The X-ray Transform and its Application in Nano Crystallography
محورهای موضوعی : فصلنامه نانوساختارهای اپتوالکترونیکیtajedin derikvand 1 , Rajab Ali Kamyabi-Gol 2 , mohammad janfada 3
1 - nternational Campus, Faculty of Mathematic Sciences, Ferdowsi University
of Mashhad
2 - Department of Pure Mathematics and Centre of Excellence in Analysis on
Algebraic Structures (CEAAS), Ferdowsi University of Mashhad
3 - Department of Pure Mathematics, Ferdowsi University of Mashhad
کلید واژه: X- ray transform, Nano crystallography, homogeneous spaces,
چکیده مقاله :
In this article a review on the definition of the X- ray transform and some of
its applications in Nano crystallography is presented. We shall show that the X- ray
transform is a special case of the Radon transform on homogeneous spaces when the
topological group E(n)- the Euclidean group - acts on ℝ2 transitively. First some
properties of the Radon transform are investigated then the relationship to texture analysis
is briefly illustrated. Finally, some of its applications in material structure detection at the
Nano scale are studied.
[1] A. Authier, “Optical properties of X-rays - dynamical diffraction,” Acta
Crystallographica A68, 40 (2012).
[2] H.J. Bunge, Mathematische Methoden der Texturanalyse, Akademie Verlag, Berlin,
1969.
[3] W. Cheney,W. Light, A Course in Approximation Theory, Brookes/Cole, Pacific
Groove, CA, 1999.
[4] S. R. Deans, The Radon transform and some of its applications, Wiley, New York,
1983.
[5] C. L. Epstein, Charles L. Introduction to the Mathematics of Medical Imaging. 2nd
ed. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2008.
[6] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton,
1995.
[7] C. Hammond. The Basics of Crystallography and Diffraction. Oxford University
Press, 1997.
[8] S. Helgason, Integral Geometry and Radon Transform, Springer, New York, 2011.
[9] J. Imhof, Determination of the Orientation Distribution Function from One Pole- ,
texture and microstructures, (1982). 5, 73–86
[10] A. B. Sekerin, Euclidean Motion Group Representations and the Singular Value
Decomposition of the Radon Transform, Integral Transforms and Spacial Functions.
00 (2005) 1-34.
[11] I. A. Vartanyants, “Coherent X-ray Diffraction Imaging of Nanostructures,”
arXiv:1304.5335 (2013).
[12] R. O. Williams, Analytical Methods for Representing Complex Textures by Biaxial
Pole Figures, (1968), 39, 4329-4335.
[13] H. Yan, and L. Li, “X-ray dynamical diffraction from single crystals with arbitrary
shape and strain field: A universal approach to modeling,” Phys. Rev. B 89, 014104
(2014).
