The homogeneous helimagnet and the liquid-solid transition

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The homogeneous helimagnet and the liquid-solid transition

J. Villain

To cite this version:

J. Villain. The homogeneous helimagnet and the liquid-solid transition. Journal de Physique, 1973,

34 (2-3), pp.211-216. �10.1051/jphys:01973003402-3021100�. �jpa-00207374�

THE HOMOGENEOUS HELIMAGNET AND ^THE LIQUID-SOLID ^TRANSITION

J. VILLAIN

Institut

Laue-Langevin,

^Grenoble

(Reçu

^{le 26}

juin 1972)

Résumé. ²⁰¹⁴ On examine la

possibilité

d’une transition avec une

susceptibilité

infinie pour q ~ 0 dans un milieu

homogène.

Dans le cadre d’un modèle

précis (§ 2)

^{on montre}

(§ 4)

que seule une

faible

singularité

peut exister, ^et

probablement

pas

d’infinité ;

^la

singularité

^forte

apparait à q

^{= 0}

et pour ^une

quantité qui

^n’est pas la

susceptibilité (§ 3).

^On

prévoit

^{un résultat}

analogue

^{dans le}

cas des

liquides (§

^{1 et}

5).

Abstract. ²⁰¹⁴ The

possibility

^{of a} transition with an infinite

susceptibility

^at

non-vanishing q

in an homogeneous ^{medium is}

investigated.

In the frame of a

precise

^model

(§ 2)

it is shown

(§

⁴⁾

that

only

^{a weak}

singularity

^can occur, and

probably

^no

infinity ;

^the strong

singularity

appears at q ^{= 0 for a}

quantity

^{which is not the}

susceptibility (§ 3).

^A similar result is

conjectured

^{for the}

compressibility

^of

liquids (§

^{1 and}

5).

Classification Physics ^Abstracts

15.00 ^- 14.80

1. Introduction. ^- It has been

suggested [1] ]

that the

pair

correlation function of ^a metastable

liquid might

^become

long

range at ^some

tempera-

ture

TL

where the

liquid

becomes unstable. More

recently,

^the

dynamical aspect

^{of this}

instability

was

investigated [2], [3] and,

very

recently, experiments

were

performed

^to check this

theory by measuring

the static and

dynamic scattering

^factors of neutrons

[4].

S(q)

^and

S(q, w). S(q)

^is

expected

^to become infinite at

TL

^{for some value} qL

of q,

^and

S(qL, w)

^is

expected

to have a

vanishing

^{width at}

TL (infinite

^{De Gennes}

narrowing).

Experimentally TL

^{could not} be reached

[4],

^and

possibly

^{there are} fundamental reasons for

that,

at least for a

simple liquid ;

^{but we do not wish to}

consider this

problem :

^we

^only

^mention

[5]

^that

symmetry

^{does not exclude a} second order transition for some well-defined _pressure at which the 3rd order terms in the

thermodynamic potential

^vanish

accidentally.

^Our

problem

^{is the}

following :

^assume

that

TL

^{can be}

reached,

then what is the

type

^{of the}

singularity ?

For T N

TL and q N qL

^the

following approximate

formula has been

suggested [1 ], [2] :

with K2 = A(T - TL). A

^{and B are} constants. The factor K at the numerator is

lacking

^{in reference}

[3],

but is _necessary because

S(g)

^{has to be}

integrable [6].

Thus, S(g)

^{turns out to be}

proportional

^to

b(q - gL)

at

T,,

^{so that the}

singularity

^of

S(q)

^{at the}

instability temperature

^{is found to be much}

sharper

^{than in}

usual transitions

[5],

^where

1 /S(q)

^remains

analytic

at

TL.

^{On the other}

hand,

^there are reasons to believe that on the

contrary, S(q)

is smoother than usual :

indeed,

^{assume that at}

TL,

^the

liquid

^becomes

locally

solid

like,

ⁱⁿ

agreement

with the ideas of references

[1 ]

and

[2].

