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Each crystallographic point group defines the (geometric) crystal class of the crystal. The point group of a crystal determines, among other things, the directional variation of physical properties that arise from its structure, including optical properties such as birefringency , or electro-optical features such as the Pockels effect .
In particular, a symmetry element can be a mirror plane, an axis of rotation (either proper and improper), or a center of inversion. [ 1 ] [ 2 ] [ 3 ] For an object such as a molecule or a crystal , a symmetry element corresponds to a set of symmetry operations, which are the rigid transformations employing the symmetry element that leave the ...
These groups are characterized by i) an n-fold proper rotation axis C n; ii) n 2-fold proper rotation axes C 2 normal to C n; iii) a mirror plane σ h normal to C n and containing the C 2 s. The D 1 h group is the same as the C 2 v group in the pyramidal groups section.
It contains also inversion center and improper rotation operations. I (the chiral icosahedral group) indicates that the group has the rotation axes of an icosahedron or dodecahedron (six 5-fold axes, ten 3-fold axes, and 15 2-fold axes). I h includes horizontal mirror planes and contains also inversion center and improper rotation operations.
For example, the C 4 axis of the square xenon tetrafluoride (XeF 4) molecule is associated with two Ĉ 4 rotations in opposite directions (90° and 270°), a Ĉ 2 rotation (180°) and Ĉ 1 (0° or 360°). Because Ĉ 1 is equivalent to Ê, Ŝ 1 to σ and Ŝ 2 to î, all symmetry operations can be classified as either proper or improper rotations.
Then a scale is strictly proper if i < j implies class(i) < class(j). It is proper if i ≤ j implies class(i) ≤ class(j). Strict propriety implies propriety but a proper scale need not be strictly proper; an example is the diatonic scale in equal temperament, where the tritone interval belongs both to the class of the fourth (as an augmented ...
That equals about 12 ounces of regular beer (at 5% alcohol; some light beers have less) or 5 ounces of wine (at 12% alcohol) or 1.5 ounces of distilled spirits (at 40% alcohol or 80 proof).
In chemistry, chirality usually refers to molecules. Two mirror images of a chiral molecule are called enantiomers or optical isomers. Pairs of enantiomers are often designated as "right-", "left-handed" or, if they have no bias, "achiral". As polarized light passes through a chiral molecule, the plane of polarization, when viewed along the ...