Unit cell vector

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

Figure 10.3. Schematic of cluster with 19 chains in a Phase II (alternating rows of left- and right-handed helices). Projected unit cell vectors a and h are shown.

Fig. 438 Mode] for a structure giving rise to the diffraction pattern in Fig. 4.39 (Phoon et al. 1993). (a) Hexagonal ABC structure (not close-packed) (b) top view (d) front view, showing lattice vectors and some of the lattice planes (c) orientation of unit cell vectors (ajb,c) with respect to the flow direction, rotation axis and the direction of the neutron beam (e) the Bragg diffraction pattern from a twinned structure.

Figure 8.1 Notation of a cutting plane by Miller indices. The three-dimensional crystal is described by the three-dimensional unit cell vectors di, 02, and 03. The indicated plane intersects the crystal axes at the coordinates (3,1,2). The inverse is (, j, ). The smallest possible multiplicator to obtain integers is 6. This leads to the Miller indices (263).

The room-temperature crystal structure of (NH3)K3C60 was first determined by Rosseinsky et al. [17]. Ammoniation of K3C60 leads to an orthorhombic distortion of the fee structure with one K+ and one NH3 per octahedral site. The C60 units are orientationally ordered with their two-fold axes aligned with the unit cell vectors (space group Fmmm), while K+ and NH3 are oppositely displaced from the site centre along the [110] direction and the K+-NH3 pairs are randomly oriented along one of four K+-NH3 directions. 

T and H components are the same as in the yttrium oxyfluorides etc., but now they occur as strips intergrown in each layer the entire structure is divided by anti-phase boundaries perpendicular to the layers, with a slip vector R equal to half the unit-cell vector in the layer-stacking direction. These structures (often slightly monoclinic ) are CC types, with finite incommensiurate portions. They will be considered more fully in Chap. 6 below. Meanwhile, we will simply point out that anti-phase-boundary structures of this sort are strictly limited to ternaries with two cations and one anion. The related ternaries considered earlier in this section - those containing one cation and two anions - do not have these boundaries they are truly non-commensurate. This difference we take to be significant. 

A complete discussion of surface structure includes the use of a shorthand notation describing the location of surface atoms. This abbreviated representation of the surface is based upon a projection of the bnlk structure on to the surface plane. From this projection the surface stmcture is described in terms of the unit cell vectors of the bulk material (a, b) ... 

The surface and bulk unit cell vectors represent the periodicity which allows a translation operation to generate an infinite array of atoms in the surface or bulk structure. The coefficients m, mu, m2, and m2i define a matrix which describes the transformation of the bulk unit vectors into the surface unit vectors. For example, the simplest surface structure occurs when the snrface maintains the same nnit cell as the bnlk. In this case the unit cell matrix would be... 

For simplicity, fractional coordinates are used to describe the lattice positions in terms of crystallographic axes, o, b, and c. For instance, the fractional coordinates are (1/2,1/2,1/2) for an object perfectly in the middle of a rmit cell, midway between all three crystallographic axes. To characterize crystallographic planes, integers known as Miller indices are used. These numbers are in the format (hkl), and correspond to the interception of unit cell vectors at (alh,blk,cll). Figure 2.8 illustrates... 

Crystal system Unit cell vector lengths Unit cell vector angles... 

It is noteworthy to point out that two sequential screw-axis or glide-plane operations will yield the original object that has been translated along one of the unit cell vectors. For example, a 63 axis yields an identical orientation of the molecule only after 6 repeated applications - 4.5 unit cells away (i.e., 6 x 3/4 = 4.5). However, since glide planes feature a mirror plane prior to translation, the first operation will... 

Figure 6.48. Illustration of the honeycomb 2D graphene network, with possible unit cell vector indices n,m). The dotted lines indicate the chirality range of tubules, from 0 = 0 (zigzag) to = 30° (armchair). For 0 values between 0 and 30°, the formed tubules are designated as chiral SWNTs. The electrical conductivities (metallic or semiconducting) are also indicated for each chiral vector. The number appearing below some of the vector indices are the number of distinct caps that may be joined to the n,m) SWNT. Also shown is an example of how a (5,2) SWNT is formed. The vectors AB and A B which are perpendicular to the chiral vector (AA are superimposed by folding the graphene sheet. Hence, the diameter of the SWNT becomes the distance between AB and A B axes. Reprinted from Dresselhaus, M. S. Dresselhaus, G. Eklund, R C. Science ofFullerenes and Carbon Nanotubes. Copyright 1996, with permission from Elsevier.

An alternate way of considering crystal planes, indexed as described earlier, is to consider the number of times a set of parallel planes intersects each crystal axis within one unit cell. The set of hkl planes cuts the a axis at h positions, the b axis at k positions, and the c axis at I positions [Figure 2.12(b)]. Ralph Steadman suggests To determine hkl move along the whole length of the unit cell vector a. Count the number of spaces crossed, and this is h. By spaces we mean spaces between planes. In Figure 2.12 hkl = 426 = 213, because, in usual crystallographic practice, any common divisor is factored out. In X-ray diffraction, described in Chapter 3, (426) is used to denote the second-order diffracted beam from a (213) plane. 

If the orientation of the principal optical directions can be found with respect to the unit cell vectors of the crystal, the orientations of the molecules in the unit cell, especially those with very anisotropic shapes and considerable unsaturation, may be found. This information was useful in the determination of the shape and size of the steroid nucleus (see Figure 1.11, Chapter 1). The relationship of molecular shape to refractive index is listed in Table 5.2, and the different refractive indices of naphthalene are shown in Figure 5.11. 

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