Simple considerations on the nuclear relaxation of diamagnetic impurities in ferromagnetic metals

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Simple considerations on the nuclear relaxation of diamagnetic impurities in ferromagnetic metals

F. Hartmann-Boutron, D. Spanjaard

To cite this version:

F. Hartmann-Boutron, D. Spanjaard. Simple considerations on the nuclear relaxation of dia- magnetic impurities in ferromagnetic metals. Journal de Physique, 1973, 34 (2-3), pp.217-227.

�10.1051/jphys:01973003402-3021700�. �jpa-00207375�

217

SIMPLE CONSIDERATIONS ON THE NUCLEAR RELAXATION

OF DIAMAGNETIC IMPURITIES IN FERROMAGNETIC METALS

F. HARTMANN-BOUTRON and D. SPANJAARD

Laboratoire de

Physique

des Solides

(*),

Université

Paris-Sud,

^Centre

d’Orsay, 91-Orsay (Reçu

^{le 31 août}

1972)

Résumé. ²⁰¹⁴ La relaxation nucléaire

d’impuretés diamagnétiques

dissoutes dans des métaux

ferromagnétiques

peut être influencée par des _processus d’ordre élevé mettant en

jeu

les excitations des électrons

magnétiques.

Nous étudions ce

problème

^à l’aide d’un modèle très

simple (polarisation uniforme),

^{en utilisant un} formalisme de

susceptibilités

^{et en} écrivant des

équations

^{de mouvements}

couplées

pour les électrons

ferromagnétiques

^{et les} électrons de conduction. Nous discutons en

détail

l’interprétation physique

des différentes

susceptibilités qui apparaissent

^{dans le}

problème,

et nous faisons le raccord avec la théorie élaborée _par

Giovannini,

Heeger,

Pincus,

^Gladstone pour traiter la relaxation nucléaire de la matrice dans les métaux

diamagnétiques

^{contenant des}

impuretés paramagnétiques.

Abstract. ²⁰¹⁴ In

ferromagnetic metals, high

^order processes

involving

^the excitations of the

magnetic

electrons _may contribute to the nuclear relaxation of

diamagnetic impurities.

^{Here this}

problem

is

investigated

^on the basis of a very

simple

^model

(uniform polarization),

^{with the}

help

^{of a} suscep-

tibility

formalism and of

coupled equations

of motion for the

ferromagnetic

and conduction electrons. The

physical interpretation

of the various

susceptibilities

^{which enter the}

problem

^is

discussed at

length,

^{and a} connection is made with the

Giovannini,

Heeger,

Pincus,

^Gladstone

treatment of the nuclear relaxation of the matrix in

diamagnetic

^metals

containing paramagnetic impurities.

LE JOURNAL DE PHYSIQUE 34, 1973,

Classification Physics ^Abstracts

18.54

1. Introduction. - Nuclear relaxation times of

diamagnetic impurity

nuclei in

ferromagnetic

^metals

at low

temperatures

^have

recently

^{been measured}

by

NMR and nuclear orientation

techniques [1], [2], [3].

In

principle,

this relaxation

proceeds through

^the

coupling

of the nucleus with the conduction electrons.

But one may wonder

if,

^{in addition to the}

simple Korringa

process,

higher

order processes

involving

the excitations of the

ferromagnetic system

^{cannot take}

a

part

in the relaxation.

This

problem

^{has been} considered in the

appendix

of a recent paper

by

Kontani et al.

[1] ]

^{who use a}

susceptibility

formalism and

coupled equations

^of

motion. But

they

^{do not}

explain

the transition from their _eq.

(A. 4)

^to

(A. 5) (which

^{involves a}

change

ⁱⁿ

the

notations, (A. 4) being expressed

^{in terms}

of xd

and

(A. 5)

^{in terms}

of;(d

^{with no} very

precise

^definition

of xd

^and

xd).

This transition is indeed the delicate

point

^{of this}

type

of calculation. For this reason,

our own

work, although

undertaken before the

publication

of the paper

by

Kontani et

al.,

^{will be}

presented

^{here as a comment of their}

appendix.

In the same way ^as Kontani et al. ^we consider ^an

(*) ^{Associé au CNRS.}

atom with nuclear

spin

^{I which} may be either ^a dia-

magnetic impurity

^{or an atom of the} matrix. In both cases, this

particular

^{atom will be denoted as « the}

impurity

^{». This atom is embedded in a}

ferromagnetic

metallic matrix which contains two

systems

^{of elec-}

trons : conduction electrons

(s)

^{with a}

spin density a(R),

^and

ferromagnetic

^electrons

(m)

^{which may or}

may ^not be

strictly localized, giving

^{rise to a}

spin S

per ^atom

(

^S

> 11 Oz).

