# Structure in the Event-by-Event Energy-Dependent Neutron-Gamma Multiplicity
Correlations in ^252Cf(sf)

Stefano Marin<EMAIL_ADDRESS>Department of Nuclear Engineering and
Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA  M.
Stephan Okar  Department of Nuclear Engineering and Radiological Sciences,
University of Michigan, Ann Arbor, MI 48109, USA  Eoin P. Sansevero
Department of Nuclear Engineering and Radiological Sciences, University of
Michigan, Ann Arbor, MI 48109, USA  Isabel E. Hernandez  Department of Nuclear
Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI
48109, USA  Catherine A. Ballard  Department of Nuclear Engineering and
Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA
Ramona Vogt  Nuclear and Chemical Sciences Division, Lawrence Livermore
National Laboratory, Livermore, CA 94550, USA  Department of Physics and
Astronomy Department, University of California, Davis, CA 95616, USA  Jørgen
Randrup  Nuclear Science Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Patrick Talou  Computational Physics Division, Los
Alamos National Laboratory, Los Alamos, NM 87545, USA  Amy E. Lovell
Theoretical Physics Division, Los Alamos National Laboratory, Los Alamos, NM
87545, USA  Ionel Stetcu  Theoretical Physics Division, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  Olivier Serot  CEA, DES, IRESNE, DER,
SPRC, Physics Studies Laboratory, Cadarache, F-13108 Saint-Paul-lès-Durance,
France  Olivier Litaize  CEA, DES, IRESNE, DER, SPRC, Physics Studies
Laboratory, Cadarache, F-13108 Saint-Paul-lès-Durance, France  Abdelhazize
Chebboubi  CEA, DES, IRESNE, DER, SPRC, Physics Studies Laboratory, Cadarache,
F-13108 Saint-Paul-lès-Durance, France  Shaun D. Clarke  Department of Nuclear
Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI
48109, USA  Vladimir A. Protopopescu  Oak Ridge National Laboratory, Oak
Ridge, TN 37830, USA  Sara A. Pozzi  Department of Nuclear Engineering and
Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

###### Abstract

The emission of neutrons and gamma rays by fission fragments reveal important
information about the properties of fragments immediately following scission.
The initial fragment properties, correlations between fragments, and emission
competition give rise to correlations in neutron-gamma emission. Neutron-gamma
correlations are important in nonproliferation applications because the
characterization of fissionable samples relies on the identification of
signatures in the measured radiation. Furthermore, recent theoretical and
experimental advances have proposed to explain the mechanism of angular
momentum generation in fission. In this paper, we present a novel analysis
method of neutrons and gamma rays emitted by fission fragments that allows us
to discern structure in the observed correlations. We have analyzed data
collected on ^252Cf(sf) at the Chi-Nu array at the Los Alamos Neutron Science
Center. Through our analysis of the energy-differential neutron-gamma
multiplicity covariance, we have observed enhanced neutron-gamma correlations,
corresponding to rotational band gamma-ray transitions, at gamma-ray energies
of $0.7$ and $1.2$ MeV. To shed light on the origin of this structure, we
compare the experimental data with the predictions of three model
calculations. The origin of the observed correlation structure is understood
in terms of a positive spin-energy correlation in the generation of angular
momentum in fission.

neutron-gamma multiplicity competition; fission fragment de-excitation

††preprint: LA-UR-21-20802

##  I Introduction

Heavy atomic nuclei can decay by fission, i.e., separating into two or more
large nuclear fragments. The complexities of the strong nuclear forces and the
electromagnetic interactions between hundreds of nucleons lead to a variety of
exit channels for the fission process, resulting in stochastic distributions
of both masses and kinetic energies of the produced fission fragment [1].
Large correlated variations exist also in the excitation energies and angular
momenta of the fragments [2, 3, 4], as well as in the emission of neutrons and
gamma rays [5]. The initial conditions of fission fragments cannot be directly
probed due in part to the short time scales of fragment de-excitation. To
study the condition of the fragments immediately after scission, we study the
emission of neutrons and gamma rays, since their emission is highly correlated
with the initial fragment conditions [2, 6]. Particularly important open
questions in fission modeling are the generation of angular momentum at
scission [7, 3, 8] and the sharing of excitation energy between fragments [4,
9].

In a previous publication [10], we surveyed the results of several earlier
publications on event-by-event neutron-gamma multiplicity correlations in
^252Cf(sf). After developing and applying analytic unfolding techniques to the
experimental data, we found these data to be in qualitative agreement with one
another and with the predictions of model calculations. Specifically, the
results indicate the existence of significant negative event-by-event
correlations between the neutron-gamma multiplicities. However, these
experimental results were affected by energy-dependent systematic biases.

Understanding neutron-gamma correlations is of great importance in
nonproliferation applications. The negative neutron-gamma covariance
represents a signature of fissioning special nuclear material. In this work,
we find that the energy-dependent covariance contains a more detailed
signature of the fission reaction, which changes drastically in magnitude and
sign across the examined energies. Furthermore, the characterization of the
energy-dependence of neutron-gamma correlations is an important component in
addressing systematic biases related to other observables.

