Homework Help: Help me simplify this!

1. Mar 25, 2009

nissanztt90

1. The problem statement, all variables and given/known data

$$f(E) = \left(\frac{E_c}{E} \right)^{1/2} + \frac{E}{kT}$$

Need to take second derivative with respect to E for Taylor expansion about $$E_0$$ where $$E_0 = [\frac{1}{4}E_c(kT)^2]^\frac{1}{3}$$, which is the Gamow peak.

2. Relevant equations

3. The attempt at a solution

So for the first derivative i got $$-\frac{E_c^\frac{1}{2}}{2E^\frac{3}{2}} + \frac{1}{kT}$$

I know this is correct since when i replace $$E$$ with $$E_0$$ is comes out to 0 which is correct since $$E_0$$ is a peak, plus it was a problem hint.

For the second derivative i get $$\frac{3E_c^\frac{1}{2}}{4E^\frac{5}{2}}$$

Im pretty sure my derivative is correct, but when i replace $$E$$ with $$E_0 = [\frac{1}{4}E_c(kT)^2]^\frac{1}{3}$$ and try to simply i get this...

$$\frac{3}{4} \* \frac{4^\frac{5}{6}}{[E_c(kT)^5]^\frac{1}{3}}$$

Can this be simplified any further other than the obvious numerical evaulation for 4 raised to 5/6?