PseudoVoigt¶

Description¶

The Pseudo-Voigt function is an approximation for the Voigt function, which is a convolution of Gaussian and Lorentzian function. It is often used as a peak profile in powder diffraction for cases where neither a pure Gaussian or Lorentzian function appropriately describe a peak.

Instead of convoluting those two functions, the Pseudo-Voigt function is defined as the sum of a Gaussian peak $G(x)$ and a Lorentzian peak $L(x)$ , weighted by a fourth parameter $\eta$ (values between 0 and 1) which shifts the profile more towards pure Gaussian or pure Lorentzian when approaching 1 or 0 respectively:

$pV(x) = \eta G(x) + (1 - \eta)L(x)$

Both functions share three parameters: Height (height of the peak at the maximum), PeakCentre (position of the maximum) and FWHM (full width at half maximum of the peak).

Thus the Pseudo-voigt function can be expressed as

$pV(x) = I \cdot (\eta \cdot G'(x, \Gamma) + (1 - \eta) \cdot L'(x, \Gamma))$

where $G'(x, \Gamma)$ and L’(x, Gamma) are normalized Gaussian and Lorentzian. And $\Gamma$ is FWHM.

In Fullprof notation, $H$ is used for FHWM instead of $\Gamma$ . In the code, gamma is used for FWHM in order to avoid confusion with peak height $h$ . To be in line with it, we prefer to use $\Gamma$ for FWHM here.

Native peak parameters¶

Pseudo-voigt function in Mantid has the following native parameters

Peak intensity $I$ : shared peak height between Gaussian and Lorentzian.
Peak width FWHM $\Gamma$ (or $H$ ): shared FWHM be between Gaussian and Lorentzian
Peak position $x_0$
Gaussian ratio $\eta$ : ratio of intensity of Gaussian.

From given FWHM

Gaussian part $G'(x, \Gamma)$

$G'(x, \Gamma) = a_G \cdot e^{-b_G (x - x_0)^2} = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-x_0)^2}{2\sigma^2}}$

where

$\sigma = \frac{\Gamma}{2\sqrt{2\ln(2)}}$

$a_G = \frac{2}{\Gamma}\sqrt{\frac{\ln{2}}{\pi}} = \frac{1}{\sigma\sqrt{2\pi}}$

$b_G = \frac{4\ln{2}}{\Gamma^2}$

Lorentzian part $L'(x, \Gamma)$

$L'(x) = \frac{1}{\pi} \cdot \frac{\Gamma/2}{(x-x_0)^2 + (\Gamma/2)^2}$

Thus both $G'(x)$ and $L'(x)$ are normalized.

Effective peak parameters¶

Peak height $h$ :

$h = I \cdot (\eta \cdot a_G + (1 - \eta) \cdot \frac{2}{\pi\cdot \Gamma}) = \frac{2 I}{\pi \Gamma} (1 + (\sqrt{\pi\ln{2}}-1)\eta)$

$\sigma$ :

$\sigma = \frac{\Gamma}{2\sqrt{2\ln(2)}}$

Derivative¶

With respect to mixing parameter $\eta$

$\frac{\partial pV(x)}{\partial \eta} = I \cdot [G'(x, \Gamma) - L'(x, \Gamma)]$

With respect to intensity $I$

$\frac{\partial pV(x)}{\partial I} = \eta G'(x, \Gamma) + (1-\eta) L'(x, \Gamma)$

With respect to peak centre $x_0$

$\frac{\partial pV(x)}{\partial x_0} = I \cdot [\eta \frac{\partial G'(x, \Gamma)}{\partial x_0} + (1 - \eta) \frac{\partial L'(x, \Gamma)}{\partial x_0}]$

$\frac{\partial G'(x, \Gamma)}{\partial x_0} = a_G\cdot e^{(-b_G(x-x_0)^2)} (-b_G) (-2) (x - x_0) = 2 b_G (x - x_0) G'(x, \Gamma)$

$\frac{\partial L'(x, \Gamma)}{\partial x_0} = \frac{\Gamma}{2\pi} (-1) (-2) (x - x_0) \frac{1}{[(x - x_0)^2 + \frac{\Gamma^2}{4}]^2} = \frac{(x-x_0)\Gamma}{\pi[(x - x_0)^2 + \frac{\Gamma^2}{4}]^2} = \frac{4\pi(x-x_0)}{\Gamma}[L'(x, \Gamma)]^2$