The goal of this paper is to further develop our previous analysis of neutron-
gamma correlations by studying energy-dependent correlations in the
spontaneous fission of ^252 Cf. We find that the structure of the correlations
observed in the experiment can be explained by known and predicted physical
mechanisms and indicates the existence of an energy-dependence in the
generation of angular momentum in fission.

Section II presents a brief overview of the physical mechanisms that give rise
to correlations in fission. Section III presents the mathematical formalism
required to extract and quantify the predicted correlations. Section IV
reports the experimental setup and the techniques used in experiment. In
Section V, we describe the most prominent features of the neutron-gamma
correlations and compare them with model calculations. Lastly, in Section VI,
we draw our conclusions on the origin of the observed correlations.

##  II Sources of Correlations

### Neutron and Gamma-Ray Emission

Following scission, fission fragments promptly decay by neutron emission, if
energetically possible, followed by gamma-ray emission. Two modes of gamma-ray
emission are usually distinguished [11, 5]: statistical gamma rays in the
continuum at higher intrinsic excitation energies (stat), and band-transition
gamma rays in the discrete-level region (disc). A pictorial diagram, inspired
by Ref. [11], is shown in Fig. 1. We have explicitly drawn a tilted initial
distribution of the initial fragment conditions to highlight suspected spin-
energy correlations in the formation of fission fragments [3, 7].

Figure 1: Schematic representation of neutron and gamma-ray emission from
fission fragments. The neutron separation energy is indicated by a dashed
line, and the Yrast rotational band is indicated as a dot-dashed line [12].

It is worthwhile noting the different roles played by neutron and gamma-ray
emission in fragment de-excitation. At scission, the total available
excitation energy is partitioned between the two fragments in the form of
intrinsic energy, shape deformation, and collective motion [4]. Neutron and
gamma-ray emission takes place from fully accelerated fragments, giving enough
time for the nuclear deformation to relax into excitation energy. Neutrons,
due to their binding energies, dissipate the largest portion of the fragment
intrinsic energy, i.e., the energy not stored in collective rotational motion
of nucleons. Stat gamma rays dissipate the remaining intrinsic excitation
energy until a rotational band is reached. disc gamma rays dissipate the
remaining rotational energy and angular momentum. The emission of disc gamma
rays is regulated most strongly by the feeding of rotational band levels,
which in turn depends on the initial angular momenta of the fragment.

In the remainder of this section we will investigate the sources of possible
neutron-gamma correlations. Positive neutron-gamma multiplicity correlations,
in the present context, indicate that the emission of each neutron makes it
more likely to emit a gamma ray. Negative correlations represent the opposite.

### Decay Competition

The most direct source of correlations is the competition between neutron and
gamma-ray emission. In theory, this competition could result in very strong
negative correlations between the neutron and gamma-ray multiplicity. In
practice, the two decay modes can be treated as independent apart from a
narrow intrinsic energy range near the neutron separation energy [12, 13]. The
gamma rays competing with the neutrons are stat. Furthermore, because the
fragments are less excited close to the neutron separation energy [14], we
expect these negative correlations to appear at low neutron energies.

### Resource Competition

Another significant source of neutron-gamma competition is that introduced by
the finite excitation energy and angular momentum of the fragments. In fact,
given the same excitation energy $E^{*}$, a fragment that dissipates more of
its intrinsic energy through neutron evaporation will have less energy to
dissipate through stat gamma-ray emission. The competition over the intrinsic
energy of the fragments would result in overall negative correlations between
neutrons and stat gamma rays.

Competition related to the fragment angular momentum $J$ are more complicated,
given the vector nature of the angular momentum. If neutron emission is found
to remove significant quantities of angular momentum, we expect negative
correlations between neutrons and disc gamma rays, since these are the gamma
rays mostly affected by the angular momenta of the fragments.

### Excitation Energy and Angular Momentum

Correlations between neutrons and gamma rays are expected if their emission
depends on the same fragment properties. For example, we know that more
energetic fission fragments will emit more energetic neutrons and stat gamma
rays. Therefore we expect the spectra of neutrons and gamma rays to receive
contributions from their shared dependence on the fragment temperature [14].

The interesting and still unclear mechanism for angular momentum generation
[2, 15, 16, 17, 18] can lead to correlated behavior in neutron-gamma emission.
In fact, the intrinsic excitation energy is strongly linked to the neutron
multiplicity [19], while the emission of disc gamma rays is regulated by the
angular momentum [3, 20]. Thus, if $E^{*}$ and $J$ are positively correlated,
i.e., positive spin-energy correlations, as phenomenological and theoretical
models predict [15, 21, 22], we also expect positive correlations between
neutrons and disc gamma rays.

These correlations will be especially important in this work, since disc gamma
rays have similar energies for many different probable fragments in fission,
and can thus give rise to correlated structure. Experimentally, spin-energy
correlations have not been conclusively proven. In fact their existence has
been disputed by some [2, 7], and supported by others [3, 23]. See also Refs.
[24, 25, 26]. Clearly, more experimental evidence regarding possible spin-
energy correlations is needed.

We note that gamma-ray emission is correlated with the properties of the post-
evaporation fission fragments. Therefore, the energy-spin correlations can be
counterbalanced by the dissipation of the fragment energy and angular momentum
during neutron emission.

