# Kittel Chapter 7: Empty Lattice Approximation

• ehrenfest
But this is precisely why it's an approximation.In summary, the conversation discusses the concept of the "Empty Lattice Approximation" in Kittel's solid-state physics book, where empty space is treated as a periodic potential with a chosen period. This approximation is used to create bands in the free particle energy spectrum, which can then be filled with electrons. The difference between the empty lattice approximation and the free electron fermi gas is that in the former, the bands are determined by shifting the free electron dispersion through a reciprocal lattice vector, while in the latter, there is no lattice and the fermi sphere is filled with electrons. The empty lattice approximation is also known as the reduced zone scheme and does not have any energy gaps, making it an
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?

## The Attempt at a Solution

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.

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

anyone?

Help!

ehrenfest said:
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.

olgranpappy said:
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?

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

## 1. What is the Empty Lattice Approximation in Kittel Chapter 7?

The Empty Lattice Approximation is a simplifying assumption used in the study of crystalline materials. It assumes that the atoms in a crystal are arranged in a perfect, regular lattice with no vacancies or defects. This simplifies the mathematical calculations in the study of the material's electronic and magnetic properties.

## 2. What is the significance of the Empty Lattice Approximation in materials science?

The Empty Lattice Approximation allows for easier mathematical analysis and modeling of crystalline materials, leading to a deeper understanding of their properties and potential applications. It also serves as a starting point for more complex models that take into account lattice defects and imperfections.

## 3. How does the Empty Lattice Approximation affect the electronic properties of a material?

The Empty Lattice Approximation simplifies the electronic band structure of a crystal, making it easier to calculate important properties such as the band gap and conductivity. However, it does not account for the effects of electron-electron interactions, which can be important in some materials.

## 4. Can the Empty Lattice Approximation be used for all types of crystalline materials?

The Empty Lattice Approximation is most commonly used for simple, periodic crystals such as metals and semiconductors. It may not be applicable to more complex structures such as amorphous materials or those with significant defects.

## 5. How does the Empty Lattice Approximation relate to other models used in materials science?

The Empty Lattice Approximation is often used as a starting point for more advanced models that take into account lattice defects, electron-electron interactions, and other factors. It is also commonly used in combination with other theoretical and experimental techniques to gain a more comprehensive understanding of a material's properties.

Replies
1
Views
2K
Replies
5
Views
1K
• Atomic and Condensed Matter
Replies
4
Views
2K
Replies
6
Views
20K
Replies
1
Views
3K
Replies
1
Views
2K
Replies
1
Views
5K
Replies
12
Views
3K