Crystal structures radius ratio

For LiF, the radius ratio is 0.57 and so an octahedral coordination around the Li cation is predicted for LiF, this corresponds to an NaCl type structure, in agreement with that observed. In fact each of the group 1 halides (except CsCl, CsBr and Csl) at 298 K and 1 bar pressure adopts the NaCl type structure CsCl, CsBr and Csl adopt the CsCl type structure. Radius ratio rules predict the correct structures in only some cases. They predict tetrahedral coordination for the cations in LiBr and Lil, and cubic coordination in NaF, KF, KCl, RbF, RbCl, RbBr and CsF (in addition to CsCl, CsBr and Csl). Radius ratio rules give only one prediction for any one ionic crystal, and some compounds undergo phase changes under the influence of temperature and pressure, e.g. when CsCl is sublimed onto an amorphous surface, it crystaUizes with the NaCl structure and, under high-pressure conditions, RbCl adopts a CsCl type structure. 

In AB -type structures, there are two types of atoms, ions, or molecules. The larger spheres are usually visualized to form an A-type lattice, and the smaller ones occupy some fraction of the holes (cubic, octahedral, or tetrahedral) in that lattice. Which holes are occupied is predicted by the radius ratio of the two spheres. For ionic crystals, the radius ratio is usually calculated as the radius of the cation over that of the anion. Ionic radii are derived from high-resolution X-ray studies. 

Colloidal crystals . At the end of Section 2.1.4, there is a brief account of regular, crystal-like structures formed spontaneously by two differently sized populations of hard (polymeric) spheres, typically near 0.5 nm in diameter, depositing out of a colloidal solution. Binary superlattices of composition AB2 and ABn are found. Experiment has allowed phase diagrams to be constructed, showing the crystal structures formed for a fixed radius ratio of the two populations but for variable volume fractions in solution of the two populations, and a computer simulation (Eldridge et al. 1995) has been used to examine how nearly theory and experiment match up. The agreement is not bad, but there are some unexpected differences from which lessons were learned. 

Since hydrofluoride synthesis is based on thermal treatment at relatively high temperatures, the possibility of obtaining certain fluorotantalates can be predicted according to thermal stability of the compounds. In the case of compounds whose crystal structure is made up of an octahedral complex of ions, the most important parameter is the anion-cation ratio. Therefore, it is very important to take in to account the ionic radius of the second cation in relation to the ionic radius of tantalum. Large cations, are not included in the... 

The formulated principals correlating crystal structure features with the X Nb(Ta) ratio do not take into account the impact of the second cation. Nevertheless, substitution of a second cation in compounds of similar types can change the character of the bonds within complex ions. Specifically, the decrease in the ionic radius of the second (outer-sphere) cation leads not only to a decrease in its coordination number but also to a decrease in the ionic bond component of the complex [277]. 

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-... 

Goldschmidt predicted from his empirical rule that calcium chloride would not have the fluorite structure, and he states that on investigation he has actually found it not to crystallize in the cubic system. Our theoretical deduction of the transition radius ratio allows us to predict that of the halides of magnesium, calcium, strontium and barium only calcium fluoride, strontium fluoride and chloride, and barium fluoride, chloride,... 

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-... 

The prediction may be made that the still unstudied crystal magnesium telluride, with the radius ratio 0.29, has the sphalerite or wurzite structure rather than the sodium chloride structure. 

The radius ratios for sphalerite and wurzite type crystals with eighteen-shell cations do not conform to our criterion, so that some other influence must be operative. Without doubt this is deformation. Here again it is seen that the tetrahedral structure is particularly favorable to deformation, for the observed Zn++—O distance (1.93 A.) is 0.21 A. shorter than the theoretical one, while in cadmium oxide, with the sodium chloride structure, the difference is only 0.01 A. 

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. 

The theoretical result is derived that ionic compounds MXS will crystallize with the fluorite structure if the radius ratio Rm/Rx is greater than 0.65, and with the rutile (or anatase) structure if it is less. This result is experimentally substantiated. 

It is also shown that theoretically a binary compound should have the sphalerite or wurzite structure instead of the sodium chloride structure if the radius ratio is less than 0.33. The oxide, sulfide, selenide and telluride of beryllium conform to this requirement, and are to be considered as ionic crystals. It is found, however, that such tetrahedral crystals are particularly apt to show deformation, and it is suggested that this is a tendency of the anion to share an electron pair with each cation. 

Revised Values of Double-Bond Covalent Radii.—This investigation has led to the value 1.34 A. for the carbon-carbon double-bond distance, 0.04 A. less than the value provided by the table of covalent radii.111 4 Five years ago, when this table was extended to multiple bonds, there were few reliable experimental data on which the selected values for double-bond and triple-bond radii could be based. The single-bond radii were obtained -from the study of a large number of interatomic distances found experimentally by crystal-structure and spectroscopic methods. The spectroscopic value of the triple-bond radius of nitrogen (in N2) was found to bear the ratio 0.79 to the single-bond radius, and this ratio was as-... 

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.)... 

By means of the radius ratio, we have already described the type of local environment around the ions in several types of simple crystals. For example, in the sodium chloride structure (not restricted to NaCl itself), there are six anions surrounding each cation. The sodium chloride crystal structure is shown in Figure 7.4. 

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. 

When the ratio of the radii increases, then a new type of crystal structure occurs, in which each divalent ion is surrounded by six univalent ions, and each univalent by three divalent ones. This arrangement, shown in Figure 14, is called the rutile type, named after the mineral Ti02. While CaF2 still has the structure with 8 4 coordination, the rutile structure is observed in MgF2 because of the smaller radius of the Mg2+ ion. After the rutile type there is a further reduction of the coordination to 4 2 this type of structure occurs in BeF2 and Si02 in the different modifications of silica, each silicon ion is surrounded by 4 O2" ions, and each O2 ion is between two Si4+ ions. The ionic ratio r+/r can also be decreased if,... 

It is crucial to discover the relationship between chemical compositions and hydrogen decomposition pressures of the AB5 compounds. Intermetallic compounds of lanthanide and transition metals form an interesting class of structures. The AB5 series crystallize in the hexagonal CaCus (P6/mmm) structure (see Figure 1). Generally, radius ratios (raAb) greater than 1.30 form the CaCus-type... 

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