lise.nscl.msu.edu http:"//lise.nscl.msu.edu

East Lansing 12-JUN-2001

Evaporation Calculator
Abrasion-Ablation Model

1. Introduction
2. Evaporation Calculator

Epilogue
References

Various versions of the EPAX parameterization [Sum00] have been used  to calculate projectile fragmentation cross-sections. The basic advantages of the parameterization are the following:

• Simple and fast calculations of real cross-sections using good enough approximation;
• Direct (one-step) calculation.

However, there are some disadvantages:

• No target dependence;
• The parameterization does not reproduce observed odd-even effects
• Parameterization estimations are inexact for lightest nuclei and nuclei near the drip-lines.

In further developments of the program LISE other kinds of reaction, in addition to projectile fragmentation will be introduced. One candidate is complete fusion with subsequent evaporation and/or fission. These reactions require an in built procedure for calculating light particle evaporation by exited nuclei.

This has resulted the creation of a new calculator called "Evaporation Calculator" [Chapter 2.]. Using this new tool the user can calculate the production cross-sections of different nuclei as a result of deexcitation of an excited nucleus. This calculator is based on the work by [Gai91]. The introduction of the evaporation algorithm allows  incorporation of an additional method for calculating projectile fragmentation cross section using an Abrasion-Ablation model [Chapter 3.].

The Evaporation calculator is available in the Calculations menu or via the icon in the toolbar (see Fig.1). The "Evaporation Calculator" dialog is shown in Fig.2. The production of ³²Mg nucleus from deexcitation of ⁴⁰Ar is presented in this figure as an example. The excitation energy distribution of ⁴⁰Ar in this example is gaussian, with the excitation energy window (50 MeV, 100 MeV) as the left and right points at the gaussian half-height.

The basic differences between the work [Gai91] and the LISE algorithm are the following:

• J.-J.Gaimard & K.-H.Schmidt [Gai91] - determines average values (A, I ) of a final daughter nucleus with the normal distribution of width t .
• The LISE code assumes that each nucleus is individual. The program sums all input (parent) energy distributions (including initial abrasion cross-section for fragmentation). Then it calculates output (daughter) distributions and determines the part of the input distributions that was not included in the output. The area of this part of input distribution is equal to the production cross-section of this nucleus as a result of the deexcitation of the initial nucleus (see Fig.3).

In this sense the LISE method is unique: it carries in itself the best features of Monte Carlo evaporation programs (energy distributions, odd-even effects, "individuality" of each nucleus) and one of the advantages of analytical methods [Gai91] - fast calculations.

The user can observe results of the evaporation calculator using plots of input and output energy distributions for a given nucleus (see Fig.3) and a two-dimensional plot of cross-sections (see Fig.4). The 2D-dimensional plot is a new feature in the program. The data is represented by rectangles centered at N (neutrons) and Z (protons), where the cross-section value depends on color and size of the rectangle. The user can change the display modes of rectangles (see Fig.5), and also change the scale (see Fig.6).

In Fig.3 and Fig.8 plots are shown with two green vertical lines corresponding to the minimum separation energy (left line) and the minimum sum of the separation energy and the effective Coulomb barrier (right line). The part of the distribution (A) located to the left of a line does not undergo evaporation (see Fig.8). The deexcitation proceeds by emission of g -rays. It is also assumed, that the part of the distribution between these two lines (B) does not undergo the further evaporation (see Fig.8, left plot). The effective barrier assumes a shift of the right line to the left, assuming a tunneling effect. The default value dR = 6 fm is taken from work [Ben98]. The effective Coulomb barrier is written as:

In Fig.8 (right plot) an example of a real situation including a tunneling effect is shown (this is still in a stage of development). Not all part would remain in the given fragment in this case in the B zone. The standard Coulomb barrier is used in this case:

The use of the effective barrier can have various consequences. For example, when the sum of the separation energy and the Coulomb barrier is positive, the nucleus is bound. However it is not excluded that at the same time the sum of the separation energy and the effective Coulomb barrier is negative (as the effective Coulomb barrier is less the standard barrier). In this case the nucleus is unbound and the program will show zero cross-section for its production.

This option includes calculating unbound nuclei suggesting via evaporations channels, for example, the most probable path of ³²Ne nucleus deexcitation is ³²Ne^*-> ³¹Ne-> ³⁰Ne in the ⁴⁰Ar+Be reaction. This option brings contribution for nuclei near the drip-lines. Cross-sections of ²⁴O, ²⁹F, ³⁰Ne are increased about 1-3% with this option from a test in the ⁴⁰Ar+Be reaction.

In the LISE code, a simple theory of fragmentation based upon a two-step abrasion/ablation model is presented [Wil87]. The abrasion process accounts for removal of nuclear matter in the overlap region of the colliding ions. An average transmission factor is used for the projectile and target nuclei at a given impact parameter to account for the finite meanfree path in nuclear matter. The ions are treated otherwise on a geometrical basis assuming uniform spheres. The surface distortion excitation energy of the projectile prefragment following abrasion of nucleons is calculated from the clean-cut abrasion formalism of [Gos77]. Wilson’s model also includes (see Chapter 3.1.2. Excitation energy):

• The excitation energy correction factor when large portions of the nucleus are removed in the collisions;
• The transfer of kinetic energy of relative motion across the intersecting boundary of the two ions (friction).

User may access the abrasion-ablation setting via the "Evaporation settings" dialog (see Fig.7).

There are restrictions in the geometrical abrasion-ablation model when light targets are used. For example, it is impossible to get a prefragment mass less than 22 in the ⁴⁰Ar+Be reaction. In other words the target cannot in this geometrical assumption make an aperture in a particle more than the volume proportional to mass 18 nucleons. Further prefragment deexcitation will not allow to be lowered more than on 5 units, though in experiments all spectrum of elements inclusive up to hydrogen is observed (see Fig.9). Corrections have been incorporated in the program LISE to solve this restriction. The code extrapolates exponentially the dependences of cross-sections and excitation energies from the prefragment mass in the break point in a lightest targets case (Fig.10).

Fig.9. Cross-sections plot for the ²⁰⁸Pb + Be reaction. LISE corrections for Geometrical Abrasion-Ablation model are switched OFF.

The user can vary excitation energy options in the "Excitation energy of prefragment" dialog (Fig.11). This dialog is available the "Options" menu or the "Evaporation settings" dialog (see Fig.7). The excitation energy calculation is based on the work of [Wil87].

Fig.11. Prefragment excitation energy settings.

The user can change coefficients of correction factors in this dialog. Original coefficients in the work [Wil87] for correction factor expression (17) are 15 and 25. We assume, that these coefficients are erroneous. Real values should be in the range 1.5 and 2.5 correspondingly. For example, if to suggest a projectile half-part abrasion the correction factor will be equal to 14.75! Comparing excitation energy with the work [Gai91], more real vale of the correction factor should be about 2.

Two additional methods of excitation energy calculation are planed in the next version. The first method is based on the diabatic model [Gai91] (convolutions of several linear distributions), the second one is a simplified approach from the first method:

Cross-sections calculated by the abrasion-ablation model may be used for other calculations into the program (see Fig.12). Cross-sections are saved in the memory (see Fig.1) and will be recalculated if the projectile (target) is changed.

All remarks, comments for the Evaporation calculator and Abrasion-Ablation model are welcome!