Relativistic effects in heavy-ion Coulomb scattering

The role of relativistic corrections in heavy-ion Coulomb scattering at intermediate energies ($E_{\text{lab}}\gtrsim 50$ MeV/nucleon) is investigated by numerically solving a full set of coupled equations. We compare two methods: (a) one involving an exact account of interaction retardation and (b) a method based on the expansion of effective Lagrangians in powers of the ion velocities, $v/c$. Our study makes it possible to infer the relevance of kinematic corrections, retardation, and magnetic interactions such as the Darwin force. We show that analytical formulas are able to describe all aspects of experimental interest in relativistic effects in heavy-ion Coulomb scattering at intermediate energies without having to solve numerically the coupled equations.

Figure 1

Absolute value of the relative difference (in percent) between the methods of Matzdorf et al. [16] and Aguiar et al. [17], with the nonrelativistic scattering angle ${\mathrm{\Theta }}^{\text{NR}}=2arctan(q_p{q}_t/\mu v_{\infty }^{2}b)$ for ${}^{208}\mathrm{Pb}+{}^{208}\mathrm{Pb}$ collisions at the laboratory energy of 100 MeV/nucleon. The dashed line is a numerical calculation following the method of Aguiar et al. [17] considering relativistic corrections up to order ${(v/c)}^4$, and the solid line is a numerical calculation for the corresponding method of Matzdorf et al. [16]. The horizontal axis represents the impact parameter $b$ (in fm).

Figure 2

Same as Fig. 1, but for a collision at grazing impact parameter $b=R_P+R_T$, as a function of the laboratory energy $E_{\text{lab}}$ (in MeV/nucleon).

Figure 3

Relative difference (in percent) between the methods of Matzdorf et al. [16] and of Aguiar et al. [17] with the nonrelativistic Rutherford scattering cross section, $d{\sigma }^{\text{NR}}/d\mathrm{\Omega }$, for ${}^{208}\mathrm{Pb}+{}^{208}\mathrm{Pb}$ collisions at the laboratory energy of 100 MeV/nucleon. The dashed line is a numerical calculation following the method of Aguiar et al. [17] considering relativistic corrections up to order ${(v/c)}^4$, and the solid line is a numerical calculation for the corresponding method of Matzdorf et al. [16]. The horizontal axis represents the center-of-mass scattering angle $\mathrm{\Theta }$ (in degrees).

Figure 4

Same as in Fig. 3, but for a collision at grazing impact parameter $b=R_P+R_T$, as a function of laboratory energy $E_{\text{lab}}$ (in MeV/nucleon).

Figure 5

Relative difference (in percent) between the analytical formulas proposed by Matzdorf et al. [16] and by Aguiar et al. [17] with the nonrelativistic Rutherford scattering cross section, $d{\sigma }^{\text{NR}}/d\mathrm{\Omega }$, for ${}^{17}\mathrm{O}+{}^{208}\mathrm{Pb}$ collisions at the laboratory energy of 100 MeV/nucleon. The dashed line uses the analytical equation (9) and the solid line is for a numerical calculation of the coupled equations following Ref. [17] considering relativistic corrections up to order ${(v/c)}^2$. The dotted line is for the analytical formula (7) predicted by Matzdorf et al. [16], which agrees within less than 0.1% with the exact results (not shown).

Figure 6

Relative difference (in percent) of the distance of closest approach $b_c$ for a given impact parameter $b$ obtained with the full relativistic calculations and with the analytical formula (10), as a function of laboratory energy in MeV/nucleon and for different projectile target systems. We use the grazing impact parameter $b=R_p+R_t$.