# A tensor decomposition model for evaluating isotopic yield in neutron-
induced fission

Qufei Song  Long Zhu  Jun Su  Corresponding author<EMAIL_ADDRESS>Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen
University, Zhuhai 519082, China  Hui Guo  School of Nuclear Science and
Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

###### Abstract

Background

Due to the complex multidimensional dependence, the prediction and evaluation
of independent fission yield distributions have always been a challenge.

Purpose

Considering the complex multidimensional dependence and high missing rate of
independent yield data, this work applies the tensor decomposition algorithm
to the prediction of independent fission yields.

Methods

After constructing yield tensors with three dimensions for 851 fission
products and filling the tensors with the independent yield data from the
ENDF/B-VIII.0 database, the tensor decomposition algorithm is applied to
predict the independent isotopic yield in fission, which resulting in the
Fission Yield Tensor Decomposition (FYTD) model.

Results

The fission yields of 235U and 239Pu are set as missing values and then
predicted. The predictions for 235U fissions by the FYTD model agree with the
ENDF/B-VIII.0 data and are better than those by the Talys and BNN+Talys
models. Furthermore, we predict not only the mass distribution but also the
isotopic yields in the fissions. For fast neutron-induced fission of 239Pu,
98% predictions of the isotopic yields by the FYTD model agree with the
ENDF/B-VIII.0 data within 1 order of magnitude. The fission yields of 238Np,
243Am, and 236Np that do not exist in the ENDF/B-VIII.0 database are predicted
and compared with those in the JEFF-3.3 database, as well as the experimental
data. Good agreement demonstrates the predictive ability of the FYTD model for
the target nucleus dependence. The scalability of the yield tensor
decomposition model over the incident neutron energy degrees of freedom is
examined. After adding a set of 2 MeV neutron-induced 239Pu fission yield data
into the yield tensor, the 2 MeV neutron-induced fission yields of 235U and
239Pu are predicted. The comparison with the experimental data shows that the
predictions are similar to those by the GEF model in the peak area but more
accurate in the valley area. Finally, the yields of the ratio of isomeric
states and neutron excess of the products as a function of product charge
number are also studied.

Conclusions

The FYTD model can capture the multi-dimensional dependence of the fission
yield data and make reasonable predictions. The FYTD model has scalability in
the energy dimension and can predict the yields of the ratio of isomeric
states.

##  I Introduction

The splitting of a heavy nucleus into two or more intermediate-mass nuclei is
called fission. Although it has been more than 80 years since its discovery
Hahn and Strassmann (1939); Meitner and Frisch (1939), the research on the
fission is still a hot topic and challenge. On the one hand, the nuclear
fission is an extremely complex process, which is the movement of the quantum
multi-body system composed of all nucleons in the nucleus in the
multidimensional space. The exploration of its mechanism is very helpful to
the development of nuclear physics fields such as nuclear structure, nuclear
reaction and super-heavy nuclear research Schunck and Robledo (2016); Bender
_et al._ (2020); Hamilton  _et al._ (2013); Pei  _et al._ (2009).The nuclear
fission also attracts attentions in astrophysics and particle physics because
it plays an important role in the formation of elements in the rapid neutron
capture process (r-process) of nucleosynthesis and the production of reactor
neutrinos Eichler  _et al._ (2015); Mueller  _et al._ (2011). On the other
hand, the nuclear fission is also significant application fields. The huge
energy released in the fission process makes fission play an important role in
both energy and military fields. Neutrons and various radioisotopes produced
by fission are used in various fields such as biology, chemistry and medicine.
Therefore, in order to make scientific use of fission, the research on the
nuclear fission is widely concerned in the field of nuclear engineering and
technology Bernstein  _et al._ (2019).

The fission product yield (FPY) is an important observable in the nuclear
fission. Generally speaking, the nuclear fission is followed by the $\beta$
decay of the unstable fragments. The FPY is divided into the independent and
cumulative cases, which are distinguished by counting the products before or
after the $\beta$ decay. The independent fission product yield (IFPY) can
reflect the information of fission process from the macro and micro
perspectives, provide important observations for the research and modeling of
fission process Ramos  _et al._ (2019). However, experimental measurements of
the IFPY are difficult and hence the available data is generally incomplete
and have large uncertainties Denschlag (1986). In major nuclear data
libraries, such as ENDF/B Brown  _et al._ (2018), JEFF Kellett  _et al._
(2009), and JENDL Shibata  _et al._ (2011), complete evaluations of IFPY are
not available for some certain actinides and only available for three neutron
incident energy points (0.0253 eV, 0.5 MeV and 14 MeV). Therefore, theoretical
predictions of the IFPY are still necessary.

Due to the complexity of the quantum many body problem and the nuclear force
problem, the deep understanding and simulation of the fission process is still
one of the most challenging tasks in nuclear physics Schunck and Robledo
(2016); Bender  _et al._ (2020). Nowadays, the microscopic nuclear fission
models, such as the time-dependent Hartree-Fock-Bogoliubov method Bulgac  _et
al._ (2016) and the time-dependent generator coordinate method Regnier  _et
al._ (2016); Younes  _et al._ (2019), have made important progress. The
fission process can be regarded as a movement of Brownian particles walking on
the multi-dimensional potential energy surface. Various macroscopic-
microscopic models based on the multi-dimensional potential energy surface are
also widely used in the calculation of the fission yield Randrup and Möller
(2011); Randrup  _et al._ (2011); Pomorski  _et al._ (2017); Liu  _et al._
(2019). For practical applications requiring higher precision,
phenomenological methods are more widely used. For example, the multi-Gaussian
semi-empirical formula Ramos  _et al._ (2019); Fang  _et al._ (2021), the
Brosa model Brosa  _et al._ (1990) and the GEF (GEneral description of Fission
observables) model Schmidt  _et al._ (2016) have all achieved considerable
success in evaluating fission yield data. The prediction of the GEF model is
also referenced in the JEFF database to further improve the fission yield data
in the database.

