the reciprocal lattice construction
The reciprocal lattice construction is the actual step-by-step recipe for turning a real crystal lattice into its reciprocal partner. It comes in two flavours that give identical answers: a hands-on geometric recipe you could draw with a ruler, and an algebraic one you would use in a computer. Learning both is worth it, because the geometric picture builds intuition while the algebra does the real work.
The geometric recipe reads like a craft instruction. For each family of planes (hkl) in the crystal, do three things: draw the plane's normal (the perpendicular direction) starting from a chosen origin; measure the interplanar spacing d_hkl; and place a point on that normal at a distance of exactly 1/d_hkl from the origin. Label that point (hkl). Repeat for every family of planes. The cloud of points you get — one per plane family — is the reciprocal lattice, and remarkably it always turns out to be a perfect lattice itself. The algebraic recipe skips the drawing: compute the three star vectors a* = (b x c)/V, b* = (c x a)/V, c* = (a x b)/V from the real cell, then every reciprocal point is g = h a* + k b* + l c* for integer h, k, l.
The construction is worth doing by hand once, because it makes three otherwise-mysterious facts obvious. First, WHY each reciprocal point stands for a plane family: you literally placed it using that family's normal and spacing. Second, WHY closely spaced planes give far-out points: small d means large 1/d. Third, WHY the reciprocal of a reciprocal lattice is the original direct lattice: apply the same recipe twice and the two inversions cancel. One caution for beginners: you must use the plane spacing d, which for centred lattices can be smaller than the naive cell-face spacing — get d right and the construction is foolproof; get it wrong and your reciprocal lattice will be off by a centering factor.
Build the reciprocal point (110) for a cubic crystal with a = 4 angstrom. Its plane spacing is d_110 = 4/sqrt(1^2+1^2) = 2.83 angstrom, its normal points along [110], and you place a point at 1/2.83 = 0.354 angstrom^-1 out along [110]. Check against the algebra: |g_110| = sqrt(1+1)/4 = 0.354 angstrom^-1. The two recipes agree.
Geometric recipe (normal, then 1/d out) and algebraic recipe (h a* + k b* + l c*) give the same point.
Use the true interplanar spacing d_hkl, not the cell-edge divided by an index. For centred lattices some (hkl) planes are more closely spaced than the naive geometry suggests, and getting d right is the whole game.