Scatter from an Atom
| A neutral atom of atomic number Z consists of Z electron. So it is expected that the scatter of an atom would just be a multiple of the free electron result. The problem is though that the electrons surrounding an atom do not represent a point scatterer. If they did the scattered amplitude would simply be Zye. Instead an atomic scattering factor, f, is introduced to account for the distribution of the electron density re(r) about the scattering origin. The development of an equation for f involves studying the scatter from small volume elements, dV, containing electron density, re(r), located a radius r from the origin. |
| This picture should be familiar. The idea is to compare the scatter from an element dV located at the origin and at r from the origin. |
| As normal scattering the the path
difference can be inferred from the above diagrams at various
observation (scattering) angles. In (a) there no path difference
and the scattering from dV is the same throughout the atom. In (b)
there is a small difference (DA-BC) and in (c) it's actually DA+BC.
In cases (b) and (c) the interference will be destructive (more so in
(c) than in (b)) and the scattered amplitude will be smaller.
The exact formulation of an analytical result for f involves a formal integration of the scattering from all the electron density in the total volume of the atom. |
| Changing to spherical coordinates the following integral results |
| This can only be integrated if there is a
suitable expression for re(r).
Expressions for the electron density boil down to solving the Schrödinger
equation for the electron probability. This can only be done
exactly in the instance of a hydrogen atom. For other atoms approximation can be
formulated (i.e. Hartree self-consistent field theory).
Results of f(sin(q)/l) are tabulated in the International Tables. |