Saturday, January 19, 2013

Electron Diffraction and Reciprocal/Fourier Sapces

After several lectures in other classes where the use of electron diffraction was described in terms of reciprocal spaces, (the Fourier transform of position as opposed to position itself), I finally saw a great explanation of why we work in the reciprocal space to learn about the structure of crystals and other materials in plain position space.  The diagram shown below, (picture 1 on Google+), sums it all up.

Put very simply, there's a very clean relationship for how an electron is diffracted based on the electron's momentum wave which is the Fourier reciprocal of the probability vs. position wave in quantum mechanics.  However, to write this relationship down in its cleanest form, you first have to describe the diffracting media in the reciprocal space as well.  Hence, the emphasis on the reciprocal space even though results are often finally translated back to position space for human consumption.

The atoms that form the cell structure in crystals are distributed periodically.  There positions can be written using basis vectors determined by the type of the crystal cell and the position of the next cell over can be inferred by the periodicity.  It's this structure that lends itself to being Fourier transformed.

The electron wave function is diffracted by this grid of atoms that forms the cell structure, (scattering points).  The magnitude of the momentum of each electron can't be changed, only the direction of the momentum, and then only by the magnitude of the reciprocal space vector, (labeled G), that joins one scattering point to the next, (see the diagram above).  The angle between the momentum and the G vector is denoted by phi and the angular distance between spots on a diffraction pattern can be calculated using, (picture 2).

Here's a question.  Does the angle between momentum and G have to be transformed back into position space to correspond to the observed diffraction pattern?  I'm guessing not since the change in direction of the momentum variable directly corresponds to where that portion of the electron beam 'landed'.  In this case, we're actually looking at the deflection of the momentum vector on the screen.  Another great reason that lectures about electron diffraction seem to always start in momentum space!

Picture of the Day:
From 1/19/13

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