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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
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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