Amber E. Chappars's pictureAmber E. Chappars

Marietta College

Research Professor - Dr. Cathrine Mader

Supported by the NSF-REU

Exploring the Three-Dimensional Ising Model

Nuclear systems are believed to undergo phase transitions. However, because of the small size of the nuclear system and our lack of understanding of the nuclear interaction, a complete thermodynamic model does not exist. Instead, we must use comparisons with simpler models, such as the Ising model. We developed a program to study the Ising model, which has a well-understood phase transition. We hope to build on this and, through comparison of cluster distributions, gain insight into the nuclear phase transition.

The Ising model represents paramagnetism in a solid material. Paramagnetic materials are essentially made up of tiny bar magnets, called magnetic dipoles. Interactions between the individual magnetic dipoles cause them to line up, causing strong magnetic fields. In atomic material, magnetic dipoles are caused by the spinning of the atomic particles, and can be referred to as spins. In the Ising model, these spins are arranged in a three-dimensional lattice, with one spin per lattice point. Spins cannot move around the lattice, but flip freely between up and down states. Each spin is able to interact with its nearest neighbors the six spins that are closest in the lattice, one in each direction along each of the three dimensions.

With no outside influences, all of the spins would always align, forming a perfect magnet. However, raising the temperature of the model causes the energy of the system to increase, therefore flipping some of the spins. The system can be described by a set of thermal properties that are temperature-dependent; these are energy, magnetization, specific heat or heat capacity, and magnetic susceptibility. At specific temperature called the critical temperature, the model shows characteristics of a phase transition. The phase transition in the Ising Model is a transition between an ordered and a non-ordered state. This is similar to the transition between an orderly liquid and a non-orderly gas in any physical material. In physical material, the transition creates a point in the graph of temperature v. energy where the temperature does not change. This discontinuity is called the critical point. The Ising model shows similar discontinuities.

Computer representations of the model were created using FORTRAN programs. The program began with an ordered system, meaning all spins were in the same direction. Disorder was randomly introduced, based on the temperature the higher the temperature; the more disordered the system became. The computer ran until the model reached equilibrium, and then began to collect data about this state. However, equilibrium is not a single configuration; rather, a large number of configurations are possible. The computer sampled these configurations and calculated average thermodynamic values.

The thermodynamic properties form a basis for our understanding of the critical phenomena in the model, which has a critical point at 4.5115. The first thermodynamic property is energy. The equation for energy is the sum over all nearest neighbor pairs, with spins equaling either positive or negative one, and multiplied by a factor J, which describes the strength of the interaction between spins. As the temperature rises, the energy increases from a relatively low energy to a relatively high one, with a steep curve near the critical point. Over a broad temperature scale, the graph of energy ranges between -1 close to zero temperature to almost zero at very high temperatures, with the values of energy in relation to the number of spins.

The second property, magnetization, is the sum of all individual spins. This represents how many spins are pointed in the same direction, and is comparable to the amount of order in the system. As the temperature increases, and therefore more randomness is introduced to the system, the magnetization decreases. Once again, the steepest slope of the graph is found near the critical point. Over a broad temperature scale, the graph of magnetization would range between exactly 1 at low temperatures and close to 0 at high.

Specific heat, or heat capacity, is essentially the first derivative of the energy. Specific heat is comparable to the amount of energy needed to raise the temperature by one unit. At approximately the critical point, a spike in the specific heat is found, which is essentially a discontinuity in the graph. Magnetic susceptibility, the first derivative of the magnetization, shows similar properties. Once again, at the critical point, there is a discontinuity in the graph of magnetic susceptibility in the form of a sharp peak. It is interesting to note that this graph is much smoother than the graph of specific heat. Because properties relating to the magnetization tend to converge faster than those based on energy, they are more commonly studied and used as indicators of critical phenomena in the model.

The theory behind the Ising model assumes that the lattice size is infinite. As we decrease the size of the lattice, it becomes easier to manage and understand, but key features of the model are lost. In significantly smaller lattice sizes, the peaks in specific heat and magnetic susceptibility become more rounded, and the critical point is much less distinct. Additionally, the peaks in both graphs become shifted to the left. The energy is more sensitive to the size effects, and so shifts further.

Another property of the model is that groups of aligned spins form magnetic domains, called clusters, similar to those found in magnetic solids, such as iron. The general size of these clusters changes as a function of temperature, with interesting properties during the phase transition. Below the critical point, most of the clusters are found at the extreme ends of the graph. Large clusters represent the bulk of the material, and mid-sized clusters appear less frequently. Increasing the temperature to the critical point, the distribution becomes relatively even over the middle range of sizes. As the temperature is increased past the critical point, the large clusters are broken into smaller ones, and there is a large decrease in the amount of large clusters.

Finally, we have seen that different ways of handling the edges, or boundaries, can cause dramatic changes in our results. The edges of the model require special attention because while most spins should have six nearest neighbors, spins at the edge have as few as three. In smaller lattices, where we have proportionally more edge spins than middle spins, our handling of the edges is particularly important. Different ways of handling the edges are called boundary conditions. We looked at two types of boundary conditions: free and periodic.

With free boundary conditions, the edges are surrounded by empty space. The free condition is similar to a nuclear system. With periodic boundary conditions, each spin is given its full six neighbors. A spin at the edge wraps around to a spin on the other edge, and a spin on the corner wraps to other corners. Periodic boundary conditions better represent the infinite system that the model is based on. In graphs of these conditions for relatively small lattice sizes, the critical temperature in the periodic graph is much closer to the expected value of 4.5115, while the free graph is shifted further to the left. If the size of the lattice was increased, the graph from free conditions should shift until it eventually became identical to the periodic graph.

In conclusion, we have a working model that produces accurate results and has been reasonably optimized. Tests still remain to be run involving boundary conditions, including the effects of larger lattice sizes and the third type of condition, fixed. We now need to continue studies of cluster distributions in comparison with nuclear systems. Hopefully, we will gain insight into a possible phase transition in nuclear matter.

Click to see a slide show of Amber Chappar's work.