Pearsons classification

As noted in section 6.2, when the material of interest is an intermetallic alloy, the solution of its crystal structure may be simplified because intermetallics often form series of isostructural compounds. In contrast to conventional inorganic and molecular compounds, stoichiometries of the majority of intermetallic phases are not restricted by normal valence and oxidation states of atoms and ions therefore, crystal structures of metallic alloy phases are conveniently coded using the classification suggested by W.B. Pearson. According to Pearson, each type of the crystal structure is assigned a specific code (symbol), which is constructed from three components as follows  

Pearson s classification is insensitive to both chemical compositions and stoichiometries of metallic alloys. It is quite useful because all known intermetallic crystal structures are grouped according to their structural symbols, which are quite simple. Thus, once the symmetry and the content of the unit cell of a new alloy phase have been established, it only makes sense to search for potentially isostructural compounds among those that have identical Pearson s symbols. 

Pearson, Handbook of lattice spacings and structures of metals, vol. 2, Pergamon Press, New York (1967) W.B. Pearson, The crystal chemistry and physics of metals and alloys, Wiley-Interscience, New York (1972). 

Crystal system Bravais lattice First two parts of Pearson s symbol 

Tens of thousands of intermetallic phases have been systematized and classified using Pearson s symbols. They are listed in a source commonly known as Pearson s Handbook, which is periodically updated and published by ASM Intemational. The Handbook also provides detailed information about the coordinates of atoms in unit cells of all known structure types of metals, alloys and related phases, which makes it a valuable tool in the structure solution of metallic materials. 

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This chapter is principally concerned with ligand systems which contain a carbon atom linked to two oxygen atoms. In the vast majority of the complexes to be considered the donor atom is the relatively small, less polarizable oxygen atom, i.e. hard under the Pearson classification. As a consequence these ligands are seldom found in conjunction with low oxidation state metals. Considerable advances have been made in this area in recent years. The most notable of these has been the work of Cotton, who has made extensive use of the ability of such ligands to bring metal... 

Drobot et al. (3 ) have studied the thermal decomposition of Mol in evacuated sealed ampoules at various temperatures. X-ray diffraction and chemical analyses of the condensed decomposition products showed the existence of a homogeneity range in which the atomic ratio of I/Mo varied from 3.00 to 2.5. The crystal structure of Mol- is 0PI6 ( ) in the Pearson classification system. 

TABLE 2.17 The Categories of Pearson Classification With -the Total Number of Atoms in the Unit Cell ... 

Accordingly, the Pearson classification is merely reductive than complete, as long as no parameter is specified, for example, the number of atoms per unit cell, which however varies. Overall, the Pearson indexing serves for an immediate view of the type of structure, at the same time indicating stoichiometric information for the unit cell. 

TABLE 2.19 The Diagrammatic/Projective Representation of the Space Groups and the Pearson Classification for Representative Examples of Crystals Primitives ... 

Miller-Bravais indices Pearson classification point groups reducing cell method Schoenflies notation space groups stereographic projection symmetry perturbations Weiss zone law Wulff map... 

Pearson Classification of Lewis Acids Represented by Metal Ions... 

According with Pearson, classification of acids and bases as hard and soft needs the recourse to the concept of strength although litde insight this... 

In this context, the actual hard and soft classification scheme will be in next compared and discussed against the traditional Pearson classification for a limited, however representative, series of molecular Lewis acids and bases as displayed in the Tables 3.1 and 3.2, respectively. 

TABLES 3.2 Qualitative Pearson Classification of Lewis Bases Tested (Pearson, 1963, 1997) ... 

The FD chain relationships confirm that the Pearson classification is only partly fulfilled by marking in bold the cases when the actual analysis fits with Pearson one, clearly appears that in the case of acids, in each Pearson classes (soft, borderline, and hard) only one acid from the computed set is recovered for bases only those classified as soft are here recovered as such although in an enlarged set. Therefore, the percentage of actual/Pearson approaches goes to 33% for both acids and bases considered apart of some internal ordering relativity. 

The relationship 5.20 is shown in Figure 5.13. It is difficult to assess the relative importance of model and experimental errors in these correlations. However, it is clear that the Ai7(Bl2)-Av(I—CN) correlation is less family dependent than the pAlBi2- i (I I) and p7fBicN-Av(I—CN) relationships shown in Figures 5.12 and 5.10. Hence the relationships 5.19 and 5.20 may support the use of Av(I—CN) as a spectroscopic scale of soft affinity. Indeed, diiodine is the archetype of soft Lewis acids in the Pearson classification since it has a very low absolute hardness (rj = 3.4 eV). Moreover, Av(I—CN) values obey the HSAB principle (soft acids prefer soft bases) since they decrease with the absolute hardness of the donor atom (in a given column of the periodic table), as shown in Table 5.28. 

Fig. 43. The crystal structures of ScRhSij, NbCoBj and LuRuB2, all having the same Pearson classification symbol and space group.

According to Yatsimirskii, group (2) and (3) species are equivalent to Pearson s hard acids and bases, and group (4), (5) and (6) species correspond to Pearson s soft acids and bases. This classification is of more value than HSAB theory to our subject. It can be seen that all cementforming anions come from group (3) and cations from groups (3), (4) and (5). Thus, the bonding in cement matrices formed from cation-anion combinations is not purely a but contains some n character. 

