### 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

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

### Kids R Kapable

Just a little note to concerned ‘grownups’ everywhere.  If you look at a kid—and I mean really look—I don’t mean notice a person shorter than you, I mean make eye contact, notice their facial expression and observe their body language—If you look at a kid, don’t assume they need your help unless they’re obviously distressed, or ask for it.  You might think this is difficult call to make.  You might think, not having kids of your own, that you’re unable to make this determination.  You are.  You do in fact, already have the skills even if you’ve never been around kids  It’s a remarkably simple call to make, just use the exact same criteria you would for determining if an adult was in distress.  Because, guess what, kids and adults are in fact the same species of animal and communicate in the same way.  Honest.  If someone—adult or child—doesn’t need your help, feel free to say hello, give a wave, give a smile, but don’t—do not—try to force help on anyone that doesn’t want or need it.

Y…