Abstract
We analytically study the problem of pore detection and certification in bulk objects by means of radiography. For an absorbent sample, the optimum thickness for pore imaging and detection is expressed in terms of the linear attenuation coefficient of the material. This can be used to maximize the signal-to-noise ratio by tuning the photon energy of the incident monochromatic beam. The problem is more complicated for transparent objects. An evident approach is radiography in coherent beams; in this case, we use a simple model allowing to find the field structure of the transmitted beam on the backside of the sample and beyond in the outer half space in terms of few dimensionless parameters, including the Fresnel number F = a2/λz, where a is the pore radius, λ is the wavelength, z is the distance from the back side of the sample to the detector, and the phase number Φ = akδ, with k = 2π/λ and δ being the bulk material decrement. The detailed analysis of this field structure is performed that can be used to find the optimum position of a detector revealing the pores parameters from the intensity distribution measured. We present the numerical results for a Gaussian type of the pore shape function and provide the software to calculate the space field structure for other pore shape functions. The stationary phase method in higher orders, used here to simplify the Fresnel integral, can be applied to extend the obtained results to 3D geometry. The suggested qualitative picture of the formation of images of pores as phase objects complements modern methods of monitoring porous-sensitive materials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Nagai, C. S. Musgrave, and W. Nazarov, Phys. Plasmas, 25, 030501 (2018).
G. L. Messing and A. J. Stevenson, Science, 322, 383 (2008).
J. Rouquerol and K. Sing, Adsorption by Powders and Porous Solids: Principles, Methodology and Application, Academic Press, London (1999).
S. Lowell, J. E. Shields, M. A. Thomas, et al., Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density, Springer, New York (2004).
L. M. Anovitz and D. R. Cole, Rev. Mineral. Geochem., 80, 61 (2015).
Yu. B. Melnichenko, Structural Characterization of Porous Materials Using SAS, Small-Angle Scattering from Confined and Interfacial Fluids, Springer, Switzerland (2016), pp. 139–171.
C. Jacobson, X-Ray Microscopy, Cambridge University Press (2020).
S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, Materials, 5, 937 (2012).
F. Pfeiffer, Nat. Photon., 12, 9 (2018).
E. Tsai, J. Billaud, D. F. Sanchez, et al., Science, 11, 356 (2019).
D. Paganin, Coherent X-Ray Optics, Oxford Series on Synchrotron Radiation, Oxford University Press (2006).
D. M. Paganin and D. Pelliccia, “Tutorials on X-ray Phase Contrast Imaging: Some Fundamentals and Some Conjectures on Future Developments,” https://arxiv.org/abs/1902.00364 (2019).
A. Snigirev, I. Snigireva, V. Kohn, et al., Rev. Sci. Instrum., 66, 5486 (1995).
T. S. Argunova and V. G. Kohn, Phys. Usp., 62, 602 (2019).
J. Rodenburg and A. Maiden, “Ptychography,” in: P. W. Hawkes and J. C. H. Spence (Eds.), Springer Handbook of Microscopy, Springer Handbooks, Springer Nature, Switzerland (2019).
N. L. Popov, I. A. Artyukov, A. V. Vinogradov, et al., Usp. Phyz. Nauk, 190, 766 (2020).
I. Schelokov, T. Weitkamp, and A. Snigirev, Opt. Commun., 213, 247 (2002).
[http://77.51.206.116:65000/fiber_web/x-ray_porosimetry.aspx].
The Quest for Quantitative Microscopy, Nat. Meth., 9, 627 (2012); https://doi.org/10.1038/nmeth.2102.
A. Erdelyi, Asymptotic Expansions, Dover (2012).
A. Papoulis, Systems and Transforms with Applications in Optics, Krieger Pub., Malabar, USA (1981).
F. W. J. Olver, Introduction to Asymtotics and Special Functions, Academic Press, New York (1974).
V. Guillemin and S. Sternberg, Geometric Asymptotics, American Mathematical Society (1990), Ch. 1.
M. V. Fedoryuk, Method of Steepest Descent [in Russian], URSS Publishing Co., Moscow (2015).
N. N. Bogoliubov and D. V. Shirkov, Quantum Fields, Benjamin/Cummings Pub., London (1982).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schelokov, I.A., Popov, N.L. & Vinogradov, A.V. Analytical Approach to the Theory of X-Ray Observation of Pores in Bulk Materials. J Russ Laser Res 42, 32–44 (2021). https://doi.org/10.1007/s10946-020-09927-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10946-020-09927-0