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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition · 356 pages

ISBN: 9780201380279

Chapter 8: Q 8.28 (page 354)

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.

Short Answer

Expert verified

Hence the codes and diagrams are given.

Step by step solution

01

Given information

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,

02

Explanation

This exercise's code can be seen below.

03

Conclusion

As can be seen in the diagram above, the system becomes highly magnetic as the temperature is lowered. That is, the M = 0 mean value flips to a non-zero value.

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Most popular questions from this chapter

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: s_is_j. 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

$c (r) = \bar{s_{i} s_{j}} - {\bar{s_{i}}}^{2}$

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 a gas of molecules whose interaction energy $u (r)$ u is infinite for $r < r_{0}$ and negative for $r > r_{0}$ , with a minimum value of $- u_{0}$ . Suppose further that $k T ≫ u_{0}$ , so you can approximate the Boltzmann factor for $r > r_{0}$ using $e^{x} \approx 1 + x$ . Show that under these conditions the second virial coefficient has the form $B (T) = b - (a / k T)$ , the same as what you found for a van der Waals gas in Problem 1.17. Write the van der Waals constants $a$ and $b$ in terms of $r_{0}$ and $u (r)$ , and discuss the results briefly.

Consider an Ising model in the presence of an external magnetic field B, which gives each dipole an additional energy of $- μ_{B}$ B if it points up and $+ μ_{B}$ B if it points down (where $μ_{B}$ is the dipole's magnetic moment). Analyse this system using the mean field approximation to find the analogue of equation 8.50. Study the solutions of the equation graphically, and discuss the magnetisation of this system as a function of both the external field strength and the temperature. Sketch the region in the T-B plane for which the equation has three solutions.

Draw all the diagrams, connected or disconnected, representing terms in the configuration integral with four factors of $f_{i j}$ . You should find 11 diagrams in total, of which five are connected.

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 around $T = 4.5$ .

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