With respect to peak width $\Gamma$

$\frac{\partial pV(x)}{\partial \Gamma} = I \cdot [\eta \frac{\partial G'(x, \Gamma)}{\partial \Gamma} + (1 - \eta) \frac{\partial L'(x, \Gamma)}{\partial \Gamma}]$

For Gaussian part:

$\frac{\partial G'(x, \Gamma)}{\partial \Gamma} = \frac{\partial a_G}{\partial \Gamma} e^{-b_G(x-x_0)^2} + a_G \frac{\partial e^{-b_G(x-x_0)^2}}{\partial \Gamma} = t_1 + t_2$

$t_1 = \frac{-1}{\Gamma} a_G e^{-b_G(x-x_0)^2} = \frac{-1}{\Gamma} G'(x, \Gamma)$

$t_2 = a_G e^{-b_G(x-x_0)^2} (-1) (x-x_0)^2 \frac{\partial b_G}{\partial \Gamma} = G'(x, \Gamma) (-1) (x-x_0)^2 \frac{-2}{\Gamma} b_G = \frac{2 b_G (x-x_0)^2 G'(x, \Gamma)}{\Gamma}$

For Lorentzian part:

$\frac{\partial L'(x, \Gamma)}{\partial \Gamma} = \frac{1}{\pi} \frac{\partial (\Gamma/2)}{\partial \Gamma}\frac{1}{(x-x_0)^2 + (\Gamma/2)^2} + \frac{\Gamma}{2}\frac{\partial \frac{1}{(x-x_0)^2 + (\Gamma/2)^2}}{\partial \Gamma} = t_3 + t_4$

$t_3 = \frac{1}{2\pi} \frac{1}{(x-x_0)^2 + (\Gamma/2)^2} = \frac{L'(x, \Gamma)}{\Gamma}$

$t_4 = \frac{\Gamma}{2\pi}\frac{-1}{[(x-x_0)^2 + (\Gamma/2)^2]^2} \frac{\Gamma}{2} = -\pi[L'(x, \Gamma)]^2$

Set peak parameters¶

Peak parameters can be estimated from observation. But some peak parameters are correlated, because peak height is not a basic parameter of Pseudo-voigt.

Here is the summary:

Peak width (FWHM $\Gamma$ ): Peak height will be re-calculated.
Peak intensity: Peak height will be re-calculated.
Peak height: Peak intensity, mixing pamameter or FWHM can be re-calculated depending on user’s choice.
Peak centre: No other parameter will be affected.
Mixing parameter $\eta$ : Peak height will be re-calculated.

Estimating mixing parameter¶

Mixing parameter $eta$ can be estimated from the observed value of peak’s height, FWHM and intensity.

About previous implementation¶

Before Mantid release v3.14, the equation of Pseudo-Voigt is defined as

$pV(x) = h \cdot [\eta \cdot \exp(-\frac{(x-x_0)^2}{-2\sigma^2}) + (1-\eta)\frac{(\Gamma/2)^2}{(x-x_0)^2 + (\Gamma/2)^2}]$

This equation has several issues:

It does not have normalized Gaussian and Lorentzian.
At $x = x_0$ , $pV(x_0) = h$ . By this definition, the mixing ratio factor $\eta$ between Gaussian and Lorentzian is the the intensity ratio at $x = x_0$ . But it does not make sense with other $x$ value. According to the literature or manual (Fullprof and GSAS), $\eta$ shall be the ratio of the intensities between Gaussian and Lorentzian.

The figure below shows data together with a fitted Pseudo-Voigt function, as well as Gaussian and Lorentzian with equal parameters. The mixing parameter for that example is 0.7, which means that the function is behaving more like a Gaussian.

Comparison of Pseudo-Voigt function with Gaussian and Lorentzian profiles.

Properties (fitting parameters)¶

Name	Default	Description
Mixing	0.5
Intensity	1.0
PeakCentre	0.0
FWHM	1.0

Categories: FitFunctions | Peak

Source¶

C++ header: PseudoVoigt.h (last modified: 2020-03-20)

C++ source: PseudoVoigt.cpp (last modified: 2020-03-20)