The traditional phenomenological models mainly rely on the least-squares
adjustment of various parameters. They can describe the existing data in some
regions well. But as fission modes evolve, the prediction capability of this
kind of models may be insufficient when the available experimental data are
very sparse Schmidt and Jurado (2018); Wang  _et al._ (2019). Recently, thanks
to the powerful ability to learn from existing data and make predictions, the
machine learning algorithms have been used in various studies in the nuclear
physics community. For example, the Bayesian neural network (BNN) algorithm
has achieved certain success in prediction of nuclear mass Niu and Liang
(2018); Utama  _et al._ (2016a), nuclear charge radius Utama  _et al._
(2016b), spallation reaction product cross section Ma  _et al._ (2020); Song
_et al._ (2022), neutron and proton drip lines Neufcourt  _et al._ (2019,
2020), etc. Likewise, various machine learning algorithms have been applied to
the study and prediction of fission yields. Lovell et al. used Mixture Density
Networks to learn parameters of Gaussian functions to predict fission product
yields Lovell  _et al._ (2019). Wang et al. predicted the mass distribution of
fission yield by combining BNN and talys models Wang  _et al._ (2019). Qiao et
al. used BNN model to predict the charge distribution of 239Pu fission yield
Qiao  _et al._ (2021).

The tensor decomposition algorithm is a standard technique to capture the
multi-dimensional structural dependence. Compared with the traditional
interpolation and fitting methods, the tensor decomposition algorithm has a
strong ability to extract the information hidden in the original data, making
it possible to impute the sparse tensor, which makes it widely used in image
processing, data mining and other fields Liu  _et al._ (2013); Chen  _et al._
(2019, 2017). Considering the complex multidimensional dependence and high
missing rate of independent yield data, this work applied the tensor
decomposition algorithm to the prediction of independent fission yields, and
established the Fission Yield Tensor Decomposition (FYTD) model. The paper is
organized as follows. In Sec. II, the establishment of FYTD model is
described. In Sec. III, FYTD model will be applied to multiple yield
prediction for verification. Finally, Sec. IV presents conclusions and
perspectives for future studies.

##  II Theoretical framework

###  II.1 Tensorization of fission product yield

The neutron-induced fission is briefly introduced by taking a typical
235U(n,f) reaction as an example:

${}^{235}_{92}\text{U}+\text{n}\longrightarrow^{236}_{92}\text{U}^{*}\longrightarrow^{92}_{36}\text{Kr}+^{141}_{56}\text{Ba}+\text{3n}.$
(1)

A specific target nucleus, such as 235U, can be represented by its number of
protons $Z_{t}$ and number of neutrons $N_{t}$. The target nucleus forms an
excited composite nucleus after the neutron incident with energy $E_{n}$.
After fission of composite nucleus, two primary fission products, one light
and one heavy, are produced and several prompt neutrons are released. Thus,
the neutron-induced fission yield of a given isotope with number of protons
$Z_{p}$ and number of neutrons $N_{p}$ depends on the neutron incident energy
and target nuclei. In addition, a considerable part of the fission products
are not in the ground state but in the isomeric state. In order to distinguish
these products, the Fission Product State (FPS) needs to be considered.

Therefore, when tensorizing the neutron-induced fission product yield data,
six dimensions can be considered, including $E_{n}$, $Z_{t}$, $N_{t}$,
$Z_{p}$, $N_{p}$ and FPS. Among six seven dimensions, the dimensions related
to the target nucleus and product are discrete, and only the neutron incident
energy $E_{n}$ is continuous. Therefore, $E_{n}$ is discretized into thermal
neutron (0.0253 eV), fast neutron (0.5 MeV) and high-energy neutron (14 MeV).

Nowadays, due to the difficulty of IFPY experimental measurement and the lack
of existing data, this yield tensor will have a very high degree of
missingness. Imputation of such sparse tensors has always been a challenge.
Similar scenes also appear in image processing in the computer field. A color
image can be thought of as a multidimensional tensor containing information
about the pixel location dimension, as well as the pixel color dimension. In
order to deal with color images with high missing rate and high signal-to-
noise ratio, tensor decomposition algorithms are widely used Liu  _et al._
(2013); Chen  _et al._ (2019, 2017). Taking the image in the upper left corner
of Fig. 1 as an example, 70$\%$ of the pixels in this image are missing. We
tensorized the image and applied the BGCP algorithm to capture the
multidimensional information of the remaining pixels. Finally, as shown in the
image in the upper right corner of Fig. 1, tensor decomposition can
effectively impute the missing pixels.

However, it is not feasible to directly build a 6D tensor and impute it. Even
based on the ENDF/B-VIII.0 database with the most abundant IFPY data, the
missing rate of this 6D tensor can still be as high as 98 $\%$. Excessive
tensor construction range will not only bring about a high missing rate, but
also incorporate some non-existent extreme neutron-rich nuclei, proton-rich
nuclei, and isomeric states into the tensor, which will make some elements of
the tensor physically meaningless. Taking into account this problem, this work
proposes a method for constructing 3D tensors. The 3D yield tensors with
dimensions $Z_{t}$, $N_{t}-Z_{t}$ and $E_{n}$ were constructed separately for
851 products with relatively rich existing data in ENDF/B-VIII.0 database. The
range of the tensor is chosen as $Z_{t}$ = 90-96, $N_{t}-Z_{t}$ = 47-54 and
$E_{n}$ = 0.0253 eV, 0.5 MeV, 14 MeV. By filling these 3D tensors with the
ENDF/B-VIII.0 data of 45 fission systems for 24 target nuclei of
227,229,232Th, 231Pa, 232,233,234,236,237,238U, 237,238Np,
238,239,240,241,242Pu, 241,243Am and 242,243,245,246Cm, the missing rate can
be reduced to 73$\%$, and the tensor will not contain any non-existing target
nuclei, product nuclei and isomeric states.

Now, let $Y_{ijk}$ represent the exciting yield in ENDF/B-VIII.0 database,
where $i,j,k$ represents $Z_{t}$, $N_{t}-Z_{t}$ and $E_{n}$ respectively. For
example, $Y_{3,5,1}$ is entry with $i=3(Z_{t}=92)$, $j=5(N_{t}-Z_{t}=51)$ and
$k=1(E_{n}=0.0253eV)$, which represents yield in 0.0253 eV neutron induced
fission of 235U. The magnitude of the independent yield is very small, even up
to 10-18. Therefore, it is necessary to fill the tensor with the logarithm of
the yield, in case this data is too small for numerical calculation. This
approach works well for products with big variation, allowing the algorithm to
capture magnitude changes in yield well. But for products with small yield
variation, the difference is even smaller after logarithmization, and it is
difficult for the algorithm to capture their difference. Therefore, to solve
this problem, we analyzed the degree of dispersion of 851 product yield data
under different fission systems. In this work the degree of dispersion is
defined as the standard deviation of the logarithm values of the existing
yield data:

$\sigma_{\text{yield
}}=\sqrt{\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}b_{ijk}\left[\frac{\ln\left(Y_{ijk}\right)-\overline{\ln(Y)}}{\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}b_{ijk}}\right]},$
(2)

where $b_{ijk}$ is 1 for the exciting yield data and 0 for the missing yield
data. If $\sigma_{\text{yield }}>1$, its magnitude varies greatly, and the
yield is logarithmized during filling; otherwise, the yield remains linear.

