Relationship between scattering vector

The X-ray and neutron scattering processes provide relatively direct spatial information on atomic motions via detennination of the wave vector transferred between the photon/neutron and the sample this is a Fourier transfonn relationship between wave vectors in reciprocal space and position vectors in real space. Neutrons, by virtue of the possibility of resolving their energy transfers, can also give infonnation on the time dependence of the motions involved. 

Figure 8.1 Relationship between wave vector, momentum transfer, and energy transfer on inelastic scattering.

Fractal behavior is reflected in a power-law relationship between scattered intensity I and scattering vector Q (Q = 2 Jt/A sin 20). Therefore, in a log—log plot of scattered intensity versus scattering vector, fractality is observed as a hnear region of the scattering curve the... 

Figure 7.5. Relationship between symmetrical (

reflection geometry. Bold bars symbolize the sample in symmetrical (dashed) and asymmetrical (solid) geometry. Incident and scattered beam are shown by dashed-dotted arrows, the incident angle is a = 0 + scattering vector s. For the tilted sample the sample-fixed scattering vector S3 is indicated (after [84])... 

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 ... 

PLS with a matrix Y instead of a vector y is called PLS2. The purpose of data evaluation can still be to create calibration models for a prediction of the y-variables from the x-variables in PLS2 the models for the various y-variables are connected. In a geometric interpretation (Figure 4.25), the m-dimensional x-space is projected on to a small number of PLS-x-components (summarizing the x-variables), and the -dimensional y-space is projected on to a small number of PLS-y-components (summarizing the y-variables). The x- and the y-components are related pairwise by maximum covariance of the scores, and represent a part of the relationship between X and Y. Scatter plots with the x-scores or the y-scores are projections of... 

Figure 4.1 The vector relationship between the incident beam vector ko, the scattered beam vector k and the scattering vector Q. The directions of the vectors correspond to their real directions, and the magnitudes of k o and k i, are both 1/...

Figure 10. Ewald sphere construction showing the relationship between the reciprocal space position vector D, the reciprocal space coordinates (R,Z), and the angles n and which define the direction of a scattered ray...

Figure 6. Sketches of the relationship between the energy E and wave vector k, when electron scattering with the periodic lattice is taken into account, (a) Nearly free electrons the scattering is relatively weak, the free electron model is approximately valid for k < n/d. (b) Strong interaction the free electron model is not valid. Tiny bands are separated by broad band gaps.

The scattering vector, Q, describes the relationship between the incident (ki) and scattered (k ) wave vectors. Q has the dimensions of (length)" and is normally quoted in units of and is the independent variable of a SANS experiment. 

From Equation (2), we deduce that diffraction is observed only when the indices h, k, l in d take integral values. These reciprocal space vectors form a lattice, the reciprocal lattice, and the mathematical relationship between the real and reciprocal lattices (and between other aspects of the diffraction pattern) is a FT, as we will explain below. The interpretation of the Ewald construction is that diffraction is observed when the scattering vector s-s0 is equal to a reciprocal space vector A bki with integral indices h, k, l. This occurs whenever such a... 

FIGURE 14.19 Relationship between the incident and scattered wave vectors, the scattering angle 0 and the scattering wave vector q. ... 

FIGURE 1 Generalized representation of a neutron-scattering experiment. Incident and scattered energies are designated by E0. B and incident and scattered momenta (wave vector) by k9, k. Expressions giving the relationship between moments and energy are also shown. 

Fortunately, there exists the 90A-scattering geometry (see Fig. 9.2 b) which leaves the acoustic wave vector independent of the refractive index (for isotropic samples). For the scattering geometries shown in Fig. 9.2, the relationships between the sound velocity, the sound frequency, and the sound wavelength are given by Eqs. (20, 21, 22) respectively, with sin(qi/2) and v °(T) defined in Eqs. 

Figure 4.8. Relationships between vectors Si and Sq and a scattering angle.

Equation 4.70 is referred as the Vector Bragg Equation which expresses the relationship between H, a vector characterizing a crystal plane, and s, a vector characterizing the scattering geometry, for constructive interference to occur. Such a vector equation implies two conditions ... 

Figure 4.15 Diagram illustrating relationships between vectors in fluorescence and Raman scattering.

These are nonzero only for particular relationships between the scattering wave vector q = ko — k and the vector 0 of the helix, i.e., selection rules have appeared. As usual, they follow from the application of laws for the conservation of energy and momentum to the case of the scattering of light by a structure with a specific symmetry. 

Fig. 7.8. The relationship between the momentum transfer, Q, the scattered wave vector k and the incident wave vector ko in a neutron-scattering event.

Figure 5. The relationship between the scattering vector, Q, and the flow geometry...

It is important to note that the incident and scattered fields are specified relative to different sets of basis vectors. The relationship between the incident and scattered fields is conveniently written in the following matrix form... 

Because these structure factors depend on the particular geometry of the substrate, there is no simple relationship between them and the polymer density profile (z). There is, however, one limiting case of interest, when the surface of the substrate is sharp and well-defined and when its curvature is smaller than the scattering vector q, then the surface can be considered as flat at the scale q . In this regime, the partial structure factors reach an asymptotic limit depending only on the structure of the DOlvmer layer and on the area per unit volume of the solid phase,... 

The relationship between the physical observables and molecular motions is thus rather more complicated for NMR than for depolarized light scattering. We follow here parts of the discussion of Dais and Spyros(28). A given chemical bond has a time-dependent orientation vector S2(t). Corresponding to the orientation vector are a series of Wigner rotation matrices which are related to the... 

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