#  Fast trajectory reconstruction techniques
for the large acceptance magnetic spectrometer VAMOS++

A. Lemasson<EMAIL_ADDRESS>M. Rejmund  GANIL, CEA/DRF -
CNRS/IN2P3, Bd Henri Becquerel, BP 55027, F-14076 Caen Cedex 5, France

###### Abstract

The large angular and momentum acceptance magnetic spectrometer VAMOS++, at
GANIL, France, is frequently used for nuclear structure and reaction dynamics
studies. It provides an event-by-event identification of heavy ions produced
in nuclear reactions at beam energies around the Coulomb barrier. The highly
non-linear ion optics of VAMOS++ requires the use of the heavy ion trajectory
reconstruction methods in the spectrometer to obtain the high-resolution
definition of the measured atomic mass number. Three different trajectory
reconstruction methods, developed and used for VAMOS++, are presented in this
work. The performances obtained, in terms of resolution of reconstructed
atomic mass number, are demonstrated and discussed using a single data-set of
fission fragments detected in the spectrometer.

††journal: Nuclear Instruments and Methods in Physics Research Section A

##  1 Introduction

Magnetic spectrometers are very powerful tools to identify and characterize
nuclei produced in nuclear collisions. In the recent years, large acceptance
magnetic spectrometers like VAMOS++ [1, 2] at GANIL and PRISMA [3] at LNL
Legnaro have significantly contributed to the advances in the domain of
nuclear structure and reaction mechanism studies. They allow the isotopic
identification and therefore the clean selection of reaction products on an
event-by-event basis. The above-mentioned spectrometers are particularly
suited for reaction at the beam energies near the Coulomb barrier, where a
large angular and momentum acceptance is necessary for efficient collection of
the nuclear reaction products of interest. However, a large angular and
momentum acceptance is related to large sizes of optical elements, composing
the spectrometer, which often result in a design that exhibits a highly non-
linear optics. Therefore, ray-tracing techniques for a trajectory
reconstruction [1, 2], are required, leading to the determination of the
magnetic rigidity and related observables for the detected heavy ions.

The isotopic identification, in terms of atomic mass number $A$ and atomic
number $Z$, is the key issue as far as the magnetic spectrometers are
concerned. The basic measurement provided by a magnetic spectrometer is the
magnetic rigidity $B\rho$ of the ion, it is proportional to the position in
the dispersive plane of the dipole magnet. The magnetic rigidity is related to
the velocity $v$ and mass-over-charge ratio $(A/q)$, where $q$ is the atomic
charge state. The measurement of the velocity $v$ allows to obtain the
corresponding mass-over-charge ratio $(A/q)$. The measurement of the atomic
mass number $A$ requires the measurement of the atomic charge state $q$.
Numerous spectrometers are used at high beam energies, from few tens to
hundreds of MeV/A, so that the heavy ions of interest can be obtained in a
fully stripped charge state, that is $q=Z$. Note that the energy necessary to
reach the fully stripped charge state for a given atomic number $Z$ increases
with $Z$. In this case the measurement of the correlation between the energy
loss of the ion $\Delta E$ and its time-of-flight $t$ or velocity $v$, is
often sufficient to determine the atomic charge state $q=Z$ and thus also the
corresponding atomic mass number $A$ [4, 5]. However, at energies near the
Coulomb barrier, heavy ions have lower atomic charge state $q<Z$, with wide
statistical distributions. In this case, the measurement of the total energy
$E_{tot}$ of the incoming ion is necessary. The relation between the total
energy $E_{tot}$ and atomic mass number $A$ combined with the measurement of
magnetic rigidity $B\rho$ allows to determine the atomic charge state $q$, for
each ion individually. Due to the large angular and momentum acceptance, the
measurement of the correlation between the energy loss $\Delta E$ and the
total energy $E_{tot}$ or velocity $v$ are necessary to obtain the atomic
number $Z$ [6]. Additionally, at lower beam energies, it is mandatory to
minimize the effective thickness of every detector, through which the heavy
ions are passing, the window thickness in particular, to reduce the related
unmeasured energy losses.

VAMOS++ has been fruitfully used for a wide range of experiments covering (i)
particle and $\gamma$-ray spectroscopy using direct transfer [7, 8, 9, 10, 11,
12, 13, 14, 15, 16], multi-nucleon transfer: [17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33], fission [34, 35, 36, 37, 38, 39, 40, 6,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54], and Coulomb
excitation [55] (ii) fission [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66] and
reaction dynamics [67, 68, 69, 70, 71, 72].

