Tetrahedral radii

Tetrahedral Radii.—In crystals with the diamond, sphalerite, and wurtzite arrangements (Figs. 7-4, 7-5, and 7-6) each atom is surrounded tetrahedrally by four other atoms. If the atoms are those of fourth- 

—The arrangement of the carbon atoms in the diamond crystal. Each atom has four near neighbors, which are arranged about it at the corners of a regular tetrahedron. 

and the sphalerite or the wurtzite arrangement (or both) by the compounds listed in Table 7-14. 

In all of these crystals it is probable that the bonds are covalent bonds with some ionic character. In ZnS, for example, the extreme covalent structure 

The tetrahedral radius sums are seen to agree closely with the experimentally determined values of the interatomic distances (which were, of course, used in their derivation), the average deviation being 0.01 A (Table 7-14). For CuF, BeO, AIN, and SiC the observed bond lengths are significantly less than the sum of the radii the difference is with little doubt due to partial ionic character. The radii haw biv. n chose , in such a way as not to need this correction m other olux i 

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Standard Tetrahedral Radii. Table III, which is closely similar to the table published byHuggins1) Jf... 

The values in Table VI were obtained in the following way. Values for C, Si, Ge, and Sn are the same as in Table III, for the tetrahedral configuration is the normal one for these atoms. Radii for F, Cl, Br, and I were taken as one-half the band-spectral values for the equilibrium separation in the diatomic molecules of these substances. Inasmuch as these radii for F and Cl are numerically the same as the tetrahedral radii for these atoms, the values for N, 0, P, and S given in Table III were also accepted as normal-valence radii for these atoms. The differences of 0.03 A between the normal-valence radius and the tetrahedral radius for Br and... 

N, 0, F, P, S, and Cl the bond orbitals for normal valence compounds lead to about the same radii as tetrahedral orbitals, whereas in atoms below these in the periodic system normal valence bonds involve orbitals which approach p-orbitals rather closely, and so lead to weaker bonds, and to radii larger than the tetrahedral radii. This effect should be observed in Br, Se, and As, but not in Ge, and in I, Te, and Sb, but not Sn. For this reason we have added 0.03 A to the tetrahedral radii for As and Se and... 

S—8) etc., equals the sum of the standard tetrahedral radii plus 0.04 A. In the C18 type crystals all (M—X) distances between neighboring atoms were assumed equal. In CoAsS and NiAsS the following were assumed Uq0 = = 0. (As—S)... 

It is interesting that a straight line drawn through the tetrahedral radii passes through the metallic radius for calcium this suggests that the metallic bonding orbitals for calcium are sp orbitals, and that those for scandium begin to involve d-orbital hybridization. 

The very long period is closely similar to the second long period, except for the interpolation of the rare-earth metals. It is interesting that a straight line can be passed through the points for barium, the two bivalent rare-earth metals, and the tetrahedral radii of the heavier elements. 

Equations (11a) and (12a) are based upon the values 1-399 and 1-430 for R sp3) for tin and lead, respectively, with slopes as indicated by the tetrahedral radii for the heavier elements. The value 1-430 for quadrivalent lead was obtained by adding to the radius for quadrivalent tin the difference 0-031A between the Mg-Pb distance in Mg2Pb and the Mg-Sn distance in Mg2Sn. The simple form of equations (116) and (126) was assumed in analogy with (106), and the equal values of the constant were obtained from the observed distances for silver and gold, both assumed to have valence 5 . 

Table 7-14.—Comparison of Observed Interatomic Distances in B3 and B4 Crystals with Sums of Tetrahedral Radii...

The tetrahedral radii for first-row and second-row elements are identical with the normal single-bond covalent radii given in Table 7-2. For the heavier atoms there are small differences, amounting to 0.03 A for bromine and 0.05 A for iodine. It is possible that these differences are due to the difference in the nature of the bond orbitals in tetrahedral and normal covalent compounds. 

The selenium-selenium and tellurium-tellurium distances observed in the manganese compounds provide further evidence for this structural interpretation. They correspond to radii that agree more satisfactorily with the normal-valence radii than with the tetrahedral radii of the nonmetallic atoms, indicating that the atoms are not forming tetrahedral covalent bonds ... 

Fig. 11-9.—Metallic radii for the elements oi the first long period and the second long period. Values of octahedral radii and tetrahedral radii are also represented.

The complex interactions in the systems in question are evidently due to the different covalent tetrahedral radii of the cores of the atoms and the different electron affinities N (or different electronegativities) of zinc, cadmium, and indium cations [1,13], which alter the close packing of sulfur in different ways (Table 1). 

TABLE 1. Tetrahedral Radii of Atomic Cores rT, Electron Affinities N, and Electronegativities of Cations and Anions... 

The covalent tetrahedral radii of Zn and Cd atoms are different but nevertheless these atoms always occupy the tetrahedral voids forming the sp bonds, which differ in the asymmetric distributions of their electron densities because of the different electron affinities of the atomic cores. 

Summing up, we have shown how the formation of ternary compounds with a tetrahedral coordination of the atoms depends on the chemical interaction, the ratio of the sums of the tetrahedral radii, and the nature of the chemical bonds between the atoms forming these conqmunds. 

Lewis and Kossel s Legacy Structure and Bonding in Main-Group Compounds Table 6 Standard tetrahedral radii of main-group elements (in pm) [56]... 

The standard tetrahedral radii obtained by Pauling and Huggins have sometimes been referred to as tetrahedral covalent radii, but we prefer the original notation since we wish to reserve the term covalent for bonds to which each atom provides one electron. We shall return to this point in Sect. 9. Since these radii can only be used to predict bond distances between atoms that are both tetrahedrally coordinated, they are of limited utility. They were nevertheless of great interest at the time since they were the first radii that provided a quantitative illustration of the decrease of atomic bonding radii across the short periods of the periodic table and their increase as a group is descended. 

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