We analyze if the nucleon structure of the $alpha$ decaying nucleus 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 distribution of nucleons in the $alpha$ decaying nuclear system. We conclude the following: (1) After inclusion of the nucleon structure into 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 $^{106}{rm Te}$ nucleus ($Q_{alpha}=4.29$ MeV, $T_{1/2}$=70 mks) 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 into 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 $Z$, (b) direction to nuclei with larger $Q_{alpha}$-values.
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