Peak Fitting and Background Models

Peak Fitting & Background Models

The peak fitting algorithms, baseline estimation engines, and background decomposition methodologies implemented in python-cmat adhere to established analytical practices in high-resolution semiconductor $\gamma$-ray spectroscopy.

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1. Photopeak Models (1D & 2D)

While ideal charge generation and preamplifier electronics produce symmetric Gaussian photopeaks, real High-Purity Germanium (HPGe) and Ge(Li) detectors exhibit low-energy tailing caused by:

• Incomplete Charge Collection & Hole Trapping: Lower drift mobility and radiation damage defect centers trap charge carriers before collection, attenuating pulse height.
• Ballistic Deficit: Charge collection rise times varying with interaction location, occasionally exceeding amplifier shaping times.
• Small-Angle Compton Scattering: Photons scattering in cryostat windows, dead layers, or target holders prior to full absorption.

python-cmat provides three analytical models for 1D projections and 2D coincidence peak fitting:

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Option 1: Standard Symmetric Gaussian (gaussian)

A pure Gaussian photopeak sitting on a linear baseline:

$$G(x) = H \exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)$$

• Net Peak Area: $$A_{\text{net}} = \sqrt{2\pi} H \sigma \approx 2.5066 \cdot H \cdot \sigma$$
• Full Width at Half Maximum (FWHM): $$\text{FWHM} = 2\sqrt{2\ln 2} \sigma \approx 2.35482 \cdot \sigma$$
• Best For: Fast preliminary fits, well-shaped scintillation detector peaks (LaBr$_3$, NaI), or HPGe peaks with negligible trapping.

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Option 2: RadWare / SAMPO Piecewise Left Exponential Tail (gaussian_tail)

The classic formulation established in RadWare (gf3), SAMPO, and xtrackn (Helmer & Lee, 1980; Radford, 1995; Routti & Prussin, 1969). The exponential tail joins the Gaussian at $z = (x - \mu)/\sigma = -\alpha$ with continuous amplitude and continuous first derivative:

$$P(x) = \begin{cases} H \exp\left(-\frac{1}{2} z^2\right), & z \ge -\alpha \ H \exp\left(\frac{1}{2}\alpha^2 + \alpha z\right), & z < -\alpha \end{cases}$$

• Net Peak Area: $$A_{\text{net}} = H \sigma \left[ \sqrt{\frac{\pi}{2}}\left(1 + \text{erf}\left(\frac{\alpha}{\sqrt{2}}\right)\right) + \frac{1}{\alpha}\exp\left(-\frac{1}{2}\alpha^2\right) \right]$$
• Parameters: $\alpha &gt; 0$ defines the tail join distance in units of $\sigma$ below centroid $\mu$.
• Best For: Standard HPGe spectroscopy with mild low-energy tailing.

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Option 3: Hypermet Analytical Model (hypermet)

The rigorous analytical formulation combining a symmetric Gaussian $G(x)$, an analytical Exponentially Modified Gaussian (EMG) convolved tail $T(x)$, and an error-function Compton step $S(x)$ (Phillips & Marlow, 1976; Campbell & Maxwell, 1993):

$$P(x) = G(x) + T(x) + S(x)$$

where:

$$G(x) = H \exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)$$

$$T(x) = \frac{f_T}{2\beta} \exp\left[\frac{x-\mu}{\beta} + \frac{\sigma^2}{2\beta^2}\right] \text{erfc}\left[\frac{x-\mu}{\sqrt{2}\sigma} + \frac{\sigma}{\sqrt{2}\beta}\right]$$

$$S(x) = \frac{A_S}{2} \text{erfc}\left[\frac{x-\mu}{\sqrt{2}\sigma}\right]$$

• Parameters:
  • $H$: Gaussian peak amplitude.
  • $\mu, \sigma$: Centroid and Gaussian dispersion width.
  • $f_T$: Integrated area of the convolved exponential tail.
  • $\beta$: Exponential decay slope length (channels).
  • $A_S$: Amplitude of the Compton step function.
• Total Net Peak Area: $$A_{\text{net}} = \sqrt{2\pi} H \sigma + f_T$$
• In 2D coincidence matrices, the true coincidence peak is evaluated as the 2D tensor product $H \cdot P_X(x) \cdot P_Y(y)$ with total net volume: $$V_{\text{fit}} = H (\sqrt{2\pi}\sigma_x + \eta_{Tx})(\sqrt{2\pi}\sigma_y + \eta_{Ty})$$

