Effective solid angle

The effective solid angle of the collected Raman radiation of a sample in a rectangular cell is only about 40 % of the solid angle of a suitable high-aperture optical system (Fig. 3.5-6 a, b). 

Figure 3.5-6 Arrangement of a, a rectangular cell and b, a spherical cell at the focus of the entrance lens of a Raman spectrometer. The effective solid angle is by a factor of 2.5 larger for a spherical cell compared to the rectangular cell. FR focal range, SM surface mirror.

In these equations and hereafter we use simplified symbols for quantities referring to the P-branch line v, 7—l- c—I, /. Namely, Xvy instead of Xu,y-i etc. The second term in (2) represents the rate of spontaneous emission into the oscillating cavity modes. 1 is approximately the effective solid angle subtended by the mirrors after several reflections. (Alternatively, e is the fraction of stable transverse modes.) After threshold eS j is negligible, eS j- Xvj- i i i " i important only before threshold —as a source of noise photons to trigger-on the lasing process. The spontaneous emission terms in (1) are given by =A j+iN y, A j is the Einstein coefficient. In infrared lasers where typically A ) 10s. S, ... 

Otot(x,y) IS the effective solid angle subtended by the seven Pmts at the (x,y) event position... 

In Chapter 6, I discussed the effect of sample geometry on count rate. Once again, it would be useful to consider a practical example. Table 8.4 lists the peak areas measured when the same amount of Eu was counted as a point source and when distributed in water and in sand. The distributed sources were 13 mm in diameter and 20 mm high and measured on the cap of a 45 % p-type HPGe detector. As one would expect, there is an obvious overall loss of count rate due to the lower effective solid angle of the distributed sources and a more pronounced loss of count rate in the low energy peaks. 

Under carefully controlled conditions, wavenumber measurements may be precise to 0.01 cm (discussed below), but usually only when a sample is left undisturbed in the sample compartment. Even if the sample is simply removed and reinserted between measurements, the repeatability is often worse than 0.01 cm . Several reasons can be advanced to explain why band shifts occur. First, the temperature of the sample may change between measurements, which leads to small spectral shifts. Second, it was noted in Section 2.6 that changes in the effective solid angle of the beam through the interferometer can lead to small wavenumber shifts. Because the cell may represent a field (Jacquinot) stop, if a cell is not placed in exactly the same position for successive measurements, bands will appear to shift from one measurement to the next. Furthermore, if the cell is slightly tilted and the angle changes appreciably from one measurement to the next, the beam may be refracted to a different position on the detector, which also shifts the wavenumber scale. Loose or insecure sample mounts should be avoided if users require the wavenumbers of absorption band maxima to be repeatable to better than 0.1 cm . ... 

The attainable particle current density per solid angle of the beam (ions pm s sr ) is an inherent property of the ion source, the so-called brightness. Because of this, reduction of the beam diameter is effected by reducing the beam current. 

Dose is related to the amount of radiation energy absorbed by people or equipment. If the radiation comes from a small volume compared with the exposure distance, it is idealized as a point source (Figure 8.3-4). Radiation source, S, emits particles at a constant rate equally in all directions (isotropic). The number of particles that impact the area is S t Tr where Tr is a geometric effect that corrects for the spreading of the radiation according to ratio of the area exposed to the area of a sphere at this distance i.e. the solid angle - subtended by the receptor (equation 8.3-4). 

FIG. 7 Effective contact angle of the aqueous KOH droplets on HOPG and mica as a function of droplet height. Solid lines correspond to fits obtained using the disjoining pressure given by Eq. (18). 

Consider continuous radiation with specific intensity I incident normally on a uniform slab with a source function 5 = Bv(Tex) unit volume per unit solid angle to the volume absorption coefficient Kp and is equal to the Planck function Bv of an excitation temperature Tcx obtained by force-fitting the ratio of upper to lower state atomic level populations to the Boltzmann formula, Eq. (3.4). For the interstellar medium at optical and UV wavelengths, effectively S = 0. 

One simple way of quantifying steric effects in actinide complexes [the cone angle factor (caf) approach] uses the sum of the solid angles subtended by the ligands to the centre of the metal... 

Guezi, I.A. and Wendt, M. (2006) An improved method for the computation of ligand steric effects based on solid angles. Dalton Trans., 3991. 

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