SPECTROSCOPYVerified Reference Parity

Pseudo-Voigt Profile

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

Primary Disciplines:SpectroscopyCrystallography (XRD)Materials ScienceXPS

Mathematical Formulation

$$y = y_0 + A \left[ \eta \frac{2}{\pi \gamma} \frac{1}{1 + 4\left(\frac{x - \mu}{\gamma}\right)^2} + (1 - \eta) \frac{2\sqrt{\ln 2}}{\sqrt{\pi} \gamma} \exp\left( -4\ln 2 \left(\frac{x - \mu}{\gamma}\right)^2 \right) \right]$$
Plaintext:y = y0 + A * [eta * L(x, mu, gamma) + (1 - eta) * G(x, mu, gamma)]

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 intensitymin(y)
$A$Integrated Peak AreaIntensity · x-unitA > 0Total integrated intensity under the combined profile(max(y) - y0) * gamma_init
$\mu$Peak CenterDiffraction Angle (2θ) / Wavenumber (cm⁻¹)min(x) <= mu <= max(x)Centroid of the diffraction reflection or vibrational modex[argmax(y)]
$\gamma$Shared FWHM2θ degrees / cm⁻¹gamma > 0Full Width at Half Maximum shared by both Gaussian and Lorentzian componentsMeasured width at (max(y) - y0)/2
$\eta$Lorentzian Fraction (Shape Factor)Dimensionless (0 to 1)0 <= eta <= 1Fractional Lorentzian character. η = 0 is pure Gaussian; η = 1 is pure Lorentzian.0.5 (equal initial mixture)

When to Choose This Model

  • Fitting XRD diffraction patterns to separate crystallite size (Lorentzian) from microstrain (Gaussian)
  • High-throughput deconvolution of overlapping Raman and FTIR bands where full convolution is too slow
  • X-ray photoelectron spectroscopy (XPS) core level deconvolution with mixed instrumental and core-hole lifetimes

Typical Scientific Applications

  • XRD Rietveld structural analysis
  • Raman line shape and crystallite confinement analysis
  • XPS surface chemical state deconvolution
  • Synchrotron radiation line shape characterization

Expected Fit Profile & Curve Morphology

Shape: peak
Fitted ModelRaw Data

Example Dataset & Expected Fit Output

Synthetic 21-point XRD diffraction peak (eta = 0.52) with mixed crystallite and microstrain broadening

R²:0.9993RMSE:1.8400

Expected Converged Parameters

  • y020.450
  • A194.200
  • mu40.000
  • gamma0.620
  • eta0.520

Sample Experimental Vectors (15 points)

#x (Independent)y (Observed)
138.00021.400
238.50025.600
339.00038.900
439.40082.400
539.600142.800
639.800235.100
739.900284.600
840.000305.200
+ 7 more points available in AltaiPlot preset

Python / SciPy Reference Implementation

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

import numpy as np
from scipy.optimize import curve_fit

def pseudo_voigt_model(x, y0, A, mu, gamma, eta):
    # Normalized Gaussian with FWHM gamma
    g_part = (2.0 * np.sqrt(np.log(2)) / (np.sqrt(np.pi) * gamma)) * np.exp(-4.0 * np.log(2) * ((x - mu) / gamma)**2)
    # Normalized Lorentzian with FWHM gamma
    l_part = (2.0 / (np.pi * gamma)) / (1.0 + 4.0 * ((x - mu) / gamma)**2)
    return y0 + A * (eta * l_part + (1.0 - eta) * g_part)

y0_init = np.min(y_data)
mu_init = x_data[np.argmax(y_data)]
peak_height = np.max(y_data) - y0_init
half_idx = np.where(y_data >= y0_init + peak_height / 2.0)[0]
gamma_init = max(np.max(x_data[half_idx]) - np.min(x_data[half_idx]), 1e-4) if len(half_idx) > 1 else (np.max(x_data) - np.min(x_data)) / 10.0
A_init = peak_height * gamma_init * 1.2

p0 = [y0_init, A_init, mu_init, gamma_init, 0.5]
bounds = ([-np.inf, 0, np.min(x_data), 1e-12, 0.0], [np.inf, np.inf, np.max(x_data), np.inf, 1.0])

popt, pcov = curve_fit(pseudo_voigt_model, x_data, y_data, p0=p0, bounds=bounds)
Numerical Verification Tolerance: Max Relative Error < 5.2e-6Verified: 2026-08-16

Comparative & Alternative Models

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

Alternative Model

Gaussian Peak Profile

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

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

Lorentzian (Cauchy-Lorentz) Profile

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

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

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

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 (Pseudo-Voigt Profile)

What is the physical meaning of the eta parameter in XRD analysis?

In X-ray diffraction, eta represents the fraction of Lorentzian profile character. Higher eta values indicate dominant finite crystallite size broadening (Scherrer effect), while lower eta values indicate dominant microstrain broadening from lattice defects.

Scientific References & Citations

  • Ida, T., Ando, M., & Toraya, H. (2000). Extended pseudo-Voigt function for approximating the Voigt profile. Journal of Applied Crystallography, 33(6), 1311-1316.[Source / DOI]