# How Does the Ising Model Apply to Lattice Gas Systems?

• Jacques_Leen
In summary, the author's idea was to model the system with a lattice gas. However, before addressing the points, he had difficulty understanding the text. His idea was that each site could host 2 atoms and given that each atom could interact with 2 other atoms, he had 3 possible states for each site. He then summed the energy in each of these states and got the grand free energy. However, when he corrected for the terms ##J## which refer to the case when two atoms occupy one adsorption site, he got a different result. The grand free energy scales in an extensive fashion with the number of sites, if there is no interaction.
Jacques_Leen
Homework Statement
Let a surface be in contact with a gas at temperature ##T##. The surface consists of ##N## sites, which can absorb atoms. Each site can absorb up to ##2## atoms. The binding Energy gain for each atom absorbed is ##-\varepsilon##. If ##2## atoms are absorbed the amount of Energy gained is reduced by a term ##J## due to coupling.

1) Find the ##\mathcal{H}##
2) let ##N=1## then write the grand partition function ##Z_g##
3) let ##N=1## then find the average number ##\langle n \rangle## of atoms absorbed
4) let ##N > 1 \text{ and } J =0## write ##Z_g##
5) let ##N > 1 \text{ and } J =0##, find ##\langle n \rangle##
Relevant Equations
$$\mathcal{H} = - \lambda \sum_{\langle i,j \rangle} N_i N_j$$

$$Z_g = \sum_{i=1}^{N} e^{-\beta (n_i \mu - E_i)}$$

$$\langle n \rangle = \frac{\partial \Phi_G}{\mu}$$
Hi everyone,

even before addressing the following points I have a serious issue in understandig the text of the Exercise.My idea was to model this system with a lattice gas. Given that each site can host 2 atoms I have 3 possibilities for each site: I'll call'em ##S_{11} S_{00}## and ## S_{10}##. The terms ##S_{00}## do not contribute, whereas energies associated to ##S_{11}## and ## S_{10}## are ##2\varepsilon −J## and ##\varepsilon##. In order to determine the ##\mathcal{H}## I now have to take into account the coupling constants associated to every neighbors sites, i.e. ##k_1,k_2,k_3## because there are 3 possible combination that allow for a coupling element given that the terms which contribute are ##S_{11}## and ## S_{10}##.*

This seems over complicated though and I fear I am missing something in the assignement. Any sort of insight or help is more than welocome.*I hope the notation now is fine ... I swear it wal ok in the preview mode but somehow it got all mixed up

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To my mind, the coupling ##J## refers to the case when two atoms occupy one adsorption site, i.e., there is no interaction between atoms occupying different adsorption sites. So there is no need for Ising-model considerations.

You're right,
but I have to admit that the text here war really not clear enough (at least to my opinion). BTW I foung a similar problem in the Statistical and Thermal Physics by Gould and Tobochnik (here). The problem here is an extension of example 4.5* in chapter 4. I will discuss my solution later

* In the book the authors explicitly state there is no coupling between the elements of the lattice

A rather important notation remark before I discuss my solution,
$$\langle n \rangle = - \frac{\partial \Phi_G}{\partial \mu} .$$
I seem to have skipped both the sign and the differentiation in the statement.

1) If I have ##N_{11}## sites hosting two atoms and ##N_{10}## absorbing one then the ##\mathcal{H}## should be:
$$\mathcal{H}= N_{11} (2 \varepsilon - J) + N_{10}(\varepsilon)$$
2) Hence if there is a total of ##N=1## sites then there are 3 possible states
$$S_{11} \rightarrow E = 2 \varepsilon - J \text{ } S_{10} \rightarrow E = \varepsilon \text{ } S_{00} \rightarrow E = 0$$
summing on those 3 states I obtain
$$Z_G = 1 + e^{- \beta (2 \varepsilon - J - 2 \mu)} + e^{- \beta (\varepsilon - \mu)}$$

3) from ## \langle n \rangle = - \frac{\partial \Phi_G}{\partial \mu} ## and ## \Phi_G = -1/\beta \ln(Z_G)## I get:
$$\langle n \rangle = 1/\beta \frac{\partial \ln(Z_G) }{\partial \mu}$$
which yields:
$$\langle n \rangle = \frac{e^{- \beta (\varepsilon - \mu) }+ 2e^{- \beta (2 \varepsilon - J - 2 \mu)} }{Z_G}$$

4-5) the procedure is similar as to points 2-3 disregarding ##J## and accounting for the terms ##N_{11}## and ##N_{10}##. I'll post the solution:

$$Z_G = 1 + e^{- \beta (2N_{11} \varepsilon - 2 N_{11}\mu)} + e^{- \beta (N_{10}\varepsilon - N_{10} \mu)}$$
$$\langle n \rangle = \frac{2N_{11}e^{- \beta (2 N_{11} \varepsilon - 2 \mu)} + N_{10}e^{- \beta (N_{10}\varepsilon - \mu)}}{Z_G}$$

criticism of any sort, corrections if you see any mistake is more than welcome

Lord Jestocost said:
I was thinking about my solution to the problem for ##N >1## sites this morning and realized it was all wrong. Thanks for the hint! Grand Free Energy scales in an extensive fashion with the number of sites, if there is no interaction. As such if
$$Z_{G1} = 1 + e^{ \beta (2 \varepsilon +2 \mu)} + e^{\beta (\varepsilon + \mu)}$$
in the ##J=0## limit, then:
$$Z_{GN} = (1 + e^{ \beta (2 \varepsilon +2 \mu)} + e^{\beta (\varepsilon + \mu)}) ^N$$
and
$$\langle n \rangle = N \frac{e^{\beta (\varepsilon + \mu) }+ 2e^{ \beta (2 \varepsilon + 2 \mu)} }{Z_{GN}}$$

I also corrected the signs, for I have misinterpreted the exchange of Energy from the text

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## 1. What is the Ising Model for a Lattice Gas?

The Ising Model for a Lattice Gas is a mathematical model used to study the behavior of particles in a lattice structure. It was originally developed to study the behavior of magnetic materials, but has also been applied to other systems such as gases and fluids.

## 2. How does the Ising Model work?

The Ising Model works by assigning a spin (up or down) to each particle in a lattice. These spins interact with their nearest neighbors according to a set of rules, which can be adjusted to simulate different types of interactions. Based on these interactions, the model predicts the behavior of the system as a whole.

## 3. What are some applications of the Ising Model?

The Ising Model has been used in a variety of fields, including physics, chemistry, biology, and economics. It has been used to study phase transitions, magnetism, and the behavior of complex systems.

## 4. What are the limitations of the Ising Model?

One limitation of the Ising Model is that it assumes a simplified system in which particles only interact with their nearest neighbors. In reality, particles may interact with particles farther away as well, which can significantly impact the behavior of the system. Additionally, the model does not take into account factors such as temperature or external forces.

## 5. How is the Ising Model related to statistical mechanics?

The Ising Model is a statistical mechanics model, meaning it uses statistical methods to study the behavior of a system with a large number of particles. It is based on the principles of thermodynamics and probability, and can be used to make predictions about the behavior of a system at the microscopic level.

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