The neutron flux is the product of the number of neutrons per unit volume and the neutron speed. It has the physical significance of being the total distance traveled in unit time by all the neutrons present in unit volume. It seems reasonable that the rate of reaction of neutrons should be proportional to the distance they travel in unit time. The flux has the dimensions of neutrons per square centimeter per second. Typical values of the flux in nuclear reactors range from around 10u to 1014 л/(cm2 — s).

To specify completely the neutron activity and to choose the proper cross sections for calculating the reaction rate constant, it is necessary to know the distribution of neutron concentration, or neutron flux, with respect to energy. In a thermal reactor the distribution of neutrons in thermal equilibrium with nuclei at an absolute temperature T is similar to the distribution of gas molecules in thermal equilibrium and can be approximated by the Maxwell- Boltzmann distribution

(2.44)

where nM(v) dv = number of thermal neutrons per unit volume with speeds between v and v + dv

nM = total number of thermal neutrons per unit volume m = mass of neutron

к = Boltzmann’s constant, 1.38054 X 10-23 J/K
The most probable speed v0 is that for which nM(v) is a maximum, or

(2.45)

For neutrons in thermal equilibrium at 20°C, the most probable speed from Eq. (2.45) is 2200 m/s.

The neutron kinetic energy E is related to the neutron speed by

From Eqs. (2.45) and (2.46) the energy E0 at the most probable speed is

E0 = kT (2.47)

and for thermal neutrons at 20°C, E0 has the value of 0.0253 eV.

By means of Eq. (2.46), the speed distribution, Eq. (2.44), can be transformed into an energy distribution,

nM(E) dE = nM Ш3/2 ЕУ2є-е^т dE (2.48)

where nM(E)dE is the number of neutrons per unit volume with energies between E and E + dE.

The distributions presented in Eqs. (2.44) and (2.48) can be written in dimensionless form in terms of the most probable speed u0 and the energy E0 at the most probable speed as follows:

пм(Фо) = J_ M2e-W

nM n/tt Vo) K ’

The left side of Eq. (2.48) is the fraction of the total thermal neutrons that have a speed ratio v/v0, per unit increment in speed ratio v/v0. Similarly,

nM(E/E0) _ _2_ пм ^/n

Dimensionless flux distributions may be obtained by multiplying the neutron density distribu­tions by the neutron speed ratio vjv0:

The dimensionless neutron density and flux distributions, Eqs. (2.49) to (2.52), are plotted in Figs. 2.10 and 2.11.