Discussion on the onset of antiferromagnetism in the hubbard model

HAL Id: jpa-00208544

https://hal.archives-ouvertes.fr/jpa-00208544

Submitted on 1 Jan 1976

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.

Discussion on the onset of antiferromagnetism in the hubbard model

F. Brouers, F. Ducastelle, J. Giner

To cite this version:

F. Brouers, F. Ducastelle, J. Giner. Discussion on the onset of antiferromagnetism in the hubbard model. Journal de Physique, 1976, 37 (12), pp.1427-1436. �10.1051/jphys:0197600370120142700�.

�jpa-00208544�

1427

DISCUSSION ON THE ONSET OF ANTIFERROMAGNETISM

IN

THE

HUBBARD MODEL

F. BROUERS

Laboratoire de

Physique

^des

Solides,

Université

Paris-Sud,

⁹¹⁴⁰⁵

Orsay,

^France

F. DUCASTELLE

Office National d’Etudes et de Recherches

Aérospatiales,

⁹²³²⁰

Chatillon,

^France and J. GINER

Institut de

Physique,

Université de

Liège,

^Sart

Tilman,

⁴⁰⁰⁰

Liège, Belgium (Reçu

^{le 31 mai}

1976, accepté

^le

30 juillet 1976)

Résumé. ²⁰¹⁴ Nous calculons la réponse à ^un champ magnétique périodique ^{de la} phase désordonnée du modèle de Hubbard dans l’approximation

d’analogie d’alliage.

^Nous étudions le comportement de la susceptibilité ^{et en} particulier ^nous recherchons une éventuelle divergence ^{de la} susceptibilité

pour ^{un vecteur} d’onde

correspondant

^{à un ordre}

antiferromagnétique.

^{Nous ne trouvons aucune}

instabilité _pour une bande à moitié

pleine

^ou presqu’à ^moitié pleine.

Abstract. ²⁰¹⁴ We calculate the response ^{to a} periodical ^external magnetic field of the magnetically

disordered phase ^of the Hubbard model in the alloy

analogy

approximation. ^We investigate ^the

behaviour of the

q-dependent

susceptibility ^{and in} particular ^{we look for a}

possible divergence

^{of the}

susceptibility

^{for a} wave-vector

corresponding

^{to an}

antiferromagnetic

ordering. ^{We do not find} any

instability ^{for a} half-filled or nearly half-filled band.

LE JOURNAL DE PHYSIQUE ^TOME 37, ^DTCEMBRE 1976,

Classification Physics ^Abstracts

8.516

1.

Introduction.

^-

Recently

^several papers

[1-6]

have been devoted to the discussion of the

possibility

of

ferromagnetic

^and

antiferromagnetic

^solutions

and their

stability

^with respect ^{to the}

paramagnetic

solutions of the Hubbard model used in the discussion of the

magnetic

^and

conductivity properties

^{of narrow}

band materials.

It is now well established that the Hubbard III

approximation [7]

^{does not}

provide

^a stable ferro-

magnetic

solution. As was shown in

Velicky et

^al.

[8],

the Hubbard III solution which includes

only

^the

scattering

corrections is identical to the coherent

potential approximation (CPA)

^{of the}

theory

^of

alloys.

Fukuyama

and Ehrenreich

[1]

have shown that the static

spin susceptibility

of the Hubbard model calculated in the CPA remains finite for _any

simple

band model and carrier number

except possibly

in the case of a

precisely

half filled

split

^{band. Brouers}

and Ducastelle

[2]

have shown that this statement is more

general

^{and should be}

true if , ^the

^Hubbard

model could be solved

exactly ^within

^the

alloy

^ana-

logy approximation.

^{This statement} is therefore true for _any extension of the

CPA,

for instance the mole- cular CPA.

In

particular

^since Weller and Gobsch

[9]

^have

shown that for a

half-filled

band and in the _parama-

gnetic phase

the full Hubbard III solution

including

resonant correction can be written as a solution of the CPA with a continuous distribution function for the stochastic

variables,

the conclusions of ref.

[2]

are also valid in that case.

