Bragg construction

Figure Bl.8.2. Bragg s law. Wlien X = 2d sin 0, there is strong, constructive interference. (B) THE RECIPROCAL LATTICE...

Figure Bl.8.3. Ewald s reciprocal lattice construction for the solution of the Bragg equation. If Sj-s. is a vector of the reciprocal lattice, Bragg s law is satisfied for the corresponding planes. This occurs if a reciprocal lattice point lies on the surface of a sphere with radius 1/X whose centre is at -s..

When there is constructive interference from X rays scattered by the atomic planes in a crystal, a diffraction peak is observed. The condition for constructive interference from planes with spacing dhkl is given by Bragg s law. 

Figure 1 Bragg diffraction. A reflected neutron wavefront (D, Dj) making an angle 6 wKh planes of atoms will show constructive interference (a Bragg peak maxima) whan the difference in path length between Df and (2CT) equals an integral number of wavelengths X. From the construction, XB = d sin 6.

X-ray diffraction occurs in the elastic scattering of X-ray photons by atoms in a periodic lattice. The scattered monochromatic X-rays that are in phase give constructive interference. Figure 4.4 illustrates how diffraction of X-rays by crystal planes allows one to derive lattice spacings by using the Bragg relation ... 

Figure 4.4. X-rays scattered by atoms in an ordered lattice interfere constructively in directions given by Bragg s law. The angles of maximum intensity enable one to calculate the spacings between the lattice planes and allow furthermore for phase identification. Diffractograms are measured as a function of the angle 26. When the sample is a...

If one measures the angles, 26, under which constructively interfering X-rays leave the crystal, the Bragg relation (1) gives the corresponding lattice spacings, which are characteristic for a particular compound. 

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the sphere of reflection, is reciprocal to the wavelength of X-ray radiation—that is, IX. The reciprocal lattice rotates exactly as the crystal. The direction of the beam diffracted from the crystal is parallel to MP in Figure 3.7 and corresponds to the orientation of the reciprocal lattice. The reciprocal space vector S(h k I) = OP(M/) is perpendicular to the reflecting plane hkl, as defined for the vector S. This leads to the fulfillment of Bragg s law as S(hkI) = 2(sin ())/X = 1 Id. 

Bragg s model is illustrated in Fig. 7.2, where the horizontal lines represent successive planes of the set. The spacing of these planes (denoted as d) is the perpendicular distance between them. The path difference is (BC) + (CD), and for constructive interference this path length must equal (nA). But since (BC) + (CD) must be equal to (2d sin 9), one deduces an expression universally known as Bragg s law ... 

Unlike the case of diffraction of light by a ruled grating, the diffraction of x-rays by a crystalline solid leads to the observation that constructive interference (i.e., reflection) occurs only at the critical Bragg angles. When reflection does occur, it is stated that the plane in question is reflecting in the nth order, or that one observes nth order diffraction for that particular crystal plane. Therefore, one will observe an x-ray scattering response for every plane defined by a unique Miller index of (h k l). 

Figure 5.7 Derivation of Bragg s law of X-ray diffraction. Parallel X-rays strike the surface at an angle 0, and are reflected from successive planes of crystals of interplanar spacing d. The path difference between reflections from successive planes is given by AB + BC, which, by geometry, is equal to 2dsin0. For constructive interference, this must be equal to a whole number of wavelengths of the incoming radiation.

FIGURE 10.4 An illustration of d, 0, and d sin0 in Bragg s law. The distance traveled by the x-ray reflected from the second plane is greater than that reflected from the first plane by 2d sin 9 in order for constructive interference to occur and a light intensity to be observed at the detector. 

For gas targets (atomic and molecular) the theory yields quite reasonable predictions of proton stopping cross sections as compared with experiment. Moreover, since chemical binding effects are naturally incorporated in the theory, the construction of tables of the velocity-dependence of CAB contributions to 5 for different compounds allows - once and for all - the estimate of 5 for protons in materials with similar CAB components without resource to Bragg s additivity rule. 

What happens if the set of (hkl) lattice planes is not exactly at the Bragg orientation As shown on figure 2b, the position of the two spots is hardly affected but the intensity of the diffracted beam is strongly modified. This behavior can be explained by means of the Ewald sphere construction. 

The above discussion has in effect been for materials with zero absorption, but this affects only the intensities. The construction of the dispersion surface and the wavevector matching are all performed on the real part of the wavevectors. When absorption is considered, the reflectivity in the Bragg case falls below 100% but it can still be over 99% for a low-absorption material such as silicon. 

Figure 4.16 The selection of tie-points on the dispersion surface for the reflection (Bragg) case, using the construction of Figure 4.13...

Figure 7.9 Ewald constructions, (a) at the Bragg condition, (b) off the Bragg condition...

Bragg s law is a special case of the Laue equations which define the condition for diffraction (constructive interference) to occur ... 

When a crystalline material is placed in a monochromatic X-ray beam, the resulting diffraction or scattering pattern depends on the sample shape or form. Simplistically, if the sample is a powder, then the scattering pattern appears as sharp concentric circles. If the sample is a fiber, then the scattering pattern appears as sharp arcs falling on layered lines. Each maximum arises from the constructive interference between the X-rays scattered by a set of parallel planes within the crystal [3]. Rewriting Bragg s law yields... 

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