Question

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 $\epsilon$) 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).

Short answer

The total energy =$-4\epsilon$

Step-by-step solution

Step 1. Given information

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

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=

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

$=-4\epsilon$

Thus, the total energy$=-4\epsilon$