Structure Amplitudes

So far we discussed all prefactors in Eq. 2.65, which were dependent on multiple parameters except for the crystal structure of the material. The only remaining term here is the structure factor, Fhmf, which is the square of the absolute value of the so-called structure amplitude, F /. It is this factor that includes multiple contributions, which are determined by the distribution of atoms in the unit cell and other structural features. 

When the unit cell contains only one atom, the resulting diffracted intensity is only a function of the scattering ability of this atom (see section 2.5.2). However, when the unit cell contains many atoms and they have different scattering ability, the amplitude of the scattered wave is given by a complex function, which is called the structure amplitude  

Taking into account Eq. 2.88, the previous Eq. 2.87 can be written in the expanded form as  

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The amplitude and therefore the intensity, of the scattered radiation is detennined by extending the Fourier transfomi of equation (B 1.8.11 over the entire crystal and Bragg s law expresses die fact that this transfomi has values significantly different from zero only at the nodes of the reciprocal lattice. The amplitude varies, however, from node to node, depending on the transfomi of the contents of the unit cell. This leads to an expression for the structure amplitude, denoted by F(hld), of the fomi... 

The two exponential tenns are complex conjugates of one another, so that all structure amplitudes must be real and their phases can therefore be only zero or n. (Nearly 40% of all known structures belong to monoclinic space group Pl c. The systematic absences of (OlcO) reflections when A is odd and of (liOl) reflections when / is odd identify this space group and show tiiat it is centrosyimnetric.) Even in the absence of a definitive set of systematic absences it is still possible to infer the (probable) presence of a centre of synnnetry. A J C Wilson [21] first observed that the probability distribution of the magnitudes of the structure amplitudes would be different if the amplitudes were constrained to be real from that if they could be complex. Wilson and co-workers established a procedure by which the frequencies of suitably scaled values of F could be compared with the tlieoretical distributions for centrosymmetric and noncentrosymmetric structures. (Note that Wilson named the statistical distributions centric and acentric. These were not intended to be synonyms for centrosyimnetric and noncentrosynnnetric, but they have come to be used that way.)... 

Referring to figure Bl.8.5 the radii of the tliree circles are the magnitudes of the observed structure amplitudes of a reflection from the native protein, and of the same reflection from two heavy-atom derivatives, dl and d2- We assume that we have been able to detemiine the heavy-atom positions in the derivatives and hl and h2 are the calculated heavy-atom contributions to the structure amplitudes of the derivatives. The centres of the derivative circles are at points - hl and - h2 in the complex plane, and the three circles intersect at one point, which is therefore the complex value of The phases for as many reflections as possible can then be... 

Square of structure amplitude Number of electrons per unit cell ... 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

The distance of each reflection from the center of the pattern is a function of the fiber-to-film distance, as well as the unit-cell dimensions. Therefore, by measuring the positions of the reflections, it is possible to determine the unit-cell dimensions and, subsequently, index (or assign Miller indices to) all the reflections. Their intensities are measured with a microdensitometer or digitized with a scanner and then processed.8-10 After applying appropriate geometrical corrections for Lorentz and polarization effects, the observed structure amplitudes are computed. This experimental X-ray data set is crucial for the determination and refinement of molecular and packing models, and also for the adjudication of alternatives. 

The crystallinity of organic pigment powders makes X-ray diffraction analysis the single most important technique to determine crystal modifications. The reflexions that are recorded at various angles from the direction of the incident beam are a function of the unit cell dimensions and are expected to reflect the symmetry and the geometry of the crystal lattice. The intensity of the reflected beam, on the other hand, is largely controlled by the content of the unit cell in other words, since it is indicative of the structural amplitudes and parameters and the electron density distribution, it provides the basis for true structural determination [32],... 

EDSA of thin polycrystalline films has several advantages First of all the availability of a wide beam (100-400 pm in diameter) which irradiates a large area with a large amount of micro-crystals of different orientations [1, 2]. This results into a special t5q)e of diffraction patterns (DP) (see Fig.l). Thus it is possible to extract from a single DP a full 3D data set of structure amplitudes. That allows one to perform a detailed structure analysis with good resolution for determining structure parameters, reconstruction of ESP and electron density. 

