- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Materials

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Conditional diffusion machine-learning framework for mapping valence electron distribution from convergent beam electron diffraction

Lijun Wu1,*, Dmitrii Torbunov2, Yihui Ren2, and Yimei Zhu1,†

  • *Contact author: ljwu@bnl.gov
  • †Contact author: zhu@bnl.gov
PDF

Phys. Rev. Materials 10, 053802 – Published 11 May, 2026

DOI: https://doi.org/10.1103/qdpk-6mnv

Abstract

Quantitative convergent beam electron diffraction (CBED) enables determination of aspherical valence electron distributions through refinement of low-order structure factors, which are highly sensitive to chemical bonding and charge density variations. However, conventional quantitative CBED (QCBED) requires solving a highly nonlinear inverse problem with many coupled parameters, and computationally intensive dynamical diffraction calculations, making it time-consuming and difficult to apply to complex systems. More broadly, reconstructing charge density and orbital electron distribution from diffraction data has long been a central challenge in both x-ray and electron crystallography. Here, we introduce an artificial-intelligence (AI)-based framework that replaces traditional refinement with a data-driven inverse solver. Using a large synthetic CBED dataset generated by Bloch-wave simulations, we train a conditional diffusion model to directly infer crystal structural parameters and multipole density formalism parameters, and hence valence electron distributions, from CBED patterns alone. By learning from forward simulations across realistic parameter space, the model effectively solves the inverse problem. Compared with direct regression approaches, the diffusion-based framework provides posterior parameter distributions for rigorous uncertainty quantification while preserving quantitative fidelity and reducing analysis time by orders of magnitude. By eliminating the need for external single-crystal x-ray diffraction data and complex nonlinear refinement, this approach enables practical, high-throughput, and in situ quantitative CBED, enabling real-time mapping of valence electron distributions and their correlation with functional responses in quantum and energy materials.

View figure in article

Physics Subject Headings (PhySH)

Collections

This article appears in the following collection:

New Insights into Functional Materials through Advanced Electron Microscopy

The Editors of Physical Review Materials are pleased to present the Collection on New Insights into Functional Materials through Advanced Electron Microscopy, highlighting cutting-edge microscopy techniques and the extraordinary advances in materials science and engineering that they enable. The Collection is being guest-edited by Joanne Etheridge from Monash University (Australia) and Yimei Zhu from Brookhaven National Laboratory (USA). Every article published in this collection underwent a rigorous peer review process, adhering to the same high standards applied to all papers. The Physical Review Materials editorial team managed the peer review and made all editorial decisions.

Article Text

References (39)

