# Neutron capture-induced nuclear recoils as background for CENS measurements
at reactors

A.J. Biffl  Corresponding author<EMAIL_ADDRESS>Department of
Physics, University of Colorado Denver, Denver, Colorado 80217, USA  A.
Gevorgian  Department of Physics, University of Colorado Denver, Denver,
Colorado 80217, USA  K. Harris  Department of Physics, University of Colorado
Denver, Denver, Colorado 80217, USA  A.N. Villano  Corresponding author:
<EMAIL_ADDRESS>Department of Physics, University of Colorado
Denver, Denver, Colorado 80217, USA

###### Abstract

Nuclear reactors represent a promising neutrino source for CENS (coherent-
elastic neutrino-nucleus scattering) searches. However, reactor sites also
come with high ambient neutron flux. Neutron capture-induced nuclear recoils
can create a spectrum that strongly overlaps the CENS signal for recoils
$\lesssim$ 100 eV for nuclear reactor measurements in silicon or germanium
detectors. This background can be particularly critical for low-power research
reactors providing a moderate neutrino flux. In this work we quantify the
impact of this background and show that, for a measurement 10 m from a 1 MW
reactor, the effective thermal neutron flux should be kept below $\sim$
7$\times$ 10-4 n/cm2s so that the CENS events can be measured at least at a
5$\sigma$ level with germanium detectors in 100 kg yr exposure time. This flux
corresponds to 60% of the sea-level flux but needs to be achieved in a
nominally high-flux (reactor) environment. Improved detector resolution can
help the measurements, but the thermal flux is the key parameter for the
sensitivity of the experiment. For silicon detectors, the constraint is even
stronger and thermal neutron fluxes must be near an order of magnitude lower.
This constraint highlights the need of an effective thermal neutron mitigation
strategy for future low threshold CENS searches. In particular, the neutron
capture-induced background can be efficiently reduced by active veto systems
tagging the deexcitation gamma following the capture.

††preprint:

##  I Introduction

Coherent Elastic Neutrino Nucleus Scattering (CENS) is the transfer of
momentum between a neutrino and a nucleus as a whole via neutral current
exchange. This process is experimentally significant for detecting low energy
($\lesssim$ MeV) neutrinos due to its large cross section, two to four orders
of magnitude larger than the commonly used inverse beta decay (IBD) [1, 2].
This allows CENS detectors to be much smaller than their IBD counterparts.

The CENS process has already been measured at an accelerator site by the
COHERENT Collaboration using the Spallation Neutron Source at Oak Ridge
National Laboratory as a neutrino source [3]. The accelerator provides
neutrinos of a few tens of MeV, and other measurements using this type of
source are planned, for example the upcoming European Spallation Source [2].
Likewise, recent studies show potential for measurements at reactor sites,
where the sources of MeV antineutrinos are commercial or research fission
reactors [4, 5, 6, 7, 8, 9, 10, 11].

The trade-off for CENS’ relatively high interaction rate is that it is
exceptionally difficult to detect because of the tiny (sub-keV) recoil of the
target nuclei, even despite recent advances in detector resolution near
threshold, for example in silicon [12, 13]. One major hurdle is that
distinguishing a nuclear recoil event caused by CENS from one caused by other
sources is difficult. In the vicinity of nuclear reactors, a large reactogenic
thermal neutron flux could cause enough neutron-capture-induced recoil events
to hide the CENS signal. A capture of a neutron via the (n,$\gamma$) process
produces one or more $\sim$MeV scale prompt gamma rays and a prompt nuclear
recoil at the sub-keV scale — coinciding with the energy scale of events from
reactor neutrinos.

The recoil spectrum from neutron capture on silicon has been recently measured
[14] and creates nuclear recoil events down to $\sim$100 eV. This paper
investigates the thermal neutron background mitigations needed to measure a
CENS signal in the vicinity of a nuclear reactor.

