Inclusive Cross Section

8 Cross Section Measurement 4.8.1 Inclusive Cross Section 

The inclusive fc-quark production cross-section a is calculated according to the formula 

The efflciency of the trigger and offline selection is obtained from the MC simulation as 

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According to Baltz et al (2001) the inclusive cross-section (summing over all numbers of pairs) can be formulated in terms of the retarded electron propagator and... 

Figure 11.27 Schematic diagrams illustrating the distribution of inclusions (cross-sectional view) in composites formed from (a) a mixture of the inclusions and powder matrix prepared by ball-milling and (b) coated inclusions prepared by chemical precipitation.

Finally, we note that in the literature the inclusive cross-section (15.3.16) for eN e X is sometimes written in terms of and a ... 

The inclusive cross-section will be given by a formula closely analogous to (15.3.1) in which the following replacements must be made ... 

As we turn up the interaction, EIMP becomes important. We must certainly modify our treatment of the K-shell electron and treat its excitation to all orders in the Born series. However, now we have to worry about the role that the other electrons play. Should we not introduce Pauli blocking operators in the Bom series Now we can make a hole in the L-shell, should we not allow this as a possible final state for the K-shell electron The answer to these questions is yes, if we wish to calculate exdusive cross sections, in perturbation theory. However it is not necessary to use many-electron perturbation theory, and we are only interested in an inclusive cross section. 

Because the projectile provides a perturbation which is treated as a sum of single particle perturbations, each orbital electronic wave function, develops independently from the others the correct description of the system is given by forming an appropriate antisymmetrized product of these time dependent orbitals. To calculate the required cross section we have to project onto all multielectron states that have a K-shell vacancy, and sum the resulting probabilities. The simple single electron theorem results. There is no role played by the passive electrons in the IPM except in defining the initial Fermi sea. For example, to attempt to include statistical correlation in a classical calculation of an inclusive cross section would be unproductive. 

Fig. 9. Cucurbituril inclusion chemistry (a) tridimensional stmcture of cucurbit[6]uril (b) conjectured cross-sectional representation of a host—guest...

The analysis given in Macbeth (M4) continues by assuming that the term dh effectively represents the cross-sectional geometry of the normal rod-bundle data. If this assumption is correct, then the general correlation [Eq. (18)], may be applied by representing A and C as functions of G and dh for a given pressure. It was found that simple power functions were adequate, and a correlation was obtained by computer optimization which predicted 97% of the vertical-upflow normal data (Nos. 6-15 inclusive in Table V) to within 12%. 

Fig. 29. Schematic representation of the longitudinal cross-section of the inclusion channel for the simple alcohol inclusions of 1 with MeOH, EtOH, and 2-PrOH 2). Hatched triangles and dotted squares represent polar areas (cf. Fig. 19, type Ila), while the rest is of apolar property...

Fig. 31. Approximation of van der Waals cross-sections of inclusion channels in 1 alcohol clathrates21 (dimensions are in A hatched regions represent O atoms of the host matrix continous solid lines indicate surfaces of apolar attribute) (a) 1 MeOH (1 2) (approximately parallel to the 0(I -Cul vectors, cf. Fig. 17a) (b) 1 2-PrOH (1 2) (orientation as before) (c) 1 2-BuOH (1 1) (through a center of symmetry at 1,1/2,1/2, cf. Fig. 30c non-zero electron density contours) (d) 1 ethylene glycol (1 1) (in the plane of the C—C single bonds of a guest molecule, indicated by projected stick models non-zero electron density contours)...

Thiourea canal inclusion compounds 19 26) have a wider diameter than those formed by urea, such that n-alkanes are not included but that molecules of cross-section approximately 5.8-6.8 A are trapped 64). Thus many inclusion compounds have been reported between thiourea and branched alkanes or cyclic molecules. Of special interest are the inclusion compounds with cyclohexane derivatives and the recent studies carried out on the preferred conformation(s) of the ring in the restricted environment of the thiourea canal. 

Fig. 14. The Vh-amylose polyiodide inclusion compound shown as a simplified projection on the a, b plane. Guest water molecules are shown as filled circles in the interstitial sites. The guest polyiodide chains are situated inside the helical amylose chains and their cross-sections are shown here as stippled circles...

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. 

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