Towards energy discretization for muon scattering tomography in GEANT4 simulations: A discrete probabilistic approach

In this study, by attempting to eliminate the disadvantageous complexity of the existing particle generators, we present a discrete probabilistic scheme adapted for the discrete energy spectra in the GEANT4 simulations. In our multi-binned approach, we initially compute the discrete probabilities for each energy bin, the number of which is flexible depending on the computational goal, and we solely satisfy the imperative condition that requires the sum of the discrete probabilities to be the unity. Regarding the implementation in GEANT4, we construct a one-dimensional probability grid that consists of sub-cells equaling the number of the energy bin, and each cell represents the discrete probability of each energy bin by fulfilling the unity condition. Through uniformly generating random numbers between 0 and 1, we assign the discrete energy in accordance with the associated generated random number that corresponds to a specific cell in the probability grid. This probabilistic methodology does not only permits us to discretize the continuous energy spectra based on the Monte Carlo generators, but it also gives a unique access to utilize the experimental energy spectra measured at the distinct particle flux values. Ergo, we initially perform our simulations by discretizing the muon energy spectrum acquired via the CRY generator over the energy interval between 0 and 8 GeV along with the measurements from the BESS spectrometer and we determine the average scattering angle, the root-mean-square of the scattering angle, and the number of the muon absorption by using a series of slabs consisting of aluminum, copper, iron, lead, and uranium. Eventually, we express a computational strategy in the GEANT4 simulations that grants us the ability to verify as well as to modify the energy spectrum depending on the nature of the information source in addition to the exceptional tracking speed.


Introduction
The wide angular distribution [1] of the incoming cosmic ray muons in connection with either incident angle or azimuthal angle is a challenging trait that leads to a drastic particle loss in the course of parametric computations through the GEANT4 [2] simulations associated with the muon tomography [3][4][5] since the tomographic configurations as well as the target geometries also influence the processable number of the detected particles apart from the generation strategies.To further detail, the basic parameters such as the scattering angle, the particle displacement, and the particle absorption owing to the volume-of-interest (VOI) de facto dictate the particle penetration through the multiple sections of the tomographic setup in addition to the VOI.Hence, a number of the loss cases notably come into effect unless the calculation conditions are fulfilled, and not only the computation statistics as well as the numerical outcomes but the initial assumptions like the energy spectrum are also perturbed since the VOI accepts a significantly lower number of particles in the instance of the substantial particle loss.While a number of source biasing techniques [6] are offered by MCNP6 [7,8] in the black box format under the class of non-analogue Monte Carlo simulations, the GEANT4 simulations are usually constrained to the existing particle generators or the general particle source (GPS) unless G4ParticleGun is favored.Motivated by the excessive particle loss and its effect on the computation time as well as the characteristic parameters identified in the muon tomography, we set forth in the present study a scheme that is hinged on the particle generation through the planar restriction by means of the vectorial construction over our tomographic setup consisting of plastic scintillators manufactured from polyvinyl toluene with the dimensions of 100×0.4×100cm 3 .This study is organized as follows.In section 2, we elucidate our methodology based on the restrictive planes and we express our characteristic parameters as well as our simulation features in section 3.While we disclose our simulation outcomes in section 4, we draw our conclusions in section 5.

Generation via planar restriction
To begin with, we principally exhibit two planar restrictive schemes to be adapted in GEANT4 as illustrated in Fig. 1 where (a) shows the particle generation from a fixed point as well as the direction restriction by means of a restrictive pseudo-plane, whereas (b) demonstrates the randomly picked up particles from a generative plane, the directions of which are projected into a similar restrictive plane.In order to practically outline the present methodology that is initially described in Fig. 1(a), the particle location in cm on the central point at height=85 cm is listed as written in Subsequently, the confined location in cm on any restrictive plane of 2L × 2D cm 2 is noted as shown in Here, G4UniformRand() is the uniform random number generator between 0 and 1, which is pre-defined in GEANT4.Then, by constructing a vector from the generative point to the restrictive plane, we obtain Thus, the selective momentum direction, i.e.P = (P x , P y , P z ), is The latter scheme that assumes a planar generation as delineated in Fig. 1(b) entails particle locations in cm on the generative plane of 2L × 2D cm 2 as written in As performed in Eq. 2 for the previous scheme, the limited locations in cm on any restrictive plane of 2L × 2D cm 2 are selected from Additionally, via a vector construction between two planes, we acquire anew Therefore, the selective momentum direction denoted by P = (P x , P y , P z ) is again The initial particle positions and the selective momentum directions are incorporated by using G4ParticleGun.
The simulation previews through both the restrictive schemes are displayed in Fig. 2 where (a) indicates the particles generated from a fixed point, while (b) presents the randomly generated particles from a fixed plane.It is worth mentioning that neither generation points/planes nor restrictive planes are subject to any limitation in terms of shape, size, or location since our recent concept is preferred in the first instance for the sake of simplicity.On top of this, it is also possible to favor different distributions especially already implemented in GEANT4, e.g.Gauss or Poisson distribution depending on the envisaged application.

