Scattering by a unit cell

Assuming that the Bragg law is satisfied, we wish to find the intensity of the beam diffracted by a crystal as a function of atom position. Since the crystal is merely a repetition of the fundamental unit cell, it is enough to consider the way in which the arrangement of atoms within a single unit cell affects the diffracted intensity. 

Qualitatively, the effect is similar to the scattering from an atom, discussed in the previous section. There we found that phase differences occur in the waves scattered by the individual electrons, for any direction of scattering except the extreme forward direction. Similarly, the waves scattered by the individual atoms of a unit cell are not necessarily in phase except in the forward direction, and we must now determine how the phase difference depends on the arrangement of the atoms. 

This problem is most simply approached by finding the phase difference between waves scattered by an atom at the origin and another atom whose position is variable in the jc direction only. For convenience, consider an orthogonal unit cell, a section of which is shown in Fig. 4-8. Take atom A as the origin and let diffraction occur from the (AGO) planes shown as heavy lines in the drawing. This means that the Bragg law is satisfied for this reflection and that the path difference between ray 2 and ray 1, is given by 

How is this reflection affected by x-rays scattered in the same direction by atom B, located at a distance x from y4 Note that only this direction need be considered since only in this direction is the Bragg law satisfied for the AGO reflection. Clearly, the path difference between ray 3 and ray 1, will be less 

Phase differences may be expressed in angular measure as well as in wavelength two rays, differing in path length by one whole wavelength, are said to differ in phase by 360°, or In radians. If the path difference is d, then the phase difference 0 in radians is given by 

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Equation (3.25), which was derived for the intensity of the radiation scattered by a unit cell, contains the atomic scattering factor / . This factor depends principally on the nature of the radiation (and therefore the scattering mechanism), the nature of the scattering point, and the scattering angle. 

To obtain the total intensity of radiation scattered by a unit cell, the scattering of all of the atoms in the unit cell must be combined. This is carried out by adding together the waves scattered from each set of (hkl) planes independently, to obtain a value called the structure factor, F(hkl), for each hkl plane. It is calculated in the following way. 

The X-ray intensity scattered by a unit cell is given by the modulus of ... 

We now consider the relative intensity of a wave diffracted by a unit cell situated at a point in the space lattice. Each atom in the unit cell scatters incident radiation, and the contribution of the unit cell as a whole is the resultant of these separate waves. This resultant amplitude is called the structure factor, or F(hkl). Referring to Figure 3.1, let the scatter-... 

As with the isomorphous replacement technique it is necessary to identify the positions, the x, y, z coordinates of the anomalous scatterers. This can be done by anomalous difference Patterson maps, which are Patterson syntheses that use the anomalous differences Fhki — F—h—k—i as coefficients (Blow and Rossmann, 1961). These maps are interpreted identically to isomorphous difference Patterson maps (see Chapter 9). Rapidly surpassing Patterson approaches, particularly for selenomethionine problems and others where the number of anomalous scatterers tends to be large, are direct methods (see below). These are strictly mathematical methods that have proved to be surprisingly effective in revealing the constellation of anomalous scatterers in a unit cell. 

We now see that the problem of scattering from a unit cell resolves itself into one of adding waves of different phase and amplitude in order to find the resultant wave. Waves scattered by all the atoms of the unit cell, including the one at the origin, must be added. The most convenient way of carrying out this summation is by expressing each wave as a complex exponential function. 

MIR), requires the introduction of new x-ray scatterers into the unit cell of the crystal. These additions should be heavy atoms (so that they make a significant contribution to the diffraction pattern) there should not be too many of them (so that their positions can be located) and they should not change the structure of the molecule or of the crystal cell—in other words, the crystals should be isomorphous. In practice, isomorphous replacement is usually done by diffusing different heavy-metal complexes into the channels of preformed protein crystals. With luck the protein molecules expose side chains in these solvent channels, such as SH groups, that are able to bind heavy metals. It is also possible to replace endogenous light metals in metal-loproteins with heavier ones, e.g., zinc by mercury or calcium by samarium. 

The prindple of a LEED experiment is shown schematically in Fig. 4.26. The primary electron beam impinges on a crystal with a unit cell described by vectors ai and Uj. The (00) beam is reflected direcdy back into the electron gun and can not be observed unless the crystal is tilted. The LEED image is congruent with the reciprocal lattice described by two vectors, and 02". The kinematic theory of scattering relates the redprocal lattice vectors to the real-space lattice through the following relations... 

Expressions (3.42) and (3.43) show that the Fourier transform of a direct-space spherical harmonic function is a reciprocal-space spherical harmonic function with the same /, in. This is summarized in the statement that the spherical harmonic functions are Fourier-transform invariant. It means, for example, that a dipolar density described by the function dl0, oriented along the c axis of a unit cell, will not contribute to the scattering of the (hkO) reflections, for which H is in the a b plane, which is a nodal plane of the function dU)((l, y). 

The resultant of the waves scattered by all the atoms in the unit cell, in the direction of the hkl reflection, is called the structure factor, F /, and is dependent on both the position of each atom and its scattering factor. It is given by the general expression for j atoms in a unit cell... 

So far we have used the scattering factor, /, to take account of the effect of structure within the unit cell. It is instructive to make use of Equation (2.5) instead. At one extreme the case of all the scattering power in a unit cell being located at one point leads to Equation (2.9) with /being a constant. In this extreme case, all orders of diffraction have the same amplitude. As the electron density within the unit cell is represented by more spread out and realistic functions, so the higher order terms in Equation (2.9) decrease more rapidly with increasing values of Q. Thus in the extreme case represented by... 

