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

[eqn]

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

[eqn]

At the center x0 the value of the Gaussian is

[eqn]

The first derivative has a peak-to-peak distance of [eqn].

[eqn]
The integral of the Gaussian lineshape function is
[eqn]
Lorentzian

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

[eqn]

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

[eqn]

At the center x0 the value of the Lorentzian is

[eqn]

The first derivative has a peak-to-peak distance of [eqn].

[eqn]
The integral of the Lorentzian lineshape function is
[eqn]
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.