Static properties of a random one-dimensional magnet

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Static properties of a random one-dimensional magnet

M. F. Thorpe

To cite this version:

M. F. Thorpe. Static properties of a random one-dimensional magnet. Journal de Physique, 1975, 36

(12), pp.1177-1181. �10.1051/jphys:0197500360120117700�. �jpa-00208363�

LE ^{JOURNAL DE} PHYSIQUE

STATIC PROPERTIES OF A RANDOM ONE-DIMENSIONAL MAGNET

M. F. THORPE

(*)

Institut

Laue-Langevin,

^B.P.

156,

38042 Grenoble

Cedex,

^France

(Reçu

^le

24 juin ^1975, accepté

^le

30 juillet 1975)

Résumé. ²⁰¹⁴ Nous donnons un calcul exact des

propriétés

thermodynamiques d’une chaine aléatoire

classique

d’Heisenberg

comportant ² types d’atomes. La

dépendance

^{de la}

susceptibilité

^{avec le}

vecteur d’onde est aussi obtenue. Des résultats numériques ^{détaillés sont} donnés pour le ^cas parti-

culier d’un alliage magnétique-non magnétique.

Abstract. ²⁰¹⁴ The thermodynamic

properties

^{of a random classical} Heisenberg chain with two kinds of atoms are calculated exactly. ^{The wave vector}

dependent susceptibility

is also obtained. Detailed numerical results are given ^{for the}

special

^{case of a}

magnetic-nonmagnetic

alloy.

Classification

Physics ^Abstracts

8.510

1.

Thermodynamics.

^{- We}

study

the static _pro-

perties

^{of a classical}

Heisenberg

linear chain with two

kinds

^{of atom A and B}

arranged randomly

^with

concentration c and 1 ^- _c,

where the nearest

neighbour exchange parameter Ji

may be

JAA, JAB

⁼

JBA

^or

JBB depending

^{on the}

nature of the atoms at sites i and i + 1. The unit vectors

Si

may be

thought

^{of as} the limit of quantum mechanical

spin operators

^as S -->

oo, h --->

^{0 and}

hS ---> 1

[1], [2].

The classical

Heisenberg

^linear chain with

only

^one

kind of atom was first solved

by

^Fisher

[1].

^{This solu-}

tion utilizes the

expansion’

^.

where

- .&.

The

Y,m(Si)

^{are the usual}

spherical

harmonies and the

P,(x)

^are

Legendre’ polynomials.

Our notation follows that of reference

[2].

^The

partition

^function

may be written

where we

integrate

^{over ail the}

spin

coordinates of a

chain

with free

^ends.

Using

^the

expansion ⁽²⁾

^{and the}

orthogonality property

^{of the}

spherical ^harmonics,

it is _easy to see that :

where the

integration

^over

Si picks

^out

YOO(SL)

^and

the

subsequent integrations

^also

pick

^out

only

^the

YOO(Si).

^{The result}

(5)

^{is true for}

a finite

chain whereas in the case of a chain with

periodic boundary

^condi-

tions,

^{it is} necessary ^to take the

thermodynamic

^limit

in order to obtain

(5).

^{The free} energy from

(5)

^is

F = - kTN In

À,o({3J) . (6)

It is almost trivial to

generalize

^{this result to a}

random chain. The

expansion (2)

^{now contains a J}

which

depends

^on

position. However,

^{the crucial}

spherical

^harmonic

part,

which arises from the

isotropy

of the

spin interactions,

^is

unchanged.

The result

(6)

becomes

which can be written more

explicitly

^{for our case}

We can evaluate the

À,o({3J)

^{from eq.}

(3)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197500360120117700

1178

Other

thermodynamic quantities

^{that can be derived}

from

(8)

^{such as} the energy and the

entropy

^{have simi-}

lar forms.

The result

(8)

^{is of much wider}

validity

^{than the}

classical

Heisenberg

linear chain. A similar result is obtained for _any model in which an

expansion

^{of the}

type

(2)

^{can be made} in which the

positive eigenvector

^is

independent of the

interaction parameters. ^For

example,

if the classical

spin Si

^is

replaced by

^a

scalar ui

^{= + 1}

to

give

^a

spin 2 Ising

^chain

[3],

^the

positive eigenvector YOO(Si)

^is

replaced by

^the

positive eigenvector

^{of the}

2 x 2 transfer matrix and the

corresponding eigen-

value becomes

and the result

(7)

^would

apply

^{to the mixed} system

[4].