^{For a solid}

S(q) _ Mft d(q - t),

^{where t}

r

are

reciprocal

^{lattice vectors. For a}

given

orientation of the solid germs, ^one

expects,

^at

TL :

by analogy

with the usual Ornstein-Zernike

formula ;

but in the

present

^{case it is} necessary ^to average

on the orientations of ⁱ for a

given 1 I 1

⁼ qL :

This

simple

^minded

argument

^{will now be}

put

into a more

sophisticated

^form.

However,

^rather

than

arguing

^{on the difhcult case of a}

liquid,

^{it is}

convenient to define a

precise

^model

^which, although

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01973003402-3021100

212

probably academic,

^{seems to have} features similar to those attributed to

liquids

ⁱⁿ references

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

2. The

planar helimagnet

^with

isotropic

interactions in a

homogeneous

^{médium. -} We consider a conti- nuous, 2-dimensional vector field in the 3-dimensional space, with energy :

where

J(r)

^{is a} function of r

= 1 r only.

It is assumed

that m(r) 1

⁼

^1,

^{so that}

^m(r)

^is

completely

^defined

by

^its

polar angle cp(r) [7].

In the absence of further

prescription,

the model has ^no

phase

transition : indeed it is the limit of an

increasing density

^of

vanishingly

^{small moments with}

vanishingly

^small

individual

susceptibility,

^so that there is no

phase

transition as can be seen from molecular field _appro-

ximation,

which in this case is

rigorous

because of the

large

number of

neighbours [8].

^{As a}

phase

transition is

desired,

the Fourier transform

mq

^of

m(r)

will be assumed to vanish

for q greater

^than

some cut off _qo > QL To ^sum _up, the

prescriptions

are :

In _any

physical analogue

^of the model

(see § 6),

condition

(4b)

expresses the fact that the number of

degrees

of freedom is of the order of the number of atoms.

Conditions

(4a)

^and

(4b)

^are

compatible :

^{it is}

indeed sufficient to find

mx(r) satisfying mx(r)

¹

and relation

(4b),

^then

M2 x(r)

^satisfies

(4b),

^therefore

m2 y =

^{1 -}

^mx

also satisfies

(4b)

^{and so does} my.

2.1 Low TEMPERATURES. ^- We start with the

assumption (to

^be

justified below)

^{that the}

ground

state is ^a

helimagnetic

structure :

For a

weakly perturbed configuration :

the _energy is :

Ô(Pk

^and

5(k)

^{are the} Fourier transforms of

ôg(r)

^and

J(r) :

The bracket in the above

expression

^is

positive (or

^{zero for k =}

0)

and therefore the

helimagnetic

structure is stable at T ⁼

0,

and remains stable at low

temperature

^if

(4b)

^is

satisfied ;

qL is the maximum of

l(k).

2.2 HIGH TEMPERATURES : : MOLECULAR FIELD APPROXIMATION

(MFA). - Let xq

^{be the}

staggered susceptibility

^{of the}

non-interacting system, only

submitted to conditions

(4) :

where

hq

^{is the}

^qth

^Fourier

component

^of the external field. The effect of Je can be taken into account

by adding

^{a molecular field 2}

3(q) mq

> ^to

h :

with

The

system

^described

by

conditions

(4)

^is

roughly equivalent

^{to an} array

of cells

^{of side}

^llqo

^with

magnetization

^{of order}

1/qo3 ; thé

individual suscep-

tibility

of each cell is

given by

^a Curie law : xo ~

f3/qg

and a similar law is

expected

^for

xq :

It is seen from

(5)

^and

(6)

^{that the Néel}

temperature

is

where qL

^maximizes

3(q). Neglecting

^{the weak}

possible

dependence

^of

Xo

^{on q,}

⁽⁵⁾

reads for T N

TN

^{and so that :}

where V is the total volume and

mq the

^Fourier

transform of

m(r),

^{and where}

vanishes at

TN.

The factor K of eq.

(1)

^is

lacking

at the numerator of eq.

(7),

^thus

violating

^{the sum}

rule

2.3 SPHERICAL MODEL. ^- The

spherical

^model

[9], [10]

^accounts for eq.

