Bragg Diffraction

Explore ideal elastic scattering from parallel crystal planes and the Bragg condition for constructive interference.

Concept Overview

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.

Mathematical Definition

The geometric path difference between two rays reflecting off adjacent parallel planes separated by distance d is given by 2d sin(θ). For constructive interference to occur, this path difference must equal an integer number of wavelengths nλ.

nλ = 2d sin(θ_B)

n: The integer order of reflection (1, 2, 3...)
λ: The wavelength of the incident wave
d: The perpendicular distance between adjacent crystal planes
θ_B: The Bragg angle, measured from the crystal plane (not the normal)

Worked Example

Let's verify the Bragg angle for a specific crystal and X-ray wavelength. Consider a first-order reflection (n = 1) from planes separated by d = 0.282 nm (like NaCl) using an X-ray of wavelength λ = 0.150 nm. We use the formula to find the angle at which the path difference exactly matches the wavelength.

nλ = 2d sin(θB) 1(0.150 nm) = 2(0.282 nm) sin(θB) 0.150 = 0.564 sin(θB) sin(θB) = 0.265957 θB = arcsin(0.265957) ≈ 15.42°

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:

Set Plane spacing d to 0.2 nm.
Set Wavelength lambda to 0.2 nm.
Set Reflection order n to 1.
Notice the required Bragg angle is calculated as 30.000000°.
Set the Probe angle theta to 45. The Path Difference becomes 0.282843 nm.
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.
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.
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.
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.
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.
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.
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.

Model Limitations

The interactive visualization provides a geometric interpretation of Bragg's Law but has several simplifications compared to real-world physics experiments:

Intensities and Widths: The model strictly predicts allowed geometric angles (delta functions in ideal infinite crystals). It does not compute diffraction peak widths (which depend on crystal size via the Scherrer equation) or reflection intensities (which depend on the atomic structure factor).
Elastic Scattering Only: It assumes perfectly elastic scattering (no Compton scattering or phonon interactions where energy is lost).
2D Projection: Real crystals are 3-dimensional with many intersecting planes (described by Miller indices). This model isolates a single set of parallel planes in 2D space. Angles are measured from the planes, not their normal. The drawn upper and lower sites are vertically aligned, and perpendicular wavefront projections give two extra legs of d sin(theta). The grazing and backscattering drawings are limits, not a ray-traced experiment. Existence of a selected order is compared using the exact decimal inputs; inverse sine and probe-path values are rounded floating-point approximations. A nonzero mismatch alone is not a prediction of complete destructive interference.

References

For more information, consult the foundational physics texts and resources which outline the derivations of Bragg's law:

OpenStax University Physics Volume 3: 4.6 X-ray Diffraction

Bragg Diffraction