Quasicrystals |

**In General** > 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.

* __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];
Janssen PRP(88);
Maddox Nat(89)jul,
DiVincenzo Nat(89)aug;
Stephens & Goldman SA(91)apr;
Goldman et al AS(96);
Pelantová & Masáková mp/06-proc [mathematical models];
news SA(09)oct [more normal than assumed];
Allouche & Meyer CRP(14)-a1401,
Adiceam a1604 [mathematical properties];
Grimm & Kramer a1906 [rev].

@ __Books__: Amann et al ed-88;
Jaric ed-88;
Senechal 95;
Janot 97;
Di Vincenzo & Steinhardt ed-99;
Barber 08 [r CP(10)];
Baake & Grimm 13,
17;
Steinhardt 19.

@ __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];
Kraus et al PRL(12)
+ Quandt Phy(12) [and topological insulators];
Gambaudo & Vignolo NJP(14)-a1309 [Brillouin zone labelling].

**Examples in Nature, Experiments, Applications**
> s.a. bose-einstein condensates; Dimer Models.

* __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.

@ __In nature__: Steinhardt & Bindi RPP(12)
+ news nat(12)aug,
pw(12)aug [samples from meteorite];
news sn(16)dec [in Russian meteorite].

@ __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].

@ __Experiments__: news pw(11)nov [new type made of hard triangular bipyramids];
news PhysOrg(13)oct
[new form of 12-sided quasicrystal accidentally discovered].

@ __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].

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send feedback and suggestions to bombelli at olemiss.edu – modified 25 jun 2019