Why can we use periodic boundary conditions?

[henry wang]

(Mentor note: moved here from noon homework thread hence no template)

I was studying vibration of a one-dimensional monatomic chain and the textbook used periodic boundary condition (PBC).
I wanted to justify the use of PBC, so I came up with this:

• atoms deep inside the crystal sees an infinite crystal with periodic potential
• atoms near the surface of the system do not see an infinite crystal nor periodic potential.
• however there aren't many atoms near the surface so they contribute little to the property of the whole system.
• therefore PBC can be used since the majority of the atoms sees periodic potential.

Is my understanding valid? And are there any mathematical way of justifying the PBC?

 

[hokhani]

(Mentor note: moved here from noon homework thread hence no template)

I was studying vibration of a one-dimensional monatomic chain and the textbook used periodic boundary condition (PBC).
I wanted to justify the use of PBC, so I came up with this:

• atoms deep inside the crystal sees an infinite crystal with periodic potential
• atoms near the surface of the system do not see an infinite crystal nor periodic potential.
• however there aren't many atoms near the surface so they contribute little to the property of the whole system.
• therefore PBC can be used since the majority of the atoms sees periodic potential.

Is my understanding valid? And are there any mathematical way of justifying the PBC?

To justify PBC, please consider the following:
In reality, the crystal size is limited at its ends. If one were to describe the waves at the crystal ends, the waves must have nodes there and so the waves are obliged to be standing (not travelling) and one can not correctly deal with the problem. To overcome this issue, one can suppose the PBC conditions to correctly deal with the traveling waves.