The Solution of the Eigenvector Problem in Synchrotron Radiation Based Anomalous Small-Angle X-Ray Scattering

Affiliation(s)

Institute of Soft Matter and Functional Materials, Helmholtz-Zentrum-Berlin, Berlin, Germany.

Institute of Soft Matter and Functional Materials, Helmholtz-Zentrum-Berlin, Berlin, Germany.

ABSTRACT

In the last three decades Synchrotron radiation became an indispensable experimental tool for chemical and structural analysis of nano-scaled properties in solid state physics, chemistry, materials science and life science thereby rendering the explanation of the macroscopic behavior of the materials and systems under investigation. Especially the techniques known as Anomalous Small-Angle X-ray Scattering provide deep insight into the materials structural architecture according to the different chemical components on lengths scales starting just above the atomic scale (≈1 nm) up to several 100 nm. The techniques sensitivity to the different chemical components makes use of the energy dependence of the atomic scattering factors, which are different for all chemical elements, thereby disentangling the nanostructure of the different chemical components by the signature of the elemental X-ray absorption edges i.e. by employing synchrotron radiation. The paper wants to focus on the application of an algorithm from linear algebra in the field of synchrotron radiation. It provides a closer look to the algebraic prerequisites, which govern the system of linear equations established by these experimental techniques and its solution by solving the eigenvector problem. The pair correlation functions of the so-called basic scattering functions are expressed as a linear combination of eigenvectors.

In the last three decades Synchrotron radiation became an indispensable experimental tool for chemical and structural analysis of nano-scaled properties in solid state physics, chemistry, materials science and life science thereby rendering the explanation of the macroscopic behavior of the materials and systems under investigation. Especially the techniques known as Anomalous Small-Angle X-ray Scattering provide deep insight into the materials structural architecture according to the different chemical components on lengths scales starting just above the atomic scale (≈1 nm) up to several 100 nm. The techniques sensitivity to the different chemical components makes use of the energy dependence of the atomic scattering factors, which are different for all chemical elements, thereby disentangling the nanostructure of the different chemical components by the signature of the elemental X-ray absorption edges i.e. by employing synchrotron radiation. The paper wants to focus on the application of an algorithm from linear algebra in the field of synchrotron radiation. It provides a closer look to the algebraic prerequisites, which govern the system of linear equations established by these experimental techniques and its solution by solving the eigenvector problem. The pair correlation functions of the so-called basic scattering functions are expressed as a linear combination of eigenvectors.

KEYWORDS

Matrix Inversion; Eigenvalues; Eigenvectors; Pair Correlation Functions; Basic Scattering Functions

Matrix Inversion; Eigenvalues; Eigenvectors; Pair Correlation Functions; Basic Scattering Functions

Cite this paper

G. Goerigk, "The Solution of the Eigenvector Problem in Synchrotron Radiation Based Anomalous Small-Angle X-Ray Scattering,"*Advances in Linear Algebra & Matrix Theory*, Vol. 3 No. 4, 2013, pp. 59-68. doi: 10.4236/alamt.2013.34012.

G. Goerigk, "The Solution of the Eigenvector Problem in Synchrotron Radiation Based Anomalous Small-Angle X-Ray Scattering,"

References

[1] A. Guinier and G. Fournet, “Small-Angle Scattering of X-rays,” Wiley, New York, 1955.

[2] O. Glatter and O. Kratky, “Small-Angle X-Ray Scattering,” Academic Press, London, 1982.

[3] O. Kratky, “The World of Negligible Dimensions and Small Angle Diffraction of X-Rays and Neutrons on Biological Macromolecules,” Acta Leopoldina, Vol. 55, No. 256, 1983, pp. 6-72.

