8 Mössbauer Spectroscopy in External Magnetic Fields
383
with H ext an external field, H DM the demagnetizing field, and H L the Lorentz field.
The Fermi contact field H c is due to the Fermi contact interaction of the nuclear
moment with the spin density at the nucleus [12]. This density comes partly from
an unbalanced spin density of s-electrons at the nucleus and partly from conduction
electrons [13, 14]. The latter one comes from polarization produced by exchange
interactions with the 3d electrons and by admixture with the 3d band. For ionic iron
compounds H c is large and negative. With increasing covalency H c decreases in
ionic compounds [15]. The orbital field H O arises from unquenched orbital angular
momentum of the parent atom for high spin ferric compounds. It is zero, because
ferric iron is an S-state ion (
6 S). In ferrous iron this term can be large and of opposite
sign to the Fermi contact field H c . The dipole field H d is caused by the arrangement
of the atomic moments in the vicinity of the Mössbauer nucleus. For most iron compounds this term is smaller than the Fermi contact and the orbital term. Whereas the
Fermi field can be assumed to be independent of the crystal symmetry, because of
the polarization of the inner s-shells, the orbital field and the dipole field are strongly
symmetry dependent. The demagnetizing field H DM = −DM reduces the hyperfine
field. It depends on the demagnetization factor D, which depends on the shape of
the sample and the magnetization M. In contrast to measurements on bulk microcrystalline materials, where H DM is negligible small, because of the multidomain
structure, it can become dominant in case of monodomain nanoparticles, especially
if the hyperfine field is small [16]. The usual Lorentz field H L = 4π/3 for cubic symmetry has to be modified by small residue H L for noncubic symmetry. Because of
the quenched orbital moment in high-spin ferric compounds H O = 0, the hyperfine
field at low temperatures is always negative and rather large. In ferrous compounds
where H O can be large, the sign of the observed hyperfine field can be positive or
negative. Best way to determine the sign is to apply a large external magnetic field (2
T − 5 T) and to observe the field dependence of the measured hyperfine field. In that
way Hanna et al. [1] have shown for the first time that in α-Fe the hyperfine field is
negative rather than positive, although at that time theory predicted a positive sign.
Detailed description of the nature of the hyperfine field can be found in [8, 12, 17,
18].
By the nuclear Zeeman effect the degeneration of the ground and exited states are
lifted and the levels split into 2I + 1 energetically different levels. In case of
57 Fe
(Fig. 8.1) the exited state (I = 3/2) splits into four and the ground state (I = 1/2)
into two levels. This gives eight different transition energies, from which two are
forbidden because of the selection rules for magnetic dipole transitions: I = ±1
and m = 0 or ±1. As a result the Mössbauer line splits into a sextet (Fig. 8.2). The
spacing of the outermost lines is the so-called hyperfine field B h f . From the intensity
ratio of the six lines information about the orientation of the magnetic field at the
nucleus can be obtained, because the intensity of the six lines depend on the angle θ m
between direction of the magnetic field at the nucleus and the γ -ray direction. For the
±3/2 → ±1/2 transitions intensity is given by
3
4
(1 + cos
2
θ m ), for ±
1
2
→ ±
1
2
it is
given by sin
2
θ m , and for ∓
1
2
→ ±
1
2
it is given by
1
4
(1 + cos
2
θ m ). According to these
equations the second and fifth line vanish, if the field at the nucleus is parallel to the
383
with H ext an external field, H DM the demagnetizing field, and H L the Lorentz field.
The Fermi contact field H c is due to the Fermi contact interaction of the nuclear
moment with the spin density at the nucleus [12]. This density comes partly from
an unbalanced spin density of s-electrons at the nucleus and partly from conduction
electrons [13, 14]. The latter one comes from polarization produced by exchange
interactions with the 3d electrons and by admixture with the 3d band. For ionic iron
compounds H c is large and negative. With increasing covalency H c decreases in
ionic compounds [15]. The orbital field H O arises from unquenched orbital angular
momentum of the parent atom for high spin ferric compounds. It is zero, because
ferric iron is an S-state ion (
6 S). In ferrous iron this term can be large and of opposite
sign to the Fermi contact field H c . The dipole field H d is caused by the arrangement
of the atomic moments in the vicinity of the Mössbauer nucleus. For most iron compounds this term is smaller than the Fermi contact and the orbital term. Whereas the
Fermi field can be assumed to be independent of the crystal symmetry, because of
the polarization of the inner s-shells, the orbital field and the dipole field are strongly
symmetry dependent. The demagnetizing field H DM = −DM reduces the hyperfine
field. It depends on the demagnetization factor D, which depends on the shape of
the sample and the magnetization M. In contrast to measurements on bulk microcrystalline materials, where H DM is negligible small, because of the multidomain
structure, it can become dominant in case of monodomain nanoparticles, especially
if the hyperfine field is small [16]. The usual Lorentz field H L = 4π/3 for cubic symmetry has to be modified by small residue H L for noncubic symmetry. Because of
the quenched orbital moment in high-spin ferric compounds H O = 0, the hyperfine
field at low temperatures is always negative and rather large. In ferrous compounds
where H O can be large, the sign of the observed hyperfine field can be positive or
negative. Best way to determine the sign is to apply a large external magnetic field (2
T − 5 T) and to observe the field dependence of the measured hyperfine field. In that
way Hanna et al. [1] have shown for the first time that in α-Fe the hyperfine field is
negative rather than positive, although at that time theory predicted a positive sign.
Detailed description of the nature of the hyperfine field can be found in [8, 12, 17,
18].
By the nuclear Zeeman effect the degeneration of the ground and exited states are
lifted and the levels split into 2I + 1 energetically different levels. In case of
57 Fe
(Fig. 8.1) the exited state (I = 3/2) splits into four and the ground state (I = 1/2)
into two levels. This gives eight different transition energies, from which two are
forbidden because of the selection rules for magnetic dipole transitions: I = ±1
and m = 0 or ±1. As a result the Mössbauer line splits into a sextet (Fig. 8.2). The
spacing of the outermost lines is the so-called hyperfine field B h f . From the intensity
ratio of the six lines information about the orientation of the magnetic field at the
nucleus can be obtained, because the intensity of the six lines depend on the angle θ m
between direction of the magnetic field at the nucleus and the γ -ray direction. For the
±3/2 → ±1/2 transitions intensity is given by
3
4
(1 + cos
2
θ m ), for ±
1
2
→ ±
1
2
it is
given by sin
2
θ m , and for ∓
1
2
→ ±
1
2
it is given by
1
4
(1 + cos
2
θ m ). According to these
equations the second and fifth line vanish, if the field at the nucleus is parallel to the