Densest local sphere-packing diversity. II. Application to three dimensions

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 83, Issue 1, 2011, Pages

  Hopkins, Adam B ^a   Stillinger, Frank H ^a   Torquato, Salvatore ^a  

^a PRINCETON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COORDINATION SHELLS; ENERGY CONFIGURATIONS; EUCLIDEAN SPACES; FINITE SUBSETS; ICOSAHEDRAL SYMMETRY; LENNARD JONES; NONOVERLAPPING SPHERES; NUCLEATION THEORY; PAIR CORRELATION FUNCTIONS; PAIR POTENTIAL; REALIZABILITY CONDITIONS; SIMILARITY METRICS; SPHERE PACKINGS; THREE DIMENSIONS; UPPER BOUND;

PACKING;

SPHERES;

EID: 79951709638     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.83.011304     Document Type: Article

References (43)

1. A. B. Hopkins, F. H. Stillinger, and S. Torquato, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.3372627 51, 043302 (2010).
2. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer, Berlin, 1998).
3. K. Schütte and B. L. van der Waerden, Math. Ann. MAANA3 0025-5831 10.1007/BF01343127 125, 325 (1953).
4. J. Leech, Math. Gaz. MAANA3 0025-5572 10.2307/3610264 40, 22 (1956).
5. K. Schütte and B. L. van der Waerden, Math Ann. MAANA3 0025-5831 10.1007/BF02054944 123, 96 (1951).
6. A. B. Hopkins, F. H. Stillinger, and S. Torquato, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.81.041305 81, 041305 (2010).
7. S. Torquato, T. M. Truskett, and P. G. Debenedetti, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.84.2064 84, 2064 (2000).
8. S. Torqua (Pubitemid 35338422)
9. B. D. Lubachevsky and R. L. Graham, Disc. Comp. Geom. MAANA3 0179-5376 10.1007/PL00009314 18, 179 (1997).
10. A. Lenard, Arch. Ration. Mech. Anal. 10.1007/BF00251602 59, 219 (1975).
11. S. Torquato and F. Stillinger, J. Phys. Chem. B JPCBFK 1520-6106 10.1021/jp0208687 106, 8354 (2002). (Pubitemid 35382943)
12. T. Kuna, J. L. Lebowitz, and E. R. Speer, J. Stat. Phys. JSTPBS 0022-4715 10.1007/s10955-007-9393-y 129, 417 (2007).
13. S. Torquato and F. H. Stillinger, Exp. Math. 15, 307 (2006).
14. A. B. Hopkins, F. H. Stillinger, and S. Torquato, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.79.031123 79, 031123 (2009).
15. max realizability condition to improve upon the best-known upper bounds [16] on the maximum infinite-volume packing fraction φ * ∞ for identical nonoverlapping spheres in various dimensions, where the method employed by Cohn and Elkies in [16] relies only on particle pair information.
16. H. Cohn and N. Elkies, Ann. Math. ANMAAH 0003-486X 10.4007/annals.2003. 157.689 157, 689 (2003).
17. Supplementary materials can be found at [http://cherrypit.princeton.edu].
18. H. Pfoertner, Densest Packings of Spheres in a Sphere [http://www.randomwalk.de/sphere/insphr/spheresinsphr.html].
19. W. Barlow, Nature (London) NATUAS 0028-0836 10.1038/029186a0 29, 186 (1883).
20. H. Cohn and A. Kumar, Ann. Math. ANMAAH 0003-486X 10.4007/annals.2009. 170.1003 170, 1003 (2009).
21. min (N - 1) is N = 60.
22. T. C. Hales, Ann. Math. ANMAAH 0003-486X 10.4007/annals.2005.162.1065 162, 1065 (2005).
23. R. M. L. Tammes, Rec. Trv. Bot. Neerl. 27, 1 (1930).
24. N. J. A. Sloane, R. H. Hardin, and W. D. Smith, Tables of Spherical Codes [www.research.att.com/njas/packings/].
25. min (216)] = 130, and 130 is the last number for which optimal spherical codes are presented in Ref. [25].
26. The Barlow packings consist of translated planes (layers) of contacting spheres with centers arranged on the sites of a triangular lattice, where the layers are stacked on top of one another. In the stackings, there are three possible translations relative to the central layer (the layer including the central sphere), denoted A, B, and C, obeying the rule that no layer can be consecutively repeated. We have found that it is only the positions of the A layers relative to the central layer that are relevant in determining the coordination shells about the central sphere. The fcc Barlow stacking, denoted by repeating ABC layers, is, thus, indistinguishable to the similarity metric from any other stacking with an A layer repeated every three layers.
27. S. C. Mau and D. A. Huse, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.59.4396 59, 4396 (1999).
28. M. R. Hoare and P. Pal, Adv. Phys. ADPHAH 0001-8732 10.1080/ 00018737100101231 20, 161 (1971).
29. J. A. Northby, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.453492 87, 6166 (1987).
30. J. P. K. Doye and D. J. Wales, Chem. Phys. Lett. 10.1016/S0009-2614(95) 01223-0 247, 339 (1995).
31. D. J. Wales and J. P. K. Doyes, J. Phys. Chem. A JPCAFH 1089-5639 10.1021/jp970984n 101, 5111 (1997).
32. Y. Xiang, H. Jiang, W. Cai, and X. Shao, J. Phys. Chem. A JPCAFH 1089-5639 10.1021/jp037780t 108, 3586 (2004).
33. D. J. Wales, J. P. K. Doye, A. Dullweber, M. P. Hodges, F. Y. Naumkin, F. Calvo, J. Hernandez-Rojas, and T. F. Middleton, The Cambridge Cluster Database [http://www-wales.ch.cam.ac.uk/CCD.html].
34. LJ are - 733. 110 71 and - 790. 278 12 [33].
35. N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway, Disc. Comp. Geom. JPCAFH 0179-5376 10.1007/BF02570704 14, 237 (1995).
36. In the diagrams of paper I, e.g., for DLP optimal packings in R 2 for N = 24, N = 45, and N = 95 disks, it is difficult to perceive that the disks forming a cavity around the central disk are not all the same radial distance from the central disk. However, although the difference in radial distances between these disks is small, it is far larger than the precision to which the spatial positions of sphere centers are calculated by the algorithm.
37. O. R. Musin, Ann. Math. ANMAAH 0003-486X 10.4007/annals.2008.168.1 168, 1 (2008).
38. H. J. Schöpe, G. Bryant, and W. van Megen, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.96.175701 96, 175701 (2006).
39. H. J. Schöpe, G. Bryant, and W. van Megen, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.2760207 127, 084505 (2007). (Pubitemid 47354012)
40. T. Schilling, H. J. Schöpe, M. Oettel, G. Opletal, and I. Snook, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.105.025701 105, 025701 (2010).
41. T. Kawasaki and H. Tanaka, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.1001040107 107, 14036 (2010).
42. J. A. van Meel, B. Charbonneau, A. Fortini, and P. Charbonneau, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.80.061110 80, 061110 (2009).
43. T. M. Truskett, S. Torquato, S. Sastry, P. G. Debenedetti, and F. H. Stillinger, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.58.3083 58, 3083 (1998).