Probing nucleon effective mass splitting with light particle emission

Abstract

The main objective of this study was to investigate the impact of effective mass splitting on heavy-ion-collision observables. We first analyzed correlations between different nuclear matter parameters obtained from 119 effective Skyrme interaction sets. The values of the correlation coefficients illustrate that the magnitude of effective mass splitting is crucial for tight constraints on the symmetry energy via heavy-ion collisions. The $^{86}$Kr + $^{208}$Pb system at beam energies ranging from 25 to 200A MeV was simulated within the framework of the improved quantum molecular dynamics model (ImQMD-Sky). Our calculations show that the slopes of the spectra of $\ln$[Y(n)/Y(p)] and $\ln$[Y(t)/Y($^3$He)], which are the logarithms of the neutron to proton and triton to helium-3 yield ratios, are directly related to effective mass splitting and can be used to probe the effective mass splitting.

Appendix 1: Single-particle potential

For the Skyrme interaction, the single-particle potential in uniform nuclear matter can be written as the summation of the local and nonlocal (momentum-dependent) parts as follows:

$$\begin{aligned} V_q=V_q^\textrm{loc}+V_{q}^\textrm{md}. \end{aligned}$$

(A.1)

Based on the definition of the single-particle potential, $V_q$ should be obtained from the derivatives of the net energy E of the system with respect to the number of particles. For the local part, $V_q^\text {loc}$ is

$$\begin{aligned} \begin{aligned} V^\textrm{loc}_q (\rho , \delta )&= \frac{ \partial u_\textrm{loc }( \rho , \delta ) }{ \partial \rho _q } \\&= \alpha \frac{\rho }{ \rho _0 } + \beta \frac{ \rho ^\eta }{ \rho _0^\eta } + (\eta - 1) B_\textrm{sym} \frac{\rho ^\eta }{\rho _0^\eta }\delta ^2\\&\quad \pm 2 \left( A_\textrm{sym} \frac{\rho }{\rho _0}+ B_\textrm{sym} \left( \frac{\rho }{\rho _0} \right) ^{\eta } \right) \delta , \end{aligned} \end{aligned}$$

(A.2)

where ‘$+$’ is for neutrons and ‘−’ for protons. The nonlocal part of the single-particle potential depends not only on the position but also on the momentum, which can be obtained by taking the functional derivative of the energy density with respect to the phase-space distribution function of protons or neutrons $f_q(r,p)$:

$$\begin{aligned} \begin{aligned} V^\textrm{md}_q(\rho ,\delta ,p)&= \frac{\delta u_\textrm{md}}{\delta f_q} \\&= 2(C_0\rho +D_0\rho _q)p^2+ 2\hbar ^2C_0\tau +2\hbar ^2D_0\tau _q, \end{aligned} \end{aligned}$$

(A.3)

where $\tau$ is the kinetic energy density and is the summation of the kinetic energy densities of neutrons and protons (i.e.,$\tau =\tau _\text{n}+\tau _\text{p}$) and $\tau _q=\frac{3}{5}k_{q,F}^2\rho _q$, with $k_{q,F}=(3\pi ^2\rho _q)^{1/3}$.