##  II  CENS and capture spectra

We used the model given by Mueller [15] for the four principle fissionable
isotopes as the model for reactor antineutrino emissions. We attempted to
emulate the expected emissions of a low-enriched ($\sim 20\%$ 235U) reactor
fuel composition, i.e. that employed by the MINER Collaboration at the Nuclear
Science Center at Texas A&M University [16]. We assume neutrino emissions
consistent with the MINER study [11], i.e., dominated by 235U (96.7%) with
smaller fractions of 238U and 239Pu (1.3 and 2.0% respectively), and
negligible ($\lesssim 0.1$%) 241Pu.

We tested the target materials silicon and germanium with natural isotope
distributions. The reactor spectrum was convoluted with the known CENS cross
section [2] using the form factor model given by [17] to yield a recoil energy
spectrum for CENS events. The form factor was quite close to 1, with a largest
deviation from Ge at a recoil energy of around 1 keV, where it had a value of
0.9895.

To calculate the expected recoil energy spectrum from neutron captures, we use
nrCascadeSim, a publicly available dedicated simulation tool developed by two
of the authors here for nuclear recoils resulting from neutron captures [18].
This software was used to generate a large sample of capture-induced recoil
energies from which we construct a probability density function (PDF) to
sample. The code takes into account the most probable multistep cascades in
the deexcitation process following capture and the possibility of decay-in-
flight for the intermediate atoms/ions within the cascade.

##  III Assumed detector configuration

Recoil datasets were simulated with a total of 3,000 (10,000) CENS events,
corresponding to approximately a 100 kg yr exposure time of natural silicon
(germanium) detectors, at the CENS event rate calculated by [11] for detector
deployment 10 m from the reactor core. Events that produce energies below the
detection threshold are removed when applying the resolution model. Detectors
were modeled with a range of different effective resolutions (defined here as
resolution $\sigma$ at a recoil energy of 50 eV). All detector resolutions
obeyed a resolution function of the form:

$\sigma(E)=\sqrt{\sigma_{0}^{2}+AE}$ (1)

where $E$ is the recoil energy, $\sigma_{0}$ is the baseline resolution, and
$A$ is a detector-specific factor which is varied to achieve a given effective
resolution. We also consider several values of $\sigma_{0}$: 1 eV, 5 eV, 10
eV, and 25 eV. Table 1 shows resolution criteria that are either future
targets or have already been achieved for several experimental efforts. We can
see that the current best demonstrated solid-target baseline resolution comes
from the $\nu$-cleus Collaboration at 3.7 eV [19]–between our two lowest
baseline resolution points and probably achievable in the near future for Si
and Ge. Going by the table it is seen that a 25 eV baseline resolution is
probably on the horizon for many detectors. For the solid detectors in the
table we also see that the typical threshold is a factor of 3–9 larger than
the baseline resolution–roughly in line with our requirement that events have
energies $\geq$5$\sigma_{0}$ (see below).

Experiment |  Detector type | Threshold |  Baseline resolution
---|---|---|---
CONNIE[20, 21] | Si CCD | 28 eVee222achieved | 5.5 eVee222achieved
CONUS[22] | Ge PPC | 200 eVee222achieved | 25 eVee222achieved
Dresden-II[4] | Ge PPC | 200 eVee222achieved | 33 eVee222achieved
MINER[11] | Si/Ge cal |  $\sim 20$ eV111target | 5 eV111target
RED-100[23, 24] | Xe 2PS | 300 eV111target | $\cdots$
RICOCHET[25, 26] | Si/Ge bol |  $\sim 50$ eV111target | 17 eV222achieved
TEXONO[27, 28] | Ge PPC | 300-400 eVee111target |  45 eVee222achieved
$\nu$-cleus[19] |  CaWO4 / |  |
Al2O3 cal | 19.7 eV222achieved | 3.7 eV222achieved |
$\nu$GeN[29] | Ge PPC |  $350$ eV222achieved | 93 eV222achieved
Table 1: Summary of CENS searches at reactors. Nonstandard abbreviations used
are as follows. cal: Calorimeter. 2PS: Two-phase scintillator. bol: Bolometer.
Note also eVee refers to electron-equivalent (ionization) energy; other
energies are heat.

