# Double-GEM based thermal neutron detector prototype

L. A. Serra Filho11footnotetext: Corresponding author. R. Felix dos Santos  G.
G. A. de Souza  M. M. M. Paulino  F. A. Souza  M. Moralles  H. Natal da Luz
M. Bregant  M. G. Munhoz  Chung-Chuan Lai  Carina Höglund  Per-Olof Svensson
Linda Robinson  and Richard Hall-Wilton

###### Abstract

The Helium-3 shortage and the growing interest in neutron science constitute a
driving factor in developing new neutron detection technologies. In this work,
we report the development of a double-GEM detector prototype that uses a 10B4C
layer as a neutron converter material. GEANT4 simulations were performed
predicting an efficiency of $3.14(10)\text{\,}\mathrm{\char 37\relax}$,
agreeing within $2.7\,\sigma$ with the experimental and analytic detection
efficiencies obtained by the detector when tested in a
$41.8\text{\,}\mathrm{meV}$ thermal neutron beam. The detector is position
sensitive, equipped with a $256$+$256$ strip readout connected to resistive
chains, and achieves a spatial resolution better than $3\text{\,}\mathrm{mm}$.
The gain stability over time was also measured with a fluctuation of about
$0.2\text{\,}\mathrm{\char 37\relax}\text{\,}{\mathrm{h}}^{-1}$ of the signal
amplitude. A simple data acquisition with only $5$ electronic channels is
sufficient to operate this detector.

##  1 Introduction

Several advances in scattering techniques inspired the interest in neutron
science, which provides different applications in areas such as chemistry,
physics, biology, medicine and engineering research. These advances are
strongly dependent on detector developments, which has also grown in the last
years, along with the capability to obtain bright monochromatic neutron
sources, whether in large or small facilities [1, 2, 3, 4].

The 3He proportional counter is one of the most widely used type of neutron
detectors, still widely used. Beyond its historical tradition, it presents
several advantages, such as the large neutron capture cross-section ($\sigma=$
$5325\text{\,}\mathrm{b}$ for $25\text{\,}\mathrm{meV}$ [5]), low sensitivity
to gamma-rays, and the possibility of working on a wide neutron energy range
in the low energy region [1]. However, due to a shortage of this gas since the
early 2000s [6], the scientific community keeps searching for alternative
solutions with $\ce{{}^{3}He}$-free detectors [7], employing 113Cd , 157Gd ,
10B, and 6Li [8, 9, 10, 11, 12] as neutron converters. Nevertheless,
substituting 3He has the price of reducing the neutron detection efficiency or
increasing the detection complexity.

One versatile alternative for neutron detection consists of using one of the
referred neutron converter isotopes with the Gas Electron Multiplier (GEM)
[13], a microstructure widely used to detect charged particles that can cover
large sensitive areas, present fair energy and position resolutions as well as
robustness. These aspects consolidated these microstructures, which are
already in use in several applications, such as high energy physics [14, 15,
16], muon tomography [17], X-rays fluorescence imaging [18], and, more
recently, neutron detection [19, 20, 21, 22].

This work presents a double-GEM detector which uses 10B4C as a neutron
converter, deposited on its aluminum cathode. The ionizing products generated
by the 10B(n,$\alpha$)7Li capture reaction, shown in Eq. 1, are a critical
factor in detector’s design. Given that 10B has a relatively large neutron
capture cross section of $3870\text{\,}\mathrm{b}$ for
$25\text{\,}\mathrm{meV}$ [5] and about $19\text{\,}\mathrm{\char 37\relax}$
natural abundance [23], one of its compounds, 10B4C, is particularly
interesting for our purposes, considering it also presents good wear
resistance besides chemical and thermal stability [24].

n + ^10B 1pt1cm {7Li ($840\text{\,}\mathrm{keV}$) + 4He
($1470\text{\,}\mathrm{keV}$) + γ ($480\text{\,}\mathrm{keV}$)  (94%)7Li
($1015\text{\,}\mathrm{keV}$) + 4He ($1775\text{\,}\mathrm{keV}$)  (6%)

This prototype can be easily assembled and offers the possibility to cover
large areas, an essential requirement for some applications. The detector was
tested in the IEA-R1 nuclear research reactor [25], at the Nuclear and Energy
Research Institute (IPEN) in São Paulo, Brazil, using a monochromatic thermal
neutron beam. Its versatility resulted from quickly obtaining neutron position
sensitivity using a few electronics channels and presenting gain stability.

##  2 Experimental setup

Our double-GEM detector is composed of a stack with a
$0.5\text{\,}\mathrm{mm}$ thick aluminum cathode coated with amorphous
enriched boron carbide (10B4C ), followed by two GEM foils, provided by CERN.

The deposition of the 10B4C film on the Al cathode was done by direct-current
magnetron sputtering in ESS Linköping Detector Coatings Workshop, located at
Linköping, Sweden. The facility is equipped with an industrial deposition unit
(CC800/9, CemeCon AG) and has been dedicated to the development and production
of B4C coatings as the solid converting layers in neutron detectors. Details
about the deposition machine and characterization of the B4C thin films for
neutron detectors can be found in [26, 27, 24].

In this work, two 10B4C targets (10B enrichment $>$ 95 wt.%) were installed on
sputtering cathodes in the deposition machine. The Al plate was attached to
another Al backing plate to achieve 10B4C coating on only one surface, where
the whole assembly was mounted to an electrically grounded sample table (see
Fig. 1). The system was first pumped down to a primary vacuum pressure of
$1.5\text{\times}{10}^{-4}\text{\,}\mathrm{P}\mathrm{a}$
($1.1\text{\times}{10}^{-6}\text{\,}\mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$)
while the Al plate was heated up to
$280\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $3\text{\,}\mathrm{h}$ mainly
to remove the water residuals on the surfaces. After an additional
$2\text{\,}\mathrm{h}$ cooling, the plate was then treated with radio
frequency plasma etching in $0.35\text{\,}\mathrm{P}\mathrm{a}$ of pure Ar to
improve the coating’s adhesion by removing native oxides on the substrate
surface and increasing surface roughness. Thereafter, the system was filled up
with $0.8\text{\,}\mathrm{P}\mathrm{a}$
($6\text{\,}\mathrm{m}\mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$) of Ar before
the beginning of the 10B4C deposition. During the deposition, the sample table
was continuously rotated to obtain a better thickness uniformity. A polished
Si reference was mounted at an equivalent position of the Al plate in the same
deposition run for thickness measurement of the coating and the thickness of
10B4C coating was measured to be approximately
$2.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by a profilometer (DektakXT,
Bruker).

