Structure Determination and Ionic Radii

Estimated nearest-neighbor distance, Madelung energy, and (Eq- 13-5), for KCl in three diflerent structures. 

Notice that each row in Table 13-7 corresponds to a single electronic configuration, the same as that of the inert gas that appears within the shaded column. There is a steady decrease in the ionic radii in each row as we move to the 

Ionic radii in A for closed-shell configurations, due to Zachariasen, 

The Iluorite structure of Cah 2 was given in Section 13-A. Use the appropriate inert-gas overlap interaction from Table 13-3 (use the values based upon Table 12-2) and the electrostatic energy from Table 1.3-1 to estimate the nearest-neighbor distance of CaFa and compare with experiment. You can ignore the d term if you like it makes only a one-percent correction. Compare also with the sum of ionic radii. 

By using the same interactions mentioned in Problem 13-1, estimate the bulk modulus from evaluated at the observed bond length. (Notice that you need to recalculate the ion density for the fluorite structure.) Notice the relative size of the contributions to the bulk modulus made by the d, d and electrostatic terms. Compare the total with the observed bulk modulus, B = (c, + 2c,2)/3. 

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How do we connect the model to the three essential properties of an atom An observable, such as size, depends on the square of the absolute magnitude of the wave function. Size, then, must be associated with the coefficient of the exponent of the radial function, but it is hard to say a priori when its value is small enough to be neglected. Hence, numerical values are obtained empirically under conditions where the atom of interest interacts with another atom. Rich sources of such information are solid-state structure determinations, but interaction with an atom of a tip of an STM constitutes another source. Consequently, different measures of size exist and reflect different types of interaction van der Waals radii (non-bonding), covalent radii (shared-electron bonding) and ionic radii (electrostatic bonding resulting from complete electron transfer) are the common ones. 

The accumulation of lattice constants gave rise to a growing Hbrary of interatomic (and interionic) distances, providing atomic and ionic radii. In 1929 Pauling published five principles (rules) that formed the first rational basis for understanding aystal structures. For example, the ratio of the ionic radii of cations to anions determines coordination number in crystals coordination number 6 for each chlorine and sodium ion in NaCl coordination number 8 for each ion in CsCl. 

The covalent and ionic radii of transition metals (TM) have also been determined, and we show some of them in Appendix 9.A.I. These data allow you to construct the complete 3D structure of TM complexes. It is important to remember that all the radii are averaged over many bonds, and therefore, we cannot expect that our predictions will be perfectly accurate for every molecule, ionic material, or TM complex we try to construct with Tables 9.1, 9.2 and 9.A.I. Such usage will provide, however, a very helpful way to imagine the structure of matter and acquire a more thorough... 

Chemical properties of elements are determined by the valence electronic structure, oxidation states, ionic radii, and coordination number. As already described in OSect. 18.2.1, the oxidation states of the actinide elements are more variable than those of the lanthanides. 

Takemura K (2007) Pressure scales and hydrostaticity. High Pressure Res 27 465 72 Jenei Zs, liermann HP, Cynn H et al (2011) Structural phase transition in vanadium at high pressure and high temperature influence of nonhydrostatic conditions. Phys Rev B 83 054101 Batsanov SS (2004) Determination of ionic radii from metal compiessibUities. J Struct Chem 45 896-899... 

The crystal structures of Hf 2 (OH) 2 (S0O 3 (H2O) i, (14) and Ce2(0H)2(S0i,)3 (H20)it (14) also have been determined and found to be isomorphous to the zirconium compound. The cell constants for this series of four isomorphous compounds reflect the effect of the ionic radii on the dimensions of the unit cell. The values for these cell constants are in Table II. Thus, the cell constants for the zirconium and hafnium compounds are nearly identical and smaller than the cell constants for the cerium and plutonium compounds which are also nearly identical. This trend is exactly that followed by the ionic radii of these elements. 

