Stopping Cross Sections

The total electronic energy-loss cross-section S_e is calculated from the electronic energy transfers Q_e(b) for each impact parameter b using the integral

S_e = 2 π INT[ b Q_e(b) db ]

from b=0 to infinity, where Q_e(b) is extrapolated towards b = 0 as well as towards b -> infinity for every target-electron shell.

To convert the total electronic energy-loss cross-section (in units of eV cm²/atom) into a stopping power dE/dx (in units of eV/cm), the stopping cross-section has to be multiplied by the atomic target density (in atoms/cm³).


Three reference test results are given in the following and may be checked with your copy of the program:


1) 300 keV/u H + H
(these are the default settings at program start)

Z_p = 1
E_p/M_p = 300 keV/u
Z_t = 1
I_Bethe = 15.05 eV
convolution-approximation model = UCA
screening = charge-state scan
shell correction = checked
barkas binary   = checked
accuracy = 3E-4

S_e = 2.902E-15 eV cm²


2) 1000 keV/u H + Ag

Z_p = 1
E_p/M_p = 1000 keV/u
Z_t = 47
I_Bethe = 470.0 eV
convolution-approximation model = UCA
screening = mean charge state
shell correction = checked
barkas binary   = checked
accuracy = 1E-3

S_e =  1.827E-14 eV cm²


3) 300 keV/u H^+ + H
(perturbation theory PCA for bare protons on atomic hydrogen)

Z_p = 1
E_p/M_p = 300 keV/u
Z_t = 1
convolution-approximation model = PCA
screening = none
numerical oscillator strengths

(select file ”Osc1.dat” supplied with this package in subdirectory OscH)
shell correction = checked
barkas binary   = unchecked
accuracy = 1E-4

S_e =  2.909E-15 eV cm²



A comparison between PCA (calculated with above settings, but with accuracy = 5E-6 and without relativistic correction) and PWBA results (calculated with our accurate PWBA code) for the energy loss due to ionization and excitation by bare protons at different energies is shown in the table below. The last column gives the result of the simple Bethe formula (the asymptotic high-energy solution of the PWBA). All these non-relativistic energy-loss cross-sections are given in units of 1E-17 eV cm². The column PCA(sc) shows the results with and the column PCA without the shell correction. It is seen that the forced shell-effect renormalization reduces the PCA result by a factor of two at 10 keV/u and yields a small uncertainty of only 1.3% in comparison to the accurate PWBA result.

Theoretical energy-loss cross-sections in units of 10-17 eV cm²
Energy/Mass PCA PCA(sc) PWBA Bethe formula
10 keV/u 1204 576.7 572.2 894.7
30 keV/u 1081 812.4 801.7 1168
100 keV/u 622.6 572.6 573.7 636.3
300 keV/u 298.5 290.8 291.8 299.1
1 MeV/u 118.5 117.6 117.8 118.3
3 MeV/u 48.22 48.10 48.09 48.14
10 MeV/u 17.30 17.30 17.30 17.30
30 MeV/u 6.639 6.639 6.638 6.636