Structure factor scattering intensity

The structure factor and intensity of scattering (5.25) have a spherically symmetric Lorentzian form centered at the zero wavevector q = 0. The fuU width oti the half a maximum (FWHM) is equal to 2k=2/. ... 

The correlation fiinction G(/) quantifies the density fluctuations in a fluid. Characteristically, density fluctuations scatter light (or any radiation, like neutrons, with which they can couple). Then, if a radiation of wavelength X is incident on the fluid, the intensity of radiation scattered through an angle 0 is proportional to the structure factor... 

The time-dependent structure factor S k,t), which is proportional to the intensity I k,t) measured in an elastic scattering experiment, is a measure of the strength of the spatial correlations in the ordering system with wavenumber k at time t. It exliibits a peak whose position is inversely proportional to the average domain size. As the system phase separates (orders) the peak moves towards increasingly smaller wavenumbers (see figure A3.3.3. 

The Q and ft) dependence of neutron scattering structure factors contains infonnation on the geometry, amplitudes, and time scales of all the motions in which the scatterers participate that are resolved by the instrument. Motions that are slow relative to the time scale of the measurement give rise to a 8-function elastic peak at ft) = 0, whereas diffusive motions lead to quasielastic broadening of the central peak and vibrational motions attenuate the intensity of the spectrum. It is useful to express the structure factors in a form that permits the contributions from vibrational and diffusive motions to be isolated. Assuming that vibrational and diffusive motions are decoupled, we can write the measured structure factor as... 

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

When low-temperature studies are performed, the maximum resolution is imposed by data collection geometry and fall-off of the scattered intensities below the noise level, rather than by negligible high-resolution structure factor amplitudes. Use of Ag Ka radiation would for example allow measurement of diffracted intensities up to 0.35 A for amino-acid crystals below 30 K [40]. Similarly, model calculations show that noise-free structure factors computed from atomic core electrons would be still non-zero up to O.lA. 

Projectors often arise in attempts to describe experiments within the structure of Quantum Mechanics. For example, in the case of the coherent scattering of X-rays by crystals the ideal measured intensities are given by the square of the structure factors... 

Unlike the wave function, the electron density can be experimentally determined via X-ray diffraction because X-rays are scattered by electrons. A diffraction experiment yields an angular pattern of scattered X-ray beam intensities from which structure factors can be obtained after careful data processing. The structure factors F(H), where H are indices denoting a particular scattering direction, are the Fourier transform of the unit cell electron density. Therefore we can obtain p(r) experimentally via ... 

For Comparison Notions of Normal Scattering. As the electron density is assumed to be a real quantity, it directly follows the central symmetry of scattering patterns known by the name Friedel s law [244], Friedel pairs are Bragg reflections hkl and hkl that are related by central symmetry. Concerning their scattering amplitudes, Friedel pairs have equal amplitude Aha = A-g and opposite phase (phki = -scattering intensity the phase information on the structure factor is lost. 

Rayleigh ratio of scattering intensity at scattering angle 0 particle scattering factor = normalized molecular structure factor... 

The method of strueture analysis developed by the Soviet group was based on the kinematieal approximation that ED intensity is directly related (proportional) to the square of structure factor amplitudes. The same method had also been applied by Cowley in Melbourne for solving a few structures. In 1957 Cowley and Moodie introdueed the -beam dynamical diffraction theory to the seattering of eleetrons by atoms and crystals. This theory provided the basis of multi-sliee ealeulations whieh enabled the simulation of dynamieal intensities of eleetron diffraetion patterns, and later electron microscope images. The theory showed that if dynamical scattering is signifieant, intensities of eleetron diffraetion are usually not related to strueture faetors in a simple way. Sinee that day, the fear of dynamical effects has hampered efforts to analyze struetures by eleetron diffraction. 

The summation is taken over all unit cells in the crystal. The overall scattering amphtude J is given by A=F J (where F is the structure factor for the hkl reflection) and the intensity 1 by the square of the amplitude... 

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