# Recent Advances in Ultrafast Structural Techniques
## 3 Structural Techniques Involving Ultrashort Electron Pulses
### Temporal Resolution in the Absence of Repulsive Interactions  
For decades the main challenge has been to break the ps-temporal resolution barrier in FED experiments. Electrons are not only fermions but also charged particles, and therefore it is extremely challenging to keep them together and preserve the temporal resolution. Coulomb repulsion or space-charge effects lengthen the pulse during its propagation towards the sample. In fact, the instrument response function of FED setups is given by the electron pulse duration at the sample position (the pump pulse is usually much shorter). The use of single-electron or low-density electron sources resulted in the first and simplest alternative to reduce space-charge effects and preserve the temporal resolution [176; 177; 212; 213]. However, there are only few samples capable of withstanding the repetitive photoexcitation conditions necessary to build a statistically meaningful (high SNR) diffraction pattern. Time-resolved electron diffraction experiments employing single-electron bursts rely on the accumulation of multiple events averaged over \(\sim\)10\({}^{8}\) pump-probe cycles to obtain a diffraction pattern with sufficient quality. To collect the data in a reasonable amount of time, single-electron sources often require the use of high repetition rate lasers \(>\)100 kHz. This procedure leads to accumulated heating effects that can significantly deteriorate the sample and even introduce artifacts, see supplementary information in Reference [41]. Therefore, it is essential to reach both ultrashort and ultrabright electron pulses to enable the study of structural dynamics in susceptible solid-state molecular materials. High-repetition electron sources are then best suited for gas phase experiments (molecular beams) in which the sample is refreshed between successive excitation pulses. Nevertheless, it is interesting to consider the limiting case in which the interactions among electrons within the bunch are switched off. This limit offers an idea of the \"optimal\" temporal resolution achievable by a DC FED instrument, such as the one shown above in Figure 13. Neglecting relativistic effects, the longitudinal broadening (\(\Delta\mathrm{I}\)) of the electron pulse can be approximated by \(\Delta\mathrm{I}\ =\ \mathrm{t_{t}}\ \Delta\mathrm{v_{0}}\), where \(\Delta\mathrm{v_{0}}\) is the initial velocity spread of the photoemitted electrons and \(\mathrm{t_{t}}\) is the electron pulse transit time from the cathode to the sample plane \(\mathrm{t_{t}}\)= \(\left(2\mathrm{d_{a}}+\mathrm{d_{s}}\right)\mathrm{v_{z}}^{-1}\), where \(\mathrm{d_{a}}\) and \(\mathrm{d_{s}}\) are the cathode-anode and anode-sample distances, respectively, and \(\mathrm{v_{z}}\) is the electron speed in the propagation direction after exiting the anode aperture. Considering that the temporal pulse broadening (\(\Delta\mathrm{t_{p}}\)) can be calculated as \(\Delta\mathrm{t_{p}}\ \approx\ \Delta\mathrm{I}\ \mathrm{v_{z}}^{-1}\), we arrive to the following expression, [4; 5] (Equation (13)):  
\[\Delta\mathrm{t_{p}}\ \approx\ \frac{\Delta\mathrm{I}}{\mathrm{v_{z}}}\ \approx\ \frac{\mathrm{t_{t}}\ \Delta\mathrm{v_{0}}}{\mathrm{v_{z}}}\ \approx\ \frac{\left(2\mathrm{d_{a}}+\mathrm{d_{s}}\right)\ \Delta\mathrm{v_{0}}}{\mathrm{v_{z}}^{2}}\ \approx\ \left(\frac{\mathrm{d_{a}}}{\mathrm{e}\ \mathrm{V}}+\frac{\mathrm{d_{s}}}{2\ \mathrm{e}\ \mathrm{V}}\right)\Delta\mathrm{p_{0}} \tag{13}\]  
where V is the applied DC voltage that satisfies \(\frac{\mathrm{m_{0}}\mathrm{v_{z}}^{2}}{2}=\ \mathrm{e}\ \mathrm{V}\), and \(\Delta\mathrm{p_{0}}\) is the initial momentum spread, \(\Delta\mathrm{p_{0}}\)\(=\ \mathrm{m_{0}}\ \Delta\mathrm{v_{0}}\). Although Equation (13) suitably approximates the temporal spread gained in the cathode-anode region through a very simple and straightforward derivation, it was found to over exaggerate the pulse broadening occurring in the drift region.