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

Comments

More Cowbell! Record Production using Google Forms and Charts

First, the what : This article shows how to embed a new Google Form into any web page. To demonstrate ths, a chart and form that allow blog readers to control the recording levels of each instrument in Blue Oyster Cult's "(Don't Fear) The Reaper" is used. HTML code from the Google version of the form included on this page is shown and the parts that need to be modified are highlighted. Next, the why : Google recently released an e-mail form feature that allows users of Google Documents to create an e-mail a form that automatically places each user's input into an associated spreadsheet. As it turns out, with a little bit of work, the forms that are created by Google Docs can be embedded into any web page. Now, The Goods: Click on the instrument you want turned up, click the submit button and then refresh the page. Through the magic of Google Forms as soon as you click on submit and refresh this web page, the data chart will update immediately. Turn up the:

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

Now available as a Kindle ebook for 99 cents ! Get a spiffy ebook, and fund more physics 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

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 sim

Copasetic Flow

Search This Blog