Mössbauer SpectroscopyThe basis of the Mössbauer effect is the recoilless resonance absorption of grays: a gray emitted in the deexcitation of a nuclear excited state can be absorbed by another nucleus of the same kind and excite it. By definition, resonance absorption can only take place if the emission energy matches the absorption energy. The energy distribution N(E) of the gradiation emitted by a free nucleus has a Lorentzian shape with a full width at half maximum (FWHM) determined by Heisenberg's uncertainty principle, known as the natural linewidth of the nuclear transition G_{0 }:
Since the gray carries a momentum that has to be conserved, this will be transferred to the free emitting nucleus which will recoil with an energy:
Due to the nuclear recoil, the emission and absorption spectra of free nuclei will be shifted with respect to each other by the amount E_{a}E_{em}=2E_{R}, where
The lifetimes t_{0} of lowlying nuclear levels can be quite long (several ns), therefore
Consequently E_{R} >> G_{0} and If the emitter and absorber nuclei are bound in a solid, this problem may be circumvented. As
the nuclei cannot recoil freely (E_{R} << E_{B}). They can only exchange energy with the lattice by creating or annihilating phonons while the gray transition occurs. Although the typical energies of phonon excitations are of the same order of magnitude as the recoil energy of a free nucleus, there is a defined probability f_{LM} (the recoilless fraction or LambMössbauer factor) that no lattice excitations take place, but the phonon state of the crystal stays unaltered during the gray emission or absorption. This process is called a zerophonon gtransition. The momentum of the recoil is here absorbed by the whole crystal: as its mass is considerably larger than that of the single nucleus (~10^{20} times), the recoil energy on the emitting or absorbing nucleus is negligible, and the emission takes place at E=E_{0}. A nucleus of the same kind can then absorb the gray resonantly. The recoilless emission and absorption of grays Basic Principle / Experimental setupThe energy spread of the grays emitted by a nucleus without recoil, given by the natural linewidth G_{0} is typically one to two orders of magnitude smaller than the splitting of the nuclear energy levels due to hyperfine interactions. If one can shift continuously the energy of the emitter nuclei over the energy range of these interactions, it is possible to study the the dependence of resonance absorption This is the basic principle of Mössbauer spectroscopy (MS). In a typical transmission experiment, the energy of the grays emitted by a radioactive source containing the Mössbauer nuclei is shifted over a broad energy range (up to ~ meV) via a Doppler shift obtained by moving the source with respect to the absorber. The energy of the grays emitted when the source has a relative velocity v is:
It shows a single absorption line whose minimum is at the velocity at which the energy of the grays emitted by the source exactly matches the energy difference between ground and excited state in the absorber. In the approximation of an infinitely thin absorber, the absorption line I(v) is a Lorentzian (like N(E) and s(E) defined in eqs. (3) and (4)), with a full width at half maximum (FWHM) equal to 2G_{0}:
Some usefull formulasThe emission spectrum of a single line source
The probability for a gquantum with frequency w
The cross section per atom contains two additive parts.
The second part of the cross section per atom concerns
For several G_{0} around the resonance energy this part is
The spectrum for a single line absorber
The effective thickness of the absorber is defined as
If we neglect the source width, for comparison with nuclear resonance scattering (NRS) where a nearly ideal (impulse) source is provided, then the transmitted intensity can be written as