[4] G. Goerigk, R. Schweins, K. Huber and M. Ballauff, “The Distribution of Sr2+ Counterions around Polyacrylate Chains Analyzed by Anomalous Small-Angle X-Ray Scattering,” Europhysics Letters, Vol. 66, No. 3, 2004, pp. 331-337. http://dx.doi.org/10.1209/epl/i2003-10215-y

[5] G. Goerigk and D. L. Williamson, “Temperature Induced Differences in the Nanostructure of Hot-Wire Deposited Silicon-Germanium Alloys Analyzed by Anomalous Small-Angle X-Ray Scattering,” Journal of Applied Physics, Vol. 99, No. 8, 2006, Article ID: 084309. http://dx.doi.org/10.1063/1.2187088

[6] A. Bota, Z. Varga and G. Goerigk, “Biological Systems as Nanoreactors: Anomalous Small-Angle Scattering Study of the CdS Nanoparticle Formation in Multilamellar Vesicles,” The Journal of Physical Chemistry B, Vol. 111, No. 8, 2007, pp. 1911-1915. http://dx.doi.org/10.1021/jp067772n

[7] Z. Varga, A. Bota and G. Goerigk, “Localization of Dihalogenated Phenols in Vesicle Systems Determined by Contrast Variation X-ray Scattering,” Journal of Applied Crystallography, Vol. 40, No. 1, 2007, pp. 205-208. http://dx.doi.org/10.1107/S0021889807001987

[8] G. Goerigk, K. Huber and R. Schweins, “Probing the Extent of the Sr2+ Ion Condensation to Anionic Polyacrylate Coils: A Quantitative Anomalous Small-Angle X-Ray Scattering Study,” Journal of Chemical Physics, Vol. 127, No. 15, 2007, pp. 154908-1-154908-8. http://dx.doi.org/10.1063/1.2787008

[9] A. Bota, Z. Varga and G. Goerigk, “Structural Description of the Nickel Part of a Raney-Type Catalyst by Using Anomalous Small-Angle X-Ray Scattering,” The Journal of Physical Chemistry C, Vol. 112, No. 12, 2008, pp. 4427-4429. http://dx.doi.org/10.1021/jp800237b

[10] G. Goerigk and N. Mattern, “Critical Scattering of Ni– Nb–Y Metallic Glasses Probed by Quantitative Anomalous Small-Angle X-Ray Scattering,” Acta Materialia, Vol. 57, No. 12, 2009, pp. 3652-3661. http://dx.doi.org/10.1016/j.actamat.2009.04.028

[11] G. Goerigk and N. Mattern, “Spinodal Decomposition in Ni-Nb-Y Metallic Glasses Analyzed by Quantitative Anomalous Small-Angle X-Ray Scattering,” Journal of Physics: Conference Series, Vol. 247, 2010, Article ID: 012022. http://iopscience.iop.org/1742-6596/247/1/012022

[12] G. Goerigk, K. Huber, N. Mattern and D. L. Williamson, “Quantitative Anomalous Small-Angle X-Ray Scattering —The Determination of Chemical Concentrations in Nano-Scaled Phases,” European Physical Journal—Special Topics, Vol. 208, No. 1, 2012, pp. 259-274. http://dx.doi.org/10.1140/epjst/e2012-01623-2

[13] I. Akiba, A. Takechi, M. Sakou, M. Handa, Y. Shinohara, Y. Amemiya, N. Yagi and K. Sakurai, “Anomalous Small-Angle X-Ray Scattering Study of Structure of Polymer Micelles Having Bromines in Hydrophobic Core,” Macromolecules, Vol. 45, No. 15, 2012, pp. 6150-6157. http://dx.doi.org/10.1021/ma300461d

[14] G. Goerigk, “The Impact of the Turing Number on Quantitative ASAXS Measurements of Ternary Alloys,” JOM, Vol. 65, No. 1, 2013, pp. 44-53. http://dx.doi.org/10.1007/s11837-012-0451-9

[15] S. Lages, G. Goerigk and K. Huber, “SAXS and ASAXS on Dilute Sodium Polyacrylate Chains Decorated with Lead Ions,” Macromolecules, Vol. 46, No. 9, 2013, pp. 3570-3580. http://dx.doi.org/10.1021/ma400427d

