Mössbauer spectroscopy

Mössbauer spectroscopy is a spectroscopic technique based on the Mössbauer effect. This effect, discovered by Rudolf Mössbauer in 1958, consists of the nearly recoil-free emission and absorption of nuclear gamma rays in solids. The consequent nuclear spectroscopy method is exquisitely sensitive to small changes in the chemical environment of certain nuclei.
Typically, three types of nuclear interactions may be observed: the isomer shift due to differences in nearby electron densities, quadrupole splitting due to atomic-scale electric field gradients; and magnetic splitting due to non-nuclear magnetic fields. Due to the high energy and extremely narrow line widths of nuclear gamma rays, Mössbauer spectroscopy is a highly sensitive technique in terms of energy resolution, capable of detecting changes of just a few parts in 10¹¹.

Basic principle

Just as a gun recoils when a bullet is fired, conservation of momentum requires a nucleus to recoil during the emission or absorption of a gamma ray. If a nucleus at rest emits a gamma ray, the energy of the gamma ray is slightly less than the natural energy of the transition, but in order for a nucleus at rest to absorb a gamma ray, the gamma ray's energy must be slightly greater than the natural energy because in both cases energy is lost to recoil. This means that nuclear resonance is unobservable with free nuclei because the shift in energy is too great, and the emission and absorption spectra have no significant overlap.
Nuclei in a solid crystal, however, are not free to recoil because they are bound in place in the crystal lattice. When a nucleus in a solid emits or absorbs a gamma ray, some energy can still be lost as recoil energy, but in this case, it always occurs in discrete packets called phonons. Any whole number of phonons can be emitted, including zero, which is known as a "recoil-free" event. In this case, the conservation of momentum is satisfied by the momentum of the crystal as a whole, so practically no energy is lost.
Mössbauer found that a significant fraction of emission and absorption events will be recoil-free, which is quantified using the Lamb–Mössbauer factor. This fact is what makes Mössbauer spectroscopy possible, because it means that gamma rays emitted by one nucleus can be resonantly absorbed by a sample containing nuclei of the same isotope, and this absorption can be measured.
The recoil fraction of the Mössbauer absorption is analyzed by nuclear resonance vibrational spectroscopy.

Typical method

In its most common form, Mössbauer absorption spectroscopy, a solid sample is exposed to a beam of gamma radiation, and a detector measures the intensity of the beam transmitted through the sample. The atoms in the source emitting the gamma rays must be of the same isotope as the atoms in the sample absorbing them.
If the emitting and absorbing nuclei were in identical chemical environments, the nuclear transition energies would be exactly equal and resonant absorption would be observed with both materials at rest. The difference in chemical environments, however, causes the nuclear energy levels to shift in a few different ways, as described below. Although these energy shifts are tiny, the extremely narrow spectral linewidths of gamma rays for some radionuclides make the small energy shifts correspond to large changes in absorbance. To bring the two nuclei back into resonance, it is necessary to change the energy of the gamma ray slightly, and in practice, this is always done using the Doppler shift.
During Mössbauer absorption spectroscopy, the source is accelerated through a range of velocities using a linear motor to produce a Doppler effect and scan the gamma-ray energy through a given range. A typical range of velocities for ⁵⁷Fe, for example, can be ±.
In the resulting spectra, gamma ray intensity is plotted as a function of the source velocity. At velocities corresponding to the resonant energy levels of the sample, a fraction of the gamma rays are absorbed, resulting in a drop in the measured intensity and a corresponding dip in the spectrum. The number, positions, and intensities of the dips provide information about the chemical environment of the absorbing nuclei and can be used to characterize the sample.

Selecting a suitable source

Suitable gamma-ray sources consist of a radioactive parent that decays to the desired isotope. For example, the source for ⁵⁷Fe consists of ⁵⁷Co, which decays by electron capture to an excited state of ⁵⁷Fe, which in turn decays to a ground state via a series of gamma-ray emissions that include the one exhibiting the Mössbauer effect. The radioactive cobalt is prepared on a foil, often of rhodium. Ideally the parent isotope will have a convenient half-life. Also, the gamma-ray energy should be relatively low, otherwise the system will have a low recoil-free fraction resulting in a poor signal-to-noise ratio and requiring long collection times. The periodic table below indicates those elements having an isotope suitable for Mössbauer spectroscopy. Of these, ⁵⁷Fe is by far the most common element studied using the technique, although ¹²⁹I, ¹¹⁹Sn, and ¹²¹Sb are also frequently studied.

