

tion. This

problem

^{is to} be solved for _any acoustic power, in order to treat at the same time the saturation and the non-saturation cases of the

spin population.

Theoretically

^{this is} easy : the

coupled

^evolution

equations

of the elastic variables and of the

spin

variables are solved

simultaneously.

A macroscopical description ^{tor the} spm system ^is

adopted,

^{and we shall}

speak

^{in terms of its}

magnetiza-

tion

or, equivalently,

^{in terms of mean}

spin operators.

We start with a

Lagrangian density

which includes both a

purely

elastic and a

magnetoelastic

^{term :}

u is the elastic

displacement (along

^{the x}

direction)

^and

X is the

spin density.

In this

paragraph

^{we will}

always

^write

Gx

^for

{

^«

(G’x )2 » 11/2.

^In the model

developed above,

the contributions to

JU

from shear and

compressive

waves were not

separated ; however,

^{when we consider} the

propagation

^{of an elastic} wave, this is necessary ; this remark will be useful in

correlating

^{our results}

with

expérimental

data when it is

possible

^to

distinguish

between the

coupling

^constants.

The effective

coupling

^constant

Gx

^{and the mean}

spin operator

^{s as defined}

by :

have been introduced in the

Lagrangian density

^{in the}

following

^manner.

The total

magnetization

^{is the sum of}

elementary magnetizations of spin packets

^{with Larmor}

frequency

roo

(where hmo

⁼

2 E).

^{Each of these}

packets

^is

coupled

^{to the}

phonons by

^{a mean}

coupling

^constant

depending

^on wo and the total

magnetoelastic

energy is

proportional

^{to :}

s(mo)

^{is the}

elementary magnetization ;

f (coo)

^{is a} normalized function :

which is related to the

spectral density n(E)

^introduced

by Phillipps [6] :

In

fact,

^the

coupling

^constant

Gx

^is

only weakly dependent (logarithmically)

^on roo

(eq. (21)); ^therefore, Gx(coo) ~-- Gx

^can be removed from the

integral,

and we have

More

generally,

^{the mean}

spin operator,

^{which is}

proportional

^{to the total}

magnetization,

^{is defined}

by

At thermal

equilibrium,

^{one has :}

Such a

description

^{in terms of mean}

operators (total

or

elementary)

is valid because there are a

large

^number

of

spins

^{in a} volume whose dimensions ^are of the order of an ultrasonic

wavelength ;

^more

explicitly :

Applying

^the

Euler-Lagrange equation

^{to the}

Lagrangian density,

^{we obtain :}

which is the

propagation equation

of the elastic wave

coupled

^{to the}

spins.

vL ⁼

(Clp)112

is the sound

velocity

without

coupling.

In this

equation

^{we have}

deliberatly neglected

^the

term

since it does not oscillate at a

frequency

^{close to the}

acoustic

frequency.

Despite

the results obtained in the

previous

^para-

graph (Gaussian

^line

shape),

^we shall take the beha-

viour of the

magnetization

^{to be}

governed by

^Bloch

equations

with relaxation times

Ti

^and

T2 given by

eq.

(17)

^and

(34). Indeed,

^{we shall set}

T 1

⁼

Tl 1

^and

T 2

⁼

Ti 1

and write :

where a is the strain.

The formulas

(39)

^{are the}

equations

of evolution of the

magnetization s(mo) (mo is omitted)

in the labo-

ratory

^frame.

As is

customary

ⁱⁿ

spin

resonance, ^we transform to a

rotating

frame where

spin

^{variables are indicated}

by

^a

tilde.

However,

because of the

propagative

^character

of the

perturbation,

^{we have to choose a}

rotating

frame which varies from

point

^to

point,

^{in order to}

be in

phase

^{with the wave}

[15].

We write the

displacement

associated with the wave

The attenuation and the

change

^of

phase velocity

are taken into account since the

amplitude

^{A is a}

function of x

and t,

^{and the}

phase

lp is a function of x.

The desired transformation is :

Neglecting

^all

rapidly oscillating

^{terms, we obtain :}

with the

approximation

of weak attenuation and small

change

^of

velocity :

In the

following,

^we

put :

We now retum to the

propagation

eq.