In this paper, three different trajectory reconstruction methods elaborated
for the VAMOS++ magnetic spectrometer throughout the years of operation are
described. The achieved performances will be discussed in terms of resolution
of the reconstructed atomic mass number.

##  2 VAMOS++ spectrometer

The layout of the VAMOS++ spectrometer is presented in Fig. 1(a). The optical
elements of the VAMOS++ spectrometer [1] consist of two large aperture
magnetic quadrupoles (Q1, Q2) for focusing heavy ions vertically and
horizontally, respectively and a magnetic dipole (D) for dispersing heavy ions
horizontally. The Wien Filter (WF) placed in-between the two magnetic
quadrupoles and the magnetic dipole, is not used in the present work.

Due to the large angular and momentum acceptance of VAMOS++ the corresponding
ion-optics is highly non-linear, as will be demonstrated in Sec. 3, and a
direct measurement of horizontal ion coordinate at the dispersive focal plane
is insufficient to determine the magnetic rigidity $B\rho$ of the transmitted
heavy ions. Therefore, more complex measurements of final coordinates, using
the heavy ion tracking detectors, is necessary to perform ray-tracing and
determine the magnetic rigidity $B\rho$, the velocity vector $\vec{v}$ and the
trajectory length $l$. Typically, the focal plane detection system of VAMOS++
included two heavy ion tracking detectors, either secondary electron detectors
(SeD) [73, 1] or drift chambers [1, 2] , providing the measurement of
horizontal and vertical coordinates ($x_{f}$, $\theta_{f}$, $y_{f}$,
$\phi_{f}$), where the index $f$ refers to final focal plane coordinates.
Note, that the final focal plane coordinates were obtained on the image focal
plane placed $7600$ mm away from the target. Additionally, an ionization
chamber, either standalone [6] or combined with a plastic detector [7] or
silicon wall detectors [1, 2] were used used to provide the measurement of the
energy loss and total energy correlation $\Delta E-E_{tot}$. The stop for the
time-of-flight measurement was provided by the plastic detector [7], secondary
electron detectors (SeD) [73, 1] or multi-wire proportional counter (MWPC) [2,
6]. For the start for the time-of-flight ($t$) measurement, during the early
years of operation, the cyclotron radio-frequency was used [1]. Since 2011,
the multi-wire proportional counter, at the entrance of VAMOS++, (MWPC) [2]
and since 2016 the dual position sensitive MWPC telescope (DPS-MWPC) [74] were
used. The DPS-MWPC in addition to the timing signal provided the measurement
of initial horizontal and vertical coordinates at the target ($x_{i}$,
$\theta_{i}$, $y_{i}$, $\phi_{i}$), where the index $i$ refers to initial
coordinates at the target.

Figure 1:  (a) Schematic view of the elements of the VAMOS++ spectrometer. The
optical elements: two magnetic quadrupoles Q1 and Q2 for a vertical and
horizontal focussing, Wien Filter WF (unused) and magnetic dipole D for
horizontal dispersion. (b) Schematic view of the VAMOS++ detection system.
$\theta$ is the angle between the $z$-axis and the projection of the velocity
vector of the trajectory on the $xz$ plane (symmetry plane), $\phi$ refers to
the angle between the velocity vector and its projection on the $xz$ plane. A
dual position sensitive multi-wire proportional counter telescope (DPS-MWPC)
is placed at the entrance of the spectrometer. At the focal plane, a multi
wire proportional counter focal plane (MWPCFP), two drift chambers and
segmented ionization chambers. Two Mylar windows used to isolate the detection
gases are also shown.

Schematic view of the VAMOS++ detection system, used to obtain the data
presented in this paper [6], is shown in Fig. 1(b). The DPS-MWPC telescope was
placed at the entrance of VAMOS++. It consists of a pair of MWPCs, placed in a
common gas volume. Each of MWPCs provides a precise timing signal as well as
horizontal $(x)$ and vertical $(y)$ position. Therefore, a combination of all
position measurements results in a determination of the interaction position
at the target $(x_{i},y_{i})$ and the corresponding scattering angles
$(\theta_{i},\phi_{i})$. The focal plane detection system of VAMOS++ [2], has
an active area of $1000\times 150$ mm2 and consists of multi wire proportional
counter focal plane (MWPCFP), two drift chambers and segmented ionization
chamber. The MWPCFP, is 20-fold segmented in horizontal direction into
sections and provides the timing signal for the measurement the time-of-flight
($t$). Each drift chamber measures the horizontal and vertical position that
are used to obtain position $(x_{f},y_{f})$ and angles $(\theta_{f},\phi_{f})$
at the image focal plane.

