Mössbauer spectroscopy: Basic principles

Published:January 01, 2004
Abstract
Among the various spectroscopic methods which today are applied in geochemistry and mineralogy, Mössbauer spectroscopy plays an important role for mainly two reasons: First, the high resolution and accuracy of the method enables quantitative measurements by the detection of very small energy differences. Second, although the applicability of Mössbauer spectroscopy is limited to a relatively small number of isotopes, the most suitable and common Mössbauer active element, iron, belongs to the five most abundant elements of the earth, and is by far the most abundant transition element. Accordingly, many of the important rockforming or ore minerals contain iron as a main or substitutional ion and much important petrological and geochemical information may be obtained by the study of iron, using the Mössbauer effect. For instance, the oxygen fugacity fO_{2} is a very important parameter in rocks and ore forming processes. Changing Fe^{2+}/Fe^{3+} ratios in Febearing minerals document varying oxygen fugacities during their formation and their subsequent geological history. The Mössbauer effect is particularly well suited to study special properties of transition metals (such as Fe), e.g. changing oxidation and spin states, sitedependent electrical fields, magnetic hyperfine interactions etc. Therefore, most of the Mössbauer studies in geochemistry and mineralogy are made on ^{57}Fe. Similarly, this paper deals mainly with Mössbauer spectroscopy on ^{57}Fe, which is the Mössbauer active Fe isotope with 2.17% natural abundance. However, there are a number of other Mössbauer isotopes, such as ^{119}Sn, ^{121}Sb, ^{197}Au etc., which have been investigated successfully with regard to geochemical as well as crystal chemical applications.
Introduction
Among the various spectroscopic methods which today are applied in geochemistry and mineralogy, Mössbauer spectroscopy plays an important role for mainly two reasons: First, the high resolution and accuracy of the method enables quantitative measurements by the detection of very small energy differences. Second, although the applicability of Mössbauer spectroscopy is limited to a relatively small number of isotopes, the most suitable and common Mössbauer active element, iron, belongs to the five most abundant elements of the earth, and is by far the most abundant transition element. Accordingly, many of the important rockforming or ore minerals contain iron as a main or substitutional ion and much important petrological and geochemical information may be obtained by the study of iron, using the Mössbauer effect. For instance, the oxygen fugacity fO_{2} is a very important parameter in rocks and ore forming processes. Changing Fe^{2+}/Fe^{3+} ratios in Febearing minerals document varying oxygen fugacities during their formation and their subsequent geological history. The Mössbauer effect is particularly well suited to study special properties of transition metals (such as Fe), e.g. changing oxidation and spin states, sitedependent electrical fields, magnetic hyperfine interactions etc. Therefore, most of the Mössbauer studies in geochemistry and mineralogy are made on ^{57}Fe. Similarly, this paper deals mainly with Mössbauer spectroscopy on ^{57}Fe, which is the Mössbauer active Fe isotope with 2.17% natural abundance. However, there are a number of other Mössbauer isotopes, such as ^{119}Sn, ^{121}Sb, ^{197}Au etc., which have been investigated successfully with regard to geochemical as well as crystal chemical applications.
It is impossible to report in a relatively short paper such as this one all the different aspects of Mössbauer spectroscopy. For this purpose, the reader is referred to the more detailed textbooks of Wertheim (1964), Greenwood & Gibb (1971), Bancroft (1973), Gonser (1975), Gütlich et al. (1978), Shenoy & Wagner (1978), Marfunin (1979), Long (1989), Long & Grandjean (1996), etc. More information on current research can be obtained from the Mössbauer Effect Data Center (MEDC), headed by J.G. Stevens in Asheville, N.C., USA. He has collected all publications on Mössbauer spectroscopy and edits the monthly Mössbauer Effect Reference and Data Journal as well as many data sets including one on minerals and related compounds. The main conference on Mössbauer spectroscopy is the International Conference on the Applications of the Mössbauer Effect (ICAME) which is organised every two years and which always has a session on Earth and Planetary Sciences, Mineralogy and Archaeology. For more information on the application of Mössbauer spectroscopy to mineralogy the reader is referred to the following chapter (McCammon, 2004) in this book.
The Mössbauer effect
The recoilless emission and resonant absorption of γ radiation by atomic nuclei in solids is called Mössbauer effect (Mössbauer, 1958). During the emission of a γ quantum by an atomic nucleus with the mass M in a gas or a fluid the γ quantum looses the recoil energy E_{R} which is transmitted to the emitting nucleus:
where M is the mass of the emitting nucleus, c is the velocity of light, and E_{γ} is the recoilfree energy of the γ ray. This equation shows that E_{R} increases with the square of E_{γ} and decreases with the mass M of the emitting atom. During the emission process, the energy of the γ quantum is decreased by E_{R} In order to achieve absorption of the γ quantum by another atomic nuclei in a gas or a liquid, the energy of the γ quantum has to be increased by E_{R} As a consequence, emission and absorption lines are separated by 2E_{R} and recoilless emission as well as resonant absorption of γ radiation by atoms in a gas or in a liquid are not possible, taking the nearly vanishing half width/intensity ratio of 10^{–15}into account (Fig. 1).
