# Simulations and analysis tools for charge-exchange $(d,{}^{2}\text{He})$
reactions in inverse kinematics with the AT-TPC

S. Giraud<EMAIL_ADDRESS>J.C. Zamora<EMAIL_ADDRESS>R.G.T. Zegers
Y. Ayyad  D. Bazin  W. Mittig  A. Carls  M. DeNudt  Z. Rahman  Facility for
Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824,
USA  Department of Physics and Astronomy, Michigan State University, East
Lansing, Michigan 48824-1321, USA  IGFAE, Universidade de Santiago de
Compostela, E-15782 Santiago de Compostela, Spain

###### Abstract

Charge-exchange $(d,{}^{2}\text{He})$ reactions in inverse kinematics at
intermediate energies are a very promising method to investigate the Gamow-
Teller transition strength in unstable nuclei. A simulation and analysis
software based on the attpcroot package was developed to study these type of
reactions with the active-target time projection chamber (AT-TPC). The
simulation routines provide a realistic detector response that can be used to
understand and benchmark experimental data. Analysis tools and correction
routines can be developed and tested from simulations in attpcroot, because
they are processed in the same way as the real data. In particular, we study
the feasibility of using coincidences with beam-like particles to
unambiguously identify the $(d,{}^{2}\text{He})$ reaction channel, and to
develop a kinematic fitting routine for future applications. More technically,
the impact of space-charge effects in the track reconstruction, and a possible
correction method are investigated in detail. This analysis and simulation
package constitutes an essential part of the software development for the
fast-beams program with the AT-TPC.

###### keywords:

AT-TPC, charge-exchange reactions , simulation , $(d,{}^{2}\mathrm{He})$

††journal: NIM A

##  1 Introduction

During the last few decades, Time Projection Chambers (TPC) have been
successfully used as large-volume tracking detectors in many particle physics
experiments, e.g., TOPAZ [1], STAR [2] or ALICE [3]. However, in the recent
years, the operation of TPCs in Active Target (AT) mode have gained a great
interest in the nuclear physics community. These type of devices allow the use
of a target medium as a tracking-detection system with a large solid angle
coverage and low-energy detection thresholds, which make them ideal for
experiments with rare-isotope beams in inverse kinematics.

Currently, many facilities around the world are investing a great effort in
the development of active target TPCs as a fundamental part for future
research programs [4, 5, 6, 7, 8, 9]. One of these projects is the already
operating Active-Target Time Projection Chamber (AT-TPC) [8] at the Facility
for Rare Isotope Beams (FRIB). The AT-TPC has a cylindrical geometry with
dimensions of 100 cm $\times$ ø 50 cm . The gas volume is enclosed by a
cathode plate and a Micromegas electron amplifier plate, as shown in Fig. 1.
The system is designed to take the beam particles impinging along the symmetry
axis of the active volume and to detect the nuclear reactions induced in their
path. The beam particles and the reaction products ionize gas atoms while
traversing the active volume and generate electrons. Upon applying an uniform
electric field, ionization electrons produced by charged particles along their
tracks drift towards the Micromegas sensor plane at a constant velocity. The
high segmentation of the Micromegas plane ($\sim 10^{4}$ triangular pads)
makes possible to obtain the energy loss and a two-dimensional image of the
track, while the third dimension (along the beam axis) is taken from the drift
time of the electrons. A hole (ø 3 cm) in the central part of the Micromegas
plane creates an insensitive region around the beam axis that allows
relatively high beam intensities (see Fig. 1). This is important for
performing experiments in combination with a magnetic spectrometer that
enables measurements of heavy residues in coincidence with the tracks in the
AT-TPC. The active region of the AT-TPC is separated from the vacuum of the
beam line and the spectrometer by 12 $\mu$m polyamide windows.

Figure 1:  Schematic view of the AT-TPC with a $(d,{}^{2}\text{He})$ reaction.
The active volume, filled with a pure D2 gas, is enclosed by a cylindrical
chamber of 100 cm long with a diameter of 50 cm. The beam impinges along the
central axis. The Micromegas plane (upstream) and the cathode plane
(downstream) have a hole in the central region that allows the beam to enter
and the heavy residue to exit the AT-TPC.

Charge-exchange $(d,{}^{2}\text{He})$ reactions at intermediate energies are
well known to be a strong Gamow-Teller (GT) transition filter ($\Delta
T,\Delta S=1$) from experiments in forward kinematics [10, 11, 12, 13, 14].
The $(d,{}^{2}\text{He})$ probe in inverse kinematics is a very promising
method to study the GT transition strength in far-from-stability nuclei. The
output channel, 2He, is an unbound system that decays into two protons. In
order to achieve this type of experiments with unstable beams, it is necessary
to operate the AT-TPC with a pure deuterium gas, which is used as a target and
a tracking medium for the two protons originating in the output channel of the
$(d,{}^{2}\text{He})$ reactions.

In this work, we have developed a simulation and analysis software based on
the attpcroot package [15, 16] to study $(d,{}^{2}\text{He})$ experiments with
the AT-TPC. The simulations offer an ideal method to benchmark and to
understand the experimental data taken with this system. The realistic
detector response provided by the simulations enables a testing method for the
analysis tools that are applied to the experimental data. This paper is
organized as follows: in Section 2, we give an overview of the simulation
routines implemented in this work, Section 3 shows a few remarkable advantages
of using coincidences with beam-like particles, space-charge effects and a
correction method are discussed in Section 4, and Section 5 presents the
conclusions.

