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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

Use the cluster expansion to write the total energy of a monatomic nonideal gas in terms of a sum of diagrams. Keeping only the first diagram, show that the energy is approximatelyU≈32NkT+N2V·2π∫0∞r2u(r)e-βu(r)drUse a computer to evaluate this integral numerically, as a function of T, for the Lennard-Jones potential. Plot the temperature-dependent part of the correction term, and explain the shape of the graph physically. Discuss the correction to the heat capacity at constant volume, and compute this correction numerically for argon at room temperature and atmospheric pressure.

In this problem you will use the mean field approximation to analyse the behaviour of the Ising model near the critical point.

(a) Prove that, when x≪1,tanhx≈x-13x3

(b) Use the result of part (a) to find an expression for the magnetisation of the Ising model, in the mean field approximation, when T is very close to the critical temperature. You should find M∝Tc-Tβ¯,whereβ(not to be confused with 1/kT) is a critical exponent, analogous to the f defined for a fluid in Problem 5.55. Onsager's exact solution shows that β=1/8in two dimensions, while experiments and more sophisticated approximations show that β≈1/3in three dimensions. The mean field approximation, however, predicts a larger value.

(c) The magnetic susceptibility χis defined as χ≡(∂M/∂B)T. The behaviour of this quantity near the critical point is conventionally written as χ∝T-Tc-γ , where y is another critical exponent. Find the value of in the mean field approximation, and show that it does not depend on whether T is slightly above or slightly below Te. (The exact value of y in two dimensions turns out to be 7/4, while in three dimensions γ≈1.24.)

At T = 0, equation 8.50 says that s¯=1. Work out the first temperature-dependent correction to this value, in the limit β∈n≫1. Compare to the low-temperature behaviour of a real ferromagnet, treated in Problem 7.64.

Consider an Ising model of just two elementary dipoles, whose mutual interaction energy is ±ϵ. Enumerate the states of this system and write down their Boltzmann factors. Calculate the partition function. Find the probabilities of finding the dipoles parallel and antiparallel, and plot these probabilities as a function of kT/ϵ. Also calculate and plot the average energy of the system. At what temperatures are you more likely to find both dipoles pointing up than to find one up and one down?

Draw all the diagrams, connected or disconnected, representing terms in the configuration integral with four factors of fij. You should find 11 diagrams in total, of which five are connected.
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