Different levels of ambient thermal neutron flux were surveyed between about
$\sim 10^{-7}$ n/cm2sand $\sim 10^{-3}$ n/cm2s. The flux range is motivated by
where this analysis produces high significance (5$\sigma$) in the presence of
modest “other” background (see below). The flux range includes the approximate
behavior of the Dresden-II [4] published thermal neutron flux — 0.25 n/cm2s—
if we account for an approximately 3000 times higher reactor output power of
that commercial reactor. The fluxes were translated to a total capture event
count via the neutron capture cross section with natural silicon and germanium
and assuming an exposure of 100 kg yr. The capture events were then sampled
from the output PDF of nrCascadeSim v1.4.2. All sampled recoils with energy
$\leq 5\sigma_{0}$ were eliminated to emulate data collection with a finite
energy threshold event trigger.

Thermal neutrons are not the only nuclear recoil backgrounds possible in CENS
experiments, so we include an “other” nuclear recoil background with shape
$\propto E^{-0.9}$, consistent with neutron-scatter background data given by
[16], and very similar to the $E^{-1.2}$ form mentioned in [5]. From MINER’s
2017 background measurement they expect a combined 100 events/kg/day in
nuclear recoil and electron recoil backgrounds but only 5-20 events/kg/day in
CENS events [16]. The MINER authors acknowledge the need to control background
and other experiments have demonstrated backgrounds with between 40-60% of
what is expected from the CENS process [10, 30]. Given this information we
normalize our nuclear recoil background to 60% of the CENS rate below about 1
keV. We do not add any further electron recoil background because it is often
diluted to higher electron equivalent energies in ionization-sensitive
detectors. Furthermore, the electron recoils in some cases can be
discriminated [31, 32, 33]. This background model is expected to represent
approximately the maximum acceptable background to a typical CENS search.
Other variations of our fits have been completed in the release of our code
for this analysis [34].

##  IV Statistical Comparison

The combined sample of recoil energies from the three sources (CENS, neutron
capture, and other backgrounds), were separated back into CENS and neutron-
capture components by binned maximum-likelihood fits with the likelihood
function:

$\mathcal{L}=e^{-n_{\text{total}}}\prod_{j=1}^{N_{\text{bins}}}\frac{1}{C_{j}!}\big{[}n_{\nu}P_{\nu,j}+n_{c}P_{c,j}+n_{b}P_{b,j}\big{]}^{C_{j}},$
(2)

where $C_{j}$ is the observed number of events in bin $j$, $n_{\nu}$, $n_{c}$,
and $n_{b}$ are the fitted number of CENS, capture, and background events,
respectively, $n_{\text{total}}$ is the sum of these, and $P_{\nu,j}$,
$P_{c,j}$, and $P_{b,j}$ are the probability of an event of each type lying in
bin $j$ (all normalized such that they sum to unity over all bins).

We employ a likelihood ratio test on $\mathcal{L}/\mathcal{L}^{\prime}$, where
$\mathcal{L}^{\prime}$ is a similar likelihood function but without the CENS
contribution–with the parameter $n_{\nu}$ in Eq. (2) set to zero.

The two likelihoods are compared using Wilks’ theorem, which states that
$2\ln(\mathcal{L}/\mathcal{L}^{\prime})$ is distributed as a chi-squared
random variable with, in this case, one degree of freedom [35]. This leads
immediately to a probability for comparing the nested models, i.e., the
probability that the combined model of Eq. (2) is a better model than the
simple model with $n_{\nu}$$\equiv$0\. We quantify the confidence level as the
Z-value ($\sigma$ value) corresponding to the symmetric (two-sided) normal
error integral with the same probability, i.e.,
$\sqrt{2\ln(\mathcal{L}/\mathcal{L}^{\prime})}$.

