As the title.
When studying the energy band of electrons by the famous A&M, I come to the confusion that the Born-von-Karman boundary seems to change the topology of materials. So whether the following corollaries, like crystal momentum and energy band, depend on this topology? If it is, it would be more befuddling since real materials do not share this topology. If it is not, are there any alternative ways to prove it?
Sincere appreciation in advance for every answer.
For calculation of any quantity that is measurable locally in a big sample (such as heat capacity per unit volume, compressibility, etc etc), the boundary conditions should not matter since you could just focus on part of the sample and the boundaries are far away. So you can just use whatever is convenient (BvK usually). However, if you were worried that there might be a problem, you could try using other boundary conditions and almost always you can succeed in doing the same calculation, but you have to do a bit more work. And indeed, the results do come out the same for almost anything you might try to calculate. Yes, strictly speaking BvK is changing the topology of the sample to something strange. And there ARE exceptional cases where the boundary conditions do matter, but these are pretty rare and typically you know when you have this situation. It is a good exercise to try calculating the heat capacity per unit volume of a free electron gas with particle-in-a-box boundary conditions and show it comes out the same as BvK boundary conditions.
