LambMössbauer

The Lamb-Mössbauer factor, also known as the recoil-free fraction, is a key concept in Mössbauer spectroscopy. It denotes the probability that a nucleus embedded in a solid emits or absorbs a gamma ray without recoil. Only recoil-free events produce the sharp resonant lines that make Mössbauer spectroscopy possible, so the Lamb-Mössbauer factor directly determines the intensity of the Mössbauer signal.

Formally, the recoil-free fraction f is often written as f = exp(-k^2 ⟨u^2⟩), where k = Eγ/(ħc) is the gamma-ray wave number and ⟨u^2⟩ is the mean-square displacement of the nucleus in the lattice due to thermal vibrations. Equivalently, the factor is sometimes described as f = exp(-2W), with W a lattice-vibration parameter. The value of f depends on both the gamma-ray energy and the temperature through the lattice dynamics; heavier nuclei and stiffer lattices, or lower temperatures, typically yield larger f.

Temperature dependence arises because atomic vibrations grow with temperature, increasing ⟨u^2⟩ and thus decreasing f. In practice, the Debye model often provides a convenient approximation to describe f(T) by relating ⟨u^2⟩ to the phonon spectrum and the Debye temperature of the material.

Historical and naming notes: the factor is named to honor Willis Lamb, whose quantum electrodynamics work influenced radiative processes, and Rudolf Mössbauer, who discovered recoil-free gamma emission in solids. In spectroscopy, f directly affects the apparent intensity of the Mössbauer line and thus the sensitivity to hyperfine interactions such as isomer shifts, quadrupole, and magnetic splitting. Related concepts include the Debye-Waller factor and other Debye-model treatments of lattice dynamics.