 # 3) In this problem we want to calculate a hysteresis curve in a mor...

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3) In this problem we want to calculate a hysteresis curve in a more realistic way. Suppose we have a square lattice and the energy density is given by the sum of the Zeeman energy and a 4-fold anisotropy energy. y U = - M H - K M a) Assuming K is positive, draw the easy and hard axes on the picture. X b) If there is a field H along the X axis, write the energy density in terms of the angle 0 c) Write the equation that you get if you are looking for the lowest energy. d) To make things simple numerically, set M = K = 1. In principle, it would be nice to have the situation where we put in an H value and derive a 0 value. But mathematically it is really hard to do this. Instead, let's put in a 0 value and find H. So, solve for H (we can divide by sin 0) e) Now, write a program, or calculate by hand, to find the information in the following table Angle H Mx = Mcos(0) Energy (degrees) 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Plot Mx (the component of the field along H) as a function of H. You should get something like this Hysteresis curve 1.5 1 0.5 M -0.5 -1 -1.5 -3 -2 -1 1 2 3 H You will notice this doesn't look like anything we have seen. The problem is that there are multiple values for Mx at each field. f) In the table that you filled in above, you also calculate the energy. To eliminate some of the multiple values, check which dots in the curve above have the highest energy....and eliminate those. g) Draw the new curve, with the high energy dots eliminated. Can you make it look the a standard hysteresis curve now?

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