One dot, one whole family of planes
Guide 1 laid out the problem: a real crystal owns hundreds of plane families — (100), (111), (231), on and on — each with its own spacing and its own tilt, and keeping track of them all in ordinary space is a bookkeeping nightmare. Guide 2 then forged three brand-new vectors, a*, b* and c*, out of the direct cell. This guide is where all that machinery finally earns its keep, and it does so through one crisp sentence you should carry with you for the rest of the ladder: each point of the reciprocal lattice stands for an entire family of parallel planes in the real crystal.
Sit with how strange and how tidy this is. A family of planes is a sprawling, infinite deck of parallel sheets stacked through all of space; the reciprocal point that represents it is a single dot. The reciprocal lattice trades the unwieldy stack for one clean address. And the trade is inside-out on purpose: widely spaced planes (large d) sit CLOSE to the origin, because 1/d is small; tightly spaced planes (small d) sit FAR out. This is exactly the crystal's shadow we promised in guide 1 — the shadow that a diffraction camera photographs directly, so that reading the film means reading this map of dots.
The vector that reaches the dot: g_hkl
Every reciprocal point has an address you build exactly the way you build a real lattice point — but from the starred basis. The reciprocal lattice vector is g_hkl = h times a* + k times b* + l times c*. Feed in the Miller indices (hkl) of a plane family, and out comes the vector that points from the origin straight to that family's dot. The integers that were plane labels in real space have become plain coordinates in reciprocal space — which is why this particular vector gets its own name, the g-vector.
Put real numbers on it. Take a simple cubic crystal with edge a = 4 angstrom. Because the reciprocal of a cubic cell is itself cubic, a*, b* and c* are mutually perpendicular with length 1/a = 0.25 per angstrom. Now walk a few points. g_(100) = 1 times a*, length 0.25, so d_100 = 1/0.25 = 4 angstrom — the full cell edge, exactly as it must be. g_(200) = 2 times a*, length 0.50, so d_200 = 2 angstrom. g_(110) = a* + b*, length 0.25 times the square root of 2 = 0.354, so d_110 = 2.83 angstrom = 4 divided by the square root of 2. Every answer matches the cubic rule d_hkl = a divided by the square root of (h^2 + k^2 + l^2) — but here it dropped out of one vector length instead of a formula.
Notice something that trips up newcomers: (200) is a genuine, distinct reciprocal point, sitting twice as far out as (100), even though 'the (200) planes' are just the (100) sheets counted twice as finely (half the spacing). Real space frowns on common factors, but reciprocal space welcomes them — and for a good reason. That extra dot is precisely the second-order of reflection off the (100) family: what Bragg's law writes as n = 2 for the (100) planes, the reciprocal lattice quietly folds into the single label (200). One of the quiet elegances of reciprocal space is that Bragg's clumsy 'order' n simply disappears into the indices.
Why the direction is the normal, and the length is 1/d
These two facts are not decreed by fiat; they fall straight out of how a* was built in guide 2, where a* was made perpendicular to the b-c plane, and each starred vector obeys a*·a = 1 while a*·b = 0 and a*·c = 0. Here is the direction fact in one line. Run a vector between two of the plane's axis-crossings — say from where the nearest (hkl) plane meets the a-axis, at a/h, to where it meets the b-axis, at b/k. Dot that vector with g. Because a*·a = 1 and b*·b = 1 while all the cross terms vanish, each end contributes exactly 1, and g times (b/k minus a/h) = 1 minus 1 = 0. A zero dot product means perpendicular — so g is at right angles to a line lying in the plane, and repeating for a second such line pins it as the plane normal.
The length fact is just as clean. The spacing d_hkl is the perpendicular distance from the origin to that nearest plane, and you get it by projecting the crossing point a/h onto the unit normal g/|g|. That projection collapses — using a·a* = 1 once more — to precisely 1/|g|. So d = 1/|g|, which is the same as |g| = 1/d. This is the inverse-size relationship living at the scale of a single family: pack the sheets closer (smaller d) and the reciprocal vector grows longer. And notice the reward for the abstraction — in real space the plane-spacing equation needs a different, gnarlier form for every crystal system, but |g_hkl| = 1/d holds automatically in ALL of them, even the fully skewed triclinic case. That generality is the whole reason crystallographers route their geometry through reciprocal space.
