Chapter 8.2, Problem 15P

For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four “neighbours”—above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of $\epsilon$ ) for the particular state of the $4\times 4$ square lattice shown in Figure 8.4?

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

To determine

Total energy in terms of $\epsilon$ .

Explanation of Solution

Introduction:

Draw a diagram to show one Ising model on $\left(4\times 4\right)$ lattice

  

Here, each dipole has four nearest neighboring dipoles except dipoles on edges.

The lattice in above diagram has 14 nearest neighboring dipoles in between parallel dipoles as well as 19 neighboring dipoles in between anti-parallel dipoles in total.

Write the expression of total interaction energy $U$ of the lattice

  $U=\left(10\right)\left(\epsilon \right)+\left(14\right)\left(-\epsilon \right)$

Simplify the above expression

  $U=-4\epsilon$

Conclusion:

Thus, the total energy is $-4\epsilon$ in terms of $\epsilon$ .