Volume reflection and channeling of ultrarelativistic protons in germanium bent single crystals

The paper is devoted to the investigation of volume reflection and channeling processes of ultrarelativistic positive charged particles moving in germanium single crystals. We demonstrate that the choice of atomic potential on the basis of the Hartree-Fock method and the correct choice of the Debye temperature allow us to describe the above mentioned processes in a good agreement with the recent experiments. Moreover, the universal form of equations for volume reflection presented in the paper gives a true description of the process at a wide range of particle energies. Standing on this study we make predictions for the mean angle reflection (as a function of the bending radius) of positive and negative particles for germanium (110) and (111) crystallographic planes.

Figure 1

The calculated universal $\mathrm{\Xi }(R/R_0)=⟨\alpha ⟩/{\theta }_c$-functions for positively charged particles and for normalized potentials for (110) plane of germanium which were shown in insert (a). The calculations were done for the Hartree-Fock potential at $T_D=290°\mathrm{K}$ and at $T_D=360°\mathrm{K}$ (the curves 1 and 2, correspondingly) and for the Moliere potential at $T_D=360°\mathrm{K}$ and at $T_D=285°\mathrm{K}$ (the curves 3 and 4, correspondingly). All the curves are functions of generalized variable $\kappa =R/R_0$, The small circles are the experimental data [9] presented for corresponding values of generalized variables $\mathrm{\Xi }$ and $\kappa$. The table 1 contains the necessary (for such presentation) quantities ${\theta }_c$ and $U_0$. The experimental errors are less than $\pm 0.02$ (for $\mathrm{\Xi }$-variable). The curve number 5 [in insert (a)] corresponds to (110) plane of silicon and it is presented for comparison.

Figure 2

The calculated universal functions $\mathrm{\Xi }(R/R_0)=⟨\alpha ⟩/{\theta }_c$ for germanium and silicon single crystals. The Hartree-Fock potentials were used at $T_D=290°\mathrm{K}$ and 640° K for germanium and silicon, respectively. The upper curves are results for positively charged particles and lower ones are results for negatively charged particles. Three points demonstrated experimental data for germanium crystal [8, 9] for $R=2.3$, 8.2 and 15 meters and $l_0=2\mathrm{mm}$ (from left to right, correspondingly). The first and second experimental points correspond to the curve number 1, and the third points correspond to the curve number 3. $R_0[\mathrm{m}]=K_R{E}_0[\mathrm{GeV}]$, where $K_R=0.0061$, 0.01 for (110) and (111) planes in germanium and $K_R=0.009$, 0.014 for (110) and (111) planes in silicon, respectively.