2.1 Model construction with GEANT4

GEANT4 is a Monte Carlo simulation software widely used in the field of nuclear physics and particle physics [38]. Because we concentrate on the simulation of the MWDC array, only the magnetic dipole with a detailed field map, the MWDC array, and the eTOF wall are constructed based on GEANT4, as shown in Fig. 2. The gap size is 180 (L)×220 (W)×100 (H) cm³, the volume size of the homogeneous field (1 % variance) is 100 (L)×110 (W)×80 (H) cm³. The other geometric information can be found in [20]. The field is calculated in parallel with the design of the dipole. The field map is shown in Fig. 3. Here, the z-direction is defined as the beam direction, and the y-direction is defined as the vertical direction, as shown in Fig. 1. The field in the vicinity of the first layer of the MWDC closest to the dipole is at the level of 500 G, which is about 10% of the central region and is ignored in the tracking procedure.

Fig. 2.Full-screen View

(Color online) Forward tracking system and dipole of the CEE constructed with GEANT4

Fig. 3Full-screen View

(Color online) Two-dimensional magnetic field distribution(a); the projection of the field of By at z=0(b); and projection of the field of By at x=0(c).

The event generator is designed based on the experimental data of heavy-ion collisions in the same energy region. As the output of the generator, the multiplicity and phase-space distribution of the particles, including pions, protons, and light charged particles, is linearly interpolated from the phase-space distribution of the same species measured in C+C, Ni+Ni, and Au+Au at beam energies from 400 to 1200 MeV/u [39]. Because the yields and width of the rapidity distribution vary monotonously with the beam energy and with of the system’s size, respectively, both longitudinally and transversely, the interpolation generates a nearly true phase distribution of the charged products in the CSR energy region. This method can circumvent the deficiency of the transport model, which usually overestimates the nucleon yield while underestimating the yield of clusters. An energy of 600 MeV/u is selected because it is the typical energy of the CEE in the HIRCL-CSR. The trajectories of the charged particles originating from the collision at the target position in the magnetic field are tracked. The responses of the MWDC and the eTOF wall are simulated by recording the energy deposition and firing position in the sensitive volume of the detector. The digitized data mainly include the ID of the fired wires, the drift radius, and the energy loss in the corresponding drift cell and the timing when the hit appears in the eTOF wall.

2.2 Track Finding and Reconstruction Based on Simulation Data

As shown in Fig. 4, the drift chamber used in this work consists of three layers of wires in different orientations. The first layer is vertical (X orientation), and the second and third layers are rotated clockwise and counterclockwise by 30 degrees (U and V orientations, respectively). The normal planes perpendicular to these three orientations are marked as the X_⊥, U_⊥, and V_⊥, respectively. The distance between two adjacent layers is 4 cm, and the size of one drift cell is 1 cm × 1 cm.

Fig. 4Full-screen View

(Color online) XUV sense wire direction of multi-wire drift chamber. The planes perpendicular to the X, U, and V wires are denoted as X_⊥, U_⊥, and V_⊥, respectively

Based on the simulation data, an algorithm for track finding and track reconstruction was developed. In general, track finding and reconstruction can be processed in the following steps depending on the specific geometric characteristics of the detector used [40].

1. Hit information recognition: The hit spatial coordinates and signal heights are extracted from the original digitized fire information based on the geometric database of the detector.

2. Local pattern recognition: In each detector, various hits that may belong to the same track are recognized.

3. Global pattern recognition: The track elements found in the local pattern recognition stage are combined into candidate tracks.

4. Track fitting: The candidate tracks are fitted. If the fitting quality meets our accuracy requirement, it is considered to be a real track; otherwise, the candidate is rejected and released to the data for the next iteration of finding and fitting.

Figure 5a describes how the hit is recognized in the local pattern recognition stage. Three different directions of the fired wires of the same drift chamber are considered to form an isosceles triangle using the geometry of the detector. The red lines in the left panel represent the wire recording a fire, and the black dots represent the approximate original position of the hit on the wire plane. If the base length d of this triangle is less than a specific threshold value (65 mm in this work), the three wires are considered to form a hit. This value is related to the MWDC parameters. We used tracks of simulation data to accumulate this value, which can cover more than 95% of the events. Following this recognition, several hits can be found in each single drift chamber. Figure 5(b) presents a schematic of the global pattern recognition procedure. The red dotted line in the figure represents the plane in a certain direction in each drift chamber, the black dot (the geometric center of the triangle) represents the hit found in the local pattern recognition in the corresponding chamber. We fit each combination (linear fit) of two hits from the first two chambers in a given plane and prolong the fit to the third plane to match the third hit. As shown by the solid black line, if the distance of the hit in the third chamber to the intersection is within a certain window, the three hits are considered to form a candidate track; otherwise the finding procedure goes to the next iteration of hit candidate recognition. At this stage, the algorithm aims at approximately assigning the hits to a certain track; thus, the drift time and drift length information are not used, and only the geometric characteristics of the fired wire are considered.

