55
SECTION 5.5. CRYSTAL-FIELD-INDUCED ANISOTROPY
that the exchange splitting of the
level is much larger than the overall crystal-field
splitting,
being the ground state. The situation in Fig. 5.5.1 a corresponds to
or to
since in crystal-field theory we have chosen the
along the uniaxial
direction. The situation in Fig. 5.5.1b corresponds to
so that we may write
Rewriting the Hamiltonian in Eq. (5.5.1) for both situations leads to
where
large
The Hamiltonian in Eq. (5.5.2) is already in diagonal form. Since we have chosen
enough, the ground state is of course
One may easily obtain the ground-state energy by calculating
In order to find the ground-state energy for
one has to diagonalize the Hamiltonian
in Eq. (5.5.3). This is a laborious procedure since the
operator will admix all states
differing by
see Table 5.2.1). It can be shown that the
ground-state wave function
is of the type
We will not further investigate this wave function except by stating that, owing to the
predominance of
it corresponds to an expectation value
which is
almost equal to
In fact, almost the full moment is obtained along the x-direction (at
zero Kelvin). This means that the magnetic energy contribution is almost equal for the two
cases (last terms of Eqs. 5.5.2 and 5.5.3).
On the other hand, one may notice that
so that the crystal-field contribution in Eq. (5.5.3) is strongly reduced when the moments point into the x-direction. The
energies associated with the Hamiltonians in Eqs. (5.5.2) and (5.5.3) can now be written as
x
It will be clear that
is lower than
for
For
the situation with the
moments pointing along the -direction is energetically favorable. These results can be
summarized by saying that for a given crystal field
the 4f-charge cloud
adapts its orientation and shape in a way to minimize the electrostatic interaction with the
crystal field. If the isotropic exchange fields experienced by the 4f moments are strong
enough, one obtains the full moment
(or at least a value very close
to it), but the direction of this moment depends on the sign of
SECTION 5.5. CRYSTAL-FIELD-INDUCED ANISOTROPY
that the exchange splitting of the
level is much larger than the overall crystal-field
splitting,
being the ground state. The situation in Fig. 5.5.1 a corresponds to
or to
since in crystal-field theory we have chosen the
along the uniaxial
direction. The situation in Fig. 5.5.1b corresponds to
so that we may write
Rewriting the Hamiltonian in Eq. (5.5.1) for both situations leads to
where
large
The Hamiltonian in Eq. (5.5.2) is already in diagonal form. Since we have chosen
enough, the ground state is of course
One may easily obtain the ground-state energy by calculating
In order to find the ground-state energy for
one has to diagonalize the Hamiltonian
in Eq. (5.5.3). This is a laborious procedure since the
operator will admix all states
differing by
see Table 5.2.1). It can be shown that the
ground-state wave function
is of the type
We will not further investigate this wave function except by stating that, owing to the
predominance of
it corresponds to an expectation value
which is
almost equal to
In fact, almost the full moment is obtained along the x-direction (at
zero Kelvin). This means that the magnetic energy contribution is almost equal for the two
cases (last terms of Eqs. 5.5.2 and 5.5.3).
On the other hand, one may notice that
so that the crystal-field contribution in Eq. (5.5.3) is strongly reduced when the moments point into the x-direction. The
energies associated with the Hamiltonians in Eqs. (5.5.2) and (5.5.3) can now be written as
x
It will be clear that
is lower than
for
For
the situation with the
moments pointing along the -direction is energetically favorable. These results can be
summarized by saying that for a given crystal field
the 4f-charge cloud
adapts its orientation and shape in a way to minimize the electrostatic interaction with the
crystal field. If the isotropic exchange fields experienced by the 4f moments are strong
enough, one obtains the full moment
(or at least a value very close
to it), but the direction of this moment depends on the sign of