the Ewald sphere
/ AY-vahlt /
The Ewald sphere is a drawing — a clever geometric picture, invented by Paul Ewald, that tells you at a glance which crystal planes will diffract for a given beam and crystal orientation. It lives in reciprocal space, sitting right on top of the reciprocal lattice, and it converts the whole question of 'will a reflection happen?' into the simple visual test 'does a reciprocal lattice point lie on this sphere?'. It is the single most-drawn diagram in diffraction.
Here is how to build it. Draw the reciprocal lattice. Pick the point where the incoming beam enters as a reference and draw the beam's wave-vector pointing at the origin of the reciprocal lattice; the beam has length 1/lambda (one over the wavelength, in the crystallographic convention). Now draw a sphere of radius 1/lambda centred at the tail of that vector, so the sphere passes through the reciprocal-lattice origin. The rule is then exact and beautiful: a diffracted beam appears for every reciprocal lattice point that happens to lie ON the surface of this sphere. When a point (hkl) touches the sphere, the vector from the origin to that point is g_hkl, and the geometry of the triangle reproduces Bragg's law exactly — the Ewald sphere and Bragg's law are two pictures of the same condition.
The construction explains, in one image, several facts that otherwise seem like separate rules. A single stationary crystal in a monochromatic beam usually has very few reciprocal points sitting exactly on the sphere, so you must ROTATE the crystal (sweeping points through the sphere) or use a RANGE of wavelengths (fattening the sphere into a shell) to catch many reflections — that is the difference between the Laue, rotation, and powder methods. A shorter wavelength makes the radius 1/lambda larger, so the sphere flattens and sweeps more points into range, which is why electrons (tiny wavelength, huge radius, nearly flat sphere) show whole planes of the reciprocal lattice at once as a spot pattern. Honest caveat: the sphere is a construction, not a physical object, and it assumes an ideal infinite crystal; real crystals have slightly fuzzy reciprocal points, which is actually helpful because it widens the window in which a reflection can be caught.
With copper K-alpha X-rays, lambda = 1.54 angstrom, so the Ewald sphere radius is 1/lambda = 0.65 angstrom^-1. A reciprocal point at g = 0.5 angstrom^-1 (that is, d = 2 angstrom) can reach the sphere, and the triangle gives sin(theta) = (g/2)/(1/lambda) = (0.25)/(0.65) = 0.385, so theta = 22.6 degrees — the same Bragg angle you get from n lambda = 2 d sin theta.
A reflection fires only when a reciprocal point sits on the sphere of radius 1/lambda; the same triangle is Bragg's law.
The Ewald sphere is drawn in reciprocal space, not real space, and its radius is 1/lambda (or 2 pi/lambda in the physics convention), not the crystal size. A common mistake is to picture it around the crystal in real space — it is centred in reciprocal space and passes through the reciprocal-lattice origin.