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: Specific heat of solid of one dimensional quartic oscillators

  1. Jul 21, 2009 #1
    1. The problem statement, all variables and given/known data

    A system consists of N very weakly interacting particles at temperature T sufficiently high so that classical stat mech is applicable. Each particle has mass M and is free to perform one dimensional oscillations about its equilibrium position. Calculate the heat capacity of this system of particles if the restoring force is proportional to x^3.

    2. Relevant equations

    spring constant = q
    energy of one oscillator E = p^2 / 2m + qx^4 / 4

    partition function: Z = integral ( exp(-BE ) dx dp
    both integrals from -inf to +inf

    where B = 1/kT

    3. The attempt at a solution

    Cv = N d/dT (average E)

    average E = - d / dB [ ln Z ]
    Z = integral [exp (-Bp^2/2m) dp ] * integral [exp (-Bqx^4/4) dx ]

    the first integral is simply sqrt(pi * 2m / B )
    I have no idea how to find the integral of exp(-x^4), so I can't find this partition function.
  2. jcsd
  3. Jul 22, 2009 #2


    User Avatar
    Homework Helper
    Gold Member

    Try the substitution [itex]t=\frac{\beta q x^4}{4}[/itex] and make use of the gamma function:

    [tex]\Gamma(z)\equiv \int_0^\infty t^{z-1}e^{-t}dt[/tex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook