.. _func-PseudoVoigt:

===========
PseudoVoigt
===========

.. index:: 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 :math:`G(x)` and a Lorentzian peak :math:`L(x)`, weighted by a fourth parameter :math:`\eta` (values between 0 and 1) which shifts the profile more towards pure Gaussian or pure Lorentzian when approaching 1 or 0 respectively:

.. math:: 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

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

where :math:`G'(x, \Gamma)` and `L'(x, \Gamma)` are normalized Gaussian and Lorentzian.
And :math:`\Gamma` is FWHM.

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


Native peak parameters
++++++++++++++++++++++

Pseudo-voigt function in Mantid has the following native parameters

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

From given FWHM

**Gaussian part** :math:`G'(x, \Gamma)`

.. math:: 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

.. math:: \sigma = \frac{\Gamma}{2\sqrt{2\ln(2)}}

.. math:: a_G = \frac{2}{\Gamma}\sqrt{\frac{\ln{2}}{\pi}} = \frac{1}{\sigma\sqrt{2\pi}}

.. math:: b_G = \frac{4\ln{2}}{\Gamma^2}


**Lorentzian part** :math:`L'(x, \Gamma)`

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

Thus both :math:`G'(x)` and :math:`L'(x)` are normalized.


Effective peak parameters
+++++++++++++++++++++++++

- Peak height :math:`h`:

.. math:: 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)

- :math:`\sigma`:

.. math:: \sigma = \frac{\Gamma}{2\sqrt{2\ln(2)}}


Derivative
++++++++++

- With respect to mixing parameter :math:`\eta`

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


- With respect to intensity :math:`I`

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

- With respect to peak centre :math:`x_0`

.. math:: \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}]

.. math:: \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)

.. math:: \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 :math:`\Gamma`

.. math:: \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:

.. math:: \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

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

.. math:: 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:

.. math:: \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

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

.. math:: 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 :math:`\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 :math:`\eta`: Peak height will be re-calculated.


Estimating mixing parameter
+++++++++++++++++++++++++++

Mixing parameter :math:`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

.. math:: 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:

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

.. figure:: /images/PseudoVoigt.png
   :alt: Comparison of Pseudo-Voigt function with Gaussian and Lorentzian profiles.

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