# Deriving optical- and acoustical branches

1. Jan 6, 2014

### Elekko

On the book "Introduction to Solid State Physics" by Kittel, on page 98 he derived the roots for optical and acoustical branches for the equation:

$$M_1 M_2 \omega^4-2C(M_1+M_2)\omega^2+2C^2(1-cos(Ka))=0$$

where the roots are:

$$\omega^2=2C(\frac{1}{M_1}+\frac{1}{M_2})$$ and
$$\omega^2=\frac{\frac{1}{2}C}{M_1+M_2}K^2 a^2$$

I'm wondering how he actually found these roots since he skipped the details? He only mentions the trigonometric identity can be set to zero... how are the roots found?

2. Jan 6, 2014

### TSny

Note that these are the roots only for the limiting case of $Ka << 1$. As Kittel states, in this case you can let $\cos(Ka) \approx 1-\frac{1}{2}K^2a^2$.

Make this approximation and note that you have a quadratic equation in $\omega^2$.