At this point, 851 yield tensors containing ENDF/B-VIII.0 data and missing
elements were obtained. Let $\mathcal{Y}$ represent these missing tensor and
$\hat{\mathcal{Y}}\in\mathbb{R}^{I\times J\times K}$ the physical reality of
the yield data. In fact, the physical reality value $\hat{Y}_{ijk}$ cannot be
known, only some exciting observations with uncertainties $Y_{ijk}$ are
available. In this work, BGCP tensor decomposition model algorithm Chen  _et
al._ (2019) will be applied to estimate the physical reality
$\hat{\mathcal{Y}}$ according to the existing observation $\mathcal{Y}$.

###  II.2 Bayesian Gaussian CANDECOMP/PARAFAC tensor decomposition

A detailed introduction to the Bayesian Gaussian CANDECOMP/PARAFAC (BGCP)
algorithm can be found in Ref. Chen  _et al._ (2019). Here is a brief
introduction to the application of BGCP in this work. It is assumed that the
uncertainty of each exciting yield data $Y_{ijk}$ follows an independent
Gaussian distribution,

$Y_{ijk}\sim\mathcal{N}\left(\hat{Y}_{ijk},\tau_{\epsilon}^{-1}\right),$ (3)

where $\tau_{\epsilon}$ is the precision. In real-world applications the
expectation of yield $\hat{Y}_{ijk}$ is unknown and replaced with the
estimated yield, which is the entry of the estimated tensor
$\hat{\mathcal{Y}}$. The CP decomposition is applied to calculate the
estimation $\hat{\mathcal{Y}}$ :

$\hat{\mathcal{Y}}=\sum_{n=1}^{r}\bm{z}^{(n)}\circ\bm{d}^{(n)}\circ\bm{e}^{(n)},$
(4)

where $\bm{z}^{(n)}\in\mathbb{R}^{I},\bm{d}^{(n)}\in\mathbb{R}^{J}$, and
$\bm{e}^{(n)}\in\mathbb{R}^{K}$ are respectively the $n$-th column vector of
the factor matrices $\bm{Z}\in\mathbb{R}^{I\times r}$,
$\bm{D}\in\mathbb{R}^{J\times r}$, and $\bm{E}\in\mathbb{R}^{K\times r}$. The
symbol $\circ$ represents the outer product.

The prior distribution of the row vectors of the factor matrix $\bm{Z}$ is the
multivariate Gaussian

$z_{i}\sim\mathcal{N}\left[\bm{\mu}_{i}^{(z)},\left(\bm{\Lambda}_{i}^{(z)}\right)^{-1}\right]$
(5)

where the hyper-parameter $\bm{\mu}^{(z)}\in\mathbb{R}^{r}$ expresses the
expectation, and $\bm{\Lambda}^{(z)}\in\mathbb{R}^{r\times r}$ indicates the
width of the distribution. The likelihood function can be written as

$\mathcal{L}\left(Y_{ijk}\mid
z_{i},\bm{d}_{j},\bm{e}_{k},\tau_{\epsilon}\right)\propto\exp\left\\{-\frac{\tau_{\epsilon}}{2}\left[Y_{ijk}-\left(z_{i}\right)^{T}\left(\bm{d}_{j}\circledast\bm{e}_{k}\right)\right]^{2}\right\\}$
(6)

where $\circledast$ is the Hadamard product.

Then, taking $z_{i}$ as an example, according to Bayesian theorem, the
posterior distribution of $z_{i}$ after observing $Y_{ijk}$ is:

$\displaystyle\mathcal{L}\left(z_{i}\mid
Y_{ijk},d_{j},e_{k},\tau_{\varepsilon}\right)$
$\displaystyle\propto\mathcal{L}\left(Y_{ijk}\mid
z_{i},d_{j},e_{k},\tau_{\varepsilon}\right)\times\operatorname{Pr}\left(z_{i}\right)$
(7)
$\displaystyle\propto\exp\left\\{-\frac{\tau_{\varepsilon}}{2}\left[Y_{ijk}-\left(z_{i}\right)^{T}\left(d_{j}\circledast
e_{k}\right)\right]^{2}\right\\}\times\mathcal{N}\left[\mu_{i}^{(z)},\left(\Lambda_{i}^{(z)}\right)^{-1}\right].$

Then the posterior values of the hyper-parameters $\bm{\mu}^{(z)}$ and
$\bm{\Lambda}^{(z)}$ are given as

$\begin{gathered}\widehat{\Lambda}_{i}^{(z)}=\Lambda_{i}^{(z)}+\Delta\Lambda_{i}^{(z)},\\\
\Delta\Lambda_{i}^{(z)}=\tau_{\varepsilon}\left(d_{j}\otimes
e_{k}\right)\left(d_{j}^{\otimes}e_{k}\right)^{T}.\\\
\hat{\mu}_{i}^{(z)}=\mu_{i}^{(z)}+\Delta\mu_{i}^{(z)},\\\
\Delta\mu_{i}^{(z)}=\left(\widehat{\Lambda}_{i}^{(z)}\right)^{-1}\left(d_{j}\circledast
e_{k}\right)\tau_{\varepsilon}\left[Y_{ijk}-\left(d_{j}\circledast
e_{k}\right)^{T}\mu_{i}^{(z)}\right].\end{gathered}$ (8)

The contribution of the exciting yield data to the hyper-parameter is
equivalent, and likelihood function of all exciting yield data is

