# Estimating angular momenta of fission fragments from isomeric yield ratios
using TALYS

Z. Gao<EMAIL_ADDRESS>A. Solders<EMAIL_ADDRESS>A.
Al-Adili  S. Cannarozzo  M. Lantz  S. Pomp  Department of Physics and
Astronomy, Uppsala University, 75120 Uppsala, Sweden

###### Abstract

Background

Angular momenta of fission fragments considerably higher than that of the
fissioning nucleus have been observed in many experiments, raising the
question of how these high angular momenta are generated. J. Wilson et al.
have proposed a model for the angular momentum as a function of the mass of
fission products based on the assumption that the angular momentum is
generated from the collective motion of nucleons in the ruptured neck of the
fissioning system. This assumption has caused a lot of debate in the
community.

Purpose

To estimate the angular momenta of fission fragments based on the observed
isomeric yield ratios in 25-MeV proton-induced fission of 238U and to compare
these using Wilson’s model.

Method

A surrogate model of the fission code GEF has been developed to generate
properties of primary fission fragments. Based on the excitation energy and
angular momentum of fission fragments from GEF, an energy versus angular
momentum matrix is reconstructed using a set of parameters. With such matrices
as input, the reaction code TALYS is used to calculate the de-excitation of
the fission fragments, including the population of the isomers, from which the
isomeric yield ratios are obtained. By varying one of the parameters, the
root-mean-square angular momentum ($J_{rms}$), which determines the angular
momentum distribution of the matrix, $J_{rms}$-dependent isomeric yield ratios
are obtained. Considering all primary fission fragments contributing to the
isomeric yield ratio for a given fission product, the average angular momentum
of those fragments is estimated.

Results

Data of 31 isomeric yield ratios in 25-MeV proton-induced fission of 238U were
analysed. As a result, the average $J_{rms}$, equivalent to average angular
momentum $J^{av}$, with uncertainties of 24 fission products, are presented.
However, no conclusive result for the angular momentum could be obtained in
seven cases. Considering the neutron emissions of the primary fission
fragments, the average angular momentum as a function of the average mass
number of the primary fission fragment is presented.

Conclusion

A mass dependency of the average angular momentum is observed in the proton-
induced fission of 238U. Moreover, the average angular momenta for mass
numbers larger than 131 could be described by the parameterisation proposed by
J. Wilson. However, the average angular momenta of 130Sn and 131Te can not be
described by Wilson’s model, which suggests a different lower limit for the
validation of the parameterization in the model. In general, higher average
angular momenta for A$\geq$ 132 are observed in the present work compared to
those from Wilson et al. This is likely due to the higher excitation energy of
the fissioning nuclei in this work compared to Wilson’s. Furthermore,
systematic measurements of the average angular momenta of fission products in
the symmetric mass region are presented for the first time. In this region, a
decreasing trend with mass number is observed, which can not be explained by
Wilson’s proposal. A different mechanism is needed to explain this
observation.

††preprint: APS/123-QED

##  I Introduction

Since the discovery of nuclear fission, considerable effort has been made to
understand and describe this phenomenon. The scissioning nucleus, being a
quantum many-body system, can not easily be described on the microscopic
level, although considerable advancements have been made in the past decades
[1, 2], and as a complement, macroscopic and phenomenological models are often
used [3, 4, 5, 6, 7]. To optimize and validate these models, accurate
experimental data from a broad range of fission reactions and observables are
needed. One of the long-standing questions in fission dynamics is the large
angular momenta of the fission fragments. Angular momentum of fission
fragments have been derived from experimental observables via indirect methods
[8, 9, 10, 11, 12, 13, 14, 15], and the results show that the root-mean-square
angular momenta typically are around 6 - 7 $\hbar$. This is, in many cases,
considerably higher than those of the fissioning nuclei, which raises the
question of how these large angular momenta are generated in the fission
process.

J. Wilson et al. [15] have proposed that the angular momentum is generated
after scission from the collective motion of nucleons in the ruptured neck of
the fissioning system. This conclusion is based on observed average angular
momenta deduced from $\gamma$-ray spectroscopy, which seems to suggest that
the angular momentum of a fission fragment is independent of that of the
complementary fragment. Wilson et al. also propose a model for the angular
momentum as a function of mass for fission products in the low and high mass
peaks. However, there is no systematic study, and hence no model, for the
angular momenta of fission fragments in the symmetric mass region.

In nuclear fission, the prompt energy release, in which the fission fragments
(FF) emits prompt neutrons and $\gamma$-rays to form fission products (FP),
happens on a timescale of 10-14 s [16]. This makes direct observations of the
properties of the fragments, including excitation energies and angular
momenta, impossible with current techniques. Hence, different methods have
been developed to deduce the angular momenta from other fission observables.

For example, in the Manchester Spin Method (MSM) [10], the angular momentum is
derived from the intensity difference between the ingoing and outgoing
$\gamma$-ray transitions of one excited state. As a result, the average
angular momenta can be deduced from $\gamma$-ray spectroscopy data [15].

