Question

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 $-u_0$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 $u_0$, 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 $u_0\to 4ϵ$and $\mu \to 2{\mu }_{\mathrm{B}}B-8ϵ$. (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?

Short answer

Therefore,

$Z_N=\sum \exp \left(-\beta H_1\right)·\exp \left[\beta \left(\mu +2u_0\right)N_L\right]$

Step-by-step solution

Given information

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 $-u_0$to the energy for each pair of neighbouring sites that are both occupied.

Explanation

Hamiltonian of this system is:

$H=-u_0\sum _{⟨i,j⟩}{n}_i{n}_j$

Hamiltonian of in the Ising model is:

$H_I=-ϵ\sum _{⟨i,j⟩}{s}_i{s}_j-2{\mu }_{0}B\sum s$

Because there can only be one particle in a single lattice site, n_i =0,1, and it can have spin s₁ = -1 or s₁ = 1, we can write following relation for the occupation of a lattice site and the spin:

$s_i=2n_i-1n_i=\frac{s_i+1}{2}$

Plugging this expression into the first Hamiltonian, we get:

$$\begin{gathered} H=-u_0\sum _{⟨i,j⟩}\frac{s_i+1}{2}·\frac{s_j+1}{2} \\ =\frac{-u_0}{4}\sum _{⟨i,j⟩}\left(s_i+1\right)\left(s_j+1\right) \\ =\frac{-u_0}{4}\sum _{⟨i,j⟩}\left(s_i{s}_j+s_i+s_j+1\right) \\ =\frac{-u_0}{4}\sum _{⟨i,j⟩}{s}_i{s}_j-\frac{u_0·\xi }{2}\sum s_i+\text{const} \end{gathered}$$

$\xi$is the number of nearest neighbours, and for a square lattice all sites have the same number of neighbours: $\xi =4$

By comparison with the Hamiltonian of the Ising model, we can see that$ϵ=u_0/4\text{or}{u}_0=4ϵ$

Grand canonical Hamiltonian is:

$$\begin{gathered} H-\mu N=\frac{-u_0}{4}\sum _{⟨i,j⟩}{s}_i{s}_j-2u_0\sum s_i+\text{const.}-\mu \sum s_i \\ H-\mu N=\frac{-u_0}{4}\sum _{⟨i,j⟩}{s}_i{s}_j-\left(\mu +2u_0\right)\sum s_i+\text{const} \end{gathered}$$

Where, $\mu$is chemical potential

Explanation

Now we can see that we will get the Hamiltonian of the Ising model if we Use substitutions: $u_0=4ϵand2{\mu }_{0}B=\mu +8ϵ$

Rewriting the equation:

$H-\mu N=H_1-\left(\mu +2u_0\right)N_L$

Where N_L is the total number of lattice sites

Grand partition function is:

$Z_N=\sum \exp [-\beta (H-\mu N\left)\right]$

Canonical partition function of the Ising model is:$Z_1=\sum \exp \left(-\beta H_1\right)$. So these two partition functions are related:

$$\begin{gathered} Z_N=\sum \exp [-\beta (H-\mu N\left)\right] \\ {Z}_N=Z_1·\exp \left[\beta \left(\mu +2u_0\right)N_L\right] \end{gathered}$$

If the relation between occupation of the lattice site and the spin is:

$s_i=2n_i-1$

We can see that for n_i = 0 (empty site) we will have s₁= -1. So, magnetic state of s_i = -1 will correspond to the low density of lattice gas.