# TFF (v.4.1): A Mathematica Notebook for the Calculation of One- and Two-Neutron Stripping and Pick-Up Nuclear Reactions

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## Abstract

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## 1. Introduction

`typewriter`fonts are used to express file names, variables, functions and commands, while mathematical formulas are in italics as usual.

## 2. Physics Background

`samegeom`, is used to calculate a transmission factor as a function of the impact parameter,

`occB`and

`occb`in the code, contain all needed nuclear structure information and must be externally supplied. They can be taken from shell model configurations, Hartree–Fock calculations, Bardeen-Cooper-Schrieffer (BCS)-type calculations, etc.

`Q1p`in the input file). The two-neutron transfer reaction requires a ground-state (g.s.) to g.s. Q-value, called

`Q2p`, that must be supplied in the

`input.nb`file. The coupled channel equation for the two-neutron case is solved numerically, and the results are interpolated real and imaginary parts of the transfer amplitudes (that are displayed as a function of the integration time). These are used to calculate and plot transfer probabilities, ${P}_{2}=\Sigma \mid {a}_{2}{\mid}^{2}$, as a function of the impact parameter. The probabilities are folded with the transmission factor (that has been calculated in the trajectory section) to obtain transfer cross-sections as a function of b as in Equation (9). One should notice the important role of the absorption factor in determining (cutting) the differential cross-section distributions at low impact parameters. There is no theoretical derivation for this parameter, which controls the depth of the imaginary potential with respect to the real ion-ion potential, and the usual practical solution is to take it close to 1/2 or 1/3. Clearly, little changes in this parameters affect the final calculations, possibly more than the little imprecision in the wavefunctions or binding energies.

#### Angular Distribution

## 3. Flowchart Diagram and User Instructions

`input.nb`that has the Mathematica notebook format, as well. The comments and example clarify each entry.

#### 3.1. Directories and Files

`wrkdir`, must be specified in the first cell. We usually employ the scheme of naming the directory with the reactants of the form ${Z}_{A}{A}_{A}$_${Z}_{a}{A}_{a}$, for example

`208Pb_18O`or

`112Sn_14C`, using the charges and masses of A and a in this order. This directory contains a copy of the input file called

`input.nb`. This file can of course be renamed, and more than one file can be present in each directory because one might want, for instance, to run the code with different linear combinations of single-particle (s.p.) states, or with different absorption factors, or different potentials, etc. The

`input.nb`file specifies which single-particle states must be used and the names of the files containing the radial wavefunctions as a function of r. These files must be stored in the

`Waves`directory because several reactions (read different working directories) might want to access the same standard single-particle wavefunctions. Of course, by using properly-chosen labeling schemes, one can have different versions for the same single-particle state (for instance, one might want to investigate what happens if a given s.p. state is obtained from different programs or obtained at the same energy, but with different diffusivities and potential depths, etc.). We use a natural naming scheme for wavefunction files:

`AZz_nl(2j)%2.dat`, where

`A`is the mass number,

`Zz`is the chemical symbol and

`nl(2j)`are principal, orbital and total angular momentum quantum numbers. The outcomes of the calculations, i.e., form factors and differential cross-sections, are written in

`wrkdir`. Form factors, once computed, are also read from that directory (therefore, these files should not be moved or renamed).

#### 3.2. Block 1

`.dat`, but

`.txt`or any other suitable extension would work, as well). These files can be read in a variety of ways, but we suggest that the wavefunction files should be set up with two columns, one for the radial variable in fm and one for the normalized wavefunction. The spacing in the radial variable is not important, nor must it necessarily be constant, because these wavefunctions are later interpolated. Next, spins, orbital and total angular momenta are read. These depend on the channel being studied and must correspond to the supplied wavefunctions. Constants are in nuclear units; masses must be given (both integers and actual masses) in atomic mass units. The potential depths used to generate the single-particle wavefunctions must be supplied in

`V0B`and

`V0b`. Then, the spin-orbit potential must also correspond to the potential used to generate the single-particle wavefunctions. Single-particle data are collected in a formatted table after reading the

`input.nb`file. Notice that one can keep several different input entry files ready and just change the file name in the

`filetoberead`string. The shell model potentials are then defined, and then, an ion-ion potential is defined following the Akyüz–Winther prescription, with an imaginary part of the same geometry: this is controlled by the absorption factor

`absf`, usually set to 1/2. The geometry might also be altered, by setting

`samegeom = False`, and introducing the proper values for the Akyüz–Winther parameters of the imaginary part. Alternatively, by setting

`akyuz = False`, the ion-ion potential is taken as a generic Woods–Saxon potential, and the parameters are read from the

`input.nb`file. The results, in terms of cross-sections, are significantly altered by small changes in the absorption because the transmission factor is folded with the probability in Blocks 6 and 7. The potential for the reactants is then plotted, and the height and position of the barrier are calculated. The ion-ion potential is also calculated for final and intermediate reaction products, because this is used in the computation of the $\gamma $ exponents entering in the coupled channel equations.

