Scattering vector, definition

The general definition of a projection has been given on p. 23 in Eq. (2.37). For the purpose of illustration let us write down an example. If s = (Si,Sj,Sk) is a representation of the scattering vector in orthogonal Cartesian coordinates, then the aforementioned ID projection is... 

Fig. 5.1. Time dependence of a typical classical trajectory in the photodissociation of C1CN — Cl 4- CN(j). (a) R(t) (solid line) is the Cl-CN separation and 7(t) (dashed line) is the orientation angle of CN with respect to the scattering vector R (for the definition of coordinates see Figure 3.1). (b) The angular momentum j(t) of CN (solid line) and the torque F1 = -dV/dj (dashed line).

In choosing beam optics to measure xrd-rsm, one must consider resolution function in the reciprocal space. The resolution function is defined by the incident beam divergence and the acceptance window of the diffracted beam side optic. Figure 6.3 schematically shows the definition of the resolution function in the reciprocal space. The X-ray detector is located at the tip of the scattering vector H in the reciprocal space. The incident beam divergence 5u> and the acceptance window of the diffracted beam optic 520 define the resolution function (gray area in Figure 6.3) in the reciprocal space. The form of the obtained diffracted intensity distribution of the crystal by xrd-rsm is defined by the convolution of the resolution function and the reciprocal lattice point of the crystal. Therefore, a resolution function smaller than... 

In order to introduce some notation and definitions, we consider a polymer block with n monomers of segment length b. Defining Ti as the interdistance between two monomers i and j in that block, and Q as the scattering wave-number (often called scattering vector ), various types of correlations can be considered depending on the number of summations involved ... 

Consider a polycrystalline sample placed at a point O. A crystal within this sample diffracts for a family of planes (hkl) if and only if the reciprocal lattice point associated with this family is on the Ewald sphere with radius 1/X. By definition, the tip of the vector kg =(l/A,)So is the origin of the reciprocal lattice and the scattering vector S, which hnks point C with lattice point hkl, has a norm equal to 1/dhki. For all the crystals contained in the sample which diffract for a given family of planes (hkl), we observe a set of lattice points hkl located on a sphere with radius 1/dhn, which is centered in C. By definition, the directions [hkl] are normal to the (hkl) planes and therefore the distribution map of the hkl lattice points on the surface of this sphere is the hkl pole figure. This configuration is shown in Figure... 

FtGURE 19 Diffraction from parallel planes (top), and definition of the scattering vector Q (bottom). 

Figure 1.7 Definition of the scattering vector q and its magnitude q. The angle between incident and scattered beams is 20 and y is the wavelength of the radiation. The wavevector of the incident beam is denoted Iq and that of the scattered beam ks...

This definition conforms to the rule that the density fluctuation background is expanded in even powers of the scattering vector [46, 60]. After the subtraction, the discrete intensity is multiplied by 47t 5. This multiplication is equivalent to the Laplacian edge-enhancement operator, as is evidenced by double application of the derivative theorem. 

Note that alternative definitions and terminologies for the scattering vector can be found throughout the literature, with the scattering angle sometimes being referred to as 0 instead of... 

The definition of the absolute value s of the scattering vector in eqn [2] has some resemblance to Bragg s law, which, in this notation, can simply be written as... 

Because of the inverse relationship between interatomic distances and the directions in which constructive interference between the scattered electrons occurs, the separation between LEED spots is large when interatomic distances are small and vice versa the LEED pattern has the same form as the so-called reciprocal lattice. This concept plays an important role in the interpretation of diffraction experiments as well as in understanding the electronic or vibrational band structure of solids. In two dimensions the construction of the reciprocal lattice is simple. If a surface lattice is characterized by two base vectors a and a2, the reciprocal lattice follows from the definition of the reciprocal lattice vectors a and a2 ... 

A definite but small anisotropy of the differential quenching cross section is observed when the electric vector EA of the exciting laser light is rotated in the scattering plane. It follows equation (VI. 1), where 9n has to be replaced by ip + (w/2). 

From the formulas listed, the components Px., Pr and Pz. of the photoelectron s spin-polarization vector P defined in the /, z detector frame can be calculated (for the definition of P see Fig. 1.5 and equ. (9.15)). If tr stands for transverse, long for longitudinal, and if the subscripts and 1 indicate the component within or perpendicular to the scattering plane, respectively, one gets1"... 

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