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Statistical mechanics: Particles with spin 
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#1
Mar512, 11:55 PM

P: 61

1. The problem statement, all variables and given/known data
We have N particles, each of which can either be spinup ([itex]s_i = 1[/itex]) or spindown ([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 spinup 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. 


#2
Mar612, 01:31 PM

P: 321

leave Z as the summation Z = Ʃ e^{Eiβ} where β = 1/K_{B}T
so ∂ln(Z)/dβ = (1/Z)(∂Z/∂β) = [E_{i}Ʃe^{Eiβ}]/Z i think b) is supposed to be (Z_{1})^{N} sorry yeah your b) is right 


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