Differential scattering cross section measurement

Angular light scattering measurements are sometimes classified as either absolute or relative. In an absolute measurement 7s//(, which is directly related to the differential scattering cross section (13.2), is determined in a relative measurement the irradiance is referred to some arbitrary scattering angle, say 10°, so that (assuming azimuthal symmetry)... 

Fig. 3.15. Schematic illustration of the crossed-beam apparatus developed by Hyder et al. (1986) for the measurement of positron elastic differential scattering cross sections. Reprinted from Physical Review Letters 57, Hyder et al, Positron differential elastic scattering cross section measurements for argon, 2252-2255, copyright 1986 by the American Physical Society.

Hyder, G.M.A., Dababneh, M.S., Hseih, Y.-F., Kauppila, W.E., Kwan, C.K., Mahdavi-Hezaveh, M. and Stein, T.S. (1986). Positron differential elastic-scattering cross-section measurements for argon. Phys. Rev. Lett. 57 2252-2255. 

Figure 5 - Small angle neutron differential scattering cross section (ooo) measured from a sample consisting of sheets of PIP (N = 23000) and deuterated 1,2-PBD (N = 3200) which were in contact for 162 hours at 52°C. The scattering contrast significantly exceeds the incoherent background (—) determined from measurements on the individual polymers, evidencing the thermodynamic miscibility of the blend.

Fig. 27. Measured total diffuse differential scattering cross-section of single crystal sapphire and ruby (1.26% atomic Cr). A = 6.0A. The statistical error is indicated. Total counting time was 48 hours [after Ref. (/ 7)]...

Hug and Surbeck 35) have proposed the use of Ada. = daL — daR, the difference of the Raman differential scattering cross sections in left and right circularly polarized incident light. This is the Raman equivalent of the circular dichroism Ae = el — eR, where s is the decadic molar extinction coefficient and, unlike the measured (but not the theoretical) IR — IL which depends on both sample and instrumental factors, is solely a molecular parameter. They introduced a chirality number q defined by... 

But what is measured in fact in neutron scattering is the differential scattering cross-section, dafcIQ. (q), which is defined as the number of neutrons scattered per second towards a detector in a certain direction per incident beam flux and solid angle. In the case of a liquid or a glass sample for which the average structure is isotropic, only the vector norms (r = r and q= q ) are relevant. 

Some of the earliest potentials computed by the SRS variant of SAPT were for Ar-H2 [149] and for He-HF [150,151]. An application of the latter potential in a calculation of differential scattering cross sections [152] and comparison with experiment shows that this potential is very accurate, also in the repulsive region. Some other SAPT results are for Ar-HF [153], Ne-HCN [154], CO2 dimer [155], and for the water dimer [129,156]. The accuracy of the water pair potential was tested [130,131] by a calculation of the various tunnehng splittings caused by hydrogen bond rearrangement processes in the water dimer and comparison with high resolution spectroscopic data [132,133]. Other complexes studied are He-CO [157,158], and Ne-CO [159]. The pair potentials of He-CO and Ne-CO were applied in calculations of the rotationally resolved infrared spectra of these complexes measured in Refs. [160,161]. They were employed [162-165] in theoretical and experimental studies of the state-to-state rotationally inelastic He-CO and Ne-CO collision cross sections and rate constants. It was reaffirmed that both potentials are accurate, especially the one for He-CO. 

Suppose an incident plane wave of flux Jo irradiates a sample, from which the scattered spherical wave emanates in all directions (see Figure 1.2). Our task is then to measure the flux J of the scattered ray as a function of the scattering direction and to interpret this information to learn about the structure of the sample. Since under a given experimental condition J will increase or decrease in proportion with Jq, we are really interested in the ratio J/Jq as a function of the scattering direction. Because Jo refers to a plane wave and J to a spherical wave, the ratio J/Jq has dimension of area per solid angle. In the neutron-scattering community the ratio J/Jq is invariably referred to as the differential scattering cross section... 

The geometry of a neutron diffraction experiment is shown in Fig. 3.2, where a neutron of incident wavevector ki is scattered into a state with wavevector kf. In the case of nnclear scattering, the scattering centre is the atontic nucleus. The essential information that is determined in a nentron (or X-ray) diffraction measurement is the differential scattering cross-section ... 

The following examples will emphasize the importance of placing intensity data on an absolute scale, typically in the form of a differential scattering cross section dS/df2(2), in units of cm As explained in Section 7.3.1, the equivalent quantity for LS is the Rayleigh ratio [15, 66, 69], and, while the use of absolute units is not essential for the measurement of the spatial dimensions (e.g. determining the Rg of a polymer coil), it forms a valuable diagnostic tool for the detection of artifacts, to which scattering techniques are sometimes vulnerable. 

In the early days of chemical dynamics (the 1960 s) the study of elastic scattering was an important project, both experimentally and theoretically, for developing the tools to be used to study more interesting processes, e.g., elastic scattering of the rare gas atoms. Y.T.Lee s (1986 Nobel Prize with M. Polanyi and D. Herschbach) measurements of the differential scattering cross sections allowed the definitive determination of the intermolecular potential energy function V(r) of essentially all the rare gas atoms with each other... 

After a short description of the experimental setup in the previous section, we now focus on the basic mathematical description of the intensity measured in a SAXS experiment. X-rays are electromagnetic waves. Assume an incident plane wave with flux 7o, which is scattered by the electrons in the sample. The scattered spherical waves interfere with each other, resulting in an angle-dependent flux J of scattered radiation. The flux of the incident plane wave corresponds to energy transmitted per unit area per unit time and the flux J of the scattered radiation to energy transmitted per unit solid angle per unit time. Now the differential scattering cross section or scattered intensity is defined as the ratio... 

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