# Structure of Manganese Oxide Nanoparticles Extracted via Pair Distribution Functions
## 3 Results and Discussion
### An Illustrative Analysis of the MnO Data  
Manganese can exist in the form of a variety of stable oxides (MnO, Mn\({}_{3}\)O\({}_{4}\), Mn\({}_{2}\)O\({}_{3}\), MnO\({}_{2}\)) [41; 42], which crystallize in different types of structures. Associated with this wide diversity of crystal  
Figure 2: \(l(Q)\) data for the MnO nanoparticle film sample. forms, defect chemistry, morphology, porosity and textures, manganese oxides exhibit a variety of distinct electrochemical properties. For example, MnO\({}_{2}\) exists in six different polymorphs (pyrolusite, ramsdellite, hollandite, intergrowth, spinel, and layered), all of which share basic structural features--small Mn\({}^{4+}\) ions in a spin-polarized 3d\({}^{3}\) configuration and large, highly polarizable O\({}^{2-}\) ions in a spin-unpolarized 2p\({}^{6}\) configuration, which are arranged in corner- and edge-sharing MnO\({}_{6}\) octahedra. These octahedral units are characteristic for each oxidation state of manganese oxide and play a crucial role in determining the electrochemical properties of various oxides. MnO crystallizes in the so-called rock-salt structure, which is a face-centered cubic (fcc) lattice with a 6:6 octahedral coordination. The experimental lattice constant at room temperature is \(a\) = 4.444 A [43]. The first three coordination shells for Mn-O distances are (1/2)\(a\) = 2.22 A, (\(\surd\)3/2)\(a\) = 3.84 A, and (\(\surd\)5/2)\(a\) = 4.97 A, and the corresponding Mn-Mn (or O-O) distances are (\(\surd\)2/2)\(a\) = 3.14 A, \(a\) = 4.44 A and (\(\surd\)(3/2))\(a\) = 5.44 A. We obtained the PDF by using PDFgetX3 [44], which is a command-line utility for extracting atomic pair distribution functions from X-ray diffraction data. Data up to \(Q\) = 25.00 A\({}^{-1}\) were used using the Fourier transform, giving a real-space resolution of \(\Delta r\) \(\approx\) 0.25 A. The PDF data were further analyzed using the DiffPy-CMI package [45], which is a library of Python modules for robust modelling of nanostructures in crystals, nanomaterials, and amorphous materials. Figure 3 shows the PDF for MnO obtained by converting the experimental \(I(Q)\) data of Figure 2. The coherent scattering domain size is seen to be only about 5 A, with the peaks attenuating rapidly at larger distances. This feature may reflect the glass-like film assembly because the particles are crystalline. Figure 4 compares the PDFs for bulk MnO (cubic rock salt structure) and four different MnO\({}_{2}\) polymorphs. MnO and MnO\({}_{2}\) can be distinguished easily by examining the positions of the first two peaks in the PDFs, which are from the Mn-O (first peak) and Mn-Mn/O-O bonds (second peak). The bonding in different MnO\({}_{2}\) structures is similar at short range, but differences can be seen in the medium-range bonds. Similarities reflect the presence of MnO\({}_{6}\) building blocks that are common to all the MnO\({}_{2}\) polymorphs, while differences arise in the details of how these MnO\({}_{6}\) octahedra are arranged in space. MnO\({}_{6}\) octahedra have been studied earlier by [PERSON] et al. using EXAFS (Extended X-ray absorption fine structure) [46]. Figure 3: PDF for MnO nanoparticle film sample obtained from the data given in Figure 2.  
Figure 5 shows the computed partial PDFs for bulk MnO. The calculated total PDF shown in Figure 6 is the sum of these partial PDFs. Partial PDFs help in assigning the peaks with their corresponding bonds. Notably, the PDF method can be used to assess the degree of structural coherence in a sample in terms of the experimental PDF. In particular, positions of peaks in the PDF can be related to the distribution of characteristic inter-atomic pair distances in the sample (In our case, the data are seen to be sensitive to only the first few pair distances). A PDF can be easily computed for any model configuration of atoms and thus allows convenient testing and refining of three-dimensional structure models of materials with varying degrees of structural coherence.