Atomic cross section, incoherent

In most cases, samples are a mixture of isotopes j with different scattering length bj and it is assumed that they are randomly distributed over the sample. Such random distribution of different isotopes produces incoherent cross-section. In analogy with isotope incoherence, spin incoherence is also observed. For nuclei with spin I l, the interaction depends on the orientation between neutron and nuclear spins scattering lengths and b for parallel and untiparallel spin, respectively, are different and the orientations of spins are randomly distributed in the nuclei even if they are the same kinds of nuclei. Then, incoherent and coherent atomic cross-sections are given by... 

Also, because of inelastic effects due to torsion, rotation, and vibration, the effective H incoherent cross section is a function of the particular chemical group (methyl, hydroxyl, etc.) in which the proton is situated [68]. This is illustrated in Table 7.2, which shows the total H-atom cross section in various liquids and polymers, which is dominated by the incoherent component (cTinc), and hence is a strong function of k and only approaches 80 bams at X 4.5 A [66, 67]. Note that none of these values corresponds to crtot from Table 7.2. 

The total atomic cross-section <7totai can be expressed as the sum of the cross-sections of the different elementary processes - that is, the photoelectric effect (r), coherent (Rayleigh) scattering (cTooh). incoherent (Compton scattering) (Omooh). photonudear absorption Oph.n and pair production, which is to be distinguished in nudear-field pah-production (k- ) and dectron-field pair production (iCe) (see Equation 4.3). 

Figure 9.17b schematically represents a cross-sectional view of the surface of a solid and represents the topmost layer of atoms by shaded circles. The open circles represent molecules in an ordered pattern on the solid substrate. Since the adsorbed molecules are ordered, their structure on the surface is characterized by what is called a supernet. Suppose we define <50 as the characteristic spacing of the substrate and 8S the equivalent quantity for the supernet. Then the two arrangements in Figure 9.17b are described by the ratios 5/60 = 4/1 and 8s/80 = 4/3. Building on the notion of reciprocal distances as developed in the discussion of Figure 9.15, it follows that the adsorbed layer with 8/80 = 4/1 should produce spots with a separation that is 1/4 that of the substrate. Likewise, for the case when 8s/80 = 4/3, a pattern of spots with a separation that is 3/4 that of the substrate is predicted. Thus, if the substrate produces spots at, say, 0 and 1, extra spots would be expected at 1/4, 2/4, and 3/4 for the 6/60 = 4 case, and at 3/4, 6/4, and 9/4 when 8s/80 = 4/3. The cases illustrated here are called coincident structures since the two patterns coincide periodically. When there is no correlation between two structures, they are said to be incoherent. 

Incoherent inelastic neutron scattering is efficient in detecting M—H vibrations because the intensity of scattering is proportional to the square of the atomic vibrational amplitude (oc 1/atomic mass) and the scattering cross section (moderately high for H). HCo(CO)4 and H3M3(CO)12 (M = Mn and Re) have been studied, for example.93... 

The first factor in square brackets represents the Thomson cross-section for scattering from a free electron. The second square bracket describes the atomic arrangement of electrons through the atomic form factor, F, and incoherent scatter function, S. Finally, the last square bracket contains the factor s(x), the molecular interference function that describes the modification to the atomic scattering cross-section induced by the spatial arrangement of atoms in their molecules. 

J H Hubbell, W J Veigele, E A Briggs, R T Brown, D T Cromer and R J Howerton (1975) Atomic form factors, incoherent scattering functions and photon scattering cross-sections. J. Phys. Chem. Ref. Data 4, 471 Errata in 1977, 6, 615. 

Table 2 Bound scattering lengths, i>(fm) and cross section for selected isotopes and for selected naturally occurring isotopic mixtures of the elements u(hams, 1 bam = 100 fm ). Z, atomic number A, mass number I, spin of the nuclear groimd state i>coh> bine, coherent and incoherent scattering lengths ffa, ffeCh, coherent and incoherent cross sections ffa, absorption cross section for 2.2 km s neutrons ...

Figure 1. Relative coherent and incoherent scattering cross-sections for some zeolite atomic species.

For a system of different atoms randomly distributed on a lattice, the incoherent cross-section arises from the incoherent scattering of the individual elements and from the disorder scattering due to the random distribution of the different atoms. This latter cross-section is... 

For nuclear incoherent scattering from a non-magnetic system with randomly oriented nuclear spins, the cross-sections per atom are independent of the neutron polarization direction and the spin-flip scattering cross section is twice that for non spin-flip scattering ... 

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