Figure 1: The assembly of the Al cathode plate (highlighted with the dotted
box) on the Al backing plate for deposition.

In addition to the deposition of the converter layer, the geometrical
disposition of the elements within the detector is a key factor regarding the
signal we want to measure. The drift, transfer, and induction regions of our
prototype are represented in Fig. 2. The drift region has
$2\text{\,}\mathrm{mm}$ thickness and is biased with $100\text{\,}\mathrm{V}$,
while the transfer and induction regions have both $1\text{\,}\mathrm{mm}$
thickness and are biased with $300\text{\,}\mathrm{V}$ and
$400\text{\,}\mathrm{V}$, respectively. The detector works under an open flow
of an Ar/CO2 (90/10) gas mixture, with a flow of
$6\text{\,}\mathrm{L}\mathrm{/}\mathrm{h}$ through the
$0.85\text{\,}\mathrm{L}$ volume aluminum chamber. The readout plane, produced
at CERN, consists of $256$+$256$ strips in X and Y directions, covering a
$100\text{\,}{\mathrm{cm}}^{2}$ sensitive area.

Figure 2: Detector scheme, showing the drift, transfer and induction regions.
Not to scale. Figure adapted from [28].

The thickness of the drift region determines the detector’s sensitivity to
background gamma rays. In this context, a common requirement for thermal
neutron detectors is to be insensitive to these gammas. Preliminary tests
comparing $20\text{\,}\mathrm{mm}$ and $2\text{\,}\mathrm{mm}$ drift regions
have shown that the latter detected five times fewer background events
(approximately $3\text{\,}\mathrm{Hz}$), measured with the neutron beam
shutter closed. The share of events within the background range varied from
$56\text{\,}\mathrm{\char 37\relax}$ for the $20\text{\,}\mathrm{mm}$ drift
region to $18\text{\,}\mathrm{\char 37\relax}$ for the $2\text{\,}\mathrm{mm}$
region. These tests were taken into account to determine the current
$100\text{\,}\mathrm{keV}$ threshold, which allows the detector to operate
above the gamma contamination region. Therefore, we operated the detector
using the $2\text{\,}\mathrm{mm}$ drift region, which is possible because the
reaction products’ high ionization power generates a large number of primary
electrons. That way, even a tiny segment of their tracks is enough to provide
a significant signal.

The signal multiplied by the GEM stack is collected by $256$+$256$ strips,
which are interconnected by a resistive chain composed of
$60\text{\,}\mathrm{\SIUnitSymbolOhm}{}$ Surface Mount Device (SMD) resistors
with $0.1\text{\,}\mathrm{\char 37\relax}$ precision, as shown in Fig. 3.
There are five signals collected to reconstruct a neutron image: two signals
for the X direction, two for the Y direction and the charge signal, collected
from the bottom electrode of the last GEM. The charge signal is used as the
trigger for the data acquisition, as shown in Fig. 4, since it is position-
independent, coming directly from the multiplication. Considering the expected
extraction efficiency $\epsilon_{\text{ext}}$ for the last GEM [29], the
charge signal collected at its bottom is given by $1-\epsilon_{\text{ext}}$,
which is approximately $64\text{\,}\mathrm{\char 37\relax}$ of the multiplied
charge for our configuration.

Figure 3: Printed Circuit Board (PCB) resistive chain to fit the detector
signal output. Each board has 128 SMD resistors and a PANASONIC®130-pin
connector. Since the readout has 256 strips for each direction, two plates are
used for each direction. Figure from [29]. Figure 4: Signal acquisition setup
scheme. The trigger circuit is pictured in blue. Figure adapted from [28].

In order to reconstruct the event’s position, we use Eq. 2.1 [18], where $Q$
is the charge information collected from the bottom of the last GEM, and
$L_{x}$ and $L_{y}$ are the lengths of the sensitive area for each direction.
As indicated in Fig. 4, $X_{1}$ and $X_{2}$ are the signals collected at each
end of the resistive chain in the X direction, while $Y_{1}$ and $Y_{2}$ are
the equivalent signals in the Y direction.

$X_{*}=L_{x}\frac{X_{2}-X_{1}}{Q}\qquad Y_{*}=L_{y}\frac{Y_{2}-Y_{1}}{Q}$
(2.1)

As mentioned before, the detector was tested at the IEA-R1 nuclear reactor
using the same neutron beam of the AURORA diffractometer [30] (see Fig. 5).
This monochromatic neutron beam is selected out of a collimated radial neutron
beam from the core of the reactor, which operates at
$4.5\text{\,}\mathrm{MW}$. A multi-blade focusing silicon monochromator is
used to select these neutrons with a fixed wavelength of
$1.399\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ ($41.80\text{\,}\mathrm{meV}$)
providing a $(6.22\pm
0.19)\text{\times}{10}^{4}\text{\,}\mathrm{n}\,\mathrm{c}\mathrm{m}^{-2}\mathrm{s}^{-1}$
flux ($\phi$), measured by the Nuclear Metrology Laboratory at IPEN with the
${}^{197}\mathrm{Au}(n,\gamma)^{198}\mathrm{Au}$ reaction, using a
$12\text{\,}\mathrm{mm}$ diameter thin gold foil, centered at the beam outlet
(letter E of Fig. 5).

