New approach for obtaining information on the many-nucleon structure in $\alpha$ decay from accompanying bremsstrahlung emission

We analyze the nucleon structure of the $\alpha$-decaying nucleus to see if it can be visible in the experimental bremsstrahlung spectra of the emitted photons which accompany such a decay. We develop a new formalism of the bremsstrahlung model taking into account the distribution of nucleons in the $\alpha$-decaying nuclear system. We conclude the following. (1) After inclusion of the nucleon structure in the model the calculated bremsstrahlung spectrum is changed very slowly for a majority of the $\alpha$-decaying nuclei. However, we have observed that visible changes really exist for the ${}^{106}\mathrm{Te}$ nucleus ($Q_{\alpha }=4.29$ MeV, $T_{1/2}=70$ μs) even for the energy of the emitted photons up to 1 MeV. This nucleus is a good candidate for future experimental study of this task. (2) Inclusion of the nucleon structure in the model increases the bremsstrahlung probability of the emitted photons. (3) We find the following tendencies for obtaining the nuclei, which have bremsstrahlung spectra more sensitive to the nucleon structure: (a) direction to nuclei with smaller $Z$ and (b) direction to nuclei with larger $Q_{\alpha }$ values.

Figure 1

(a) The bremsstrahlung probabilities of the emitted photons in the $\alpha$ decay of the ${}^{210}\mathrm{Po}$ nucleus and experimental data [53] [in calculations ${\mathbf{\rho }}_{i,0}=1.7$ fm for the $\alpha$ particle and ${\theta }_f={90}^{\circ }$ are used; ${\theta }_f$ is the angle between direction of the $\alpha$ particle motion (or its tunneling) after emission of photon and the direction of the photon emission]. Here the experimental data given by open circles are extracted from Ref. [53], the solid green line represents the calculations without the included nucleon structure, the dashed blue line represents the calculations with the included nucleon structure, the dash-dotted red line represents calculations for the pointlike $\alpha$ particle taken from Ref. [57] (see line 6 in Fig. 1 in that paper). The spectrum for calculations with inclusion of the many-nucleon structure is located above the spectrum without this structure, but such a difference is practically not visible. (b) The difference of the functions of errors, $\mathrm{\Delta }\varepsilon \left(E_k\right)$, defined in Eqs. (24) and that obtained after comparative analysis between the new calculations with and without the nucleon structure and experimental data given in panel (a).

Figure 2

The bremsstrahlung probabilities of the emitted photons at the $\alpha$ decay of the nuclei ${}^{219}\mathrm{Pa}$ (a) and ${}^{106}\mathrm{Te}$ and ${}^{110}\mathrm{Te}$ (b) [in calculations ${\mathbf{\rho }}_{i,0}=1.7$ fm for the $\alpha$ particle and ${\theta }_f={90}^{\circ }$ are used; ${\theta }_f$ is the angle between the direction of the $\alpha$ particle motion (or its tunneling) after emission of photon and the direction of the photon emission]. (a) The dashed red line is for calculations for ${}^{219}\mathrm{Pa}$ with included nucleon structure, the solid blue line is for calculations for ${}^{219}\mathrm{Pa}$ without included nucleon structure. (b) The dash-double dotted red line is for calculations for ${}^{106}\mathrm{Te}$ with included nucleon structure, the solid blue line is for calculations for ${}^{106}\mathrm{Te}$ without nucleon structure, the dash-dotted green line is for calculations for ${}^{110}\mathrm{Te}$ with included nucleon structure, and the dashed brown line is for calculations for ${}^{110}\mathrm{Te}$ without included nucleon structure. The curve for each nucleus after inclusion of the nucleon structure is located above in comparison with the curve without this structure. The nucleon structure is visible more strongly in the spectra for nuclei with higher $Q_{\alpha }$ values and smaller $Z$.

Figure 3

Ratio ${\tilde{Z}}_{\mathrm{eff}}^{(\mathrm{dip},\mathrm{micro})}/{\tilde{Z}}_{\mathrm{eff}}^{(\mathrm{dip},\mathrm{no}-\mathrm{micro})}$ between the effective charges calculated with inclusion of many-nucleon structure and without it in dependence on the energy of the emitted photon [here ${\tilde{Z}}_{\mathrm{eff}}^{(\mathrm{dip},\mathrm{micro})}$ is the effective charge with the included many-nucleon structure defined in Eq. (21) and ${\tilde{Z}}_{\mathrm{eff}}^{(\mathrm{dip},\mathrm{no}-\mathrm{micro})}$ is the effective charge calculated without such a structure, i.e., for the pointlike $\alpha$ particle and nucleus]. One can see that the role of the many-nucleon description in the calculation of the effective charge ${\tilde{Z}}_{\mathrm{eff}}^{(\mathrm{dip},\mathrm{micro})}$ is increased with increasing energy of the emitted photon. This explains why the bremsstrahlung spectra with the included many-nucleon structure are larger than the bremsstrahlung spectra without such a structure in Figs. 2 and 2, and such a difference only increased with increasing energy of the emitted photons.

Figure 4

The bremsstrahlung probabilities of the emitted photons during capture of proton by the ${}^{121}\mathrm{Sb}$ nucleus at the incident proton energy of 10 MeV (a), 50 MeV (b), 100 MeV (c), and 300 MeV (d) in the laboratory frame. Here solid blue lines are for calculations with many-nucleon structure and dashed red lines are for calculations without included nucleon structure. One can see that the curve for the nucleus after inclusion of the many-nucleon structure at each proton incident energy is located above in comparison with curve without such a structure. From the figures one can estimate how the bremsstrahlung probability (for the same photon energy) is increased with increasing energy of the incident proton. In all figures one can see that the bremsstrahlung probabilities for emission of the hard photons at kinematic limit (defined by the energy of the incident proton) tends to zero.