# Structure of Manganese Oxide Nanoparticles Extracted via Pair Distribution Functions
## 1 Introduction  
Among the metal oxides, manganese oxide plays a vital role in technological applications. Manganese oxide microstructures and nanostructures are used, for example, in waste-water treatment, catalysis, biomedicine, air-purification sensors, super-capacitors, and rechargeable batteries [1; 2; 3; 4; 5; 6; 7; 8]. Transition metal oxides, especially MnO and MnO\({}_{2}\), are good candidates for anodes in batteries as they provide high capacity and low cost and are also environmentally friendly [9; 10; 11]. X-ray scattering techniques are useful for characterizing the sizes of crystals and particles as well as their crystallographic phases, which taken together control their physical properties. These techniques are generally non-destructive and measure properties averaged over an ensemble of many particles. Most crystal structures have been determined via X-ray diffraction (XRD) experiments. The scattering pattern from a crystal typically consists of several tens or even hundreds of Bragg reflections from various atomic planes in the lattice. Crystal structures can be specified in terms of just a few variables, such as the unit cell parameters and the positions of the symmetry-independent atoms. Direct imaging techniques such as electron microscopy also provide unique opportunities for structure analysis at a more local scale and thus complement diffraction techniques [12; 13]. A transmission electron microscope (TEM), for example, can provide multidimensional analysiscapabilities for characterizing nanoparticles and can be used to image nanoscale features with atomic resolution. Electron microscopy normally measures a two-dimensional (2D) projection of the three-dimensional (3D) structure, although the 3D structure can be obtained via electron tomography [14; 15; 16]. The increasing complexity of nanomaterials has driven the development of even more advanced characterization techniques combined with tomography. Energy-dispersive X-ray (EDS) spectroscopy [17; 18] and electron energy loss spectroscopy (EELS) [19] are examples of such advances, which have enabled the investigation of the morphologies, compositions, and oxidation states of nanomaterials. X-ray scattering of nanoparticles is much weaker than that of bulk materials due to their small volume and limited coherence. In order to increase the signal, samples must be measured either over long times or with high-flux sources. For nanostructures, diffraction patterns often show only a few Bragg reflections, which makes conventional crystallography challenging. Moreover, the assumptions underlying the analysis neglect information on structural disorder due to high frustrated surface contributions. The key is to recognize that a large amount of information here is stored in diffuse scattering, which is normally not taken into account in conventional X-ray analysis. It is clear that other methods are needed for a complete understanding of the structure of nanoparticles. High-energy XRD (HE-XRD) measurements combined with a Pair Distribution Function (PDF) analysis can yield valuable information on the short-range atomic ordering of nanocrystalline materials with some degree of structural coherence and periodicity. A large body of literature [20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37] details how the PDF analysis can enable a complete structural investigation of a nanostructured material. Application to nanomaterials, however, has become practical only recently, with the advent of high-flux and high-energy synchrotron radiation sources. Our study is motivated by the growing interest in applying high-energy X-ray scattering techniques to investigate nanostructures. We emphasize that the PDF method analyzes all the scattered intensity and it can therefore probe deviations from perfect crystallinity. This is important because nanomaterials can be ordered and possess periodic atomic arrangements like ordinary crystals, but they can also be disordered and/or have nonperiodic atomic arrangements like glasses.