# A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources
## 5 Final Comments
Going back to \(\psi\) and \(\varsigma\), we note that they are quantum parameters, but their ratio defines the laser strength equivalent parameter, namely  
\[K_{l}=\frac{\psi}{\varsigma}. \tag{60}\]  
If we combine Equation (60) with the second of Equation (57), we end up with the following relationship  
\[\frac{\Delta\omega_{s}}{\omega_{s}}=4\frac{\gamma}{\sqrt{1+\frac{K_{l}^{2}}{2} }}\frac{\psi}{K_{l}}, \tag{61}\]  
which is a \"measure\" of the quantum frequency shift in terms of the electric field size, energy and laser strength parameter, having included also the effect of the transverse motion in the definition of the relativistic factor (see Equation (4)). Equation (61) is useful to specify different region of interest regarding the relative weight of quantum or non-linear intensity effects in the CBS process (see Figure 15 and Ref. [54]). According to the Equation (56), \(\psi=1\) corresponds to the Sauter-Schwinger critical field yielding the creation of an \(e^{\pm}\) pair [54; 55]. In Figure 15a, we have reported \(\Delta\omega_{s}/\omega_{s}\) vs. \(K_{l}\) for different values of \(\psi\). In correspondence with \(\psi=1\), the laser electric field is equivalent to \(\mathcal{E}_{ss}\), for lower values the insurgence of quantum effects becomes less significant. We have also marked the regions specifying the different regimes (Thomson, Compton, linear and non-linear). In Figure 15b, we have shown a comparison with the chart reported in Ref. [58], reporting the different zones of interest and the historical evolution in terms of laser power (hence of \(K_{l}\)), allowing the experimental benchmarking of these regimes. It should be noted that Figure 15a has been drawn for \(\gamma=100\); for larger values, the lines are shifted towards larger quantum shifts values, thus allowing the access to the quantum regime with lower \(K_{l}\) values. Figure 14: CBS differential cross section vs. \(\varepsilon_{2}\) (given in GeV) for different energies of the laser pump (\(1.2\ eV\) (continuous), \(120\ eV\) (dot-dash), \(1200\ keV\) (dash)). We have commented on the possibility of dealing with quantum effects in CBS scattering, but it is evident that can they be hardly detected in X-ray source devices. Most of the relevant effect is observed in higher energy photon scattering, which opens a so reach phenomenology and new effects in non linear QED [47; 48; 49; 50; 54; 55; 57], requiring a more appropriate and thorough discussion, for which there is not time in the present context. Before concluding, we like to comment further on the CBS spectral broadening. We remind the reader therefore that, collecting the scattering photons on a suitably collimated angle, a narrow band or nearly mono-chromatic beam could be available, as a consequence of the restriction of photons at larger angles and therefore at larger wavelengths. However, the spatial, angular and energetic distribution of the electrons introduce the broadening, which combines with the bandwidth associated with the collimator. For this reason the total spectral bandwidth will be provided by  
\[\varsigma=\frac{\hbar\omega_{I}}{m_{e}c^{2}}=\frac{\Lambda_{e}}{\Lambda_{I}}, \tag{56}\]0  
where [44]  
\[\varsigma=\frac{\hbar\omega_{I}}{m_{e}c^{2}}=\frac{\Lambda_{e}}{\Lambda_{I}}, \tag{56}\]1  
is the broadening induced by the collimator with an angular acceptance \(\Phi\). It is evident that a large detection angle implies either a broadened spectral bandwidth or a larger number of collected photons. A general criterion to find a compromise, compatible with a specific application, between the two options is that of choosing  
\[\varsigma=\frac{\hbar\omega_{I}}{m_{e}c^{2}}=\frac{\Lambda_{e}}{\Lambda_{I}}, \tag{56}\]2  
For further comments on the problem of line broadening the reader is addressed to [42]. In this article, we have introduced a number of tools useful for the design of CBS X-ray sources, we have provided comments on the different regimes of operation, including strong field and we touched on quantum corrections.