Thermodynamics of Alternating Quantum-Classical Spin Chains Showing Isotropic Nearest Neighbor Couplings

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Thermodynamics of Alternating Quantum-Classical Spin Chains Showing Isotropic Nearest Neighbor Couplings

Jacques Curély, Roland Georges

To cite this version:

Jacques Curély, Roland Georges. Thermodynamics of Alternating Quantum-Classical Spin Chains

Showing Isotropic Nearest Neighbor Couplings. Journal de Physique I, EDP Sciences, 1995, 5 (4),

pp.485-499. �10.1051/jp1:1995142�. �jpa-00247074�

Classification Physics Abstracts

75.50G 75.10D 75.10H

Thermodynamics of Alternating Quantum.Classical Spin Chains

Showing Isotropic Nearest Neighbor Couplings

Jacques Curély

(~,*) ^{and Roland}

Georges

_(~)

(~) Laboratoire de Cristallographie et de Physique Cristalline, Université àe Boràeaux1, 351 Cours de la Libération, 33405 Talence Cedex, France

(~) Laboratoire de Chimie du Solide, Université de Bordeaux1, 351 Cours de la Libération,

33405 Talence Cedex, France

(Received

15 November 1994, accepted 19 December 1994)

Résumé. Un traitement général est proposé _pour résoudre le problème des chaînes ferri-

magnétiques constituées de deux sous-réseaux

(S,s)

et caractérisées _par des couplages isotropes

entre premiers voisins ainsi _que deux paramètres d'échange. En conséquence, deux _cas physiques

intéressants sont examinés _{: 1) les} chaînes sont exclusivement composées de couplages ferromag- nétiques _ou

antiferromagnétiques

ii) les chaînes présentent _une alternance régulière de _ces deux types de couplages, Des expressions littérales sont données _pour la fonction de partition _en champ nul, les corrélations _spin-spin ainsi que la susceptibilité. Le comportement ^basse tem-

pérature ^{de la} susceptibilité est étudié

au moyen de la longueur de corrélation. En particulier, dans le _cas de la compensation des moments magnétiques, il est montré _que

ce comportement

est principalement décrit _par la compétition entre la divergence de la longueur de corrélation et l'annulation du moment magnétique de la cellule élémentaire. Pour finir, _nous rappelons _un test

expérimental _{qui a} initié _ce modèle théorique _: il _concerne le composé

Mncu(obp)(H20)3.H20

[où

obp=oxamidobis(propionato)]

Abstract. We propose a general treatment for solving the _case of ferrimagnetic chaius made up of _two sublattices

(S,s)

and characterized by isotropic couphngs between nearest neighbors,

as well _as two exchange pararneters. Therefore, two _cases of physical interest _are examined:

i)

the chains _are exclusively composed of ferromagnetic _or antiferromagnetic couplings; ii) trie chains show _a regular alternation of these two types of couplings. Closed-form expressions of trie ?ero-field partition function, trie spiu-spin correlations, _as well _as the susceptibility, _are given. The low-temperature behavior of the susceptibility _is studied by _means of the correlation

length. In particular, _in the magnetic moment compensation, _we show that this behavior _is mainly described by the competition between the divergence of the correlation length and the

eva~lesce~lce of the mag~letic moment per ^unit cell. F1~lally _we recall _an experimental test which

bas imtiated this theoretical mortel: it _co~lcer~ls the compound

M~ICU(obp)(H20)3.H20 [where obp=oxamidobis(propionato)].

(*)AU correspondence should be addressed to Jacques Curély

© Les Editions de Physique 1995

1. Introduction

Oue-dimeusioual

maguetism

is _a field where theoretical and experime~ltal studies have stimu- lated each other

[1-4].

^The reason for this stroug interest is that the

specific

behavior of such ID

materials

(notably

with _an infinite

critical-temperature domain)

and its

ability

to sometimes be solved

exactly,

allow for _a

possible

^{check of}

experimental

results with the

proposed specific

models

[5-7].

To

briefly

_{sum up} trie

history

of this

field,

_{we can say}

that,

at

first,

trie stud- ied

compounds

_were

regular

homometallic

airains,

in which trie

magnetic

centers _are

equally spaced along

trie chain

[8-10].

Then

alternating

homometallic chains characterized

by

two intrachain

exchange

parameters

appeared [11-13].

More

recently,

the first bimetallic chains

were described

[14-18] and,

because of the _new

possibilities

offered

by

molecular

chemistry engineering, alternating

bimetallic chains _were

synthetized. Among

these

materials,

the _so- called

ferrimagnetic

chains offer many

interesting

features: their solid-state _structure exhibits

well-separated magnetic

chains

along

which two cationic

species

_occupy host sites,

generally

with

antiferromagnetic

nearest

neighbor couplings,

thus

providing

the main basic conditions for

ferrimagnetism

and

considerably increasing

the number of nontrivial situations

[19-21].

From _a theoretical

point

of _view, the necessity ^of

interpreting

the

experimental magnetic properties

of the chains described above

begins

with the determination of the eoEective

spin

Hamiltonian. In _a

previous

_{paper [22]} we have established the

general

conditions which _must be

obeyed by

the chain Hamiltonian for

allowing

_an

analytical

recurrent treatment: in

partic-

ular the basic condition is that all the involved _spm operators commute. Then the isotropic

or

anisotropic

nature of

coupling

with nearest spin

neighbors plays

an

important

role. In this paper we

exclusively

locus _on chains

showing isotropic couplings.

When the chain is

only

_com-

posed

of quantum spin moments, trie Hamiltoman

always

contains

non-commutating

spin _oper- ators; therefore

approximate techniques

_are

required:

spin-wave

theory

_[23],

high-temperature

serres expansions

[24-26],

Green's function

approaches

_{[27], or} numerical

extrapolations

from

exact calculations _on finite

length

chains

[28-32]. However,

for trie

purely

classical _case, trie commutation condition is fulfilled.

Taking

into account this aspect, Fisher bas given closed- form expressions for trie main

thermodynamical

^functions of interest [33] ^and

Stanley

bas _gen- eralized this work to classical

spins

of

arbitrary dimensionality

[34].

Following

_a scheme similar to

Fisher's,

Seiden [35] ^bas solved trie

problem

of _a chain

showing

_an alternation of _a quantum

spm ^s =

1/2

with _a classical _one: trie presence of classical spins allows _one to separate each unit cell operator

exp(-flH~)

and _a relevant rotation in trie _{spm space} authorizes the calculation of

the trace for each operator.

At the end of the

eighties

_we

generalized

Seiden's model to

arbitrary

_spm quantum num-

bers

[19,36,37] but,

_so far,

only

the broad outlines have been

presented;

_more

recently, using

the

sanie

principles,

_we have

proposed

_a model in which whole quantum

subsystems

alternate with

classical

spins

[38]. In this paper, m Section 2, _we recall the basic

principles

but now we detail the method. In Section 3 _we

study

the

low-temperature

behavior of

(S, s)N

chains. In

particular

we examine the _cases where the chain is

exclusively composed

of

ferromagnetic

_or

antiferromag-

netic

couplings

_or is characterized

by

the

regular

alternation of these _two types of

couplings.

We

specifically

discuss the compensation case

(no

net moment in the

ground state)

which also reveals subtle aspects ^of both short- _and

long-range orderings. Finally,

_we recall the interpre- tation of the experimental results obtained _on the

compound Mncu(obp)(H20)3.H20 [where

obp=oxamidobis(propionato)]

because _it has imtiated the present theoretical work

[19, 37].

2. General Cousiderations

Let _us consider _a

Heisenberg exchange coupling

chain submitted to _an externat

magnetic

field B

applied along

the _z-axis of quantization

(see Fig. l).

Thus for _a

(2N +1) spin

chain Soso

S~s~ SNSN the most

general

Hamiltonian _may be written

N-1 N

H =

£ _H)~

₊

£Hf~~, ₍₁₎

1=O i=O

with

H)~

=

JS~.s~,

S~ ₌

(1+ a)S~

₊

(1- a)S~+1, (2)

HÎ'~~

₌

-(GSI

+

gs/)B. (3)

S~ is the _z component ^{of trie classical} vector operator S~

(trie spin

quantum number S is

large enough

for

[SIS)]

_to be

negl1glble ^compared

to

S$Sf-classical

_spm

approximation);

_g and G

are trie associated Landé factors:

they

characterize trie

magnetic

ions of trie unit cell. J refers to trie

exchange

interaction between trie _nearest

neighbors only:

in _our

writing

J > 0 denotes

an

antiferromagnetic coupling.

Note also that _we shall restrict the discussion _to the

positive

Si-1

''d,

Fig, i. Structure of _a quantum-classical _sp1~l cha1~l.

values of _a without

loosing generality:

this is due to the fact that

changing

the

sign

^of all the a's is

equivalent

to _reverse the order of the

magnetic

ion seque~lce;

consequently

the

magnetic

chain

properties

_are

unchanged.