Fig. 5Full-screen View

(Color online) Schematic view of the local (a) and global(b) pattern recognition.

In the fitting phase, the fitting process is divided into two steps, which are fitted using the least squares method, and the drift length is carefully considered.

1. Two-dimensional fitting: This step is performed to fit the track in each U_⊥, X_⊥, and V_⊥ plane. Fig.6 depicts the three drift signals in a single plane. Considering the X_⊥ plane as an example, fitting the candidate track in this plane is equivalent to finding the common tangent line of three circles. This procedure can be derived analytically, assuming that the line is approximately perpendicular to the sense wire plane. Under this assumption, if we construct

Fig. 6Full-screen View

Two-dimensional fitting to three circles with the radius being the drift length recorded on each individual wire.

the fitting procedure actually searches the values in the vicinity of minimum χ². Here, the Z’ direction refers to the direction perpendicular to the MWDC plane. The slope k and the intercept b, assuming they are small enough, can be deduced by

where ri (i=1,2,3) is the radius of each circle, and ($x_i^{\text{'}}$,$z_i^{\text{'}}$) (i=1,2,3) are the coordinates of the center of each circle in the local coordinate system fixed to the MWDC plane. It should be pointed out that the assumption that the slope and the intercept of the common tangent are small enough can always be met by coordinate transformation.

After the two-dimensional fitting procedure in all three planes, the fitted lines are used as the initial condition in the three-dimensional fitting, as described below.

2. Three-dimensional fitting: The function of the line in three-dimensional space is fitted to obtain the real track. Here, we assume that the residual magnetic fields in the locations of the MWDCs are negligible and the track forms a straight line penetrating the MWDCs, which can be expressed as [41]

In the ideal case, this track is identical to the three lines given by the two-dimension fitting in the three planes U_⊥, X_⊥, and V_⊥. In reality, however, the three lines do not necessarily coincide perfectly; thus, the real track is again a fitted line in the three-dimensional coordinate space minimizing

where αi is the angle between the wire and the vertical direction. Taking the partial derivative of χ² with respect to the parameters a, b, c, d and ignoring the high-order small quantity, one obtains

(5)

where a, b, c, d are the parameters of the line in which the track is located; $x_i^{\text{'}}$ and $z_i^{\text{'}}$ (i=1-9) are the intersection coordinates of the perpendicular line and the circle, respectively, where the perpendicular line extends from the point where the wire is located to the line obtained by two-dimensional fitting, and the circle is centered at the location of the fired wire and has a radius equal to the drift length, as depicted in Fig. 7.

Fig. 7Full-screen View

The position relationship of a fitted track and a drift circle. The cross point ($x_i^{\text{'}}$,$z_i^{\text{'}}$) is marked.

Figure 8 presents the distribution of the tracking reside, defined as the difference between the drift length and the distance from the wire to the track in each X_⊥, U_⊥, and V_⊥ plane. The results are accumulated for 8985 tracks containing all the light charged particles from the simulated data. Through a double Gaussian fitting, we obtain an average root mean square (RMS) of the residual distribution of 29 μm. It should be pointed out that this result includes the effects of multiple scattering in air and other materials of the detectors, treated automatically in the simulation. The uncertainties caused by the detection are not considered. To check the track finding efficiency in a high multiplicity environment, we simulated 500 events randomly with each event containing ten tracks. It was found that, with the current track finding procedure, more than 90 % of the tracks can be reconstructed, and the residue is not deteriorated.

Fig. 8Full-screen View

(Color online) Tracking reside distribution in GEANT4 simulation and reconstruction. The blue dotted curves are the two Gaussian fits and the solid red curve is the sum.

2.3 Momentum reconstruction and particle identification

The final step is the reconstruction of the particle momentum with the magnetic field. We developed a track reversal algorithm that does not require an ideal homogeneity of the field. The particle is reversely incident into the magnetic field along the reconstructed linear track. The deflection in the magnetic field is calculated step by step until the track passes through the transverse plane o-xy at the target position. The ratio of momentum to charge, p/q (the charge is fixed at 1 in this work), is iteratively adjusted according to the position of the particle relative to the target. The iteration is ceased when the particle hits the target point within a given precision, taken as the beam spot size (10 mm in this work). In order to make achieve fast iteration and a reasonable momentum, the initial value of the momentum is calculated analytically by assuming that the magnetic field is uniform in the area of the dipole in the y-direction. Figure 9 shows the projection of the track in the XZ plane. The red dot represents the target point; the black frame is the magnetic field region; the solid red line is the reconstructed track in the form of a straight line from the MWDC array measurement, and the red dotted line is the extension of the track into the magnetic field. The solid blue and green lines represent the position where the track is reversed to the o-xy plane with a particle momentum being too small or too large, respectively.

