The Voight function is the convultion. The Pseudo-Voigt function is the linear combination of the two. Following to your key words you have to Combine the Density functions (Dichtefunktion)of the Gaussian and the lorentzian Profil. I Thin I have mentioned in my nanoscale publication. (in Detail in Book Strahleninduzierte Härtung von auf Acrylsäureestern basierte...

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For both of them you have to use the same FWHM (a combined one) The factor before the functions have to be correlated.

Dear Michael Schmitt Sir,

Thank you for comment. We know Pseudo-Voigt is a linear combination of Lorentzian and Gaussian. But I'm searching a way to add two Pseudo-Voigt functions (βf and βg) into one. As I mentioned above the addition of two Lorentzian and gaussian functions separately.

Please reconsider your question! What is beta? Is it really the functions or is it some parameters describing some properties of these functions? Usually in many parts of the literature beta refers to the integral breadth. So first we have to understand your question.

Dear Andreas Leineweber Sir,

Thank you for comment. Basically βf and βg etc. are the different instrumental factors which are participating to calculate overall instrumental broadening. So, we want to add these factors in Pseudo-Voigt manner because we assume that the curve come from XRD analysis is Pseudo-Voigt function.

Thanks and Regards

Lakshaman Kumar

This problem can be solved but it is complicated to explain, and I am not sure, whether you yourself really understand your question.

The parameters describing width and shape of a pseudo Voigt function are typically the full width at half maximum (FWHM), or alternatively the integral breadth (β) of the full function, as well as a weighing factor eta. The latter comes from the fact that a pseudoVoig is sum of an equal-area Gaussian and Lorentzian of exaxtly the same FWHM (and β). The weighting factor is (1-eta) for the Gaussian and eta for the Lorentzian. FWHM_G (or β_G; I will continue in terms of FWHM, because in my own works I used mainly FWHM) and FWHM_L are, however, not the FWHM values of this Gaussian and Lorentzian composing the pseudoVoigt (I just said that these are typically taken equal upon composing a pseudoVoigt). What is typically meant with FWHM_G and FWHM_L in the context of convoluting or deconvoluting peak-shape functions is the corresponding values pertaining to a Voigt function, i.e. a Gaussian and a Lorentzian convoluted with each other, APPROXIMATING the pseudoVoigt. So these factors are typically parameters of Voigt functions and not of pseudoVoigt fuctions. For Voigt function, also the mentioned additivity of the FWHM_L and of the squared FWHM_G are valid.

Due to the computational effort to calculate (true) Voigt functions they are often approximated by pseudoVoigt functions. This requires, however, a mapping of (FWHM_G, FWHM_L)_Voigt (FWHM, eta)_pseudoVoigt. Then you pretend to work with Voigt functions (which are easily convoluted using the mentioned additivity rules) but actually approximate them by pseudoVoigt functions. In XRD business this is e.g. done in the Thomposon-Cox-Hastings version of the pseudoVoigt function, and there a possible mapping is described (Thompson, Cox, Hastings, J. Appl. Crystallogr, 1987). Independently, Delhez, Mittemeijer et al have described this in the early 1980s in a series of papers; there, also the integral breadth has been used. I have myself referred to this issue (and to the papers) in 2004 (J. Appl. Crystallogr.) and in a review in 2011 (Z. Kristallogr.). I confess, that it also took me some time what is done upon doing convolutions with pseudoVoigt functions, but there is no other way to dig through it.