Comparison of potential models of nucleus-nucleus bremsstrahlung

At low photon energies, the potential models of nucleus-nucleus bremsstrahlung are based on electric transition multipole operators, which are derived either only from the nuclear current or only from the charge density by making the long-wavelength approximation and using the Siegert theorem. In the latter case, the bremsstrahlung matrix elements are divergent and some regularization techniques are used to obtain finite values for the bremsstrahlung cross sections. From an extension of the Siegert theorem, which is not based on the long-wavelength approximation, a new potential model of nucleus-nucleus bremsstrahlung is developed. Only convergent integrals are included in this approach. Formal links between bremsstrahlung cross sections obtained in these different models are made. Furthermore, three different ways to calculate the regularized matrix elements are discussed and criticized. Some prescriptions for a proper implementation of the regularization are deduced. A numerical comparison between the different models is done by applying them to the $\alpha +\alpha$ bremsstrahlung.

Figure 1

Schematic view of the contour of integration in the complex $z$ plane.

Figure 2

Bremsstrahlung cross sections $d\sigma /d{E}_{\gamma }$ corresponding to the operator ${\mathcal{M}}_{\lambda \mu }^E$ (full lines), to the operator ${\tilde{\mathcal{M}}}_{\lambda \mu }^{E\left(\mathrm{S}\right)}$ (dashed lines), and to the operator ${\tilde{\mathcal{M}}}_{\lambda \mu }^{E(\mathrm{S},\mathrm{LWA})}$ (dots) as a function of the initial energy $E_i$ for three photon energies: $E_{\gamma }=1$, 5, and 15 MeV. Differences between full lines, dashed lines, and dots are nearly indistinguishable at the scale of the figure.

Figure 3

Bremsstrahlung cross sections $d\sigma /d{E}_{\gamma }$ at $E_{\gamma }=1$, 5, and 15 MeV at the long-wavelength approximation. Full lines correspond to $\epsilon \to 0$, dashed lines correspond to ${\epsilon }_0=0.001$ ${\mathrm{fm}}^{-1}$ for $E_{\gamma }=1$ MeV and to ${\epsilon }_0=0.01$ ${\mathrm{fm}}^{-1}$ for $E_{\gamma }=5$ and 15 MeV, and dots correspond to ${\epsilon }_0=0.002$ ${\mathrm{fm}}^{-1}$ for $E_{\gamma }=1$ MeV and to ${\epsilon }_0=0.02$ ${\mathrm{fm}}^{-1}$ for $E_{\gamma }=5$ and 15 MeV. The Gaussian regularization function is considered.

Figure 4

Bremsstrahlung cross sections $d\sigma /d{E}_{\gamma }$ at $E_{\gamma }=15$ MeV at the long-wavelength approximation. Full lines correspond to $\epsilon \to 0$, dashed lines correspond to ${\epsilon }_0=0.01$ ${\mathrm{fm}}^{-1}$, and dots correspond to ${\epsilon }_0=0.02$ ${\mathrm{fm}}^{-1}$. The exponential regularization function is considered.

Figure 5

Bremsstrahlung cross sections $d\sigma /d{E}_{\gamma }$ at $E_{\gamma }=1$ MeV at the long-wavelength approximation by considering only the $4^{+}\to 2^{+}$ transition. They are evaluated for $\epsilon \to 0$ (full), ${\epsilon }_0=0.001$ ${\mathrm{fm}}^{-1}$ (dash), ${\epsilon }_0=0.002$ ${\mathrm{fm}}^{-1}$ (dot), ${\epsilon }_0=0.005$ ${\mathrm{fm}}^{-1}$ (dash-dot), ${\epsilon }_0=0.01$ ${\mathrm{fm}}^{-1}$ (dash-dash), and ${\epsilon }_0=0.02$ ${\mathrm{fm}}^{-1}$ (dash-dot-dot). The Gaussian regularization function is considered.