Crystals

In General > s.a. [condensed matter]; Defects; statistical mechanics.
$ Def: Any material whose diffraction pattern is essentially discrete (International Union of Crystallography, 1992)!
* Crystallization process: Landau-Ginzburg model.
* Symmetries: An n-fold symmetry axis can be consistent with translation invariance only for n = 1, 2, 3, 4, 6 (in 4D).
@ General references: Born & Huang 54 [dynamics of lattices]; Glusker & Trueblood 85.
@ And quantum theory: Grushevskaya & Gurskii qp/06 [many-particle effects on electron states]; Bottesi & Zemba JSM(08)-a0801 [effective theories for electrons]; > s.a. modified coherent states.
@ Symmetries: Baake JPA(97) [color symmetries]; Michel PRP(01); Nardone JMP(06).
> Specific examples: see Water and Ice.

Quasicrystals > s.a. cohomology; Penrose Tiling; Phonon; Root Lattice.
* Idea: A new atomic structure, neither crystalline (in the sense of periodic) nor glassy, with long-range translational and orientational order; Theoretically, the inspiration came from the Penrose tiling.
* Experiment: 3D non-periodic tilings with icosahedral structure have been seen first by D Shechtman of NBS in 1983 [@ Shechtman et al PRL(84)], from diffraction patterns produced by some alloys of Al and Mn, and others; The golden mean appears all over the place in the patterns.
* Structure: The structure function does not factorize into an intrinsic part and a geometric part, as for regular lattices, and the definition of the unit cell is not arbitrary as in the regular case; Roughly, they are made of "clusters of clusters" of atoms, where individual clusters (typically with a magic number of atoms, like 13 or 55) are more tightly packed than in crystals.
* Properties: Extremely low electrical and thermal resistivity (worse than glass); Harder than steel; Friction and stickiness lower than Teflon (and can be still lowered); The only drawback is that they are brittle – ok if used as thin films on substrates; All of this probably is a consequence of their tight-cluster structure.
@ General references: Levine & Steinhardt PRL(84); Mermin & Troian PRL(85) [mean-field theory]; Amann et al ed-88; Janssen PRP(88); Jaric ed-88; Maddox Nat(89)jul, DiVincenzo Nat(89)aug; Stephens & Goldman SA(91)apr; Senechal 95; Goldman et al AS(96); Janot 97; Di Vincenzo & Steinhardt ed-99; Pelantová & Masáková mp/06-in [mathematical models]; news SA(09)oct [more normal than assumed].
@ Uses, applications: news pw(07)feb [in Islamic art]; news pw(07)mar [as filter for terahertz light].
@ And physics: Albuquerque & Cottam PRP(03) [elementary excitations]; Monreal et al IJMPA(08)-a0804 [quantum particle and effective non-commutative geometry].
@ Related topics, types: Lifshitz RMP(97) [colored]; Fisher & Rabson mp/01 [classification with group cohomology]; Gouliaev cm/01/ACA [analytic]; Lifshitz FP(03) [without forbidden symmetries]; Cornwell PS(04) [icosahedral]; Masáková et al JPA(05) [Voronoi and Delaunay tiles]; Au-Yang & Perk mp/06-in [projections from 5D]; Böröczky et al JGP(06) [combinatorial properties]; Fujita a0906/ACA [decagonal, inflation rules].

Related Topics > s.a. optical technology [photonic crystals]; matter [crystallization, mathematical models]; particles [propagation].
* Liquid crystals: They consist of rod shaped molecules with the ability to polarize light; An applied voltage lines up the rods and shuts off or turns on transmitted light; 1999, seen to be able to produce sound.
@ Liquid crystals: in Landau & Lifshitz 86; de Gennes & Prost 95; Ondris-Crawford et al AJP(95)sep-RL; > s.a. topological defects.
@ In higher dimensions: Parisi JSP(08) [compact regular lattices].
@ Other topics: Michel & Mozrzymas in(78), in Nash & Sen, ch8 [Morse theory and symmetry breaking]; Nussbaum AJP(00)oct [Bravais lattice]; Rabson et al FP(03) [and cohomology]; Libbrecht pw(08)jan [snowflakes]; > s.a. Ewald Construction [reciprocal lattice].

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