##  2 Simulation of $(d,{}^{2}\text{He})$ reactions

Simulation routines of $(d,{}^{2}\text{He})$ reactions were developed using
the attpcroot package [15, 16] written in the C++ programming language.
attpcroot uses the fairroot framework [17] which was developed for the
analysis of experiments at FAIR and later redesigned for other experimental
setup. The attpcroot package provides consistent analysis tools for both
simulated and experimental data. The simulation part of this package comprises
three main stages: event generator, digitization and reconstruction. The
latter stage is also common for the data analysis part of the package.

###  2.1 Event generator

The simulation events are generated with the Virtual Monte-Carlo (VMC) package
[18] that serves as an interface to the geant4 toolkit [19]. The sensitive
volume of the AT-TPC is defined as a cylindrical geometry filled with a
deuterium gas (D2) that serves simultaneously as target and tracking medium
for the charged particles involved in the reaction. Thus, the event generator
class creates the particles that are transported by geant4 and stores the hit
information in a class that handles the simulated data points.

Figure 2:  Double-differential cross section of the reaction
${}^{14}\text{O}(d,{}^{2}\text{He}){}^{14}\text{N}$ at 100 MeV/u (populating
the 14N $1_{1}^{+}$ state at 3.9 MeV) as a function of the scattering angle
($\theta_{\text{c.m.}}$) and the relative energy ($\varepsilon_{pp}$). This
cross section was calculated using the code accba [20].

The $(d,{}^{2}\text{He})$ events are generated along the active volume using a
realistic beam emittance based on the ion-optical properties of the beam line
at the S800 spectrometer at FRIB [21]. Three particles are produced in the
final channel of each reaction: two protons (from the ${}^{2}\text{He}$ decay)
and an ejectile. A routine to simulate the in-flight decay of the residues
after the reaction is also included in the code. In order to have a consistent
treatment of the $(d,{}^{2}\text{He})$ reaction, a relativistic three-body
kinematics is used to generate the particles in the output channel. An
important part of the $(d,{}^{2}\text{He})$ kinematics is the description of
the relative energy between the two protons ($\varepsilon_{pp}$) in the
${}^{2}\text{He}$ frame. $\varepsilon_{pp}$ is calculated with the code accba
(Adiabatic Coupled-Channels Born Approximation) [20], which has been
successfully used to reproduce $(d,{}^{2}\text{He})$ data from experiments in
forward kinematics [13, 22, 23]. Fig. 2 shows the double-differential cross
section for the ${}^{14}\text{O}(d,{}^{2}\text{He}){}^{14}\text{N}$ reaction
(populating the 14N $1_{1}^{+}$ state at 3.9 MeV) as a function of
$\varepsilon_{pp}$ and $\theta_{\text{c.m.}}$ (center-of-mass angle)
calculated with the code accba. As expected for a GT transition, the
differential cross section peaks at forward scattering angles. The relative
energy between the two protons $\varepsilon_{pp}$ is distributed from 0 up to
5 MeV. These angular and energy distributions are included as an input to the
event generator to accordingly determine the detection efficiency as a
function of $\theta_{\text{c.m.}}$ and $\varepsilon_{pp}$.

###  2.2 Digitization

The digitization stage uses the generated tracks and simulates the detector
response. This part is divided in two subroutines: electron diffusion and
electronics response. In the electron diffusion subroutine, the number of
ionization electrons is calculated for every hit based on the energy
deposition and the average ionization energy for the gas target. Subsequently,
the drift time of the electrons to reach the pad plane is calculated using a
drift velocity obtained from magboltz [24], which simulates the electron
transport in a certain gas under the influence of an electromagnetic field. It
is important to mention that the uncertainty on the reconstructed excitation
energy of the ejectile depends on the electron diffusion in the gas volume.
For example, a large diffusion parameter produces thicker proton tracks which
degrades the fitting precision. The corresponding parameters were set to
realistic values achieved with the AT-TPC operated with a pure D2 gas. Effects
on the electron drift due to field distortions can be simulated in detail by
using electron transport routines in this method, as explained in Section 4.
Fig. 3 shows an example of a simulated $(d,{}^{2}\text{He})$ event and its
respective projection on the pad plane. Each proton hit produces an electron
cloud that drifts in the gas volume towards the detector pixels in the pad
plane. In particular, measurements of proton tracks in the AT-TPC require a
high gain in the electronics, which can enhance also the detection of
$\delta$-rays. The output of this subroutine is coupled to the electronics
response module that simulates a signal for each pad. In this part, an
electron avalanche generates a pulse shape by using a realistic response of
the Micromegas and the GET (general electronics for TPCs) [25] system.

Figure 3:  A simulated $(d,{}^{2}\text{He})$ event with the AT-TPC. Note that
the pad plane has a 3 cm aperture in the central region that allows the beam-
like particles pass through the active volume.

