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East Lansing 07-Mar-2002

Fusion residue transmission

1. How to calculate fusion residue transmission (step by step)

2. Characteristics of fusion residue calculations

3. Other topics

References

A new fusion-evaporation model "LisFus" for the fast calculations of fusion residue cross sections has been developed. This model is based on the Bass fusion cross-section algorithm [Bas74], the evaporation cascade code of Gaimard and Schmidt’s Abrasion-Ablation model [Gai91], and the transport integral theory [Baz94]. The user has also the possibility to use fusion residue cross sections calculated by the program PACE, which has been incorporated in the LISE package. The code PACE [Gav80] (Projection Angular-momentum Coupled Evaporation) is a modified version of JULIAN - the Hillman-Eyal evaporation code using a Monte-Carlo method coupling angular momentum. The program LISE has now the possibility to calculate the transmission of fusion residues through a fragment separator. The transmission calculations have to be performed following the steps described below:

1. Choosing reaction type
2. Beam, Target, Residue settings
3. Options
4. Residue excitation function
5. Setting of residue ion charge
6. Target thickness
7. Separation

Choose the "Fusion → Residue" option in the "Production mechanism" dialog (menu "Options"). There are no other settings for the "Fusion → Residue" reaction in this. At this point the SETUP window will show:

The user can input an expected thickness target value if he or she is sure that this thickness will be satisfactory for this beam-target-residue combination, but it is recommended to calculate a thickness value by following the next steps.

For fusion residue reactions it is strongly recommended to SET the "Charge State" option ON! To decrease the calculation time (fusion calculations take more time than fragmentation) it is recommended to use the following values:

The "Prefragment search" part of this dialog will be blocked when the reaction "Fusion→residue" has been chosen. For fast calculations it is recommended to use

1.4. Residue excitation function The next task is to determinate the excitation energy value of the compound nucleus corresponding to the maximum of the residue production cross section. To do this the user can load the subroutine to plot the excitation function one of two possible ways: from the "Fusion-residue information window" dialog (see Fig.1, item A), or from the "Plots → Cross section distributions" menu. If the reaction mode "Fusion → Residue" is selected, the "Cross sections" dialog (menu "Options"), and the "Cross section Plot" dialog (menu "Plots", see Fig.2) will correspond to the fusion-residue reaction. As an example the excitation function of the 42Ca residue from the 49Ti compound is shown in Fig.3 (it takes only 20 seconds on average to perform these calculations!!!). Note: It is possible to skip this step if the "Cross sections from file" option in the "Preferences" dialog is set, and set the input cross section value for the chosen residue independent from energy in the "Cross sections" dialog (menu "Options"). The program will automatically try to calculate "LisFus" fusion-residue cross sections and show them in the "Cross sections" dialog, but the user can break out of the calculation by clicking Cancel in the "Choose a range to calculate" dialog and input his or her own cross section value.

Fig.1. The Fusion-Residue information window. It is activated by clicking the “C”ompound button in the SETUP window (see chapter 1.1.).

Fig.2. The "Fusion cross section plot" dialog. To obtain the excitation function of a residue the user has to choose the options "Energy dependence" and "Calculate a residue excitation function — YES". The recommended number of points for the fusion plot is 32.

The subroutine "Optimal target plot" (menu "Calculations") is used to get the maximum rate of the chosen residue. The target thickness calculations for the residue ⁴²Ca^19+ 19+ are shown in Fig.5. The user can click the button to put the calculated values into the program settings.

This step is the most time consuming. A benchmark table (Table 1) for various configurations is given below. A Pentium III (600 MHz, 128MB RAM) computer has been used for the calculations.

The final step is the calculation of the selection of the chosen residual by the separation devices to maximize transmission and purity. In today’s version, four methods of separation are available in the program LISE: magnetic rigidity, energy loss in a wedge located in the dispersive focal plane, Wien velocity filter, and combination of a Wien filter and a dipole. The most popular method of separation for small energies is velocity separation (velocity filter). At small energies using energy loss in a wedge is ruled out. In the future electrostatic and gas-filled separators will be available in the program LISE. A comparison of different methods of selection is shown in Table 2 and Fig.6, where it is apparent that the "Velocity Filter + Dipole D6" (VAMOS) method gives the best results: best purification and highest intensity of the chosen residue.

