Hemihedral symmetry

Crystals composed of the R and S enantiomers of the same racemic mixture must be related by mirror symmetry in terms of both their internal structure and external shape. Enantiomorphous crystals may be sorted visually only if the crystals develop recognizable hemihedral faces. [Opposite (hid) and (hkl) crystal faces are hemihedral if their surface structures are not related to each other by symmetry other than translation, in which case the crystal structure is polar along a vector joining the two faces. Under such circumstances the hemihedral (hkl) and (hkl) faces may not be morphologically equivalent.] It is well known that Pasteur s discovery of enantiomorphism through die asymmetric shape of die crystals of racemic sodium ammonium tartrate was due in part to a confluence of favorable circumstances. In the cold climate of Paris, Pasteur obtained crystals in the form of conglomerates. These crystals were large and exhibited easily seen hemihedral faces. In contrast, at temperatures above 27°C sodium ammonium tartrate forms a racemic compound. 

Til, Til, and lTT) is reduced, while that of Til (andT 11, llT, and lTl) is not, and the resulting crystal is entirely bounded by the first-mentioned set of planes and thus has a hemihedral form. To produce an effect of this sort, molecules of the dissolved impurity need not be entirely without symmetry, but they must lack planes of symmetry, inversion axes, and a centre of symmetry. 

The symmetry of the model of a molecule or of a molecular ensemble depends on the conditions of the relevant physical (or chemical) measurement, and may vary for the same system according to time scale of observation and instrumental sensitivity. Whether the model of a chemical system is chiral or achiral may therefore depend on the conditions of observation. There is no ambiguity when chirality properties are observed the hemihedrality of quartz crystals, the optical rotation of hexahelicene, and the enantiospecificity of hog-kidney acylase, for example, are all unmistakable manifestations of an underlying structural chirality. On the other hand, achirality is not so simply implied by the absence of such observations. 

Ihsteur begins his first lecture by discussing the precedents that led up to his research and then defines hemihedral crystals. These are cubical crystals with four little facets inclined at the same angle to the adjacent surfaces and arranged alternately so the same edge of the cube does not contain two facets (Fig. 4). Under these conditions, no point or plane of symmetry exists in the cube. 

At this point we should recall that there are seven crystal systems. All crystals of the same crystal system have lattices with the same orientation symmetry. In each of the seven crystal systems, the group which defines it is the one with the highest order, and is termed holohedral. Within the same system, all the groups which are not holohedral are termed merihedral and, in particular, hemihedral for the case where the group order is half that of the holohedral. Those which possess a centre of symmetry are called centrosymmetric hemihedral, whilst those which lack an inversion centre are enantio-morphic hemihedral. 

Each crystal system contains several classes that exhibit only a partial symmetry for instance, only one-half or one-quarter of the maximum number of faces permitted by the symmetry may have been developed. The holohedral class is that which has the maximum number of similar faces, i.e. possesses the highest degree of symmetry. In the hemihedral class only half this number of faces have been developed, and in the tetrahedral class only one-quarter have been developed. For example, the regular tetrahedron (4 faces) is the hemihedral form of the holohedral octahedron (8 faces) and the wedge-shaped sphenoid is the hemihedral form of the tetragonal bipyramid Figure 1.9). 

Electric dipole transition can take place only between the states of different parities. In the free ion, therefore, direct inner shell transitions d d and /— / are forbidden. For the ions in crystals, however, this prohibition is removed by the mixing of orbitals having opposite parity. Such mixing may be induced by a) absence of a center of S5mimetry of the crystal field (existence of the hemihedral part of the crystal field) and b) destruction of the center of symmetry by odd vibrations. This was first pointed out by Van Vleck. In most salts of the 3d group ions the mechanism (b) plays the dominant role for the optical absorptions caused by the transitions between d" states. ... 

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