###  2.3 Reconstruction

The event reconstruction and further tasks are the same for both simulated and
experimental data. After the digitization, a point cloud (collection of hits
in the 3D space) for each reaction in the gas volume is extracted. The
analysis of these point clouds is performed with a pattern recognition and
fitting routine that searches $(d,{}^{2}\text{He})$ events and extracts the
kinematical information such as scattering angle, momentum, and excitation
energy. Tracking algorithms allow to identify the points that belong to the
particle tracks (inliers) within a entire point cloud and reject the noise
points (outliers). A tracking algorithm based on ransac (RANdom SAmple
Consensus) has been successfully employed to analyze data from the AT-TPC
[26]. However, ransac is rather sensitive to the inlier-outlier threshold and
it requires a prior parameter fine tuning each time for a specific
application. Also, a common problem of the algorithm is that it fails when
describing data that contain multiple structures, such as point clouds with
several tracks that have a significant amount of pseudo-outliers. A few
improved versions of RANSAC-like algorithms were developed for the
reconstruction of $(d,{}^{2}\text{He})$ events [27]. In these routines, the
process of model verification is optimized by using a probabilistic approach.
For instance, the candidate trajectories (model hypotheses) are evaluated on
basis of an error probability distribution. This reduces the outlier
sensitivity of ransac and provides a more robust method to identify particle
tracks in point clouds with a large fraction of outliers. Also, a modification
in the sampling process based on the relative distance and deposited charge
was included to preferentially generate more useful ransac hypotheses. The
random sampling is constructed in a way that neighboring points are selected
with higher probability by assuming a Gaussian distribution for the relative
distance between points. Similarly, a probability density distribution using
the total charge of the point cloud allows to select more efficiently the
points that have a larger energy deposited in gas [27]. Finally, the tracking
algorithm is coupled to a least squares routine that improves the ransac
output and provides a better estimate of the reaction vertex and scattering
angles.

For $(d,{}^{2}\text{He})$ experiments, the AT-TPC operates without a magnetic
field. Thus, the particle tracks are 3D line trajectories. The tracking
algorithm fits all particle tracks in each event and identifies
$(d,{}^{2}\text{He})$ reactions as two protons emitted with the same vertex
(see Fig. 1). The momenta of the recoiling protons ($\bm{P}_{px}$ with
$x=1,2$) are extracted by combining the track fitting result and the
respective track length. The proton energy in the gas can be directly
extracted from stopping power tables calculated with the code srim [28]. This
information is sufficient for the reconstruction of a ${}^{2}\text{He}$
particle. The first step is to obtain the momentum of ${}^{2}\text{He}$ from
the proton tracks

$\bm{P}_{{}^{2}{\text{He}}}=\bm{P}_{p1}+\bm{P}_{p2}.$ (1)

Then, the relative energy between the two protons is extracted from the
invariant-mass calculation in the ${}^{2}\text{He}$ frame

$\varepsilon_{pp}=\sqrt{E_{{}^{2}\text{He}}^{2}-\bm{P}^{2}_{{}^{2}{\text{He}}}}-2m_{p},$
(2)

where $E_{{}^{2}\text{He}}$ is the total energy of the ${}^{2}\text{He}$
particle and $m_{p}$ the proton mass.

To ensure that two protons are in the spin-singlet (${}^{1}\text{S}_{0}$)
state [29], and the $(d,{}^{2}\text{He})$ reaction proceeds exclusively with
the transfer of spin $\Delta S=1$, it is preferred to constrain the analysis
to small values of $\varepsilon_{pp}$. In forward kinematics experiments,
small values of $\varepsilon_{pp}$ are selected through the limited acceptance
of the spectrometers used to detect the two protons or through software cuts
in the measured $\varepsilon_{pp}$ distribution [10, 30, 14]. However, in
inverse kinematics, the integrated $\varepsilon_{pp}$ range depends on the
detection efficiency and acceptance of the detector system. At small
$\theta_{\text{c.m.}}$ the momentum transfer of the reaction is also small,
resulting in a low kinetic energy of the virtual ${}^{2}\text{He}$ particle.
Consequently, the two protons from the decay of ${}^{2}\text{He}$ will also
have small kinetic energies. The reconstruction of the ${}^{2}\text{He}$
particles is limited by the size of the insensitive region of the AT-TPC (see
Fig. 3). In this case, the two proton tracks are detected when the relative
energy $\varepsilon_{pp}$ is sufficiently high. This is shown in Fig. 4(b):
relative energies between 1 and 2.5 MeV are probed for
$\theta_{\text{c.m.}}\approx 0^{\circ}$. At larger $\theta_{\text{c.m.}}$, the
momentum transfer increases, which is reflected in larger kinetic energies for
the ${}^{2}\text{He}$ particle (also for the two protons). If the energy of
the two protons becomes too high, they escape from the active volume of the
AT-TPC and the reconstruction of the event is not possible. Therefore, at high
$\theta_{\text{c.m.}}$, only events with $\varepsilon_{pp}\lesssim 1$ MeV can
be reconstructed, as shown in Fig. 4(b). As the differential cross section for
a particular transition depends on both $\theta_{\text{c.m.}}$ and
$\varepsilon_{pp}$ (see Fig. 2), a $\varepsilon_{pp}$ acceptance correction as
a function of $\theta_{\text{c.m.}}$ must be taken into account for extracting
the differential cross section and the GT transition strength. The simulations
presented here are essential for this correction.