Fig.6. dE-E plots for different types of selection. The most intense peak on all pictures corresponds to the ion ⁴²Ca¹⁹⁺.

The yield of reaction products (I_sec) selected by a fragment separator depends from the primary beam intensity (I_beam), the number of atoms in the target (N), the production cross section (CS), and the transmission (coef) through the fragment separator: I_sec = I_beam× CS× N× coef. Assuming I_beam and N are constant the remaining parameters to calculate are the cross section of residue production and the transmission. In order to calculate the transmission it is necessary to know:

This chapter is devoted to the principles of cross section and kinematics distribution calculations.

The residue velocity after reaction (fusion and evaporation of light particles) is assumed to be equal to the compound nucleus recoil velocity. A Maxwell distribution of velocities is used to calculate the root-mean-square velocity after evaporation of light particles. To simplify calculations and to get the result in an analytical form it was assumed that:

• Each step represent only one-nucleon evaporation
• The excitation energy of the daughter nucleus on each step is a delta function
• The separation energy of one nucleon is equal to (ME_compound - ME_residue)/(A_compound-A_residue)

In this case it is possible to consider the final distribution as the convolution of (A_compound-A_residue) gaussian distributions with . Thus the width of the final gaussian distribution is equal to  and the root-mean square velocity is determined as (the parallel and perpendicular components of the momentum distribution are   respectively).

Fig.7. Fragment of the PACE4 result window. The results of the root-mean square velocity subroutine are shown in a red rounded rectangle.

The initial excitation energy of the compound nuclei is taken as the average of excitation energy between the beginning and the end of the target. A root-mean-square velocity calculation subroutine has been added to the PACE4 program for the "beam & target" mode to check this assumption (see Fig.7).

Comparisons between the LISE and PACE4 calculations were made for different combinations of "beam & target & excitation energy". The residue velocity calculations performed by the PACE4 program using a Monte Carlo method are close to the LISE simplified model (see examples in Fig.8). Some discrepancies are observed for high excitation energies of heavy compounds (see the right bottom plot in Fig.8) but for the fast calculations performed in the LISE program the agreement is acceptable. The results from the PACE program were used only for nuclei which output exceeded three hundred events.

Fig.8. The widths of velocity distribution versus the mass number of residue calculated by the PACE4 code (red rectangle) and the LISE program (black curve) for three different reactions ¹²C+¹²C (left), ⁴⁸Ca+⁹Be (middle), and ⁴⁸Ca+¹²⁴Sn (right) at several excitation energies.

Angular distributions in the center of mass system are assumed isotropic. The calculated widths of longitudinal and transverse momentum distributions are used with the formula from the work [Bor83] to get the angular distributions of residues.

The program assumes that the targets used are thin enough and that the cross section of residue formation is constant within the target thickness. Since the calculation of the cross section is the most time consuming, this assumption greatly shortens the overall calculation time. The program is performing the following steps to calculate the cross section of a fusion residue:

• The target is divided into N slices (N is the dimension of the evaporation distribution defined in the "Search of prefragment and evaporation options" dialog)
• The energy of the primary beam (E_i) is calculated for each slice i of target thickness due to energy loss
• The excitation energy of the compound nucleus (E_i^*) is calculated from the primary beam energy (E_i);
• The fusion cross section (s _i) is calculated;
• The s (E^*) distribution (N+1 points) is filled with values of s _i and E_i^*;
• The area of this distribution is normalized to the average value 
• This excitation function of compound nucleus is used as initial distribution to calculate the cross sections of residues with the help of the Abrasion-Ablation evaporation subroutine.

It is possible to see the evolution of the excitation functions using the Evaporation calculator in the mode "Excited nucleus evaporation".

In order to avoid long calculation times the program asks a minimum atomic number (Z) down to which to calculate the cross sections. The user can interrupt the calculations at any time by pressing the Escape key. In the case of the optimum thickness calculation the program automatically considers the cascade only down to the nucleus of interest. All values of calculated cross sections are kept in memory. The number of kept cross sections is displayed in the "Setup" window . These cross sections are not saved in the LIZ-file and are erased from memory if the following changes are made:

• Type of reaction, projectile or target
• Coefficients used for evaporation calculation
• Number of evaporation channels
• Dimension of evaporation distributions
• If the target thickness or the beam energy have been changed by at least 2%

The main advantage of the LISE code calculations ("LisFus") of residue formation cross section is the speed. The user can get the excitation function in a short time (typically less than one minute, if the residue is close to the compound nucleus, and the dimension of distributions is not to high). Only such type of fast calculations is suitable for the computation of the transmission and yield of fusion residues in a reasonable time.