The detector (letter A of Fig. 5) and the preamplifiers’ box (letter C of Fig.
5) were mounted over an aluminum plate. This setup was fixed to a X-Y-Z
precision table, that allowed the fine tuning of the detector’s position with
respect to the beam.

Figure 5: Position of the detector (A) relatively to the beam. The following
elements are also indicated: ground shielding of the y-direction resistive
chains (B), preamplifiers box (C), high voltage inputs (D), and the beam
outlet (E).

##  3 Results and discussion

In this section, we present the main experimental results concerning the
characterization of the detector: the spatial resolution obtained by two
different methods, the detector efficiency, and the gain stability. We also
discuss how these results relate to the previous expectations obtained with
simulations.

###  3.1 Position calibration and resolution

To perform the spatial calibration of the detector, we used handmade cadmium
masks which has a high neutron absorption cross section. Holes of different
diameters were drilled in three masks. The positions and areas of the holes
were measured using an image obtained with a table scanner. One of the
produced masks in shown in Fig. 6, with its $15$ holes, and its neutron image,
obtained with the mask centered with respect to the detector’s sensitive area.

Figure 6: Left: digital image of the mask obtained with a table scanner.
Right: calibrated neutron image.

The calibration is given by a linear dependency between the coordinates of
certain holes in arbitrary units and their coordinates in known distances. The
chosen holes are colinear in the X and Y directions, normally the ones in the
central column for Y direction and the central row for X direction. We
measured the detector spatial resolution by two methods: edge and point spread
functions. The first one consists of using edges to fit step functions, as
shown in Fig. 7.

The data was fitted with the complementary error function

$\operatorname{erfc}(p)=1-\frac{2}{\sqrt{\pi}}\int_{0}^{p}e^{-t^{2}}dt,$ (3.1)

where

$p(x)=\frac{x-\mu}{\sqrt{2}\sigma}.$ (3.2)

This function provides the same parameters of interest ($\sigma$ and $\mu$) as
the edge spread function (ESF). In this work, we define the position
resolution as the FWHM of the Gaussian (or the 10-to-90 % distance of the
ESF), i. e., 2.35$\sigma$.

Figure 7: Left: neutron image of one of the masks, with a selected region of
an edge marked as the red rectangle. Right: Complementary error function
($\operatorname{erfc}$) adjusted in the y-projection of the chosen area, the
spatial resolution obtained is $3.0\pm 0.1\text{\,}\mathrm{mm}$. Figure
adapted from [28].

We measured the position resolution with the edge spread function in different
regions of the detector’s sensitive area and along X and Y coordinates. The
results are shown in Fig. 8.

Figure 8: The points where the spatial resolution was measured using the edge
spread function. The set of spatial resolution values resulted in an average
value of $2.8\pm 0.3\text{\,}\mathrm{mm}$ for the X direction and $3.0\pm
0.1\text{\,}\mathrm{mm}$ for the Y direction, compatible with each other. We
considered the uncertainties of the average spatial resolution for each
direction as the standard deviation of each set (X and Y). Consequently, it is
reasonable to consider the detector’s resolution as the average of the
resolutions for both directions, which results $2.9\pm
0.2\text{\,}\mathrm{mm}$.

The second method relies on the fact that the FWHM of an ideally punctual hole
converges to the spatial resolution of the detector, which is the width of the
point spread function (PSF). The projection $P_{meas}$ obtained for a hole
with a given diameter is the convolution of the PSF, for which it is safe to
assume a 2D Gaussian function, with the projection of a circular area. This
projection over the X direction can be written as

$P_{meas}(x)=(P_{x}\circledast
g)(x):=\int_{-\infty}^{\infty}f(\tau)g(x-\tau,\sigma)d\tau,$ (3.3)

where $g$ is the Gaussian function and $P_{x}$ the projection of a circular
area over the X direction. Fitting this convolution for holes of several
diameters allows us to estimate the spatial resolution. The convolution shown
in Eq. 3.3 is computed numerically and the fit222the fitted parameter is the
resolution itself, since it is linearly dependent on $\sigma$. is done using
the Least Square Method. The result is shown in Fig. 9.

Figure 9: Convolution fit for holes with different diameters, resulting in the
spatial resolution in the X (left) and Y (right) directions. We also present
the absolute residuals (Abs. Res.), i.e. the difference between the
experimental data and the fit, for each plot. Figures adapted from [28].

Although compatible with the $3\text{\,}\mathrm{mm}$ obtained with the ESF,
the values obtained with this method are slightly better. This can be
explained by the fact that the FWHM assigned to each diameter is the average
of the FWHM for several runs using holes of that diameter.

The values obtained with the two different techniques result in a spatial
resolution compatible with at least $3\text{\,}\mathrm{mm}$. This spatial
resolution is compatible with the obtained by similar detectors [21, 22] and
is mostly limited by the fact that our detector uses only five electronic
channels, which inherently only allows the estimation of the centroid of the
track’s charge distribution, which is at a certain distance from the neutron
capture position. This distance is minimized when the maximum length of the
track is truncated, such as in the case of $2\text{\,}\mathrm{mm}$ drift. It
would be possible to strongly enhance the spatial resolution by operating the
detector in the $\mu$TPC mode (such as in [31]), however, it would be
necessary to use a much more sophisticated acquisition system.

