1. Not finding help here? Sign up for a free 30min 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!

Statistical mechanics: Particles with spin

  1. Mar 5, 2012 #1
    1. The problem statement, all variables and given/known data

    We have N particles, each of which can either be spin-up ([itex]s_i = 1[/itex]) or spin-down ([itex]s_i = -1[/itex]) with [itex]i = 1, 2, 3....N[/itex]. The particles are in fixed position, don't interact and because they are in a magnetic field with strength B, the energy of the system is given by:

    [tex]E(s_1, ...., s_n) = -mB \sum_{i=1}^{N} s_i[/tex]

    with m > 0 the magnetic moment of the particles. The temperature is T.

    a) Calculate the canonic partition function for N = 1 and the chance that this particle is in spin-up state [itex]P_+[/itex].

    b) For any N, calculate the number of microstates [itex]\Omega(N)[/itex], the Helmholtz free energy F(N,T) and the average energy per particle U(N, T)/N

    3. The attempt at a solution

    a) [tex]Z_1 = e^{-\beta m B} + e^{\beta m B} = 2 \cosh{\beta m B}[/tex]
    [tex]P_+ = \frac{e^{-\beta m B}}{2 \cosh{\beta m B}}[/tex]

    b) The number of possible microstates is [itex]\Omega(N) = 2^N[/itex], correct?

    I know that [itex]U = -\frac{\partial \ln Z}{\partial \beta}[/itex], but I'm not sure how to calculate Z here.
    Last edited: Mar 6, 2012
  2. jcsd
  3. Mar 6, 2012 #2
    leave Z as the summation Z = Ʃ e-Eiβ where β = 1/KBT

    so ∂ln(Z)/dβ = (1/Z)(∂Z/∂β) = [-EiƩe-Eiβ]/Z

    i think b) is supposed to be (Z1)N sorry yeah your b) is right
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Statistical mechanics: Particles with spin
  1. Statistical Mechanics (Replies: 3)

  2. Statistical Mechanics (Replies: 2)