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2024 | OriginalPaper | Chapter

11. Diffraction and Reciprocal Space

Author : Rick Ubic

Published in: Crystallography and Crystal Chemistry

Publisher: Springer International Publishing

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Abstract

This chapter introduces the concept of diffraction as a way of linking measurable macroscopic phenomena to microscopic ones.

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previous chapter Other Classifications of Structures

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Fraunhofer diffraction is the type of diffraction that occurs when the Fresnel number, F = a²/Lλ << 1. Under this condition, the diffraction pattern is independent of the distance to the screen and depends only on the angles (θ) to the screen from the aperture (i.e., only its apparent magnification can be altered by adjusting L).

Recall that \( \left(\frac{\sin \left(2\upbeta \right)}{2\ \sin \left(\upbeta \right)}\right)=\cos \left(\upbeta \right) \).

Recall that the lattice points (and therefore atoms) in successive lattice planes need not align on top of each other.

Bragg made the connection between diffraction and reflection by observing the elliptical shape of diffraction spots, which he noticed was also characteristic of the reflections of beams of light from a mirror.

It often seems that this convention is “more honor’d in the breach than the observance.” Indeed, it is astonishing how many publishing scientists are not aware of the difference between Miller indices and Laue indices, and even most diffraction texts seem to completely side-step the issue. One particularly venerated text only adds to the confusion by using (hkl) in the text body but hkl in the figures!

Curiously, Thomson studiously avoided the word “electron,” which had been coined by G.J. Stoney in 1891 (more on that in Chap. 13).

Of course, they were both right!

Recall that e^ix = cos x + i sin x

Note that ∣F∣ is the modulus of the complex F, that is, its magnitude on the Argand diagram. If F = A + iB, then ∣F∣ = A² + B² = (A + iB)(A − iB).

The possibility of double diffraction is dependent upon the curvature of the Ewald sphere, which is λ; therefore, the curvature for 200 kV electrons (λ = 0.025 Å) is ~62 times larger than that for CuKα₁ x-rays λ = 1.540562 Å)!

In this context, “formal” means that this notation has the form of a determinant, but does not strictly adhere to the definition. It is merely a convenient way to express the expansion of the cross product.

W. F. a. P. K. Max von Laue, “Eine quantitative Prüfung der Theorie für die Interferenz-Erscheinungen bei Röntgenstrahlen,” Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften, pp. 363–373, 1912.

W. K. a. G. Möllenstedt, “Elektroneninterferenzen im konvergenten Bündel,” Annalen der Physik, vol. 428, no. 2, pp. 113–140, 1939.CrossRef

H. E. Armstrong, “Poor Common Salt!,” Nature, vol. 120, p. 478, 1927.CrossRef

H. E. Armstrong, “Hydrolysis, Hydrolation and Hydronation as Determinants of the Properties of Aqueous Solutions,” Proceedings of the Royal Society, vol. A81, pp. 80–95, 1908.

Title: Diffraction and Reciprocal Space
Author: Rick Ubic
Publisher: Springer International Publishing
Book: Crystallography and Crystal Chemistry
Print ISBN: 978-3-031-49751-3

Electronic ISBN: 978-3-031-49752-0

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-49752-0_11

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