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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?

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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.

Diamond structure

See, e.g., Ashcroft and Mermin, 1976, Chapter 4.

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