There is 1 atom split between the 8 corners. Since we assume that the height, c, is also a, its volume becomes. The rhombus is just two equilateral triangles put together, so it has an area of. The volume of the primitive cell is the area of the rhombus, times the height. This is a parallelpiped with angles of 60º, 120º, and 90º. To make the calculations simpler, I will use the primitive cell.
If we assume that the c/a ratio = 1, then it is possible to calculate the simple hexagonal APF for an ideal simple hexagonal unit cell. If you wanted to describe the simple hexagonal crystal with math, you would describe the cell with the primitive vectors: Indeed, in simple hexagonal silicon, the c/a ratio is ~1. In simple hexagonal crystals, however, there is no reason for c to be different than a and b. In the most common hexagonal crystal, Hexagonal Close-Packed (HCP), the c/a ratio is about 1.6. The c axis can be larger, smaller, or equal to the a axis–which defines the c/a ratio in hexagonal crystals. Since we use the hard sphere model, where close-packed atoms touch, this distance is twice the atomic radius. This primitive cell has 3 independent axes, so that’s how we define our lattice constants.īy definition, axes a and b are the same length (so we only talk about a) and have an angle of 120º to each other. However, we can also define a “primitive” simple hexagonal cell, which is ⅓ of the size of the regular cell. The conventional simple hexagonal lattice is a hexagonal prism. If the c/a ratio is smaller than 1 (c is smaller than a), the CN will only be 2.įor simple hexagonal silicon, the c/a ratio is almost one, so the CN ~ 8. If the c/a ratio is 1 (so all 3 lattice parameters have the same length), the CN = 8. This, however, assumes that the hexagonal unit cell has a longer vertical axis–in other words, lattice constants a and b are the same length (by definition, this is true), but lattice constant c is longer. There are also 2 atoms above and below, but if these are farther, the CN is still just 6. That means there are 6 nearest neighbors (NNs) in the plane. The simple hexagonal unit cell is close-packed within the planes. If the structure were ordered so that one atom always occupied a specific positions, this would be a different crystal structure (and would be called an intermetallic compound, rather than an alloy). Remember that these are disordered structures–even though there are two elements, each element occupies a random position in the simple hexagonal crystal. Other alloys that can take the simple hexagonal crystal structure are BiIn, CdSn 19, In 7Sb 3, and InSb. The phases γ Sn-In, γ Sn-Hg, β Sn-Hg, and β Sn-Cd can also have this crystal structure. Of course, the prototype for the crystal structure is γ-HgSn 6-10, which is a mercury tin alloy (substitutional solid solution) with mostly tin and about 10-17 at.% mercury. Simple hexagonal is a rare crystal structure which does not naturally occur in any pure elements (although it is possible to create a metastable, high pressure phase of silicon with the simple hexagonal crystal structure). Common Examples of Simple Hexagonal MaterialsĬommon Examples of Simple Hexagonal Materials.
You can reference my articles on the FCC, BCC, or HCP crystals which are aimed at beginners in materials science and explain terms like APF and CN. If you are new to materials science and don’t understand something in this article, I’ve made a similar article for many common crystal structures. Since this is an unusual crystal structure, I will assume that you are a somewhat advanced student in materials science, and are looking for quick facts. Simple hexagonal has 3 atoms per unit cell (or 1 per primitive cell), lattice constant a = 2R, Coordination Number CN = 6 (or 8), and Atomic Packing Factor APF = 61%. It is an uncommon crystal structure that doesn’t show up in any pure elements, but it does exist in some alloys. The Simple Hexagonal (SH) unit cell can be imagined as a hexagonal prism with an atom on all 12 corners, and both faces. You may see the simple hexagonal unit cell drawn in the conventional way, or the primitive way. The prototype for the simple hexagonal crystal is γ-HgSn 6-10. The simple hexagonal crystal belongs to space group #191 or P6/mmm, Strukturbericht A f, and Pearson symbol hP1.
There are many other crystal structures which also have a hexagonal Bravais lattice (such as the very common hexagonal close-packed), and simple hexagonal is just one of these. The Simple Hexagonal (abbreviated SH in this article) is a crystal structure which has a single-atom basis on the hexagonal Bravais lattice.
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