In early attempts, a statistical method based on the Rayleigh distribution
[17, 18], used to describe the angular momentum distribution of the excited
nucleus, was used to derive the angular momenta from measured isomeric yield
ratios [8, 9, 12, 11]. The isomeric yield ratio (IYR), in general terms
referring to the relative yield of different long-lived states of a fission
product, is in this context more specifically defined as the relative yield of
the high spin state to the total yield of the fission product.

The Madland-England model [19], which is commonly used in data evaluations to
predict IYR that have not yet been measured, is based on the statistical
method. In the Madland-England model, the angular momenta, assumed to be the
same for all fragments from a specific fission reaction, is related to the IYR
through the spin divider $J_{c}$, which is determined from the spins of the
isomeric states. The model ignores angular momentum removal by neutrons and
gammas. Although the model can be used to estimate certain trends in the IYRs,
it has been shown to be too crude to give reliable results for angular
momentum determinations [20, 21].

The GEneral description of Fission observables (GEF) model is a publicly
available, open-source code which models the whole fission process based on
nuclear data, while at the same time conserving the fundamental laws of
physics [3]. In GEF, a simplified de-excitation process of the primary FFs
determines the correlation of the IYR with the angular momenta, using the same
assumption as the Madland-England model. While the internal parameters of GEF
are optimized to reproduce fission observables from neutron-induced and
spontaneous fission, the predictability of observables from other fission
reactions, such as proton-induced fission, is less good [21]. Furthermore, in
cases where GEF fails to reproduce experimental observations, it is not
straightforward for the user to tune the relevant parameters, for example, to
obtain estimates of the angular momentum of the FFs.

Other fission models, such as MCHF [4], CGMF [5], FIFRELIN [6] and FREYA [7],
use the Hauser-Feshbach formalism [22] to describe the de-excitation of the
fission fragments, linking the angular momenta to the IYRs. However, these
models also use assumptions and model parameters to describe the path from
compound nuclei to fission fragments, which are not validated with the nuclear
data from proton-induced fission. In this study, GEF is only used to provide
the initial states of the fission fragments, which will be tuned in the
following calculations. Hence, GEF is good enough to be employed in this
study.

In an alternative approach to link the angular momenta to the measured IYRs,
the reaction code TALYS [23], which also uses the Hauser-Feshbach formalism
for the deexcitation, has been used to calculate the yields of the isomers
starting from different angular momentum distributions [24, 13, 14]. In these
studies, a clear and strong correlation between the angular momentum and IYR
has been observed. Based on that correlation, the average root-mean-square
angular momentum $J_{rms}$, determining the angular momentum distribution, has
been deduced. However, also in this approach, several assumptions and
simplifications are made that might affect the results. For example, the
contribution from each FF to the production of the isomer is not calculated.
Without that information, it is too simple that three FFs or even fewer are
considered in calculating the isomeric yield ratio. The calculated IYR from
each FF is compared with the experimental value to deduce the $J_{rms}$, and
the fission yields are not considered in calculating the average $J_{rms}$.
Furthermore, the excitation energy of the excited FFs is either modelled using
only the mean energy or as a Gaussian distribution. In both cases, the
excitation energy is assumed to be independent of the angular momentum
distribution. This means that when the value of $J_{rms}$ is increased to an
extreme value, the population of the FFs might exceed the yrast line, which is
nonphysical.

In order to have a more realistic description of the angular momentum and
energy distributions of the FFs, TALYS has been coupled to GEF. In this
approach, TALYS is used to calculate the yield of the isomers, starting with
the properties of the fission fragments extracted from GEF simulations. The
results were evaluated by comparing the derived IYR with measured values, and
the conclusion was that the angular momenta of the FFs from GEF was
underestimated [21].

The present work builds on the GEF + TALYS approach by developing a surrogate
model for GEF, in which the $J_{rms}$ of the angular momentum distributions
for the FFs can be varied without crossing the yrast line into an unphysical
region. The goal is to estimate the most likely angular momenta of the fission
fragments contributing to the yields of the isomers in the 31 measured IYRs
from 25 MeV proton-induced fission of 238U [21].

##  II Method

###  II.1 Surrogate model

In the development of the model, GEF 2023/1.2 was used to provide the
excitation energy (Ex) and angular momentum (J) distributions of the primary
fission fragments in proton-induced fission of 238U at 25 MeV. Based on this
GEF data, the energy vs angular momentum matrix (Ex-J) of the excited FFs was
parameterised. These parameters could then be used to reconstruct the matrix
and use it as input to TALYS, in order to derive the IYR. Thanks to the
parameterisation, the parameter determining the angular momentum distribution
of the input matrix could be tuned to reproduce the measured IYR.

####  II.1.1 GEF: Ex-J matrix

Due to the high incident proton energy, up to three neutrons can be emitted
before scission, resulting in four paths of fission in GEF. This pre-scission
neutron emission changes the scissioning nucleus (SN) as well as its
excitation energy. Hence, the properties of the primary fragments produced
from different scissioning nuclei will be different.

