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Chapter 8: Q. 8.16 (page 341)
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 total time required is
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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 to 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 , 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 and . (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?
For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four "neighbors"-above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ) for the particular state of the square lattice shown in Figure 8.4?
Figure 8.4. One particular state of an Ising model on a square lattice (Problem 8.15).
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
(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 (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 in two dimensions, while experiments and more sophisticated approximations show that in three dimensions. The mean field approximation, however, predicts a larger value.
(c) The magnetic susceptibility is defined as . The behaviour of this quantity near the critical point is conventionally written as , 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 .)
The critical temperature of iron is . Use this value to make a rough estimate of the dipole-dipole interaction energy , in electron-volts.
Consider an Ising model of just two elementary dipoles, whose mutual interaction energy is . 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 . 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?
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