Once the ${}^{2}\text{He}$ particle is reconstructed, a missing-mass
calculation is performed to extract the excited states populated by in the
reaction. Fig. 4(c) shows the reconstructed kinematic plot for the
${}^{14}\text{O}(d,{}^{2}\text{He}){}^{14}\text{N}$ reaction
($Q_{\text{g.s.}}=3.7$ MeV) and its respective projection in the missing mass
(Fig. 4(d)). Simulations of $(d,{}^{2}\text{He})$ reactions populating the
ground state (g.s.) and excitation energies of 10 and 20 MeV are assumed in
order to test the reconstruction routines.

Figure 4:  a) Simulated 14O(d,2He)14N event, at a beam energy of 100 MeV/u,
with the respective track fitting and vertex reconstruction. b) Internal
energy of 2He as function of the center-of-mass scattering angle. The red
dashed lines indicate the particle ranges of 60 mm and 235 mm, respectively.
The points above the 235 mm range correspond to tracks with scattering angles
different than 90°. c) Kinematics for the 14O(d,2He)14N reaction,
reconstructed 2He scattering angle as function of 2He kinetic energy. Three
excited states in 14N were assumed in this example: Ex= 0, 10 and 20 MeV. The
red lines are theoretical kinematics lines at 10 MeV excitation energy for
$\varepsilon_{pp}$ values of 0 and 2.5 MeV. The black dashed lines indicate
center-of-mass scattering angles (0.5°, 3.5°and 7°). d) Reconstructed missing
mass (excitation energy) of 14N, for ground state, 10 and 20 MeV inputs. The
distributions were normalized. The peak at 10 MeV has a full width at half
maximum (FWHM) of about $1.6$ MeV.

The broadening of the kinematic lines is not only due to the energy and angle
resolutions of the reconstruction, the distribution of the internal 2He energy
also contributes to it (e.g., about $\sim 30\%$ in the g.s. Fig. 4(c)). The
reconstructed scattering angle and kinetic energy of the 2He particle at
$E_{x}=10$ MeV (see Fig. 4(c)) seems to deviate from two-body kinematics
calculation at $\theta_{cm}\gtrsim 6$°, this is explained by the limited
acceptance at large scattering angles (Fig. 4(b)) and the resolution of the
reconstruction. The resolution of the excitation-energy peaks (Fig. 4(d))
depends on the kinematics of the reaction. For instance, the resolution of the
ground-state transition in the present example is dominated by the uncertainty
in the reconstructed scattering angle of the 2He particle, whereas at higher
excitation energies the resolution depends on both the kinetic energy and the
scattering angle. The large acceptance of the AT-TPC provides a good
kinematical reconstruction on almost the entire angular range. Due to the
relative good angular resolution (better than $1.5^{\circ}$ in
$\theta_{\text{lab}}$) achieved by the tracking algorithm, the excited states
are easily separable up to about $8^{\circ}$ in the center-of-mass system with
an energy resolution of about 1.6 MeV (FWHM).

##  3 Coincidences with beam-like particles

Complete kinematics measurements including the detection of beam-like
particles provide a strong selectivity of the reaction mechanism and also
reduce significantly the background in the experimental data. Therefore, a
routine that includes a phase-space generator was implemented to properly
simulate the in-flight decay and momentum of the residues. As the reaction
ejectiles in inverse kinematics are boosted to forward angles in the
laboratory system, the AT-TPC is coupled with the S800 spectrometer at FRIB
[21]. Then, the $(d,{}^{2}\text{He})$ events can be correlated with the
respective particle identification in the spectrometer focal plane detectors.
For instance, Fig. 5(a) shows the simulated dispersive and non-dispersive
angles of the beam-like particles accepted in the S800 spectrometer assuming
an incident 14O beam at 100 MeV/u. Also, from the momentum of the accepted
residues it is possible to obtain the respective magnetic rigidity ($B\rho$).
Fig. 5(b) shows the simulated $B\rho$ distributions for the
${}^{14}\text{O}(d,{}^{2}\text{He}){}^{14}\text{N}$ reaction including several
decay channels such as $p$, $n$, $np$ and $\alpha$.

Figure 5:  (a) Simulated dispersive and non-dispersive angles of the ejectiles
in the 14O(d,2He)14N reaction at the target location. (b) Magnetic rigidity of
the ejectiles 14N and several decay products.

