Differential oscillator strengths

The parameter e0 was chosen for best agreement with the experimental data of Opal et al.52 at = 500 eV. Jain and Khare applied this equation to the calculation of ionization cross sections for C02, CO, HzO, CH4, and NH3 and achieved fairly good agreement with experiment for all cases except for CO, where the cross section was too low, though the ionization efficiency curve still exhibited the correct shape. The main limitation of this method, which it has in common with the BED theory, is the inclusion of the differential oscillator strengths for the target molecule which restricts the number of systems to which it can be applied. 

FIGURE 2.7 Differential oscillator strength distribution (DOSD) in the vapor and liquid phases of water. Reproduced from La Verne and Mozumder (1986), with permission of Am. Chem. Soc. ... 

Dipole oscillator strengths form important input into all stopping models based on Bethe or Bohr theory. Emphasis has frequently been on total /-values which show only little sensitivity to the specific input. More important are differential oscillator-strength spectra, in particular at projectile speeds where inner-shell excitation channels are closed. Spectra bundled into principal or subshells [60] are sufficient for many purposes, but the best available tabulations are based on analysis of optical data rather than on theory, and such data are unavailable for numerous elements and compounds [61]. 

Figure 11 A Platzman plot, the ratio of the experimental single differential cross section for electron emission from helium by 1-MeV protons to the corresponding Rutherford cross sections plotted as a function of RjE. The experimental cross sections are from Ref. 54 and the differential oscillator strength is taken from Ref. 43.

The property of a molecule which gives the most information about the electronic continuum states is the differential oscillator strength in the continuum absorption region, which is defined by... 

Fig. 30. The total differential oscillator strength for benzene including the structure factor but neglecting any vibrational effects. The cross-hatched portion of the figure represents the transition to a continuum orbital of e2ll symmetry, while the remainder is for a transition to an elu orbital. The positions of higher ionization thresholds are indicated.

Fig. 31. The total differential oscillator strength for benzene calculated without including the structure factor.

For clarity of notation in discussion, we now denote the differential oscillator strength df0m(0)/d for the ionization process (0 - m) by df(0)/d as given in 2.18). The notation of the type df0m(0) is used in Sections H.C and II.D since this is more commonly employed in scattering theory. 

All these quantities are equivalent since they all depend on the same transition matrix element, although their units are not the same. The / value has the advantage of being a dimensionless quantity. With broad band illumination, the appropriate quantities are those which are integrated over the spectral feature, such as the / value or the Einstein coefficient. With narrow band illumination (i.e. a monochromatic source narrower than the spectral feature), it is appropriate to use a quantity which is defined point by point within the line profile, such as the absorption coefficient, the cross section, or the differential oscillator strength df/dE. 

Fig. 4.2. Plot of the differential oscillator strength df/dE as a function of energy for the principal series of Na (experimental data) each box represents one transition, arranged (see the previous figure) so that the df/dE graph has no gaps. A smooth curve then connects the points shown, and should, in this example, tend to a straight line. Since the measurements were made in the presence of a magnetic field, the very high Rydberg members appear to lose intensity because satellite lines emerge, as explained in chapter 10 (after M. Nawaz et al [166]).

Fig. 4.11. Differential oscillator strength plot for Sr, showing measurements for high Rydberg members up to n = 28 (note this is the same Rydberg series as shown in fig. 2.2 - after J.-P. Connerade et al. [163]).

Figure 7.18 Differential oscillator strengths obtained from band and continuum measurements of the v" = 0 — 1 progressions of the B3S — X3SJ system of O2 near the dissociaton limit (d.l.). Continuity is observed across the limit (from Lewis, et al., 198 ).

The quantity df /dQ, called the differential oscillator strength, is simply proportional to the optical absorption coefficient, e. Since the logarithmic term varies only slowly with g, we see that the probability for resonant transitions is roughly proportional to (dfldQ)/Q or to s hv)/hv. The transitions described by equation 2 are called optical or resonant transitions. 

The differential oscillator strength distributions have been determined in detail and over a wide range of energy for a number of molecules in the gas phase. An important feature of these df /dQ distributions is that a maximum is always found around twice the ionization potential, i.e. around 20 eV for hydrocarbons. In Figure 1 we show the... 

In the equation the quantity df/dQ is the so-called differential oscillation strength. dfldQ is proportional to the optical absorption coefficient. The logarithmic term varies only slowly with Q, therefore the probability of resonant transition is roughly proportional to the dfldQ)/Q ratio, i.e., it is proportional to the ratio of optical absorption coefficient and the transition energy. If the optical absorption spectrum is accurately known, this equation gives a possibility for the calculation of the transition probabilities and also the yields of the different excited molecules (Hatano 1968, 1999 Makarov and Polak 1970 Kouchi and Hatano 2004). 

It is not so obvious how well Born-Bethe ideas are satisfied in describing excitations in solids with their extended band states. However, the structure of the basic result (eq. (3)) with the partitioning into a Rutherford cross section and a form factor depending on momentum transfer remains intact. All that is required is for the differential oscillator strength to be reinterpreted in terms of a response function for the system e(q,AE), the Fourier transform of the space/time dielectric response e(r,t). 

Insights into which electrons in the solid are contributing to which features can be gained by examining the differential oscillator strength d/(0, A )/d(A ). Integrating with respect to A gives... 

The total differential oscillator strength for transitions from the initial bound state into all channels is ... 

Some of the above approximations can be removed from future calculations. The full treatment of NON can be realized numerically. Actually, in our code [63] for differential oscillator strength, NON is fully treated following the formalism of King et al. [64,65]. Also, the orthogonality between (j> and V oE can be checked and evaluated by modifying the current code [61]. 

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