Bragg Diffraction

Bragg diffraction (or Bragg scattering) occurs when electromagnetic radiation, typically X-rays, scatters from the regular array of atoms in a crystal lattice. W.L. Bragg found that the crystal can be modeled as a set of parallel planes, separated by a distance d. When waves strike these planes at a glancing angle θ, constructive interference occurs if the extra path length traveled by waves reflecting from successive planes is an integer multiple of the wavelength.

Experiment: Visualizing Path Mismatch

You can test the Bragg condition in the Bragg Diffraction module. Let's explore how a mismatched angle affects the path difference and verification outcomes:

1. Set Plane spacing d to 0.2 nm.
2. Set Wavelength lambda to 0.2 nm.
3. Set Reflection order n to 1.
4. Notice the required Bragg angle is calculated as 30.000000°.
5. Set the Probe angle theta to 45. The Path Difference becomes 0.282843 nm.
6. The Mismatch is the path difference minus nλ: 0.082843 nm. This nonzero mismatch misses the selected-order Bragg condition; this model does not compute the resulting scattered intensity.
7. Restore Probe angle theta to 30. With d = 0.2, lambda = 0.2 and n = 1, the path is 0.2 nm and signed mismatch is 0 up to floating-point roundoff.
8. Set Reflection order n to 2 and Probe angle theta to 90. The required angle is 90 degrees, path difference 0.4 nm and mismatch 0. This is the backscattering limit.
9. Set Plane spacing d to 0.3, Wavelength lambda to 0.1, Reflection order n to 3 and Probe angle theta to 30. Read required angle 30 degrees, path difference 0.3 nm and mismatch approximately 0.
10. Set Plane spacing d and Wavelength lambda both to 0.2, keep n = 3. The chosen order has no real angle because 0.6 nm exceeds 2d = 0.4 nm; the inverse sine is not clamped.
11. Set Reflection order n to 1 and Probe angle theta to 0. The grazing-limit path difference is 0 and mismatch is -0.2 nm. Set Plane spacing d to 0: the input error replaces all dependent outputs. Restore 0.2 to recover.
12. Set Plane spacing d to 0.15, Wavelength lambda to 0.1, Reflection order n to 3 and Probe angle theta to 90. Exact raw-decimal equality n lambda = 2d gives a real 90-degree boundary even though naive binary multiplication can round 0.1 × 3 slightly above 0.3.