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\heading{Ex 6030: Thermionic emission of electrons } 
\auname{Dotan Davidovich} 

{\bf The problem:}

\Dn

A piece of metal ("Cathode") is placed inside a vacuum metal tube ("Anode"). The cathode has a work function $W$ and surface $A$ and is held at temperature $T$ .
\Dn

Calculate the saturation current $I_s$ ($V \rightarrow \infty$) .
\Dn

a)For $ T \ll W $
\Dn

b)For $ T \gg W $
\Dn

\includegraphics[scale = 0.6]{6030pic}
\includegraphics[scale = 0.6]{6030pic2}
\Dn

{\bf The solution:}
\Dn

a) Under the assumption that the electron gas inside the metal is in a state of quasi-static thermal equilibrium we use the velocity distribution: 

\[\tag{1} F_{(v)} = 2\cdot L^3 \cdot (\frac{m}{2\pi})^3\cdot f(\frac{1}{2}mv^2 - \mu)\cdot d^3v \]

Now, understanding that only incident electrons with kinetic energy larger than the potential barrier will escape ($V_z \geq \sqrt{\frac{2\cdot (W+\mu_F)}{m}} $) and using the incident flux on a wall perpendicular to the Z axis , we get:
\[\tag{2} J = \int 2\cdot \frac{F_{(v)}}{L^3}\cdot V_z = 2\cdot (\frac{m}{2\pi})^3\cdot \int_{V_{Z_{min}}}^\infty V_z dV_z \cdot \int_0^\infty \frac{2\pi V'dV'}{e^{-\beta(\frac{1}{2}mv_z^2-\mu)}\cdot e^{\beta\frac{1}{2}mv'^2} + 1}   \] 

Integrating over the $x-y$ velocities and using a dimensionless variable $x=\beta W$ :

\[\tag{3} J = 2\cdot \frac{m^2}{(2\pi)^3}\cdot 2\pi \cdot T \cdot \int_{V_{Z_{min}}}^\infty  ln(1+ e^{-\beta(\frac{1}{2}mv_z^2-\mu)}) V_zdV_z = \frac{4\pi m}{(2\pi)^3} \cdot T^2 \cdot \int_{\beta W}^\infty ln(1+e^{-x})dx    \]

For $ T \ll W $ we use $ ln(1+e^{-x}) \approx e^{-x}  $ :

\[\tag{4} J=\frac{4\pi m T^2}{(2\pi)^3}\cdot e^{-\beta W}  \] 

The current density is  : $ e\cdot J $ , and for the saturation current all emitted electrons make it to the Anode :

\[\tag{5} I_s=e\cdot J \cdot A = e\cdot A \cdot \frac{4\pi m T^2}{(2\pi)^3}\cdot e^{-\beta W} \]

\Dn

b) For $ T \gg W $ , realizing we are now in the classical regime :

\[\tag{6}  \mu=Tln(n \cdot \lambda^3)   \]

Subsituting into (4) we get :

\[\tag{7}     J=n\cdot (\frac{ T}{2\pi m})^\frac{1}{2}\cdot e^{-\beta\frac{1}{2}mv_z^2}     \]




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