# Uncertainty evalutation through data modelling for dimensional nanoscale measurements

## TL;DR

This paper introduces a Bayesian data modelling approach to evaluate measurement uncertainty in nanoparticle dimensional measurements, enabling comprehensive uncertainty quantification without relying on detailed physical models.

## Contribution

It proposes a novel statistical method for uncertainty evaluation in nanoparticle measurements using Bayesian modelling and calibration curves, applicable when physical models are incomplete.

## Key findings

- Generated a probability density function for nanoparticle dimensions.
- Successfully applied the method to AFM measurements with nano-gratings.
- Provided a comprehensive uncertainty quantification framework for nanoscale measurements.

## Abstract

A major bottleneck in nanoparticle measurements is the lack of comparability. Comparability of measurement results is obtained by metrological traceability, which is obtained by calibration. In the present work the calibration of dimensional nanoparticle measurements is performed through the construction of a calibration curve by comparison of measured reference standards to their certified value. Subsequently, a general approach is proposed to perform a measurement uncertainty evaluation for a measured quantity when no comprehensive physical model is available, by statistically modelling appropriately selected measurement data. The experimental data is collected so that the influence of relevant parameters can be assessed by fitting a mixed model to the data. Furthermore, this model allows to generate a probability density function (PDF) for the concerned measured quantity. Applying this methodology to dimensional nanoparticle measurements leads to a PDF for a measured dimensional quantity of the nanoparticles. A PDF for the measurand, which is the certified counterpart of that measured dimensional quantity, can then be extracted by reporting a PDF for the measured dimensional quantity on the calibration curve. The PDF for the measurand grasps its total measurement uncertainty. Working in a fully Bayesian framework is natural due to the instrinsic caracter of the quantity of interest: the distribution of size rather than the size of one single particle. The developed methodology is applied to the particular case where dimensional nanoparticle measurements are performed using an atomic force microscope (AFM). The reference standards used to build a calibration curve are nano-gratings with step heights covering the application range of the calibration curve.