The identification of a beam-like particle in coincidence with two protons
emerging from the same vertex is a stringent condition that suppresses the
background. As discussed before, $\delta$-rays are expected to be a
significant contribution in the point clouds. These points are an important
source of background that can lead to a wrong identification of
$(d,{}^{2}\text{He})$ events. In this case, the integrated charge and number
of points of each track are substantially smaller than the ones expected for a
real proton track. Fig. 6 (insert) shows that these spurious tracks are easily
suppressed by the integrated charge condition. Random coincidences due to
multiple-scattering events or scattering close to a $(d,{}^{2}\text{He})$
reaction vertex can also be a source of background. However, the probability
of occurrence for such events is very small, even for beam rates at the level
of $R=10^{6}$ pps. For example, the probability to have an elastic scattering
[14O$(d,d)$] event during a time window of $W=100\times 10^{-6}$ s (typical
drift time of the electrons for a $L=100$ cm long drift volume) can be modeled
by assuming a Poisson probability distribution with $\lambda=L\rho\sigma
WR\approx 8.7\times 10^{-3}$. In this case, $\rho\approx 1.74\times 10^{19}$
cm-3 is an estimate value of the density of atoms in the gas target, and the
scattering cross section $\sigma\sim 50\times 10^{-27}$ cm2
($\theta_{\text{c.m.}}\lesssim$ 8 deg). Using the Poisson distribution with
this information, one finds that the probability to have at least one deuteron
elastically scattered within the same time window of a $(d,{}^{2}\text{He})$
reaction is in the order of 1%. Therefore, the background contribution due to
an accidental detection of two deuterons elastically scattered, with vertices
within $\sim 1$ cm [similar condition as the two protons of a
$(d,{}^{2}\text{He}$) reaction], is reduced by a factor 100, which makes this
source of background negligible.

Figure 6:  Reconstructed 14N ground-state energy distribution in the
($(d,{}^{2}\text{He}$) reaction (grey peak). The contribution due to
stochastic noise fitting is shown in red. The inserted plot represents a
distribution of the integrated charge per track and the small bump below
$5\times 10^{3}$ (arb. unit) is associated to the red background in the
excitation energy distribution.

###  3.1 Kinematic fitting

In complete kinematic measurements, the resolution of the reconstructed
quantities can be improved by using a mathematical procedure called kinematic
fitting (KF). KF is a widely used technique in high-energy physics [31].
Recently, this method has been proposed for the analysis of nuclear reactions
at low energy [32]. The goal of the kinematic fitting is to reduce the
uncertainties from the measured quantities by minimizing the reaction
kinematics using a few constraints such as the momentum and energy
conservation. The solution method implemented in this work is based on the
kinematic fitting programs fit70 [33] and kwfit [31].

In the particular case of a $(d,{}^{2}\text{He})$ reaction, the input channel
corresponds to a projectile and a deuterium target, while the output channel
is composed by two protons and an ejectile. Each particle have a four-momentum
$\alpha_{i}=(p_{xi},p_{yi},p_{zi},E_{i})$ vector that is extracted from the
observables of an experiment. The deuteron target is at rest in the laboratory
frame, and therefore, this information is only included as a reaction
constraint. Let $\alpha$ represent the column vector that contains the four-
momentum vectors of the projectile and the three particles in the output
channel

$\displaystyle\alpha$ $\displaystyle=\begin{pmatrix}\alpha_{1}\\\
\alpha_{2}\\\ \alpha_{3}\\\ \alpha_{4}\end{pmatrix}$ (3)

The constraints assumed in the KF are the momentum (3D) and the total energy
conservation as

${\bf H}(\alpha)\equiv 0,\hskip 8.5359pt\text{where}\hskip 8.5359pt{\bf
H}=(H_{1},H_{2},H_{3},H_{4}).$ (4)

The equation of above can be expanded around a convenient point $\alpha_{A}$
that produces a set of linear equations [31]

$0=\frac{\partial{\bf H}(\alpha_{A})}{\partial\alpha}(\alpha-\alpha_{A})+{\bf
H}(\alpha_{A})={\bf D}\delta\alpha+{\bf d},$ (5)

where ${\bf D}$ is a matrix of dimension $4\times 4$ that contains the partial
derivatives of the constraint parameters $D_{ij}=\frac{\partial
Hi}{\partial\alpha_{j}}$, and ${\bf d}$ is a vector of constraints evaluated
at the point $\alpha_{A}$. The fitting technique is based on the Lagrange
multipliers method, in which the $\chi^{2}$, given by

$\chi^{2}=(\alpha-\alpha_{0})^{T}V_{\alpha
0}^{-1}(\alpha-\alpha_{0})+2\lambda^{T}({\bf D}\delta\alpha+{\bf d}),$ (6)

is minimized. Here, $V_{\alpha 0}$ is the covariance matrix and $\lambda$ is a
vector of Lagrange multipliers. A routine for minimizing Eq. (6) was
implemented in the code. The resulting parameters $\alpha$ are improved
(compared to the input values $\alpha_{0}$) by the momentum and energy
constraints of the reaction. In order to avoid problems with the
reconstruction of relative energy $\varepsilon_{pp}$, an anticorrelation
between the two protons in the output channel was defined in the covariance
matrix. The KF was tested with the reconstruction of two peaks at 5 and 10
MeV, as shown in Fig. 7. As the beam particles interact with the gas in the
insensitive region, no information of the beam momentum is possible to be
extracted from the point clouds. For the KF, it was assumed the beam particle
impinging the AT-TPC in the normal direction with a momentum resolution
(defined in the covariance matrix) of 0.1%. The momentum of the ejectile
particle was extracted directly from the event generator with the respective
acceptance provided by the S800 spectrometer, as shown in Fig. 5. As explained
in Section 2.3, the energy resolution achieved from the tracking is about 1.6
MeV, but with the kinematic fitting the resolution is improved to a value of
0.8 MeV. Thus, the minimization routine corrects for the uncertainty in the
energy and angles of the proton tracks by using energy and momentum
conservation. Therefore, a significant improvement in the reconstruction of
the invariant mass is expected for complete kinematic measurements. The KF
routine presented here can also be implemented in the analysis of future
experiments. In this case, it might be required the use of tracking detectors
before the AT-TPC in order to extract with a better precision the projectile
momentum and reaction vertex. Also, it would be necessary to investigate in
more detail the reconstruction of virtual particles emerging from the decay of
the residues of $(d,{}^{2}\text{He})$ reactions.

