1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Help me simplify this!

  1. Mar 25, 2009 #1
    1. The problem statement, all variables and given/known data

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

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


    2. Relevant equations



    3. The attempt at a solution

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

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

    For the second derivative i get [tex]\frac{3E_c^\frac{1}{2}}{4E^\frac{5}{2}}[/tex]

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

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

    Can this be simplified any further other than the obvious numerical evaulation for 4 raised to 5/6?
     
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted