Editing Photocathode (section)

=== Mean Transverse Energy (MTE) and Thermal Emittance ===
For some applications, the initial momentum distribution of emitted electrons is important and the [[mean transverse energy]] (MTE) and thermal emittance are popular metrics for this.  The MTE is the variance of the transverse momentum in a direction along the photocathode's surface and is most commonly reported in units of milli-electron volts.<ref>Bradley, D. J., Allenson, M. B., & Holeman, B. R. (1977). The transverse energy of electrons emitted from GaAs photocathodes. ''Journal of Physics D: Applied Physics'', ''10''(1), 111–125. {{doi|10.1088/0022-3727/10/1/013}}</ref>

<math display=block>\text{MTE} = \frac{\langle p_{\perp}^2 \rangle}{2m_e}</math>

In high brightness photoinjectors, the MTE helps to determine the initial [[Beam emittance|emittance]] of the beam which is the area in phase space occupied by the electrons.<ref>Bazarov, I. V., Dunham, B. M., Li, Y., Liu, X., Ouzounov, D. G., Sinclair, C. K., Hannon, F., & Miyajima, T. (2008). Thermal emittance and response time measurements of negative electron affinity photocathodes. ''Journal of Applied Physics'', ''103''(5), 054901. {{doi|10.1063/1.2838209}}</ref>  The emittance (<math>\varepsilon</math>) can be calculated from MTE and the laser spot size on the photocathode (<math>\sigma_x</math>) using the following equation.

<math display=block>
\varepsilon = \sigma_x\sqrt{\frac{\text{MTE}}{m_ec^2}}
</math>

where <math>m_ec^2</math> is the rest mass of an electron.  In commonly used units, this is as follows.

<math display=block>
\overset{[\mu\text{m}]}{\varepsilon} \approx \overset{[\mu\text{m}]}{\sigma_x} \sqrt{\frac{ \overset{ [\text{meV}] }{ \text{MTE} }}{511 \times 10^6}}
</math>

Because of the scaling of transverse emittance with MTE, it is sometimes useful to write the equation in terms of a new quantity called the thermal emittance.<ref>Yamamoto, N., Yamamoto, M., Kuwahara, M., Sakai, R., Morino, T., Tamagaki, K., Mano, A., Utsu, A., Okumi, S., Nakanishi, T., Kuriki, M., Bo, C., Ujihara, T., & Takeda, Y. (2007). Thermal emittance measurements for electron beams produced from bulk and superlattice negative electron affinity photocathodes. ''Journal of Applied Physics'', ''102''(2), 024904. {{doi|10.1063/1.2756376}}</ref>  The thermal emittance is derived from MTE using the following equation.

<math display=block>
\varepsilon_{\text{th}} = \sqrt{\frac{\text{MTE}}{m_ec^2}}
</math>

It is most often expressed in the ratio um/mm to express the growth of emittance in units of um as the laser spot grows (measured in units of mm).

An equivalent definition of MTE is the temperature of electrons emitted in vacuum.<ref>Musumeci et al. (2018). “Advances in Bright Electron Sources.” https://doi.org/10.1016/j.nima.2018.03.019</ref> The MTE of electrons emitted from commonly used photocathodes, such as polycrystalline metals, is limited by the excess energy (the difference between the energy of the incident photons and the photocathode's work function) provided to the electrons. To limit MTE, photocathodes are often operated near the photoemission threshold, where the excess energy tends to zero. In this limit, the majority of photoemission comes from the tail of the Fermi distribution. Therefore, MTE is thermally limited to <math>k_BT</math>, where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the temperature of electrons in the solid.<ref>Siddharth Karkare, S., Adhikari, G., Schroeder, W. A., Nangoi, J. K., Arias, T., Maxson, J., and Padmore, H. (2020). “Ultracold Electrons via Near-Threshold Photoemission from Single-Crystal Cu(100)." Phys. Rev. Lett. 125, 054801. </ref> 

Due to conservation of transverse momentum and energy in the photoemission process, the MTE of a clean, atomically-ordered, single crystalline photocathode is determined by the material's band structure. An ideal band structure for low MTEs is one that does not allow photoemission from large transverse momentum states. <ref>Parzyck et al. (2022). “Single-Crystal Alkali Antimonide Photocathodes.” Phys. Rev. Lett. 128, 114801.</ref>

Outside of accelerator physics, MTE and thermal emittance play a role in the resolution of proximity-focused imaging devices that use photocathodes.<ref>Martinelli, R. U. (1973). Effects of Cathode Bumpiness on the Spatial Resolution of Proximity Focused Image Tubes. ''Applied Optics'', ''12''(8), 1841. {{doi|10.1364/AO.12.001841}}</ref>  This is important for applications such as image intensifiers, wavelength converters, and the now obsolete image tubes.