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In the previous pages, we have seen that diffraction can be understood in reciprocal space, and that a reflection $hkl$hkl corresponds to a reciprocal lattice vector $\mathbf{g}_{hkl}$ghkl​. However, the fact that a reflection “can occur” is not the same as that reflection being “actually strong.” What determines the reflection intensity is the arrangement of atoms within the unit cell. The quantity that summarizes this information is the structure factor $F(hkl)$F(hkl).

Definition of the Structure Factor

Consider a unit cell containing $N$N atoms. The $j$j-th atom is located at fractional coordinates $(x_j, y_j, z_j)$(xj​,yj​,zj​) and has an atomic scattering factor $f_j$fj​. The structure factor for a reflection $hkl$hkl is given by$$
F(hkl)= \sum_{j=1}^{N} f_j \exp[2\pi i (hx_j+ky_j+lz_j)]
$$This is the sum of scattered waves from each atom expressed as complex numbers. The term $hx_j+ky_j+lz_j$hxj​+kyj​+lzj​ appearing in the exponent represents the phase difference that each atom contributes to that reflection. As this formula shows, reflection intensity is not simply determined by whether there are “many atoms.” Depending on the arrangement of atoms within the unit cell, scattered waves may constructively interfere or cancel each other out. Therefore, even for the same crystal, the intensity varies greatly from one reflection to another.

The observed diffraction intensity $I(hkl)$I(hkl) is proportional to $|F(hkl)|^2$∣F(hkl)∣2. It is expressed more precisely as$$
I(hkl)=S \cdot L \cdot P \cdot A \cdot |F(hkl)|^2
$$where $S$S is the scale factor, $L$L is the Lorentz factor, $P$P is the polarization factor, and $A$A is the absorption correction. In practice, temperature factors (Debye-Waller factor) and multiplicity are also included, so the observed intensity results from a product of many correction terms applied to the squared modulus of the structure factor.

Example: Body-Centered Cubic (BCC)

As a simple example, consider a BCC structure. The unit cell contains atoms at the origin $(0,0,0)$(0,0,0) and at the body center $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$(21​,21​,21​). In this case$$

$$That is, reflections with an odd sum are completely extinguished. This is the most basic example of the systematic absences characteristic of a body-centered lattice.

Similarly, when there is a specific regularity in the atomic arrangement within the unit cell, as in face-centered cubic or diamond structures, certain types of reflections become systematically weak or completely vanish. The systematic organization of such “systematic absences arising from symmetry and translations” is the topic of reflection conditions, which will be covered on the next page.

Note that the structure factor $F(hkl)$F(hkl) is a complex number. What can be directly observed in diffraction experiments is $|F(hkl)|^2$∣F(hkl)∣2, and the phase itself is lost. This is the starting point of the phase problem in crystal structure analysis. However, in this series, we will first organize reflection conditions and systematic absences, and then apply them to problems such as determining crystal structures and identifying space groups.