4 From Small Molecules to Complex Systems: A Survey of Chemical …
185
levels. Using the dipole selection rules possible transitions can then be determined.
However, not only one Mössbauer spectrum is calculated, but one for each of the states
split by zero field splitting [35]. The occupancy of these 2S + 1 states depends on the
temperature T, so that the spectra are still weighted with a temperature-dependent
factor. Here the borderline cases of fast or slow relaxation of the electronic spin are
usually taken into account. The limit of slow relaxation holds if the nuclear Larmor
frequency
S
↔
A
is considerably slower than the relaxation rates of the electronic spin
system S. In this case the calculated Mössbauer spectra for each eigenfunction
ϕ j
need to be weighted with the Boltzmann factor p j and finally summed up:
p j =
exp
−
E j
k B T
K
exp
−
E K
k B T
(4.5)
If, on the contrary, the spin transition rates are considerably larger than the nuclear
larmor frequency the fast relaxation limit applies. In this case the spin expectation
value in Eq. 4.4 is replaced by the thermal average of the 2S + 1 spin expectation
values:
S
f ast
=
j
S
j
exp
−
E j
k B T
K
exp
−
E K
k B T
(4.6)
Now only a single Mössbauer spectrum is being calculated with a hyperfine field
which is the average of the hyperfine fields caused by population of the 2S + 1
electronic sublevels. If the electronic relaxation rate is comparable to the Larmor
frequency complicated Mössbauer spectra with relatively broad lines arise. For the
spectroscopist who is not interested in the determination of electronic relaxation
rates via Mössbauer spectroscopy it is advisable to push the system under study to
the slow relaxation limit by performing Mössbauer experiments around liquid He
temperatures. But it should also be mentioned here that there are suitable theoretical
models to treat the case of intermediate relaxation. Examples on myoglobin have
been treated by Winkler et al. [36]. Also for the case of iron containing magnetic
nanoparticles it might be interesting to determine iron dependent spin relaxation rates
which can be performed by the model of Blume and Tjon [37].
It should be noted that the procedure discussed above is sufficient for the calculation of non-interacting paramagnetic iron sites in a single crystal of a chemical
complex (or a protein) when the unit cell has only one molecule. Only in this
case there exists exactly one orientation of the molecular frames with respect to
the external magnetic field B. However, since the large majority of Mössbauer spectroscopic studies are performed on powder samples or in the case of iron proteins
on frozen solution samples it is necessary to calculate one Mössbauer spectrum for
every molecular orientation. This so called powder averaging is done numerically by
185
levels. Using the dipole selection rules possible transitions can then be determined.
However, not only one Mössbauer spectrum is calculated, but one for each of the states
split by zero field splitting [35]. The occupancy of these 2S + 1 states depends on the
temperature T, so that the spectra are still weighted with a temperature-dependent
factor. Here the borderline cases of fast or slow relaxation of the electronic spin are
usually taken into account. The limit of slow relaxation holds if the nuclear Larmor
frequency
S
↔
A
is considerably slower than the relaxation rates of the electronic spin
system S. In this case the calculated Mössbauer spectra for each eigenfunction
ϕ j
need to be weighted with the Boltzmann factor p j and finally summed up:
p j =
exp
−
E j
k B T
K
exp
−
E K
k B T
(4.5)
If, on the contrary, the spin transition rates are considerably larger than the nuclear
larmor frequency the fast relaxation limit applies. In this case the spin expectation
value in Eq. 4.4 is replaced by the thermal average of the 2S + 1 spin expectation
values:
S
f ast
=
j
S
j
exp
−
E j
k B T
K
exp
−
E K
k B T
(4.6)
Now only a single Mössbauer spectrum is being calculated with a hyperfine field
which is the average of the hyperfine fields caused by population of the 2S + 1
electronic sublevels. If the electronic relaxation rate is comparable to the Larmor
frequency complicated Mössbauer spectra with relatively broad lines arise. For the
spectroscopist who is not interested in the determination of electronic relaxation
rates via Mössbauer spectroscopy it is advisable to push the system under study to
the slow relaxation limit by performing Mössbauer experiments around liquid He
temperatures. But it should also be mentioned here that there are suitable theoretical
models to treat the case of intermediate relaxation. Examples on myoglobin have
been treated by Winkler et al. [36]. Also for the case of iron containing magnetic
nanoparticles it might be interesting to determine iron dependent spin relaxation rates
which can be performed by the model of Blume and Tjon [37].
It should be noted that the procedure discussed above is sufficient for the calculation of non-interacting paramagnetic iron sites in a single crystal of a chemical
complex (or a protein) when the unit cell has only one molecule. Only in this
case there exists exactly one orientation of the molecular frames with respect to
the external magnetic field B. However, since the large majority of Mössbauer spectroscopic studies are performed on powder samples or in the case of iron proteins
on frozen solution samples it is necessary to calculate one Mössbauer spectrum for
every molecular orientation. This so called powder averaging is done numerically by