Magnetocaloric effect
The magnetocaloric effect (hereafter MCE), is the cooling or heating in varying
magnetic fields. It may be enhanced in molecule-based materials.
Besides the fundamental interest in the MCE properties of novel materials, this
effect is of technological importance since it can be used for cooling applications
according to a process known as adiabatic demagnetization.
Specific heat experiments are particularly suitable for studying the MCE, since
this effect is intimately related to the thermodynamics.
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The
principle
associated
of
principle
MCE
and
of
the
adiabatic
demagnetization is easily understood:
The material, assumed to be a
(super)paramagnet with spin S per
formula unit, is initially in state A(Ti,Hi),
at temperature Ti and field Hi.
If we change the magnetic field under adiabatic conditions (Stot = 0), the magnetic
entropy change (Sm) must be compensated for by an equal but opposite change of
the entropy associated with the lattice, with a change in material temperature .
2
If we make an adiabatic field change in the
H f < Hi
magnetic field from Hi to Hf this brings the
system
to
state
B(Tf,Hf)
with
the
temperature change Tad = Tf – Ti
(horizontal arrow). If alternatively the
magnetic field is isothermally changed to Hf
in a reversible process, the system goes to
state C(Tf,Hf) with the magnetic entropy
change Sm (vertical arrow).
It is easy to see that if the magnetic change H reduces the entropy (Sm< 0), then
Tad is positive, whereas if H is such that Sm > 0, then Tad < 0
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Nanostructured materials made of superparamagnetic clusters are particularly
appealing in terms of MCE, because the usually large magnetic moments of the
clusters are easily polarized by the applied field providing large magnetic entropy
changes.
A very interesting situation is represented by molecular magnetic clusters in which
a total spin can be defined for each individual cluster as a result of dominant
intracluster interactions.
Example:
Calculated Sm for an isolated magnetic particle
with S = 10 and varying axial anisotropy D = 0.5,
1.5 and 3.0 K upon a field change H = (0 -1) T.
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Supramolecular chemistry provides a potentially attractive way of assembling and
maintaining controlled nanostructures.
The magnitude of the magnetic interactions between the molecular building-blocks
is crucial to determine the properties of such materials.
The main parameter in this discussion is the energy level separation between
different spin states that gets enlarged by the reduced size of the molecular units.
At variance with bulk materials, for which energy levels constitute a continuum band,
these systems can be considered in between the classical and quantum world.
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Specific heat measurements
The specific heat C of a given material, that is the heat capacity per unit mass
(or mole), is defined as the temperature derivative of its internal energy,
C = dU/dT.
Its determination thus provides integrated information of all the energy levels of
the magnetic system. Because in a typical molecular cluster there are energy
levels split in the range 1–102 cm-1 (1 cm-1 = 1.43 K = 12.2 eV), anomalies in the
specific heat are easily visible at liquid He temperature.
One important feature that characterizes the
specific heat technique with
respect to, e.g., magnetization or NMR, is that it can be employed in zero
applied field; no magnetic perturbation is therefore introduced in the system.
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Spin–spin correlations, that is ordering phenomena, are ideally investigated
using specific heat experiments. Given the fact that the interaction energies for
molecular spins are usually weak, phase transitions may only occur at very low
temperatures, typically below 1 K.
In these cases, the application of any external—even small—magnetic field may
wash out the collective phenomena and the use of a noninvasive technique,
such as specific heat, is particularly suitable, if not necessary.
The specific heat is also directly related to the magnetic entropy S of the system
through
the search for
. This quantity is currently receiving increasing attention in
possible applications of molecular magnets as refrigerant
materials.
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On the other hand, an external magnetic field in the range 10-1 – 10 T gives rise
to a Zeeman splitting of around one Kelvin (8.6 10-5 eV) to a molecular spin, so
the anomaly in the specific heat can be consequently shifted in a controlled
manner.
In addition, the dependence of C on magnetic fields gives information on the
magnetic states, i.e., whether they are ‘‘classical’’ spin states or quantum
superpositions of these.
Specific heat experiments are currently carried out to detect fine energy
splittings in molecular magnets in a versatile way and with an accuracy that is
comparable to that of spectroscopic techniques.
As we shall see below, specific heat experiments provide a privileged tool to
investigate time-dependent phenomena in the quantum regime of such systems.
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In order to investigate the quantum behavior of molecular spin clusters, all the
measurements must be performed at very low temperature.
The QD-PPMS, is an automated workstation that allows to perform a variety of
experiments requiring fine setting of the thermal control in the temperature range
1.9−400 K. Using the 3He refrigerator the temperature available goes down to 0.3 K.
A superconducting coil allows to apply a magnetic field up to ±7 T.
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To measure the specific heat it is necessary to read the response of the system in
terms of temperature changes after a controlled quantity of heat has been
supplied.
The instrument apt to realize this kind of experiment is the calorimeter, that
essentially is constituted by a platform with an heater to provide the heat Q(t) and
a thermometer to read the temperature T(t).
Four or more wires hold the calorimeter and they
constitute the electrical and the thermal link at
the same time. For this reason, the size and the
material of which these wires are made need to
be carefully chosen according to the temperature
range and the size of the sample.
10
All measurements are done in High Vacuum (< 10-5 mbar) to reduce heat
transfer by the surrounding gas molecules
In principle, the sample should be thermally decoupled from the surrounding, and
the experiments can be seriously affected by non-perfectly adiabatic conditions.