If, however, the emitting nucleus is incorporated into a solid crystal, the recoil energy may be annihilated because M → ∞ implies E_{R} → 0, so that recoilfree emission and resonant absorption now become possible. At a given temperature, there is a certain portion f of the emitting nuclei where γ rays are emitted without loss of energy of the γ quantum (for instance by the excitation of lattice vibrations). This portion f is called the recoilfree fraction (DebyeWaller factor, LambMössbauer factor) and is given by:
In these equations is the component of the mean square vibrational amplitude of the emitting nucleus in the direction of the γ ray, λ is the wavelength of the γ quantum, ħ is Planck's constant divided by 2π, and E_{γ}= (2πħc)/λ is the recoilfree energy of the γ quantum. The recoilfree fraction depends on both a nuclear property, i.e. E_{v,} and a crystal property, i.e.The recoilfree fraction f becomes larger with decreasing E_{γ}. For instance, f = 0.91 for the 14.413 keV transition of ^{57}Fe and f = 0.06 for the 129 keV transition of ^{191}Ir (Gütlich et al., 1978). With decreasing f distinctly increases. Because usually decreases at low temperature, f increases with falling temperature; this is one reason why lowtemperature measurements with the application of cryostats are very important in Mössbauer spectroscopy.
Another important property of the Mössbauer effect is the extremely small line width Γ of the γ ray compared to its total energy E_{γ}. This line width is related to the lifetime of the nuclear excited state via the Heisenberg uncertainty principle with respect to energy and time. The uncertainty in energy δE corresponds to the lifetime & of the nuclear state and appears also as the line width γ of the γ ray. The two are related by the equation of Weisskopf & Wigner (1930):
Hence, the mean lifetime ι of the excited nuclear state is directly related to the width Γ of the relevant transition line. As an example, the first excited state of ^{57}Fe has a mean lifetime ι = 1.43 × 10^{–7} s, yielding a line width Γ = 4.55 × 10^{–9} eV, which is exceedingly small compared to the energy of the corresponding γ radiation E_{γ} = 14.413 keV. This small line width enables the detection of small changes of the γ radiation energy on the order of 10^{–12} to 10^{–14} eV, which is in the range of nuclear hyperfine interactions, i.e. interactions between the nuclear moments and the surrounding electric or magnetic fields.
Weisskopf & Wigner (1930) also derived that the spectral line has Lorentzian or BreitWigner form and follows the formula
Nuclear transitions from an excited state (e) to the ground state (g), or vice versa, involve all possible energies within the range of AE. The transition probability or intensity as a function of energy, I(E), therefore yields a spectral line centred around the most probable transition energy E_{γ} (Fig. 2).
Suitable Mössbauer isotopes are those with a relatively low energy of the γ radiation E_{γ} and therefore a high recoil free fraction f, as well as a comparatively long mean lifetime ι of the first excited nuclear level and therefore a small half width Γ The most advantageous combination is found in ^{57}Fe which has been used in more experiments than all the other isotopes shown in Table 1. However, other Mössbauer isotopes, successfully used in geochemistry and mineralogy, are e.g. ^{119}Sn, ^{121}Sb, ^{197}Au etc. (Table 1).
It should also be mentioned that during the last years much progress has been made by using synchrotron radiation for the excitation of lowenergy nuclear resonances, according to the original proposal by Ruby (1974). The socalled nuclear resonant forward scattering (NFS) of synchrotron radiation is a timedifferential technique. Advantages of synchrotron radiation are time structure, coherence, highly collimated intensity (small beam size), polarisation, tuneability of energy etc. The new technique is far more sensitive for hyperfine parameters (isomer shift, quadrupole splitting, magnetic hyperfine interaction) compared to conventional Mössbauer spectroscopy and complements the conventional Mössbauer spectroscopy in the energy domain, but will certainly not replace it. Nuclear scattering of synchrotron radiation also includes nuclear inelastic scattering, which enables to determine phonon spectra by the simultaneous nuclear resonance absorption of a photon and creation or annihilation of one (or more) phonon(s). NFS will open a new field of applications in mineralogy and geochemistry. More details on NFS can be obtained from Gerdau & de Waard (1999).
