Solving Phonon Excitation in Diatomic Lattices

In summary, the conversation discusses proving that for small k, in an acoustic mode of vibration in a diatomic lattice, \omega \propto k and finding the constant of proportionality. The equations and attempts at a solution are provided, with the final result being \omega^2 = \frac{2T}{\sqrt{Mm}}k. However, there are some errors in the steps taken, including changing a minus to a plus in the square root and not using the correct expansion for \sqrt(1-x).
  • #1
aabb009
2
0

Homework Statement


Ok, I need to show that in an acoustic mode of vibration in a diatomic lattice, for small [tex]k[/tex], [tex]\omega \propto k[/tex], and find the constant of proportionality.

Homework Equations


[tex]A_1\left(\omega^2M-\frac{2T}{a}\right)+A_2\left(\frac{2T}{a}cos(ka)\right)=0[/tex]
, and
[tex]A_1\left(\frac{2T}{a}cos(ka)\right)+A_2\left(\omega^2m-\frac{2T}{a}\right)=0[/tex]
hence:
[tex]\omega^2 = \frac{T}{a}\left[\frac{1}{M} + \frac{1}{m}\right] - \frac{T}{a}\left[\left(\frac{1}{M}+\frac{1}{m}\right)^2-\frac{4sin^2(ka)}{Mm}\right]^{1/2}[/tex]


The Attempt at a Solution


I work through it, but repeatedly find that [tex]\omega^2 \propto k[/tex], and I can't see anyway of getting a [tex]k^2[/tex] factor on the right.
[tex]\omega^2 = \frac{T}{a}\left[\left(\frac{1}{M} + \frac{1}{m}\right) - \sqrt{\left(\frac{1}{M}+\frac{1}{m}\right)^2-\frac{4sin^2(ka)}{Mm}}\right][/tex]
[tex]\omega^2 = \frac{T}{a}\left[\left(\frac{1}{M} + \frac{1}{m}\right) - \sqrt{\left(\frac{1}{M}+\frac{1}{m}\right)^2\left[1-\frac{Mm4sin^2(ka)}{(M+m)^2}\right]}\right][/tex]
[tex]\omega^2 = \frac{T}{a}\left(\frac{M+m}{Mm}\right)\left[1 - \sqrt{\left[1-\frac{Mm4sin^2(ka)}{(M+m)^2}\right]}\right][/tex]
with small angle approximation we get:
[tex]\omega^2 = \frac{T}{a}\left(\frac{M+m}{Mm}\right)\left[1 - \sqrt{1+\frac{4Mmk^2a^2}{(M+m)^2}}\right][/tex]
[tex]\omega^2 = \frac{T}{a}\left(\frac{M+m}{Mm}\right)\left(1-1+\frac{2\sqrt{Mm}ka}{m+M}\right)[/tex]
hence
[tex]\omega^2 = \frac{2T}{\sqrt{Mm}}k[/tex]

Where am I going wrong? I don't see any way to prove this.
 
Physics news on Phys.org
  • #2
aabb009 said:

Homework Statement


Ok, I need to show that in an acoustic mode of vibration in a diatomic lattice, for small [tex]k[/tex], [tex]\omega \propto k[/tex], and find the constant of proportionality.

Homework Equations


[tex]A_1\left(\omega^2M-\frac{2T}{a}\right)+A_2\left(\frac{2T}{a}cos(ka)\right)=0[/tex]
, and
[tex]A_1\left(\frac{2T}{a}cos(ka)\right)+A_2\left(\omega^2m-\frac{2T}{a}\right)=0[/tex]
hence:
[tex]\omega^2 = \frac{T}{a}\left[\frac{1}{M} + \frac{1}{m}\right] - \frac{T}{a}\left[\left(\frac{1}{M}+\frac{1}{m}\right)^2-\frac{4sin^2(ka)}{Mm}\right]^{1/2}[/tex]

The Attempt at a Solution


I work through it, but repeatedly find that [tex]\omega^2 \propto k[/tex], and I can't see anyway of getting a [tex]k^2[/tex] factor on the right.
[tex]\omega^2 = \frac{T}{a}\left[\left(\frac{1}{M} + \frac{1}{m}\right) - \sqrt{\left(\frac{1}{M}+\frac{1}{m}\right)^2-\frac{4sin^2(ka)}{Mm}}\right][/tex]
[tex]\omega^2 = \frac{T}{a}\left[\left(\frac{1}{M} + \frac{1}{m}\right) - \sqrt{\left(\frac{1}{M}+\frac{1}{m}\right)^2\left[1-\frac{Mm4sin^2(ka)}{(M+m)^2}\right]}\right][/tex]
[tex]\omega^2 = \frac{T}{a}\left(\frac{M+m}{Mm}\right)\left[1 - \sqrt{\left[1-\frac{Mm4sin^2(ka)}{(M+m)^2}\right]}\right][/tex]
the next step is wrong, you changed a minus into a plus between the terms in the sqrt
with small angle approximation we get:
[tex]\omega^2 = \frac{T}{a}\left(\frac{M+m}{Mm}\right)\left[1 - \sqrt{1+\frac{4Mmk^2a^2}{(M+m)^2}}\right][/tex]
the next step is wrong. you didn't use the right expansion of sqrt(1-x). use
[tex]
\sqrt(1-x)\approx 1-x/2
[/tex]
[tex]\omega^2 = \frac{T}{a}\left(\frac{M+m}{Mm}\right)\left(1-1+\frac{2\sqrt{Mm}ka}{m+M}\right)[/tex]
hence
[tex]\omega^2 = \frac{2T}{\sqrt{Mm}}k[/tex]

Where am I going wrong? I don't see any way to prove this.
 
Last edited:

FAQ: Solving Phonon Excitation in Diatomic Lattices

What is a phonon?

A phonon is a collective excitation or vibration in a crystal lattice structure, consisting of a quantized unit of vibrational energy. It is considered the fundamental unit of thermal energy in solids and plays a crucial role in various physical properties of materials.

How are phonons excited in diatomic lattices?

Phonons in diatomic lattices are excited through the transfer of energy between neighboring atoms. This can occur through various mechanisms such as thermal energy, external forces, or defects in the lattice structure.

What techniques are used to solve phonon excitation in diatomic lattices?

There are several techniques used to solve phonon excitation in diatomic lattices, including analytical methods such as the harmonic approximation and numerical methods such as molecular dynamics simulations. These techniques involve solving the lattice's equations of motion and calculating the phonon dispersion relation.

How do phonons affect the thermal conductivity of materials?

Phonons play a significant role in the thermal conductivity of materials by carrying heat energy through the lattice. The phonon mean free path, which is the average distance traveled by a phonon before scattering, is a crucial factor in determining thermal conductivity.

What applications rely on an understanding of phonon excitation in diatomic lattices?

Understanding phonon excitation in diatomic lattices is essential in various fields, including materials science, solid-state physics, and engineering. It is crucial for designing and improving materials with desired thermal and mechanical properties, such as in thermoelectric devices and heat management systems.

Back
Top