\sect{Adsorption of polar molecules to a surface}

Consider a 2D adsorbing surface in equilibrium with a 3D gas of atoms that have a temperature $T$
and a chemical potential $\mu$. On the surface there are $M$ sites. Each site can absorb at most one atom. 
At the adsorption site an atom forms an electric dipole $d$ that can be oriented at any direction away from the surface (see figure).
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In the presence of a perpendicular electric field $\mathcal{E}$ 
the dipole has energy is ${-\mathcal{E} d \cos(\theta)}$, 
where ${|\theta|<\pi/2}$ is the angle between $d$ and $\mathcal{E}$.

\Dn

(a) Calculate the grand partition function $\mathcal{Z}(\beta,\mu,\mathcal{E})$

\Dn

(b) Derive the average number $N$ of absorbed atoms.

\Dn

(c) Use the formal approach to define the average
polarization $D$ as the expectation value of
a system observable. Derive the state equation for $D$.

\Dn

(d) What are the results in the limit $\mathcal{E}\rightarrow0$, 
and in particular what is the ratio $D/N$. 
Explain how this result can be obtained without going through the formal derivation. 


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