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2. Channel Bin Centering Convention (! al centro del canale)

In multichannel analyzer (MCA/ADC) spectroscopy and the standard GASPware / xtrackn suite, an integer histogram bin $k$ represents counts collected in the channel interval $[k, k+1)$ with its physical center located at:

$$\text{Physical Center} = k + 0.5$$

python-cmat strictly adheres to this Fortran xtrackn convention ("al centro del canale", src/xtrack/trackn.F):

• All 1D and 2D continuous peak fitting routines evaluate analytical models on bin centers ($x_i = k + 0.5, y_j = l + 0.5$).
• Calibrated peak energies $E(\mu) = a_0 + a_1 \mu + a_2 \mu^2$ are evaluated directly at the continuous centroid $\mu$, guaranteeing exact 1:1 energy calibration agreement with xtrackn.
• Peak curves and centroid marker stems naturally align with the exact center of histogram bins on both browser canvases and vector PDF exports.

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3. Self-Consistent 2D $\gamma$-$\gamma$ Background Decomposition (Gamba & Morhác)

In 2D $\gamma$-$\gamma$ coincidence spectroscopy, counts in the vicinity of a coincidence peak $(E_d, E_f)$ are composed of four distinct topological components (Gamba et al., NIM A 928 (2019) 93–103; Morhác et al., NIM A 401 (1997) 113):

1. $p|p^t$ (True Net Coincidence Peak): Genuine correlated full-energy cascade events ($E_d \otimes E_f$), modeled as a 2D peak profile $H \cdot P_X(x) \cdot P_Y(y)$.
2. $p|bg$ (Det 1 Peak with Det 2 Continuum / Random Ridge): Full energy in Det 1 ($E_d$) and Compton continuum (or randoms) in Det 2 $\rightarrow$ forms a vertical cross-ridge $R_x \cdot P_X(x)$.
3. $bg|p$ (Det 2 Peak with Det 1 Continuum / Random Ridge): Full energy in Det 2 ($E_f$) and Compton continuum (or randoms) in Det 1 $\rightarrow$ forms a horizontal cross-ridge $R_y \cdot P_Y(y)$.
4. $bg|bg$ (2D Compton Continuum & Accidental Coincidences): Compton continuum and time-uncorrelated random coincidences in both detectors $\rightarrow$ modeled as a 2D planar baseline $b_0 + b_x(x - x_c) + b_y(y - y_c)$.

No User-Tuned Random Fraction Required

Because accidental (time-uncorrelated) coincidences naturally distribute into the exact same topological structures (FEP–Random into ridges, Compton–Random into continuum, Random–Random into the local plane), no arbitrary user-tuned random fraction is required.

The 2D fitting engine extracts all parameters simultaneously via Levenberg-Marquardt optimization with Poisson statistical weights ($w_{ij} = 1 / \sqrt{\max(1, M_{ij})}$), and calculates the discrete Gamba net area:

$$n^t_{p|p} = n^m_{p|p} - n^m_{p|bg} - n^m_{bg|p} + n^m_{bg|bg}$$

and the Peak-to-Total-Background ratio:

$$\Pi = \frac{n^t_{p|p}}{n^m_{p|p}}$$

Gamba 4-Component 3D Coincidence Slicing (3D Matrix Volumes)

When analyzing 3D matrix volumes (e.g. $\gamma$-$\gamma$-$\text{Rings}$, $\gamma$-$\gamma$-$\Delta t$, or 3-fold symmetric cubes) in cmat3d_webviewer.py, fitting a 2D coincidence peak on any orthogonal plane automatically performs discrete Gamba background decomposition on the 3rd axis ($z$).

The 3D volume is sliced over the 2D Peak ($P|P$) ROI, the two orthogonal Compton/random cross-ridges ($P|BG$ and $BG|P$), and the 4 corner continuum regions ($BG|BG$):

• $\text{Spec}{P|P}(z) = \sum{(x,y) \in \text{ROI}_{P|P}} V(x, y, z)$
• $\text{Spec}{P|BG}(z) = \sum{(x,y) \in \text{ROI}_{P|BG}} V(x, y, z)$
• $\text{Spec}{BG|P}(z) = \sum{(x,y) \in \text{ROI}_{BG|P}} V(x, y, z)$
• $\text{Spec}{BG|BG}(z) = \sum{(x,y) \in \text{ROI}_{BG|BG}} V(x, y, z)$

