\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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