# Recent Advances in Ultrafast Structural Techniques
## 3 Structural Techniques Involving Ultrashort Electron Pulses
### Transverse Electron Beam Properties: Transverse Emittance and Coherence Length  
Regardless of the chosen definition of brightness, a reality check shows that even the most advanced electron sources present a \(B_{\text{6D}}\) that is several orders of magnitude below the quantum limit, \(B_{\text{Q}}\). There is still a lot of room for improvement and in most practical applications the transverse and longitudinal phase spaces are weakly coupled; therefore it is more convenient to treat them separately. The transverse electron beam coherence length (\(L_{\text{x}}\ \propto\ \text{\varepsilon}_{\text{rx}}\)\({}^{-1}\)) is defined as follows, (Equation (6)):  
\[L_{\text{x}}=\frac{\text{\lambda}_{\text{dB}}}{2\pi\ \text{\sigma}_{\text{ \sigma}_{\text{\Theta}}}}=\frac{\text{\lambda}_{\text{dB}}\ \text{p}}{2\pi\ \text{\sigma}_{\text{Px}}}=\frac{\text{h}}{ \text{\sigma}_{\text{Px}}} \tag{6}\]  
where \(\text{\lambda}_{\text{dB}}\) is the de Broglie wavelength and \(\text{\sigma}_{\text{\theta}}\) the rms angular spread or variance. \(L_{\text{x}}\) (or \(\text{\varepsilon}_{\text{rx}}\)) is an essential electron beam parameter. For instance, it is known that in XFELs the value of \(\text{\varepsilon}_{\text{rx}}\) at the entrance of the undulator corresponds to the wavelength of the X-rays produced (\(\lambda\)) according to, (Equation (7)):  
\[\frac{\text{\varepsilon}_{\text{rx}}}{\text{\beta}\text{\gamma}}\ \lesssim\ \frac{\text{\lambda}}{4\pi}\ \beta\ =\ \frac{\text{v}}{\text{c}} \tag{7}\]  
where v is the electron speed, and \(\text{\gamma}\) is the Lorentz factor. Equation (7) links electron beam and X-ray beam properties and highlights the importance of attaining more intense and coherent electron sources. Thus, by reducing \(\text{\varepsilon}_{\text{rx}}\) (or increasing \(L_{\text{x}}\)) one could in principle achieve shorter photon wavelengths or reduce the length of the linac [208]. The design and implementation of better and more efficient photocathodes is currently an area of active research. Generally speaking, the spatial resolution provided by a given photon or electron source is determined by its transverse (\(L_{\text{x}}\)) and longitudinal or temporal (\(L_{\text{I}}\)) coherences. The expression for longitudinal coherence of photons is easily deduced by replacing \(\text{\lambda}_{\text{dB}}\) by \(\lambda\) in Equation (6). Thus, for photons, (Equation (8)):  
\[L_{\text{x}}=\frac{\text{\lambda}}{2\pi\ \text{\sigma}_{\text{\theta}}} \tag{8}\]  
and (Equation (9)):  
\[L_{\text{I}}=\frac{\text{\lambda}^{2}}{2\ \left|\text{\Delta}\right|} \tag{9}\]Then, given \(\mathrm{E}\ =\ \frac{\mathrm{h}\mathrm{c}}{\lambda}\), and therefore \(\frac{\left|\Delta\mathrm{E}\right|}{\mathrm{E}}=\frac{\left|\Delta\lambda \right|}{\lambda}\); we obtain (Equation (10)):  
\[L_{\mathrm{I}}=\frac{\lambda\ \mathrm{E}}{2\left|\Delta\mathrm{E}\right|} \tag{10}\]  
On the other hand, \(L_{\mathrm{I}}\) for electrons becomes, (Equation (11)):  
\[L_{\mathrm{I}}=\frac{\lambda_{\mathrm{dB}}{}^{2}}{2\left|\Delta\lambda_{ \mathrm{dB}}\right|} \tag{11}\]  
and considering \(\lambda_{\mathrm{dB}}\ \approx\ \lambda_{\mathrm{dB}}\mathrm{p}_{\mathrm{z}}\ \approx\ \mathrm{h}\), we obtain \(\left|\Delta\mathrm{p}_{\mathrm{z}}\right|\ \approx\ \frac{\mathrm{h}\ \left|\Delta \lambda_{\mathrm{dB}}\right|}{\lambda_{\mathrm{dB}}{}^{2}}\) and, (Equation (12)):  
\[L_{\mathrm{I}}\ \approx\ \frac{\mathrm{h}}{2\left|\Delta\mathrm{p}_{\mathrm{z}}\right|} \tag{12}\]  
Note that assuming \(\left|\Delta\mathrm{p}_{\mathrm{z}}\right|\ \approx\ \left|\Delta\mathrm{p}_{ \mathrm{x}}\right|\) and combining Equations (6) and (12), we obtained \(L_{\mathrm{x}}\approx L_{\mathrm{I}}\), which seems to indicate that transverse and longitudinal coherences should have a similar effect on the spatial resolution of X-ray and electron diffractometers. However, the small de Broglie wavelength or small scattering angles (\(\theta\ \ll\ 1\)) of keV-electrons makes the effect of \(L_{\mathrm{I}}\) negligible.