Trie

partition

function

ZN(B)

for trie

(2N +1)

_spm chain _may be written

ZN

(B)

=

/ _dsoTrs~.. / _ds~Trs~ / _dSNTrs~

exp

(-

^fl

_(f(H)~

⁺

^Hf~~

⁺

^(~~l)

,

~_~

(4)

where

Trs~

represents ^trie trace

operation applied

to trie matrix associated to trie spm op-

erator _s~. Trie

knowledge

^of trie partition function

ZN(B)

allows _one to define trie

parallel

magnetization

Ji4N and trie zero-field

susceptibility

_{XN as}

~~ ^~

~ÎÎ _~~~'

~~

IZÎÎ(0) ~~Î~Î~~

~_~

' ~~~

where

ZN(0)

is trie partition ^function

expressed

for B

= 0. Note

that,

in order to bave _a clearer

insight,

_we ^shall

preferably

consider _X, trie

susceptibility

^referred to trie unit

cell,

and trie

product XT

normalized to its infinite temperature value

(tue

Curie

constant): (xT)n.

First _we focus _on tue calculation of tue zero-field

partition

function.

Tuer,

in equation

(4),

trie Hamiltoman involved in trie

exponential

argument ^is reduced to trie

exchange

_one

H/~;

moreover because of trie

commutativity

property of trie operators

H/~

_we bave

N-1

IN-1

exp

-fl £ _H)~

=

fl

_exp

_{(-flHf). (6)}

i=o 1=0

Consequently,

for each

site1,

_we bave _to evaluate trie trace of trie operator

exp(-flH)~). Using

trie fact that vector Si defined in equation

(2)

_can be chosen _{as a} local axis of

quantization

_we bave

Trs~ (exp (-flH)~))

=

~

_exP

(aÀ/1+

~Î C°S

Ôi,~+l)

' ~~~

~

~~

s

with

~

^{i ~2}

=

-PJ 2(1+ a2),

1J ⁼ ~

(8)

(note that,

_as S is _a unit vector, S bas been

dropped).

As this trace

only depends

_on tue

angle 8~,~+i

^between vectors S~ and

S~+i,

it becomes

possible

to

expand

it _on tue infinite basis of

spuerical

uarmonics

+ce +é

Trs~(exp(-flH1~)) =L L Aé(À,~,s)G~(Si)n~~(Si+1), (9)

with

Ag(À,1~,

s)

= 27r

£

^~~

^~

_exp

(aÀfi) _{Pg(z)dz (10)}

wuere

Pg(z)

is _a

Legendre polynomial.

Tuer

using

equation

(9)

in

equation (4) expressed

in tue zero-field limit _as well _as tue

ortuogonality

condition to wuicu tue spuerical uarmonics

obey,

tue values _m

= 0 and t

= 0 _are selected and _we bave

finally ZN(0)

=

4R(Ao(À,

_~,

s))'~+~ (II)

with

~~~>,~,~~

=

) f _£

_~

(u«,~

i)

_exp

_(u«,~)

_>u«,~

=

«Àfi.

_~~~~

~

~~-~ ~=+i ^«

At this step it must be noticed that trie calculation of

ZN(B)

involves _a further contribution

(1.e.,

^trie

magnetic

one described

by Hf~~)

in each

exponential

_argument: thus trie square

root which _appears in

equation (7)

now contains other terms

exclusively depending

_on B and trie _z components _{of S~ and}

S~+i (as

^{trie externat field is}

applied along

trie

z-axisi.

Conse-

quently

it becomes

impossible

to

expand

each trace of

exp(-flH~)

_as in

equation (9)

and trie

magnetization Ji4N

uas _no closed-form expression.

However,

for

calculating

tue

susceptibility

_{x, it is}

always

possible to express it witu tue

uelp

of

spin-spin

correlations.

Considering equations (1)-(5)

and

using

tue fact tuat we deal witu

an

isotropic excuange coupling

^between nearest

spin neigubors,

tue

susceptibility

_x referred _to tue unit cell _can be written _as

x = jG~ ^<

(si

_{)~ >}

+g~

_<

(si

_{)~ >}

+

£ _(G2

_<

_sjsj

_>

_+g2

_<

_sjsj

_>

_{+Gg j< sjsj}

_{> + <}

_sisj >j) (13)

j#i

Tue calculation of _<

(SI

₎₂ > and _<

(SI

)2 > is

obvious;

_as for tue _various correlations

<

SIS]

>, <

s)sj

>, <

SIS]

> and <

s]Sj

>, tue work is similar _to tuat _{one encoun-} tered for

evaluating

tue zero-field

partition

function

ZN(0);

it just ^dioEers

by

trie introduction

of _extra terms at rows1 and

j.

If

SI

is introduced _at row1 _we

expand

it _as

SI

=

~~f°(S~). ₍₁₄₎

Then,

trie

orthogonality

condition and trie

specific properties

^of

spherical

harmonics allow _one to select trie values _m

= 0 and t

= 1. Trie introduction of operator

sj imposes

trie evaluation

of trie trace of trie operator

sjexp (-flHj~).

In _a first step _{we express} trie operator

sj

on

trie

orthogonal

basis

(i'j,j'~,k'~)

in wuich

k'j

is orientated

along

trie direction of _vector

Sj (cf, Eq. (2));

let

§(,

_@j ^and

§j

^{be trie} new operators;

noting

that

§(

^and

§j

^bave no

diagonal

elements and _owmg to

equation (2)

trie calculation of this trace reduces _to

Trs~ (sj exp(- flJSj

_sj

))

"

kj k'jT~SJ

_(~~ ~~P

(~fl~~J

_~Î~

'

