Understanding the relevance of frequency
(as distinct from position) is crucial when one encounters
resonant systems in nature. The reciprocal lattice is similarly
useful when one encounters crystals. In the figure below,
the *direct lattice* is on
the left while the corresponding *reciprocal lattice* (frequency-space
transform) is on the right. (Note: hit reload if both left and right models
below don't load on the first pass.) The balls in the direct lattice
correspond to atoms or molecules with spacing measured in distance
units like meters or Angstroms, while the balls in the reciprocal
lattice correspond to spots in a power spectrum (or in diffraction)
with spacing measured in reciprocal-distance units e.g.
reciprocal-Angstroms.

The left model shows a simple-cubic lattice a few unit-cells
across. An "Ewald slice" (or diffraction pattern) of its
corresponding reciprocal lattice is on the right. That bright central
spot in the reciprocal space model is oft referred to as the "DC peak"
or "unscattered beam". Also note that square or angle brackets are
typically used for direct space vector indices, while round or
curly brackets are used for reciprocal space (Miller) indices.
The set of planes defined by the red atoms in direct space
gives rise to the reciprocal lattice spots (perpendicular to that
plane) in cyan on the right (best seen in <111>_view).
What are the Miller indices of that plane?
How about the Miller indices of the dark yellow spot also
active in <111>_view?
The buttons underneath allow you to modify the model. What happens
to the reciprocal lattice when you *add* stuff in direct space?
What if you shrink the spacing between scattering centers?
Is the reciprocal lattice of a simple-cubic lattice also simple-cubic?
How about the reciprocal lattice of a body-centered cubic crystal?
How many three-fold symmetric zones do these lattices exhibit?

Reciprocal lattice spots show up in diffraction experiments when they intersect the Ewald sphere. For more on this, and more direct/reciprocal comparisons, check out our notes on the r2 loop.