Experimental
The principal setup of a Mössbauer experiment with ^{57}Fe is shown in Figure 3. A radioactive single line source (^{57}Co/Rh) provides a γ radiation with energy of 14.413 keV by the decay of ^{57}Co to ^{57}Fe and via the transition from the first excited state of the Fe nucleus with spin 3/2 to the ground state of the Fe nucleus with spin 1/2. This γ radiation has in principle exactly the energy needed to excite the ^{57}Fe nuclei in the absorber, to go from the ground state to the first excited state. If resonant absorption takes place, a detector (usually a proportional counter) behind the absorber observes a decrease of the count rate. In general, resonance absorption is not observed because the nuclear levels of the Fe nuclei in the absorber are shifted or even split by the socalled hyperfine interactions. To restore resonance absorption, the source is moved by an electromechanical drive system relative to the absorber. This causes a Doppler shift of the energy of the emitted γ radiation, which has the same order of magnitude as the shifts and splitting of the nuclear levels due to the hyperfine interactions in the absorber. The drive is operated at constant acceleration with a symmetric velocity shape in conjunction with a multichannel analyser (usually 1024 channels) operated at “time mode” so that each channel corresponds to a distinct velocity range. The count rate registered as a function of the source velocity (mm/s) in a multichannel analyser represents a Mössbauer spectrum. The velocity scale is often calibrated with the spectrum of a foil of metallic iron at room temperature. The whole measurement is repeated electromechanically so often that the counts in the channels in the offresonant region (background) are about 1 million per channel. Experiments are usually performed with stationary absorber whereas the moving source is kept at ambient conditions. The absorber can be cooled, heated, kept under high pressure or brought into an external magnetic field.
Evaluation
The setup described above produces two mirrorsymmetric spectra (512 channels each), which usually are folded and then evaluated with computer fitting programs based on leastsquares minimisation methods, generally assuming Lorentzian line shape (Eqn. 5). The fit gives the intensities (areas), positions, and half widths of the resonant absorption lines as exactly as possible. In most of these computer programs, the weighted mean square deviation χ^{2} between the experimental data points and the corresponding theoretical values has to be minimised:
A more sophisticated method evaluates the spectra by diagonalisation of the Hamiltonian for mixed magnetic and electrostatic interaction and fits directly the hyperfine parameters isomer shift δ, quadrupole splitting δE_{Q}and magnetic field H of a subspectrum, as well as the corresponding relative intensities (areas) and half widths (e.g. software MOESALZ by Lottermoser et al., 1993). Another method to analyse spectra is the fitting of quadrupole splitting distributions (QSDs) or hyperfine field distributions (HFDs) using the Voigtbased method of Rancourt & Ping (1991) and the software RECOIL by Lagarec and Rancourt which is distributed by Intelligent Scientific Applications Inc., Canada.
Hyperfine interactions
Hyperfine interactions between the nuclear moments and surrounding electric and/or magnetic fields perturb the nuclear energy levels and are of great importance in Mössbauer spectroscopy, because they provide the main chemical, physical, and crystallographic information. These perturbations can shift the nuclear energy levels as in the case of electric monopole interaction or can split degenerate nuclear levels into sublevels as in the case of electric quadrupole or magnetic dipole interaction. In practice, only these 3 kinds of hyperfine interactions are considered in Mössbauer spectroscopy.
Electric monopole interaction (isomer shift δ)
The electric monopole interaction is an electrostatic interaction between the positively charged nucleus of finite size and the negative electronic charge and causes a shift of the nuclear levels; this is called isomer shift δ_{0} (or IS_{0}), because the effect depends on the difference in the averaged nuclear radii of the ground (g) and isomeric excited (e) states. The term “chemical shift” has also been used.
The energy difference δ E between the electrostatic potential of a point nucleus and a finite one in the electric field of the surrounding electrons is
with Ze being the nuclear charge, –eψ(0)^{2} is the electronic charge density at the nucleus, and R_{i} the radius of the nucleus.
This equation relates the electrostatic energy of the nucleus to its radius, which will in general be different for each nuclear state. Observations, however, are not made on the location of individual nuclear levels but on the γ ray resulting from transitions between two such levels. The energy of the γ ray represents the difference in electrostatic energy of the nucleus in two different states of excitation, which differ in nuclear radius. The expression for the change in energy of the γ ray due to the nuclear electrostatic interaction is therefore the difference of two terms evaluated from Equation 7 for the nucleus in the excited state δE _{e} and the nucleus in the ground state δE_{g}:
The energy shift of Equation 8 is very small when considered in terms of the absolute value of energy of the γ quantum. However, in Mössbauer spectroscopy the nuclear transition energy in a source is compared with that in an absorber, which are different because of different electronic and crystallographic environment in the source and the absorber (Fig. 1). By this method and using a convenient compound as a standard, small differences in the energy of the γ rays can be detected without knowledge of the absolute value of the standard. For example, the energy of the γ ray of ^{57}Fe is 14.413 keV (± 10 eV), but one can measure energy differences as small as 10^{–10} eV.
