In General > s.a. condensed matter / statistical mechanical systems; X Rays [crystallography].
$ 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: Glusker & Trueblood 10; Sidebottom 12 [r PT(13)may].
@ Dynamics: Born & Huang 54 [dynamics of lattices]; Gerstle 15 [peridynamics, computational]; > s.a. Defects; Elasticity.
@ And quantum theory: Grushevskaya & Gurskii qp/06 [many-particle effects on electron states]; Bottesi & Zemba JSM(08)-a0801 [effective theories for electrons]; Gajda et al EPL(16)-a1511 [Pauli crystals, from the exclusion principle]; > s.a. modified coherent states.
@ Symmetries, types: Baake JPA(97) [color symmetries]; Michel PRP(01); Nardone JMP(06); Bradley & Cracknell 10 [mathematical theory]; Allen a1006 [hexagonal, four-index notation]; Hitzer a1306-proc [history]; Honari & Mohades a1601 [symmetries and acoustic spectrum]; Alexandradinata et al PRX(16) [extended classification using generalized symmetries that include quasimomentum translations].
> Specific examples: see Water and Ice; Wigner Crystal.

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; The name "quasicrystal" was coined by P J Steinhardt.
* In nature: A meteorite fragment kept in a Florence Museum holds a grain of quasicrystal; P J Steinhardt and others found more where that one came from in Russia.
* Experiment: 3D non-periodic tilings with icosahedral structure have been seen first by Dan Shechtman of NBS in 1983 [@ Shechtman et al PRL(84)], from diffraction patterns produced by some alloys of Al and Mn, then made by An-Pang Tsai in Japan in 1987, and others; The golden mean appears all over the place in the patterns; We now know that they are not a rare exception, and they can be found in many alloy systems.
* 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-proc [mathematical models]; Barber 08 [r CP(10)]; news SA(09)oct [more normal than assumed]; Allouche & Meyer CRP(14)-a1401, Adiceam a1604 [mathematical properties].
@ Uses, applications: news pw(07)feb [in Islamic art]; news pw(07)mar [as filter for terahertz light]; news pw(12)jan [in ancient Islamic architecture].
@ And physics: Albuquerque & Cottam PRP(03) [elementary excitations]; Monreal et al IJMPA(08)-a0804 [quantum particle and effective non-commutative geometry]; Colli & Mariano PLA(11) [corrections to the standard formulation of quasicrystal linear elasticity].
@ In nature: Steinhardt & Bindi RPP(12) + news nat(12)aug, pw(12)aug [samples from meteorite]; news sn(16)dec [in Russian meteorite].
@ 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-conf [projections from 5D]; Böröczky et al JGP(06) [combinatorial properties]; Fujita ACA(09)-a0906 [decagonal, inflation rules]; news pw(11)nov [new type made of hard triangular bipyramids]; Kraus et al PRL(12) + Quandt Phy(12) [and topological insulators]; Gambaudo & Vignolo NJP(14)-a1309 [Brillouin zone labelling]; news PhysOrg(13)oct [new form of 12-sided quasicrystal accidentally discovered].

Time Crystals
* Idea: A previously unknown state of matter proposed by Wilczek in 2012, whose structure would repeat periodically but in time rather than in space, as an example of spontaneous breaking of time-translation symmetry; The trick to finding an example is to find a system in in its ground state which is nevertheless in motion, which means it would be an example of perpetual motion machine (without breaking fundamental laws because it would not be possible to extract energy from them); The closest that modern technology has come to a time crystal is a current-carrying superconductor, but the actual systems set up so far to not qualify as true time crystals.
@ General references: Wilczek PRL(12)-a1202, comment Bruno PRL(13)-a1210 [quantum time crystals] + Zakrzewski Phy(12) + Coleman Nat(13)jan; news SA(13)feb, sf(13)apr; Nozières a1306, Bruno PRL(13)-a1306 + news PhysOrg(13)aug [impossibility of spontaneous rotation]; Watanabe & Oshikawa PRL(15)-a1410 + news PhysOrg(15)jul [no-go result]; Castillo et al a1410 [quantum fluctuations]; Strocchi & Heissenberg a1605 [existence of quantum time crystals]; Yao et al a1608; Khemani et al a1612 [new definition and applications]; Richerme Phy(16) [on an approach to non-equilibrium/driven tieme crystals]; news ls(17)mar [experiments]; Sacha & Zakrzewski a1704 [rev]; > s.a. classical systems [classical time crystals].
@ Floquet systems: Else et al PRL(16)-a1603 + news PhysOrg(16)sep [spontaneously broken time-translation symmetry]; Zhang et al a1609 [observation].
@ Special models: Li et al PRL(12)-a1206 + news pw(12)jul [4D spacetime crystals of trapped ions]; news ns(12)jul, csm(12)sep [computer based on a time crystal could outlive the universe]; Borzdov a1410 [electromagnetic]; Smolyaninov EJTP-a1501 [metamaterial model].

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 produces a nematic order (directional rather than positional) that shuts off or turns on transmitted light; 1999, seen to be able to produce sound; > s.a. condensed matter [soft matter].
* Quantum crystals: Crystals are made up of atoms (like He) or molecules (like H) that are so light that nuclear quantum effects become important, and new, nonclassical theories are needed to describe them; The lightness of the atoms or molecules gives them, according to quantum mechanics, a large zero-point motion, which results in delocalization of bonds between them; > s.a. Plasticity.
@ In higher dimensions: Parisi JSP(08) [compact regular lattices].
@ Liquid crystals: in Landau & Lifshitz v7; de Gennes & Prost 95; Ondris-Crawford et al AJP(95)sep [RL]; Dunmur & Sluckin 11 [history]; Blinov 11 [structure and properties]; Ravnik Phy(13) [and Hopf fibrations]; news pw(14)jan [living liquid crystals]; Mila Phy(17) [magnetic analog for spins in a copper oxide]; > s.a. Defects; topological defects.
@ Other topics: Michel & Mozrzymas in(78), in Nash & Sen 83, ch8 [Morse theory and symmetry breaking]; Nussbaum AJP(00)oct [Bravais lattice]; Rabson et al FP(03) [and cohomology]; Libbrecht pw(08)jan [snowflakes]; news wisc(13)apr [vaterite is composed of two interspersed crystal structures]; Taioli et al a1511 [non-Euclidean crystallographic group]; > s.a. Ewald Construction [reciprocal lattice]; Foam [polycrystals]; metamaterials [granular crystals].

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