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\sect{Fermi gas in 2D+3D connected boxes with gravitation}

Consider a mesoscopic box that has dimensions ${L\times L\times \ell}$, 
such that ${\ell \ll L}$. In the box there are ${N}$ spin~$1/2$ electrons.   
The mass of an electron is~$\mass$. 
In items (a-d) assume that the temperature is ${T=0}$. 
Express your answers using ${\mass, L, \ell, g, T}$.

\begin {itemize}

\item[(a)]
Describe the single particle density of states. Specify the energy range 
over which it is the same as for a two dimensional box.

\item[(b)]
Find the fermi energy ${E_{F}}$ assuming that it is in the range 
defined above. What is the maximum number ${N_{max}}$ of electrons 
that can be accommodated without violating the 2D description? 

\item[(c)]
Assuming ${N<N_{max}}$ find the pressure ${P}$ on the side walls 
of the box, and the force  ${F}$ on the horizontal walls. 

\end {itemize}

The box is attached to a tank that has dimensions ${L \times L \times L}$, 
that is placed at height ${D}$ relative to the box. 
Assume very strong  gravitation field with acceleration constant~$g$.

\begin {itemize}

\item[(d)]
Assume ${N=N_{max}}$.  
What is the minimum height ${D_{min}}$ to place the tank 
such that all of the electrons stay in the box?

\item[(e)]
Assume ${N=N_{max}}$ and ${D=D_{min}}$. 
The temperature of the system is raised a little bit. 
As a result some of the particles that were in the 2D box 
are transferred to the tank. Estimate their number ${N'}$. 
You can neglect the effect of the temperature on the chemical potential. 

\end {itemize}

\[
\int_{0}^{\infty}\frac{x^{\frac{1}{2}}}{e^{x}+1}dx=\Gamma\left(\frac{3}{2}\right)\left(1-\frac{1}{\sqrt{2}}\right)\zeta\left(\frac{3}{2}\right)=0.678
\]


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