~~~~

with

,

(1+ o)Sj

₊

(1- a)Sj~i

kj

k

j ⁼

/2

_{(1 +} 02 ₊

(1

^a2 cas

8~,j+i (16)

Then

expanding Sj

and

Sj~i

with

equation (14)

we bave

Trs

(s j ^exp(-flJSj sj))

=

-PJ

^~~

(l

₊

a)Yi°(Sj

₊ (1

a)Yi°(Sj+1)

? 3

x

f «~~P 1"~~/~

^{+ ~ ~°~}

~JIJ+~ (17)

~~_~

~/i

_{+ ~ Cos}

e~,~+i

In this expression trie summation _{over a}

only depends

_on trie

angle 8j,j+i

between vectors

Sj

and

Sj+1 Tuerefore,

_as for

equation (7),

it _can be

expanded

_on tue infinite basis of

spuerical

uarmomcs. We obtain _an expression ^similar to tuat given

by equation (9)

in wuicu

trie coefficient Ag(À,1~,

s)

is _now

replaced by

_a coefficient Bg(À,1~,

s)

defined _as follows

Bé(À1/

_S)

= 2Sr

Î

«~

/l~ ~~ilw~

Pé(z)dz. (18)

Then,

tue

orthogonality

condition and tue

specific properties

of

spuerical

uarmonics allow _one to select tue values _m

= 0 and t

= 1.

Finally

_we uave

<

SIS]

>=

PJ-~,

_<

SIS(

>=

QIPJ-~,

<

sis]

>=

Q-IPJ-~-i,

<

sis]

>=

QIQ-IPJ-~-i, (19)

wuere

p ~

Ai(À>11>S) Ao(À>11>

Sl'

(i

₊

ea)Bo(À,

_~,

s)

₊

(i ea)Bi

_{(À, ~,}

s)

~~

" ~~~

Ao(À,~, s)

^{'~ " *~'}

(~°)

with

~il~~~l~

_~~

"

A ^~Î~

~

+i1

^{~~~'~ ~~~~}

^~~~ "~~~~~~~~

_~~~

x exp

(u«,~)

,

(21)

BO (À,1~, s =

~ f _£

e exp (u«,~

,

(22)

~ _~l

«=-s~=+1

Bi (À, _~,

s)

=

j _{£ £ j}

lui,~

2u«,~ + 2

«~À~)

exp (u«,~

(23)

~Î~

~=+~

In these previous expressions, À,1~,

Ao(À,i~, si

and u«,~ are

given by equations (8)

and

(12),

_re-

spectively.

Then using trie expressions of trie _various

spin-spin

correlations in trie

susceptibility

definition

(cf. Eq. (13))

_we bave for _a unit cell

where _r is trie ratio of trie

magnetic

moment

magnitudes

_per unit cell

r=

~

(25)

95

Finally

it must be noticed tuat trie evaluation of trie

susceptibility just

necessitates tue knowl-

edge

^of tue four coefficients

Ao, Ai,

BO and El

Previously defined,

wuatever tue value of tue

spin

quantum number _s. Note also that all tuese coefficients _can be

easily

calculated in

trie

classical-spin approximation by

_assuming trie

following

transformation

However,

when _s becomes

infinite,

in order to

keep

_a

physical meaning,

trie quantum

spin

moment

magnitude

_{gs as} well _as trie

exchange

_energy Js _are constrained to remain finite

through

_a convenient

adaptation

_{of g and}

J, respectively.

Another parameter of

significant importance,

in trie present context, ^{is trie} correlation

length

defined _as follows

+ce ~/2

£'l~ _{Î< ~Î~Î+n >Î}

"

j ~fl~c (27)

£ Î~ ~Î~Î+n >Î

n=0

Then, taking

into _account trie

expression

of _<

SI S/~~

>

(cf. Eq. (19))

it is

easily

shown that i

jpj(i

₊

(P(i 1/~

(~~) f= à (i-ipi)2

3.

Study

of the

Low-Temperature

Behavior

The

low-temperature study

is

mainly

conditioned

by

trie classification of trie parameter a with respect to

unity.