$\displaystyle\mathcal{L}\left(\mathcal{Y}\mid\bm{Z},\bm{D},\bm{E},\tau_{\epsilon}\right)\propto$
$\displaystyle\prod_{i=1}^{I}\prod_{j=1}^{J}\prod_{k=1}^{K}\left(\tau_{\epsilon}\right)^{1/2}$
(9)
$\displaystyle\times\exp\left[-\frac{\tau_{\epsilon}}{2}b_{ijk}\left(Y_{ijk}-\hat{Y}_{ijk}\right)^{2}\right]$

where $b_{ijk}$ is 1 for the exciting yield data and 0 for the missing yield
data. Placing a conjugate $\Gamma$ prior to the precision $\tau_{\epsilon}$,

$\tau_{\epsilon}\sim\Gamma\left(a_{0},b_{0}\right)$ (10)

The posterior values of the hyper-parameters $a_{0}$ and $b_{0}$ are given as

$\displaystyle\hat{a}_{0}=\frac{1}{2}\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}b_{ijk}+a_{0}$
(11)
$\displaystyle\hat{b}_{0}=\frac{1}{2}\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}\left(Y_{ijk}-\hat{Y}_{ijk}\right)^{2}+b_{0}$

Based on Eq. (11), each exciting yield data contributes to the increase of
$\frac{1}{2}$ in $\hat{a}_{0}$, and
$\frac{1}{2}\left(Y_{ijk}-\hat{Y}_{ijk}\right)^{2}$ in $\hat{b}_{0}$. Cases
for the subscripts $j$ and $k$ are similar.

Figure 1:  (Color online) Model framework presentation and example.

In Fig. 1, the above method is illustrated and a example is given. In brief,
for a specific fission product, we denote its yields in different fission
systems by $\hat{\mathcal{Y}}$. According to the $\mathrm{CP}$ decomposition,
the tensor $\hat{\mathcal{Y}}$ is expressed as the outer product of the factor
matrices $\bm{Z},\bm{D}$, and $\bm{E}$. The prior distributions of the factor
matrices are assumed to be multivariate Gaussians. With the exciting observed
yield data, the posterior values of the factor matrices and their
distributions can be calculated using Bayesian inference and iteration. Taking
the prediction of 236Np fission yield induced by 0.5 MeV neutrons as an
example, the information captured by the algorithm comes from the existing
data on the same dimension, the same $Z_{t}$ data (237Np and 238Np data), the
same $N_{t}-Z_{t}$ data (234U and 238Pu yield data) and all yield data under
0.5 MeV. Finally, the predicted fission yield is reconstituted with the factor
matrices $\bm{Z}$, $\bm{D}$, and $\bm{E}$.

Without considering ternary fission, the products produced by each fission
should be two kinds, so the sum of all fission product yields should also be
2. However, after tensor reconstruction, the sum of the yields obtained is not
strictly 2 due to the precision of numerical calculation, and the yield of
completion is always slightly smaller, such as 1.96, 1.98, etc. Therefore, it
is necessary to perform a certain physical correction on the value. In this
paper, the mass of the fission system minus the number of prompt neutrons is
divided by two, and this value is used as the standard to judge whether the
fission products belong to light nuclei or heavy nuclei, and the yields of
light nuclei and heavy nuclei are normalized to 1, respectively. After
normalization, the predicted yield data is finally obtained.

##  III Results and discussions

In order to quantitatively evaluate the prediction of FYTD model, the Root
Mean Square Error (RMSE) and $\chi_{N}^{2}$ are used to measure the deviation
between prediction and the ENDF/B-VIII.0 data. For the convenience of
presentation, ENDF/B-VIII.0 is abbreviated as ENDF/B in the following text and
figures. For a fission system, the RMSE is calculated by evaluating the
deviation of the predicted results of its 851 products from the ENDF/B:

$RMSE=\sqrt{\frac{1}{N}\sum_{p=1}^{N}\left[log(\widehat{Y}^{p})-log(Y^{p})\right]^{2}},$
(12)

where N = 851, $\widehat{Y}^{p}$ represents the FYTD model prediction of the
yield of the p-th product and $Y^{p}$ represents the corresponding ENDF/B
data.

Considering most current fission yield prediction works and experimental
measurements mainly focus on the mass distribution of the product, in order to
compare with other models and data, as defined in Ref. Wang  _et al._ (2019),
use $\chi_{N}^{2}$ to measure the deviation of the predicted value of the mass
distribution from the ENDF/B data:

$\chi_{N}^{2}=\frac{1}{N}\sum_{p=1}^{N}\left(\widehat{Y}^{p}-Y^{p}\right)^{2}$
(13)

Here N = 107, which means the range of statistics is $\mathrm{A}=66-172$ for a
total of 107 mass points.

The evaluation of RMSE and $\chi_{N}^{2}$ have different emphases. In
calculation of RMSE, the magnitude difference between the predicted value of
each product and the ENDF/B data is considered, which can globally evaluate
the accuracy of magnitude prediction. Therefore it does not ignore the
contribution of some products with small yield values. In contrast,
$\chi_{N}^{2}$ focuses more on evaluating the accuracy of peak area
predictions.

###  III.1 ENDF/B-VIII.0 data prediction

In this part, the ENDF/B yield data for each fission system are sequentially
removed and predicted by FYTD model to systematically analyze the learning and
prediction ability of FYTD model. Table 1 shows the RMSE of the FYTD model
when predicting each fission system. In general, it can be seen that after
removing the data to be predicted, the more remaining data in the learning set
with the same dimension as the data to be predicted, the smaller the RMSE. For
example, the RMSE of the prediction data for U, Pu, and Cm are mostly small,
because even if the learning data of a fission system is removed, there are
still many existing data in the same dimension that can capture information
during prediction. For 229Th and 232Th, after removing their data from the
learning set, there are few existing data that can be referenced when
predicting them, resulting in a high RMSE. It can be found that the largest
RMSE occurs at 227Th and 232U. This is because after the data of 227Th was
removed, the data of j = 1 ($N_{t}-Z_{t}$ = 47) does not exist in the learning
set at all. Therefore, when making predictions, it is completely impossible to
capture the information of this dimension, resulting in unreliable predictions
and extremely large RMSE. We believe that such predictions lacking dimensional
information is unreliable and should be avoided when predicting using FYTD
model. And in all following predictions, the situation will not occur, all
calculations will avoid this problem.

