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The energy of an X-ray transition from the level $n$ to the level $m$ can be written in the form: $$ \newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\f}[2]{\frac{ #1}{ #2}} \newcommand{\l}[0]{\left(} \newcommand{\r}[0]{\right)} \newcommand{\mean}[1]{\langle #1 \rangle}\newcommand{\e}[0]{\varepsilon} \newcommand{\ket}[1]{\left|#1\right>} \Delta E=R_\infty \l \f{(Z-\sigma_n)^2}{n^2}-\f{(Z-\sigma_m)^2}{m^2}\r$$ I know that here $\sigma_n$ and $\sigma_m$ represent 'screening factors' (roughly independent of $Z$) and take account of the screening of nuclear charge by other electrons, but I am yet to find an explanation of why this form can be and is chosen with screening factors independent of $Z$. And in what cases the screening factors can be used. Please can someone explain.

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The expression is semi-empirical, and intended to reproduce the famous Balmer formula for the lines of hydrogen: $$ \Delta E\sim R_\infty\left(\frac{1}{n^2}-\frac{1}{m^2}\right)\, . $$ The Balmer formula can be obtained using solutions to the Schrödinger equation for the hydrogen atom and generally for single-electron atoms with $Z$ protons, where energy levels $$ E_n\sim R_\infty \frac{Z^2}{n^2}\, . $$ Phenomenologically, electrons in the outer shells do not see the full nuclear charge $Z$, but net charge smaller than $Z$ since the inner electrons "screen" or partially cancel the charge of the nucleus, creating a smaller net electric potential (still assumed to be central in nature if the screening is from electrons in closed atomic shells). This is the origin of the $Z-\sigma_n$ factor.

Since the $\sigma_n$ depends mostly on inner electrons (provided their orbitals are more or less completely inside the outer electron orbitals), this quantity does not depend very much on $Z$.

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