Factor groups atomic number

The redox behaviour of Th, Pa and U is of the kind expected for d-transition elements which is why, prior to the 1940s, these elements were commonly placed respectively in groups 4, 5 and 6 of the periodic table. Behaviour obviously like that of the lanthanides is not evident until the second half of the series. However, even the early actinides resemble the lanthanides in showing close similarities with each other and gradual variations in properties, providing comparisons are restricted to those properties which do not entail a change in oxidation state. The smooth variation with atomic number found for stability constants, for instance, is like that of the lanthanides rather than the d-transition elements, as is the smooth variation in ionic radii noted in Fig. 31.4. This last factor is responsible for the close similarity in the structures of many actinide and lanthanide compounds especially noticeable in the 4-3 oxidation state for which... 

Methods for treating the factor group vibrations have been given by Davydov (25), as well as by Bhagavantam and Venkatarayudu (22). A simple analysis is possible through what is known as the correlation method (20, 26, 27) by which one is able to write the irreducible representations and thereby classify k=0 phonons directly and simply. The number of A = 0 phonons is 3 N, where N is taken to be the number of atoms in the entire unit cell. However, there are only 3 N-3 optically active phonons because the acoustic vibrations have approximately... 

To derive factor group (space group) selection rules, it is necessary to utilize X-ray data for a molecule from a literature source or from Wyckoffs (54) Crystal Structures. The factor group and site symmetries of the ion, molecule, or atoms must be available, as well as the number of molecules per unit cell reduced to a primitive unit cell. 

Analysis of the symmetry of the chemical species under study (i.e. the point group for a free molecule, the space and factor groups for a crystal [44]), according to the site symmetry of every atom, allows the determination of the irreducible representation of the total modes and, after the subtraction of the translational and rotational modes (the acoustic modes for the crystals), the irreducible representation of the vibrational (or optical ) modes can be obtained. This means that the number of vibrational modes belonging to the symmetry species associated with the molecular or crystal symmetry can be counted. Consequently, the number of active modes can be counted, according to the symmetry selection rules of the different techniques (in particular IR and Raman). 

---