2 Mössbauer Spectra of Samples with Slow Paramagnetic Relaxation
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6.2 Moăssbauer Spectra of Samples with Slow Paramagnetic Relaxation
and
b
b X
2Jij S~j ;
H^ex ¼ ÀS~i Á
203
(6.5)
j6¼i
where m0 is the vacuum permeability, ~
rij is the vector connecting the ions i and j,
and Jij is the exchange coupling constant.
Finally, the spin Hamiltonian also contains contributions from the magnetic and
quadrupole hyperfine interactions, H^hf and H^Q where
b
b
H^hf ¼ S~Á Ae Á ~
I
(6.6)
and
H^Q ¼
i
eQVzz h ^2 1
2
2
I z 3II ỵ 1ị ỵ 13I^x I^y Þ
4Ið2I À 1Þ
(6.7)
where I is the nuclear spin, A~ is a hyperfine interaction tensor (which for Fe3+ in
the high-spin state often can be approximated by a scalar), Q is the nuclear
quadrupole moment, Vzz is the z component of the electric field gradient (EFG),
and is the asymmetry parameter.
The crystal field interaction gives rise to an energy splitting into a number
of Kramers doublets. In the case of high-spin Fe3+ with spin S ¼ 5=2, there are
three Kramers doublets, each of which give rise to separate contributions in the
Moăssbauer spectra of samples with slow paramagnetic relaxation. For l ¼ 0 and
a ¼ 0, they can be labeled jỈ1=2i, jỈ3=2i and jỈ5=2i.
In the interpretation of the Moăssbauer spectra of samples with slow paramagnetic relaxation, it is important to realize that the eigenstates of the electronic
system depend critically on the perturbations that lift the degeneracy of the
Kramers doublets and on the relative size of the crystal field splitting and
the Zeeman splitting. This can be illustrated by comparing spectra obtained in
different applied magnetic fields. Figure 6.1 shows the magnetic field dependence of
Moăssbauer spectra at 4.5 K of Fe3+ ions with very slow relaxation [12]. The sample
was a frozen aqueous solution with 0.03 M Fe(NO3)3 and an appropriate amount
of ionic glass-formers, such that it was possible for the sample to be frozen to
an amorphous state with a homogeneous distribution of [Fe(H2O)6]3+ complexes
[13, 14]. This is essential to ensure slow spin–spin relaxation (see Sect. 4.2),
because the solubility of Fe3+ in crystalline ice is very low, and if crystallization
of ice takes place during freezing, the iron ions will be concentrated in the regions
between the ice crystals and this gives rise to fast spin–spin relaxation [13]. The
magnetic fields indicated in Fig. 6.1 were applied perpendicular to the g-ray
direction. At the smallest fields, H^Z , H^dd and H^ex are negligible, and the hyperfine
interaction alone lifts the degeneracy of the Kramers doublets. The spectra,
obtained in applied fields of 0.0001 and 0.0020 T, essentially consist of an asymmetric component superimposed on two almost identical symmetric sextets. The
asymmetric component is due to one of the Kramers doublets for which the
204
6 Magnetic Relaxation Phenomena
0
0.050 T
4
8
0
0
0.0001 T
8
0
8
0
0.150 T
4
0.0020 T
4
Absorption (%)
Absorption (%)
0.100 T
4
4
8
8
0
0.225 T
4
8
0
0.0125 T
4
0
8
0.620 T
4
0
0.0250 T
4
8
8
–16 –12 –8
–4
0
Velocity (mm
4
s–1)
8
12
16
–16 –12 –8 –4
0
4
Velocity (mm
8
12 16
s–1)
Fig. 6.1 Moăssbauer spectra of an amorphous frozen aqueous solution of 0.03 M Fe(NO3)3,
obtained at 4.5 K with various applied transverse magnetic fields. The bar diagrams indicate
theoretical line positions of the spectral components. The lines are fits to the experimental data.
