Lamb-Mössbauer factor

The Mössbauer effect is a recoilless absorption and emission of a photon. The recoilless absorption and emission can occur if the recoil energy can not excite lattice vibrations and the particular nucleus does not change its state. The probability for a such process is defined by the Lamb-Mössbauer factor $f_{LM}$. Assuming a continuous phonon spectrum with the density of states $\rho_{P}$

$$\begin{array}{lll} \rho_{P}=\frac{9N\omega^{2}}{\omega_{D}^{3}} &... ... && \nonumber \\ \rho_{P}=0 & \textnormal{for} & \omega>\omega_{D} \end{array}$$

where $\omega_{D}$ is the Debye frequency, N is the number of eigenfrequencies in the solid, and $\omega$ is the phonon frequency. The Debye model can be used to approximate the value of the Lamb-Mössbauer factor [26].

$$f_{LM}=e^{-\frac{3E_{R}}{k_{B}\Theta_{D}} \bigg{(} 1+4\frac{T^{2}}{\Theta_{D}^{2}}\int_{0}^{\frac{\Theta_{D}}{T}}\frac{x dx}{e^{x}-1}\bigg{)}}$$ (35)

The integral in the upper equation can be found in reference [29]. The following approximation for high temperatures (T $\geq\Theta_{D}$) can be used

$$f_{LM}=e^{-\frac{6E_{R}T}{k_{B}\Theta_{B}^{2}}}.$$ (36)

For diffusion experiments, where usually higher temperatures are necessary, the Lamb-Mössbauer factor, which determines the effect-to-background ratio in NRS and QMS experiments , causes often additional difficulties due to its decrease with increased temperature.