As an example, Figure 1 shows the Ex-J matrix of the excited FF 133Te*
produced from the scissioning nucleus 237Np* obtained from GEF, in color-
filled contours. In the left and upper panels, the matrix is projected to the
energy axis and angular momentum axis, respectively.

Figure 1: Ex-J matrix of the FF 133Te* produced from the scissioning nucleus
237Np* obtained from GEF, in color-filled contours. In the main panel, black
points and error bars represent means and standard deviations of energies of
the nuclei with a specific angular momentum. The red points show a fit to the
average energy versus angular momentum. Red error bars are the standard
deviation of energies, which are the same for all angular momentum. The left
panel shows the excitation energy distribution from GEF in black points and a
fitted normal distribution, N(19.6 MeV, 9.6 MeV), in red. The upper panel
shows the angular momentum distribution from GEF in black points and a fit of
the Rayleigh distribution, giving $J_{rms}=9.2~{}\hbar$.

In GEF, the angular momentum distribution of a particular FF follows the
Rayleigh distribution [17, 18],

$P(J)=\frac{2\cdot J+1}{2\sigma^{2}}e^{\frac{-J(J+1)}{2\sigma^{2}}},$ (1)

in which the only parameter, the spin cut-off $\sigma$, determines the mean
and the width of the distribution. As a result, the root-mean-square angular
momentum ($J_{rms}$) from such a distribution is given by,

$J^{2}_{rms}=2\sigma^{2}-\sqrt{0.5\pi}\sigma+0.25.$ (2)

In the literature, this is generally approximated by
$J_{rms}\approx\sqrt{2}\sigma$, despite it being off by approximately 0.5
$\hbar$. As seen in the upper panel of Figure 1, the angular momentum
distribution is well described by a Rayleigh distribution with
$J_{rms}=9.2~{}\hbar$.

The energy distribution, shown in black points in the left panel of Figure 1,
is reasonably well described by a normal distribution N(19.6 MeV, 9.6 MeV),
except for the energy above 30 MeV. Such a distribution is good enough to
serve as a surrogate model.

If, instead, the data is analysed for one value of the angular momentum at a
time, the resulting energy distributions are well described by normal
distributions. The means and standard deviations of these distributions are
presented as black points with error bars in the main panel of Figure 1. It
can be noticed that the standard deviations of the individual distributions
are approximately the same, and can hence be replaced by their average value
$\bar{\sigma}_{E}$.

From Figure 1, it is observed that the mean energy for each angular momenta
($\bar{E}_{ex}$) correlates with the angular momenta. Following the
parameterisation suggested by Al-Adili  _et al._ [25] the relations ship can
be described by

$\bar{E}_{ex}(J)=C\cdot J(J+1)+\bar{E}_{offset},$ (3)

in which $C$ and $\bar{E}_{offset}$ are free parameters. In the case of 133Te,
the correlation is written as $\bar{E}_{ex}=0.037\cdot J(J+1)+16.0$ MeV.

To summarise, a set of four parameters: $J_{rms}$, $C$, $\bar{E}_{offset}$,
and $\bar{\sigma}_{E}$, are needed to describe the Ex-J matrix of a single FF
from one particular scissioning nucleus. Considering the four possible
scissioning nucleus in the fission of 238U at 25 MeV, a total of 20
parameters, also including the four probabilities for each scissioning nuclei,
are needed to describe the Ex-J matrix. As an example, the values of 20
parameters describing the four Ex-J matrices for 133Te* are listed in Table 1.

Table 1: Parameters describing the Ex-J matrix of the excited FF 133Te* obtained from GEF. $P_{SN}$ represents the probability of producing the FF 133Te* from that particular scissioning nucleus. SN | $P_{SN}$ | $J_{rms}$ | $C$ | $\bar{E}_{offset}$ | $\bar{\sigma_{E}}$
---|---|---|---|---|---
|  | ($\hbar$) |  | MeV | MeV
239Np∗ | 0.10 | 10.8 | 0.025 | 35.1 | 9.4
238Np∗ | 0.20 | 10.7 | 0.027 | 34.1 | 11.2
237Np∗ | 0.37 | 9.2 | 0.037 | 16.0 | 7.8
236Np∗ | 0.33 | 7.6 | 0.036 | 7.6 | 4.9

####  II.1.2 Reconstruction of the Ex-J matrix

The Ex-J matrix of any excited FF could be easily reconstructed using the
parameters obtained from the GEF data. However, as the aim is to construct a
surrogate model in which the angular momentum can be easily varied, having
four different values for $J_{rms}$ is cumbersome. As a simplification, the
four different values of $J_{rms}$ presented in Table 1 are replaced by a
single value. This value is obtained by fitting the total angular momentum
distribution, from all scissioning nuclei, with a single Rayleigh
distribution. Together with the values of the other parameters ($C$,
$\bar{E}_{offset}$ and $\bar{\sigma_{E}}$ in Table 1) this is used to
reconstruct the Ex-J matrices from the different scissioning nuclei. Finally,
the four reconstructed matrices are weighted with their respective
probabilities ($P_{SN}$ in Table 1) and added into one single matrix.

