X-rays scattered by a crystal shine only where the geometry allows — where the path difference between rays bouncing off adjacent atomic planes is a whole number of wavelengths. That single condition, nλ = 2d sin θ, turned the crystal into a ruler you read with light. The wavelength and spacing go in, the engine solves the angles, the diffraction peaks come out. The blue team builds and defends it; the red team tries to break it.
source W.H. & W.L. Bragg, The Reflection of X-rays by Crystals, Proc. R. Soc. A 88, 428–438 (1913) — doi:10.1098/rspa.1913.0040. Rendered, not quoted.
A ray reflecting off the second plane travels an extra 2d sin θ — the two short legs into and out of the deeper plane. The two beams add constructively only when that extra path is a whole wavelength:
nλ = 2d sin θ, integer n. Solve for the angle: θn = arcsin(nλ / 2d). Since sin θ ≤ 1, an order exists only while nλ ≤ 2d — that ceiling is why a crystal gives a finite set of spots, not a smear.
Orders for the current crystal (live):
| order n | sin θ | θ (deg) | reflects? |
|---|
Bragg's law measures θ; the crystal geometry supplies d. For a cubic lattice of edge a, the plane family (h k l) is spaced d = a / √(h²+k²+l²).
So nλ = 2d sin θ is the ruler that reads the-miller-indices' spacings: each measured angle names a set of planes and returns its d. Bragg (1913) hands the peaks; Miller hands the labels. Each sphere is the next one's premise.
The blue team's live check: sweep orders and crystals, solve each angle, and confirm the true path difference 2d sin θ equals nλ to machine precision. If red drops the factor of two, this badge is where it shows.
Three numbers define a reflection: the X-ray wavelength λ (Cu Kα ≈ 1.54 Å), the plane spacing d (how far apart the atomic mirrors sit), and the order n (how many whole wavelengths of path difference).
| symbol | is | role |
|---|---|---|
| λ | wavelength | the light's ruler-tick |
| d | plane spacing | what we want to read |
| θ | glancing angle | measured off the plane |
| n | order | whole wavelengths of path |
θ is measured from the plane, not the normal — that is what puts the factor of 2 in the law. Feed these into the panel below.
Every value is computed on the spot from nλ = 2d sin θ — the angle, the path difference and the cutoff are never looked up.
What the machine produces, proven: for each order n, the glancing angle θn = arcsin(nλ / 2d) — and a hard verdict on whether that order exists at all, since no reflection is possible once nλ > 2d. Smaller d pushes every peak to a larger angle; that inverse is how the pattern encodes the lattice.
The blue team's witness (left) confirms 2d sin θ = nλ live; the red team (right) tries to make it wrong.
It is also kinematic: it assumes each photon scatters once. In a large perfect crystal, dynamical effects (extinction, multiple scattering) shift and reshape the peaks. The law is the skeleton of diffraction, not the whole body.
"X-rays reflect off crystal planes like light off a mirror." Kept, corrected. The "reflection" is constructive interference of X-rays scattered by every atom; Bragg's mirror is an exact geometric equivalence for the angle, not a literal specular bounce.
"The law is nλ = d sin θ." Cut. The path difference is two legs — 2d sin θ. Drop the 2 and every angle and the λ≤2d cutoff are wrong. (That is exactly the tamper below.)
"W.L. Bragg derived it with his father." Cut. The relation is W. L. Bragg's (the son, 1912); the spectrometer and the 1913 papers are the joint work; the 1915 Nobel was shared.
The red team's move: drop the factor of two — compute angles from nλ = d sin θ instead of 2d sin θ. The peaks land at the wrong angles and the λ≤2d cutoff breaks. The blue team's witness (window 7) is watching.
Drop the 2 and the solved angle no longer satisfies the real physics 2d sin θ = nλ; the witness recomputes the true path difference, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.