The ionization chamber, is five-fold segmented in horizontal direction to
improve its counting rate capabilities. It is also eight-fold segmented in
depth. The atomic number $Z$ of the heavy ion is obtained from the correlation
of the measured $\Delta E$ and $E_{tot}$, see also Ref. [6].

The complete identification of the heavy ion using VAMOS++ spectrometer is
obtained from the following relations:

$\displaystyle v$
$\displaystyle=\frac{l}{t},\beta=\frac{v}{c},\gamma=\frac{1}{\sqrt{1-\beta^{2}}}$
$\displaystyle(A/q)$ $\displaystyle=\frac{B\rho}{3.107\cdot\beta\cdot\gamma}$
$\displaystyle A_{(E_{tot},\gamma)}$
$\displaystyle=\frac{E_{tot}}{931.494\cdot(\gamma-1)}$ $\displaystyle q_{int}$
$\displaystyle=\left\lfloor\dfrac{A_{(E_{tot},\gamma)}}{(A/q)}+0.5\right\rfloor$
$\displaystyle A$ $\displaystyle=(A/q)\cdot q_{int}$

where $v$ corresponds to the velocity in cm/ns, $l$ the trajectory length in
cm, $t$ the time-of-flight in ns, $c$ the speed of light, $B\rho$ the magnetic
rigidity in Tm, $E_{tot}$ the total energy in MeV, $A_{(E_{tot},\gamma)}$, is
the atomic mass measured from total energy and velocity with resolution
arising from the total energy resolution, $(A/q)$ the mass-over-charge ratio,
$q_{int}$ is the integer value of the atomic charge state and $\left\lfloor
x+0.5\right\rfloor$ is the nearest integer value of x, $A$ the atomic mass
number.

##  3 Ion-optics and ray tracing

The trajectory of an ion along the spectrometer, can be described using the
standard ion optics formalism using a six parameter vector
$\vec{t}=(x,\theta,y,\phi,l,\delta)$ defined relative to a reference
trajectory vector $\vec{t}_{0}=(0,0,0,0,l_{0},1)$ for an ion with the
reference magnetic rigidity $B\rho_{0}$. The parameters $x$ and $y$ correspond
to two transverse distances from the reference trajectory, see Fig. 1(b),
$\theta$ is the angle between the $z$-axis and the projection of the velocity
vector of the trajectory on the $xz$ plane (symmetry plane), $\phi$ refers to
the angle between the velocity vector and its projection on the $xz$ plane,
$\delta=B\rho/{B\rho_{0}}$ defines the relative magnetic rigidity and $l$ is
the path length from the target to the image focal plane.

Figure 2:  Aberrations of the VAMOS++ spectrometer: calculated angle
($\theta_{f}$) as a function of the positions ($x_{f}$) at the image focal
plane of VAMOS++ illustrating effects of aberrations in position and angle.
The figure shows the final positions of the trajectories for discrete relative
rigidity ($\delta=B\rho/B\rho_{0}$) varied in $2\%$ step and horizontal angle
$\theta_{i}$ by $10$ mrad. The arrow shows the increasing B$\rho$ direction.
The trajectories, whose vertical angle $\phi_{i}=0$ mrad are indicated by
triangles, $\phi_{i}=\pm 50$ mrad by circles and $\phi_{i}=\pm 100$ mrad by
squares. The arrow across the figure represents the positions of central rays
with $\theta_{i}=0$ mrad. The inset shows a zoom in the region of low $B\rho$.
The horizontal position of the physical segments of the MWPCFP located in the
focal plane is also indicated.

