Semiclassical method of analysis and estimation of the orbital binding energies in many-electron atoms and ions

Orbital binding energies in the ground state of many-electron elements obtained in experiments or in quantum-mechanical„ calculations are studied. Their dependences on the atomic number and on the degree of ionization are analyzed. The Bohr–Sommerfeld semiclassical quantization condition is used and filled-shell orbital binding energy approximate scaling is shown. The scaling is similar to the one in the Thomas–Fermi model, but with two other functions-coefficients. The effective method of the demonstration of binding energies in a large number of atoms through these two functions is proposed. As a result, special features of the elements of the main and transition groups and the influence of relativistic effects are vividly manifested. Simple interpolation expressions are built for the two functions. One can use them to estimate orbital binding energies in the filled shells of many-electron atoms and ions to within 10% for middle elements and from 10% to 30% for heavy ones. The estimate can be used as the initial approximation in precessional atomic computations and also for rough calculations of the ionization cross sections of many-electron atoms and ions by electrons and heavy particles, failing more precise data.