#### 3.3. Block 2

`Waves`directory. Of course, the path to this directory can be changed, as well as the files’ names. This block (and subsequent Block 3) must be read at first run for the calculation of form factors, but can be skipped in subsequent calculations, if the form factors have been stored.

#### 3.4. Block 3

`ListPlot`, for example) and set the variable

`interpolationRadius`to a proper value. Depending on the size of the nuclei and especially on the position of the nodes of the integrand (that somewhat reflects the nodes of the wavefunctions), we have used values of the order of 20 fm for ${}^{17}$O and ${}^{207}$Pb and up to 50 fm for more weakly-bound isotopes, the idea being that

`interpolationRadius`is bigger than the last node, and therefore, the interpolation is made in the exponentially-decaying region. Notice that even a very tightly-bound nucleus might have a weakly-bound and broadly-extended orbital close to the threshold. The results of form factor calculations are written to conventionally-named files for each particular channel. The naming scheme used here is

`FF_x${}_{1}$y${}_{1}$z${}_{1}$%2-x${}_{2}$y${}_{2}$z${}_{2}$%2_lam = $\lambda $.dat`, where x${}_{i}$y${}_{i}$z${}_{i}$ corresponds to the set of quantum numbers $n\ell \left(2j\right)$ of the initial and final single-particle states and $\lambda $ takes all possible values (remember that normally, the larger the $\lambda $, the larger the form factor value). Since the symbol / is not allowed in the file name, the %2 notation allows a straightforward reading, for example 1d3%2 or 2s1%2, etc.

#### 3.5. Block 4

#### 3.6. Block 5

`Elab`in the code (alternatively, the center of mass bombarding energy can be given by inverting the formula in the cell where energy is defined), the initial and final distances,

`rin, rf`, that are usually taken as equal and large (typically 100 fm). They correspond to the disk of integration of the equations of motion. The trajectory is integrated in a time interval that goes from

`−tmax`to

`+tmax`, repeatedly, for a large number of different impact parameters. Some of these parameters are read in the

`input.nb`file, and others can be adjusted in the main program itself, for example the integration time and disk radii. Notice that, for computational purposes, an integer impact parameter,

`m`, is introduced. It has been decided that

`m`$=b/10$, where b is in fm. The trajectory is solved for all $b=0.1,0.2,0.3,\dots $ up to ${b}_{max}\sim 20$ fm. This value corresponds to

`mmax`= 200 in the code. This value can be increased at will (at the price of slowing the calculations), for instance when one thinks that 20 fm is not enough to take into account the grazing distance if the two interacting nuclei are very large (which might happen for super-heavy or very weakly-bound orbitals). The result is a set of numerical solutions or interpolations (called

`Eq`), labeled by

`m`, that contain all of the trajectory information, distance, angle and conjugate momenta: $r\left(t\right),\varphi \left(t\right),{p}_{r}\left(t\right),{p}_{\varphi}\left(t\right)$, which can be evaluated and plotted. Notice that ${p}_{\varphi}$, which is connected to the total angular momentum, is a constant of motion, as it is expected for the motion in a plane. For completeness, we give also a plot of the trajectory in the $x,y$ reference frame.

#### 3.7. Block 6

`m`in the

`NDSolve`cell. Amplitudes are then plotted, and the probability is calculated and plotted. These plots are nonsensical at very low impact parameters, as explained above, and this region can be neglected. Based on the latter and on the transmission factor, the differential cross-section $d\sigma /db$ is calculated, plotted and integrated to give the total cross-section in mb . The differential cross-section can also be written to a file with the last cell.

#### Updates

`extSmat.ph`, with a very generic format, namely two numbers and a string (that can also be empty), the numbers being the real and imaginary part of the S-matrix as calculated by some other program (in particular, FRESCO can be used). The phase shifts are then calculated and used instead of the one calculated in the program. This possibility must be chosen in the input file by setting the flag

`external = True`.

#### 3.8. Block 7

`NDSolve`cell integrates the second order coupled equations along the trajectory to get the excitation amplitudes. This calculation might take some time depending on the number of channels and on the computer performances, but for typical case (10–20 channels), it runs in a few minutes on an Intel

^{®}Core™ i7-2640M CPU at 2.80 GHz (two cores, four logical processors) with 8 Gbytes of RAM (Windows 7 O.S.). Several amplitudes and probabilities are plotted, and a number of numerical checks is proposed. Finally, the probability is folded with the transmission factor, and the cross-sections (differential and total) are calculated and plotted. The limit of validity of the numerical subroutine that solves the coupled differential equations is very small and should not, likely, be of any practical relevance. The code gives plots of $d\sigma /db$ for each pair of pure (single-particle)${}^{2}$ configurations and then gives the cross-section for the special configuration given by

`occb`and

`occB`coefficients. Notice that the values for pure configurations are unaffected by these coefficients, and the users can change the occupation amplitudes on-the-fly in the input file, run the Input evaluation cell in Block 1 and then re-run the Cross-sect for chosen configuration cell in Block 7, without running other cells.

