  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.