The result

(7)

^{would not} be obtained in cases where the

positive eigenvector depends

^on the interaction parameters ^{as for}

example

^{in the}

Ising

^model

just

discussed with an extemal field

tenn h ui

^added

;

to the Hamiltonian. In this case the

positive eigen-

vector

depends

^{on the ratio}

Ji/h

^which

prevents

^closure

being exploited

in any

simple

^way.

°

Notice that the

sign

^{of J enters}

through (9)

^or

(10)

which are both even functions of J. The

properties

^of

these random chains in zero field are therefore inde-

pendent

^{of the}

sign

^{of J. Of course this is not true for}

the wave vector

dependent susceptibility

^{to be dis-}

cussed in the next section.

2. The wave vector

dépendent susceptibility.

^-

The cross-section for neutron

scattering S(g)

^{in the}

quasielastic approximation

^is

just given by

^{the wave}

vector

dependent susceptibility (apart

^{from a fac-}

tor

kT) [5]

For a linear

chain,

^this

expression depends only

^on

the

component

^of q

parallel

^to the chain which we

will denote

by

q. If the interatomic

spacing

^{is a, then :}

As shown in

[2],

^a

general

correlation function between

operators

oci,

oc’,,

^at

sites i,

^{i + r} may be written

In the present ^{case we are} interested _{in ai} ⁼

SZ

^and

rx¡+r

⁼

Siz+r

^{so that}

only

^{the 1 =}

1,

^{m = 0 term} contributes and

and

S(q)

^becomes

where ] r is

^necessary because the summation over r

goes ^over all

positive

^and

negative integers.

^{In the}

case of a pure chain with

only

^{one atomic}

species,

this sum is

easily

^{evaluated to}

give

where we have written u ⁼

Â1/Â0

^{which may be}

evaluated

directly

^from

(3)

^to

give

For a random

chain,

this formalism _goes

through except

^{that there are three}

separate

^{u, i.e.} uAA, UAB =UBA’

and UBB

corresponding

^{to the}

appropriate pair

^of

atoms. The correlation function

Si. Si,,>

^{for a}

sequence ABABAA... ABB now becomes

The

problem

^{is to} insert this

expression

^into

(12)

^and

perform

^the summations over the random chain.

This

procedure

^is

possible

^{in the}

general

^{case but}

particularly simple

for the dilute

magnetic

^chain.

3. Dilute

magnetic

^{chain. - We} dilute the chain with a concentration 1 ^{- c} of

non-magnetic

^atoms

so that uAA ^{= u} and _UAB ⁼ _UBA ⁼ _UBB ⁼ 0. The free energy

(8)

^becomes

where J is the

coupling

^{between the}

magnetic

^atoms

and the free _energy is

proportional

^to the number of

near

neighbour pairs

^of

magnetic

^atoms.

In order to calculate the

susceptibility,

^{it is easier}

to do a

configurational

average rather than the sum

over i in _eq.

(12).

^{This is an}

entirely

^correct

procedure

in the

thermodynamic

limit that we are interested in.

In this case it is rather

simple

^{and the}

susceptibility

per

magnetic

^{atom is}

This

expression

^{is the same as}

⁽¹⁶⁾

^{but with u - cu.}

1As T --+ 0

and q - 0,

^the

susceptibility S(q) diverges

only if c

^{= 1 as we would} expect. ^{It is}

possible

^to say that there is a

phase

transition at zero temperature in the _pure system. The introduction of

non-magnetic

atoms breaks _up the system ^into

finite, non-interacting

sections and there is no

ordering

^{even at zero tem-}

perature.

