## 3 Electron-Atom Bremsstrahlung and Coulomb Focusing  
Using the second-order nonrelativistic perturbation analysis [8; 14], the differential electron-ion bremsstrahlung cross section \(d^{2}\sigma_{b}\) can be written as:  
\[d^{2}\sigma_{b}\,=\,d\sigma_{C}\cdot dW_{\omega}\,, \tag{11}\]  
where \(d\sigma_{C}(q)\) is the differential elastic scattering cross section:  
\[d\sigma_{C}(q)\,=\,\frac{1}{2\pi\hbar v_{0}^{2}}\big{|}\,\overline{V}({\bf q} )\,\big{|}^{2}q\,dq\,, \tag{12}\]  
\(v_{0}\) is the initial velocity of the projectile electron, \(\overline{V}({\bf q})\) is the Fourier transformation of the interaction potential \(V({\bf r})\):  
\[\overline{V}({\bf q})\,=\,\int d^{3}{\bf r}\,e^{-i{\bf q}\cdot{\bf r}}\,V({\bf r })\,, \tag{13}\]  
where \({\bf q}(\,=\,{\bf k}_{0}\,-\,{\bf k}_{f})\) is the momentum transfer, and \({\bf k}_{0}\) and \({\bf k}_{f}\) are the wave vectors of the initial and final states of the projectile electron, respectively. Here, \(dW_{\omega}/d\Omega\) is the differential probability of emitting a photon of frequency between \(\omega\) and \(\omega\,+\,d\omega\) in the solid angle \(d\Omega\):  
\[\frac{dW_{\omega}}{d\Omega}\,=\,\frac{\alpha}{4\pi^{2}}\,\Lambda^{2}\sum_{ \hat{\bf e}}\big{|}\,\hat{\bf e}\cdot{\bf q}\,\big{|}^{2}\,\frac{d\omega}{ \omega}\,, \tag{14}\]  
where \(\Lambda\) is the Compton wave number given by \(\Lambda\,=\,\hbar/mc\), \(\alpha\) is the fine structure constant, and \(\hat{\bf e}\) is the unit photon polarization vector. By integrating over the directions of the radiation photon in Equation (14), we obtain the bremsstrahlung cross section in the form:  
\[d^{2}\sigma_{b}(q)\,=\,\frac{1}{3\pi^{2}\beta_{0}^{2}}\frac{\alpha}{\left(mc \right)^{2}}\,\big{|}\,\overline{V}({\bf q})\,\big{|}^{2}q^{3}\,dq\,\frac{d \omega}{\omega}\,, \tag{15}\]  
since the summation over polarizations gives the angular distribution factor \(\sin^{2}\theta\), where \(\theta\) is the angle between \({\bf k}_{0}\) and \({\bf q}\). In Equation (15), the quantity \(\beta_{0}\) is defined as \(\beta_{0}\,=\,v_{0}/c\). In the nonrelativistic Born approximation, it is known that the domain of applicability is \(v_{0}\,>\,Zac\) since the projectile energy \(E_{0}(\,\equiv\,mv_{0}^{2}/2)\) is greater than \(Z^{2}Ry\), where \(Ry(\,=\,me^{4}/2\hbar^{2}\,\int\,\approx\,13.6\,{\rm eV})\) is the Rydberg constant. It has been also known that the nonrelativistic Bethe-Heitler formula is invalid for the final state of the projectile electron near the cutoff region \(mv_{0}^{2}/2\,\int\,\approx\,\hbar\omega\) owing to the inaccuracy of the Born approximation for \(v_{0}\,\approx\,Zac\). Therefore, the nonrelativistic Bethe-Heitler formula must be corrected for hard spectral photon energies. In order to correct the Bethe-Heitler cross section, we must consider the motion of the initial and final states of the projectile electron in the external field of the target ion using a continuum wave function for the Coulomb potential since the final Coulomb wave function must be different from the initial Coulomb wave function due to the momentum transfer and the energy loss of the initial projectile electron. It has been shown that the Coulomb correction to the nonrelativistic Bethe-Heitler bremsstrahlung formula [1; 3] using the Born approximation is obtained by the Elwert-Sommerfeld factor which is given by the ratio of the absolute square of the Coulomb wave functions at infinity \((r\,\rightarrow\,\infty)\) and at the origin \(({\bf r}\,=\,0)\).