Kinematics of a cascade decay for the precision measurement of ²⁹{rm Si} binding energy

Comparison of a Penning trap and a flat-crystal spectrometer experiments gives a direct test of $E=mc^2$. The result is $1-\Delta mc^2/E =(-1.4 \pm 4.4) \times 10^{-7}$ for $^{29}{\rm Si}$ and $^{33}{\rm S}$. The dominant uncertainty is on the $\gamma$-ray measurement in neutron capture reactions, and the secondary $\gamma$-ray has the uncertainty 4.0 eV for $^{29}{\rm Si}$. We calculated the Doppler effect of the secondary $\gamma$-ray as $-646.9 \cos \theta$ eV from the relativistic energy momentum relation of the $^{28}{\rm Si}(n,\gamma)^{29}{\rm Si}$ reaction. This corresponds to the full wave of half maximum of 431.3 eV. The error 4.0 eV comes mainly from the Bragg angle measurement between the centroids of the linewidths which means that only the most probable part of the whole data has been considered. It is necessary to confirm the assumption of the isotropy for the object to measure. We discussed a coincidence measurement as one of the methods to overcome the assumption.