In our prototype design for a power transmission laser, outlined in [[Prototype free-electron maser design]], we chose to utilize a photocathode as an electron source, since it is generally much easier to build than a thermocathode[^1]. However, the ultimate design for a space-based prototype (see [[A realistic space-based prototype]]) will use a thermocathode (also known as a "hot cathode"), which is much more efficient[^2]. We will now proceed to give a description of the physics underlying thermocathodes, as well as considerations for our design.
### Cathode heating and thermionic emission
To have free electrons in the first place, electrons must be ejected out of a material by some mechanism. One approach is **thermionic emission** - heating the cathode until electrons gain enough kinetic energy to escape from its surface. Another approach is **photoelectric emission** - shining light on the cathode, and using the elastic collisions between photons and electrons to eject them from the material. Combining the two approaches results in **photon-enhanced thermionic emission (PETE)**, which makes use of both effects to improve efficiency.
From [[Free-electron maser physics]], we know that the current density $J(x, y)$ on a cathode at temperature $T$ is given by:
$
J = A_G T^2 e^{-W/k_BT}
$
Where $k_B$ is the Boltzmann constant, $T$ is the temperature, $W$ is the work function of the material, and $A_G$ is a material constant, sometimes known as the *Richardson constant*. Since $A_G$ and $W$ are fixed by material properties, the only variable contribution to the thermionic current is from the temperature. For realistic cathodes, which have specific operating temperatures at which they are most efficient, tables of their current density are available. For instance, consider the below table[^3]:
| Material | Operating temperature | Emission efficacy ($\eta_e$) |
| ------------------------------------------------------------ | --------------------- | ---------------------------- |
| Pure tungsten | 2500 K | 5 mA/W |
| Thoriated tungsten | 2000 K | 100 mA/W |
| Barium aluminate | 1300 K | 400 mA/W |
| Oxide-coated (e.g. nickel cathode with barium oxide coating) | 1000 K | 500 mA/W |
A rough trend is that more efficient modern thermocathodes typically operate at lower temperatures (and usually last longer). In particular, oxide-coated thermocathodes are among the most efficient designs. The output current of the thermocathode $I_c$ can be found from the emission efficacy $\eta_e$ and the power density $S$ of the incident solar radiation via:
$
I_c = \eta_e S \oint dA
$
Where the integration is done over the surface of the thermocathode. Note that in our specific case, the intensity $S$ of the incident solar radiation is *inversely proportional* to the cathode's cross-sectional area $A_c$ (by $S = \frac{P_{T}}{A_{c}}$, where $P_T = I_\odot A_m$ is the total power collected by the solar mirrors, $I_\odot$ is the [solar constant](https://en.wikipedia.org/wiki/Solar_constant), and $A_m$ is the mirror surface area). This means, for instance, that if the total sunlight incident upon 1-meter solar mirror is focused on a circular thermocathode of a radius of $\text{3 cm}$, then the incident intensity is around $S = \text{1.5 MW}/ \mathrm{m}^2$. This is slightly less than some small nuclear reactor designs, but still substantially high. In less formal language, such a small cathode would be _blazingly hot_. Indeed, we can calculate its temperature via the Stefan-Boltzmann law $T = (S/\sigma)^{1/4}$ where $\sigma = 5.67037 \times 10^{-8}\ \mathrm{W\cdot K^{-4} m^{-2}}$ is the Stefan-Boltzmann constant, which gives us around $T = \text{2,200 K}$ - a temperature far higher than the working temperature of all non-tungsten cathodes on our table. A cathode with a larger surface area would be cooler and thus can operate at the optimal temperatures of the more efficient oxide-coated cathodes, improving efficiency greatly.
### Modelling thermocathodes
> **TODO:** In process of updating next section with this: https://physics.stackexchange.com/questions/866924/applicability-of-modelling-electromagnetic-heating-with-inhomogeneous-heat-equat
We may model the temperature $T(\mathbf{x}, t)$ of a continuously-heated cathode caused by an incident electric field $\mathbf{E}(\mathbf{x}, t)$ with the **inhomogeneous heat equation**[^1]:
$
\rho c_p \frac{\partial T}{\partial t} - \nabla \cdot (k\nabla T) = \sigma |\mathbf{E}|^2 - \mu (T^4 - v^4)
$
> **Note:** This same PDE can be used to model the effects of atmospheric heating of microwaves (if we assume that air is a generally homogeneous medium).
In steady state conditions (that is, after the material reaches thermal equilibrium) and constant $k$, this reduces to the **nonlinear Poisson equation**:
$
\nabla^2 T = -\frac{1}{k}\left[\sigma |\mathbf{E}|^2 - \mu (T^4 - v^4)\right]
$
Note that due to the nonlinear $T^4$ term, this PDE can only be solved numerically unless if $\mu$ is small.
Where $T_0$ is the **surface heat density** describing the amount of heat supplied to the cathode per unit area, $T_\text{out}$ is the temperature of the surroundings, $\mu$ is a material constant, and $\sigma$[^3] describes the **thermal response** of the material to the electric field[^2]. Here, we can set $\sigma = 1$ for simplicity (it's the qualitative characteristics of the solution that we're interested in, _not_ the precise temperature values or units).
#### Impact of material-specific properties
After discussing the temperature-dependence of the current density from a heated cathode, we can now focus on the material-specific properties that also influence the current density.
As we discussed previously, one substantial material property is the **work function** of the material the cathode is made of. The work function is significant because it represents the *effective potential* that electrons have to overcome before they have enough kinetic energy to escape from the surface of the cathode. Different metals used for cathodes have different work functions, ranging from $\pu{2 - 6 eV}$, as can be seen in the below table:
_Source: [XPS library](https://xpslibrary.com/work-functions-of-elements/)_
[^1]: Indeed, a key inspiration was the use of photocathodes for classroom demonstrations of the [photoelectric effect](https://en.wikipedia.org/wiki/Photoelectric_effect), which typically uses UV light and a metal with a low work function.
[^2]: Technically it uses a PETE cathode (photon-enhanced thermionic emission cathode).
[^3]: The table is adapted from [this Wikipedia article](https://en.wikipedia.org/wiki/Hot_cathode#Transmitting_tube_hot_cathode_characteristics). The materials are ranked from lowest to highest efficiency.