Crystal Lattices, the Reciprocal Lattice and Bloch's Theorem
A perfect crystal repeats, so its electron states can be labelled by a wavevector and written as a plane wave times a function with the lattice's own period. The reciprocal lattice is the bookkeeping that makes diffraction and band structure the same statement.
What a learner can do afterwards
- Constructs a reciprocal lattice from a simple direct lattice
- States Bloch's theorem and says what the crystal wavevector labels
- Connects the diffraction condition to the boundary of the first Brillouin zone
1 · Read
Over 90 percent of solids are crystalline: atoms sit in a pattern repeating in all three directions, since orderly packing maximizes attraction. The smallest chunk still showing the full repetition is the unit cell, and stacking copies rebuilds the whole crystal. Rock salt cleaves into cubes for this reason, and pure silicon crystals are the base of electronics.
X-rays have wavelengths near atomic plane spacing, so a crystal acts as a three-dimensional grating scattering beams into sharp spots. Measuring the spots reveals plane spacing and arrangement. That diffraction pattern is a picture of the reciprocal lattice, the flipped bookkeeping of the direct one. For a row of atoms spaced a apart, the reciprocal lattice is a row of points spaced 2 pi over a: close atoms, wide points.
Bloch's theorem classifies electron states in a periodic potential: each state takes a crystal wavevector label and reads as a plane wave times a function with the lattice period. Two routes reach bands: nudge the free electron gas with the ion lattice, or overlap atomic orbitals until they blur. The wavevector is the label periodicity provides.
Diffraction and band edges follow from the same condition. The Bragg condition says equal-energy states differ by a reciprocal lattice vector; those wavevectors lie on the zone boundaries, the nearest edging the first zone. When gaps open there, read them as diffraction striking the electron waves.
Repeating atoms give a unit cell, diffraction photographs its reciprocal, and Bloch labels electrons by wavevector to the zone edge.
Adapted in part from OpenStax Chemistry 2e, section 10.6 (CC BY-NC-SA 4.0), openstax.org · license
Rewritten readings were written for this dataset.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.