###  3.2 Detector efficiency and gain stability

####  3.2.1 Analytic Calculation

It is possible to obtain the expected detection efficiency $\varepsilon_{d}$
from analytical calculations. The first step is to compute the capture
efficiency $\varepsilon_{c}$, which depends on the macroscopic cross-section
$\Sigma$ for the converter material and its thickness $d=$ $2.2\pm
0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$, where the uncertainty should
comprise eventual irregularities of the deposition process. Taking the
converter $(^{10}\mathrm{B}_{4}\mathrm{C})$ density $\rho=$
$2.3\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$ [32], its neutron capture
cross section for $41.8\text{\,}\mathrm{meV}$ neutrons $\sigma=$ $2994\pm
9.\text{\,}\mathrm{b}$ [5], its molar mass
$\mathrm{M}_{\mathrm{B}_{4}\mathrm{C}}=$
$52\text{\,}\mathrm{g}\text{\,}{\mathrm{mol}}^{-1}$, and the stoichiometry $a$
of 10B in a 10B4C converter layer, which is $4$, one can obtain the number of
target nuclei of the converter:

$N=a\frac{\rho}{\text{M}}N_{A}=1.065\times 10^{23}\,\text{cm}^{-3}.$ (3.4)

The result of Eq. 3.4 is used to compute the macroscopic cross-section [33]
for the converter layer:

$\Sigma=N\sigma=(3.19\pm 0.01)\times 10^{-2}\,\mu\text{m}^{-1}.$ (3.5)

The neutron beam intensity after crossing our converter layer is

$I_{\text{beam}}=e^{-\Sigma d}=(93.2\pm 0.3)\%.$ (3.6)

Therefore, the capture efficiency is given by the percentage of absorbed
neutrons:

$\varepsilon_{c}=1-e^{-\Sigma d}=(6.8\pm 0.3)\%.$ (3.7)

When the neutron is captured too far from the lower side of the converter
layer (at the boundary with the gaseous region), it might not be detected
because the products of the capture reaction do not have enough energy to exit
the layer, as pictured in Fig. 10 (case A). Therefore, such products have to
fulfill a set of geometrical requisites that depend on the capture’s position,
which are shown in Fig. 10 (case B and C). Finally, the products that enter
the drift region with too low energy are also not detected since there is an
energy threshold to handle electrical noise and other acquisition system
specifications.

Figure 10: The spheres of radius $\tau_{\alpha}$ and $\tau_{Li}$ define the
accessible region for the $\alpha$ and 7Li nuclei, respectively, inside the
converter layer (A). Given certain capture point, only the products within the
solid angle $\Omega$ reaches the gas (B). The solid angle $\Omega$ is a
function of the angle $\text{cos}^{-1}(x/\tau)$, where $x$ is the smallest
distance between the capture point and the drift region (C). Figure adapted
from [28].

In order to account for the constrains discussed above, the values for
$\tau_{\alpha}$ and $\tau_{\text{Li}}$, for both possible decay channels of
Eq. 1, were evaluated from direct integration of the average dE/dx plot
generated through SRIM [34]. It is also necessary to consider the energy
threshold, which was done by finding $\tau$ for which the average remaining
energy for the given particle is equal to the threshold used in the
experiment, of $100\pm 10.\text{\,}\mathrm{keV}$. This process is shown in
Fig. 11.

Figure 11: Remaining energy for each 10B(n,$\alpha$)7Li reaction product as a
function of depth for a theoretically infinite 10B4C converter layer. The
$\tau$ values used for the calculations are the depth at which the particle’s
remaining energy equals the threshold. This condition gives:
$\tau_{\alpha_{1}}$($1775\text{\,}\mathrm{keV}$) $=$
$3.97\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$\tau_{\alpha_{2}}$($1470\text{\,}\mathrm{keV}$) $=$
$3.13\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$\tau_{\mathrm{Li_{1}}}$($1015\text{\,}\mathrm{keV}$) $=$
$1.54\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$\tau_{\mathrm{Li_{2}}}$($840\text{\,}\mathrm{keV}$) $=$
$1.31\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

The considerations above are translated to the calculation for the detection
efficiency $\varepsilon_{d}$ as follows:

$\varepsilon_{d}=F\varepsilon_{c}=F(1-e^{-\Sigma d}),$ (3.8)

where $F$ is the share of capture products emitted within the solid angle
$\Omega$ of Fig. 10B, and is computed as

$F=\sum_{\text{all
products}}F(d,\tau_{i})=\sum_{\tau_{i}=\tau_{\alpha_{1}},\tau_{\alpha_{2}},\tau_{\mathrm{Li}_{1}},\tau_{\mathrm{Li}_{2}}}\int_{0}^{d}\int_{0}^{\text{cos}^{-1}(x/\tau_{i})}d\Omega
dx\approx 0.477,$ (3.9)

for a boron carbide layer of thickness $2.2\text{\,}\mathrm{\SIUnitSymbolMicro
m}$, where $\tau_{i}$ are the path length for each product:
$\tau_{\alpha_{1}}=$ $3.97\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$\tau_{\alpha_{2}}=$ $3.13\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
$\tau_{\mathrm{Li_{1}}}=$ $1.54\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$\tau_{\mathrm{Li_{2}}}=$ $1.31\text{\,}\mathrm{\SIUnitSymbolMicro m}$. As
result, we obtain the neutron detection efficiency

$\varepsilon_{d}=(3.22\pm 0.14)\,\%.$ (3.10)

####  3.2.2 Simulations

Monte Carlo simulations play an important role in the development and
optimization of this kind of detector. For this purpose, GEANT4 is a powerful
tool to describe the detector and for the simulation of the interaction of
particles with matter [35], allowing the simulation of the nuclear reaction,
like Eq. 1, as well as the transport of the reaction products inside the
detector.

The detector was simulated with GEANT4 using the physics list QGSP_BERT_HP
(which has high precision models for low energy neutrons [36]), according to
the device described in section 2. The simulation consisted in a parallel
uniform beam with ${10}^{6}$ neutrons of
$1.399\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ ($41.80\,\mathrm{meV}$),
hitting the detector in a direction perpendicular to the converter layer.

