An Ising-type model for spm conversion ... model is that it provides an excellent
agreement with bath the conversion curve ..... and analysis of experimental data.
2. ..... contribution of the oscillators cari be numerically obtained by integrating. ).
A Simple Ising-Like Model for Spin Conversion Including Molecular Vibrations A. Bousseksou, H. Constant-Machado, F. Varret
To cite this version: A. Bousseksou, H. Constant-Machado, F. Varret. A Simple Ising-Like Model for Spin Conversion Including Molecular Vibrations. Journal de Physique I, EDP Sciences, 1995, 5 (6), pp.747-760. .
HAL Id: jpa-00247099 https://hal.archives-ouvertes.fr/jpa-00247099 Submitted on 1 Jan 1995
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Phys.
J.
I
France
(1995)
5
747-760
1995,
JUNE
747
PAGE
Classification
Physics
Abstracts 63.20K
31.30N
75.10H
Ising-Like
Simple
A
33.40
Molecular
Vibrations
A.
Bousseksou
(1),
(~
Laboratoire
de
Cédex, France Département (~) 05,
Cedex
(~)
Chimie
Coordination
de
Spin
for
(~,*),
Constant-Macuado
H.
de
Model
C.N.R.S.,
du
Physiques (**),
Recherches
(~)
Varret
F.
Including
Conversion
205
Route
Université
Narbonne,
de
Pierre
Curie,
Marie
et
Toulouse
31077
Paris
75252
France
Laboratoire
Magnétisme
de
d'optique (***
et
Université
Saint-Quentin,
Versailles
de
78035
,
Cédex,
Versailles
(Received
Abstract.
An
model
for
Each
two
studied.
agreement
change
upou
with
accepted
aud
possible frequencies
two
entropy
revised
Ising-type
been
has
excellent
an
1994,
Julre
6
vibrations
haviug
France
spm level
wf/, w(/.
bath
the
Jalruary1995)
9
explicitly accounting
conversion,
The
system
is
advantage
major
conversion
and
curve
with
associated
of this
calorimetric
for
intrarnolecular
harmouic
p
is
model
data,
oscillators
provides particular the it
that m
conversion.
spin
Arrhenius plots, the predictions of trie model are accurately compared to of Môssbaùer experimental data on the spin equilibrium A very accurate curves. investigation by Jacobi, Spiering and Gütlich, provides evidence for a small eflect typical for essentially monitored vibrations. A navet example is given where trie conversion by curve is vibrations; this origmates from the extremely small value of the energy gap between HS aud LS electrovibrational groundstates.
help
the
With the
literature
However, can
be
most
in
Then
used.
impficitely
cases, as trie model
accounting
for
a
first
to
intramolecular
a
the
low
siInple
frequency
approximation
two-levellnodel
with
additional
trie
of
model
degeueracies
vibrations.
Introduction
1.
Among tue tueoretical mortels developed uigh spin (HS) and low spin (LS) states, widely used because of tueir practical (*)
on
leave
(**) CNRS (***) CNRS
©
approach,
reduces
Les
URA
Universidad n° 71
URA
n°
from
Editions
de
Central
1531
Physique
1995
of
to
tue
describe
tue
two-state
convenience.
Venezuela
tuermodynamics models
iii
or
of
two-level
conversion models
between
[2-4]
are
JOURNAL
748
Tue
description iii
macroscopic
two-state
rium, according
reversible
tue
to
considers
I
N°6
"spin isomers"
two
equilib-
cuemical
in
reaction:
LS
(AS°, AH°)
HS
=
of cooperativity, AS°, AH° do vibrations importance of molecular
absence
In
PHYSIQUE
DE
(1)
depend on tue molar the spin conversion
no
fractions
nHs>
nLs.
problem was first recognized by Sorài and Seki [5] on tue basis of calorimetric Indeed, tue purely measurements. models [1-4] resulted electronic in a value of tue molar change entropy upon conversion, Asejectr 't Asspin oniy 13.4 J K~~ mol~~, far below ail experimental data, ranging from Tue missing entropy was attributed vibrational elfects, mostly of 35 to 80 J K~~ mol~~. to intramolecular origin. Furtuer studies provided evidence for spectacular changes in tue vibrational spectra, togetuer witu a minor change in tue Debye temperature, observed by Môssbauer data are given in tue references [7,8]. reviews of ail tue corresponding spectroscopy [6]. Recent model explicitly accounting for vibrations introduced Tue first by Zimmermann and was Kônig [9]. Its main goal was to qualrtitatively account for trie molar entropy increase upon spin conversion, AS. Tue model includes ligand field and spin-orbit coupling, but tuese ingredients vibration assumed to follow tue Debye model, did not yield any specific elfect. Tue modes are irrespectively of tueir intra- or intermolecular cuaracter; tue Debye temperature is assumed Tue
in
"
to
~/8[~
vary as tuus
n(~,
(36 J.K~~ mol~~). models tue
Anotuer and
tions our
lattice
a
knowledge,
nature
In
Tuis
an
using 8Ls
in
K, 8Hs
"140
tue
paper
same
results
contribution
a
coupling through
intermolecular
tue
model is based
vibronic
8[~)nHs.
