Crystal model with periodic boundary conditions

[Mechdude]

user meopemuk mentioned this:
In the case of a crystal model with periodic boundary conditions, basis translation vectors e1 and e2 are very large (presumably infinite), which means that basis vectors of the reciprocal lattice k1 and k2 are very small, so the distribution of k-points is very dense (presumably continuous).

here : meopeuk
i do not get his argument, is there a place where i can find a thorough treatment of the thinking behind translation vectors being huge for periodic crystal boundary conditions.

 

[M Quack]

Boundary conditions are necessary for the systematic mathematical treatment of crystals. The distance of the boundaries (size of the box in which the calculations are carried out) does not have to be large.

But in order to realistically treat real-world crystals it should be. Unit cell sizes are mostly a few nanometers or less, and physical crystals usually are ~10 micrometers or millimeters at least. So the box has to be 10000 or 1 million unit cells - which is almost as good as infinite, and in many ways infinite is preferable, because then reciprocal space becomes continuous rather than discrete.

This approximation breaks down when you start working with nanomaterials, where the crystal "grains" are only a few unit cells large. Then ... interesting... things happen.

 

[Mechdude]

Boundary conditions are necessary for the systematic mathematical treatment of crystals. The distance of the boundaries (size of the box in which the calculations are carried out) does not have to be large.

But in order to realistically treat real-world crystals it should be. Unit cell sizes are mostly a few nanometers or less, and physical crystals usually are ~10 micrometers or millimeters at least. So the box has to be 10000 or 1 million unit cells - which is almost as good as infinite, and in many ways infinite is preferable, because then reciprocal space becomes continuous rather than discrete.

This approximation breaks down when you start working with nanomaterials, where the crystal "grains" are only a few unit cells large. Then ... interesting... things happen.

Thanks for the reply M Quack,
so meopemuk used this physical reason to come up with the concept of large (relative to unit cells) translation vectors? and the corresponding small reciprocal vectors?
mechdude.