Taking 14 different thicknesses ranging from
$0.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of the boron carbide converter
layer, we evaluate the thermal neutrons detection efficiency by counting the
charged particles that leave the boron converter and reach the gaseous volume.
For a nominal thickness of $2.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$, an
efficiency of $3.14\pm 0.10\text{\,}\mathrm{\char 37\relax}$ was obtained.
While the maximum efficiency was achieved with a thickness of
$2.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ resulting in $3.17\pm
0.10\text{\,}\mathrm{\char 37\relax}$, as shown in Fig. 12. We did not
considered the experimental threshold ($100\text{\,}\mathrm{keV}$) when
evaluating the theoretical efficiency in this case.

Figure 12: Detection efficiency for one layer of boron carbide, simulated with
GEANT4. The fit for maximum efficiency is an arbitrary function adjusted just
to guide the eye within the interval that comprises the data around the peak
(continuous line).

Using the low-energy electromagnetic model Livermore [36], we simulated the
deposited energy by the reaction products inside the $2\text{\,}\mathrm{mm}$
of the drift region. This simulation is used as an energy pre-calibration for
the experimental spectrum since both present the same primary structure,
consisting of an energy peak around $600\text{\,}\mathrm{keV}$, as shown in
Fig. 13. We did not include the experimental energy threshold
($100\text{\,}\mathrm{keV}$) of our system nor the charge multiplication
process in the simulation, which means considering that the acquired energy is
equal to the energy deposited by the reaction products in the absorption
region of the detector.

Figure 13: Experimental energy spectrum compared with the GEANT4 simulation.
The simulation does not include the effect of the detector finite energy
resolution and was normalized to the maximum value of the experimental energy
distribution. Figure from [28].

####  3.2.3 Experimental measurements

The detection efficiency is evaluated by dividing the number of detected
neutrons $N_{det}$ by the total number of neutrons $N_{tot}$ hitting the
detector. We calculated the total number of neutrons from the neutron flux
$\phi$, measured as described in section 2, the area $A$ selected from the
beam and the time length $\Delta t$ of the measurement. The neutron flux was
corrected using the counts of a fission chamber installed at the neutron beam
outlet using as reference the number of counts registered by this same chamber
during the neutron flux calibration process. With these values, the
experimental neutron detection efficiency is given by

$\varepsilon=\frac{N_{det}}{N_{tot}}=\frac{N_{det}}{{\phi A\Delta t}}.$ (3.11)

Using cadmium masks with squared holes, we selected different areas from the
beam: $0.36\text{\,}{\mathrm{cm}}^{2}$ and $1.44\text{\,}{\mathrm{cm}}^{2}$,
at the same position where the flux measurement was taken. The total number of
detected neutrons $N_{det}$ was counted with negligible uncertainties at these
rates. We obtained $2.75\pm 0.26\text{\,}\mathrm{\char 37\relax}$ detection
efficiency for the $0.36\text{\,}{\mathrm{cm}}^{2}$ mask and $2.57\pm
0.15\text{\,}\mathrm{\char 37\relax}$ for the
$1.44\text{\,}{\mathrm{cm}}^{2}$, which are compatible within $1\,\sigma$ with
each other.

The gain stability was monitored in real-time, in order to provide necessary
corrections to the image reconstruction in offline data analysis. A gain shift
would imply changes in efficiency since the energy spectrum would be displaced
from its original position with respect to the energy threshold. Besides that,
changes in the gain over time may bring the detector to an unstable operation
in terms of discharge probability.

The data shown in Fig. 14 present a method to analyze the gain stability: the
energy channel of the run was divided into batches of $4\text{\,}\mathrm{min}$
each (separated by the dashed vertical lines) and plotted over time. Then, we
locate the spectrum peak using a peak identification algorithm [37] and fit a
Gaussian to it, registering its center as the cross marker in Fig. 14. Slow
gain changes due to environmental changes are easily seen as a tendency in the
sequence of the points, whose energy spectrum can be drifted by a reasonable
amount as it happens in the case reported by [18]. The tendency analysis can
be carried out by fitting the behaviour of the difference between each peak
position and the average position of all peaks, which we labelled as
$\Delta_{\text{peak}}$ in the Fig. 14.

There is a slight drift of the spectrum’s peak within about
$2.5\text{\,}\mathrm{h}$, as the fit result of Fig. 14 suggests. However, this
drift is smaller than $10$ ADC channels, corresponding to about
$0.2\text{\,}\mathrm{\char 37\relax}\text{\,}{\mathrm{h}}^{-1}$ change, which
is not statistically sufficient to be interpreted as a reasonable change.
Also, at the beginning of the operation, some slow gain stabilization of these
detectors are expected, as reported in other works [38, 39, 40, 41, 42]. The
monitoring remains essential since it can be used to correct the detector’s
efficiency if significant changes happen, using the detector’s current rate
and the neutron monitor fission chamber.

Figure 14: Temporal stability analysis with the tendency analysis. This run
has 4793145 events and was divided into batches of $4\text{\,}\mathrm{min}$
($123\,403$ entries each), generating 39 peaks. The average of the peak
position was $897\pm 2.$ ADC channels. The fit for $\Delta_{\text{peak}}$
resulted an angular coefficient $-0.028\pm\,0.009$ ADC channel/min and a
linear coefficient equals to $2.2\pm\,0.8$ ADC channels.

##  4 Final Considerations

A neutron detector was built with a single boron carbide layer and a resistive
charge division readout. This detector shows the feasibility of a position-
sensitive detector that can be easily mounted in a neutron beam line based on
simple standard electronics. The detector also operates above of the
background gamma rays sensitive region (registering no events when the neutron
beam is off), an essential feature for its use in environments like nuclear
reactor research facilities. If needed, the gamma sensitivity can be
controlled by changing the drift region, and a new energy pre-calibration can
be quickly obtained with new simulation results.

