Of course, our imaginary ideal crystal differs from real-world crystals in many ways. One way that real crystals differ from our idealized concept is that atoms are constantly in motion. Usually, this motion is such that the atoms vibrate around some point in the structure so that the time-averaged positions are the same from one unit cell to the next. Since the overall amount of vibration is determined by the temperature of the material, this is called thermal motion but this motion does not completely disappear at lower temperatures and if 0 K were achievable, atoms would still have what is called zero-point motion. These instantaneous deviations from ideal siting are called dynamic disorder. Another expected deviation from our ideal model is that atoms may not be in exactly the same positions from one unit cell to the next; this is known as static disorder. The effect of this is indistinguishable in diffraction measurements from thermal motion unless this is studied as a factor of temperature, though other measurements, such as neutron inelastic scattering will be sensitive to the difference.
The intensities of reflections are lowered because this thermal motion and static displacements spreads out the siting of the nucleus or electrons over some region and this lowers the intensities of the higher Q reflections for the same reason as why f (Q) decreases with angle in x-rays. This was noted and explained very early in the history of crystallography with what is known as the Debye-Waller factor, which relates this lowering of intensities to the mean atomic displacements averaged over all atoms, but a better formulation, since atoms can have very different amounts of motion depending on their mass and bonding environment, is to formulate the structure factor equation with a term for each atom in this fashion:
\[F_{hkl}=\sum _j^N f_j \exp [-2\pi i(hx_j + ky_j + lz_j)] \exp [-2\pi i(\vec {Q}\cdot \vec {U_j})]\]
The term that has been added, \(\exp [-2\pi i(\vec {Q}\cdot \vec {U_j})]\), defines an atomic displacement parameter for atom j, as a tensor \(\vec {U_j}\), noting that the subscript j allows different values or even levels of expansion for each atom. The term atomic displacement parameter, models how the atom is distributed away from its average position, due to both static and dynamic disorder. This an important concept in crystallography and is commonly abbreviated as ADP, particularly in the single-crystal structure field. The term thermal parameter used to be used for what we now call atomic displacement parameter, but since ADPs incorporate both static and dynamic disorder and not all dynamic disorder is thermal, APD is a much better term than thermal parameter.
Note that different forms can be used for \(\vec {U_j}\). The simplest form uses a scalar isotropic term, \(U_{iso,j}\), where the final term in the structure factor equation becomes \(\exp [-8\pi ^2U_{iso,j}\sin ^2{\theta }/\lambda ^2)]\). Note that \(U_{iso,j}\) is expressed in Å\({}^2\) and is scaled so that \(\sqrt {3 U_{iso,j}}\) is the root-mean-square displacement for atom j.
There are also more complex descriptions for ADPs: In addition to the scalar \(\rm U_{iso}\) description, which describes the displacements as an isotropic sphere, crystallographers also commonly use a symmetrical 3x3 tensor to describe an anisotropic ellipsoidal distribution where atomic displacements have different amplitudes in different directions. The most common way this is done is with six anisotropic atomic displacement terms, \(\rm U_{ij}\) (where i and j can be 1, 2 or 3; but note that \(\rm U_{ij} = U_{ji}\) so only six unique terms are needed). The \(\rm U_{ij}\) terms describe an ellipsoid that has major axes that will not align with a, b & c when the \(i \neq j\) terms are non-zero. In the anisotropic case this term becomes \(\exp [-2\pi ^2(h^2\operatorname {a}^{\ast 2}U_{11,j}+k^2\operatorname {b}^{\ast 2}U_{22,j}+\ l^2\operatorname {c}^{\ast 2}U_{33,j}+2hk\operatorname {a}^\ast \operatorname {b}^\ast U_{12,j}+2kl\operatorname {b}^\ast \operatorname {c}^\ast U_{23,j}+2hl\operatorname {a}^\ast \operatorname {c}^\ast U_{13,j})]\) and the \(\rm U_{ii}\) (diagonal) terms are again the squares of the root-mean-square displacements. This description is known to crystallographers as anisotropic ADP’s but this ignores that there are exist more complex descriptions of atomic displacement than \(\rm U_{ij}\) with more than six terms, these are usually referred to as anharmonic displacements. It is fairly uncommon for powder diffraction data to be sufficiently sensitive to atomic displacements that \(\rm U_{ij}\) terms can be determined. It is quite rare (though not unheard-of) for even higher order cumulants to be used in crystallography than \(\rm U_{ij}\) terms. In any case, GSAS-II only offers \(\rm U_{iso}\) and \(\rm U_{ij}\) descriptions for atomic displacements.
While it is historically common to refer to the \(\rm U_{iso}\) and \(\rm U_{ij}\) terms as “thermal parameters”, this usage should be avoided since these terms indicate both static and dynamic displacements of atoms from their mean sites and thus are not only thermal. Nonetheless, in most materials ADPs will scale with temperature and it is common to collect diffraction data at low temperatures to reduce them. Reduced ADP values provide stronger scattering at high Q, usually leading to more accurate diffraction structure determinations. I say usually here, because sometimes materials transform to lower symmetry at low temperature, which requires description of the material with a complex structure (and different).
Other deviations from ideal behavior that are sometimes seen include: An atom may be missing from some unit cells, known as a vacancy, or a unit cell may have extra atoms, known as interstitials. The composition may also vary from one box to another, where for example, in zeolites where aluminum and silicon share roles in building up the infinite structure of the material together, in some unit cells a silicon atom is located at a location but in other unit cells an aluminum atom will occupy that location in the structure. Note that since Al normally has slightly longer bonds than Si, one should also expect a bit of static disorder in that region of the box.
Entropy considerations mean that no crystal can be completely perfect and that there will be locations where the unit cells will not stack perfectly. These are known as defects or dislocations. There can be boundaries where the unit cells are not in exact alignment with each other, likely due to defects of some sort or regions with different composition. Even though on average the unit cells are all aligned across the entire crystal, the crystal can be considered as made up of smaller blocks where within each block the alignment is nearly perfect. This character for a crystal is called its mosaic behavior. Most crystals have some degree of mosaic character, and its presence is very much desired for single-crystal diffraction measurements.
Finally, we have used here a classical picture of a crystal with only three-dimensional ordering. As mentioned, it is possible to have modulated structures, where atoms are displaced within the unit cells following regular patterns that require at least one extra dimension of ordering to describe the structure. The structure must then be described using four or more directions, which is known as supersymmetry. Finally, quasicrystals require a description in six-dimensional space. GSAS-II does allow treatment of four dimensional superspace structures, but not five or six-dimensional space.