Periodic Boundary Conditions proof

In summary, when modeling bloch waves in a solid, one must assume periodic boundary conditions in order to obtain wave solutions. This can be done by choosing L to be the size of the elementary cell, or by assuming that the boundary conditions become unimportant when L is much larger than the elementary cell.
  • #1
Wminus
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Hi! When we model bloch-waves in a solid we assume that there exist some kind of periodic boundary conditions such that the wave function is periodic. In 1D, ##\psi(x)## repeats itself for every ##L##, ##\psi(x) = \psi(x+L)##, such as here:
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OK, fine, we get pretty wave solutions if we assume the existence of the PBC. But what ##L##? As far as I know the only repeating unit in a crystal is the Wieger Seitz cell, which is sized on the atomic scale.

Is ##L## just the wavelength?
 

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  • #2
No, L is the size of your crystal, so e.g. 1cm. In the end, this size doesn't matter to much and you often take ##L \to \infty##.
 
  • #3
DrDu said:
No, L is the size of your crystal, so e.g. 1cm. In the end, this size doesn't matter to much and you often take ##L \to \infty##.

How can that be? if that is true, it means ##\psi## is a standing wave... But bloch waves are traveling!
 
  • #4
That's why you use periodic boundary conditions, the waves then move on a circle and you can have left and right moving waves.
 
  • #5
this makes no sense physically. if you have a crystal cube of 1cm, you are assuming that its left side is connected to its right?
 
  • #6
No, it doesn't, but there is a theorem by Wigner that the influence of the boundary conditions on the states vanishes like 1/N, where N is the number of particles (or elementary cells). So if you are interested in the bulk states only, you can pick the boundary conditions as seem convenient.
 
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  • #7
DrDu said:
No, it doesn't, but there is a theorem by Wigner that the influence of the boundary conditions on the states vanishes like 1/N, where N is the number of particles (or elementary cells). So if you are interested in the bulk states only, you can pick the boundary conditions as seem convenient.
Ah, OK, I see. So you can pick whatever PBC you like? What about if L = length of unit cell?
 
  • #8
No, I said that the boundary conditions become unimportant when L is much larger than the elementary cell.
 
  • #9
DrDu said:
No, I said that the boundary conditions become unimportant when L is much larger than the elementary cell.

Yes you did, my apologies. And thanks for the help!
 

1. What are periodic boundary conditions?

Periodic boundary conditions are a set of boundary conditions used in simulations to mimic an infinite system. This means that the system is repeated periodically, allowing for the simulation of a larger system without needing to model every single atom or molecule.

2. Why are periodic boundary conditions used in simulations?

Periodic boundary conditions are used in simulations because they allow for the study of larger systems without increasing the computational cost. They also help to avoid edge effects and simulate bulk behavior.

3. How are periodic boundary conditions implemented in simulations?

Periodic boundary conditions are implemented by replicating the simulation cell in all three dimensions. This means that when an atom or molecule crosses the boundary of the cell, it reappears on the other side as if the system is repeating itself. This is typically done using minimum image convention.

4. What is the purpose of using minimum image convention in periodic boundary conditions?

Minimum image convention is used in periodic boundary conditions to ensure that the shortest distance between two particles is calculated correctly. This is important because without it, there may be artifacts or incorrect results in the simulation due to particles interacting with their periodic images.

5. Are there any limitations to using periodic boundary conditions?

While periodic boundary conditions are a useful tool in simulations, there are some limitations. They assume that the system is infinitely repeating, which may not always be realistic. They also may not accurately capture surface effects or interactions with external fields. It is important to carefully consider the use of periodic boundary conditions and their potential impact on the results of a simulation.

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