Bragg reflections crystal lattice scattering

The change in the intensity with temperature is calculated with the temperature factor. This change is produced by the crystal lattice vibrations, that is, the scattering atoms or ions vibrate around their standard positions as was previously explained (see Section 1.4) consequently, as the crystal temperature increases, the intensity of the Bragg-reflected beams decreases without affecting the peak positions [25], Debye and Waller were the first to study the effect of thermal vibration on the intensities of the diffraction maxima. They showed that thermal vibrations do not break up the coherent diffraction this effect merely reduces the intensity of the peaks by an exponential correction factor, named the temperature factor, D(0) [2,26], given by... 

Two of the more direct techniques used in the study of lattice dynamics of crystals have been the scattering of neutrons and of x-rays from crystals. In addition, the phonon vibrational spectrum can be inferred from careful analysis of measurements of specific heat and elastic constants. In studies of Bragg reflection of x-rays (which involves no loss of energy to the lattice), it was found that temperature has a strong influence on the intensity of the reflected lines. The intensity of the scattered x-rays as a function of temperature can be expressed by I (T) = IQ e"2Tr(r) where 2W(T) is called the Debye-Waller factor. Similarly in the Mossbauer effect, gamma rays are emitted or absorbed without loss of energy and without change in the quantum state of the lattice by... 

In the Bragg formulation of diffraction we thus refer to reflections from lattice planes and can ignore the positions of the atoms. The Laue formulation of diffraction, on the other hand, considers only diffraction from atoms but can be shown to be equivalent to the Bragg formulation. The two formulations are compared in Fig. 2B for planes with Miller indices (110). What is important in diffraction is the difference in path length between x-rays scattered from two atoms. The distance si + s2 in the Laue formulation is the same as the distance 2s shown for the Bragg formulation. The Laue approach is by far the more useful one for complicated problems and leads to the concept of the reciprocal lattice (Blaurock, 1982 Warren, 1969) and the reciprocal lattice vector S = Q 14n that makes it possible to create a representation of the crystal lattice in reciprocal space. 

Normal Bragg reflections are produced as a result of the interaction of X rays with the electrons of atoms that are arranged on a regular periodic lattice. Diffuse scattering is due to crystal lattices that depart from a regular periodic character. " When atoms vibrate in a crystal they must affect their neighbors and, because of their sizes, are more likely... 

As we established earlier, a powder diffraction pattern is one-dimensional but the associated reciprocal lattice is three-dimensional. This translates into scattering from multiple reciprocal lattice vectors at identical Bragg angles. Consider two points in a reciprocal lattice, 00/ and 00/. By examining Eqs. 2.29 to 2.34 it is easy to see that in any crystal system l/c/ (00/) = l/c/ (00/). Thus, Bragg reflections from these two reciprocal lattice points will be observed at exactly the same Bragg angle. 

This is the familiar formulation of Bragg s law for a three-dimensional point lattice. It says that the Fourier transform of a point lattice is absolutely discrete and periodic in diffraction space, and that we can predict when a nonzero diffraction intensity will appear for any family of planes hkl, and what the angle of incidence and reflection 0 must be in order for an intensity to appear. Bragg s law, notice, is completely independent of atoms, or molecules, or unit cell contents. The law is imposed by the periodicity of the crystal lattice, and it strictly governs where we may observe any nonzero intensity in diffraction space. It tells us when the resultant waves produced by the scattering of all of the atoms in the many individual unit cells, each represented by a single lattice point, are exactly in phase. 

This is known as Bragg s Law and describes the fact that the path differences of the X rays scattered from parallel lattice planes hkl are an integral number of wavelengths. If A and dhkl are known, values of dm maybe determined. When an X-ray beam strikes a crystal, diffraction will occur when, and only when, Bragg s Law is satisfied. The spacing between lattice planes dw is a function of the unit cell dimensions and the indices h,kj of those crystal planes, so that if 2 Qhki is measured for several different Bragg reflections (with different hkl values), the unit-cell dimensions can be found. 

BRAGG SCATTERING PROM OPTICAL LATTICES Just as x-rays may be Bragg reflected from crystal lattices where both the lattice spacings and the x-ray wavelength are of the order of... 

The Bragg points of the finite crystal are connected by streaks of intensity along the surface normal direction in the reciprocal lattice. At the minimum position, midway between Bragg reflections, the scattered intensity is similar to the scattering expected from a single monolayer. The streaks of intensity are known as crystal-truncation rods (CTRs) as they arise from... 

---