Scattering and diffraction. The structure factor
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The structure factors, F(hkl), are the fundamental quantities on which the function of electron density depends. These are very important magnitudes, since the maxima of the electron density function, ρ(xyz), show the location of the atoms, that is, the internal structure of the crystals.

The structure factors represent the diffracted waves, which when colliding with a photographic plate, or a detector, leave their mark in the form of well-defined spots that form the diffraction pattern. Therefore, from an experimental point of view, a crystal structure is defined by as many structure factors as spots contained in the diffraction pattern.

The structure factors, in terms of their amplitudes and phases, in the context of  crystal diffraction.

Sinusoidal representation of a diffracted wave

Analytically speaking, each structure factor can be considered as a vector, with its amplitude and phase (referred to an arbitrary origin of phases), and represents the total wave resulting from the co-operative dispersion (diffraction), caused by all the atoms of the cell, in a given direction of space.

The graphical representation of diffraction waves as vectors is equivalent to considering that the waves can also be represented as complex numbers. The real and imaginary parts of it correspond to the projections of the wave amplitude on the axes of the representation. The phase is the angle that forms the vector with the horizontal axis, which acts as the origin to which phases are referred.




In other words, as we shall see below, a structure factor, F(hkl), is the resultant of all waves scattered in the direction of the hkl reflection by the n atoms contained in the unit cell:


Each of these waves shows an amplitude proportional to the atomic scattering factor, ƒj, that measures the X-ray scattering power of each atom.

The atomic scattering factor is independent of the position of the atom in the unit cell. It depends only on the type of atom and the direction of scattering, so that it reaches a maximum in the same direction of the incident X-rays, and decreases as a function of the angle of departure. The variation of the scattering factor of the carbon atom is shown in the left drawing. At the angular value corresponding to (sin θ) / λ = 0, the magnitude of the atomic scattering factor is always equal to the total number of electrons in the atom, but decreases strongly as the angular value increases.

 In a first approach the scattering power of the different atoms does not depend on the wavelength of the X-ray radiation. However, there are side effects that make them different...

If the incident X-ray radiation has a frequency close to the natural oscillation frequency of the electrons of a given atom, there occurs the so-called anomalous dispersion, which modifies the atomic dispersion factor, ƒj (=ƒ0), so that its expression is modified with two terms, ƒ' and ƒ'', which represent the real and imaginary components, respectively, of the anomalous fraction of the atomic scattering factor...
 As mentioned above, a structure factor, F(hkl), is the resultant of all waves scattered in the direction of the hkl reflection by the n atoms contained in the unit cell. Its mathematical expression must therefore take into account the scattering from every atom contained in it.

But let's see how we can get the mathematical expression that defines it ...

Analytical expression of the phase
Suppose a crystal formed by the repetition of the atomic model constituted by the pair of atoms (red and blue) shown in the left figure. Of course, any crystalline model can be decomposed into as many simple lattices as atoms (look at the two lattices drawn below, red and blue).

Any crystalline model can be decomposed into simple lattices...



 If the Bragg's Law is true, the phase difference between the two reflected red beams will be 0° (= 360° = 2π radians). And the same will happen for the two reflected blue beams.

However, since there is a separation between the two lattices, there will be a phase shift between red and blue waves. and therefore the total diffracted intensity will be less than the arithmetic sum of both, red + blue intensities.

The resulting amplitude (ie the diffraction intensity) is controlled by the separation between the two lattices (ie by the shape of the motif being repeated), while the resulting diffraction geometry is the same as the one produced by of each single lattice. The diffraction geometry depends only on the lattice geometry.





The red X-ray beams, which are being reflected on the red planes of indices hk, fulfill the Bragg's Law, and in the same manner will behave the blue beams on the blue planes hk.

Said in other words, if Bragg's Law is fulfilled, the phase shift between waves reflected on planes of the same color, shifted (a/h) in the direction of the a-axis, must be 2π = 360º. For the same reason, the phase difference due to the plane separation (b/k) in the direction f the b-axis must also be 2π = 360º.

But the beams reflected on the blue planes show a phase shift respect to the red ones by a quantity (ΔΦ radians) that depends on the separation between the two lattices...






In fact, this offset can be easily calculated using the following "rule of three" proportions, applied to the three independent space directions:
a/h ..... 2π           b/k ..... 2π        c/l ..... 2π
x   ..... ΔΦa          y   ..... ΔΦb            z   ..... ΔΦ
ΔΦa = 2π h x/a
ΔΦb = 2π k y/b
 ΔΦc = 2π l z/c

Combining the three phase shifts, and generalizing to the three dimensions:
ΔΦ = 2π (h x/a + k y/b + l z/c)

Finally, taking fractional coordinates (that is, assuming x=x/a, y=y/b, z=z/c) and replacing  ΔΦ by Φ:
Φ = 2π (h x + k y + l z)  radians      (Formula 1)


Analytical expression of the structure factor
 
Once the mathematical expression of the phase is established in terms of the shape of the crystalographic model (Formula 1), let's see how to arrive at the analytical expression of the structure factor...
 
Suppose that ƒ1 represents the dispersion of the red atoms, and ƒ2 for the blue atoms (figure on the left). The total resultant dispersion of both atom types will be F(hkl)...

Writing it using vector notation...
F(hkl) = ƒ1 + ƒ2   
According to the scheme shown on the left, the module of this vector sum will be:

 
and its phase, referred to an arbitary phase origin:


Generalizing now for all atoms, and taking into account the general expression for the phase (Formula 1, above), the module of the structure factor will be:
         (Formula 2)

and its relative phase:


We have used the vector graphic representation to deal with diffraction waves, and this is equivalent to considering that waves can be represented as complex numbers. In this type of representation, the real and imaginary parts correspond to the projections of the wave amplitude on the cartesian axes, and the phase is the angle that forms the vector with the horizontal axis, which acts as an origin to which phases are referred.

Therefore, and taking into account Formula 2, the complex expression for the structure factor will be:


which, according to Euler's formula, can also be written as:

Formula 3.  The structure factor as a complex number


Evaluation of structure factors

If we know the internal structure of any crystal, ie the types of atoms (ƒj) that constitute it, and the positions (x,y,z) of all atoms (n) contained in the unit cell, we can immediatly calculate the structure factors, F(hkl), that define the crystal. To do this, it is enough to apply Formula 3, which actually involves calculating the inverse Fourier transform of the electron density function:

Fórmula 4.  The electron density function defined at the point (x, y, z) in the unit cell

The structure factors calculated with Formula 3 above, ie from the known atomic structure, are represented by vectors (modules and phases) and their numerical values, corresponding to the so-called absolute scale, since they are calculated with the dispersion factors (ƒj) that depend on the atomic numbers of the atoms existing in the unit cell.

However, the conventional situation is the opposite one. That is, we normally pretend to solve Formula 4, to determine the structure of the crystal by solving the function of electronic density at each point of the unit cell. And for this purpose we have to measure experimentally the structure factors using the X-ray diffraction. However, we must remember that experimentally we can only measure their modules, and therefore we have to face the so-called phase problem.

The modules of the experimental structure factors are related to the intensities of the diffracted beams, but these are in a relative scale, since they depend on multiple experimental aspects, such as the crystal dimensions and the brightness of the primary X-ray beam.

But let's go back...
 
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