Figure 1 shows example fits for different combinations of effective
resolutions and neutron fluxes in silicon. Resolutions of 10 eV and 35 eV are
used in conjunction with neutron fluxes of $4.36\times 10^{-5}$ n/cm2s and
$9.77\times 10^{-5}$ n/cm2s with $\sigma_{0}=1$ eV. The black points show the
histogram of our toy model data for given CENS, thermal flux, and background
contributions. The toy model data is smeared using our hypothetical resolution
functions and plotted for recoil energies between 0 eV and 1800 eV. Solid blue
and orange curves show the fitted PDFs for with- and without-CENS fits. The
dashed blue curve shows the CENS contribution to the with-CENS fit. The
confidence level of each fit is given in the upper left corner of each plot,
red if less than $5\sigma$ and blue if greater.

In Figure 1 the leftmost and rightmost plots show the two methods by which the
CENS signal can be detected, either by improving resolution so the CENS signal
can be discerned on the low-energy downslope (leftmost plot), or by reducing
the neutron flux so that the peak caused by CENS events is clearly visible
over the signal from capture events (rightmost plot). The center plot shows a
scenario where the thermal neutron flux is too high to produce a 5$\sigma$
CENS detection with the assumed livetime. Figure 2 shows the analogous fit
plots for germanium. In the germanium case we use resolutions of 20 eV and 45
eV with thermal neutron fluxes of $5.52\times 10^{-4}$ n/cm2s and $1.75\times
10^{-4}$ n/cm2s. The thermal neutron fluxes used for germanium are generally
higher than for silicon because germanium is more resilient to the flux —
although there are more capture events the capture spectrum overlaps less
strongly than in silicon.

Figure 1: (Color online) Likelihood fitting examples for three different
combinations of resolution and neutron flux for $\sigma_{0}=$1 eV in silicon.
The left two plots show a flux of $9.77\times 10^{-5}$ n/cm2s, corresponding
to $\sim 1.1$ M capture events, and the right is lower flux, $4.36\times
10^{-5}$ n/cm2s, corresponding to $\sim 500,000$ capture events. The right two
plots show data with an effective resolution of 35 eV, and the left plot shows
an effective resolution of 10 eV. The black points show the histogram of our
toy model data for given CENS, thermal flux, and background contributions.
Solid blue and orange curves show the fitted PDFs for with- and without-CENS
fits. The dashed blue curve shows the CENS contribution to the with-CENS fit.
Figure 2: (Color online) Likelihood fitting examples for three different
combinations of resolution and neutron flux for $\sigma_{0}=$1 eV in
germanium. The left two plots show a flux of $5.52\times 10^{-4}$ n/cm2s,
corresponding to $\sim 31.6$ M capture events, and the right is lower flux,
$1.75\times 10^{-4}$ n/cm2s, corresponding to $\sim 10$ M capture events. The
right two plots show data with an effective resolution of 45 eV, and the left
plot shows an effective resolution of 20 eV. The black points show the
histogram of our toy model data for given CENS, thermal flux, and background
contributions. Solid blue and orange curves show the fitted PDFs for with- and
without-CENS fits. The dashed blue curve shows the CENS contribution to the
with-CENS fit.

##  V Results and Discussion

Figure 3 shows the results for 100 kg yr exposure in Si (left) and Ge (right)
for $\sigma_{0}=1$ eV. To produce this plot, we generated a grid of 25
resolution values and 25 flux values and completed fits on that grid for each
material. The contour plot is a smoothing of those fit results. The vertical
axis shows ambient thermal neutron flux and the horizontal axis shows the
effective resolution. Colors correspond to confidence levels. The two
horizontal broken lines delineate a few reference flux levels — the gray
dashed line shows the generally accepted ambient neutron flux at sea level, 4
cm-2 hr-1 [36]. The gray dotted line shows 1 part in $10^{9}$ of the neutron
flux measured at the MINER facility, 5.8$\times 10^{7}$ cm-2 s-1 [16],
extrapolated to ten meters from the reactor core. As reference, the two lines
correspond to approximately 13,000,000 (63,000,000) and 7,500,000 (37,000,000)
neutron capture events in Si (Ge), respectively.

