# A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources
## 1 Introduction
If we replace the number of electrons with the bunch charge divided by the electron elementary charge (\(N_{e}=Q_{e}/e\)) and the number of photons in the laser pulse as the laser energy divided by the single photon energy, we find from Equation (8)  
Figure 2.5  
Figure 2.6  
Figure 2.7  
Figure 2.8  
Figure 2.9  
It should be noted that in SI units \(k\) reads  
\[\mathcal{E}_{s}=\frac{(1-\beta\,\cos(\vartheta_{i}))\mathcal{E}_{l}}{(1-\beta \cos(\vartheta_{f}))+\frac{\mathcal{E}_{l}}{E_{e}}(1-\cos(\vartheta_{f}- \vartheta_{i}))}. \tag{1}\]0As already underlined, the key parameters of our discussion are the bunch charge, the laser energy, the transverse cross sections of laser/electron bunches and the collision repetition frequency. Regarding the design strategy, we cannot define (at least on the basis of the elements provided so far) any effective optimization criteria, except that of keeping the largest values of \(Q_{e}\), \(\bar{E}_{l}\) and \(f\) and the smallest values of \(\sigma_{e,l}\). Furthermore, regarding the collision frequency we note that the combined use of recirculated beams and of laser pulse stacking in high Q-cavities provides values exceeding the tens of \(MHz\) (for more details see Refs. [27; 28]). Albeit fairly elementary, what we have accomplished so far is sufficient to get preliminary indications regarding the working point of an X-ray CBS source. In order to operate in the hard X-ray region (7-8 \(KeV\)) with laser wavelength of 1 \(\upmu\)m, an e-beam with 25 \(MeV\) energy is sufficient. In order to proceed we use Equation (10) to specify the amount of charge density to get a definite X-ray flux. If we assume that the laser and electron r.m.s. beam transverse sections are matched (\(d\cong 1\)), we find  
\[\mathcal{E}_{s}=\frac{(1-\beta\,\cos(\vartheta_{i}))\mathcal{E}_{l}}{(1-\beta \cos(\vartheta_{f}))+\frac{\mathcal{E}_{l}}{E_{e}}(1-\cos(\vartheta_{f}- \vartheta_{i}))}. \tag{1}\]1  
In Table 1 we have reported the normalized emittance (\(\varepsilon_{n}\)), namely the electron beam phase space area at the cathode. The corresponding value when the beam is brought to relativistic energies is \(\varepsilon_{n}/\gamma\). It should furthermore be noted that \(\beta_{T}\) (the Twiss parameter) and \(\varepsilon_{n}\) reported in the table refers to radial and vertical components. Furthermore, using for the other quantities on the r.h.s. of Equation (12) the values reported in Table 1[29; 30; 31; 32], we end up with \(\overline{Q_{e}}\big{[}C/m^{2}\big{]}\cong 0.025\), which is a reasonable value, as we will see below. Taking into account that the e-beam transverse sections are linked to emittance and beam parameters by the relationship  
\[\mathcal{E}_{s}=\frac{(1-\beta\,\cos(\vartheta_{i}))\mathcal{E}_{l}}{(1-\beta \cos(\vartheta_{f}))+\frac{\mathcal{E}_{l}}{E_{e}}(1-\cos(\vartheta_{f}- \vartheta_{i}))}. \tag{1}\]2  
we can derive from Equation (12) the total charge as  
\[\mathcal{E}_{s}=\frac{(1-\beta\,\cos(\vartheta_{i}))\mathcal{E}_{l}}{(1-\beta \cos(\vartheta_{f}))+\frac{\mathcal{E}_{l}}{E_{e}}(1-\cos(\vartheta_{f}- \vartheta_{i}))}. \tag{1}\]3  
The use of the parameters in Table 1 yields for the amount of the bunch charge \(Q_{e}=1.579\)\(pC\), consistent with the assumptions in ref. [29]. A key point, loosely mentioned in the previous discussion, is the striking request on the smallness of the electron and laser beam sections at the interaction point (IP). This can be realized by exploiting beams with small normalized emittance and small beta parameters. The realizations of these conditions pass through two bottlenecks: the design of an adequate RF gun producing an electron beam with a small emittance at the cathode and an extremely  
\begin{table}
\begin{tabular}{c c} \hline \hline \(N_{x}[s^{-1}]\) & \(1.6\times 10^{14}\) \\ \hline \(f[Hz]\) & \(10^{8}\) \\ \hline \(\bar{\bar{E}}_{l}[J]\) & \(10^{-2}\) \\ \hline \(\lambda_{l}[m]\) & \(10^{-6}\) \\ \hline \(\gamma\) & \(50\)  
challenging beam transport at the IP (see below for further comments).