Figure 2 shows the reconstructed Ex-J matrix of the FF 133Te*, which is a
weighted sum of four matrices. Each matrix is reconstructed using a set of
three parameters: $C$, $\bar{E}_{offset}$, and $\bar{\sigma}_{E}$ from Table
1, and $J_{rms}$ = 9.0 $\hbar$. Projections of the matrix to the energy and
angular momentum axes are presented in red curves in the left and upper
panels, respectively. The corresponding projections of the Ex-J matrix from
GEF are presented as black lines. The projected angular momentum of the matrix
is a good approximation of the GEF data. This means that the single values of
$J_{rms}$ could be used to replace the four values in Table 1, without serious
loss of information.

Figure 2: Reconstructed Ex-J matrix of the primary FF 133Te, for all four
scissioning nuclei. The upper and left panels show the angular momentum
distribution and the excitation energy distribution from GEF in black dots and
corresponding projections of the reconstructed matrix as red curves.

To study how the angular momentum affects the de-excitation of the FF
$J_{rms}$ is varied, and the resulting matrices are fed to TALYS. As an
example, Figure 3 shows a reconstructed matrix with a $J_{rms}$ of
$15.0~{}\hbar$. In the upper panel of Figure 3 it can be seen that the
reconstructed angular momentum distribution changes as expected. At the same
time, the energy distribution is also significantly changed (left panel of
Figure 3) due to the dependency on the angular momentum described by Eq 3.

Figure 3: Reconstructed Ex-J matrix with $J_{rms}=15.0~{}\hbar$ for the FF
133Te* based on the output from GEF. The upper and left panels show the
angular momentum distribution and the excitation energy distribution from GEF
in black dots and corresponding projections of the reconstructed matrix by red
curves.

###  II.2 De-excitation with TALYS

TALYS [23] is a general nuclear reaction code commonly used in data
evaluation. In this study, the latest version, TALYS 1.96 [26], is used to
calculate the de-excitation of excited FFs generated by the surrogate model.

The calculations in TALYS were, for each fragment, initialized with a
reconstructed Ex-J matrix from the surrogate model. However, TALYS is
inherently limited to a maximum of 31 discrete values of the angular momenta
in the initialisation of the excited nucleus. This means that for large values
of $J_{rms}$, the distribution will be cut. As an example, in the case of
133Te, for a $J_{rms}$ of 20 $\hbar$, the integrated probability of spins
exceeding the limit ($J~{}=~{}30.5~{}\hbar$) is about 10 $\%$, and as a
result, the spin distribution will deviate from the Rayleigh distribution. To
avoid large systematic effects of this limitation in TALYS, $J_{rms}$ was
limited to a maximum of 25 $\hbar$ in this study.

To predict the level densities in the continuum above the highest discrete
energy level, and to supplement the discrete levels whenever the level scheme
is incomplete, the Back-shifted Fermi gas Model (BFM) [27] was used as the
default option, In earlier studies, impacts of the choice of level density
model on the calculated IYRs have been studied [24, 13]. Different level
density models were therefore tested, and the BFM model showed better
performance than the other models in most cases. However, for cadmium, indium
and antimony, the microscopic level densities calculated by S. Goriely showed
a better performance and were therefore used instead.

As pointed out by V. Rakopoulos [14], and also observed in the current work,
the knowledge of the discrete level scheme can have a significant impact on
the calculation of the IYRs in TALYS. Hence, the level schemes used in the
calculations were updated with the latest experimental data retrieved from the
RIPL-3 database [28]. In all calculations, TALYS was set up to use all
experimentally known levels up to the fortieth level, which is the upper limit
in TALYS.

Provided with an input file, TALYS calculates the de-excitation of the excited
nucleus and prints the results to an output file. From this output, the
population of the long-lived high-spin ($P_{hs}$) and low-spin ($P_{ls}$)
states are extracted. For the FF 133Te*, three of the products after de-
excitation, for which experimental data of the IYRs are available for
comparison, are 129,131,133Te. Figure 4 shows the population of these three
products as a function of $J_{rms}$ in colored solid curves, while the black
curve shows the total population of the three products. Considering that the
population is normalized to the population of the initial fragment, about half
of the FFs 133Te∗ de-excites to either of these three products. Hence,
observations from these three FPs provide an important part of the information
necessary to estimate the $J_{rms}$ of the FF 133Te*.

Figure 4: Populations of the fission products from the de-excitation of 133Te*
as a function of $J_{rms}$.

Figure 4 also shows that the population of the high spin state has a different
dependency on $J_{rms}$ compared to the total population. As a result, the
IYR, representing the relative yield of two states, will depend on the
$J_{rms}$. This dependency will be discussed further in the next section.

Furthermore, it can be observed in Figure 4 how the summed population of two
states of 129Te increases with $J_{rms}$, while the population of 133Te
decreases. The population of 131Te increases up to about $10.6~{}\hbar$, and
then starts to decrease. Such dependencies are not surprising, considering
that the excitation energy of the fragment increases with angular momentum
(see Eq. 3), leading to larger neutron emission.

