The Ising Model Describes Magnetization

You can magnetize certain materials, such as iron, by putting them into a magnetic field. Below a certain critical temperature, such materials have a net magnetic moment. They are magnetized. But if you heat them above that temperature, they lose magnetization. What is the microscopic basis for this behavior  

The net magnetic moment in the up direction is represented by the number of upward arrows (t) minus the number of downward arrows ( ). 

In the Ising model, each atomic magnet occupies one lattice site with a spin that is either up or down. This model is named for physicist E Ising, who solved it in his Ph.D. thesis work with W Lenz in 1925. If you model each spin as an f or I on a one-dimensional lattice, the partition function can be computed in the same way as the partition fimction for a sequence of W or C units in the helix-coil model. The difference between the one-dimensional Ising model and the helix-coil model is just the statistical weights. For the magnet model, the weights are cj(tt) = and (tl) = dUt) = where 

The partition function of this magnet model is found by growing the linear sequence of spins in the same way as you did for the helix-coil model. For a single site, the partition function is Qi = 2, because there are two states t or i. To get the two-site partition function, you multiply Qi by ie + e - ), because the second arrow can point either in the same direction as the hrst or in the opposite direction. In terms of the hyperbolic cosine function, cosh(x) = (e + e )/2, you have 

Every site that you add to the one-dimensional array of spins multiplies the partition function by a factor of (e- + e - ), so the partition function for a linear lattice of N magnetic spins is 

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