Schneider and Drchal

[4]

have shown

by using

energy considerations that in the strong correlation limit the Hubbard III

scattering

correction

approxi-

mation

(CPA) yields

^an unstable satured ferroma-

gnetic

^{state for}

nearly

half-filled band. For n 1

only

^the

paramagnetic

^solution

exists,

ⁱⁿ agreement with the

previous

^results.

Although

^{it is not} known if the exact solution of the Hubbard Hamiltonian

yields

any stable ferro-

magnetic

solution for _any electron band

model,

the results of the

alloy analogy approximation

^of

that model are in agreement ^{with Van Vleck’s}

[10]

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197600370120142700

statement that itinerant

ferromagnetism

^{is due essen-}

tially

^{to the} intra-atomic

exchange

interaction between electrons of

degenerate

d-orbitals. Brouers and Ducas- telle

[2]

have considered a model with orbital

dege-

neracy and

analyzing

the behaviour of the static

spin susceptibility

have shown that in that case an insta-

bility

^{of the}

susceptibility

and therefore a ferroma-

gnetic

^{solution was}

possible.

These conclusions confirm and extend

previous

calculations of Schneider

et al.

[11]

who reached similar conclusions in the Hubbard I

approximation [12]

and in the strong correlation limit.

The

possibility

^{of an}

antiferromagnetic

^solution

of the

single-band

Hubbard model remains also a

controversial

question although

^a certain number of results have been obtained and discussed

recently.

Mehrotra and Viswanathan

[5]

have shown that the Hubbard

decoupling

^{scheme is not able to}

yield

a stable

antiferromagnetic

^state.

By

^contrast

Gupta

et al.

[6]

have obtained an

antiferromagnetic

^solution

by calculating

the order parameter

self-consistently

within the two sublattice CPA

(Brouers et

^al.

[13])

of the Hubbard

alloy analogy approximation.

Meyer

^and Schweitzer

[14]

^have introduced a

spin dependent

band shift

following

^{Roth’s formalism}

[15]

and found two

antiferromagnetic

^{solutions in a small}

region

around the half-filled band for s.c. and b.c.c.

structures but the solutions are unstable toward

paramagnetism

^and

ferromagnetism.

In this _paper, we report the result of a calculation of the _response of the

magnetically

disordered

phase

of the

alloy analogy

^{model to a}

periodical

^external

magnetic

^{field. We}

investigate

^the

properties

^{of the}

q-dependent susceptibility

^{and in}

particular

^{we look}

for a

possible divergence

^{of the}

susceptibility

^{for a}

wave-vector

corresponding

^{to an}

antiferromagnetic ordering.

^{We do not find} any

instability

^{for a half-}

filled

single

^band.

2. Model. ^- The model

employed

^to

investigate

the itinerant

magnetism

^{in narrow band}

binary alloys

is

represented by

the Hubbard Hamiltonian

written in second

quantization using

^{a Wannier basis.}

The operator

a’

^and ana ^are the creation and annihi- lation operators ^{for an} electron with

spin

^{Q at the}

site n and

fin, = a ’

a,,,. As was first

pointed

^out

by

Hubbard

[7],

the motion of electrons with

spin s

^{= ± Q}

can be described

approximately by

^the

following alloy

Hamiltonian

where _t;na is

equal

^{to e’ = U or c,’}

0, depending

on whether an electron with

spin a

^{is at site n}

or not. At a fixed instant in time the electrons with

opposite spins

^are

regarded

^as

occupying given positions

^on

randomly

distributed lattice sites. One

neglects

^the

dynamical

effects of

opposite spin

^elec-

trons and one

replaces

^{the time} average which arises because of the continuous rearrangement ^of

spins by

^a

configuration

^{average in the}

alloy problem.

Hamiltonian

(2)

^can therefore be viewed as

describing

two

interdependent binary alloys A1-n-a Bn-a’

^Here

na is the average number of electrons _per site with

spin a :

nna

).

The relative concentration of atoms with energy levels

9

is 1 ^- _n-a or 1 _{- n,,} and with energy level t!3 is _n-a or n,,

corresponding

^{to whether}

the electrons have

spin

^{Q or - Q}

respectively.

^{It is}

assumed for the _purpose of

calculating

the motion of J

spin

electrons that electrons of

spin -

^{Q are}

fixed at the lattice sites and vice-versa.

