Electron Scattering Cross Sections Results and Discussion: : Comparison with theoretical and empirical models
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Page 3 of 3 3.3 Comparison with theoretical and empirical models Joshipura and Vinodkumar30, 31 analytically presented the total electron scattering cross section as a function of electron energy using a two parameter formula for 10-electron system molecules H2O, NH3, and CH4 for energies higher than 100 eV. In this expression the total cross section (
where E is the electron energy in keV, A and B are parameters that depend on the molecular properties of the target gas, and ao is the Bohr radius. In order to compare the present cross sections with this analytical formalism, logarithmic values of present experimental cross sections are scaled as a function of logarithmic energy and given in Figures 3.08 and 3.09 for CH4 and NH3 respectively. In the same figures the cross sections produced in other laboratories are given for comparison. It is evident from these figures that the present cross sections, as well as those of Garcia and Manero17, follow the general trend of equation (3.1) for the entire energy range. However, the cross sections reported by Zecca et al.16, 21 fall below the general trend of equation (3.1) at energies higher than about 1000 eV. Using the present experimental cross sections in the form given in equation (3.1), the two parameters A and B are found to be 1.80 and 0.80 for CH4, and 1.60 and 0.79 for NH3. To compare the present cross sections produced by SiH4 and PH3 molecules with the analytical expression of Joshipura and Vinodkumar30, 31, values of present experimental cross sections were scaled as a function of logarithmic energy and given in Figures 3.10 and 3.11 for SiH4 and PH3, respectively. It is clear from these figures that the present cross sections are in fair agreement with the general trend of equation (3.1). Using the present experimental cross sections in the form given in the equation (3.1), the two parameters A and B are found to be, respectively, 11.0 and 0.72 for SiH4, and 10.7 and 0.71 for PH3. Garcia and Manero19 studied the energy dependence of the total cross sections in their experimental measurements using the analytical expression by Joshipura and Vinodkumar30, 31. They found their measurements agree with the general trend predicted by the analytical expression (3.1). The fitting parameters A and B were obtained from their experimental results for several molecules. They found the B parameter takes a similar value for all of the studied molecules. The average value of this parameter is 0.78 ± 0.02. Using this observation, they proposed an empirical expression for cross section at intermediate electron energies for molecules with 10 - 22 electrons. In this empirical expression cross section is related to energy E (in keV),
where Z is the number of electrons in the target molecule, a is the polarizability of the target molecules, and Figure 3.08 The variation of total cross sections with energy for CH4 in log-log scale. The circles are the present measurements. The squares and triangles are, respectively, the measurement by Zecca et al.16 and Garcia and Manero17. The solid line is the general trend predicted by Joshipura and Vinodkumar30, 31.
Figure 3.09 The variation of TCS with energy for NH3 in log-log scale. The circles are the present measurements. The squares and triangles are, respectively, the measurement by Garcia and Manero22 and Zecca et al.21. The solid line is the general trend predicted by Joshipura and Vinodkumar30, 31.
Figure 3.10 The variation of TCS with energy for SiH4 in log-log scale. The circles are the present measurements. The squares and triangles are, respectively, the measurements by Zecca, Karwasz, and Brusa 21 and Sueoka, Mori, and Hamada23. The solid line is the general trend predicted by Joshipura and Vinodkumar30, 31.
Figure 3.11 The variation of TCS with energy for PH3 in log-log scale. The circles are the present measurements. The solid line is the general trend predicted by Joshipura and Vinodkumar30, 31. The Bethe theory defines the total inelastic cross section, and the Born approximation gives the total elastic cross section. The combined Bethe-Born32 theory gives an asymptotic formula for the total scattering cross section of fast moving charged particles. This theory is characterized by a few parameters that permit evaluation of where E is the incident energy in eV, R is the Rydberg energy,
and provided the values of the constants Table 3.2 Jain and Baluja24 calculated values of the constants
Figure 3.12 Total electron scattering cross sections of CH4 in 10-20 m2. The circles are the present measurements. The crosses and triangles are, respectively, the measurement by Zecca et al.16 and Garcia and Manero17. The solid line and the dashed line are, respectively, the theoretical predictions by Jain and Baluja24 (Bethe-Born theory) and predictions by an empirical model of Garcia and Manero19.