### Inter-Fragment Correlations

While all of the sources of correlations discussed above are related to
neutrons and gamma rays from the same fragment, it is also possible to have
correlations between emissions from opposite fragments. The strongest source
of inter-fragment correlations are those related to the initial excitation
energy and angular momentum of the fragments. If the fragments are assumed to
have strongly correlated initial conditions, we expect the correlations
between neutrons and gamma rays originating from opposite fragments (inter-
fragment) to be of the same order of magnitude as those between particles
originating in the same fragment (intra-fragment).

Because inter-fragment correlations do not suffer from internal decay
competition or resource competition, we expect inter-fragment correlations to
lack negative sources. Stated in an equivalent way, we expect inter-fragment
correlations to be positive overall.

A noteworthy special case is mentioned here. If the initial fragment
conditions are strongly correlated to one another but one of the fragments
emits significantly fewer neutrons than the other, we expect inter-fragment
correlations to be large. In fact, the angular momentum of the fragment
emitting fewer neutrons would be less changed by neutron emission.

### Summary

We summarize the discussion of the possible origins of neutron-gamma
correlations in Table 1. For each source discussed, we specify whether we
expect it to be introducing positive or negative correlations. We also
distinguish, in the $\gamma$-type column, whether the correlations arise
between neutrons and stat gamma rays or disc gamma rays. Lastly, we indicate
whether these correlations can also arise from inter-fragment correlations.

Table 1:  Summary of the expected sources of neutron-gamma correlations Source | Sign of Correlation |  $\gamma$ \- type | Inter-Fragment
---|---|---|---
Decay competition | Negative | stat | No
Energy competition | Negative | stat | No
Angular momentum competition | Negative | disc | No
Fragment temperature | Overall neutral | stat | Yes
Spin-Energy correlation | Positive | disc | Yes

##  III Analysis

Having discussed why and where we expect to see correlations, we need to
quantify their magnitude. In this section we discuss a new type of statistical
analysis of radiation data, the normalized differential multiplicity
covariance.

### Covariance

Let us define the event-by-event neutron and gamma multiplicities, $N_{n}$ and
$N_{\gamma}$. We investigate the correlations in the neutron-gamma emission by
analyzing the multiplicity covariance $\text{cov}(N_{n},N_{\gamma})$. This
quantity describes the linear correlations in the event-by-event fluctuations
of the neutron and gamma multiplicities with respect to the mean values. We
have already shown in our previous study [10] that the neutron-. disc
multiplicity covariance is negative. Decay competition has a a negligible
effect on the correlations integrated over all energies. Using Table 1, we
conclude that energy and angular momentum competition dominate over the other
sources of correlations. To investigate the other sources, we analyze the
dependence of the neutron-gamma covariance on the energies of the outgoing
particles.

### Differentials

The energy-dependent covariance is introduced as the covariance between the
neutron and gamma-ray event-by-event spectra,

$\text{cov}\left(\frac{dN_{n}}{dE_{n}},\frac{dN_{\gamma}}{dE_{\gamma}}\right)=\frac{\partial^{2}\text{cov}\left(N_{n},N_{\gamma}\right)}{\partial
E_{n}\partial E_{\gamma}},$ (1)

where $E_{n}$ and $E_{\gamma}$ are the neutron and gamma-ray energies. The
arguments of the covariance in the left-hand side of Eq (1) are the event-by-
event multiplicities differentiated by spectra, i.e., the number of particles
emitted with a specified energy, listed for each fission event. The right-hand
side of Eq. (1) then follows from the bilinearity of the covariance operator.
The energy-differentiated covariance is then interpreted as the correlations
between neutron and gamma rays multiplicities, if only neutrons and gamma rays
of energies $E_{n}$ and $E_{\gamma}$ are counted. The differentiated
covariance allows us to investigate how the emission of neutrons of a
specified energy affects the emission of gamma rays with another specified
energy. For sufficiently narrow energy bins, the event-by-event multiplicity
in each bin also tends to zero. For sufficiently small multiplicities, the
correlations approach linearity and are described appropriately by the
covariance.

### Covariance Scaling

While the differential covariance describes the correlations between neutrons
and gamma rays, it also scales with the multiplicity in each energy bin. To
compare the strength of the correlations across energies, we normalize the
covariance by the independent spectra, i.e.,

$C_{E_{n},E_{\gamma}}=\frac{\partial^{2}\text{cov}\left(N_{n},N_{\gamma}\right)}{\partial
E_{n}\partial E_{\gamma}}\left(\frac{d\langle
N_{n}\rangle}{dE_{n}}\frac{d\langle
N_{\gamma}\rangle}{dE_{\gamma}}\right)^{-1}.$ (2)

With this choice of normalization, $C_{E_{n},E_{\gamma}}$ satisfies

$C_{E_{n},E_{\gamma}}\geq-1,$ (3)

where the equality holds only in the case where the emission of a neutron of
energy $E_{n}$ completely precludes the emission of a gamma ray of energy
$E_{\gamma}$, and vice versa.

