Solid-angle corrections

The result states that the convolution of two Lorentzian functions with the width parameters 5t and S2 yields again a Lorentzian function with 5 = 5 + S2. 

Starting with the parametrization for two-electron emission given in equ. (4.68), and using equ. (4.67) which relates the bipolar spherical harmonics to spherical harmonics attached to the actual directions ka = a, pa and Kb = b, pb of electron emission, the triple differential cross section for two-electron emission can be rewritten as 

The observed intensity C(Ka, Kb) of true coincidences is then characterized by the settings of both spectrometers at Ka = a, a and Kb = b, Db, respectively, where these angles refer to the principal rays. Assuming constant detection efficiencies, the intensity C(Ka, Kb) then follows from an integration over the corresponding acceptance angles  

Each of the two spherical harmonics refers to the laboratory frame, but with the transformations into the detector frames described in equ. (8.92a), one gets1  

These solid-angle factors Qkq can be calculated easily because all quantities refer to the detector frame alone. In particular, for k = 0 one gets 

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The amplitude of the signal, A, is also determined. It depends on spins, multipolarities, and eventually on mixing ratios of the transitions involved, that is, entirely on nuclear parameters, and thus is a well-known quantity in practice, it is further reduced by solid angle correction factors. If more than one structure at the site of the PAC probe is present in the sample, the relative amplitudes will, in general, directly reflect the relative population of the different sites. 

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). 

The equilibrium constants can be approximated by ratios of ion currents in some instances otherwise, the currents are converted to partial pressures by comparison with the evaporation of known amounts of a standard material. Various geometric corrections (K) such as the solid angle subtended by the sample at the orifice, the Clausing factor for orifice geometry, molecular cross-section (o-), which control ionization efficiency, and detector efficiency are included in the general relationship... 

Figure 1.8 Example of the spectral flux Ny of the undulator/wiggler radiation measured at the X-Al beam line at NSLS (National Synchrotron Light Source, Brookhaven National Laboratory, LISA) with the undulator parameters K = 1.50, X = 8 cm, N = 35, for a 500 mA beam current, with a 0.1% bandpass and a solid angle of 1 mrad2. The values are corrected for the beamline/monochromator efficiency and the photodiode detector response the dip at 4.4 nm is an artifact due to carbon contamination of the optical elements. From [BRA89], (Reproduced with permission from Review of Scientific Instruments.)...

The radiant power P in watts (W) is the energy of a beam that reaches a given area per unit time. The intensity is the radiant power-per-unit solid angle. Both quantities are proportional to the square of the amplitude of the electric field (see Figure 24-lb). Although it is not strictly correct, radiant power and intensity are frequently used interchangeably. 

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