Seven-dimensional Trajectory Reconstruction for VAMOS++

Technical Summary: Seven-dimensional Trajectory Reconstruction for VAMOS++

Problem Statement
The VAMOS++ magnetic spectrometer is a critical tool for isotopic identification in nuclear structure and reaction mechanism research near the Coulomb barrier. Its operation relies on measuring the magnetic rigidity ($B\rho$) and the trajectory length ($l$) of ions. Standard trajectory reconstruction methods, such as the established four-dimensional (4D) mapping technique, assume a point-like beam interaction volume. This assumption holds for experiments using thin targets with narrow beam spots. However, this approach becomes a significant limitation for experiments involving large beam spot sizes or extended gaseous target volumes, such as those planned for studying fission dynamics. In these scenarios, the three-dimensional position of the beam interaction point within the target cannot be neglected, as variations in the interaction coordinates (particularly the horizontal position $x_i$) induce substantial optical aberrations and trajectory length changes that the 4D method cannot adequately resolve. Extending the 4D matrix-based method to include these additional dimensions is computationally infeasible due to the excessive size of the required matrices.

Methodology
To address the high-dimensional, non-linear nature of the ion optics in VAMOS++, the authors developed a seven-dimensional (7D) trajectory reconstruction method utilizing artificial deep neural networks (DNNs).

• Input and Output: The DNN takes seven input coordinates: the beam interaction position within the target volume ($x_i, y_i, z_i$), the horizontal and vertical angles of the outgoing reaction products ($\theta_i, \phi_i$), and the final horizontal position and angle at the focal plane ($x_f, \theta_f$). The network outputs the relative magnetic rigidity ($\delta = B\rho/B\rho_0$) and the relative trajectory length ($\epsilon = l_f/l_0$).
• Training Dataset: The network was trained on a theoretical dataset of $2 \times 10^8$ ion trajectories generated using the standard magnetic ray-tracing code ZGOUBI. The dataset covered the full operational range of the spectrometer, including relative rigidities $\delta \in [0.7, 1.4]$ and interaction volumes defined by $x_b \in [-3, 3]$ mm, $y_b \in [-6, 6]$ mm, and $z_b \in [-70, 50]$ mm.
• Network Architecture: A dense deep feed-forward neural network architecture was selected. Through an evaluation of complexity versus performance, the optimal architecture was determined to be $N_l \times N_u = 6 \times 32$ (6 layers with 32 neurons each), resulting in approximately 5,569 trainable parameters. This architecture was chosen to balance resolution with computational efficiency.
• Training Workflow: The training process involved monitoring the root mean square deviation (RMSD) for $\delta$ and $\epsilon$. The network was validated against experimental thin-target data to ensure the Full Width at Half Maximum (FWHM) of the reconstructed atomic mass number ($A$) was optimized. Training was terminated when further convergence did not yield experimental improvements, avoiding overfitting.

Key Contributions

1. Development of 7D Reconstruction: The paper presents a novel method capable of handling the seven-dimensional non-linear problem of ion trajectory reconstruction, explicitly incorporating the 3D beam interaction coordinates.
2. Deep Learning Implementation: It demonstrates the efficacy of deep neural networks in modeling complex, non-linear ion optics where traditional matrix-based methods fail due to dimensionality constraints.
3. Efficiency: The proposed 7DNN method achieves a processing speed of $1.4 \times 10^6$ events per second with a memory footprint of approximately 22 kB, making it suitable for both online and offline analysis. This is a significant reduction in memory usage compared to the 4D method, which requires approximately 1 GB for its compressed matrices.

Results

• Intrinsic Precision: When applied to the complete theoretical dataset generated by ZGOUBI, the method achieved an intrinsic reconstruction precision of FWHM($\delta_c - \delta_r$) = 0.65‰ and FWHM($\epsilon_c - \epsilon_r$) = 0.18‰. The precision remained consistent across various interaction coordinates ($x_i, z_i$) and focal plane parameters ($x_f, \theta_f$).
• Experimental Validation: The method was validated using experimental data from the E826 GANIL experiment (thin-target fusion-fission reactions). The reconstructed atomic mass number resolution, FWHM($A$)/$A$, ranged from 4‰ for heavy fission fragments to 7‰ for light fragments.
• Comparison with 4D Method: In the experimental validation, the 7DNN method demonstrated performance comparable to the standard 4D method for light fission fragments and a slight advantage for heavy fission fragments.
• Detector Resolution Impact: Simulations indicated that while the DNN's intrinsic resolution is high, the final mass resolution is dominated by detector resolutions, particularly the time-of-flight ($t$) and the horizontal angles ($\theta_i, \theta_f$). When all detector resolutions were included in the simulation, the predicted FWHM($A$)/$A$ values (6.32‰ for $A=80$ and 4.17‰ for $A=160$) closely matched the experimental results.

Significance and Future Outlook
The paper claims that the 7DNN method represents a significant advancement for VAMOS++, offering resolution comparable to or superior to previously employed methods while enabling the analysis of experiments with extended interaction volumes. The authors highlight that this capability is essential for upcoming programs utilizing voluminous gas targets, where the spectrometer's optical axis will be rotated relative to the beam axis, necessitating the explicit inclusion of 3D interaction coordinates. Furthermore, the speed and efficiency of the model suggest its utility in rapid ion transmission simulations, potentially facilitating the development of a "digital twin" for the spectrometer to aid in absolute cross-section determinations and fission yield measurements. The authors emphasize that the method is specifically designed to overcome the limitations of point-like beam assumptions in large-acceptance spectrometers.