#### Updates

## 4. Model Calculations and Examples

`wrkdir`is set to be

`16O_208Pb`. The package comes with additional text files in the

`Waves`directory, for the radial wavefunctions for the two nuclei, namely

`17O_1d5%2.dat`and

`17O_2s1%2.dat`, with a self-explaining naming scheme, that contain two columns of data each (r and ${R}_{n\ell j}\left(r\right)$) for single particle wavefunctions of a neutron in the field of ${}^{16}$O and

`207Pb_3p1%2.dat`for the neutron hole in the ground state of ${}^{207}$Pb. In the actual Waves directory, there are also other s.p. states of lead that can be used to run more complicated cases (see below). The scheme for the two-neutron reaction with the two proposed channels is illustrated in Figure 2. Of course, this is just a very simplified model scheme; a proper calculation should entail at least the three lowest states of ${}^{207}$Pb with all possible values of $\lambda $, and the absorption factor should be carefully chosen in order to reproduce the data. When every variable, from paths and file names to quantum numbers and reactions parameters, is properly set, the first task is to compute and store form factors for all possible channels, which in general include not only the combinations $1{d}_{5/2}\to 3{p}_{1/2}$ and $2{s}_{1/2}\to 3{p}_{1/2}$, but also the possible $\lambda $’s. Fortunately, here is the case for a single value of $\lambda $ in each case, $\lambda =3$ and $\lambda =1$ , respectively. We also furnish, as a mean of comparison, the form factors for these two channels in the files

`FF_1d5%2-3p1%2_lam=3.dat`and

`FF_2s1%2-3p1%2_lam=1.dat`. The former is plotted in logarithmic scale in Figure 3. This information can be used to check Blocks 1–3 along the flowchart of Figure 1.

`occb[1]`= 1), should give 0.267993 mb. Clearly, these numbers do not perfectly match the published data in [13], because there has been no detailed search on the optimal modeling of potentials, wavefunctions, ion-ion interaction, absorption factors, and so on. In fact, it is probably necessary to reduce the absorption factor to about 1/3 and adjust the potentials’ geometry to match the root mean square radii. Notice that the input file contains three states for lead, but the parameter

`nstb`has been set to one, so that only the g.s. is read. The reader might want to set it to three and run the code in order to see a more complicated case. If everything has run properly, taking all spectroscopic amplitudes equal to unity for simplicity, the final outcome should be ${\sigma}_{2n}=0.68384$ mb.

#### 4.1. Practical Suggestions

`input.nb`file; there is no re-ordering or sorting of any type. The channel’s numbers (for the particular configuration) simply count the different $\lambda $’s in ascending order. These numbers are useful when the reaction calculations are performed in Blocks 6 and 7, and the user should return to this table for the interpretation of the results. It is often useful to know the Coulomb barrier height before choosing a laboratory or center of mass energy in the input file: one could run Block 1 with a dummy energy datum, inspect the barrier height from the plot potentials cell, correct and save the

`input.nb`file and then run again the input evaluation cell. Always remember to inspect the bombarding energy that is given in the trajectory parameters, auxiliary functions cell in Block 5. The Example cells in Blocks 6 and 7 run with all entries for channels equal to one, because this ensures that they will always run. Of course, the user might want to check another channel and a different impact parameter (remember that m is ten times b). In these cells, $\mu =\lambda $ for simplicity.

#### 4.2. Operating Systems

## 5. Conclusions and Warnings

`rin`). It might be the case that at high energies, the velocity is quite high, and there is no time for multiple interactions between target and projectile (in that case, one should try and reduce the time steps). Another possibility is the appearance of orbiting phenomena, for a certain given impact parameter, which are usually quite small and tend to be washed out by the transmission factor distribution. To keep this under control, the cell trajectory integration in Block 5 plots the time of closest approach as a function of the impact parameter: a cusp behavior might mark an incomplete orbiting. Each case must be examined anew, and a proper check of the subroutine that solves the coupled differential equation should be performed by inspecting the outcome of the integration.

#### 5.1. Approximations and Limitations

#### 5.1.1. No Recoil

#### 5.1.2. Semiclassical Trajectories

`rin`.