(Reprinted with permission from [12]; copyright 1977 by Elsevier)
hyperfine interaction cannot be described in terms of an effective magnetic hyperfine field, because the eigenstates are combined electronic and nuclear states [1, 9,
12]. At applied magnetic fields of 0.0125 and 0.0250 T, the spectra still show the
presence of the two nearly identical sextets, but the asymmetric component has
transformed to a symmetric sextet with a relatively small splitting. This is because
the Zeeman splitting with these applied fields is large compared to the hyperfine
interaction such that the degeneracy of the Kramers doublets is lifted by the Zeeman
interaction. With increasing applied fields, one can follow the evolution of the
spectra when the Zeeman energy becomes comparable to, or larger than, the
crystal field splitting. From fits of the whole series of spectra in Fig. 6.1, it was
possible to estimate the crystal field parameters D ﬃ 0.20 cmÀ1, l ﬃ 0.2, and
a ﬃ 0.017 cmÀ1 [12].
When the Zeeman energy is large compared to the crystal field interaction,
the electronic wave functions are approximately the jSz i states with energy
6.3 Moăssbauer Relaxation Spectra
205
E ﬃ gmB BSz . In this case, the magnetic hyperfine field is given by Bhf ﬃ a0 Sz ,
where a0 is a constant, which for Fe3+ in the high-spin state is on the order of
20 T. The magnetic splitting of the Moăssbauer spectrum is then proportional to the
vector sum of the magnetic hyperfine field and the applied magnetic field. At
low temperatures, the populations of the electronic states differ, especially if
large magnetic fields are applied, and this is reflected in the relative areas of the
sextets [15].
Studies of the field dependence of Moăssbauer spectra from other dilute frozen
solutions with more than one type of complex have demonstrated that Moăssbauer
spectroscopy not only allows the determination of the crystal field parameters but
can also give additional information about the relative amounts of different complexes such as [Fe(H2O)6ÀnCln]3Àn in frozen aqueous solutions containing Fe3+ and
different concentrations of ClÀ [14]. It has been found that the lower symmetry of
complexes with different ligands leads to increased crystal field splittings. The
technique has also been used to study crystal field interaction in biological samples
[5], and in a number of inorganic compounds with low iron concentration.
Recently, it has been demonstrated that nuclear forward scattering of synchrotron
radiation can also be used for studies of crystal field interactions in samples with
slow paramagnetic relaxation [16].
6.3
Moăssbauer Relaxation Spectra
The Moăssbauer line shape in the presence of magnetic relaxation has been the
subject of many theoretical studies [9, 1726]. Moăssbauer spectra are very sensitive
to relaxation effects when the relaxation time is of the same order of magnitude
as the nuclear Larmor precession time in the magnetic hyperfine field. In 57Fe
Moăssbauer spectroscopy studies of Fe3+ with magnetic hyperfine fields on the order
of 45–60 T, the Larmor precession time is on the order of nanoseconds. The detailed
spectral shape depends on the way in which the hyperfine field fluctuates. In the
simple case of longitudinal relaxation with a magnetic hyperfine field that can
~hf , the spectral shape can be calculated
~hf and B
assume only the two values ỵ B
with relatively simple models [9, 22]. Theoretical spectra calculated by Wickman
et al. [9] are shown in Fig. 6.2. For relaxation times t ) tM , the spectra consist of
sextets with narrow lines. The lines start to broaden for relaxation times on the
order of the mean life time of the excited nuclear state ($140 ns for 57Fe). With
decreasing relaxation time, the lines become further broadened and later they
collapse pair-wise, first lines 3 and 4, then lines 2 and 5, and finally lines 1 and 6.
