Crystal Lattices, the Reciprocal Lattice and Bloch's Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Electrons in a repeating lattice

Science · Matter & Materials · ages 23-24
Name ______________________   Date ____________

  1. What does Bloch's theorem say about an electron in a periodic potential?

    • A free particle ignoring the lattice
    • An orbit around one nucleus only
    • Plane wave times lattice-periodic function, labelled by wavevector

  2. What is a unit cell?

    • The smallest repeating chunk of the pattern
    • A single atom with no neighbours
    • The whole laboratory sample

  3. The reciprocal lattice matters only for diffraction.

    Circle one:   True   False

  4. What does the crystal wavevector label?

    • Camera exposure settings
    • Bloch electron states
    • Unit cell paint colors

  5. What are the sharp spots in an X-ray pattern?

    • Shadows of the sample holder
    • A picture of the reciprocal lattice
    • Stains on the detector

  6. Why do X-rays reveal crystal structure?

    • Their wavelength matches plane spacing
    • They melt the crystal surface
    • They carry electric charge

  7. Lena files every Bloch state by position alone. What is wrong?

    • Positions are forbidden inside crystals
    • She dropped the wavevector label periodicity provides
    • Cells cannot repeat in three dimensions

  8. What is the Bragg condition in reciprocal language?

    • A melted spot on the film
    • A random camera angle
    • A reciprocal vector connecting two states
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Electrons in a repeating lattice W1-mt_nbNnHpGnZI-s1

  1. Plane wave times lattice-periodic function, labelled by wavevector · Periodicity forces the labelled plane-wave-times-periodic form.
  2. The smallest repeating chunk of the pattern · Stacking unit cells rebuilds the entire crystal.
  3. False · It also organizes band structure and zone boundaries.
  4. Bloch electron states · Bloch states carry the wavevector as their address.
  5. A picture of the reciprocal lattice · Spot positions encode plane spacings through the flipped lattice.
  6. Their wavelength matches plane spacing · Matched wavelengths turn the crystal into a readable grating.
  7. She dropped the wavevector label periodicity provides · Position repeats, so it cannot distinguish the states.
  8. A reciprocal vector connecting two states · That connection is the zone boundary where gaps open.
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