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north / south
The Ising Model
The Ising Model provides a simple way of describing how a magnetic material responds to thermal energy and an external magnetic field. This two-dimensional version of the Ising Model contains 10,000 different sites that represent the domains of the magnetic material. In this model, each domain has a corresponding spin of north or south. The spins can be thought of as the poles of a bar magnet. The north spins attract south spins and vice versa.
The model assigns a value of +1 or -1 to the spins north and south respectively. The direction of the spins influences the total potential energy of the system. The total potential energy (U) is computed by:
U = -Î å Jij*Ji'j' - B å Jij
where Î is a constant found by the multiplication of the neighboring domain sites. For example, if one site was north (+1) and a neighboring site was south (-1), the Î is -1. The Î is found by looking at the four neighboring sties (for example: -1*1*1*1 gives us a Î value of -1). The B parameter is a constant representing a external B field.
The magnetism changes in the material depending on the thermal energy surrounding it. The Ising model uses the random spin-flip method (Metropolis Algorithm) to determine the new magnetism. This means that we randomly choose one of the domains and change its spin. We then calculate the change in potential energy caused by this random flip and determine if this random flip should be accepted. We determine this with the following equation:
r = e-DE/T
where T is the temperature of the external heat source. DE refers to the change in potential energy caused by the single random flip. If DE is less than zero, then the flip is always accepted. If DE is greater than zero, then a random number is generated ranging form 0 to 1. If r is less than our random number, than the flip is accepted, otherwise, the spin is returned to its original direction.
Magnetization
We can then calculate the net magnetization of the material by finding the
difference between the total number of sites pointing north and the total number
of sites pointing south. The following equation shows the percentage of
the material that is magnetized.
M = (å north - å south) / ( # of domains)2 *100
In the current example, you can see how magnetism changes as the temperature increases. The example used here starts with a completely magnetized material with all the domains point south. As the temperature increases, we see domains begin to point north. Finally, at a very high temperature, we lose nearly all of our magnetism. At low temperature clusters of similar spin domains form, and at zero there is no change in magnetism because it is at absolute zero.
Phase Change
By looking at the standard deviation of the
magnetization at different temperatures, we can find where the phase change
occurs for this material. The phase change occurs when there are high
levels of standard deviation of the magnetism. The program must be run for
a long-time in order to see the expected levels of standard deviations at
different temperatures. For this material, we see the phase change occur
at about 2.5 kT/J.
This data-graph shows approximately where the phase change occurs for this material. The peaks of each vertical line represent the approximate standard deviation of the magnetism at that temperature. If we find the temperature corresponding to the highest peak, then we can find the temperature of the phase change for our material. It occurs approximately between 2.3 and 2.6.
Energy
A magnetic material always favors a decrease in
energy. For this reason, the energy of a system randomly decreases over
time. Since magnetism is dependent on energy, we see that clusters of
similar spins form at low temperatures, but at high temperatures the domains are
fairly evenly scattered. As the temperature of the system is increased, we
see an increase in the potential energy of the magnetic material. This
occurs because our material is absorbing the external thermal energy.
This graph demonstrates how the potential energy of the system changes at a specific temperature. Over time the system loses potential energy because the system always favors a net decrease in energy.
Applet Instructions
Getting Started
Click the right mouse button to get a popup-menu. From this menu, choose options. You must chose an initial design for our material. Choose one of the Initialize buttons to set this. All north means that each domain's spin is pointed up (+1). All south sets each domain's spin down (-1). Press start to begin the simulation. By right-clicking the mouse and choosing Animation you can see the actual Ising Model take place.
- BLUE means up (north).
- RED means down (south).
You can change the temperature of the system by using either the slider at the bottom of the applet (it ranges from 0-4 energy units) or by entering a number in the box next to the slider. You can enter only positive number in this box since the program uses units similar to the Kelvin scale (no negative values).
Step Size
You can chose a Step Size from the Options panel. This changes how fast the program simulates the changing magnetism in the material.
Single Step: program randomly chooses one domain, decides if the flip should be accepted, and updates the screen image.
Ising Step: program theoretically does the spin/flip method for every domain and then updates the screen image.
B Field
You can chose an external magnetic field (B) to influence the magnetism in the material. Use the slider to change the external field from -1 to +1. A positive external magnetic field influences domains in the material to point in the direction of the B field. A negative B field does just the opposite.
Graphs
At the bottom of the Options panel are the graph options. Chose a checkbox to display the corresponding graph. You can either uncheck the box, or click the X button on a graph window, to close a graph window. The Clear Graphs button clears all of the graph's information.