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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 蔚) for the particular state of the 4脳4square lattice shown in Figure 8.4?

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

Short Answer

Expert verified

The total energy =-4蔚

Step by step solution

01

Step 1. Given information

The 4脳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=10蔚+14(-蔚)

=-4蔚

Thus, the total energy=-4蔚

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Most popular questions from this chapter

Problem 8.10. Use a computer to calculate and plot the second virial coefficient for a gas of molecules interacting via the Lennard-Jones potential, for values of kT/u0 ranging from 1to 7. On the same graph, plot the data for nitrogen given in Problem 1.17, choosing the parameters r0 and u0so as to obtain a good fit.

Draw all the connected diagrams containing four dots. There are six diagrams in total; be careful to avoid drawing two diagrams that look superficially different but are actually the same. Which of the diagrams would remain connected if any single dot were removed?

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.

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.

Problem 8.13. 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 approximately
U鈮�32NkT+N2V路2蟺鈭�0鈭�r2u(r)e-尾u(r)dr
Use 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.
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