0
$\begingroup$

I am doing a topology in condensed matter course and they begin with a sentence ( link here )

Actually, we can even solve the problem of an electron in an N site ring (triangle being 𝑁=3). The trick to doing this is a neat theorem called Bloch’s theorem. Bloch’s theorem is the key to understanding electrons in a crystal. The defining property of a crystal is that the atomic positions repeat in a periodic manner in space. We account for ALL the atoms in the crystal by first identifying a finite group of orbitals called the unit-cell.

As far as I learned, a unit cell consists of atoms, and by translating it we can recreate the whole lattice. As an orbital, I understand the solution of Schrodinger's equation. But I don't know how to understand or imagine the sentence

identifying a finite group of orbitals called the unit-cell.

Because I was thinking in my mind about some structures with points, balls meaning atoms, not orbitals.

$\endgroup$
1
  • $\begingroup$ The point is that if there is a crystal lattice, with associated symmetries, the solutions to the electron wavefunctions will also have those associated symmetries, and spatially will be reflected by having a set of orbitals within a unit cell. $\endgroup$
    – Jon Custer
    Apr 20, 2023 at 15:05

1 Answer 1

0
$\begingroup$

The text is correct, but incredibly cryptic. People who study too much of mathematics sometimes forget how to talk to humans.

Let us start with the standard stuff. A crystal is a lattice of repeating unit cells. Computer calculations is the least computer effort and time if you use primitive unit cells, but often for humans to see the symmetries, we choose a slightly bigger unit cell that makes the symmetries much easier to see. Inside the unit cells are ions and their electrons' orbitals. The ions are finitely charged, so if you just care about the filled electron orbitals and a few, low excitation energy, empty electron orbitals, then those are a finite set of electron orbitals.

Thus, you can also equivalently say that the finite set of electron orbitals can be identified as a unit cell.

That is almost exactly what the author stated. Sigh.

You should be aware that crystal symmetries will slightly alter them. i.e. they should not be exactly the atomic orbitals, but rather atomic-like orbitals with the crystal's symmetries. Which are automatically fulfilled if you expand the atomic orbitals in a dense Fourier basis to high energies, and then discarding the parts that are not compatible with the crystal symmetries. Then normalise over again. Things work quite nicely.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.