Diffracted waves

If a magnetic field is applied to the crystal, tlie domains become aligned and the nuclear and magnetic wavelets do interfere with one another. Then the amplitude of the diffracted wave depends on the orientation of the neutron spin. In special cases, Coq 2 08 example, the interference may be totally destructive... 

Reconstmction of the object wave is achieved by illumination of the developed hologram with the reference wave as shown in Figure 3a. The diffracted wave amphtude from the hologram is given by equation 3, where the first term represents the attenuated reference wave after passage through the hologram. 

Figure 18.10 The diffracted waves from the protein part (ted) and from the heavy metals (green) interfere with each other in crystals of a heavy-atom derivative. If this interference is positive as illustrated in (a), the intensity of the spot from the heavy-atom derivative (blue) crystal will be stronger than that of the protein (red) alone (larger amplitude). If the interference is negative as in (b). the reverse is true (smaller amplitude).

The wavelike character of electrons was confirmed by showing that they could be diffracted. The experiment was first performed in 1925 by two American scientists, Clinton Davisson and Lester Germer, who directed a beam of fast electrons at a single crystal of nickel. The regular array of atoms in the crystal, with centers separated by 250 pm, acts as a grid that diffracts waves and a diffraction pattern was observed (Lig. 1.21). Since then, some molecules have been shown to undergo... 

From the relative intensities of the two diffracted waves one can deduce the absolute direction of the vector W —> Y with respect to the b axis. This type of reasoning was exploited by Nishikawa and Matsukawa (14) in 1928 and independently by Coster, Knol, and Prins (15) in 1930 to determine the absolute polarity of successive layers of zinc and sulfur in a polar crystal of zinc sulfide. In the zinc sulfide crystal, planes of zinc and sulfur alternate parallel to the face (111), as shown in Scheme 3 the distance between the close pairs of zinc and sulfur planes is one-quarter of the whole 111 spacing. 

In crystallography, the difiiraction of the individual atoms within the crystal interacts with the diffracted waves from the crystal, or reciprocal lattice. This lattice represents all the points in the crystal (x,y,z) as points in the reciprocal lattice (h,k,l). The result is that a crystal gives a diffraction pattern only at the lattice points of the crystal (actually the reciprocal lattice points) (O Figure 22-2). The positions of the spots or reflections on the image are determined hy the dimensions of the crystal lattice. The intensity of each spot is determined hy the nature and arrangement of the atoms with the smallest unit, the unit cell. Every diffracted beam that results in a reflection is made up of beams diffracted from all the atoms within the unit cell, and the intensity of each spot can be calculated from the sum of all the waves diffracted from all the atoms. Therefore, the intensity of each reflection contains information about the entire atomic structure within the unit cell. 

The wave function v /ex (r) of electrons at the exit face of the object can be considered as a planar source of spherical waves according to the Huygens principle. The amplitude of diffracted wave in the direction given by the reciprocal vector g is given by the Fourier transformation of the object function, i.e. 

Up to now, the dynamical approach developed by Takagi is applied in electron diffraction structure analysis for accurate diagnosis of a thick distorted nanocrystals. Within this approach, the amplitudes of the transmitted and diffracted waves i given by the following Takagi-... 

Fig. 12.10. Time-resolved S(f, y) along a line perpendicular to a crack in glass, scanning across the crack (a) some distance from the end of the crack (b) 75 //m from the end of the crack. As in Fig. 9.3(b), the horizontal axis is time t the vertical axis is y, and the value of S(t, y) is indicated by the intensity, with mid-grey as zero and dark and light as negative and positive values of S. In both figures, the first echo (seen as the first stripy vertical bar) is the geometric reflection from the surface of the specimen, and the second echo (seen as the second stripy vertical bar) is the Rayleigh reflection ( 7.2). The patterns forming a x are the reflections from the near and the far sides of the crack, which cross over when the lens is directly above the crack. In (b), where the scan passes quite near to the tip of the crack, the hyperbolic pattern is due to the crack-tip-diffracted wave (Weaver et al. 1989).

X-ray diffraction uses the elastic scattering of X-rays from structures that have long-range order. Diffracted waves from different atoms in the structure can interfere with each other... 

Obviously a structure is rot determined instantly with X-ray diffraction techniques Rather the structure is instantaneous in Ihc sense that the lime period over which the diffracted wave interact with the electrons of the molecule is miinilcsmully short with respect Id the freuuuncy of ulomic morons. 

There are two approaches to the solution of the phase problem that have remained in favor. The first is based on the tremendously important discovery or Patterson in the 1930s ihal the Fourier summation of Eq. 3. with (he experimentally known quantities F2 (htl> replacing F(hkl) leads nol to a map of scattering density, but to a map of all interatomic vectors. The second approach involves the use of so-called direct methods developed principally by Karie and Hauptman of the U.S. Naval Research Laboratory and which led to the award of the 1985 Nobel Prize in Chemistry. Building upon earlier proposals that (he relative intensities of the spots in a diffraction pattern contain information about a crystal phase. Hauptman and Karie developed a mathematical means of extracting the information. A fundamental proposition of (heir direct method is that if thrice intense spots in the pattern have positions whose coordinates add up to zero, their relative phases will cancel out. Compulations done with many triads of spots yield probable phases for a significant number of diffracted waves and further mathematical analysis leads lo a likely solution for the structure of the molecule as a whole. 

If one examines the series in Equation (2.9) it will be seen that the terms will tend to reinforce whenever Q=2vl/d, where / denotes an integer. These values of Q define a one-dimensional reciprocal lattice and whenever Q takes on one of these values the diffracted waves will reinforce. These values of Q correspond to the familiar Bragg reflections given by 2d sin 0 =/A. It should be noted that for finite values of N the reciprocal lattice points have a finite width. 

This holds (at least formally) also for diffraction leaky waves (like those considered by the Florence group), since diffracted waves can be thought of as evanescent waves with imaginary wavevectors, which represent small departures from the propagation vector of the undiffracted principal wave. 

Up to this point we did not make any specific assumptions about the real space lattice. It could contain more than one atom per lattice point and more than more than one type of atoms. In such a case the lattice would be described using a Bravais lattice plus a basis (see Section 8.2.2. To obtain the intensity of the diffracted wave for crystals with a basis, we simply have to sum up the contributions from all scattering points within the unit cell. The scattering probability for a crystal of N unit cells with an electron density ne(r) is proportional to ... 

The methods used for complete structure determination cannot be reviewed in any detail here. It is well known that the central problem is that of determining the relative phase relations of the diffracted waves. As only the intensities can be observed, the problem may be approached by trial and error calculations until a good measure of agreement with the observed values is obtained. This method has in fact been used for a great many of the aromatic structures dealt with in this article. The rigid molecular frameworks with at most only a few degrees of freedom lend themselves to such a treatment. A more systematic approach along the same lines is possible by the method of Fourier transforms (Lipson and Taylor, 1958). 

Figure 6.3 Schematic of the resolution function in the reciprocal space. Two vectors ki and kd represent the incident and the diffracted wave vectors and H represents the scattering vector. The divergence of the incident X-ray and the acceptance window of the diffracted beam side optic are represented as Suj and 6(29).

The structure factor, which is nothing but the wave function of the density, cannot be measured directly and the intensity of the diffracted wave I = F2(hkl), does not contain the phase information required for Fourier synthesis of the density. 

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