When _{a <} 1, trie

exchange

interactions

J(1+ a)

and

J(1- a),

involved per unit

cell,

uave tue _same

sign

(1.e. ^{tuat of}

J);

if J _< 0 ail tue

couplings

_are

ferromagnetic

and,

at 0

K,

ail tue

spin

moments are oriented towards tue _same direction

(see Fig. 2a);

_on

tue contrary, if J _> 0 all tue

couplings

_are

antiferromagnetic and,

at 0

K,

two consecutive

spin

moments are oriented

along opposite

directions

(see Fig. 2b).

^Wuen a > 1, tue involved

excuange

interactions

J(1+ a)

and

J(1 ai

_now uave opposite

signs

(1.e. tuat of J and

-J);

in other words _we uave _a

regular

alternation of

ferromagnetic

and

antiferromagnetic couplings

and trie chain shows _a

vanishing global

moment at 0 K

(see Fig. 2c).

Note that, when _a

= 1,

we deal with _a dimerized chain.

J<0 a<1 J>0 a<1 j>0 ^a>1

j/[°. j/

*'~'"'

C~Î))

~-fi

^ho

~_j

oeil S'-' oeil si.,

a b

c

Fig. 2. Structures of the chair grou~ld states for _o < i

(Fig.

2a: J _< 0;

Fig.

2b: J > 0) a~ld for

a > 1

(Fig.

2c: J _> 0).

The low-temperature behavior

study

of trie

susceptibility

necessitates trie evaluation of _quan- tities

P, Qi

and

Q-i

_m that _range of temperatures; more

exactly,

because of their respective

definitions,

their behavior is

given by

that of trie ratios Ai

/Ao, Bi/Ao

and

Bo/Ao

which bave been introduced in tue

previous

section

(cf. Eqs. (12), (21)-(23) ).

As tuese coefficients contain

exponential

terms, ^tueir

low~temperature

^beuavior is

mainly given by

^tuese factors cuaracter- ized

by

tue maximum argument 2fl(

J(s (note tuat,

_as two consecutive arguments

only

differ

by

tue

quantity -2fl(J(,

tuis

approximation

remains valid wuen tue temperature is

plainly

lower tuan

(J(

). ^{All tue main results of} interest uave been summarized in Table I.

Whatever tue value of _o with respect to

unity,

in trie

low-temperature

_range,

(P(

tends _to

unity according

to _a T-law.

Consequently,

trie examination of

equation (28)

allows _one to say that trie correlation

length (

behaves _as

(1- (P( )~l

and thus

diverges according

to a T~l-law

(see

Tab.

I).

Tuis aspect uas been

previously

encouutered

[19,

33,

36,

_38] aud _{appears as} tue

signature

of

isotropic coupliugs iuvolviug

classical spiu moments

alteruatiug

witu quantum

ones

(or

dioEerent classical

ones).

Note tuat, wuen _a

=

(case

of dimerized

cuains),

P vanisues and the correlation

length

uas _{no more}

physical

interest. Thus the

low~temperature

behavior of trie

susceptibility

is

mainly given by

^{that of}

Il P)~l

_{on condition} that trie _numerator of trie fraction

containing

trie term P

(cf. Eq. (24))

does not vanish; at this step it must be

noted

that,

in trie

low-temperature

domain, this _numerator is neither _{more nor} less than trie square of trie

magnetic

moment M per ^unit cell. In other words trie

low~temperature

behavior of

XT

is

mainly given by

that of

(M2.

Therefore trie chain _can be considered _{as an}

assembly

of

quasi-rigid quasi-mdependent blocks,

each _one of

length (

and moment M per unit cell.

Table I.

Low-temperature

behauior

of

the

s~sceptibility

and the correlatiue q~antities

of

main interest

for

uario~s

significant

values

of

the

eichange

parameter a.

a<1

~ _~

fllJls (1~ °~)

~

fllJls

^~

(2fllJlS)~

^~

")

'

~~ ~ _~ ~'

Q~ J

(1

eo 1

Î

^~

_(J(

_{1 +}

eo

2fl(J(s

^~

(2fl(J(s)2

^{~ '}

~s li'o f+~~')'~li-a~), aST-°;

x r~

~ )~~

lJls Ii _a~) (r j)

~ ~ ~

+$ ^3r~ _Ii

a~)

+

4(r13a~

1) ²

lsa~ i))

,

asT-0.

Table I.

(continued)

OE#

P=0

VT;

~~ --j~)~,

e=+1, asT-0.

(=0 VT;

X '~

~~)~~

r~

+

~ _2jr)

,

as T _- 0.

o>1

P-(i-o~~~iù+...l _aST-0.

