×

zbMATH — the first resource for mathematics

Objective structures. (English) Zbl 1120.74312
Summary: An objective atomic structure is a collection of atoms represented by mass points or ions for which every atom sees precisely the same atomic environment, up to rotation (or, more generally, orthogonal transformations) and translation. An objective molecular structure is a collection of molecules in which corresponding atoms in each molecule see precisely the same environment up to orthogonal transformation and translation. Many of the most actively studied structures in science satisfy these conditions, including an arbitrary ordered periodic crystal lattice, the tails and also the capsids of certain viruses, carbon nanotubes, many of the common proteins and \(C_{60}\). A single crystal rod that has been bent and twisted into helical form also satisfies the conditions in a certain sense. The quantum mechanical significance of objective structures is described and some general methods for generating such structures are developed. Using these methods, some unexpected objective structures are revealed. Methods for simplified atomic level calculations of the energy, equilibrium and dynamics of these structures are given.

MSC:
74A25 Molecular, statistical, and kinetic theories in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Allen, M.P.; Tildesley, D.J., Computer simulation of liquids, (1987), Oxford University Press Oxford · Zbl 0703.68099
[2] Arroyo, M.; Belytschko, T., Finite crystal elasticity of carbon nanotubes based on the exponential cauchy – born rule, Phys. rev. B, 69, 115415, (2004)
[3] Braig, K.; Otwinowski, Z.; Hegde, R.; Boisvert, D.; Joahimiak, A.; Horwich, A.L.; Sigler, P.B., Crystal structure of groel at \(2.8 Å\), Nature, 371, 578-586, (1994)
[4] Branden, C.; Tooze, J., Introduction to protein structure, (1999), Garland Publishing New York
[5] Cahill, K., Helices in biomolecules, Phys. rev. E, 72, 062901, (2005)
[6] Caspar, D.L.D.; Klug, A., Physical principles in the construction of regular viruses, Cold spring harbor symp. quant. biol., 27, 1-24, (1962)
[7] Chouaieb, N.; Maddocks, J.H., Kirchhoff’s problem of helical equilibria of uniform rods, J. elasticity, 77, 221-247, (2004) · Zbl 1071.74031
[8] Crane, H.R., Principles and problems of biological growth, Sci. mon., 70, 376-389, (1950)
[9] Crick, F.H.C.; Watson, J.D., The structure of small viruses, Nature, 177, 473-475, (1956)
[10] Dolbilin, N.P.; Lagarias, J.C.; Senechal, M., Multiregular point systems, Geometry: discrete comput. geometry, 20, 477-498, (1998) · Zbl 0915.51017
[11] Ericksen, J.L., Special topics in elastostatics, (), 189-244 · Zbl 0475.73017
[12] Ericksen, J.L., The Cauchy and Born hypotheses for crystals, (), 61-77 · Zbl 0567.73112
[13] Falk, W.; James, R.D., An elasticity theory for self-assembled protein lattices with application to the martensitic phase transition in bacteriophage T4 tail sheath, Phys. rev. E, 73, 011917, (2006)
[14] Friesecke, G.; Theil, F., Validity and failure of the cauchy – born hypothesis in a two-dimensional mass-spring lattice, J. nonlinear sci., 12, 5, 445-478, (2002) · Zbl 1084.74501
[15] Hahn, T. (Ed.), 2003. In: Kopsky, V., Litvin, D.B. (Eds.), International Tables for Crystallography, vols. A, E. Kluwer Academic Publishers, Dordrecht, Boston.
[16] Harris, W.F.; Scriven, L.E., Cylindrical crystals, contractile mechanisms of bacteriophages and the possible role of dislocations in contraction, J. theor. biol., 27, 233-257, (1970)
[17] James, R.D., Dumitrica, T., 2006. Objective molecular dynamics, preprint.
[18] Kostyuchenko, V.A., Leiman, P.G., Rossmann, M.G., 2006. Three-dimensional CryoEM reconstruction of extended tail of bacteriophage T4, in preparation.
[19] Kovács, F.; Tarnai, T.; Guest, S.D.; Fowler, P.W., Double-link expandohedra: a mechanical model for expansion of a virus, Proc. R. soc. London ser. A, 460, 3191-3202, (2004) · Zbl 1063.92030
[20] Ksaneva, K.P., Electron microscopy of virulent phages for streptococcus lactis, Appl. environ. microbiol., 31, 590-601, (1976)
[21] Leiman, P.G.; Chipman, P.R.; Kostyuchenko, V.A.; Mesyanzhinoz, V.V.; Rossmann, M.G., Three-dimensional rearrangement of proteins in the tail of bacteriophage T4 on infection of its host, Cell, 118, 419-429, (2004)
[22] Nikulin, V.V.; Shafarevich, I.R., Geometries and groups, (1987), Springer Berlin, Heidelberg, (translated from the Russian by M. Reid) · Zbl 0621.51001
[23] Parrinello, M.; Rahman, A., Crystal structure and pair potentials: a molecular dynamics study, Phys. rev. lett., 45, 14, 1196-1199, (1980)
[24] Rabe, K.M.; Waghmare, U.V., Localized basis for effective Hamiltonians: lattice Wannier functions, Phys. rev. B, 52, 13236-13246, (1995)
[25] Rabe, K.M.; Waghmare, U.V., Ferroelectric phase transitions from first principles, J. phys. chem. solids, 57, 1397-1403, (1996)
[26] Shih, W.M.; Quispe, J.D.; Joyce, G.F., A 1.7-kilobase single-stranded DNA that folds into a nanoscale octahedron, Nature, 427, 618-621, (2004)
[27] Stubbs, G., Fibre diffraction studies of filamentous viruses, Rep. prog. phys., 64, 1389-1425, (2001)
[28] Tadmor, E.B.; Ortiz, M.; Phillips, R., Quasicontinuum analysis of defects in solids, Philos. mag. A, 73, 1529-1563, (1996)
[29] Toukan, K.; Carrion, F.; Yip, S., Molecular dynamics study of structural instability of two-dimensional lattices, J. appl. phys., 56, 5, 1455-1461, (1983)
[30] Toupin, R.A., St. Venant’s principle, Arch. ration. mech. anal., 18, 83-96, (1965) · Zbl 0203.26803
[31] Vainshtein, B.K., Fundamentals of crystals: symmetry and methods of structural crystallography, (1994), Springer Berlin, Heidelberg
[32] Weinan, E., Ming, P., 2006. Cauchy-Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal., in press. · Zbl 1106.74019
[33] Zanzotto, G., The cauchy – born hypothesis, nonlinear elasticity and mechanical twinning in crystals, Acta. crystallogr. A, 52, 839-849, (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.