Figure 3.13 Total electron scattering cross sections of NH3 in 10-20 m2. The circles are the present measurements. The squares and triangles are, respectively, the measurement by Garcia and Manero22 and Zecca et al.21. The solid line and the dashed line are, respectively, predictions by an empirical model of Garcia and Manero19 and the theoretical predictions by Jain and Baluja24 (Bethe-Born theory).
Figure 3.14 Total electron scattering cross sections of SiH4 in 10-20 m2. The circles are the present measurements. The squares and triangles are, respectively, the measurements by Zecca, Karwasz, and Brusa 21 and Sueoka, Mori, and Hamada23. The solid line is the predicted by an empirical model of Garcia and Manero19. The dotted line is the theoretical prediction by Jain and Baluja24 (Bethe-Born theory).
Figure 3.15 Total electron scattering cross sections of PH3 in 10-20 m2. The circles are the present measurements. The solid line and the dashed line are, respectively, the theoretical prediction by Jain and Baluja24 (Bethe-Born theory) and the prediction by an empirical model of Garcia and Manero19. From these figures it is evident that the theoretical prediction made in the Bethe-Born approximation (SCOP method) was lower than the current experimental cross sections. For CH4 molecules, the deviation was 19% at lower energies and systematically increased with increasing energy up to 29% at 4000 eV. For NH3 molecules, the deviation was 15% at lower energies and systematically increased with increasing energy up to 24% at 4000 eV. However, for SiH4 there is closer agreement between the Bethe-Born predictions and the experimental observation at energies below 300 eV. But for energies higher than 300 eV, Bethe-Born predictions are 15% - 24% lower than the experimental observations. For PH3 molecules, the highest deviation of 60% occurs at 100 eV and energies between 500 – 2000 eV showed a 15% deviation that increased up to 23% at 3500 eV. From these comparisons, the present total cross section measurements and the measurements by other experimental groups have shown that the Bethe-Born approximation (SCOP method) generally under-predicts the total scattering cross sections. Next, the general formalism of equation (3.4) is used to compare the present cross section with the Bethe-Born theory in the form of the Bethe plot: E Figure 3.16 The Bethe plot of CH4 cross sections. The circles are the present measurements. The squares and triangles are, respectively, the measurement by Garcia and Manero17 and Zecca et al.16. Figure 3.17 The Bethe plot of NH3 cross sections. The circles are the present measurements. The squares and triangles are, respectively, the measurement by Garcia and Manero22 and Zecca et al.21. Figure 3.18 The Bethe plot of SiH4 cross sections. The circles are the present measurements. The squares and triangles are, respectively, the measurement by Zecca, Karwasz, and Brusa 21 and Sueoka, Mori, and Hamada23. It is reasonable to predict some sort of pattern between the cross section ratios CH4:NH3 and SiH4:PH3 because the number of electrons in CH4 is the same as that in NH3, while the number of electrons in SiH4 is the same as that in PH3. Further, the molecular symmetry of SiH4 is identical to that of CH4, while the molecular symmetry of PH3 is identical to NH3. In order to investigate similarities between the cross sections of these two pairs of molecules, we have normalized the cross sections of CH4 and NH3 molecules, and the cross sections of SiH4 and PH3 molecules. These are displayed in Figures 3.20 and 3.21. It is clear from Figure 3.20 that the CH4 molecule has a higher cross section than the NH3 molecule over the entire energy range. Further, the ratio of cross sections between these two molecules is independent of the energy. In contrast, as can be seen from Figure 3.21 the ratio of cross sections between SiH4 and PH3 depends on energy. At energies below 1000 eV, the SiH4 cross section is greater than that of NH3, while at higher energies both molecules have the same cross section. Figure 3.20 Cross sections of the CH4 molecule normalized to the cross section of the NH3 molecule with the associated error level. Straight solid line is the cross sections of NH3 molecule and diamonds are the cross section of CH4 molecule.
Figure 3.21 Cross sections of the SiH4 molecule normalized to the cross section of the PH3 molecule with the associated error level. Straight solid line is the cross sections of PH3 molecule and diamonds are the cross section of SiH4 molecule. | |||||||||||||||||||||||||
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