## 2 Mean Effective Charge  
The electron-atom interaction potential \(V_{e-a}(\mathbf{r})\) for the electron-atom bremsstrahlung process is represented by:  
\[V_{e-a}(\mathbf{r})\,=\,-\,\frac{Ze^{2}}{r}\,-\,\int d^{3}\mathbf{r}^{\prime} \,\frac{e\rho_{e}(\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,, \tag{1}\]  
where \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) denote the position of the projectile electron with respect to the center of the target atom and the position of the bound electron, respectively, \(Z\) is the charge number of the nucleus, \(e\) is the electron charge, and \(\rho_{b}(\mathbf{r}^{\prime})[\,=\,-\,en_{b}(\mathbf{r}^{\prime})]\) is the bound electron density of the target atom with \(n_{b}(\mathbf{r}^{\prime})\) being the bound electron number density. Then, the Fourier transformation \(\widetilde{V}_{e-a}(\mathbf{q})\) of Equation (1) is obtained as:  
\[ \begin{array}{rl}\widetilde{V}_{e-a}(\mathbf{q})&=\,\int d^{3}\mathbf{r}\,e ^{-i\,\mathbf{q}\cdot\mathbf{r}}\,V_{e-a}(\mathbf{r})\\ &=\,-\,\frac{4\pi\rho^{2}}{q^{2}}\left[Z\,-\,F_{a}(\mathbf{q})\right],\end{array}  \tag{2}\]  
where \(F_{a}(\mathbf{q})\) denote the atomic form factor due to the distribution of bound atomic electrons given by \(F_{a}(\mathbf{q})\,=\,\int d^{3}\mathbf{r}^{\prime}\,e^{-i\,\mathbf{q}\cdot \mathbf{r}}n_{b}(\mathbf{r}^{\prime})\). It is obvious that \(F_{a}(\mathbf{q})\) is zero in the electron-ion bremsstrahlung process since there are no atomic electrons in the target ion. The Thomas-Fermi model [22] is very useful to study the process in the many-electron atoms and the collision dynamics in the neutral atoms. Using the typical parameters in the Thomas-Fermi model, \(F_{a}(\overline{q})\) is represented by:  
\[ \begin{array}{rl}F_{a}(\overline{q})&=\,\frac{4\pi}{q}\,\frac{b^{2}}{Z^{2/3} }\int_{0}^{\infty}dx\,x\sin(\overline{q}x)\,n_{b}(x)\\ &=\,Z[1\,-\,\overline{q}\,H(\overline{q})]\,,\end{array}  \tag{3}\]  
where the parameter \(b\,=\,(1/2)(3\pi/4)^{2/3}a_{0}\), \(a_{0}(\,=\,\hbar^{2}/me^{2})\) is the Bohr radius of the hydrogen atom, \(\hbar\) is the Planck constant divided by \(2\pi\), \(m\) is the electron mass, \(\overline{q}\,(\,\equiv\,\,\,\,\,  
momentum transfer, \(x\,(\,\equiv\,r^{\prime}Z^{1/3}/b)\) is the dimensionless distance, and \(n_{b}(x)\) is the Thomas-Fermi number density given by:  
\[n_{b}(x)\;=\;\frac{Z^{2}}{4\pi b^{3}}\bigg{[}\frac{X(x)}{x}\bigg{]}^{3/2}\,, \tag{4}\]  
with \(X(x)\) being the solution of the Thomas-Fermi equation, \(d^{2}X/dx^{2}\;=\;X^{3/2}/x^{1/2}\) (with the boundary condition \(X(0)\;=\;1\)), and \(H(\overline{q})\) is the screening function of the atomic electrons in the Thomas-Fermi scheme defined by:  
\[H(\overline{q})\;=\;\int_{0}^{\infty}dx\,\sin(\overline{q}x)X(x)\,. \tag{5}\]  
The approximate Thomas-Fermi solution can be often given in the form of single-exponential Mott-Massey solution [9; 16], such as:  
\[X_{MM}(x)\;\cong\;e^{-sx}\,, \tag{6}\]  
where \(s\cong 0.66\). Since the boundary conditions [16] for the Thomas-Fermi equation, \(d^{2}X/dx^{2}\;=\;X^{3/2}/x^{1/2}\), are known as \(X(0)\;=\;1\) and \(X(\infty)\;=\;0\), the single-exponential approximate Mott-Massey solution, \(X_{MM}(x)\;\cong\;e^{-sx}\), would be reasonably reliable to investigate the atomic collision and radiation processes including many-electron neutral atoms. By taking of the [PERSON]'s Thomas-Fermi solution given by Equation (6), the atomic form factor is obtained as:  
\[F_{a}(\overline{q})\;=\;Z\bigg{(}1\,-\,\frac{\overline{q}^{2}}{\overline{q}^{ 2}\,+\,s^{2}}\bigg{)}\,. \tag{7}\]  
Now, Equation (2) based on [PERSON]'s single-exponential Thomas-Fermi solution becomes:  
\[\overline{V}^{\prime}_{e-a}(\overline{q})\;=\;-\,\frac{4\pi Ze^{2}}{\overline{ q}^{2}}\bigg{(}\frac{b}{Z^{1/3}}\bigg{)}^{2}\frac{\overline{q}^{2}}{\overline{q}^{ 2}\,+\,s^{2}}\,. \tag{8}\]  
Hence, the effective charge of the target atom can take the form:  
\[ \begin{array}{ll}Z_{eff}(\overline{q})&=\;Z\,-\,F_{a}(\overline{q})\\ &=\;Z\frac{\overline{q}^{2}}{\overline{q}^{2}\,+\,s^{2}}\,.\end{array}