Modifying TALYS to Implement Custom Nuclear Level Densities

Malachi Tolman

Department of Physics, Brigham Young University-Idaho

April 3, 2014

BRIGHAM YOUNG UNIVERSITY-IDAHO

DEPARTMENT APPROVAL

of a senior thesis submitted by

Malachi Tolman

Date Kevin Kelley

Date David Oliphant

Date Richard Datwyler

1

Abstract

Many codes exist that model nuclear reactions. Many of them use different models to simulate reactions. We

are working to use a more modern code (TALYS) which incorporates newer models to replicate calculations

made by an older code (STAPRE). This project focused on modifying TALYS to read in nuclear level

densities from a file and implement them correctly. We hoped that if TALYS used the same level densities

as STAPRE, that it would produce results within 5% of what STAPRE made. However, modifying only the

level densities did not produce the desired results. After adjusting several input parameters, we still find

large differences between the two codes.

2

Contents

1 Introduction 5

1.1 Computational Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Level Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 STAPRE and TALYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Process 9

2.1 Editing TALYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Generating the Level Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Processing Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Results 13

3.1 Level Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 S-wave Radiation Widths and gnorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Discrete Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Conclusion and Future Research 23

3

List of Figures

3.1 % difference vs. A for (n, γ) reactions: custom level density . . . . . . . . . . . . . . . . . . . 14

3.2 (n, 2n) cross sections for custom level density . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 (n, γ) reactions for S-wave radiation modification . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 (n, 2n) cross sections for S-wave radiation modification . . . . . . . . . . . . . . . . . . . . . . 16

3.5 (n, γ) reactions for gnorm flagged inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.6 (n, 2n) cross sections for gnorm flagged inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.7 Fe-52 cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.8 % difference vs. STAPRE cross sections for (n, γ) reactions: gnorm . . . . . . . . . . . . . . . 19

3.9 %difference vs. STAPRE cross sections for (n, 2n) reactions: gnorm . . . . . . . . . . . . . . 19

3.10 Cross sections of Ca-42 for (n, γ) reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.11 Cross sections of (n, 2n) reactions for Ca-42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.12 % difference for (n, γ) reactions for discrete level modification . . . . . . . . . . . . . . . . . . 22

3.13 % difference for (n, 2n) cross sections for discrete level modification . . . . . . . . . . . . . . . 22

4

Chapter 1

Introduction

1.1 Computational Nuclear Physics

An understanding of nuclear reactions aids in a number of applications. It helps us understand nulear energy,

solar processes, and explosive yield via underground weapons testing. But nuclear reactions are complicated

processes that involve a lot of different parameters. And theory has yet to be able to accurately model

a great number of them. Computational nuclear physics seeks to first determine whether a given theory

matches experimental data, then, if it works, it can be used to predict the behavior of processes that can’t

be measured.

A measurement of prime concern is a quantity called a “cross section”. During a particle-nucleus reaction,

cross sections measure the frequency of a given outcome occurring. For example, when firing neutrons at an

iron nucleus, there is a given chance that the nucleus will absorb the neutron, then launch it out again. How

often this occurrs is related to the cross section of that given reaction. Cross-sections vary according to the

energy of the incident particle, so they are normally plotted as a function of incident energy.

1.2 Level Densities

A number of details about the incoming particle and the target nucleus affects the outcome of a particle-

nuclear reaction. The bulk of this research project focused on the level density of different nuclei and their

5

effect on cross sections. To understand level densities, it must first be understood that protons and neutrons

in the nucleus exist in discrete energy levels just like the electrons that surround the nucleus. Due to being

in a bound quantum state, the nucleons can only exist at certain energy levels that have a given spacing

(similar to, n=1 electrons versus n=2). But at high excitation energies, the spacing between each energy level

becomes small enough that the discrete spectrum of energy levels can be modeled continously. From here

it is natural to not talk of individual levels, but instead in terms of level densities. Level densities measure

how many energy levels per MeV exist in the nucleus at a given excitation energy. Different isotopes have

varying energy levels due to factors such as spin, parity, and binding energy.

Level densities have a large impact on cross sections, as illustrated by the famed Hauser-Fechbach [1]

model

σ(E|E′) = σc(E)E′σc(E

′)DR−1(E − E′)∫ E

0

E′′σc(E′′)DR−1(E − E′)dE′′

. (1.1)

Here σc represents the cross section of forming the compound nucleus (the nucleus with the absorbed neutron

before it ejects it). DR−1 is the level density of the target nucleus, demonstrating the dependence that cross

sections have on this parameter.

