The Structure-Factor Equation

We have indicated that interference and reinforcement effects depend both on the positions of atoms in a structure and the number of electrons associated with each atom. A quantitative treatment of these effects makes use of the important structure-factor equation which represents the addition of waves (sine and cosine functions) from each atom within a unit cell. All waves are of the same lengths but amplitudes and phases may differ. The structure-factor equation, to be given here but not derived, deals with the relative intensities of the reflected rays rather than with the absolute amplitudes or intensities of reflected rays (which depend on the amplitudes and intensities of the x-rays used as a source). The relative intensity, /, of a ray of indices hkl, from a set of planes hkl, is 

The structure-factor equation may be applied to crystalline copper as follows the unit cell is face-centered cubic if we consider copper atoms to be in the corners (coordinates 0, 0, 0), there must be atoms also at the positions i, i, 0 0, i and 0, All atoms, being the same, have the 

This is now the relative intensity of any reflected x-ray beam (A, k and l being the beam indices). Recalling that cos (rnr) may be either 1 or — 1, depending upon whether n is even or odd, it may be proved (Exercise 8) that only those reflections will show for which A, k, and l are all odd or all even. The value of /Cu diminishes as the beam indices increase thus, aside from the systematic absences mentioned, the beams of high index tend to be fainter than the beams of low index. 

One may similarly use the structure-factor equation to predict characteristic absences or faintnesses or reflections from more complicated systems if there is more than one type of atom, the values for atomic scattering factors (/ values) must, of course, also be taken into consideration. 

Descriptions of the elements of symmetry used by crystallographers will not be given here. It is emphasized, however, that knowledge concerning 

---

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 structure—factor equation implies, and correctly so, that each reflection on the film is the result of diffractive contributions from all atoms in the unit cell. That is, every atom in the unit cell contributes to every reflection in the diffraction pattern. The structure factor is a wave created by the superposition of many individual waves, each resulting from diffraction by an individual atom. 

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. 

Show from the structure-factor equation for crystalline copper, as developed on page 320, that only those reflections will show for which h, k, and l are all odd or all even. 

Using the structure-factor equation (5) show that for a body-centered cubic cell all reflections will be absent for which the sum of the indices is odd. 

If a unit cell has a center of symmetry at the origin of coordinates, for every atom at point u, v, w, there will be a corresponding atom at point 1-u, l t>, l w. Show that under these conditions, the sine terms in the structure-factor equation total zero. 

For a crystal assumed to be infinite in extent, F(/i) exists only at reciprocal lattice points. The structure factor Equation (21), written in terms of the indices of reflexion and fractional coordinates within the unit cell g(xyz) is... 

Because the diffraction experiment involves the average of a very large number of unit cells (of the order of 10 in a crystal used for X-ray diffraction analysis), minor static displacements of atoms closely simulate the effects of vibrations on the scattering power of the average atom. In addition, if an atom moves from one disordered position to another, it will be frozen in time during the X-ray diffraction experiment. This means that atomic motion and spatial disorder are difficult to separate from each other by simple experimental measurements of intensity falloff as a function of sm6/X. For this reason, atomic displacement parameter is considered a more suitable term than the terms that have been used historically, such as temperature factor, thermal parameter, or vibration parameter for each of the correction factors included in the structure factor equation. A displacement parameter may be isotropic (with equal displacements in all directions) or anisotropic (with different values in different directions in the crystal). 

FIGURE 5.14 The curves represent the scattering power of various atoms, ranging from atomic number 1 to atomic number 90, as a function of sin2 0/X2, that is, as a function of resolution. Values for the scattering factor can be read directly from the graphs (or from tables) and used in the structure factor equation to calculate Fh/a ... 

Therefore, to better approximate the real scattering efficiency of each atom, the scattering factor fj in the structure factor equation is multiplied by an exponential term that effectively reduces the fj as a function of sin 9, where 9 is the Bragg angle. The new term has the form... 

If a crystal contains a center of symmetry among its space group elements, then for every atom at point xj = (xj, yj, zj), there is a corresponding atom at —xj = (—xj, —yj, —Zj). The structure factor equation for Fjt will therefore contain a term... 

In the structure factor equation the coefficients in the summation were all nonzero electron densities fj occurring at Xj, yj, Zj, which is really p(xj, yj, zj), and Fhki were the entities being calculated. Hence the coefficients in the electron density equation yielding p(x, y, z) must be the reciprocal space entities Fhki Finally, to keep units consistent, and the mathematics consistent with Monsieur Fourier, the sign of the imaginary term must be changed to minus, and the constant V must be inverted to 1/V, the volume of the reciprocal unit cell. Thus the electron density equation assumes the form. 

