Uniaxial groups

The uniaxial groups are of two kinds, those that contain case (b) 1E, 2E representations which are CCs of one another (called lower symmetry groups), and those with no case (b) representations (termed upper symmetry groups ). For the latter, (x y) span an E irreducible representation while z is the basis of non-degenerate A or B representation. Consequently,... 

Any extended part of a linear polymer molecule exhibits a strong anisotropy of many properties since its atoms and atomic groups are oriented and the macromolecule is actually a one-dimensional crystal . The parallel packing of these parts during the formation of a uniaxially oriented polymer substance imparts the anisotropie properties of a single molecule to the whole polymeric material. 

The first group of tests is carried out on specimens generally fabricated into a dumb-bell shape, with forces applied uniaxially. The usual apparatus consists of a machine with a pair of jaws, which during the test are moved relative to each other, either together or apart, in a controlled manner. A chart recorder is employed to give a permanent record of the results obtained, so that the force at fracture can be determined. Whether this kind of set up measures tensile, compressive, or flexural strength depends on how the sample is oriented between the jaws, and on the direction that the jaws are set to travel relative to one another. 

The mathematical expression of N(6, q>, i//) is complex but, fortunately, it can be simplified for systems displaying some symmetry. Two levels of symmetry have to be considered. The first is relative to the statistical distribution of structural units orientation. For example, if the distribution is centrosymmetric, all the D(mn coefficients are equal to 0 for odd ( values. Since this is almost always the case, only u(mn coefficients with even t will be considered herein. In addition, if the (X, Y), (Y, Z), and (X, Z) planes are all statistical symmetry elements, m should also be even otherwise = 0 [1]. In this chapter, biaxial and uniaxial statistical symmetries are more specifically considered. The second type of symmetry is inherent to the structural unit itself. For example, the structural units may have an orthorhombic symmetry (point group symmetry D2) which requires that n is even otherwise <>tmn = 0 [1], In this theoretical section, we will detail the equations of orientation for structural units that exhibit a cylindrical symmetry (cigar-like or rod-like), i.e., with no preferred orientation around the Oz-axis. In this case, the ODF is independent of t/z, leading to n — 0. More complex cases have been treated elsewhere [1,4]... 

Given the efforts in this group and others (Table 1) to form the Cd based II-VI compounds, studies of the formation of Cd atomic layers are of great interest. The most detailed structural studies of Cd UPD have, thus far, been published by Gewirth et al. [270-272]. They have obtained in-situ STM images of uniaxial structures formed during the UPD of Cd on Au(lll), from 0.1 M sulfuric acid solutions. They have also performed extensive chronocoulometric and quartz crystal microbalance (QCM) studies of Cd UPD from sulfate. They have concluded that the structures observed with STM were the result of interactions between deposited Cd and the sulfate electrolyte. However, they do not rule out a contribution from surface reconstructions in accounting for the observed structures. 

The molecular planes of the two molecules in the unit cell are nearly parallel to each other and the binary axis which makes the actual 2 point group symmetry close to mm2 symmetry. The crystal is thus near to uniaxial and the "quasi-optical axis" close to the direction perpendicular to the molecular planes (assuming that the... 

Recent work by this group has focused upon elastic deformation of these helices and their response to uniaxial forces.163 Under sufficient tension, a low-pitch helix reversibly separates into a straight domain with a pitch angle of 90° and a helical domain with a pitch angle of 16.5°, as shown in Figure 5.44. 

As pointed out in Section 6.3, for the (CH)3 case, all the essential symmetry properties of the LCAOs we seek are determined by the operations of the uniaxial rotational subgroup, Cft. When the set of six pn orbitals is used as the basis for a representation of the group C6, the following results are obtained ... 

Note first that y(E) = 6 while all other characters are zero. The reason is that the operation E transforms each into itself while every rotation operation necessarily shifts every 0, to a different place. Clearly this kind of result will be obtained for any /z-membered ring in a pure rotation group C . Second, note that the only way to add up characters of irreducible representations so as to obtain y = 6 for E and y - 0 for every operation other than E is to sum each column of the character table. From the basic properties of the irreducible representations of the uniaxial pure rotation groups (see Section 4.5), this is a general property for all C groups. Thus, the results just obtained for the benzene molecule merely illustrate the following general rule ... 

The second expression is simply a list of the six /s, in numerical order, each multiplied by the character for one of the six operations, in the conventional order , CA, Cl,..., C. This must be true for each and every representation. Hence, the sets of characters of the group are the coefficients of the LCAO-MOs. The argument is obviously a general one and applies to all cyclic (CH) systems belonging to the point groups Dnh, each of which has a uniaxial pure rotation subgroup, C . 

This chapter is concerned with the influence of mechanical stress upon the chemical processes in solids. The most important properties to consider are elasticity and plasticity. We wish, for example, to understand how reaction kinetics and transport in crystalline systems respond to homogeneous or inhomogeneous elastic and plastic deformations [A.P. Chupakhin, et al. (1987)]. An example of such a process influenced by stress is the photoisomerization of a [Co(NH3)5N02]C12 crystal set under a (uniaxial) chemical load [E.V. Boldyreva, A. A. Sidelnikov (1987)]. The kinetics of the isomerization of the N02 group is noticeably different when the crystal is not stressed. An example of the influence of an inhomogeneous stress field on transport is the redistribution of solute atoms or point defects around dislocations created by plastic deformation. 

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