#### 3.1. Computational Details

Note that the intensity of diffuse scattering, which is related to short and medium range order, is orders of magnitude smaller than that of the Bragg peaks. Fourier transformation of experimental total scattering data gives the pair distribution function, which is a direct probe of interatomic distances in real space. The window in

Q-space, which depends on the radiation source, defines the accuracy of the data in the real space. For elastic scattering, the diffraction vector,

Q, has a magnitude

where

λ is the wavelength of incident photons and 2

$\theta $ is the scattering angle. Since sin

$\theta $ ≤ 1, the experimentally accessible range of

Q is limited to less than 4π/λ. For example, the CuK

_{α} radiation, which is a common radiation source, has a wavelength of 1.54 Å with a

Q-range of about 8 Å

^{−1}, but for investigating nanoparticles, a

Q_{max} larger than 20 Å

^{−1} is needed. For our MnO data here, we have

Q_{max} = 25 Å

^{−1}, which enables us to achieve relatively good real-space accuracy.

We will use the PDF approach as a ‘small-box modelling’, where one considers relatively small unit cells with periodic boundary conditions. For periodic model structures, the PDF,

G(

r), can be computed in real space as:

where

b_{i,j} are related to thermal movements of the scattering atoms,

r_{ij} are the relative positions of the scattering atoms,

N is the number of particles and

ρ_{0} is the particle density.

The PDF provides atomic-scale structural insights from the distribution of atom–atom distances ranging from the local coordination scale to several nanometers. In addition to atom–pair separations (peak positions), the PDF also provides direct information about coordination number (integrated peak intensity), static and dynamic disorder (peak shape), and the coherent scattering-domain size (peak attenuation). Modelling the PDF for known periodic structures allows us to assess the effects of modifying the lattice constant, thermal vibration parameters, atomic positions and site occupancies. In addition, PDF-dependent shape parameters can also be used for extracting the radius of a spherically shaped nanoparticle.

#### 3.2. 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 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 Å [

43]. The first three coordination shells for Mn–O distances are (1/2)

a = 2.22 Å, (√3/2)

a = 3.84 Å, and (√5/2)

a = 4.97 Å, and the corresponding Mn–Mn (or O–O) distances are (√2/2)

a = 3.14 Å,

a = 4.44 Å and (√(3/2))

a = 5.44 Å.

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 Å

^{−1} were used using the Fourier transform, giving a real-space resolution of Δ

r ≈ 0.25 Å. 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 Å, 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 Belli et al. using EXAFS (Extended X-ray absorption fine structure) [

46].

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. In this way, periodic and non-periodic models can be evaluated on the same footing and different levels of structural information can be extracted. For crystalline samples, atomic structure can be determined completely, but for less ordered samples, the information obtained is more limited.

Figure 5 shows that the experimental data have the highest intensity at the Mn–O distance, indicating that we have MnO

_{6} octahedral building blocks or at least distorted/strained octahedral units. The arrangement of the neighboring octahedra is probably quite “random”, resulting in destructive signals. A simple fitting to the PDF data has been done in order to extract the lattice constant for the MnO data. The result of the analysis is shown in

Figure 6, and the fitted structure is illustrated in

Figure 7 with the lattice constant given by

a = 4.56 Å. A fit based on a hypothetical 57-atom cluster is discussed in the

Supplementary Material.

Our results are quite different from those for synthetic crystalline MnO

_{2} birnessite obtained by Petkov [

25], where sharp Bragg peaks are fitted with a model based on a hexagonal lattice rather than the cubic model invoked in our case. Petkov [

25] has also shown that very diffuse XRD patterns for bacterial and fungal MnO

_{x} can be described with monoclinic and triclinic lattices.