Unit cell filling

The initial configuration is set up by building the field 0(r) for a unit cell first on a small cubic lattice, A = 3 or 5, analogously to a two-component, AB, molecular crystal. The value of the field 0(r) = at the point r = (f, 7, k)h on the lattice is set to 1 if, in the molecular crystal, an atom A is in this place if there is an atom B, 0, is set to —1 if there is an empty place, j is set to 0. Fig. 2 shows the initial configuration used to build the field 0(r) for the simple cubic-phase unit cell. Filled black circles represent atoms of type A and hollow circles represent atoms of type B. In this case all sites are occupied by atoms A or B. 

Figure 1.1 Defects in crystalline solids (a) point defects (interstitials) (b) a linear defect (edge dislocation) (c) a planar defect (antiphase boundary) (d) a volume defect (precipitate) (e) unit cell (filled) of a structure containing point defects (vacancies) and (/) unit cell (filled) of a defect-free structure containing ordered vacancies. ...

Unit cell Parallelepiped defined by basis vectors a, b, and c. Unit cell fills space under translation. 

Fig. 17. Epistilbite. (a) Temperature dependences of unit-cell volume (filled triangle) and water molecules content per unit cell (filled circle), (b) Unit-cell parameters of the monoclinic structure, (c) Individual H2O molecules per unit cell [03C1].

The otiier type of noncrystalline solid was discovered in the 1980s in certain rapidly cooled alloy systems. D Shechtman and coworkers [15] observed electron diffraction patterns with sharp spots with fivefold rotational synnnetry, a syimnetry that had been, until that time, assumed to be impossible. It is easy to show that it is impossible to fill two- or tliree-dimensional space with identical objects that have rotational symmetries of orders other than two, tliree, four or six, and it had been assumed that the long-range periodicity necessary to produce a diffraction pattern with sharp spots could only exist in materials made by the stacking of identical unit cells. The materials that produced these diffraction patterns, but clearly could not be crystals, became known as quasicrystals. 

In the face-centred cubic structure tirere are four atoms per unit cell, 8x1/8 cube corners and 6x1/2 face centres. There are also four octahedral holes, one body centre and 12 x 1 /4 on each cube edge. When all of the holes are filled the overall composition is thus 1 1, metal to interstitial. In the same metal structure there are eight cube corners where tetrahedral sites occur at the 1/4, 1/4, 1/4 positions. When these are all filled there is a 1 2 metal to interstititial ratio. The transition metals can therefore form monocarbides, niU ides and oxides with the octahedrally coordinated interstitial atoms, and dihydrides with the tetrahedral coordination of the hydrogen atoms. 

Let us now look at the c.p.h. unit cell as shown in Fig. 5.4. A view looking down the vertical axis reveals the ABA stacking of close-packed planes. We build up our c.p.h. crystal by adding hexagonal building blocks to one another hexagonal blocks also stack so that they fill space. Here, again, we can use the unit cell concept to open up views of the various types of planes. 

Many ionic compounds are considered to pack in such as way that the anions form a close-packed lattice in which the metal cations fill holes or interstitial sites left between the anions. These lattices, however, may not necessarily he as tightly packed as the label close-packed implies. The radius of an F ion is approximately 133 pm. The edge distances of the cubic unit cells of LiF, NaF, KF, RbF, and CsF, all of which... 

The uncertainties given are calculated standard deviations. Analysis of the interatomic distances yields a selfconsistent interpretation in which Zni is assumed to be quinquevalent and Znn quadrivalent, while Na may have a valence of unity or one as high as lj, the excess over unity being suggested by the interatomic distances and being, if real, presumably a consequence of electron transfer. A valence electron number of approximately 432 per unit cell is obtained, which is in good agreement with the value 428-48 predicted on the basis of a filled Brillouin polyhedron defined by the forms 444, 640, and 800. ... 

Fig. 3.—Parallel packing arrangement of the 2-fold helices of cellulose I (1). (a) Stereo view of two unit cells approximately normal to the ac-plane. The two comer chains (open bonds) in the back, separated by a, form a hydrogen-bonded sheet. The center chain is drawn in filled bonds. All hydrogen bonds are drawn in dashed lines in this and the remaining diagrams, (b) Projection of the unit cell along the c-axis, with a down and b across the page. No hydrogen bonds are present between the comer and center chains.

Fig. 8.—Packing arrangement of four symmetry-related 2-fold helices of mannan II (6). (a) Stereo view of two unit cells approximately normal to flic frc-plane. The two chains in the back (open bonds) and the two in the front (filled bonds) are linked successively by 6-0H-- 0-6 bonds. The front and back chains, both at left and right, are further connected by 0-2 -1V -0-2 bridges, (h) Projection of the unit cell along the c-axis the a-axis is down the page. This highlights the two sets of interchain hydrogen bonds between antiparallel chains, distinguished by filled and open bonds. The crossed circles are water molecules at special positions.

Fig. 9. — Antiparallel packing arrangement of the 3-fold helices of (1— 4)-(3-D-xylan (7). (a) Stereo view of two unit cells roughly normal to the helix axis and along the short diagonal of the ab-plane. The two helices, distinguished by filled and open bonds, are connected via water (crossed circles) bridges. Cellulose type 3-0H-0-5 hydrogen bonds stabilize each helix, (b) A view of the unit cell projected along the r-axis highlights that the closeness of the water molecules to the helix axis enables them to link adjacent helices.

Fig. 12.—Packing arrangement of shallow, 6-fold, V-amylose (10) helices, (a) Stereo view of two unit cells approximately normal to the fee-plane. The helix at the center (filled bonds) is antiparallel to the two helices at the comers in the back (open bonds). Intrachain hydrogen bonds (3-OH - - 0-2 and 6-OH 0-3) are shown in thin lines.

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