100% Gaussian and 100% Lorentzian


Examples of 100% Gaussian and 100% Lorentzian peak-shapes used to Peak-fit O (1s) signal



Peak-shape functions[edit]


Comparison of Gaussian (red) and Lorentzian (blue) synthetic peak-shapes. The HWHM (w/2) is 1.



Plot of the centered synthetic Voigt peak-shape for four cases.

Each case has a full width at half-maximum of very nearly 3.6. The black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively.



Lorentzian peak-shape function can be represented as

where L signifies a Lorentzian function standardized, for spectroscopic purposes, to a maximum value of 1;[note 1] {\displaystyle x} is a subsidiary variable defined as

where p0 is the position of the maximum (corresponding to the transition energy E), p is a position, and w is the full width at half maximum(FWHM), the width of the curve when the intensity is half the maximum intensity (this occurs at the points p = p0±w/2). The unit of p0p and w is typically wavenumber or frequency. The variable x is dimensionless and is zero at p=p0.


The Gaussian peak-shape has the standardized form,

The subsidiary variable, x, is defined in the same way as for a Lorentzian shape. Both this function and the Lorentzian have a maximum value of 1 at x = 0 and a value of 1/2 at x=±1.


The third peak-shape that has a theoretical basis is the Voigt function, a convolution of a Gaussian and a Lorentzian,

{\displaystyle V(x;\sigma ,\gamma )=\int _{-\infty }^{\infty }G(x’;\sigma )L(x-x’;\gamma )\,dx’,}

where σ and γ are half-widths. The computation of a Voigt function and its derivatives are more complicated than a Gaussian or Lorentzian.[4]