##  IV Experiment

### Experiment

We analyze the data of Marcath et al. [27] collected using the Chi-Nu array at
the Los Alamos neutron science center (LANSCE). We have analyzed $1.7\times
10^{10}$ ^252Cf spontaneous fission events, each tagged by an ionization
chamber fabricated at Oak Ridge National Laboratory [28]. Due to hardware
limitations, only $42$ of the $54$ EJ-309 organic scintillation detectors were
active during the measurement. Each detector was placed at a distance of
approximately $1$ m from the fission source. Coincidences between the EJ-309
detectors and the fission chamber are employed to measure the neutron and
gamma-ray multiplicities, $D_{n}$ and $D_{\gamma}$ respectively, detected in
each event. For more information on this experiment, see Refs [27, 29, 10].

### Simulation and Response

We performed a high-fidelity simulation of the system response using MCNPX-
PoliMi [30, 31], a Monte Carlo-based radiation transport code. In the
simulation, the response of each detector is individually modeled, taking into
account any differences in distance to source, light output, and calibration.
We introduce a small perturbation in the light outputs and the array geometry
in the simulations to better reproduce the reference evaluated neutron
spectrum after radiation transport. The parameters of the light output
response for each detector, modeled analytically as exponential functions
[32], are slightly perturbed from the literature values to better fit the
reference spectrum. Clearly, the unfolding procedure is systematically biased
to reproduce the known ^252Cf(sf) neutron spectrum. No corrections are applied
to reproduce the gamma-ray spectrum. The analysis of the simulated system
response is presented in Fig. 2. The absolute detection efficiencies,
$\epsilon_{n}$ and $\epsilon_{\gamma}$, are not uniform across the emitted
particle spectra.

Figure 2: Simulated energy response of the Chi-Nu array to (a) neutrons and
(b) gamma rays. The solid red line, referencing the right-hand axis, shows the
evaluated spectrum included in the standard MCNPX-PoliMi californium source
[30, 31]. The vertical red and horizontal dashed black lines represent
respectively the mean emission energy and mean system efficiency. The grey
bands indicate the regions outside the energy acceptance of the system.

### Unfolding Matrices

In experiment, we do not have access to the emitted multiplicities, $N_{n}$
and $N_{\gamma}$, or the emitted neutron and gamma-ray energies $E_{n}$ and
$E_{\gamma}$. Rather, we have access to detected multiplicities $D_{n}$ and
$D_{\gamma}$, differentiated with respect to time of flight energy $T_{n}$ for
neutrons, and energy deposited by gamma rays $F_{\gamma}$. Due to the large
distance between detectors and source, time of flight leads to accurate
assignment of the energy up to $E_{n}\approx 7$ MeV. However, because gamma
rays interact through Compton scattering with the liquid organic scintillator,
only a fraction of the energy is deposited in each gamma-ray interaction.

In our simulations, we calculate the matrix $dE_{n}/dT_{n}$ describing, for
each neutron emitted with energy $E_{n}$, the probability of assigning it an
energy of $T_{n}$. Similarly, we calculate $dE_{\gamma}/dF_{\gamma}$ for the
probability of a gamma ray of energy $E_{\gamma}$ to deposit energy
$F_{\gamma}$. Energy and time resolution, determined from the experimental
data, have been applied in simulation to smooth out all calculated results.

The unfolding of the measured neutron-gamma correlations and spectra is
performed using the response matrices and the measured neutron-gamma
correlations,

$\frac{\partial^{2}\text{cov}(N_{n},N_{\gamma})}{\partial E_{n}\partial
E_{\gamma}}=\frac{\partial^{2}\text{cov}(D_{n},D_{\gamma})}{\partial
T_{n}\partial
F_{\gamma}}\left(\epsilon_{n}\frac{dE_{n}}{dT_{n}}\right)^{-1}\left(\epsilon_{\gamma}\frac{dE_{\gamma}}{dF_{\gamma}}\right)^{-1}\
,$ (4) $\frac{d\langle N_{n}\rangle}{dE_{n}}=\frac{d\langle
D_{n}\rangle}{dT_{n}}\left(\epsilon_{n}\frac{dE_{n}}{dT_{n}}\right)^{-1}$ (5a)
$\frac{d\langle N_{\gamma}\rangle}{dE_{\gamma}}=\frac{d\langle
D_{\gamma}\rangle}{dF_{\gamma}}\left(\epsilon_{\gamma}\frac{dE_{\gamma}}{dF_{\gamma}}\right)^{-1}\
.$ (5b)

### Regularization

To reduce the effects of noise in the unfolded distributions, we have applied
Tikhonov regularization [33]. In this unfolding procedure, regularization
parameters $\alpha_{n}$ and $\alpha_{\gamma}$ are introduced. The unfolding
procedure is optimized and validated on the measured spectra, where we have
determined the values of and uncertainties on $\alpha_{n}$ and
$\alpha_{\gamma}$. By comparison with the evaluated spectra, we have
determined the effective acceptance regions in the unfolded distributions,
$1.2\leq E_{n}\leq 8$ MeV and $0.22\leq E_{\gamma}\leq 3.2$ MeV. These
acceptance regions, indicated by the areas between the shaded bands in Fig. 2,
cover $66\%$ and $78\%$ of the neutron and gamma-ray spectra, respectively.
The results of the spectral unfolding procedure are shown in Fig. 3.

Figure 3: Measured spectra and unfolded spectra after Tikhonov regularization.
The same reference spectra given in Fig. 2 are also shown in red.

