In the case of solid moderators it is necessary to consider the exchange of energy between the neutron and the crystal. Energy can only be transferred in discrete amounts corresponding to the vibrational quanta of the crystal. A quantum of vibrational energy of a crystal is called phonon and in the energy exchange with neutrons the crystal absorbs or emits phonons. In the case of graphite the carbon atoms
are arranged in a hexagonal pattern in planar sheets. The valence bond between atoms acts almost completely within the separate sheets. Binding between different sheets is weak. This gives rise to a very anisotropic structure and one must distinguish between the phonon frequency spectra parallel and perpendicular to the lattice planes, рц(ы) and р±(ш), where the probability of phonons having frequencies between ш and ш + da is ри(ы) du> and p±{u>) du>, respectively.

The scattering cross-section consists of two parts: coherent and incoherent. The first contains interference effects between the neutron waves scattered from the various nuclei, analogous to X-ray diffraction. In graphite these interference effects are noticeable even in material consisting of randomly oriented crystals, but since they largely occur in that part of the scattering cross-section referring to interactions without neutron energy change they need not be treated in neutron thermalization theory, which even for graphite may be quite accurately performed in the incoherent approximation.

Using the incoherent approximation one can obtain (see ref. 2)

a(E’ -> E, cos во) = ^ VlF J Є ‘ * ‘ dt (6.9)

where оь is the bound atom scattering cross-section

with ov free atom cross-section, A ratio of scatterer to neutron mass, hк is the neutron momentum change vector

йк = m( v’ — v)

h reduced Plank constant, m neutron mass, v neutron velocity.

The scattering function (k, t) relates the change in neutron momentum to the change in quantum state of the scatterer, averaged over all orientations of the scattering material with respect to the incident neutrons.

For graphite a model developed by Parks gives13,4’

*(",f) = exp{^/0 l —^[(h + lXe-^-D + hCe^’-DldldcoJ (6.11)

with

p(cO, l) = l2p±(co) + (1 — l2)pu((d) (6.12)

M mass of the scattering atom

where / is the cosine of the angle between к and the normal to the lattice planes, and h the average number of oscillator quanta excited at the existing temperature T,

1

ft — ^ htnIkT _ j —

Substitution of (6.11) in (6.9) gives the scattering cross-sections. The difficulty of performing the time integration is avoided by a series expansion of х(к, t) called phonon expansion. These equations are programmed in various codes (e. g. SUMMIT,<5> LEAP’6,7’). The scattering kernel to be used in spectrum calculations is then obtained as

cr(E’ -* E) = f a(E’ -> E, cos во) dCl = 2tt f a(E’ ->

The frequency spectra p±(ш) and рц(ш) used in these kernel calculations for graphite are usually those obtained by Yoshimori and Kitano.’8’ Their theory is based on the assumption of four types of forces between the carbon atoms. Two act wholly within the basal planes, the third is determined by the displacement perpendicular to this plane relative to the three nearest neighbours from the same plane. The fourth is the only coupling assumed to exist between different planes and is related to the axial component of the displacement relative to the nearest neighbours in the axial direction. Comparisons of experimental and theoretical results for neutron spectra in poisoned graphite at several temperatures’3’ have shown that this model is quite adequate for all reactor physics purposes.