[16] M. Sakou, A. Takechi, S. Murakami, K. Sakurai and I. Akiba, “Study of the Internal Structure of Polymer Micelles by Anomalous Small-Angle X-Ray Scattering at Two Edges,” Journal of Applied Crystallography, Vol. 46, No. 5, 2013, pp. 1407-1413. http://dx.doi.org/10.1107/S0021889813022450

[17] H. B. Stuhrmann, “Resonance Scattering in Macromolecular Structure Research,” Advances in Polymer Science, Vol. 67, 1985, pp. 123-163. http://dx.doi.org/10.1007/BFb0016608

[18] H. B. Stuhrmann, “Anomale R?ntgenstreuung zur Erforschung makromolekularer Strukturen,” Macromolecular Chemistry, Vol. 183, No. 10, 1982, pp. 2501-2514. http://dx.doi.org/10.1002/macp.1982.021831020?

[19] J. Westlake, “A Handbook of Numerical Matrix Inversion and Solution of Linear Equations,” Wiley, New York, 1968.

[20] J. P. Simon, O. Lyon and D. de Fontaine, “A Comparison of the Merits of Isotopic-substitution in Neutron Small-Angle Scattering and Anomalous X-Ray Scattering for the Evaluation of Partial Structure Functions in a Ternary Alloy,” Journal of Applied Crystallography, Vol. 18, 1985, pp. 230-236. http://dx.doi.org/10.1107/S0021889885010196

[21] D. T. Cromer and D. Liberman, Journal of Chemical Physics, Vol. 53, No. 5, 1970, pp. 1891-1898. http://dx.doi.org/10.1063/1.1674266

[22] D. T. Cromer and D. Liberman, Acta Crystallographica, Vol. A37, 1981, pp. 267-268.

[1] A. Guinier and G. Fournet, “Small-Angle Scattering of X-rays,” Wiley, New York, 1955.

[2] O. Glatter and O. Kratky, “Small-Angle X-Ray Scattering,” Academic Press, London, 1982.

[3] O. Kratky, “The World of Negligible Dimensions and Small Angle Diffraction of X-Rays and Neutrons on Biological Macromolecules,” Acta Leopoldina, Vol. 55, No. 256, 1983, pp. 6-72.

[4] G. Goerigk, R. Schweins, K. Huber and M. Ballauff, “The Distribution of Sr2+ Counterions around Polyacrylate Chains Analyzed by Anomalous Small-Angle X-Ray Scattering,” Europhysics Letters, Vol. 66, No. 3, 2004, pp. 331-337. http://dx.doi.org/10.1209/epl/i2003-10215-y

[5] G. Goerigk and D. L. Williamson, “Temperature Induced Differences in the Nanostructure of Hot-Wire Deposited Silicon-Germanium Alloys Analyzed by Anomalous Small-Angle X-Ray Scattering,” Journal of Applied Physics, Vol. 99, No. 8, 2006, Article ID: 084309. http://dx.doi.org/10.1063/1.2187088

[6] A. Bota, Z. Varga and G. Goerigk, “Biological Systems as Nanoreactors: Anomalous Small-Angle Scattering Study of the CdS Nanoparticle Formation in Multilamellar Vesicles,” The Journal of Physical Chemistry B, Vol. 111, No. 8, 2007, pp. 1911-1915. http://dx.doi.org/10.1021/jp067772n

[7] Z. Varga, A. Bota and G. Goerigk, “Localization of Dihalogenated Phenols in Vesicle Systems Determined by Contrast Variation X-ray Scattering,” Journal of Applied Crystallography, Vol. 40, No. 1, 2007, pp. 205-208. http://dx.doi.org/10.1107/S0021889807001987

[8] G. Goerigk, K. Huber and R. Schweins, “Probing the Extent of the Sr2+ Ion Condensation to Anionic Polyacrylate Coils: A Quantitative Anomalous Small-Angle X-Ray Scattering Study,” Journal of Chemical Physics, Vol. 127, No. 15, 2007, pp. 154908-1-154908-8. http://dx.doi.org/10.1063/1.2787008

