Overview
The Allred group studies fundamental aspects of bonding in transition metal solids. Of particular interest are emergent phenomena, where the physical properties emerge indirectly from the collective interactions among atoms. These sort of effects tend to fall into the broad class of compounds known as quantum materials, where the properties emerge from complex and/or unconventional electronic states.
Our approach to better understanding bonding in these systems combines two main activities: crystallography/diffraction/scattering and the chemical application of group theory to phase transitions.
Tools
We rely on diverse diffraction methods, tailored to the problem at hand. We make good use of conventional powder and single crystal x-ray diffraction, before we progress to one of the many user facilities at national laboratories. Our recent work has made heavy use of total scattering (x-ray and neutron) applied to single crystals, which allows us to make detailed pictures of the local short-range and average long-range atomic structures simultaneously.
Total Scattering on single crystals
Total scattering can be thought of as the process of “Reciprocal space mapping” for the purpose of developing a detailed picture of the local environment of atoms in a lattice. The ideas behind this draw on a few different disciplines, so it is good to define terms.
Crystallography is usually presented as a tool applied to the long-range ordering which is manifested through diffraction in the form of Bragg peaks. However, there is always another, diffuse component to the diffraction that originates from short-range correlations in the material. Total scattering is the analysis of both portions of the scattering together. That is:
Total = Bragg (Long Range) + Diffuse (Short Range)
The total scattering can be thought of as occupying a hypothetical space, with each intensity corresponding to a point in this space. Each point in space corresponds to the three-dimensional (3D) momentum vector of the diffracted particle (x-ray, neutron, electron, etc.), which has units of inverse length. Thus, the diffraction measurements are mapping out was is usually known as either “Momentum Space” or “Reciprocal Space”.
The relative position of diffraction features tells us something about the periodicity of a structural correlation (such as the relative positions of atoms with respect to each other), while their intensities and shapes give information about what sort of correlation is being represented. How long range is it (correlation length)? Is it from atom type ordering or atom position ordering (substitutional vs. size effect scattering)?
A proven tool showing the power of total scattering analysis is the pair distOne example of a powerful application of total scattering measurements is the pair distribution function (PDF) analysis method. When applied to polycrystalline powders, the PDF method is a tried-and-true approach to uncovering ways in which a standard crystal structure solution might mislead the crystallographer about the actual atom-scale environment. Consider an example of a crystal wherein certain bonds alternate between long and short in a very disordered fashion. The crystal structure solution depends only on the Bragg peaks and may show uniformly medium medium bond lengths. The PDF method will also include the diffuse scattering portion of the total scattering, which is what is needed to resolve the two different bond length types.
3D-ΔPDF
One drawback of powder PDF is that the reciprocal space map is not available, and so the directions of correlations are also missing. To recover the 3D reciprocal space map, a single crystal is required. This approach has its own special challenges, though, and is too new to have a set of standards to follow.
In the course of total scattering measurements, our group has found several opportunities to explore this new technique. The ability to distinguish between different crystallographic axes as proven especially important in uncovering the nature of the local bonding.
Chemical Applications of Group Theory
The way a human or computer might talk about atom displacements during a structural transition is not always the most elegant way. For example, most crystal structure models depend on x,y,z coordinates of individual atoms, which correspond only to the translational vectors of the crystal and need not correspond to any meaningful distortion mode.
Borrowing from group theory, one can construct irreducible representations (irreps) of the atomic distortions in a lattice. Irreps have proven to be a much more natural and elegant way of describing structural distortions in a solid, since they tend to map to collective atomic displacements oriented to the point group symmetry of the atomic position. The irrep modes can often be directly tied to important parameters in phase transition theories such as Landau theory.
Our group has found that it is almost always helpful to use the irrep notation when possible. We have found that the number of parameters in a disordered structural model can be reduced by orders of magnitude by using an irrep basis, for example.