Compton coefficient

The probability that Compton scattering will occur is called the Compton coefficient or the Compton cross section. It is a complicated function of the photon energy, but it may be written in the form... 

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). 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

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).

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

The coefficients of the small-p expansion of IIo(p), Eq. (5.40), have been extracted by fitting to experimental Compton profiles for both atoms and molecules [167]. Table V.l gives a flavor of the tense confrontation between... 

Complexes, see also specific type in solution, structures, see X-ray diffraction n-Complexes, 4 178-184 Complex formation constant, outersphere, 43 46, 55 electrovalent interaction in, 3 269-270 Compressibility coefficient of activation, 42 9 Comproportionation constants, class II mixed-valence complexes, 41 290-292 Comproportionation equilibrium, 41 280-281 Compton effect, 3 172 Conantokins, calcium binding, 46 470-471 Concanavalin A, 36 61, 46 308 Concensus motif, 47 451 Concentration-proportional titrations of poly-metalates, 19 250, 251, 254 Condensation... 

According to Evans (1995), differentiation of features within the materials is possible because p at each point directly depends on the electron density of the material in that point (pe), the atomic number (Z) of the chemical components of the materials in that point, and the energy of the incoming X-ray beam (/0). In particular, the linear attenuation coefficient can be approximately considered as the sum of the Compton scatter and photoelectric contributions ... 

D. Hardacre, C. Seddon, K. R. Compton, R. G. Voltammetry of oxygen in the room-temperature ionic liquids l-ethyl-3-methylimida-zolium bis(triflyl)imide and HexEt3N+ TfiN one-electron reduction to form superoxide. Steady-state and transient behavior in the same cyclic voltammogram resulting from widely different diffusion coefficients of oxygen and superoxide. J. Phys. Chem. A 2003,... 

This conclusion appears to be in conflict with the relative magnitudes of the coherent and Compton interaction coefficients, illustrated for an organic explosive such as trinitrotoluene (TNT) in Fig. 3. It is apparent that the cross-section for Compton scatter dominates over the energy range above 10 keV. 

Fig. 3. Compton and coherent scatter interaction coefficients from XCOM program [ ].

The existence of such a coefficient was recognized by Maxwell j1 numerous measurements under a variety of conditions have been made by Soddy and Berry,2 Knudsen,3 Langmuir,4 Blodgett and Langmuir,6 Michels,6 Compton and Lamar,7 Archer,8 Gregory,9 Alty,10 and others. 

Ti(acac)2 was rapidly and quantitatively analyzed by X-ray fluorescence (XRF) spectroscopy. Fe(acac)2 was similarly determined by XRF with correction for Compton scattering. An instrumental geometrical factor and an equivalent wavelength were obtained experimentally, while all the other factors were calculated with the mass absorption coefficients of Fe ". K and XRF spectra of Cr(acac)3 and other Cr compounds were measured with a Bragg spectrometer. The relative intensities of the 3, K 2, K / and K lines with respect to the K line confirm the chemical effect on the intensity... 

The total absorption coefficient fx is given approximately by the sum of the partial absorption coefficients due to the photoeffect (/ ph), the Compton effect ( c) and pair formation ( p) ... 

The attenuation coefficient for a beam of gamma rays is related to the number of gamma rays removed from the beam, either by absorption or scattering. For the Compton effect, the absorption cross section is determined by the energy absorbed by the electron, which is the total collision energy minus the average scattered photon... 

Vol. III. Physical and Chemical Tables (1962). Includes data on characteristic wavelengths, absorption coefficients, atomic scattering factors, Compton scattering, etc. Also treatments of intensity measurements, texture determination, particle size broadening, small angle scattering, and radiation hazards. 

Absorption means diminution of coherent x-ray intensity in the crystal through inelastic processes such as atomic absorption and fluorescence, photoelectron emission, and Compton effect extinction means intensity diminution due to loss through diffraction by fortuitously oriented mosaic blocks. The simple extinction expression due to Darwin, given in Eq. (18), is only a rough approximation more accurate treatments will be mentioned in what follows. In Eq. (17) the absorption factor is expressed in terms of the linear absorption coefficient /inn (calculated from tabulated values of the elemental atomic or mass absorption coefficients, updated values of which will appear in Vol. IV of International Tables,2 the path length f, of the incident ray from the crystal surface to the point of diffraction r, and the path length t2 of the diffracted ray from that point to the crystal surface. 

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 detection efficiency of a detector is another important property in PET technology. Since it is desirable to have shorter scan times and low tracer activity for administration, the detector must detect as many of the emitted photons as possible. The 511-keV photons interact with detector material by either photoelectric absorption or Compton scattering, as discussed in Chap. 1. Thus, the photons are attenuated (absorbed and scattered) by these two processes in the detector, and the fraction of incident 7 rays that are attenuated is determined by the linear attenuation coefficient (/x) given in Chap. 1 and gives the detection efficiency. At 511 keV, /x = 0.92 cm-1 for bismuth germanate (BGO), 0.87 cur1 for lutetium oxyorthosilicate (LSO), and 0.34 cm-1 for Nal(Tl) (Melcher, 2000). Consequently, to have similar detection efficiency, Nal(Tl) detectors must be more than twice as thick as BGO and LSO detectors. 

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