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Diffraction and Reciprocal Space

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Crystallography and Crystal Chemistry

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Abstract

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

The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them.

—W.L. Bragg

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Notes

  1. 1.

    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).

  2. 2.

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

  3. 3.

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

  4. 4.

    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.

  5. 5.

    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!

  6. 6.

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

  7. 7.

    Of course, they were both right!

  8. 8.

    Recall that eix = cos x + i sin x

  9. 9.

    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).

  10. 10.

    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)!

  11. 11.

    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.

Works Cited

  1. 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.

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  4. 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.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Micron School of Materials Science and Engineering, Boise State University, Boise, ID, USA

    Rick Ubic

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  1. Rick Ubic
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Review Questions

Review Questions

  1. 1.

    Briefly define/explain the term Laue class.

  2. 2.

    What direction is common to both planes (241) and (331) in a monoclinic unit cell with a = 3.7 Å, b = 4.5 Å, c = 7.7 Å, β = 84.3°?

  3. 3.

    Given a monoclinic unit cell with lattice constants a = 2 Å, b = 3 Å, c = 5 Å, and β = 94°, use the reciprocal metric tensor to calculate:

    1. (a)

      The interplanar spacing of (120) planes

    2. (b)

      The angle between (120) and (112) planes

  4. 4.

    Calculate the structure factor, F, for the following reflections of an ideal cubic perovskite (ABX3) in terms of fA, fB, fX. Your answers should have no imaginary parts.

$$ {\displaystyle \begin{array}{l}100\\ {}110\\ {}111\\ {}200\end{array}} $$

Which is likely to be the weakest of the four?

The atomic positions are:

$$ {\displaystyle \begin{array}{lll}\mathrm{A}& & 0,0,0\\ {}\mathrm{B}& & \frac{1}{2},\kern3pt \frac{1}{2},\kern3pt \frac{1}{2}\\ {}\mathrm{X}& & \frac{1}{2},\kern3pt \frac{1}{2},\kern3pt 0\\ {}& & \frac{1}{2},\kern3pt 0,\kern3pt \frac{1}{2}\\ {}& & 0,\kern3pt \frac{1}{2},\kern3pt \frac{1}{2}\end{array}} $$
  1. 5.

    For the electron diffraction pattern shown below:

    1. (a)

      What are the indices of the reflection marked hkl?

    2. (b)

      What is the zone axis?

    3. (c)

      What can you conclude about the structure based on the systematic absences and the relative spot intensities?

  2. 6.

    Given a unit cell with lattice constants a = 1 Å, b = 2 Å, c = 3 Å, α = 70°, β = 80°, γ = 100°, calculate the reciprocal lattice constants a*, b*, c*, α*, β*, and γ*.

  3. 7.

    The three real-space planar patterns which gave rise to the diffraction patterns below have identical lattice constants.

    1. (a)

      What are the systematic absences (if any) in each case?

    2. (b)

      Explain the difference in appearance of these diffraction patterns in terms of possible structural differences between the three real-space planar patterns from which they were obtained.

    3. (c)

      Suggest a possible planar system and plane group for the real-space patterns from which each of these diffraction patterns was obtained.

      3 electron diffraction patterns of the x-ray beam passing through the sample. It exhibits the observation of different phases.

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Ubic, R. (2024). Diffraction and Reciprocal Space. In: Crystallography and Crystal Chemistry. Springer, Cham. https://doi.org/10.1007/978-3-031-49752-0_11

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