For ray-tracing purposes, a set of vectors in the initial phase space starting
at the origin $x_{i},y_{i}=0$ (assuming a point-like source) with well defined
angles ($\theta_{i_{k}},\phi_{i_{k}}$) and relative magnetic rigidity
($\delta_{i_{k}}$),
$\vec{t}_{i_{k}}=(0,\theta_{i_{k}},0,\phi_{i_{k}},0,\delta_{i_{k}})$, were
generated. These vectors exceed a maximum angular and momentum acceptance of
the spectrometer and are defined within the range $\Delta\theta,\Delta\phi=\pm
160$ mrad and $\Delta\delta=\pm 0.3$. The generated density of initial vectors
depends on the targeted resolution of the trajectory reconstruction
($\theta,\phi,\delta,l$) and the trajectory reconstruction method applied (see
below). The trajectories of the ions in VAMOS++ were calculated using ray-
tracing code ZGOUBI [75]. Realistic field descriptions for each magnetic
element were considered in the calculation by incorporating median plane field
maps issued from the 3D field maps generated using electromagnetic computation
code TOSCA [76]. The code ZGOUBI traces an ion through a system of magnetic
fields and calculates the final coordinates numerically by integrating the
equation of motion in a magnetic field. The final coordinates of the
trajectories are calculated on the image focal plane located $7600$ mm away
from the target. For each trajectory, its initial trajectory vector
trajectory, $\vec{t}_{i}=(0,\theta_{i},0,\phi_{i},0,\delta_{i})$ and final
trajectory vector
$\vec{t}_{f}=(x_{f},\theta_{f},y_{f},\phi_{f},l_{f},\delta_{f})$, are stored,
provided that the trajectory reaches the image plane of VAMOS++. It should be
noted that $\delta_{i}=\delta_{f}$.

The optics of the VAMOS spectrometer was designed to have the horizontal plane
as dispersive plane. In the first order, the dispersion is $\sim
2\text{cm}/\%$. The horizontal coordinates in the dispersive plane are
dominantly decoupled and thus independent of the vertical coordinates. This
can be seen in the Fig. 2, where the dependence on vertical coordinate
($\phi_{i}$) is limited. The horizontal position of the beam at the target, in
the first order, results in a proportional position displacement in the focal
plane ($\delta_{x_{i}}\propto\delta_{x_{f}}$). Therefore, in the first order,
the horizontal size of the beam spot of $\sim 2~{}mm$ would result in $\sim
1\text{\textperthousand}$ uncertainty in the magnetic rigidity.

To illustrate the non-linearity in the ion-optics of VAMOS++ the calculated
$\theta_{f}$ as a function of $x_{f}$ in the image focal plane of VAMOS++ is
shown in Fig. 2. Overall effect of aberrations in position and angle can be
seen in the figure. The impact on the horizontal image, due to the non-null
vertical coordinate $\phi_{i}$, is also indicated. The triangles corresponding
to the trajectories with $\phi_{i}=0$ mrad delimit a full $B\rho-\theta_{i}$
acceptance phase space of VAMOS++. The circles $\phi_{i}=\pm 50$ mrad and
squares $\phi_{i}=\pm 100$ mrad indicate the complex 3-dimensional
$(B\rho-\theta_{i}-\phi_{i})$ acceptance phase space. It is interesting to
notice, that in the region of the low relative $B\rho$

  * 1.

in the first order $B\rho$ is proportional to $x_{f}$,

  * 2.

the ions with large $|\phi_{i}|$ are accepted for small $|\theta_{i}|$ and are
progressively lost for larger $|\theta_{i}|$.

In the region of the high relative $B\rho$

  * 1.

in the first order $B\rho$ is proportional to $\theta_{f}$,

  * 2.

the ions with large $|\phi_{i}|$ are accepted for large $|\theta_{i}|$ and are
progressively lost for smaller $|\theta_{i}|$.

Finally, the difference in position between the corresponding points
$\phi_{i}=0,\pm 50,\pm 100$ mrad compared to the $B\rho$ step of $2\%$
illustrate the importance of the vertical coordinate in the determination of
$\delta B\rho$ resolution in the few per mille limit.

##  4 Trajectory reconstruction

The goal of the trajectory reconstruction methods described in this section is
to provide the high resolution vector
$\vec{t}_{rec}=(\theta_{i},\phi_{i},\delta_{i},l_{f})$ from measured final
or/and initial coordinates on an even-by-event basis. Additionally, the
reconstruction methods should be implemented for efficient on-line analysis of
data. In the following sections, three different trajectory reconstruction
methods will be described and their performances, applied to the same
experimental data set, will be discussed.

