Crystal systems and unit cells

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• Chapter 13: Solids and Modern Materials.

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• Motivating questions: What holds solids together? Why is diamond hard but graphite soft (both pure carbon)? Why is graphite a conductor while diamond is insulating? Why do metals conduct but glass does not?

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• X-ray crystallography uses X-rays to probe the regular arrangement of atoms in crystalline solids (crystal lattice) via diffraction detectable by an X-ray detector.

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• X-rays are electromagnetic radiation with short wavelength and high energy: $E = h\nu = \frac{hc}{\lambda}$ where h = 6.63\times10^{-34}\,\text{J·s}, $c = 3.00\times10^{8}\,\text{m/s}$, and $\lambda$ is the wavelength.

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• X-ray scattering from atoms can be constructive or destructive interference depending on path differences.

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• Constructive interference follows Bragg condition: $n\lambda = 2d\sin\theta$
  • $d$ = distance between atoms, $\lambda$ = X-ray wavelength, $n$ = diffraction order.

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• Perspective matters: multiple viewpoints help determine crystal structure.

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• Crystalline solids: regular, long-range order (e.g., quartz).
• Amorphous solids: disorder (e.g., glass).

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• X-ray crystallography reveals crystal lattice. The smallest repeating unit is the unit cell, which repeats to form the macroscopic crystal. Unit cells are classified by symmetry.

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• Fundamental types of unit cells (crystal systems):
  • Cubic: $$a=b=c,\