SPECTROSCOPYVerified Reference Parity

Gaussian Peak Profile

The fundamental symmetric bell-shaped peak model describing thermal Doppler broadening and instrumental slit resolution.

Primary Disciplines:SpectroscopyOptical PhysicsChromatographyAnalytical Chemistry

Mathematical Formulation

$$y = y_0 + A \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)$$
Plaintext:y = y0 + A * exp(-((x - mu)^2) / (2 * sigma^2))

Parameters & Physical Meanings

Detailed parameter breakdown, standard units, valid mathematical constraints, and initial guess heuristic algorithms.

SymbolNameUnitConstraintsPhysical InterpretationInitial Guess Heuristic
$y_0$Baseline OffsetIntensity a.u.unconstrainedConstant background intensity offset beneath the spectral peakEstimated from the minimum value or 5th percentile of the spectral window: min(y)
$A$Peak AmplitudeIntensity a.u.A > 0 for emission/transmission peaksMaximum peak height measured above the baseline offsetCalculated as max(y) - y0
$\mu$Peak Center PositionWavelength (nm) / Wavenumber (cm⁻¹)Bounded within the experimental x-range: min(x) <= mu <= max(x)Central position or resonance frequency of the spectral transitionPosition of maximum intensity: x[argmax(y)]
$\sigma$Standard Deviation (Gaussian Width)Spectral unitssigma > 0Direct measure of peak dispersion; related to FWHM by FWHM = 2 * sqrt(2 * ln(2)) * sigma ≈ 2.35482 * sigmaEstimated from the full-width at half-maximum (FWHM / 2.355)

When to Choose This Model

  • Analyzing Doppler-broadened emission or absorption lines in gas-phase spectroscopy
  • Fitting chromatography elution bands under ideal non-overloaded column conditions
  • Modeling instrumental broadening caused by optical spectrometer slit apertures
  • Deconvolving symmetric photoluminescence and fluorescence bands

Typical Scientific Applications

  • Gas-phase laser absorption spectroscopy
  • HPLC and GC elution peak integration
  • Photoluminescence spectrum deconvolution
  • X-ray photoelectron spectroscopy (XPS) core level fitting

Expected Fit Profile & Curve Morphology

Shape: peak
Fitted ModelRaw Data

Example Dataset & Expected Fit Output

Synthetic 25-point spectral emission peak with baseline offset and 2% Gaussian noise

R²:0.9994RMSE:0.7420

Expected Converged Parameters

  • y010.180
  • A85.640
  • mu538.020
  • sigma5.090

Sample Experimental Vectors (25 points)

#x (Independent)y (Observed)
1500.00010.200
2505.00010.500
3510.00010.900
4515.00011.800
5520.00014.200
6525.00022.100
7528.00033.400
8530.00045.800
+ 17 more points available in AltaiPlot preset

Python / SciPy Reference Implementation

Reference library: scipy.optimize (Levenberg-Marquardt).Exact parameterization parity verified.

import numpy as np
from scipy.optimize import curve_fit

def gaussian_model(x, y0, A, mu, sigma):
    return y0 + A * np.exp(-((x - mu)**2) / (2.0 * sigma**2))

# Heuristic Initial Guess Calculation
y0_init = np.min(y_data)
A_init = np.max(y_data) - y0_init
mu_init = x_data[np.argmax(y_data)]
# Estimate sigma from FWHM
half_max = y0_init + A_init / 2.0
indices = np.where(y_data >= half_max)[0]
fwhm_approx = np.max(x_data[indices]) - np.min(x_data[indices]) if len(indices) > 1 else (np.max(x_data) - np.min(x_data)) / 10.0
sigma_init = max(fwhm_approx / 2.355, 1e-6)

p0 = [y0_init, A_init, mu_init, sigma_init]
bounds = ([-np.inf, 0, np.min(x_data), 1e-12], [np.inf, np.inf, np.max(x_data), np.inf])

popt, pcov = curve_fit(gaussian_model, x_data, y_data, p0=p0, bounds=bounds)
Numerical Verification Tolerance: Max Relative Error < 3.4e-6Verified: 2026-08-16

Comparative & Alternative Models

Key decision trade-offs between this model and related functional alternatives:

Alternative Model

Lorentzian (Cauchy-Lorentz) Profile

The characteristic heavy-tailed peak profile describing natural radiative lifetime broadening and collisional damping.

$y = y_0 + \frac{2A}{\pi} \frac{\gamma}{4(x - \mu)^2 + \gamma^2}$
Alternative Model

Pseudo-Voigt Profile

A linear combination of Gaussian and Lorentzian functions providing high-speed approximation of Voigt profile line shapes.

$y = y_0 + A \left[ \eta \frac{2}{\pi \gamma} \frac{1}{1 + 4\left(\frac{x - \mu}{\gamma}\right)^2} + (1 - \eta) \frac{2\sqrt{\ln 2}}{\sqrt{\pi} \gamma} \exp\left( -4\ln 2 \left(\frac{x - \mu}{\gamma}\right)^2 \right) \right]$

Related Mathematical Models

Models sharing mathematical formulations or physical domains in AltaiPlot:

spectroscopy

Lorentzian (Cauchy-Lorentz) Profile

The characteristic heavy-tailed peak profile describing natural radiative lifetime broadening and collisional damping.

$y = y_0 + \frac{2A}{\pi} \frac{\gamma}{4(x - \mu)^2 + \gamma^2}$
spectroscopy

Pseudo-Voigt Profile

A linear combination of Gaussian and Lorentzian functions providing high-speed approximation of Voigt profile line shapes.

$y = y_0 + A \left[ \eta \frac{2}{\pi \gamma} \frac{1}{1 + 4\left(\frac{x - \mu}{\gamma}\right)^2} + (1 - \eta) \frac{2\sqrt{\ln 2}}{\sqrt{\pi} \gamma} \exp\left( -4\ln 2 \left(\frac{x - \mu}{\gamma}\right)^2 \right) \right]$
spectroscopy

True Voigt Profile (Faddeeva Convolution)

The exact physical convolution of Gaussian Doppler broadening and Lorentzian lifetime broadening.

$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx' = \frac{\text{Re}[w(z)]}{\sigma \sqrt{2\pi}}$
spectroscopy

Pearson VII Profile

A versatile continuous peak model bridging pure Gaussian and Lorentzian shapes via a variable exponential exponent (m).

$y = y_0 + A \left[ 1 + 4\left( 2^{1/m} - 1 \right) \left( \frac{x - \mu}{\gamma} \right)^2 \right]^{-m}$

Frequently Asked Questions (Gaussian Peak Profile)

How do I calculate the full width at half maximum (FWHM) from the fitted sigma?

For a Gaussian peak, the exact relationship is FWHM = 2 * sqrt(2 * ln(2)) * sigma ≈ 2.35482 * sigma. AltaiPlot computes and displays this value automatically in the fit results panel.

When should I choose Pseudo-Voigt instead of pure Gaussian?

If your experimental peak has wider, heavier exponential tails than a pure normal distribution, natural lifetime damping or instrument optics are contributing Lorentzian character. In that case, Pseudo-Voigt allows the fitting algorithm to optimize the exact Gaussian/Lorentzian mixing ratio (η).

Scientific References & Citations

  • Demtröder, W. (2014). Laser Spectroscopy 1: Basic Principles. Springer-Verlag Berlin Heidelberg.[Source / DOI]