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Chapter 8: Q 8.25 (page 353)

In Problem 8.15 you manually computed the energy of a particular state of a 4 x 4 square lattice. Repeat that computation, but this time apply periodic boundary conditions.

Short Answer

Expert verified

Therefore, the energy is 4J.

Step by step solution

01

Given information

In Problem 8.15 you manually computed the energy of a particular state of a 4 x 4 square lattice. Apply periodic boundary conditions.

02

Explanation

In the Ising model, a 4x4 square lattice has 16 sites. When periodic boundary conditions are applied, the energy of a certain state is given by:

Where spins are equal to:

Skν=+1Skμ=-1Siν=Siμ

Energy is equal to:

role="math" localid="1647835907601" E=-J·∑i,jSiν·Skν+J·∑i,jSiν·Skμ=-J·(-2)+J·(+2)E=4J

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Most popular questions from this chapter

By changing variables as in the text, express the diagram in equation 8.18 in terms of the same integral as in the equation8.31. Do the same for the last two diagrams in the first line of the equation8.20. Which diagrams cannot be written in terms of this basic integral?

Implement the ising program on your favourite computer, using your favourite programming language. Run it for various lattice sizes and temperatures and observe the results. In particular:

(a) Run the program with a 20 x 20 lattice at T = 10, 5, 4, 3, and 2.5, for at least 100 iterations per dipole per run. At each temperature make a rough estimate of the size of the largest clusters.

(b) Repeat part (a) for a 40 x 40 lattice. Are the cluster sizes any different? Explain. (c) Run the program with a 20 x 20 lattice at T = 2, 1.5, and 1. Estimate the average magnetisation (as a percentage of total saturation) at each of these temperatures. Disregard runs in which the system gets stuck in a metastable state with two domains.

(d) Run the program with a 10x 10 lattice at T = 2.5. Watch it run for 100,000 iterations or so. Describe and explain the behaviour.

(e) Use successively larger lattices to estimate the typical cluster size at temperatures from 2.5 down to 2.27 (the critical temperature). The closer you are to the critical temperature, the larger a lattice you'll need and the longer the program will have to run. Quit when you realise that there are better ways to spend your time. Is it plausible that the cluster size goes to infinity as the temperature approaches the critical temperature?

To quantify the clustering of alignments within an Ising magnet, we define a quantity called the correlation function, c(r). Take any two dipoles i and j, separated by a distance r, and compute the product of their states: sisj. This product is 1 if the dipoles are parallel and -1 if the dipoles are antiparallel. Now average this quantity over all pairs that are separated by a fixed distance r, to |obtain a measure of the tendency of dipoles to be "correlated" over this distance. Finally, to remove the effect of any overall magnetisation of the system, subtract off the square of the average s. Written as an equation, then, the correlation function is

c(r)=sisj¯-si¯2

where it is understood that the first term averages over all pairs at the fixed distance r. Technically, the averages should also be taken over all possible states of the system, but don't do this yet.

(a) Add a routine to the ising program to compute the correlation function for the current state of the lattice, averaging over all pairs separated either vertically or horizontally (but not diagonally) by r units of distance, where r varies from 1 to half the lattice size. Have the program execute this routine periodically and plot the results as a bar graph.

(b) Run this program at a variety of temperatures, above, below, and near the critical point. Use a lattice size of at least 20, preferably larger (especially near the critical point). Describe the behaviour of the correlation function at each temperature.

(c) Now add code to compute the average correlation function over the duration of a run. (However, it's best to let the system "equilibrate" to a typical state before you begin accumulating averages.) The correlation length is defined as the distance over which the correlation function decreases by a factor of e. Estimate the correlation length at each temperature, and plot graph of the correlation length vs.

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