###  II.3 Angular momentum

From the results of the TALYS simulations, the populations of the high spin
and low spin states are used to calculate the IYRs,

$R=\frac{P_{hs}}{P_{hs}+P_{ls}}.$ (4)

As described above, multiple neutron emission is considered in the de-
excitation. Hence, each FP can be produced from multiple excited FFs. As an
example, the derived IYRs of 131Te, as a function of $J_{rms}$ of the
contributing FFs, are presented in Figure 5.

Figure 5: Calculated IYRs of the FP 131Te from all excited FFs, 131-139Te*,
contributing to the production of 131Te. The horizontal line and grey region
represent the measured IYR with uncertainties of the FP 131Te [21]. The
vertical dashed line marks the value of Jrms where the average curve match the
measured ratio.

As seen in Figure 5, the calculated IYRs from different initial FFs show a
similar dependency on the $J_{rms}$, indicating that common factors affect the
calculated IYR. One such factor is the level scheme, including the two long-
lived states, of 131Te. On the other hand, the observed variations between the
curves could indicate that the FFs have different impacts on IYR.

Considering all contributing FFs, one can calculate the weighted average IYR
as a function of $J_{rms}$

$\bar{R}(J_{rms})=\frac{\sum^{N}_{i=0}P_{hs}\cdot
Y_{i}}{\sum^{N}_{i=0}(P_{hs}+P_{ls})\cdot Y_{i}},$ (5)

where $N$ is the number of possible contributors, and $Y_{i}$ is the
independent fission yield of the contributor, obtained from the GEF
simulation.

As an example, the calculated average IYRs of the FP 131Te are presented in
black points in Figure 5. As seen, the average IYR matches the experimental
ratio at a $J_{rms}$ of $12.3~{}\hbar$. Furthermore, the calculated IYRs cross
the experimental ratio minus (plus) one sigma at a $J_{rms}$ of $11.5~{}\hbar$
($13.4~{}\hbar$). Hence, a value of $12.3^{1.1}_{0.8}~{}\hbar$ is assigned as
the most likely $J_{rms}$ of the FFs de-exciting to the FP 131Te.

The procedure described above is repeated for all 31 IYRs. However, in some
cases the calculations do not reproduce the experimental value for any
$J_{rms}$ in the range 0.2 to 25 $\hbar$. Figure 6 shows the example of the
IYR of the FP 121Cd, obtained from the de-excitations of the FFs 121-129Cd*.
As seen, the average IYRs do not reach the experimental ratio for any
$J_{rms}$, although some of the ratios for single FFs ($A={126-129}$)
reproduce the experimental ratio.

Figure 6: Calculated IYRs of the FP 121Cd from all excited FFs, 121-129Cd*,
contributing to the production of 121Cd. The horizontal line and grey region
represent the measured IYR with uncertainties of the FP 121Cd [21].

If we assume that the experimentally determined IYR is correct, there are at
least three other possible reasons for the failure of the model to reproduce
the experimental data. First of all, besides $J_{rms}$, the values of the
other parameters ($C$, $\bar{E}_{offset}$, $\bar{\sigma_{E}}$) also affect the
energy and angular momentum population of the FFs. In principle, also these
parameters could be optimized against experimental data. However, this would
require a much larger data set, probably including neutron and gamma
multiplicities, in order not to have an underdetermined problem. Secondly, the
de-excitation process, as described by the Hauser-Feshbach approach in TALYS,
might need adjustment. Thirdly, and probably the most likely reason, is the
lack of experimental excitation levels, and poor knowledge of de-excitation
paths toward the long-lived states of the FP. In the case of 121Cd, the number
of known levels in the complete scheme is 9, indicating a poor knowledge of
the nuclear structure.

Furthermore, several studies [29, 30, 31, 32] have shown that a parameter
(rspincut) in TALYS, as a multiplication to the spin cut-off parameter, can
have a substantial impact on the calculated isomeric yield ratios for light
particle-induced reactions. This parameter is used to adjust the predicted
spin distribution from the level density models before applying it to the
angular momentum distribution. In these studies, the suggested value of the
rspincut parameter is about 0.5, which reduces the width of angular momentum
distribution. This value was tested with the surrogate model, but no
significant change could be observed. This can be understood, as in default
TALYS calculations, the level density model and rspincut determine the angular
momentum distribution of the excited nucleus formed in the reactions, while in
our surrogate model, it is determined by the $Ex-J$ matrix, and hence
independent of the parameters in TALYS.

###  II.4 Primary contributor

As seen in Figure 5 and 6, the IYR as a function of $J_{rms}$ behaves
differently for different fission fragments. In forming the weighted average
ratios in Equation 5 the relative contribution from each fragment is taken
into account. Although all possible contributors are included, the ratio is
largely determined by two to four of the contributors. This can be shown by
studying the relative contributions, which can be calculated by multiplying
the independent yield of each fragment with the fraction of that fragment that
decays to the studied FP. The result for 131Te and 121Cd is presented in Table
2. At the same time, the average number of neutrons emitted from these
fragments de-exciting to a specific fission product can be calculated.

