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Model-free refinement in EXAFS analysis of CdS magic size clusters

Andrei Sapelkin, Queen Mary University of London, UK

Conexs 2020, Newcastle


Centre for Condensed Matter and Materials Physics

Acknowledgements

Prof. David Dunstan, QMUL, UK Prof. Stanislav Yurchenko, Bauman Moscow State Technical University, RussiaDr. Nikita Krychkov, Bauman Moscow State Technical University, RussiaMs. Lei Tan, QMUL, UKMs. Ying Liu, QMUL, UKProf. Martin Dove, QMUL, UKProf. Chris J. Pickard, University of Cambridge, UKProf. Kui Yu, Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, P. R. China

Quantum Dots

http://www.sigmaaldrich.com/technical-documents/articles/materials-science/nanomaterials/quantum-dots.html


Quantum Dots: regular vs magic size



Magic Size Clusters

Two CdS Magic Sized Clusters

[1] B. Zhang, T. Zhu, M. Ou, N. Rowell, H. Fan, J. Han, L. Tan, M. T. Dove, Y. Ren, X. Zuo, S. Han, J. Zeng, and K. Yu, “Thermally-induced reversible structural isomerization in colloidal semiconductor CdS magic-size clusters,” Nat. Commun., vol. 9, no. 1, pp. 1–10, 2018.

M311 and M322

Atomic structure of CdS

TEM can and does anneal small samples.

K. Yu, M. Z. Hu, R. Wang, M. Le Piolet, M. Frotey, M. B. Zaman, X. Wu, D. M. Leek, Y. Tao, D. Wilkinson, and C. Li, “Thermodynamic equilibrium-drivenformation of single-sized nanocrystals: Reaction media tuning CdSe magic-sized versus regular quantum dots,” J. Phys. Chem. C, vol. 114, no. 8, pp. 3329–3339, 2010.

Atomic structure of CdS

Just like in many non-periodic systems XRD is a challenge. Mass-spec givesa rough idea of cluster size

K. Yu, M. Z. Hu, R. Wang, M. Le Piolet, M. Frotey, M. B. Zaman, X. Wu, D. M. Leek, Y. Tao, D. Wilkinson, and C. Li, “Thermodynamic equilibrium-drivenformation of single-sized nanocrystals: Reaction media tuning CdSe magic-sized versus regular quantum dots,” J. Phys. Chem. C, vol. 114, no. 8, pp. 3329–3339, 2010.



Science results and our results

L Tan et al, Nanoscale, 2019, 11, 21900–21908



Current options to obtain a structural model

• Random structure search by energy minimisation (e.g. AIRSS code)

• Reversed Monte Carlo (e.g. RMCprofile)

L Tan et al, Nanoscale, 2019, 11, 21900–21908

L Tan et al, J. Phys. Chem. C 2019, 123, 48, 29370-29378



X-ray scattering: Pair distribution function

L. Tan et al, Nanoscale, 2019, 11, 21900–21908



Science results and our results

Williamson et al., Science 363, 731–735 (2019)



CdS magic size clusters: x-ray absorption



Magic size clusters: x-ray absorption

https://arxiv.org/abs/1806.03274



CdS magic size clusters: x-ray absorptionBulk R343

M311 M322

Cd-Cd

Cd-Cd



Can we convert these signal differences into structural

models? If so, how?


EXAFS quality of fit: what can we expect?

c-Ge x2


The EXAFS equation and the model-free fit

•Treat data analysis as a function fitting routine

•Provide reasonable initial guess

•Utilise ML and full Bayesian fitting


Model-free refinement using LS/ML followed by full Bayesian analysis

What is Bayesian analysis and why use it:

• For a given dataset (i.e. 𝜇𝜇) find such parameter distribution (i.e. 𝐱𝐱) so that to maximise probability

𝑃𝑃(𝐱𝐱|𝜇𝜇) = 𝑃𝑃(𝜇𝜇|𝐱𝐱)𝑃𝑃(𝐱𝐱)𝑃𝑃(𝜇𝜇)

• Allows to incorporate prior information and yields parameter distributions• Flexible, so can be generalised to other techniques (e.g. PDF, Raman etc.) for

multiple refinement.


Model-free refinement using LS/ML followed by full Bayesian analysis

However


Model-free refinement using ML followed by full Bayesian analysis• ML doesn’t discriminate between meaningful and meaningless parameters (but can be

used to generate priors).

• The way to overcome this is Bayesian approach. Using FEFF and custom refinement code we calculate likelihood L as a function of the parameters rather than looking for its maximum.

• Multiplying by the probability distribution function of the parameters and integrating over parameter space gives the Bayes Factor (BF). But this can’t be easily done for more than a few (e.g. 3-4) parameters on a typical desktop/laptop because parameters are correlated resulting is complex multidimensional integrals.

• We need parameters that are uncorrelated (orthogonal) so that N-dimensional integral can be reduced to N one-dimensional integrals. This can be done by diagonalising the covariance matrix.

• Set the new parameter space and express the model to be fitted in terms of these new parameters.

• Express the residuals in terms of these new parameters.

• Calculate the full Bayes Factor (FBF) as a product of multiple one-dimensional integrals rather than a multidimensional integral.


Model-free refinement using ML algorithm


Model-free refinement using ML algorithm to generate priors

Cd-O

Cd-S

Cd-Cd


Model-free refinement using ML algorithm to generate priors


Results so far following Bayesian analysis

•Cd-to-S ratio is around 2.04.

•CdS/CdO ratio is around 7.75/3.79.

•The following models are consistent with the calculations: Cd40S21 or Cd39S24 or Cd38S27.

•This is to be compared with the AIRSS results: (CdS)nnanoclusters, n = 34,28,32,30 and 29 (J. Phys. Chem. C 2019, 123, 48, 29370-29378) and “educated guess” of Cd37S20 (see Science 363, 731–735, 2019)

•These will need to be checked for self-consistency: model produced and EXAFS signal is calculated.


Overall workflow

So far Where we want to be


Outstanding issues

•The form of the Debye-Waller factor when disorder is large

•Limitations due to theoretical scattering amplitudes and phaseshifts



Summary

‣ Full Bayesian analysis routine has been developed that allows to include prior information and inform about “overfitting”

‣ Full Bayesian analysis routine is being developed to include background‣ This will be followed by extension to include other experimental methods (e.g. PDF)‣ Extension is now in progress for systems with large anharmonicity effects and

disordered systems


EXAFS mean-square relative displacement


Correlation and Debye-Waller factor

J. Chem. Phys. 140, 134502 (2014)


Pair correlation function

density normalisation requires particle to be on the corresponding node

with

and

Thus, the idea is in factorising delocalisation operator through delocalisation operators between the nodes.


Delocalisation and pair interactions

The latter can be written as follows for Gaussian disorder and Einstein approx.

At finite temperatures


Model systems: MD and 3D


Model systems: 2D

Slide Number 1Slide Number 2Quantum DotsQuantum Dots: regular vs magic sizeMagic Size ClustersAtomic structure of CdS Atomic structure of CdS Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 25Slide Number 26EXAFS quality of fit: what can we expect?Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39EXAFS mean-square relative displacementCorrelation and Debye-Waller factorPair correlation functionSlide Number 43Slide Number 44Slide Number 45

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