Fig. 9Full-screen View

(Color online) Schematic view of the track reversal algorithm in momentum reconstruction.

In the case where the deflection angle ∂θ of each step is sufficiently small, a sufficiently small step size can be obtained, as shown in Fig. 10. The magnetic field will change the position and the direction of the momentum of the moving charged particles. Considering the momentum in the x-direction and the magnetic field in the y-direction as an example, the displacement in the z-direction is

Fig. 10Full-screen View

(Color online) The momentum in the x direction produces a displacement and the momentum direction changes under the action of the magnetic field B_y in the y-direction.

and the displacement in the x-direction is

the momentum change in the z-direction is

The displacement and change in the direction of the momentum produced by the momentum in other directions under the action of the magnetic field can be obtained similarly. The displacement and change in the direction of the momentum are accumulated to obtain a change in the position and direction of the movement in one sufficiently small step.

In the case of a detector array with a spatial resolution of 29 μm, the beam spot size is assumed to be 10 mm. The difference between the reconstructed momentum and the Monte-Carlo truth track momentum is shown in Fig. 11. Here, 300 μm hit position residues and a TOF resolution of 60 ps are incorporated in the simulation. It is shown that the final momentum resolution is 4 %. Using this momentum resolution and taking into account a TOF resolution of about 60 ps, one can arrive at a clear particle identification, as shown in Fig. 12, where the events of Sn+Sn at 600 MeV/u are used.

Fig. 11Full-screen View

(Color online) Momentum resolution defined by the error between the Monte-Carlo truth input and the reconstruction value. The blue dotted curves are the two Gaussian fits and the solid red curve is the sum.

Fig. 12Full-screen View

(Color online) Particle identification achieved in GEANT4 simulation, considering 300 μm tracking residues and a TOF resolution of 60 ps. [20]

3.1 Experimental situation

In order to further verify the feasibility of the forward tracking system of the CEE, a prototype system was built. As shown in Fig. 13, the system consists of three drift chambers with a sensitive area of 4040 cm². The distance between two contiguous signal wires of the same layer is 20 mm, which is also the distance between two contiguous layers. The distance between two contiguous drift chambers is 199 mm. Two plastic scintillators with sizes of 500×150×15 mm³ in front of and behind the drift chamber array, respectively, are installed to provide the start timing of the track and to trigger the data acquisition. The entire unit is installed on a rigid aluminum frame.

Fig. 13Full-screen View

(Color online) Schematic view of the prototype system (a), and the prototype system mounted for the beam test (b).

The working gas of the drift chamber is a mixture of argon (90%) and carbon dioxide(10%). The preamplifier is designed based on a current-amplification scheme [42], and the data acquisition system is based on Flash-ADC [43] requiring a trigger signal from the coincidence of the four PMT signals of the scintillators.

The beam test of the prototype was carried out in the E3 test beam at the Institute of High Energy Physics, Chinese Academy of Science. The incident beam is generated by a 2.5 GeV primary electron beam hitting a Beryllium target. The momentum of the secondary beams, which are mixed mainly with protons, pions, and very few mixed events, are selected by a magnetic dipole behind the primary target station. The beam used in the prototype test has a momentum of 400 MeV/c or 500 MeV/c, and a count rate of about 100/min.

3.2 Analysis and discussion of results

Figure 14 presents the correlation between the TOF and the energy loss measured by the two scintillators, with only a few components presented. The TOF is recorded by the two scintillators. Several components are visible in the plot. The two main peaks are produced by protons and pions. Pions are situated in the region with very small energy loss for the minimum ionization feature. Figure 15 depicts the correlations between the energies in the two plastic scintillators. Pions and protons with similar P/Q can be seen in the plot, as well as the two-proton events with very small P/Q. The random coincidence between the single proton and double protons are visible. By imposing a window cut on the two plots, the pure single proton events can be selected. As a result, with a total of 155525 events, we obtained 102127 single proton events. With these selected one-proton events, one can count the multiplicity of the fired wires M_w in each layer, as shown in Fig. 16. Clearly, with the clean one-proton events, the multiplicity distribution peaks at 1. An extension to M_w>1 is visible, indicating that, in some layers, the neighboring wires may also be fired simultaneously. From the multiplicity distribution, with a threshold of M_w>=1, one can compute the efficiency of each layer, as summarized in Table 1. The average efficiency is between 70% and 80%, with an exception in the V1 wire plane. The firing efficiency is not high enough for the protons at this energy, because the working voltage is lowered to avoid the high leakage current arising from the high air humidity present at the time of the experiment (conducted in summer). In addition, one wire of the V1 layer is out of order, causing an even lower efficiency in this layer, as indicated in the third column in Table 1.

