Ising Model for a Lattice Gas

[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

 

[Lord Jestocost]

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.

 

[Jacques_Leen]

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

 

[Jacques_Leen]

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]

Have a look at:
http://mtdevans.com/projects/physics-problems/absorption-of-molecules-onto-a-surface/
or
Exam Solutions: Advanced Statistical Physics Part II: Problems ...

 

[Jacques_Leen]

Have a look at:
http://mtdevans.com/projects/physics-problems/absorption-of-molecules-onto-a-surface/
or
Exam Solutions: Advanced Statistical Physics Part II: Problems ...

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