Orientational sampling schemes based on four dimensional: Polytopes

Symmetry

Volumn 2, Issue 3, 2010, Pages 1423-1449

  Mamone, Salvatore ^a   Pileio, Giuseppe ^a   Levitt, Malcolm H ^a  

^a UNIVERSITY OF SOUTHAMPTON   (United Kingdom)

Author keywords

Group theory; Orientational sampling; Polychora; Polytope; Powder averaging; Schl afli symbols; Solid state nuclear magnetic resonance

Indexed keywords

EID: 79953317754     PISSN: None     EISSN: 20738994     Source Type: Journal    
DOI: 10.3390/sym2031423     Document Type: Article

References (33)

1. Mehring, M. High Resolution NMR in Solids, 2nd ed.; NMR-Basic Principles and Progress; Springer: Berlin, Germany, 1982; Volume 11.
2. Levitt, M.H. Spin Dynamics. Basics of Nuclear Magnetic Resonance, 2nd ed.; Wiley: Chichester, UK, 2007.
3. Bak, M.; Nielsen, N.C. REPULSION, A Novel Approach to Efficient Powder Averaging in Solid-State NMR. J. Magn. Reson. 1997, 125, 132-139.
4. Edén, M.; Levitt, M. H. Computation of Orientational Averages in Solid-State NMR by Gaussian Spherical Quadrature. J. Magn. Reson. 1998, 132, 220-239.
5. Edén, M. Computer simulations in solid-state NMR. III. Powder averaging. Concepts Magn. Reson. 2003, 18A, 24-55.
6. Stevensson, B.; Edén, M. Efficient orientational averaging by the extension of Lebedev grids via regularized octahedral symmetry expansion. J. Magn. Reson. 2006, 181, 162-176.
7. Suter, D.; Pearson, J.G. Experimental Classification of Multi-Spin Coherence under the Full Rotation Group. Chem. Phys. Lett. 1988, 144, 328-332.
8. van Beek, J.D.; Carravetta, M.; Antonioli, G.C.; Levitt, M.H. Spherical tensor analysis of nuclear magnetic resonance signals. J. Chem. Phys. 2005, 122, 244510.
9. Pileio, G.; Levitt, M.H. Isotropic filtering using polyhedral phase cycles: Application to singlet state NMR. J. Magn. Reson. 2008, 191, 148-155.
10. Varshalovich, D.A.; Moskalev, A.N.; Khersonskǐ i, V.K. Quantum theory of angular momentum; World Scientific Publishing: Teaneck, NJ, USA, 1988.
11. Samoson, A.; Sun, B.Q.; Pines, A. New angles in motional averaging. In Pulsed Magnetic Resonance: NMR, ESR, and Optics, A recognition of E. L. Hahn; Bagguley, D.M.S., Ed.; Oxford Science Publications: Oxford, UK, 1992.
12. Emsley, J.L.; Schmidt-Rohr, K.; Pines, A. Group theory and NMR dynamics. In NMR and More: In Honour of Anatole Abragam; Goldman, M., Por, M., Eds.; Les Editions de Physique: Paris, France, 1994.
13. Lebedev, V.I. Quadratures on a Sphere. Zh. Vychisl. Mat. mat. Fiz. 1975, 15, 48.
14. Lebedev, V.I. Quadratures on a Sphere. Zh. Vychisl. Mat. mat. Fiz. 1976, 16, 293.
15. Lebedev, V.I.; Skorokhodov, A.L. Quadrature Formulas of Orders 41, 47 and 53 for the Sphere. Russian Acad. Sci. Dokl. Math. 1992, 45, 587.
16. Weissenstein, E.W. Regular Polychoron. http://mathworld.wolfram.com/RegularPolychoron.html. (accessed on 6 July 2010)
17. Coxeter, H.S.M. Regular Polytopes; Methuen: London, UK, 1948.
18. Delsarte, P.; Goethals, J.M.; Jacob, S.J. Spherical codes and designs. Geom. Dedicata 1977, 6, 363-388.
19. Du Val, P. Homographies, quaternions and rotations; Oxford University Press: Oxford, UK, 1964.
20. Frobenius, F. Über die Charaktere der symmetrischen Gruppe. Sitzungberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin 1900, 516-534.
21. Veysseyre, R.; Weigel, D.; Phan, T.; Effantin, J.M. Crystallography, geometry and physics in higher dimensions. II. Point symmetry of holohedries of the two hypercubic crystal systems in four-dimensional space. Acta Cryst. Sec. A 1984, 40, 331-337.
22. Kondo, T. The characters of the Weyl group of type F4. J. Fac. Sci. Univ. Tokyo Sect. I 1965, 11, 145-153.
23. Grove, L.C. The characters of the hecatonicosahedroidal group. J. Reine Angew. Math. 1974, 265, 160-169.
24. Levitt, M.H. An Orientational Sampling Scheme for Magnetic Resonance based on a Four-Dimensional Polytope. In Future Directions in Magnetic Resonance; Khetrapal, C.L., Ed.; Springer: Berlin, Germany, 2010.
25. Armstrong, M.A. Groups and Symmetry; Sringer-Verlag: New York, NY, USA, 1988.
26. Herstein, I.N. Topics in algebra, 2nd ed.; Xerox College Publishing: Lexington, KY, USA, 1975.
27. Cotton, A. Chemical Applications of Group Theory, 3rd ed.; Wiley: New York, NY, USA, 1990.
28. Hamermesh, M. Group theory and its application to physical problems; Addison-Wesley: London, UK, 1962.
29. Altmann, S.L. Rotations, quaternions, and double groups; Oxford University Press: New York, NY, USA, 1986.
30. Wigner, E.P. Group theory and its application to the quantum mechanics of atomic spectra; Academic Press: New York, NY, USA, 1971.
31. Wolfram, S. Mathematica: A System for Doing Mathematics by Computer. Addison-Wesley: New York, NY, USA, 1991.
32. Cohn, H.; Kumar, A. Universally optimal distribution of points on spheres. J. Amer. Math. Soc. 2007, 20, 99-148.
33. Altschuler, E.L.; Perez-Garrido, A.; Stong, R. A Novel Symmetric Four Dimensional Polytope Found Using Optimization Strategies Inspired by Thomson's Problem of Charges on a Sphere. 2008, arXiv:physics/0601139v1