The magnetic splitting of a pair of lines collapses at a critical relaxation time
tcr ðme ; mg Þ [22]
tcr ðme ; mg ị ẳ
h
;
ge me gg mg ịmN Bhf
(6.8)
206
6 Magnetic Relaxation Phenomena
τ = 1 × 10–8
I1
I2
I3
τ = 1 × 10–12
Itot
Intensity
τ = 2.5 × 10–8
–0.8
–0.4
0
τ = 1 × 10–10
τ = 7.5 × 10–8
τ = 1 × 10–9
τ = 1 × 10–6
τ = 5 × 10–9
0.4
0.8
–0.8
–0.4
0
0.4
0.8
Velocity (cm s–1)
Fig. 6.2 Theoretical 57Fe Moăssbauer relaxation spectra for longitudinal relaxation with the
indicated relaxation times and with a hyperfine field that can assume the values Ỉ55 T. The
symmetry direction of the axially symmetric EFG is assumed parallel to the magnetic hyperfine
field. (Reprinted with permission from [9]; copyright 1966 by the American Physical Society)
where g and m are the nuclear g-factors and the z components of the nuclear spins,
respectively, and the subscripts e and g refer to the excited and the ground states.
mN is the nuclear magneton. For Bhf ¼ 55 T, lines 1 and 6, lines 2 and 5, and lines 3
and 4 collapse for tcr ¼ 1:6 ns, tcr ¼ 2:7 ns and tcr ¼ 9:8 ns, respectively. Thus,
there is not a uniquely defined critical relaxation time for which the magnetic
splitting of the whole spectrum collapses.
For relaxation times t ≲ 1 ns, the spectra can be described as three Lorentzian
lines with different line width, and for relaxation times around 10À10 s, the spectra
appear as asymmetric doublets with line widths that decrease with decreasing
relaxation time. In the theoretical spectra in Fig. 6.2, the EFG was assumed uniaxial
6.3 Moăssbauer Relaxation Spectra
207
with the symmetry direction parallel to the magnetic hyperfine field. In this case, the
line corresponding to the Ỉ 1=2 ! Ỉ3=2 transition is broader than the line
corresponding to the Ỉ 1=2 ! Ỉ1=2 and the ặ 1=2 ! ầ1=2 transitions. If the
symmetry direction of the EFG is perpendicular to the fluctuating magnetic hyperfine field, the asymmetry of the doublet spectra is opposite [22].
The simple model with a hyperfine field that fluctuates between only two values
~hf may be a fair approximation in some studies of superparamag~hf and B
ỵB
netic relaxation in magnetic nanoparticles (see Sect. 6.5) but in the case of paramagnetic relaxation, the magnetic hyperfine field can usually assume several
different values. As discussed in Sect. 6.2, Fe3+ in the high-spin state with spin
S ¼ 5=2 has six electronic states with different magnetic hyperfine interactions. If a
large magnetic field is applied, such that H^Z is predominant in the spin Hamiltonian,
the electronic states are the jSz i states, and the relaxation is still longitudinal, but the
magnetic hyperfine field can fluctuate between the six different values with
Bhf ¼ a0 Sz . In other cases, the hyperfine field may fluctuate in different directions
and this makes calculations of relaxation spectra more complex [1, 23–26]. The
evolution of the spectra as a function of relaxation time is qualitatively similar to
that of the longitudinal case, but the detailed line shape of the spectra depends on
the way in which the magnetic hyperfine field fluctuates.
In many paramagnetic samples, such as ferric compounds with moderate dipole
and exchange interaction between the Fe3+ ions, the paramagnetic relaxation results
in spectra consisting of singlets or doublets with broadened lines [2, 3] due to
relaxation times in the range 10À11–10À9 s. Doublet spectra are usually asymmetrically broadened, and the asymmetry depends on the way in which the hyperfine
field fluctuates relative to the EFG (e.g. longitudinal, isotropic, or isotropic transverse [26]). If the sample is exposed to a magnetic field, the directions of the
fluctuating hyperfine fields are changed. Therefore, magnetic fields can have a
significant influence on the asymmetry of the spectra [2, 3]. In FeCl3·6H2O,
the Fe3+ ions are surrounded by four water molecules and two ClÀ ions in the
trans positions. This leads to a nearly axial symmetry around the Fe3+ ions. The
Moăssbauer spectrum of polycrystalline FeCl3·6H2O at 78 K in a small field
(Fig. 6.3, upper spectrum) is an asymmetric doublet, because the hyperfine field
fluctuates longitudinally along the symmetry direction of the EFG. When a field of
1.3 T is applied, the hyperfine field fluctuates in directions that are determined by
the combined effect of the crystal field and the applied magnetic field. The angles
between the hyperfine fields and the symmetry direction of the EFG are then more
random and this gives rise to a more symmetric spectrum (Fig. 6.3, lower spectrum).
