Wave-vector Bragg

The wave-vector Bragg equation describes diffraction phenomenon and can be derived using the reciprocal lattice approach as shown in the next section. 

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.

The vibrational excitations have a wave vector q that is measured from a Brillouin zone center (Bragg peak) located at t, a reciprocal lattice vector. 

Fig. 20. Effect of an uniaxial pressure on the relative magnetic Bragg peak intensities associated with the three equivalent k-vectors for UN, UAs and USb single crystals. The uniaxial pressure is applied along the (001) direction. Full circles correspond to magnetic peaks Ii, associated with a wave vector perpendicular to the stress and open circles to magnetic peaks I, associated to a wave vector parallel to the stress. (Rossat-Mignod et al. )...

Exercise 16.3-2 Show that if k is the wave vector of incident radiation (X-ray or neutron) or the wave vector of a particle or quasiparticle, then eq. (25) leads to the Bragg diffraction condition. 

Equations (1) are the von Laue conditions, which apply to the reflection of a plane wave in a crystal. Because of eqs. (1), the momentum normal to the surface changes abruptly from hk to the negative of this value when k terminates on a face of the BZ (Bragg reflection). At a general point in the BZ the wave vector k + bm cannot be distinguished from the equivalent wave vector k, and consequently... 

How are the Laue condition and the Bragg condition connected In Fig. A.3 the wave vectors of the incident and outgoing radiation and the scattering vector are drawn for the Bragg reflection of Fig. A.l. We can conclude that for specular reflection, the scattering vector lattice plane. Its length is given by... 

Figure A.3 Incident wave vector ki, outgoing wave vector kf, and scattering vector q for Bragg reflection (Fig. A.l) The length of the scattering vector is given by q = 2 ki sin = f sin0.

Such electrons will be reflected backward and forward between two (adjacent) core ions (Bragg reflection). Due to the periodic lattice, electron propagation is prohibited in the x direction when the wave vector is equal (or close) to kx =... 

The in-plane structure of the interface is probed by measuring the scattered intensity as a function of the angle 26 (fig. 3.59). or, equivalently, as a function of the horizontal component of the wave-vector, If the molecules in the interface are arranged in a two-dimensional periodic structure, diffraction occurs when coincides with a reciprocal lattice vector fulfilling the Bragg condition... 

Fig. 1. Powder x-ray profiles of solid Ceo at atmospheric pressure (top) and 1.2-GPa hydrostatic pressure (bottom). Dots are experimental points (approximately 70 per point), and the solid curves are least-squares fits to an fee structurewith adjustable lattice constant a. The fitted relative intensities have no physical significance in this simple model. The scattered wave vector Q = 4TTsin6/X, where 0 is the Bragg angle for these profiles wavelength = 0.71 A. Indexing of the strongest peaks is indicated. The high-Q shoulder on the (311) is the weak (222) reflection the low-Q shoulder on the (111), observed to some extent in all our nominally pure Cfio samples, is presently unidentified. The variable intensity of this shoulder has litde eflfect on the lattice constant of a particular sample, so we can safely conclude that it has no effect on the compressibility derived from the present data.

For crystalline samples under plane-wave illumination, a diffraction pattern is observed as a spot pattern. The individual spots depend on the crystal orientation, its structure factor, and obey Bragg s law. Bragg s law states that the difference between the scattered k and the incident ko wave vector is equal to a vector g of the reciprocal lattice ... 

High-resolution transmission electron microscopy can be understood as a general information-transfer process. The incident electron wave, which for HRTEM is ideally a plane wave with its wave vector parallel to a zone axis of the crystal, is diffracted by the crystal and transferred to the exit plane of the specimen. The electron wave at the exit plane contains the structure information of the illuminated specimen area in both the phase and the amplitude.. This exit-plane wave is transferred, however affected by the objective lens, to the recording device. To describe this information transfer in the microscope, it is advantageous to work in Fourier space with the spatial frequency of the electron wave as the relevant variable. For a crystal, the frequency spectrum of the exit-plane wave is dominated by a few discrete values, which are given by the most strongly excited Bloch states, respectively, by the Bragg-diffracted beams. 

As mentioned above, the XSW field arises from the interference between the coherently related incident and Bragg-diffracted beams from the surface of a perfect crystal. In the vicinity of a Bragg reflection (Fig. 24A-B), an incident plane wave (with wave vector fco) and a reflected wave (with wave vector kfi) interfere to generate a standing wave with a periodicity equivalent to that of the (h, k, 1) diffracting planes. The ratio of the electric field amplitudes of the reflected and incident waves is given by... 

This is the Bragg condition. It specifies the wave vector component of the dielectric constant fluctuation that will give rise to scattering at an angle d. 

To understand this, consider a free electron with energy, e, and wave vector, k. As shown on the next page in 5.3.1., e varies with k. This curve is a potential energy diagram of the free electron. Note that we have converted sin 9 (the linear sinusoidal function) into jt (the radial function). If we apply a mono-atomic lattice, arranged in a linear manner, and having a lattice constant of a, to our electron, we can show that Bragg reflection occurs at ... 

In the following discussion, we will assume the most common case of cr-polarized symmetrical Bragg diffraction from a semi-infinite crystal with 1° < 9b < 89°. Figure 2 shows the case of -polarization with the vector directions of the two E-fields pointing perpendicular to the scattering plane defined by the two wave vectors. The incident and exit angles of the two wave vectors with respect to the surface are equivalent for a symmetric reflection. 

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