Abstract
Crystals are variously defined as an infinite array of atoms in space, a lattice plus a basis, or even any solid having an essentially discrete diffraction pattern. This chapter develops these ideas and introduces the concepts of the unit cell, lattice planes, and directions.
May crystals give you power!
— Isabel Walbourne.
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Notes
- 1.
A diaper, in this context, is a repeating geometric or floral pattern used to decorate a surface (originally a linen or cotton fabric woven in a repeating pattern of small diamonds). Early cloth diapers (or nappies) consisted of soft tissue cut into geometric shapes, a process known as diapering. The word “diaper” was later applied to the cloth used to make diapers and ultimately to the diapers themselves.
- 2.
In 2014, Shechtman ran for the presidency of Israel but received only one vote in the Knesset and so lost to Reuven Rivlin from the Likud party. Afterward, the Israeli press and humorists paraphrased Pauling, calling him a “quasipresident” in reference to the “quasi-scientist” quote.
- 3.
This was the year William Hallowes Miller was born!
- 4.
Two integers are coprime (or relatively prime) if there is no integer other than 1 and − 1 that evenly divides both of them (that is, their greatest common divisor is 1). For example, 2 and 3 are coprime, but 2 and 4 are not.
- 5.
Whewell also invented the term “scientist” in 1833 in response to a challenge by the poet S.T. Coleridge. Before then they were called either “men of science” or “natural philosophers.” He later coined the term “physicist” in 1840.
- 6.
Confusingly, these are frequently referred to as families of planes.
- 7.
The choice of whether to call these planes {100}, {010}, or {001} is arbitrary, but {010} is seldom chosen.
- 8.
The choice of whether to call these planes {110}, {101}, or {011} is arbitrary (negative indices are generally avoided unless they are crystallographically necessary).
- 9.
They have different structure factors (see Chap. 12).
- 10.
The choice of which of these three notations to use is arbitrary, but <010> is rarely chosen.
Works Cited
C. S. Barrett, Structure of Metals, 2nd ed., New York: McGraw-Hill, 1952, p. 1.
C. S. Weiss, über eine verbesserte Methode für die Bezeichung der verschiedenen Flächen eines Krystallisationssystemes; nebst Bemerkungen über den Zustand von Polarisierung der Seiten in den Linien der krystallinischen Structur, Berlin: Realschul-Buchhandlung, 1817, pp. 286–336.
W. H. Miller, A Treatise on Crystallography, London: John W. Parker, 1839.
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Review Questions
Review Questions
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1.
What are quasi-periodic structures?
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2.
Briefly define/explain each of these terms. Add a sketch if convenient: lattice, basis, crystal, unit cell, basis vectors, and lattice parameters.
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3.
Use the parallelepiped rule to calculate the volume of a triclinic unit cell with basis vectors defined as:
\( \boldsymbol{a}=\left(1\hat{i}+2\hat{j}+1\hat{k}\right)\ \mathring{\mathrm{A}}, \) \( \boldsymbol{b}=\left(2\hat{i}+1\hat{j}+3\hat{k}\right)\ \mathring{\mathrm{A}}, \) and \( \boldsymbol{c}=\left(4\hat{i}+1\hat{j}+1\hat{k}\right)\ \mathring{\mathrm{A}} \).
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4.
What is the angle between the [124] and \( \left[32\overline{1}\right] \) directions in a unit cell with lattice constants a = 5 Å, b = 7 Å, c = 9 Å, α = β = γ = 90°?
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5.
Identify the Miller indices of the planes depicted below:
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6.
Identify both the Miller indices and Miller-Bravais indices of the planes depicted below:
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7.
Calculate the theoretical density of CaTiO3 given that a = 5.4425 Å, b = 7.6410 Å, c = 5.3805 Å, α = β = γ = 90°, and Z = 4. Assume that MCa = 40.078 g/mole, MTi = 47.867 g/mole, and MO = 15.9994 g/mole.
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Ubic, R. (2024). Crystals. In: Crystallography and Crystal Chemistry. Springer, Cham. https://doi.org/10.1007/978-3-031-49752-0_2
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DOI: https://doi.org/10.1007/978-3-031-49752-0_2
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