Four FFs, 132-135Te*, together contribute to more than 80% of the production
of the FP 131Te, which is why the curve for the average IYR is close to the
corresponding curves in Figure 5. For the production of 121Cd, the primary
contributors are 123-126Cd*.

Table 2: Contributions to the production of 131Te and 121Cd in the TALYS calculation. Contributions larger then 10 % are marked in bold font. Te | A | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139
---|---|---|---|---|---|---|---|---|---|---
Contr. | (%) | 1.7 | 14.6 | 31.7 | 25.3 | 17.1 | 7.7 | 1.9 | 0.2 | 0.01
Cd | A | 121 | 122 | 123 | 124 | 125 | 126 | 127 | 128 | 129
Contr. | (%) | 0.39 | 4.2 | 18.8 | 37.8 | 26.5 | 10.4 | 1.7 | 0.1 | 0.003

##  III Results and discussion

When experimental determinations of angular momentum in fission are discussed
in the literature, the deduced values are usually associated with the fission
products in question, even though it is the angular momentum of the initial
fragment that is sought. In this study, we are also unable to associate an
estimated $J_{rms}$ to a specific FF, but rather to a range of FF isotopes.
With Table 2 in mind, the deduced value of $12.3^{1.1}_{0.8}~{}\hbar$ can for
example be considered a reasonable estimate of the $J_{rms}$ for FFs of
Tellurium isotopes in the mass range $132-135$.

To simplify the discussion and comparison with earlier studies, the results
will, in the succeeding, be presented as a function of the FP for which the
IYR is measured. Using the surrogate model and experimental values on the IYR
[13, 14, 21], the average $J_{rms}$ have been estimated for 31 FPs in proton-
induced fission of 238U, and the results are presented for each FP in Table 3.

As mentioned in the introduction, the approach to derive $J_{rms}$ described
in this study builds on earlier work. To compare the methods, the 10 values of
$J_{rms}$ presented by V. Rakopoulos _et al._ [13, 14] are also presented in
Table 3.

Table 3: Estimateed values of $J_{rms}$ with lower and upper limits from the experimental IYRs in the proton-induced fission of 238U at 25 MeV [13, 14, 21]. Blank positions mean that the calculations could not reproduce the experimental values for any $J_{rms}$ below 25 $\hbar$. The last column is the values reported by V. Rakopoulos _et al._ Nuclide | IYR | $J^{av}_{rms}$ | Lower | Upper |  $J^{av}_{rms}$ ($\hbar$)
---|---|---|---|---|---
|  [13, 14, 21] | ($\hbar$) | ($\hbar$) | ($\hbar$) | [13, 14]
102Nb | 0.735(21) |  |  |  |
119Cd | 0.871(15) | 15.7 | 1.4 | 1.9 | 12.3(5)
121Cd | 0.867(4) |  |  |  | 14.7(1)
123Cd | 0.876(7) | 12.3 | 0.6 | 0.7 | 15.7(2)
125Cd | 0.902(8) | 10.4 | 0.6 | 0.7 |
127Cd | 0.87(4) | 12 | 3 | 12 |
119In | 0.978(15) | 22.3 | 6.9 | 2.1 | 26.4(4)
121In | 0.971(11) |  |  |  | 25.1(5)
123In | 0.958(2) |  |  |  | 21.2(2)
125In | 0.950(3) | 12.7 | 0.6 | 0.7 | 15.9(9)
126In | 0.574(16) | 8.6 | 0.3 | 0.3 |
127In | 0.921(2) | 12.9 | 0.3 | 0.3 | 9.5(2)
128In | 0.58(4) |  |  |  |
129In | 0.884(8) |  |  |  |
128Sn | 0.580(20) | 11.1 | 0.5 | 0.5 | 7.9(4)
129Sn | 0.777(20) | 8.0 | 0.5 | 0.7 |
130Sn | 0.540(20) | 10.8 | 0.9 | 1.2 | 6.4(2)
131Sn | 0.681(12) |  |  |  |
129Sb | 0.539(21) | 10.4 | 0.4 | 0.4 |
132Sb | 0.433(29) | 8.5 | 0.5 | 0.6 |
134Sb | 0.625(10) | 7.6 | 0.4 | 0.40 |
129Te | 0.832(4) | 23.3 | 0.9 | 1.0 |
131Te | 0.867(14) | 12.3 | 0.8 | 1.1 |
133Te | 0.794(2) | 8.39 | 0.06 | 0.06 |
132I | 0.542(20) | 8.2 | 0.3 | 0.3 |
133I | 0.27(3) | 7.3 | 0.4 | 0.4 |
134I | 0.639(5) | 9.4 | 0.1 | 0.1 |
136I | 0.730(20) | 10.3 | 1.0 | 2.2 |
133Xe | 0.618(13) | 5.8 | 0.2 | 0.2 |
135Xe | 0.729(7) | 8.2 | 0.2 | 0.2 |
138Cs | 0.799(17) | 8.7 | 0.5 | 0.6 |

As mentioned in Section II.2 De-excitation with TALYS, there is a limit in
TALYS of 30 discrete spin values. Hence, if a FF is initialized with a
$J_{rms}$ larger than about $20~{}\hbar$ the angular momentum distribution
will be cut. As a result, values of $J_{rms}$ larger than $20~{}\hbar$ should
be considered less trustworthy.

