Hexagonal coordinate system

FIGURE 3.6 In (a), the disposition of asymmetric units related by a 4j screw axis is illustrated. As with the 2i screw axis, continued application of the symmetry operator in a crystal simply generates asymmetric units in adjacent unit cells which were already present due to unit cell translations. In (b), a 61 screw axis produces six identical asymmetric units whose equivalent positions are specified according to a hexagonal coordinate system. It follows that such a symmetry axis could only be compatible with a unit cell having a hexagonal face (i.e., a hexagonal prism). 

Fig. 2.15. Primitive rhombohedral cell (space group / 32) in hexagonal coordinate system. The hexagonal cell has lattice points at 0,0,0 2/3,1/3,1/3 l/3,2/3,2/3. The threefold axes are indicated by triangles. The twofold axes, whose directions are indicated by broad arrows, occur in pairs separated by Cjj/2 the numbers give the fractional Zh coordinates...

MILLER-BRAVAIS INDICES FOR HEXAGONAL COORDINATE SYSTEMS... 

A complication arises from the action of a threefold axis in a hexagonal coordinate system. In this case, the indices of equivalent faces are not obtained simply by permutations and sign changes of (hkl). To circumvent this problem, we use four indices (hkil) defined with respect to the vectors ai,a2,a3 perpendicular to the threefold axis (Fig. 2.37), and c parallel to the threefold axis. For /i, fc> 0, the length cut by the plane on a3 is negative, hence we obtain ... 

Fig. 2.37. Indices hkil) with respect to a hexagonal coordinate system (325/), (532/), (253/).

A consequence of Neumann s symmetry principle is that direct tensor Onsager coefficients (such as in the diffusivity tensor) must be symmetric. This is equivalent to the addition of a center of symmetry (an inversion center) to a material s point group. Thus, the direct tensor properties of crystalline materials must have one of the point symmetries of the 11 Laue groups. Neumann s principle can impose additional relationships between the diffusivity tensor coefficients Dij in Eq. 4.57. For a hexagonal crystal, the diffusivity tensor in the principal coordinate system has the form... 

For a crystal having a hexagonal symmetry, a set of four numbers, [uvtw], named the Miller-Bravais coordinate system (see Figure 1.5), is used to describe the crystallographic directions, where the first three numbers, that is, u, v, t, are projections along the axes at, a2, and a3, describing the basal plane of the hexagonal structure, and w is the projection in the z-direction [2,3],... 

Although the coordinate systems in these two divisions are of different symmetry, hexagonal axes may be used in the description of rhombohedral crystals and vice versa.)... 

As in the earlier three-dimensional example, where it was convenient to use hexagonal axes involving linear combinations of the rhombohedral basis vectors, it may also be useful here to use an alternative coordinate system to bring out certain symmetry properties. For molecules with a Tj frame, one choice is to take ... 

Figure 4.15 First Brillouin zone of the hexagonal close-packed crystal lattice, kx.ky, kz are the axes of the Cartesian coordinate system in fe-space. The symmetry points and symmetry lines are indicated. See Table 4.1 for details.

Fig. 9.1 The tilted Cartesian coordinate system with 60° angle between x and y axes and the tilted network in the x y-plane. A polyhex corresponding to the carbon skeleton of dibenzo[b,g] phenanthrene is placed onto this network in such a way that x,y intersections appear in the centre of each of its hexagons...

For uniaxial liquid crystals like nematic mesophases, and lamellar and hexagonal amphiphilic phases, the system is cylindrically symmetric with respect to an axis called the director. It is convenient to perform the transformation from the molecular coordinate system to the laboratory system via the director coordinate system (D). For a sample which is macroscopically aligned so that the director has the same direction throughout the sample we then have... 

In-situ x-ray scattering measurements. X-ray measurements were conducted on beam-line 7-2 at the Stanford Synchrotron Radiation Laboratory (SSRL) and on beam-line X-20A at the National Synchrotron Light Source (NSLS) using monochromatic X-ray beams of wavelengths 1.240 A and 1.409 A, respectively. In each case, the detector resolution was defined by 2 mm (vertical) and 8 mm (horizontal) slit at distance of 0.5 m from the sample position. The (111) face is indexed to a hexagonal lattice described in Reference [18]. Using this scheme, the surface normal lies along the (00L) ex direction, and the (H00)j, j and (OKO)j vectors lie in the plane of the surface and subtend 60°. In this coordinate system (0,0,3X, = (1,1,1,) ,bie and (0,1,2), = (0,0,2), . 

In hexagonal crystals, where the base plane exhibits a 120° symmetry, the Miller-Bravais system is used to specify planes and directions, using a coordinate system with four axes Three of these, with angles of 120° between them, lie in the base plane and are equivalent, the fourth is perpendicular to the base plane [hkil] (figure B.2). In this way, the symmetry of the crystal is reflected in the indices. The first three indices obey the additional constraint... 

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