# Are Denser Packings of Spheres than Closest Packings Possible How Many Closest Packings of Spheres Exist.

код для вставкиСкачатьHIGHLIGHTS Are Denser Packings of Spheres than Closest Packings Possible? How Many Closest Packings of Spheres Exist? By Ulrich Miiller* In 1611 Johannes Kepler wrote a humorous 24-page epistle which was a New Year’s present to his benefactor in Prague, the Imperial Councilor Wackher von Wackenfels.[’I Here he proposed an explanation of the hexagonal shape of snowflakes: Water vapor, once it feels the penetrating coldness, freezes into little spheres with some definite size which he calls vapor spheres (sphaere vapidae). The vapor spheres in contact with one another adopt the arrangement that we nowadays call cubic closest packing of spheres and which we know to be the structural principle of many metals and of the solid noble gases. For this packing Kepler asserted, “It is the tightest possible; in no other arrangement can more spheres be accommodated in the same vessel.”[21The term closest packing of spheres shows that chemists, physicists, and crystallographers have long intuitively accepted Kepler’s assertion; however, it was not proven mathematically until 1991 by M! Y. H~iang.‘~] For periodically ordered packings of spheres, i.e. lattice packings, Gauss furnished the proof of the highest packing density. although he did not mention packings of spheres at However, he introduced the concept of a lattice, and by relating it to the Seeber’s theorems[5] on number theory he stated: the volume V of the (primitive) unit cell of a lattice will always satisfy the condition V@ 2 abc if the face and body diagonals of the cell are not shorter than the lattice parameters u, h, and c. The upper limit of the space filling of a lattice packing of equal spheres follows from this condition as was shown by Minkowski.[61The primitive unit cell of the cubic closest packing of spheres fulfilling Gauss’s condition is a rhombohedra1 cell with the cell edges a = b = c = 2r and the volume V = $a3@ ( r = radius of a sphere). This volume is exactly equal to the limiting value of the Gauss condition: V l / z = ( $ a 3 f i ) f i 2 a3 (it also is identical for any other closest packing of spheres). A crystal lattice having a smaller unit cell volume, i.e. with greater packing density, is therefore not possible for a packing of equal spheres. This statement, however, is valid only when there is a crystal lattice. Just as chemists presently have caught the buckyball fever, crystallographers are fiercily discussing the topic of quasi crystals. Quasi crystals have symmetry (e.g. fivefold rotation axes or icosahedral symmetry), and there is a near order around the atoms, but they have no three-dimensional periodical order. Could a quasi-crystalline packing of spheres exist having a higher degree of space filling than a cubic closest packing of spheres? Since the twelve spheres surrounding a central sphere of the same size in an icosahedral arrangement are not in contact with each other, a larger space filling seems possible at first (the coordination number in a closest packing of spheres is 12). Mathematicians have repeatedly furnished [*] Prof. Dr. U. Muller FB Chemie der Universitit Hans-Meerwein-Strasse, D-W-3550 Marburg (FRG) Angrn. Clirrn. lnt. Ed. Engl. 31 (1992) N o . 6 proofs with upper limits for the space filling, with values that have been decreasing over the years in the third or fourth decimal place, e.g. 0.77964 (1958), 0.77844 (1986), and finally 0.77836 (1988). There have also been repeated attempts to design nonperiodical packings of spheres, for instance with fivefold symmetry[’] or consisting of shells with icosahedral symmetry,[’] but not even the space filling of a closest packing of spheres was ever achieved. The central idea of Hsiang’s proof concerns the domains of influence of the spheres (also called the Wirkungsbereich, Voronoi domain, Dirichlet domain, or Wigner-Seitz cell). The domain of influence