- #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.