In samples with negligible quadrupole interaction, isotropic relaxation with
relaxation times around 10À11–10À9 s results in spectra consisting of a single,
broad Lorentzian line [2, 3, 21, 26], but if the relaxation is longitudinal, the
spectrum consists of a superposition of three singlets with line widths (FWHM)
[2, 3]
G ẳ G0 ỵ ge me gg mg ị2 m2N hÀ1
tl ;
(6.9)
208
6 Magnetic Relaxation Phenomena
0
0.001 T
ABSORPTION (%)
5
10
0
1.3 T
5
10
–6
–4
–2
0
2
4
6
Velocity (mm s1)
Fig. 6.3 Moăssbauer spectra of FeCl3ã6H2O at 78 K in applied magnetic fields of 0.001 and 1.3 T
a
0
2
ABSORPTION (%)
4
6
8
0
b
1
2
3
4
5
–10
–5
0
+5
+10
Velocity (mm s1)
Fig. 6.4 Moăssbauer spectra of NH4Fe(SO4)2ã12H2O at 113 K. (a) In zero applied magnetic field;
(b) In a transverse magnetic field of 0.8 T
6.3 Moăssbauer Relaxation Spectra
209
where G0 is the line width when the relaxation broadening is negligible and tl
is the spin correlation time for longitudinal relaxation. Figure 6.4 shows the
experimental Moăssbauer spectra of ferric alum (NH4Fe(SO4)2 12H2O), which
illustrate this [2, 3]. The spectrum (a) was obtained in zero applied magnetic field
and is well fitted with a single Lorentzian line indicating isotropic relaxation. This
is in accordance with the very small crystal field splitting in ferric alum resulting in
a spin Hamiltonian in which the dipole interaction is predominant [2, 3]. When a
magnetic field of 0.8 T is applied (Fig. 6.4b), the predominant term in the spin
Hamiltonian is the Zeeman term (6.3), and the hyperfine field then fluctuates
parallel and antiparallel to the applied field. As shown in Fig. 6.4, the spectrum
can be fitted with broad and narrow components with line widths given by (6.9). By
use of a polarized source of 57Co in metallic iron, it is possible to separate the three
components with different line widths and such measurements have confirmed the
interpretation [27]. Similar data have been obtained for Fe(ClO4)3·6H2O and cubic
(NH4)3FeF6 [28].
The aforementioned examples illustrate that singlet and doublet spectra with line
broadening due to relaxation effects may be very sensitive to applied magnetic
fields. Furthermore, as discussed in Sect. 6.4, the relaxation time may also be
influenced by magnetic fields. Therefore, application of magnetic fields can be
very useful to distinguish between line broadening due to relaxation effects and line
broadening due to, for example, distributions of isomer shifts and quadrupole
splittings.
If large magnetic fields are applied to a paramagnetic sample at low temperatures, the induced magnetization is proportional to the appropriate Brillouin
function. The magnetic field at the nucleus can then be described as a sum
~hf (which is proportional to the Brillouin
of the average hyperfine field B
~
~f ðtÞ.
function), the applied field B, and a fluctuating part of the hyperfine field, B
À11
À9
If the paramagnetic relaxation time is on the order of 10 –10 s, the spectra
will show a line broadening that is proportional to the relaxation time and to
hB2f ðtÞi, which decreases with increasing applied fields [19, 29]. The broadening
differs for the nuclear transitions, such that lines 1 and 6 will be most broadened
and lines 3 and 4 least broadened. Furthermore, the line positions will be slightly
shifted relative to the line positions in spectra of samples with infinitely fast
relaxation because of relaxation effects [19, 29]. For example, Fig. 6.5 shows
Moăssbauer spectra of ferric alum obtained at 4.2 K in the indicated magnetic
fields applied parallel to the g-ray direction [29]. Lines 2 and 5 have negligible
intensity because the magnetic field at the nuclei is parallel to the g-ray direction.