###  4.1 The Polynomial approach

In the early years of operation of VAMOS++, the polynomial expansion method
was used for the trajectory reconstruction. For each of the experimentally
detected ions, the measured coordinates in the image focal plane form a vector
of final coordinates $\vec{t}_{exp}=(x_{f},\theta_{f},y_{f},\phi_{f})$. Let
$m=\,\theta_{i},\,\phi_{i},\,\delta_{i},l_{f}$ denote four coordinates to be
reconstructed forming a vector
$\vec{t}_{rec}=(\theta_{i},\phi_{i},\delta_{i},l_{f})$. Each of the
coordinates $m$ can be expressed as independent non-linear function
$m=F_{m}(x_{f},\theta_{f},y_{f},\phi_{f})$. It used only the final focal plane
coordinates. Its $7^{th}$ order implementation is described in Ref. [1]. This
method was suitable for measurements involving the relatively light ions
$A<70$ [17, 18, 19]. For heavier ions the $10^{th}$ order polynomial expansion
method was introduced in $2008$. The $10^{th}$ order polynomial expansion
method will be detailed in this section.

The four non-linear inverse transfer functions $F_{m}$ can be expressed as a
$10^{th}$ order polynomial of four variables
$(x_{f},\theta_{f},y_{f},\phi_{f})$, measured exclusively in the image focal
plane of VAMOS++. The inverse transfer function $F_{m}$ can be expressed as:

$\begin{array}[]{l}F_{m}=\displaystyle\sum_{i,j,k,l=0}^{i+j+k+l=10}C_{m_{ijkl}}(x_{f})^{i}(\theta_{f})^{j}(y_{f})^{k}(\phi_{f})^{l}\end{array}$
(1)

where the coefficients $C_{m_{ijkl}}$ are related to the properties of the
inverse transfer map of the system. The unknown coefficients $C_{m_{ijkl}}$
can be determined numerically. A computer program was developed for this
purpose. It uses the set of trajectories computed by ZGOUBI, as described in
Sec. 3, and determines the best converged solution for $C_{m_{ijkl}}$ by
fitting the polynomial expression in an iterative procedure. The initial
trajectories have been calculated in steps of $d\theta_{i}=10$ mrad,
$d\phi_{i}=10$ mrad and $d\delta_{i}=5\times 10^{-3}$. It should be noted that
due to mid plane symmetry in the system, the coefficients $C_{{m}_{ijkl}}$ are
null for $\theta_{i}$, $\delta_{i}$ and $l_{f}$ for odd values of $k+l$ and
for $\phi_{i}$ for even values of $k+l$. The remaining number of non-null
coefficients for $\theta_{i}$, $\delta_{i}$ and $l_{f}$ is $511$ and for
$\phi_{i}$ is $490$. Once the $C_{m_{ijkl}}$ coefficients were determined,
they were used for the reconstruction algorithm to map the measured final
coordinates data on an event-by-event basis. Since the algorithm is
independent of any optics code once the coefficients are fixed, it can easily
be adopted in both on-line and offline event identification. It should be
noted that the VAMOS++ vertical magnification is of about $7$ in first order
and that the finite vertical beam spot size induces a large uncertainty of the
reconstructed values of $\theta_{i}$, $\delta_{i}$ and $l_{f}$. Therefore, the
reconstruction of $\theta_{i}$, $\delta_{i}$ and $l_{f}$ were obtained solely
from $F_{m}(x_{f},\theta_{f},0,0)$ while the reconstruction of $\phi_{i}$ was
obtained from $F_{m}(x_{f},\theta_{f},y_{f},\phi_{f})$.

###  4.2 Two-dimensional (2D) mapping

The $10^{th}$ order polynomial approach, despite its complexity, is unable to
account for all the details of the aberrations with a single set of
coefficients for the whole focal plane. This results in reduced resolution of
the reconstructed $\theta_{i}$, $\delta_{i}$ and $l_{f}$. Attempts to overcome
this limitation by sub-dividing the focal plane phase space for the polynomial
approach did not result in significant improvement in the atomic mass
resolution. Therefore, the two-dimensional (2D) mapping approach was
introduced in 2011 while increasing the size of the focal plane detection [2].

Let the coordinates $m=\theta_{i},\delta_{i},l_{f}$ be reconstructed using
only the final image focal plane coordinates $x_{f},\theta_{f}$ and the
reconstruction of the coordinate $\phi_{i}$ remain as in the polynomial
approach, Sec. 4.1. The values of each of the $m$ coordinates can be stored in
the two-dimensional array $M_{m}$, as unsigned $2$-byte integer, with the
numerical precision of $10^{-3}$, $1$ mrad and $1$ mm for $\delta_{i}$ ,
$\theta_{i}$ and $l_{f}$, respectively. The dimensions of each array $M_{m}$
were chosen to be $1100\times 550$ corresponding to steps of $1$ mm and $1$
mrad for the coordinates $x_{f}$ and $\theta_{f}$, respectively. Every of the
arrays $M_{m}$ occupies $\sim 1.15$ Mb of RAM. The initial trajectories have
been calculated in steps of $d\theta_{i}=0.05$ mrad, $d\phi_{i}=20$ mrad and
$d\delta_{i}=2\times 10^{-4}$, for initial coordinates. The step size have
been chosen to ensure a continuity of the arrays $M_{m}$. For each calculated
trajectory the $\delta_{i}$, $\theta_{i}$ and $l_{f}$ coordinates were stored
in the $M_{m}$ arrays, as a function of $x_{f}$ and $\theta_{f}$ coordinates,
as follows:

