# Maxwell-Boltzmann Distribution

If a large number of neutrons are introduced into an infinite, non-absorbing scattering medium, that is a medium from which neutrons are neither lost by escape nor by absorption, a state of thermal equilibrium will be attained. In this state, the probabilities are equal (that a neutron will gain or lose energy in a collision with a scattering nucleus). The kinetic energies of the neutrons would then be represented by the Maxwell-Boltzmann distribution commonly referred to as the 'Maxwellian' distribution. Strictly speaking, such a distribution can be attained only if the scattering nuclei are not bound but are free to move. It will be seen shortly that this and other required conditions are not satisfied in an actual reactor system; nevertheless, it is useful to assume as a first approximation that the neutrons become thermalized to the extent that the Maxwellian distribution exists.

The Maxwell-Boltzmann distribution law can be derived from the kinetic theory of gases or by the methods of statistical mechanics. For the present purpose, the kinetic energy distribution of neutrons in thermal equilibrium at the absolute (kelvin) temperature *T* may be expressed by

where

*dn* is the number of neutrons with energies in the range from *E* to *E* + *dE n* is the total number of neutrons in the system *k* is the Boltzmann constant

The equation may be written in a slightly different form by letting *n*(*E*) represent the number of neutrons of energy *E* per unit energy interval. Then *n*(*E)dE* is the number if neutrons having energies in the range from *E* to *E* + *dE,* which is equivalent to *dn* in Equation 1.17. The latter may thus be written as

where the left side represents the fraction of the neutrons having energies within a unit energy interval at energy E. The right side of Equation 1.18 can be evaluated for various *E *at a given temperature and the Maxwellian distribution curve is obtained in this manner, indicating the variation of n(E*)/n* with the kinetic energy *E* of the neutrons.

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