Table 1:  ENDF/B-VIII.0 data contained in the learning set of this work and prediction RMSE RMSE | 0.0253 eV | 0.5 MeV | 14 MeV
---|---|---|---
227Th | 2.034 |  |
229Th | 1.095 |  |
232Th |  | 0.942 | 0.866
231Pa |  | 1.005 |
232U | 1.874 |  |
233U | 0.598 | 0.473 | 0.411
234U |  | 0.659 | 0.244
235U | 0.622 | 0.552 | 0.260
236U |  | 0.391 | 0.271
237U |  | 0.419 |
238U |  | 0.712 | 0.683
237Np | 0.466 | 0.612 | 0.622
238Np |  | 0.607 |
238Pu |  | 0.768 |
239Pu | 0.429 | 0.281 | 0.390
240Pu | 0.284 | 0.326 | 0.554
241Pu | 0.652 | 0.424 |
242Pu | 0.451 | 0.482 | 1.170
241Am | 0.560 | 0.495 | 0.498
243Am |  | 0.638 |
242Cm |  | 0.839 |
243Cm | 0.683 | 0.521 |
244Cm |  | 0.443 |
245Cm | 0.779 |  |
246Cm |  | 0.637 |

###  III.2 235U and 239Pu fission yield prediction

Figure 2:  (Color online) 235U fission yield mass distribution predicted by
various models and corresponding ENDF/B-VIII.0 data (red dot), at neutron
incident energies of (a) 14 MeV and (b) 0.5 MeV. The black solid curves, blue
dashed curves and dash-dotted curves represent the predictions of FYTD,
BNN+TALYS and TALYS models respectively. BNN+TALYS and TALYS predictions are
taken from Ref. Wang  _et al._ (2019).

In this part, all 235U or 239Pu yield under the three energy points are set as
missing values, and the FYTD model is used to reproduce the 235U or 239Pu
yield data. Fig. 2 shows the predicted mass distribution of the products in
235U fission at 14 MeV and 0.5 MeV. In BNN+TALYS model, JENDL-4.0 data was
used for model training, and this paper uses the ENDF/B-VIII.0 data. However,
the fission yield data in JENDL database actually refers to that in ENDF/B
database, the fission yield data in these two database are almost the same. In
addition, the prediction by the BNN+TALYS model are also made when all 235U
yield data in the learning set are removed. Thus, it is very suitable to
compare with the prediction by the FYTD model. It can be seen from Fig. 2 that
the TALYS model has a large deviation from the ENDF/B data at the heavy
nucleus peak, and also fails to predict the neutron energy dependence of
valley yields. After combining the trained BNN algorithm, the BNN+Talys model
can correct these deviations. However, the BNN+Talys model still has defects.
In the valley area (A=116$\thicksim$128) of Fig. 2(b), the yield here is low,
around $4\times 10^{-4}$, but the BNN+TALYS model significantly overestimated
this value, predicting a value around $2\times 10^{-3}$. This phenomenon also
appears in the area of A=104$\thicksim$120 in Fig. 2(a). In general, the FYTD
model performs better, it predicts the yields in the peak areas well, and
successfully predicts the neutron energy dependence of the valley yields, only
slightly underestimates the yields around A = 109. The accuracy of the FYTD
model prediction can also be proved by $\chi_{N}^{2}$ in Table. 2.

This result proves that when all 235U yield data in the learning set are
removed, the FYTD model can better capture information from other heavy
nucleus yield data and predict the 235U yield data. At the same time, it can
also capture the neutron energy dependence information of yield data and
predict the variation trend of yield with neutron energy. Compared with
algorithms such as BNN, the FYTD model preform better in fission yield
evaluating and predicting.

Table 2:  The validation errors $\chi_{N}^{2}$ of various models. Errors for TALYS, BNN-40 and BNN-40 + TALYS are taken from Ref. Wang  _et al._ (2019) Models | Validation $\chi_{N}^{2}(\times 10^{-5})$
---|---
TALYS Koning and Rochman (2012) | 8.334
BNN-40 Wang  _et al._ (2019) | 1.640
BNN-40 + TALYS Wang  _et al._ (2019) | 1.134
FYTD | 0.701
Figure 3:  (Color online) Logarithm of the ratio of FYTD prediction to
ENDF/B-VIII.0 data (logarithmic error) for 0.5 MeV neutron induced 239Pu
fission. (a) is the distribution of errors and (b) is the count of errors of
different sizes. Each dot represents a product, the ordinate is the prediction
error of the product, and the abscissa is the yield of the product.

Compared with the BNN and BNN+TALYS model, which only predicted the mass
distribution of the yield, the FYTD model can also predict the isotopic
distribution of the product. In order to more comprehensively show the
difference between the predicted yield and the ENDF/B-VIII.0 data of 851
fission products, Fig. 3 shows the logarithmic error distribution of the fast
neutron induced fission product yield of 239Pu and the RMSE of the predicted
results. A dotted line with zero error is marked in the two figures. The
closer the dot is to the line, the smaller the error. The RMSE of the
prediction result at this time is 0.395, which is little higher than the 0.281
in Table. 1. This is understandable, because the learning set of this
prediction result removes all the data of all three energy points of 239Pu,
while the test in Table. 1 only removes the data of one energy point. It can
be seen from Fig. 3(a) that the greater the yield value of the product, the
higher the accuracy of the prediction. This confirms that the method of
constructing tensors with partial logarithmic and partial linear coordinates
can make nuclide prediction with large yield more accurate. Fig. 3(b) shows
the count of errors of different sizes. It can be seen that the logarithmic
error of 98 $\%$ isotopes is within $\pm$ 1\. that is, the difference between
the predicted yield and ENDF/B data is within 1 order of magnitude. And for 88
$\%$ nuclides, the difference between the predicted yield and ENDF/B data is
within 0.5 order of magnitude. However, it can be seen from Fig. 3(a) that
239Pu fast neutron-induced fission yield data spans 16 orders of magnitude
from $10^{-18}$ to $10^{-3}$, this prediction of isotope distribution by the
FYTD model is relatively satisfactory.

###  III.3 238Np, 243Am and 236Np fission yield prediction

After several verification, it is proved that the FYTD model has good learning
and prediction ability for ENDF/B data. On this basis, the FYTD model is
applied to the real data prediction to predict the missing data in the ENDF/B
database. In order to verify whether the predicted results are reliable, the
predicted yield data will be compared with JEFF database and experimental
data.