For a

given configuration,

^{one can define a}

Green’s

function

and the _average number of electrons with

spin

^{+ a}

is

given by

where N is the number of sites. The _average Green’s function

being

^a function of the concentration _n-a is

given by (at

^{T = 0}

K) :

The Fermi level is defined

by

^{a third}

equation

where n is the _average number of electrons _per site.

3.

Spin susceptibility.

^{- The}

paramagnetic

^static

spin susceptibility corresponding

^to Hamiltonian

(2)

has been discussed in ref.

[2].

^{Here we} consider the

q-dependent susceptibility.

^A

periodic

external field induces in the

paramagnetic phase periodic

^variations

in the number of _{up and} down

spin

electrons. In the

alloy analogy

model this

corresponds

^to concentration

variations.

^{If we start} from the disordered _parama-

gnetic phase,

^{we can} calculate the

spin susceptibility corresponding

^to

periodic spin dependent

^concen-

tration _{nn -u} in the

alloy analogy approximation.

This

corresponds

^to concentration variations

1

1429

induced

by

^an infinitesimal external field

The

q-dependent spin susceptibility

^reads

where _J-lB is the Bohr magneton ^and

This

gives

^for

X(q) :

In the

alloy analogy

^model

n-,(q) plays

the role of a

concentration. One has

6G,(q)

^{is the Fourier} transform of

6 ( n ) I Gin>

^and

since

in the

alloy analogy approximation

^{where Z is the}

self-energy

operator which is site

diagonal

^{in the CPA}

One has therefore

If we define the Fourier transform

The

q-dependent spin

^enhanced

susceptibility

^is

defined

by

X’(q)

^is the unenhanced

susceptibility

^and

where

This allows a calculation of a

possible instability

of the

paramagnetic susceptibility

^{for a}

given

q.

To calculate

K(q),

^we

expand

^the self-consistent

equation

^{as a} function of the concentration fluctua- tions

bc,,

⁼ cn -

co

^on each site _n,

c’ being

^the

concentration in the

completely

disordered _para-

magnetic

^state.

To achieve that programme, ^{we use an} extension of the CPA to

inhomogeneous

systems ^discussed in

appendix

^{1 and we} vary the self-consistent _equa-

tions tn >

^{= 0 with} respect ^to the concentration fluctuations

ðCn (appendix 2).

One obtains for

bc(q)

^the

expression

^{in terms of}

complete

^disorder

quantities

where

and

This

gives

4. Calculation of

K(q).

^{- This}

quantity

^{has been}

calculated for two models :

a)

^The

simplest

^{one, a.} one-electron band with nearest

neighbour

interactions on a BCC

(similar

to ref.

[6]) ^{CICs ;}

b)

^{The model}

density

^{of states}

simulating

^{the BCC}

structure

already

introduced for the calculation of

ordering

energy

[16, 17].

^{In that} case, ^we calculate

K(q)

^{for the wave vector}

By introducing

^{a second}

neighbour

interaction one can avoid

pathological

difficulties related to the

occurrence of a gap in model

a).

The first case is

trivial,

^{one needs}

A( q, E)

^which

is obtained from its definition

(16)

If one divides the sublattice into ^two sublattices a

and fl

^{and if n is on} sublattice a :

For

with

I k,, >

^{is the Bloch state built on} the sublattice a ; k

belongs

^{to the Brillouin zone of the s.c. lattice.}

In the first

model,

the Green’s function is

given by

where a a(p) ^{is the}

self-energy corresponding

^to sublattice

rx(P)

and determined from the self-consistent _equa- tion

(A8)

^and

s(k)

^{is the}

dispersion

^relation

corresponding

^{to the BCC} structure. One gets

immediately

^{in the}

completely

disordered case

where ax

⁼ up

where we have used the definition of the

density

^{of states}

If one notices that the

diagonal

Green’s function

one has

In the second

model,

^{the band structure} is charac- terized

by

the values of the _energy

levels 8;

^{and the}

values of the

hopping integrals

^{between atoms of the}

same

(Wma,nP)

^and of different sublattices

(Wma,nP).

In the BCC structure, the transfer

integrals

^between

first

Wap

and second

Wem

^nearest

neighbours

^{are of the}

same order of

magnitude

^{and determine}

mainly

the band structure of the _pure metals. We

keep only

these two

integrals

^{and we do not take into account}

the

degeneracy

of the d-states.