In order to test the validity of the approximations used in the LisFus model, comparisons with calculations from the PACE program were made for the reaction ⁴⁰Ar + ⁹Be at various energies of the primary beam. The maximum angular momentum was taken from the optical model used in the PACE4 calculations. The calculations for the ⁴⁹Ti excited compound nucleus were carried out at a maximum spin of 0.5 h. The excitation functions of some residues from the reaction 40Ar + 9Be are shown in Fig.9. Overall there is a quite good agreement between the calculations from the LisFus model and the PACE4 program except for the titanium isotopes. All excitation functions of the LisFus model for titanium isotopes have a systematic shift in the region of low excitation energies. This can be explained by the fact that at small excitation energies the program PACE predicts the deexcitation of nuclei by emission of gamma-rays, whereas in the LisFus model this decay channel is not included, as the only evaporation channels are emission of light particles.

If the primary beam is among the transmitted in fusion residue mode, then the LISE program automatically switches to the fragmentation mode to calculate the transmission of the primary beam. It then returns in the fusion residue mode to get the transmission of this nucleus assuming it arose from the de-excitation of the compound nucleus.

The user can get the transmission window corresponding to the primary beam by clicking the right button of the mouse on the primary beam cell in the table of nuclides (see Fig.10). Both types of reactions can be seen, assuming that their transmission is not equal to zero. A label indicating what kind of reaction is responsible for the given charge states of the nucleus has been added after the transmission parameter number 24 (see the rounded rectangle in Fig.10). This option has been made especially to prevent from sending the primary beam into the detectors. The magnetic rigidity distributions of the transmitted charge states of ⁴⁰Ar ion are shown in Fig.11.

It is also interesting to compare the outputs of fragmentation and fusion-evaporation reaction products and their energy distribution not only for the case of the primary beam, but for other nuclei as well. Performing this comparison requires the following conditions:

•   The energy of the primary beam should neither be too high, so that the cross section for fusion-evaporation is still significant, nor too low, so that projectile fragmentation can still occur

•   The overlap region between fragmentation and fusion-evaporation corresponds usually to a residue far enough from the compound nucleus, where the excitation energy of the compound is high

•   Both reactions coexist only for light targets where the compound is not too far from the primary beam

•   The nucleus of interest to be produced from projectile fragmentation must have less neutrons and protons than the primary beam

Fig.12. Energy distributions of ³²Al ions after the target in the reaction ⁴⁰Ar (40 MeV/u) + Be (75 mg/cm2). The energy distribution for the fragmentation reaction is shown by the black curve, and for the fusion residue reaction by the red curve.

As an example the reaction ⁴⁰Ar (40MeV/u) + ⁹Be (75mg/cm²) -› ³²Al has been chosen. The energy distributions of the ³²Al ions after the target for both types of reactions are shown in Fig.12. For this case the products from both reactions can be well separated in the experiment. The velocity of the projectile fragment was assumed to be equal to the beam velocity and the Goldhaber momentum distributions was used to calculate the ³²Al energy distribution for the fragmentation reaction.

It is now possible to build two-dimensional plots of cross sections, transmission and database characteristics. Before plotting the data the user can choose in the dialogs what kind of plot he or she wishes. An example of a two-dimensional plot is shown in Fig.13.

A considerable change has taken place in the calculation of the daughter nuclei excitation functions of the evaporation cascade model. In the previous versions it was assumed that each i-th point of an intermediate nucleus excitation function had only one projection in the daughter nucleus excitation function:

,             /1/

where S is the separation energy, B is the reduced coulomb barrier in the case of a charge particle evaporation, and t is the temperature of the daughter nucleus. The value in this i-th point of the daughter distribution was equal to the multiplication of the i-th value of the intermediate nucleus and the probability for this daughter evaporation channel:

Stopping power functions (dE/dx) have been incorporated in the lise_xls.dll  library and can be loaded as a function in Excel through the LISE.xls  file (see Fig.15).

Fig.15. Fragment of the Excel window. The block with new functions “Stopping power” is indicated by the red rounded rectangle.

Fruitful discussions with A.Andreev, and M.Toennessen are gratefully acknowledged.