The average of the obtained thermal neutron detection efficiency is $2.66\pm
0.15\text{\,}\mathrm{\char 37\relax}$, which agrees within 2.7 $\sigma$ with
the prediction of $3.22\pm 0.14\text{\,}\mathrm{\char 37\relax}$ and with the
preliminary simulation of $3.14\pm 0.10\text{\,}\mathrm{\char 37\relax}$, and
can be reduced by altering the layer thickness allowing different
applications, such as beam monitors. At the current configuration, one can use
the detector to measure the beam profile given that the position resolution
obtained was at least $3\text{\,}\mathrm{m}\mathrm{m}$. There is also the
possibility to raise the efficiency by adding new converter layers over the
GEMs, as seen in other works [43, 19, 20]. For future works and following that
same idea, it is possible to use boron coated Thick-GEMs, operating with
unitary effective gain, raising the thermal neutron detection efficiency.

## Acknowledgments

This work was supported by CNPq grant 156767/2019-8.
H. Natal da Luz acknowledges project GAČR GA21-21801S (Czech Science
Foundation).

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On the log-hyperconvexity index of pseudoconvex H$\ddot{o}$lder domains in
$\mathbb{C}^{n}$

Tianlong Yu

Abstract: In this note we prove that every bounded pseudoconvex domain in
$\mathbb{C}^{n}$ with H$\ddot{o}$lder boundary has positive log-hyperconvexity
index which gives a positive answer to a question posed by Boyong Chen.

Introduction:

A bounded pseudoconvex domain $\Omega$ in $\mathbb{C}^{n}$ is called
hyperconvex if there exists $0>\phi\in PSH(\Omega)$ such that for every c<0,
$\phi^{-1}([-\infty,c))$ is relative compact in $\Omega$. In [7] Diederich and
Fornaess showed that every bounded pseudoconvex domain in $\mathbb{C}^{n}$
with $C^{2}$ boundary is hyperconvex. Actually they proved that there exists a
smooth psh function which is comparable to -$d^{\alpha}$, for some
$1>\alpha>0$, where d is the boundary distance. In [9] Kerzman and Rosay
showed that every bounded pseudoconvex domain in $\mathbb{C}^{n}$ with $C^{1}$
boundary is hyperconvex. In [6] Demailly showed that every bounded
pseudoconvex domain in $\mathbb{C}^{n}$ with Lipchitz boundary is hyperconvex.
Actually he proved that there exists a smooth psh function which is comparable
to $(logd)^{-1}$. In [8] Harrington showed that every bounded pseudoconvex
domain in $\mathbb{C}^{n}$ with Lipchitz boundary also admits a psh function
which is comparable to -$d^{\alpha}$, for some $1>\alpha>0$. In [2], by
combining the localization principle for hyperconvex domains in
$\mathbb{C}^{n}$ proved in [9] and a new estimate of relative extremal
function, Chen showed that every bounded pseudoconvex domain in
$\mathbb{C}^{n}$ with H$\ddot{o}$lder boundary is hyperconvex. Actually he
proved that locally there exists a psh function(the local relative extremal
function) which has a lower bound comparable to $-(-logd)^{-\tau}$ , for some
$1>\tau>0$. In [3], by combining the results in [2] and a new psh Hopf lemma
and the patching methods proved in [5] , Chen and Xiong obtained a global
lower bound of the global relative extremal function in the H$\ddot{o}$lder
boundary case.

In [4] , as the hyperconvexity index introduced by Chen[1], Chen and Zheng
introduced the concept of log-hyperconvexity index as follows:

######  Definition 1.

For a bounded domain $\Omega$ in $\mathbb{C}^{n}$, define

$\alpha_{l}(\Omega)=sup\\{\alpha\geq 0;\exists 0>\phi\in C(\Omega)\cap
PSH(\Omega)s.t.-\phi\leq C(-logd)^{-\alpha}$near the boundary$\\}$

as the log-hyperconvexity index of $\Omega$.

Therefore the main results in [2] read as every bounded pseudoconvex domain in
$\mathbb{C}^{n}$ with H$\ddot{o}$lder boundary locally has a positive log-
hyperconvexity index. And in [2][3][4], Chen raised the question that whether
every bounded pseudoconvex domain in $\mathbb{C}^{n}$ with H$\ddot{o}$lder
boundary has a (global) positive log-hyperconvexity index. In this note, we
give a positive answer to this question by establishing the following:

######  Theorem 1.

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^{n}$ with
H$\ddot{o}$lder boundary. Then for some $\tau>0$, there exists a continuous
psh function $w(z)$ on $\Omega$ and $M_{1},M_{2}>0$ such that

$-\frac{M_{1}}{(-logd(z))^{\tau}}\frac{1}{log(-logd(z))}<w(z)<-\frac{M_{2}}{(-logd(z))^{\tau}}\frac{1}{log(-logd(z))}$.

The proof of the theorem is a standard psh patching argument as used by
Kerzman-Rosay[9], Demailly[6], Coltiou[5], Harrington[8], here we work twice
the patching process, once locally and once globally.

Proof of the theorem:

Choose finite points $p_{j}\in\partial\Omega,0.1>r_{j}>0$ such that
$\partial\Omega\subseteq\cup B(p_{j},r_{j})$ and the following holds after
local holomorphic coordinate changes:

$B(p_{j},10r_{j})\cap\Omega=\\{z\in
B(p_{j},10r_{j})$|Im$z_{n}$<$g_{j}$($z^{\prime}$,Re$z_{n})\\}$,

where $z^{\prime}=(z_{1},\cdots,z_{n-1})$ and $g_{j}$ is a H$\ddot{o}$lder
continuous function defined on
$\\{(z^{\prime},x_{n})\in\mathbb{C}^{n-1}\times\mathbb{R}||z^{\prime}|^{2}+x_{n}^{2}<100r_{j}^{2}\\}$
with a H$\ddot{o}$lder index $\beta_{j}\in(0,1)$ and satisfies
$g_{j}(0^{\prime},0)=0$.(Here we assume $p_{j}=0$ in local coordinates)

For t>0, let $\Omega_{t,j}=\\{z\in
B(p_{j},10r_{j})$|Im$z_{n}$<$g_{j}$($z^{\prime}$,Re$z_{n}$)+t$\\}$ which are
also pseudoconvex domains containing
$B(p_{j},10r_{j})\cap\Omega=:\Omega_{0,j}$.

