Pair production coefficient

The probability for pair production to occur, called the pair production coefficient or cross section is a complicated function of E and Z (see Evans and Roy Reed). It may be written in the form... 

Figure 5.4 Pair-production coefficient for lead as a function of...

Absorption Coefficient—Fractional absorption of the energy of an unscattered beam of x- or gamma-radiation per unit thickness (linear absorption coefficient), per unit mass (mass absorption coefficient), or per atom (atomic absorption coefficient) of absorber, due to transfer of energy to the absorber. The total absorption coefficient is the sum of individual energy absorption processes (see Compton Effect, Photoelectric Effect, and Pair Production). 

Figure 5 X-ray mass attenuation coefficients for aluminum as a function of photon energy. At low energies, photoelectric absorption predominates. At higher energy, incoherent (Compton) scatter becomes almost the exclusive contributing mode. Eventually, pair production dominates at very high energies (above 10 MeV).

Figure 7 The atomic number dependence of attenuation in different energy domains. This is illustrated by how the mass attenuation coefficient of the predominating mode (photoelectric (PE) at 50 keV, incoherent (Inc) at 500 keVand pair production (PP) at 5 MeV) varies with Z.

If [X] is the concentration of AH in the aqueous phase (Figure 8), the concentration of AH in the octanol is P times this. The aqueous-phase concentration of the ion is [X] times the degree of dissociation. Multiplying this product by the ion-pair partition coefficient Pj (or P -) gives the concentration of the ion pair in the octanol. The actual amount of species in a phase is given by its concentration times the volume of the phase. At the pK, the amount of neutral species equals the amount of ionized species. Setting the sum of the two terms in the left quadrants equal to the sum of the two terms on the right, one can derive Equation 20. The equivalent e ression for bases is Equation 21. 

It should be noted that the nuclear spin states of the pairs (12) may be identified with those of the pair products c which were previously denoted K and L in cases of simple first-order NMR spectra of the products. In other cases, the coefficients of the expansion of the spin functions Xjc of the products in terms of the Xfc... 

In Fig. 5, the relationship between the Townsend ionization coefficients (ion pair production) for air at atmospheric pressure at different temperatures and field strengths is given. 

Attenuation of 7 Radiations. When 7 radiations pass through the absorber medium, they undergo one or a combination of the above three processes (photoelectric, Compton, and pair production) depending on their energy, or they are transmitted out of the absorber without any interaction. The combined effect of the three processes is called the attenuation of the 7 radiations (Fig. 1.9). For a 7 radiation passing through an absorber, the linear attenuation coefficient (fie) of the 7 radiation is given by... 

Figure 1.10. Linear attenuation coefficient of 7 rays of different energies in water (equivalent to body tissue). The relative contributions of photoelectric, Compton scattering, and pair production processes are illustrated.

The linear attenuation coefficient is the sum of the probabilities of interaction per unit path length by each of the three scattering and absorption processes photoelectric effect, Compton effect, and electron-positron pair production. The reciprocal of p is defined as the mean-free path, which is the average distance the photon travels in an absorber before an interaction takes place. 

Fig. 1(a). Mass attenuation coefficient of X-ray and 7-photons in water, aluminium, iron and lead [2], (Note the relatively small dependence on chemical structure except at low energies, due to the photoelectric effect and, at high energies, to pair production, (b) Total energy absorption due to photoelectric effect, Compton scattering and pair production [2],... 

In the case of photoelectric effect, assumption (1) is valid. For pair production, however, it is questionable because only the ener r - 1.022 MeV is given to the electron-positron pair. The rest of the energy, equal to 1.022 MeV, is taken by the two annihilation gammas, and it may not be deposited in the medium. There are cases when Eq. 4.65 is modified to account for this effect. Gamma absorption coefficients, as defined by Eq. 4.65, are given in App. D. 

Gamma-ray absorption occurs as illustrated in Figure 6.16 by four different processes coherent scattering, photoelectric effect, Compton effect, and pair production. For each of these processes, a partial coefficient can be expressed ... 

It was shown previously that the log( 1 //f) spectra of many agricultural products and food can be reproduced from the first 50 pair of Fourier coefficients including the mean term [4,5]. It seems logical to assume that if these coefficients can be used to reproduce the original spectra with such accuracy composition measurements could be made with those same coefficients. We will now demonstrate that calibration equations can be developed from the first six pairs of coefficients with no loss in accuracy. 

Attenuation by scattering or pair-production processes is not treated at all in the present Volume. Attenuation by ionization of inner shells is treated in the Section on Ionization Processes and X-Ray Production , pp. 217/20. Compilations of individual and total photon cross sections (attenuation coefficients) for the whole range of photon energies were cited in, e.g., Tungsten Suppl. Vol. A2, 1987, pp. 69/77. 

Linear attenuation coefficient Probabilities per unit path length that the X-ray photon will be removed from the beam. This includes the effects of photoelectric absorption, Compton scatter, and pair production. 

A low ion pair yield of products resulting from hydride transfer reactions is also noted when the additive molecules are unsaturated. Table I indicates, however, that hydride transfer reactions between alkyl ions and olefins do occur to some extent. The reduced yield can be accounted for by the occurrence of two additional reactions between alkyl ions and unsaturated hydrocarbon molecules—namely, proton transfer and condensation reactions, both of which will be discussed later. The total reaction rate of an ion with an olefin is much higher than reaction with a saturated molecule of comparable size. For example, the propyl ion reacts with cyclopentene and cyclohexene at rates which are, respectively, 3.05 and 3.07 times greater than the rate of hydride transfer with cyclobutane. This observation can probably be accounted for by a higher collision cross-section and /or a transmission coefficient for reaction which is close to unity. 

To demonstrate mass action, we show that for any possible reaction the activity product Q matches the equilibrium constant K. This step is most easily accomplished by computing log Q as the sum of the products of the reaction coefficients and log activities of the corresponding species. The reaction for the sodium-sulfate ion pair, for example,... 

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