Crystal symmetries dense packing

Order and dense packing are relative in tire context of tliese systems and depend on tire point of view. Usually tire tenn order is used in connection witli translational symmetry in molecular stmctures, i.e. in a two-dimensional monolayer witli a crystal stmcture. Dense packing in organic layers is connected witli tire density of crystalline polyetliylene. 

Low Miller index surfaces of metallic single crystals are the most commonly used substrates in LEED investigations. The reasons for their widespread use are that they have the lowest surface free energy and therefore are the most stable, have the highest rotational symmetry and are the most densely packed. Also, in the case of transition metals and semiconductors they are chemically less reactive than the higher Miller index crystal faces. 

The top layer of a (111) surface actually has sixfold symmetry, but the rotational symmetry of the top layers together is threefold. Since the near surface region can influence where gases adsorb on the surface and the LEED intensities exhibit threefold rotational symmetry at normal incidence, the (111) surface will be considered to have threefold rotational symmetry. Although most of the adsorption studies have been carried out on fee and bcc crystals, there have been several studies reported on hep crystals. For hep metals the basal or (0001) plane is the surface most frequently studied by LEED investigations and it is the most densely packed plane having threefold rotational symmetry. 

The symmetry of the crystal structure is a direct consequence of dense packing. The densest packing is when each building element makes the maximum number of contacts in the structure. First, the packing of equal spheres in atomic and ionic systems will be discussed. Then molecular packing will be considered. Only characteristic features and examples will be dealt with here, following Kitaigorodskii [38]. ... 

The densely packed organic thin films of alkanes tend to exhibit alterations of the network structures, depending on the parity of the number of the methylene units—a result called the odd-even effectP Binary mixtures of primary linear alcohols that differ by one methylene group show ideal mixing if the shorter alcohol possesses an odd number of carbon atoms. In contrast, if the shorter component is even numbered, co-crystals are formed. The appearance of mixing (random or ordered) on the surface rather than phase separation is ascribed to the similar symmetries of the unit cells of the pure components. ... 

As noted earlier, the factors that affect crystal solubility differ substantially from those that affect other solubility boundaries. Foremost among these are the shape of molecules (symmetry and stereochemical structure) and conformational freedom. The thermodjmamic stability of the crystal phase is strongly influenced by the ability of molecules to pack densely together in this phase. Structural features which inhibit this packing may dramatically alter the temperature range of the crystal solubility. An historic example is the effect of the cis double bond in oleates, which lowers the Krafft eutectic significantly relative to that of stearates. The trans double bond has a smaller effect, which parallels the influence... 

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