# Fourier Transforms

Fri Jun 30 16:07:21 EST 1995

```Dear Toby,
The other responses pretty well describe what the Fourier transform is
(or how to look it up).  Just in case that is not your question, but rather
you want to know why the Fourier transform of the electron density is simply
related to the diffraction amplitudes, I think I can help.
The x-ray scattering amplitude can be thought of as having no struc-
ture.  That is, any point charge will scatter the x-ray in the same manner (of
course proportionally with the magnitude of the charge).  In diffraction, all
the x-rays scattered in one direction end up in a particular spot, and each
direction is an eigenfunction of momentum.  Thus, the scattering can be viewed
as going from one momentum eigenfunction to the sum of other momentum eigen-
functions.  The momentum eigenfunction in the co-ordinate representation is
exp(ik.r), i.e. a plane wave.  In the momentum representation, the eigenfunc-
tion is a delta-function.  In the momentum representation, the incident delta
function is scattered by the Fourier transform of the electron density to a sum
of other delta functions, and, since the scattering still has no structure, the
scattering must be proportional to the structure factors, which are the Fourier
components of the electron density.  If this was not your question, delete this