The black lines represent $5\sigma$ contours, approximately 99.99994%
confidence in the presence of the CENS signal, for three different values of
$\sigma_{0}$ (the 1 eV contour is the contour for the pictured colormap data).
Each contour was yielded after smoothing the data by six iterations through a
Jacobi relaxation scheme. Above each line, it is unlikely that the CENS signal
can be extracted from the data given that baseline resolution.

Figure 3: (Color online) Confidence level for varying ambient thermal neutron
flux (vertical axis) and effective resolution (horizontal axis) for a baseline
resolution of 1 eV in Si (left) and Ge (right). Horizontal lines mark fluxes
corresponding the accepted ambient sea level flux (dashed) and one part in
$10^{9}$ of the measured flux at the MINER facility [16] (dotted). Black
contour lines representing $5\sigma$ confidence levels for different values of
the baseline resolution $\sigma_{0}$ are also shown. The value of $\sigma_{0}$
corresponding to each contour is given by the label on each contour.

Up to statistical fluctuations, the confidence increases monotonically moving
toward lower fluxes and better resolutions, as would be expected. This is the
case for all baseline resolutions probed.

Many interesting features can be seen in this data. For $\sigma_{0}=1$ eV,
below an effective resolution of $\sim 20$ eV, improving resolution has a
dramatic effect on tolerance to ambient neutrons. An improvement in effective
resolution from 10 eV to 20 eV is enough to allow measurement of CENS in an
ambient flux three times larger. However, at larger baseline resolutions, the
improvement with effective resolution is much smaller, and at $\sigma_{0}=10$
eV, changing the effective resolution does almost nothing. This is an
instructive, and promising, result, emphasizing that the feasibility of CENS
measurements gets ever greater as detector technologies improve, but that
decreasing the thermal neutron flux of current experiments may be a vital step
in achieving these measurements.

Further, even with drastic detector improvements, thermal neutrons could still
represent a substantial background in the vicinity of nuclear reactors. It can
be seen in Figure 3 that the measured MINER reactor-adjacent flux needs to be
brought down to one part in a billion to begin seeing a CENS signal at a
baseline resolution of only 1 eV, and must be brought down even further at
$\sigma_{0}=10$ eV. Note that the flux measurement by MINER [16] was early in
the campaign. Many materials like boron, cadmium, gadolinium, polyethylene,
and more can be used to bring down thermal fluxes by large factors. The
thermal neutron background situation is somewhat better in germanium detectors
owing to the larger CENS cross section and lower capture rates at low energy.

We have done our analysis in terms of nuclear recoil energies but many
detectors will measure the ionization caused by nuclear recoils instead. This
will tend to shift all the nuclear recoil events toward lower energies, as the
ionization yield is below unity and typically lower at lower energies [37].
However, we expect our results to be unaffected as the shift is largely the
same for both capture and neutrino induced recoils. The resolution and
detector threshold would also have to be adapted to the specific case.

In germanium detectors the thermal neutron flux can be measured _in situ_ and
compared with our results. The flux measurement is based on the rates of one
of the X-ray emissions from electron capture of 71Ge. After several half-lives
($T_{1/2}$=11.4 days) the ambient neutron flux will be equal to the rate of
the $K$-shell electron capture (EC) line divided by a quantity which is the
product of the branching of that line, the macroscopic cross section of 70Ge
capture, and the volume of the detector.

