beta_shape_pnpi/docs/corrections/07-detector-response.md

---

title: "08 – Detector Response Convolution" date: 2026-05-01 tags:

• beta-spectrum/detector-response status: active aliases: [Convolution, Response Matrix, Smearing] cssclasses: [] related_components: 00-beta-spectrum-overview, 06-radiative-corrections

---

Detector Response Convolution — M(E_j) = \\sum_i \\Phi(W_i) \\cdot R(E_j; W_i) \\cdot \\Delta W_i

Overview

Detector response convolution bridges the gap between theoretical beta spectra and experimentally measured data. A theoretical spectrum \\Phi(W) assumes perfect energy resolution, while real detectors smear the true energy distribution due to finite resolution, charge collection inefficiencies, and electronic noise.

The convolution operation:

M_j = \\sum_i \\Phi(W_i) \\cdot R(E_j; W_i) \\cdot \\Delta W_i

where:

• \\Phi(W_i) — theoretical differential spectrum at true energy W_i
• R(E_j; W_i) — detector response probability: probability that an electron of true energy W_i deposits energy E_j in channel j
• \\Delta W_i — energy bin width at W_i
• M_j — predicted measured counts in detector channel j

Two Modes of Operation

Tabulated Response

Receive a pre-computed response matrix from Monte Carlo simulations (GEANT4, EGSNRC) or dedicated calibration measurements. The matrix has shape (N\_{\\text{calib}}, N\_{\\text{channels}}), where each row i is the detector response to mono-energetic electrons at calibration energy E_i.

from beta_spectrum import DetectorResponse

# Load from MC simulation output
detector = DetectorResponse.from_mc_simulation(
    channel_energies=channel_grid,
    response_matrix=mc_response,
    calibration_energies=mc_energies,
)

Analytical Model

Parameterized model with Gaussian core + exponential tail, suitable for quick estimates when no tabulated response is available.

detector = DetectorResponse.from_gaussian_params(
    channel_energy_range=(1.0, 1.6),  # in m_e units
    n_channels=4096,
    sigma_a=0.0,       # energy-independent resolution (m_e)
    sigma_b=0.0,       # sqrt(E) contribution (m_e^{-1/2})
    tail_fraction=0.0, # fraction of events in exponential tail
    tau=0.01,          # decay constant of tail (m_e)
    model="gaussian",  # "gaussian" | "gaussian_tail" | "tikhonov"
    fano_factor=0.0,   # Fano factor (0.12 for Si, ~0 for MMC)
)

Analytical Response Models

Gaussian Core

Pure Gaussian peak with energy-dependent resolution:

G(E; E_0, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp\\left(-\\frac{(E - E_0)^2}{2\\sigma^2}\\right)

Gaussian + Exponential Tail

Models low-energy tailing from charge collection inefficiencies in semiconductor detectors (standard for Si-PIPS detectors):

f(E) = (1 - f\_{\\text{tail}}) \\cdot G(E; E_0, \\sigma) + f\_{\\text{tail}} \\cdot T(E; E_0, \\sigma, \\tau)

where T is the Tikhonov function:

T(E) = \\frac{1}{2\\tau} \\exp\\left(\\frac{\\tau(E - E_0)}{2} + \\frac{\\sigma^2\\tau^2}{8}\\right) \\cdot \\text{erfc}\\left(\\frac{E - E_0}{\\sqrt{2}\\sigma} + \\frac{\\sigma\\tau}{\\sqrt{2}}\\right)

Energy-Dependent Resolution

\\sigma(E) = \\sqrt{\\sigma_a^2 + (\\sigma_b \\sqrt{E})^2 + F \\cdot E \\cdot w}

• \\sigma_a — energy-independent contribution (electronic noise, etc.)
• \\sigma_b — \\sqrt{E} term (statistical fluctuations in charge creation)
• F — Fano factor (\\approx 0.12 for Si, \\approx 0 for MMCs)
• w — mean ionization energy (\\approx 3.6 eV for Si, converted to m_e units)

Calculator Implementation

[!tip] Code mapping Module: beta_spectrum/components/detector_response.py → DetectorResponse class

