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Ising-like model for the two-step spin-crossover
Azzedine Bousseksou, J. Nasser, J. Linares, K. Boukheddaden, F. Varret
To cite this version:
Azzedine Bousseksou, J. Nasser, J. Linares, K. Boukheddaden, F. Varret. Ising-like model for
the two-step spin-crossover. Journal de Physique I, EDP Sciences, 1992, 2 (7), pp.1381-1403.
�10.1051/jp1:1992217�. �jpa-00246629�
J.
Pbys.
I
France
2
(1992)
1381-1403
JULY
1992,
PAGE
1381
Classification
Physics Abstracts
75,10H
05.50
31.30N
76.80
Ising-like
model for the
two-step
spin-crossover
A.
Bousseksou(~),
J. Nasser(~,~), J.
Linares(~),
K.
Boukheddaden(~)
and
F.
Varret(~)
(~)
D4partement de Recherches
Physiques, UA CNRS
71,
Universit4
Pierre
et
Marie Curie,
F-75252 Paris
Cedex
05,
France
(~)
Service de Physique du Solide
et
de R4sonance Magn4tique, CEN Saclay, F-91191 Gif-sur-
Yvette Cedex, France
(Received
10
April
1991,
revised
28
November
1991
and
17
February1992,
accepted
27
March
1992)
Rdsum4. Nous
avons
analys4
un
modble
de
type
Ising, h deux
sous-r4seaux
couplds
"an-
tiierromagn4tiquement"
,
dans l'approximation du champ
moyen.
Ce
modble
permet
de bien
reproduire les transitions
de spin "en deux
(tapes",
dont
nous
donnons
une
d6finition prdcise
Lorsque
les
deux
sous-r£seaux
sont
4quivalents, il implique
une
brisure spontande
de
sym4trie
qui
peut
intervenir
dans
un
domaine de
temp6rature
limits
par
deux "temp£ratures de N4el",
De plus,
lorsqu'ils
sont
in6quivalents,
il
pr4dit
le
renversement
simultan4 de I' "aimantation" des
deux sous-r4seaux
pour une
valeur "caract4ristique"
de
la temp4rature. Nous
avons
analys4
en
d4tail l'ensemble de
ces
eiets. Ce mod+le
nous a
permis d'ajuster
et
de discuter les rdsultats
ex-
p4rimenta~1x
disponibles
concemant
[Fe(2
pic)3]Cf2
EtOH
et
Fe"
[5N02
Sal
N(1,
4,
7, 10)].
Abstract.
We
have
analyzed
an
Ising-like model,in
the
mean-field
approach,
involving
two
"antiierromagnetically"
coupled sublattices. This model simulates the so-called "two-step" spin-
crossover
transition, for which
a
precise
definition
is
given. If both sublattices
are
equivalent, it
implies
a
spontaneous
breaking ofsymmetry which
may occur
within
a
temperature
range
limited
by
two
"N4el temp6ratures". It,
also
predicts
a
simultaneous reversal
of
the magnetization of
the sublattices
(it
they
are
unequivalent)
at
a
"characteristic" value of
temperature.
These
features
are
analyzed simultaneously with
some
details. The
present
model fits and explains
well the available experimental data concerning
[Fe(2 pic)3]Cf2 EtOH
and
Fe"[5N02
Sal
N(1,
4,
7,10)].
1.
Introduction.
Recently
an
'~unusual" low-spin
=
high-spin
(LS
4
HS)
conversion
of
Fe~~ has been observed
in
some
molecular
solids
[1-3].
In
these compounds, the fraction
nHs
of molecules in the
HS
state
increases
with
temperature
in
two steps
with
a
plateau
of
a
few
Kelvins.
In
this
report
we
propose
a
model which simulates such
a
behaviour
This
model
is
based
on
the approach
JOURNAL DE
PHYSIQUE I
-T.
2, N' 7, JULY
1902 50
1382
JOURNAL
DE PHYSIQUE I
N°7
of discrete
levels
[4]
which offers
a
fruitful
comparison
to
magnetic models.
