Interference, constructive

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

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

FIGURE 2 2 Interference between waves (a) Constructive interference occurs when two waves combine in phase with each other The amplitude of the resulting wave at each point is the sum of the amplitudes of the original waves (b) Destructive interference decreases the amplitude when two waves are out of phase with each other... 

Grating Equation. The light incident on each groove is diffracted or spread out over a range of angles, and in certain directions reinforcement or constructive interference occurs, as stated in the grating formula ... 

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 Simplistic schematic illustration of the scattering mechanism upon which X-ray photoelectron diffraction (XPD) is based. An intensity increase is expected in the forward scattering direction, where the scattered and primary waves constructively interfere.

Figure 4 Interference pettern created when regularly spaced atoms scatter an incident plane wave. A spherical wave emanates from each atom diffracted beams form at the directions of constructive interference between these waves. The mirror reflection—the (00) beam—and the first- and second-order diffracted beams are shown.

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.

The chaiacteristic feature of valence bond theory is that it pictures a covalent bond between two atoms in tenns of an in-phase overlap of a half-filled orbital of one atom with a half-filled orbital of the other, illustrated for the case of H2 in Figure 2.3. Two hydrogen atoms, each containing an electron in a I5 orbital, combine so that their orbitals overlap to give a new orbital associated with both of them. In-phase orbital overlap (constructive interference) increases the probability of finding an electron in the region between the two nuclei where it feels the attractive force of both of them. 

FIGURE 1.20 (a i Constructive interference. The two component waves (left) are "in phase" in the sense that their peaks and troughs coincide. The resultant (right) has an amplitude that is the sum of the amplitudes of the components. The wavelength of the radiation is not changed by interference, only the amplitude is changed, (b) Destructive interference. The two component waves are "out of phase" in the sense that the troughs of one coincide with the peaks of the other. The resultant has a much lower amplitude than either component. 

When two or more waves pass through the same region of space, the phenomenon of interference is observed as an increase or a decrease in the total amplitude of the wave (recall Fig. 1.20). Constructive interference, an increase in the total amplitude of the wave, occurs when the peaks of one wave coincide with the peaks of another wave. If the waves are electromagnetic radiation, the increased amplitude corresponds to an increased intensity of the radiation. Destructive interference, a decrease in the total amplitude of the waves, occurs when the peaks of one wave coincide with the troughs of the other wave it results in a reduction in intensity. 

In an x-ray diffraction experiment on a single crystal of sodium chloride, with the use of radiation from a copper source (X = 154 pm), constructive interference was observed at 0 = 11.2°. What is the spacing of the layers responsible for the diffraction ... 

The phase-twisted peak shapes (or mixed absorption-dispersion peak shape) is shown in Fig. 3.9. Such peak shapes arise by the overlapping of the absorptive and dispersive contributions in the peak. The center of the peak contains mainly the absorptive component, while as we move away from the center there is an increasing dispersive component. Such mixed phases in peaks reduce the signal-to-noise ratio complicated interference effects can arise when such lines lie close to one another. Overlap between positive regions of two different peaks can mutually reinforce the lines (constructive interference), while overlap between positive and negative lobes can mutually cancel the signals in the region of overlap (destructive interference). 

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 ... 

With powdered samples, an image of diffraction lines occurs because a small fraction of the powder particles will be oriented such that, by chance, a certain crystal plane is at the angle 6 to the incident beam for constructive interference (see Fig. 4.4). 

A crystal therefore acts as a three-dimensional diffraction grating for these x-rays, and three equations (the Laue equations) must be satisfied if there is to be constructive interference of these monochromatic x-rays. 

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