Bragg's law
/ Bragg -> BRAG /
Imagine standing in a canyon between many parallel cliff walls, all evenly spaced, and clapping once. Each wall throws back an echo. From most listening positions the echoes arrive at a jumble of times and blur together; but stand at just the right angle and every echo arrives perfectly in step, stacking into one loud crack. Bragg's law is the rule for those special angles when the echoes are X-rays and the cliff walls are evenly spaced planes of atoms in a crystal.
W. L. Bragg pictured a crystal as stacks of parallel atomic planes, a distance d apart, each acting as a faint half-silvered mirror. A wave reflecting off the second plane travels an extra distance compared with one reflecting off the first; geometry shows that extra distance is 2 d sin theta, where theta is the glancing angle between the beam and the planes. The two reflected waves add up (constructive interference) only when that extra distance is a whole number of wavelengths. That is Bragg's law: n lambda = 2 d sin theta, with n = 1, 2, 3, ... the order. Turn the crystal until the equation is satisfied and a bright reflected beam flashes out.
A quick number: for planes spaced d = 2 angstrom probed with copper X-rays (lambda = 1.54 angstrom), the first-order (n = 1) beam appears at theta = arcsin(1.54 / (2 x 2)) = arcsin(0.385) = 22.6 degrees. Measure that angle and you have measured d — the whole trick behind reading a crystal's spacings from its diffraction angles. Honest caveat: atomic planes are not really mirrors, and nothing literally reflects. Bragg's law is a wonderfully convenient picture that gives exactly the right answer; the fully rigorous statement is the Laue condition, and the two are provably equivalent.
Rock salt (NaCl) has (200) planes spaced d = 2.82 angstrom. With copper radiation (lambda = 1.54 angstrom), the first-order reflection sits at theta = arcsin(1.54 / (2 x 2.82)) = 15.9 degrees, so the detector, placed at 2 theta, registers a peak at 31.8 degrees.
Known lambda plus a measured angle yields d; this is how a diffractometer converts peak positions into a table of plane spacings.
The n and the d are not independent labels: the nth-order reflection from (hkl) planes is customarily relabelled the first-order reflection from (nh nk nl). So the 'second order of (100)' becomes simply '(200)', and n is quietly absorbed into the indices.