Actually the wires of the thermometer and of the heater, as well as the sample
holder, are responsible for the thermal leak. Different models can be adopted to
study the system.
Model 1)
The sample, with heat capacity Cs , is
thermally linked to the thermometer and
heater through the sample holder. The
empty calorimeter has a small but finite
heat capacity
Cadd = Cholder + Cthermo +
Cheater, where add stands for addenda.
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In a first approximation, we consider these three objects in excellent thermal
contact and linked to the thermal bath through a controlled thermal impedance
having thermal conductivity K1.
The approach to thermal equilibrium requires that energy is exchanged between
the spins and the lattice, which constitutes the thermal reservoir.
The so-called single-molecule magnets (hereafter SMMs) relax to equilibrium by
flipping the macroscopic spins over the classical anisotropy barrier if the thermal
energy is sufficiently large.
Below a certain blocking
temperature, the relaxation slows drastically and
quantum tunnelling through the anisotropy barrier is seen to contribute to the
dynamics of the macroscopic spins.
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If a controlled heat quantity Q(t) is provided by the heater to the sample and to
the holder block the response of the system is given in terms of temperature
changes T(t).
The relevant quantity is the total heat capacity Ctot = Cs + Cadd.
The heat Q(t) also flows to the thermal bath through the thermal conductivity of
the wires so the energy balance for this system is:
where the term on the left is the rate of heat dissipated by the heater, the first
term on the right is the heat flowing from the system to the thermal bath and the
second term on the right is the heat used for the temperature variation of the
system (sample + addenda). T0 is the fixed temperature of the thermal bath while
T1(t) is the actual temperature of the system (sample + addenda) that is time
dependent.
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Different solutions of previous equation are possible with a convenient choice of
the thermal conductivity K1 and of the time dependence of the heat Q(t).
1) If dQ(t)/dt = Q0(t - t0) (heat pulse at t = t0 for a very short time) and K1 is very
small (adiabatic condition), the solution is T1 - T0 = Q0/Ctot. Thus, controlling the
value of Q0 and measuring the temperature variation T1 - T0 , one may directly
estimate the heat capacity Ctot = Q0/(T1 - T0 ).
This describes
the adiabatic methods and it coincides with the definition of
specific heat
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2) If dQ(t)/dt = P (t - t0) (step-like heat signal starting at t = t0, where P is a
constant with dimension of power) and K1 is finite, then the solution of the
equation is
T - T0 = PK1-1 exp(-(t - t0)/t1,
where t1 = Ctot/K1 is the relaxation time.
This method, known as the relaxation method, it is widely used for automatized
measurements.
3) If dQ(t)/dt = p0cos2(t) (sinusoidal current through the heater with angular
frequency ) and K1 is finite, then the steady state solution is
T - T0 = p0 (4  C)-1  sin(2  t) + p0(2K1)-1
where  is a correcting factors that in optimal condition is close to 1.
This method is known as the ac method. It has some drawbacks thus it is preferred
when (i) the knowledge of the absolute value is not essential, and (ii) very small
heat signals are requested (for instance to detect phase transition anomalies).
15
Organic materials have often a bad thermal conductivity, so the approximation of
previous equation is no longer valid. A more realistic model of the calorimeter
stage is shown in the figure, where a finite K2 is considered, that accounts for both
the physical link between sample and holder block, and the rate of internal
equilibrium of the sample; establishing an internal relaxation time τ2 = Cs/K2.
16
To this model it corresponds the new equation:
where Cp and Tp are the heat capacity and temperature of the platform. Cs and
T1 are the heat capacity and temperature of the sample. To get the solution of
this equation one may use curve fitting method (CFM). The procedure is rather
complicate. In the limit τ1 > τ2, the temperature rise (or decay) following a heat
pulse is essentially given by the sum of two exponentials, when the constant
heating power is on or off:
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Specific heat of molecular antiferromagnetic wheels
The specific heat of molecule-based materials is the result of several contributions.
Even at cryogenic temperatures the lattice contribution Clatt is still substantial and,
since it grows very fast with temperature, at  10 K Clatt becomes already
predominant, masking most of the features of the magnetic contribution Cm .
Furthermore, the details of the spectrum of the quantized lattice vibrations
(phonons) can be quite complex due to the huge number of independent vibration
modes.
One usually applies the approximation of an idealized simplified crystal with 3
acoustic branches and (3N - 3) optical branches. The three acoustic branches are
treated within the Debye model, giving the so-called Debye contribution CD, while
the optical modes are described by the Einstein model giving the corresponding CE
contribution.
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Molecular rings often have a planar shape with
an axial symmetry of the spin arrangement.
The dominant antiferromagnetic Heisenberg
Cr8: AF, S = 0
exchange coupling between nearest neighbour
spin centres, in combination with an even
number of spins, provides a singlet ground
state at low temperature and zero field for the
isometallic wheels.
Cr7Ni: S-para, S = ½
The appeal of these AF rings comes first from the energy separation between the
lowest lying states that range between a few Kelvin and a few tens of Kelvin, so
that these levels are well isolated below liquid helium temperature.
19
Attività pratica : Misure del calore specifico di
microcristalli di
SMM a
temperature criogeiche
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