Fig. 14Full-screen View

(Color online) Scattering plot of the TOF signals and the energy loss in the scintillators.

Fig. 15Full-screen View

(Color online) Scattering plot of the energy loss signals in the two scintillators.

Fig. 16Full-screen View

(Color online) Fired wire multiplicity distribution in the 9 sense wire planes.

Unlike in the GEANT4 simulations, in the real test, the firing efficiency of each layer is less than 100%. Thus, the algorithm based on the simulation data has to be improved to overcome the inefficiency of the chambers. In particular, in the hit recognition, it is no longer required to have three wires being fired. The incident particle may only fire one or two wires in a certain chamber. Clearly, in order to construct a real track in three-dimensional space, the minimum hit multiplicity is 2, and the minimum wire multiplicity is 5. The multiplicity refers to the number of firing wires included in the candidate track. Using the improved searching method, the search efficiency was higher than 99%.

Figure 17 depicts the collinear characteristics of the hits in the three chambers. Panel (a) displays the schematic pattern of the fired wires viewed from different planes, e.g., the X_⊥ plane. The blue dotted lines represent the sense wire planes X of the three chambers, the red dots represent the fired wires in the candidate track, and the solid black lines represent the two track segments connecting fired wires in the first and second two chambers. It can be expected that the three fired wires viewed in the normal plane X_⊥, shall lie close to a straight line, and the angle between the two track segments satisfies α₁₂ 180°. The panel (b) of Fig. 17 presents the distribution of α₁₂. It is shown that the main peak appears in the vicinity of 180° with some discrete values on the left side. This is a natural broadening effect because the broadening of the beam spot is larger than the size of one single drift cell, and the drift length was not applied in this plot.

Fig. 17Full-screen View

(Color online) (a) Fired wire pattern in the perpendicular planes X_⊥, U_⊥, and V_⊥; (b) Angle between the two track segments shown in (a).

The tracking fitting algorithm for the data is the same as that in the simulation. However, in the experiment, the processing of the signal acquired by the flash ADC system has to be treated carefully. Digital signal filtering has been introduced to enhance the signal-to-noise ratio. The details of the technique are described in [43]. Furthermore, the relationship between the drift length L_d and the drift time T_d, which is determined by the time difference between the arrival of the signals of the drift chamber and scintillators, has to be calibrated before the fitting procedure. The calibration method is described in [31].

The results in Fig. 17 suggest that the track patterns are accurate and further track fitting can be processed. Because there are only three chambers and nine layers of sense wires, in order to access the tracking quality with the maximum redundancy, the tracks with nine wires fired, with one wire on each plane, are further selected to calculate the tracking residue, as shown in Fig. 18. Using a double Gaussian fit for the main peaks between ±1 mm, the weighted average of the standard deviations of the two Gaussian peaks is found to be 295 μm.

Fig. 18Full-screen View

(Color online) Track residue distribution for the tracks with nine wires fired. The blue dotted curves are the two Gaussian fits and the solid red curve is the sum

The uncertainty of 295 μm results from many sources of uncertainty, including those from the TOF broadening of the beam, the space-time relation, the signal distortion, and the mechanical accuracy in the assembly of the chamber. Except for the mechanical accuracy, all other terms, summing up to 150 μm [44], can be evaluated in the test of electronics and data analysis. For the mechanic accuracy, it is found that, unlike in the calibration test where the displacement is reasonably small and can be corrected, the relative mechanical displacement of the wire planes are comparable with the residue. It is also found that the calibration of the displacement does not bring significant improvement. Therefore, it is very important to improve the mechanical assembling accuracy in the future mass production of chambers.

The redundancy of tracking is an important factor. Because there are three chambers in a row of the MWDC array prototype, there should be one layer of sense wire plane for a given orientation (X, U, or V) in each chamber to reduce the total number of electronic devices by 50%. The ability of tracking may not be influenced as predicted by the simulation. However, in real experiments, the redundancy of the tracking is reduced due to the non-perfectness of the detector or the signal, particularly for the tracks with a fired wire multiplicity of less than nine. Thus, we suggest resuming, as traditionally done, the design of two sense wire planes displaced by half the size of the drift cell for each direction of sense wire in each chamber.