2.3 Further Removal of the Degeneracy of the N-electron States
43
In systems with an even number of electrons and both D and E different from
zero, the zero-field splitting completely removes the degeneracy of the ground state
manifold. On the contrary, the levels in systems with an odd number of electrons
remain doubly degenerate at zero-field, often referred to as Kramers doublets. For
systems with integer spin moment, the wave function of the lowest level is dominated
by the M S = 0 determinant when D is positive. This means that the projection of
the spin moment on the magnetic z-axis is (practically) zero, while the projection
on the x–y plane is maximal; the system has easy plane magnetism. When D is
negative, the largest contributions to the lowest level arise from the determinants
with M S =± M Smax , and hence, maximal projection of the spin moment on the z
axis. This is known as easy-axis magnetism. The same applies for half-integer spin
moment systems.
2.6 Demonstrate that the degeneracy of the M S =± 1/2 sub-levels of the
S = 1/2 manifold cannot be removed without an external magnetic field.
Hint: Calculate the 1/2, ±1/2| ˆ
H ZFS |1/2, ±1/2 matrix elements.
2.3.2 Splitting in an External Magnetic Field
The last column in Fig. 2.1 shows how an external magnetic field H affects the
energies of the M S sublevels of the electronic manifolds of a paramagnetic material.
This effect is described by the Zeeman Hamiltonian
ˆ
H ZE = μ B H · ( ˆ
L + g e ˆ
S)
(2.23)
When spin-orbit coupling is neglected and the ground state has no orbital moment,
the expression reduces to its isotropic spin-only form
ˆ
H ZE = μ B g e H · ˆ
S
(2.24)
Defining the field direction as the z-axis, the Hamiltonian reduces to μ B g e H ˆ
S z and
the energies of the M S sublevels vary linearly with the field strength as
E n = M S μ B g e H
(2.25)
Typical examples of such paramagnetic systems are organic radicals where spin-orbit
coupling plays a minor role, but Eq. 2.25 can also be used to describe the evolution of
the energy of the M S sublevels of the ground state in 3d transition metal complexes
when the zero-field splitting is absent (S = 1/2) or significantly larger than the effect
of the external field and the spin-orbit interaction with excited states is small. This
splitting of the energy levels with the external field is not only at the very origin
43
In systems with an even number of electrons and both D and E different from
zero, the zero-field splitting completely removes the degeneracy of the ground state
manifold. On the contrary, the levels in systems with an odd number of electrons
remain doubly degenerate at zero-field, often referred to as Kramers doublets. For
systems with integer spin moment, the wave function of the lowest level is dominated
by the M S = 0 determinant when D is positive. This means that the projection of
the spin moment on the magnetic z-axis is (practically) zero, while the projection
on the x–y plane is maximal; the system has easy plane magnetism. When D is
negative, the largest contributions to the lowest level arise from the determinants
with M S =± M Smax , and hence, maximal projection of the spin moment on the z
axis. This is known as easy-axis magnetism. The same applies for half-integer spin
moment systems.
2.6 Demonstrate that the degeneracy of the M S =± 1/2 sub-levels of the
S = 1/2 manifold cannot be removed without an external magnetic field.
Hint: Calculate the 1/2, ±1/2| ˆ
H ZFS |1/2, ±1/2 matrix elements.
2.3.2 Splitting in an External Magnetic Field
The last column in Fig. 2.1 shows how an external magnetic field H affects the
energies of the M S sublevels of the electronic manifolds of a paramagnetic material.
This effect is described by the Zeeman Hamiltonian
ˆ
H ZE = μ B H · ( ˆ
L + g e ˆ
S)
(2.23)
When spin-orbit coupling is neglected and the ground state has no orbital moment,
the expression reduces to its isotropic spin-only form
ˆ
H ZE = μ B g e H · ˆ
S
(2.24)
Defining the field direction as the z-axis, the Hamiltonian reduces to μ B g e H ˆ
S z and
the energies of the M S sublevels vary linearly with the field strength as
E n = M S μ B g e H
(2.25)
Typical examples of such paramagnetic systems are organic radicals where spin-orbit
coupling plays a minor role, but Eq. 2.25 can also be used to describe the evolution of
the energy of the M S sublevels of the ground state in 3d transition metal complexes
when the zero-field splitting is absent (S = 1/2) or significantly larger than the effect
of the external field and the spin-orbit interaction with excited states is small. This
splitting of the energy levels with the external field is not only at the very origin