^In

^practice,

^these

^magnetic

electrons will be d electrons in transition metals and

f

electrons in rare earth metals.

Between the two

types

^of

electrons,

there exists an

exchange coupling :

where

Nv

^{is the number of}

magnetic

^{atoms per unit}

volume and the Fourier

components 6q, Sq

^of

^6(R)

^and

S(R)

^{are defined as in reference}

[1] ;

q lies in the first Brillouin zone.

The

hyperfine

hamiltonian of the

impurity

may be written as :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01973003402-3021700

where K is non zero

only

^{when the}

impurity

^{is an}

atom of the matrix. In view of the

present uncertainty concerning

^the

origin

^{of the}

hyperfine

^{fields on}

impu-

rities in metals

[4], [5], [6],

A will be considered ^as

an

adjustable parameter.

If we admit that the presence of the

impurity

^does

not

greatly

alter the

properties

^{of the} matrix and

assuming

^{a uniform}

polarization

^model

(1),

^the

nuclear resonance

frequency

^is

given by :

or, since

hw)n

where

N,(EF)

^{is the}

^density

^{of s} electrons per ^atom and _per

spin

^at the Fermi level

(N(EF)

^{= 3}

^{p/4 EF}

for _p electrons per ^atom and a free electron

band).

Let us now consider two small transverse fictitious fields

hqw bqro

^{at circular}

^frequency

^w, the first of which

only

^{acts on s}

electrons,

the second one

only acting

^{on m electrons}

By

definition of the effective transverse

susceptibilities

~ ~ ~

xm, xsm

^and

xe,

^we shall write :

where the response

aqw, Sqw

^{must be}

^computed

^{in the}

presence of all electronic interations

existing

^{in the}

metal,

^{and the}

susceptibilities

^{have the dimension}

(Ny g2 uB2/energy).

In terms of these

susceptibilities,

^{the nuclear}

relaxation time of the nuclear

spin

^{1 is}

given by [1] :

2.

Computation

^and

interprétation

of the effective

susceptibilities.

^- 2.1 GENERAL FORMULAS. ^- We have

~ ~ ~

now to calculate

2e, xsm

^and

xm.

^{We have}

already

mentioned the existence of a

coupling Jcsm

^between

electrons s and m. In the case where the

magnetic

electrons

belong

^{to rare earth atoms}

(and

^{are in fact}

4 f

electrons),

^this

coupling

^is

responsible

^{for the}

apparition

^of

magnetic

^{order and no other}

coupling

has to be introduced. On the

contrary,

^{in the iron} group the ^onset of

magnetism

is attributed

mainly

to d-d interactions which may be

approximately represented

^{as an} internal molecular field with _compo- nent q, ^m

given by

Taking everything

^into

^account,

^{we may thus write}

for the transverse motion of the

spins

in the presence of electronic

interactions,

^two

coupled equations :

From which we deduce that :

(1) ^{In the} presence of the impurity ^the properties ^{of the} matrix no longer exhibit translational invariance, contrary to what we are assuming ^{here for} simplicity. ^{But we} hope that our model, although very crude, is sufficient to display ^the main physical phenomena of interest to us.

when U ⁼

0,

^these

equations

^{are identical to}

(A.4)

of Kontani et al.

[1] except

^{for the}

change

^{in notations}

(xd

>

Xm, xd >

xm,

Xe >

xe, xe >

ie) .

We must now indicate what _Xm and _Xe are. In

fact,

these are not

simple

^electron

susceptibilities

^since

they

^must

[7]

^be calculated in the presence of the

longitudinal

effects of

Jcsm

^{and U.}

2.2 RARE EARTH METALS

( f ELECTRONS).

^{- For}

simplicity

^{let us first consider the case of} well localized

ferromagnetic spins

with U = 0

(for example gado-

linium

metal). Then,

ⁱⁿ the presence of ^an external

longitudinal

^field

Ho,

^the

susceptibility Xf( wB)

ⁱⁿ eq.

(8) (m

⁼

f)

^is

given by :

where the third term in the numerator

represents

^the

effect,

^{on one of the localized}

spins S,

^{of the}

exchange

field

resulting

^{from the}

polarization

^{of the s band due}

to all other

magnetic

^ions.