Figure 4:  (Color online) Fission yield mass distribution predicted by FYTD
model (black solid curves) and corresponding JEFF-3.3 data (red dots), for (a)
243Am + n at 0.0253 eV and (b) 238Np + n at 0.0253 eV.

Taking 243Am and 238Np as examples, their fission yield data under thermal
neutron is missing in the ENDF/B-VIII.0 database, but they exists in the
JEFF-3.3 database. The predicted fission yield will be compared with the data
in JEFF-3.3. shows the comparison of yield mass distribution. It can be seen
from Fig. 4(a) that the prediction of 243Am by the FYTD model is basically
within the error range of JEFF data. Only at the edge A=80$\thicksim$85, there
is a certain deviation between the prediction results and the JEFF data. For
prediction of 238Np in Fig. 4(b), there is a certain difference of the light
nucleus peak from JEFF data.

From the above two comparisons, it can be seen that the prediction of the FYTD
model for the yield data of 243Am and 238Np under thermal neutron are
consistent with the JEFF data, but this is based on the existing data. As can
be seen from Table 1, 243Am and 238Np have no data under thermal neutrons but
under fast neutrons in ENDF/B database. It is relatively easy for the FYTD
model to predict the data of a target nucleus at one energy points when the
data at other energy point is known. However, the yield data of some target
nuclei are very scarce, such as 236Np. Its independent yield data are missing
in both ENDF/B and JEFF databases, and there are only a few experimental
measurement data of chain yield. To predict the yield data of 236Np, it is
necessary to capture information from other target nuclei, which tests the
model’s ability to learn and predict the dependence of target nucleus.

Figure 5:  (Color online) Fission yield mass distribution of 236Np + n at
0.0253 eV predicted by FYTD model (black solid curves) and two sets of
corresponding experimental data (red and green dots) from Ref. Lindner and
Seegmiller (1990); Gindler  _et al._ (1981). The inset panel on the upper left
shows the valley areas in logarithmic scale.

Figrue 5 show the mass distribution of fission products in thermal fission of
236Np measured by the two experimental groups and the corresponding
predictions by the FYTD model. The panel on the upper left is added to show
the valley areas in logarithmic scale. It can be seen that the predictions by
the FYTD model at heavy nucleus peak agree with the two sets of experimental
data, and there is a slight deviation near A = 100 at light nucleus peak. As
for the valley value, the predicted magnitude is consistent with the
experimental data. This result proves the prediction ability of the FYTD model
on the target nucleus dependence of fission yield.

###  III.4 Prediction of 2 MeV neutron-induced fission yield

The comparison and verification above proved the FYTD model ability to learn
and predict neutron energy dependence and target nucleus dependence. However,
there are few energy points in the yield data in the current database, a
reliable model needs to be able to predict data at more energy points. And the
above verification is also based on three energy points. In order to verify
the scalability of the FYTD model over the incident neutron energy degrees of
freedom, the only fission yield data under 2 MeV (239Pu + n at 2 MeV) in
ENDF/B-VIII.0 database is included in tensor construction. We test whether the
model can predict the yield data of the remaining heavy nucleus at 2 MeV based
on this yield data of 239Pu + n at 2 MeV.

Figure 6:  (Color online) Fission yield mass distribution for (a) 235U and (b)
229Th at 2 MeV predicted by FYTD model (black dots), GEF model (green dots)
and corresponding experimental data (red dots) from Ref. Glendenin  _et al._
(1981); Agarwal  _et al._ (2008). The inset panel shows the valley areas in
logarithmic scale.

The predictions of yield at 2 MeV by the FYTD model are compared with the
experimental data and the predictions of the GEF model. The data of the GEF
model comes from the database GEFY-6.2. Figrue 6 shows the mass distribution
of the products in 235U and 229Th fissions at 2 MeV. It can be seen form Fig.
6(a) that both predictions by the FYTD and GEF models agree well with
experimental data in the peak area. In the valley area shown in the inset
panel, the prediction by the FYTD model is more consistent with the
experimental data than that by the GEF model. This may be due to the generally
low yield in the valley area and the lack of experimental data in this area,
resulting in a slightly insufficient predictive ability of traditional
phenomenological models such as GEF.

For predictions for 229Th in Fig. 6(b), no completely strict experimental data
for 229Th at 2 MeV are published. The experimental data at 1.9 MeV was used
for comparison. According to energy dependence, it can be simply speculated
that the data at 1.9 MeV should be slightly higher than the data at 2 MeV in
the peak area, and lower in the valley area. The peak predicted by the FYTD
model is just slightly lower than the experimental value and higher than that
predicted by the GEF model. Overall, the predictions by the FYTD model are
also more consistent with the experimental data than the predictions of the
GEF model.

Figure 7:  (Color online) Neutron excess as a function of product charge
number for 235U + n fission at different energies, in which the 2 MeV results
are predicted by the FYTD model. Spherical shell Z = 50, N =82 and deformed
shells Z = 44, N = 64 are marked by dashed lines.

Figures 7 and 8 verify the predicted fission yield induced by 2 MeV neutron of
235U from a more physical point of view. Figure 7 shows the neutron excess of
235U fission products induced by neutrons of different energies. The data of 2
MeV are the prediction by the FYTD model, and the rest are for ENDF/B data.
The positions of some shells are marked in the figure. The definition of
neutron excess is referred to the Ref. Ramos  _et al._ (2019) and is defined
as the ratio of the average neutron number to the proton number of the
product. This quantity can reflect the neutron proton composition of the
fission product and the influence of shell effect on fission. From the data at
0.0253 eV, 0.5 MeV and 14 MeV in the figure, it can be observed that the
neutron proton composition of the product is greatly affected by the shells of
Z = 50, N = 82 and Z = 44, N = 64. However, with the increase of neutron
incident energy, the excitation energy of fission system increases, and the
curve gradually flattens. This is because the increase of excitation energy
will hinder the influence of shell effect, and the increase of valley area in
yield mass distribution is also caused by this reason. It can be seen from the
figure that the predicted data under 2 MeV neutron incidence generally
conforms to this law. In most areas, the predictions are between 0.5 MeV and
14 MeV data, and only deviate at Z = 43. This verification better proved that
FYTD model learn and predict the impact of excitation energy on yield.

Figure 8:  (Color online) Isomeric ratios predicted by FYTD model (black dots)
and GEF model (green dots) for several fission products from 235U + n at 2
MeV.

