The Reciprocal Lattice

the inverse-size relationship

The single rule you must internalise about reciprocal space is that it turns sizes inside out: big in real space means small in reciprocal space, and small means big. A crystal with a large, roomy unit cell has a small, tightly packed reciprocal lattice; a crystal with a tiny cell has a widely spread reciprocal lattice. It is the same trade-off you feel with a zoom lens — the more you spread a picture out, the finer the detail you can resolve, but the smaller the field of view.

In numbers it is direct: a reciprocal edge is one over the corresponding real edge (a* = 1/a for a rectangular cell), and the reciprocal cell volume is one over the real volume (V* = 1/V). A real cell 10 angstrom on a side gives a reciprocal edge of 0.1 angstrom^-1; a 2 angstrom cell gives 0.5 angstrom^-1, five times larger. The rule also works direction by direction inside one crystal: a cell that is long along c and short along a produces a reciprocal cell that is SHORT along c* and long along a*. Long real, short reciprocal — always.

This is not an abstract nicety; it governs real experiments. Large-unit-cell materials (proteins, zeolites, framework structures) have their reciprocal points crammed close, so their diffraction spots crowd together and demand high angular resolution. More strikingly, the rule extends to FINITE size: a crystal that is only a few dozen atoms thick in one direction is 'short' in real space, so its reciprocal points smear OUT into short rods (relrods) or broaden into fuzzy spots in that direction. This is exactly why nanocrystals give broad diffraction peaks — the Scherrer equation, which reads crystallite size from peak width, is the inverse-size relationship applied to particle dimensions.

A gold nanoparticle 5 nm across contains about 20 atomic planes along one direction. Being 'short' in real space, its (111) diffraction peak broadens; the Scherrer equation turns that width back into the 5 nm size. A millimetre-sized single crystal, effectively infinite, gives razor-sharp points instead.

Small crystals (short in real space) give broad reciprocal features; that width IS the inverse-size rule at work.

The rule cuts both ways at once: making the real cell smaller enlarges the whole reciprocal lattice, while making a crystal thinner smears its reciprocal points. Do not confuse cell size (spot spacing) with crystal size (spot sharpness) — they are different applications of the same principle.

Also called
reciprocity of scalebig-real-small-reciprocal rule大小反比關係