How to calculate level densities still undergoes changes, today. One common model, as used by Gilbert

and Cameron [2], is the Fermi-gas model which treats the level density as an exponential of the energy of

the nucleus

ρ(E) =

√π

12

exp(2√aU)

a1/4U5/4

1√2πσ

. (1.2)

This model is not valid for nuclei with energies less than 10 MeV, so another model is used at lower excitation

energies, which has the form

N(E) = exp[(E − E0)/T ],

ρ = N(E)/T.

(1.3)

This is the Constant Temperature Model. The parameters E0 and T in equation (1.3) are determined by

matching this equation to the Fermi-gas model at a given energy by point and slope. This matching has to

be adjusted manually case by case to ensure that the integrated level density matches known spectroscopic

levels.

The parameter a in (1.2) changes depending on which model is used to calculate it. For example, Iljinov

6

and Mebel [3] proposed a model where a is dependent on energy and a shell correction function giving

a(U,Z,N) = a(A)[1 + δWg(Z,N)f(U −∆)/(U −∆)]. (1.4)

Here, a is a function of the nucleus’s surface area, δWg is a nuclear mass formula, and f(U) describes

the energy dependence of the parameter with U being the excitation energy and ∆ being the pairing energy.

This prescription has been used for a global fit of a to all isotopes used in modeling solar reactions [4]. It

can also be fit to smaller regions of the periodic table [5].

Yet another parameter that level densities heavily rely upon is the spin-cutoff parameter, the σ in the

Fermi-gas model (1.2). Gilbert and Cameron first proposed that σ be modeled as

σ2 = g⟨m2⟩t, (1.5)

where g is the sum of the neutron and proton single-particle level spacings, m is the familiar magnetic

quantum number, and t is the temperature of the nucleus given by t =√

(U/a). Other prescriptions suggest

σ2 = 0.01496λA5/3

√U

a. (1.6)

Here, λ is related to the effective moment of inertia of the nucleus.

Each time a prescription for level density parameters is modified, editing a multi-thousand line code to

accomodate such changes can be cumbersome. The ideal solution to such is to have an option where level

densities as a function of energy are tabulated offline and then read in by the reaction code. The program

then interpolates the data between the given points, and the data from such is used for nuclear level densities

in calculating the cross sections.

1.3 STAPRE and TALYS

Many different codes have been written to simulate nuclear reactions. They use models such as the ones

listed above to model the factors that go into formulating a cross section. Lawrence Livermore National

Laboratory worked on a project to produce cross sections of certain isotopes to help with weapon testing [5].

They used a code named STAPRE [6] to predict what the cross sections of certain nuclear decays would be.

7

However, STAPRE is somewhat difficult to use and very difficult to modify. This is due to its lack of

comments and poor documentation for tens of thousands of lines of code. TALYS [7] is a modularized,

well-commented, and well documented code, making it easier to modify. It uses more recent models than

STAPRE, so it would be ideal to switch over to using TALYS. However, it was uncertain as to whether

TALYS could produce the same results that STAPRE did given the same inputs. Our objective is to see

just how close TALYS’s cross sections came to STAPRE’s (and experimental data) using the same inputs

that STAPRE used in previous calculations.

But TALYS treats spin-dependence differently than STAPRE does when it comes to computing level

densities. So instead of reprogramming TALYS to be able to handle this dependence, we decided to simply

add an option to read level densities from a table and interpolate the data. In this regard, level densities

could be calculated in a separate program, then TALYS would simply read them in. This project focused on

creating that switch and testing the results. However, after that alone failed to provide satisfactory results,

other minor switches that were already built into TALYS were explored as well.

8

Chapter 2

Process

2.1 Editing TALYS

My first task was to edit TALYS so that it could read in and utilize level density tables. We already knew that

TALYS had a switch for using different level density models. But upon inspection, I found that out of five

pre-made options, two used different tables of tabulated level densities already available in the source code.

The second of the two, the Hilaire-Goriely tables, were more robust in that they included more energies and

also accounted for positive and negative parities. So I utilized the features that accommodated this option

as much as possible when including the switch to read-in level densities from a file.

After reading over the documentation, I started creating the new option in the file input2.f. TALYS reads

in the user’s command file and parses each line into arrays of characters. The six input files located in the

source code search the beginning of each of these arrays for key words. Each key word should be followed by

any number of other commands that specify what is to be done with said key word. Thus, it is in input2.f

that it checks for the key word “ldmod”, the command used to specify which level density model is to be

used.