##  V Results

The unfolding procedure introduced in the last section is applied to the
experimental data. We have analyzed the data using energy bins of width $0.2$
MeV for $E_{n}$ and $0.08$ MeV for $E_{\gamma}$. We present our experimental
determination of $C_{E_{n},E_{\gamma}}$ in Fig. 4. We have included systematic
errors associated with the unfolding by including experimental curves above
and below the indicated fixed $E_{n}$. Systematic biases associated with
background, particle misclassification, detector blinding, and inelastic
gamma-ray production are removed according to the procedure outlined in Ref.
[10]. To facilitate the visualization of the data, we have included a
horizontal dashed line indicating the average of the correlations across all
$E_{n}$ and $E_{\gamma}$.

We observe structure in the form of two enhancements at $E_{\gamma}\approx
0.7$ and $1.2$ MeV. A downslope develops at the lowest energies, extending
down to the rejection region at $<0.3$ MeV. Higher-energy correlation
enhancements start to appear with increasingly higher gates on neutron
outgoing energies, notably at $E_{\gamma}\approx 1.7$ MeV.

Figure 4: Partial derivative analysis of the correlations. In each panel, the
fixed neutron energy is indicated, and a green band represents the average of
the experiment over all energies. The experimental data are shown in black,
the error bars are given based on variations within neighboring $E_{n}$
slices. The horizontal green line represents the correlations averaged across
all energies.

The experimental results are compared to the CGMF [5, 34], FIFRELIN [11, 26,
35], and FREYA [36, 37] model calculations. The release versions of CGMF and
FREYA are used here. FIFRELIN results are shown for an energy independent
(constant) and an energy dependent ($E^{*}$-dependent) model for the
generation of angular momentum in the fragments [38]. A Gaussian smoothing
informed by the energy resolution of the system is applied to all model
calculations to be consistent with the data analysis. We note that the goal of
the current analysis is not to compare any individual model calculations to
the data, but rather to determine which features of the data differ from all
the model calculations and can therefore offer insights into the fission
mechanism. The feature of the correlations we focus on in this work are the
correlations enhancements at $E_{\gamma}=0.7$ and $1.2$ MeV.

### Enhanced Correlation Structure

The two correlation enhancements at $0.7$ and $1.2$ MeV appears to arise from
disc gamma rays positively correlating to the emission of neutrons. The gamma
rays giving rise to these enhancements cannot be assigned to a single fragment
level scheme, but are rather emergent behaviors from the combination of many
transitions of similar energies. We note that much more work, both
experimental and theoretical, is needed in order to quantify the spin-energy
correlations.

The enhancement at $1.2$ MeV corresponds to transitions from quasi-spherical
nuclei near ^132 Sn. The enhancement at $0.7$ MeV receives contributions from
both light and heavy fragments, and it is due to electric quadrupole Yrast
transitions with initial spins $4\hbar$, $6\hbar$, and $8\hbar$. The predicted
valley at $E_{\gamma}<0.4$ MeV is due to competition between neutron emission
and disc gamma rays.

The correlation enhancement at $E_{\gamma}=0.7$ MeV indicates the existence of
positive spin-energy correlations. In fact, the emission of neutrons
correlates strongly with the intrinsic excitation energy of the fragment.
Enhancements at $E_{\gamma}\approx 0.7$ MeV indicate that the high-spin,
$J\approx 4-8\ \hbar$ rotational bands are more populated with each neutron
emission. This hypothesis is confirmed by comparing FIFRELIN calculations
performed with and without spin-energy coupling. In the calculations, both
shown in Fig. 4, we have found that the amplitude of the enhancement is
strengthened significantly when the spiin-energy coupling is considered. We
have not attempted to quantify the strength of the spin-energy correlations
from the experimental data, but this could be pursued in future work.
Qualitatively, we can say that the observation of the enhanced correlations
indicates the presence of spin-energy correlations in the generation of
angular momentum in fission fragments.

The enhanced correlations at $E_{\gamma}=1.2$ MeV reveals less about the spin-
energy correlations and more about biasing of the sample. In fact, the main
contributor to this enhancement are low-spin transitions in the spherical
fragment region. We note that neutron outgoing energies from near-symmetric
fissions are generally higher than for the most probable splits [19]. The
observation of high-energy neutrons in the lab frame thus biases the fission
sample more strongly towards symmetric fission, and is thus positively
correlated with high-energy disc gamma-ray emission.

### Other Features

Two other notable features of the energy dependent correlations are their
behaviors at the higher and lower end of the gamma-ray spectrum. At high
$E_{\gamma}$, the correlations are most representative of stat emission, since
these gamma rays have a harder spectrum than those from disc emission. In the
calculations, we find that the intra-fragment contributions to the
correlations involving stat gamma rays are predominantly negative. Therefore,
we might interpret positive correlations at high $E_{n}$ and $E_{\gamma}$ as
originating from inter-fragment correlations, and thus conclude from the
experiment that inter-fragment correlations are not as strong as intra-
fragment correlations. This in turn would lead to the conclusion that the
initial states of the fragments, specifically the intrinsic excitation energy,
between the two fragments, are not strongly correlated.

