crystalssemiconductor-physics
Silicon has a valence 4 (electron shells: 2, 8, 4), hence having 4 neighbors in the 3D space, which for me should appear as a mesh of regular tetrahedrons, with the Si atoms at the centre.
But silicon's crystal structure is Face Centered Diamond Cubic.
From a tetrahedral mesh, I can obtain a hexahedral mesh by proper selection of vertex… But why isn't the mesh defined as tetrahedrons, which could be more intuitive? Or it is indeed a hexahedral mesh?
Four digit Miller index is sometimes used for hexagonal lattices. The idea is to have useful property: permutation of indexes gives an equivalent direction.
It is mentioned in wikipedia, by the way. So you could start from there.
Four-digit Miller indexes are never used for cubic structure. If you could provide the citation where you saw them the answer could be better.
There are two separate issues here.
Firstly the smallest possible unit cell for a crystal is called the primitive unit cell. However in many cases this is an awkward shape and it's easier to use a bigger unit cell that contains more than one primitive unit cell. The FCC unit cell is one of these non-primitive (I'm not sure what the actual term is) unit cells.
Secondly even the primitive unit cell does not necessarily contain just a single atom. The corners of the primitive cell are marked by atoms that have the same environment, but the silicon structure contains two different environments for the silicon. Have a look at this picture of the silicon structure:
If you look at the atom I've outlined in red and the atom I've outlined in green you'll see they are in different environments. If you look at the upper two bonds you'll see that for the red atom the V formed by the bonds projects out of the screen. For the green atom the V lies in the plane of the screen. In effect you get the red atom by rotating the green atom 90ยบ about a vertical axis.
So even the primitive cell contains two silicon atoms, and the compound FCC cell contains four primitive cells. Despite this, every silicon atom is indeed at the centre of a tetrahedron.
Best Answer
The silicon lattice has a diamond structure where each Si atom has four nearest neighbors connected by a covalent bond forming tetrahedra that are periodic in space as can be seen in the picture. Thus one tetrahedron represents a possible primitive unit cell of the crystal whose translational repetition generates the crystal lattice.
Each tetrahedron contains two Si atoms, one in the center and a quarter on each corner of the tetrahedron. However, different unit cells can be used to describe the same crystal lattice. It is easier to visualize the diamond lattice by the depicted conventional (not primitive) cubic unit cell with side length $a$. It can be seen that this corresponds to two interpenetrating face centered cubic Bravais point lattices that are displaced with respect to each other by a quarter of the cube's diagonal.
From the perspective of crystal structure, the Si atoms belonging to these two sub-lattices are different, although they are chemically identical. It can be seen in the picture that a corner atom has a nearest neighbor in one direction of the diagonal but not in the other direction. Thus the crystal needs two Si atoms per primitive unit cell whose choice is not unique. One possible primitive unit cell is, as you suspected, the tetrahedron. Possible basis vectors for this primitive unit cells of the face centered lattice are the translations from one corner of the conventional cubic unit cell to its adjacent quadratic face centers.
See, e.g., Ashcroft and Mermin, 1976, Chapter 4.