# Longitudinal acoustic phonons: why does the dielectric function vanish?

1. Sep 10, 2008

### FranzDiCoccio

Hi everybody,

In chapter 10 of Kittel's "Introduction to Solid State Physics" it is said that the zeros of the dielectric functions determine the frequency of the longitudinal modes of oscillation, $$\epsilon(\omega_L) = 0$$, Eq. (17).
Am I missing something or this is actually an "unproved claim" as far as Kittel's book is concerned?
I mean, just below Eq (17), at page 276, there is a very short reference to depolarization, "to be discussed below". As far as I understand such a discussion is in the section "screening and phonons in metals" (page 286), where Eq. (17) is given for granted...

Hence, in summary, $$\epsilon(\omega_L) = 0$$ seems to me just a piece of information that Kittel gives us, without proving it. Or is it too obvious for proving?
Maybe I'm missing something... Sometimes one misses evident points when looking from a too short distance...
Has anyone any illuminating comment?

I have taken a look at more advanced textbooks, and, as far as I understand, it is a matter of poles and resonances..

Thanks a lot

Last edited: Sep 10, 2008
2. Sep 11, 2008

### Physics Monkey

Hi Franz,

I don't think its obvious (and what is really?). One way to see the connection between zeros of the dielectric function and longitudinal waves is to consider a Maxwell equation in medium. Written in momentum space Gauss' law looks like $$k \cdot D = 0$$ or $$\epsilon k \cdot E = 0$$. What this equation tells you is that in the absence of "free charge" we must either have $$k\cdot E =0$$ or $$\epsilon = 0$$. Thus longitudinal waves $$k \cdot E \neq 0$$ require $$\epsilon = 0$$.

Hope this helps!

Last edited: Sep 11, 2008
3. Sep 12, 2008

### FranzDiCoccio

Hi Physics Monkey,

Put in this form, Kittel's statement is still perhaps not obvious, but becomes quite intuitive, at least at a first sight. I wonder why Kittel didn't include such an illuminating comment based on simple Fourier analysis and vector calculus...

Anyway, this argument seems to suggest that what matters is transverse vs longitudinal.
But I suspect that optical vs acoustical has a role as well...

Indeed, I had found the argument you provide in the book by Ashcroft and Mermin (Ch. 27), and a similar one in Chapter 3 of Kittel's Quantum Theory of Solids. In both cases the subject is optical phonons, where one has a depolarization field.

Does this argument apply also to acoustic phonons? I do not see that. I'm asking because Kittel explicitly refers to acoustic phonons, at least in my copy of the 7th edition of his ItSSP book (page 286).

4. Sep 12, 2008

### Minich

Yes.
"optical vs acoustical" hasn't a role in "poles"...

As to referencing Kittel it depends )
17 equation in my case is: (see attachment)

#### Attached Files:

• ###### kittel-17.jpg
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5. Sep 12, 2008

### FranzDiCoccio

Physics Monkey's argument is extremely clear to me. Actually so clear
that I'm surprised that, as far as I can see, it is nowhere to be found in a basic book like Kittel's
"Introduction to Solid State Physics". But I could have missed it..
Anyway, in other books, I find it only in connection with long wavelength optical modes, and the polarization field thereof.

I'll take a closer look to more advanced treatments.

I'm not sure I get that. Are you suggesting that switching to another book could be a wise choice?
Anyway I mentioned that I was referring to the 7th edition of Kittel ITSSP. Is your attachment from the same edition?

cheers :)

6. Sep 12, 2008

### Minich

Excuse me, please. I live in Russia and I had never been outside former USSR. Russian old literature on this subject is rather huge (in russian) and contains the answer to your question. But after the cold war and after the greate criminal revolution in Russia we have very poor scientific literature in russian :(((
So, i think that this question had been solved many years ago and contemporary editors mechanically adopt it.
By the way, when was the first edition of Kittel book?
I like Kittel very much, especially for his "elementary" "Statistical mechanics".
May be you are more familiar with contemporary books. But Quantum Mechanics and EMF theory hardly changed since 1930 years, except some countable on fingers of one hand exclusions.
I don't know the exact edition )
To my mind the first is the first forever especially love )

To get closer to the ground, i love V. Fock(the head of my department in Leningrad State University), Kittel, Haken, Dicke, Tamm (see Theory of Electricity with the answer to your question, he is the Nobel prize winner, once i have met with his son speaking about job, his son was the leader of the expedition to Everest), Davydov, Maslov, Arnold,... and i don't like very much Landau, Halatnikov and other pathological members of this club (except Migdal and some other members).

As a matter of fact, as a winner of schoolchildren competition in Moscow region in 1969 i received as a prize the book (among other 15 books) "Theory of superconductivity" of 1960 edition (i was 16 then). It was a translation to russian all significant works on this subject: Frohlich, BCS work, Cooper review, Valatin,....

Is it a mystery?

It's my handbook now )

Old books are closer to the physical meaning. New books are closer to a business success.

Last edited: Sep 12, 2008
7. Sep 13, 2008

Minich,