Bragg gave you an angle; now meet the vector
Guide 2 handed you Bragg's law, n lambda = 2 d sin theta, and it is unbeatable for quick arithmetic: shine monochromatic X-rays at a crystal and one family of planes 'reflects' only at the special Bragg angle where the echoes from successive planes fall exactly one wavelength out of step and add up — echoes from evenly spaced cliff walls that only reinforce at certain angles. But that tidy picture quietly hides two things. It pretends atoms are smooth mirror-planes (they are not — every atom scatters in all directions), and it can only think about one plane family at a time. The Laue–Ewald picture we build now keeps every atom honest, handles all the families in one stroke, and lives in reciprocal space — the shadow world the previous rung already built for you.
Start by drawing the wave itself. Represent the incoming monochromatic beam by a wavevector k_0 that points along the beam and has length 1/lambda. Elastic scattering does not change the wavelength, so the scattered wave gets its own wavevector k — same length 1/lambda, but pointing off in a new direction. The difference between them is the star of this guide, the scattering vector s = k minus k_0: literally how much the wave's marching direction got kicked. A short exercise in geometry gives its length, |s| = 2 sin theta / lambda, where the full turn 2 theta between the incident and scattered beams is the two-theta scattering angle. Every question of the form 'did this wave diffract?' is now packed into that one arrow.
Bragg's angle and this guide's vector are two dialects for one physical fact — wave interference from the whole crystal. Bragg picks a set of 'reflecting' planes and asks for the angle that makes their echoes add; the scattering vector refuses to pick any plane and instead measures the kick s directly, then asks which kicks make the entire crystal's scattering reinforce. Same physics, two accents. Keep both: Bragg for fast angle sums, the vector for seeing the whole pattern at once.
The Laue conditions: three bull's-eyes at once
Von Laue put the demand for constructive interference directly to the lattice. Two lattice points one lattice vector apart scatter in step only if the path difference between them is a whole number of wavelengths. Insist on that along each of the three cell edges a, b, c and you get the three Laue conditions: a·s = h, b·s = k, c·s = l, with h, k, l all integers (yes, that middle k is the Miller index, a different creature from the wavevector). Any one of the three is easy to satisfy on its own. The whole difficulty — and the whole richness — is that all three must hold for the very same scattering vector s at the very same moment. Three bull's-eyes with a single arrow.
There is exactly one kind of vector that hits all three integer conditions at once, and the reciprocal rung already forged it: the g-vector g_hkl = h times a* + k times b* + l times c*. So the three Laue conditions fuse into one breathtakingly compact sentence — the diffraction condition itself: s = g_hkl. Constructive interference happens if and only if the scattering vector equals a vector of the reciprocal lattice. Every reflection the crystal can ever produce is labelled by the same integers (hkl) that name a reciprocal-lattice vector, a reciprocal point, and a plane family — one triple of numbers doing three jobs.
See how large an upgrade this is over Bragg. Bragg took one family at a time and needed you to already know its spacing before hunting the angle. The condition s = g takes all families in a single equation, uses nothing but the reciprocal lattice you build once from the cell, and quietly swallows Bragg's clumsy 'order' n — as guide rec-3 showed, the second order of reflection off (100) is just the ordinary reciprocal point (200). The positions of every possible reflection have become pure geometry: build the reciprocal lattice, and you are holding the complete menu of what can diffract, before a single photon is fired.
The Ewald sphere: a compass for diffraction
Ewald turned the equation s = g into a picture you can literally draw, and it is the single most useful diagram in diffraction. Lay down the reciprocal lattice and mark its origin O. Draw the incident wavevector k_0 (length 1/lambda) so that its head lands ON the origin, with its tail at a point C one radius behind. Now sweep out a sphere of radius 1/lambda centred on C — the Ewald sphere. Since the scattered k also has length 1/lambda and also starts at C, its head must lie somewhere on that sphere. The diffraction condition s = k minus k_0 = g then reads as one crisp geometric rule: a reflection fires exactly when a reciprocal-lattice point happens to sit on the sphere's surface, and the diffracted beam shoots from C straight out through that point.
EWALD CONSTRUCTION (2D slice: the sphere is drawn as a circle)
_____ P = reciprocal point (hkl)
__--'' o on the sphere -> it reflects
_-'' /|
/ / |
/ k/ | g_hkl = s = k - k_0 (runs O -> P)
k_0 / / |
incident / / |
beam ----> C ----------- O ----+ . . . . reciprocal lattice
\ 1/lambda | (origin at O)
\ (radius) |
\ |
'-_ |
''--..._____|
C = sphere centre, the shared tail of k_0 and k; |k_0| = |k| = 1/lambda
O = reciprocal-lattice origin, where k_0 ends (ALWAYS on the sphere)
P = a reciprocal point; a reflection fires only when P lands on the sphere
diffraction condition: s = k - k_0 = g_hkl
The picture teaches three things at a glance. First, the origin O is always on the sphere (s = 0, the straight-through undiffracted beam), so it never counts as a reflection. Second, any other reciprocal point that touches the surface reflects, while points sitting inside or outside stay dark. Third — and this is the working heart of a diffractometer — spin the crystal and its whole reciprocal lattice rotates rigidly about O, carrying point after point briefly through the surface, each crossing flashing a spot. That is exactly what a single-crystal instrument does: rotate to sweep reciprocal points across the Ewald sphere and catch every flash on a detector.
Bragg and Laue are the same statement
The two pictures are not rivals; they are one equation in two costumes, and it takes only a line to show it. The reciprocal rung established that |g_hkl| = 1/d_hkl and that g points along the plane normal of the (hkl) family. The scattering vector s has length 2 sin theta / lambda and, by its construction as k minus k_0 of equal-length wavevectors, it too points along that same normal, bisecting the incident and scattered beams. So the single condition s = g matches BOTH direction (both are the plane normal) AND length: 2 sin theta / lambda = 1/d, which rearranges instantly to lambda = 2 d sin theta. That is Bragg's law, first order — and higher orders are simply larger-index reciprocal points, exactly as before. The interplanar spacing d you plug into Bragg is nothing but the reciprocal of the length of the g-vector Ewald draws.
Put real numbers on the equivalence. Take a plane spacing d = 2 angstrom and Cu K-alpha radiation, lambda = 1.54 angstrom. Bragg the fast way: theta = arcsin(1.54 / (2 times 2)) = arcsin(0.385) = 22.6 degrees, so the detector sits at 2 theta = 45.2 degrees. Ewald the geometric way: the reciprocal point lives at |g| = 1/d = 0.5 per angstrom from O, and the sphere radius is 1/lambda = 1/1.54 = 0.649 per angstrom. A chord of length 0.5 subtended in that circle gives sin theta = (|g| / 2) / (1/lambda) = |g| times lambda / 2 = 0.5 times 1.54 / 2 = 0.385, so theta = 22.6 degrees. Identical answer, reached two ways — proof that the Bragg angle and the Ewald geometry are the same fact.
Why the wavelength must match the atoms — and what the sphere leaves out
The Ewald sphere makes the wavelength rule obvious in one look. The farthest a reciprocal point can sit and still touch the sphere is its diameter, 2/lambda, so ONLY families with 1/d less than or equal to 2/lambda — that is, d greater than or equal to lambda/2 — can ever reflect. Try visible light, lambda around 500 nm = 5000 angstrom: the sphere radius 1/lambda = 0.0002 per angstrom is minuscule, nowhere near the nearest reciprocal point at about 0.5 per angstrom, so the sphere never reaches a single point and a crystal simply does not diffract visible light. Switch to X-rays near 1 angstrom and the sphere swells enough to catch a whole crowd of points. That, in one sentence, is the wavelength requirement: to read atomic structure you need a wavelength comparable to the interatomic spacing.
Now the honest flip side. A monochromatic beam on a STATIONARY single crystal almost never lands a reciprocal point exactly on the sphere — an exact touch is a coincidence of measure zero — so, left alone, you usually see nothing at all. Crystallographers get around this in three classic ways. Rotate the crystal, so its reciprocal points sweep through the sphere one by one (the rotating-crystal method). Grind it to a fine powder, so every orientation is present at once and each family's full ring of points meets the sphere somewhere, giving cones of diffracted rays. Or use a spread of wavelengths at once (the original Laue method), so a whole nest of spheres of different radii catches many points on a single fixed crystal.
One more case is worth meeting because it looks so different. For electrons at 100 to 200 keV the wavelength is tiny, lambda around 0.025 to 0.037 angstrom, so the sphere radius 1/lambda is enormous — roughly 27 to 40 per angstrom — and over the reciprocal lattice its surface is nearly a flat plane. It slices a whole layer of reciprocal points at once, which is why a selected-area electron-diffraction pattern looks like a direct photograph of a 2D section of the reciprocal lattice — a beautiful, literal example of the pattern being the reciprocal lattice. The honest catch, as always: electrons interact with matter far more strongly than X-rays, so samples must be ultrathin and the clean rule 'intensity proportional to |F|^2' breaks down under dynamical scattering.