Inserting (12)

ⁱⁿ

(9),

^we

get :

where

xe(q, OJ)

^{must be the} transverse

susceptibility

^of

the ^s electrons in the presence of the

exchange JCsf interaction,

^which introduces a

splitting

²

Jsf S

between the up and down

spin

^electrons

As is well known :

so

that, when q

^{= ce =}

0,

^{the two last terms in the} denominator of

(13)

cancel out, ^as

they

^{should in}

order to preserve the rotational invariance of the aS

system.

Let us now consider the case

q = 0,

^{and assume for}

simplicity

^{that in the absence of}

exchange (Jesf) coupl- ing

^{the s} electrons behave like free electrons.

Then,

the

imaginary part

^of

xe(q, w)

^{is different from zero}

only

^when

[1] :

with

it is

given by [1]

As concerns the real

part (2),

^since

liWn/EF’

h(OnIJ/f « 1, it depends only

very

slightly

^{on ce for}

w - wn.. For small _q, one has that

Consequently

This

represents

^the

susceptibility

^{of a}

ferromagnetic assembly

^of

spins

^{in the} presence of

spin

^{waves with}

dispersion

^relation

More

generally

^for

arbitrary

q, it _may be shown that since

Jsf SjEF 1,

^{the real}

part

^of

differs

only slightly

^{from the} value obtained

by making Jsf

⁼

0,

^{Q) =}

0,

ⁱⁿ

/e(00)

^and

xe(q, w). Then,

we may write :

in which _xo is the free

(unpolarized)

^electron suscep-

tibility.

^{This shows that} whatever the value _{of q} the

spin

^wave

dispersion

relation is

given by :

But it is well known that in a

magnetic system

^with

exchange

hamiltonian :

the

spin

^wave

frequencies

^are

equal

^to

[8] :

where

J(q)

^is the Fourier transform of

Jil.

In the case we are

considering

^here

[9],

^{the ferro-}

magnetic spins

^are

coupled by

the Ruderman-Kittel

(RKKY)

mechanism for which the

exchange

^hamil-

(2) ^See Appendix.

tonian in the traditional

approach

may be written

as

[10] :

so that :

Inserting (26)

^into

(24),

^we

get again

eq.

(22).

Having

^{thus checked the internal}

consistency

^{of our}

results,

^{we must however notice that the}

rigourous spin

^wave

dispersion

^{relation is not}

given by (22),

but must be obtained

by equating

^{to zero the real}

part

of the dénominator of

(13)

This is an

implicit equation

^{in ce, which in addition}

to the

spin

^wave branch may have other roots

probably

associated with

higher energy electronic

excitations

[ 11 ].

As concerns the

imaginary part,

^{which becomes non}

zero

when q

>

qm;n,

^it

corresponds

^{to the}

damping

of the

spin

^waves

by

the Stoner excitations of the conduction electrons. It is

interesting

^to notice that this

damping

^is

proportional

^{to the}

frequency

^of

excitation of these

spin

^waves.

For small co

where

F(q/kF),

the Lindhard

function,

^{is for}

small q :

Then :

where Aco is the

damping

^{of the}

spin

^wave

(excited

at

ro)

^and coq ^{its circular}

frequency,

^{i. e. for}

small q

When the

spin

waves are excited at their own fre- quency it follows from

(30)

^{that :}

In normal rare earth

metals,

^{there are three conduction}

electrons per ^atom.

Specific

^heat measurements lead to

N(EF) N 2jeV jat ;

^{for a} free electron like band this would

correspond

^to

EF N

¹ eV. On the other

hand, approximating

the Fermi surface

by

^a

sphere

^{we have}

that

kF

^{= 0.8 A. U. = 1.5 x}

¹⁰⁸ cm - 1, and, assuming

a

parabolic

band with effective mass m* ⁼ _{3.1 me,}

we arrive at

EF N

^{2.5 eV}

[12]. Finally,

^Freeman

et al.

[13]

^{have estimated}

EF -

^{3.4 eV.}

Anyhow, if, following

^{Sano et al.}

[5],

^we

set ( 2 Jsm 1 equal

^{to 0.15}

eV,

it appears that

A,w/wq ^{1 ;}

^this

shows that the

spin

^{waves retain their}

meaning

^when

q >

qsmin.

As concerns the

spin

^wave

energies (eq. (31))

taking N(EF) N 2/eV/at, Jsf N

^0.075

eV, S N 7/2

we arrive at :

This _may be

compared

^{with the neutron} diffraction results for

gadolinium [14].

^{In the c}

direction,

^when

the

spin

^wave energy is of order 8 meV. Formula

(33)

would lead to

hco, -

^{8.6 meV.}

But, then,

^{at the}

edge

of the Brillouin zone

(q

⁼

2 nlc),

^{we would have}

hwq ~

34 meV instead of the observed 14 meV.

