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Explicit formulas are obtained for atomic form factors describing the energy loss and momentum transfer of a charged particle to electrons in the subshells of the L and M shells. The Bethe-Born approximation is used with hydrogenic wave functions. Application is made to the Lm ionization cross section for protons incident on copper in the energy range from 0.5 to 1.7 MeV.

Radial integrals have been calculated under the one-electron hydrogenic model. Two different values of the effective charge parameter, one for the initial state and the other for the final state, are retained in these formulae. The model is able to reasonably reproduce the existing dipole oscillator strength values with little effort. The dipole oscillator strength values are given for many ions for the first time.

The proton stopping cross section of liquid water for the energy range from 40 keV to 10 MeV is calculated by applying the modified local-plasma model and employing a simple model of liquid water. The calculated stopping cross section of liquid water is about 5.6% to 14% lower than the calculated vapor-state results for the range of 80 to 500 keV and is about 8.5% to. 13.4% lower than measured vapor-state results. The present results agree well with the measurements for ice crystals. The mechanism of this physical-state effect is also presented.