Acoustic and Optical Branches for Waves on a Diatomic Row of Masses

[mamela]

Dispersion Relation:

w²=B(1/m+1/M)+-SQRT[(1/m+1/M)²-(4sin²ka/mM)]


For waves on a diatomic row of masses, what is the physical meaning of acoustic and optical branches of this dispersion relation? What is the expression for the maximum frequency of the optical branch?

Really stuck on this question and can't find ANYTHING on the internet!

Please help!

 

[olgranpappy]

Dispersion Relation:

w²=B(1/m+1/M)+-SQRT[(1/m+1/M)²-(4sin²ka/mM)]


For waves on a diatomic row of masses, what is the physical meaning of acoustic and optical branches of this dispersion relation? What is the expression for the maximum frequency of the optical branch?

Really stuck on this question and can't find ANYTHING on the internet!

Please help!

For the first part probably you are required to discuss how the masses vibrate differently for each mode. In which mode are the two masses of the unit cell pushing against one another and in which mode are them pushing with one another. Which mode reduces to a uniform translation of the lattice as k->0? Etc.

As for the second question. The optical branch is the higher energy branch... so how would you find the maximum value?

 

[inempty]

the mode of an acoustic branch means vibration when atomics in the same cell will always have identical velocity. So there are only 3 acoustic branches in the 3d space (also, 2 in the 2d space and so on). on the other hand, the mode of an optical branch signifies that atomics in a single cell will move relatively. We can find with the long-lambda limit, the optical vibration will become motion that positive particles have relative to negative particles collectively and then can be described by Huang equations. In conclusion, acoustic vibrations describe the motion of cells and optical ones describe the relative motion in cells.

 

[Rajini]

acoustic: intermolecular vibration or lattice vibration.
here all molecules will vibrate. For eg., translational (3), rotational (3) and acoustic (3). It is also called sound waves. These appear below ~10 meV or ~80 cm^-1 in the vibrational spectrum. In some context acoustic can be imagined as translational. See Solid state physics by Kittle.
optical : Einstein modes or intramolecular vibrations or molecular vibrations. Often called genuine normal modes of vibration. Each atom of a molecule will vibrate relatively to other atoms, i.e. optical modes are within a molecule. Theoretically one molecule is enough to get all optical modes.
hope this helps.

 

[Tipi]

I've made the calculation in the solution of a final exam. Look at the first question in this file :

http://www.forumphysique.com/levesque/Site/Matière_Condensée_files/A05-PHY6505-final-SOL.pdf

It's in french but, as usual, the maths are... in math. In the document, Case 1 is the limit where k << pi/a which correspond to the branches near k=0. Case 2 give where the branches cut k=pi/a. The lower branch (Fig. 2) is almost linear in k, exactly like the sound wave dispersion relation, so it's the acoustic branch. The upper branch is the optical one because the long wavelength can interact with light in ionic crystals.

It corresponds to my solution of the problem 2 (Diatomic linear chain) of Ashcroft/Mermin.

Cheers,

TP

 

[mamela]

I understand better the optical and acoustic branches now, thanks.

TiPi: I understand that the optical and acoustic branches correspond to different solutions for k, but how can you deduce that cos ka = 1 + ((ka)²/2) if coska is much less than PI/a?

Is it correct to say then that the maximum optical frequency is taking the positive square root in the equation and taking the case when sin² = 0?

 

[Tipi]

how can you deduce that cos ka = 1 + ((ka)²/2) if coska is much less than PI/a?

Taylor serie around ka = 0 :

$$\cos ka = \frac{\cos 0}{0!} - \frac{ka\sin(0)}{1!} - \frac{(ka)^2\cos(0)}{2!} + ...$$

Note that you misreproduced my statement, it's a minus before the second order term.

Is it correct to say then that the maximum optical frequency is taking the positive square root in the equation and taking the case when sin² = 0?

For this system, yes, because the maximum is at k = 0.

 

[Gokul43201]

Hi mamela,
This is a standard textbook problem (I remember it from Ashcroft & Mermin, but it probably appears in most every Solid State text), and therefore the thread is probably more appropriate for the Homework & Coursework Forum. Please keep this in mind for similar questions in the future.

Hi Tipi,
Please see the forum guidelines for helping with coursework/textbook problems. We do not encourage giving out complete solutions to such questions, and prefer that members offer tutorial assistance via hints and guidance (see olgranpappy's post).

Forum guidelines: https://www.physicsforums.com/showthread.php?t=414380