I made an extra conditional in the block of code involving ldmod to check for “6” among the options

following “ldmod” in the input file. If it does, then it reads in the next “word” as the file destination name.

Because of the limitations of Fortran’s string concatenation, it needs to know the length of the filename. For

9

this, I used an algorithm that was utilized in machine.f to count the number of characters in the cstring. I

also created a check to see if the file name ends with a slash, since the program would require one later on.

If it did not, I added it on the end. As standard with the other models, a check is made to see if the atomic

number and atomic mass were provided. If they weren’t, then TALYS sets a flag to apply the option to all

isotopes. In order for this block of code to work, I added the variables “ldfilename” and “lenldfn” to the file

talys.cmb. These store the file name and length of the file name.

My next step was to find all the cases where ldmod is used and make sure it functioned as desired. As

mentioned, I planned on having the custom table be read-in and utilized the same way that the Hilaire-

Goriely tables were used. I ran the “grep” command to find all the files where such was used. In a few

cases, such as under the files densityout.f, densitytable.f, and input3.f, I didn’t quite understand the meaning

behind the variables being flagged. But I made them flag everything for ldmod 6 as they would for ldmod 5.

Most other cases, the changes were minor and easily understood. The first happened in checkvalue. I had

to increase the range from five to six to make sure that an error message wasn’t produced. In a former case

of densityout, I edited the conditional to include ldmod 6 and write a string stating that a user- input table

is being used. And in strucinitial, a check for ldmod 5 was changed to happen for ldmod 6 as well. This

file determined how many energies existed for each isotope on the table. Thus while the program read and

skipped over certain isotopes, it knew how many lines to pass over.

The file densitytable.f dealt largely with the level density model. A first modification to this file happened

where parity was flagged. I included ldmod 6 in this check so that way positive and negative parity could be

accounted for in separate tables if the user desired. Next the destination file had to be concatenated from

the inputted filename to point to specific files. TALYS works by having all isotopes organized into separate

files by atomic number. So the destination file points to a directory that contains these files. But next in

densitytable was a check for fission barriers. If fission barriers were being used and ldmod 6 was specified, I

changed the level density model to five and outputted a message saying that ldmod 6 did not support fission

barriers. Finally, as previously mentioned, a final flag involving parity was made. So I made it include ldmod

6.

The last modification that had to be made was in densitypar. Again, the destination directory had to

10

be changed to account for the inputted file name. But also here there resided a curious list of parameters.

Upon inspecting the documentation, we found that they were a set of parameters that would shift the level

densities so that the resulting cross sections would match empirical data. In order to be consistent with the

project of getting TALYS to reproduce results using the same inputs as those from STAPRE, I turned these

off in the case of ldmod 6.

2.2 Generating the Level Densities

Once we knew that TALYS could read in level densities from a file, my next task was to create files that had

tabulated level densities at given energy levels which were consistent with the models used in STAPRE. To

do this, I modified a program named rld9. This program takes Fermi-gas parameters from a file, and fits the

integrated level densities to known spectroscopic (low-lying) levels. It then prints level density parameters

in a format that STAPRE can read.

However, the file format for this program had been changed since it was last used to test level densities

for STAPRE. So I had to take the old prms files and convert them into a format compatible with the newer

version. An odd bug was occurring with reading in the old tables that disallowed the ISYMM parameter to

be read in as an integer. A mismatched data type would occur whenever I attempted to read it in through

either the ifstream or scanf functions. But after using the atoi function in c++, I was able to get the tables

to convert correctly.

Once I accomplished that, I modified rld9 to output level density tables in a format compatible with

TALYS. This happened mostly within the file “fitcts” in the rld9 source code. If the user presses “w” during

the program, instead of only writing the parameters of the currently selected isotope, I changed it to write

level densities for every isotope in the prms file. It organized the isotopes into separate files by atomic

number as per the format that TALYS uses. These separate directories could then be copied and moved to

run with TALYS.

11

2.3 Processing Cross Sections

The final step was to calculate the cross sections for all the reactions included in the sets calculated using

STAPRE. This was done using resources at the Fulton supercomputing lab at Brigham Young University

[8].

I first had to modify a program designed to run TALYS on a parallel architecture. The program, suitably

named “runtalys”, required minimal modifications. I made one change in the file config.c in the source

code to name the variable LEVELDENSITYDIR instead of the previous LEVELDENSITYFILE. The main

change happened in writeinp.c, where I changed the old file read in option to one that suited the directory

destination name. Since this program was modified from one that ran STAPRE, I commented out a great

amount of code that was designed to write out the level density tables from the program.