The lines are fits to a relaxation model and the bar diagrams indicate the expected
line position in the case of infinitely fast relaxation [29].
For very short magnetic relaxation times, t ≲ 10À12 s, the line broadening due
to relaxation is negligible, and the magnetic splitting of the spectra is proportional
to the average value of the magnetic field at the nucleus, i.e. it vanishes when the
average magnetic field at the nucleus is zero.
Relaxation phenomena can also be studied by nuclear forward scattering of
synchrotron radiation [16, 30]. This is discussed in Chap. 9.
210
6 Magnetic Relaxation Phenomena
Fig. 6.5 Moăssbauer spectra of NH4Fe(SO4)2ã12H2O at 4.2 K and with the indicated magnetic
fields applied parallel to the g-ray direction. The lines indicate fits in accordance with a theoretical
relaxation model [19, 29]. The bar diagrams indicate the theoretical line positions in the case of
infinitely fast relaxation. (Adapted from [29]; copyright 1973 by Springer-Verlag)
6.4
Paramagnetic Relaxation Processes
In paramagnetic materials, the relaxation frequency is in general determined by
contributions from both spin–lattice relaxation and spin–spin relaxation. Spin–
lattice relaxation processes can conveniently be studied in samples with low concentrations of paramagnetic ions because this results in slow spin–spin relaxation.
Spin–spin relaxation processes can be investigated at low temperatures where
the spin–lattice relaxation is negligible. Paramagnetic relaxation processes have
6.4 Paramagnetic Relaxation Processes
211
conventionally been studied by AC susceptibility measurements with time scale
typically in the range from $1 to $107 s. Because Moăssbauer spectroscopy has a
time scale on the order of nanoseconds, this technique makes it possible to study
spin–spin and spin–lattice relaxation processes that cannot be studied by AC
susceptibility measurements. The following illustrates how Moăssbauer spectroscopy can give information about spin–lattice and spin–spin relaxation processes.
6.4.1
Spin–Lattice Relaxation
As discussed in Sect. 6.2, the electronic states of a paramagnetic ion are determined
by the spin Hamiltonian, (6.1). At finite temperatures, the crystal field is modulated
because of thermal oscillations of the ligands. This results in spin–lattice relaxation,
i.e. transitions between the electronic eigenstates induced by interactions between the
ionic spin and the phonons [10, 11, 31, 32]. The spin–lattice relaxation frequency
increases with increasing temperature because of the temperature dependence of the
population of the phonon states. For high-spin Fe3+, the coupling between the spin
and the lattice is weak because of the spherical symmetry of the 6S ground state. This
leads to small crystal field splittings and relatively long spin–lattice relaxation times,
tsl , which are often in the range where they can be studied by Moăssbauer spectroscopy, even at room temperature. For nonspherical ions, like Fe2+, the coupling to the
lattice is stronger resulting in short spin–lattice relaxation times and the Moăssbauer
spectra may be influenced by relaxation effects only at low temperatures.
Several types of spin–lattice relaxation processes have been described in the
literature [31]. Here a brief overview of some of the most important ones is given.
The simplest spin–lattice process is the direct process in which a spin transition is
accompanied by the creation or annihilation of a single phonon such that the
electronic spin transition energy, D, is exchanged by the phonon energy, hoq .
Using the Debye model for the phonon spectrum, one finds for kB T ) D that
2
tÀ1
sl / D T:
(6.10)
If the Zeeman splitting is large compared to the crystal field splitting, this leads
2
to tÀ1
sl / B T. Usually, the direct process is important only compared to other
spin–lattice processes at low temperatures, because only low-energy phonons with
hoq ¼ D contribute to the direct process.