Let $d_{t,j}$be the boundary distance of
$\Omega_{t,j},C_{t,j}=d(\partial\Omega_{0,j}\cap
B(p_{j},9r_{j}),\partial\Omega_{t,j}\cap B(p_{j},9.5r_{j}))>0$. Then as in the
proof of the main theorem in [2], we can find $\gamma_{j}>1$ which depends on
$\beta_{j}$, such that $t^{\gamma_{j}}<C_{t,j}<t$ for all small t. We may
assume $\gamma_{j}=\gamma$ for all j.

In the following we first work locally in $B(p_{j},2r_{j})$, so we omit j for
the sake of convinience. So $\Omega=\Omega_{0,j}\cap
B(p_{j},2r_{j});d_{t}=d_{t,j};C_{t}=C_{t,j}$.

let
$v_{n}(z)=\frac{-logd_{t_{n}}(z)-(1+\epsilon_{0})(-logC_{t_{n}})}{a_{n}(-logC_{t_{n}})}$
, $z\in\Omega$. $\Omega_{n}=\\{z\in\Omega|t_{n+1}^{\prime}\leq
d(z)<t_{n}^{\prime}\\}$, where $a_{n},\epsilon_{0},t_{n},t_{n}^{\prime}$ are
positive constants which will be determined later. We define
$v(z)=max\\{v_{1}(z),\cdots,v_{n}(z)\\}$ for $z\in\Omega_{n}$.

We want to choose $t_{n},t_{n}^{\prime},a_{n},\epsilon_{0}$ properly such that
v(z) is a well-defined psh function on $\Omega$. For that purpose, we only
need to check $v_{n}(z)>v_{n+1}(z)$, when $d(z)=t_{n+1}^{\prime}$.

We have
$v_{n+1}(z)<-\frac{1}{a_{n+1}}+\frac{log(t_{n+1}^{\prime}+C_{t_{n+1}})}{a_{n+1}logC_{t_{n+1}}}<-\frac{1}{a_{n+1}}+\frac{log(t_{n+1}^{\prime}+t_{n+1})}{a_{n+1}logt_{n+1}}$;

$v_{n}(z)>-\frac{1+\epsilon_{0}}{a_{n}}+\frac{log(t_{n+1}^{\prime}+t_{n})}{a_{n}logC_{t_{n}}}>-\frac{1+\epsilon_{0}}{a_{n}}+\frac{log(t_{n+1}^{\prime}+t_{n})}{a_{n}log(t_{n}^{\gamma})}$.

Here we use the monotonicity of the function $\frac{log(x+c)}{logx}$. Now we
fix $t_{1}\in(0,1)$, let
$t_{n+1}=t_{n}^{2\gamma}=\cdots=t_{1}^{(2\gamma)^{n}},t_{n+1}^{\prime}=t_{n}$.

Then we have
$\frac{log(t_{n+1}^{\prime}+t_{n+1})}{logt_{n+1}}<\frac{log((t_{1})^{(2\gamma)^{n-1}})}{logt_{n+1}}=\frac{1}{2\gamma}<1$;

$\frac{log(t_{n+1}^{\prime}+t_{n})}{log(t_{n}^{\gamma})}>\frac{log2}{(2\gamma)^{n-1}\gamma(logt_{1})}+\frac{1}{\gamma}$.

Hence for $\epsilon_{0}<<1$, n sufficiently large, we can find $\lambda>1$,
such that for $a_{n}=\lambda^{n}$, we have $v_{n}(z)>v_{n+1}(z)$, when
$d(z)=t_{n+1}^{\prime}$.

Now for $z\in\Omega_{n},v(z)<-\frac{\epsilon_{0}}{\lambda^{n}}$ and
$logd(z)\in[(2\gamma)^{n-1}logt_{1},(2\gamma)^{n-2}logt_{1})$, on the other
hand, for $z\in\Omega_{n},v(z)>v_{n}(z)>-\frac{1+\epsilon_{0}}{\lambda^{n}}$,
so there exists $C_{1},C_{2}>0,\tau^{\prime}>0$ such that

$-\frac{C_{2}}{(-logd(z))^{\tau^{\prime}}}<v(z)<-\frac{C_{1}}{(-logd(z))^{\tau^{\prime}}}$,
for all $z\in\Omega$.

Now we again work locally for fixed j, but with positive t. In other words, we
replace the above $\Omega$ by $\Omega_{t,j}\cap B(p_{j},2r_{j})$ which are
also H$\ddot{o}$lder domains with the same H$\ddot{o}$lder index as
$\Omega_{0,j}$. Therefore, after some rescaling, we get, by the above
arguments, a family of psh functions $v_{t,j}$ defined on $\Omega_{0,j}\cap
B(p_{j},2r_{j})$ for all small positive t, such that there exists C>1,
$\tau_{j}>0$, independant of t, with the following estimates hold on
$\Omega_{0,j}\cap B(p_{j},2r_{j})$:

$-\frac{C}{(-logd_{t,j}(z))^{\tau_{j}}}<v_{t,j}(z)<-\frac{1}{(-logd_{t,j}(z))^{\tau_{j}}}$.

Let $\tau_{j_{0}}=min\\{\tau_{j}\\}$, and
$w_{t,j}(z)=-(-v_{t,j}(z))^{\frac{\tau_{j_{0}}}{\tau_{j}}}$, then $w_{t,j}(z)$
are psh functions and

$-\frac{C^{\frac{\tau_{j_{0}}}{\tau_{j}}}}{(-logd_{t,j}(z))^{\tau_{j_{0}}}}<w_{t,j}(z)<-\frac{1}{(-logd_{t,j}(z))^{\tau_{j_{0}}}}$.

