# Briet Wigner Resonance formula

The Breit-Wigner formula, also known as the Breit-Wigner resonance formula, is a mathematical expression used to describe the probability of a resonance occurring in a particle physics experiment. It is named after Gregory Breit and Eugene Wigner, who developed the formula in the 1930s.

Let us consider only s wave (l=0) neutrons as projectile of low energy and expecting a single related resonance level to derive a single Briet Wigner formula ignoring intrinsic spins of interacting particles. Thus, consideration must be given to low energy (n-p) scattering. In partial wave analysis, we have observed that phase shifts in n-p scattering is related to scattering length ak and effective range r and written as

Here k, δ and ak are related, the form of which determines shape of resonance. We introduce ak by

for all value of k.

If there is no reaction, a(k) is real and σ(r) =0. If σ(r) ≠0, a(k) is in general complex. In the partial wave analysis, the scattering amplitude nl of the ith partial wave is related with phase shift δl by the relation,

For low energy scattering, only s wave (l=0) is involved and for elastic scattering

Dropping the subscript 0, n = (cos δ + i sin δ)²

Putting a(k) = a in eq 2

cot δ= - 1/ ak

or, tan δ = - ak

cos δ = 1 / √ 1 + a²k²

sin δ = - 1 / √ 1 + a²k²

As we know the scattering cross section and reaction cross section for the channel of ith partial wave are

For real a; σr = 0

At resonance, phase shift is nearly π/2 so that

1/ a (Eo) = 0 where Eo is energy at which resonance occurs.

Expanding 1/a(E) about Eo we may write

which is the required expression for Briet Wigner Resonance formula.

The Breit Wigner formula is commonly used to fit experimental data in particle physics experiments, where resonances are observed in the cross-sections of various reactions. The formula can also be applied in other areas of physics, such as nuclear physics and condensed matter physics, where resonant phenomena occur.

This note is a part of the Physics Repository.