Orientation function definition

In this review the definition of orientation and orientation functions or orientation averages will be considered in detail. This will be followed by a comprehensive account of the information which can be obtained by three spectroscopic techniques, infra-red and Raman spectroscopy and broad line nuclear magnetic resonance. The use of polarized fluorescence will not be discussed here, but is the subject of a contemporary review article by the author and J. H. Nobbs 1. The present review will be completed by consideration of the information which has been obtained on the development of molecular orientation in polyethylene terephthalate and poly(tetramethylene terephthalate) where there are also clearly defined changes in the conformation of the molecule. In this paper, particular attention will be given to the characterization of biaxially oriented films. Previous reviews of this subject have been given by the author and his colleagues, but have been concerned with discussion of results for uniaxially oriented systems only2,3). 

The sign of the volume depends on the chosen orientation of the 3D space considered, according to Definition 3S of orientation. Using this sign of the volume determinant, we dehne the orientation function as follows ... 

Definition (Orientation function on a molecule in space) Consider 3D space and suppose we have chosen its orientation as described above. We consider a molecule of n atoms placed in 3D space and the (") subsets ao.ai.Oj.aj) of four atoms. 

The definition raises many questions, which we discuss below. Two medn problems eue when the orientation function has a value of 0, and the enormous number n of quadruples. 

The second main problem is the reduction of the number of quadruples to be considered. We could, of course, write down the orientation function by listing the function values for all quadruples. However, the number of all quadruples is huge already for medium-sized n and thus this seems impractical. We will show that it is not necessary to write down the orientation of each quadruple explicitly, as the orientations are interdependent and some are obtained easily from others. This follows from the definition using the volume determinant. Moreover, not all orientations of quadruples are of equal relevance. We may restrict our investigations to a selection of the most relevant orientations, and by so doing we can describe a conformation with various levels of accuracy. 

One fascinating and very useful feature of minimal Radon partitions is that they can be obtained from the orientation function, i.e. on a purely combinatorial level, without considering coordinates. For this purpose we introduce abstract minimal Radon partitions for an abstract orientation function. It will turn out that a pair A, B is a minimal Radon partition of a conformation if and only if it is an abstract minimal Radon partition of the corresponding orientation function. Before giving a formal definition of abstract minimal Radon partition, we first consider an example ... 

Definition (Abstract minimal Radon partition) An abstract minimal Radon partition of an abstract orientation function over atoms 0,...,n I) is a pair (A.B) of disjoint subsets of the set of atoms,... 

As a prelude to the discussion it is necessary to consider the definition of orientation in terms of the Euler angles, and the definition ofan orientation distribution function in terms ofan expansion ofLegendre functions. These definitions set the scene for examining the information which can be obtained from different spectroscopic techniques. In this review, infra-red and Raman spectroscopy and nuclear magnetic resonance, will be considered. 

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The value objective function is oriented at the company s profit and loss definitions. Guiding principle is to only use value parameters that can be found in the cost controlling of the company signed by controlling. Penalty costs and without currency and weighting factors being applied to steer optimization results but having no actual financial impact - as it can be often found in supply chain optimization models - do not meet this requirement. 

Fig. 2. Stopping cross section for protons impinging on oriented H2O as a function of proton velocity. See Fig. 1 for the definition of target orientations.

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