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3.8.4 The Mössbauer effect
The Mössbauer effect refers to the recoil-free emission of a γ-ray from the excited state of a nucleus and the resonant absorption of the γ-ray by the ground state of an identical nucleus. Whilst in principle any nucleus could exhibit these effects, in practice the number of Mössbauer nuclides is limited by practical considerations arising from the physical properties of the specific nucleus, its progenitor or parent nucleus and the nature of the solid state. Isolated nuclei will not exhibit the Mössbauer effect because their motions cause Doppler shifts in the energies of the γ-ray and the peak resonant energy producing a mis-match in these energies. In a solid the emission and absorption may be accompanied by emission or absorption of phonons again causing a mis-match in energies.
To obtain a large recoil-free fraction, f, the Mössbauer nuclide must be locked into a crystal or metallic matrix such that momentum recoil on emission or absorption of a γ-ray is taken up by the matrix as a whole, i.e. there is no accompanying phonon emission or absorption. To give an acceptable recoil-free fraction the element must have a high Debye temperature, ΘD.
The Mössbauer effect may be used directly or indirectly in a variety of studies to determine, e.g., the state of valency, changes in site symmetry arising through a variety of causes—thermal, strain, alloying, catalysis, etc.—or to investigate the possible structure of a molecule by comparison with known structures. The fields of application include biology, chemistry and chemical engineering, geology and mineralogy, medicine, metallurgy and physics.
Because these effects bring different energy changes in different materials it is rare for the source to emit a γ-ray of the precise energy required for absorption. The energy of the source is therefore varied by moving it. The change in energy, ΔE, is given by the Doppler shift
where v is the velocity of the source (or absorber) and c is the velocity of light in free space.
Energies may be converted to equivalent velocities for a given nuclide using the expression
The physical requirements for a Mössbauer nuclide are that it should have a resonant absorption cross-section which is large relative to other absorption cross-sections and that the energy of the transition between the ground and excited states should have a very small energy spread. The resonant cross-section is dependent on the inverse square of the energy and the internal conversion coefficient. The theoretical emission and absorption profiles follow a distribution given by the expression
where E0 and σ0 are the peak energy and absorption (or emission) cross-section, σ is the cross-section at energy E and Γ is the full line width at half the peak value. The peak cross-section is given by the expression
where I and I* are the spins of the ground and the excited states of the nucleus and α is the internal conversion coefficient.
The energy spread, ΔE, i.e. the line width, Γ, for emission or absorption is inversely related to the lifetime, τ (and half-life, t1/2 = τ.ln 2), by the Heisenberg uncertainty principle
with corresponding velocity spread v = 137.0 t1/2−1Eγ−1 mm s−1, Eγ in keV and t1/2 in ns. The theoretical compounded line width for emission and absorption is approximately 2Γ. In practice, the line width may be slightly greater owing to the physical thickness of both the source and absorber and self-absorption in the source.
For practical as well as economic reasons the half-life of the parent nuclide must be long. Any practical nucleus must therefore have a low energy (1–100 keV), an excited state with a small internal conversion coefficient and a lifetime in the range 10−10 < τ < 10−6 seconds. The use of synchrotron radiation as a source for Mössbauer experiments has been investigated but it was considered that the practical difficulties and economic aspects were not favourable.
Mössbauer spectroscopy provides information about the environment of the emitting or absorbing nucleus through minute changes to the nuclear energy levels resulting from three effects—the isomer shift, the electric quadrupole splitting and the magnetic hyperfine splitting.
The isomer shift is due to the changes in the electron density at the nucleus when an atom is incorporated into a solid or molecule. The changes in energy of the ground and excited states may not necessarily be the same. The isomer shift results from the difference between the two. A similar, but smaller, change may be brought about if samples of the same material are studied at different temperatures. The isomer shift should always be quoted with respect to a standard material.
The electric quadrupole splitting results from the interaction between the nuclear quadrupole and the crystal or ligand fields. The interaction is such that only nuclear spin states greater than 1/2 are split into different energy levels irrespective of sign. The quadrupole splitting is a measure of the site symmetry in a crystal or the nature of the ligands in molecular structures.
The magnetic hyperfine (or Zeeman) splitting results from the action of the magnetic moments of the ground and excited states of the nucleus with a magnetic field (internal or external). The splitting of the levels depends on the g-factors for the ground and excited states. These are given by the expressions 2gμNIzI −1B and 2g*μNIz*I* −1B, where 2μN = 6.3049 ×10−8 eVT −1, g and g* are the ratios of the nuclear magnetic moment to the nuclear magneton, μN, for the ground and excited states and B is the magnetic field at the nucleus in tesla. The maximum energy spread is given by the expression ΔE = 6.3049 × 10−8 × ( |g*I*zI*−1 | + | gIzI−1 | ).B eV, subject to the selection rules Iz* = Iz or Iz* = Iz ± 1 (although in cases where both a magnetic field and an electric quadrupole field are present, mixing of states may cause some relaxation of these rules).
The energy spread for a 57Fe nucleus in an iron foil with internal field 33.0 T at the nucleus is calculated as follows. The spins of the ground and excited states are 1/2− and 3/2+ and the g-factors +0.090 69 and −0.154 9 respectively. The maximum value for |g*Iz*I* −1| is 0.154 9 and for |gIzI−1| 0.090 69. The value of the energy spread is as follows:
The figure illustrates these three effects relative to a free 57Fe nucleus, a frequently used nuclide and one with a relatively simple energy diagram. In cases where both an electric field and a magnetic field are present the changes to the energy levels depend not only on the magnitude of the fields but also on their relative orientation. It is not possible to construct a generalised diagram. The observed spectra may provide information about the relative orientation of these fields within the crystal or molecule, particularly studies with oriented or single crystals. Whilst most work is performed with a transmission geometry, some research is carried out using scattering geometries or photoelectron spectroscopy, the latter providing information about surface and near-surface nuclei.
In practice the source is usually moved either with constant acceleration or with constant velocity. The calibration of the source velocity may be performed by recording the Mössbauer spectrum of an absorber with known parameters. Interferometric techniques have also been used to calibrate the velocity. Since the energy changes are small it is customary to quote the isomer shift and quadrupole splitting in mm s−1 since velocities are usually in that range. Typical energies for the various physical phenomena relating to Mössbauer spectroscopy are listed in Table 1.
Energy level diagram for 57Fe
The number of known Mössbauer nuclides is small compared with the total number of known isotopes. Some elements, e.g. iron and tin, are most frequently studied because of their practical value. About twenty nuclides have been cited in recent literature with total annual publications running into thousands. The pertinent parameters for these isotopes are listed in Table 2.
Table 2: Relevant parameters for commonly used Mössbauer nuclides
N. N. Greenwood and T. C. Gibb (1971) Mössbauer
Spectroscopy, Chapman & Hall.
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