If we

neglect

^{the transfer}

integrals W,,fl

^which

couple

^both

sublattices,

^{we obtain two}

tight binding

bands centered on the

energies 8; (i

⁼

A, B)

^whose

energies

^and

eigenstates

^are

respectively given by

8«(k) (or ap(k)) and I qi,,,(k) > (°r I ql,6(k) >).

^{The com-}

plete

Hamiltonian is then

given

^{the basis}

{ t/J a’ t/J p } ^by

where the summation is extended over the Brillouin

zone of the cubic

crystal.

^Moreover for the numerical

computation,

^{we will}

replace

^the

coupling

^function

by

^an average value

independent

^{of k.}

1431

This kind of

approximation

^has

already

^{been used}

in the case of s-d

mixing

^effects

[18] ;

^the correspon- dance is established if we

replace

^a

and fl by

^{s and d.}

In that model

A(q)

is obtained from its defini- tion

(17).

^We consider the wave-vector

To calculate

equation (26)

we need

From the definition of the Green’s function

one gets

immediately

and

In the

complete

disordered case a,, ⁼ a, and ^{we assume}

Using

the definition of the

density

^{of states}

(29),

^one

gets

The function

K(q)

^{has been} calculated in both models

using

^for

go(E) a semi-elliptic

function for

a half-filled band and for

nearly

half-filled band.

We have found

numerically

^that

K(q*)

^{is such that}

- 1

K(q*)

^{0. In} the half-filled

band,

^{one can}

show

analytically (Appendix 3)

^that

K(q*) --+ -

¹

when U --+ oo. In the same limit

x(q*)

^{tends to 0}

as

U-’.

In

figure

^{1 we} have drawn

x(q*)

^{in terms of U for}

the first model. It

diverges

^{at U = 0 due to} the pre-

sence of a gap in the BCC one-band and decreases

slowly

^{with U.}

From our

calculations,

^{one can} conclude that a

perturbative

^treatment

starting

^{from the} parama-

gnetic

^solution of the CPA

alloy analogy approxi-

mation Hubbard model does not exhibit _any

tendancy

towards

antiferromagnetism.

This result contredicts

a recent self-consistent calculation of

Gupta et

^al.

[6]

who claims that an

antiferromagnetic

solution is

possible

^{in the}

non-degenerate

Hubbard model within the

alloy analogy approximation

^{when U is}

larger

than some critical value. As our calculations are not

incompatible

^{with a}

possible

first order transition

we have redone their calculations and we do not find any

antiferromagnetic

solutioliq We discuss this

point

^{in ref.}

[19].

FIG. l.

The results of this _paper as well as the conclusions of ref.

[2]

^indicate

clearly

the limitations of the

alloy analogy approximation.

^{It is a} well known exact result of the half-filled band Hubbard model that in the limit of _very

large U,

^it

approaches

the Heisen-

berg

^{model. In that}

regime

the interatomic electron

correlation at

neighbouring

^sites

produces

^{the ten-}

dancy

^towards

antiferromagnetism.

For that _reason, it is

hardly expected

^{that the}

Hubbard

alloy analogy approximation yields

^a

meaningful

result in this

regime

where localized moments are formed and the method of this _paper should be restricted to the metallic

regime.

^{As it is}

believed

generally

^{that an}

instability

towards anti-

ferromagnetism

^{will not occur}

prior

^{to the} appearance of localized moments, the conclusions obtained in this _paper are in

agreement

^{with this}

general

expec- tation.

Appendix

^{1 :} Concentration fluctuations in disor- dered

alloys.

^{- In} the usual CPA for

homogeneous alloy

cn ⁼ c, and the _energy levels are

only dependent

on the nature of the considered ^atom i ; the effective Hamiltonian

JCO

is then determined

by

^a site inde-

pendent

^effective

potential (or self-energy)

^localized

on each site n

This.

self-energy

^is determined

by

the condition that the

density

^{of states} per ^{atom on} site n and for the average

medium g(E) >’

^is

equal

^{to the} average of the

partial (or conditional)

densities of states

which are the densities of states

averaged

^{over all}

the

configurations

for which the site n is

occupied by

^{an atom i :}

Z’ can also be obtained

by

the condition that the electrons of the _average medium are not scattered

on the _average

by

^atoms

sitting

^{on n with}

respective probabilities

^c’

(i

⁼

^A, B)

the t-matrix

elements ti

^are

given by

in

^terms of the function F

n n >

^where

Ul 0) = _{(E -} Jeeff) -1

^is the Green’s function for the

averaged

medium and

c(O) = n E (0) 1 n >.