###  III.1 Comparison of methods

Although there are many similarities between the method described by V.
Rakopoulos _et al._ [13, 14] and the one presented in this paper, there are
also important differences. In the method of Rakopoulos, the excitation energy
of the FFs is described by a single mean energy with a width of 0.5 MeV,
instead of the more realistic energy distribution used in this study. This
simplification also ignores the correlation between the excitation energy and
angular momentum, here described by Eq (3), and as a result, the FF population
might extend past the yrast line. Furthermore, multi-fission channels were
ignored and the yields of the FFs were not considered in the calculation of
the weighted $J_{rms}$ of the FFs.

Another difference between the two models is that the previous method uses the
Skyrme-Hartree-Fock-Bogoluybov table, containing microscopic level densities,
to calculate the IYRs of the cadmium, indium and tin isotopes. In this study,
the back-shifted Fermi gas model (BFM) is instead adopted, as it was shown to
have the best overall performance.

Due to the aforementioned differences between the two methods, it is not
surprising that discrepancies are observed in Table 3. In a few cases (121Cd
and 121,123In), the model by Rakopoulos could reproduce the measured IYR,
while the present method fails. The reasons for this have been discussed in
section II.2. For 121Cd, the IYR from the surrogate model could be increased
by reducing the value of parameters $C$ and $\bar{\sigma_{E}}$ in the
reconstruction of the excitation matrices, leading to a match between the
calculated and measured IYR. On the other hand, in the case of 121,123In it
does not matter how much the other parameters of the surrogate model are
varied, the resulting IYRs will not match the experimental IYR for any
$J_{rms}$. The reason could be in the assumptions of the model, for example,
in how the matrix is reconstructed. However, before drawing any conclusions,
the level schemes of 121,123In should be carefully examined. In the cases of
125,127Cd the situation is reversed. Here, the IYRs could not be reproduced by
the previous method, while the surrogate model produces $J_{rms}$-values, as
seen in Table 3.

Figure 7 shows the derived average $J_{rms}$ as a function of mass number for
the measured cadmium, indium and tin isotopes with the two methods. With the
surrogate model, the derived average $J_{rms}$ values range from about 10
$\hbar$ to 16 $\hbar$, except for 119In where the value is larger. With
Rakopoulos’ method, a larger variation in the average $J_{rms}$ is observed,
ranging from 6 $\hbar$ to 26 $\hbar$.

Figure 7: A comparison of the deduced average $J_{rms}$ as a function of mass
number for the cadmium, indium and tin isotopes with the two methods.

###  III.2 Average angular momenta

J. Wilson et al. have suggested that the angular momenta of fission fragments
are generated from the collective motion of nucleons in the ruptured neck of
the fissioning system [15]. Based on the Rayleigh distribution and the Fermi
gas model, an equation to describe the observed mass dependency of the average
angular momentum is proposed:

$J^{av}=cA^{1/4}_{N}A^{7/12}_{F},$ (6)

in which $c$ is a free parameter and $A_{F}$ is the fragment mass. $A_{N}$ is
a parameter that for fission fragments near the doubly magic 132Sn is
$A_{F}-130$, while for light fragments near the doubly magic 78Ni it is
$A_{F}-78$. In their paper, J. Wilson et al. fit this equation to their data
for fast (average energy 1.9 MeV) neutron-induced fission of 238U. However,
they do not present any data, and there is also no description or prediction
of the $J^{av}(A)$ in the symmetry region.

To compare the results of this study with the results by Wilson et al., the
root-mean-square angular momenta ($J_{rms}$) in Table 3 have been converted to
average angular momenta ($J^{av}$). The result is presented in Figure 8 as a
function of the mass number of the FP. For comparison, the experimental values
and fit from the paper by Wilson et al. are included in the same figure.

Equation (6) was also fitted to the proton-induced data for A larger than 131,
and is shown as a red curve in Figure 8. The model describes the data rather
well. However, the average angular momenta associated to 130Sn and 131Te
cannot be described by Equation (6). The failure to describe these nuclides
might indicate that Eq. 6 is not valid for mass numbers close to 130. This
observation is also supported by the fact that the model predicts $J^{av}$ to
be zero for A = 130. On the other hand, it could be argued that the minimum
angular momentum should be observed for the most spherical fission products
around the doubly magic 132Sn.

Figure 8: Average angular momentum as a function of mass number of the
resulting fission product. The colored points present data from proton-induced
fission of 238U, and the red curve is fit to these data for A greater than
131. The red dashed line is obtained from a linear fit to the data for A less
than 133.
The black points and curves present the average angular momenta for the
fission fragments in the fast neutron-induced fission of 238U, and a fit to
the data from J. Wilson et al. [15].

