Coherently elastic neutron scattering

In the study of chain dimensions, we are interested in coherently elastic neutron scattering only, that is, d/i = 0 and k - ko- The experimental data needed are da/dO and S(Q), not d cj/dO d and S(Q, ca). The differential corss section do/dH and the scattering law 5coh(Q) are related in a very simplified form  

The subscripts 1 and 2 refer to one substance dissolved in another substance (including one solid sample mixed with another solid sample) coh refers to the coherent scattering, and V is the partial molar volume. The scattering law 5 (Q) here is the Fourier transform of the density fluctuation correlation function [instead of G(r, f)] of the scattering centers  

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It can be shown (see, e.g.. Squires 1996) that coherent elastic (neutron) scattering from a crystal occurs only when the scattering vector is equal to a valid reciprocal lattice vector (note that for arbitrary arrangements, this requirement is not fulfilled and therefore, the special -eligible - orientations of the crystal have to be found) ... 

The prerequisite for an experimental test of a molecular model by quasi-elastic neutron scattering is the calculation of the dynamic structure factors resulting from it. As outlined in Section 2 two different correlation functions may be determined by means of neutron scattering. In the case of coherent scattering, all partial waves emanating from different scattering centers are capable of interference the Fourier transform of the pair-correlation function is measured Eq. (4a). In contrast, incoherent scattering, where the interferences from partial waves of different scatterers are destructive, measures the self-correlation function [Eq. (4b)]. 

Coherent quasi-elastic neutron scattering due to the pair-correlation of atomic motions is more complex than incoherent scattering from single particle motion. Therefore, the theory of coherent scattering function is not yet fully developed. However, approximate expressions for the coherent scattering function exist (incoherent approximation). 

In another important class of experiments one measures the coherent elastic magnetic scattering of neutrons by a paramagnetic crystal in an external magnetic field H. The degenerate ion state is split into (2/ + 1) levels whose probability of occupation is... 

Bragg s law (and therefore, the coherent elastic differential cross section) can be exploited for determining the precise crystalline structure. Obviously, neutron scattering from crystals is not restricted to the coherent elastic contribution scattering observed in between Bragg peaks is called diffuse scattering, which is generally connected to some sort of disorder (thermal. 

The quasi-elastic neutron scattering from H2O and D2O, in harmotome, has been further measured in samples with 2, 4, and 12 water molecules per unit cell at 294 K < T < 423 K [98S1]. From the observed coherent scattering relaxation time for D-D correlations, 2 to 11 ps were obtained. Fitting the incoherent quasi-elastic peaks from H2O to a two site jump model, yielded residence times of the protons in the range from 4 to 6 ps [98S1], close to the previous reported values. 

Quasi-elastic Neutron Scattering. Coherent and incoherent inelastic neutron scattering are unique experimental techniques to characterize molecular motions on a time scale between 10 and 10 s. The continued development of high resolution inelastic scattering techniques in the past two decades (157-159) enables measurement of the dynamic structure factor S(Q, co) and the... 

Bragg scattering Coherent elastic scattering of monochromatic neutrons by a set of crystal planes. 

The elastic coherent cross section for nuclear and magnetic neutron scattering may be written... 

Distances between scatterers of (0.60 0.06) A estimated from the interference fringes do not correspond to any visible distance between protons in the crystal structure. However, they correspond closely to distances between projections of proton positions onto directions corresponding to normal coordinates (namely, x, y or z as shown in Fig. 1). It appears that neutrons are scattered coherently by lines of protons quite visible in the crystal structure (see Fig. 17). The elastic scattering function presented in Fig. 16 corresponds to a double-quantum-slits experiment. The interference fringes are quite different from those observed for neutrons scattered by double-classical-slits [Zeilinger 1988]. It can be concluded that quantum entanglement is not limited to single pairs. 

Despite the great incoherent cross-section of the H-atom, elastic coherent neutron scattering experiments may be used at a lower resolution to detect the presence of clusters of H2O molecules. The method used is that of small-angle neutron scattering, often labeled by its acronym SANS . In such an experiment, the scattered intensity of a sample is recorded as a fnnction of the angle 6 between the wave vectors of the scattered and initial waves or, equivalently, as a function of the amplitude Q of the wave vector difference Q = k -ic2 0f the scattered beam defined by and incident beam defined by ki (the wave vector defines the direction of propagation of a wave, as seen in eq. (3.6)). As can be seen in Figure 11.1, these quantities are related by the equation ... 

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