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

TL;DR

This paper compares Penning trap and spectrometer measurements to test Einstein's mass-energy equivalence for silicon-29, analyzing gamma-ray uncertainties and Doppler effects to improve precision in nuclear binding energy measurements.

Contribution

It introduces a detailed analysis of gamma-ray measurement uncertainties and Doppler effects in the context of high-precision nuclear mass measurements for $^{29}{ m Si}$.

Findings

The mass-energy equivalence for $^{29}{ m Si}$ is confirmed within experimental uncertainty.

Doppler effect calculations indicate a significant impact on gamma-ray energy measurements.

Measurement uncertainties are primarily due to gamma-ray detection and Bragg angle measurement.

Abstract

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.