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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

    gabbagabbahey

    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]
     
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