The Ewald Sphere and the Brillouin Zone

Where the rung leaves off

Four guides in, the reciprocal lattice is no longer strange. Guide 1 explained why we even want one: Bragg's law and the directions of diffracted beams are clumsy to handle in ordinary space, but they turn clean the moment we build a second lattice whose points stand for whole families of planes. Guide 2 constructed the reciprocal axes a*, b*, c* from the real cell edges, each perpendicular to a pair of real axes and scaled so its length is an inverse spacing. Guide 3 gave every reciprocal point (hkl) a crisp meaning -- it points along the normal of the (hkl) planes and sits a distance 1/d_hkl from the origin. And guide 4 revealed the deep identity: the reciprocal lattice is the Fourier transform of the direct one, which is why a small real cell blooms into a large reciprocal cell.

This closing guide cashes all of that in. We will assemble two objects that turn the reciprocal lattice from bookkeeping into a working instrument. The first is the Ewald sphere, a single geometric picture that predicts exactly which reflections a real diffraction experiment will catch and where they will fly out. The second is the Brillouin zone, which is nothing more exotic than the Wigner-Seitz cell of the reciprocal lattice -- the same fair-share construction you met in real space, done one lattice over. And along the way we will earn the slogan that makes crystallographers think backwards: a diffraction pattern is a direct photograph of the reciprocal lattice.

The whole apparatus in one picture

Let us hold the finished machine in view. The reciprocal lattice is a second, infinite lattice living in reciprocal space, spanned by the reciprocal axes a*, b*, c*, with a point at every whole-number combination h a* + k b* + l c*. The vector from the origin to the point (hkl) is the reciprocal lattice vector g_hkl, and it carries two facts at once: its direction is the normal to the (hkl) planes, and its length is exactly 1/d_hkl, the reciprocal of their spacing. Close-spaced planes (small d) sit far out; wide-spaced planes (large d) sit near the origin. That single object -- a point whose position encodes both the tilt and the spacing of a family of planes -- is the workhorse of everything that follows.

Two quick sizes keep us honest. Because reciprocal lengths are inverse real lengths, the reciprocal cell volume is simply V* = 1 / V, the inverse of the real cell's volume. Take a cubic crystal with edge a = 4 angstrom: then a* = 1/4 = 0.25 per angstrom, and V* = 1/(4^3) = 1/64 per angstrom cubed. Shrink the real cell to a = 2 angstrom and a* doubles to 0.5 per angstrom -- the inverse-size relationship in action. This is why proteins, with their huge cells of 100 angstrom or more, throw diffraction patterns whose spots crowd close together, while a tiny metal cell flings its spots far apart.

The Ewald sphere: a machine for predicting reflections

When does a beam actually diffract? The answer is the Laue condition, and it is beautifully blunt: a diffracted beam appears exactly when the scattering vector -- the change in wavevector, k_out minus k_in -- equals some reciprocal lattice vector g_hkl. Because the scattering is elastic, k_in and k_out have the same length, 1/lambda; only their direction changes. Paul Ewald's insight was to draw this. Lay down the reciprocal lattice, then build a sphere of radius 1/lambda positioned so that the incoming beam's wavevector ends on the origin (000). The rule becomes visual: a reflection fires for every reciprocal point that happens to lie on the sphere's surface.

1. Draw the incident wavevector k_in, length 1/lambda, pointing along the beam, with its tip landing on the reciprocal-lattice origin (000).
2. From the tail of k_in, sweep out a sphere of radius 1/lambda -- the Ewald sphere. By construction the origin (000) already sits on its surface.
3. Scan the reciprocal lattice: wherever another point (hkl) also touches the sphere, draw k_out from the sphere's centre to that point. That is a real diffracted beam, and g_hkl = k_out - k_in is satisfied.
4. Points off the sphere give nothing. To collect more reflections, rotate the crystal so points sweep through the surface, or use many wavelengths, or grind the sample to a powder so every orientation is present at once.

  Ewald sphere -- a 2-D slice (crystallographer's 1/lambda units)

    reciprocal lattice:   .   .   .   .   .   .   .
                          .   .   .   .   .   .   .