Another feature of the metal ions that are typically involved in cementitious bonding in AB cements is that most of them fall into the category of hard in Pearson s Hard and Soft Acids and Bases scheme (Pearson, 1963). The underlying principle of this classification is that bases may be divided into two categories, namely those that are polarizable or soft, and those that are non-polarizable or hard. Lewis acids too may be essentially divided into hard and soft, depending on polarizability. From these classifications emerges the useful generalization that hard acids prefer to associate with hayd bases and soft acids prefer to associate with soft bases (see Section 2.3.7). 

They indicated that the softness parameter may reasonably be considered as a quantitative measure of the softness of metal ions and is consistent with the HSAB principle by Pearson (1963, 1968). Wood et al. (1987) have shown experimentally that the relative solubilities of the metals in H20-NaCl-C02 solutions from 200°C to 350°C are consistent with the HSAB principle in chloride-poor solutions, the soft ions Au" " and Ag+ prefer to combine with the soft bisulfide ligand the borderline ions Fe +, Zn +, Pb +, Sb + and Bi- + prefer water, hydroxyl, carbonate or bicarbonate ligands, and the extremely hard Mo + bonds only to the hard anions OH and. Tables 1.23 and 1.24 show the classification of metals and ligands according to the HSAB principle of Ahrland et al. (1958), Pearson (1963, 1968) (Table 1.23) and softness parameter of Yamada and Tanaka (1975) (Table 1.24). Compari.son of Table 1.22 with Tables 1.23 and 1.24 makes it evident that the metals associated with the gold-silver deposits have a relatively soft character, whereas those associated with the base-metal deposits have a relatively hard (or borderline) character. For example, metals that tend to form hard acids (Mn +, Ga +, In- +, Fe +, Sn " ", MoO +, WO " ", CO2) and borderline acids (Fe +, Zn +, Pb +, Sb +) are enriched in the base-metal deposits, whereas metals that tend to form soft acids... 

The concept of hard and soft acids and bases ( HSAB ) should also be mentioned here. This is not a new theory of acids and bases but represents a useful classification of Lewis acids and bases from the point of view of their reactivity, as introduced by R. G. Pearson. 

This concept was introduced qualitatively in the late 1950s and early 1960s by Pearson, in the framework of his classification of Lewis acids and bases, leading to the introduction of the hard and soft acids and bases (HSAB) principle [19-21]. This principle states that hard acids prefer to bond to hard bases and soft acids to soft bases. In many contributions, the factor of 1/2 is omitted. The inverse of the hardness was introduced as the softness S=l/rj [22]. A third quantity, which can be expressed as a derivative with respect to the number of electrons is the Fukui function, was introduced by Parr and Yang [23,24] ... 

All structures in this family contain icosahedra and at least one other coordination type. Frank and Kasper demonstrated that structures formed by the interpenetration of the four polyhedra contain planar or approximately planar layers of atoms primary layers made by tessellation of triangles with hexagons and/or pentagons, and intervening secondary layers of triangles and/or squares. For a classification and coding of the nets and of their stacking, see Pearson (1972) and also Shoemaker and Shoemaker (1969) or Frank and Kasper (1958, 1959). 

Intermetallic phases could be classified following the most important factor which controls their crystal structure (Pearson 1972, Girgis 1983, Hafher 1989, Westbrook and Fleischer 1995, Cahn and Haasen 1996). As a recapitulation of these points we may again refer to Pearson (1972), who underlined the role of the following factors in the classification of intermetallic phases ... 

The ratio, C/E, gives a quantitative order of relative hardness or "softness for the various Lewis acids and agrees fairly well with the qualitative classification of Pearson (2). The adds which do not follow the qualitative classification are BF3 and SO2. As mentioned above, the parameters for BF3 were determined from data limited to oxygen donors. The qualitative ordering of SO 2 is incorrect, Emd as will be shown shortly when strong interactions are compared with weak ones, the procedures... 

The remaining exceptions concern the lanthanide series, where samarium at room temperature has a particular hexagonal structure and especially the lower actinides uranium, neptunium, and plutonium. Here the departure from simple symmetry is particularly pronounced. Comparing these three elements with other metals having partly filled inner shells (transition elements and lanthanides), U, Pu, Np have the lowest symmetry at room temperature, normal pressure. This particular crystallographic character is the reason why Pearson did not succeed to fit the alpha forms of U, Pu, and Np, as well as gamma-Pu into his comprehensive classification of metallic structures and treated them as idiosyncratic structures . Recent theoretical considerations reveal that the appearance of low symmetries in the actinide series is intimately linked to the behaviour of the 5f electrons. 

Table 3.20 Pearson s Classification of Lewis Acids and Bases...

Many systems of notation and classification have been proposed. The well-known books by R. W. G. Wyckoff, A. F. Wells, F. C. Phillips, L. Bragg, M. J. Buerger, L. V. Azakoff, D. M Adams, and W. B. Pearson (Appendix A, Further Reading) have discussed these proposals. These proposals include close packing of atoms, nets, or prism connections, stacking of coordination polyhedra and even a crystal-algebra method. Application of most of these proposals requires familiarity with the features of many structures. Only specialists can be expected to have... 

It would be closing the eyes to half of the content of Pearson s classification to consider only systems where complex formation constants have (or can) been measured. If one wants to stabilize unusually low oxidation numbers, it is well-known that the best chances are to select the ligands among the class H-, 1 ,... 

For the chemist, there is no doubt whatsoever that the softness of the central atoms do not follow the a values, though this is valid for the halides. Thus, Cs(I) is more polarizable than Ag(I), Ba(II) more than Cd(II) and La(III) more than In(III) in disagreement with Pearson s classification. Said in other words, the chemical bonding involves far stronger perturbations than the linear electric fields inducing the a-polarizability. 

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