The precision structure analysis of crystals and quantitative reconstruction of ESP requires the maximum possible set of structure amplitudes, which would provide a precise scaling factor, information on the thermal atomic motion and the high resolution of the ESP with good statistical accuracy being at least 1-2%. Such an experimental data set can be collected by an electron diffractometer which was designed on the basis of EMR-102 [3] (Fig. 3). 

Depending on the type of the DP there exist different formulas for the transmission case which relate the intensity with the structure amplitude in the kinematical approximation [1,2], The key-formulas for integral intensity are ... 

The most important question for the calculation of the structure amplitudes from the intensities is that for the validity of the kinematical approximation. Due to the strong interaction of fast electrons with matter the effects of dynamical scattering become more pronounced with increasing size of the microcrystallites in the film. In order to justify application of the kinematical equations it is necessary that the diffracted intensity is much less... 

Aside a few specimens with predominantly kinematical scattering, many specimens investigated by EDSA show pronounced d5mamical scattering. In these cases suitable corrections must be applied to link lOhki I and observed Ihki- For the latter case one has to use successive approximations, i.e. evaluation of parameters from weak kinematical reflections which are then used to apply dynamical corrections to strong dynamical reflections. Without such corrections the residual / -factor of the structure amplitudes is usually about 20 % or more, while suitable corrections lead to / -factors in the range of 5 - 2 %. 

Where p (r) is the electron density of each pseudo atom, Pcore(r) and pvai ( r) are the core and spherical densities of the valence electron shells, Pvai and Pim (multipoles) describe the electron shell occupations, k and k denote the spherical deformation and y (r/r) is a geometrical function. The parameters K, k , Pvai and Pim are refined during adjustment of the experimental and models structure amplitudes. 

This equation can be transformed by Fourier conversion and Mott formulate the relation determining an electron structure amplitude ... 

The values of the ESP at the nuclear positions, as obtained from the electron and Hartree-Fock structure amplitudes for the mentioned crystals (using a K-model and corrected on self-potential) are given in table 2. An analysis shows that the experimental values of the ESP are near to the ab initio calculated values. However, both set of values in crystals differ from their analogs for the free atoms [5]. It was shown earlier (Schwarz M.E. Chem. Phys. Lett. 1970, 6, 631) that this difference in the electrostatic potentials in the nuclear positions correlates well with the binding energy of Is-electrons. So an ED-data in principle contains an information on the bonding in crystals, which is usually obtaining by photoelectron spectroscopy. 

It should be noted that the experimental set of structure amplitudes for Ge was obtained up to sinOA = 1.72 A. This allowed not only to refine more exactly the scale and temperature factors but also provided high resolution in the electron density, the ESP maps and the Laplacian of the electron density. For example the inner electron shells of the core in Ge can be seen in figure 11. 

Figure 7. Projection of the electrostatic potential along (I I0)-direction for spinel, using the set of theoretical structure amplitudes in the angle range (sin0/X) up to 1.6A (h=k=l=l). The atomic positions are signed by names of atoms. It can be noted that Li-position intensities are too -weak, but distinguishable. The projections were calculated by means of JANA 2000 program (and also by means of AREN, PROMETHEUS).

The structure amplitude, F. In an ammonium chloride crystal the unit cell is a cube containing one NH4 and one Cl ion. If the centre of a chlorine ion is taken as the corner of the unit cell, then the ammonium ion lies in the centre of the cell (Fig. 110). 

The ammonium chloride crystal forms a particularly simple example of the effect of atomic arrangement on the intensities of the various reflections. The structure amplitude will be treated more generally in a later section,... 

Angle factors. The two factors already mentioned—the crystal structure amplitude and the factor for the number of similar planes— give a general idea of the reason for the variation of intensity from one arc to another in the powder photograph of ammonium chloride. We have now to consider the general diminution with increasing angle. 

