Many-electron atoms
high-energy core states and transitions

The following exercises focus on the core-electron excitations of many-electron atoms / ions.

They must be solved based on some (usually approximate) quantum-mechanical treatment (Hartree-Fock approximation to Schrödinger's wave equation, and consequent effective-Z approximation...).

The nuclear charge is assumed to be Z q_e.

a₀ = ℏ²/(m_e e²) is the Bohr radius and
E_Ha = e²/a₀ is the Hartree energy.
Here m_e is the electron mass and e² = q_e² /(4πε₀) is the electromagnetic characteristic coupling constant.

For many-electron atoms reduced-mass corrections are irrelevant and conceptually meaningless.

Relativistic effects (e.g. spin-orbit coupling) are usually important, because they scale as Z⁴. By comparison the effects of external magnetic fields are always "weak" (and usually irrelevant) for core-hole problems.

1. Estimate the energy of the absorption K edge in Cr (Z=24).
   RESULT: Z^eff_n=1=22 yields approx. 6580 eV (experimentally it is 5989.2 eV).
2. Estimate the energy and wavelength of K_α emission in Se (Z=34). Estimate the separation between the K->L_II and the K->L_III lines.
   RESULT: 11970 eV (exper.: 11224 eV). 39 eV (with a spin-orbit effective Z^eff-SO_n=2=34 -3.5=30.5).