Download 6_1_Unique Magnetic Center

6.1
Magnetic Properties of Molecular Species with a Unique Magnetic Center
So far, we have looked at how to derive the Curie Law and how to derive the Van Vleck
equation. Now we will start looking at when and how to apply these to specific cases.
We will start with the simplest case first: single molecules with one magnetic center
i.e. simple mononuclear transition metal species....e.g. Mn(acac)2(OH2)2
•
For the simplest case, we will assume that the 2S+1Γ ground state has no first-order
angular momentum (so we can use S instead of J) and has a large separation in energy
from the first excited states (so we can neglect coupling between the ground and
excited states and assume isotropic g and isotropic magnetic properties)....
...high spin MnII is a good example.
We are also ignoring the ligand field, thus treating the molecule as isolated Mn2+!
It is convenient to choose the 2S+1Γ ground state as the energy origin (i.e. set to zero)
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In the absence of an applied field, the 2S+1 spin degeneracy is retained.
For example, for hs-MnII, the ground state is S = 5/2.
There are 2S+1 = 6 spin microstates given by MS = ±5/2, ±3/2, ±1/2
These all have the same energy in the absence of an applied field (and ignoring the
ligand field).
•
When a magnetic field H is applied, the energies of each spin microstate depend on
En = M S gβH
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These energies En are linear in H (as can be seen above), therefore En(2) vanishes and
we can use the simplified form of the Van Vleck formula
2
χ=
N A ∑ En(1) exp(− En( 0) kT )
n
kT ∑ exp(− En( 0) kT )
n
•
Since we have chosen the ground state (n = 0) energy to be zero (E0(0) = 0) and all the
microstates are degenerate in zero field, then En(0) = 0
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So, from the power series En = En(0) + En(1)H + En(2)H2 + ...
En = 0 + En(1)H + 0
En = MSgβH
Therefore En(1) = MSgβ
and
MS = (-S, 1-S, ..., S-1, S)
Section 6.1 - 1
•
So
χ=
χ=
N Ag 2β 2
kT
M S =+ S
∑
2
M S /( 2 S + 1)
M S =− S
N Ag 2β 2
S ( S + 1)
3kT
THE CURIE LAW!
THE CURIE LAW IS APPLICABLE TO SYSTEMS IN WHICH:
1. H/T << 1.5 T/K
2. the material in question is comprised of non-interacting paramagnetic species
with only one paramagnetic center apiece
3. the material in question has no first-order angular momentum and a large
separation in energy from the first excited states
MAGNETIZATION
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It is important to keep in mind that the Curie Law is only valid when H/kT is small
enough. The molar magnetization M is then linear in H. In other words, the
approximation χ = M/H is valid.
Thought experiment:
At weak applied magnetic field, the measured molar magnetization responds in a linear
manner to the applied field. In other words, as the field is increased by a certain amount,
the material becomes more magnetized by a proportional amount.
At some point, however, there is going to be a limit to how magnetized a sample material
can get. There are only so many unpaired electrons per molecule. At a strong enough
field, all the unpaired electrons are forced to align with the field and this is the maximum
amount of magnetization possible for the sample material. Any stronger field won’t make
a difference. Thus a SATURATION MAGNETIZATION is possible for a material.
Another way of stating this is that the applied field lifts the degeneracy of the spin
microstates. At a strong enough field, the energy difference between spin microstates is
so large that only the
lowest microstate can be
populated (according to
the Boltzmann
distribution) at the
applied temperature.
Section 6.1 - 2
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So, when H/kT becomes large (i.e. at high field and low temperature), χ ≠ M/H. We
have to used χ = ∂M/∂H.
