Three-fold rotation axis

Because of the orientational freedom, plastic crystals usually crystallize in cubic structures (Table 4.2). It is significant that cubic structures are adopted even when the molecular symmetry is incompatible with the cubic crystal symmetry. For example, t-butyl chloride in the plastic crystalline state has a fee structure even though the isolated molecule has a three-fold rotation axis which is incompatible with the cubic structure. Such apparent discrepancies between the lattice symmetry and molecular symmetry provide clear indications of the rotational disorder in the plastic crystalline state. It should, however, be remarked that molecular rotation in plastic crystals is rarely free rather it appears that there is more than one minimum potential energy configuration which allows the molecules to tumble rapidly from one orientation to another, the different orientations being random in the plastic crystal. 

The symmetry elements that intersect at a Wyckoff position determine its site symmetry. For example, a Wyckoff position that lies on the intersection of two mirror planes has mm2 (C2v) symmetry (Fig. 10.6(a)) while one that lies at the intersection of a mirror plane and a three-fold rotation axis along its normal has 3/m (Csh) symmetry (Fig. 10.6(b)). An atom lying on a general position has no symmetry other than a one-fold axis which is represented by the symbol 1 (Ci). 

An example of a molecule with a three-fold rotation axis is the conformation of. vym-1,3,5-triethylcyclohexane shown in Figure B.l. Note that all molecules possess a trivial Ci axis (indeed, an infinite number of them). Note also that if we choose a Cartesian coordinate system where the proper rotation axis is the z axis, and if the rotation axis is two-fold, then for every atom found at position (x,y,z) where x and y are not simultaneously equal to 0 (i.e., not on the z axis itself) there will be an identical atom at position (—x,—y,z). If the rotation axis is four-fold, there will be an identical atom at the three positions (—x,y,z), (x,—y,z), and (—x,—y,z). Note finally that for linear molecules the axis of the molecule is a proper symmetry axis of infinite order, i.e., Cao-... 

An alternative view of an octahedron down a three-fold rotation axis... 

The structure reveals that the PtFg octahedra, one of which is represented in Fig. 1, are compressed along the three-fold rotation axis along which the dioxygenyl ions lie. 

This trigonal distortion of the MFg ion is similar to that previously observed in the rhombohedral potassium hexafluoro-osmate(v) structured Each ion has twelve fluorine neighbours, six in a puckered ring and almost coplanar with the ion, the other six in sets of three, one above and one below this plane. The situation of the dioxygenyl ion with its long axis coincident with the three-fold rotation axis is more acceptable on the grounds of close packing than the alternative random orientation (equivalent to rotation) of the ion. Apart from the non-spherical nature of the cation, the structure is almost identical with that proposed for the potassium hexafluoroantimonate(v). ... 

Symbols of finite crystallographic symmetry elements and their graphical representations are listed in Table 1.4. The fiill name of a symmetry element is formed by adding "N-fold" to the words "rotation axis" or "inversion axis". The numeral N generally corresponds to the total number of objects generated by the element, and it is also known as the order or the multiplicity of the symmetry element. Orders of axes are found in columns two and four in Table 1.4, for example, a three-fold rotation axis or a fourfold inversion axis. 

Furthermore, as we will see in sections 1.5.3 and 1.5.5, below, transformations performed by the three-fold inversion and the six-fold inversion axes can be represented by two independent simple symmetry elements. In the case of the three-fold inversion axis, 3, these are the threefold rotation axis and the center of inversion acting independently, and in the case of the six-fold inversion axis, 6, the two independent symmetry elements are the mirror plane and the three-fold rotation axis perpendicular to the plane, as denoted in Table 1.4. The remaining four-fold inversion axis, 4, is a unique symmetry element (section 1.5.4), which cannot be represented by any pair of independently acting symmetry elements. 

Three-fold rotation axis and three-fold inversion axis... 

It is easy to see that the six symmetrically equivalent objects are related to one another by both the simple three-fold rotation axis and the center of inversion. Hence, the three-fold inversion axis is not only the result of two simultaneous operations (3 and 1), Iwt it is also the result of two independent operations. In other words, 3 is identical to 3 then 1. 

The six-fold inversion axis Figure 1.15, right) also produces six symmetrically equivalent objects. Similar to the three-fold inversion axis, this symmetry element can be represented by two independent simple symmetry elements the first one is the three-fold rotation axis, which connects pyramids 1-3-5 and 2-4-6, and the second one is the mirror plane perpendicular to the three-fold rotation axis, which connects pyramids 1-4, 2-5, and 3-6. As an exercise, try to obtain all six symmetrically equivalent pyramids starting from the pyramid 1 as the original object by applying 60° rotations followed by immediate inversions. Keep in mind that objects are not retained in the intermediate positions because the six-fold rotation and inversion act simultaneously. 

Symmetry operations due to the presence of the three-fold rotation axis along the body diagonal of a cube in the [111] direction are described by... 

In general, g =l/ , where n is the multiplicity of the symmetry element which causes the overlap of the corresponding atoms. When the culprits are a mirror plane, a two fold rotation axis or a center of inversion, n = l and g = 0.5. For a three fold rotation axis = 3 and = 1/3, and so on (Figure... 

Figure 2.52. The illustration of forbidden overlaps as a result of an atom being too close to a finite symmetry element mirror plane (left), three-fold rotation axis (middle), and four-fold rotation axis (right). Assuming that there are no defects in a crystal lattice, these distributions require g" = 1/2,1/3 and 1/4, respectively.

This set of unit cells shows that there is no six-fold rotational axis along the c-axis (remember that the top view looks down the c-axis). However, there is a three-fold rotational axis, and this is sufQcient to place wurtzite in the hexagonal system. 

Fig. 5.4 Three-fold rotation axis of ammonia (projected on the plane of the page)...

In parallel with the Vi activation, also the splitting of the V3 vibration occurs upon coordination. In the free CCb " ion, the V3 vibration is doubly degenerate (Fig. 9, bottom-left quadrant). Doubly degenerate vibrations occur only in molecules possessing an axis higher than twofold, which is the case of the Dsh symmetry, having a three-fold rotational axis (see... 

Fig. 9, upper-left quadrant C3 - three-fold axis) (Nakamoto, 1997). The lowering of the symmetiy of the carbonate ion from Dsh to either C2 or Cs, which means loss of the equivalence of the three C-O bonds in the CQ32- and, therefore, loss of the three-fold rotational axis, leads to the sep>aration (sphtting) of the doubly degenerate vibrations (Fig. 9, bottom-right quadrant). 

First we view the optically active tris(chelate) complexes down the three-fold rotational axis. 

EXAMPLE 13-2 BH3, like ammonia, has three planes of reflection symmetry that contain the three-fold rotational axis. However, BH3, being planar, also has a reflection operation through the molecular plane. Is this reflection operation in the... 

SOLUTION A three-fold improper axis coincident with the regular three-fold rotational axis appears to have the correct symmetry characteristics It rotates each hydrogen to the next position and then interchanges the spaces above and below the plane. Do we need it in the group Only if there is no other single operation that accomplishes the same thing. There is no such operation, so this improper axis is indeed a nonredundant symmetry element. ... 

Fig. 3.11 The three minimum configuration for the regular hexagon of points (a) has a one-fold rotational axis of symmetry (b) has a three-fold rotational axis of symmetry (c) has a two-fold rotational axis of symmetry, (a) has the smallest path length.

---