SPECTROSCOPYVerified Reference Parity

Pearson VII Profile

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

Primary Disciplines:Crystallography (XRD)Materials ScienceSpectroscopy

Mathematical Formulation

$$y = y_0 + A \left[ 1 + 4\left( 2^{1/m} - 1 \right) \left( \frac{x - \mu}{\gamma} \right)^2 \right]^{-m}$$
Plaintext:y = y0 + A * [1 + 4 * (2^(1/m) - 1) * ((x - mu) / gamma)^2]^(-m)

Parameters & Physical Meanings

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

SymbolNameUnitConstraintsPhysical InterpretationInitial Guess Heuristic
$y_0$Baseline OffsetIntensityunconstrainedConstant background intensitymin(y)
$A$Peak AmplitudeIntensityA > 0Maximum peak height above baselinemax(y) - y0
$\mu$Peak Center Position2θ / cm⁻¹min(x) <= mu <= max(x)Center position of peak reflectionx[argmax(y)]
$\gamma$FWHM Peak WidthSpectral unitsgamma > 0Full Width at Half MaximumWidth at (max(y) - y0)/2
$m$Shape Parameter (Pearson Exponent)Dimensionlessm >= 0.5Controls tail heaviness: m = 1 is Lorentzian; m >= 10 approximates Gaussian.1.5

When to Choose This Model

  • Fitting asymmetric or heavy-tailed XRD reflections where Pseudo-Voigt does not capture tail behavior
  • Empirical fitting of complex amorphous or polymer diffraction halos

Typical Scientific Applications

  • XRD line profile analysis
  • Polymer crystallinity determination

Expected Fit Profile & Curve Morphology

Shape: peak
Fitted ModelRaw Data

Example Dataset & Expected Fit Output

Synthetic 21-point XRD profile with Pearson exponent m = 1.45 and FWHM = 0.85 degrees

R²:0.9994RMSE:1.4200

Expected Converged Parameters

  • y011.850
  • A273.400
  • mu28.450
  • gamma0.840
  • m1.460

Sample Experimental Vectors (17 points)

#x (Independent)y (Observed)
127.00012.400
227.50016.800
327.80028.900
428.00058.400
528.10094.200
628.200165.100
728.300248.600
828.400285.200
+ 9 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 pearson_vii_model(x, y0, A, mu, gamma, m):
    c1 = 4.0 * (2.0**(1.0 / m) - 1.0)
    return y0 + A * (1.0 + c1 * ((x - mu) / gamma)**2)**(-m)

p0 = [np.min(y_data), np.max(y_data)-np.min(y_data), x_data[np.argmax(y_data)], 1.0, 1.5]
popt, pcov = curve_fit(pearson_vii_model, x_data, y_data, p0=p0)
Numerical Verification Tolerance: Max Relative Error < 3.9e-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

True Voigt Profile (Faddeeva Convolution)

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

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

Frequently Asked Questions (Pearson VII Profile)

How does Pearson VII compare to Pseudo-Voigt?

While Pseudo-Voigt is a linear sum of Gaussian and Lorentzian functions, Pearson VII is an exponential power model that can describe sub-Lorentzian peaks (m < 1) and continuous tail shapes.

Scientific References & Citations

  • Hall, M. M., et al. (1977). The Pearson VII distribution function in X-ray profile analysis. Journal of Applied Crystallography, 10(1), 66-68.[Source / DOI]