J. Wilson et al. do not present any data for the angular momentum of fission
products in the symmetric mass region (105 $\leq$ A $<$ 130). Here our data
reveal that the average angular momenta decrease with increasing mass number
in contrast with the predicted mass dependency by J. Wilson. In Wilson’s
model, the average momentum is generated from the collective motion of
nucleons in the ruptured neck, which always increases from the starting mass
number (78 or 130 or any other values). Hence, it is impossible to predict a
decreasing trend of the average angular momentum with Wilson’s model. There is
a need for a different mechanism to explain the observed decreasing trend of
the average angular momentum in the symmetry region.

It might be worth mentioning that a linear fit to the data gives an average
decrease by $0.3\pm 0.2~{}\hbar$ per atomic mass unit, though there is not a
reasonable guess on what it means by now.

###  III.3 Neutron emission

From the TALYS output of the surrogate model, information on the neutron
multiplicity for each fission fragment can be extracted. From this
information, the average neutron multiplicity for each fragment decaying to a
certain fission product can be derived. In figure 9 this is plotted together
with the same information obtained directly from the GEF model.

Figure 9: The average number of neutrons emitted from the primary fission
fragments de-exciting to a specific fission product as derived from TALYS,
compared with the same information from GEF.

In Figure 9, a similar behaviour of the two models is observed. Although the
neutron multiplicity in proton-induced fission in GEF is not validated using
experimental data and should hence not be taken as a benchmark, the agreement
builds some confidence in the surrogate model. The discrepancies that can be
observed are likely due to the different ways of dealing with the de-
excitation process and different assumptions of angular momenta of the fission
fragments.

###  III.4 Angular momentum of the primary fission fragment

Considering that the derived angular momentum is a property of the initial
fragments, it would be interesting to study how this varies with, for example,
the fragment mass. Using the neutron emission information of the de-excitation
from TALYS, shown in Figure 9, the average mass number of the fission
fragments de-exciting to a certain fission product was calculated. As a
result, the fission product mass number in Figure 8 is shifted to the most
probable FF mass number, as presented in Figure 10.

Figure 10: The average angular momentum as a function of the average mass
number of the primary fission fragments de-exciting to the observed fission
products. The dotted line represents a fitting to the angular momentum as a
function of the primary fission fragments in the symmetry region.

Assuming the same mass dependence of the angular momentum of the fission
fragments as for the fission products, the data in Figure 10 has been fitted
with Equation 6 for A $>$ 134 and with a straight line in the symmetric mass
region. The decreasing trend of the average angular momentum for A $<$ 134 is
similar to that for A $<$ 132 in Figure 8. However, the minimum of the average
angular momenta is shifted by approximately two mass units towards higher
masses. This shift makes sense, considering that the neutron multiplicity in
the proton-induced fission of 238U at 20 MeV is about 2 [33], which is similar
to the fissioning system in this study.

##  IV Conclusion and outlook

A surrogate model of GEF that can be used to generate energy vs angular
momentum matrices describing excited fission fragments has been developed.
Using TALYS to calculate the de-excitation of the fragments while varying the
parameters of the model, the impact of the state of primary FFs on the IYRs in
nuclear fission has been investigated.

31 IYRs in proton-induced fission of 238U at 25 MeV are compared with the
calculated ratios from the model. Based on that comparison, 24 estimates of
the root-mean-square angular momentum, $J_{rms}$, have been estimated. For
some fission products, the estimated $J_{rms}$ are compared with the values
derived by V. Rakopoulos, and differences between the two methods are
discussed.

It has been shown that the mass dependence of average angular momentum
$J^{av}$ for the nuclides with mass numbers larger than 131 could be
reasonably well described by the model suggested by J. Wilson _et al._ [15].
This dependence is based on the assumption that the angular momentum is
generated from the collective motion of nucleons in the ruptured neck of the
fissioning system.

In the symmetric mass region, a decreasing trend of the average angular
momentum of the fission products is observed for the first time. This
observation could not be explained by J. Wilson’s proposal, and needs a
different mechanism to model the angular momentum generation in the symmetry
region.

To link the average angular momentum to the primary fission fragments, the
number of emitted neutrons is calculated for each fragment. This way, the
average angular momentum as a function of the average mass number of the
primary fission fragment is obtained. In this description, the minimum of the
angular momentum is close to mass number 134, deviating from the shell closure
at 132 by two units.

Considering the fact that the surrogate model fails to reproduce the
experimental data in some cases shows that the model needs further
investigation and development. In particular, the impact of the (un)known
levels scheme of the fission products, and possible dependencies of the IYR on
parameters other than the angular momentum, are worth future studies. Also,
the effect of other parameters in TALYS, such as rspincut possibly related to
the momentum of inertia [26], should be investigated in order to improve the
IYR calculation.

##  V Acknowledgement

This work was supported by the Swedish research council Vetenskapsrådet (Ref.
No. 2017-06481), the European Commission within the Seventh Framework
Programme through Fission-2013-CHANDA (Project No. 605203), the Swedish
Radiation Safety Authority (SSM).

The computations were enabled by resources in project NAISS 2023/22-271
provided by the National Academic Infrastructure for Supercomputing in Sweden
(NAISS) at UPPMAX, funded by the Swedish Research Council through grant
agreement no. 2022-06725.

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