Mott's scattering cross-section formula

[Demon117]

Homework Statement

We were asked to derive the Mott's scattering cross section. Given by

$$\sigma(\theta)=(\frac{1}{4k^{4}}) (\frac{1}{(\sin\frac{\theta}{2})^{4}} - \frac{1}{(\cos\frac{\theta}{2})^{4}}\cos[\frac{2}{k}\ln(\cot\frac{\theta}{2})])$$

I get it into this form (that was easy, lengthy but easy) and then we're suppose to use these units:
$\mu=\frac{m}{2}$, $v=\frac{\hbar k}{m}$, and $\alpha = e^{2}$

to show that it is actually:

$$\sigma(\theta)=(\frac{e^{2}}{mv^{2}})^{2} (\frac{1}{(\sin\frac{\theta}{2})^{4}} - \frac{1}{(\cos\frac{\theta}{2})^{4}} \cos[\frac{e^{2}}{\hbar v}\ln(\cot\frac{\theta}{2})])$$

So, either I have completely forgotten how to do dimensional analysis or this equation as written cannot be possible since the units inside the cosine do not reduce to unity. Any suggestion on how to do this?

 

[vela]

How do the units on the first version of the equation work out? They don't look right there, either.