autuors
"
bave
130
the
K,
suown
reacues tue
tue
to
phonons. tue
formal
free
Tue
energy
total
r-
molar
magnitude correct equivalence between
treated in tue mean-field approximation, and macroscopic two states models tuermodynamics of regular solutions in tue Bragg-Williams approximation. vibrations model exphcitly including bas been developed by Kambara [loi; tuis
two-level based
on
to
calculated
variation
entropy
(8[~
+
equivalent
of the
Recently,
it
bas
Kambara's used
electronic
witu
state
botu
intramolecular
vibra-
yields qualitatively correct results; uowever, to fit experimental data, probably because of the detailed
model to
also
calculations.
proposed
we
been
not
needed
coupling of tue
tue
on
strain.
[4]
taking advantage AS° =
of the R Ln
relation:
formai ~~~
(2)
gLs
tue degeneracies associated "Effective" witu tue two spin states. values et gLs are degeneracy ratio much higuer tuan tue pure spin value (2s +1), are needed to reacu experimental AS values. This increase in degeneracies merely accounts for the density of
where
gHs
of tue tue
vibrational
states
mucu
higher
in the HS
thon in tue LS
state.
Values
of tue
ratio
as
high
as
good lits iii] of tue experimental nHs(T) curves, mucu improved for tueir -= high-temperature branches, while trie experimental values of tue molar entropy were correctly beuind such a procedure will be analyzed uere. obtained. Tue approximations intramolecular vibrations, since tue intermolecular Here we shall deal only witu tue vibration been estimated contribution compared to tue [5,6] to be of minor to AS bas importance, 500
2000
led to
intramolecular
contribution.
Tue
vibrational
elfects Will be
cooperativity. Tuis
reported is organized of tue
as
follows: calorimetric
Section
2:
brief
Section
3:
Section
4:
description of tue "Ising-like" model; tue model including vibrations; short
review
aspects;
better
cuaracterized
in
absence
of
SIMPLE
A
N°6
ISING-LII~E
MODEL
FOR
CONVERSION
SPIN
749
~
~ ,
T
, , , ,
/ , '
, '
SLS
,,
/
ÎS
/
'
~
, /
/ /
,
,'
ÔCj~°rural)
'
/
/
/ ,
T
T
Ç
T
b)
a) Fig.
a)
l.
~~°~~~~
thermal
associated
variations
ÎÎ
°~
Section
5:
Section
6:
~~'
It is
LS
pure
molar
and
in
entropy
HS
trie
conversion,
of spilr
case
b) corresponding
states.
variation
of
compared to the specific heat,
trie
results; discussion
useful
the spin
of the
~'
Calorimetric
2.
variation with
to
and
analysis of experimental
Aspect
precisely define the molar
conversion
data.
as
a
reversible
increase
entropy
reaction,
chemical
àsjT)
s~sjT)
=
conversion
upon
then
AS(T). Considering
(per mole):
s~sjT)
j3)
SHS(T), SLS(T)
the molar entropies the compound would have at T if temperature are entirely in tue uigu-spin or low-spin state, respectively. Very likely, tue molar entropy of tue compound, S(T), is between SLS(T) and SHS(T); in any case, wuen a complete spin it varies from tue low-spin value at low conversion temperature to tue uigu-spin value occurs, This is schematized corresponding specific heat curve in Figure la. Trie at uigh temperature. where
it
were
is
suown The
in
Figure 16, given
equilibrium
)
as
temperature
~~ =
Tc
dT
T.
vs.
defined
by
> nHs
"
nLs
consequently:
AH(Tc)
"
"
is
such
that
AG(Tc)
2
TORS(Tc
=
0;
(4)
"normal" Tue regimes Ii.e., tuose of tue pure LS or HS states), can be extrapolated to tue from eacu side; tuis leads to tue discontinuity ACJ°~~l~' evidenced equilibrium temperature cuaracterized by a discontinuity of by Sorài and Seki [5] in tue case of a true spin transition, nHs(T). Trie discontinuity in tue normal specific ueat provided tue first experimental evidence for tue electron-vibrational of tue spin conversion. nature ~~ AS(Tc) is easily deduced from tue plot of Wuen tue spin T. conversion vs. is complete, Tuis
is
demonstrated
in tue
following
T
way:
JOURNAL
750
PHYSIQUE
DE
I
N°6
ASITC) SHslTc) SLS(Tc) s(Tc) s~s(Tc) + s~s(Tc) s(Tc) "
=
Assuming S(T')
SHS(T') (complete
=
S(0)
=
for T'
conversion
Tc)
»
SLS(0) (tuird tuermodynamical law)
tuen:
/
AS(Tc)
~~
=
(dS(T)
dSLs(T))
+
/
(5)
The
3.