(38)

^{and we}

obtain :

We then have to solve the

following

^five

coupled equations :

Such a

problem

^is

beyond

^the scope of this

article,

and we limit ourselves ^to the search for

stationary solutions,

^i.e. solutions in which all time derivatives

are zero

This

type

of solution describes the case where the duration of a wave train is

long compared

^{to the}

characteristic times

Ti

^and

T2 ;

^this may ^or may ^not be the case

experimentally.

The

system

^to be resolved is then :

The last three

equations

^are first solved as functions of _so. We find :

These results must be substituted into the first two

equations of (41 ).

^{Since our main concem in} this paper is with the ultrasonic

attenuation,

^we concentrate on the first

equation only

and obtain :

The

quantities f (coo),

so,

Tl, T2 in

^this

integral depend only weakly

^on roo in

comparison

^{with the}

resonant

denominator,

^{so that}

they

may be taken out of the

integral.

We thus obtain :

Setting

with

we

finally

^{have :}

The solution of this differential

equation,

^{with the}

condition A ⁼

Ao

^{for x =}

^0,

^is

given by

Two

limiting

^{cases are} of interest :

The solution then become

or

This is the non-saturation limit and

ln.s

is the usual attenuation

length.

The solution is :

where E is the ultrasonic

intensity.

One can define an attenuation

length

for the acoustic power

by :

It is

easily

^seen from the differential

equation

^{that :}

Defining

^a critical ultrasonic

intensity

^or

amplitude Ac by :

we have

In order to facilitate the

comparison

^with

experi-

mental

data,

^{it is useful to} write the critical condi- tion

(46)

^{as a} function of the ultrasonic flux

It then can be rewritten as

and in the saturated

regime,

^{we have :}

or

This result indicates

that ls 1

^is

weakly temperature dependent (ls 1 N T- 1/2) further,

^{in the}

region

^of

high temperatures (1iro kT),

^we

predict

^that

1.-..l

varies

aSW2 (which

^{is a} well-known

result),

^{whereas in}

the saturated

regime ls 1

^{must increase as}

ro3,

^instead

of

w4.

These are two differences with the standard

theory [9].

, 6.

Comparison

^with ultrasonic

experiments.

^{- An}

ultrasonic wave

propagating through

^an

amorphous

material suffers two

types

^of attenuation.

First,

the elastic wave modulates the

spin popula-

tion and a non-resonant attenuation occurs

[16] ;

^it

originates

^{in the non} reversible

part

^of

the modulation

and is related to

Tl, essentially.

’

The same

phenomenon

has been described and observed for

paramagnetic

systems

[17, 18].

Secondly,

^{there is a resonant} attenuation due to transitions between the two

levels;

its order of

magnitude

has been calculated in the

previous

para-

graph (eq. (43)).

Experimentally,

^{these two} contributions are

easily distinguished :

^{the first one is}

independent

^{of the}

acoustical

intensity,

whereas the second is

subject

^to

saturation,

^as has been shown.

Several observations have been

reported

^{of the}

non-linear character of ultrasonic attenuation on

different

amorphous

^materials

[9, 10], [19, 20, 21, 22].

We shall compare ^our

theory

^{with the} paper

by

Arnold et al.

[9]

^which

provides

many

quantitative

measurements; up ^{to now}

only experiments

^with

longitudinal

ultrasonic waves have been

reported ;

and information can therefore be deduced

only

^for

the

longitudinal coupling

^constants.

6.1 EVALUATION oF

Gx

^AND

Tl.

^- The effective

coupling

^constant

Gx

^can be deduced

independently

from the value of the non-saturated attenuation :

1 __1 ... B n 2 ...2

Using

^the

following

numerical values taken from

specific

^heat measurements and observations of the ultrasonic attenuation :

one obtains

With this value the relaxation times ^can be

easily

calculated :

6.2 EVALUATION or

T2

^AND

Gz.

^{- In order to}

evaluate

T2,

^{the formula}

giving

the critical acoustic