438
M. Reissner
after a time ¯
τ on might be explained by taking into account the influence of entropy
(Fig. 8.62). In zero field the energy of a correlated iron atom is lower than the one of
an atom in the paramagnetic state due to the exchange interaction of the atom with
neighbouring atoms. On the other side the entropy is much lower for an ensemble of
correlated iron atoms than for the same number of atoms in the paramagnetic state.
If the energy difference between the two states is not large, the entropy cannot be
neglected. In an applied field the probability of states being occupied such that the Fe
moment is parallel or almost parallel to B a increases, but all magnetic substates are
allowed in paramagnetic atoms and transitions between different spin directions take
place so fast that only a resulting B ind is visible in the Mössbauer spectra. The energy
of the resulting state is shifted to lower values by B a < μ Fe >, where < μ Fe > is the
thermal average of the Fe magnetic moment of an uncorrelated Fe atom. For correlated Fe atoms there will be an axial anisotropy inherent in the exchange interaction
which permits only parallel or antiparallel orientation of μ cl , like in superparamagnetic systems [166]. The state with direction of the resulting moment μ cl parallel to
B a is shifted by −μ cl B a to lower values, and for antiparallel alignment by +μ cl B a
to higher values. μ cl is the effective magnetic moment of a correlated region. If the
magnetic moment of a correlated region changes its direction from close to parallel to close to antiparallel with respect to B a , the high entropy of the paramagnetic
state would favour its occupation, even if the energy of the correlated state was
comparable to that of the paramagnetic one (Fig. 8.62). In this picture a decay of
correlations is possible. In this model the ratio of the occupation probabilities of the
two states can be written as p on / p of f = ex p(−S/k B + E/k B T + μ cl B a /k B T ),
where it is assumed that μ Fe << μ cl , S is the entropy difference and E is
the energy difference of the two states for an Fe atom. The equilibrium condition p on W on→of f = p of f W of f →on and the definitions of τ on = 1/W on→of f and
τ of f = 1/W of f →on give p on / p of f = τ on /τ of f . This allows to get information about
S, E, and μ cl from the obtained τ on /τ of f values. Fits of field and temperature
dependence of τ on /τ of f are only possible if S, E, and μ cl are assumed to be
field and temperature independent (Fig. 8.63). Although fits are not perfect, results
of values of S, E, and μ cl should be good enough to get estimation of the order
of correct values. The values of S are between 1.3 and 3.2 k B , the ones of E are
between 1 × 10
−21 and 1.7 × 10
−21 J (corresponding to 72 and 123 K) and those of
μ cl are between 6.2 and 8.5 μ B for all samples. Assuming an Fe moment of 1.77 μ B
as obtained for YFe 2 [63] and parallel alignment of the Fe moments in the correlated
regions, gives only three to five Fe atoms per cluster, in good agreement with results
of magnetic measurements.
8.6 Conclusion
The presented examples cover a broad range of different magnetic phenomena, investigated in much detail by application of external magnetic fields. In case of Ga
substituted Co ferrites it was shown that the magnetically determined weak ferro-
M. Reissner
after a time ¯
τ on might be explained by taking into account the influence of entropy
(Fig. 8.62). In zero field the energy of a correlated iron atom is lower than the one of
an atom in the paramagnetic state due to the exchange interaction of the atom with
neighbouring atoms. On the other side the entropy is much lower for an ensemble of
correlated iron atoms than for the same number of atoms in the paramagnetic state.
If the energy difference between the two states is not large, the entropy cannot be
neglected. In an applied field the probability of states being occupied such that the Fe
moment is parallel or almost parallel to B a increases, but all magnetic substates are
allowed in paramagnetic atoms and transitions between different spin directions take
place so fast that only a resulting B ind is visible in the Mössbauer spectra. The energy
of the resulting state is shifted to lower values by B a < μ Fe >, where < μ Fe > is the
thermal average of the Fe magnetic moment of an uncorrelated Fe atom. For correlated Fe atoms there will be an axial anisotropy inherent in the exchange interaction
which permits only parallel or antiparallel orientation of μ cl , like in superparamagnetic systems [166]. The state with direction of the resulting moment μ cl parallel to
B a is shifted by −μ cl B a to lower values, and for antiparallel alignment by +μ cl B a
to higher values. μ cl is the effective magnetic moment of a correlated region. If the
magnetic moment of a correlated region changes its direction from close to parallel to close to antiparallel with respect to B a , the high entropy of the paramagnetic
state would favour its occupation, even if the energy of the correlated state was
comparable to that of the paramagnetic one (Fig. 8.62). In this picture a decay of
correlations is possible. In this model the ratio of the occupation probabilities of the
two states can be written as p on / p of f = ex p(−S/k B + E/k B T + μ cl B a /k B T ),
where it is assumed that μ Fe << μ cl , S is the entropy difference and E is
the energy difference of the two states for an Fe atom. The equilibrium condition p on W on→of f = p of f W of f →on and the definitions of τ on = 1/W on→of f and
τ of f = 1/W of f →on give p on / p of f = τ on /τ of f . This allows to get information about
S, E, and μ cl from the obtained τ on /τ of f values. Fits of field and temperature
dependence of τ on /τ of f are only possible if S, E, and μ cl are assumed to be
field and temperature independent (Fig. 8.63). Although fits are not perfect, results
of values of S, E, and μ cl should be good enough to get estimation of the order
of correct values. The values of S are between 1.3 and 3.2 k B , the ones of E are
between 1 × 10
−21 and 1.7 × 10
−21 J (corresponding to 72 and 123 K) and those of
μ cl are between 6.2 and 8.5 μ B for all samples. Assuming an Fe moment of 1.77 μ B
as obtained for YFe 2 [63] and parallel alignment of the Fe moments in the correlated
regions, gives only three to five Fe atoms per cluster, in good agreement with results
of magnetic measurements.
8.6 Conclusion
The presented examples cover a broad range of different magnetic phenomena, investigated in much detail by application of external magnetic fields. In case of Ga
substituted Co ferrites it was shown that the magnetically determined weak ferro-