The net coincidence spectrum on the 3rd axis is constructed using the discrete Gamba formula:

$$\text{Spec}_{\text{net}}(z) = \text{Spec}_{P|P}(z) - \frac{N_{P|BG}}{N_{BG|BG \text{ (1)}}} \text{Spec}_{BG|P}(z) - \frac{N_{BG|P}}{N_{BG|BG \text{ (2)}}} \text{Spec}_{P|BG}(z) + \frac{N_{BG|BG \text{ (corr)}}}{N_{BG|BG \text{ (all)}}} \text{Spec}_{BG|BG}(z)$$

This cleanly eliminates both Compton ridges and time-uncorrelated random events from the 3rd-axis projection, providing pure net lifetime curves or detector ring distributions.

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4. 1D Multi-Gate Slicing & Normalized Background Subtraction (xtrackn W/X/Z)

Following the classic GASPware xtrackn gate methodology:

• Peak Gate Windows ($W$): Arbitrary consecutive peak windows ${ [W_{k,\min}, W_{k,\max}] }{k=1}^K$ defined using W. Total peak width: $$\Delta W{\text{tot}} = \sum_{k=1}^K (W_{k,\max} - W_{k,\min} + 1)$$
• Background Gate Windows ($X$): Arbitrary consecutive background windows ${ [X_{m,\min}, X_{m,\max}] }{m=1}^M$ defined using X. Total background width: $$\Delta X{\text{tot}} = \sum_{m=1}^M (X_{m,\max} - X_{m,\min} + 1)$$
• Coincidence Slicing: Slicing matrix $M$ along the gated axis produces 1D projection vectors: $$S_{W_k}(i) = \sum_{j = W_{k,\min}}^{W_{k,\max}} M_{i, j}, \quad S_{X_m}(i) = \sum_{j = X_{m,\min}}^{X_{m,\max}} M_{i, j}$$
• Normalized Background Subtraction: When background regions are present ($\Delta X_{\text{tot}} &gt; 0$), the net coincidence spectrum $S_{\text{net}}$ is: $$\text{Scale} = \frac{\Delta W_{\text{tot}}}{\Delta X_{\text{tot}}}$$ $$S_{\text{net}}(i) = \sum_{k=1}^K S_{W_k}(i) - \text{Scale} \cdot \sum_{m=1}^M S_{X_m}(i)$$
• Visual Overlays & Negative Baseline:
  • Gated peak windows render in translucent green (#22c55e30), background in amber (#f59e0b30).
  • The 2D matrix renders matching vertical/horizontal band overlays across gated channels.
  • A dashed zero-baseline is drawn on the gated 1D projection when over-subtracted continuum channels fluctuate below zero counts.

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5. Continuous Wavelet Transform (CWT) Photopeak Search

Automated peak finding in $\gamma$-ray spectra is challenged by Poisson statistical noise, varying peak widths, and steep Compton scattering continuum baselines. python-cmat implements Continuous Wavelet Transform (CWT) multi-scale peak detection (Du et al., 2006):

Ricker Wavelet (Mexican Hat)

$$\psi(t) = \frac{2}{\sqrt{3\sigma}\pi^{1/4}} \left(1 - \left(\frac{t}{\sigma}\right)^2\right) \exp\left(-\frac{t^2}{2\sigma^2}\right)$$

Because $\int_{-\infty}^{\infty} \psi(t) dt = 0$, convolving a spectrum with $\psi(t)$ at characteristic scale $s \approx \text{FWHM} / 2.355$ effectively subtracts polynomial continuum baselines while transforming photopeaks into pronounced local maxima in wavelet scale space.

Wavelet Ridge Lines & Prominence

Wavelet coefficients $C(s, \tau) = \frac{1}{\sqrt{s}} \int x(t) \psi\left(\frac{t-\tau}{s}\right) dt$ are evaluated across multiple scales surrounding the expected peak FWHM. Photopeaks create continuous ridge lines across adjacent scales with high signal-to-noise ratio ($SNR = C(s, \tau) / \sigma_{\text{noise}}$), cleanly separating real doublets and weak transitions from random channel-to-channel Poisson fluctuations.

