Energy mean excitation

Since reaction may also yield electronically-excited products when AG° is sufficiently negative, we include this reaction, as we did in (20). The mean excitation energy used for the formation of the electronically-excited Ru(III) product is 1.76 eV (20). As has been explained elsewhere (20, 28), the formation of the other possible electronically excited products is, in most cases at least, less probable. The same V(r) was... 

More interesting, perhaps, is the observation that St(c) is very close to the total stopping cross section, This would imply that in the Bethe formulation, the mean excitation energy associated with the random orientation and that associated... 

The orbital mean excitation energies, ly given in equation (3) are defined by requiring that the total mean excitation energy is written as... 

One of the first consequences of the above ideas was the development of the Orbital Local Plasma Approximation (OLPA) by Meltzer et al. [37-39]. The main ingredients in the OLPA consist in approximating the orbital weight factors by the orbital occupation numbers and adapting the Lindhard-Scharff Local Plasma Approximation (LPA) [10-12] to an orbital scheme whereby the orbital mean excitation energy was originally defined as [37,38]... 

Calculation of OLPA-TFD(l/8)W orbital and total mean excitation energies... 

Using the obtained values of the orbital parameters contained in Table 1 and those reported in Ref. [52] for all atoms with 2corresponding orbital and total mean excitation energies using equations (5), (9), (28), and (29). Table 2 shows the results calculated here (TFDW) as compared to those by Oddershede and Sabin (OS) and Meltzer, Sabin, and Trickey (MST). In general, the Is, 2s, 3s, and 3p orbital mean excitation energies obtained through the TFDW method are closer to the OS values as compared to the MST estimates. Major differences are observed for the 2p, 3d, 4s, and 4p orbitals, with the MST values closer to those by OS except for the 4s orbital for atoms from Zn to Kr. 

For purposes of comparison with the other calculations, the set of calculated orbital mean excitation energies shown in Table 1 were grouped in shell-hke contributions as suggested by Chen et al. [62], i.e.. 

Figure 5 shows the mean excitation energies of the K, L, and M shells as a function of atomic number obtained in this work (TFD(1/8)W) (solid circles) as compared to calculations by Oddershede and Sabin (OS) [33] (open squares) and the original OLPA method developed by MST [37,38] based on the LSDA-DFT (open diamonds). It seems clear from this figure that the predictions of the TFD-(l/8)W-based OLPA method for the shell (and orbital) mean excitation energies are in reasonable agreement with the other - more elaborate - calculations. We note that all the OLPA-based calculations use y, = 1. 

Table 2. Orbital (4) and total (/) mean excitation energies (in eV) for atoms with 2[Pg.350]

Fig. 5. Mean excitation energy (eV) of the K, L, M shells and 4s p orbitals for He through Kr obtained through equation (31) and 4 values from Table 2. ( ) this work ( ) Oddershede and Sabin [33] (O) Meltzer et al [37,38].

Using the orbital implementation of the kinetic theory together with the derived TFDW mean excitation energies obtained in the previous section, we have calculated the proton stopping as a function of projectile velocity for a selected number of elemental target materials in the gas phase (O, Ar, and Br) and in the condensed phase (Si, Ni, and Ga), as an example. Figures 7 and 8 show the results of this calculation (continuous curves) as compared with the predictions by OS [33] (solid circles) and with the corresponding empirical fit to experimental values... 

So far we have given evidence for the flexibility and adequacy of the orbital implementation of the KT by Oddershede and Sabin to study atomic and molecular stopping through different approaches. A major breakthrough based on this theory is the OLPA put forward by Meltzer et al., since it incorporates the simplicity - yet physically sound basis - of the LPA for the calculation of mean excitation energies. 

Table 3. Optimized orbital exponents orbital (4), and total (/) mean excitation energies of Si for confinement radii R (a.u.) = 4,6, and o°... 

The orbital implementation of the KT has established a firm theoretieal basis for the CAB formalism whereby ehemieal binding effects on proton stopping cross sections may be estimated. Furthermore, we have given evidence on the flexibility of this theory to allow for the incorporation of alternative descriptions of orbital and total mean excitation energies - such as the OLPA scheme - which may be adapted for the study of energy-loss problems in matter under different conditions (i.e., gases, solids, and matter under high pressure). 

Finally - and equally important - Jens contribution to the formal treatment of GOS based on the polarization propagator method and Bethe sum rules has been shown to provide a correct quantum description of the excitation spectra and momentum transfer in the study of the stopping cross section within the Bethe-Bloch theory. Of particular interest is the correct description of the mean excitation energy within the polarization propagator for atomic and molecular compounds. This motivated the study of the GOS in the RPA approximation and in the presence of a static electromagnetic field to ensure the validity of the sum rules. 

The parameters in Eq. (1) illustrate the primary dependence of stopping power on the square of the projectile charge z, the velocity of the projectile /i, and the density of target electrons Z/A. The mean excitation energy 7, a major target parameter, is incorporated in the first term of L fi) given as... 

Equation (3) incorporates relativistic effects, effects of target density, and corrections to account for binding of inner-shell electrons, as well as the mean excitation energy C/Z is determined from the shell corrections, S/2 is the density correction, Ifj accounts for the maximum energy that can be transferred in a single collision with a free electron, m/M is the ratio of the electron mass to the projectile mass, and mc is the electron rest energy. If the value in the bracket in Eq. (4) is set to unity, the maximum energy transfer for protons... 

In the local response model each electron density volume element is separately characterized by a two-parameter formula giving electric dipole oscillator strength as a function of frequency [12]. One of the two parameters is fixed by the oscillator strength sum rule, while the other is an effective mean excitation energy, taken to be the plasma energy huip by Andersson et al [9]. This model requires introduction of a low-density cutoff of the dipole response, because a... 

The coefficients ArS are Franck-Condon-type factors for an irregular system they can be taken as random numbers multiplied by a Gaussian function with spectral width AE centered around the mean excitation energy. 

Figure 32 Dissociation rates of rotationally cold NO2 as function of the mean excitation energy. Thick solid line phase space theory (PST). The two vertical dashed lines are reaction thresholds for NO2 starting in its lowest rotational level (i.e., Do) and its first excited rotational level. Reprinted, with permission of the American Institute of Physics, from Ref. 275.

A = mean excitation energy a = average isotropic polarizability r = unit vector along the distance between point dipoles. 

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