New approach to determine proton-nucleus interactions from experimental bremsstrahlung data

A new approach is presented to determine the proton-nucleus interactions from the analysis of the accompanying photon bremsstrahlung. We study the scattering of $p+{}^{208}\mathrm{Pb}$ at the proton incident energies of 140 and 145 MeV, and the scattering of $p+{}^{12}\mathrm{C},p+{}^{58}\mathrm{Ni},p+{}^{107}\mathrm{Ag}$, and $p+{}^{197}\mathrm{Au}$ at the proton incident energy of 190 MeV. The model determines contributions of the coherent emission (formed by an interaction between the scattering proton and nucleus as a whole without the internal many-nucleon structure), incoherent emission (formed by interactions between the scattering proton and nucleus with the internal many-nucleon structure), and transition between them in dependence on the photon energy. The radius parameter of the proton-nucleus potential for these reactions is extracted from the experimental bremsstrahlung data analysis. We explain the hump-shaped plateau in the intermediate- and high-energy regions of the spectra by the essential presence of the incoherent emission, while at low energies the coherent emission predominates which produces the logarithmic shape spectrum. We provide our predictions (in absolute scale) for the angular distribution of the bremsstrahlung photons in order to test our model, results, and analysis in further experiments.

Figure 1

(a) The bremsstrahlung cross sections for $p+{}^{208}\mathrm{Pb}$ calculated on the basis of the first matrix element $p_1$ in Eq. (35) at the incident proton energy of 140 MeV depending on the parameter $r_R$ in comparison with the experimental data of Edington and Rose [1] at $\theta ={90}^{\circ }$. One can see a slow sensitivity of the calculated spectra from the parameter $r_R$. In particular, such a sensitivity should be present after normalization of all calculated spectra on the same experimental point. (b) Estimated errors obtained by a method of minimization given by formula (48) depending on values of the parameter $r_R$. One can see that there is a visible minimum which indicates the presence of optimal values for $r_R$, at which agreement between theory and experimental data is the highest.

Figure 2

(a) The bremsstrahlung cross sections for $p+{}^{208}\mathrm{Pb}$ calculated on the basis of the first matrix element $p_1$ in Eq. (35) at the incident proton energy of 145 MeV depending on the parameter $r_R$ in comparison with experimental data of Clayton et al. [6, 9] at $\theta ={90}^{\circ }$. Once again we obtain a slow sensitivity of the spectra on the parameter $r_R$. One can see a clear difference between the logarithmic form of the calculated spectra and hump-shaped behavior in experimental data. (b) Estimated errors obtained by a method of minimization depending on values of the parameter $r_R$. One can see that there is a visible minimum which indicates the presence of optimal values for $r_R$, at which agreement between theory and experimental data is the highest.

Figure 3

(a) The calculated bremsstrahlung cross sections for $p+{}^{208}\mathrm{Pb}$ at 140 MeV on the basis of the second matrix element $p_2$ in Eq. (36) at different values of the parameter $r_R$ in comparison with experimental data of Edington and Rose [1] at $\theta ={90}^{\circ }$. (b) The estimated errors obtained by a method of minimization depending on values of the parameter $r_R$.

Figure 4

(a) The calculated bremsstrahlung cross sections for $p+{}^{208}\mathrm{Pb}$ at 140 MeV on the basis of the third matrix element $p_3$ in Eq. (37) at different values of the parameter $r_R$ in comparison with experimental data of Edington and Rose [1] at $\theta ={90}^{\circ }$. (b) The estimated errors obtained by a method of minimization depending on values of the parameter $r_R$.

Figure 5

The bremsstrahlung cross sections for $p+{}^{208}\mathrm{Pb}$ at 145 MeV of the proton energy calculated on the basis of the last matrix element $p_4$ in Eq. (44) depending on the parameter $f$ in comparison with the experimental data of Clayton et al. [6] at $\theta ={90}^{\circ }$. The best agreement between theory and experimental data is obtained for the factor $f=0.00018$ (used parameters of potential: $r_R=r_C=1.17$ fm).

Figure 6

The bremsstrahlung cross sections for $p{+}^{12}\mathrm{C}$ (a) and $p+{}^{197}\mathrm{Au}$ (b) at 190 MeV of the proton energy calculated on the basis of the last matrix element $p_4$ in Eq. (44) depending on the parameter $f$ in comparison with the experimental data of van Goethem et al. [11] at the angle of $\theta ={75}^{\circ }$ (all calculated spectra are normalized on the second point of experimental data, used parameters of potential are $r_R=r_C=0.95$ fm). Once again, we obtain a visible sensitivity of the spectra on the parameter $f$ characterized contribution of the emission caused by the dynamics of the nucleons of the nucleus. The best agreement between theory and experimental data is observed for $p{+}^{12}\mathrm{C}$ at the factor $f=0.01$ [see purple solid line 5 in (a)] and $p+{}^{197}\mathrm{Au}$ at the factor $f=0.0007$ [see blue solid line 5 in (b)]. This result demonstrates that the inclusion of the dynamics of nucleons and its connection with spin properties of the scattering proton into the model allows us to describe (and, so, to explain) the existing plateau in the spectra of the bremsstrahlung photons.

Figure 7

The bremsstrahlung cross sections for $p+{}^{58}\mathrm{Ni}$ (a) and $p+{}^{107}\mathrm{Ag}$ (b) at 190 MeV of the proton energy calculated on the basis of the last matrix element $p_4$ in Eq. (44) depending on the parameter $f$ in comparison with the experimental data of van Goethem et al. [11] at the angle of $\theta ={75}^{\circ }$ (the experimental data are normalized on the geometrical cross section, all calculated spectra are normalized on the second point of experimental data, used parameters of potential are $r_R=r_C=0.95$ fm, in all figures we use the renormalized factor $h=f\times {10}^4$). The best agreement between theory and experimental data is observed for $p+{}^{58}\mathrm{Ni}$ at the factor $f=0.002$ [see purple solid line 5 in (a)] and $p+{}^{107}\mathrm{Ag}$ at the factor $f=0.0012$ [see purple solid line 7 in (b)].

Figure 8

(a) The ratio between the incoherent emission (formed by interactions between the scattering proton and internal momenta of nucleons of nucleus, denoted as $d{\sigma }_1/d{E}_{\gamma }$) and the coherent emission (formed by interaction between the scattering proton and nucleus as whole without its internal many-nucleon structure, denoted as $d{\sigma }_2/d{E}_{\gamma }$) depending on the energy of emitted photons for $p+{}^{197}\mathrm{Au}$ at $E_{\mathrm{p}}=190$ MeV and the angle of $\theta ={75}^{\circ }$ [used parameters of potential are $r_R=r_C=0.95$ fm, $f=0.0007$ according to results in Fig. 6]. (b) The angular distribution of the emitted photons for $p+{}^{197}\mathrm{Au}$ at $E_{\mathrm{p}}=190$ MeV in comparison with the experimental data of van Goethem et al. [11] measured at the angle of $\theta ={75}^{\circ }$ [$f=0.0007$ according to results in Fig. 6].