In this problem you will use the mean field approximation to analyze the behavior of the Ising model near the critical point.
((a)) Prove that, when x≪1, tanhx≈x−31x3.
((b)) Use the result of part (a) to find an expression for the magnetization of the Ising model, in the mean field approximation, when T is very close to the critical temperature. You should find M∝(Tc−T)β, where β (not to be confused with 1/kT) is a critical exponent, analogous to the β defined for a fluid in Problem 5.55. Onsager's exact solution shows that β=1/8 in two dimensions, while experiments and more sophisticated approximations show that β≈1/3 in three dimensions. The mean field approximation, however, predicts a larger value.
((c)) The magnetic susceptibility χ is defined as χ≡(∂M/∂B)T. The behavior of this quantity near the critical point is conventionally written as χ∝(T−Tc)−γ, where γ is another critical exponent. Find the value of γ in the mean field approximation, and show that it does not depend on whether T is slightly above or slightly below Tc. (The exact value of γ in two dimensions turns out to be 7/4, while in three dimensions γ≈1.24.)