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6. Automated Multi-Peak Fitting with Peak-Aware Background Estimation

Pressing H (or clicking Fit All [H]) executes automated multi-peak fitting across all visible candidate photopeaks on display. The continuum baseline is estimated using a Peak-Aware Continuum Average engine designed specifically to handle both high-statistics projections and low-statistics / coincidence-gated spectra without noise-clipping bias:

The Low-Statistics Under-Fit Problem & Solution

Standard morphological baseline filters like plain SNIP rely on concave lower-envelope clipping ($v_i = \min(v_i, \frac{v_{i-p} + v_{i+p}}{2})$). In low-statistics spectra or background-subtracted coincidence cuts ($S_{\text{net}} = S_{\text{gross}} - k S_{\text{bg}}$), large noise grass fluctuations cause plain SNIP to clip down to the lower noise envelope ($\approx \bar{y} - 2\sigma$), severely under-fitting the continuum and creating artificial false peaks.

Because the peak search routine (P) has already identified candidate photopeaks, channels outside the candidate peak clusters are known a priori to be pure continuum:

1. Peak Exclusion Masking: $$\text{ROI}_{\text{cl}} = \left[ \min(\text{cluster}) - 2.0 \times \text{FWHM}, \ \max(\text{cluster}) + 2.0 \times \text{FWHM} \right]$$
2. Continuum Rolling Average: Across pure background channels ($\mathbb{E}[\epsilon_i] = 0$), the continuum is smoothed using a rolling Bartlett filter with window $W \approx 5.0 \times \text{FWHM}$. This estimates the central expectation value ($\bar{y}$) directly through the center of the noise grass ($&lt; 0.1$ count bias), with full tolerance for negative count bins.
3. Cluster Baseline Bridging: Across each peak cluster, the baseline connects the smoothed left continuum to the smoothed right continuum linearly without clipping or artificial bowing.
4. Dense Forest Fallback: If peak coverage exceeds 92% of the display window, the engine automatically falls back to SNIP.

Multiplet Clustering & Adaptive ROIs

Adjacent peaks separated by $\le 3.2 \times \text{FWHM}$ are grouped into multiplets. Each cluster is fitted simultaneously with Levenberg-Marquardt optimization using variance weights $w_i = 1 / \sqrt{\max(1.0, |y_i|)}$, supporting Gaussian, RadWare / SAMPO, or Hypermet profiles. Centroids are constrained within $\pm 2.5$ channels of markers to prevent line migration.

Manual Peak Marking (G) & Multiplet Deconvolution (V)

• G: Snaps to the local peak summit within $\pm 2$ channels and computes parabolic sub-channel centroid refinement ($\delta = \frac{1}{2} \frac{y_0 - y_2}{y_0 - 2y_1 + y_2}$). Hovering an existing marker within 1.2 channels deletes it.
• V: Pressing V anywhere within a multiplet cluster automatically fits all components simultaneously with full parameter covariance error propagation.

Region-Constrained Fitting with Straight-Line Background (R)

Pressing R twice sets left and right region limits $[R_{\text{left}}, R_{\text{right}}]$. The continuum level is calculated by averaging 1 FWHM on either side: $$\bar{y}{\text{left}} = \frac{1}{w} \sum{c = R_{\text{left}} - w}^{R_{\text{left}} - 1} y[c], \quad \bar{y}{\text{right}} = \frac{1}{w} \sum{c = R_{\text{right}} + 1}^{R_{\text{right}} + w} y[c] \quad (w = \text{round}(\text{FWHM}))$$ Pressing V inside the region fits all peaks within the boundary on top of this straight-line background.

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7. Fit Results File Logging

To streamline spectroscopic reporting and automate data export for external calibration programs, python-cmat supports real-time appending of all 1D and 2D peak fit results to a clean, fixed-width text log.

Fixed-Width Column Formatting

To ensure clean vertical alignment across text editors and avoid tab misalignment for large peak counts (10+ digits with uncertainties), columns are right-aligned with fixed character widths:

• 1D Fits:

  # 1D Fits:     Energy(err)               Net_Area(err)               FWHM(err)          Chi2    Peak_to_BG
             1164.257(0.003)             339769.5(735.4)            3.060(0.006)        144.37          5.84
               1164.76(0.00)             339769.5(735.4)              3.06(0.01)        144.37          5.84
  

  (Energy and FWHM display in keV when calibrated or channels when uncalibrated).

