Introduction to neutron reflectometry fitting

Introduction to neutron reflectometry fitting

The aim of this course, and the companion lecture, are to give an introduction to the following subjects:

  • the Fourier transform and how a Born approximation approach may be used to analyse neutron reflectometry data;

  • the logic of model-dependent analysis;

  • neutron reflectometry “slab models” and their traditional parameterisation;

  • reparameterisation of these models to include chemical and physical insight; and

  • the process and problems associated with fitting in a model-dependent analysis procedure.

It is assumed that this reader has been introduced to the technique of neutron reflectometry, such as what this technique can be used to study and the data collection methodology.

This material was developed by Andrew McCluskey from the European Spallation Source. If you have any questions, please get in touch. Thanks to Drs Maximilian Skoda, Andrew Caruana, Stephen Hall, and Andrew Nelson for feedback on this material.

Note

In this course, we make use of the Python programming language heavily to show mathematics and plot figures. The aim is that the course should not require knowledge of Python to understand the content. If you are not comfortable with Python, feel free to skip the code blocks, but make sure to pay attention to the plots that are produced.

Bibliography

Some particularly useful books and papers for reflectometry analysis, and data analysis in general include:

  • Elementary Scattering Theory: For X-ray and Neutron Users by Devinder Sivia [Siv11];

  • Current Opinion in Colloid & Interface Science, 42, 2019 covers a range of applications in soft and biological matter including [Lak19, Sko19, WC19];

  • Some interesting reviews of magnetic reflectometry analysis include [FM05, TZ07, ZTBT07]; and

  • Data Analysis: A Bayesian Tutorial by Devinder Sivia and John Skilling [SJ06].

This list is not exhaustive and we suggest searching for and reading relevant work in your field once you understand the basics.

Abeles48

F. Abelès. Sur la propagation des ondes électromagnétiques dans les milieux sratifiés. Ann. Phys., 12(3):504–520, 1948. doi:10.1051/anphys/194812030504.

BP63

T. Bayes and R. Price. An essay towards solving a problem in the doctrine of chances. Phil. Trans. R. Soc. Lond, 53:370–418, 1763. doi:10.1098/rstl.1763.0053.

Bjorck11

M. Björck. Fitting with differential evolution: an introduction and evaluation. J. Appl. Crystallogr., 44(6):1198–1204, 2011. doi:10.1107/s0021889811041446.

CSS+18

R. A. Campbell, Y. Saaka, Y. Shao, Y. Gerelli, R. Cubitt, E. Nazaruk, D. Matyszewska, and M. J. Lawrence. Structure of surfactant and phospholipid monolayers at the air/water interface modeled from neutron reflectivity data. J. Colloid Interface Sci., 531:98–108, 2018. doi:10.1016/j.jcis.2018.07.022.

FM05

M. R. Fitzsimmons and C. F. Majkrzak. Applications of polarized neutron reflectometry to studies of artificially structured magnetic materials. In Y. Zhou, editor, Neutron, X-Rays and Light. Scattering Methods Applied to Soft Condensed Matter, pages 107–155. Springer, 2005.

JPSC21

L. H. John, G. M. Preston, M. S. P. Sansom, and L. A. Clifton. Large scale model lipid membrane movement induced by a cation switch. J. Colloid Interface Sci., 596:297–311, 2021. doi:10.1016/j.jcis.2021.03.078.

Lak19

J. H. Lakey. Recent advances in neutron reflectivity studies of biological membranes. Curr. Opin. Colloid Interface Sci., 42:33–40, 2019. doi:10.1016/j.cocis.2019.02.012.

LR99

M. R. Lovell and R. M. Richardson. Analysis methods in neutron and x-ray reflectometry. Curr. Opin. Colloid Interface Sci., 4(3):197–204, 1999. doi:10.1016/S1359-0294(99)00039-4.