of a sphere is the polyheron made up of the set of planes perpendicular to the lines that connect the sphere under consideration to the surrounding spheres and passing through the midpoints of these lines. (For spheres in contact with each other these are the tangent planes between the spheres.) Kepler described the domain of influence of a sphere in the cubic closest packing by considering the seeds in a pomegranate. Initially the seeds are round, but later as the rind hardens and the seeds continue to grow, they become tightly packed and squeezed together, filling the remaining space and finally taking the shape of rhombic dodecahedra (Fig. 1). In an arbitrary packing of spheres the domains of influence fill the space completely, so that the space filling results from the ratio of the sphere volume to the mean volume of all domains of influence. The domain of influence of the cubic closest packing of spheres, the rhombic dodecahedron, has a volume of 4l,h when the sphere radius is r = I . With the sphere volume f n we thus calculate the well-known degree of space filling of the cubic closest packing of spheres to be 47c/(41/2) = kn@ = 0.7404805. The smallest possible local domain of influence of a single sphere results when it is coordinated i c ~ s a h e d r a l l y ;this [~~ domain of influence is a pentagonal dodecahedron (Fig. 1). It has a volume of 101/2(65 - 291/J), which yields a local space filling of 0.7546974. This value had been suspected to be the upper limit for sphere packings but was never proven.”] Since space cannot be completely filled by pentagonal dodecahedra, other spheres must have different domains of influence. What is the minimum mean volume of all domains of influence? Fig. 1 . Rhombic dodecahedron (left) and pentagonal dodecahedron (right) enclosing spheres of equal size. VCH K~rlagsgesellsrhafimbH, W-6940 Weinheim. 1992 OS70-0833/92/0606-0727 $3.50+.2S/O 727 Tahle 1. Number of possible stacking variants of closest packings of spheres depending on the space group type 1151. N N P6Jn7mr P6,mc P3ml P6m2 P3ml ~~~~~~ 2 3 4 5 6 7 8 9 10 11 12 24 36 48 1 [dl - ~ 1 - 1 - 2 - ~ - - - 2 - 3 - - - 3 30 252 2 040 1 70 3514 173740 12 990 65 268 4 192200 6 RSn7 ~ ~ - 1 3 2 3 5 11 11 984 65 240 4192080 - 3 2 10 15 57 208 159005154 488 667 525 580 of layers per packet R3m total ~ - - 1 1 [bl - - - - 1 - 1 1 1 2 3 6 7 16 21 43 59 290 159 139498 488676086380 - 1 = number - 1 6 28 120 - 2 42 620 [a] Hexagonal closest packing. [b] Cubic closest packing. space group F m h In addressing this question Hsiang considered a second shell of spheres, consisting of 42 to 44 spheres around the 12 spheres of the first shell; including the central sphere this amounts to a total of 55 to 57 spheres. His impressive goal was to analyze all possible configurations and to derive the domains of influence of the spheres of the first shell. Most configurations could be eliminated because the volumes are too large; for other configurations it had to be shown that they are close to certain key configurations. The distinction of these cases makes up the greater part of the proof which extends over more than 150 pages. First a new method for estimating the volume of local domains of influence was used to prove that the global space filling cannot exceed the local space filling in a pentagonal dodecahedron. Then after introducing the concept of semiglobal space-filling, he derived the maximum global space-filling; it is 0.7404805. Hsiang first circulated his proof among mathematicians who now have the opportunity to check its correctness. In other circles his work became known by two communications," 0 %111 Another problem that had never been solved was the question of the possible number of different closest packings of spheres that all have the same space filling as cubic closest packing. For stacks of hexagonal layers of spheres, any stacking sequence of layers with the positions A , B, and C is possible as long as no two layers of the