It is
well known that, for
an
isolated
molecule, the
LS
state
is lower in
energy
than the
HS
state
if the ligand field is
stronger
than the
mean
spin-pairing
energy
The
LS
@
HS
conversion is
thermally possible
if
the
energy
distance
6
between
the ES and
the
LS
states
is
of
the
same
order
as
the thermal
energy
kT
[5],
In
the discrete level approach
(adapted
here
from
[4]),
for
the
isolated
molecule, the Hamil-
tonian is
taken
as:
~
2~"
where
«
is
a
fictitious spin which has
two
eigen-values,
~
l,
corresponding
to
the LS
and
HS
states
respectively. The degeneracies
ga, gb
of the LS, HS
states
are an
essential
element of
the
model;
gb
is
larger
than
ga
because of electronic
and vibrational
properties. In this
formalism,
the
spin conversion is
characterized by
the
mean
value of
«:
jai
=
-nLs
+
nHs
with
nLs
+
nHs
"
1,
SO
that
nHs
=
((i
+
(«)).
at
0 K,
(a)=
-I,
all the
molecules
are
in the LS
state
(nHs
#
0);
at
T
-
oc,
(«)
-
~~
~
~~
ci
I
if
gb
»
ga;
nearly all the molecules
are
in the
HS
state
9a
+
gb
(nHs
t
I).
A
characteristic
temperature
for
the spin
conversion,
Tc,
is defined
by kTcLn
(~~)
=
6
,
9a
which
corresponds
to
(«)
=
0 (nHs
#
0.S).
In the
scope
of the
present
analysis, it is pointed
out
that the
steepest
(and major)
part
of
the thermal variation
of
(«),
occurs
around
Tc.
In
the
crystal,
the interaction
W;j
between
molecules
I and
j
is
a
function of both spins
«;
and
«j
The general
development of
this function
to
the
second order
is:
Wij"A;J(";+«j)+J;j«;.«j
So the total Hamiltonian
is:
7i
=
£
jA;«;
+
£
J;j
«;
«j,
S
I",jl
where
£
is
a
sum over
sites, and
£
a
sum over
pairs of sites.
;
j;,jj
In
the expression
of 7i,
the values
of
the
parameters
A;
are
directly
related
to
the
crystal
field
on
the site I.
This field is the
sum
of
the
ligand field
of the molecule
I
and of the
crystal
field
created
by the other
molecules
of the crystal.
The
parameters
A;
and
J;j
are
considered here
as
phenomenological,
and
for simplicity
we
shall consider A;
=
A for
all
sites;
then
7i
is
an
Ising-like Hamiltonian with field
(the
difference
with the
true
Ising
model is
ga
#
gb).
The
case
of inequivalent
sites
will
be
described
by
unequal
J;j
values.
It
is worth noticing here that
a
twc-step
curve can
be
trivially
obtained
by considering
two
independent sublattices
with unequal values
of
A;. The
present
paper,
considering equal A;, is then
focused
on
the
role of the intermolecular
interactions in the
double-step
transitions.
N°7
MODEL FOR TWO-STEP
SPIN-CROSSOVER
1383
Wajnflas2 and Pick
[4]
have studied, in
the mean-field approximation,
the "ferromagnetic"
case
of
a
single
sublattice characterized by
a
single
"exchange"
parameter
J
<
0.
They
found
that the crystal displays
a
first
order
transition
at
the
temperature
Tc
,
now
defined
as
k Tc
Ln(~~)
=
A,
if the interaction
parameter
(J( is
larger than the threshold value k
Tci
ga
typical plots of
(«(T))
are
reproduced in
section
2, figure I.
The
occurrence
of such
a
transition
is clearly
due
to
the degeneracies
gb
#
ga,
since
the ferromagnetic Ising model under field
does
not
lead
to
a
phase transition (for
gb
-
ga,
7~
-
oc).
On the
other
hand, it is remarkable that
the first-order
transition is
not
accompanied
by
a
symmetry
change in the
crystal;
this is because
symmetry
is already broken by the
"field"
(parameter
A).
In
the
present
study,
we
analyse the "antiferromagnetic"
and
"ferrimagnetic"
cases,
involving
two
sublattices,
also in
the mean-field
approach.
