Waves electron density

Metal nanoparticles have been used for many applications because of their unique characteristics, even before they were visualized as small particles of nano-meter order by using a transmission electron microscope [118]. For example, colored glasses, which gained in popularity in medieval times, contain nanoparticles of noble metals. These colors originate from the SPR of metal nanoparticles, which is the resonance phenomenon of surface electron density wave with incident light wave at the metal surface [119]. Since this resonance is sensitive to the dielectric constant of surrounding media, the phenomenon has... 

The electron density in a crystal precisely fits the definition of a periodic function in which an exact repeat occurs at regularly fixed intervals in any direction (the crystal lattice translations). Therefore the electron density in a crystal with a periodicity d can be described by a Fourier synthesis in which each component cosine wave (which we will call an electron-density wave) has a periodicity (i.e., wavelength) d/n, and the amplitude of the rath-order Bragg reflection. 

Each electron-density wave provides a component for summation to give the electron-density map, shown in Figure 6.11. If the electron density of a crystal could be described precisely by a single cosine wave that repeats three times in the unit cell dimension d, then the electron density has a periodicity of d/3 and the diffraction pattern will have intensity only in the third order (only one diffracted beam, 3 0 0). This is the electron-density wave that is used in the summation that gives an electron-density map if there is only one term because only one Bragg reflection is ob-... 

FIGURE 6.9. Contributions to terms in a Fourier synthesis, (a) Individual terms from 000 to 10 00 h =10, k = 0, I = 0) are represented with positive areas shaded. These ten electron-density waves combine to give the electron density shown at the bottom of the diagram. This electron density is dependent on the phases (-f == 0°, - = 180 ) which are 000 + 100 200 300 -h 400 500 600 + 700 800... 

Electron-density waves hkl lie perpendicular to the sets of crystal lattice planes with Miller indices hkl. The wavelength of the electron-density wave is the spacing of hkl crystal lattice planes, i.e., The amplitude and the relative phase of the electron-... 

FIGURE 6.11. The meaning of electron-density waves from the Bragg reflections 100, 200, and 300, and their summation to give an electron-density map with peaks at atomic positions (PD means path difference). These electron-density waves have amplitudes and relative phases that depend on the atomic arrangement in the a direction. When the corresponding electron-density map is calculated, it should contain peaks in the positions corresponding to actual atomic positions. 

FIGURE 6.15. The Fourier transforms (FTs) of the 200, 300, and 500 electron-density waves. Shown on the left is one unit cell and an electron-density wave, and, on the right, its Fourier transform (Bragg reflection). 

The Fourier transform thus provides the bridge between Bragg reflections and the electron-density map. The Bragg reflection, order n, contributes to the electron-density map (Equation 6.3) an electron-density wave with a periodicity d/n. For example, the Bragg reflection 300 represents an electron-density wave that repeats three times in the a direction of the unit cell and has an amplitude F(300). If only the 300 Bragg reflection is observed, then the electron density repeats three times... 

FIGURE 6.16. Summing Fourier transforms of the Fourier components of an electron-density map to get a representation of a diffraction pattern, (a) The Fourier transforms of three electron-density waves (200, 300, and 500). (b) The sum of the electron-density waves in (a) (with relative phases 200, a ici — 180°, 300, a/,ki = 0°, and 500. Ohjci = 180°) give a diffraction pattern with intensity at 200, 300, and 500, but no phase information. The sign of the results of the Fourier transformation are lost when the values of F hkl) are calculated. 

FIGURE 6.18. Overall scheme of a crystal structure determination in one dimension. (a) Atomic structure, (b) Bragg reflections that are measured, (c) phases assigned to give electron-density waves with the correct phases, a(hkl), and amplitudes. I F(hkl) I, (d) the sununation of density waves to give an electron-density map, and (e this electron-density map has peaks at atomic positions (compare with the situation in (a), the true atomic arrangement). 

Electron-density wave A term in the Fourier summation of waves of different amplitudes and frequencies to give the electron density in the crystal. Each electron-density wave represents the contribution of a single structure factor F(hkl) to the total electron density map. 

FIGURE 8.1. Four waves with the same amplitude and periodicity are combined in three different ways [(a), (b), and (c)] as a result of different relative phase angles In each case the result of the Fourier synthesis (addition of waves) is different. Shown at the top of this Figure is a crystallographer with information on amplitudes and periodicities of the electron-density waves to be summed (on cardboard strips), but no information on relative phases (how to align the cardboard strips). 

Constraints of direct methods. The phase angles can be estimated by statistical methods that are based on the concept that the electron density is never negative, and that it consists of isolated, sharp peaks at atomic positions. The statistical methods for combining electron-density waves subject to these conditions are called direct methods. They make it possible to derive phases for a set of structure factors when only information on the magnitudes of F hkl) is available. At present this is the method of choice for small molecules. 