Several problems exist with the above conclusion, which is premature at this
stage, and will need to be investigated further. First of all, as mentioned
for the enhancement at $1.2$ MeV, gating on higher neutron energies does not
only bias towards higher energy fragments, but also towards symmetric fission.
Therefore, the correlations shown here are systematically biased by this
selection. Second, in the calculations, we find that one of the largest
sources of positive correlations in the high $E_{\gamma}$ region are
continuum-discrete transitions. Because these effects cannot be disentangled
from the inter-fragment correaltions, we do not attempt to draw conclusions
based on this region.

At low $E_{\gamma}$, the correlations in the data are more negative than the
average. The models predict this feature to originate from inter-fragment
correlations, specifically between neutrons from the light fragment and disc
gamma rays from the heavy fragment.

##  VI Conclusion

We have presented an investigation of the neutron-gamma correlations in ^252
Cf(sf). The first result of this study is the development and application of
the normalized differential multiplicity covariance, a powerful tool in the
analysis of radiation correlations. We have analyzed the correlations between
neutrons and gamma rays using the normalized differential multiplicity
covariance. This analysis yielded previously unobserved structure in the
neutron-gamma correlations. Specifically, we have presented experimental
evidence of enhancements in neutron-gamma correlations around the gamma-ray
energies of $0.7$ and $1.2$ MeV. We have compared the experimental data to
model calculations, to determine the physical origins of the observed
structure. The appearance of enhancements in the correlations can likely be
explained by the correlations between excitation energy and angular momentum.
Further theoretical and experimental studies will be needed to quantify the
strength of these correlations.

For nonproliferation applications, the observation of structure in the
neutron-gamma correlations represent a further signature of special nuclear
material. The presence of positive correlations enhancements at
$E_{\gamma}\approx 0.7$ and $1.2$ MeV, and negative correlations elsewhere,
can be exploited by energy-sensitive systems to identify the presence of
fissioning material, and distinguish it from other radioactive sources.
Preliminary calculations of ^239 Pu$(\text{n}_{\text{th}},\text{f})$ have
shown that the correlation structure is predicted to persists in this reaction
as well.

An experiment involving event-by-event neutron-gamma correlations in
coincidence with a fission fragment detector capable of measuring masses and
kinetic energies is being planned. Such an experiment would further our
understanding of energy and angular momentum correlations in fission and shed
light on the decay competition of neutrons and gamma rays near the neutron
separation energy. Furthermore, there still remains significant disagreement
in the literature about correlations between neutron-gamma competition and
fragment properties. The planned experiment will investigate these
correlations, perhaps leading to a reconciliation of previous results.

###### Acknowledgements.

S.M. thanks the experimental group at LANSCE-LANL and M.J. Marcath for sharing
the experimental data used in this analysis. This work was in part supported
by the Office of Defense Nuclear Nonproliferation Research & Development (DNN
R&D), National Nuclear Security Administration, US Department of Energy. This
research was funded in-part by the Consortium for Verification Technology
under Department of Energy National Nuclear Security Administration award DE-
NA0002534. The work of V.A.P. was performed under the auspices of UT-Battelle,
LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy.
The work of P.T., A.L., and I.S. is carried out under the auspices of the
National Nuclear Security Administration of the U.S. Department of Energy at
Los Alamos National Laboratory under Contract No. 89233218CNA000001. The work
of R.V. was performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under Contract DE-AC52- 07NA27344. J.
R. acknowledges support from the Office of Nuclear Physics in the U.S.
Department of Energy under Contract DE-AC02-05CH11231.