• 2D Coincidence Fits (Gamba Decomposition):

  # 2D Fits:    Energy1(err)                Energy2(err)               Net_Area(err)             Gamba_Area(err)              FWHM1(err)              FWHM2(err)          Chi2    Peak_to_BG
             1164.215(0.052)            1345.264(99.304)                32.8(9123.4)                  25.6(16.1)            3.002(0.110)          0.636(119.698)          1.23          0.01
               1164.71(0.05)              1345.76(99.30)                32.8(9123.4)                  25.6(16.1)              3.00(0.11)            0.64(119.70)          1.23          0.01
  

Enabling & Controls

• Web Viewer GUI: Click ON / OFF toggle in the sidebar Peak Fitting control group or set a custom filename.
• CLI Startup Flag: python3 cmat_webviewer.py --fit-log [filename] matrix.cmat
• Headless Shell / Macro: fit_log on [filename], fit_log off, fit_log status
• Default Filename: If no filename is specified, an automatic timestamped file (fit_results_YYYYMMDD_HHMMSS.txt) is generated to prevent overwriting previous analyses.

Automated Isotope Identification

Fit results logs can be automatically analyzed and matched against evaluated nuclear structure data to identify parent isotopes and 2D cascades using the ENSDF Isotope Identification tool.

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8. Automatic 2D Coincidence Peak Search & Compton Ridge Discrimination

Detecting true physical $\gamma$-$\gamma$ coincidence peaks in a large 2D matrix ($4096 \times 4096$) is complicated by two major sources of false positives:

1. Compton Scattering Cross-Ridges ($p|bg$ and $bg|p$): A high-intensity photopeak in Det 1 coincident with a continuous Compton scattering event in Det 2 creates a continuous vertical line across the entire matrix. Any noise fluctuation along this ridge resembles a 2D peak.
2. Detector Cross-Talk & Scattering Artifacts: Single gamma rays scattering from one detector into another produce diagonal ridges ($\rho_{xy} \approx -1, x + y = \text{const}$).

python-cmat implements a dedicated 4-stage hybrid 2D search engine to discriminate genuine coincidences:

Stage 1: 1D Projection Seeding

Continuous Wavelet Transform (CWT) peak searches are executed across both 1D total projections (Det 1 and Det 2) down to a user-selectable signal-to-noise ratio threshold ($\text{SNR}_{\text{min}}$): $$\mathcal{S}_X = { x_1, x_2, \dots, x_M }, \quad \mathcal{S}_Y = { y_1, y_2, \dots, y_N }$$ The search grid evaluates candidate pairs $(x_i, y_j) \in \mathcal{S}_X \times \mathcal{S}_Y$ bounded within local subregions $\Omega = [x_i - w_x, x_i + w_x] \times [y_j - w_y, y_j + w_y]$.

Stage 2: Local 4-Component Gamba Decomposition

For each candidate pair $(x_i, y_j)$, the local 2D subregion is decomposed using the Gamba & Morhác model: $$M(x, y) = H \cdot P_X(x) P_Y(y) + R_x P_X(x) + R_y P_Y(y) + [b_0 + b_x(x - x_i) + b_y(y - y_j)]$$ Extracting the gross volume $n_{p|p}^m$, vertical ridge $n_{p|bg}^m$, horizontal ridge $n_{bg|p}^m$, continuum $n_{bg|bg}^m$, and net coincidence volume $n_{p|p}^t$.

Stage 3: Compton Ridge & Cross-Talk Rejection

Candidate peaks are subjected to physical validation filters:

• Gamba Ratio Filter: The peak-to-total-background ratio $\Pi = n_{p|p}^t / n_{p|p}^m$ must exceed the user-defined threshold ($\Pi_{\text{min}}$, default $0.05$). Pure Compton ridges have $n_{p|p}^t \approx 0 \implies \Pi \approx 0$ and are cleanly rejected.
• Statistical Significance: The net peak height $H$ must exceed $\text{SNR}{\text{threshold}} \cdot \sqrt{\max(1, B{\text{local}}})$.
• Covariance Ellipse Filter: The 2D cross-correlation coefficient $|\rho_{xy}| &lt; 0.85$ rejects diagonal scattering cross-talk.
• Width Bounding: Fitted dispersions must satisfy physical detector bounds ($0.4 \times \text{FWHM}_0 \le \text{FWHM} \le 3.5 \times \text{FWHM}_0$).

Stage 4: 2D Non-Maximum Suppression (NMS) & Levenberg-Marquardt Fitting

Surviving candidates are clustered within $2.5 \times \text{FWHM}$. The strongest local maximum is refined with full Levenberg-Marquardt optimization and parameter covariance estimation. For symmetric matrices ($M_{ij} = M_{ji}$), candidate pairs across the diagonal $(x, y) \leftrightarrow (y, x)$ are deduplicated.