Analysis of Mössbauer spectra

As described above, Mössbauer spectroscopy has an extremely fine energy resolution and can detect even subtle changes in the nuclear environment of the relevant atoms. Typically, there are three types of nuclear interactions that are observed: isomeric shift, quadrupole splitting, and hyperfine magnetic splitting.

Isomer shift

Isomer shift is a relative measure describing a shift in the resonance energy of a nucleus due to the transition of electrons within its s orbitals. The whole spectrum is shifted in either a positive or negative direction depending upon the s electron charge density in the nucleus. This change arises due to alterations in the electrostatic response between the non-zero probability s orbital electrons and the non-zero volume nucleus they orbit.
Only electrons in s orbitals have a non-zero probability of being found in the nucleus. However, p, d, and f electrons may influence the s electron density through a screening effect.
Isomer shift can be expressed using the formula below, where K is a nuclear constant, the difference between R_e² and R_g² is the effective nuclear charge radius difference between excited state and the ground state, and the difference between _a and _b is the electron density difference in the nucleus. The Chemical Isomer shift as described here does not change with temperature, however, Mössbauer spectra do have a temperature sensitivity due to a relativistic effect known as the second-order Doppler effect. Generally, the impact of this effect is small, and the IUPAC standard allows the Isomer Shift to be reported without correcting for it.
The physical meaning of this equation can be clarified using examples:

While an increase in s-electron density in ⁵⁷Fe spectrum gives a negative shift because the change in the effective nuclear charge is negative, an increase in s-electron density in ¹¹⁹Sn gives a positive shift due to a positive change in overall nuclear charge.
Oxidised ferric ions have lower isomer shifts than ferrous ions because s-electron density at the nucleus of ferric ions is greater due to a weaker screening effect by d electrons.

The isomer shift is useful for determining oxidation state, valency states, electron shielding and the electron-drawing power of electronegative groups.

Quadrupole splitting

reflects the interaction between the nuclear energy levels and the surrounding electric field gradient. Nuclei in states with non-spherical charge distributions, i.e. all those with a spin quantum number greater than 1/2, may have a nuclear quadrupole moment. In this case, an asymmetrical electric field splits the nuclear energy levels.
In the case of an isotope with a I = 3/2 excited state, such as ⁵⁷Fe or ¹¹⁹Sn, the excited state is split into two substates m_I = ±1/2 and m_I = ±3/2. The ground-to-excited state transitions appear as two specific peaks in a spectrum, sometimes referred to as a "doublet". Quadrupole splitting is measured as the separation between these two peaks and reflects the character of the electric field at the nucleus.
The quadrupole splitting can be used for determining the oxidation state, spin state, site symmetry, and the arrangement of ligands.

Magnetic hyperfine splitting

Magnetic hyperfine splitting is a result of the interaction between the nucleus and a surrounding magnetic field. A nucleus with spin I splits into 2I + 1 sub-energy levels in the presence of a magnetic field. For example, the first excited state of the ⁵⁷Fe nucleus with spin state I = 3/2 will split into 4 non-degenerate sub-states with m_I values of +3/2, +1/2, −1/2 and −3/2. The equally-spaced splits are said to be hyperfine, being on the order of 10⁻⁷ eV. The selection rule for magnetic dipole transitions means that transitions between the excited state and ground state can only occur where m_I changes by 0 or 1 or −1. This gives 6 possible for a 3/2 to 1/2 transition.
The extent of splitting is proportional to the magnetic field strength at the nucleus, which in turn depends on the electron distribution of the nucleus. The splitting can be measured, for instance, with a sample foil placed between an oscillating source and a photon detector, resulting in an absorption spectrum, as illustrated in Fig. 4. The magnetic field can be determined from the spacing between the peaks if the quantum "g-factors" of the nuclear states are known. In ferromagnetic materials, including many iron compounds, the natural internal magnetic fields are quite strong and their effects dominate the spectra.