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Under this circumstance, we have to go back to our fundamental expression in
molecular magnetism:
∂E
⎞ exp⎛ − En ⎞
N A ∑ ⎛⎜ n
⎟
⎜
∂
H
kT ⎟⎠
⎠
⎝
n ⎝
M=
∑n exp⎛⎜⎝ − En kT ⎞⎟⎠
•
In order to make this expression useful, we will first define a partition function Z as
−E
Z = ∑ exp ⎛⎜ n ⎞⎟
kT ⎠
⎝
n
We can now write:
⎛ − ∂E
⎜
∑
⎞ ⎝
∂ ln Z ⎛ 1
= ⎜ ⎟
∂H
⎝ kT ⎠
n
n
⎞ exp⎛ − En ⎞
⎜
∂H ⎟⎠
kT ⎟⎠
⎝
∑ exp⎛⎜⎝ − E
n
n
⎞
kT ⎟⎠
⎛ 1 ⎞⎛ M ⎞
⎟⎟
= ⎜ ⎟⎜⎜
⎝ kT ⎠⎝ N A ⎠
which leads to
⎛ ∂ ln Z ⎞
M = N AkT ⎜
⎟
⎝ ∂H ⎠
•
⎛ ∂ 2 ln Z ⎞
⎟
2 ⎟
⎝ ∂H ⎠
and
χ = N AkT ⎜⎜
For a spin-only system, with non-interacting spin sites and no coupling between the
ground and excited states (i.e. the same system we just determined could be described
by the Curie Law when H/T << 1.5 T/K)
Z=
+S
∑ exp⎛⎜⎝ − M
S
gβH
M S =− S
⎞
kT ⎟⎠
⎡ (2 S + 1)x ⎤
sinh ⎢
⎥⎦
2
⎣
=
sinh(x / 2 )
where x = gβH/kT
Now we can rewrite ∂ ln Z ∂H ...
Section 6.1 - 3
∂ ln Z ⎛ gβ ⎞⎛
⎡ (2 S + 1)x ⎤
⎛ x ⎞ ⎞
= ⎜
− coth⎜ ⎟ ⎟
⎟⎜ (2 S + 1)coth ⎢
⎥
∂H
2
⎝ 2kT ⎠⎝
⎣
⎦
⎝ 2 ⎠ ⎠
⎛ gβ ⎞⎛
⎡ (2 S + 1) y ⎤
⎛ y ⎞ ⎞
= ⎜
− coth⎜ ⎟ ⎟
⎟⎜ (2 S + 1)coth ⎢
⎥
⎝ 2kT ⎠⎝
⎣ 2 S ⎦
⎝ 2 S ⎠ ⎠
•
The molar magnetization M is then
M = N A gβSBS ( y )
BS ( y ) =
•
where y = gβSH/kT
where BS(y) is the Brillouin function defined by
2S + 1
⎛ 2S + 1 ⎞ 1
⎛ 1
coth⎜
y ⎟ −
coth⎜
2S
⎝ 2S
⎠ 2S
⎝ 2S
⎞
y ⎟
⎠
We can check that for H/kT and y small, BS(y) may be replaced by
BS ( y ) =
y ( 2 S + 1)
+ term in y 3 + !
3S
such that, in this approximation, χ = M/H is effectively given by the Curie law.
On the contrary, when H/kT becomes very large, BS(y) tends to unity and M tends to
the saturation magnetization value MS
M S = N A gβS
***If the saturation magnetization is expressed in Nβ units,
its value is simply given by gS.***
The variations of M
in Nβ units
for g = 2 and
different values of S
are shown.
Section 6.1 - 4
Now, let’s move on to the next simplest case: weakly interacting molecules
•
Again, let us assume that the molecules each have only one magnetic center
i.e. simple mononuclear transition metal species as before....e.g. Mn(acac)2(OH2)2
The only difference is that we are now going to look at what happens if the molecules are
not totally magnetically isolated from one another.
This is, in fact, a much more realistic scenario. It is rarely the case that paramagnetic
molecules packed in a crystal lattice in the solid state have absolutely no intermolecular
interactions.
•
The phenomenon of interaction between magnetic centers is an important aspect to
molecular magnetism.
•
Let’s look at how the Curie Law can be modified to account for weak intermolecular
interactions...
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We can treat this problem using the “molecular field approximation” by adding a
perturbation to the Zeeman term (the term that describes the energy of microstates as
a function of applied magnetic field).
•
In other words, we will treat the magnetic field that a molecule in a crystal
experiences as a result of unpaired electrons in surrounding molecules as part of the
total magnetic field it experiences.
•
Here, we come back to the idea that, in a quantum mechanical system, we can derive
the energies if we use the correct Hamiltonian.