^This

scheme can be extended to

spatially varying

^concen-

tration _cn. The

density

^{of states for the} average

medium gn(E)

^and the conditional densities of states

gn(E)

^{are now site}

dependent

^and

(A3)

^is

replaced by

similar conditions

The effective medium is defined

by

^site

dependent potentials

^{localized on the}

corresponding

^{sites n}

The

potentials a,,

^{are obtained in terms of the}

scattering

operators

by

^{the condition}

where the brackets mean that we

perform

^the average

according

^to

(A6)

^{and where the}

scattering

operators

tn

^{are defined}

by

Finally Fn = n I Gin> in ^(A9)

is related to

the

Green’s function G ⁼

(E - JCeff) ^1.

The function G and therefor E and

Jeeff depend

^{on the} energy E.

The _average

density

^{of states} per ^atom

g(E)

is obtained from

Fn by

The conditional densities of states

gn(E)

^{are related}

to

Fn by

The

equivalence

between the conditions

(A6)

^and

(A8)

can be

easily

obtained from

(A 11).

The

previous

^{formal treatment can be}

applied

^to

the

study

^of

partially

^{ordered state. The state of an}

ordered

crystal

is characterized

by m

different values of the concentration

corresponding

^{to the m different}

sublattices. The

quantities

a.

and g.

^{are determined}

by

^the

previous equations

^and

by

^{the m} self-consistent

equations (A8).

This scheme has been

previously

used for models of

ordering

ⁱⁿ

simple

^structures

where m

corresponds

^{to a}

and fl

^{the two} sublattices of an alternant lattice

[13].

Appendix

^{2. - We} calculate the variation to first order of the

spatially varying

self-consistent _equa- tion

(A6)

^{due to} concentration

(order parameter)

variation

For a

binary alloy,

^{one has}

t(O) and c(O) refer to the

complete

^disorder.

1433

From

taking

^account of the relation

one obtains

immediately

and since

averaging

where we have

put Fn

⁼

^F,

^the

diagonal

^Green’s

function

corresponding

^to

complete

^disorder.

Introducing

the Fourier transform

we have for each wave-vector q :

where

and

G(R)

⁼

GRO, 0 being

^a

particular

^site of the lattice.

The Fourier transform of condition

(A 12) yields

or

i.e.

if we define

since for a

binary alloy dcn - - dct

⁼

^den

^{one has}

with and

Appendix

^{3. - In this}

appendix

^{we want to calculate}

the limit

of x(q*)

^{for U --+ oo.} To’this end we determine in this limit the

quantities K(q*)

^and

x°(q*).

^{We use}

the moment method discussed in the paper of

Velicky

et al.

[8]

^{which we} first recall

briefly.

We consider ^a

binary alloy Ai-c B,

characterized

by

^the

potentials

^{I;A = 0 SB =} 6 in the limit 6 -+ ^oo

(Fig. 2).

We want to calculate the

integral

^{of a}

quantity

over the states of the A-subband. We can write

FIG. 3.

e is a contour around the cut ^« A » on the real axis

(Fig. 3).

^{We assume that}

qJ(z)

^{has no other}

singularity

than the cuts on the real axis and that

qJ(z)

^{tends to}

zero faster than Z-l when z - oo. One has therefore the

spectral representation

If one defines the

density ql(E) = - 1m _{E + ,}

one can write

If o y

( )

_7E

P( )

Expanding

both sides in z - ’ and

identifying equivalent

^{terms one gets}

the moments of the total band in terms of the moments of the subbands

in the limit 6 --+ oo the moments are obtained

by

identification. One can obtain

pg

^{which is}

equal

^{to the}

integral

This

technique

^{can be used to} calculate the

asymptotic

^{form of}

equation (23) integrated

^{over subband A}

One has first to

expand

^the

integrand.

