Coherent and Incoherent Scattering Length

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 ...

Coherent and incoherent scattering lengths and cross-sections. 

In terms of the coherent and incoherent scattering lengths, bcoh and bmc, defined as... 

A simple example of the coherent and incoherent scattering amplitudes is illustrated by the case of hydrogen. The coherent scattering length is defined by... 

To recognize island formation one takes advantage of the difference between coherent and incoherent diffraction from a set of N identical scatterers. If the waves scattered off the individual scatterers are incoherent in their phases, the observed intensity will be proportional to N (addition of intensities). If, however, these scattered waves are coherent, the intensity will be proportional to N (addition of amplitudes). Incoherence occurs either when the incident wave arrives with incoherent phases at different scatterers, which occurs in practice for scatterers separated by at least the coherence length of the incident beam, or when the scatterers themselves are located incoherently, i.e. are disordered. 

To an incident beam of neutrons the atoms of chorine are a random mixture of Cl and Cl. We calculate the proportions of neutrons scattered coherently and incoherently from chlorine (75% Cl, 25% Cl) with coherent scattering lengths, icoh (m) from Appendix 1. The average scattering length, which provides the coherent signal, is... 

G is corrected by the subtraction of a constant p to exclude a 8(q) term from S(q,u>). To establish the relationship in Eq. (2.75) the nuclear scattering lengths have to be expressed in terms of the coherent and the incoherent scattering lengths. For this purpose the following equation can be used... 

Neutrons are scattered isotropically from individual nuclei, whereas, for LS and SAXS, the scattering originates in the electron cloud, so the atomic form factors are in principle (2-dependent. However, the variation is small in practice (<1% for Q < 0.1 A ) and is usually neglected for SAXS and LS [36]. The Thompson-scattering amplitude of a classical electron is rj = 0.282 x 10 cm [65], so the X-ray scattering length of an atom, /, is proportional to the atomic number (/ = rjZ) and increases with the number of electrons per atom. For neutrons, values of b vary from isotope to isotope (see below). If the nucleus has nonzero spin, it can interact with the neutron spin, and the total cross section (atot) splits into coherent and incoherent components as explained below. 

Table 8.1 Coherent scattering lengths and incoherent scattering cross-sections for selected nuclei...

The first term arises from the ordered arrangement of the scattering lengths, the other from the disordered components. The differential cross-section can then be written as the sum of a coherent and incoherent part ... 

The neutron scattering cross section of an element can be broken into its coherent and incoherent components (there is also an absorption cross section which is umelated to scattering which we will not address here). Hie cross section reflects the number of neutrons scattered per second from the element divided by the intensity of the incident neutron beam. For coherent scattering events, there is a spatial correlation between the scatterings from different nuclei of the same type (with the same scattering length density). Hiese spatial correlations allow us to determine the Van Hove or the pair-pair correlation function, that is, the spatial correlations between the different atoms. For incoherent scattering events, this spatial correlation... 

During an NFS experiment with a sample that contains more than one kind of scatterer (i.e., HS and LS isomer), the superposition of forward scattered waves could occur coherently or incoherently. Longitudinal scattering is always coherent, because there is no path-length difference for nuclei located along the X-ray beam. 

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