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# 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 , the nucleus-nucleus interaction barrier , and the center-of-mass energy of the colliding system :
 (16.2)

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

where = 1.3 fm, is the electron charge and , are the atomic numbers of the target and projectile nuclei. is the interaction radius which in the Kox formula is divided into volume and surface terms:

and 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 . Therefore can depend only on the volume of the projectile and target nuclei:

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

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

The surface component () 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 , and are obtained using a minimizing procedure with the experimental data. In this procedure the parameters and were fixed while was allowed to vary only with the beam energy per nucleon. The best fit is provided by = 1.1 fm and with the corresponding values of 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 only in chart and figure form, which is not suitable for Monte Carlo calculations. Therefore a simple analytical function is used to calculate in Geant4. The function is:

where 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:

 (16.3)

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

where , and are

In Ref. [3] as well, no functional form for is given. Hence the same simple analytical function is used by Geant4 to derive values.

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

where , , and are given by

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.

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