Structure factor, Fhkl

The three-dimensional periodic electron-density distribution in a single crystal can be represented by a three-dimensional Fourier series with the so-called structure factors Fhkl as Fourier coefficients ... 

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 ... 

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. 

I have stated that both structure factors and electron density can be expressed as Fourier series. A structure factor describes one diffracted X-ray, which produces one reflection received at the detector. A structure factor Fhkl can be written as a Fourier series in which each term gives the contribution of one atom to the reflection hkl [see Fig. 2.15 and Eq. (2.3)]. Here is a single term, called an atomic structure factor fhkl, in such a series, representing the contribution of the single atom j to reflection hkl ... 

Each diffracted ray is a complicated wave, the sum of diffractive contributions from all atoms in the unit cell. For a unit cell containing n atoms, the structure factor Fhkl is the sum of all the atomic fhkl values for individual atoms. Thus, in parallel with Eq. (2.3), we write the structure factor for reflection Fhkl as follows ... 

When we describe structure factors and electron density as Fourier series, we find that they are intimately related. The electron density is the Fourier transform of the structure factors, which means that we can convert the crystallographic data into an image of the unit cell and its contents. One necessary piece of information is, however, missing for each structure factor. We can measure only the intensity Ihkl of each reflection, not the complete structure factor Fhkl. What is the relationship between them It can be shown that the amplitude of structure factor Fhkl is proportional to the square root of... 

Equation (5.18) tells us how to calculate p(jc,y,z) simply construct a Fourier series using the structure factors Fhkl. For each term in the series, h, k, and 1 are the indices of reflection hkl, and Fhkl is the structure factor that describes the reflection. Each structure factor Fhkl is a complete description of a diffracted ray recorded as reflection hkl. Being a wave equation, Fhkl must specify frequency, amplitude, and phase. Its frequency is that of the X-ray source. Its amplitude is proportional to (- j /)1/2, the square root of the measured intensity Ihkl of reflectionhkl. Its phase is unknown and is the only additional information the crystallographer needs in order to compute p(x,y,z) and thus... 

A representation of structure factors on this plane must include the two properties we need in order to construct p(x,y,z) amplitude and phase. Crystallog-raphers represent each structure factor as a complex vector, that is, a vector (not a point) on the plane of complex numbers. The length of this vector represents the amplitude of the structure factor. Thus the length of the vector representing structure factor Fhkl is proportional to The second prop-... 

In Chapter 4, Section HI.G, I mentioned Friedel s law, that lhkl = h k i-It will be helpful for later discussions to look at the vector representations of pairs of structure factors Fhkl and F h k l, which are called Friedel pairs. Even though hkl and l h k l are equal, Fhkl and F h k l are not. The structure factors of Friedel pairs have opposite phases, as shown in Fig. 6.3. 

Suppose we are able to locate a heavy atom in the unit cell of derivative crystals. Recall that Eq. (5.15) gives us the means to calculate the structure factors Fhkl for a known structure. This calculation gives us not just the amplitudes but the complete structure factors, including each of their phases. So we can compute the amplitudes and phases of our simple structure, the heavy atom in the protein unit cell. Now consider a single reflection hkl as it appears in the native and derivative data. Let the structure factor of the native reflection be Fp. Let the structure factor of the corresponding derivative reflection be FHp. Finally, let FH be the structure factor for the heavy atom itself, which we can compute if we can locate the heavy atom. 

The factors that are included when calculating the intensity of a powder diffraction peak in a Bragg-Brentano geometry for a pure sample, composed of three-dimensional crystallites with a parallelepiped form, are the structure factor Fhkl 2=l/ TS )l2, the multiplicity factor, mm, the Lorentz polarization factor, LP(0), the absorption factor, A, the temperature factor, D(0), and the particle-size broadening factor, Bp(0). Then, the line intensity of a powder x-ray diffraction pattern is given by [20-22,24-26]... 

Thus, each measured intensity (Ihkl) can be reduced to structure factor amplitude (Fhkl) with unknown phases, where is proportional to the square root of fhkl- Each structure factor amplitude (Fhkl) its... 

In words, the structure factor that describes reflection hkl is a Fourier series in which each term is the contribution of one atom, treated as a simple sphere of electron density. So the contribution of each atom j to Fhkl depends on (1) what element it is. which determines jf, the amplitude of the contribution, and (2) its position in the unit cell (Xj, yj, z-)> which establishes the phase of its contribution. 

Superficially, except for the sign change (in the exponential term) that accompanies the transform operation, this equation appears identical to Eq. (5.9), a general three-dimensional Fourier series. But here, each Fhkl is not just one of many simple numerical amplitudes for a standard set of component waves in a Fourier series. Instead, each Fhkl is a structure factor, itself a Fourier series, describing a specific reflection in the diffraction pattern. ("Curiouser and curiouser," said Alice.)... 

Because Fhkl is a periodic function, it possesses amplitude, frequency, and phase. It is a diffracted X ray, so its frequency is that of the X-ray source. The amplitude of Fhkl is proportional to the square root of the reflection intensity lhkl, so structure amplitudes are directly obtainable from measured reflection intensities. But the phase of Fhkl is not directly obtainable from a single measurement of the reflection intensity. In order to compute p(x,y,z) from the structure factors, we must obtain, in addition to the intensity of each reflection, the phase of each diffracted ray. In Chapter 6,1 will present an expression for p(x,y,z) as a Fourier series in which the phases are explicit, and I will discuss means of obtaining phases. This is one of the most difficult problems in crystallography. For now, on the assumption that the phases can be obtained, and thus that complete structure factors are obtainable, I will consider further the implications of Eqs. (5.15) (structure factors F expressed in terms of atoms), (5.16) [structure factors in terms of p(x,y,z)], and (5.18) [p(x,y,z) in terms of structure factors]. 

Figure 6.3 Structure factors of a Friedel pair. F k ( is the reflection of Fhkl in the real axis.

Values of are calculated and stored for each layer line, where Fhkl, .the, structure factor for the reflection with Miller indices h,k and Z. The effects of matrix scattering are approximated by a modification of the atomic scattering factors according to the expression... 

The output is the integrated intensity value for that particular reflection. FHKL - This is the structure factor calculating program. The input is a list of hkl s and intensity values. The output consists of E values and phase angles to be used as input to the electron density program. ELECDEN - Calculates the electron density and contours the E-map on a Tektronix 4662 digital plotter. PATTERSON - Used to calculate three-dimensional Patterson maps. 

Fhkl is the static structure factor. The corresponding spectrum is a series of Dirac peaks in well defined q directions the Bragg reflections. The symmetry of the lattice and the atom positions in the elementary cell is given by the analysis of this diffraction pattern (position and intensity of each peak) [10]. 

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