Once the program was modified, I imported all of the edited TALYS code, the discrete level density

tables, and the new runtalys program into my account with Fulton. I created separate scripts for each batch

of isotopes according to the separate groups of discrete level densities created by rld9. I then submitted the

batches. There were over 600 isotopes (each with 32 reaction channels) that were processed. Each isotope

took about five minutes to calculate the cross sections. Before I copied the results from my account to my

hardrive, I removed a number of files from the output directory that I wouldn’t use in my analysis.

12

Chapter 3

Results

3.1 Level Density

After the cross sections were processed, it was easiest to observe the difference between the TALYS and

STAPRE cross sections by taking a percentage difference between the two at a given energy for each isotope.

To observe the (n, γ) cross sections, I created a script that went through every TALYS and STAPRE cross

section at 30 keV and listed their respective cross sections at that point in a list. A percentage difference

between the two was calculated by taking the TALYS value minus the STAPRE value divided by the STAPRE

value. Then, I used gnuplot to plot the percentage difference against the atomic number. We chose to observe

the difference at this particular energy since it corresponds to the typical temperature in stellar reactions.

The same process was done for (n, 2n) reactions, except the difference was analyzed at 14.1 MeV. The

reason that this energy was used was because it is the temperature of neutrons emitted in deuterium-tritium

fusion, a process that power thermonuclear weapons and fusion reactors.

The first results of this process can be seen below. As mentioned, the first set of cross sections were to

see if TALYS could replicate STAPRE’s results by using the same discrete level densities. The results were

as to be expected for only trying one switch. After seeing that some cross sections were up to 300% different,

we tried some different inputs.

13

Figure 3.1: Percentage difference vs. atomic number of (n, γ) reactions for custom level density. The

percentage difference is represented in decimal form.

Figure 3.2: (n, 2n) cross sections for custom level density

14

3.2 S-wave Radiation Widths and gnorm

The next switch to be analyzed was using the same S-wave radiation widths. This was already built into

TALYS, and the file that STAPRE used was already converted into the appropriate format for TALYS and

ready to be used. Importing the file into runtalys took only one line of declaring which file should be read in

for the S-wave radiation widths. This would only have a profound effect on the (n, γ) cross sections. After

the cross sections were processed, the same means of plotting the results were used. They can be seen in

figures 3.3 and 3.4.

As can be seen, this had no affect on the cross sections. However, we quickly realized that one other

parameter had to be entered for the radiation widths to have any meaning. So the cross sections were repro-

cessed using a flag named “gnorm”. This caused all the radiation width calculations to actually normalize

to the inputted parameters. The results of such were analyzed and can be seen in figures 3.5 and 3.6.

This had a profound affect on the ng cross sections. They still were not brought to be within the 5-10%

range that was hoped for, but at least it came down to 80% from the original 300% difference.

One thing to note was how the percentage difference depended on the cross section itself, since if the

cross sections themselves were small at that given energy, then a small modification to the cross section

would induce a drastic change in the percentage difference. In the case of the (n, 2n) reactions, the threshold

energy sometimes existed above the energy level we were measuring at. This meant that some cross sections

would be zero, although both STAPRE and TALYS printed the smallest number they could to prevent errors

associated with calculating using zeros.

To see how this overall affected the results, I generated plots of the percentage difference against the

STAPRE cross section. Doing such showed that for the (n, 2n) reactions, this affect was taking place, since

the largest percent differences only occurred in regions where the cross section was the smallest. But for

(n, γ) reactions, this wasn’t entirely the case. The most extreme cases occurred where the cross section was

small, but other cases that were outside the desired target range occurred where the cross section was largest.

As a final check, we observed what kind of isotopes were experiencing the largest difference in cross

sections between the two codes. If the isotopes creating the large percentage difference were unstable, then

there would be no way to tell if TALYS was not matching empirical data. As originally noticed, almost all of

15

Figure 3.3: (n, γ) reactions for S-wave radiation modification

Figure 3.4: (n, 2n) cross sections for S-wave radiation modification

16

Figure 3.5: (n, γ) reactions for gnorm flagged inputs

Figure 3.6: (n, 2n) cross sections for gnorm flagged inputs

17

Figure 3.7: Cross-sections generated by TALYS and STAPRE for iron-52. The bar represents the energy at

which the percentage difference was being measured, and the actual cross sections that were measured at

that point can be seen in the key. This was an obvious case of how the shifted peaks in the cross sections

affected the analysis of percentage difference for (n, 2n) reactions.