~~

- -5

~

(l

^{+ ~~}

_+,.,)

, 5 # +1 _aS T _- 0.

s

(J(

1 + en

2afl(J(s

^'

(

m~

~~~~~

(a~

₁₎

, as T _- 0;

20

~ ~

~ÎÎ~~

~

ÎÎ~Î~~ ^~(~(~

'

" ~

~ ~'

When _{a <} 1, all the

couplings

_are

ferromagnetic (J

<

0)

_or

antiferromagnetic (J

_>

0);

for

r

#

1 the chain shows _a

global

moment. If _we examine the

general low-temperature

expansion of the

susceptibility

_{given in} Table I, the first term appears ^to be the dominant _one; thus the

product XT

behaves _as

XT

+~

~i~~~

iris(i a~) (r ll~

aS T _- ° _a < i> r

#

1

(29)

This behavior _is illustrated

by

the _curves of

Figure

3 where _we have considered the value

a =

0.50;

_as

expected

trie

divergence

law is enhanced if

couplings

_are

ferromagnetic (the

factor

r -1 is

replaced by

_r

+1).

For

antiferromagnetic couplings,

in trie

compensation

_case

(r

=

1),

trie chain bas _no net moment at 0 K: The behavior of

XT

is given

by

trie

competition

between the

divergence

^{of trie} correlation

length ( (T~14aw)

and trie evanescence of

M,

trie

magnetic

moment per unit cell

(T.law).

As the factor _r

J/(J(

involved in

equation (29) vanishes,

it is necessary ^to consider further _terms of the

low-temperature

expansion of the

susceptibility

(Tab. I). Then,

under this

condition,

it appears that the

product XT

shows _a constant limit

ix T)~

~x

m

O.50

r

J

i.o

O.O 1,O 2JO kaT/'Jl

Fig. 3. Thermal variations of the product

(xT)n

for

a

(S,1/2)

N chai~l show1~lg isotropic coupl1~lgs

for several values of _r a~ld J (g = 2, _a =

0.50).

XT

_-

))~~

⁺

~~~~j)jj °~~T

_+..

, as T _- 0, _a < 1, r = 1.

(30)

BS S

At this step ^it must be also noted that the

T~vamshing

law of M is characteristic of

isotropic coupled

_spm chains

showing

_an alternation of classical and quantum

spin

moments

(or

dioEerent classical

ones).

This behavior which recalls _a Curie law has _{no common}

point

^with trie _param~

agnetic

behavior of free _moments. In

particular

trie

possibility

of confusion with

paramagnetic impurities

must trot been

ignored

when _one tries to

interpret experimental

data.

Practically,

trie

rigorous

moment

compensation (r

₌

1)

is seldom encountered but trie examination of trie

curves of

Figure

4 allows _oue to say

that,

for

observiug

a very differeut behavior with respect to

the Curie law, it is necessary to consider temperatures doser and closer to 0 K if the ratio _r of

lx T)n

OE o.so

O,60

~

O.gO o.95 1.oo o.50

O,40 ^'

O.Oo o,10 O,20

kaT/J

Fig. 4. Thermal variations of the product

(xT)n

for

a

(S,1/2)N

chain show1~lg isotropic couplings

for several values of _r

(J

> 0, _{g =} 2, _{a #} O.50).

the

magnetic

moment

magnitudes

is closer to

unity.

When _{a >} 1, ^the

couplings

with _nearest

neighbors

_are

alternatively ferromagnetic

and

antiferromaguetic.

Whatever the

sigu

of

J,

the chain

ground

state at 0 K is characterized

by

_a

vanishing

moment. This _{case appears to} be very similar to the

previous

_one where _a < 1, J > 0 and _r

= 1.

Thus,

in the

low.temperature

range, we hàve

~~

_~

ÉÎ~

^~

~Î(~~~

)ÎÎÎ~

~ ~

' ~~ ~ _~ ~' _° ~ ~' ~~~~

Therefore,

the

product XT

follows _a Curie law in spite ^{of the absence of free}

magnetic

moments and whatever the

sign

of J. Moreover the

T~vanishing

law of

XT

is due _to the fact

that,

in the

low-temperature domain, XT precisely

behaves _as

(M~,

where

( diverges

_as ^T~l

(see

Tab.

I)

and M vanishes _as T.

However,

in contrast with the _{case o <} 1, the ratio _r has _no

major

effect. It must be

only

noticed that the initial

slope

of the thermal variation of

XT

vanishes

without any

change

of

sign

if _r

=

lia.

All these aspects _are illustrated

by

the _curves of

Figure

5 where _we bave considered trie value _o

= 1.50. At this step ^{it is} of

gieat

interest to detail trie

deep

_reasons which _are

responsible

for such _a behavior because

they

involve _{a more} subtle mechanism.

By

_using trie low~temperature behavior of

P, Qi

and

Q-i (see

Tab.

I)

in

equation (24)

it is

easily

shown that (1

P)~l

_no

longer diverges;

in

addition,

if _we consider all trie correlations

concerning

trie different classical moments,

they exactly

compensate trie

classical self correlation.

