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Chapter 8: Q 8.31 (page 355)

Modify the Ising program to simulate a three-dimensional Ising model with a simple cubic lattice. In whatever way you can, try to show that this system has a critical point at aroundT=4.5.

Short Answer

Expert verified

Therefore, we used the Ising program to simulate a three-dimensional Ising model with a simple cubic lattice.

Step by step solution

01

Given information

Modify the Ising program to simulate a three-dimensional Ising model with a simple cubic lattice. System has a critical point at around T=4.5.

02

Explanation

Code for 3D Ising model

03

Explanation

Because there were only 40points in this temperature range, the graph is "pointy." The number of iterations for these graphs was 5000, therefore the creation of these plots took a lengthy time.

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Most popular questions from this chapter

Modify the ising program to compute the total magnetisation (that is, the sum of all the s values) for each iteration, and to tally how often each possible magnetisation value occurs during a run, plotting the results as a histogram. Run the program for a 5 x 5 lattice at a variety of temperatures, and discuss the results. Sketch a graph of the most likely magnetisation value as a function of temperature. If your computer is fast enough, repeat for a 10 x 10 lattice.

Consider an Ising model of 100 elementary dipoles. Suppose you wish to calculate the partition function for this system, using a computer that can compute one billion terms of the partition function per second. How long must you wait for the answer?

The Ising model can be used to simulate other systems besides ferromagnets; examples include anti ferromagnets, binary alloys, and even fluids. The Ising model of a fluid is called a lattice gas. We imagine that space is divided into a lattice of sites, each of which can be either occupied by a gas molecule or unoccupied. The system has no kinetic energy, and the only potential energy comes from interactions of molecules on adjacent sites. Specifically, there is a contribution of -u0to the energy for each pair of neighbouring sites that are both occupied.

(a) Write down a formula for the grand partition function for this system, as a function of u0, T, and p.

(b) Rearrange your formula to show that it is identical, up to a multiplicative factor that does not depend on the state of the system, to the ordinary partition function for an Ising ferromagnet in the presence of an external magnetic field B, provided that you make the replacements u0→4ϵand μ→2μBB-8ϵ. (Note that is the chemical potential of the gas while uB is the magnetic moment of a dipole in the magnet.)

(c) Discuss the implications. Which states of the magnet correspond to low density states of the lattice gas? Which states of the magnet correspond to high-density states in which the gas has condensed into a liquid? What shape does this model predict for the liquid-gas phase boundary in the P-T plane?

The critical temperature of iron is 1043K. Use this value to make a rough estimate of the dipole-dipole interaction energy ε, in electron-volts.

For each of the diagrams shown in equation 8.20, write down the corresponding formula in terms of f-functions, and explain why the symmetry factor gives the correct overall coefficient.
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