Kittel Chapter 7: Empty Lattice Approximation

[ehrenfest]

[SOLVED] kittel chapter 7

Homework Statement

This question refers to Kittel's solid-state physics book. I have edition 8.

In this chapter, there is a section called the "Empty Lattice Approximation". Can someone explain what the title of that chapter means i.e. in what sense is that lattice empty, where is that used, why is that approximation necessary?

Homework Equations

The Attempt at a Solution

 

[olgranpappy]

I don't have Kittel in front of me, but I think he just means to treat empty space as a periodic potential (which it is...) with whatever period you want (say, 'a'). One can break up the free particle energy spectrum and shift it along reciprocal "lattice" vectors ((2 Pi)/a, or whatever) and get something that looks like a band structure for the free particle.

 

[ehrenfest]

What is the difference between the empty lattice approximation and the free electron fermi gas?

 

[ehrenfest]

anyone?

 

[ehrenfest]

Help!

 

[olgranpappy]

What is the difference between the empty lattice approximation and the free electron fermi gas?

in the empty lattice approximation you pretend there is still a lattice so you get bands--bands determined by shifting the free electron dispersion through a reciprocal lattice vector--and you fill up the bands till you get the number of electrons you want.

in the free electron gas there is no lattice and you just fill up the fermi sphere till you get the number of electrons you want.

 

[ehrenfest]

in the empty lattice approximation you pretend there is still a lattice so you get bands--bands determined by shifting the free electron dispersion through a reciprocal lattice vector--and you fill up the bands till you get the number of electrons you want.

in the free electron gas there is no lattice and you just fill up the fermi sphere till you get the number of electrons you want.

So the empty lattice approximation IS the reduced zone scheme, correct? I guess, I just don't see at all how that represents the band structure because then the dispersion relation is continuous at the BZ boundary in the empty lattice approximation That is, it just bounces back and forth between the boundaries. I thought that the point of an energy band was that it the dispersion relation WAS NOT CONTINUOUS. What I am saying is that there are no energy gaps in the empty lattice approximation and isn't that what we are interested in?

 

[olgranpappy]

I don't know what *we* are interested in. But, yes, in the empty lattice there are no gaps.