Single-scattering approximation

In the first Bom approximation, the interaction between the photons and the scattering system is weak and no excited states are involved in the elastic scattering process. Furthermore, there is no rescattering of the scattered wave, that is, the single-scattering approximation is valid. In the Feynman diagrams (Fig. 1.2), there is only one point of interaction for first-Born-approximation processes. 

Using the single scattering approximation and assuming that the atoms are very small (so that the incident wavefront is effectively plane) it is found that the oscillatory component of the EXAFS is given by 123,125... 

A model for simulating the extinction spectra during the aggregation process was developed in our work [20]. We assume that a cluster suspension is dilute so that the single scattering approximation is valid. This means that the... 

Thus, the amplitude of the secondary electron emission from the eore level within the secondary electron single-scattering approximation can be presented as the sum T (p) - - + Tqj p), where the direct emission into the final... 

The extended fine structure of the secondary electron spectrum is a result of secondary electron emission ftom an atom when the direct secondary electron emission interferes with the emission of secondary electrons scattered by the nearest neighbors of the ionized atom. In the previous section we have considered the amplitudes of these processes in the single-scattering approximation. 

Initially, we will focus on the mesoscopic description associated with the radiative transfer equation. Then, we will introduce the single-scattering approximation and two macroscopic approximations the PI approximation and two-flux approximation. AH of these discussions are based on the configuration shown in Fig. 6. Collimated emission and Lambertian emission wiU also be considered in the discussion later they correspond to the direct component and the diffuse component of solar radiation, respectively. Throughout our study, the biomass concentration Cx is homogeneous in the reaction volume V (assumption of perfect mixing), and the emission phenomena in V are negligible. The concentration Cx is selected close to the optimum for the operation of the photobioreactor the local photon absorption rate. 4 at the rear of the photobioreactor is close to the compensation point A.C (see Section 5 and chapter Industrial Photobioreactors and Scale-up Concepts by Pruvost et al.). 

Pa —4 3nd the single scattering approximation is used. The arrow indicates the part of the distribution that is due to the ballistic photons, ie, the arrow represents a Dirac distribution. This illustration does not allow for analysis of the ratio of ballistic to scattered photons, but we invite the reader to see Fig. 14. 

Implementation of the Single Scattering Approximation for an Equivalent Transport Problem Application to a Flat-Plate Photobioreactor... 

Fig. 13 presents the angular distribution of L resulting from the singlescattering approximation for the equivalent transport problem a, and Ph 4 as well as the reference solution produced by the Monte Carlo method for and k xt and the phase function of C. reinhardtii. In the reference situation, at the location in question (zq = 3 cm), the ballistic beam is completely attenuated all the photons have undergone at least one scattering event but deviated very htde from their incident direction (see Section 3.2). This situation results in a complex angular distribution centered around the incident direction (see Fig. 13A). In our equivalent transport problem, this complex distribution is replaced by the sum of a Dirac distribution (contribution of the ballistic photons, ie, 75% of the photons in the present case, see Fig. 14) and a relatively broad distribution (contribution of the scattered photons) that is simply modeled as Eqs. (62) and (63) under the single-scattering approximation (see Fig. 13B). The angular distribution of the scattered intensity at different locations is shown in Fig. 15.

The irradiance field obtained with the single-scattering approximation is shown in Fig. 14. Panel (A) shows the proportion of baUistic and scattered photons within the reaction volume. As indicated by the intermediate value of the transport optical thickness e = 1.1, the scattered photons are in the minority but cannot be disregarded. Fig. 14B shows comparison between... 

In contrast to the single-scattering approximation (previous paragraph), which is mesoscopic, below we adopt a macroscopic point of view. In this section and in Section 3.5, angular distribution of intensity is xed a priori, and the radiative transfer equation is integrated over all propagation directions in order to formulate a closed equation for the irradiance. 

The single-scattering approximation applied to the equivalent transport problem for analysis of coUimated illumination in the case of nonreflecting surfaces Eqs. (65) to (68). 

Under the single scattering approximation, the intensities of L3 and L2 lines scale in proportion to the specimen thickness, thus, their ratio L3/L2 has little... 

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