SPECTROSCOPYVerified Reference Parity

True Voigt Profile (Faddeeva Convolution)

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

Primary Disciplines:SpectroscopyAstrophysicsAtmospheric PhysicsPlasma Physics

Mathematical Formulation

$$V(x; \sigma, \gamma) = \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) \, dx' = \frac{\text{Re}[w(z)]}{\sigma \sqrt{2\pi}}$$
Plaintext:V(x) = Re[wofz(z)] / (sigma * sqrt(2*pi)) where z = (x - mu + i*gamma) / (sigma * sqrt(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 spectral background intensitymin(y)
$A$Integrated Peak AreaIntensity · x-unitA > 0Total integrated transition probability(max(y) - y0) * (sigma_init + gamma_init)
$\mu$Peak Center Positionnm / cm⁻¹min(x) <= mu <= max(x)Transition resonance frequencyx[argmax(y)]
$\sigma$Gaussian Width ComponentSpectral unitssigma > 0Thermal Doppler broadening width0.5 * (FWHM / 2.355)
$\gamma$Lorentzian HWHM ComponentSpectral unitsgamma > 0Collisional and radiative lifetime half-width0.5 * (FWHM / 2.0)

When to Choose This Model

  • Atmospheric trace gas remote sensing and satellite absorption spectroscopy
  • Stellar absorption line profile deconvolution in astrophysics
  • Laser-induced breakdown spectroscopy (LIBS) plasma diagnostics

Typical Scientific Applications

  • Atmospheric FTIR line fitting
  • Plasma emission spectroscopy
  • Stellar atmosphere spectral analysis

Expected Fit Profile & Curve Morphology

Shape: peak
Fitted ModelRaw Data

Example Dataset & Expected Fit Output

Synthetic 21-point atmospheric absorption line with sigma = 1.2 cm⁻¹ (Doppler) and gamma = 1.0 cm⁻¹ (Lorentzian)

R²:0.9996RMSE:0.4800

Expected Converged Parameters

  • y00.980
  • A245.800
  • mu1005.010
  • sigma1.210
  • gamma0.980

Sample Experimental Vectors (17 points)

#x (Independent)y (Observed)
11000.0001.200
21001.0001.800
31002.0003.400
41003.0008.900
51003.50015.600
61004.00028.400
71004.50045.200
81004.80054.800
+ 9 more points available in AltaiPlot preset

Python / SciPy Reference Implementation

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

import numpy as np
from scipy.special import wofz
from scipy.optimize import curve_fit

def voigt_model(x, y0, A, mu, sigma, gamma):
    z = ((x - mu) + 1j * gamma) / (sigma * np.sqrt(2.0))
    v_profile = np.real(wofz(z)) / (sigma * np.sqrt(2.0 * np.pi))
    return y0 + A * v_profile

p0 = [np.min(y_data), np.max(y_data)-np.min(y_data), x_data[np.argmax(y_data)], 1.0, 1.0]
popt, pcov = curve_fit(voigt_model, x_data, y_data, p0=p0)
Numerical Verification Tolerance: Max Relative Error < 4.2e-6Verified: 2026-08-16

Comparative & Alternative Models

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

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

Gaussian Peak Profile

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

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

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 (True Voigt Profile (Faddeeva Convolution))

When should I use True Voigt over Pseudo-Voigt?

True Voigt is essential when exact physical parameters (such as true gas kinetic temperature from sigma and natural lifetime from gamma) must be isolated with zero approximation error.

Scientific References & Citations

  • Armstrong, B. H. (1967). Spectrum line profiles: The Voigt function. Journal of Quantitative Spectroscopy and Radiative Transfer, 7(1), 61-88.[Source / DOI]