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Condensed Matter Physics


Crystallographic restriction

Crystallographic restriction: six-fold symmetry conforms.
Crystallographic restriction: five-fold symmetry doesn't conform.

crystallographic restriction translational periodicity aperiodic lattice six-fold symmetry (60o) possible without overlaps five-fold symmetry (72o) would lead to atoms encroaching on each other

Five-fold symmetry in nature

Buckminsterfullerene molecule.
Image © Michael Ströck via Wikimedia.

Five-fold symmetry is possible in finite and non-periodic structures such as Buckminsterfullerene, C60

Aperiodic tilings

A quasicrystalline Penrose tiling.
Image from Inductiveload via Wikimedia.

tilings self-similarity inflation - deflation Tilings made from two different types of tile can be regular but aperiodic.

A periodic tiling of Penrose tiles.
Image from Geometry guy via Wikimedia.

But not all such tilings are aperiodic: The arrangement shown here has ordinary crystalline translational periodicity.


Single-grain quasicrystal.
Image from Ames Laboratory via Wikimedia.

Crystal - quasicrystal single crystal - single-grain quasicrystal

A phason flip in an aperiodic tiling.
Image from A Szallas, A Jagannathan, S Wessel; Phys Rev B 79 (2009) 172406.

phonons - phasons

Image © DTR via Wikimedia.
Image © DTR via Wikimedia.

Electron diffraction pattern of a ZnMgHo quasicrystal.
Image © Materialscientist via Wikimedia under this CC licence.

electron diffraction 10-fold (or 5-fold or 8-fold) symmetry periodic in the other two directions Dan Shechtman, Nobel prize for chemistry, 2011 icosahedral quasi-crystals aperiodic in all three dimensions

Defects in tilings.
Image © KJ Strandburg, PR Dressel, Argonne National Laboratory; Phys Rev B 41 (1990) 2469.

defects exist in quasicrystals as they do in crystals. The diagram above is a simulation showing the disorder of a two-dimensional quasi-crystal (tiling) increasing as the temperature rises.

Ternary Co-Al-Fe phase diagram showing extent of quasicrystalline phase.
Image © V Raghavan, J Phase Equil Diff 31 (2010) 449.

Many quasicrystalline materials are ternary alloys, often involving aluminium. The phase diagram shows the small area of thermodynamic stability of the quasi-crystalline phase in the system Al-Cu-Fe (dark red). Adjacent to it are two-phase regions where the quasi-crystal phase is in equilibrium with various crystalline compounds (light red).


Materials Physics lecture: crystallographic phase transitions