It is
known
[6]
that the antiferromagnetic Ising
model
under magnetic
field
can
lead
to
a
para-antiferromagnetic
phase
transition. In
the spin-
conversion
system,
the corresponding
effect
is
a
spontaneous
breaking of
symmetry
between
the sublattices
(when
they
are
structurally
equivalent),
which
has been briefly described by
Bari and Sivardibre
[7].
We
report
here
a
detailed analysis of the
syrnrnetry
breaking and of
the
twc-step
character
of
the transition, the
latter point being the initial
goal
of the
present
study.
The
present
paper
contains the description
and analysis
of
the model
(Sect.
2),
the
com-
parison with
a
previous
study
of
Bari and
Sivardibre
[7]
(Sect.
3)
and the application of the
present
model
to
the available
experimental data
(Sect. 4).
The
latter section is also devoted
to
the conclusion.
For convenience,
we
shall refer
to
Wajnsflasz
and Pick, Bari and Sivardilre,
as
W-P, B-S
respectively.
The
notation is
used
are
presented
on a
recapitulary table
(Tab. I),
which also
summarizes the
results
of
the
present
analysis
Table
I.
List of notations and
summary
of
the
results
of the
present
analysis.
Notation
~
9a
9b
Tc
(Eq.
(4)): simultaneous
sign
change of
(«Al
,
(«B) (if
JA
#
JB)
and sign change of
(«)
T~~,T(~:
sign changes of
(«A)
T~~,
T(~:
sign
changes of
(«B)
if
(JAI
>
lJBl
T~~,
T£~
exist if JAB
>
(JAB )s
~°CB
~
~°CA
~
~°~A
~
~°~B
T~~,
T£~
have
no
physical
relevance
T~~
#
T£~
define the
twc-step
character
two
possibilities:
(~)
~°CB
~
~C
< T~B
(II)
TCB
~
~°~B
~
~°C
the
continuous
/
discontinuous
character
of conversions
changes:
for
T~~
at
(JAB)di if JB(
<
kTc
for
T£~ at (JAB)da
if
JB(
>
kTc
1384
JOURNAL
DE PHYSIQUE I N°7
JB
j~- j~-
j~-.
j~+ j~+ j~+
CA CB C' CA
CB
C
Tp,
T$
:
limiting
temperatures
of
possible
asymmetrical solutions.
In
principle:
Tp
<
T~
<
T$
<
T$
or
Tp
<
T$
<
Tc
if
T~,
T£
#
Tc
do
not
exist.
2,
Analysis
of
the model.
We
start
with the Hamiltonian:
7i
=
£
(A
«I
+
£
J;j«;
«j
(I)
"
I",jl
where
the fictitious
spins
«
have eigenvalues
~
l,
associated with degeneracies
ga
#
gb.
We
consider
two
sublattices A,B containing
the
same
number
of
molecules, such that:
"~~~~
(l
+
(~A,B))
'~HS
(n#
+'~#)
~(~+
(~))>
where
(«)
=
(«Al
+
(«B)
Let
JA, JB, JAB
be
the
intra- and inter-sublattice
interaction
parameters
(including
the
num-
ber of
neighbours).
In
the
present
study
we
found that the
onset
of
a
double-step transition
required
intra-
and inter-sublattice interactions,
respectively 'ferromagnetic" and "antiferrc-
magnetic"
:
JA>
JB
<
0,
JAB
>
0.
In
the mean-field approach
7i
is replaced
by
one-site Hamiltonians:
~~
'~
~'~ ~~~
~'~~
~
~~~ ~'~~~
~~~~
I~B
"
"B
+
"B
(JB
("B)
+
JAB
("Al
(~~)
Equations
(2a,2b)
are
formally
similar
to
the
mean~field
equation
derived
by
Wajnflasz
and
Pick
[4]
in the
one-sublattice model: the crystal-field
value A is merely
shifted by
the
constant
value
2 JAB
<
«B,A
>,
respectively. The mean-field
equations,
derived
from
(2a,2b)
are:
~'~~
l~li~-
/(~lll(~~~/~(~~~
~~~~
and
~'~~
l~/r~~-
/(~ll~($~ll~(~~~
~~~
where
r
is the
ratio
gb/ga,
and
fl
=
(kT)~l
The expression of F,
the free
energy,
can
be
found in the
Appendix.