# Line shapes

The most common line shapes in EPR are Gaussian and Lorentzian. In addition, Dysonians and Voigtians are used.

Gaussian

The formula for a Gaussian absorption lineshape normalized so that its integral is 1 is




where x0 is the line centre and is the distance between the inflection points. It is related to the FWHM (full width at half height) via




At the center x0 the value of the Gaussian is




The first derivative has a peak-to-peak distance of .



The integral of the Gaussian lineshape function is


Lorentzian

The formula for a Lorentzian absorption lineshape normalized so that its integral is 1 is




where x0 is the center and is the distance between the inflection points. It is related to the FWHM (full width at half height) via




At the center x0 the value of the Lorentzian is




The first derivative has a peak-to-peak distance of .



The integral of the Lorentzian lineshape function is


Voigtian and Pseudovoigtian

The Voigt line shape is the convolution of Lorentzian and a Gaussian line shape. It cannot be expresed in closed analytical form. It can be approximated by a linear combination of a Lorentzian and a Gaussian, a so-called pseudo-Voigt function.