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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

$$pV(x)=I\cdot (\eta \cdot G^{\prime }(x,\mathrm{\Gamma })+(1-\eta )\cdot L^{\prime }(x,\mathrm{\Gamma }))$$

where $G^{\prime }(x,\mathrm{\Gamma })$ and L’(x, Gamma) are normalized Gaussian and Lorentzian. And $\mathrm{\Gamma }$ is FWHM.

In Fullprof notation, $H$ is used for FHWM instead of $\mathrm{\Gamma }$. 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 $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ (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 $G^{\prime }(x,\mathrm{\Gamma })$

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

where

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

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

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

Lorentzian part $L^{\prime }(x,\mathrm{\Gamma })$

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

Thus both $G^{\prime }(x)$ and $L^{\prime }(x)$ are normalized.

Effective peak parameters

• Peak height $h$:

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

• $\sigma$:

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

Derivative

• With respect to mixing parameter $\eta$

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

• With respect to intensity $I$

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

• With respect to peak centre $x_0$

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

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

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

• With respect to peak width $\mathrm{\Gamma }$

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

For Gaussian part:

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

$$t_1=\frac{-1}{\mathrm{\Gamma }}{a}_G{e}^{-b_G(x-x_0{)}^2}=\frac{-1}{\mathrm{\Gamma }}{G}^{\prime }(x,\mathrm{\Gamma })$$

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

For Lorentzian part:

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

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

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

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 $\mathrm{\Gamma }$): 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

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

This equation has several issues:

1. It does not have normalized Gaussian and Lorentzian.

2. At $x=x_0$, $pV(x_0)=h$. By this definition, the mixing ratio factor $\eta$ between Gaussian and Lorentzian is 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.

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

C++ source: PseudoVoigt.cpp