                   _______________
                 /                 \        <-- radius = 1 / lambda
     k_in       /                   \
  ---------->  C ------------------- O  .   .   .   .   .
                 \                 /  \
                  \               /    \  g_hkl
                   \_____________/      *  (hkl) sits ON the sphere
                                         \
                                          --->  k_out   (DIFFRACTS)

   O = reciprocal origin (000)      C = centre of the Ewald sphere
   A reflection fires ONLY when a reciprocal point touches the sphere.
   Bragg emerges:  |g| = 1/d = (2/lambda) sin(theta)  ->  lambda = 2 d sin(theta)

Put numbers on it with copper K-alpha radiation, lambda = 1.54 angstrom, so the sphere's radius is 1/1.54 = 0.649 per angstrom. A family of planes with spacing d = 2 angstrom has |g| = 1/d = 0.5 per angstrom. The origin and that point both sit on the sphere, and a little isosceles-triangle geometry gives |g| = (2/lambda) sin(theta), so sin(theta) = |g| times lambda / 2 = lambda / (2 d) = 1.54 / 4 = 0.385, and theta = 22.6 degrees (the beam bends by 2 theta = 45.2 degrees). Rearranged, that reads lambda = 2 d sin(theta) -- this is Bragg's law, exactly. The Ewald sphere is not a new law; it is Bragg's law drawn as a picture in reciprocal space.

The Brillouin zone: a Wigner-Seitz cell in reciprocal space

Now the second object, and it costs almost nothing new. Back on the lattice rung you built the Wigner-Seitz cell of a real lattice: pick a lattice point, draw lines to its neighbours, cut each with a perpendicular bisecting plane, and keep the little region closer to your point than to any other. It is the most honest single-point unit cell -- every scrap of space assigned to its nearest lattice point. Now do the identical construction, but on the reciprocal lattice, around its origin. The region you carve out is the first Brillouin zone. Same fair-share recipe, one lattice over into reciprocal space.

A quick example keeps it concrete. Draw g-vectors from the origin to the nearest reciprocal points, bisect each perpendicularly, and the smallest cell those planes fence off is the zone. For a simple-cubic reciprocal lattice the first zone is just a cube. It gets prettier when the lattices swap partners: an FCC crystal has a BCC reciprocal lattice, whose first Brillouin zone is a truncated octahedron -- the same fourteen-faced solid that turns up all over close-packed geometry. The shape is always a faithful fingerprint of the reciprocal lattice's symmetry, which is the crystal's own symmetry seen from the other side.

Why crystallographers live in reciprocal space

Now the payoff that reorganises how you think. Run a diffraction experiment and each beam that flashes out corresponds to one reciprocal point (hkl) that crossed the Ewald sphere. Record where all the beams land and you are, quite literally, mapping the reciprocal lattice: a single crystal gives a lattice of sharp spots, a direct picture of the reciprocal lattice; a powder, with grains in every orientation, smears those spots into rings. The camera never photographs the atoms directly -- it photographs their reciprocal shadow, the very lattice this whole rung has been building.

But the pattern speaks in two voices, and separating them is the master skill. The positions of the spots fix the geometry of the reciprocal lattice, and therefore the unit cell -- its edges, angles, and symmetry. The intensities of the spots are a separate story: each reciprocal point is weighted by |F_hkl|^2, the structure factor, which depends on which atoms sit where inside the cell. So positions give you the box (the lattice), intensities give you the furniture (the motif). It is the crystal-equals-lattice-plus-motif split from the very first rung, now read straight off a photograph.

Two honesties close the rung. First, the pattern records intensities, |F|^2 -- and squaring throws away the phase of each wave. You cannot simply run the transform backwards from spots to an electron-density map; recovering the lost phases is the famous phase problem, the reason solving a structure takes cleverness and not just a camera. Second, the Ewald sphere's radius depends brutally on the probe. X-rays give a gently curved sphere that threads only a few points at a time, so you rotate the crystal to sweep through them. Fast electrons have wavelengths near 0.04 angstrom, a radius near 25 per angstrom -- a sphere so vast it is almost flat, slicing a whole plane of reciprocal points at once. That is why an electron-diffraction pattern in a microscope looks like a clean 2D net; the price is that electrons scatter so strongly they demand ultrathin samples and invite the dynamical scattering that muddles the simple intensity rules. And with that, the reciprocal lattice hands you off, fully armed, to the diffraction rung that puts it to work.