$\displaystyle k$ $\displaystyle=$ $\displaystyle\lfloor x_{f}+600+0.5\rfloor$
$\displaystyle l$ $\displaystyle=$
$\displaystyle\lfloor\theta_{f}+200+0.5\rfloor$ $\displaystyle
M_{\delta_{i}}[k][l]$ $\displaystyle=$ $\displaystyle\lfloor\delta_{i}\times
1000+0.5\rfloor$ $\displaystyle M_{\theta_{i}}[k][l]$ $\displaystyle=$
$\displaystyle\lfloor\theta_{i}+200+0.5\rfloor$ $\displaystyle
M_{l_{f}}[k][l]$ $\displaystyle=$ $\displaystyle\lfloor l_{f}+0.5\rfloor$

where: $x_{f}$ and $l_{f}$ are in mm and $\theta_{i}$ and $\theta_{f}$ are in
mrad. The inverse procedure to obtain reconstructed parameters from measured
($x_{f}$, $\theta_{f}$) is straightforward.

###  4.3 Four-dimensional (4D) mapping

The further improvement of the reconstruction quality was one of the reasons
to build the DPS-MWPC [74], shown in Fig. 1(b). DPS-MWPC placed at the
entrance of VAMOS++ provides the two-fold time measurement as well as two-fold
vertical and horizontal position leading thus the scattering angles
$\theta_{i},\phi_{i}$ and the interaction point at the target $x_{i},y_{i}$.
Typical resolutions of the reconstructed angles and positions on the target
were reported in Ref. [74] to be $\sigma=1.1~{}(1)$ mrad and
$\sigma=239~{}(30)\mu$m um respectively.

The four-dimensional (4D) mapping method, introduced in 2016, is an extension
of the 2D mapping, where in addition the final image plane coordinates
$x_{f},\theta_{f}$, the initial coordinates $\phi_{i},\theta_{i}$ will also be
used. The reconstructed coordinates will be $m=\delta_{i},l_{f}$. The values
of each of the m coordinates can be stored in the four-dimensional array
$M_{m}$, as unsigned $2$-byte integer, with a numerical precision of $10^{-3}$
and $1$ mm for $\delta_{i}$ and $l_{f}$, respectively The dimensions of each
array $M_{m}$ were chosen to be $960\times 450\times 180\times 260$, steps of
$1$ mm, $1$ mrad, $2$ mrad and $1$ mrad, for coordinates $x_{f}$,
$\theta_{f}$, $\phi_{i}$ and $\theta_{i}$, respectively. Both arrays $M_{m}$
have a total memory requirement of $\sim 75$ Gb. The initial trajectories have
been calculated in steps of $d\theta_{i}=0.05$ mrad, $d\phi_{i}=2$ mrad and
$d\delta_{i}=1\times 10^{-4}$, for initial coordinates. The step size has been
chosen such to guarantee a continuity of the arrays $M_{m}$. For each
calculated trajectory the $\delta_{i}$ and $l_{f}$ were stored in the $M_{m}$
arrays as a function of the $x_{f}$, $\theta_{f}$, $\phi_{i}$ and $\theta_{i}$
coordinates, as follows:

