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2024 | OriginalPaper | Buchkapitel

3. Point Symmetry

verfasst von : Rick Ubic

Erschienen in: Crystallography and Crystal Chemistry

Verlag: Springer International Publishing

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Abstract

Symmetry may seem like a self-evident property of an object – you know it when you see it – but for the study of crystallography a rigorous definition is required, so here goes: Symmetry is that property possessed by an object when some operation (translation, rotation, reflection, and/or inversion) leaves it indistinguishable from its original position. In other words, symmetry is a demonstration of the invariance of the object to the operation. Topics such as electronic bands, phonon dispersion, vibrational spectroscopy (e.g., Raman and IR), and the properties of crystalline materials generally require a good understanding of crystal symmetry.

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Vorheriges Kapitel Crystals

Nächstes Kapitel Crystal Systems

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Shubnikov fell victim to the Stalinist purges of the 1930s at the age of just 36.

cos(a + b) = cos(a)cos(b) – sin(a)sin(b) and sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

The terms enantiomer and enantiomorph are used interchangeably. Both are derived from the Greek ἐναντίος (enantíos), meaning “opposite.” The terms enantiomer and chirality are mainly applied to molecules, whereas the term enantiomorph is preferred in crystallography.

The notation was introduced in 1907 by German chemist Hermann Emil Fischer and is now obsolete for all molecules except sugars and amino acids. Working with glyceraldehyde, he named one variant D (from the Latin dextro, meaning “on the right”) and the other L (from the Latin laevo, meaning “on the left”). The modern notation would be R (from the Latin rectus, meaning “right”) and S (from the Latin sinister, meaning “left”).

The designation is related to the position of the OH group (for sugars) or NH₂ group (for amino acids) on the bottommost chiral centre when the molecule is drawn with the most oxidised carbon at the top.

but not for lack of trying. The US company Richardson-Merrell requested approval on no fewer than six occasions but was refused each time because the company failed to provide clinical evidence to refute reports of patients who developed nerve damage in their limbs after long-term thalidomide use. The FDA finally approved the drug for some other uses in 1998.

In a group containing elements a, b, c… elements a and b are conjugate if a = c⁻¹bc. Every element is necessarily conjugate with itself (e.g., let c = E).

“Spiegel” is German for “mirror.” These groups contain only an n-fold rotoreflection axis, where n must be even because when n is odd an n-fold rotoreflection axis is equivalent to a combination of an n-fold rotation axis and a perpendicular mirror plane: S_n = C_nh.

These are sometimes referred to as reps, but the term irreps is much more widely accepted; therefore, to avoid confusion, reps will be used in reference to reducible representations, while irreps will be used for irreducible ones.

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I cheated because I already knew what the character table looked like.

In a group containing elements a, b, c… elements a and b are conjugate if a = c⁻¹bc. Every element is necessarily conjugate with itself (e.g., let c = E).

It is left to the reader to prove that \( {E}^{-1}{\upsigma}_{\mathrm{v}}E={C}_2^{-1}{\upsigma}_{\mathrm{v}}{C}_2={\upsigma}_{\mathrm{v}}\ne {\upsigma_v}^{\prime } \).

An overtone band is the spectral band that occurs in a vibrational spectrum of a molecule when the molecule makes a transition from the ground state (v = 0) to the second excited state (v = 2), where v is the vibrational quantum number for the molecule. Overtone bands are multiples of the fundamental absorption frequency; that is, the first overtone is approximately twice the energy of the fundamental.

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The correct orientation of the site group symmetry elements can be easily inferred by placing an atom in a general position x,y,z and generating all the symmetry-related positions with crystal-simulation software.

G. Placzek, “Rayleigh-Streuung und Raman-Effekt,” in Handbuch der Radiologie, E. Marx, Ed., Leipzig, Akademische Ver-lagsgesellschaft M.B.H., 1934, p. 205–374.

“Application Note – The Raman Spectrum of Water,” StellarNet Inc., 14 August 2019. [Online]. Available: https://www.stellarnet.us/application-note-raman-spectrum-water/.

G. Behrens, L. Kuhn, R. Ubic and A. Heuer, “Raman spectra of vateritic calcium carbonate,” Spectroscopy Letters, vol. 28, no. 6, pp. 983–995, 1995.CrossRef

W. G. Fately, F. R. Dollish, N. T. McDevitt and F. F. Bentley, Infrared and Raman Selection Rules for Molecular and Lattice Vibrations: The Correlation Method, New York: John Wiley and Sons, 1972.

G. Behrens, “The Martensitic Transformation in Zirconia, PhD Thesis, Case Western Reserve University,” 1993.

Titel: Point Symmetry
verfasst von: Rick Ubic
Verlag: Springer International Publishing
Buch: Crystallography and Crystal Chemistry
Print ISBN: 978-3-031-49751-3

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

Copyright-Jahr: 2024
DOI: https://doi.org/10.1007/978-3-031-49752-0_3

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