RECIPROCAL LATTICE ROW - the a* direction of a cubic crystal
a = 4 angstrom, a* = 1/a = 0.25 per angstrom
origin (100) (200) (300) (400)
|-----------o-----------o-----------o-----------o-----> a*
0 0.25 0.50 0.75 1.00 (per angstrom)
= 1/a = 2/a = 3/a = 4/a
distance from origin = |g_hkl| = 1 / d_hkl
(100): 0.25 -> d = 4.00 angstrom (the full cell edge)
(200): 0.50 -> d = 2.00 angstrom (2nd order off (100))
(300): 0.75 -> d = 1.33 angstrom
equal 1/a steps along the row = one plane family plus its higher orders
Small cell, big shadow: the reciprocal unit cell
Collect the g-vectors for every integer triple (hkl) and the dots they land on form a complete lattice in their own right — the reciprocal lattice, every one of whose reciprocal lattice points is the address of a real plane family. That lattice is spanned by its own reciprocal unit cell, the little box built on a*, b*, c*. And its size is fixed by an exact and beautiful rule: the reciprocal cell volume is V* = 1/V, the plain inverse of the direct cell volume (in the crystallographer's convention we have been using, which carries no factor of 2 pi).
That single equation carries the headline of the whole rung: small real cell, big reciprocal cell, and vice versa. Shrink a real edge to a short a and its partner a* = 1/a grows long, flinging the reciprocal dots far apart; stretch the real cell out and the reciprocal dots crowd in tight. This is Fourier duality wearing a crystallographer's hat, and it is the topic of the very next guide: the reciprocal lattice IS the Fourier transform of the direct lattice, and Fourier transforms always swap tight-for-wide — a narrow slit throws a broad diffraction fringe, a fat object casts a fine one. Hold that thought; guide 4 makes it precise.
Two honest caveats before we climb on, because the map is idealized. First, a reciprocal 'point' is truly point-like only for an infinite perfect crystal; a real, finite crystal smears each point into a tiny blob whose width scales as 1/(crystal size) — that smear is the seed of Scherrer peak-broadening you will meet later. Second, and more important: the reciprocal lattice fixes only WHERE diffraction spots can appear — their positions are pure lattice geometry. How BRIGHT each spot is comes from the motif sitting on each lattice point, a separate quantity (the structure factor) covered in the next rung. Positions from the lattice, intensities from the motif: it is the lattice-versus-crystal split from the very start of this ladder, now speaking in reciprocal space.
Why crystallographers live in reciprocal space
Here is the deep reason the whole shadow world is worth building: a diffraction pattern is a direct photograph of the reciprocal lattice. That is not a loose analogy — it is the literal content of the diffraction pattern as reciprocal lattice. Each spot on a single-crystal film sits exactly where a reciprocal point projects, and its brightness paints in what we call the weighted reciprocal lattice — the same grid of dots, each dot dimmed or brightened by its intensity. You do not compute the reciprocal lattice and then hunt for it in the data; the camera simply hands it to you, and reading the film IS reading the map of dots.
Two famous constructions make their home in exactly this space, and both wait for you in guide 5. The Ewald sphere is a sphere drawn in reciprocal space: wherever its surface happens to slice through a reciprocal point, a diffracted beam flashes out, so it is nothing more than the geometric rule for 'which plane families are satisfying Bragg's law right now.' And the Brillouin zone is no exotic beast either — it is simply the Wigner-Seitz cell of the reciprocal lattice, the very same 'region closer to this point than to any other' cell you already built for the real lattice, now drawn in reciprocal space, where it becomes the natural playground of electron bands and lattice vibrations.