$\displaystyle k$ $\displaystyle=$ $\displaystyle\lfloor x_{f}+550+0.5\rfloor$
$\displaystyle l$ $\displaystyle=$
$\displaystyle\lfloor\theta_{f}+200+0.5\rfloor$ $\displaystyle m$
$\displaystyle=$ $\displaystyle\lfloor\phi_{i}/2+90+0.5\rfloor$ $\displaystyle
n$ $\displaystyle=$ $\displaystyle\lfloor\theta_{i}+130+0.5\rfloor$
$\displaystyle M_{\delta_{i}}[k][l][m][n]$ $\displaystyle=$
$\displaystyle\lfloor\delta_{i}\times 1000+0.5\rfloor$ $\displaystyle
M_{l_{f}}[k][l][m][n]$ $\displaystyle=$ $\displaystyle\lfloor
l_{f}+0.5\rfloor$

where: $x_{f}$ and $l_{f}$ are in mm and $\theta_{i}$, $\theta_{f}$ and
$\phi_{i}$ are in mrad. VAMOS++ acceptance phase space makes the $M_{m}$
arrays relatively sparse. The application of a zero suppression algorithm,
compressed row storage (CRS) [77] allows to reduce the total required memory
size from $\sim 75$ Gb to $\sim 1$ Gb. To further facilitate the memory usage,
the compressed arrays can be stored in a binary format on disk and read into a
permanent shared memory segment made available for several analysis
programs/processes at the same time.

It should be noted that, contrary to the polynomial and two-dimensional
methods, the four-dimensional method does not ensure that every input vector
($x_{f}$,$\theta_{f}$,$\phi_{i}$,$\theta_{i}$) results in a valid $M_{m}$.
This is due to the limited phase space of the spectrometer. This feature will
be discussed in the following section and outlook.

##  5 Experimental results

In this section, the different methods are applied to a common benchmark
experimental dataset and the performances in terms of reconstructed atomic
mass are compared and discussed. The experimental dataset arise from an
experiment performed at GANIL, where the fission fragments were produced in
fusion and transfer induced fission reactions using a 238U beam at the energy
of $6.2$ MeV/u on a 9Be target ($1.6$ and $5$ $\mu$m thick). A schematic view
of the experimental setup is shown in Fig. 1(a). The VAMOS++ spectrometer was
placed at $20^{\circ}$ relative to the beam axis. Further detail can be found
in Ref. [6].

The results will be first illustrated using the data obtained from the
physical section number $4$ of the MWPC detector of the focal plane. In Fig. 3
the correlation between the atomic charge state $q$ and the mass-over-charge
ratio $(A/q)$ is shown. Panels (a), (b) and (c) show the comparison of the
results of the polynomial approach, the two-dimensional and four-dimensional
mapping, respectively. The gradual improvement of the quality of the
identification can be seen. It should be noted that improved (A/q)
reconstruction translates in an improved charge state resolution. The spectra
of atomic mass number $A$ obtained from four-dimensional mapping method, in
red, is compared to that obtained from polynomial approach in blue in Fig.
3(d) and to that obtained using two-dimensional mapping in blue in Fig. 3(e).
It can be seen from the figures that atomic mass resolution ($\Delta
A_{\scriptscriptstyle FWHM}/A$) is significantly improved by using the four-
dimensional mapping. The resolution for $A=100$ is found to be
$5\text{\textperthousand}$ for the four-dimensional mapping compared to
$7.5\text{\textperthousand}$ for the polynomial approach and
$6\text{\textperthousand}$ for the two-dimensional mapping. This correspond
respectively to 50% and 20% improvement of the atomic mass resolution for this
region of the VAMOS++ focal plane.

Figure 3:  Atomic charge state $(q)$ as a function of mass-over charge ratio
$(A/q)$ for physical section number $4$ of the MWPCFP ($250mm<x_{f}<300mm$)
using different reconstruction methods a) polynomial approach, b) 2D mapping,
c) 4D mapping. The atomic mass number $A$ using 4D mapping in red compared to
d) polynomial approach and e) 2D mapping methods in blue.

It can be seen in Fig. 2 that the optical aberrations of VAMOS++ change as a
function of the magnetic rigidity $B\rho$ of the ion and thus also as a
function of the $x_{f}$ image plane coordinate. It has been observed that the
differences between the results of different reconstruction methods are the
largest in the region of low relative $B\rho$. In section number $10$ (see
Fig. 2), polynomial approach and two-dimensional mapping were found to be
equivalent and the four-dimensional resulted in a 10% improvement yielding to
an atomic mass resolution of $6\text{\textperthousand}$ for $A=100$. In
section number 16, the three approaches are found to give equivalent atomic
mass resolution. This can be explained by the fact that for higher magnetic
rigidity, the ions have in average higher velocity and the contribution of the
time of flight resolution in the atomic mass resolution dominates.

