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\sect{Adsorbtion of polar molecules to a surface}

A large number $n$ of identical mass $\mass$ atoms are bounded 
within a surface that has ${M}$ adsorbtion centers.  
Each adsorbtion center can connect one atom, 
such that a polar molecule $AB$ is created. 
The dipole moment of each molecule is $d$, 
and it can be oriented either vertically (1 possible orientation)
or horizontally (4 possible orientations). 
The binding energy is ${\epsilon_0}$. 
Additionally a vertical electric field ${\mathcal{E}}$ is applied.
The interaction energy between the field and 
the dipole is ${-\vec{\mathcal{E}} \cdot \vec{d}}$.
The polarization of the system is defined via 
the expression for the work, ${dW=-D d\mathcal{E}}$.

\begin {itemize}

\item[(1)]
Find the canonical partition function $Z_n(\beta)$ of the system.

\item[(2)]
Derive an expression for the chemical potential $\mu(T;n)$.

\item[(3)]
Given $\mu$, deduce what is the coverage $\langle n \rangle$.

\item[(4)]
Re-derive the expression for $\langle n \rangle$ 
using the grand canonical partition function $\mathcal{Z}(\beta,\mu)$.

\item[(5)]
Calculate the polarization $D(\mathcal{E})$ of the system. 

\end {itemize}


{\em Remarks:} 
In items (1-2) it is assumed the the system is closed
with a given number $n$ of adsorbed atoms. 
Hence it is treated within the framework of the canonical ensemble. 
In items (3-4) the system is in equilibrium with a gas 
of atoms: the chemical potential $\mu$ is given, 
and the average $\langle n \rangle$ should be 
calculated using the grand-canonical formalism. 
In item (5) it is requested to verify that the same result 
is obtained in the canonical and in the grand-canonical treatments. 


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