Three-dimensional electron waves, crystals

A three-dimensional (3D) piece of metal can be considered as a crystal of infinite extension in the directions x, y and z with standing waves with the wave numbers k, ky and k, each being occupied with two electrons as a maximum. In a piece of bulk metal the energy differences Sk y are so small that A->0, identical with quasi-free continuously distributed electrons. Since the energy of free electrons varies with the square of the wave numbers, its dependence on k describes a parabola. Figure 4a shows these relations. 

An x-ray analysis will measure the diffraction pattern (positions and intensities) and the phases of the waves that formed each spot in the pattern. These parameters combined result in a three-dimensional image of the electron clouds of the molecule, known as an electron density map. A molecular model of the sequence of amino acids, which must be previously identified, is fitted to the electron density map and a series of refinements are performed. A complication arises if disorder or thermal motion exist in areas of the protein crystal this makes it difficult or impossible to discern the three-dimensional structure (Perczel et al. 2003). 

In inorganic semiconductor crystals with three-dimensional system of conjugated bonds (for example, for the system of sp3 bonds in crystals with tetrahedral cells [27]) delocalization of electron/hole wave functions sharply increases and, accordingly, exchange interaction between these particles decreases. A distance between electron and hole in such bulk crystals, which... 

We want to know the number of points allowed in k-space. In onedimensional space, the segment between successive nx values is simply 2n/L in two dimensions, the area between successive nx and ny points is (2n /L)2 in three dimensions, it is the volume 2%/L)3. If the crystal has volume V, then the three-dimensional region of k-space of volume X will contain X/(2n/L)3 = XV/8n3k values (points) in other words, the k-space density will be V/8n3. We now fill the volume V with electrons with free-wave solutions (each with two possible spin angular momentum projection eigenvalues h/2 or h/2). Let us fill all N electrons, lowest-energy first, within a defined sphere of radius kF (called the Fermi wavevector) the number of k values allowed within this sphere will be... 

CRYSTALS AS WAVES OF ELECTRONS IN THREE-DIMENSIONAL SPACE... 

From Figure 3.2, a crystal emerges as a virtually infinite array of identical unit cells that repeat in three-dimensional space in a completely periodic manner. Like a simple sine wave in one dimension, it repeats itself identically after a period of a, b, or c along each of the three axes. A crystal is in fact a three-dimensional periodic function in space, a three-dimensional wave. The period of the wave in each direction is one unit cell translation, and the value of the function at any point xj, yj, Zj within the cell, or period, is the density of electrons at that point, which we designate p(xj, yj, Zj). 

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. 

A EXPERIMENTAL FIGURE 3-38 X-ray crystallography provides diffraction data from which the three-dimensional structure of a protein can be determined, (a) Basic components of an x-ray crystallographic determination. When a narrow beam of x-rays strikes a crystal, part of it passes straight through and the rest is scattered (diffracted) in various directions. The intensity of the diffracted waves is recorded on an x-ray film or with a solid-state electronic detector, (b) X-ray diffraction pattern for a topoisomerase crystal collected on a solid-state detector. From complex analyses of patterns like this one, the location of every atom in a protein can be determined. [Part (a) adapted from L. Stryer, 1995, Biochemistry, 4th ed., W. H. Freeman and Company, p. 64 part (b) courtesy of J. Berger.]... 

When such features exist, they are penetrated by the electron beam so the material is represented by a three-dimensional point lattice and diffraction only occurs when the Ewald sphere intersects a point. This produces a transmission-type spot pattern. For smooth surfaces, the diffraction pattern appears as a set of streaks normal to the shadow edge on the fluorescent screen, due to the interaction of the Ewald sphere with the rods projecting orthogonally to the plane of the two-dimensional reciprocal lattice of the surface. The reciprocal lattice points are drawn out into rods because of the very small beam penetration into the crystal (2—5 atomic layers). We would emphasize, however, that despite contrary statements in the literature, the appearance of a streaked pattern is a necessary but not sufficient condition by which to define an atomically flat surface. Several other factors, such as the size of the crystal surface region over which the primary wave field is coherent and thermal diffuse scattering effects (electron—phonon interactions) can influence the intensity modulation along the streaks. 

Figure 2.8 (a) A three-dimensional crystal in the real-space Lx, Ly, Lz are dimensions of the crystal, (b) The electron wave vector k in fe-space. The components of fe are kx = TtnxjLx] ky — JUMy/Lyl kz ---... 

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