Bond length differences between HS and LS isomers have been determined for a number of iron(II), iron(III) and cobalt(II) complexes on the basis of multiple temperature X-ray diffraction structure studies [6]. The available results have been collected in Table 17. Average values for the bond length changes characteristic for a particular transition-metal ion have been extracted from these data and are obtained as AR 0.17 A for iron(II) complexes, AR 0.13 A for iron(III) complexes, and AR = 0.06 A for cobalt(II) complexes. These values may be compared with the differences of ionic radii between the HS and LS forms of iron(II), iron(III) and cobalt(II) which were estimated some time ago [184] as 0.16, 0.095, and 0.085 A, respectively. 

There are two forms of zinc sulfide that have structures known as wurtzite and zinc blende. These structures are shown in Figures 7.7a and 7.7b. Using the ionic radii shown in Table 7.4, we determine the radius ratio for ZnS to be 0.39, and as expected there are four sulfide ions surrounding each zinc ion in a tetrahedral arrangement. Zinc has a valence of 2 in zinc sulfide, so each bond must be 1/2 in character because four such bonds must satisfy the valence of 2. Because the sulfide ion also has a valence of 2, there must be four bonds to each sulfide ion. Therefore, both of the stmctures known for zinc sulfide have a tetrahedral arrangement of cations around each anion and a tetrahedral arrangement of anions around each cation. The difference between the structures is in the way in which the ions are arranged in layers that have different structures. 

These three structures are the predominant structures of metals, the exceptions being found mainly in such heavy metals as plutonium. Table 6.1 shows the structure in a sequence of the Periodic Groups, and gives a value of the distance of closest approach of two atoms in the metal. This latter may be viewed as representing the atomic size if the atoms are treated as hard spheres. Alternatively it may be treated as an inter-nuclear distance which is determined by the electronic structure of the metal atoms. In the free-electron model of metals, the structure is described as an ordered array of metallic ions immersed in a continuum of free or unbound electrons. A comparison of the ionic radius with the inter-nuclear distance shows that some metals, such as the alkali metals are empty i.e. the ions are small compared with the hard sphere model, while some such as copper are full with the ionic radius being close to the inter-nuclear distance in the metal. A consideration of ionic radii will be made later in the ionic structures of oxides. 

When comparing ionic porosity of different minerals, for self-consistency, the same set of ionic radii should be used, and the same temperature and pressure should be adopted to calculate the molar volume of the mineral. Table 3-3 lists the ionic porosity of some minerals. It can be seen that among the commonly encountered minerals, garnet and zircon have the lowest ionic porosity, and feldspars and quartz have the highest ionic porosity. More accurate calculation of IP may use actual X-ray data of average inter-ionic distance and determine the ionic radius in each structure. 

It is also possible to determine accurate electron density maps for the ionic crystal structures using X-ray crystallography. Such a map is shown for NaCl and LiF in Figure 1.45. The electron density contours fall to a minimum—although not to zero—in between the nuclei and it is suggested that this minimum position should be taken as the radius position for each ion. These experimentally determined ionic radii are often called crystal radii the values are somewhat different from the older sets and tend to make the anions smaller and the cations bigger than previously. The most comprehensive set of radii has been compiled by... 

Since the electron distribution function for an ion extends indefi-finitely, it is evident that no single characteristic size can be assigned to it. Instead, the apparent ionic radius will depend upon the physical property under discussion and will differ for different properties. We are interested in ionic radii such that the sum of two radii (with certain corrections when necessary) is equal to the equilibrium distance between the corresponding ions in contact in a crystal. It will be shown later that the equilibrium interionic distance for two ions is determined not only by the nature of the electron distributions for the ions, as shown in Figure 13-1, but also by the structure of the crystal and the ratio of radii of cation and anion. We take as our standard crystals those with the sodium chloride arrangement, with the ratio of radii of cation and anion about 0.75 and with the amount of ionic character of the bonds about the same as in the alkali halogenides, and calculate crystal radii of ions such that the sum of two radii gives the equilibrium interionic distance in a standard crystal. 

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