Structure Characterisation

In this thesis, powder diffraction was predominately used to identify phases present in polycrystalline samples, and also determine the crystal structures of individual phases. The unique symmetry and elemental characteristics of each phase leads to a fingerprint pattern of reflections that can be identified in a mixture. The weight ratio of phases within the mixture also scales with the intensity of the Bragg peaks observed, and so some information on purity can also be obtained.

For phases in which the underlying crystal structure producing the diffraction pattern is not well known, the Rietveld method288 _{can be used to extract crystallographic information. This method} addresses powder specific structure solution problems such as overlapping peaks, and

crystallite shape effects. It uses least squares refinement to fit the measured pattern Yi0 at each point ‘i’ due to the ‘hth_{’ reflection, with a simulated pattern Yic:}

𝑌𝑌𝑖𝑖𝑖𝑖 =𝐼𝐼0�(𝑘𝑘ℎ𝐹𝐹ℎ2𝑚𝑚ℎ𝐿𝐿ℎ𝑃𝑃(∆ℎ) +𝐼𝐼𝑏𝑏) (2.1)

where I0 is the incident intensity, kh is the phase scale factor, Fh2 _{is the structure factor for the} reflection h, mh is the multiplicity of that reflection, Lh encompasses correction factors such as preferred orientation, P(Δh) is the function for peak shape and Ib is the background intensity. Simplified equation adapted from [289_].

The fit of the models is evaluated in this thesis using the metric ‘weighted profile residual’ (Rwp) which includes a statistical weight wi,[290_{] and is defined below:}

𝑅𝑅𝑤𝑤𝑤𝑤 = ��∑ 𝑤𝑤𝑖𝑖(𝑌𝑌𝑖𝑖0− 𝑌𝑌𝑖𝑖𝑖𝑖) 2

∑ 𝑤𝑤𝑖𝑖(𝑌𝑌𝑖𝑖0)2 � (2.2)

Several different radiation sources are used in this thesis to achieve different resolution and contrast in the diffraction data. These radiation sources and their experimental parameters are detailed below.

2.2.2 X-ray Powder Diffraction

The X-ray Powder Diffraction (XRPD) method involves using X-rays as a radiation source to determine structural information on powdered polycrystalline samples. X-rays are relatively easy to generate, and this is an accessible lab-based technique to monitor the purity of materials as they are synthesized and conduct structure refinement.

In this thesis, XRPD data were collected using a PANalytical Empyrean X-ray Powder Diffractometer with a Cu Kα1/α2 radiation source (λ = 1.5406 and 1.5444 Å). Powdered samples were loaded on silicon zero-background plates with a small amount of ethanol and dried. Data were collected over an appropriate angular range and scan rate for resolution in the pattern (typically 5 – 90° 2θ for an hour). Basic phase matching was conducted with HighScore Plus291_{. Refinement was conducted with Jana2006}292 _{using pseudo-Voigt peak} shapes, and model data were sourced from the International Crystal Structure Database293_. Further specific detail about refined parameters is given within the relevant chapters.

In Chapter 6, crystallite size (D, nm) is estimated from the diffraction data through the Scherrer294 _equation:

𝐷𝐷= _𝛽𝛽cos𝐾𝐾𝐾𝐾_𝜃𝜃 (2.3)

where K is the shape factor (taken = 0.9 in this investigation), λ is the radiation wavelength, β

is the line broadening at full width half maximum (FWHM) fitted via Jana2006, and θ is the Bragg angle for the relevant peak.

While XRPD is used extensively in this thesis, it is important to note that as X-rays scatter from the electrons of atoms, scattering strength scales linearly with atomic number. This can lead to issues refining crystal structure that contain light elements, or neighbouring elements on the periodic table. Some elements will also fluoresce X-rays if they are of similar atomic weight to the element used as the X-ray source, resulting in a high background (and low signal to noise ratio). For this reason (where possible) X-ray diffraction was complemented with other diffraction techniques.

2.2.3 Neutron Powder Diffraction

Unlike X-rays, neutrons scatter from the nuclei of atoms through the strong nuclear force. This interaction varies non-monotonically with increasing atomic number and is sensitive to an element’s isotope. Electron magnetic moments can also scatter neutrons, as neutrons possess a spin. In this thesis, the neutron powder diffraction technique is used to obtain contrast between elements of a similar atomic mass and information on magnetic structure.