MBD+98

C. F. Majkrzak, N. F. Berk, J. A. Dura, S. K. Satija, A. Karin, J. Pedulla, and R. D. Deslattes. Phase determination and inversion in specular neutron reflectometry. Physica B, 248(1–4):338–342, 1998. doi:10.1016/S0921-4526(98)00260-9.

MKH10

J. Mayer, K. Khairy, and J. Howard. Drawing an elephant with four complex parameters. Am. J. Phys., 78(6):648–649, 2010. doi:10.1119/1.3254017.

MSFE+19

A. R. McCluskey, A. Sanchez-Fernandez, K. J. Edler, S. C. Parker, A. J. Jackson, R. A. Campbell, and T. Arnold. Bayesian determination of the effect of a deep eutectic solvent on the structure of lipid monolayers. Phys. Chem. Chem. Phys., 21(11):6133–6141, 2019. doi:10.1039/c9cp00203k.

NP19

A. R. J. Nelson and S. W. Prescott. Refnx: neutron and x-ray reflectometry analysis in python. J. Appl. Crystallogr., 52(1):193–200, 2019. doi:10.1107/s1600576718017296.

NevotC80

L. Névot and P. Croce. Caractérisation des surfaces par réflexion rasante de rayons x. application à l\textquotesingle étude du polissage de quelques verres silicates. Rev. Phys. Appl. (Paris), 15(3):761–779, 1980. doi:10.1051/rphysap:01980001503076100.

Par54

L. G. Parratt. Surface studies of solids by total reflection of x-rays. Phys. Rev., 95(2):359–369, 1954. doi:10.1103/physrev.95.359.

Siv11

D. S. Sivia. Elementary Scattering Theory: For X-Ray and Neutron Users. Oxford University Press, 2011. ISBN 978-0-19-922868-3.

SJ06

D. S. Sivia and Skilling J. Data Analysis: A Bayesian tutorial. Oxford University Press, 2 edition, 2006. ISBN 978-0-19-856832-2.

Sko19

M. W. A. Skoda. Recent developments in the application of x-ray and neutron reflectivity to soft-matter systems. Curr. Opin. Colloid Interface Sci., 42:41–54, 2019. doi:10.1016/j.cocis.2019.03.003.

SP97

R. Storn and K. Price. Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim., 11(4):341–359, 1997. doi:10.1023/a:1008202821328.

Tan80

C. Tanford. The Hydrophobic Effect: Formation of Micelles and Biological Membranes. John Wiley & Sons, 2 edition, 1980. ISBN 978-0-471-04893-0.

TZ07

B. P. Toperverg and H. Zabel. Neutron Scattering in Nanomagnetism. In D. L. Price and F. Fernandez-Alonso, editors, Neutron Scattering – Magnetic and Quantum Phenomena, pages 339–434. Academic Press, 2007.

TKH+19

B. W. Treece, P. A. Kienzle, D. P. Hoogerheide, C. F. Majkrzak, M. Lösche, and F. Heinrich. Optimization of reflectometry experiments using information theory. J. Appl. Crystallogr., 52(1):47–59, 2019. doi:10.1107/s1600576718017016.

WC19

R. J. L. Welbourn and S. M. Clarke. New insights into the solid-liquid interface exploiting neutron reflectivity. Curr. Opin. Colloid Interface Sci., 42:87–98, 2019. doi:10.1016/j.cocis.2019.03.007.

WPMB99

M. Wormington, C. Panaccione, K. M. Matney, and D. K. Bowen. Characterization of structures from x-ray scattering data using genetic algorithms. Phil. Trans. R. Soc. Lond. A, 357(1761):2827–2848, 1999. doi:10.1098/rsta.1999.0469.

ZTBT07

H. Zabel, K. Theis-Bröhl, and B. P. Toperverg. Polarized Neutron Reflectivity and Scattering from Magnetic nanostructures and spintronic materials. In H. Kronmüller, S. Parkin, R. Wiesendanger, and G. Guntherodt, editors, Handbook of Magnetism and Advanced Magnetic Materials. John Wiley & Sons, 2007.