Figure 7:  Missing-mass reconstruction before (dashed line) and after (solid
line) the kinematic fitting. Arbitrary states at 5 and 10 MeV excitation
energy were assumed to demonstrate the performance of the technique.

##  4 Space-charge effects

###  4.1 Charge build-up

Space-charge effects due to ion accumulation inside the AT-TPC are expected to
be present at high beam intensities. A large number of electron-ion pairs
[$N(r,z)$] are produced in the gas volume along the beam path. While electrons
drift relatively fast (tens of $\sim\mu$s) to the pad plane, the drift time of
ions is a few orders of magnitude slower. It means that the beam is
continuously ionizing the gas target producing a charge build-up in the
central region of the AT-TPC. This accumulated charge can produce a distortion
of the electric field and also affect the electron drift. Experimental
evidences of such space-charge effects in the AT-TPC have recently reported
[34]. Assuming a constant beam rate $R$, the production rate of electron-ion
pairs per unit volume can be roughly approximated by

$\dot{N}=\left(\frac{dE}{dx}\right)\frac{R}{wA},$ (7)

where $\left(\frac{dE}{dx}\right)$ is the beam stopping power in the gas, $A$
is the beam spot area and $w=37$ eV is the mean ionization energy for $D_{2}$
[35]. The charge density in the AT-TPC volume at a time $t$ is given by [36]

$\rho(x,y,z,t)=\int_{0}^{t}e\dot{N}(x,y,z,t^{\prime})dt^{\prime},$ (8)

with $e$ the electric charge unit. As an example, a 56Ni beam at 100 MeV/u and
rate of $10^{5}$ pps interacting with the AT-TPC filled with a deuterium gas
at 500 Torr are assumed. A realistic beam profile of a Gaussian distribution
of $\sigma=0.5$ cm is assumed. The accumulated charge density after an
integration time of $t_{\text{max}}=L/u^{+}$ ($L$ is the AT-TPC length and
$u^{+}$ is the ion drift velocity) can yield up to 3 pC/cm3 around the beam
axis, as is shown in Fig. 8(a). In order to have a rough estimation of the
electric field produced by this charge density, a radial field component is
calculated using Gauss’ law in the approximation of an infinite cylinder

$E(r)=\frac{e}{\varepsilon_{0}}\left(\frac{dE}{dx}\right)\frac{Rt_{\text{max}}}{w}\left(\frac{1-\exp\left(-\frac{r^{2}}{2\sigma^{2}}\right)}{r}\right),$
(9)

where $\varepsilon_{0}$ is the permittivity of free space. A more rigorous
solution can be obtained by integrating the corresponding Green’s function, as
explained in Ref. [37]. The resulting effect of this field component is a
distortion of the electron drift distance. In order to evaluate such effects,
it is necessary to simulate the electron transport by solving the Langevin
equation numerically [36, 37]. An electron transport routine was implemented
in the code in the pre-digitization module. An example of a
$(d,{}^{2}\text{He})$ reaction in a distorted electric field is shown in Fig.
8(b). As can be noticed, the tracks are mostly affected in the central region
of the AT-TPC. These space-charge effects may cause problems in the event
reconstruction, in particular, for the extraction of scattering angles and
reaction vertex. The radial distortion is defined as the relative distance
between the projected initial point of the electron $(x_{0},y_{0})$ and the
final point on the pad plane $(x_{f},y_{f})$. Depending on the drift distance,
the radial distortion can increase because of the electron transport. For
instance, one can see the distortion at different initial distances relative
to the beam axis $(r)$ in Fig. 8(c). The space-charge effects are stronger
near the beam axis, and they become negligible for electrons produced close to
the wall. For an initial distance of $r=5$ cm, the distortion can be as big as
3 cm, after a drift distance of 100 cm, which is approximately the equivalent
distance of 6 pads. The space-charge effects have also a significant
dependence on the beam particle due to the amount of electrons produced in the
beam energy loss. The higher the beam charge, the stronger the radial
distortion is. Fig. 8(d) shows an example of the radial distortion at $r=10$
cm for different beams. As expected, the space-charge effects are higher for
56Ni which are about a factor 4 stronger than for 24Mg.

Figure 8:  (a) Charge distribution assumed in the model to describe the ion
build-up. The distribution is a Gaussian profile extended along the active
volume. (b) Simulated ($d$,2He) event including space-charge effects. An
electric field component in the radial direction distort the tracks close to
the beam axis. (c) Radial distortion as a function of the drift distance. The
effects are stronger close to the beam axis. (d) Radial distortion as a
function of the drift distance for three ion beams. The distortion effect
increase with the ion-beam charge.