Each diffracted X ray that arrives at the film to produce a recorded reflection can also be described as the sum of the contributions of all scatterers in the unit cell. The sum that describes a diffracted ray is called a structure-factor equation. The computed sum for the reflection hkl is called the structure factor Fhkl. As / will show in Chapter 4, the structure—factor equation can be written in several different ways. For example, one useful form is a sum in which each term describes diffraction by one atom in the unit cell, and thus the series contains the same number of terms as the number of atoms. 

The scattering amplitude is given by the absolute value of the following expression, which is a Fourier transform of the total electron-density distribution in a unit cell and is called a crystal structure factor ... 

X-ray and neutron diffraction patterns can be detected when a wave is scattered by a periodic structure of atoms in an ordered array such as a crystal or a fiber. The diffraction patterns can be interpreted directly to give information about the size of the unit cell, information about the symmetry of the molecule, and, in the case of fibers, information about periodicity. The determination of the complete structure of a molecule requires the phase information as well as the intensity and frequency information. The phase can be determined using the method of multiple isomor-phous replacement where heavy metals or groups containing heavy element are incorporated into the diffracting crystals. The final coordinates of biomacromolecules are then deduced using knowledge about the primary structure and are refined by processes that include comparisons of calculated and observed diffraction patterns. Three-dimensional structures of proteins and their complexes (Blundell and Johnson, 1976), nucleic acids, and viruses have been determined by X-ray and neutron diffractions. 

Thus, for a unit cell containing n atoms that scatter with amplitudes /i /2 "./n and phases v>, v>2. > V n. the resultant diffracted amplitude is given by... 

Since the interaction of hard X-rays with matter is small, the kinematical approximation of single scattering is valid in most cases, except for perfect crystals near Bragg scattering. The intensity scattered by a block-shaped crystal with N, q and N, unit cells along the three crystal axes defined by the vectors Uj, a and a, takes the form ... 

The structure factor F hkl) is the Fourier transform of the unit cell contents sampled at reciprocal lattice points, hkl. The structure factor amplitude (magnitude) F is the ratio of the amplitude of the radiation scattered in a particular direction by the contents of one unit cell to that scattered by a single electron at the origin of the unit cell under the same conditions (see Chapter 3). The first report of the structure factor expression was given by Arnold Sommerfeld at a Solvay Conference. The structure factor F has both a magnitude F(hkl) and a phase rel-... 

When we have a large number of individual waves, like those produced by the scattering of X-rays from families of planes, or from all of the unit cells in a crystal, or from all of the atoms within a unit cell, we are ultimately interested in knowing how all of the waves add together to yield a resultant wave that we can observe, characterize, and use. Waves are more complicated to sum than simple quantities like mass or temperature because they have not only an amplitude, a scaler, but also a phase angle 0 with respect to one another. This must be taken into account when waves are combined. As will be seen below, waves share identical mathematical properties with vectors (and with complex numbers, which are really nothing but vectors in two dimensions). 

To this point we have been interested in the scattered waves, or X rays from atoms that combine to yield the observed diffraction from a crystal. Because the waves all have the same wavelength, we could ignore frequency in our discussions. In X-ray crystallography, however, we are equally interested in understanding how the waves diffracted by a crystal can be transformed and summed, in a symmetrical process, to produce the electron density in a unit cell. 

The continuous distribution of waves F in any and all directions k, for an incident direction o, is the diffraction pattern of the two points xo, yo, zo and x, y, z. This simple diffraction pattern, the physical distribution of resultant waves in space arising from the scattering of two points is, in mathematical terms, the Fourier transform of the two points. If more points, each designated by a subscript j, are added to the set, as in Figure 5.2, as we might have for atoms comprising molecules in a unit cell, then the formulation of the... 

The wave produced by all of the atoms in a unit cell, when Bragg s law is satisfied, will simply be the sum of the waves scattered by the individual atoms within the cell. To obtain the diffracted wave produced by the entire crystal (which is what we measure as a diffraction intensity), we need only sum the waves scattered by each atom in one unit cell and then multiply by the total number of unit cells, N, in the crystal. What are the amplitudes and relative phases of these individual waves with respect to one another, or to some other reference wave ... 

We have now shown, by three different approaches, that if one knows the atomic coordinates xj, yj, Zj of all of the atoms j in a unit cell, and their scattering factors fj, then one can precisely predict the amplitude and phase of the resultant wave scattered by a specific family of planes hkl. We can calculate this for any and all families of planes in the crystal, hence the amplitude and phase can be calculated for every structure factor in the X-ray diffraction pattern. Given the structure of a crystal, namely the coordinates of the atoms in the unit cell, we can predict the entire diffraction pattern, the entire Fourier transform of the crystal. This is an enormously powerful statement. It means that if, by some means, we can deduce the positions of the atoms in a crystal structure, then we can immediately check the correctness of that deduction by seeing how well we can predict the relative values of the intensities in the diffraction pattern. 

The resultant wave scattered by all the atoms of the unit cell is called the structure factor because it describes how the atom arrangement, given by uvw for each atom, affects the scattered beam. The structure factor, designated by the symbol F, is obtained by simply adding together all the waves scattered by the individual atoms. If a unit cell contains atoms 1, 2, 3,..., A, with fractional coordinates i Vi w, 2 2 3 t a M s. and atomic scattering factors then... 

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