\sect{Polar adsorption of particles to a surface}
Consider an $M$ site system in an equilibrium with gas of particles that have mass $\mass$.
The chemical potential of the gas is~$\mu$ and its temperature is~$T$.
A particle can bind to a site. Each site can absorb at most one atom.
The binding energy is~$\varepsilon$, and the length of the bonds is~$a$.
In such state it behaves as a rotor that has moment of inertia $I=ma^2$, and a dipole moment $qa$.
The polarization can be in any direction away from the surface ($2\pi$ staradians). \\
{\bf Tip:} The kinetic part in a rotor Hamiltonian is
%
\[ \frac{1}{2I} \left[ p_{\theta}^2 + \frac{p_{\varphi}^2}{\sin^2(\theta)} \right] \]
% 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
(1) Calculate the partition function $Z_{\perp}(\beta,f)$ for an occupied site,
assuming electric field $f$ perpendicular to the surface.
\Dn
(2) Calculate the partition function $Z_{\parallel}(\beta,f)$ for an occupied site,
assuming electric field $f$ parallel to the surface.
\Dn
(3) Express the $M$ site grand partition function $\mathcal{Z}(\beta,\mu,f)$ in terms of $Z$.
Additionally, write an explicit expression for zero field.
\Dn
(4) Express the average number $N$ of adsorbed particles in terms of $Z$.
Additionally, write an explicit expression for zero field.
\Dn
(5) Find a leading order expression for the average polarization $D/N$ for weak perpendicular~$f$.
\Dn
(6) Find a leading order expression for the average polarization $D/N$ for weak parallel~$f$.
\Dn
(*) Tip: one can use a shortcut in the calculation of $Z$, bypassing the integration over the momentum variables.
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