• Analytical models: Gaussian core, Gaussian + exponential tail (Tikhonov)
• Tabulated mode: Linear interpolation from MC-calculated response matrix
• Convolution: DetectorResponse.convolve(W, spectrum, normalize=True)
• Integration with BetaSpectrum: BetaSpectrum.convolve_with_detector(detector, W, config)
• Integration with BetaSpectrumAnalyzer: BetaSpectrumAnalyzer.convolved_spectrum(detector, normalize)

API Usage

from beta_spectrum import (
    BetaSpectrum, SpectrumConfig, DetectorResponse,
    BetaSpectrumAnalyzer, T_to_W
)

# 1. Create theoretical spectrum for 99Tc
config = SpectrumConfig(
    Z_parent=43, Z_daughter=44, A_number=99,
    endpoint_MeV=0.294  # 99Tc Q-value
)
spectrum = BetaSpectrum.from_config(config)
W, kinetic_MeV = spectrum.get_energy_grid(config)

# 2. Define detector response (sigma = 1.0 keV in m_e units)
sigma_1keV_me = 1.0 / 511.0  # convert keV to m_e units
W0 = T_to_W(0.294)
detector = DetectorResponse.from_gaussian_params(
    channel_energy_range=(1.0, W0 + 0.05),
    n_channels=512,
    sigma_a=sigma_1keV_me,
    sigma_b=0.0,
    tail_fraction=0.0,
    model="gaussian",
    fano_factor=0.0,
)

# 3. Convolve
convolved = spectrum.convolve_with_detector(detector, W=W, config=config)

# 4. Or use the Analyzer
analyzer = BetaSpectrumAnalyzer(spectrum, config)
convolved_normalized = analyzer.convolved_spectrum(detector, normalize=True)

Convolution Algorithm

The convolution proceeds in three steps:

1. Build response matrix: For each energy bin W_i, compute the detector response R(E_j; W_i) across all channels j. Result: matrix of shape (N\_{\\text{bins}}, N\_{\\text{channels}}).

2. Weight by spectrum: Multiply each response row by the theoretical spectrum value \\Phi(W_i) and bin width \\Delta W_i.

3. Sum over energies: For each channel j, sum contributions from all energy bins:

   M_j = \\sum_i \\Phi(W_i) \\cdot R(E_j; W_i) \\cdot \\Delta W_i = (R^T \\cdot (\\Phi \\cdot \\Delta W))\_j

The bin width is computed as the half-bin average: \\Delta W_i = \\frac{1}{2}(\\Delta W\_{i-1/2} + \\Delta W\_{i+1/2}).

[!note] Normalization Each response function R(E; W_i) is normalized to unit area before convolution. This ensures that convolving a mono-energetic line produces a response with the same total counts. The final convolved spectrum preserves the integral of the theoretical spectrum.

Energy Units

All detector response computations use total energy in units of $m_e c^2$ (natural units, m_e = 1):

• m_e c^2 = 510.998950 keV
• Conversion: W = 1 + T / (m_e c^2) where T is kinetic energy
• A resolution of \\sigma = 1.0 keV corresponds to \\sigma = 1.0 / 511.0 \\approx 0.00196 in m_e units

[!warning] Unit consistency The detector channel energy range and the spectrum energy grid must be in the same units (total energy in m_e units). Use T_to_W() from beta_spectrum.utils to convert kinetic energy in MeV to total energy in m_e units.

Physical Interpretation

The convolution operation smears the sharp theoretical spectrum into a realistic measured spectrum:

• Peak broadening: Finite energy resolution widens spectral features
• Endpoint smearing: The sharp cutoff at W_0 becomes a gradual fall-off
• Tail effects: Charge collection inefficiencies create low-energy tails (modeled by gaussian_tail)
• Count conservation: Total number of events is preserved (integral of convolved spectrum = integral of theoretical spectrum)

References

• Hayen et al., RevModPhys 90, 015008 (2017) — Comprehensive review of beta-decay theory
• Tikhonov, Soviet Physics Doklady 8, 591 (1964) — Original formulation of the response function
• Sigmund, Phys. Rev. 130, 1497 (1963) — Low-energy tailing in semiconductor detectors