# Kittel 8th ed. Problem 6.6: Frequency Dependence of Conductivity

• ehrenfest
In summary, the conversation revolves around problem 6.6 in Kittel's Solid-state physics book, which discusses the frequency dependence of electrical conductivity. The main question is about the significance of the variable omega (\omega) and its relationship to the conductivity equation (43). There is also a discussion about the steady state and the role of collisions in electron motion. The conversation ends with a mention of a different derivation of equation (42) and a suggestion to consult the TA for further clarification.
ehrenfest

## Homework Statement

Please stop reading unless you have Kittel's Solid-state physics book. I have the 8th edition.

I am working on problem 6.6 "Frequency dependence of the electrical conductivity". Could someone just give me a total rewording of what is going on in this question? For example, what is $\omega$ and and why doesn't the $\omega$ dependence of $\sigma$ show up in equation (43)?

## The Attempt at a Solution

$\omega = 1/\tau$ I guess. Tau is the mean time between collisions, right?

eq 43 is just sigma(0)

Look how $\omega$ is defined in problem 7 on same page.

Eq 43 was derived under steady state, i.e infinite time, and the steady state is is given by eq 41.

In problem 7, they say that omega is the frequency. But what exactly is it the frequency of?
Also, in the paragraph above (42), the first two sentences say:

Because of collisions of electrons with impurities, lattice imperfections, and phonons, the displaced sphere may be maintained in a steady state in an electric field. If the collision time is tau, the displacement of the Fermi sphere in the steady state is given by (41) with tau=t.

The first sentence makes absolutely no sense to time. It would make more sense if you changed "may" to "may not". The second sentence makes some sense...but their use of he word steady state is confusing. Can you define that for me? Can you translate what they are saying?

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as I said, the frequency omega is 1/tau (according to my book!)

so it is the frequence beteen "collisions", number of collisions per unit time.

So now you can solve problem 6 right?

Because of collisions of electrons with impurities, lattice imperfections, and phonons, the displaced sphere may be maintained in a steady in an electric field.

I don't know what would make most sence, the most important thing is that you understand the second one, and what's happening in fig 10. "The instrinsic" motion of electrons are random and the modulus of their k is large. When you supply a small force (electric field), you get a small net displacement of the fermisphere in k-space. The electrons still moves randomly, but EACH has get a small delta(k) in the same direction as all the other electrons.

I think that the first sencence means that it is not guranteed that the fermisphere will evolve, since many things can happen with the electrons. That you will see in the chapters about semiconductors and band theory of solids etc.

It really does not make sense that omega is 1/tau because then there is no reason why they would have a tau*omega in the expression below. I actually found a solution online that says it is the frequency of the electric field. What I am trying to figure out is how they arrived at that conclusion!

EDIT: I am starting to understand now--see the section on Motion in Magnetic Fields; do you think this is the cyclotron frequency?

BTW, Kittel's derivation of equation (42) makes absolutely 0 sense. He skips about ten steps in that derivation. See here for a real derivation: http://www.tf.uni-kiel.de/matwis/amat/elmat_en/kap_1/backbone/r1_3_3.html; tau is actually 1/2 of the time between collisions; thus it is the time required to reach the drift velocity after a collision; Kittel fails to explain any of that or even define tau correctly; that is why I hate him

BTW: the equation used in this problem comes from equations (50) and (51)

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OK.

Well that can explain why there is an imaginary unit there aswell..

are these problems in your course curriculum? Have you asked your teacher? You said that he aint helping you. Or is it that you don't understand his explanations?

If your teacher's not helping you at all, maybe you should talk to the principal or similar..

No I don't think it's the cyclotron frq, then there would be a c as subsctript.

His "explanations" are simply recitations of what is in the book. He would have no idea how to do this problem (without the solution manual). But I will ask the TA on monday. Thanks for your help.

LOL good teacher :)

We did no exercise from Kittel in my solid course, just so you know, I was just brainstorming how to solve that problem.

Good luck!

## 1. What is the significance of problem 6.6 in the Kittel 8th edition?

Problem 6.6 in the Kittel 8th edition is significant because it explores the frequency dependence of conductivity, which is an important concept in the study of electrical conductivity and materials science.

## 2. How does conductivity change with frequency in this problem?

In this problem, conductivity is found to decrease with increasing frequency. This is because at higher frequencies, the electrons in the material do not have enough time to respond to the changing electric field, leading to a decrease in conductivity.

## 3. What is the physical explanation for the frequency dependence of conductivity?

The physical explanation for the frequency dependence of conductivity lies in the behavior of electrons in a material. At lower frequencies, the electrons have more time to respond to the electric field and contribute to conductivity. However, at higher frequencies, the electrons cannot keep up with the changing electric field, leading to a decrease in conductivity.

## 4. How does the frequency dependence of conductivity vary between different types of materials?

The frequency dependence of conductivity can vary greatly between different types of materials. For example, metals tend to have a strong frequency dependence, while insulators have a much weaker frequency dependence. This is due to the differences in the behavior of electrons in these materials.

## 5. How is the frequency dependence of conductivity measured in experiments?

The frequency dependence of conductivity can be measured in experiments by using techniques such as impedance spectroscopy or dielectric spectroscopy. These techniques involve applying an alternating electric field to a material and measuring the resulting conductivity at different frequencies.

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