Hexagonal lattice unit cell

Similar measurements were performed using freeze-fracture electron microscopy. Two major sets of fracture planes were identified. One set, parallel to the (2 2 0) planes, yielded a square lattice similar to that seen in the thin-section study. The other set, parallel to the (1 1 1) planes, yielded an hexagonal lattice. Unit cell sizes, calculated on the basis of the diamond lattice, were very similar to those obtained from the thin-section electronmicrographs. 

In this diagram, a series of hexagon-shaped planes are shown which are orthogonal, or 90 degrees, to each of the corners of the cubic cell. Each plane connects to cuiother plane (here shown as a rectangle) on each fiace of the unit-cell. Thus, the faces of the lattice unit-cell and those of the reciprocal unit-cell can be seen to lie on the same pltme while those at the corners lie at right angles to the corners. 

The direct lattice and reciprocal lattice unit cells are marked on the crystal pattern of a planar hexagonal net in Figure 16.10, using eqs. (13), (14), and (18). The scales chosen for... 

Fig. 15. Intensity profiles along the equator of the bony fish muscle low angle X-ray diffraction pattern from muscles at rest (A), fully active (B), and in rigor (C). The indexing in (A) is based on the hexagonal A-band lattice, and the arrows indicate peaks that come from the Z-band. (C) to (F) are computed electron density maps based on the amplitudes of the A-band peaks in (A) to (A), respectively. The simple lattice unit cell is outlined in (D). (From Harford and Squire, 1997.)...

Bravais showed in 1850 that all three-dimensional lattices can be classified into 14 distinct types, namely the fourteen Bravais lattices, the unit cells of which are displayed in Fig. 9.2.3. Primitive lattices are given the symbol P. The symbol C denotes a C face centered lattice which has additional lattice points at the centers of a pair of opposite faces defined by the a and b axes likewise the symbol A or B describes a lattice centered at the corresponding A or B face. When the lattice has all faces centered, the symbol F is used. The symbol I is applicable when an additional lattice point is located at the center of the unit cell. The symbol R is used for a rhombohedral lattice, which is based on a rhombohedral unit cell (with a = b = c and a = ft = y 90°) in the older literature. Nowadays the rhombohedral lattice is generally referred to as a hexagonal unit cell that has additional lattice points at (2/3,1 /3, /s) and (V3,2/3,2/3) in the conventional obverse setting, or ( /3,2/3, ) and (2/3, /3,2/3) in the alternative reverse setting. In Fig. 9.2.3 both the primitive rhombohedral (.R) and obverse triple hexagonal (HR) unit cells are shown for the rhombohedral lattice. 

The rocksalt structure consists in two interpenetrating fee lattices of anions and cations, in which all atoms are in an octahedral environment. It is met in alkaline-earth oxides (MgO, CaO, SrO, BaO) and in some transition metal oxides like TiO, VO, MnO, FeO, CoO, NiO, etc, with cations in a 4-2 oxidation state. The non-polar surfaces of lowest Miller indices are the (100) and (110) surfaces they have neutral layers, with as many cations as oxygen ions, and their outermost atoms are 5- and 4-fold coordinated, respectively. Actually, planar surfaces can only be produced along the (100) orientation. The polar direction of lowest indices is (111) it has an hexagonal 2D unit cell, three-fold coordinated surface atoms and equidistant layers of either metal or oxygen composition. 

A primitive unit cell is a lattice unit cell that contains only one lattice point. The four primitive plane lattice unit cells are labelled p oblique, imp), rectangular, op), square, (tp) and hexagonal, (hp). They are normally drawn with a lattice... 

Fig. 118. Atomic stmcture of the fuUy relaxed 2x1 zigzag chain structure of one monolayer of Sb on Ge(lll). Atoms are labeled as in Table 70. (a) Top view of the surface showing Sb atoms and the first layer of Ge atoms. The lattice vectors of the hexagonal 1x1 unit cell are shown, together with the 2x1 surface unit ceU. (b) Side view of the surface in the (110) plane.

FIGURE 1.54 Hexagonal graphite unit cell. When the origin is set on a carbon atom, the positions of the four atoms of the cell, measured from a lattice point, are (0,0,0) ... 

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 BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

The many commercially attractive properties of acetal resins are due in large part to the inherent high crystallinity of the base polymers. Values reported for percentage crystallinity (x ray, density) range from 60 to 77%. The lower values are typical of copolymer. Poly oxymethylene most commonly crystallizes in a hexagonal unit cell (9) with the polymer chains in a 9/5 helix (10,11). An orthorhombic unit cell has also been reported (9). The oxyethylene units in copolymers of trioxane and ethylene oxide can be incorporated in the crystal lattice (12). The nominal value of the melting point of homopolymer is 175°C, that of the copolymer is 165°C. Other thermal properties, which depend substantially on the crystallization or melting of the polymer, are Hsted in Table 1. See also reference 13. 

Doping of alkali-metals into CNTs has been examined [11]. The X-ray powder diffraction (XRD) patterns of the K- or Rb-doped CNTs show that alkali-metals are intercalated between the CNT layers. The hexagonal unit cell is essentially the same as that of the stage-1 alkali-metal intercalated graphite ACg (A=K, Rb). For a sample doped with Rb, the observed lattice parameter of the perpendicular... 

LaNis has the CaCu, structure, space group P6/mmm [26] the hexagonal metal lattice is shown in Fig. 4. The crystal structure of LaNi5D7 has been determined 127, 28] and is illustrated in Fig. 5. There are three types of interstitial D sites La2Ni4 tetrahedra, and NiA tetrahedra. The unit cell is doubled along the oaxis because of the formation of a superlattice which is a consequence of long-range correlations between occupied and unoccupied... 

Figure 5.1. Unit cells of the face-centered cubic (fee), body-centered cubic (bcc), and hexagonally closed packed (hep) lattices.

In two dimensions, five different lattices exist, see Fig. 5.6. One recognizes the hexagonal Bravais lattice as the unit cell of the cubic (111) and hep (001) surfaces, the centered rectangular cell as the unit cell of the bcc and fee (110) surfaces, and... 

Fig. 4.12 (a) CdSe wurtzite unit cell (b) schematic illustration of a hexagonal (wurtzite) CdSe basal plane on a (111) section of the gold lattice, emphasizing the 2 3 lattice match. Note the [111] Au//(0001)CdSe orientation, with the CdSe a-directions aligned along the (llO)Au. The outlined rhombus indicates the projection of a CdSe unit cell. (Adapted from [112])... 

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