The result

(20)

^{takes a}

particularly simple

^{form in}

the critical

region

^defined

by

which is

with the dimensionless correlation

length j given by

and

This critical

region

^{as defined}

by (21)

^{is broader}

than the usual one which adds

qaj >

^{1 to}

(21),

^but

is somewhat more

appropriate

^{in this case. At zero}

temperature the correlation

length

^is

given by

The results derived in this section could be

applied

to TMMC

[5]

which contains Mn++ chains with S

= 2

and which could

probably

^be

doped

^with

a

nonmagnetic

^{ion such as Zn + + . The}

theory

^{of the}

previous paragraph

^{could be}

applied

^{if we correct}

for the finite

spin.

This has been done

by Hutchings

et al.

[5]

^{for the} pure ^case

by inserting

^{a factor}

S(S + 1 )

in with the

exchange

^{and with}

S(q).

^{Insomuch as this}

just

introduces a scale

factor,

^{it cannot}

easily

^be

checked

experimentally

^{and it is not} clear that

S(S + 1 )

is to be

preferred

^over

^S2

^{with the}

exchange. Using

the notation of

Hutchings et

^al.

[4]

^{where the}

exchange

2

Jnn

^is

negative

^{we have to}

put

in the

equations

^{of this}

^section,

^{so that with}

we have

and

where

These results are those of

[5]

^with u ---> cu. Notice that K and B are not

independent

but related at all

concentrations

by

^.

The critical

scattering

^{occurs around}

Q

⁼

1,

^{and not}

at

Q

⁼

0,

because of the

antiferromagnetic coupling.

In

figure

^{1 we show a}

plot

^{of x}

against

^T for various concentrations. The ^{c =} 1

theory

^and

experimental points

^are taken from

[5].

^{It can be seen that} the inverse correlation

length

^increases

rapidly

upon dilution.

From eq.

(23)

^{we can see that the}

slope

^{of K -}

1/e;

against

^{T is}

govemed by

^{a factor}

(1

⁺

c)/2 lc

^which

only

^increases

by 6 %

^{as c} goes from 1 to 0.5. The

departure

^from

linearity

^{that is}

apparent

^{but small} in

figure

¹

corresponds

^to

going

^{out of the critical}

region

^{as we} have defined it.

FIG. 1. ^- The inverse correlation length ^K plotted against tempe-

rature for various Mn concentrations in dilute Mn linear chains.

The c ⁼ 1 data points ^{and curve are from work on TMMC} [5].

1180

4. The

general

^{case. - For the}

general

^{case we}

consider two

magnetic species A,

^{B with concentra-} tions _c, 1 ^{- c}

leading

^{to three}

exchange

interactions

JAA, J AB JBA,

^and

JBB

^{and therefore to three u}

functions _{UAA, UAB} ⁼ _UBA, and _uBB. We must insert sequences like

(18)

^{into the}

expression

^{for the sus-}

ceptibility (12)

and do the summations. Instead of

summing

^{over all}

sites i,

it is convenient to

replace

£ ^by

^a

configuration

average. We therefore define

where zero is some

arbitrary

^{référence site. The} super-

scripts

^on C denote that ^a

configuration

average has been

performed

^over the sites

1, 2, 3, ..., 1-

^{1 but}

site 0 is an A atom and site 1 is also an A atom.

Clearly

we may define similar

quantities C, AB

⁼

CBA

^and

CBB.

^These

quantities

^are useful because

they obey simple

recursion relations

for 1 >

1,

with the initial conditions

In

writing

^down

(27)

^{we have}

performed

^{the confi-}

guration

average ^{over atom} 1. Similar

equations

^can

be written for

CBB

^and

CHA merely by interchanging

A --> B and c H 1 ^- c in

(27)

^and

(28).

These

equations

^{are most}

easily

^solved

by

^intro-

ducing

^a

parameter x

^to find the normal modes of

(27).

we choose

and

solving

^this

equation

^{for x, we} get ^{two roots}

x+,

so that

with

Wè can now solve _eq.

(32)

^for

Ci%j

^and

CAB

sepa-

rately using

the initial conditions

(28)

With these results we are now in a

position

^to evaluate the

configurationally averaged susceptibility

^{for an A}

type

^{atom at the}

origin

The

corresponding quantity SB(q)

^{for a B}

type

^{atom at the}

origin

may be obtained from

(35) by letting

^{A H B}

and c ^H 1 ^- c. The

complete susceptibility

^{is then}

given by

The

susceptibility S(q)

^will

only diverge

^{at zero}

temperature

if either all the interactions ^are

ferromagnetic

or all the interactions are

antiferromagnetic.