In Chapter 3 specifically, data from neutron diffraction experiments are presented. The high resolution powder diffractometer ‘Echidna’295 _{at the Australian Centre for Neutron Scattering} was used to collect room temperature powder diffraction patterns. A monochromatic neutron beam of wavelength of 1.622 Å was used and the data were fitted using Jana2006. These samples were then measured on a different high intensity, medium resolution diffractometer called ‘Polaris’296 _{at the ISIS Muon and Neutron Source (Rutherford-Appleton labs, United} Kingdom). With this instrument, room temperature and 500 K data were obtained in situ using a double-skinned resistive element furnace from a sample sealed in a 1.5 mm glass capillary. A polychromatic neutron beam produced from a spallation source was used, and the data were processed and refined using Mantid297 _{and GSAS}298, 299 _{respectively.}

2.2.4 Electron Diffraction

An electron beam can also be used as a radiation source for diffraction studies. A convenient way to obtain information is by using Transmission Electron Microscopy (TEM). With this method an electron beam can be passed through very small (ideally) individual crystallites and an intensity pattern can be collected on the other side.

In electron diffraction, the electrons scatter off the electrostatic Coulomb potential. This interaction is very strong and as a consequence, multiple diffraction events can occur before the electrons leave the sample. For this reason, unlike with neutrons and X-rays, refinement of the intensity is not meaningful. However, due to the strong interaction weak reflections and non-Bragg intensity features can be seen in Electron Diffraction Patterns (EDPs). In this thesis, electron diffraction is used to supplement diffraction information from the aforementioned methods.

EDPs presented in Chapters 3, and 4 and were collected using a 200 kV JEOL2100F FEGTEM Transmission Electron Microscope at the Centre for Advanced Microscopy, ANU.

Patterns were collected from samples ground in n-butanol and dropped onto a holey carbon film on a copper grid. This gave a well dispersed array of crystallites that could be individually isolated for analysis.

2.2.5 Structure Validation from Simulation 2.2.5.1 Bond Valence Sum

In order to validate crystal structures determined through Rietveld refinement, one can calculate the bond valence sum of ions in the structure. The sum (V) of bond valences (vi) from each bond around an atom should add to its formal valence state per Pauling’s principles for electrostatic valence300_{. The bond valence of each bond to atom i is calculated as:}

𝑣𝑣𝑖𝑖 = exp�𝑅𝑅0−_𝑏𝑏 𝑅𝑅𝑖𝑖� (2.4)

where R0 is the ideal bond length (calculated such that the atom i will have an exact valence), Ri is the experimentally determined bond length and b is an empirical constant (typically 0.37 Å301_{). This comparison to empirical data allows one to judge if a refinement is reasonable based} on data from other structures with the same elements and valence states.

Large deviations are expected to make the structure ‘unstable’, and an overall parameter called the ‘global instability index’ (GII) can be calculated:

𝐺𝐺𝐼𝐼𝐼𝐼= �∑𝑁𝑁𝑖𝑖=1_𝑁𝑁𝑑𝑑𝑖𝑖2 (2.5) where di is the deviation between the bond valence sum and the formal valence state of atom i, and N is all atoms in the structure. The GII of the structure has been correlated before with the total energy calculated by density functional theory302_.

After refinement of a crystal structure, a ‘.cif’ file was generated and input into the web program ‘SoftBV’301, 303 _{for evaluation of each ion's bond valence sums and the overall GII. Deviations} lower (‘underbonding’) and higher (‘overbonding’) than the ideal valence were considered points within the structure for further refinement and/or local structure investigation.

2.2.5.2 Density Functional Theory

Evaluation of the total energy of condensed matter systems can be performed using computational quantum mechanical means. In this thesis density functional theory (DFT) is used to calculate low energy crystal structures and magnetic configurations. This method uses the idea that the system's total energy can be determined in terms of functionals of the electron density in the system304_{. The DFT method is limited by the use of an exchange-correlation} approximation when calculating total energy. Given metal oxide systems such as the ones under investigation are strongly correlated, the choice of approximation will influence the results, but these calculations are a convenient method for predicting the properties of large periodic systems nonetheless.

In this thesis the code package WIEN2k305 _{was used to perform the DFT calculations} associated with Chapter 3. The GGA+U306 _{approach was used, with the PBE-GGA exchange} correlation functional and the on-sight repulsion U and on-sight exchange J from the literature58_{. Radial muffin tin sizes of Bi (2.3 a.u.), Fe (1.94 a.u.), Cr (1.94 a.u.), and O (1.67} a.u.) were used to simulate the atomic potentials. A grid of 64 k points was applied, and the product of the muffin tin size and maximum k vector was set to 8 to optimize computational time. The starting geometry was a single unit cell of BFCO (though converted to P1 space group for magnetic spin analysis) which had 6 iron/chromium sites, and forces were converged at <1 mRy/Bohr to attain a relaxed geometry. Both a double perovskite unit cell and a unit cell with a single antisite defect were calculated and compared for total energy and magnetic moment distribution. Initial alignments of the magnetic moments were set to be antiparallel (to form an antiferromagnetic structure).

While diffraction and simulation give information about the crystalline nature of the phases within produced samples, more information on the chemical nature and morphology can be obtained by other means. The following section details spectroscopy and microscopy techniques to further characterise the synthesized materials.

2.3

Composition and Morphology Characterisation