The gauge-invariant description of the alpha-alpha bremsstrahlung with initial and final state interactions included

Oct 15, 2020, 6:15 PM


Poster report Section 2. Experimental and theoretical studies of nuclear reactions. Poster session 2 (part 3)


Adam Arslanaliev


One of motivations in studying the $\alpha+\alpha\rightarrow\alpha+\alpha+ \gamma$ bremsstrahlung is to get a supplementary information on a strong part of the alpha-alpha interaction. Our departure point in describing this reaction is to use the Fock-Weyl criterion and a generalization of the Siegert theorem [1,2]. Along the guideline we obtain the gauge-independent bremsstrahlung amplitude in a nonrelativistic cluster picture. This amplitude can be expressed through the alpha particle form factor $F_{CH}(q)$ and the three dimensional overlap integral $I(\textbf{k}',\textbf{k};\textbf{q}) = \langle\chi^{(-)}_\textbf{k'}|e^{-i\frac12\textbf{qr}}|\chi^{(+)}_\textbf{k}\rangle$, where the "distorted" wave $\chi^{(-)}_\textbf{k'}$ ($\chi^{(+)}_\textbf{k}$) describes the $\alpha$-$\alpha$ scattering in the final (entrance) channel. The corresponding interaction operator $V = V_{S} + V_{C}$ consists of the strong nuclear interaction between alpha particles $V_S$, while $V_C$ describes the Coulomb repulsion between them. Such a consideration leads to the division $I = I_C + I_{CS}$ with the Coulomb integral $I_C$ responsible for the Coulomb bremsstrahlung and the mixed Coulomb-strong one $I_{CS} = I - I_C$ (cf. [3]). In its turn, the Coulomb integral is given by the analytical expression [4], while the integral $I_{CS}$ can be reduced to the summation of its partial wave expansion with the simple radial integrals. A distinctive feature of our approach is to provide the convergence of the expansion. The numerical calculations of the radial integrals are performed with help of the contour integration method [5]. In order to demonstrate to which extent the obtained results depend on the choice of the model interaction $V_S$ we show in Fig.1 the cross section $d\sigma = d^5\sigma/dE_\gamma d\Omega_{1i}d\Omega_{1f}$ for the coplanar kinematics in which one of the outgoing alphas is detected in coincidence with the emitted photon. In such a kinematics all momenta have a coplanar disposal, where the photon momentum is directed along the Z-axis and the rest lie in the XZ-plane, viz.,$\hat{\textbf{k}}_{1i}=(\theta_{1i},0),\hat{\textbf{k}}_{1f}=(\theta_{1f},\pi)$. We see, first, that the strong interaction effects can be dominant to compared the pure Coulomb interaction and, second, measurements of such a correlation function can bring a supplementary information on the strong part of the interaction between alpha particles.
[1] A. Shebeko, Sov. J. Nucl. Phys ${\bf 49}$, 30 (1989).
[2] A. Shebeko, Phys. At. Nucl. ${\bf 77}$, 518 (2014).
[3] D. Baye, C. Sauwens, P. Descouvemont and S. Keller, Nucl. Phys. A ${\bf 529}$, 467 (1991)
[4] M. Gravielle, J. Miraglia, Comp. Phys. Comm. ${\bf 69}$, 53 (1992)
[5] C. Vincent, H. Fortune, Phys. Rev. C ${\bf 2}$, 782 (1970)
[6] B. Buck, \textit{et al.}, Nucl. Phys. A ${\bf 275}$, 246 (1977)
[7] S. Ali, A.R. Bodmer, Nucl. Phys ${\bf 80}$, 99 (1966)

The $\alpha$-$\alpha$ bremsstrahlung cross section versus the angle between $\textbf{k}_{1f}$ and $\textbf{k}_\gamma$ at the  different values of the incident energies $E_i$ calculated for potential from [6] (solid curves), potential from [7] (dotted) and Coulomb one (dashed).

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