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9. Nuclear Lifetime & Half-Life Fitting (halflife.py)

In fast-timing nuclear spectroscopy (e.g. LaBr$_3$(Ce) scintillators, BaF$2$, TAC, TDC, or digital CFD timestamp time-difference $\Delta T$ matrices), nuclear level half-lives ($T{1/2}$) are extracted by convolving the instrumental prompt response function (IRF) with the physical exponential decay.

Analytical Physics Model

The combined model convolves a Gaussian prompt response (representing the instrument time resolution with dispersion $\sigma_g = \text{FWHM} / 2.35482$) with an exponential radioactive decay:

$$F(x) = \text{BG} + \frac{A}{\lambda} \cdot \mathcal{M}(x; x_0, \sigma_g, \lambda)$$

Where:

• $\lambda = \frac{\ln 2}{T_{1/2}}$ is the decay constant ($\text{ch}^{-1}$).
• For positive half-lives ($T_{1/2} &gt; 0$, right-sided decay tail $x &gt; x_0$): $$F(x) = \text{BG} + \frac{A}{2} \exp\left( \frac{\sigma_g^2 \lambda^2}{2} - \lambda(x - x_0) \right) \text{erfc}\left( \frac{\sigma_g \lambda - (x - x_0)/\sigma_g}{\sqrt{2}} \right)$$
• For negative half-lives ($T_{1/2} &lt; 0$, inverted / left-sided decay tail $x &lt; x_0$): $$F(x) = \text{BG} + \frac{A}{2} \exp\left( \frac{\sigma_g^2 \lambda^2}{2} + \lambda(x - x_0) \right) \text{erfc}\left( \frac{\sigma_g \lambda + (x - x_0)/\sigma_g}{\sqrt{2}} \right)$$
• For prompt-only resolution calibration ($T_{1/2} = 0$): $$F(x) = \text{BG} + A \exp\left(-\frac{(x - x_0)^2}{2\sigma_g^2}\right)$$
• For pure exponential decay without prompt broadening ($\text{FWHM} = 0$): $$F(x) = \text{BG} + A \exp(-\lambda |x - x_0|) \cdot \Theta(\pm(x - x_0))$$

Numerical Stability via Scaled Complementary Error Function

Direct evaluation of $\exp(\dots) \text{erfc}(u)$ can suffer from severe floating-point underflow or overflow for large $u$. python-cmat implements numerically stable evaluation using scipy.special.erfcx:

$$\exp(z) \text{erfc}(u) = \text{erfcx}(u) \cdot \exp\left( z - u^2 \right) = \text{erfcx}(u) \cdot \exp\left( -\left(\frac{x - x_0}{\sqrt{2}\sigma_g}\right)^2 \right)$$

For deep tail regions ($u &lt; -25$), the standard asymptotic transformation $\text{erfc}(u) = 2 - \text{erfc}(-u)$ prevents overflow.

Background Profiling ($\chi^2$ Scan)

Because the background offset and exponential decay slope can exhibit strong mathematical covariance in truncated fit ranges, halflife.py provides automated $\chi^2$ profiling over fixed background grids:

1. Fix background to a sequence of values $B_k \in [B_{\text{min}}, B_{\text{max}}]$.
2. Perform bounded Levenberg-Marquardt / Trust Region Reflective optimization over free parameters $(T_{1/2}, \text{FWHM}, x_0, A)$.
3. Record $\chi^2(B_k)$ and locate the global parabolic minimum $\chi^2_{\text{min}}$.
4. Calculate $1\sigma$ confidence limits from the profile crossing points $\chi^2(B) = \chi^2_{\text{min}} + 1.0$.

Error Propagation

For coincidence-gated time spectra $Y = W - s \cdot B$, the statistical variance per bin is propagated as: $$\sigma_{Y, i} = \sqrt{\max(|Y_i|, 1.0) \cdot (1.0 + s)}$$ and for raw un-gated time spectra: $$\sigma_{Y, i} = \sqrt{\max(Y_i, 1.0)}$$ These uncertainties are utilized as inverse-variance weights $w_i = 1 / \sigma_{Y, i}$ in the minimization objective: $$\chi^2 = \sum_{i} \left(\frac{y_i - F(x_i)}{\sigma_i}\right)^2$$