Second Order Doppler Shift (SODS)

The Second Order Doppler Shift (SODS) is a temperature-dependent effect on the center shift of a Mössbauer spectrum. Above 0 $\ensuremath{\unskip\,\mathrm{K}}$ atoms in a lattice oscillate about their mean position. The frequency of this oscillation is of the order of $10^{12}\ensuremath{\unskip\,\mathrm{Hz}}$ meaning that the average displacement during the lifetime of a Mössbauer event is zero. However, the second term in the Doppler shift depends on $v^{2}$ leading to the mean square displacement being non-zero. This energy shift is given by

$$\frac{\delta{}E_{\gamma}}{E_{\gamma}} = -\frac{ \langle{}v^{2}\rangle }{ 2c^{2} }$$ (2.7)

For $^{57}$Fe in the high temperature limit this gives a change of +0.07 $\unskip\,\mathrm{mms}$$^{-1}$ for a decrease of 100 $\unskip\,\mathrm{K}$.[5]