## References

  * Brosa  _et al._ [1990] U. Brosa, S. Grossmann, and A. Müller, Nuclear scission, Physics Reports 197, 167 (1990).
  * Wilhelmy  _et al._ [1972] J. B. Wilhelmy, E. Cheifetz, R. C. Jared, S. G. Thompson, H. R. Bowman, and J. O. Rasmussen, Angular momentum of primary products formed in the spontaneous fission of ${}^{252}\mathrm{Cf}$, Phys. Rev. C 5, 2041 (1972).
  * Nifenecker  _et al._ [1972] H. Nifenecker, C. Signarbieux, M. Ribrag, J. Poitou, and J. Matuszek, Gamma-neutron competition in the de-excitation mechanism of the fission fragments of 252Cf, Nuclear Physics A 189, 285 (1972).
  * Schmidt and Jurado [2011] K. H. Schmidt and B. Jurado, Final excitation energy of fission fragments, Phys. Rev. C 83, 061601 (2011).
  * Talou  _et al._ [2018] P. Talou, R. Vogt, J. Randrup, M. E. Rising, S. A. Pozzi, J. Verbeke, M. T. Andrews, S. D. Clarke, P. Jaffke, M. Jandel, T. Kawano, M. J. Marcath, K. Meierbachtol, L. Nakae, G. Rusev, A. Sood, I. Stetcu, and C. Walker, Correlated prompt fission data in transport simulations, The European Physical Journal A 54, 9 (2018).
  * Hoffman [1964] M. M. Hoffman, Directional correlation of fission fragments and prompt gamma rays associated with thermal neutron fission, Phys. Rev. 133, B714 (1964).
  * Schmid-Fabian [1988] R. Schmid-Fabian,  _Investigation of the Spontaneous Fission of Cf-252 at the Heidelberg-Darmstadt Crystal-Ball-Spectrometer_ , Ph.D. thesis, University of Heidelberg (1988).
  * Brosa [1985] U. Brosa, Sawtooth curve of neutron multiplicity, Phys. Rev. C 32, 1438 (1985).
  * Signarbieux  _et al._ [1972] C. Signarbieux, J. Poitou, M. Ribrag, and J. Matuszek, Correlation entre les energies d’excitation des deux fragments complementaires dans la fission spontanee de 252cf, Physics Letters B 39, 503 (1972).
  * Marin  _et al._ [2020] S. Marin, V. A. Protopopescu, R. Vogt, M. J. Marcath, S. Okar, M. Y. Hua, P. Talou, P. F. Schuster, S. D. Clarke, and S. A. Pozzi, Event-by-event neutron–photon multiplicity correlations in 252cf(sf), Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 968, 163907 (2020).
  * Litaize and Serot [2010] O. Litaize and O. Serot, Investigation of phenomenological models for the monte carlo simulation of the prompt fission neutron and $\gamma$ emission, Phys. Rev. C 82, 054616 (2010).
  * Thomas and Grover [1967] T. D. Thomas and J. R. Grover, Angular momentum effects in the gamma-ray de-excitation of fission fragments, Phys. Rev. 159, 980 (1967).
  * Spyrou  _et al._ [2016] A. Spyrou, S. N. Liddick, F. Naqvi, B. P. Crider, A. C. Dombos, D. L. Bleuel, B. A. Brown, A. Couture, L. Crespo Campo, M. Guttormsen, A. C. Larsen, R. Lewis, P. Möller, S. Mosby, M. R. Mumpower, G. Perdikakis, C. J. Prokop, T. Renstrøm, S. Siem, S. J. Quinn, and S. Valenta, Strong neutron-$\gamma$ competition above the neutron threshold in the decay of ${}^{70}\mathrm{Co}$, Phys. Rev. Lett. 117, 142701 (2016).
  * Blatt and Weisskopf [1952] J. M. Blatt and V. F. Weisskopf,  _Theoretical Nuclear Physics_, Dover Books on Physics Series (Dover Publications, 1952).
  * Bonneau  _et al._ [2007] L. Bonneau, P. Quentin, and I. N. Mikhailov, Scission configurations and their implication in fission-fragment angular momenta, Phys. Rev. C 75, 064313 (2007).
  * Rakopoulos  _et al._ [2018] V. Rakopoulos, M. Lantz, A. Solders, A. Al-Adili, A. Mattera, L. Canete, T. Eronen, D. Gorelov, A. Jokinen, A. Kankainen, V. S. Kolhinen, I. D. Moore, D. A. Nesterenko, H. Penttilä, I. Pohjalainen, S. Rinta-Antila, V. Simutkin, M. Vilén, A. Voss, and S. Pomp, First isomeric yield ratio measurements by direct ion counting and implications for the angular momentum of the primary fission fragments, Phys. Rev. C 98, 024612 (2018).
  * Wilson  _et al._ [2021] J. N. Wilson, D. Thisse, M. Lebois, N. Jovančević, D. Gjestvang, R. Canavan, M. Rudigier, D. Étasse, R.-B. Gerst, L. Gaudefroy, E. Adamska, P. Adsley, A. Algora, M. Babo, K. Belvedere, J. Benito, G. Benzoni, A. Blazhev, A. Boso, S. Bottoni, M. Bunce, R. Chakma, N. Cieplicka-Oryńczak, S. Courtin, M. L. Cortés, P. Davies, C. Delafosse, M. Fallot, B. Fornal, L. Fraile, A. Gottardo, V. Guadilla, G. Häfner, K. Hauschild, M. Heine, C. Henrich, I. Homm, F. Ibrahim, Ł. W. Iskra, P. Ivanov, S. Jazrawi, A. Korgul, P. Koseoglou, T. Kröll, T. Kurtukian-Nieto, L. Le Meur, S. Leoni, J. Ljungvall, A. Lopez-Martens, R. Lozeva, I. Matea, K. Miernik, J. Nemer, S. Oberstedt, W. Paulsen, M. Piersa, Y. Popovitch, C. Porzio, L. Qi, D. Ralet, P. H. Regan, K. Rezynkina, V. Sánchez-Tembleque, S. Siem, C. Schmitt, P.-A. Söderström, C. Sürder, G. Tocabens, V. Vedia, D. Verney, N. Warr, B. Wasilewska, J. Wiederhold, M. Yavahchova, F. Zeiser, and S. Ziliani, Angular momentum generation in nuclear fission, Nature 590, 566 (2021).
