$\alpha$-clustering effect on flows of direct photons in heavy-ion collisions

Abstract

In this study, we reconstruct the $\gamma$-photon energy spectrum, which is in good agreement with the experimental data of $^{86}$Kr + $^{12}$C at E/A = 44 MeV within the framework of the modified EQMD model. The directed and elliptic flows of free protons and direct photons were investigated by considering the $\alpha$-clustering structure of $^{12}$C. Compared with free protons, direct photon flows provide clearer information about the early stage of a nuclear reaction. The difference in the collective flows between different configurations of $^{12}$C is observed in this study. This indicates that the collective flows of direct photons are sensitive to the initial configuration. Therefore, the $\gamma$ bremsstrahlung process might be taken as an alternative probe to investigate the $\alpha$-clustering structure in a light nucleus from heavy-ion collisions within the Fermi-energy region.

Appendix

Strictly speaking, one should sample adequate times according to the Wigner function of a nucleon as follows:

$$\begin{aligned} \begin{aligned} w_i({{\mathbf {r}}})=&\rho _{i}({\mathbf {r}}) \\ =&\left( \frac{1}{\pi \lambda _i}\right) ^\frac{3}{2}\exp \left\{ -\frac{1}{\lambda _i}\left( \mathbf {r-{\mathbf {R}}_i}\right) ^2 \right\}, \\ w_i({{\mathbf {p}}})=&\frac{w_i({\mathbf {r}},{\mathbf {p}})}{\rho _{i}({\mathbf {r}})} \\ =&\left( \frac{\lambda _i}{\pi \hbar ^2}\right) ^\frac{3}{2}\exp \left\{ -\frac{\lambda _i}{\hbar ^2}\left[ {\mathbf {p}}-\left( {\mathbf {P}}_i+\delta _i\hbar {\mathbf {R}}_i-\delta _i\hbar {\mathbf {r}}\right) \right] ^2\right\} . \end{aligned} \end{aligned}$$

(17)

Herein, $w_i({\mathbf {r}})$ is the probability that the ith nucleon is at point ${\mathbf {r}}$, and $w_i({\mathbf {p}})$ represents the probability of this nucleon with momentum ${\mathbf {p}}$ when its position is known at ${\mathbf {r}}$. It is easy to obtain certain quantities from Eq.17 as follows:

$$\begin{aligned} \begin{aligned} \overline{{\mathbf {r}}}\,=\,&{\mathbf {R}}_i, \\ \overline{{\mathbf {r}}^2}\,=\,&{\mathbf {R}}_i^2+\frac{3}{2}\lambda _i, \\ \overline{{\mathbf {p}}}\,=\,&{\mathbf {P}}_i, \\ \overline{{\mathbf {p}}^2}\,=\,&\overline{\left[ {\mathbf {P}}_i-\delta _i\hbar \left( {\mathbf {r}}-{\mathbf {R}}_i\right) \right] ^2}+\frac{3\hbar ^2}{2\lambda _i}\\ \,=\,&{\mathbf {P}}_i^2+\overline{\delta ^2_i\hbar ^2\left( {\mathbf {r}}-{\mathbf {R}}_i\right) ^2}+\frac{3\hbar ^2}{2\lambda _i}\\ \,=\,&{\mathbf {P}}_i^2+\frac{3\hbar ^2}{2\lambda _i}\left( 1+\lambda _i^2\delta _i^2\right). \end{aligned} \end{aligned}$$

(18)

We also consider the zero-pointer kinetic energy into momentum sampling, our formulation is as follows:

$$\begin{aligned} \begin{aligned} \Delta {\mathbf {p}}\,=\,&{\mathbf {p}}-{\mathbf {P}}_i, \\ {\mathbf {p}}_i=&{\mathbf {P}}_i+\sqrt{1-\frac{1}{M_i}}\times \Delta {\mathbf {p}}. \end{aligned} \end{aligned}$$

(19)

Here, ${\mathbf {p}}_i$ is the momentum of the nucleon after its zero-point kinetic energy is considered.

If one samples the coordinate and momentum adequately, the following can be easily proven:

$$\begin{aligned} \begin{aligned} \overline{{\mathbf {p}}_i^2}=&\overline{\left( {\mathbf {P}}_i+\sqrt{1-\frac{1}{M_i}}\times \Delta {\mathbf {p}}\right) ^2}\\ =&{\mathbf {P}}_i^2+\left( 1-\frac{1}{M_i}\right) \times \frac{3\hbar ^2}{2\lambda _i}\left( 1+\lambda _i^2\delta _i^2\right), \\ \overline{T_i}=&\frac{{\mathbf {P}}_i^2}{2m_i}+\left( 1-\frac{1}{M_i}\right) \times \frac{3\hbar ^2}{4m_i\lambda _i}\left( 1+\lambda _i^2\delta _i^2\right) \\ =&\frac{{\mathbf {P}}_i^2}{2m_i}+\frac{3\hbar ^2\left( 1+\lambda _i^2\delta _i^2\right) }{4m_i\lambda _i}-T_\mathrm {zero}. \end{aligned} \end{aligned}$$

(20)

Thus far, this proves that the kinetic energy obtained by this sampling method is consistent with the expected kinetic energy described by the Hamilton of the EQMD model. This part of the kinetic energy is dominant, namely in the “Fermi-motion” question in the study by Maruyama et al. [11]. In the actual calculation, it should be noted that there is an appropriate simplification for simulation purposes.