Then a law :

giving

values of order of 15-20 meV at the

edges

^{of the}

Brillouin zone would seem more

appropriate

^{to a}

rough representation

^{of the}

spin

^wave

dispersion

^{law in}

this

hexagonal

metal in which the

energies

^{of the}

spin

waves are

fairly anisotropic

and far from

simple parabolic looking

^curves.

Besides the term

Dq 2,

^the

spin

^wave energy contains

a term g,uB

Ho.

^{The value}

of q

for which the two contributions ^are

equal

^{is such that :}

Assuming

^{that :}

(which corresponds

^to

we find :

This is to be

compared

with the value of

qsnin. Taking Jsf N

^0.075

eV,

^{S =}

7/2, EF

^{= 2 eV}

(corresponding

to

NS(EF) N 1/eV/at,

^{which would be} consistent with eq.

(34))

^one

^{gets :}

With

EF

^{= 1}

eV,

^we would have obtained ^a value twice as

big.

^But

Jsf

^{is not} well known

either,

^{so that}

we cannot

expect

^to

get precise

^estimates.

2.3 TRANSITION METALS

(d ELECTRONS).

^{- In this}

case, the

magnetic

d electrons must be treated ^{as a} band and U "# 0.

Taking

^{into account the}

longitudinal

effects the

energies

of up and down

spin

^{electrons are}

given,

^{as functions of the}

unperturbed energies by :

in which

nd, nd,

^are the numbers of d electronç,with

spins 11

per ^atom.

Xd(q, w)

^{must be the} transverse

susceptibility

^{of the}

d electrons calculated

using expressions (38)

^{for the}

energies [7].

Notice that

nd t

^and nd! ^{are not}

arbitrary

^{but must be}

determined in a self-consistent way

using

the relation :

and

expressions (38)

^{for the}

energies (it

^is

^possible

to relate the ^Q > ^{term to}

nd t - nd 1 :

The condition for the appearance of

ferromagnetism

is then found to be :

in

agreement

^{with the} classical result when one makes

Jsd

^{= 0}

(Nd(EF)

⁼

density

^{of d states} per ^atom and per

spin).

In

practice,

^{U is taken to have an} effective value of 1.5 eV

[7],

and the occurrence of

ferromagnetism

appears

mainly

as a consequence of the dd interaction ^{term as} is

usually

^admitted.

Assuming

^that nd r -

nd,

^{has been} determined self-

consistently,

^{let us set} nd t - nd 1 = 2 S. Since

Xd(q, w)

must be

computed

^{from the}

energies (38),

it follows that

(approximating

^very

roughly

the d band

by

^{a free}

electron

band),

^Im

xd(q, w)

will be different from zero

only

^{if :}

with

It is difficult to

predict

any real order of

magnitude

^of

qmin

^{since the real d band is}

quite

^{different from a free}

electron band

[15]. Nevertheless,

^formulas

(15)

^and

(43) together

with the numerical estimates of

Jsd

^{and U}

suggest

^that

qmin

^and

qm;n

^{must have}

comparable

^orders

of

magnitude

^with

qm;n

>

qmin.

Let us now come back to the

study

^of

xd.

^{As in the}

case of rare

earths,

it may be checked

that,

^for

small q :

This shows the existence of

spin

^{waves with a}

disper-

sion law :

When q increases,

^the

dispersion

law becomes less

simple

^{and the}

spin

^{waves are}

damped.

^{As a matter of}

fact in eq.

(39)

-

When q

>

qm;n, xe

^is different from _zero, this

corresponds

^to the relaxation of the

spin

^waves

by

^the

Stoner excitations of the conduction electrons.

-

When q

>

qmin, xd = ^0,

^this

corresponds

^{to the}

merging

^{of the}

spin

^{wave branch into the} continuum of Stoner excitations of the d electrons.

D and

qm;n

^can be obtained

experimentally by

^neu-

tron diffraction

[16].

^In

iron,

^{D is found to be of order}

280

meV . A2.

As for

qmin

^it

corresponds

^{to the}

steep,

decrease in diffused

intensity

^which

accompanies

^the

merging

^{of the}

spin

^waves into the Stoner excitations.

This occurs when

hco,

⁼

Dq2 =

^{85 meV}

[16].