In the first case all the u will

approach

^{1 as T - 0 and}

S(q

⁼

0)

will

diverge

whilst in the second case all the u will

approach -

^{1 as T - 0 and}

S(q

⁼

nla)

^will

diverge.

^For

mixed

ferromagnetic

^and

antiferromagnetic interactions,

^the

system

will still order as T -> 0 but the suscep-

tibility

^that

couples

^{to the order}

^parameter

^{will not have a}

spatial periodicity

^described

by

^a wavevector q.

Of course

S(q)

^can still be defined and measured in these cases and is

given by (35)

^and

(36).

Most

experiments

^{do not measure}

spin-spin

correlations but rather

magnetisation-magnetisation

^corre-

lations. The

theory

^{that we} have worked out can

easily

be modified to include this if we

put Mi

= gA

Si

^or

gB

Si depending

^on whether the site i is

occupied by

^{an A or B} atom, then eq.

(12), (35)

^and

(36)

^become

Rather than write out more

explicitly

^the

lengthy expression

^for

M(q),

^{we will}

only

^{do so for a few}

special

cases.

When q

⁼

0,

^{we have}

This

clearly diverges

^{at T = 0 for all}

ferromagnetic couplings,

^i.e. UAA ⁼ UAB ⁼ UBA = UBB ⁼ 1. At

high

^tem-

peratures

this becomes

that is the _average

squared

^moment.

(Combining

this with the

1/kT

factor that we have not been

carrying,

^we get ^the

expected

^Curie

law.)

^In the critical

region

^defined

by (21)

^we may write

M(q)

^{in a} form similar to

(22)

where

and

The appearance of ^{the B tenn} is a néw

feature;

it does not occur in the pure ^or dilute

magnetic

systems. ^It reflects the disorder

directly

^{and could}

be

quite sizeable,

^{as of course in}

applying

^{this work}

to real

s stems with

^finite

spins, appropriate spin

factors

S(S ^{+ 1)}

^{must be}

incorporated

^into

the g

factors.

5. Conclusions. ^- We havé shown that the static

properties

of random classical

Heisenberg

^linear

chains can be worked out

by exploiting

^the

indepen-

dence of the

positive eigenvector

^{from the}

exchange

interactions. It should be

possible

^to

apply

^these

results to real systems ^if

appropriate spin

^{factors are}

incorporated

^{into the}

exchange

interactors and

the g

factors.

However, although

quantum corrections do not seem to be

important

^{for Mn++}

(S

⁼

2),

^this

may ^not be the case for

Ni ^(S

⁼

1). Finally

^we

note that in many mixed systems, ^{the relation}

is

obeyed [6].

If this is also the case in linear

chains,

then from _eq.

(23a)

^the square ^root of the inverse correlation

length

should be linear in the concen-

tration.

Acknowledgments.

^{- 1} should like to thank S. W.

Lovesey

^{for a} useful conversation and the ILL for their

hospitality

^{in the summer of 1975.}

Note added in proof ^This problem has also been considered by ^T. Tonegawa, H. Shiba and P. Pincus, Phys. ^Rev.11 (1975) ^{4683. The}

work here corresponds ^{here to their} quenched ^limit. They gave ^not considered _q dependent quantities ^{but our} expressions ^{for the} pair ^corre-

lations are in agreement.

References [1] FISHER, ^M. E., ^{Am. J.} Phys. ³² (1964) ^343.

[2] THORPE, M. F. and BLUME, M., Phys. ^{Rev. B 5} (1972) ^1961.

[3] ^{See for} example MATTIS, D. in The Theory of Magnetism (Harper

and Row, London) 1965, p. 236.

[4] WORTIS, M., Phys. ^{Rev. B 10} (1974) ^4665.

[5] HUTCHINGS, M. T., SHIRANE, G., BIRGENEAU, ^{R. J. and} HOLT, S. L., Phys. ^{Rev. B 5} (1972) ^1999.

[6] COWLEY, ^{R. A. and} BUYERS, ^{W. J. L., Rev. Mod.} Phys. ⁴⁴ (1972)

406.