Kox and Shen Formulae

Both the Kox and Shen formulae are based on the strong absorption model. They express the total reaction cross section in terms of the interaction radius $R$, the nucleus-nucleus interaction barrier $B$, and the center-of-mass energy of the colliding system $E_{CM}$:

$$\sigma_{R} = \pi R^{2}[1-\frac{B}{E_{CM}}].$$ (16.2)

Kox formula: Here $B$ is the Coulomb barrier ($B_c$) of the projectile-target system and is given by

$$\begin{eqnarray*} B_{c}=\frac{Z_{t}Z_{p}e^{2}}{r_{C}(A^{1/3}_{t}+A^{1/3}_{p})}, \end{eqnarray*}$$

where $r_{C}$ = 1.3 fm, $e$ is the electron charge and $Z_t$, $Z_p$ are the atomic numbers of the target and projectile nuclei. $R$ is the interaction radius $R_{int}$ which in the Kox formula is divided into volume and surface terms:

$$\begin{eqnarray*} R_{int}=R_{vol}+R_{surf} . \end{eqnarray*}$$

$R_{vol}$ and $R_{surf}$ correspond to the energy-independent and energy-dependent components of the reactions, respectively. Collisions which have relatively small impact parameters are independent of both energy and mass number. These core collisions are parameterized by $R_{vol}$. Therefore $R_{vol}$ can depend only on the volume of the projectile and target nuclei:

$$\begin{eqnarray*} R_{vol}=r_{0}(A^{1/3}_{t}+A^{1/3}_{p}) . \end{eqnarray*}$$

The second term of the interaction radius is a nuclear surface contribution and is parameterized by

$$\begin{eqnarray*} R_{surf}=r_{0}[a\frac{A^{1/3}_{t}A^{1/3}_{p}}{A^{1/3}_{t}+A^{1/3}_{p}}-c]+D. \end{eqnarray*}$$

The first term in brackets is the mass asymmetry which is related to the volume overlap of the projectile and target. The second term $c$ is an energy-dependent parameter which takes into account increasing surface transparency as the projectile energy increases. $D$ is the neutron-excess which becomes important in collisions of heavy or neutron-rich targets. It is given by

$$\begin{eqnarray*} D=\frac{5(A_{t}-Z_{t})Z_{p}}{A_{p}A_{r}}. \end{eqnarray*}$$

The surface component ($R_{surf}$) of the interaction radius is actually not part of the simple framework of the strong absorption model, but a better reproduction of experimental results is possible when it is used.

The parameters $r_0$, $a$ and $c$ are obtained using a $\chi^{2}$ minimizing procedure with the experimental data. In this procedure the parameters $r_{0}$ and $a$ were fixed while $c$ was allowed to vary only with the beam energy per nucleon. The best $\chi^{2}$ fit is provided by $r_{0}$ = 1.1 fm and $a = 1.85$ with the corresponding values of $c$ listed in Table III and shown in Fig. 12 of Ref. [2] as a function of beam energy per nucleon. This reference presents the values of $c$ only in chart and figure form, which is not suitable for Monte Carlo calculations. Therefore a simple analytical function is used to calculate $c$ in Geant4. The function is:

$$\begin{eqnarray*} c=-\frac{10}{x^{5}}+2.0 \mbox{ } \rm {for} \mbox{ } x \ge 1.5 \end{eqnarray*}$$

$$\begin{eqnarray*} c=(-\frac{10}{1.5^{5}}+2.0)\times(\frac{x}{1.5})^{3} \mbox{ } \rm {for} \mbox{ } x < 1.5 , \end{eqnarray*}$$

$$\begin{eqnarray*} x=log(KE) , \end{eqnarray*}$$

where $KE$ is the projectile kinetic energy in units of MeV/nucleon in the laboratory system.

Shen formula: as mentioned earlier, this formula is also based on the strong absorption model, therefore it has a structure similar to the Kox formula:

$$\sigma_{R} = 10\pi R^{2}[1-\frac{B}{E_{CM}}].$$ (16.3)

However, different parameterized forms for $R$ and $B$ are applied. The interaction radius $R$ is given by

$$\begin{eqnarray*} R=r_{0}[A^{1/3}_{t}+A^{1/3}_{p}+1.85\frac{A^{1/3}_{t}A^{1/3}_{... ...-1/3}_{CM}\frac{A^{1/3}_{t}A^{1/3}_{p}}{A^{1/3}_{t}+A^{1/3}_{p}} \end{eqnarray*}$$

where $\alpha$, $\beta$ and $r_0$ are

$$\begin{eqnarray*} \alpha = 1 fm \end{eqnarray*}$$

$$\begin{eqnarray*} \beta = 0.176MeV^{1/3} \cdot fm \end{eqnarray*}$$

$$\begin{eqnarray*} r_{0}= 1.1 fm \end{eqnarray*}$$

In Ref. [3] as well, no functional form for $C'(KE)$ is given. Hence the same simple analytical function is used by Geant4 to derive $c$ values.

The second term $B$ is called the nuclear-nuclear interaction barrier in the Shen formula and is given by

$$\begin{eqnarray*} B=\frac{1.44Z_{t}Z_{p}}{r}-b\frac{R_{t}R_{p}}{R_{t}+R_{p}} (MeV) \end{eqnarray*}$$

where $r$, $b$, $R_t$ and $R_p$ are given by

$$\begin{eqnarray*} r=R_{t}+R_{p}+3.2fm \end{eqnarray*}$$

$$\begin{eqnarray*} b=1MeV\cdot fm^{-1} \end{eqnarray*}$$

$$\begin{eqnarray*} R_{i}=1.12A^{1/3}_{i} -0.94A^{-1/3}_{i} ~ (i=t,p) \end{eqnarray*}$$

The difference between the Kox and Shen formulae appears at energies below 30 MeV/nucleon. In this region the Shen formula shows better agreement with the experimental data in most cases.