But,

for trie quantum moments, trie self correlation

s(s +1)

is not

exactly compensated by

ail trie correlations

involving

different quantum spin moments

(-s~).

Therefore _we deal with _a

purely

quantum

effect,

the importance of which weakens when _s becomes infinite (1.e, ^if we tend to the classical _spin

approximation).

Nevertheless it remains to determine if the classical moments

(which

alternate with the

(X TIn

_{TIn a= I.So}

a ₌ 1.50

r=1.o r=2~o

s

i-o

O.O '

O.O ig

kaT/J

Fig. 5. Thermal variations of the product

(xT)n

for _a

(S,s)N

chain showing isotropic couplings

for several values of _s (J > 0, _{gs =} 1, _{o =} 1.50, _{r =} 2.0); the dashed fine corresponds to the value of

(xT)n

_in the classical _spm approximation

(inset:

J _> 0, _{gs #} 1, _a = 1.50, r =

1.0).

quantum

ones) play

_an

important'role

in this effect. For

examining

this

specific

point, we

have achieved trie exact numerical calculation of trie properties of _a tetramer made _up of four quantum spin moments

(see

inset of

Fig. 6)

[36]; the

exchange

term is described

by

equation

(2)

in which S~ and

S~+i

are here

replaced by

_si and _s2

(respectively

_s2 and

si)

if _s~ is

S'i (respectively s'2).

As the

exchange

_energy J is

positive

and _a > 1, one cari say that trie

coupling

is

ferromagnetic

^for

pairs (si, s'2

and

(s2, s'i ),

and

antiferromagnetic

for

pairs (si, s'i

and

(s2,s'2).

Trie spin quantum numbers _are such _{as s =}

1/2

for _si and _s2i

S'i

and

s'2

_are

cuaracterized

by

^s'

(wuen

^s' becomes infinite classical spin

approximation

this numerical work uas been

adapted conveniently).

In

Figure

6 _we uave

reported

tue thermal variations of xn for varions values of SI and for _a

= 1.50 and _r

=

2.0;

note tuat trie Landé factors _as well _as tue

exchange

_energy J bave been

adapted

_so that

g's'

and Js'

respectively

remain

unchanged

wuen s' is modified. Tuus _we deal witu _a short

cyclic

cuain tue cuaracteristics of wuicu _are very close to tuat of tue

previous (S,1/2)N

_spin cuain wuen s' becomes infinite. Trie tetramer

ground

state is _a

singlet

_one but, when trie value of s' increases, trie

density

of

multiplet

states increases in trie close

neighborhood

of trie

ground

state. Under these

conditions,

trie

susceptibility

shows _{a more} and _{more narrow} maximum for _a temperature ^{which is doser} and doser to 0 K, as s' increases; trier xn

rapidly

decreases _{to a constant} value, at 0

K,

which

corresponds

to trie Van Vleck contribution. When s' becomes infinite

(classical

_spin

approximation)

_{x is} maximum _at 0 K. This _cari be

directly

observed _on trie _curves of

Figure

6.

Under

addition,

_{one cari} notice that trie

non-vanishing

value of x increases when trie ratio

s'là strongly

deviates from

unity. Therefore,

for trie infinite spin

chain,

trie

divergence

of _{x con} be

interpreted

as an enhancement of trie Van Vleck contribution to trie

susceptibility,

due to trie fact that trie quantum

spin

coherence _occurs at

lengths

which become infinite.

Tuus, though

we uave considered _a finite

lengtu cuain,

tuis low temperature

study

confirms tuat tue classical cuaracter of spin moments

altemating

witu quantum ones can

deeply modify

tue

susceptibility

beuavior.

Now _we

briefly

recall below _an

experimental

illustration

previously publisued

wuicu uas imtiated tuis tueoretical model [19]. ^It concerns tue

interpretation

of tue

magnetic

properties of trie

compound Mncu(obp)(H20)3.H20

where

obp

is _an abbreviation for trie

ligand

_oxa-

~n

~l lo5Ô ~'

~

S~

S~

1/2

0E5 kaT/Jss'

Fig. 6. Thermal variations of the susceptibility _xn for _a

(s',1/2)

tetramer showing isotropic _cou- plmgs for several values of

s'(J

_{> 0, g}

= 2,

g's'

= 2, _{a #} 1.50, _{r #}

2.0);

the dashed fine corresponds

to the value of _{xn in} the classical sp1~l approXimatio~1 (1~lset: tetramer structure),

