liquidcrystal第二部分.pdf

2) Quasicrystals The Nobel Prize for Chemistry was awarded in 2011 to Daniel Shechtman for a discovery that genuinely shook the foundations of crystallography: in the words of the Nobel Committee, “His discovery of quasicrystals revealed a new principle for packing of atoms and molecules,” which “led to a paradigm shift within chemistry.” Quasicrystals are found in amorphous metals, usually obtained by quenching. The diffraction pattern of a quasicrystal contains sharp spots but with “forbidden” symmetries such as 10-fold rotation axes. These properties indicate that they are aperiodic, suggesting a connection to tilings in mathematics. Quasicrystals: the evidence A quasicrystal is a solid with conventional crystalline properties, but exhibiting a point-group symmetry inconsistent with translational periodicity. Like crystals, quasicrystals ● have discrete diffraction patterns, ● crystallize into polyhedral forms, ● have long-range orientational order, all indicating that their structure is not random. The unconventional (5-fold) symmetries mean the discrete diffraction pattern is not a reciprocal periodic lattice: this must be a solid that is ordered but not periodic – only quasiperiodic. The discovery of such quasicrystals in 1982 contradicted a long-held belief that all crystals should be periodic arrangements of atoms or molecules. It is easy to show that in two and three dimensions the possible rotations that superimpose an infinitely repeating periodic structure on itself are limited to angles of 360°/n, where n can be only 1, 2, 3, 4, or 6. The proof proceeds by constructing vectors from the basis vectors of the proposed n-fold symmetric point, e.g. a m = [cos(2πm/5), sin(2πm/5)], m = 0, 1, 2, …, n-1, for n = 5. Now a 1 +a 4 = a 0 /φ, φ = (1+√5)/2 1, a contradiction. The three-dimensional analog is the inability of dodecahedra to fill space. – combinations of the allowed rotations lead to only 32 point gro