Photoabsorption and Charge Oscillation of the Thomas—Fermi Atom

The complications of calculating the photoabsorption cross section of atoms for frequencies in the far ultraviolet and soft x-ray regions motivate an analysis of this problem on the basis of the Bloch semiclassical model of hydrodynamic charge oscillation in the neutral Thomas-Fermi atom. The hydrodynamic modes of oscillation of the neutral Thomas-Fermi atom have a continuous frequency spectrum. The resulting photoabsorption cross section is a continuous function of frequency which scales with atomic number $Z$ and in this sense is a universal cross section, approximately applicable to all heavy atoms. Numerical solutions of the normal mode functions of dipole charge oscillation are used to calculate the photoabsorption cross section in a range of photon energies $0.816 \mathrm{Z}\mathrm{eV}<ħ\omega <272\mathrm{} \mathrm{Z}\mathrm{eV}$, the range where the model is expected to be most realistic. In this range the semiclassical hydrodynamic cross section agrees with experimental data for the noble gases as well as could be expected for a cross section applicable to all atoms. The model cross section, extended to zero and infinite frequencies by analytical calculation, checks the sum rule to within 2%; but gives a value $I=4.95\mathrm{}$ $Z$ eV for the logarithmic mean excitation energy of stopping power formulas. The unrealistically low value of $\frac{I}{Z}$ results because the Thomas-Fermi atom exaggerates the number of electrons which absorb at low frequencies. Use of the hydrodynamic cross section in the approximate range of validity of this model $ħ\omega >0.816\mathrm{}$ Z eV gives $I=12.4\mathrm{}{e}^{\beta }$ Z eV, where $\beta$ depends upon the oscillator strength below $ħ\omega =0.816\mathrm{}$ Z eV which cannot be satisfactorily determined from the Bloch model. Used within its limitations the Bloch model of hydrodynamic oscillation in the statistical atom provides a useful method of estimating photoabsorption cross sections and could possibly be applied to other atomic processes.