Coincidence unit cell

In practice the distinction between rational and irrational values of 1/6 is unimportant, because LEED cannot distinguish between unit cells larger than the coherence distance of the electron beam ( 100 A). It is customary to designate as incommensurate any overlayer that produces a coincidence unit cell larger than the LEED coherence distance. In fact, a truly incommensurate overlayer is impossible, since it could only occur in the limit of vanishing adsorbate-substrate forces parallel to the surface. 

In the SC and SS structures, the nodes of the component lattices of A and B coincide at multiples of the basic vector(s) and there is a coincidence ( super -) lattice, and its coincidence unit cell. The former represents a set of coinciding nodes of the true component lattices, and in the ideal case - with structurally-independent layer sets - it is not connected with common structural changes (modulation) in them. For growth considerations it is important that the SC and SS (as well as CC) structures have a coincidence mesh (coincidence net) parallel to the layers, whereas the IS structures have only a coincidence row in the layer plane. 

The FCC structure is illustrated in figure Al.3.2. Metallic elements such as calcium, nickel, and copper fonu in the FCC structure, as well as some of the inert gases. The conventional unit cell of the FCC structure is cubic with the lengdi of the edge given by the lattice parameter, a. There are four atoms in the conventional cell. In the primitive unit cell, there is only one atom. This atom coincides with the lattice pomts. The lattice vectors for the primitive cell are given by... 

The two forms differ by the way they pack, a direct result being the different tilt angle of their molecular axis (24" and 30" for the low-temperature and high-temperature form, respectively). Another important difference is the fact that the inversion center of the molecule coincides with a center of symmetry of the unit cell in the HT form, whereas it does not in the LT form 84J. Direct consequences of this feature have not yet been identified. It will be of course of great interest to know what would be its influence on charge transport properties. 

Characterization of the samples by TGA and CHN analysis shows that the template was effectively removed (C < 0.2 wt%). Small-angle X-ray scattering data of the calcined solid shows a reduction in the unit cell due to thermal shrinkage, while the values for the Fenton samples coincide with the starting precursor. Our approach therefore completely preserves the unit cell corresponding to the diameter of the micelles contained in the mesophase. 

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

Figure 3.4. The crystal systems and the Bravais lattices illustrated by a unit cell of each. All the points which, within a unit cell, are equivalent to each other and to the cell origin are shown. Notice that, in the primitive lattices the unit cell edges are coincident with the smallest equivalence distances. For the rhombohedral lattice, described in terms of hexagonal axis, the symbol hR is used instead of a symbol such as rP. In the construction of the so-called Pearson symbol ( 3.6.3), oS and mS will be used instead of oC and mC.

The 3D potential map was examined section by section perpendicular to the c-axis. There are totally 6 layers stacked along the c axis in each unit cell. Only two of these 6 layers are unique, one flat layer occurring twice (at z = 0.25 and 0.75) and one puckered layer occurring four times (at z 0.10, 0.40, 0.60 and 0.90). Sections corresponding to the flat (F) and puckered (P) layers are shown in Figs. 6a and b, respectively. The flat layers coincide with mirror planes. The stacking sequence is PFP (PFP" ) , where P relates to P via a mirror reflection on the flat layer, and the (PFP ) block is related to the PFP block by a 63 operation along the c axis. 

As a result, the same band structure must result for the H atom chain, irrespective of whether it is based on the wave functions of N H atoms or of A/2 H, molecules. In fact, the curve of Fig. 10.4 coincides with the curve of Fig. 10.6. The apparent difference has to do with the doubling of the lattice constant from a to a = 2a. As we see from equation (10.4), tire same wave functions yi/f, result for k = 0 and for k = 2nla, the same ones for k = ii/a and for k = in/a, etc. Whereas the curve in Fig. 10.4 runs steadily upwards from k = 0 to k = n/a, in Fig. 10.6 it only runs until k = nl 2a) = n/d, then it continues upwards from right to left. We can obtain the one plot from the other by folding the diagram, as shown in tire lower part of Fig. 10.6. Ihe folding can be continued triplication of the unit cell requires two folds, etc. 

Crystals, except those belonging to the cubic system, are anisotropic in this respect the force of repulsion varies with the orientation of the crystal with respect to the direction of the field. The graph representing vectorialiy the diamagnetic susceptibility in all directions in a crystal is an ellipsoid, whose orientation with respect to the unit cell is restricted by symmetry in exactly the same way as that of the optical indicatrix. Thus, for uniaxial crystals the magnetic ellipsoid an ellipsoid of revolution whose unique axis coincides with the threefold, fourfold, or sixfold axis of the crystal for orthorhombic crystals the ellipsoid has three unequal axes which necessarily coincide with the three axes of the crystal for monoclinie crystals the only restriction is that one of the principal axes of the magnetic ellipsoid must coincide with the b axis of the crystal while for triclinic crystals the orientation of the ellipsoid is not restricted in any way. 

Here (3 is a function of vp, the number of sites active toward recombination in the recombination sphere. (In [110] the concentration and recombination volumes were expressed in units of the volume v0 of the unit cell, and n0 coincides with the fraction of sites or interstiatial sites occupied respectively by vacancies or interstitial atoms.) We note that, in the model being discussed, the cell itself in which a vacancy occurs is considered inactive with respect to recombination of an interstitial on it. 

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