Transfer and multi-nucleon transfer reactions are essential tools for probing nuclear structure and reaction dynamics, requiring precise determination of the identity, energy, and emission angles of reaction products. The increasing granularity of modern silicon telescope arrays enhances experimental capabilities but challenges detector calibration, as conventional channel-by-channel approaches become inefficient and difficult to scale. In this work, we present a fully automated, physics-informed calibration framework based on neural networks, specifically designed for highly segmented silicon detector arrays. The method formulates calibration as a global optimization problem, in which detector gains and geometrical corrections are determined simultaneously by minimizing the width of the reconstructed excitation energy under two-body kinematics constraints. The approach relies exclusively on experimental data and well-established physical principles, without requiring explicit modeling of detector response. A distinctive feature is the use of multiple neural network sub-models sharing a common loss function with embedded physics constraints, enabling coherent and self-consistent calibration across all detector channels. This strategy ensures scalability, robustness, and reproducibility, making it particularly suitable for next-generation detector systems with increasing complexity. The performance of the method is demonstrated using experimental data from the Particle-Identification Silicon-Telescope Array (PISTA) in high-resolution fission studies in inverse kinematics. The results show excellent agreement with theoretical kinematics, high-quality particle identification, and a significant improvement in calibration efficiency. The proposed framework provides a general and adaptable solution for the calibration of complex detector systems in modern nuclear physics experiments.
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Figure 1: PISTA array and experimental setup: (a) The schematic front view of the segmentation of the trapezoidal telescope of PISTA located at the azimuthal angle ϕ=90∘ . This diagram illustrates 91 horizontal strips of the ΔE stage and 57 vertical strips of the Eres stage. In the reference frame of the telescope, the z axis aligns with the telescope’s normal. The origin is positioned at 105.5 mm, and the z axis intersects the detector’s plane at the centers of the middle horizontal and vertical strips. The red lines represent the coordinates for the constant polar laboratory angle θ in increments of 5 degrees, while the blue lines indicate those for the constant azimuthal angle ϕ in increments of 10 degrees. (b) The schematic top view of the experimental setup. The 12C target is positioned in the xy plane at the origin of the coordinate system, while the 238U beam axis aligns with the z axis. The PISTA telescopes encircle the target at a distance of 105.5 mm and a polar angle of θ=45∘ . The corresponding telescope’s normal is indicated by dashed line. Eight telescopes are placed at the azimuthal angle ϕ every 45∘ . The dual position-sensitive Multi-Wire Proportional Chamber (DPS-MWPC) placed at the entrance of the VAMOS++ spectrometer, rotated in the xz plane by 20∘ relative to the beam axis is also shown; its normal is indicated by dashed line. Inverse kinematics is exemplified by the 238 U+ 12 C reaction. The target-like 12 C is detected in the PISTA array, providing its energy and emission angle, while the fission of the projectile-like 238 U produces two fission fragments, of which one ( FF1 ) is detected and identified in the VAMOS++ spectrometer. The measured properties of the detected target-like product are used to reconstruct the reaction kinematics.
Figure 2: Initial identification charts: Two-dimensional energy correlation plots between the raw energy loss ( ΔE ) and the raw residual energy ( Eres ), resulting in correlated bands for each atomic number and atomic mass number. (a) Complete ΔE and Eres correlation, including the elements: He, Li, Be, B, C, N, and O, registered by the telescope located at ϕ=135∘ . (b) Zoom on the isotopes of C, where only the data from every sixth ΔE strip are shown to highlight the ΔE energy range registered by the individual strips.
Figure 3: PISTA neural networks: Neural networks model consisting of three independent sub-models trained to provide ΔE and Eres gains along with telescope placement corrections in terms of the vertical shift δy and the polar angle shift δθ . The architecture of each of the sub-models is given in the form Nl×Nu+Nou comprising Nl layers and Nu units (neurons) per layer, followed by Nou output units. The inputs of the model are indicated in red, the inputs to the loss function are indicated in blue and the gradient’s feedback obtained for a loss are indicated in green.
Figure 4: Algorithmic structure of the physics-informed loss function. The network outputs are mapped to physical parameters, followed by energy calibration. The processing then bifurcates depending on the event type: experimental events undergo full kinematic reconstruction, while calibration events enforce an energy anchor. Both branches contribute to the final RMSD loss.
Figure 5: PISTA neural network calibration results: Two-dimensional correlation plots: (a) The energy ( Etot=ΔE+Eres ) as a function of the polar angle ( θ ) for elastic scattering of 238 U on 12 C recorded in the telescope situated at ϕ=135∘ . The solid red line represents the corresponding calculated correlation, leading to an excitation energy E∗=0 MeV. (b) The reconstructed excitation energy ( E∗ ) as a function of the polar angle ( θ ) for elastic scattering recorded in the telescope situated at ϕ=180∘ . One-dimensional spectra of the reconstructed excitation energy ( E∗ ): (c) For three telescopes located at ϕ=90∘,135∘ , and 180∘ and (d) For each of the three telescopes, located at ϕ=90∘ (black), 135∘ (blue), and 180∘ (red), normalized to the same peak height.
Figure 6: Particle identification: Two-dimensional energy correlation plots between the energy loss ( ΔE cos( α )) and the residual energy ( Eres ) for three telescopes located at ϕ=90∘,135∘ , and 180∘ : (a) The spectra without requiring a coincidence with VAMOS++ and (b) The spectrum obtained in coincidence with any ion detected in VAMOS++ spectrometer. The angle ( α ) is the angle between the particle’s trajectory and the detector’s normal. One-dimensional spectrum of the particle identification (c) For three telescopes located at ϕ=90∘,135∘ , and 180∘ and (d) For each of the three telescopes, located at ϕ=90∘ (black), 135∘ (blue), and 180∘ (red).