###  4.2 Recombination

Electron recombination is also an important space-charge effect that needs to
be taken into account in the simulation. For this effect, the electrons
produced by ionizing particles in the gas volume may recombine with nearby
ions due to electromagnetic interaction. For high-ionization densities, the
recombination effect can induce significant losses in the collected charge.
Two types of recombination processes are possible in the AT-TPC, columnar or
volume recombination [35]. The columnar process is when an ion-electron pair
produced in the same track recombines, while the volume process is a more
general case where the ions are recombined with electrons from different
tracks. Due to the relative strong fields and highly ionizing particles used
in the AT-TPC, the columnar recombination is expected to be negligible in
comparison with the volume recombination [34, 35]. In the latter case, the
recombination depends on the ionization rate $\dot{N}$, which can be
approximated by Eq. (7). Thus, the recombination effect is beam-rate
dependent. The time scale required for an ion-electron pair to recombine
depends on the intrinsic properties of the gas expressed by the recombination
coefficient $\alpha$, which for a deuterium gas is $\alpha\sim 2.5\times
10^{-6}$ cm3/s [38].

The recombination loss in ionization chambers (parallel plate chambers) can be
estimated as [39]

$f=\frac{\alpha\dot{N}L^{2}}{6\mu^{2}E^{2}},$ (10)

where $\mu$ is the ion mobility and $E$ is the strength of the electric field.
Eq. 10 provides an estimate value for the loss in charge collection assuming a
constant beam rate impinging the gas target. As an example, Table 1 shows the
recombination loss for three beams at 100 MeV/u and $10^{5}$ pps interacting
with a deuterium gas at 500 Torr and field strength of 500 V/cm.

Table 1: Recombination loss and electron-ion production rate in a deuterium gas at 500 Torr and 500 V/cm. Three different beams were assumed at 100 MeV/u and $10^{5}$ pps.  Ion Beam |  $\dot{N}$ [cm3/s] |  $f$ [%]
---|---|---
12C |  $5.5\times 10^{7}$ |  0.5
24Mg |  $2.1\times 10^{8}$ |  1.7
56Ni |  $1.1\times 10^{9}$ |  9.1

As can be noticed, the recombination increases with the beam energy loss in
the gas. As the recombination is expected to occur most likely at near the
beam region, the absolute effect may be a reduction in the space-charge build-
up that distorts the electric field. For beam intensities above $10^{6}$ pps,
the recombination effect becomes significant and it is expected that part of
the information of the tracks is lost [34].

###  4.3 Correction

The space-charge effects, mentioned in 4.1 and 4.2 can be corrected for using
a distortion map of the full active volume generated from simulations. As the
beam impinges in central region of the active volume, space-charge effects can
be assumed to be axially symmetric. This reduces the extraction of a
distortion map to a two-dimensional problem. An electron transport routine
including charge build-up and recombination effects was used to generate and
transport electrons at different initial positions along the whole active
volume. The relative distance of the electron cloud on the pad plane with
respect to a given initial electron position defines the distortion. The
distortion map was applied directly to the geometrical coordinates of each hit
in order to restore the particle tracks. Fig. 9(a) shows a comparison of the
reconstructed missing-mass energy for simulations with and without space-
charge effects, and after the correction based on the distortion map. As can
be seen, the energy resolution is strongly affected by the space charge mostly
at high excitation energies. This effect can be related with a wrong
determination of the reaction vertex and scattering angle of the protons.
Similarly, the reconstruction efficiency of the 2He particle is reduced by
about 13% when the space-charge effects are included, i.e., 87% of the tracks
can be restored with the present correction. This reduction in efficiency is
because a few short tracks lose a fraction of their points in the central dead
region (pad plane hole). However, with the correction, one is able to restore
the energy resolution of the peaks. The minimum distance between the fitted
proton tracks in the central region gives an estimate of how well the vertex
can be reconstructed, see Fig. 9(b). The space-charge effects produce a wider
distribution of the minimum distance between the two proton tracks that leads
to a larger uncertainty in the determination of the vertex point. For this
study, we chose the minimum distance between tracks to be less than 1 cm,
which corresponds to about $2\sigma$ of the distorted track distribution. The
correction improves the vertex reconstruction, which also impacts the missing-
mass energy, as shown in Fig. 9(a). A similar procedure can be employed to
correct experimental data. In this case, the distortion map might be extracted
from simulations or from the same data by quantifying the distortion of the
tracks in the full active volume.

Figure 9:  Simulated 14O($d,^{2}$He) data with and without space-charge
effects and the respective correction using a distortion map. (a)
Reconstructed missing-mass energy. (b) Distribution of the minimum distance
between the proton tracks.

##  5 Summary

A simulation and analysis code has been developed to study charge-exchange
$(d,{}^{2}\text{He})$ reactions in inverse kinematics using the AT-TPC. The
simulation package provides a realistic detector response that allows for
processing the simulation output in the same manner as the experimental data.
Dedicated subroutines for the event generator, digitization and reconstruction
of nuclear reactions are used for simulating experiments with the AT-TPC. In
particular, the event generator of $(d,{}^{2}\text{He})$ reactions takes into
account the production of two recoiling protons at low-momentum transfer with
a relative energy, based on adiabatic coupled-channel calculations that are
known to reproduce the dependence of the $(d,{}^{2}\text{He})$ differential
cross section on $\varepsilon_{pp}$ and $\theta_{cm}$. A point cloud of the
reaction is obtained from the digitization of the proton tracks in the active
gas volume. Pattern recognition and fitting algorithms are used for the
analysis of the point clouds and for reconstructing the $(d,{}^{2}\text{He})$
reactions. The missing-mass energy is very well reconstructed independently of
the geometrical acceptance of the detector.