~~ l~n _"

~~~

B~

~ ^(l~iz)

#

~~

K

Ci

Î(1-iz)m _8.6K 1* 27.7 K

12 o.ô90 Ci

omé~

_{~ cU~~}

~ c _, o

ON .H

Mnculobp)lHzo)3.Hzo

a) b)

Fig. 7. Structure of the ferrimag~letic chairs in the compou~ld

Mncu(obp)(H20)3.H20 (Fig.

7a

from Ref, [19]); experimental results

(dotted

fine) ^and theoretical

curve for the thermal variations of

trie product

(xT)n

for

a powder of

Mncu(obp) (H20)3.H20 (Fig.

7b from Ref. [37]).

midobis(propionato).

Trie structure of this

compound

is shown in

Figure

7a. It _cari be described with the

help

of well.isolated chains

(from

_a

magnetic

point ^of

view)

characterized

by

the

alternation of cations Mn~+

(s'

=

5/2)

and Cu~+

(s

=

1/2);

note

that,

_{as soon as a}

spin

quantum ^{number reaches the} value

5/2,

trie dassical spin

approximation

is

usually employed; consequently,

one cari say that the present

compound

is characterized

by (S,1/2)N

chains. Within trie

chain,

the Mn and

Cu _{atoms are}

bridged by

an oxamido _group

Oi-02-Ci-C2-Ni-N2

with _a Mn.Cu

separation

of 5.452

À

_and

by

_a

carboxylato

_group

04-C8-06

with _a Mn-Cu

separation

of 6.066

À.

_Two distinct

exchange

parameters are

expected;

_moreover the nature of host

sites,

_on the _one

hand,

and trie involved

ions,

_on the other

hand,

allow _one to consider that trie interactions between

nearest

neighbors

_are of trie

Heisenberg

type. Trie best

fitting

of

susceptibility

measurements

(see Fig. 7b)

is obtained for trie

couple

of values

J(1+ a)

= 46.8 K and

J(1- a)

= 8,6

K,

thus

leading

to _{a =} 0.69 and J

= 27.7 K. It is reasonable to attribute trie strong value of the

exchange

_energy to trie oxamido

bridge

_[7,

14,

là,

I?i.

It is remarkable that trie value of the

exchange

_energy between Mn~+ and Cu~+ has

already

been determined in structures for which these ions, ^{considered in the} same host sites, _are

bridged by

oxamato

(36 K)

and oxalato

(26 K)

_groups, deduced from oxamido

by respectively substituting

_{one or} both NR groups

by

oxygen ^atoms

[7,17,39,40]. Therefore,

these values _are in

perfect

_agreement with that

precisely

obtained for trie oxamido _group

(46.8 K)

if _we consider that there is _a

simple

linear variation of the

exchange

_energy with respect to the number of oxygen ^atoms involved in the

bridge

between Mn~+ and Cu~+

JOURNAL DEPHY%QUEI T5,_N°4, JIARCH 1995 18

4. Conclusion

In the present ^work we have set a

general

formulation for

solving

the statistical

problem

of _a very

large

class of one-dimensional

ferrimagnets

made _up of _two sublattices

(S, s)N

and char-

acterized

by isotropic couplings

between _nearest

neighbors

_as well _as two

exchange

parameters.

We have shown that it is

only possible

to derive

simple

dosed-form

expressions

of the zero-field

partition function,

tue

spin-spin

correlations _as well _as for tue

susceptibility.

We uave _exam-

ined tue _cases wuere tue cuain is

exclusively composed

of

ferromagnetic

_or

antiferromagnetic

couplings,

_or shows _a

regular

alternation of tuese two types ^of

couplings.

In all _cases, in the low- temperature _range, ^the chain _can be described _{as an}

assembly

of

quasi-independent quasi-rigid blocks,

eacu _one of

lengtu (,

tue correlation

length,

and moment

M,

tue

magnetic

moment per unit cell. Tuerefore tue

product XT

beuaves _as

(M~;

_in

particular,

in tue

magnetic

moment

compensation,

its beuavior is

mainly

described

by

tue

competition

between tue

divergence

of

j

and tue evanescence of M. Tuis uas allowed _us to show subtle aspects ^{of botu suort~ and}

long-range

orderings. Finally

we uave recalled tue

interpretation

of tue

experimental

results obtained _on tue

compound Mncu(obp)(H20)3.H20 [wuere obp=oxamidobis(propionato)]

be-

cause it has initiated the present theoretical work. Tue

problem

of

isotropic

^chains

showing

randomly

distributed pararneters has not been examined; the present mortel _cari be

easily generahzed by taking

into account these aspects. ^{So far} we have

just

considered the _case of

exchange couphngs randomly

distributed [19];

but,

from _an

experimental point

^of

view,

the important

problem

of cationic _{vacancies can} also be considered in _a wider

generalization

of the present work. Tuerefore _we believe that this theoretical work _can be of great

help

for fu-

ture

experimental

data discussion and the further

investigations

^of

interesting

one-dimensional

compounds.

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