18

Figure 3.8: Percentage difference vs. STAPRE cross section for ng reactions for gnorm flagged inputs

Figure 3.9: Percentage difference vs. STAPRE cross section for n2n cross sections for gnorm flagged inputs.

This set follows a tight correlation while the previous doesn’t follow a trend as well.

19

the cross sections that lied above 60% difference were unstable isotopes. However, there was a single isotope

within this range that was stable, calcium-42. The plots for such can be seen below. Here, STAPRE does

match current empirical data, while TALYS is consistently lower for both reactions.

3.3 Discrete Levels

The final attempted modification occurred with changing the discrete level files to be the same as the ones

that STAPRE used. These were placed directly into the TALYS structure directory. The analysis of the

resulting cross sections can be seen in figures 3.12 and 3.13.

These adjustments actually had a slight worsening effect. The largest percentage difference was made to

go out to 100% on the (n, γ) cross sections while the (n, 2n) cross sections remained about the same.

20

Figure 3.10: Cross sections of calcium-42 for STAPRE and TALYS. STAPRE matches empirical data on

this isotope, while TALYS is significantly lower, albeit has the same shape. The vertical bar represents the

energy at which the percentage difference was measured for (n, γ) reactions, the the values listed in the key

are the measured cross sections at that value.

Figure 3.11: Cross sections of (n, 2n) reactions for calcium-42 for STAPRE and TALYS. Like the ng plot,

TALYS has the same shape, but is lower than STAPRE.

21

Figure 3.12: % difference for (n, γ) reactions for discrete level modification

Figure 3.13: % difference for (n, 2n) cross sections for discrete level modification

22

Chapter 4

Conclusion and Future Research

As can be seen from the previous chapter, the results weren’t what we had hoped for. Although the results

from TALYS managed to be able to draw closer to those of STAPRE with a few switches, they never did

come within the encompassing 5% range.

Depending on how thorough future researchers want to be, an analysis of the exact implementation of

the level densities would make a difference in seeing exactly where things are going so differently for the

two codes. However, for now, the changes made to TALYS already have been incorporated and are being

submitted to Lawrence Livermore National Laboratory to see if they can utilize them.

Besides just the switches that this project focused on, it was noted that TALYS uses models for direct

reactions that cannot be turned off (direct reactions are a different reaction mechanism not incoporated in

STAPRE that can have a small contribution to cross sections below 20 MeV incident energy). And since

STAPRE does not consider direct reactions at all, that could be one point of discrepancy. But again, it all

depends on how thorough future researchers want to be to figure out how to account for the affect such a

thing is making.

Once the discrepancies are found, one breakthrough in nuclear modeling predicts that K-mixing could

have a larger impact on large mass nuclei cross sections than originally thought. K-mixing is the term used

to denote the selection of the z-component of angular momentum within the nucleons. In all models used so

far, the angular distribution amongst the nucleus is assumed to be spherical. However, accounting for more

23

exact angular distributions could affect cross sections up to 30%, namely among mishapen, massive nuclei

[9].

A project undertaking to include this kind of model into TALYS could be looking at years of research.

Changing the angular distribution in such a way could affect a number of parameters such as transmission

coefficients and level densities. Thus both of these and others would need to be carefully considered to

observe what changes would need to be made.

24

Bibliography

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Canadian Journal of Physics, 87, 1965.

[3] A.S. Iljinov and M.V. Mebel. Phenomenological statistical analysis of level denities, decay widthsand

lifetime of excited nuclei. Nuclear Physics, A543:517–557, 1992.

[4] Thomas Rauscher, Friedrich-Karl Thielemann, and Karl-Ludwig Kratz. Nuclear level density and the

determination of thermonuclear rates for astrophysics. The American Physical Society, 56(3), November

1997.

[5] R.D. Hoffman, K. Kelley, F.S. Dietrich, R. Bauer, and M.G. Mustafa. Modeled neutron and charged-

particle induced nuclear reaction cross sections for radiochemistry in the region of yttrium, zirconium,

niobium, and molybdenum. Lawrence Livermore National Laboratory, June 2006.

[6] M. Uhl and B. Strohmaier. Stapre a computer code for particle induced activation cross sections and

related quantities. IRK - Vienna Report, IRK 76/01, 1976.

[7] Arjan Koning, Stephane Hilaire, and Stephane Goriely. TALYS-1.4 A nuclear reaction program. Nuclear

Research and Consultancy Group (NRG), December 2011.

[8] Fulton supercomputing lab. http://marylou.byu.edu/documentation/resources.

[9] S.M. Grimes. Physical Review C, 88(2).

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