Third-Order Shape Attribute

The count of paths of three contiguous bonds, P, forms the basis for the description and quantitation of another shape attribute. This structural attribute is compared to two extreme structures defined by and Fo tbe third-order attribute, is taken from a twin star structure shown in Table 2, No. 5 for A = 8, and No. 7 for A = 9, For the linear graphs, Trble 2, No. 6 and No. 8 are the corresponding representations. In general, for any odd value of A, P, .,x = (A - 1) (A - 3)/4 and for any even value of A, = (A - 2) /4. In general, Pmi, = A - 3. A suitable algorithm in which third-order shape information, can be calculated for P, is 

The scaling factor of 4 is used in the numerator to bring the onto approximately the same numerical scale as the other kappa values. 

---

In summary, we have shown how two very similar Mo and Nb oxide mesostructured systems leads to formation of either continuous macroporous structures with extended tube lengths or free-standing torroids with limited extension in the third dimension. We attribute this difference to relative rates of condensation versus nucleation in each system. A unique TEM-induced reaction involving evolution of shape was also discussed and demonstrates that many of the new mesostructures with order on the macroscopic level may also show dynamic behavior. 

Firstly, the two peaks attributed to the third and fourth orders of basal spacing of the layered-silicate seen in the XRD spectra for the injection-moulded specimens and feedstock compounds were absent in the fibre spectra. Because the disappearance of the third and fourth order peaks could be an effect of exfoliation or because of low X-ray beam intensity, it is not possible to distinguish which of the two effects is more prevalent. However, the first and second order peaks of the hsv fibres manifested a more depressed peak shape and exhibited a lower peak intensity compared with the Isv fibres, but, in theory, the Isv fibre should exhibit a greater peak intensity because of the larger circumference and hence area exposed for beam diffraction. It is quite apparent, therefore, that the disappearance of the third and fourth order peaks, and the suppressed peak shape and intensity for the first and second order peaks, are mainly attributed to further exfoliation of the precedently intercalated layered-sihcate through an extensional deformation effect during the melt-spinning process. 

---