The Breit-Wigner formula describes how the cross section σ of nuclear reactions depends on the energy of the incident particles near a resonance value of the energy. It was proposed by the American physicists G. Breit and E. Wigner in 1936. It is sometimes also called the dispersion formula, because of its similarity to the expression that describes the dispersion of light. When an incident particle interacts with a target nucleus, a compound nucleus C may be formed which has a number of quasi-stationary energy levels. The width of a level Γ is related to the lifetime τ of the quasi-stationary state by Γ=h/τ. If the energy (in the centre-of-inertia system) is close to the energy of one of the levels of the compound nucleus, the probability of forming the compound nucleus becomes especially large and the cross sections rise sharply, forming resonance maxima. For an isolated resonance (when Γ is many times smaller than the energy distance to other resonances with the same quantum numbers) a is then determined by the Breit-Wigner formula. A similar situation arises in interactions of elementary particles if their total energy is close to the mass of an unstable elementary particle — a resonance with the corresponding quantum numbers (spin, parity, strangeness and so on).
The Breit-Wigner formula for the cross section of the reaction a+X→C→b+Y, which proceeds through a compound nucleus (or resonance) C of spin I, has the following form near the resonance energy Ε0:
The indices i and f denote the entrance and the exit channel; X is the de Broglie wavelength, Ε is the kinetic energy of the particles a and X, Ia, Ix are the spins of the particles and X; Γf and Γi are the partial widths of the level of the compound nucleus C, related to the probability of its decay (and formation) through the different channels f and i. The total width of the level is Γ=ΣiΓi (Fig.).
The cross section σ of the reaction 14C (pn)15N (from the yield of neutrons at an angle of 90°) as a function of the proton energy Εp in the laboratory coordinate system. The two maxima correspond to two levels of the compound nucleus.
Nuclear widths vary, with the excitation energy and the mass of the nucleus, from 0.1 eV to hundreds of keV. In the case of elementary particles — resonances, the widths lie in the range from a few tens of keV to hundreds of MeV.