Silicon Sampling in Calorimetry & Simulation

• Silicon sampling is a dual-application framework encompassing high-resolution calorimetry for energy reconstruction and LLM-based methods for generating synthetic population samples.
• The calorimetry approach uses both homogeneous and inhomogeneous structures with advanced calibration techniques like dE/dx and sampling fraction to achieve near-1% energy linearity.
• Extensions of silicon sampling include photonic quantum applications and wafer-level testing, where precise sensor segmentation, ASIC/FPGA integration, and bias mitigation in LLM simulations are critical.

Silicon sampling broadly encompasses two distinct technical paradigms, each characterized by rigorous statistical methodologies and specialized instrumentation: (1) silicon-based sampling calorimetry for precision energy reconstruction in particle detectors and (2) the use of LLMs as stochastic simulators (“silicon sampling” in the social sciences) for generating synthetic population samples. Both domains center on the systematic extraction of representative measurements or simulated outputs from high-dimensional silicon-based systems, grounded in controlled sampling and calibration strategies.

1. Principles and Implementation of Silicon Sampling in Calorimetry

High-granularity silicon-pad electromagnetic calorimeters employ multilayer stacks of silicon sensors interleaved with passive absorbers (typically tungsten) to sample the spatial and energy distribution of charged particles in showers. Each active silicon layer functions as a spatially segmented MIP (minimum ionizing particle) counter, capturing deposited energy at fine granularity (typical pad sizes range from 0.5–1 cm², thicknesses of 200–500 μm) (Paganis et al., 2017, Acar et al., 2020, Quast, 2017, Muhuri et al., 2019, Abramowicz et al., 13 Jan 2025, Sawan et al., 2024, Tomita et al., 2014).

Design choices include:

• Homogeneous sampling structure: Uniform absorber thickness per layer; delivers near-constant sampling fraction and simpler calibration.
• Inhomogeneous sampling structure: Varying absorber thickness toward detector back; used for cost reduction and channel-count optimization in systems like CMS-HGCAL.

2. Calibration Algorithms: dE/dx vs. Sampling Fraction

Silicon-sampling calorimeters typically reconstruct incident energy via two paradigms:

• dE/dx method: Energy per layer is estimated as $$E_{rec}(dE/dx) = (N_1 \cdot \Delta E_{passive,1}^{MIP} + \Delta E_{silicon,1}) + \sum_{i=2}^{N} [((N_{i-1} + N_i)/2) \cdot \Delta E_{passive,i}^{MIP} + \Delta E_{silicon,i}]$$, relying on calibrated average energy loss in passive layers (Paganis et al., 2017).
• Sampling-fraction (SF) approach: $$SF = \frac{\sum_{i=1}^{N} E_{active,i}}{\sum_{i=1}^{N} E_{active,i} + \sum_{i=1}^{N} E_{passive,i}}$$, then $$E_{rec}(SF) = (\sum_{i=1}^{N} E_{active,i}) \times SF_{corr}^{-1}$$, with $SF_{corr}$ corrected for shower depth $t$ per event (Paganis et al., 2017).

Key performance metrics:

• Energy resolution: Typically parameterized as $\sigma/E = a/\sqrt{E} \oplus c$, with $a$ (stochastic term) and $c$ (constant term) sensitive to both layer granularity and calibration strategy.
• Linearity and absolute scale: SF-based methods restore scale and linearity to within 1% across 20–500 GeV and reduce the constant term in inhomogeneous stacks compared to dE/dx (Paganis et al., 2017).

3. Sensor and Electronics Design for Sampling Precision

Progress in silicon sampling has focused on pad segmentation, substrate choice, edge effect mitigation, and high S/N (signal-to-noise) ratio achievement:

• Pad areas from 0.5–1.1 cm², minimal inter-pad gaps (~10–80 μm), and careful guard-ring design enhance spatial uniformity and charge collection efficiency (Abramowicz et al., 13 Jan 2025, Sawan et al., 2024, Tomita et al., 2014).
• ASICs such as FLAME and SKIROC2/2-CMS provide dynamic ranges from sub-MIP up to thousands of MIP equivalents, with sub-2 mW/channel power budgets and timing down to ~20 ps (Acar et al., 2020, Quast, 2017, Abramowicz et al., 13 Jan 2025, Tomita et al., 2014).
• FPGA-based deconvolution and zero-suppression maintain timing and amplitude fidelity for efficient data reduction (Abramowicz et al., 13 Jan 2025, 1804.01985).

4. Statistical Sampling in Silicon-based Modeling and Social Simulation

Silicon sampling extends to synthetic population simulation via LLMs (“silicon samples”) for opinion polling and cognitive studies (Chapala et al., 27 Dec 2025, Sun et al., 2024, Ong, 2024). Methodology involves:

• Demographic conditioning: Synthetic respondents $R_i = \{d_k^{(i)} \sim D_k\}_{k=1...K}$ drawn from empirical marginal distributions (e.g., race, age, ideology).
• Prompt-driven sampling: LLMs generate samples based on finely specified prompts; sample distributions are compared to empirical benchmarks using divergence measures such as Jensen-Shannon Divergence (JSD) and Kullback–Leibler divergence (Chapala et al., 27 Dec 2025, Sun et al., 2024).
• Bias and mitigation: Social Desirability Bias (SDB) can be attenuated by prompt reformulation, notably third-person neutral phrasing, improving LLM-human distributional alignment (Chapala et al., 27 Dec 2025).

5. Specialized Silicon Sampling Protocols in Physics and Engineering

Other forms of silicon-based sampling appear in photonics and materials science:

• Quantum sampling on silicon chips: On-chip boson sampling protocols (standard, scattershot, Gaussian) exploit photon generation and interference in silicon waveguide circuits, enabling benchmarking tasks such as molecular vibronic spectrum simulation (Paesani et al., 2018, Bell et al., 2019).
• Wafer-level and FPGA manufacturing testing: Spatial sampling algorithms (Random, Stratified, k-means, hybrid SDE-enabled approaches) systematically reduce test cost while maximizing predictive accuracy via Gaussian Process Regression; SDE-based hybrids (K-SDE, S-SDE) enhance spatial coverage and lower RMSD (WeiQuan et al., 4 Jun 2025).

6. Limitations, Biases, and Future Directions

Silicon sampling in both calorimetry and social simulation is subject to domain-specific limitations:

• Calorimetry: Radiation damage to n-type substrates (leakage increase post-neutron exposure), need for p-type designs for extended radiation tolerance (Sawan et al., 2024), and mechanical assembly precision (alignment <100 μm).
• LLM-based simulation: Entrenched LLM biases (harmlessness, extremity) require validation, prompt sensitivity testing, and sub-population post-stratification (Sun et al., 2024, Chapala et al., 27 Dec 2025, Ong, 2024).
• Photonics/materials: Event rates and loss in photonic sampling circuits are bottlenecks for scaling to quantum advantage (Paesani et al., 2018, Bell et al., 2019); solutions include loss mitigation, integration of quantum sources, and advanced detector arrays.

Silicon sampling models, whether for energy measurement, social simulation, or quantum benchmarking, require systematic calibration, bias analysis, and careful methodological reporting to ensure reproducibility and representative coverage. The convergence of fine-granularity detector engineering, stochastic model sampling, and advanced statistical protocols positions silicon sampling as a foundational technique across experimental physics, engineering, and computational social sciences.