# Free Electron Model: Why periodic boundary conditions and what is L ?

#### Dizzle

Free Electron Model: Why periodic boundary conditions and what is "L"?

Right, hello!

The quantum free electron model for electrons in solids (in One dimension) says we need to use periodic boundary conditions such that if Y(x) is the wavefunction, then Y(x) = Y(x+L).

Where L seems to be the width of the infinite square potential well used to derive the wavefunction and associated energy eigenvalues.

Now I realise that the periodic boundary conditions relates to the periodic lattice but, if L is the length of the 1D metal, then surely x+L is outside of the metal?

If L actually relates to some very small length inside the lattice, perhaps the size of a unit cell, then why when deriving the Fermi Energy do we use n=N/L where N is the total number of electrons in the metal and n is the number per unit length.

So my question really is, what is "L"?

Thanks

PS how do I use greek and other symbols here?

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#### ZapperZ

Staff Emeritus
2018 Award
Re: Free Electron Model: Why periodic boundary conditions and what is "L"?

Are you sure "L" is the length of the metal? It appears from the periodicity of the wavefunction that "L" is the distance between each square well, i.e. the periodicity of the lattice. That's why the wavefunction repeats itself every chance of x equaling to L.

Zz.

#### Dizzle

Re: Free Electron Model: Why periodic boundary conditions and what is "L"?

Yes that makes more sense, but if you derive the Fermi Energy in 1D, you will start with the density of states function in 1D which has an "L" in it, which appears to relate to the periodicity of the lattice as you say.

However, when you do derive the Fermi Energy by integrating the density of states up to the Fermi Energy and set this integral equal to the total number of electrons in the metal, N, and rearranging for the Fermi Energy, you will have a (N/L)^2 in your formula. n=N/L is then substituted in, where n is the number of electrons per unit length, suggesting L is the length of the metal.

Thats where I'm confused.

#### ZapperZ

Staff Emeritus
2018 Award
Re: Free Electron Model: Why periodic boundary conditions and what is "L"?

All "L's" are not created equal.

In the latter, there's no periodicity. So that "L" is related to the size of the solid. Do not focus on the symbol. Focus on what the symbol represents.

Zz.

#### Dizzle

Re: Free Electron Model: Why periodic boundary conditions and what is "L"?

Hmm yes I think you are right given what I have told you, but I think I have misunderstood something and given you wrong information.

I think my problem was with N. I think N is not the total electrons in the metal, but just the number per unit cell, which means N/L does give me the number per unit length, because L is the size of the cell.

So if N was 2 electrons per cell, and L was 1 Angstrom, then 2/1Ang = 2x10^10 electrons per unit length.

I think that makes sense. Thanks.

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