Bayesian Analysis of Ceria Nanoparticles from Line Profile Data

A Bayesian/Maximum entropy (MaxEnt) method is applied to quantify the broadening of Xray line profiles in terms of the nanocrystallite size effects in ceria. The analysis is in general agreement with transmission electron microscopy results, while demonstrating the importance of appropriate a priori information needed in the method. The analysis also identifies other microstructural effects, such as the presence of dislocations and shape anisotropic effects, which may be influencing the size distributions determined from the Bayesian/MaxEnt method. INTRODUCTION A critical step in developing functional nanoparticles is the development of methods for characterising their shape and size properties from techniques such as electron microscopy and X-ray diffraction. Furthermore, the rapid and accurate analysis of this data is particularly important in the growing nanotechnology industry. In this paper, we present the application of a Bayesian/Maximum entropy (MaxEnt) method for analysing the broadened X-ray line profiles from ceria. This method quantifies the broadening in terms of the microstructure that gives rise to it, including crystallite size distribution, shape and internal defects [1]. This process of exacting and relating the broadened line profiles to the material’s microstructure is known as solving an inverse problem. The Bayesian/MaxEnt method has been specifically developed to solve these types of problems. An important aspect of the Bayesian/MaxEnt method is its ability to consider different size distribution models, and to examine their influence on the size distribution or solution determined to be the most probable. This is achieved by defining the models as a priori information. This approach advances the analysis carried out by Armstrong et al. [2, 3], where a single lognormal distribution was applied as the model without a firm physical basis. In contrast, this study uses both lognormal and gamma distributions functions as models, and examines their influence on the size distribution determined from the Bayesian/MaxEnt method. This work is important in the development of the NIST Nanocrystallite Size Standard Reference Material (SRM 1979), where characterisation of the material’s microstructure and models is necessary in the certification of the SRM. Copyright ©JCPDS International Centre for Diffraction Data 2005, Advances in X-ray Analysis, Volume 48. 59 This document was presented at the Denver X-ray Conference (DXC) on Applications of X-ray Analysis. Sponsored by the International Centre for Diffraction Data (ICDD). This document is provided by ICDD in cooperation with the authors and presenters of the DXC for the express purpose of educating the scientific community. All copyrights for the document are retained by ICDD. Usage is restricted for the purposes of education and scientific research. DXC Website – www.dxcicdd.com ICDD Website www.icdd.com The analysis is applied to commercially available cerium oxide samples. We show that the Bayesian/MaxEnt method can determine the size distribution, while requiring a minimal number of assumptions. Transmission electron microscopy (TEM) crystallite size data was also collected to independently test the Bayesian/MaxEnt method. THEORETICAL OVERVIEW Nanoparticle size broadening of X-ray line profiles can be described in terms of the size distribution of the particles [2, 3]