Integrated cross sections

Like the analogous equation for capillary condensation (Equation (3.74) Equation (3.81) is based on the tacit assumption that the pore is of constant cross-section. Integration of Equation (3.81) over the range of the mercury penetration curve gives an expression for the surface area -4(Hg) of the walls of all the pores which have been penetrated by the mercury ... 

Plots of each of these quantities as a function of particle size would look quite different and, therefore, would tell different stories. Except for a scale factor, each of them plotted as a function of wavelength for the same particle size would be identical. In our first example of extinction (Fig. 4.6) we displayed the efficiency Qext, as we shall often do in this chapter. In Chapter 12, however, our preference switches to the extinction cross section per unit particle volume. Unnormalized extinction cross sections (strictly speaking, the differential scattering cross section integrated over the acceptance angle of the detector) are more appropriate in Section 13.5 on particle sizing. 

A final aspect of these radiation collisions, shown experimentally by Thomson et a/.,17 is that the cross section integrated over all the collisional resonances increases with the rf field. Since the mth collisional resonance has a cross section of o = oR Jm KEmvJ/oj) and 2m J2m(x) = 1, in general, 2 om > crR.17... 

Photonuclear cross section, integrated, Kern-photoeffekt, integrierter Wirkungsquerschnitt 285. 

Definitions similar to Eqs. (79) (81) also hold for differential cross sections, integral cross sections, and rate constants in the BCRLM, except that each is obtained for every choice of bending states Aj and A2. 

The dlstorted wave Born-approximation (DWBA) is a quantum technique for calculating differential cross sections, integral cross sections and product state distributions for three dimensional (3D) chemical reactions of the type... 

The proportionality constant is the product of the biomass concentration Cx and the specific light absorption coefficient (i.e., the optical cross section). Integrating from the light-exposed surface (z = 0) to any location z then delivers an exponential decrease of the photon flux density over 2 ... 

The techniques discussed in Sects. 8.2-S.4 allowed the measurement of absolute rate constants of selected collision-induced transitions, which represent integral inelastic cross sections integrated over the angular distribution and averaged over the thermal velocity distribution of the collision partners. Much more detailed information on the interaction potential can be extracted from measured differential cross sections, obtained in crossed molecular beam experiments [958, 1074, 1075]. 

Here cr(a,k) stands for the total cross section. Integration over the thickness x of the target gives... 

The detector used for the detection of y radiation in NRA is generally placed either 135° or 90° to the direction of the incident beam. Higher count rates can be obtained at forward directions. Nuclear physics experiments often use a detector angle of 55° to obtain a measured yield proportional to that expected from the reaction cross section integrated over 4n angles. This geometry is suitable for the absolute determination of elemental concentrations from published total cross-section values, but it has no advantage when reference samples are used. 

Normally, one assumes that the differential reaction cross-section does not depend on the azimuthal angle. Hence, one takes advantage of the axial symmetry of the collision and works with differential cross-sections integrated over . Thus, one uses the element of sohd angle dm = 27t sin 0 dO ... 

G.R. Martin, J.L. Smoot, and K.D. White. A comparison of surface-grab and cross sectionally integrated stream-water-quality sampling methods. Water Environ. Res., 64 866-876,1992. 

With the above substitutions, the equilibrium relations of Eqs. (8.36b) and (8.36d) as well as Eqs. (8.36c) and (8.36e) may be combined such that each resulting equation contains the axial derivative of one of the sought inclination angles of the cross-section. Integration in consideration of the clamped boundary conditions at one end, then yields... 

Model III relies on constant eddy viscosity (which is a function of gas velocity and column diameter) but predicts the power law form of the gas holdup profile. It accomplishes that by using both gas and liquid continuity equations, and be establishing that the cross-sectional integral of the slip velocity is a constant multiple of the superficial gas velocity. The agreement of model III with the velocity data of Hills is excellent as is its prediction of Hills s measured voidage profiles. 

The sign minus reflects the fact that the flux density of the particles decreases with increasing x. The proportionality coefficient Go, which depends on the collision velocity v, is named the total collision cross section. Integrating (2.1.), we obtain... 

An analogous effect occurs in the integral cross sections, integrated over all scattering angles. Fig. 11 predicts peaks in the... 

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