It follows.

that :

By comparison,

^with

to 1

Both

qmins

^and

qmin

^{are not} very small

compared

with the average radius of the Brillouin ^zones

On the other

hand,

the value

of q for which

is of order

. much smaller than

qm;n

^and

q’

Before

closing

^this

paragraph,

^{we must add a few}

remarks :

- In the

spin

^wave

dispersion

^relation g,uB

Ho

^must

in

reality

^be

replaced by g,uB(Ho

⁺

Hl - Haem)

where

HA

^{is the}

anisotropy

field and

Hdem

^{the dema-}

gnetizing

^{field. In a} multidomain

sample Hdem

⁼

Ho.

In a

uniformly magnetized sphere

-

Strictly speaking x(q)

^{defined in the reduced}

zone scheme is a sum over the

reciprocal

^{lattice vec-}

tors

[18].

This is related to the

possible

occurrence of

umklapp

processes in the relaxation calculations and also has

important bearings

^{on the}

magnetic

structures of rare

earth metals. But

assuming

^a

ferromagnetic

^{rare earth}

metal amounts

assuming

^that the maximum of the real

x(q)

^lies

at q

⁼

0,

^{as in the case of free electrons.}

3.

Application

^{to nuclear} relaxation. ^- 3.1 GENE-

RAL FORMULAS. ^- Let us now come back to eq.

(7) (9)-(11).

^{We have to extract the}

imaginary parts

~ ~ ~

of

X.,

^{Xsm and Xe}

taking

^{account of the fact that :}

Then,

^{we find that :}

which after substitution in

(9)-(11) give again

eq. A. 5 of Kontani when one sets

x’m

^{= 0.}

After

completing

^{all the}

calculations,

^we find that :

The first term is

just

the usual

Korringa

^{term. The}

last two terms arise from processes

involving

^the

magnetic

^{m electrons.}

Let us now assume that in the summation over q,

we may

neglect umklapp

processes and that

the

màin contribution comes from the

small q region.

Then,

and, dropping

^{some of the}

(q, ro)

^to

simplify

^the

aspect

of the

formulas,

eq.

(51)

^{becomes :}

Let us

temporarily neglect

^{the term in}

x’’m.

^{From the}

considerations of

paragraph

^{2 it} follows that Re

lm

may be written

approximately

^{as :}

where,

^at

least

^{at low} q,

hcoq

^{is a}

^spin

^{wave energy.}

Including anisotropy

^and

demagnetizing

^field

effects,

its

expression

^is

given by :

We shall define :

As concerns the

intégration

^over q,

xë(q, (On)

^{is diffé-}

rent from zero

only

^if

qmin

q ²

ksF.

^{Then if}

2

ksF

qB

(edge

^{of the Brillouin}

zone)

^the

integration

must run from

qmin

^{to 2}

^ksF.

^{Otherwise it must run}

from , qmin s

^to qB.

We shall

write

Then eq.

(53)

^{leads to :}

In the

high temperature limit 1 hw.1k,3 T |

^«

1,

this result is identical _{to eq.}

(A. 9)

of Kontani et al.

[1 ].

One should notice that the

Log

^{term is}

probably

overestimated since

all q

^values contribute to it so

that eq.

(52)

^{is not}

strictly

^{valid. It is}

possible

^{to check}

that the last term of _eq.

(57)

^{reduces to}

Weger’s

result

[19]

^when

ko/qmin ->

^{0 :}

in which 1 is the area of the Fermi surface and úJex =

Dja2.

Notice that in the derivation of these results

a)

we have

neglected

^the

umklapp

processes

b)

^{we have}

assumed that the

spin

^wave

spectrum

^{of the matrix is}

not

seriously perturbed by

^the presence of the diama-

gnetic impurity [20].

Therefore the

following

^estimates

will be more

qualitative

^than

quantitative.

3.2 RARE EARTH METALS. ^- 3.2.1

Diamagnetic

^im-

purity

^{in a rare earth metal.}

Then

xm -

⁰ in eq.

(53) (no

^intrinsic

damping

^of

the localized

spins) and, using :

the last bracket of _eq.

(57)

^{takes the form :}

When the anisotropy

field is weak

compared

^{with them}

external field :

ka/qm;n ~ qo/qmin ~ 1/2

^{in a field}

Ho

⁼

^10’

^Oe

(§ 2.2).

^Thus the second and third terms

begin

^{to become field}

dependent

^above

^{104 Oe.}

^As

concerns the relative orders of

magnitude, - taking Er N

²

eV, Jsf ’"

^0.075

eV, S

⁼

7/2,

^{we find that}

the three terms of eq.

