Radius ratio values

As mentioned by Zachariasen, Goldschmidt1) found that the range of radius-ratio values leading to stability of the C-modification is about... 

Table 7.5 Arrangements of Ions Predicted to Be Stable from Radius Ratio Values. ...

Table 3.1 Range of Radius Ratio Values for Various Cationic Coordination Numbers... 

This example was chosen to show that the radius ratio structure predictions work quite well except near the limiting radius ratio values. Here, r+ /r is borderline for MgS, and we predicted the wrong structure. This is not surprising, since the concept of fixed ionic radii is not well founded. 

Table 13-17.—Radius Ratio Values fob Cbtbtalb with Octahedral and Tetrahedral Coordination...

Radius ratio values relative to O2 are given in Table 4.3. The table shows that silicon (Si) exists in four-fold (tetrahedral) coordination with oxygen (O), i.e. it will fit into a tetrahedral site. This explains the existence of the Si04 tetrahedron. Octahedral sites, being larger than tetrahedral sites, accommodate cations of larger radius. However, some cations, for example strontium (Sr2+) and caesium (Cs+) (radius ratio >0.732), are too big to fit into octahedral sites. They exist in eight-fold or 12-fold coordination and usually require minerals to have an open, often cubic, structure. 

Table 4.3 Radius ratio values for cations relative to O2. From Raiswell et al. (1980).

A comparison of the limiting radius-ratio values given in Table I with the radius ratios of alkali halides in Table II shows that the observed structures are not always as predicted. According to the simple theory, the roeksalt structure should be stable only within the range 0.414 rM/rx 0.732. Thus LiCl (0.33), LiBr (0.31), and Lil (0.28) should have tetra-hedrally coordinated structures, while KF (0.98), RbF (1.09), and CsF (1.24) should have eight or twelve coordinated structures. These disagreements are perhaps not surprising in view of the crudeness of the approximations involved, but more realistic models do not lead immediately to an explanation of the persistence of the roeksalt structure not only in the alkali... 

Table 6.1. The radius-ratio values given in Table 6.1 are consistent with a CN of 6 based on the critical radius ratios given earlier in Table 5.4. The interstitial atoms are located either in an octahedral site or in the center of a trigonal prism. For the transition metals, the tetrahedral interstices in the close-packed structures are too small for C or N.

This then is the limiting radius ratio for six nearest neighbours— when the anion is said to have a co-ordination number of 6. Similar calculations give the following limiting values ... 

It was shown by Hund (Ref. 23) that for small values of n (less than 6 or 9, depending upon the assumptions made) the rutile structure can bgpome stable. However, our discussion makes it probable that the transition is actually due to the radius ratio. 

This theoretical result is completely substantiated by experiment. Goldschmidt,31 from a study of crystal structure data, observed that the radius ratio is large for fluorite type crystals, and small for those of the rutile type, and concluded as an empirical rule that this ratio is the determining factor in the choice between these structures. Using Wasastjerna s radii he decided on 0.67 as the transition ratio. He also stated that this can be explained as due to anion contact for a radius ratio smaller than about 0.74. With our radii we are able to show an even more satisfactory verification of the theoretical limit. In Table XVII are given values of the radius ratio for a large number of compounds. It is seen that the max-... 

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. 

In Table XVIII are given values of the radius ratio for the salts of beryllium, magnesium and calcium (those of barium and strontium, with the sodium chloride structure, also obviously satisfy the radius ratio criterion). It is seen that all of the sodium chloride type crystals containing eight-shell cations have radius ratios greater than the limit 0.33, and the beryl-... 

Since the repulsive forces are determined by the true sizes of ions, and not their crystal radii, the radius ratios to be used in this connection are the ratios of the univalent cation radii to univalent anion radii.12 Values of this ratio for small ions are given in Table II, together with predicted and observed coordination numbers, the agreement between which is excellent. 

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. 

For compounds of the composition MX (M = cation, X = anion) the CsCl type has the largest Madelung constant. In this structure type a Cs+ ion is in contact with eight Cl-ions in a cubic arrangement (Fig. 7.1). The Cl- ions have no contact with one another. With cations smaller than Cs+ the Cl- ions come closer together and when the radius ratio has the value of rM/rx = 0.732, the Cl- ions are in contact with each other. When rM/rx < 0.732, the Cl- ions remain in contact, but there is no more contact between anions and cations. Now another structure type is favored its Madelung constant is indeed smaller, but it again allows contact of cations with anions. This is achieved by the smaller coordination number 6 of the ions that is fulfilled in the NaCl type (Fig. 7.1). When the radius ratio becomes even smaller, the zinc blende (sphalerite) or the wurtzite type should occur, in which the ions only have the coordination number 4 (Fig. 7.1 zinc blende and wurtzite are two modifications of ZnS). 

EXO 0748-676, Cottam et al. (2002) have found absorption spectral line features, which they identify as signatures of Fe XXVI (25-time ionized hydrogenlike Fe) and Fe XXV from the n = 2 —> 3 atomic transition, and of O VIII (n = 1 —> 2 transition). All of these lines are redshifted, with a unique value of the redshift z = 0.35. Interpreting the measured redshift as due to the strong gravitational field at the surface of the compact star (thus neglecting general relativistic effects due to stellar rotation on the spectral lines (Oezel Psaltis 2003)), one obtains a relation for the stellar mass-to-radius ratio ... 

If we have N hard spheres (of radius rs) forming a close-packed polyhedron, another sphere (of smaller radius rc) can fit neatly into the central hole of the polyhedron if the radius ratio has a well-defined value (see also 3.8.1.1). The ideal radius ratio (rc/rs) for a perfect fit is 0.225.. (in a regular tetrahedron, CN 4), 0.414.. (regular octahedron CN 6), 0.528.. (Archimedean trigonal prism CN 6), 0.645... (Archimedean square antiprism CN 8), 0.732.. (cube CN 8), 0.902... (regular icosahedron CN 12), 1 (cuboctahedron and twinned cuboctahedron CN 12). 

The importance of the geometrical factor in determining the stability of these phases has been pointed out (Pearson 1972). In a simplified description, Laves phases AM2 of the MgCu2 type may be presented as cubic face-centred packing of large spheres A which form tetrahedral holes that are occupied by tetrahedra of smaller spheres M. The ideal value of the radius ratio rA/rM is 1.225. The values experimentally observed for the various Laves types range from 1.05 to 1.7. 

The rather complex structure of the compound NaZn13 was studied by Ketelaar (1937) and by Zintl and Haucke (1938). Every Na atoms is surrounded by 24 Zn atoms at the same distance. The lattice parameters of several MeZn13 compounds pertaining to this structural type are, in a first approximation, independent of the size of the alkali (or alkaline earth) metal atom. Similar consideration may be made for the MeCd13 compounds. Zintl, therefore, considered the fundamental component of this crystal structure to be a framework of Zn (or Cd) atoms with the alkali (or alkaline earth) metal atoms occupying the holes of the framework. However notice (Nevitt 1967) that in compounds MeX13 radius ratios (rMe/rx) deviating by more than about 15% from the mean value 1.54 are unfavourable for the occurrence of the structure. 

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