#### 5.1.3. Sequential 2n Transfer

#### 5.1.4. No Coupling to Other Channels

#### 5.2. Work Plan for Future

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Flowchart diagram of the code. Blocks (corresponding to groups of cells) are indicated with dashed black boxes. Inside each block, data reading is indicated as yellow cards, algorithms are indicated as blue rectangles and outputs as green parallelograms. Red rhombi define decisional branching points. Notice that Blocks 2 and 3 can be skipped when form factors are already known (i.e., when Block 2 plus Block 3 have been successfully run in a previous calculation).

**Figure 2.**Scheme of channels for the example reaction${}^{16}$O(${}^{208}$Pb, ${}^{206}$Pb)${}^{18}$O. Only two states of oxygen are used for the sake of simplicity. More accurate schemes must contain all relevant single-particle states in both target and projectile and, for each combination, all possible values of $\lambda $.

**Figure 3.**Form factor for the $1{d}_{5/2}\to 3{p}_{1/2}$ and $2{s}_{1/2}\to 3{p}_{1/2}$ transfers with $\lambda =3$ and 1 respectively as a function of the ion-ion distance r.

**Figure 4.**Transfer probability, transmission factor and the resulting differential cross-section ($d\sigma /db$) are displayed as a function of the impact parameter b for a typical case. Small changes in the imaginary potential affect the transmission factor $T\left(b\right)$ and consequently the cross-section profile of the lower panel. The present simplified example does not refer to the case discussed in Section 4.

**Figure 5.**Angular distribution ($d\sigma /d\mathsf{\Omega}$) as a function of the angle $\theta $ for the two states of ${}^{17}$O considered in the paper. This has been obtained with the parameters I3 of [13]. The small mismatch with respect to the results obtained there can be attributed to small differences in the code and can be easily fixed with small adjustments in the parameters.

**Table 1.**Summary of user-defined variables. If the index

`i`is present, it refers to the particular state from 1 to

`nstB`or

`nstb`. Target and projectile are referred to with

`B`and

`b`, while

`c`labels the channel.

Code Variables | Explanation | Units or Values |
---|---|---|

mA, ma, mB, mb | Mass | amu |

AA, Aa, AB, Ab | Mass number | - |

ZA, Za, ZB, Zb | Atomic numbers | - |

lB[i], lb[i] | Orbital angular momentum of B, b | - |

jB[i], jb[i] | Total angular momentum of B, b | - |

sB[i], sb[i] | Spin of single-particle in B, b | 1/2 |

V0A[i],V0b[i] | Potential depth (volume) in B and b | MeV |

VlsB[i],Vlsb[i] | Spin-orbit potential depth (surface)) in B, b | MeV |

a | Diffusivity | fm |

absf | Absorption factor | 0.5 (typically)) |

akyuz | Select Akyuz-Winther or optical pot. | False/True |

samegoem | Select equal geometry for W and V | False/True |

external | Select external phase shifts | False/True |

interpolationRadius | Interpolation radius | fm |

$\lambda $ | Angular momentum transfer | - |

$\nu $ | Order of recoil power expansion | 0 |

En | Center of Mass bombarding energy | MeV |

rin | radius of integration disk | fm |

rffmin, rffmax | Form factors range of r | fm |

tmax | ‘Asymptotic’ time | $\pm 40$ fm/c |

m, (mm, mdum) | Integer impact parameter, $b/10$ | fm/10 |

EB[i], Eb[i] | Neutron binding energies | MeV |

SpecB[i], Specb[i] | Single particle energies above the g.s. | MeV |

Q1p | Q-value for one-neutron transfer | MeV |

Q2p | Q-value for two-neutron transfer | MeV |

occB[i], occb[i] | Occupation amplitudes | - |

chn[B,b] | Channel number | |

$\lambda $[B,b,c] | Angular momentum transfer of channel c | - |

howmany | Number of channels | - |

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## Share and Cite

**MDPI and ACS Style**

Fortunato, L.; Inci, I.; Lay, J.-A.; Vitturi, A. TFF (v.4.1): A *Mathematica* Notebook for the Calculation of One- and Two-Neutron Stripping and Pick-Up Nuclear Reactions. *Computation* **2017**, *5*, 36.
https://doi.org/10.3390/computation5030036

**AMA Style**

Fortunato L, Inci I, Lay J-A, Vitturi A. TFF (v.4.1): A *Mathematica* Notebook for the Calculation of One- and Two-Neutron Stripping and Pick-Up Nuclear Reactions. *Computation*. 2017; 5(3):36.
https://doi.org/10.3390/computation5030036

**Chicago/Turabian Style**

Fortunato, Lorenzo, Ilyas Inci, José-Antonio Lay, and Andrea Vitturi. 2017. "TFF (v.4.1): A *Mathematica* Notebook for the Calculation of One- and Two-Neutron Stripping and Pick-Up Nuclear Reactions" *Computation* 5, no. 3: 36.
https://doi.org/10.3390/computation5030036