Scattering, elastic

Since the wavelength (energy) of the incident beam can be varied (e.g., when using a synchrotron as the photon source) the scattering vector rather than the angle is often used as the variable  

Here X is the wavelength of the incident beam, 6 is half of the scattering angle, and e is the unit vector characterizing the direction of the vector q. The latter is defined by the difference of the wave vectors of the incident and the scattered beam, k k = 2nlX) and kf, respectively, since we are dealing with elastic scattering fc = fc.  

To change the resolution in q, or to extend the q-range, the wavelength of the incident beam or the sample-detector distance can be adjusted. The colUmation of the incident beam can have either point or slit geometry. In the case of sUt geometry, the measured scattered intensity represents the convolution of the scattering pattern of the sample for point collimation with the slit profile [23]. 

In the SAS regime, the angular characteristic of the individual scatterer can be disregarded and an angle independent amplitude A, can be attributed to each wave emitted. The scattered intensity is given by the square of the sum of overall waves emitted from the sample volume irradiated (21.2) as a consequence, the phase shift information between the individual waves is lost and only the autocorrelation function of the system under investigation (in case of a two-phase media, e.g., solid and voids) can be extracted from the experimental data. As the phase information is missing, the autocorrelation function cannot be unequivocally related to a three-dimensional backbone stractore. 

Here dE/dQcoh is the coherent differential cross-section per sample volume irradiated, Vj, Psld( ) represents the autocorrelation function of the so-called scattering length density, and r is the difference between pairs of position vectors For isotropic systems, the vector in (21.2) can be simplified  

Burcham Nuclear Reactions, Levels, and Spectra of Light Nuclei. Sect. 16. 

The differential cross section for elastic scattering of a charged particle by a nucleus is given in terms of the asymptotic phase shifts di of the partial wave of angular momentum 1% by the formula (Schiff ) 

The phase shift di represents the refraction of the incident partial wave of angular momentum I by the non-CouLOMB field if all di are zero (l6.1) reduces to the Rutherford scattering formula. If the non-CouLOMB field is due to a nucleus of radius R, and if X R, then the phase shifts are important up to an Z-value given by 

The quantities Fi,Gi,F, j.es defined in Sect. 9- The separation into hard sphere scattering (phase shift (pi) and nuclear resonant scattering (phase shift i) is a mathematically convenient way of distinguishing between potential scattering 

Schiff Quantum Mechanics, p. 120. New York McGraw Hill 1949  

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

If a beam of monoenergetic ions of mass A/, is elastically scattered at an angle 6 by surface atoms of mass Mg, conservation of momentum and energy requires that... 

Buck U 1974 Inversion of molecular scattering data Rev. Mod. Phys. 46 369 Buck U 1975 Elastic scattering Adv. Chem. Phys. 30 313... 

Electrons interact with solid surfaces by elastic and inelastic scattering, and these interactions are employed in electron spectroscopy. For example, electrons that elastically scatter will diffract from a single-crystal lattice. The diffraction pattern can be used as a means of stnictural detenuination, as in FEED. Electrons scatter inelastically by inducing electronic and vibrational excitations in the surface region. These losses fonu the basis of electron energy loss spectroscopy (EELS). An incident electron can also knock out an iimer-shell, or core, electron from an atom in the solid that will, in turn, initiate an Auger process. Electrons can also be used to induce stimulated desorption, as described in section Al.7.5.6. 

Radiation probes such as neutrons, x-rays and visible light are used to see the structure of physical systems tlirough elastic scattering experunents. Inelastic scattering experiments measure both the structural and dynamical correlations that exist in a physical system. For a system which is in thennodynamic equilibrium, the molecular dynamics create spatio-temporal correlations which are the manifestation of themial fluctuations around the equilibrium state. For a condensed phase system, dynamical correlations are intimately linked to its structure. For systems in equilibrium, linear response tiieory is an appropriate framework to use to inquire on the spatio-temporal correlations resulting from thennodynamic fluctuations. Appropriate response and correlation functions emerge naturally in this framework, and the role of theory is to understand these correlation fiinctions from first principles. This is the subject of section A3.3.2. 

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. 

Approximate methods may be employed in solving tiiis set of equations for tlie (r) however, the asymptotic fonn of the solutions are obvious. For the case of elastic scattering... 

To see how this works, consider elastic scattering in a situation where the electron-target interaction can be... 

The ratio of elastically to inelastically scattered electrons and, thus, their importance for imaging or analytical work, can be calculated from basic physical principles consider the differential elastic scattering cross section... 

In SEM and STEM, all detectors record the electron current signal of tire selected interacting electrons (elastic scattering, secondary electrons) in real time. Such detectors can be designed as simple metal-plate detectors, such as the elastic dark-field detector in STEM, or as electron-sensitive PMT. For a rigorous discussion of SEM detectors see [3],... 

In the simplest case of bright-field imaging, the CTF can easily be deduced the elastically scattered electron... 

Egerton R F 1976 Measurement of inelastic/elastic scattering ratio for fast electrons and its use in the study of radiation damage Phys. Status Solid a 37 663-8... 

Figure Bl.24.4. Energy loss components for a projectile that scatters from depth t. The particle loses energy A E- via inelastic collisions with electrons along the inward path. There is energy loss A E in the elastic scattering process at depth t. There is energy lost to melastic collisions A along the outward path. For an incident energy Eq the energy of tlie exiting particle is = q - A iv - AE - A E. ...

Elastic scattering involves no pemianent changes in the internal structures (states a and P) of A and B. Both the energy rel and angular momentum L (AB) of relative motion are tiierefore all conserved. 

The rate coefficient for elastic scattering between two species with non-isothennal Maxwellian distributions is then... 

V..(R) is the static interaction for elastic scattering. The Bom scattering amplitude is a pure fiinction only of... 

Since this agrees with the first Bom differential cross section for (in)elastic scattering, Femii s Rule 2 is therefore valid to first order in the interaction F. 

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