(8) (but

^{not for the more}

^general

condition

m2(r) - 1) ^{by using}

^the

^density

^{matrix :}

where V is the volume and :

where the

Lagrange multiplier y

is determined

by :

The critical

temperature TN

⁼

1 /KB f3N,

^determined

by

^the

condition Il - 3(q,)

⁼

0,

^{is found to be} zero, in contradiction with what was shown

above,

para-

graph

^2.1.

The best way ^to describe critical

phenomena

ⁱⁿ

our model in a consistent

approximation

^{is to make}

a

change

^of

thermodynamical

^variables.

3.

Change

^of

thermodynamical

variables. ^- Rather than

m(r),

^{we wish to introduce} the variable :

In terms of the

polar angle Q(r)

^of

m(r),

eq.

(3)

reads :

The cosine can be

expanded

in powers of

u(r)

as follows :

cos 1

After

integration

^{of this}

expansion,

^the

expansion

of the

integral

contains :

a)

A zero-th order term

which will be

represented by

its Mac Laurin

expansion :

and has a minimum 1

b)

^Terms

containing

the derivatives of u,

namely :

a term

proportional

^to

(div U)2

For the sake of

simplicity,

the last 3 terms, which

are of

higher

^{order in u, will be}

neglected.

It is relevant to

express

^{in terms of the Fourier}

214

transform of

u(r) - VQ(r),

^{which is a vector with}

components :

The variable uq ⁼

qqJq

^has been introduced because

1 Uq 12 >,

^{and not}

1 qJq 2

> remains finite for q

going

^{to zero} in the limit of infinite volume. a,

y, 03BE

denote the

coordinates : x,

y, ^z.

Expressing

^{the term}

a)

^{and the first 2 terms}

b)

^as

functions of

u,,, inserting

into eq.

(9),

^and

integrating

over r and

r’,

^we

get :

C, D,

^{E are constants.}

If the ordered

phase,

^{as was said in}

paragraph 2, corresponds

^to

u(r)

> ⁼

C’,

^{the constant E}

is

expected

^{to be}

positive

^{and the}

transition,

^{in terms}

of uq, ^takes

place at q -

0. This transition is of a

very ^common

type, comparable

for instance to a

Bose

condensation,

^since uo becomes

macroscopic (proportional

^to the volume

V)

below the transition

temperature TN.

Conventional ^«

scaling theory

^»

[9], [11] ] predicts

^above

TN a

correlation function of the

following type :

where q

^{is a small}

positive

coefficient

and f is

^{a function.}

In the

appendix,

^the

thermodynamics

^{of the}

system

is

investigated

^{in the Hartree}

approximation

^and

eq.

(11)

is derived

microscopically with q

^{= 0.}

4. Correlation-funetion for

m(r).

^{- We have}

given

a consistent

description

^{of the}

system

^{in terms of}

u(r). Unfortunately

^the correlation function

(11)

is not

easily

accessible to

experiment (in

^the

^liquid-

solid

transition, u(r) represents

^{a vector}

joining

2

neighbouring

^{atoms : see next}

section).

^{It is}

therefore of interest to

investigate

the suscep-

tibility m(o) . m(r)

>. It cannot be deduced from

u(O). uer)

> without additional

assumptions,

^be-

cause

strictly speaking,

^the

knowledge

^of

higher

order

correlations,

^uuuu

>,

etc., ^is

required.

^To

approximate

^these correlations

by

^a factorization is not

satisfactory,

^{as shown in the}

appendix.

It seems

preferable

^to

apply

the ideas of the ^«

droplet

model »

[11]

^{and to assume that at}

TN

^the

system

^is constituted of ^«

droplets »

^which look somewhat like the condensed

phase. Quantitatively,

^{this idea}

is

expressed by

^the

following assumption :

P(r)

^is

roughly

^the

probability

^{that r}

belongs

^to

the same «

droplet

^{» as the}

origin. m(r)

^and

u(r)

are mean values

corresponding

^to

given

^{values of}

m(0)

^and

u(O). Comparison

^of eq.

(11)

^and

(12b) yields :

The case T >

TN

^{will not} be treated for

brevity.

At T

TN, P(r)

^{has a} finite limit for

large

^r.