Possible phase angles are constrained by these two conditions (Figure 8.3), so that relative phase determination hinges on the mathematical expressions for Fourier series. In the total Fourier synthesis involving F[hkl) with the correct value of a[hkl) these two conditions should apply. This can be appreciated by an examination of Figure 6.9 (Chapter 6), where the negative features of all the electron-density waves have disappeared in the final summation. Exceptions occur in neutron scattering where... 

Tn the determination of the crystal structure of hexamethylbenzene (Figure 1.9, Chapter 1), the Bragg reflections 730, 340 and 470 form a triplet. It was found that by placing atoms at the intersections of high positive values for the electron-density waves (plots called structure factor graphs), Kathleen Lonsdale (in 1929) was able to solve the crystal structure, shown for the 730 and 340 electron-density waves in Figure 8.9.39... 

FIGURE 8.9. Hexeimethylbenzene structure (Ref. 39), and the electron-density waves 340 and 730 (hatched). The actual crystal structure is shown (cf., Figure 1.9, Chapter 1). Note that all atoms lie on the positive intersections of these electron-density waves. 

FIGURE 9.2. The relationship between a high-amplitude electron-density wave (left, traveling vertically) and periodicity in the structure (right). The repeat unit is indicated at the far right. The 10 0 0 electron-density wave is intense (see Figure 3.16. Chapter 3) for DNA-like structures. The repeat unit is 34 A long and the distance... 

FIGURE 11.3. The sources from the Bragg reflections in the X-ray diffraction pattern of (a) unit-cell dimensions (from sin0/A values), (b) periodicities of electron-density waves (from hkl values), and (c) atomic coordinates, y, and 2 (from intensities of the Bragg reflections). 

Because electrons are concentrated around atomic nuclei, knowing p(xj, yj, Zj) for all points j is essentially the same as knowing the distribution of atoms in the unit cell, which in turn means the structure of the molecules which inhabit the unit cell. This is illustrated in Figures 3.22 and 3.23. Thus another way of looking at a crystal is that it is a three-dimensional, periodic, electron density wave that repeats in a perfectly regular manner in space. This is important because several hundred years of physics and mathematics have been focused on periodic waves and their properties, and many clever mathematical tools exist that allow us to manipulate and analyze them. 

In summary then, a crystal can be conceived of as an electron density wave in three-dimensional space, which can be resolved into a spectrum of components. The spectral components of the crystal correspond to families of planes having integral, Miller indexes, and these can, as we will see, give rise to diffracted rays. The atoms in the unit cell don t really lie on the planes, but we can adjust for that when we calculate the intensity and phase with which each family of planes scatter X rays. The diffracted ray from a single family of planes (which produces a single diffraction spot on a detector) is the Fourier transform of that family of planes. The set of all diffracted rays scattered by all of the possible families of planes having integral Miller indexes is the Fourier transform of the crystal. Thus the diffraction pattern of a crystal is its Fourier transform, and it is composed of the individual Fourier transforms of each of the families of planes that sample the unit cells. 

In the case of a periodic, three-dimensional function of x, y, z, that is, a crystal, the spectral components are the families of two-dimensional planes, each identifiable by its Miller indexes hkl. Their transforms correspond to lattice points in reciprocal space. In a sense, the planes define electron density waves in the crystal that travel in the directions of their plane normals, with frequencies inversely related to their interplanar spacings. 

Remember from Chapter 4 that the periods and frequencies of waves are reciprocally related.) Exactly those properties are expressed by their reciprocal lattice vectors h. The amplitudes of these electron density waves vary according to the distribution of atoms about the planes. Although the electron density waves in the crystal cannot be observed directly, radiation diffracted by the planes (the Fourier transforms of the electron density waves) can. Thus, while we cannot recombine directly the spectral components of the electron density in real space, the Bragg planes, we can Fourier transform the scattering functions of the planes, the Fhki, and simultaneously combine them in such a way that the end result is the same, the electron density in the unit cell. In other words, each Fhki in reciprocal, or diffraction space is the Fourier transform of one family of planes, hkl. With the electron density equation, we both add these individual Fourier transforms together in reciprocal space, and simultaneously Fourier transform the result of that summation back into real space to create the electron density. 

Can we connect this equation to the Bragg s law picture Each F /is an individual Bragg reflection hence each Bragg reflection describes part of the entire electron density distribution. The Fourier sum (or in other words inverse FT) describes the entire electron density. More accurately, the reflections sample, or are caused by, the direction- and distance-dependent properties of the electron density in the unit cell. The way it varies periodically in space can be imagined as an electron density wave . This is perhaps possible if there is no variation in y and z. In this case, Equation (10) would reduce to the one-dimensional FT (analogous to... 

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