Atomic-Radii Ratio

As mentioned in Ch. 9, Sec. 3.2, the nitrogen atom is smaller than the carbon atom and interstitial nitrides are formed more readily than the corresponding carbides (see Ch. 3). As shown in Table 10.1, the nine early 

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Table 3.3 Carbon/Metal Atomic Radii Ratio of Interstitial Carbides...

Chromium carbide, with a carbon/metal atomic-radii ratio of 0.61, is a borderline case and, strictly speaking, belongs to the intermediate class of carbides (reviewed in Ch, 2, Sec. 5.3). Yet, unlike other intermediate carbides, it meets the refractory criteria and is a material of major industrial importance. For these reasons, it is included in this book (see Ch. 7). 

Group IV Carbides Lowest carbon/metal atomic radii ratio... 

Highest carbon/metal atomic radii ratio Several compositions fee, hep, and hexagonal structures Host metal has only one structure bcc... 

The space filling is determined by the ratio of the atomic radii, which is also a controlling factor for the existence ranges of the crystal structures. Both atomic radii ratios and interatomic distances have been utilized by Pearson (1972) in drawing near-neighbor diagrams , which can be used to analyze the type of interaction among atoms and to derive some indications of the bond type. 

The carbon-metal atomic radius ratio for the room temperature form of iron, a-Fe, is 0.60, just in excess of the Hagg limit. Consequently, the ability of a-Fe to accept interstitial C is marginal only 0.022 weight % or 0.06 atom % C can be accommodated in the random solid solution known... 

Although the cellular model proposed by Miedema is rather successful for the semi-quantitative description of the thermodynamic property of intermetallic compounds and liquid alloys. The ignorance of geometrical factor makes this model to be somewhat inaccurate. In some of our work, inclusion of atomic radius ratio can give better results of computation by support vector regression. 

For halides the cation should have a charge of 2+ rather than 4+ for tetrahedral coordination. The only fluoride compound capable of containing two-coordinate F and four-coordinate cations is Bep2. For ZrF, the radius ratio rule predicts that Zr" " is eight-coordinate if all fluorine atoms are two - c o o rdinate. 

In this discussion, two mutually canceling simplifications have been made. For the transition value of the radius ratio the phenomenon of double repulsion causes the inter-atomic distances in fluorite type crystals to be increased somewhat, so that R is equal to /3Rx-5, where i has a value of about 1.05 (found experimentally in strontium chloride). Double repulsion is not operative in rutile type crystals, for which R = i M + Rx- From these equations the transition ratio is found to be (4.80/5.04)- /3i — 1 = 0.73, for t = 1.05 that is, it is increased 12%. But Ru and Rx in these equations are not the crystal radii, which we have used above, but are the univalent crystal radii multiplied by the constant of Equation 13 with z placed equal to /2, for M++X2. Hence the univalent crystal radius ratio should be used instead of the crystal radius ratio, which is about 17% smaller (for strontium chloride). Because of its simpler nature the treatment in the text has been presented it is to be emphasized that the complete agreement with the theoretical transition ratio found in Table XVII is possibly to some extent accidental, for perturbing influences might cause the transition to occur for values a few per cent, higher or lower. 

Many complex ions, such as NH4+, N(CH3)4+, PtCle", Cr(H20)3+++, etc., are roughly spherical in shape, so that they may be treated as a first approximation as spherical. Crystal radii can then be derived for them from measured inter-atomic distances although, in general, on account of the lack of complete spherical symmetry radii obtained for a given ion from crystals with different structures may show some variation. Moreover, our treatment of the relative stabilities of different structures may also be applied to complex ion crystals thus the compounds K2SnCle, Ni(NH3)3Cl2 and [N(CH3)4]2PtCl3, for example, have the fluorite structure, with the monatomic ions replaced by complex ions and, as shown in Table XVII, their radius ratios fulfil the fluorite requirement. Doubtless in many cases, however, the crystal structure is determined by the shapes of the complex ions. 

It is in some measure demonstrated that the formation of A-B and B-B contacts provides the energy for the compression of the A atoms and permits AB2 phases with radius ratios so much larger (up to 1 -67) than the ideal (1-225) to adopt the MgCu2 type structure. At radius ratios somewhat lower than the ideal, the B atoms are insufficiently compressed for A-B and A-A contacts to form. This is probably a consequence of there being twice as many B atoms as A atoms, and it results in fewer known Laves phases with radius ratios below the ideal value than above it. 

Binary and ternary structure types with isolated B atoms are listed in Table 1. In the metal borides of the formula (My, Mi ),B or T,(B, E) (M-p, M - = transition metals, E = nonmetal), the influence of the radius ratio as well as the... 

A second product is the ICE Solid-State Model Kit, developed by L. A. Mayer and G. C. Lisensky, which makes it possible to build extended three-dimensional structures Using a base with holes, templates for some 60 different structures, rods, and four sizes of spheres in radius ratios, common crystal structures can be assembled in a matter of minutes (3). Furthermore, many structures can be assembled from different perspectives by teams of students For example, the cubic NaCl unit cell can be assembled with its orientation on the face of the cube or on the body diagonal. Natural cleavage planes can be found with the kit Lifting one sphere will separate atomic planes from one another. (Contact ICE for ordering information.)... 

The zinc blende type is unknown for truly ionic compounds because there exists no pair of ions having the appropriate radius ratio. However, it is well known for compounds with considerable covalent bonding even when the zinc blende type is not to be expected according to the relative sizes of the atoms in the sense of the above-mentioned considerations. Examples are CuCl, Agl, ZnS, SiC, and GaAs. We focus in more detail on this structure type in Chapter 12. 

Alternate layers can be occupied by two different kinds of metal atom, then every pair of the face-sharing octahedra contains two different metal atoms this is the ilmenite type (FeTi03). Ilmenite is, along with perovskite, another structure type for the composition AiiMiv03. The space for the A2+ ion is larger in perovskite. Which structure type is preferred can be estimated with the aid of the ionic radius ratio r(A2+)/r(02-) < 0.7 ilmenite... 

Although the face-centred cubic structure of metals is close packed, it is still possible for atoms which are much smaller than the host metal atoms to fit into interstitial sites inside the structure, while maintaining the essential properties of metals such as electrical conductivity and heat transport. These interstitial sites are of two kinds. The octahedral interstitial sites have six metal atoms at equal distances from the site, and therefore at the apices of a regular octahedron. The tetrahedral interstitial sites have four nearest neighbour metal atoms at the apices of a regular tetrahedron. A smaller atom can just fit into the octahedral site if the radius ratio is... 

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