(59)

^{are in thé ratios :}

This must not be taken too

seriously

^since

EF

^and

Jsf

are known within a factor 2 : then the last term could be 4 or 8 trimes smaller as well. But it would still dominate over the other two.

Let us now come back to eq.

(57).

The inverse relaxation time has the form

or, at ^«

high » temperatures :

and we have found that :

If we consider two matrices with

approximately

the same

properties

^{for the} conduction band

(La, Gd),

for a

given impurity AN,(EF)

^{should be}

approximately

the same in the two matrices.

Then,

^{if one of these}

matrices is

ferromagnetic

^{and the} other not, the

high temperature

^{value of}

Ti T, independent

^of wn, should be much smaller in the

ferromagnetic metal,

^corres-

ponding

^{to a much more}

rapid

relaxation. Also

(TI)f,,,,.

^{should be a linear} function of the external field.

The low

temperature

situation is more

complicated

since in a

diamagnetic

^matrix

n" = hYn Ho;

^while

in a

ferromagnetic

^matrix

3.2.2 Atom

of the

^{matrix in}

a pure

^{rare earth metal.}

In this _case, the contribution of the K term to

liwn (eq. (52))

^is

^usually

^{dominant over that of A and}

we still

expect

^{the last term} in eq.

(57) (Weger process)

to be more

important

^{than the others. This term} may also be written

so that at ^«

high

^»

temperatures :

T,he

proportionality

^of

1/Ti

^to

oei

^{T at finite}

tempera-

tures has been checked

nicely by

^{Sano et al.}

[5]

ⁱⁿ

the case of

Dysprosium

^metal where the different Zeeman transition are

split by quadrupole

^.efllects.

The transition

probabilities

^{are not}

expected

^to

depend

on

Ho

^since

HA

is very

big (the

^{bottom of the}

spin

wave

spectrum

^{lies at about}

25 oK).

As concerns the

precise

evaluation of

Tl,

^{the inclu-}

sion of

umklapp

processes is found to be _necessary.

Similar results have been obtained in Terbium

[21].

3.3 TRANSITION ^{METALS. -} In this _case, we may more or less consider that ^we have

spin

^{waves in}

the whole Brillouin ^zone but that these

spin

^waves

become _very

strongly damped

as soon as

they

^become

degenerate

with the continuum of Stoner excitations of the d electrons

(q

>

qmin).

The

anisotropy

^fields

being usually

^small

( N ¹⁰³ Oe)

we have that

k2â ~ q6.

Then from the discussion of

paragraph 2.3,

^{it fol-}

lows that :

i. e. we cannot

expect

^{to see a}

dependence

^of

Tl

on the external

magnetic

^field.

The

dependence

^observed

by

^{Kontani et al. at a few}

kOe thus seems to be due to residual Bloch Walls and domains effects ^{- or} to arise from other

phenomena.

Turning

^{now to the}

expression (53)

^of

lIT1,

^{let us}

estimate the ratios of the contributions due to

x§

^and

X"

^{in the last term}

Assuming

^{that Re}

xd(q, w) ~ N, S/Dq 2

^and appro-

ximating

^very

roughly xd(q, w) by

^{a free electron}

expression,

^{these two} contributions will be

respectively proportional

^to

(qmax

^{= 2}

kF

^{or 2}

kdF) :

and

The ratio of these two

quantities

^{is :}

For

iron,

^{the first bracket is of order}

1/25,

^{the second}

10,

^{the third}

3/7 - 0.5,

^{the fourth 9.}

This leads to a ratio of order

unity.

^{This means that}

in

expression (63)

^the contribution of

the Xd"

^{term is}

probably comparable

^{to that of}

the x§

^{term. It is}

interesting

^to notice that in

the factor

(1 - 2 UXdlg2 uB Nv-2 is

^{the exact}

equi- valent,

^{in the}

ferromagnetic

^{case, of the}

exchange

enhancement factor invoked in relaxation theories for

nearly magnetic

^metals

[22].

The

Y"

^term

corresponds

^to indirect nuclear rela- xation

by

the Stoner excitations of the

ferromagnetic

electrons. In the absence _{of any}

precise evaluation,

we shall no

longer

consider it in detail in what follows.

3 . 3 .1

Simple diamagnetic impurities.

^- Let us consi- der the

magnitude

^of the different terms of eq.