Hence
$-\frac{\tau_{j_{0}}}{\tau_{j}}logC+\tau_{j_{0}}log(-logd_{t,j}(z))<-log(-w_{t,j}(z))<\tau_{j_{0}}log(-logd_{t,j}(z))$.

As $d(z)+t^{\gamma}<d_{t,j}(z)<d(z)+t$ for $z\in\Omega_{0,j}\cap
B(p_{j},2r_{j})$, there exists $C_{3}>0$, independant of t, j, z, such that
$0<log(-logd_{t,j}(z))-log(-log(d(z)+t))<C_{3}$ (Here we actually measure the
boundary distance in different local holomorphic coordinates, but they are
comparable, so we can use the standard Eucilidean distance d(z).)

Now, like standard patching process, we let

$w_{t}(z)=max_{z\in
B(p_{j},2r_{j})}\\{-log(-w_{t,j}(z))+\eta_{j}(z)+M|z|^{2}\\}$, where
$\eta_{j}(z)>\tau_{j_{0}}C_{3}+\frac{\tau_{j_{0}}}{\tau_{j}}logC$ when $z\in
B(p_{j},r_{j})$, with compact support in $B(p_{j},2r_{j})$, M is large enough
to make $w_{t}$ psh. By the choice of $C_{3},w_{t}$ is a well-defined psh
function on $\Omega$ (near the boundary) and, after adding some constants,
satisfies:

$-C_{4}+\tau_{j_{0}}log(-log(d(z)+t))<w_{t}(z)<\tau_{j_{0}}log(-log(d(z)+t^{\gamma}))$,
for some positive constant $C_{4}$.

Next we do the psh patching process globally in $\Omega$.

Let $\Omega_{n}^{\prime}=\\{z\in\Omega|t_{n+1}^{\prime\prime}\leq
d(z)<t_{n}^{\prime\prime}\\}$,
$w_{n}(z)=\frac{w_{t_{n}}(z)-\tau_{j_{0}}log(-log(t_{n}^{\gamma}))-\epsilon_{1}}{\tau_{j_{0}}log(-log(t_{n}^{\gamma}))b_{n}}$,
$w(z)=max\\{w_{1}(z),\cdots,w_{n}(z)\\}$ for $z\in\Omega_{n}^{\prime}$, where
$\epsilon_{1}>0$ is a fixed constant.

Again we shall choose $t_{n}^{\prime\prime},t_{n},b_{n}$ properly to make w(z)
a well-defined psh function on $\Omega$. For that purpose, we only need to
check $w_{n}(z)>w_{n+1}(z)$, when $d(z)=t_{n+1}^{\prime\prime}$.

We have
$w_{n}(z)>-\frac{1}{b_{n}}[1-\frac{\tau_{j_{0}}log(-log(d(z)+t_{n}))-\epsilon_{1}-C_{4}}{\tau_{j_{0}}log(-log(t_{n}^{\gamma}))}]$;

$w_{n+1}(z)<-\frac{1}{b_{n+1}}[1-\frac{log(-log(d(z)+t_{n+1}^{\gamma}))}{log(-log(t_{n+1}^{\gamma}))}]$.

Let k be a large enough postive integer,
$t_{n+1}^{\prime\prime}=t_{n+1}^{\frac{1}{k}}-t_{n+1}^{\gamma},t_{n+1}=t_{n}^{k}=\cdots=t_{1}^{k^{n}}$;
then when $d(z)=t_{n+1}^{\prime\prime}$ we have

$1-\frac{\tau_{j_{0}}log(-log(d(z)+t_{n}))-\epsilon_{1}-C_{4}}{\tau_{j_{0}}log(-log(t_{n}^{\gamma}))}<\frac{log(\frac{logt_{n}^{\gamma}}{log(2t_{n})})}{log(-logt_{n}^{\gamma})}+\frac{\epsilon_{1}+C_{4}}{\tau_{j_{0}}log(-log(t_{n}^{\gamma}))}$;

$1-\frac{log(-log(d(z)+t_{n+1}^{\gamma}))}{log(-log(t_{n+1}^{\gamma}))}=\frac{log(k\gamma)}{log(-logt_{n+1}^{\gamma})}=\frac{log(k\gamma)}{log(-logt_{n}^{\gamma})+logk}$;

From this estimates, we see that for large k and n, there exists
$\lambda^{\prime}>1$ such that, by letting $b_{n}=(\lambda^{\prime})^{n}$, we
have $w_{n}(z)>w_{n+1}(z)$, when $d(z)=t_{n+1}^{\prime\prime}$. Now for
$z\in\Omega_{n}^{\prime}$,

$w(z)<-\frac{\epsilon_{1}}{(\lambda^{\prime})^{n}\tau_{j_{0}}log(-log(t_{n}^{\gamma}))}$;

$w(z)>w_{n}(z)>-\frac{C_{5}}{(\lambda^{\prime})^{n}\tau_{j_{0}}log(-log(t_{n}^{\gamma}))}$
for some positive constant $C_{5}$;

and $logd(z)\in[logt_{n+1}^{\prime\prime},logt_{n}^{\prime\prime})$.
Therefore, by the choice of $t_{n},t_{n}^{\prime\prime}$, there exists
$M_{1},M_{2}>0,\tau>0$ such that

$-\frac{M_{1}}{(-logd(z))^{\tau}}\frac{1}{log(-logd(z))}<w(z)<-\frac{M_{2}}{(-logd(z))^{\tau}}\frac{1}{log(-logd(z))}$,
for all $z\in\Omega$

As all the boundary distance functions are continuous and the whole patching
process only involves maximum of finite functions, we complete the proof.

Acknowledgement:

I would like to thank my supervisor professor Xiangyu Zhou who encouraged me
to study problems related to hyperconvexity.

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Tianlong Yu: National Center for Mathematics and Interdisciplinary Sciences,
CAS, Beijing, 100190, China.

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