## Transcribed Text

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?

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.