# Recent Advances in Ultrafast Structural Techniques
## 3 Structural Techniques Involving Ultrashort Electron Pulses
### Electrons as Structural Probes: Fermionic Nature and Source Brightness  
The electron is an interesting elementary particle to play with. Its spin of one-half makes the electron a fermion and as such \"no two electrons can share the same state\" (Pauli exclusion principle). Therefore, an equivalent process to that of light amplification by stimulated emission of radiation (laser), which arises from the bosonic behaviour of photons, is not possible for electrons. One of the challenges is thus to develop an electron source with sufficient coherence and brightness to resolve the ensuing atomic motions. There is in fact a fundamental \"quantum limit\" to electron beam brightness (\(B_{\text{Q}}\)) enforced by the Heisenberg uncertainty principle. Note that the minimum angular divergence a laser beam can achieve is also given by the uncertainty principle. However, there is no fundamental boundary to laser brightness besides practical and physical constraints usually dictated by the damaging threshold of amplifying laser medium. [40] The definition of electron beam brightness (in German, \"Richstrahlwert\") was initially introduced by von Borries and Ruska [197] in 1939, (Equation (1)):  
\[B\ = \frac{\text{I}}{\text{A}_{\text{s}}\ \Omega} \tag{1}\]  
where I is the current; A\({}_{\text{s}}\) the areal spot size at focal point; and \(\Omega\) the beam divergence in sr. Equation (1) has now evolved into different quasi-equivalent forms depending on the background of the scientific community. Beam accelerator scientists prefer working in position-momentum phase space owing to the invariance of the six-dimensional (6D) phase-space density (\(\rho_{\text{6D}}\)) or volume (V\({}_{\text{6D}}\)) imposed by the Liouville's theorem (as long as beam dynamics can be described by Hamilton's equations). V\({}_{\text{6D}}\) is the volume occupied by the electron bunch in this 6D phase space (x, p\({}_{\text{x}}\), y, p\({}_{\text{y}}\), z, p\({}_{\text{z}}\)), which is proportional to the product of the normalized rms beam emittances \(\varepsilon_{\text{nx}}\varepsilon_{\text{ny}}\varepsilon_{\text{nz}}\), (Equation (2)):  
\[\text{V}_{\text{6D}}\propto\ \varepsilon_{\text{nx}}\varepsilon_{\text{ny}} \varepsilon_{\text{nz}} \tag{2}\]  
where \(\varepsilon_{\text{nq}}\)\(=\)\(\frac{1}{\text{m}_{\text{0}}\text{c}}\ \sqrt{\left\langle\text{q}^{2}\right\rangle\left\langle\text{p}_{\text{q}} \right\rangle^{2}\left\rangle-\left\langle\text{ q}\ \text{p}_{\text{q}}\right\rangle^{2}}\ \approx\ \frac{1}{\text{m}_{\text{0}}\text{c}}\ \sigma_{\text{q}}\rho_{\text{p}_{\text{q}}}\) and q = x, y, z. Hence, (Equation (3)):  
\[B_{\text{6D}}=\frac{\text{N}\,\text{e}}{\text{V}_{\text{6D}}}\ \propto\ \frac{\text{N}\,\text{e}}{\sigma_{\text{x}}\sigma_{\text{p}_{\text{x}}}\ \sigma_{\text{y}}\sigma_{\text{p}_{\text{y}}}\ \sigma_{\text{z}}\sigma_{\text{p}_{\text{z}}}} \tag{3}\]  
where \(\sigma\) are variances or rms spreads in positions x, y, z; and momenta \(\text{p}_{\text{x}^{\prime}}\), \(\text{p}_{\text{y}^{\prime}}\), \(\text{p}_{\text{z}}\). Figure 14: **300-kV DC FED apparatus at the Ultrafast electron Imaging Lab (UeIL). ** Left panel displays the 300-kV power supply, and the optical table with beam transport optics. Right panel shows the 300-kV FED instrument surrounded by a radiation enclosure. The result from Equation (3) is a true figure of merit that can be compared against the quantum degenerate limit, (Equation (4)):  
\[B_{\text{Q}}=\frac{\left(\text{m}_{\text{0}}\,\text{c}\right)^{3}2\text{e}}{\text {h}^{3}}=\frac{2\text{e}}{\text{\lambda}_{\text{c}}^{3}}\ \approx\ 10^{25}\ \text{A}\ \text{m}^{-2} \tag{4}\]  
where \(\text{\lambda}_{\text{c}}\) is known as Compton wavelength.