# Towards a quantitative theory for transmission X-ray microscopy

## TL;DR

This paper develops a model for X-ray microscopes to better understand how they capture images of tiny objects like gold nanospheres.

## Contribution

A new experimental and theoretical framework using Mie theory to evaluate TXMs with quantitative accuracy.

## Key findings

- Including both absorption and phase properties of gold is necessary for accurate TXM image modeling.
- A small tilt in condenser illumination improves qualitative agreement with measurements.
- The microscope underestimates absorption by 10–20% compared to the model predictions.

## Abstract

Transmission X-ray microscopes (TXMs) are now increasingly used for quantitative analysis of samples, most notably in the spectral analysis of materials. Validating such measurements requires quantitatively accurate models for these microscopes, but current TXM models have only been tested qualitatively. Here we develop an experimental and theoretical framework for evaluation of TXMs that uses Mie theory to compute the electric field emerging from a nanosphere. We approximate the microscope’s condenser illumination by plane waves at the mean illumination angle and the zone plate by a thin lens. We find that this model produces good qualitative agreement with our 3D measurements of 60 nm gold nanospheres, but only if both β and δ for the complex refractive index n = 1 – δ + iβ of gold are included in the model. This shows that both absorption and phase properties of the specimen influence the acquired TXM image. The qualitative agreement improves if we incorporate a small tilt into the condenser illumination relative to the optical axis, implying a small misalignment in the microscope. Finally, in quantitative comparisons, we show that the model predicts the nanosphere’s expected absorption as determined by Beer’s law, whereas the microscope underestimates this absorption by 10–20%. This surprising observation highlights the need for future work to identify the microscope feature(s) that lead to this quantitative discrepancy.