  1. J. Donatelli et al., Basic research needs for inverse methods for complex systems under uncertainty, Technical Report, US Department of Energy (USDOE), Washington, DC, Office of Science, Advanced Scientific Computing Research (ASCR) (2025), https://doi.org/10.2172/2583339.
  2. J. C. H. Spence and J. M. Zuo, Electron Microdiffraction, (Plenum Press, New York, 1992).
  3. J. M. Zuo, M. Kim, M. O'Keeffe, and J. C. H. Spence, Direct observation of d-orbital holes and Cu-Cu bonding in Cu2O, Nature (London) 401, 49 (1999).
  4. K. Tsuda and M. Tanaka, Refinement of crystal structural parameters using two-dimensional energy-filtered CBED patterns, Acta Crystallogr. Sect. A 55, 939 (1999).
  5. K. Tsuda, Y. Ogata, K. Takagi, T. Hashimoto, and M. Tanaka, et al., Refinement of crystal structural parameters and charge density using convergent-beam electron diffraction - the rhombohedral phase of LaCrO3, Acta Crystallogr. Sect. A 58, 514 (2002).
  6. J. C. Zheng, A. I. Frenkel, L. Wu, J. Hanson, W. Ku, E. S. Božin, S. J. L. Billinge, and Y. Zhu, Nanoscale disorder and local electronic properties of CaCu3Ti4O12: An integrated study of electron, neutron, and x-ray diffraction, x-ray absorption fine structure, and first-principles calculations, Phys. Rev. B 81, 144203 (2010).
  7. C. Ma, L. Wu, W.-G. Yin, H. Yang, H. Shi, Z. Wang, J. Li, C. C. Homes, and Y. Zhu, Strong coupling of the iron-quadrupole and anion-dipole polarizations in Ba(Fe1−xCox)2As2, Phys. Rev. Lett. 112, 077001 (2014).
  8. L. J. Wu, Y. M. Zhu, and J. Tafto, Test of first-principle calculations of charge transfer and electron-hole distribution in oxide superconductors by precise measurements of structure factors, Phys. Rev. B 59, 6035 (1999).
  9. T. T. Shang, D. Xiao, F. Meng, X. Rong, A. Gao, T. Lin, Z. Tang, X. Liu, X. Li, Q. Zhang, et al., Real-space measurement of orbital electron populations for Li1−xCoO2, Nat. Commun. 13, 5810 (2022).
  10. Y. Zhu, J. C. Zheng, L. Wu, A. I. Frenkel, J. Hanson, P. Northrup, and W. Ku, Nanoscale disorder in CaCu3Ti4O12: A new route to the enhanced dielectric response, Phys. Rev. Lett. 99, 037602 (2007).
  11. J. Lee, M. Carbone, and W. Yin, Machine learning the spectral function of a hole in a quantum antiferromagnet, Phys. Rev. B 107, 205132 (2023).
  12. K. Kaufmann, C. Zhu, A. S. Rosengarten, D. Maryanovsky, T. J. Harrington, E. Marin, and K. S. Vecchio, Crystal symmetry determination in electron diffraction using machine learning, Science 367, 564 (2020).
  13. J. Munshi, A. Rakowski, B. H. Savitzky, S. E. Zeltmann, J. Ciston, M. Henderson, S. Cholia, A. M. Minor, M. K. Y. Chan, and C. Ophus, Disentangling multiple scattering with deep learning: Application to strain mapping from electron diffraction patterns, npj Comput. Mater. 8, 254 (2022).
  14. J. Chen, H. Zhang, C. B. Wah, and W. Chen, Automated crystal system identification from electron diffraction patterns using multiview opinion fusion machine learning, Proc. Nat. Acad. Sci. USA 120, e2309240120 (2023).
  15. R. Yuan, J. Zhang, L, He, and J.-M. Zuo, Training artificial neural networks for precision orientation and strain mapping using 4D electron diffraction datasets, Ultramicroscopy 231, 113256 (2021).
  16. L. Tiong, J. Kim, S. S. Han, and D. Kim, Identification of crystal symmetry from noisy diffraction patterns by a shape analysis and deep learning, npj Comput. Mater. 6, 196 (2020).
  17. C. Ophus, Four-dimensional scanning transmission electron microscopy (4D-STEM): From scanning nanodiffraction to ptychography and beyond, Microsc. Microanal. 25, 563 (2019).
  18. J. Madsen, P. Liu, J. Kling, J. B. Wagner, T. W. Hansen, O. Winther, and J. Schiøtz, A deep learning approach to identify local structures in atomic-resolution transmission electron microscopy images, Adv. Theory Simul. 1, 1800037 (2018).
  19. W. Xu and J. LeBeau, A deep convolutional neural network to analyze position averaged convergent beam electron diffraction patterns, Ultramicroscopy 188, 59 (2018).