The isomer shift δ_{0} relative to a standard source can be calculated by taking the difference of Equation 8 written both for the source (E_{S}) and the absorber (E_{A}):
In Equation 9 Ze is the nuclear charge, δR = R_{e}– R_{g}, 2R = R_{e} + R_{g} (R_{e} and R_{g} differ only slightly from each other), is the selectron density at the absorber nuclei, and is the selectron density at the source nuclei (nonrelativistic case). That means, δ_{0} depends on two factors. The first factor contains only nuclear parameters, namely the difference between the radius of the nucleus in the excited (isomeric) state and that of the ground state and is constant for a particular Mössbauer isotope. Thus Equation 9 can also be written:
where the isomer shift calibration constant α = –0.25 ± 0.03 mm/s for ^{57}Fe (Grodzicki et al., 2003). The second factor in Equation 10 depends on the difference in the electronic charge density at the absorber nuclei and the source nuclei, depending on the chemical bonding, valence and spin state of the Mössbauer atom. Using always the same source, the isomer shift depends directly on the electron density at the absorber nuclei. In this case, Equation 10 can also be written as:
Wellseparated values for the isomer shift of Fe^{2+} as well as Fe^{3+} are found in ionic compounds. At first sight, this seems strange because Fe^{2+} with the electronic configuration [Ar]3d^{6} differs from Fe^{3+} with the electronic configuration [Ar]3d^{5} just by a single d electron. According to Equations 10 and 11 only the s electrons contribute to the isomer shift, because they have a finite probability at the nucleus. However, since the probability density of a 3d orbital extends into the core region, a 3d electron may be with a certain probability closer to the nucleus than a 3s electron. Accordingly, adding a d electron in going from Fe^{3+} to Fe^{2+} will reduce the attractive Coulomb potential for the 3s electron. This leads to an expansion of the 3s orbital and thus a reduced electron density at the nucleus. Because the isomer shift calibration constant α is negative for ^{57}Fe, the isomer shift δ_{0} of Fe^{2+} is distinctly larger (about 0.7–0.8 mm/s in oxides) than δ_{0} of Fe^{3+}, provided that the first coordination sphere including the coordination polyhedron remains unchanged. Typical Mössbauer spectra of Fe^{2+} and Fe^{3+} in garnets and babingtonite are displayed in Figures 6 and 7, respectively. The distances between the centroids of the doublets and the zero point of the velocity scale in mm/s correspond to the isomer shift δ = (δ_{0} + δ_{SOD}) of Fe^{2+} and Fe^{3+}. The distinct difference between δ(Fe^{2+}) and δ(Fe^{3+}) can easily be recognised.
In summary, with increasing electron density at the ^{57}Fe nucleus due to increasing valence or increasing covalent bonding character the isomer shift δ_{0} decreases. For instance, in both Fe^{2+} and Fe^{3+} compounds, δ_{0} decreases with decreasing coordination number of iron. This can be referred to increasing selectron density at the Fenucleus due to stronger orbital overlap caused by smaller average cationanion distances in a smaller coordination polyhedron with lower coordination number. However, this is true only for Mössbauer nuclei such as ^{57}Fe, which have a smaller radius in the excited state than in the ground state and consequently δR/R and therefore α are negative. For Mössbauer nuclei with δR/R positive, such as ^{119}Sn, increasing electron density at the nucleus is shown by an increase of the isomer shift.
Isomer shifts of Fe^{2+} and Fe^{3+} in babingtonite, Ca_{2}Fe^{2+}Fe^{3+}Si_{5}0_{14}OH, distinctly decrease with increasing temperature (Fig. 8). This temperature dependence of the isomer shift is mainly caused by the socalled secondorder Doppler shift δ_{SOD}, which adds to the isomer shift δ_{0} caused by the electric monopole interaction (Eqn. 10), to give the total isomer shift δ:
Therefore, the name centre shift is used for δ by some authors. The isomer shift δ is evaluated from the Mössbauer spectra and referred to the isomer shift δ of αiron at room temperature as a standard (δ_{α}_{Fe}). Considering the emitting or absorbing Mössbauer nucleus with mass M vibrating on its lattice site in the crystal, the secondorder Doppler shift δ_{SOD} depends on the force constant K and the mean square displacement of the nucleus according to the formula
Because the mean square displacement increases with temperature, δ_{SOD} increases and therefore the isomer shift δ decreases (Eqns. 12 and 13).
The electric quadrupole interaction (quadrupole splitting ΔE_{Q})
Electric quadrupole interaction takes place between a nuclear quadrupole moment Q and an electric field gradient (EFG) q arising from the surrounding charges. If the nuclear charge distribution deviates from spherical symmetry (nuclear spin I > 1/2), the nucleus has a quadrupole moment. Q is positive for an elongated nucleus (cigar) and negative for a flattened nucleus (disc). The first excited state of the Fe nucleus with spin 3/2 has a positive quadrupole moment (0.16 barn) whereas the ground state with spin 1/2 has no quadrupole moment. Q is constant for a certain Mössbauer isotope and differences in the quadrupole splitting of different compounds under constant experimental conditions can only arise from changes of the EFG.
The EFG is determined by the extranuclear charges (electrons and lattice ions) and is nonzero if the charge distribution around the nucleus deviates from cubic symmetry. The electric field gradient is the second derivative of the potential V at the nucleus of all surrounding electric charges. It is a 3 × 3 secondrank tensor and is given by
are the components of the EFG tensor. Because V_{ij} = V_{ji,} the tensor is symmetric and according to Laplace's equation it is also traceless, i.e.:In principalaxis systems, the offdiagonal elements of the EFG tensor become zero. If the principal axes are chosen in such a way that
the EFG can be described by two independent parameters: (i) the electric field gradient and (ii) the asymmetry parameter where 0 ≤ η ≤ 1 due to Equations 16 and 17.If the Mössbauer isotope is located at a structural position with the site symmetry of a 3 or 4fold symmetry axis, then V_{xx} = V_{yy} and η becomes zero so that the EFG is axially symmetric. For cubic point symmetry, the EFG is zero.
For a nuclear 1/2 → 3/2 transition, as in ^{57}Fe, the quadrupole splitting δEQ (or δ, QS) is:
and in the case of axial symmetry (η = 0)The quadrupole interaction splits the first excited level (I = 3/2) of the Fe nucleus in two sublevels. Hence, two transitions from the ground state to the split excited state are possible and the corresponding Mössbauer spectrum (Fig. 5) exhibits two resonant absorption lines separated by the quadrupole splitting δEQ. The distance between the centroid of both lines and the zero point of the velocity scale corresponds to the isomer shift 8. Both lines should have equal intensities except for the occurrence of an anisotropic recoilfree fraction f in a polycrystalline sample (GoldanskiiKaryagin effect) or in the presence of texture or relaxation.
The quadrupole splitting δE_{Q} is another important parameter for the application of the Mössbauer effect in chemistry, solid state physics and crystallography, because δEQ provides a wealth of information on the electronic and crystallographic structure.
Within the frame of crystalfield theory (CFT), deviations of the EFG from cubic symmetry may be thought to arise from two contributions:
the socalled lattice contribution, q_{lat} or resulting from charges of the surrounding ions,
the socalled valence electron contribution,q_{val} or resulting from an anisotropic electron distribution in the valence shell of the Mössbauer atom.
Therefore, V_{zz} can be expressed by the following formula:
γ_{∞} and R_{0} are the socalled Sternheimer antishielding and shielding factors, respectively, which account for the induced charge polarisation of the electrons in closed shells by and respectively. On the one hand, the electron shell of a Mössbauer atom is distorted by electrostatic interaction with the noncubic charge distribution in its surroundings, which increases This antishielding effect is accounted for by the correction factor (1 – γ_{∞}) with γ_{∞} &≈–9. On the other hand, an anisotropic charge distribution in the valence shell causes a deformation of the inner closed shells, yielding a reduction of . Therefore, has to be corrected by the shielding factor (1 – R_{0}) with R_{0}&≈ 0.10–0.15.Equation 22 is useful for the qualitative interpretation of the quadrupole splitting in Mössbauer spectra. In the case of “highspin” Fe^{3+} with the formal electron configuration [Ar] the five 3d orbitals are occupied each by one electron with the same spin direction. This spin arrangement is almost spherically symmetric and therefore is expected to be very small. However, recent nonempirical electronic structure calculations (Grodzicki et al., 1997; Keutel et al., 1999; Grodzicki & Amthauer, 2000; Grodzicki et al., 2001) have shown that even in cases where should vanish according to CFT the anisotropy of the valence shell yields the dominating contribution to V_{zz}. The most striking example is Fe^{3+} in epidote, Ca_{2}Al_{2}Fe^{3}+Si_{3}O_{12}OH. The reason is that the formally empty spindown orbitals undergo strong covalent interactions with the ligand orbitals and become partially occupied. In the case of highspin Fe^{2+} with the electron configuration [Ar] the single spindown d electron occupies the d orbital with the lowest energy with its spin antiparallel to the five other d electrons. This causes a strong asymmetry of the Fe^{2+} electron distribution around the nucleus and therefore always makes the dominant contribution to the quadrupole splitting, which usually is about an order of magnitude larger than the lattice contribution . This is the main reason why in general a quadrupole doublet assigned to highspin Fe^{2+}exhibits a large quadrupole splitting δE_{Q} (~ 3 mm/s), whereas a quadrupole doublet assigned to highspin Fe^{3+} often exhibits a smaller quadrupole splitting δE_{Q} (~ 0.5 mm/s). Typical Mössbauer spectra of a Fe^{2+}bearing pyrope garnet and a Fe^{3+}bearing andradite garnet are shown in Figure 6. The spectrum of babingtonite, Ca_{2}Fe^{2+}Fe^{3+}Si_{5}O_{14}OH, containing both Fe^{2+}as well as Fe^{3+}, is displayed in Figure 7. From the area ratios of Fe^{2+} and Fe^{3+} doublets, the Fe^{2+}/Fe^{3+} ratio can be determined with very high precision (±1%) if possible different recoilfree fraction f_{i} (i = Fe^{2+} or Fe^{3+}) and saturation effects are considered.
In compounds with octahedrally coordinated lowspin Fe^{2+} with the electron configuration [Ar] as in pyrite, the negative charge distribution around the nucleus is more symmetric and the quadrupole splitting is small; whereas in compounds with lowspin Fe^{3+} with the electron configuration [Ar] the negative charge distribution around the nucleus is more asymmetric and therefore the quadrupole splitting is larger than in highspin Fe^{3+} compounds. In general, does not exceed values of &≈ 0.50 mm/s so that only for small quadrupole splittings the valence and the lattice contribution may be of similar size. Especially, in highspin Fe^{2+} compounds, is about an order of magnitude smaller than and has opposite sign.
The quadrupole splitting of Fe^{2+} at the octahedral positions of the babingtonite structure exhibits a strong decrease with increasing temperature whereas the quadrupole splitting of Fe^{3+} is almost independent of temperature (Fig. 8). This difference in the temperature dependence of ΔE_{Q} for Fe^{2+} (highspin) and Fe^{3+} (highspin) is very characteristic. As mentioned above, the highspin Fe^{3+} cation has a (^{6}A_{1g}) electronic ground state and is spherically symmetric. Lowspin Fe^{2+} behaves similarly as a result of its (^{1}A_{1g}) configuration in the ground state. Therefore, it is unlikely that there will be any lowlying excited levels in either of these configurations that can be thermally occupied, and it is predicted and observed that the EFG in both Fe^{3+} (highspin) and Fe^{2+}(lowspin) compounds will be essentially independent of temperature.
The situation is somewhat more complex for octahedrally coordinated highspin Fe^{2+} and lowspin Fe^{3+}. For an ideal octahedron, the electronic configurations are [Ar](^{5}T_{2g}) and [Ar](^{2}T_{1g}), respectively. In general, however, the coordination polyhedra will be somewhat distorted so that the degeneracy of the t_{2g} levels is partially
or completely removed. Very often the splitting of the t_{2g} levels is of the order of 10–100 meV so that thermal occupation of lowlying excited states is possible. This leads to a more or less pronounced temperature dependence of the quadrupole splitting decreasing with increasing temperature. As an example, the quadrupole splitting of Fe^{2+} at the dodecahedral site of pyrope decreases by 0.54 mm/s with increasing temperature from 80 K to 681 K (Amthauer et al., 1976). This temperature dependence of ΔE_{Q} may in turn be used to estimate the splitting between the ground and excited electronic state although such inferences occasionally may be misleading as recent calculations on chlorite have shown (Lougear et al., 2000).
At this point of the discussion, a short comparison with another Mössbauer isotope, i.e. ^{119}Sn, should be made. The energy of the γ radiation E_{y} = 23.87 keV for ^{119}Sn and E_{y} = 14.413 keV for ^{57}Fe. Therefore, the recoilfree fraction at a given temperature f(T) is distinctly smaller for ^{119}Sn than for ^{57}Fe. This is not such a serious problem because the lower recoilfree fraction of the source and the absorber can be compensated by longer measuring times in order to obtain a ^{119}Sn Mössbauer spectrum with a sufficiently good signal/noise ratio. More difficulties arise from the differences of the mean lifetimes ι of the excited states of both isotopes, which are ι= 17.25 ns for ^{119}Sn and ι= 97.81 ns for ^{57}Fe. The calculated line width of ^{57}Fe (2Γ = 0.194 mm/s) is about three times smaller than the calculated line width of ^{119}Sn (2Γ = 0.6456 mm/s). That is the reason why smaller energy shifts and splittings due to hyperfine interactions can be much better resolved by ^{57}Fe than by ^{119}Sn Mössbauer spectroscopy. This statement can be verified by a comparison of the ^{57}Fe Mössbauer spectrum of an andradite, Ca_{3}Fe^{3+}_{2}Si_{3}O_{12}, in Figure 6 and the ^{119}Sn Mössbauer spectrum of a tinbearing andradite Ca_{3}(Fe^{3+},Sn^{4+})Si_{3}O_{12} in Figure 9. Both ions, Fe^{3+} as well as Sn^{4+}, occupy the distorted octahedra of the garnet structure, which have the point symmetry . However, the expected quadrupole splitting appears only in the quadrupole doublet of octahedral Fe^{3+}in Figure 6 but not in the single line of octahedral Sn^{4+} in Figure 9, which is somewhat broadened due to an unresolved quadrupole splitting.
Magnetic hyperfine interaction (magnetic hyperfine splitting ΔE_{M})
The magnetic hyperfine splitting ΔE_{M} arises from the interaction of the nuclear magnetic dipole moment μ and a magnetic field H at the nucleus: the Zeeman effect. Each nucleus with spin I > 0 has a nonzero magnetic dipole moment μ. There are several effects contributing to the magnetic field at the nucleus. The most important are:
The dominant contribution to the local magnetic field H arises from the socalled Fermi contact interaction. If an ion contains a partially filled valence shell with a different number of spinup and spindown electrons, as e.g. the 3d shell in highspin Fe^{2+} or Fe^{3+}, the s electrons of the ion core will be polarised due to differences in the exchange interaction between electrons with parallel and antiparallel spins. Since electrons with parallel spins repel each other more effectively than those with antiparallel spins, the spinup 3d electrons in the case of Fe^{2+} or Fe^{3+} polarise the core 3s shell in such a way that the probability density of the spinup 3s electron is shifted closer to the nucleus, resulting in a spin density at the nucleus with the same sign as the spin of the valence shell. In turn, the increased density of the spinup 3s electron at the nucleus displaces the spinup 2s electron density near the nucleus resulting in a spin density with a sign opposite to the valence spin due to the spindown 2s electron density. Since the probability density of the 2s electrons at the nucleus is much larger than that of the 3s electrons, the polarisation of the 2s shell dominates the spin density at the nucleus. The resulting Fermi contact interaction field H_{s} has, therefore, a sign opposite to the magnetic field arising from the 3d electrons of the valence shell. In metals the polarised conduction electrons also contribute to the contact interaction with the nucleus.
The orbital motion of valence electrons with the total orbital momentum quantum number L gives rise to a contribution H_{L} to the magnetic field at the nucleus. In metallic iron this term is estimated to be 7 T. H_{L} makes no contribution in the case of Fe^{3+} because L = 0.
A contribution from the socalled spindipolar field H_{D} which arises from the electron spin of the parent atom.
H_{ext} is the value of a magnetic field at the nucleus induced by an external magnetic field (up to 10 T). This contribution is effectively zero away from a large magnet.
The demagnetisation field DM and the Lorentz field (4/3 πM) are very small.
Therefore, a general expression for the magnetic field acting at the nucleus would be:
The magnetic dipole interaction can be described by the Hamiltonianwhere g is the gyromagnetic ratio, μ_{n} is the nuclear magneton and I is the nuclear spin. Diagonalisation of the firstorder perturbation matrix yields the eigenvalues E_{M} of H_{M} as
withThe magnetic dipole interaction splits the nuclear state with the spin quantum number I into 2I +1 equally spaced sublevels, which are characterised by the sign and the magnitude of the nuclear spin quantum number m_{I}. Therefore, the magnetic dipole interaction in ^{57}Fe splits the first excited state with the spin 3/2 into 4 and the ground state with the spin 1/2 into 2 sublevels, schematically displayed in Figure 10. Considering the selection rules for magnetic dipole transitions, i.e. ΔI = 1 and Δm_{I} = 0, ± 1, six transitions are allowed, which are shown in Figure 10. They are labelled from 1 to 6 with increasing energy. The corresponding Mössbauer spectrum exhibits 6 resonant absorption lines (sextet), also shifted from zero velocity by the electric monopole interaction (isomer shift). The relative intensities of these lines in a spectrum of a powder absorber without texture should be 3:2:1:1:2:3 (with increasing energy). The Mössbauer spectrum of metallic iron obtained at room temperature represents such a type of spectrum and is shown in Figure 11. This spectrum is often used to calibrate the velocity scale because the positions of the absorption lines and the local magnetic field H are known with high accuracy.
Combined electric and magnetic hyperfine interactions
The point symmetry of the Fe site in the structure of αFe at ambient P, T conditions is cubic and therefore there is no quadrupole splitting (V_{zz} = 0). The distances between the lines 1 and 2, 2 and 3, 4 and 5, 5 and 6 are equal. However, this situation is very rare. Quite often there is additional quadrupole splitting, because the symmetry of the lattice positions occupied by Fe is lower than cubic and quadrupole interaction is expected (V_{zz} ≠ 0) in addition to the electric monopole and the magnetic dipole interaction. Assuming E_{Q} ≪ E_{M}, the quadrupole interaction causes a perturbation of the magnetic dipole interaction. This situation is often observed in the Mössbauer spectra of Fe^{3+}compounds. Assuming also V_{zz} > 0, i.e. the sign of the EFG is positive, the sublevels of the excited state with m_{I} = 3/2 and m_{I} = –3/2, are shifted by the amount of 1/2AEQ to higher energies, whereas the sublevels of the excited state with m_{I} = 1/2 and m_{I} = –1/2 are shifted by the same amount to lower energies. As a consequence, the sublevels of the excited level are no longer equidistant and the lines 1, 2, 5, and 6 are typically shifted in the way shown in the right part of Figure 10. The direction of the energy shift is reversed if V_{zz} < 0. Therefore, it is possible to determine δE_{Q} as well as the sign of V_{zz} from a magnetically split Mössbauer spectrum of a powdered sample. The Mössbauer spectrum of andradite garnet, , taken at 5 K absorber temperature, shows such a type of spectrum (Fig. 12). If E_{Q} ≫ E_{M}, the situation is more complex.
In general, there is an angle θ between the main component of the electric field gradient V_{zz} and the axis of the magnetic field H as shown in Figure 13. If E_{Q} ≪ E_{M}and the EFG tensor is axially symmetric, then θ can also be determined from the Mössbauer spectra according to the following formula obtained by a firstorder perturbation treatment:
As shown in Figure 13, it is also possible to determine the angles between the crystallographic axes a, b, c and V_{zz} or H if oriented single crystals are measured and the wavevector k is parallel to a certain crystallographic direction (Lottermoser et al., 1996).
In andradite, , Fe^{3+} occupies an octahedral position with site symmetry 3 so that the EFG is axially symmetric. Andradite orders antiferromagnetically below 10 K and therefore the Mössbauer spectrum taken at 5 K absorber temperature exhibits combined magnetic and quadrupole interactions (Fig. 12). The sign of the quadrupole interaction is positive. The angle θ between V_{zz} and H evaluated from the spectrum according to Equation 29 is 55°30', i.e., within the experimental error, in fairly good agreement with the angle of 54°44' between the directions <100> and <111> in a cubic crystal. Andradite is cubic (SG Ia3d) and the 3fold axis and V_{zz} have to be parallel to <111>. Therefore, the direction of the magnetic field H at temperatures below the Néel point is most probably parallel to <100>.
The local magnetic field H at 5 K evaluated from the Mössbauer spectrum of andradite in Figure 12 is 51.7(4) T. This is a typical value for Fe^{3+} in silicates. In general, local magnetic fields of Fe^{3+} in oxidic environments are larger than those of Fe^{2+}. For comparison, the size of the magnetic field of Fe^{2+} in fayalite, Fe_{2}SiO_{4}, at 5 K is 32.28(1) T at the M1 position and 11.7(3) T at the M2 position (Lottermoser et al., 1996).
Conclusions
Mössbauer spectroscopy is a powerful method not only to determine geochemically or petrologically interesting quantities such as the Fe^{2+}/Fe^{3+} ratio in minerals, but also to provide many informations on nuclear physics, mineral physics, and crystal chemistry of minerals and cannot easily be replaced in the near future by other methods. However, for a sophisticated and quantitative interpretation of the spectra, electronic structure calculations based on the density functional theory are needed (Grodzicki, 1985; Grodzicki & Amthauer, 2000). Nuclear forward scattering using synchrotron radiation opens a new field of research in geochemistry and mineralogy.
References
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Contents
Spectroscopic methods in mineralogy
Spectroscopic methods provide information about the local structure of minerals. The methods do not depend on longrange periodicity or crystallinity. The geometric arrangement of atoms in a mineral phase is only one aspect of its constitution. Its vibrational characteristic, electronic structure and magnetic properties are of greatest importance when we consider the behaviour of minerals in dynamic processes. The characterisation of the structural and physicochemical properties of a mineral requires the application of several complementary spectroscopic techniques. However, it is one of the main aims of this School to demonstrate that different spectroscopic methods work on the same basic principles. Spectroscopic techniques represent an extremely rapidly evolving area of mineralogy and many recent research efforts are similar to those in materials science, solid state physics and chemistry. Applications to different materials of geoscientific relevance have expanded by the development of microspectroscopic techniques and by in situ measurements at low to hightemperature and highpressure conditions.