Unit cells waves scattered from

This is the most useful quantitative intensity formula that may be derived from kinematical theory, since it is applicable to thin layers and mosaic blocks. We add up the scattering from each unit cell in the same way that we added up the scattering from each atom to obtain the stractme factor, or the scattering power of the unit cell. That is, we make allowance for the phase difference r, . Q between waves scattered from unit cells located at different vectors ri from the origin. Quantitatively, this results in an interference function J, describing the interference of waves scattered from all the unit cells in the crystal, where... 

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 ... 

In order to test such an application we have calculated the spin and charge structure factors from a theoretical wave function of the iron(III)hexaaquo ion by Newton and coworkers ( ). This wave function is of double zeta quality and assumes a frozen core. Since the distribution of the a and the B electrons over the components of the split basis set is different, the calculation goes beyond the RHF approximation. A crystal was simulated by placing the complex ion in a lOxIOxlOA cubic unit cell. Atomic scattering factors appropriate for the radial dependence of the Gaussian basis set were calculated and used in the analysis. 

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. 

W L Bragg [7] observed that if a crystal was composed of copies of identical unit cells, it could then be divided in many ways into slabs with parallel, plane faces whose distributions of scattering matter were identical and that if the pathlengths travelled by waves reflected from successive, parallel planes differed by integral multiples of the... 

Unlike the wave function, the electron density can be experimentally determined via X-ray diffraction because X-rays are scattered by electrons. A diffraction experiment yields an angular pattern of scattered X-ray beam intensities from which structure factors can be obtained after careful data processing. The structure factors F(H), where H are indices denoting a particular scattering direction, are the Fourier transform of the unit cell electron density. Therefore we can obtain p(r) experimentally via ... 

Electron dynamic scattering must be considered for the interpretation of experimental diffraction intensities because of the strong electron interaction with matter for a crystal of more than 10 nm thick. For a perfect crystal with a relatively small unit cell, the Bloch wave method is the preferred way to calculate dynamic electron diffraction intensities and exit-wave functions because of its flexibility and accuracy. The multi-slice method or other similar methods are best in case of diffraction from crystals containing defects. A recent description of the multislice method can be found in [8]. 

Whereas the values of h, k, l in theory should span from — oo to + oo, they are limited in practice to those finite values which are accessible to the diffraction experiment for a given radiation. The structure factor Fhkl is the resultant of N waves scattered in the direction of the reflection hkl by the N atoms in the unit cell, as expressed by equations 2 or 3 ... 

The spherical-wave description has its own great advantages. It is best adapted to the scattering and emission by the spherical ion cores and it does not require the presence of any structural periodicity. In particular, it is well suited to the treatment of multiple scattering between different atoms within any cluster of atoms, in particular within a periodic unit cell as in LEED. It is also convenient for the treatment of fine structure arising from back-scattering by nearby atoms, as in (S)EXAFS, NEXAFS, EAPFS, etc. (i.e. in step 3 in our four-step description). 

Now we consider diffraction in terms of the path differences of waves on scattering. This will lead to an understanding of the effect on the diffraction pattern of the periodicity of the crystal from unit cell to unit cell. There are two methods that have been used to consider this, one by Max von Laue and the other by W. L. Bragg. 

Clearly, the atoms, or electron densities in the unit cells, do not lie exactly on planes, any family of planes. In reality, they lie mostly in between. Why then should imaginary planes, which these planes are, scatter X rays at all The answer is that they don t. We can, however, make the assumption that all atoms do lie on nearby planes of each family, and then compensate for the fact that they are really displaced from the planes when we calculate the diffracted wave from those planes. This is somewhat analogous to assuming all of the mass of an object is concentrated at its center of mass, or representing a complicated charge distribution by a dipole moment. 

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. 

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... 

