Diffraction techniques are making progress in tackling the difficult problem of solving the structures of nanoparticles and nanoscale materials. The great gift of x-ray crystallography has made us almost complacent in our ability to locate the three-dimensional coordinates of atoms in a crystal with a precision of around 10{sup -4} nm. However, the powerful methods of crystallography break down for structures in which order only extends over a few nanometers. In fact, as we near the one hundred year mark since the birth of crystallography, we face a resilient frontier in condensed matter physics: our inability to routinely and robustly determine the structure of complex nanostructured and amorphous materials. Knowing the structure and arrangement of atoms in a solid is so fundamental to understanding its properties that the topic routinely occupies the early chapters of every solid-state physics textbook. Yet what has become clear with the emergence of nanotechnology is that diffraction data alone may not be enough to uniquely solve the structure of nanomaterials. As part of a growing effort to incorporate the results of other techniques to constrain x-ray refinements - a method called 'complex modeling' which is a simple but elegant approach for combining information from spectroscopymore » with diffraction data to solve the structure of several amorphous and nanostructured materials. Crystallography just works, so we rarely question how and why this is so, yet understanding the physics of diffraction can be very helpful as we consider the nanostructure problem. The relationship between the electron density distribution in three dimensions (i.e., the crystal structure) and an x-ray diffraction pattern is well established: the measured intensity distribution in reciprocal space is the square of the Fourier transform of the autocorrelation function of the electron density distribution {rho}(r). The fact that we get the autocorrelation function (rather than just the density distribution) by Fourier transforming the measured intensity leaves us with a very tricky inverse problem: we have to extract the density from its autocorrelation function. The direct problem of predicting the diffraction intensity given a particular density distribution is trivial, but the inverse, unraveling from the intensity distribution the density that gives rise to it, is a highly nontrivial problem in global optimization. In crystallography, this challenging, nontrivial task is sometimes referred to as the 'phase problem.' The diffraction pattern is a wave-interference pattern, but we measure only the intensities (the squares of the waves) not the wave amplitudes. To get the amplitude, you take the square root of the intensity I, but in so doing you lose any knowledge of the phase of the wave {phi}, and half the information needed to reconstruct the density is lost. When solving such inverse problems, you hope you can start with a uniqueness theorem that reassures you that, under ideal conditions, there is only one solution: one density distribution that corresponds to the measured intensity. Then you have to establish that your data set contains sufficient information to constrain that unique solution. This is a problem from information theory that originated with Reverend Thomas Bayes work in the 18th century, and the work of Nyquist and Shannon in the 20 th century, and describes the fact that the degrees of freedom in the model must not exceed the number of pieces of independent information in the data. Finally, you need an efficient algorithm for doing the reconstruction. This is exactly how crystallography works. The information is in the form of Bragg peak intensities and the degrees of freedom are the atomic coordinates. Crystal symmetry lets us confine the model to the contents of a unit cell, rather than all of the atoms in the crystal, keeping the degrees of freedom admirably small in number. A measurement yields a multitude of Bragg peak intensities, providing ample redundant intensity information to make up for the lost phases. Finally, there are highly efficient algorithms, such as 'direct methods,' that make excellent use of the available information and constraints to find the solution quickly from a horrendously large search space. The problem is often so overconstrained that we can cavalierly throw away lots of directional information. In particular, even though Bragg peaks are orientationally averaged to a 1D function in a powder diffraction measurement, we still can get a 3D structural solution. Now it becomes easy to understand the enormous challenge of solving nanostructures: the information content in the data is degraded while the complexity of the model is much greater.« less
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Nov 7, 2023
Boris A Zakharov 
Open Access