reciprocal space

Reciprocal space is simply the arena in which the reciprocal lattice lives — an abstract space whose coordinates are measured in units of 1/length rather than length. If ordinary (direct) space is where you measure how far apart atoms sit in nanometres, reciprocal space is where you measure how RAPIDLY the crystal repeats, in cycles per nanometre. A large spacing in real space becomes a small coordinate in reciprocal space, and vice versa. It sounds strange at first, but it is the natural home of anything to do with waves and periodicity.

Concretely, a point in reciprocal space is a vector with length 1/distance. Think of it as answering the question 'how many wave crests fit in one unit of real distance?' A feature that repeats every 2 angstrom in the crystal corresponds to a reciprocal-space point at 1/2 = 0.5 angstrom^-1. This is exactly the same bookkeeping a sound engineer uses when they swap a waveform (amplitude versus time) for its spectrum (amplitude versus frequency): frequency space is the audio world's reciprocal space. Physicists often attach a factor 2 pi and call it k-space or momentum space, because a particle's momentum is proportional to 1/wavelength; it is the same space wearing a different label.

Why live there? Because diffraction, wave propagation, and electronic states are all local and simple in reciprocal space even when they are tangled in real space. A whole crystal's worth of planes collapses to a single point; the messy interference of scattered waves becomes the crisp rule 'a spot appears where a reciprocal point meets the Ewald sphere'. The honest caveat is that reciprocal space is a mathematical construction, not a place — you never travel there. But your diffraction detector effectively samples it, which is why learning to think in reciprocal space is the price of admission to crystallography.