the Bragg angle
/ Bragg -> BRAG /
When a crystal is set to diffract, there is one special tilt at which its atomic planes throw the beam out brightly. The Bragg angle, written theta, is that tilt: the angle between the incoming beam and the reflecting planes themselves at which Bragg's law is satisfied. It is the number you plug into n lambda = 2 d sin theta, and the number a diffractometer is really hunting for as it slowly rotates the sample.
The one thing beginners must get right is what theta is measured from. In optics you usually measure a ray's angle from the surface normal (the perpendicular). In the Bragg picture theta is measured from the plane itself — the glancing angle a stone skips at, not the steep angle. So a Bragg angle of 20 degrees means the beam grazes in 20 degrees above the plane, and the beam is deflected by twice that, 40 degrees, from its original path. That deflection, 2 theta, is the scattering angle the detector sits at — never confuse theta with 2 theta.
Because sin theta = n lambda / (2 d), the Bragg angle carries the physics: widely spaced planes (large d) diffract at small theta (close to the incoming beam), while finely spaced planes (small d) push theta out to large angles. And since sin theta can never exceed 1, there is a hard limit — no reflection at all unless lambda < 2 d — which is exactly why the probing wavelength must be small enough to fit the atomic spacings.
For silicon's (111) planes, d = 3.14 angstrom. With copper radiation (lambda = 1.54 angstrom), sin theta = 1.54 / (2 x 3.14) = 0.245, so theta = 14.2 degrees and the detector records the peak at 2 theta = 28.4 degrees — the strong low-angle line every silicon pattern shows.
The Bragg angle theta is half the detector angle; here 14.2 degrees at the crystal becomes 28.4 degrees at the detector.
theta is measured from the plane, not from the plane normal, and the beam is deflected through 2 theta. Reporting a peak at 'theta' when you mean '2 theta' (or vice versa) is one of the most common and costly slips in diffraction.