[9] A. Bota, Z. Varga and G. Goerigk, “Structural Description of the Nickel Part of a Raney-Type Catalyst by Using Anomalous Small-Angle X-Ray Scattering,” The Journal of Physical Chemistry C, Vol. 112, No. 12, 2008, pp. 4427-4429. http://dx.doi.org/10.1021/jp800237b

[10] G. Goerigk and N. Mattern, “Critical Scattering of Ni– Nb–Y Metallic Glasses Probed by Quantitative Anomalous Small-Angle X-Ray Scattering,” Acta Materialia, Vol. 57, No. 12, 2009, pp. 3652-3661. http://dx.doi.org/10.1016/j.actamat.2009.04.028

[11] G. Goerigk and N. Mattern, “Spinodal Decomposition in Ni-Nb-Y Metallic Glasses Analyzed by Quantitative Anomalous Small-Angle X-Ray Scattering,” Journal of Physics: Conference Series, Vol. 247, 2010, Article ID: 012022. http://iopscience.iop.org/1742-6596/247/1/012022

[12] G. Goerigk, K. Huber, N. Mattern and D. L. Williamson, “Quantitative Anomalous Small-Angle X-Ray Scattering —The Determination of Chemical Concentrations in Nano-Scaled Phases,” European Physical Journal—Special Topics, Vol. 208, No. 1, 2012, pp. 259-274. http://dx.doi.org/10.1140/epjst/e2012-01623-2

[13] I. Akiba, A. Takechi, M. Sakou, M. Handa, Y. Shinohara, Y. Amemiya, N. Yagi and K. Sakurai, “Anomalous Small-Angle X-Ray Scattering Study of Structure of Polymer Micelles Having Bromines in Hydrophobic Core,” Macromolecules, Vol. 45, No. 15, 2012, pp. 6150-6157. http://dx.doi.org/10.1021/ma300461d

[14] G. Goerigk, “The Impact of the Turing Number on Quantitative ASAXS Measurements of Ternary Alloys,” JOM, Vol. 65, No. 1, 2013, pp. 44-53. http://dx.doi.org/10.1007/s11837-012-0451-9

[15] S. Lages, G. Goerigk and K. Huber, “SAXS and ASAXS on Dilute Sodium Polyacrylate Chains Decorated with Lead Ions,” Macromolecules, Vol. 46, No. 9, 2013, pp. 3570-3580. http://dx.doi.org/10.1021/ma400427d

[16] M. Sakou, A. Takechi, S. Murakami, K. Sakurai and I. Akiba, “Study of the Internal Structure of Polymer Micelles by Anomalous Small-Angle X-Ray Scattering at Two Edges,” Journal of Applied Crystallography, Vol. 46, No. 5, 2013, pp. 1407-1413. http://dx.doi.org/10.1107/S0021889813022450

[17] H. B. Stuhrmann, “Resonance Scattering in Macromolecular Structure Research,” Advances in Polymer Science, Vol. 67, 1985, pp. 123-163. http://dx.doi.org/10.1007/BFb0016608

[18] H. B. Stuhrmann, “Anomale R?ntgenstreuung zur Erforschung makromolekularer Strukturen,” Macromolecular Chemistry, Vol. 183, No. 10, 1982, pp. 2501-2514. http://dx.doi.org/10.1002/macp.1982.021831020?

[19] J. Westlake, “A Handbook of Numerical Matrix Inversion and Solution of Linear Equations,” Wiley, New York, 1968.

[20] J. P. Simon, O. Lyon and D. de Fontaine, “A Comparison of the Merits of Isotopic-substitution in Neutron Small-Angle Scattering and Anomalous X-Ray Scattering for the Evaluation of Partial Structure Functions in a Ternary Alloy,” Journal of Applied Crystallography, Vol. 18, 1985, pp. 230-236. http://dx.doi.org/10.1107/S0021889885010196

[21] D. T. Cromer and D. Liberman, Journal of Chemical Physics, Vol. 53, No. 5, 1970, pp. 1891-1898. http://dx.doi.org/10.1063/1.1674266

[22] D. T. Cromer and D. Liberman, Acta Crystallographica, Vol. A37, 1981, pp. 267-268.