# A Collection of Formulae for the Design of Compton Back-Scattering X-ray Sources
## 2 Fixing a More Realistic Model
We carry out the integration on the transverse coordinate, without specifying (see below) \(t_{p}(z)\), we get therefore  
\[ \begin{split}&\frac{d^{2}N_{x}}{c\,dtdz}=\frac{2\pi\sigma_{Th}N_{l}N_ {e}(1+\beta_{0})}{\sigma_{l}(z)^{2}\sigma_{e}(z)^{2}A_{e}A_{l}}e^{-\frac{t_{l} (z)^{2}+t_{e}(z)^{2}}{\Delta\tau^{2}}}\int_{0}^{\infty}e^{-\left[\frac{\sigma_ {e}(z)^{2}+\sigma_{l}(z)^{2}}{\sigma_{e}(z)^{2}\sigma_{l}(z)^{2}}\right]^{2}rdr =\\ &=\frac{\pi\sigma_{Th}(1+\beta_{0})}{A_{e}A_{l}}\widetilde{L}_{0} (z),\\ &\Delta\tau_{e}=\Delta\tau_{l}=\Delta\tau,\\ &\widetilde{L}_{0}(z)=\frac{N_{l}N_{e}}{[\sigma_{e}(z)^{2}+\sigma _{l}(z)^{2}]}e^{-\frac{t_{l}(z)^{2}+t_{e}(z)^{2}}{\Delta\tau^{2}}}.\end{split}  \tag{25}\]  
The superimposed tilde denotes the luminosity divided by the collision frequency, \(\tilde{L}_{0}(z)\) is a function of the longitudinal coordinate and therefore  
\[\frac{dN_{x}}{dt}=\frac{c\pi\sigma_{Th}(1+\beta_{0})}{A_{e}A_{l}}\int_{- \infty}^{\infty}L_{0}(z)dz. \tag{26}\]  
It is important to emphasize that \(dN_{x}/dt\) is not the photon flux \(\dot{N}_{x}\) defined in the previous section. The derivative in Equation (26) refers to the microscopic time, while the superimposed \"dot\" accounts for the macroscopic time associated to the rate of collisions. In order to perform the integration, it is necessary to carefully specify the dependence of the luminosity on the z-variable. We note therefore that  
\[\sigma_{e}(z)^{2}+\sigma_{l}(z)^{2}=(\sigma_{l}^{2}+\sigma_{e}^{2})(1+\zeta^{2}),\]  
\[\zeta=\kappa\,z,\]  
\[\kappa=\frac{1}{Z_{R}}\sqrt{\frac{1+r_{T}^{2}\frac{Z_{R}^{2}}{\beta_{T}^{2}}}{1+ r_{T}^{2}}}, \tag{27}\]  
\[r_{T}=\frac{\sigma_{e}}{\sigma_{l}}.\]  
Regarding the definition of \(t_{p}(z)\), we note that an appropriate discussion, including the detector time (namely the time at which the X-ray flux is measured on a screen placed at \(z=z_{d}\)) has been developed in Ref. [33]. The definition of the laser and electron pulse time, \(t_{l}\) and \(t_{e}\), are respectively reported in Ref. [33] is more general and comprises a fairly elaborated conceptual scheme, accounting for the current time \(t\) (namely that ruling the interaction at the IP) and the detector time \(t_{d}\) (namely the time at which the scattered photons are collected at \(z_{d}\)). Since we limit ourselves to the backscattering geometry we drop such a distinction and simply set  
\[t_{e}(z)=t, \tag{28}\] \[t_{l}(z)=2\sqrt{2}\frac{z}{c},\]  
where \(2\sqrt{2}\) is just a matter of normalization, due to the definition of the laser pulse duration \(\Delta t\), linked to the bandwidth as \(  
eventually yields  
\[ \begin{split}\frac{dN_{x}}{dt}=\frac{\sigma_{Th}L_{0}}{f}\Phi_{e}^{- \left(\frac{t}{\Delta\tau}\right)^{2}},\\ \Phi=\frac{2\pi^{5/2}c(1+\beta_{0})}{A_{e}A_{l}\kappa}e^{\kappa} \Gamma\left(\frac{1}{2},\chi\right),\\ \kappa=\left(\frac{2\sqrt{2}}{\sigma_{z,e}\sqrt{\chi}}\right). \end{split}  \tag{32}\]  
The function \(\Phi\) marks the difference with respect to the results of the previous section (see Equation (8)). To appreciate its physical meaning, further algebraic manipulations are in order. We can, accordingly, introduce the flux function  
\[ \begin{split}\Phi=\frac{2}{\sqrt{\pi}\Delta\tau}\sqrt{\bar{x}}e^{ \kappa}\Gamma\left(\frac{1}{2},\chi\right),\\ \chi=\left(\frac{2\sqrt{2}}{c\Delta\tau\kappa}\right)^{2}=8r_{z} ^{2}\left[\frac{Z_{R}}{\sigma_{z,e}}\sqrt{\frac{1+r_{T}^{2}}{1+r_{T}^{2}\frac {Z_{R}^{2}}{\beta_{T}^{2}}}}\right]^{2},\\ r_{z}=\left(\frac{\sigma_{z,e}}{\sigma_{z,l}}\right),\end{split}  \tag{33}\]  
and if we integrate with respect to time, we obtain the number of scattered photons  
\[ \begin{split} N_{x}\cong 2\sigma_{Th}\frac{L_{0}}{f}F(\chi),\\ F(\chi)=\sqrt{\chi}e^{\chi}\Gamma\left(\frac{1}{2},\chi\right).\end{split}  \tag{34}\]  
The X-ray photon counting, including the rate of collisions, is therefore  
\[\dot{N}_{x}\cong 2\sigma_{Th}L_{0}F(\chi). \tag{35}\]  
It is evident that the corrections to the result of Section 1 are essentially given by the function \(F(\chi)\), depending on a single variable.