At higher temperatures, the two-phonon (Raman) processes may be predominant. In such a process, a phonon with energy
hoq is annihilated and a phonon with
hoq À hor is taken up in a transition
energy
hor is created. The energy difference
of the electronic spin. In the Debye approximation for the phonon spectrum, this
gives rise to a relaxation rate given by
Z oD 6
o exp
ho=kB Tịdo
/
;
(6.11)
t1
sl
2
exp
h
o=k
ẵ
0
B Tị 1
212
6 Magnetic Relaxation Phenomena
7
À1
2
resulting in tÀ1
sl / T for T ( yD and tsl / T for T ) yD , where yD is the Debye
hịyD .
temperature and oD ẳ kB =
If optical phonons are responsible for the Raman processes, the Einstein model
for the phonon spectrum is more appropriate. In this case, one finds
t1
sl ẳ
expyE =Tị
ẵ1 expyE =Tị2
;
(6.12)
where yE is the Einstein temperature. In large applied magnetic fields, no field
dependence of the Raman processes is expected. For T ( yE , one finds that
À1
2
tÀ1
sl / expðÀyE =TÞ and for T ≳ yE, tsl / T .
The Orbach process is a two-phonon process that takes place via population of
an excited electronic state with energy E0 . The temperature dependence of the
relaxation rate is given by
tÀ1
sl /
1
:
expðE0 =kB TÞ À 1
(6.13)
As an example of a Moăssbauer study of spinlattice relaxation, Fig. 6.6 shows
the spectra of Fe3+ in the ammonium alum, NH4(57Fe0.02Al0.98)(SO4)2·12H2O [32].
The crystal field splitting of Fe3+ in this compound is small (D % 0.024 cmÀ1). The
spectra were obtained at the indicated temperatures in a magnetic field of 1.23 T
applied perpendicular to the g-ray direction. This field is sufficiently large to ensure
that the Zeeman energy is much larger than the crystal field energy such that the
electronic states are the jSz i states. Because of the low concentration of Fe3+ ions,
spin–spin relaxation is almost negligible and the relaxation is therefore dominated
by spin–lattice relaxation. At the lowest temperatures, the spectra are magnetically
split because of slow relaxation but, as the temperature increases, the spectra
gradually collapse to a broadened singlet. The solid lines are fits to a model for
longitudinal relaxation and the spin–lattice relaxation frequencies, shown in
Fig. 6.7, were estimated from the fits. The temperature dependence of the relaxation
frequency was in accordance neither with the Debye model for direct processes nor
with the Debye model for Raman processes [32]. A fit based on the Einstein model
for Raman processes (6.12) with yE ¼ 450 K is shown by the broken curve in
Fig. 6.7. The model fits the data well at low and high temperatures but, at intermediate temperatures, there are significant deviations from the model. Moreover, it
was found that in this temperature range, the relaxation time depends on the
strength of the applied magnetic field in contrast to the expectation for a Raman
process.
Ammonium alums undergo phase transitions at Tc % 80 K. The phase transitions
result in critical lattice fluctuations which are very slow close to Tc . The contribution to the relaxation frequency, shown by the dotted line in Fig. 6.7, was
calculated using a model for direct spin–lattice relaxation processes due to interaction between the low-energy critical phonon modes and electronic spins.
6.4 Paramagnetic Relaxation Processes
213
0
1
79 K
158 K
88 K
169 K
98 K
199 K
108 K
215 K
126 K
229 K
139 K
247 K
150 K
295 K
2
3
0
1
2
3
0
1
2
Absorption (%)
3
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
4
12
8
4
0
4
8
12
12
8
4
0
4
8
12
Velocity (mm s1)
Fig. 6.6 Moăssbauer spectra of NH4(57Fe0.02Al0.98)(SO4)2ã12H2O obtained at the indicated temperatures in a transverse magnetic field of 1.23 T. The full lines are fits to a model for longitudinal
relaxation. (Reprinted with permission from [32]; copyright 1979 by the Institute of Physics)
This model also explains the unexpected magnetic field dependence of the relaxation time in a temperature range in which Raman processes are normally expected
to be predominant [32].