Armed with this all this nomenclature we can formulate a much simpler expression than the Debye equation for the scattering from a crystal, since we only need to consider the atoms in one unit cell. As was mentioned, the scattering from a perfect crystal can be described as diffraction that occurs at discrete spots and those spots can be indexed by three numbers, (h, k, l), that represent a direction in reciprocal space, \(ha^\ast +\ kb^\ast +lc^\ast \). I will use the historical (and misleading) term of “reflections” for those spots. The choice of the term reflection was a questionable choice since the physics of diffraction has nothing to do with the physics of reflections of light, such as when we see ourselves in a mirror. Nonetheless, it is a widely accepted terminology that I will not fight against.
A useful quantity for understanding reflections is called the structure factor, \(F_{hkl}\), which represents the scattering amplitude and phase for each reflection. This takes into account the scattering from every atom in the unit cell, where \[F_{hkl}=\ \sum _{ij}^{N}{f_i\exp (-2\pi i\vec {d_{hkl}^\ast } \cdot \vec {x_{j}})}\]
where \(\vec {d^*_{hkl}}\) is the vector in reciprocal space for reflection (h, k, l) and \(\vec {x_j}\) is the vector from origin to the atom location in real space. Now we can see the value of the reciprocal space construct from how it simplifies the dot product. The quantity \(\vec {d_{hkl}^\ast }\cdot \vec {x_j} = hx_j+\ ky_j+lz_j\), even for a triclinic system because between the reciprocal lattice was defined so that a*·b = a*·c = … = 0 and a*·a= … = 1. Note that \(F_{hkl}\) values will be complex numbers where the relative values of the real and imaginary components and their signs indicate a phase. The magnitude of the structure factor, \(|F_{hkl}|\), describes the strength of this particular diffraction peak. Conversion of \(|F_{hkl}|\) to the actual intensity observed for the Since hkl indicates a direction in reciprocal space, we can think of \(F_{hkl}\) as describing both the strength and phase for scattering in the hkl direction. Note that our previous definitions for reciprocal space require a*·b = a*·c = 0 and a*·a = 1 (and likewise permuted with b* and c*) and this means that (\(\vec {q_{hkl}}\cdot \vec {x_i}\)) can be simplified to (hxi + kyi + lzi). Thus, the structure factor can be rewritten as
\[F_{hkl}=\sum _j^N f_j \exp [-2\pi i(hx_j + ky_j + lz_j)]\]
Note that while the Debye equation is a double sum over all atoms in the material, this equation is a single summation over only the atoms in the unit cell. In very approximate terms, this has reduced the complexity of the computation by a factor of 1040 for the dimensions of a real sample. The structure factor in general is a complex value, where the complex part and the sign provides phase information, but when we measure reflection intensities we cannot directly observe phases. The observable intensities of reflections will depend on the square of the structure factor, as well as experimental effects depending on how the measurement is performed.