From 'we need it' to 'here is how to build it'
Guide 1 made the case: a family of parallel planes is an awkward thing to carry around — an infinite deck of sheets, described by a spacing d and a tilt — yet diffraction keeps handing us exactly those families. The escape was to shrink each whole family down to a single dot, placed along the family's normal at a distance set by 1/d. Do that for every family at once and the dots themselves form a perfectly periodic array: the reciprocal lattice. That was the promise. This guide keeps it, by building the array from scratch.
A lattice, direct or reciprocal, is fully pinned down by three basis vectors — three arrows whose whole-number combinations land on every point. In the ordinary crystal we called them a, b, c, and their integer combos ua + vb + wc reach every lattice point. So the entire job here is to manufacture three new arrows — written a*, b*, c* and read 'a-star, b-star, c-star' — such that their integer combos h a* + k b* + l c* reach every reciprocal point. Get those three right and the whole shadow world falls into place automatically.
The construction: cross products and the dual relations
The definition looks slick but every piece earns its place. Each starred vector is a cross product of the other two direct vectors, divided by the cell volume V = a . (b x c). Recall that a cross product b x c is an arrow perpendicular to both b and c, so a* points straight out of the plane spanned by b and c — which is precisely the (100) plane. In one clean stroke, a* has become the normal to the (100) planes. That is the reciprocal lattice vector machinery: build a plane normal, then scale it so its length carries the spacing.
- Compute the cell volume V = a . (b x c), the scalar triple product — geometrically, the volume of the parallelepiped that a, b, c span. It is one positive number and it is the same divisor for all three starred vectors.
- Form a* = (b x c) / V. The cross product makes it perpendicular to b and c; dividing by V fixes its length. Then cycle the letters: b* = (c x a) / V and c* = (a x b) / V.
- Check the nine dual relations. Each star dotted with its own partner gives 1 (a*.a = 1, b*.b = 1, c*.c = 1); dotted with either other direct vector it gives 0 (a*.b = 0, a*.c = 0, and so on). Those nine numbers ARE the definition — the cross-product recipe is just the fastest way to satisfy them.
BUILDING a*, b*, c* (crystallographer convention: a* . a = 1)
a* = (b x c) / V b* = (c x a) / V c* = (a x b) / V
with cell volume V = a . (b x c)
the nine dual (dot-product) relations:
a*.a = 1 a*.b = 0 a*.c = 0
b*.a = 0 b*.b = 1 b*.c = 0
c*.a = 0 c*.b = 0 c*.c = 1
geometry: a* is PERPENDICULAR to the b-c plane (the (100) plane)
|a*| = 1 / d_100 (one over the (100) plane spacing)
worked cubic example, a = 4 angstrom, so a* = b* = c* = 0.25 /A
(hkl) |g| = sqrt(h^2+k^2+l^2)/a d_hkl = a/sqrt(...) check 1/d
(100) 0.250 /A 4.00 A 0.250
(110) 0.354 /A 2.83 A 0.354
(111) 0.433 /A 2.31 A 0.433
(200) 0.500 /A 2.00 A 0.500
every reciprocal point sits at distance 1/d_hkl from the originLook at the magnitude the definition forces on a*. Its length is |b x c| / V. But |b x c| is the area of the (100) face, and V is (that face area) times (the perpendicular gap between neighbouring (100) planes). Area divided by area-times-gap leaves 1/gap — that is, |a*| = 1/d_100, one over the interplanar spacing. So a* does double duty: its direction is the (100) normal and its length is 1/d_100. The cross-product-over-volume definition was engineered, not guessed, to make both of those true at once.
The reciprocal lattice vector g_hkl: direction and length
With the three stars in hand, a single reciprocal point is just an integer combination of them: the reciprocal lattice vector g_hkl = h a* + k b* + l c*. The whole payoff of the last section now generalizes from the three special faces to every plane at once. The crisp claim — and it is the emotional center of this entire rung — is that g_hkl points along the normal of the (hkl) planes, and its length equals 1/d_hkl. The point labelled (hkl) in the shadow world is nothing but the tip of that arrow.
Run the cubic numbers to feel it. With a = 4 angstrom the stars all have length 0.25 per angstrom and sit at right angles, so g_110 = 1 a* + 1 b* + 0 c* is the diagonal of a little square of side 0.25, length sqrt(2) times 0.25 = 0.354 per angstrom. Invert it: 1/0.354 = 2.83 angstrom, which is exactly d_110 = 4/sqrt(2). The direction also checks out — g_110 leans equally on a* and b* and not at all on c*, so it points along [110], the true normal of the (110) planes in a cube. Direction and length, both correct, both for free.
The reciprocal unit cell, its volume, and inverse sizes
The three stars a*, b*, c* span their own box — the reciprocal unit cell — and stamping it out endlessly tiles all of reciprocal space, exactly as the direct cell tiles real space. Its edge lengths and angles come straight from the direct cell but they are not the same numbers: in a stretched or skewed cell the starred angles differ from the direct ones (they agree only when the direct cell is orthogonal, like our cube). This box is small, tidy, and periodic, which is the whole reason a diffraction pattern looks like a neat grid of spots rather than a tangle.
Its volume is the punchline: V* = 1/V, the reciprocal of the direct cell's volume (in the crystallographer convention). Our cube has V = 4^3 = 64 cubic angstrom, so V* = 1/64 = 0.0156 per cubic angstrom. This single equation carries the deepest intuition in the whole subject, the inverse-size relationship: a small real cell makes a large reciprocal cell, and vice versa. Double the crystal's edge to 8 angstrom and V leaps to 512, so V* collapses to 1/512 — the reciprocal points crowd eight times closer together. Big real, small reciprocal. Small real, big reciprocal.
That flip is not a coincidence of the algebra; it is the signature of a Fourier transform, and the reciprocal lattice is precisely the transform of the direct one. This is the Fourier duality view that guide 4 unfolds in full: broad features in real space map to fine, tightly-spaced features in reciprocal space, which is why a large repeat gives closely-spaced diffraction spots. For now just hold the picture — the reciprocal lattice is the crystal's shadow, and shrinking the object always spreads its shadow out.
Where the shadow world comes alive: Ewald sphere and Brillouin zone
Two famous constructions live natively in the space you just built. The first is the Ewald sphere. Draw the incident X-ray beam as an arrow of length 1/lambda ending at the reciprocal lattice's origin, then draw a sphere of radius 1/lambda around that arrow's tail. Diffraction fires from exactly those reciprocal points the sphere happens to pass through — that geometric coincidence is the diffraction condition itself, restated with no mention of angles. It is why a single still crystal usually gives almost no spots (few points touch the sphere) and why you rotate the crystal or use many wavelengths to sweep more points onto it.
The second is the Brillouin zone, and here the earlier rungs on real-space cells pay a surprise dividend. Recall the Wigner-Seitz cell: the region of a lattice closer to one chosen point than to any other, carved out by the perpendicular bisector planes between neighbours. Perform that exact construction on the RECIPROCAL lattice and the cell you carve is the first Brillouin zone. It is the natural home for electron and phonon bands — the same reciprocal geometry, now doing physics rather than diffraction. Guide 5 develops both the Ewald sphere and the Brillouin zone properly; here just register that both were waiting inside the lattice you built today.