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7.1 Zero Field Splitting and Magnetoanisotropy
Definitions:
Zero field splitting (ZFS): the lifting of the degeneracy of spin states in the absence of
an applied magnetic field
Magnetoanisotropy: the non-uniform distribution of magnetic properties in 3D space
Organic diradicals: dipole-dipole interactions
For two electrons (i.e., two magnetic dipoles) in two orbitals of different energies, six
microstates arise:
1 closed shell singlet with both electrons in the lower energy orbital
1 open shell singlet with an electron in each orbital
3 belonging to the triplet with an electron in each orbital
1 closed shell singlet with both electrons in the higher energy orbital
For all three possible singlet states, it is (obviously) not possible to split the microstate.
For the triplet state, S = 1 and MS = +1, 0, -1.
In the absence of an applied field, these microstates should be degenerate, and this is
strictly true if the two unpaired electrons are distributed in a spherical geometry.
i.e., If there is no direction in which the electrons can “move” in order to minimize the
repulsive dipole-dipole interactions, the three microstates remain at the same energy.
If, however, the molecular orbitals are NOT spherically distributed, a ZFS can arise such
that the degeneracy of these microstates is lifted due to the geometry of the molecule!
This is the case in a planar molecule, for example.
Section 7.1 - 1
The ZFS of the microstates arising from the 3Γ ground state is commonly described by
the ZFS parameters D and E.
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D and E are clearly energy terms describing the extent of ZFS splitting in the “axial”
and “equatorial” (or “rhomibic”) directions respectively.
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D is (+)ve for an oblate spin distribution, i.e., a flattening in one direction or
compression long one axis
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D is (-)ve for a prolate spin distribution, i.e., an elongation in one direction or
lengthening of one axis
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E = 0 for a triplet species with:
a) 3-fold or higher symmetry and
b) axial symmetry
EPR is the technique most used to determine D and E.
ZFS and EPR Spectroscopy – Determining D and E
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The zero-field splitting within a 2S+1Γ state WITHOUT first-order angular
momentum is expressed by the phenomenological Hamiltonian
Hˆ D = Sˆ ⋅ D⋅ Sˆ
where D is a symmetric and traceless tensor (the ZFS tensor).
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In terms of the principle axes that diagonalize the ZFS tensor…
Section 7.1 - 2
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2
2
2
Hˆ D = −XSˆ X − YSˆY − ZSˆ Z
where X is the Dxx element in the diagonalized tensor
and Y is the Dyy element
and Z is the Dzz element
Since the tensor is traceless, X + Y + Z = 0
This dipolar ZFS Hamiltonian is then simply included in the total spin
Hamiltonian, such that the total spin Hamiltonian is the sum of all the
phenomenological components. For example, if there are no other “zero order
energy” components (such as exchange coupling), then the total spin Hamiltonian is
the ZFS Hamiltonian plus the Zeeman perturbation:
Hˆ = Sˆ ⋅ D⋅ Sˆ + βSˆgH
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•
g being the g-tensor
Since D is a traceless tensor, mathematically it is possible to write the dipolar ZFS
Hamiltonian in terms of two independent constants, D and E.
Hˆ D = Sˆ ⋅ D⋅ Sˆ
2
2
2
Hˆ D = D[ Sˆ z − S(S +1) /3] + E[ Sˆ x − Sˆ y ]
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The magnitudes of D and E are related to how strongly the dipoles (i.e., the two spins
in the organic diradical example) interact.
D and E are negligible for 2 electrons localized in parts of the molecule that are very
far apart.
D and E are the axial and rhombic zero-field splitting parameters, respectively. They
are related to the principle values Duu (u =x, y, z) of the D tensor through:
D = 3Dzz /2
E = Dxx − Dyy /2
€ Typically, in an organic diradical, the values of D and E are very small.
Section 7.1 - 3