If the resonances are well separated the cross-section for a reaction x is given by the Breit-Wigner single level formula

(Е-Еа)2+ЇГ2

J = spin of target nucleus,

I = spin of compound nucleus,

E0 = resonance energy,

К = reduced de Broglie neutron wavelength,

E = energy of incoming neutron, or relative energy between neutron and target nucleus if this is not at rest (see § 3.4).

As one can see from (3.1) the cross-section has a maximum for E = E0 and reaches half of this maximum value for E — E0 = Г12, so that Г is the width at half height of the resonance (Fig. 3.1). The Breit-Wigner formula cannot be extrapolated to E-» 0 because the parameter Г„ can be considered independent of energy only in small intervals around the resonance. At very low energy

r„ = t„0Ve

This formula substituted in the Breit-Wigner expression gives the l/v behaviour of the low-energy cross-section of most isotopes. This 1 lv part can often be calculated from the known resonance parameters, but for some isotopes (e. g. 232Th) negative energy resonances (bound levels) have to be considered in order to reproduce the experimental 1 lv cross-sections.

As we have seen the scattering cross-section is composed of a fairly constant potential scattering and a strongly energy-dependent resonance scattering. In the vicinity of resonances these two terms interfere so that the scattering cross-section can be smaller than the potential part alone. In general we have

0’s O’ s, pot "l — O’ s, res 3“ O’ s, int

where ors, inl is the interference part of the scattering cross-section. Considering as a reference frame a coordinate system centred on the centre of mass of the two particles (nucleus and neutron) involved in the collision and defining p0 the cosine of the scattering angle in this reference system, we can expand the scattering cross-section in Legendre polynomials

сгЛр-о) = 2 СГ, Р,(ро), (3.3)

/=0

/ = 0 gives the isotropic component (s-wave scattering) while the component / = 1 is called p-wave scattering.

The /th partial wave of the scattering cross-section is given by:

і—— resonance———— 1 і———— interference———- 1 |—————— —potential—— 1

= —p f ІГ2 [Г„2 — 2Äà sin2 S, + 2Г(2і — E0) sin 2&] + 4ttX2(21 + 1) sin2 Si

{Л — Ло) +4І

(3.4)

with Si = phase shift associated with the potential scattering.