(57)

for a

diamagnetic impurity. Setting kO/qmin

⁼

0,

we find that the braket is

equal

^{to :}

For

iron,

^{we can}

try

^to

adopt 1 Jsd 1

^{N 1}

eV, S N 1, EF

^{= 5}

eV, Dk2F

⁼

EF/ 12, 2qmax/qmin ~

^{7.5. Then the}

three terms of the bracket _{in eq.}

(64)

^are

respectively equal

^{to :}

Therefore if we compare the relaxation of a same

well-behaved

impurity (for example Ag)

^{in a diama-}

gnetic

^{metal and in a}

ferromagnetic

metal like

iron,

^for

a

given

^{value of}

A Ns(EF),

^its relaxation should be several times faster in iron

(~

5 times faster with the above numerical

values).

These estimates ^are

only tentative,

but the existence

of such an enhancement should not be

ignored

^{in the}

theories where one compares

(Tl T)-1

^{with the} square of the

hyperfine

^field.

3.3.2 Atom

of

^{the matrix. - In that case, as well}

as for

impurities

^{of the 3 d and 4 d} groups the relaxa- tion is

thought

^{to be}

mainly

^{of orbital}

origin [22]

(in

connection with virtual bound state

filling

^{in the}

case of

impurities).

^This process has ^not been consider- ed in the above calculations.

On the other

hand,

^{if the}

Weger

process eq.

(58) happened

^{to be} dominant the

quantity Co 2 n Tl

^{T should}

be constant for ^a

given

^{matrix whatever be the}

impu-

rity (normal

^or

transition).

4. Remark. - All

throughout

^this calculation the

ferromagnetic system

^{has been treated as}

being

^at

zero

temperature.

^{At finite}

temperature,

the method which has been used would lead to a

dispersion

^term

Dq2

^{in the}

^spin

^{wave energy}

^varying

^like

S’Z

^>

i. e. of the form

(1 - aT3/2.

It is well known that in

insulating ferromagnets

the real variation is

(1 - bT5/)

and this must also hold for rare earth metals.

For this _reason,

using

^an

expression

^of

type (9)

in order to describe the behaviour of a

paramagnetic impurity

^{in a}

diamagnetic

^{metal at finite}

temperatures

cannot be

expected

^to

give

^{correct results}

(see below).

5. Relation of the

présent

considérations to the

Giovannini-Heeger-Pincus-Gladstone

calculations. ^- These authors

[23] study

the relaxation of ^a nucleus situated in a

diamagnetic

^{metallic matrix at}

point

r ⁼

0,

ⁱⁿ the presence of ^a

paramagnetic impurity

situated at

point

^{r =}

Ri*

Here we have to work in real space with _suscep- tibilities

x(r, cw)

instead of

Xq.. Nv being

the number of sites _{per unit}

volume,

we assume a

coupling

between the

impurity spin

^{and the} conduction electron

spin density

^at

point Rj’

Let us calculate the transverse response of the _sys-

tem in the presence of ^a field

density h,(co) d(r) acting

at

point r

^{= 0 on the}

spin density a

^{and a field}

h(w) acting

^at

point Rj

^{on the localized}

^spin

^S.

The

coupled equations

^{of motion are}

given by

in which

Defining

^{an effective}

susceptibility x( w)

^{for the}

impu- rity by

we find that

These

equations

show that the observable suscep-

tibility

^{of the}

impurity

^{is indeed}

x( w),

and that in the presence of this

impurity

^at

point Rj,

^{the electron}

response ^at

point

^{r to} the field h at r ⁼ 0 is

given by

In

particular

^the

susceptibility

^{to be used in}

the compu-

tation of the relaxation time of a nuclear

spin

^at

This

equation

is the strict

equivalent

of eq.

(11).

As concerns the

relationship

^between

((cv)

^and

X(w),

since

X(co)

^{must have} rotational invariance

properties, x(cv)

^{must include the}

longitudinal

^effects of the conduc- tion electrons i. e. at 0 OK

Then

which reduces to when ^{úJ =} 0 as it should. As ^a matter of fact the exis- tence of the

J2

term in eq.

(72)

^is associated with the self energy of the Ruderman-Kittel interaction. In

the static limit

(co

⁼

0),

this energy has the form

(2 J)2

x

(r

⁼

0, 0) S(S

⁺

1),

^{or, in terms of the}

x(q, w)

the form

It is

clearly

rotation invariant. On the other

hand,

^it

diverges.

This last feature has been commented

by

Kittel

[24].

The

inspection

^{of the} self energy, ^as well as the remarks of

paragraph

⁴

suggest

^{that the use of a for-} mula of

type (72)

^{in order to}

compute x(w)

^{at finite}

temperatures

^{cannot be}

expected

^to

give good

^results.

It is then better to

adopt

^an

empirical

^{and reaso-}

nable

representation

^{such as for}

example

eq.

