## 4 Coulomb Focused Bremsstrahlung Cross Section  
Using Equations (4) and (15) with the integration over the momentum transfer \(q\) for the domain \(q_{\text{min}}[\,=\,(k_{0}-k_{f})]\leq q\leq q_{\text{max}}[\,=\,(k_{0}+k_{f})]\), the electron-atom bremsstrahlung cross section [9]\((d\sigma_{b}/d\varepsilon)^{MM}_{e-a}\) per photon energy using [PERSON]'s single-exponential Thomas-Fermi solution \(X_{MM}(x)\) is given by:  
\[ \begin{array}{ll}\left(\frac{d\sigma_{b}}{d\varepsilon}\right)^{MM}_{e-a}&= &\frac{16}{3}\frac{Z^{2}\alpha r_{0}^{2}c^{2}}{\varepsilon v_{0}^{2}}\left\{ \frac{1}{2}\,\ln\left[\frac{\overline{\varepsilon}^{2}\,+\,(\sqrt{\overline{ E}_{0}}+\sqrt{\overline{E}_{0}-\overline{\varepsilon}})}{\overline{ \varepsilon}^{2}\,+\,(\sqrt{\overline{E}_{0}}-\sqrt{\overline{E}_{0}- \overline{\varepsilon}})}^{2}\right]\right.\\ &\left.+\,\frac{2\overline{\varepsilon}^{2}\,\sqrt{\overline{E}_{0}}\,\sqrt{ \overline{E}_{0}}-\overline{\varepsilon}}{(\overline{\varepsilon}^{2}\,+\,( \sqrt{\overline{E}_{0}}+\sqrt{\overline{E}_{0}-\overline{\varepsilon}})\,][ \overline{\varepsilon}^{2}\,+\,(\sqrt{\overline{E}_{0}}-\sqrt{\overline{E}_{0 }-\overline{\varepsilon}})}^{2}]\right\},\end{array}  \tag{17}\]  
where \(\varepsilon(\,=\,\,\hbar\omega)\) is the bremsstrahlung photon energy, \(r_{0}(\,=\,e^{2}/mc^{2})\) is the classical electron radius, \(\overline{\xi}\,=\,\xi Z^{1/3}\), \(\xi(\,=sa_{0}/b)\,=\,0.7455\), \(\overline{E}_{0}\,\equiv\,E_{0}/Ry\), and \(\overline{\varepsilon}\,\equiv\,\varepsilon/Ry\). For the electron-ion bremsstrahlung process [14], the expression of the bremsstrahlung cross section \(\left(d\sigma_{b}/d\varepsilon\right)_{e-i}\) is rather simple due to the omission of the screening effects:  
\[\left(\frac{d\sigma_{b}}{d\varepsilon}\right)_{e-i}\,=\,\frac{16}{3}\frac{Z^{2 }\alpha r_{0}^{2}c^{2}}{\varepsilon v_{0}^{2}}\,\ln\left(\frac{\sqrt{\overline {E}_{0}}\,+\,\sqrt{\overline{E}_{0}\,-\,\overline{\varepsilon}}}{\sqrt{ \overline{E}_{0}}\,-\,\sqrt{\overline{E}_{0}\,-\,\overline{\varepsilon}}} \right). \tag{18}\]  
However, the Coulomb focusing effect in the electron-atom bremsstrahlung process has not been investigated in the previous work of [PERSON] and [PERSON] [9], thus, the electron-atom bremsstrahlung cross section has to be corrected in the low-energy region. It is expected that Equations (6) and (15) provide the Coulomb focusing for the bremsstrahlung with neutral atoms since \(\overline{Z}_{eff}\) describes the effective charge of the neutral atom by the electron-encounter since the Coulomb focusing factor is needed for the modification of the bremsstrahlung cross section near the threshold region. The modifiedElwert-Sommerfeld Coulomb focusing factor \(f_{CF}(\overline{E}_{0},\ \overline{\varepsilon},\ \overline{Z}_{eff})\) for the electron-atom bremsstrahlung process is then given by:  
\[ \begin{array}{ll}f_{CF}(\overline{E}_{0},\ \overline{\varepsilon},\ \overline{Z}_{eff})&=\frac{ \overline{\
u}_{0}}{\overline{\
u}_{f}}\frac{1-\exp(-2\pi\overline{Z}_{eff} \alpha c/\
u_{0})}{1-\exp(-2\pi\overline{Z}_{eff}\alpha c/\
u_{f})}\\ &=\ \frac{\sqrt{E_{0}}}{\sqrt{E_{0}-\overline{\varepsilon}}}\frac{1-\exp \left[-\frac{2\pi}{\sqrt{E_{0}}}\frac{\
u^{3}}{2^{2}+\overline{\varepsilon}^ {2}}\right]}{1-\exp\left[-\frac{2\pi}{\sqrt{E_{0}-\overline{\varepsilon}}} \frac{\
u^{3}}{2^{2}+\overline{\varepsilon}^{2}}\right]}\,\end{array}  \tag{19}\]  
since the expression of the standard Elwert-Sommerfeld factor [14] is given by \(f_{CF}(\eta_{i},\ \eta_{f})=(\eta_{f}/\eta_{i})[1-\exp(-2\pi\eta_{i})]/[1-\exp(-2\pi \eta_{f})]\). Hence, Equation (19) is the universal formula for the Coulomb focusing factor for the bremsstrahlung process with neutral atoms.