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2 "bravais": the Reciprocal Lattice and X-ray Diffraction










"bravais" illustrates, in 2 dimensions, the relationships between a crystal structure and its associated reciprocal lattice. Arbitrary two-dimensional structures with one or two atoms per cell can be constructed and the corresponding reciprocal lattice displayed. bravais also simulates the x-ray diffraction for these structures illustrating the concepts of structure factor and extinction.

If you can spare the time to load the file (220K), you will see a portion of the "bravais" screen in animation. (An 85K version is also available.) The two dimensional crystal is on the left, and its diffraction pattern is on the right. In the animation, the x coordinate of one of the atoms is gradually being changed from 0 to 0.5.

The pairs of displays you can choose below show the two-dimensional crystal on the right and the corresponding x-ray diffraction pattern on the left. (If the two images appear one above the other, widen the display area by dragging the lower right corner of the panel to the right.) In these examples the two atoms in the cell are at positions [0,0] and [0.5,y]. By clicking on the "next frame" button, you can work through a series of frames in which some parameter of the system is varied. (The student, of course, has the freedom to adjust any parameter at will.) If you wish to see an image in full size, click on the image.

In the first example, the initial frame is y=0. Clicking on the "next frame" button successively changes y in steps of 0.1 until the second atom is displaced by the full lattice constant. Can you predict the diffraction pattern for y=0.5 before you get there? (If frames are incomplete or missing, "Reload" the page and wait until loading is completed before clicking.)

start (85K)

Another option in "bravais" is to change the atomic scattering amplitude f for one of the atoms. In the second sequence of frames y=0.5 and the atomic scattering factor of one atom relative to the other is changed in steps of 0.2 from f=0.0 to f=1.0. This change is indicated schematically by a change of color of one of the atoms.

start (40K)

In this last example, we've made a mistake. The x-ray pattern is NOT that appropriate for the structure displayed. How can we fix the mistake? 

       X-ray pattern                    crystal structure


Table of contents for Chapter 2 of "Simulations for Solid State Physics"
  1. Introduction
  2. Crystal lattice vs. crystal structure
  3. Lattice planes
  4. Reciprocal lattice
  5. Diffraction conditions
  6. Atomic form factor
  7. Structure factor
    1. Extinctions
    2. Atomic inequivalence
    3. Positional information
    4. Structural determination
  8. Summary
  9. Appendix: "bravais" -- the program
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