Complete expression for intensity of reflection. Perfect arid imperfect crystals. If relative intensities are being calculated, it is sufficient to multiply the structure amplitude (which is treated generally in the next section) by all the correction factors mentioned. Thus, for a powder photograph, the intensity of each arc is proportional to... 

Most actual crystals are imperfect different portions of the lattice are not quite parallel, and the crystal behaves as if it consisted of a number of blocks (of the order of 10 5 cm in diameter) whose orientation varies over several minutes or even in some cases up to half a degree. This imperfection is perhaps connected with the manner of growth in thin layers (see Chapter II and Plates I and II) each layer may be slightly wavy, and there may be cracks or impurities between the layers. Most crystals are imperfect in this way, and in structure determination it is usually safe to assume that the intensity of any reflection is proportional to the square of the structure amplitude. To make quite sure that a crystal is ideally imperfect , it may be dipped in liquid air the shock-cooling produces imperfections. 

General expression for the structure amplitude. We are interested primarily in the arrangement of the atoms in crystals and the effect of the arrangement on the intensities of diffracted X-ray beams. 

The exjsessions used are valid for crystals of all types, from cubic to tridinic the structure amplitude depends on atomic coordinates (as fractions of the unit cell edges), irrespective of the shape of the cell. 

When the general arrangement is known it is then necessary to determine precise atomic coordinates. Sometimes the positions of certain atoms are invariant—they are fixed by symmetry considerations—but in complex crystals most of the atoms are in general5 positions not restricted in any way by symmetry. The variable parameters must be determined by successive approximations here the work of calculating structure amplitudes for postulated atomic positions can be much shortened by the use of graphical methods, to be described later in this chapter. It cannot be denied, however, that the complete determination of a complex structure is a task not to be undertaken lightly the time taken must usually be reckoned in months. 

For all other crystal planes there are no simple phase relations between waves from M and those from N, and therefore no further systematic absences. Thus, the distance x of the atoms from the screw axis in the direction of the a axis is not, except by accident, a submultiple of a0i and therefore there are no systematic absences of kOO reflections. One or two of these may not appear on the photograph because the structure amplitudes happen to be very small but the point is that there are no systematic absences. The same is true for all other planes—101 for instance (Fig. 140 6), since the distance s between such a plane of atoms as NQ and the plane through P is not, except by accident, a simple submultiple of the spacing d1Q1. 

Fig. 144. Above urea (c projection). 310 and 310 planes have the same structure amplitude. Below penta-erythritol (c projection). 310 and 3l0 planes have different structure amplitudes. Note positions of atoms with respect to planes in each case.

The procedure in determining the parameters will naturally vary with circumstances, but a few general principles can be given. First, as to the most convenient method of calculation. We have seen (p. 228) that for any crystal plane hkl the contribution of each atom (coordinates xyz) to the expression for the structure amplitude consists of a cosine term/ cos 2 n hx- -ky- -lz) and a sine term/ sin 2ir hx- -1cy- -lz). Equivalent atoms (those rebated go each other by symmetry elements) have... 

The intensities of the hOO reflections depend only on the value of x (this is the only parameter along the a axis) hence the relative intensities of the various orders of 7 00 lead to the determination of x. (The results are presented in Pauling s paper in the form of curves like those for rutile in Fig. 123.) Similarly y was found from the relative intensities of the 0 0 reflections. Along the c axis there are two parameters zx and z2, but zx was isolated from z2 by considering only those hid reflections which have h odd and k odd if the expressions for the contributions of all the atoms in the cell are combined together, the complete structure amplitude for these reflections is found to be... 

It should be remembered that when a reference atom is moved, all the atoms related to it by symmetry elements move also in a manner determined by the symmetry elements and the problem is to know, for any particular reflection, the direction in which to move the reference atom so that the contribution of the whole group of related atoms either increases or decreases. This problem is best solved by the use of charts which show at a glance the magnitude of the structure amplitude for such a group of atoms for all coordinates of the reference atom. 

Graphical methods and machines for evaluating structure amplitudes. The evaluation of the structure amplitudes for a large... 

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