Figure 4:  a) Atomic mass number $A$ for events collected using a full image
focal plane of VAMOS++, in red 4D mapping method and in blue 2D mapping
method. b) Resolution of atomic mass number $\Delta A_{\scriptscriptstyle
FWHM}/A$ as a function of atomic mass number $A$ for the polynomial (green
triangles), two- (open blue squares) and four-dimensional (red circles)
reconstruction methods.

Further, the atomic mass number spectrum of fission fragments obtained for the
complete focal plane is shown in Fig. 4(a). The spectrum shown in red
correspond to the four-dimensional mapping reconstruction and the spectrum
shown in blue to that of the two-dimensional mapping. Figure 4(b) shows the
associated FWHM resolution obtained for atomic mass number $\Delta
A_{\scriptscriptstyle FWHM}/A$, in per mille (‰), as a function of $A$. A
clear improvement in width $\Delta A_{\scriptscriptstyle FWHM}$ of about $8\%$
for the four-dimensional mapping method, relative to other methods, can be
seen in the figure. The downward slope of the $\Delta A_{\scriptscriptstyle
FWHM}/A$ as a function of increasing $A$ results from nearly constant $\Delta
A_{\scriptscriptstyle FWHM}$.

Finally, the efficiency of the reconstruction method is 100% for polynomial
and two-dimensional methods. However, in the case of the four-dimensional
method, a typical efficiency of $\sim 97$% was obtained. This reduced
efficiency can be related to the highly constrained phase space arising from
the optics of the spectrometer. As an example, an extension of the beam on
target (typically $\Delta x_{\scriptscriptstyle FWHM}=1.2$ mm and $\Delta
y_{\scriptscriptstyle\scriptscriptstyle FWHM}=1.5$ mm [74]) will result in
some initial angles ($\theta_{i}$, $\phi_{i}$) residing out of the calculated
phase space assuming a point-like beam spot ($x_{i}=0$ and $y_{i}=0$). As a
consequence, for such cases, the four-dimensional method will not provide
reconstructed $\delta_{i}$ and $l_{f}$. Nevertheless, the two-dimensional
method can be used as a failover solution for these events at the cost of a
reduced resolution. The implementation of the reconstruction method accounting
for a size of the beam spot is considered for future work. It should be
however noted that, the use of such a method on an event-by-event basis, will
require an increase of the matrix size at least by a factor of $\sim 10$.

##  6 Summary and perspectives

The large angular and momentum acceptance magnetic spectrometer VAMOS++, is
particularly well suited for the studies of nuclear structure and reactions
dynamics at the beam energies near the Coulomb barrier. The main objective of
VAMOS++ is to provide on an event-by-event basis the isotopic identification
of the reaction products of interest. A high-resolution of the atomic mass
measurement requires the use of the trajectory reconstruction methods, due to
the highly non-linear ion optics of the spectrometer. Three trajectory
reconstruction methods have been developed and used in the past years

  1. 1.

polynomial approach, using $x_{f}$, $\theta_{f}$, $y_{f}$ and $\phi_{f}$
coordinates,

  2. 2.

two-dimensional (2D) mapping using $x_{f}$, $\theta_{f}$ coordinates,

  3. 3.

four-dimensional (4D) mapping $x_{f}$, $\theta_{f}$ and $\phi_{i}$ and
$\theta_{i}$ coordinates.

These methods makes use of the parameters derived from of set of trajectories
of the ions in VAMOS++ calculated using ray-tracing code ZGOUBI [75]. All
methods are fast and allow an efficient treatment of the experimental data in
an on-line and off-line analysis. The trajectory reconstruction method were
applied to a single data-set of fission fragments.

An improvement of about $8$% was obtained using 4D mapping method as compared
to the polynomial approach and two-dimensional mapping method, leading to
$\Delta A_{\scriptscriptstyle FWHM}/A$ ranging from $4.5$‰ ($\frac{1}{220}$)
for heaviest fragments to $9$‰ ($\frac{1}{110}$) for lightest fragments.

The trajectory reconstruction methods presented on this work are based on the
assumption that the beam spot size at the target is point-like, $x_{i}=0$ and
$y_{i}=0$, while the typical beam spot size is $\Delta x_{\scriptscriptstyle
FWHM}=1.2$ mm and $\Delta y_{\scriptscriptstyle\scriptscriptstyle FWHM}=1.5$
mm [74]. In future work, it is foreseen to extend the mapping method including
the event-by-event measurement of the interaction point on the target and
investigate its impact of the resolution of the reconstructed atomic mass
number.

## Acknowledgements

The authors thank B. Jacquot and D. Ramos for numerous fruitful discussions.

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