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Chapter 8: Q. 8.15 (page 341)

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 ε) for the particular state of the 4×4square lattice shown in Figure 8.4?

Figure 8.4. One particular state of an Ising model on a 4×4square lattice (Problem 8.15).

Short Answer

Expert verified

The total energy =-4ε

Step by step solution

01

Step 1. Given information

The 4×4Lattice has 24 nearest - neighbor interactions of which 10 are between anti parallel dipoles and 14 are between parallel dipoles

02

Step 2.  To find the total energy,

In the figure the anti-parallel interactions are made in solid lines whereas the parallel interactions are made with parallel dotted lines.

The total energy=

U=10ε+14(-ε)

=-4ε

Thus, the total energy=-4ε

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Most popular questions from this chapter

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.)

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.

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 in the presence of an external magnetic field B, which gives each dipole an additional energy of -μBB if it points up and +μBB if it points down (whereμB 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.

Consider a gas of "hard spheres," which do not interact at all unless their separation distance is less than r0, in which case their interaction energy is infinite. Sketch the Mayer f-function for this gas, and compute the second virial coefficient. Discuss the result briefly.
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