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[[Image:tetrahedron.png|thumb|right]] | [[Image:tetrahedron.png|thumb|right]] | ||
The '''hard tetrahedron model''' | The '''hard tetrahedron model'''. Such a structure has been put forward as a potential model for [[water]]<ref>[http://dx.doi.org/10.1080/00268979500100281 Jiri Kolafa and Ivo Nezbeda "The hard tetrahedron fluid: a model for the structure of water?", Molecular Physics '''84''' pp. 421-434 (1995)]</ref>. | ||
==Maximum packing fraction== | ==Maximum packing fraction== | ||
It has recently been shown that regular tetrahedra are able to achieve packing fractions as high as <math>\phi=0.8503</math><ref>[http://dx.doi.org/10.1038/nature08641 Amir Haji-Akbari, Michael Engel, Aaron S. Keys, Xiaoyu Zheng, Rolfe G. Petschek, Peter Palffy-Muhoray and Sharon C. Glotzer "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra", Nature '''462''' pp. 773-777 (2009)]</ref> (the [[hard sphere model |hard sphere]] packing fraction is <math>\pi/(3 \sqrt{2}) \approx 74.048%</math> <ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>). This is in stark contrast to work as recent as in 2006, where it was suggested that the "...regular tetrahedron might even be the convex body having the smallest possible packing density"<ref>[http://dx.doi.org/10.1073/pnas.0601389103 J. H. Conway and S. Torquato "Packing, tiling, and covering with tetrahedra", Proceedings of the National Academy of Sciences of the United States of America '''103''' 10612-10617 (2006)]</ref>. | It has recently been shown that regular tetrahedra are able to achieve packing fractions as high as <math>\phi=0.8503</math><ref>[http://dx.doi.org/10.1038/nature08641 Amir Haji-Akbari, Michael Engel, Aaron S. Keys, Xiaoyu Zheng, Rolfe G. Petschek, Peter Palffy-Muhoray and Sharon C. Glotzer "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra", Nature '''462''' pp. 773-777 (2009)]</ref> (the [[hard sphere model |hard sphere]] packing fraction is <math>\pi/(3 \sqrt{2}) \approx 74.048%</math> <ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>). This is in stark contrast to work as recent as in 2006, where it was suggested that the "...regular tetrahedron might even be the convex body having the smallest possible packing density"<ref>[http://dx.doi.org/10.1073/pnas.0601389103 J. H. Conway and S. Torquato "Packing, tiling, and covering with tetrahedra", Proceedings of the National Academy of Sciences of the United States of America '''103''' 10612-10617 (2006)]</ref>. | ||
==Phase diagram== | ==Phase diagram== | ||
<ref>[http:// | <ref>[http://arxiv.org/abs/1106.4765 Amir Haji-Akbari, Michael Engel, Sharon C. Glotzer "Phase Diagram of Hard Tetrahedra", arXiv:1106.4765v2 Sat, 2 Jul (2011)]</ref> | ||
==Truncated tetrahedra== | ==Truncated tetrahedra== | ||
Dimers composed of | Dimers composed of truncated tetrahedra <ref>[http://dx.doi.org/10.1103/PhysRevLett.107.155501 Joost de Graaf, René van Roij, and Marjolein Dijkstra "Dense Regular Packings of Irregular Nonconvex Particles", Physical Review Letters '''107''' 155501 (2011)]</ref> are able to achieve packing fractions as high as <math>\phi= 207/208 \approx 0.9951923</math> | ||
<ref>[http://dx.doi.org/10.1103/PhysRevLett.107.155501 Joost de Graaf, René van Roij, and Marjolein Dijkstra "Dense Regular Packings of Irregular Nonconvex Particles", Physical Review Letters '''107''' 155501 (2011)]</ref><ref>[http://dx.doi.org/10.1063/1.3653938 Yang Jiao and Salvatore Torquato "A packing of truncated tetrahedra that nearly fills all of space and its melting properties", Journal of Chemical Physics '''135''' 151101 (2011)]</ref> | <ref>[http://dx.doi.org/10.1063/1.3653938 Yang Jiao and Salvatore Torquato "A packing of truncated tetrahedra that nearly fills all of space and its melting properties", Journal of Chemical Physics '''135''' 151101 (2011)]</ref> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
'''Related reading''' | '''Related reading''' | ||
*[http://dx.doi.org/10.1103/Physics.3.37 Daan Frenkel "The tetrahedral dice are cast … and pack densely", Physics '''3''' 37 (2010)] | *[http://dx.doi.org/10.1103/Physics.3.37 Daan Frenkel "The tetrahedral dice are cast … and pack densely", Physics '''3''' 37 (2010)] | ||
[[category: models]] | [[category: models]] | ||