First-Order Shape Attribute

For this attribute of shape, described by the counts of one bond fragments, P, we selected Pm x to be the complete graph, where all atoms are bonded to each other. For any number of atoms. A, the value of = A(A - l)/2. For the Pmin structure, we have selected the linear graph where the value is = A - 1. In Table 2, entry No. 1 is the graph of P, ax, where A = 6. Entry No. 2 is the graph of P i, where A = 6. 

The shape attribute of the first order lies between the complete graph and the linear graph. This is the basis of our definition of this shape attribute. We are not considering, or numerically defining, spheres, ellipsoids, or other geometric figures. 

We proceed to derive an algorithm which yields a numerical index for any molecule with A atoms and with P,. To embrace the information for the extremes in Eq. [48], the products of ratios of P, to Pm ,x nd P, to P, n are used. We write down an index of shape of order one, using k (kappa) as the index symbol  

The 2 in the numerator is a scaling factor that makes k = A when there are no cycles in the graph of the molecule. The index, k, can be expressed in terms of the count of atoms. A  

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By selecting the models for each order of shape attribute as we have, we can quantitate these attributes from their relationship to two extreme structures associated with each model. The first-order shape attribute of a molecule with a certain value of lies between a linear graph and a complete graph, all with the same number of atoms. 

The sample deformed to 30/o reduction in the ordered state shows an S-shape behaviour within the ordered region with a drop of about 30% at 400°C. The first decrease of microhardness at about 130°C seems to be correlated with the early decrease of resistivity and, therefore, may be attributed to the recovery of a great number of deformation induced excess vacancies. 

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