^Since

G(R)

calculated in the CPA is exact up ^to

z-’,

^this

approximation

will

give

^{the first} exact terms in an

expansion ^{in 6 - ’.}

By definition,

^{one has}

differentiating

since

one can write

and Fourier

transforming

For the CICs model

q* - [1, 1, 1]

^{and we}

integrate

^{over the band A.}

One knows that

and for

q*

or

with

This

gives :

if R1

^{are the sites of the} sublattice 0 0.

1435

If we

expand

we can write

keeping only

^the

higher

^{terms in the}

potentials

ER l In the same way

We have thus :

and the summation on

Rl

ⁱⁿ

(A40)

^is

given by

^the

expansion

where

pg

is the second moment of the ordered band.

The moments of

(A40)

^are

Using

^now

equation (A29)

^to define the

pf and

^{in terms of} /In.

Solving

^{the set of}

equations

up to /l5 for ð ---+- oo

yields

^{the result}

and

As we have taken

only

^{terms z-4 in the}

expansion

^of

G,

^{one can} conclude that in this limit CPA

gives

an exact result.

We can calculate

X’(q) by

^{the same}

technique

^{for 6 -+ oo . We start from}

one has

and thus

We are interested in the first terms in

ð - 1,

^{we take}

only R,

^{for first}

neighbours.

^{To the}

highest

^{order of g in the}

z - 1

expansion,

^{we have}

Averaging G2(Ri) gives

^the

following expression

From which one determines the _{moment N} and from the

equations (A29)

up ^to the seventh moment,

we obtain for b - ^oo and

As

we have

and since here

c = 2

References

[1] FUKUYAMA, H. and EHRENREICH, H., Phys. ^{Rev. B 7} (1973) 3266.

[2] BROUERS, F. and DUCASTELLE, F., ^J. Physique ³⁶ (1975) ^851.

BROUERS, F., Phys. ^{Stat. Sol.} (b) ⁷⁶ (1976) ^145.

[3] NOLTING, W., Phys. ^{Stat. Sol.} (b) ⁶⁸ (1975) ^243.

[4] SCHNEIDER, ^{J. and} DRCHAL, V., Phys. ^{Stat. Sol.} (b) ⁶⁸ (1975)

207.

[5] MEHROTRA, C. and VISWANATHAN, K. S., Phys. ^{Rev. B 7} (1973)

559.

[6] ^{GUPTA, A.} K., EDWARDS, D. M. and HEWSON, A. C., ^J. Phys. ^C 8 (1975) 3207.

[7] HUBBARD, J., Proc. R. Soc. A 281 (1964) ^401.

[8] VELICKY, B., KIRKPATRICK, ^{S. and} EHRENREICH, H., Phys.

Rev. 175 (1968) ^747.

[9] WELLER, W. and GOBSCH, G., Phys. ^{Stat. Sol.} (b) ⁶⁰ (1973) ^783.

[10] ^{VAN VLECK, J. H., Rev. Mod.} Phys. ²⁵ (1953) ^220.

[11] SCHNEIDER, J., HEINER, ^{E. and} HAUBENREISSER, W., Phys. ^Stat.

Sol. (b) ⁵³ (1972) ^553.

[12] HUBBARD, J., Proc. R. Soc. A 276 (1963) ^238.

[13] BROUERS, F., GINER, J. and VAN DER REST, J., ^J. Phys. ^F4 (1974)

214.

[14] MEYER, ^{J. S. and} SCHWEIZER, ^J. W., AIP Conf. ^{Proc. 10} (1973)

526.

[15] ROTH, L., Phys. ^{Rev. 184} (1969) ^451.

[16] GAUTIER, F., DUCASTELLE, ^{F. and} GINER, J., ^Phil. Mag. ³¹ (1975) 1373.

[17] GINER, J., BROUERS, F., GAUTIER, ^{F. and VAN DER} REST, J., ^{to be} published.

[18] BROUERS, F. and VEDYAYEV, A. V., Phys. ^{Rev. B 5} (1972) ^348.

[19] BROUERS, F., DUCASTELLE, F., TREGLIA, G., BRAUWERS, ^M.

and GINER, J., ^J. Physique Colloq. ³⁷ (1976) ^C4-205.