  * Randrup and Vogt [2021] J. Randrup and R. Vogt, Generation of fragment angular momentum in fission (2021), arXiv:2103.14778 [nucl-th] .
  * Göök  _et al._ [2014] a. Göök, F.-J. Hambsch, and M. Vidali, Prompt neutronmultiplicity in correlation with fragments from spontaneous fission of Cf-252, Physical Review C 90, 064611 (2014).
  * Vogt and Randrup [2021] R. Vogt and J. Randrup, Angular momentum effects in fission, Phys. Rev. C 103, 014610 (2021).
  * Bertsch  _et al._ [2015] G. F. Bertsch, W. Loveland, W. Nazarewicz, and P. Talou, Benchmarking nuclear fission theory, Journal of Physics G: Nuclear and Particle Physics 42, 1 (2015), arXiv:1502.05985 .
  * Randrup and Vogt [2014] J. Randrup and R. Vogt, Refined treatment of angular momentum in the event-by-event fission model FREYA, Phys. Rev. C 89, 044601 (2014).
  * Chebboubi  _et al._ [2017] A. Chebboubi, G. Kessedjian, O. Litaize, O. Serot, H. Faust, D. Bernard, A. Blanc, U. Köster, O. Méplan, P. Mutti, and C. Sage, Kinetic energy dependence of fission fragment isomeric ratios for spherical nuclei 132sn, Physics Letters B 775, 190 (2017).
  * Fréhaut  _et al._ [1983] J. Fréhaut, A. Bertin, and R. Bois, Mesure de $N_{n}$, $N_{\gamma}$, et $E^{T}_{\gamma}$, pour la fission de 232th, 235u et 237np induite par des neutrons d’energie comprise entre 1 et 15 mev, in  _Nuclear Data for Science and Technology_ , edited by K. H. Böckhoff (Springer Netherlands, Dordrecht, 1983) pp. 78–81.
  * Wang  _et al._ [2016] T. Wang, G. Li, L. Zhu, Q. Meng, L. Wang, H. Han, W. Zhang, H. Xia, L. Hou, R. Vogt, and J. Randrup, Correlations of neutron multiplicity and $\gamma$ -ray multiplicity with fragment mass and total kinetic energy in spontaneous fission of Cf 252, Physical Review C 93, 1 (2016).
  * Thulliez  _et al._ [2019] L. Thulliez, O. Litaize, O. Serot, and A. Chebboubi, Neutron and $\gamma$ multiplicities as a function of incident neutron energy for the ${}^{237}\mathrm{Np}(n,f)$ reaction, Phys. Rev. C 100, 044616 (2019).
  * Marcath  _et al._ [2018] M. J. Marcath, R. C. Haight, R. Vogt, M. Devlin, P. Talou, I. Stetcu, J. Randrup, P. F. Schuster, S. D. Clarke, and S. A. Pozzi, Measured and simulated ${}^{252}\mathrm{{Cf}}(\mathrm{sf})$ prompt neutron-photon competition, Phys. Rev. C 97, 044622 (2018).
  * Kirsch  _et al._ [2017] L. E. Kirsch, M. Devlin, S. Mosby, and J. Gomez, A new measurement of the 6li(n,a)t cross section at mev energies using a 252cf fission chamber and 6li scintillators, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 874, 57 (2017).
  * Schuster  _et al._ [2019] P. F. Schuster, M. J. Marcath, S. Marin, S. D. Clarke, M. Devlin, R. C. Haight, R. Vogt, P. Talou, I. Stetcu, T. Kawano, J. Randrup, and S. A. Pozzi, High resolution measurement of tagged two-neutron energy and angle correlations in ${}^{252}\mathrm{Cf}$ (sf), Phys. Rev. C 100, 014605 (2019).
  * Pozzi  _et al._ [2003] S. A. Pozzi, E. Padovani, and M. Marseguerra, MCNP-PoliMi: a monte-carlo code for correlation measurements, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 513, 550 (2003).
  * Pozzi  _et al._ [2012] S. A. Pozzi, S. D. Clarke, W. J. Walsh, E. C. Miller, J. L. Dolan, M. Flaska, B. M. Wieger, A. Enqvist, E. Padovani, J. K. Mattingly, D. L. Chichester, and P. Peerani, MCNPX-PoliMi for nuclear nonproliferation applications, Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 694, 119 (2012).
  * Norsworthy  _et al._ [2017] M. A. Norsworthy, A. Poitrasson-Rivière, M. L. Ruch, S. D. Clarke, and S. A. Pozzi, Evaluation of neutron light output response functions in ej-309 organic scintillators, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 842, 20 (2017).
  * Morozov [1984] V. A. Morozov,  _Methods for Solving Incorrectly Posed Problems_ , edited by Z. Nashed (Springer-Verlag New York, 1984).
  * Talou  _et al._ [2020] P. Talou, I. Stetcu, P. Jaffke, M. E. Rising, A. E. Lovell, and T. Kawano, Fission fragment decay simulations with the cgmf code (2020), arXiv:2011.10444 [nucl-th] .
  * Litaize  _et al._ [2015] O. Litaize, O. Serot, and L. Berge, Fission modelling with FIFRELIN, The European Physical Journal A 51, 177 (2015).
  * Randrup and Vogt [2009] J. Randrup and R. Vogt, Calculation of fission observables through event-by-event simulation, Phys. Rev. C 80, 024601 (2009).
  * Verbeke  _et al._ [2018] J. Verbeke, J. Randrup, and R. Vogt, Fission reaction event yield algorithm freya 2.0.2, Computer Physics Communications 222, 263 (2018).
  * L. Thulliez  _et al._ [2016] L. Thulliez, O. Litaize, and O. Serot, Sensitivity studies of spin cut-off models on fission fragment observables, EPJ Web of Conferences 111, 10003 (2016).