In the FYTD model, the isomeric states of products can be considered, and the
corresponding data can be learned and predicted. Figure 8 shows the proportion
of isomeric states for 128Sb, 130Sb, 131Te, 132Sb, 132I, 133Te, 133Xe, 134I
and 135Xe produced in fission. It can be seen that except for 130Sb, the
predictions by the FYTD model are consistent with those by the GEF model. The
above comparison proves the powerful ability of the multi-dimensional
dependence learning, which can predict physical laws to a some extent.
Predictions for fission at 2 MeV also proves that the FYTD model is not only
applicative at 0.0253 eV, 0.5 MeV and 14 MeV, but also extend to other neutron
energy with the help of data for one target nucleus.

##  IV CONCLUSION

This work applied the tensor decomposition algorithm to the prediction of
independent fission product yields. A Fission Yield Tensor Decomposition model
was established and applied to a variety of yield predictions for
verification.

The fission yields of 235U and 239Pu are set as missing values and then
predicted. Compared with those by the Talys, BNN, and BNN+Talys models, the
predictions by the FYTD model agree better with the ENDF/B-VIII.0 data. The
predicted fission yields under fast neutron and high-energy neutron incident
are compared. It is found that the FYTD model can reproduce the neutron energy
dependence of the yield data. The data for 239Pu fission under fast neutron
incident spans 16 orders of magnitude from 10-3 to 10-18. Comparison between
the prediction and ENDF/B-VIII.0 data for 851 fission products shows that
98$\%$ of them agree with each other within 1 order of magnitude, and 88$\%$
within 0.5 order of magnitude.

By comparing the predictions with the data of 238Np and 243Am in the JEFF-3.3
database, as well as the experimental data of 236Np, the predictive ability of
the model for the target nucleus dependence of the yield data is demonstrated.
The scalability of the yield tensor decomposition model over the incident
neutron energy degrees of freedom is examined. After adding a set of 2 MeV
neutron-induced 239Pu fission yield data into the yield tensor, the 2 MeV
neutron-induced fission yields of 235U and 229Th are predicted. The comparison
with the experimental data shows that the prediction of this work is similar
to the prediction by the GEF model in the peak area and more accurate in the
valley area. At the same time, the prediction of the ratio of isomeric states
and excess of neutrons in the product also proves that the model has the
ability to learn and predict the laws of physics.

The multiple rounds of comparative verification show that the FYTD model can
capture the complex multi-dimensional dependence of fission yields and make
reasonable predictions. At the same time, the model has scalability in the
energy dimension, and also has a certain ability to predict physical laws.
Facing the demands of various nuclear energy devices and systems with complex
neutron spectrum in the future, the scalability of the FYTD model in the
energy dimension makes it potentially valuable for future application. The
predictions and evaluations of this work are based on the ENDF/B-VIII.0
database. In the future we will conduct evaluations based on the EXFOR
database.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China
under Grants No. 11875328, 12075327 and 12105170.