A few analysis tools were successfully tested with simulations. For instance,
tracking algorithms with a better selection of proton tracks were employed to
improve the detection efficiency of the 2He particle decay. Detection of beam-
like particles originating in $(d,{}^{2}\text{He})$ reactions in inverse
kinematics provides a better selectivity of the reaction channel and a strong
background suppression. In this work, we investigated the use of a kinematic
fitting procedure for improving the reconstruction of the reaction parameters.
It was shown that the energy resolution of the excited states can be improved
by about a factor 2 if beam tracking detectors are used. In the future, the
kinematic fitting tool will provide a powerful method for data analysis in
experiments with complete particle detection. Also, space-charge effects were
investigated with a model that accounts for charge build-up and ion
recombination. These effects are most relevant around the beam axis, and
generate a distortion of the particle tracks in the central region. At
relatively high beam intensities and large nuclear charge, the space-charge
effects are expected to play a significant role in the deformation of the
particle tracks. Therefore, the simulation model presented here was used to
generate distortion maps that can be applied in the future to correct
experimental data.

The analysis tools developed in this work are designed to process both
simulated and experimental data. Future $(d,{}^{2}\text{He})$ experiments in
inverse kinematics using the AT-TPC will use the code. Many of the routines
are also valid for the analysis of other type of reactions involving fast
beams such as inelastic scattering and fission.

## Acknowledgments

This work was supported by the US National Science Foundation under Grants
PHY-2209429 (Windows on the Universe: Nuclear Astrophysics at FRIB),
PHY-1430152 (JINA Center for the Evolution of the Elements). This material is
based upon work supported by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics and used resources of the Facility for Rare Isotope
Beams (FRIB), which is a DOE Office of Science User Facility, operated by
Michigan State University, under Award Number DE-SC0000661. This work has
received financial support from Xunta de Galicia (Centro singular de
investigación de Galicia accreditation 2019-2022), by European Union ERDF, and
by the “María de Maeztu” Units of Excellence program MDM-2016-0692 and the
Spanish Research State Agency. Y. A. acknowledges the support by the Spanish
Ministerio de Economía y Competitividad through the Programmes “Ramón y Cajal”
with the Grant No. RYC2019-028438-I.

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which shows that the space
$H^{3}(\mathbb{Z}^{3},K_{0}^{\\#}(\mathcal{O}_{\infty}))$ of our new invariant
naturally interpolates the set of Dixmier-Douady classes
$H^{3}(\mathbb{T}^{3},\mathbb{Z})$.

##  5 Comments

1\. To establish Conjecture 4.9,(2) for the general poly-$\mathbb{Z}$ case, it
is probably better to work on $\mathcal{O}_{\infty}$ and
$M_{\mathfrak{P}^{\infty}}$ separately instead of
$M_{\mathfrak{P}^{\infty}}\otimes\mathcal{O}_{\infty}$ in view of Corollary
3.11. In order to be able to push the induction argument in the
$\mathcal{O}_{\infty}$ case further, we need, as an induction hypothesis, a
natural splitting of the exact sequence

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Such a statement is out of reach in our brute force method, and probably
requires a homotopy theoretical interpretation of the grooup
$\mathcal{F}_{\mathcal{O}_{\infty}(G)}$. Note that a similar exact sequence

${0}$${\ker\delta_{1}}$${\bar{E}_{\mathcal{O}_{\infty}}^{1}(X)}$${H^{3}(X,\mathbb{Z})}$${0}$$\scriptstyle{\delta_{1}}$

splits because $H^{3}(X,\mathbb{Z})$ is identified with
$\bar{E}_{\mathbb{C}}^{1}(X)$.

2\. To show Conjecture 4.9,(1) in full generality, we might need a $G$-kernel
version of Gabe-Sábo’s classification theorem. A recent work of Arano-
Kitamura-Kubota’s [1] may be the first step toward it.

3\. For a non-amenable exact group $G$ satisfying the Haagerup property, we
still have a chance to get reasonable theories $\mathcal{E}_{A}(G)$ and
$\mathcal{F}_{A}(G)$ by considering only amenable actions. As there exists a
characterization of amenability of group actions in terms of central sequences
[37], it is possible to define it for $G$-kernels too.

4\. To define $\mathcal{E}_{A}(G)$ and $\mathcal{F}_{A}(G)$ in the stably
finite case, we need to assume strongly outerness for cocycle actions and
$G$-kernels. Even with this modification, we cannot expect any similar results
in the stably finite case as we have already seen it in Remark 3.13. In fact,
Matui-Sato [32] showed $\mathcal{E}_{A}(\mathbb{Z}^{2})\cong
H^{2}(G,\mathbb{R}/K_{0}(A))$ for $A=M_{\mathfrak{P}^{\infty}}$ and
$A=\mathcal{Z}$.

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