\sect{Adsorption of polar molecules to surface}

Consider a 2D adsorbing surface in equilibrium with a 3D gas of atoms that has temperature $T$
and 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).

In presence of an electric field $\mathcal{E}$ perpendicular to the surface the dipole has energy $-\mathcal{E}dcos(\theta)$ where $|\theta|<\pi/2$ is the angle between $d$ and $\mathcal{E}$.

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

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

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

(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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\includegraphics[scale=0.45]{A34.eps}
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OLD: 


\sect{Baruch's  A43.}

1. Consider a 3-dimensional gas of atoms with  a chemical potential
$\mu$ that can adsorb on any of M sites on a surface; at each site
at most one atom can be adsorbed. At the adsorption site an atom forms
an electric dipole ${\bf d}$ that can be oriented at any direction
{\em away} from the surface (see figure). In presence of an
electric field ${\bf E}$ perpendicular to the surface the dipole
has energy $-Ed\cos \theta$ where $|\theta | \leq \pi/2$ is the
angle between ${\bf d}$ and ${\bf E}$.
\begin{itemize}
\item [(a)]
Evaluate the number N of adsorbed atoms. Normalize the phase space
of each adsorbed atom to 1. Derive the limit $E\rightarrow 0$ and explain why is the
result finite, in spite of the adsorption energy being $0$ at
$E=0$.
\item [(b)]
Find the average electric dipole perpendicular to the surface.\\
\end{itemize}


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\includegraphics[scale=0.45]{A34.eps}
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