Condition for constructive interference

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. 

A real two-dimensional crystal consists of a periodic structure in not only one, but in two directions. This second periodic structure in another direction also leads to a condition for constructive interference. The conditions for both directions have to be fulfilled at the same time. Both conditions are only fulfilled where the cones of both orientations intersect. As a result constructive interference occurs only on lines starting from the point of incidence. On a detector plate we would observe spots where these lines cross the plate. 

Figure 8.20 Condition for constructive interference for a one-dimensional array of scattering centers (top). Constructive interference occurs along the cones that reflect the rotational symmetry of the one-dimensional arrangement (middle). For a two-dimensional crystal, constructive interference is obtained along lines (bottom).

Equation (A.5) is known as the Laue2 condition for constructive interference. 

Consider the diffraction from a one-dimensional row of regularly spaced atoms. From Figure 5.19 it can be seen that the condition for constructive interference is... 

X-rays have wavelengths that are comparable to the spacing between atoms. Rows of atoms cause diffraction, so if the wavelength of the X-rays is known, the spacing between atoms can be determined. Figure 3.8 shows that the condition for constructive interference, hence strong scattered X-rays, is... 

Since, the scattered ray interfere subsequently, for rays reflected by two adjacent planes, the condition for constructive interference is given by (see Figure 1.22) [22]... 

One of the important featmes of Bragg scattering is that the strength of the scattered beam depends on how well the atoms are localized at their ideal lattice locations. If the atoms are all located very close to the lattice positions defined by the interference pattern, they will all nearly satisfy the conditions for constructive interference, and the scattering will be strong. 

The condition for constructive interference corresponds to that when the scattered waves are in phase. In Fig. 3.15a, the wavefront labeled 1 would have to travel a distance AB + BC farther than the wavefront labeled 2. Thus if and when AB + BC is a multiple of the wavelength of the incident X-ray A, that is,... 

The condition for constructive interference is simply that the path length difference for the two photons is an integer multiple of wavelengths (as shown in Fig. 2A), or... 

FIGURE 316. William H. Bragg and his son William L. Bragg reversed von Laue s experiment and used x-rays to measure the distances between ions or atoms in crystals (see text), (a) Depicts the conditions for constructive interference of x-rays termed Braggs Law (b) schematic of Braggs s x-ray apparatus (WH. Bragg and W.L. Bragg, X-Rays And Crystal Structure, 4th ed., London, 1924). 

FIGURE 7 12 (a) Schematic cross section oi an Interference filter. Note that the [Pg.176]

Diffraction is described by either Bragg or von Laue equations. The Bragg equation relates as a condition for constructive interference the wavelength of the electromagnetic radiation k to the interplanar distance a and the observation angle 0... 

The condition for constructive interference in all dimensions is a set of the common intersection of these cones, defined by the simultaneous solution of the respective equations. Thus, the equations represent one form of the scattering conditions required for diffraction to occur in three dimensions... 

The condition for constructive interference is that the waves should all be in phase with one another. Figure 1.15 gives examples of waves... 

The requirements for x-ray diffraction are (1) the atomic spacing in the solid must be comparable with the wavelength of the x-rays and (2) the scattaing centers must be spatially distributed in an ordered way (e.g, the environment present in crystals). The diffraction of x-rays by crystals (Fig. 3.17) was treated by Bragg in 1912. The condition for constructive interference, giving rise to intense diffraction maxima, is known as Bragg s law ... 

The condition for constructive interference of the original and shifted wavefronts is given by... 

Now the conditions for constructive interference expressed by equation 7 allow us to calculate the net amplitude of scattered waves. The amplitude F of the electric field at P will be the sum of the amplitudes scattered from A and B, or. 

The condition for constructive interference is that the path difference hetween waves diffracted at neighboring lines (highlighted in Figure 16.2) he an integral multiple of the wavelength. This leads to (Equation 16.5) ... 

Figure 7.26 Small-angle diffraction from a stack of lamellar crystals (shaded areas) with the repeating distance (long period) d. The condition for constructive interference is expressed in the Bragg equation.

Equation (1.0) is actually the condition for constructive interference (CJiristman 1988). The incident beam of X-Rays used is normally Cu Ka. In the so-called 0-0 optical set-up, the incident X-Ray beam moves over the same angle as the detector, as shown in Fig 4 (b). In Fig 4 (a). Si and S2 are slits and L is the position of the detector. The 20 angle, which is the angular value given with corresponding peak intensities in the XRD pattern, is actually the angle of deviation and can be seen in Fig. 4 (a) marked 5. The value of 8 is 20. 

The genius of sir Lawrence Bragg provided a simple physical model of constructive scattering of X-rays by crystals, an immediate counterpart of the mathematics in term of simple phenomena. The conditions for constructive interference in a crystal will now be rewritten using Bragg s approach. 

Concepts of the reciprocal lattice and Brillouin zones are introduced in the chapter on X-rays along with a more formal freatment of the scattering of X-rays or wave-like particles. This formal treatmenf for obfaining the Laue conditions for constructive interference leads to a deeper understanding of the scattering process and gives students the ability to calculate the structure factors for various materials. The applications of X-ray diffraction to material identification, structure determination, and crystal characterization are briefly discussed. 

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