~Aoâ =
wuere
Ao tue
is tue
energy -1
values
spacing of tue
+)
,
~~
~~~~~~~
Resolving
~
~
"~~
"'
for tue ~~
canonical
7i in
uatcued
of
areas
Figure 1(b).
Jâ
< à
isolated
two-level
approacu,
written
(6)
>
system
and à
HS, respectively;
LS and
is
fictitious
a
tuen
tue
spin
operator
uigu-spin
fraction
yields:
statistics
1
tue
up
mean-field
in
2
state
spin
Î
summing
cooperativity
É
taking
(5)
Model
including
uamiltonian,
by
obtained
is
(Ising-Like)
Two-Level
two-level
Tue
AS(T~)
tuat
means
dSHs(T))
(dS(T)
Tc
0
Relation
~,
~iHs
~eq
"
iii e~£]~~
"
(7)
with
A(nHs) Equation (7)
Ao
=
formally equivalent
is
2J < à
tue
to
>=
Ao
ASO
AH
Keq
~
RT
e
=
(8)
4JnHs
law:
action
mass
+ 2J
(9)
R ,
here assuming the interaction to be purely enthalpic. By identifying equations (7) and (9), an interesting expression entropy change upon (complete) conversion: written
AS°
R Lu
=
is
obtained
for
tue
molar
~~~
(10)
gLs
A
second
rium
useful
relation
temperature
is
easily
derived
from
(nHs "1/2), uolding for Ao
"
any
equations (7) and (8), dealing witu
equilib-
the
J: ~~~
kBT~ Ln
(ii)
gLs
Conversion which
are
curves
linear
intramolecular
in
are
tue
interactions
conveniently absence are
of
visualized
interactions,
included
in
term
and
(see Fig. 2).
of
show
Arruenius
plots,
typical S-suaped
Ln
Keq
distortions
=
f(
),
T when
SIMPLE
A
N°6
ISING-LIKE
FOR
MODEL
CONVERSION
SPIN
751
b .,
as
?5
SO
J
c
'25
IOC
'7£
'50
22£
mû
'10"N
Fig.
plots computed using equation (7), positive for cooperativity coupling):
Arrhenius
2.
coupling
here
Degelreracy
defined ~~~
ratio
5, Ô(nHs
=
"
will
It
relations
be
noted
(7)
and
that
(8),
the
the
=
diflerent
values
0, 110,
150
of
the
220
K
intermolecular
(a)
froIn
(d).
to
K.
300
#
2
with
intersect
the
ordinate
(T
axis
cc), directly yields, using
-
AS°;
of
value
J
,
~
gLs
for
AS°
R Lu
=
K~~(T
cc)
-
=
R Lu
~~~
(12)
gLs
other
In
change
words, closely
trie
high
of the
values
temperature
Indeed,
experimental data are contribution), and a quasi-complete electronic to the of the electronic degeneracy ratio). excess Two-Level
4.
Each
related.
Including
Model
molecule
assumed
is
frequencies u~(j in the The partition function
LS
spin populations and trie molar entropy associate large value of AS (with respect HS
character
at
high temperature(in
Vibrations
to
behave
state,
w~(
associated
as in
with
HS
oscillators,
harmonic
with
state.
molecule
one
independent
of p
set
a
the
is
written:
A(nHs) ~(~) where
gLsE>
gHsE
spin. Usually,
the
are
expression of the
=
contribution
25 + 1
=
gHSE~ÎÎ(~)~
~
degeneracies
electronic
electronic
~~~~
gLSE~Î~(~)
"
5 is
of the
the
to
energy
for Fe"
taken
two
~~~ states.
dilference
(13)
A(nHs) between
is
the
states
mean-
of
field
dilferent
complexes.
gLSE
vibrational
The
partition
function
is
written:
fl P
2~vib(T)
"
Zvib(LJi>
T)
(14)
1=1
z(w,, T) state).
where HS
By
thermal
is
the
averaging,
partition the
function
equilibrium
of the
constant
isolated is
harmonic
obtained:
i~h
oscillator
(in
trie LS
or
JOURNAL
752
~
=
~~~
This the
to
equation
self-consistent
thermal
the
law is
action
mass
nHs(T) simple
system:
longer
no
as
N°6
~~~HS ~~~
~
~15)
numerically (by dichotomy,
resolved
is
of the
evolution
I
~~~~
gHSE2~$((T) ~LSE2~II (T)
~HS
~eq(T)
PHYSIQUE
DE
depend on temperature. transformations Straightforward algebraic
K~~(T).
or
in
as
to
3, and leads
section
with)
start
identification
The
to
yields equation (15)
of AH
and
AS
and
values
which
~~~
yield:
~(i)
P
~~~~
~~
~
w(
gLSE
~~~~
~
specific heat jump and the entropy change, between the pure HS and by dilference
The
for complete
both
calculated
LS
conversion,
spin
easily
are
states.
Results
5.
first: the is developed previously noticed [4] that
particular
A useful
For &w