(3.1)

of reference

[23]

^{i. e., in terms}

of X(ro) (not X(w))

As

^a final remark let us now go back ^to the case of ^a

diamagnetic impurity

^{in a}

ferromagnetic

^metal.

Then K ⁼ 0 in _eq.

(51). Taking

^account of eq.

(50),

^it

appears that the second line

of eq. (51) ( oc xé Xe

^Re

;.)

corresponds

^{to the}

Giovannini-Heeger type

of pro-

cess while the third line

( oc xé2

^Im

xm) corresponds

to the Benoit-de Gennes-Silhouette

type

of process

(see

^eq.

⁽⁷⁰⁾

and reference

[23]).

^This

clearly

^shows

the

complementarity

between the situations considered in reference

[23]

and in the

present

^work.

6. Conclusion. ^- In

conclusion,

^{we have shown}

that the nuclear relaxation of

diamagnetic impurities

in

ferromagnetic

^metals may be

strongly

^influenced

by

^indirect processes

involving

^{virtual or} real exci- tations of the

ferromagnetic

electrons. Relaxation

by

^real

spin

^{waves is made}

possible by

^the

damping

of the

spin

^waves

^due,

^{either to their}

coupling

^with

the conduction

electrons,

^{or to their}

merging

^into

the continuum of Stoner excitations of the

magnetic

electrons. Formulas

previously

^derived

by

^other

aLthors ^are obtained as

particular

^{cases of our}

general expression (51).

^{As concerns numerical}

values,

^the

enhancement factor of the relaxation rate

by

processes

involving spin

waves, is found to be of order one

to five in the iron group and of order one hundred in the rare earth _group.

Although

^{our very}

simple

model

certainly

^{has to be}

improved

^{in order to}

provide

more

precise estimates,

the existence of such an

enhancement should not be

ignored

^{in the}

theory

of

hyperfine

interactions in metals.

Acknowledgments.

^{- Itis a}

pleasure

^{to thank}

Dr. I. A.

Campbell

who called our attention to the work

by Sano, Kobayashi

^{and Itoh}

(ref. [5]),

^which

was the

starting point

^{of this}

study.

Appendix Computation

^{of the}

transverse, frequency dependent spin susceptibility

of the free électron gas in the presence

of ^a

longitudinal magnetic

^field

(3)

This

problem

^has

already

been considered

by

^de

Chatel and Szabo

[25],

^who

give

the result in reduced units of their own. Here we

give

the formulas for small w with notations closer to those used in the main text.

NOTATIONS. ^- Electron gas in the absence of

magnetic

^{field :} Fermi energy

EF,

^{Fermi vector}

kF,

N ⁼

kF3/3 n2 density

of electrons of both

spins.

- Electron gas in the presence of ^a

longitudinal magnetic

^field

Ho. Response

^{at circular}

frequency

^co.

- N ⁼

N+

^{+ N- where}

N+,

^{N- are the densities}

of electrons of a

given spin.

We define

Then

(3) ^In

practice

^the (large) longitudinal fields which we are

considering ^{in thé main} text are the fictitious"fields associated with the couplings involving 7sch Jsf and U They only ^act

on

the spin ^of thé çlectron, not on its ortbital motion.

with

(since

^the

Landé g

^{factor is}

negative,

^when

Ho

^is

positive strictly speaking

^the

spin ^points

^downwards

so that

N-

>

N+ ;

ⁱⁿ

the, main

^{text it}

was assumed

^for

simplicity

^{that the}

^localized spin pointed upwards,

thus

leading

^to

N+

> N- for

,Tsm

>

0).

^{We shall also}

define

IMAGINARY PART OF THE SUSCEPTIBILITY. ^- We

just give

the result for small

AIEF.

^Then

x"(q, w)

is different from zero

only

^when qm;n q 2

kF -

qmin and it is

given by

^eq.

(16)

^{of the main text.}

REAL PART OF THE SUSCEPTIBILITY. ^- In terms of

It is

possible

^{to check} that when Li ⁼

0,

^{and after}

performing

the relevant

modifications,

^this

expression

is in

agreement

with eq.

(6. 6a)

of the book

by

^Schrieffer

(Theory

^of

Superconductivity).

We have done a numerical evaluation of

x’(q, 0)

for several values of the

parameter

^{W = 2}

4/E%.

The results are

reported

ⁱⁿ

figure

^{1. It is seen that even}

for rather

large

values of 2

4 /E%, x’(q, 0) only slightly

deviates from the standard Lindhard function.

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M., ^HIOKI

T.,

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