Structure Fourier transform: electron density

First I will discuss Fourier series and the Fourier transform in general terms. I will emphasize the form of these equations and the information they contain, in the hope of helping you to interpret the equations — that is, to translate the equations into words and visual images. Then I will present the specific types of Fourier series that represent structure factors and electron density and show how the Fourier transform inter con verts them. 

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... 

The sample s structure (i.e., electron density distribution) can be calculated by inverse Fourier transformation using the amplitudes [i.e., the FQikl)] and phase factors (the exponential term) of the scattered x-rays using Eq. 2 ... 

The structure factor F(Q) in X-ray crystallography is the fourier transform of the electron density for the molecule ... 

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 ... 

Affected by multiple scattering are, in particular, porous materials with high electron density (e.g., graphite, carbon fibers). The multiple scattering of isotropic two-phase materials is treated by Luzatti [81] based on the Fourier transform theory. Perret and Ruland [31,82] generalize his theory and describe how to quantify the effect. For the simple structural model of Debye and Bueche [17], Ruland and Tompa [83] compute the effect of the inevitable multiple scattering on determined structural parameters of the studied material. 

A major application of QED is the accurate determination of crystal charge density. The scientific question here is how atoms bond to form crystals, which can be addressed by accurate measurement of crystal structure factors (Fourier transform of charge density) and from that to map electron distributions in crystals. 

Equation 1 is a discrete Fourier transform, it is discrete rather than continuous because the crystalline lattice allows us to sum over a limited set of indices, rather than integrate over structure factor space. The discrete Fourier transform is of fundamental importance in crystallography - it is the mathematical relationship that allows us to convert structure factors (i.e. amplitudes and phases) into the electron density of the crystal, and (through its inverse) to convert periodic electron density into a discrete set of structure factors. 

As discussed in the following chapter, difference electron density maps, representing Ap = pobs — pcak, are based on the Fourier transform of the complex difference structure factors AF, defined as... 

According to Eq. (1.22), the structure factor F(H) is the Fourier transform of the electron density p(r) in the crystallographic unit cell. The electron density p(r) is then obtained by the inverse Fourier transformation, or... 

The structure factor is the Fourier transform of the thermally averaged electron density ... 

So each reflection is described by an equation like this, giving us a large number of equations describing reflections in terms of the electron density. Is there any way to solve these equations for the function p(x,y,z) in terms of the measured reflections After all, structure factors like Eq. (2.4) describe the reflections in terms of p(x,y,z), which is precisely the function the crystallographer is trying to learn. I will show in Chapter 5 that a mathematical operation called the Fourier transform solves the structure-factor equations for the desired function p(x,y,z), just as if they were a set of simultaneous equations describing p(x,y,z) in terms of the amplitudes, frequencies, and phases of the reflections. 

As I stated in Chapter 2, computation of the Fourier transform is the lens-simulating operation that a computer performs to produce an image of molecules in the crystal. The Fourier transform describes precisely the mathematical relationship between an object and its diffraction pattern. The transform allows us to convert a Fourier-series description of the reflections to a Fourier-series description of the electron density. A reflection can be described by a structure-factor equation, containing one term for each atom (or each volume element) in the unit cell. In turn, the electron density is described by a Fourier series in which each term is a structure factor. The crystallographer uses the Fourier transform to convert the structure factors to p(.x,y,z), the desired electron density equation. 

Because the Fourier transform operation is reversible [Equations (5.10) and (5.11)], the electron density is in turn the transform of the structure factors, as... 

As already explained at the beginning of Sect. 2, XRD is a fundamental method to obtain information on the electron density structure of the phase. The electron density distribution is the Fourier transform of the structure factor. However, in XRD experiments squares of modulus of Fourier components are recorded, while the information about the phase of the Fourier components is lost. For the centrosym-metric structure the possible phase values are reduced to 0 and n. The amplitudes of the Fourier components can thus be taken as real numbers, positive or negative. With n detected peaks (usually less than ten in the case of B1 or B1 rev phases) this... 

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 ... 

---