If we do this, then the product of the transform of the object and the lattice becomes, as in Figure 5.4, simply the line of points, where each point serves as an identical source of a common wave corresponding to the scattering of the entire continuous object for some diffraction vector s. Although the lattice points produce a wave for any and all diffraction vectors s = (k — ko), because the waves arise from points in a lattice, the waves cancel, or sum to zero except when all the points belong to a family of planes hkl for which Bragg s law is satisfied, that is, when s = h. When this condition is met, the waves emitted from each point constructively interfere and sum in an arithmetic manner. The lattice then multiplies the resultant wave from the object, the atoms within the unit cell, by the total number of unit cells in the crystal and allows us to observe it, but only for specified values of s, namely only at those points in diffraction (Fourier transform) space where s = h. 

What is seen for one dimension is quite the same for the two- or three-dimensional cases as well. Just as the resultant wave created by the interference of the scattered waves from all of the atoms in the molecules could be considered as arising from discrete lattice points, the same is true for a real crystal. We can consider the resultant waves produced by the scattering of all of the atoms in the unit cells to simply be emerging from a single lattice point common to each cell, as in Figure 5.10. Because the contents of the unit cell are continuous and nonperiodic, their transform, or resultant waves F-s will be nonzero for all s. Because the lattice points in a crystal are discrete and periodic, however, the waves from all lattice points will constructively interfere and be observable only in certain directions according to Bragg s law, that is, when s = h. 

If Bragg s law is not satisfied for this family of planes, it doesn t matter what the distribution of electron density in the unit cell is, since no diffraction from the crystal occurs. Scattered waves from any one unit cell will be out of phase with those from all others, producing destructive interference. 

Consider further an atom that lies exactly between two planes. If the difference in phase between two successive planes is 2n, then this atom would scatter with a phase oilit/l = n. In fact any atom in the cell lying an arbitrary distance D from the nearest plane would scatter with a phase of (D/dhu)2jt. In other words, the phase angle of the wave scattered by any atom in the unit cell is simply a function of its distance from a plane of that particular family. Can we determine how far any atom is from an hkl plane and, therefore, its phase We can indeed, so long as we know the positions, the x, y, z coordinates, of the atoms in the unit cell. 

The working vector v having unit length and which is normal to the planes can be drawn. It passes not only through this unit cell but identically through all unit cells in the crystal. We use this unitary vector simply to define a direction. The phase of a wave scattered by an atom with respect to the set of planes will be 0 = 2ji(D/dhia), where D is the atom s distance from the nearest plane. But D is the projection of 3ri onto i>, that is, l) = x Ft, so that 0 = 2jt(xi v/dhu)-... 

The vector v is unit length and normal to the planes, the distance between the planes, and therefore v/d i is the reciprocal lattice vector h. Hence 0 = 2jr(Jti h). Only those points in the unit cell where there is an atom need to be considered, since the scattering from any other point will be zero. We can replace xi with xj, where xj represents the coordinates xj, yj, Zj of the j th atom. The resultant diffracted ray from the entire unit cell, then, is the sum of the waves scattered by all N of the atoms j in the unit cell ... 

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. 

The structure factors Ff,u of a crystal, and therefore the observed structure amplitudes Fhki-obs, depend only on the distribution of scattering material. Each F%u is the sum of the scattered waves from the individual atoms in the unit cell. If additional heavy atoms are introduced into the unit cell, and all else remains constant, the new resultant Fhu-deriv will be the sum of the old, native Fhki-nat plus the contribution of the wave scattered by the heavy atom. This is illustrated by a hypothetical case in Figure 8.3 and graphically in Figure 8.4. We are dealing with a phase-dependent interference phenomenon. Hence, when waves are added, the new Fhu-deriv of the derivative structure may have an amplitude either greater or less than the Fhki-nat for the native structure. 

The underlying principle of anomalous scattering methods is related to the isomorphous replacement concept of creating reference waves within crystal unit cells that interfere in some way with the resultant wave from all of the light atoms belonging to the macro-... 

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. 

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