Wikipedia defines lattice constant as physical dimension of unit cells in a crystal lattice.

Unit cell is defined as:

The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure. The geometry of the unit cell is defined as a parallelepiped.

Why that elementary pattern that repeats itself in crystals is always a parallelepiped? In real life such elementary repeating unit/part/block is very often not a parallelepiped at all!

Is this some abstraction and a "bounding parallelepiped" is meant - the one into which definitely fits whatever real elementary repeating pattern is?

  • 1
    $\begingroup$ What cases do you think at not described by a parallelepipied? Are you confusing the lattice and the basis? $\endgroup$
    – Ian Bush
    Jun 11 '18 at 17:10
  • $\begingroup$ It is not the only one but the norm for chemists/biochemists, solid state physicists define the primitive cell differently as the Wigner-Seitz cell. This has a rather polygonal look. Choose a victim point to be the centre. Connect this point to other nearby points with lines, then draw perpendicular lines at the midpoints. These latter intersect to form the primitive cell. $\endgroup$
    – porphyrin
    Jun 11 '18 at 17:26

A unit cell is always a parallelepiped because this is the definition of a unit cell. In case when you don't have a repeating unit of this kind (which might indeed be the case in real life), you don't have a crystal, also by definition. You might be dealing with a quasicrystal or something.

3D periodicity is what defines a crystal. If you have it, this means you have three non-coplanar periodic vectors, and these inevitably define a parallelepiped.


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