Mark Bowick and Alex Travesset 2001 J. Phys. A: Math. Gen. 34 1535 doi:10.1088/0305-4470/34/8/301
Mark Bowick and Alex Travesset
Show affiliationsWe study the formulation of bond-orientational order in an arbitrary two-dimensional geometry. We find that bond-orientational order is properly formulated within the framework of differential geometry with torsion. The torsion reflects the intrinsic frustration for two-dimensional crystals with arbitrary geometry. Within a Debye-Huckel approximation, torsion may be identified as the density of dislocations. Changes in the geometry of the system cause a reorganization of the torsion density that preserves bond-orientational order. As a byproduct, we are able to derive several identities involving the topology, defect density and geometric invariants such as Gaussian curvature. The formalism is used to derive the general free energy for a 2D sample of arbitrary geometry, in both the crystalline and hexatic phases. Applications to conical and spherical geometries are briefly addressed.
61.50.Ah Theory of crystal structure, crystal symmetry; calculations and modeling
61.50.Lt Crystal binding; cohesive energy
02.40.Hw Classical differential geometry
61.72.Lk Linear defects: dislocations, disclinations
61.50.Ks Crystallographic aspects of phase transformations; pressure effects
53Axx Classical differential geometry
82D25 Crystals (For crystallographic group theory, see 20H15)
Issue 8 (2 March 2001)
Received 26 May 2000, in final form 14 November 2000
Mark Bowick and Alex Travesset 2001 J. Phys. A: Math. Gen. 34 1535
Koji Asami and Katsuhisa Sekine 2007 J. Phys. D: Appl. Phys. 40 1128
Jean Macher and Jürgen Schaffner-Bielich 2005 Eur. J. Phys. 26 341
R Zhang et al 2009 J. Phys.: Conf. Ser. 191 012005
Hao Zhang et al 1999 J. Phys.: Condens. Matter 11 2035
X J Liu et al 2002 J. Phys. B: At. Mol. Opt. Phys. 35 2069
G D Rieck and K Koopmans 1964 Br. J. Appl. Phys. 15 419
N Berglund 2000 Nonlinearity 13 225
A Warczak et al 1983 J. Phys. B: At. Mol. Phys. 16 1575
John Cooper and Parvaiz Ali 2005 J. Radiol. Prot. 25 221