Towards a quantitative theory for transmission X-ray microscopy

James G. McNally, Christoph Pratsch, Stephan Werner, Stefan Rehbein, Andrew Gibbs, Jihao Wang, Thomas Lunkenbein, Peter Guttmann and Gerd Schneider

Beilstein J. Nanotechnol. 2025, 16, 1113–1128. https://doi.org/10.3762/bjnano.16.82

Supporting Information

The Supplement is a single document with multiple sections listed according to their order of citation in the main text. The different sections provide (1) detailed explanations of the different models: S1 – Mean angle condenser; S2 – Plane-wave model of the condenser; S3 – Extension of the Mie theory to oblique plane waves; S4 – Simplification of the Mie theory to the anomalous diffraction approximation; S5 – Equivalency of the Mie theory and the parabolic wave equation; S6 – Calculation of the nanosphere’s plane-wave and scattered-wave fields at the camera; S7 – Incoherent Beer’s law model; S11 – Calculation of polar tilt angle as a function of azimuthal angle in a tilted condenser; (2) calculation of specific model predictions: S8 – Effect of coherence patch size in the mean-angle condenser model; S12 – Radial intensity profiles of the tilted vs. non-tilted theory; S14 – Sensitivity of radial intensity profiles to deviations in nanosphere geometry; and (3) descriptions of experimental procedures and measurements: S9 – Rotation of the experimental data around the optical axis; S10 – Estimation of the axial tilt in the data; S13 – Structural characterization of the 60 nm gold nanospheres by SEM and TEM.

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Towards a quantitative theory for transmission X-ray microscopy

James G. McNally, Christoph Pratsch, Stephan Werner, Stefan Rehbein, Andrew Gibbs, Jihao Wang, Thomas Lunkenbein, Peter Guttmann and Gerd Schneider

Beilstein J. Nanotechnol. 2025, 16, 1113–1128. https://doi.org/10.3762/bjnano.16.82

How to Cite

McNally, J. G.; Pratsch, C.; Werner, S.; Rehbein, S.; Gibbs, A.; Wang, J.; Lunkenbein, T.; Guttmann, P.; Schneider, G. Beilstein J. Nanotechnol. 2025, 16, 1113–1128. doi:10.3762/bjnano.16.82

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TY  - JOUR
A1  - McNally, James G.
A1  - Pratsch, Christoph
A1  - Werner, Stephan
A1  - Rehbein, Stefan
A1  - Gibbs, Andrew
A1  - Wang, Jihao
A1  - Lunkenbein, Thomas
A1  - Guttmann, Peter
A1  - Schneider, Gerd
T1  - Towards a quantitative theory for transmission X-ray microscopy
JF  - Beilstein Journal of Nanotechnology
PY  - 2025///
VL  - 16
SP  - 1113
EP  - 1128
SN  - 2190-4286
DO  - 10.3762/bjnano.16.82
PB  - Beilstein-Institut
JA  - Beilstein J. Nanotechnol.
UR  - https://doi.org/10.3762/bjnano.16.82
KW  - 3D imaging
KW  - mathematical model
KW  - Mie theory
KW  - nanoparticle
KW  - transmission X-ray microscope
N2  - 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.
ER  -

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