Characteristic parameters and simulation setup
Before getting down to test our schemes, we express our characteristic parameters to be computed in the wake of the GEANT4 simulations.The average scattering angle due to the target volume and its standard deviation over N number of the non-absorbed/non-decayed muons is determined as expressed in [9][10][11] Additionally, the root-mean-square (RMS) of the scattering angle over N number of the non-absorbed/nondecayed muons is calculated by using the following expression: Along with the scattering angle, we squarely track the number of the absorbed muons within the VOI as denoted in # In−target Capture = # of muMinusCaptureAtRest in VOI (11) Last but not least, we define the particle loss entitled off-target loss as follows where # Out−scattering is the number of the scattered muons from the VOI by leaking out of the tomographic device, # Decay is the negligible number of the decayed muons into electrons/positrons, # Off−target Capture is the insignificant number of the absorbed muons outside the VOI, and # Initial Deflection is the number of muons that miss the VOI only in the case of the wide beams, which occasionally occurs due to the barriers before the VOI despite the initial restricted orientation to the VOI boundary, i.e. the tiny deflection owing to the detector layers.Our simulation features are summarized in Table 1, and we use a 80-bin discrete muon energy spectrum extracted from the CRY generator [12] between 0 and 8 GeV.The muon tracking is accomplished by G4Step, and the recorded hit positions on the detector layers are post-processed at the hand of a Python script.

Simulation outcomes
We asses our methodology over our tomographic configuration described in Fig. 1(a)-(b) and we select our set of materials and the VOI geometry in accordance with another study [13] dedicated to the muon tomography where the material list consists of aluminum, copper, iron, lead, and uranium, and the target geometry is composed of a rectangular prism with the dimensions of 40×10×40 cm 3 .As indicated in Fig. 1, we contrast three restrictive planes labeled as a, b, and c that are placed atop the VOI, amidst the VOI, and beneath the VOI, respectively.We commence with the first scheme that is based on the point -plane generation, and the simulation outcomes by using restrictive plane a are listed in Table 2.
Table 2: Point -plane scheme, restrictive plane a, thickness=10 cm.As shown in Table 2, the computed parameters including the particle loss show a characteristic tendency depending on the atomic number as well as the material density for a fixed thickness.Although the muon beam is already directed to the VOI boundary even in the case of restrictive plane a, which leads to an immoderate reduction in the particle loss compared to the conventional approaches, a remarkable number of the loss events in agreement with the intrinsic properties of the target material are still observed.
Table 3: Point -plane scheme, restrictive plane b, thickness=10 cm.In order to see the positional effect of the planar restriction, the simulation outcomes from restrictive plane b are tabulated in Table 3.In comparison with Table 2, we observe that the characteristic parameters except the particle loss slightly change when the muon beam is narrowed by using restrictive plane b; however, the particle loss manifests a minimum reduction of 31% as opposed to restrictive plane a. Whereas restrictive plane b is capable of diminishing the particle loss by a factor of order in certain cases, we still notice that the particle loss remains distinctive among the simulated materials.
By using restrictive plane c, we further decrease the incident angle and we obtain the simulation results as written down in Table 4.In comparison with Table 3, restrictive plane c yields a minuscule change in terms of the characteristic parameters containing the particle loss, which also means that the variation rate of the characteristic parameters is expected to be insignificant beyond restrictive plane c.It is noteworthy to mention that a partial transition from the particle loss to the particle absorption is perceptible according to Tables 2-4 especially if the VOI material is a potent absorber since the low-energy muons that lead to the particle loss in the wide beams typically have the absorption potential when interacting with the VOI material in the narrow beams, which also means that a certain portion o the particle loss is converted into the particle absorption in the VOI material towards restrictive plane c.In the next step, we continue with the plane -plane scheme, and Table 5 lists the simulation outcomes for restrictive plane a.In spite of the schematic change, we see that the characteristic parameters excluding the particle loss do not exhibit a significant difference.On the other hand, the particle loss via restrictive plane a within the plane -plane interplay results in the elevated values as displayed in Table 5 in contrast to Tables 2-4.
Table 6: Plane -plane scheme, restrictive plane b, thickness=10 cm.So as to demonstrate the impact of the spatial change in the planar restriction for this scheme, the simulation results via restrictive plane b are tabulated in Table 6, and we experience a similar trend compared to the pointplane scheme that induces a drastic diminution in the particle loss along with the tiny variations in the rest of the characteristic parameters.As a means to complete our quantitative investigation for the plane -plane scheme, the simulation results for restrictive plane c are listed in Table 7, and we face a close trend as opposed to Table 4, which also means that the reduction rate in the particle loss is moderated together with the very minor variations in the remaining characteristic parameters.In the long run, our last simulations are devoted to investigate the thickness effect by solely using restrictive plane b since we aim at optimizing the particle loss with an ideal angular acceptance.Thus, Table 8 shows the characteristic parameters that are acquired by means of the point -plane scheme as well as restrictive plane b for a thickness of 40 cm with the same material group.From Table 8, we numerically demonstrate that all the characteristic parameters increase as a function of thickness, and we find the most notable rise in the particle absorption.Finally, Table 9 lists the simulation results through the plane-plane scheme for the same thickness, and we see that the latter scheme is not significantly different from the initial scheme with regard to the characteristic parameters omitting a higher number of the particle loss.

Conclusion
All in all, by setting out our restrictive generation scheme, we optimize the particle loss by keeping an angular disparity that is directly dependent on the VOI geometry as well as the vertical position of the restrictive plane for a tomographic system of a finite size.Upon our simulation outcomes, we show that the particle generation by means of restrictive planes is an effective strategy that is flexible towards a variety of computational objectives in GEANT4.Into the bargain, we explicitly observe that the off-target loss is a characteristic parameter that varies in an ascending order from aluminum to uranium.

Figure 2 :
Figure 2: Simulation previews by using restrictive plane b for copper in GEANT4 (a) point -plane scheme and (b) plane -plane scheme.