From the perspective of information theory, it is not straightforward to categorize the amount of structural information in a powder diffraction pattern. From one perspective, every reflection that is within the measured data range does provide some information about the positions and ADPs for the atoms, even if it does not have intensity above the background level, but this information is compromised by the overlap of reflections. Since background levels are higher in powder diffraction than single-crystal diffraction (since single-crystal background is spread over \(4\pi \) steradians in volume while powder background is only spread in one dimension, so there will be a higher uncertainty about when a reflection intensity is small. In single-crystal diffraction, the number of observations tends to scale with the complexity of the structure, but that is not always the case with powder diffraction. The resolution and sensitivity of the instrument, as well as sample broadening effects limit the amount of information that can be measured with powders. Further, static and dynamic disorder can reduce the range where diffraction can be observed, so complex structures may provide data over a narrower range of Q than a simple structure, as much as we might wish otherwise.
With more complex structural models, one may not have enough well-determined independent observations to fit all independent parameters that are available. Better data may help this by separating more reflections or by improving signal-to-noise allowing one to see more weak peaks. Likewise, there are sometimes parameters that cannot be differentiated regardless of the data type. As an example of the former, for x-ray diffraction from a complex metal oxide, the ADP values for the O atoms, where each will have a minor impact on the overall scattering, may refine to pretty much random values. This is where constraints can help.
What constraints are is a way to reduce the complexity of a model by designating that parameters are linked together is some way. As an example, we can indicate that all O atoms in a model have the same \(\rm U_{iso}\) value, so if we have 10 O atoms, rather than having 10 \(\rm U_{iso}\) values, we now have a single parameter that dictates the \(\rm U_{iso}\) for all 10 atoms. A constraint can also be an equation. If we create a constraint that \(\rm Occ_1 + Occ_2 = 1\), where \(\rm Occ_j\) is the occupancy for atom \(j\), then what we are doing is to create a new parameter, which we can call \(\delta _{occ12}\) and rather than varying the two parameters \(\rm Occ_1\) and \(\rm Occ_2\) we vary \(\delta _{occ12}\) where the occupancies become \(\rm Occ_1-\delta _{occ12}\) and \(\rm Occ_2+\delta _{occ12}\).
Thus, in our complex metal oxide that we discussed above, where individual \(\rm U_{iso}\) values for O have a small leverage on the overall fit, what is commonly done is to introduce a constraint that all O atoms will share the same \(\rm U_{iso}\) value. This will increase how much impact the \(\rm U_{iso}\) value has on the quality of the fit, ensuring that a more accurate value is more likely and at the same time it decreases the complexity of the model. As an example for the case where parameters cannot be distinguished, consider a site that is shared by two different types of atoms. Since the coordinates or \(\rm U_{iso}\) value for each of these two atoms will contribute in exactly the same manner to every \(F_{hkl}\) value, changing will have the same effect as changing the other. The x, y and z fractional coordinates and \(\rm U_{iso}\) value for the two atoms sharing a site must be constrained to be the same.
Constraints are implemented in GSAS-II by summing the derivatives for all constrained parameters to determine the effect of changing the now grouped parameters and then when a shift is computed, all grouped parameters are set to the same value. Atom coordinates are handled slightly differently, in that shifts are applied to atom positions, but the effect is the same.
One special case of constraints are those of rigid bodies, where the geometry for a group of atoms that are refined together will be defined. Rigid bodies are sufficiently complex that they are described in a separate chapter, 26 and are implemented in another manner than what is described here.