Does any lattice or lattice shape has a periodic boundary condition?

In summary, the conditions for us to construct a periodic boundary condition(PBC) for the lattice shape in the attachments are that the left and down neighbors of each site are the BLUE site, and that the direction of the PBC is along the direction of the BOUNDARY EDGES.
  • #1
qijiongli
3
0
If not, then what are the conditions for us to construct a periodic boundary condition(PBC)?
If so, then please help me construct a PBC for the lattice shape in the attachments.

I want to ask that what lattice site m's left neighbor is and what lattice site i's down neighbor is.From the picture, both m's left and i's down neighbor is the BLUE site, but in the PBC, the neighbors are GREEN and RED correspondingly (right?). However, the GREEN site and the RED site are impossible to be the same site in the PBC(right?). So I'm confused with it.

thanks a lot
 

Attachments

  • angle_45-pbc.jpeg
    angle_45-pbc.jpeg
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  • #2
Are you joining adjacent edges or opposite edges ? Try rotating the lattice through 45 deg, working out the PBC and then rotating back.
 
  • #3
Thank you very much for your help.

Do you mean that the direction of the PBC is along the direction of the BOUNDARY EDGE( the dotted GREEN line showed in the attachment)? Then both m's left and i's down neighbor is the BLUE site (K) on the opposite edge.If so, it seems to say that all PBCs are along the direction of the BOUNDARY EDGES?
 

Attachments

  • angle_45-pbc-more.jpeg
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  • #4
I don't think I understand the problem. Have a look at the picture, I have shown a point in solid yellow and its neighbours in outlines yellow. The corners will have widely 'separated' neighbours.
 

Attachments

  • angle_45-pbc.jpg
    angle_45-pbc.jpg
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  • #5
thanks a lot!
Mentz114 said:
I don't think I understand the problem. Have a look at the picture, I have shown a point in solid yellow and its neighbours in outlines yellow. The corners will have widely 'separated' neighbours.

For the above lattice shape, it can be constructed from the square lattice by connecting the next-nearest-neighbor sites.And each site's neighbors are easy to find by the original square lattice.

But I find that it is just a spatial case. And now I want to simulate a Ising model on the following lattice, whose boundary edges are the lines connecting one of the next-next-nearest-neighbors on the square lattice(dotted lines shows). The crossing points of the solid lines will be placed spins. Under the PBC, what the RED site's right and down neighbor will be?
 

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  • angle_sita-.jpeg
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1. What is a periodic boundary condition in a lattice?

A periodic boundary condition in a lattice is a boundary condition where the physical properties of the lattice are repeated periodically across the edges of the lattice. This means that the behavior of the lattice at one edge is the same as the behavior at the opposite edge.

2. How does a periodic boundary condition affect the properties of a lattice?

A periodic boundary condition can affect the properties of a lattice in several ways. It can affect the energy levels, the band structure, and the electronic properties of the lattice. It can also affect the stability and symmetry of the lattice.

3. Can any lattice shape have a periodic boundary condition?

Yes, any lattice shape can have a periodic boundary condition as long as it has a repeating pattern. This includes common lattice shapes such as square, hexagonal, and cubic lattices, as well as more complex shapes like quasicrystals.

4. How is a periodic boundary condition different from other boundary conditions?

A periodic boundary condition is different from other boundary conditions because it allows for an infinite lattice to be simulated by using a finite unit cell. This means that the properties of the lattice can be studied without the need for an infinitely large system.

5. What are some applications of periodic boundary conditions in lattice simulations?

Periodic boundary conditions are commonly used in lattice simulations in materials science, chemistry, and physics. They are used to study the properties of crystals, polymers, and other materials. They are also used in computer simulations to model the behavior of large systems, such as in molecular dynamics simulations.

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