 Fig. 1 HCP unit cell (left) and FCC unit cell (right)  Fig. 2 Thomas Harriot in ca. 1585 first pondered the mathematics of the cannonball arrangement or cannonball stack, which has an FCC lattice. Note how the two balls facing the viewer in the second tier from the top contact the same ball in the tier below. This does not occur in an HCP lattice (the left unit cell in Fig. 1, above). Close-packing of spheres is the arranging of an infinite lattice of spheres so that they take up the greatest possible fraction of an infinite 3-dimensional space. Carl Friedrich Gauss proved that the highest average density that can be achieved by a regular lattice arrangement is . The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. Image File history File links Close_packing. ...
Image File history File links Close_packing. ...
Thomas Harriot (ca. ...
A sphere is a perfectly symmetrical geometrical object. ...
(30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...
In mathematics, the Kepler conjecture is a conjecture about sphere packing in three dimensional Euclidean space. ...
There are two regular lattices that achieve this highest average density. They are called face-centered cubic (FCC) and hexagonal close-packed (HCP), based on their symmetry. Both are based upon sheets of spheres arranged at the vertices of a triangular tiling; they differ in how the sheets are stacked upon one another. In both arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres (octahedral) and two smaller gaps surrounded by four spheres (tetrahedral). In crystallography, the cubic crystal system (or isometric crystal system) is the most symmetric of the 7 crystal systems. ...
In crystallography, the hexagonal crystal system is one of the 7 lattice point groups. ...
Sphere symmetry group o. ...
Relative to a reference layer with positioning A, two more positionings B and C are possible. Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius.
The most regular ones are:
- HCP = ABABABA
- FCC = ABCABCA
Many crystal structures are based on a close-packing of atoms, or of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very close to one another in energy, and it may be difficult to predict which form will be preferred from first principles. Quartz crystal Copper(II) sulfate and iodine crystal Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid...
The coordination number of HCP and FCC is 12 and its atomic packing factor (APF) is the number mentioned above, 0.74. In chemistry, the coordination number (c. ...
In crystallography, atomic packing factor is the fraction of volume in a crystal structure that is occupied by atoms. ...
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