## References

  * Hahn and Strassmann (1939) O. Hahn and F. Strassmann, Naturwissenschaften 27, 11 (1939).
  * Meitner and Frisch (1939) L. Meitner and O. R. Frisch, Nature 143, 239 (1939).
  * Schunck and Robledo (2016) N. Schunck and L. Robledo, Reports on Progress in Physics 79, 116301 (2016).
  * Bender  _et al._ (2020) M. Bender, R. Bernard, G. Bertsch, S. Chiba, J. Dobaczewski, N. Dubray, S. A. Giuliani, K. Hagino, D. Lacroix, Z. Li, _et al._ , Journal of Physics G: Nuclear and Particle Physics 47, 113002 (2020).
  * Hamilton  _et al._ (2013) J. Hamilton, S. Hofmann, and Y. Oganessian, Annual Review of Nuclear and Particle Science 63, 383 (2013).
  * Pei  _et al._ (2009) J. C. Pei, W. Nazarewicz, J. A. Sheikh, and A. K. Kerman, Physical Review Letters 102, 192501 (2009).
  * Eichler  _et al._ (2015) M. Eichler, A. Arcones, A. Kelic, O. Korobkin, K. Langanke, T. Marketin, G. Martinez-Pinedo, I. Panov, T. Rauscher, S. Rosswog, C. Winteler, N. T. Zinner, and F.-K. Thielemann, The Astrophysical Journal 808, 30 (2015).
  * Mueller  _et al._ (2011) T. A. Mueller, D. Lhuillier, M. Fallot, A. Letourneau, S. Cormon, M. Fechner, L. Giot, T. Lasserre, J. Martino, G. Mention, A. Porta, and F. Yermia, Physical Review C 83, 054615 (2011).
  * Bernstein  _et al._ (2019) L. A. Bernstein, D. A. Brown, A. J. Koning, B. T. Rearden, C. E. Romano, A. A. Sonzogni, A. S. Voyles, and W. Younes, Annual Review of Nuclear and Particle Science 69, 109 (2019).
  * Ramos  _et al._ (2019) D. Ramos, M. Caamaño, F. Farget, C. Rodríguez-Tajes, L. Audouin, J. Benlliure, E. Casarejos, E. Clement, D. Cortina, O. Delaune, X. Derkx, A. Dijon, D. Doré, B. Fernández-Domínguez, G. de France, A. Heinz, B. Jacquot, C. Paradela, M. Rejmund, T. Roger, M.-D. Salsac, and C. Schmitt, Physical Review C 99, 024615 (2019).
  * Denschlag (1986) H. O. Denschlag, Nuclear Science and Engineering 94, 337 (1986).
  * Brown  _et al._ (2018) D. Brown, M. Chadwick, R. Capote, A. Kahler, A. Trkov, M. Herman, A. Sonzogni, Y. Danon, A. Carlson, M. Dunn, D. Smith, G. Hale, G. Arbanas, R. Arcilla, C. Bates, B. Beck, B. Becker, F. Brown, R. Casperson, J. Conlin, D. Cullen, M.-A. Descalle, R. Firestone, T. Gaines, K. Guber, A. Hawari, J. Holmes, T. Johnson, T. Kawano, B. Kiedrowski, A. Koning, S. Kopecky, L. Leal, J. Lestone, C. Lubitz, J. Márquez Damián, C. Mattoon, E. McCutchan, S. Mughabghab, P. Navratil, D. Neudecker, G. Nobre, G. Noguere, M. Paris, M. Pigni, A. Plompen, B. Pritychenko, V. Pronyaev, D. Roubtsov, D. Rochman, P. Romano, P. Schillebeeckx, S. Simakov, M. Sin, I. Sirakov, B. Sleaford, V. Sobes, E. Soukhovitskii, I. Stetcu, P. Talou, I. Thompson, S. van der Marck, L. Welser-Sherrill, D. Wiarda, M. White, J. Wormald, R. Wright, M. Zerkle, G. Žerovnik, and Y. Zhu, Nuclear Data Sheets 148, 1 (2018).
  * Kellett  _et al._ (2009) M. A. Kellett, O. Bersillon, and R. Mills, (2009).
  * Shibata  _et al._ (2011) K. Shibata, O. Iwamoto, T. Nakagawa, N. Iwamoto, A. Ichihara, S. Kunieda, S. Chiba, K. Furutaka, N. Otuka, T. Ohsawa, T. Murata, H. Matsunobu, A. Zukeran, S. Kamada, and J.-i. Katakura, Journal of Nuclear Science and Technology 48, 1 (2011).
  * Bulgac  _et al._ (2016) A. Bulgac, P. Magierski, K. J. Roche, and I. Stetcu, Physical Review Letters 116, 122504 (2016).
  * Regnier  _et al._ (2016) D. Regnier, N. Dubray, N. Schunck, and M. Verrière, Physical Review C 93, 054611 (2016), publisher: American Physical Society.
  * Younes  _et al._ (2019) W. Younes, D. M. Gogny, and J.-F. Berger,  _A microscopic theory of fission dynamics based on the generator coordinate method_ , Vol. 950 (Springer, 2019).
  * Randrup and Möller (2011) J. Randrup and P. Möller, Physical Review Letters 106, 132503 (2011), publisher: American Physical Society.
  * Randrup  _et al._ (2011) J. Randrup, P. Möller, and A. J. Sierk, Physical Review C 84, 034613 (2011).
  * Pomorski  _et al._ (2017) K. Pomorski, F. A. Ivanyuk, and B. Nerlo-Pomorska, The European Physical Journal A 53, 59 (2017).
  * Liu  _et al._ (2019) L.-L. Liu, X.-Z. Wu, Y.-J. Chen, C.-W. Shen, Z.-X. Li, and Z.-G. Ge, Physical Review C 99, 044614 (2019).
  * Fang  _et al._ (2021) Z.-X. Fang, M. Yu, Y.-G. Huang, J.-B. Chen, J. Su, and L. Zhu, Nuclear Science and Techniques 32, 72 (2021).
  * Brosa  _et al._ (1990) U. Brosa, S. Grossmann, and A. Müller, Physics Reports 197, 167 (1990).
  * Schmidt  _et al._ (2016) K.-H. Schmidt, B. Jurado, C. Amouroux, and C. Schmitt, Nuclear Data Sheets 131, 107 (2016).
  * Schmidt and Jurado (2018) K.-H. Schmidt and B. Jurado, Reports on Progress in Physics 81, 106301 (2018).
  * Wang  _et al._ (2019) Z.-A. Wang, J. Pei, Y. Liu, and Y. Qiang, Physical Review Letters 123, 122501 (2019).
  * Niu and Liang (2018) Z. Niu and H. Liang, Physics Letters B 778, 48 (2018).
  * Utama  _et al._ (2016a) R. Utama, J. Piekarewicz, and H. B. Prosper, Physical Review C 93, 014311 (2016a).
  * Utama  _et al._ (2016b) R. Utama, W.-C. Chen, and J. Piekarewicz, Journal of Physics G: Nuclear and Particle Physics 43, 114002 (2016b).
  * Ma  _et al._ (2020) C.-W. Ma, D. Peng, H.-L. Wei, Z.-M. Niu, Y.-T. Wang, and R. Wada, Chinese Physics C 44, 014104 (2020).
  * Song  _et al._ (2022) Q.-F. Song, L. Zhu, and J. Su, Chinese Physics C 46, 074108 (2022).
  * Neufcourt  _et al._ (2019) L. Neufcourt, Y. Cao, W. Nazarewicz, E. Olsen, and F. Viens, Physical Review Letters 122, 062502 (2019).
  * Neufcourt  _et al._ (2020) L. Neufcourt, Y. Cao, S. Giuliani, W. Nazarewicz, E. Olsen, and O. B. Tarasov, Physical Review C 101, 014319 (2020).
  * Lovell  _et al._ (2019) A. Lovell, A. Mohan, P. Talou, and M. Chertkov, EPJ Web of Conferences 211, 04006 (2019).
  * Qiao  _et al._ (2021) C. Y. Qiao, J. C. Pei, Z. A. Wang, Y. Qiang, Y. J. Chen, N. C. Shu, and Z. G. Ge, Physical Review C 103, 034621 (2021).
  * Liu  _et al._ (2013) J. Liu, P. Musialski, P. Wonka, and J. Ye, IEEE Transactions on Pattern Analysis and Machine Intelligence 35, 208 (2013).
  * Chen  _et al._ (2019) X. Chen, Z. He, and L. Sun, Transportation Research Part C: Emerging Technologies 98, 73 (2019).
  * Chen  _et al._ (2017) X. Chen, Z. Han, Y. Wang, Q. Zhao, D. Meng, L. Lin, and Y. Tang, arXiv preprint arXiv:1705.06755 (2017).
  * Koning and Rochman (2012) A. J. Koning and D. Rochman, Nuclear Data Sheets Special Issue on Nuclear Reaction Data, 113, 2841 (2012).
  * Lindner and Seegmiller (1990) M. Lindner and D. W. Seegmiller, Radiochimica Acta 49, 1 (1990).
  * Gindler  _et al._ (1981) J. Gindler, L. Glendenin, E. Krapp, S. Fernandez, K. Flynn, and D. Henderson, Journal of Inorganic and Nuclear Chemistry 43, 445 (1981).
  * Glendenin  _et al._ (1981) L. E. Glendenin, J. E. Gindler, D. J. Henderson, and J. W. Meadows, Physical Review C 24, 2600 (1981).
  * Agarwal  _et al._ (2008) C. Agarwal, A. Goswami, P. C. Kalsi, S. Singh, A. Mhatre, and A. Ramaswami, Journal of Radioanalytical and Nuclear Chemistry 275, 445 (2008).