A problem remaining is that we must know the phase and amplitude of the reference wave provided by the heavy atom. We can, however, calculate both of these directly from the structure factor equation if we know the atomic number of the heavy atom, which we can read off a periodic table of the elements, and if we know its jc, y, z, position in the unit cell. Given this information, then we have at our disposal the amplitude of the unknown wave Fnat, the amplitude of the resultant wave Fderiv, the amplitude of the reference wave fuA, and the phase of the reference wave < >ha- We have the information required to solve for the phase angle of the native structure factor, the unknown wave, just as we did for the benchtop experiment above. 

It may not be obvious how we would locate the x, y, z coordinates of the heavy atom in the unit cell. Indeed it is sometimes not a simple matter to find those coordinates, but as for the heavy atom method described above, it can be achieved using Patterson methods (described in Chapter 9). As we will see later, Patterson maps were used for many years to deduce the positions of heavy atoms in small molecule crystals, and with only some modest modification they can be used to locate heavy atoms substituted into macromolecular crystals as well. Another point. It is not necessary to have only a single heavy atom in the unit cell. In fact, because of symmetry, there will almost always be several. This, however, is not a major concern. Because of the structure factor equation, even if there are many heavy atoms, we can still calculate Juki, the amplitude and phase of the ensemble. This provides just as good a reference wave as a single atom. The only complication may lie in finding the positions of multiple heavy atoms, as this becomes increasingly difficult as their number increases. 

Once a rotation and translation search has been successfully achieved, then it is possible to place the known molecule in the crystallographic unit cell of the unknown molecule so that its atoms assume the approximate coordinates of the corresponding unknown atoms. By the structure factor equation, the spatially transformed coordinates of the known molecule can be used to calculate phases for the Fuu of the unknown crystal. These phases, of course, will only be approximate because the molecules are not truly identical, yet, because they are structurally similar, the calculated phases may provide adequate estimates. These approximations can then be used as a starting point for improvement and refinement of the unknown molecules in both real and reciprocal space. As described in Chapter 10, this knowledge can ultimately guide us to the correct structure for the unknown. 

We had previously concluded from geometrical considerations that the base-centered cell would produce a 001 reflection but that the body-centered cell would not. This result is in agreement with the structure-factor equations for these two cells. A detailed examination of the geometry of all possible reflections, however, would be a very laborious process compared to the straightforward calculation of the structure factor, a calculation that yields a set of rules governing the value of F for all possible values of plane indices. 

In this case, the proper atomic scattering factors for each atom must be inserted in the structure-factor equation, which will have eight terms ... 

As discussed in Sec. 2-7, the sodium-atom positions are related by the facecentering translations and so are the chlorine-atom positions. Whenever a lattice contains common translations, the corresponding terms in the structure-factor equation can always be factored out, leading to considerable simplification. In this case we proceed as follows ... 

The student should carefully note that a lot of algebra can be eliminated, whenever a lattice is known to be centered in any way, by factoring common translations out of the structure-factor equation and inserting immediately the known values of the terms representing these translations. This shortcut procedure is illustrated for NaCl ... 

Since the structure is face-centered, we know that the structure factor will be zero for planes of mixed indices. We also know, from example (e) of Sec. 4-6, that the terms in the structure-factor equation corresponding to the face-centering translations can be factored out and the equation for unmixed indices written down at once ... 

Note that these positions involve a common translation, which may be factored out of the structure-factor equation. 

In practice the infinite sums can be reduced to a maximum number of harmonics. From Equation (32) it is easily seen that the Bessel functions diminish rapidly with the order of the satellite. The intensity of a Bragg reflection, proportional to the square of the structure factor Equation (35) or Equation (38), is then reduced due to the presence of the Bessel functions. 

The structure factor equation each atom in the unit cell contributes to each of the diffracted rays according to the following equation for the structure factor. 

Consider first the true crystalline state. Using the unit cell information, all the structure factors, equation 5.27, can be calculated and a graph can be prepared where each Bragg reflection appears with its intensity at the proper 0 location in a 0/intensity plot. A visually conspicuous, overall landscape view of the distribution of Bragg reflections in 0-space is thus obtained. This corresponds to the pattern obtained experimentally by diffraction from finely ground crystalline material, in which crystallites are oriented at... 

Certain special features to be imposed on a model may be expressed by more complicated constraint equations. We note as an example the assumption of a rigid molecule with prescribed dimensions whose position and orientation are to be refined. The position may be described by the coordinates of the centre of mass and the orientation by three Euler angles with respect to a unitary coordinate system. The atomic coordinates and thus the structure factor. Equation [1], are expressed as functions of these six parameters. The latter may then be adjusted to optimize the deviance. A similar procedure can be used to constrain the atomic displacement parameters of a molecule to rigid-body movements described by a translation tensor, a libration tensor and a transla-tion/libration-correlation tensor (TLS model). This model neglects intramolecular vibrations. 

---