Recall: The Zeeman operator ĤZE = βΣi(Îi + geŝi)·H
This accounts for the interaction between the magnetic field and the electronic
angular momenta, where Îi is the orbital momentum operator and ŝi is the spin
momentum operator of some electron i.
ge is the gyromagnetic factor of a free electron and β is the Bohr magneton.
For the time being, we are considering species in which there is no contribution from
the orbital angular momentum, so the Zeeman operator becomes:
ĤZE = gβŜ·H
and when a field is applied, the energies of the 2S+1 Zeeman components are given
by En = MSgβH
(look familiar?)
Section 6.1 - 5
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Now we must add our “molecular field” perturbation to our Zeeman term.
•
This perturbation takes the form -zJ<S>Ŝ
where <S> is the mean (average) value of the Ŝ component of the spin operator
J is the interaction parameter between two nearest neighbor magnetic species
z is the number of nearest neighbors around a given magnetic molecule in the lattice
We will discuss the concept of J in more detail later on. For now, it is sufficient to
recognize that, using this perturbation term,
when J is positive, the interaction is ferromagnetic; i.e. spins tend to align parallel
when J is negative, the interaction is antiferromagnetic; i.e. antiparallel spins
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So our total spin Hamiltonian is now:
Ĥ = gβŜ·H - zJ<S>Ŝ
The eigenvalues E(S,MS) of which are:
E(S,MS) = MS(gβH – zJ<S>)
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<S> is given by the Boltzmann distribution law
⎛ − E (S , M S ) ⎞
M S exp⎜
⎟
∑
kT
⎝
⎠
M S =− S
S =
=
+S
⎛ − E (S , M S ) ⎞
∑ exp⎜⎝ kT ⎟⎠
M S =− S
+S
⎡ M (gβH − zJ S )⎤
M S ⎢1 − S
⎥
kT
M S =− S
⎣
⎦
+S
⎡ M S (gβH − zJ S )⎤
∑
⎢1 −
⎥
kT
M S = − S ⎣
⎦
+S
∑
( gβH − zJ S )
3kT
S ( S + 1) gβH
=−
3kT − zJS ( S + 1)
= − S ( S + 1)
<S> is negative because the Zeeman components with negative MS are more
populated than those with positive MS.
•
The magnetic moment m of the electron with spin s is given by
The minus sign comes from the negative charge of the electron.
m = -geβs
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For a polyatomic system in the molecular field approximation, the magnetization M
may be expressed as
M = -NAgβ<S>
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So the magnetic susceptibility χ defined as ∂M/∂H is
N A g 2 β 2 S ( S + 1)
χ=
3kT − zJS ( S + 1)
Section 6.1 - 6
•
χ may be rewritten as
χ=
C
(T − Θ)
known as the Curie-Weiss Law
C is the Curie constant (as previously discussed)
Θ is the Weiss constant (or Weiss temperature) defined by
Θ=
zJS ( S + 1)
3k
In the framework of this model,
positive Θ indicates ferromagnetic intermolecular interactions
negative Θ indicates antiferromagnetic intermolecular interactions
In the case of positive Weiss constant (Θ > 0) the Curie-Weiss law is obviously limited to
the temperature range T > Θ.
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A plot of χ-1 = f(T) for a system obeying the Curie-Weiss law gives a straight line
with a slope of C-1. The intercept with the T axis yields both the sign and value of Θ.
Section 6.1 - 7
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Magnetic data are often presented as a plot of χT vs. T in order to facilitate
observation of small changes in susceptibility with temperature.
In these plots, a positive Θ leads to an increase and a negative Θ leads to a decrease
of χT upon cooling. The lower the temperature, the more pronounced the effect.
•
It is important to point out that deviations from the Curie Law can have origins other
than intermolecular interactions.
We will soon see that zero-field splitting (arising from spin-orbit coupling) has a
similar effect on the average magnetic susceptibility.
Furthermore, intramolecular interactions lead to a deviation from the Curie Law,
however, the effect is usually observed at higher temperatures (since the magnetic
interactions within a molecule are usually stronger than those between molecules,
leading to large splitting in energies of microstates).
Section 6.1 - 8