Crystal two-dimensional

Anthracene and Naphthalene Crystals Two-dimensional Triplet Excitons... 

WAXS measurements are based on the ratio of the intensity of the crystalline peaks and the amorphous background. Fitting the amorphous background is best done with the help of a computer. Various corrections can be applied to obtain more accurate absolute crystallinities. However, most frequently we are interested in approximate absolute but accurate relative crystallinities and the corrections are not always necessary. The peak width can give information on crystal perfection - sharper peaks indicate more perfect crystals. Two-dimensional diffraction patterns of isotropic materials show rings corresponding to the diffraction peaks. Intensity variations within the rings can be used to assess crystalline orientation. WAXS only looks at material which can be penetrated by X-rays and so the depth of the analysis is limited. The patterns for filled systems can be hard to interpret but various methods have been developed - for example for use in continuous carbon-fibre composites. 

The difficulty of such pictures is that rubbers are fundamentaly three-dimensional and, unlike crystals, two-dimensional pictures are not comprehensive. However, one can imagine a spaghettilike mixture with permanent crosslinking bonds along their length. 

CS. The true two-dimensional crystal with chains oriented vertically exists at low T and high ir in the CS phase. This structure exhibits long-range translational order. 

Fig. Vn-2. Conformation for a hypothetical two-dimensional crystal, (a) (lO)-type planes only. For a crystal of 1 cm area, the total surface firee energy is 4 x lx 250 = 1000 eigs. (b) (ll)-type planes only. For a crystal of 1-cm area, the total surface free eneigy is 4 x 1 x 225 = 900 ergs, (c) For the shape given by the Wulff construction, the total surface free energy of a 1-cm crystal is (4 x 0.32 x 250) + (4 x 0.59 x 225) = 851 ergs, (d) Wulff construction considering only (10)- and (ll)-type planes.

The surface tensions for a certain cubic crystalline substance are 7100 = 160 ergs/cm, 7110 = 140 eigs/cm, and 7210 = 7120 = 140 ergs/cm. Make a Wulff construction and determine the equilibrium shape of the crystal in the xy plane. (If the plane of the paper is the xy plane, then all the ones given are perpendicular to the paper, and the Wulff plot reduces to a two-dimensional one. Also, 7100 = 7010. etc.)... 

An enlarged view of a crystal is shown in Fig. VII-11 assume for simplicity that the crystal is two-dimensional. Assuming equilibrium shape, calculate 711 if 710 is 275 dyn/cm. Crystal habit may be changed by selective adsorption. What percentage of reduction in the value of 710 must be effected (by, say, dye adsorption selective to the face) in order that the equilibrium crystal exhibit only (10) faces Show your calculation. 

Fig. XV-5. Fluorescence micrographs illustrating morphologies of two-dimensional (2D) sireptavidin crystals at three streptavidin/avidin ratios 15/85, 25/75, 40/60 from left to ri t. Scale bar is 100 gm (From Ref. 31.)...

Surface states can be divided into those that are intrinsic to a well ordered crystal surface with two-dimensional periodicity, and those that are extrinsic [25]. Intrinsic states include those that are associated with relaxation and reconstruction. Note, however, that even in a bulk-tenuinated surface, the outemiost atoms are in a different electronic enviromuent than the substrate atoms, which can also lead to intrinsic surface states. Extrinsic surface states are associated with imperfections in the perfect order of the surface region. Extrinsic states can also be fomied by an adsorbate, as discussed below. 

When atoms, molecules, or molecular fragments adsorb onto a single-crystal surface, they often arrange themselves into an ordered pattern. Generally, the size of the adsorbate-induced two-dimensional surface unit cell is larger than that of the clean surface. The same nomenclature is used to describe the surface unit cell of an adsorbate system as is used to describe a reconstructed surface, i.e. the synmietry is given with respect to the bulk tenninated (unreconstructed) two-dimensional surface unit cell. 

Onsager L 1944 Crystal Statistics. I. A two-dimensional model with an order-disorder transition Phys. Rev. 65 117-49... 

It is relatively straightforward to detemiine the size and shape of the three- or two-dimensional unit cell of a periodic bulk or surface structure, respectively. This infonnation follows from the exit directions of diffracted beams relative to an incident beam, for a given crystal orientation measuring those exit angles detennines the unit cell quite easily. But no relative positions of atoms within the unit cell can be obtained in this maimer. To achieve that, one must measure intensities of diffracted beams and then computationally analyse those intensities in tenns of atomic positions. 

Figure B3.2.12. Schematic illustration of geometries used in the simulation of the chemisorption of a diatomic molecule on a surface (the third dimension is suppressed). The molecule is shown on a surface simulated by (A) a semi-infinite crystal, (B) a slab and an embedding region, (C) a slab with two-dimensional periodicity, (D) a slab in a siipercell geometry and (E) a cluster.

The main drawback of the chister-m-chister methods is that the embedding operators are derived from a wavefunction that does not reflect the proper periodicity of the crystal a two-dimensionally infinite wavefiinction/density with a proper band structure would be preferable. Indeed, Rosch and co-workers pointed out recently a series of problems with such chister-m-chister embedding approaches. These include the lack of marked improvement of the results over finite clusters of the same size, problems with the orbital space partitioning such that charge conservation is violated, spurious mixing of virtual orbitals into the density matrix [170], the inlierent delocalized nature of metallic orbitals [171], etc. 

As for crystals, tire elasticity of smectic and columnar phases is analysed in tenns of displacements of tire lattice witli respect to the undistorted state, described by tire field u(r). This represents tire distortion of tire layers in a smectic phase and, tluis, u(r) is a one-dimensional vector (conventionally defined along z), whereas tire columnar phase is two dimensional, so tliat u(r) is also. The symmetry of a smectic A phase leads to an elastic free energy density of tire fonn [86]... 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

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. 

In some Hquid crystal phases with the positional order just described, there is additional positional order in the two directions parallel to the planes. A snapshot of the molecules at any one time reveals that the molecular centers have a higher density around points which form a two-dimensional lattice, and that these positions are the same from layer to layer. The symmetry of this lattice can be either triangular or rectangular, and again a positional distribution function, can be defined. This function can be expanded in a two-dimensional Fourier series, with the coefficients in front of the two... 

Fig. 3. Alignment of amide dipoles in polyamide crystals (a) for a two-dimensional array of an odd nylon, nylon-7, (b) for a one-dimensional array of an odd—odd nylon, nylon-5,7 (c) for one-dimensional arrays of polyamides containing even segments an even nylon, nylon-6 an even—even nylon, nylon-6,6 ...

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