For ^a

given

^{value of}

u(O) =

^{u, the average value}

of

m(O).m(r)

^{is from}

(12a)

^and

(13) :

and

averaging

^over the orientations of u

yields :

The Fourier transform of this function is not infinite

at q

= qL,

except if il

⁼

0,

^{where the}

loga-

rithmic

singularity

^of

paragraph

^{1 is recovered.}

5.

Liquids.

^- Our model can be

transposed

^to

liquids :

^{the function}

g(r),

^{instead of}

being

^the

polar angle

^of

m(r),

^{now defines « reticular surfaces »}

by

^the

equation

^cos

qJ(r) =

^{1. It is also}

possible

^to

define 3 families of reticular surfaces

corresponding

to 3 variables

qJa(r) (a

⁼

1, 2, 3).

^{The atoms would}

be located at the intersections of 3 surfaces. A

possible

hamiltonian is :

The hamiltonian

(3)

^{is a}

simplified

version with

only

^{one function}

Q(r).

^{The ordered}

phase

^described

in

paragraph

^2.1

clearly pictures

^{a solid since the}

reticular surfaces are

parallel planes.

The

description

^{of the}

liquid phase

is very poor, due to the absence of

dynamics

^{and to} the presence of elastic

properties

^{that a}

liquid

^has

probably

^not.

It is not clear whether a more realistic

description

^of

a

liquid

^would

give

^{rise to an}

instability

^{of the}

type investigated

in this paper, but if it

does,

^{there is}

probably

^no

singularity

^{at finite q in the}

compressi- bility,

^{or a weak} one, ^as discussed in

paragraphs

¹

and 3.

Acknowledgments.

^- The author is

especially grateful

^to Pr. P. G. De

Gennes,

^B.

Jancovici,

^{P. C.}

Martin and L. Verlet for their valuable comments and

suggestions. Acknowledgments

^{are also made}

to Dr. Kleban and

Kugler,

^{and to} Pr. Schofield.

Appendix :

^{The Hartree}

approximation

The hamiltonian

(10)

^{will now be} treated in the Hartree

approximation, namely :

In this

approximation,

the correlation function

is,

^from

(10)

^and standard fluctuation

theory [5] :

The Néel

temperature TN

^{is defined}

by

^the

following

relation :

Below

TN,

uo is

given by

the minimization of

(10)

after insertion of

(A.1) :

where Ox is the direction of _uo.

Inserting (A. 4)

^into

(A. 2) yields :

where ç

^{is defined}

by

^the

following integral equation :

When ç

increases from 0 to

00, j (ç)

^decreases

from a

positive

^{value to}

0, except

^for uo -

0,

^then

f(ç)

^{= 0.}

Therefore ç

> 0 for T

TN,

^{so that the}

stability

condition

u _ q uq > > 0 is fulfilled,

ⁱⁿ

agreement

with Section 2.1.

Remarks. ^-

a)

The rotational invariance

requires ôu-L. ôu-L

> ⁼ oo, where

buô

^{is the} fluctuation

of _uo

perpendicular

^to uo >. On the other

hand, u_q.uq

> does _{not go} to zero in the direction

perpendicular

^to uo >. This

discontinuity

^is

likely

^a

spurious

result of the Hartree

approximation,

as it would show _up in ^a similar treatment of an

ordinary, crystalline ferromagnet,

^{as well.}

b)

^At

high temperature

^the

present approximation predicts _{u _ q uq}

> ~

T¥2.

This result should not be taken too

seriously

because it is not

legitimate

^to

neglect

^{terms of}

higher degree

^{in u} in eq.

(10).

If the factorization

(A.l)

^{is made in formula}

(10),

it is

possible

^to calculate

m(O). m(r)

>

[7] :

The

integrand

^is

positive,

^{so that the}

exponent

decreases when r increases and

m(o) . m(r)

> is ^a

decreasing

function of r. This result cannot be

accepted

because the same treatment

yields

^an

analogous

result for a

crystalline helimagnet,

^whereas

m(o). m(r)

> is known to

be,

^at

TN,

^an

oscillating

function of r in this case. This is the reason

why

the

droplet

^{model was} used in section 4 rather than the Hartree

approximation.

216

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