The energy E appearing in the Breit-Wigner formula (3.1) is the relative energy between target nucleus and neutron. This means that the thermal motion of the nuclei of reactor material should be considered and for the calculation of all reactions taking place in a reactor the relative energy should be used. In practice one uses effective cross-sections in which only the neutron energy appears. These effective cross-sections are, of course, temperature dependent, and must be defined in such a way as to give the proper reaction rate. Therefore we must have:

vacW(v, T) = j VrdO-(Vrd)P(vrei, T) dvrc,

where v = neutron velocity,

vr<!, = relative neutron-nucleus velocity, va = velocity of absorber nucleus,

P(prd, T) = probability of having the relative velocity vrel,

T = temperature.

Considering that and that

P(va, T) = M(va, T) Maxwellian distribution of nucleus velocities

we have

where E = energy corresponding to velocity v,

A = mass of the absorber in units of neutron mass.

One can demonstrate that if cr(vIci) follows the 1/u law, also cr^s will follow the 1/u law, independently of the temperature T. Of course <7ея-мт if T-»0°K. In the case of resonances, temperature has the effect of lowering the maximum of the resonance and broadening it at the same time (Doppler broadening).

Substituting the Breit-Wigner expressions in (3.5) one can obtain the Doppler broadened formulation of the resonance cross-section (the suffix eff used in (3.5) will be dropped here).

(Ta =-^сгоф(С, к) absorption cross-section,

Г„ . .

CTS = уг (Тоф + <T s, pot + <T s, int scattering cross-section,

with

crs, int = interference scattering, crs, pot = potential scattering.

і/»(£, K) and xU> x) are the so-called shape functions

. 2(E — Eo)

where к = ————-

Г

С ————— Г75 = тг — ratio of natural to Doppler width,

1 D

E0 = resonance energy.