  20. J. Aguiar, M. L. Gong, R. R. Unocic, T. Tasdizen, and B. D. Miller, Decoding crystallography from high-resolution electron imaging and diffraction datasets with deep learning, Sci. Adv. 5, eaaw1949 (2019).
  21. D. Torbunov, Y. Ren, L. Wu, and Y. Zhu, Parameter inference and uncertainty quantification with diffusion models: Extending CDI to 2D spatial conditioning, arXiv:2601.17224.
  22. P. Coppens, X-Ray Charge Densities and Chemical Bonding (Oxford University Press, New York, 1997).
  23. N. K. Hansen and P. Coppens, Testing aspherical atom refinements on small-molecule data sets, Acta Crystallogr. Sect. A 34, 909 (1978).
  24. F. A. Parpia, C. F. Fischer, and I. P. Grant, GRASP92: A package for large-scale relativistic atomic structure calculations, Comput. Phys. Commun. 94, 249 (1996).
  25. L. J. Wu, T. Vogt, H. Su, and J. W. Davenport, and J. Tafto, Valence-electron distribution in MgB2 by accurate diffraction measurements and first-principles calculations, Phys. Rev. B 69, 064501 (2004).
  26. T. Lippmann, P. Blaha, N. H. Andersen, H. F. Poulsen, T. Wolf, J. R. Schneider, and K.-H. Schwarz, Charge-density analysis of YBa2Cu3O6.98. Comparison of theoretical and experimental results, Acta Crystallogr. Sect. A 59, 437 (2003).
  27. E. A. Zhurova and V. G. Tsirelson, Electron density and energy density view on the atomic interactions in SrTiO3, Acta Crystallogr. Sec. B-Struct. Sci. 58, 567 (2002).
  28. K. He, X. Zhang, S. Ren and J. Sun, Deep residual learning for image recognition, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, Las Vegas, NV, 2016), pp. 770–778.
  29. I. Loshchilov and F. Hutter, Decoupled weight decay regularization, in 7th International Conference on Learning Representations (OpenReview.net, New Orleans, 2019).
  30. F. Zhu, D. Torbunov, Z. Jiang, T. Zhao, A. Yogarathnam, Y. Ren, and M. Yue, Diffusion model-based parameter estimation in dynamic power systems, Commun. Eng. (2026), doi: 10.1038/s44172-026-00670-z.
  31. J. Sohl-Dickstein, N. Maheswaranathan, and S. Ganguli, ICML'15: Proceedings of the 32nd International Conference on Machine Learning (ACM Digital, New York, 2015), Vol. 37, pp. 2256–2265.
  32. J. Ho, A. Jain, and P. Abbeel, Denoising diffusion probabilistic models, in NIPS'20: Proceedings of the 34th International Conference on Neural Information Processing Systems (ACM Digital, New York, 2020), pp. 6840–6851.
  33. T. Gneiting, F. Balabdaoui, and A. Raftery, Probabilistic forecasts, calibration and sharpness, J. R. Stat. Soc. B. Stat. Methodol. 69, 243 (2007).
  34. V. Kuleshov, N. Fenner, and S. Ermon, Accurate uncertainties for deep learning using calibrated regression, in Proceedings of the 35th International Conference on Machine Learning, Stockholm, Sweden, Proceedings of Machine Learning Research, Vol. 80 (PMLR, 2018).
  35. Y. Huang, D. Torbunov, B. Viren, H. Yu, J. Huang, M. Lin, and Y. Ren, Unpaired image translation to mitigate domain shift in liquid argon time projection chamber detector responses, Mach. Learn.-Sci. Technol. 5, 045021 (2024).
  36. H. Guan and M. Liu, Domain adaptation for medical image analysis: A survey, IEEE Trans. Biomed. Eng. 69, 1173 (2022).
  37. J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Aittala, and T. Aila, Proceedings of the 35th International Conference on Machine Learning, Stockholm, Sweden, Proceedings of Machine Learning Research, Vol. 80 (PMLR, 2018).
  38. M. Guo, Y. Wu, C. M. Hobson, Y. Su, S. Qian, E. Krueger, R. Christensen, G. Kroeschell, J. Bui, M. Chaw, L. Zhang, et al., Deep learning-based aberration compensation improves contrast and resolution in fluorescence microscopy, Nat. Commun. 16, 313 (2025).
  39. I. Lobato, T. Friedrich, and S. Van Aert, Deep convolutional neural networks to restore single-shot electron microscopy images, npj Comput. Mater. 10, 10 (2024).

Outline

Information

Phys. Rev. Materials 10, 053802– Published 11 May, 2026
  • Received 19 December 2025
  • Accepted 6 April 2026
Article part of CHORUS
Crossmark - Check for updates button

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation