In order to access this option you should select the EXPERT OPTIONS sheet (close to the upper left corner of the main program window).

Five types of projectile-screening potentials are provided in this program. All formulas in this section are given in atomic units (a.u., m_e=e=h/2π=1). The calculation of non-equilibrium energy transfers and stopping powers, with options other than “NONE” and “MEAN-CHARGE STATE”, is provided for advanced users only.

a) NONE: the pure Coulomb potential V_p = -Z_p / r is used (for bare projectiles only)

b) YUKAWA/BOHR: the Bohr-like screening potential (Yukawa potential) is used.

This potential is simply given by V_p = -Z_p / r exp(-r / λ) and corresponds to the electrostatic potential produced by a foreign charge Z_p placed in a homogeneous electron gas in the Thomas-Fermi approximation. The screening length λ (in Å) has to be given in a separate window. For solids, the polarization of the medium due to the presence of a slow projectile can be taken into account by using λ equal to the Debye screening length.

c) SINGLE-ZETA: the single-zeta screening potential is used.

This potential describes the electrostatic interaction of a target-electron with projectiles carrying one or two electrons in hydrogen-like 1s orbitals. In a separate window, values have to be given for the effective projectile charge Z_eff "seen" by the projectile electrons in the 1s shell and for the number of bound projectile electrons np.

Note that Z_eff describes the shape of the projectile orbitals. It is related to 1/λ in the Yukawa/Bohr screening-potential and has nothing to do with the effective charge as it appears in some simple stopping power scalings. Obviously, for np = 1, Z_eff should be equal to the projectile charge Z_p if excited projectile states may be neglected (a single projectile electron in the ground state is attracted by the bare projectile nucleus). For np = 2, Z_eff = Z_p - 0.3 (Slater screening) is a reasonable estimate.

d1) GENERAL (USE DHFS IS NOT SELECTED): the general screening potential is used.

This potential is given by

V_p = -(Z_p-np) / r - np Σ (a_i + b_i r) exp(-c_i r)/r

consisting of a sum over generalized single-zeta potentials that can be used to fit any electrostatic screening function for projectiles carrying many bound electrons. In a separate window, values have to be given
1.for the total number of bound projectile electrons np
2.and for the name of an ASCII file (default is general.dat) containing the coefficients a_i, b_i and c_i described in the equation above.

Each line of the file corresponds to a set of the coefficients a_i, b_i and c_i (in atomic units). An example for the Si0 (neutral Si) projectile-screening function file (with 4 single-zeta functions and np= Z_p =14) given is below.

0.10853  -4.33903     10.43863
0.16517    4.83823    7.70347
0.49188    1.35201    7.47841
0.23442    0.48662    1.60316

Note that the sum over all coefficients a_i must be equal to one and the sum over all coefficients b_i should be positive for purely repulsive potentials. For neutral projectiles one may find tabulated potentials in the literature. For charged projectiles, however, one should use one of the atomic-structure codes to obtain a numerical potential (see, e.g., the optimized effective atomic potential code by J.D. Talman as it may be downloaded from the Computer Physics Communication library, CPC). For fast projectiles (E/M >> 100 keV/u) it is advised to exclude the electron-exchange part of the potential. Furthermore, the potential should be valid for external electrons (in our case: target electrons). Usually, the mean direct electron-electron interaction potential is given for an internal electron. One may simply scale this part of the potential by a factor np/(np-1) to derive the corresponding potential for an external electron. Note also that some of the codes give potential energies in Rydberg units (1 Ry = 13.6 eV), whereas one needs atomic units (1 a.u. = 27.2 eV = 2 Ry) for the fit.

A rough estimate of the potential may also be obtained from 1s-like orbitals and effective charges Z_eff_i for each shell i. The constants a_i are then equal to the ratio of the occupation number for each shell and np. The constants b_i are roughly given by b_i = a_i * Z_eff_i / n_i, where n_i is now the main quantum number. Finally, c_i = 2 * Z_eff_i / n_i.

The file with the coefficients must be placed in the same subdirectory where CasP is running.

d2) GENERAL (USE DHFS IS SELECTED): the general screening potential is used (see above) with an automatic replacement of all screening coefficients. The coefficients for neutral atoms (np= Z_p) are obtained from a Dirac-Hartree-Fock-Slater (DHFS) potential from F.Salvat et al., Phys.Rev. A36, 467 (1987). For charged projectiles, the coefficients c_i are replaced by c_i*Z_p/np (P.Sigmund, Phys.Rev. A56, 3781 (1997)). This option is usually sufficient for all non-equilibrium energy loss cases.

e) MEAN-CHARGE STATE: a Brandt-Kitagawa projectile screening potential is used.

This potential is obtained from the mean-charge state q_mean (q_mean = Z_p - np_mean) of the ion. q_mean stems from fits to experimental charge-state results for gases as well as for solid targets, including effects of the projectile-shell structure. The mean charge state of the projectile is then used to calculate the screening length λ of the ion, considering the condition of an electron distribution according to the Brandt and Kitagawa theory (ρ ~ exp(-c r)/r). This electron density is then used to compute the electrostatic screening function.

Since this approximation involves an averaged potential (instead of averaging the energy loss over each charge state according to the charge-state distribution), it is expected to be accurate only for highly charged projectiles carrying few electrons (q_mean >> 2 or q_mean > 0.8 Z_p). Furthermore, this potential includes only a single screening constant and thus, the projectile shell-structure is smeared out.

f) CHARGE-STATE SCAN: averages the energy loss according to an approximate projectile charge-state distribution for equilibrium stopping powers. The charge-state distribution is a Gaussian centered around q_mean with a standard deviation delta_q (limited to the 4 most important charge-state fractions). These parameters have been obtained from a fit to 1400 experimental data points for a large variety of projectiles and targets (solids and gases) at energies above about 20 keV/u (see Publications). Here we distinguish between solids (including liquids) and gases in their standard phases at room temperature. For each charge state method d2) is used to describe the projectile screening (scaled Dirac-Hartree-Fock-Slater potential). This option allows also to calculate the contribution to equilibrium stopping powers due to projectile-electron loss (and electron capture).