### 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…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…