%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
%%begin{figure}
\putgraph{Ex303}
%%\caption{}\label{}
%%end{figure}