same position are adjacent; therefore, the number of stacking variants is infinite. But how many arrangements are possible for periodical stackings in which a packet of a given number of layers is repeated? First of all, we have to make a clarifying statement: if we take the cubic closest packing of spheres and consider a stacking direction inclined towards the layer plane A + B + C -+ A ..., then we have a translationary equivalent layer after one layer (one layer per packet). However, if we only consider a stacking direction perpendicular to the layers, the packet is repeated only after three layers, (ABC) (ABC) ... An inclined stacking of a packet having N layers will always result in a rhombohedra1 structure which can be described in a hexagonal setting with packets consisting of 3N layers. In the following we will only consider the perpendicular stacking direction. For a small number of layers per packet the possible stacking sequences can be written down and The method for calculating the number of possibilities for an ---f 728 $3 VCH Verlugsgesdl.schu/r mbH, W-6940 Wernhcim, 1992 arbitrary number of layers per packet was first demonstrated in 1981 by McLarnan.[l5I This approach is based on Polya's theorem,[l6](which can also be used to calculate the number of isomers of organic compounds). If, in addition, some group theory and White's extension of Polya's theorem are included,[", 'I the number of stacking variants with a given symmetry (space group) can also be calculated. Table 1 lists some of the numbers calcuhted by McLarnan. The table illustrates why it is a significant progress to be able to calculate the numbers with dependence on symmetry: The largest numbers result for low symmetries, whereas in the higher symmetry space group types P6,lmmc and R3m the numbers are relatively small. Chemically reasonable crystal structures are usually only those with higher symmetry. The method described by McLarnan["] can also be adapted for the calculation of the number of crystal structure types possible with particular space groups for any kind of compound, provided that some definite packing principle can be stated." I thank Prof. Dr. E. Krajft for helping me locate Kepler's pupers. German version: Angew. Chrm. 1992, 104. 744 [l] J. Kepler, Srrena seu de nive se.xungulu, G. Tampach. Frankfurt. 1611. Reprinted in: Johannes Keplrr -gesummelie Werke(Eds.:W. van Dyck. M. Caspar, F. Hammer), Vol. IV, C. H. Beck, Miinchen, 1941, p. 261-280. English translation: On the SiY-Cornered Snowjluke, Oxford University Press, 1966. [2] "Coaptatio fiet arctissima: ut nullo praetera ordine plures glohuli in idem vas compingi queant." [3] W. Y Hsiang, Reports PAM-530 and PAM-535. Center for Pure and Applied Mathematics, University of California, Berkeley, 1991. Bull. Bra:& iun Murh. Soc., in press. [4] C. F. Gauss, Wcrke 11. Kiinigliche Gesellschaft der Wissenschaften. Gottingen, 1876. pp. 388-~196. (51 L. A. Seeber. Unrrr.~uchrcngenuber die Eigenschu/ten der posiriveii rernui-en yuadrurrschen Formen, Freihurg, 1831. [6] H. Minkowski, Nurhr. Ges. W i v . Garringen Murh. P h u . KI. 1904, 311. [7] B. G. Bagley, Narure 1965. 208. 674; ;bid. 1970, 225. 1040. [XI A. L. Mackay, Artu Cr.wtaNogr. 1962. 15. 916. [9] L. Fejes Tbth, Lugerringen in der Ebene uuf der Kugrl and im Raum, Springer, Berlin, 1953, p. 171. [lo] I. Stewart, New Scientist 1991, 130. No. 1777. 29. [Ill N. M a x , Nature 1992, 355, 115. [12] G. Higg, Ark Kern; Minerd. Geol. B 1943, 16, 1. [13] G. S. Zhdanov. Dokl. Akud. Nuuk SSSR 1945. 4X. 40. [14] P. A. Beck, Z . Krisldlogr. 1967. 124. 101 [15] T. J. McLarnan, Z. KristuNugr. 1981, 155, 269. [16] G. Polya. Acta Murh. 1937, 68, 145. [17] D. E. White, Dircre/e Marh. 1975, 13, 277. [IS] T. J. McLarnan. Z. Krblullogr.. 1981. 1.55, 277. (191 U . Miiller. Acra C r w M o g r . Set/. B 1992. 48, in press. 0570-0833/9210606-0728 $3.50+ .25jO Angrw. Chrm. Inr. Ed. Engl. 3f (1992) No. 6

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