a reciprocal lattice vector

Just as the real crystal lattice is generated by three edge vectors a, b, c (the arrows that step you from one lattice point to the next), the reciprocal lattice is generated by three of its OWN edge vectors, written a*, b*, c* (say "a-star"). These three starred vectors are the reciprocal basis: stack whole-number amounts of them and you land on every reciprocal lattice point. They are the toolkit from which the whole shadow world is built.

The starred vectors are defined from the real ones by a rule that looks fussy but does exactly one thing — it makes each star vector point along the normal to one set of cell faces. In symbols: a* = (b x c) / V, b* = (c x a) / V, c* = (a x b) / V, where 'x' is the vector cross product and V = a . (b x c) is the volume of the real unit cell. The cross product b x c is perpendicular to both b and c, i.e. perpendicular to the (100) face — so a* is the normal to the (100) planes, and its length works out to 1/d_100. The neat summary of the whole scheme is a dot product table: a*.a = 1, b*.b = 1, c*.c = 1, but a*.b = 0, a*.c = 0, and so on (each star vector is 1 with its own partner and 0 with the others).

A subtle payoff: unless the crystal is cubic, a* is generally NOT parallel to a. In a slanted (say monoclinic) cell the reciprocal axes tilt away from the direct axes, because a* must stay perpendicular to b and c, not to itself. One convention warning that trips up newcomers: crystallographers use the definition above, giving a*.a = 1 and lengths of 1/d. Solid-state physicists insert a factor of 2 pi, so a*.a = 2 pi and lengths of 2 pi/d. Both describe the same lattice; only the scale differs. This field uses the crystallographers' 1/d convention throughout.