Thus far, this study has not discussed the possibility of vetoing events based
on outgoing capture gamma rays. In general, capture events are highly vetoable
by looking for signs of the gammas emitted during each capture. As a simple
estimate of plausible effective flux reduction, Table 2 lists the expected
effective flux $f_{\text{effective}}$ over the actual flux $f_{\text{actual}}$
as a function of $\bar{d}$, the angle-weighted average distance an emitted
gamma would travel before leaving a detector made of either silicon or
germanium. Values are based on the probability of gammas from capture event
leaving the detector without interacting, based on the same cascades
(potentially involving multiple exiting gamma rays) and cascade rates used to
generate the capture event spectra, and using gamma/nucleus cross sections
yielded via linearly interpolating data from NIST’s XCOM database [38]. These
estimates show that physically larger detectors can make great gains in
reducing the importance of ambient thermal neutrons, particularly in the case
of germanium.

$f_{\text{effective}}/f_{\text{actual}}$

$\bar{d}$ (cm) | Si | Ge
---|---|---
1 | 0.8382 | 0.442
5 | 0.4267 | 0.07027
10 | 0.1937 | 0.01251
20 | 0.04538 | 0.001166
50 | 0.001747 | 7.313e-06
100 | 7.119e-05 | 1.917e-09

Table 2: Expected effective thermal neutron flux $f_{\text{effective}}$ over
actual flux $f_{\text{actual}}$ in silicon and germanium detectors as a
function of $\bar{d}$, the average distance a gamma must travel before leaving
the detector. Reduction is based on the probability of gammas leaving the
detector without interacting.

Some assumptions about the “other” backgrounds should be noted. The low-
energy-skewed spectrum (${\propto E^{-0.9}}$) can be viewed as an
instantiation of the anomalous excess of events in low-threshold cryogenic
detectors at low energy recently measured by several collaborations [39, 16,
40, 41, 42, 43, 44, 45, 46, 30, 47, 48, 49, 50]. The origin of these events
are unclear, and a single common explanation for the excesses is unlikely,
making it difficult to assess the likely form for such an excess in our
simulated experiment. However, since we consider the nuclear recoil energy
scale these excess events are either similarly peaked to our assumed
additional background model, or they are more spread out due to the less-than-
unity quenching factor of nuclear recoils.

The last point that bears discussion is that the reactor model used here may
be a best-case scenario. A parallel track of calculations in silicon using
antineutrino spectrum data from the Daya Bay nuclear reactor complex in
Southern China [51] yielded consistently lower confidence values. This fact
highlights possible systematic uncertainties in our analysis due to
uncertainties in the reactor anti-neutrino spectrum. The precise shape and
overall normalization of this spectrum is actively being improved [52, 53],
but the current status does not preclude changes to our results based on the
form of the spectrum. This is expected to be a more significant effect than
changes based on uranium enrichment and relative fission contributions from
235U, 238U, 239Pu, and 241Pu. This is certainly a less significant effect than
the ability to veto capture events, but still should be noted.

Overall, our analysis shows that keeping the effective thermal neutron flux
(flux after veto) below $\sim$ 10-4 n/cm2s for a 1 MW reactor at 10 meters is
necessary for silicon detectors–even ones with exceptional resolutions.
Germanium is somewhat more robust for detectors with exceptional resolution,
requiring the flux to be below $\sim$ 7$\times$10-4 n/cm2s under the same
conditions. For germanium detectors with 25 eV baseline resolution (still
excellent by today’s standards) the thermal neutron flux needs to fall below
$\sim$ 10-5 n/cm2s. These rules of thumb can be roughly adapted to other
reactors by multiplying this flux by the reactor power in MW (uranium
enrichment level probably makes a sub-dominant difference) and accounting for
the distance by an inverse-square law for the neutrinos.

All of the Python code used for this analysis can be obtained via the Open
Storage Framework (OSF) data entry [34]. The PDFs for the thermal neutron-
induced nuclear recoils were obtained from our public code [18].

###### Acknowledgements.

We gratefully acknowledge support from the U.S. Department of Energy (DOE)
Office of High Energy Physics and from the National Science Foundation (NSF).
This work was supported in part by DOE Grant DE-SC0021364 and NSF Grant No.
2111090.

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