Sizes of ions

If a material behaves as a semiconductor without the addition of dopants (see below), it is an intrinsic semiconductor. 

Each of Si, Ge and ot-Sn is classed as an intrinsic semiconductor, the extent of occupation of the upper band increasing with increasing temperature. Electrons present in the upper conduction band act as charge carriers and result in the semiconductor being able to conduct electricity. Additionally, the removal of electrons from the lower valence band creates positive holes into which electrons can move, again leading to the ability to conduct charge. 

A charge carrier in a semiconductor is either a positive hole or an electron that is able to conduct electricity. 

Extrinsic semiconductors contain dopants a dopant is an impurity introduced into a semiconductor in minute amounts to enhance its electrical conductivity. 

In Ga-doped Si, the substitution of a Ga (group 13) for a Si (group 14) atom in the bulk solid produces an electron-deficient site. This introduces a discrete, unoccupied level 

As mentioned earlier, the sizes of isolated ions are ill-defined, but in condensed phases, the ions can be assigned definite sizes, because of the strong repulsion of the contagious electronic shells. In crystals, the interionic distances can be measured by x-ray and neutron diffraction with an uncertainty of a fraction of a pm and individual ionic radii (at least for monatomic and globular ions) have been assigned. These radii, r, do depend on the coordination numbers of the ions and a set for the characteristic coordination in salt crystals that are usually met have been estabhshed by Shannon and Prewitt [43,44], 

In solutions, the distances between the centers of ions and of the nearest atoms of the surrounding solvent molecules can also be measured by x-ray and neutron diffraction, but with a somewhat larger uncertainty, 2pm. In aqueous solutions, if the water molecule is assigned a constant radius r = 138 pm (one half of the experimental collision diameter), then the distances t((I -0 )/pm = 138-fr,/pm have been established by Marcus within the experimental uncertainty, with the same ionic radii as in the crystals [45, 46], These radii, as selected in Ref 6 and annotated there, are listed in Table 2.8. The distances between the centers of ions in solutions in solvents other than water and of the nearest atoms of the solvents have also been determined in some cases reported by Ohtaki and Radnai [50] and confirm the portability of the Tj values among solvents, provided the mean solvent coordination number is near that in water. 

The sizes of polyatomic (nonglobular) ions in crystals are also expressed by their thermochemical radii according to Jenkins and coworkers [47], Circular reasoning may be involved in their determination, because these radii depend on calculated lattice energies of crystals that in turn depend on the interionic distances. The assigned uncertainties of these radii are 19 pm for univalent and divalent anions increasing to twice this amount for trivalent ones and they are listed in Table 2.8 too. A further problem with these values is the use of the Goldschmidt radii for the alkali metal counterions, r°, rather than the Shannon-Prewitt ones [43,44] appropriate for ions in solution. 

A different approach to the sizes of ions (applicable to crystalline salts) is to consider their volumes rather than their radii as suggested by Jenkins et al. [48]. It is assumed that the volume of a formula unit of the salt M X is additive in the individual 

A further measure of the sizes of ions, pertaining to ions in solution, is their intrinsic molar volumes, The molar volume of a bare unsolvated ion, 4jrNJ3)r, cannot represent the intrinsic volume of the ion in solution, because of the void spaces between the solvent molecules and the ion and among themselves. Mukerjee [52] proposed for aqueous alkali metal and halide ions at 25°C a factor of A = 1.213, producing  

Deciding on the actual size of an ion in solution is one of the fundamental problems in electrolyte solutions. This is because there are several ways in which the acmal size could be defined. And so when talking about the actual size and giving it a magnitude, it is necessary to state to what entity this is referring, i.e. is it  

Before we embark upon a discussion of the structures of ionic solids, we must say something about the sizes of ions, and define the term ionic radius. The process of ionization (e.g. equation 5.4) results in a contraction of the species owing to an increase in the effective nuclear charge. Similarly, when an atom gains an electron (e.g. equation 5.5), the imbalance between the number of protons and electrons causes the anion to be larger than the original atom. 

Although from a wave-mechanical viewpoint, the radius of an individual ion has no precise physical significance, for purposes of descriptive crystallography, it is convenient to have a compilation of values obtained by partitioning measured interatomic distances in ionic compounds. Values of the ionic radius (/-jon) may be derived from X-ray diffraction data. However, experimental data only give the internuclear distance and we generally take this to be the sum of the ionic radii of the cation and anion (equation 5.6). 

Internuclear distance between a cation and the closest anion 

Equation 5.6 assumes a hard sphere model for the ions, with ions of opposite charge touching one another in the crystal lattice. Use of such an approximation means that the assignment of individual radii is somewhat arbitrary. Among many approaches to this problem we mention three. 

Pauling considered a series of alkali metal halides, each member of which contained isoelectronic ions (NaF, KCl, RbBr, Csl). In order to partition the ionic radii, he assumed 

Lande assumed that in the sohd state structures of the lithium halides, LiX, the anions were in contact with one another (see diagram 6.1 and Fignre 6.15a with the accompanying discussion). Lande took half of each anion-anion distance to be the radius of that anion, and then obtained rLj+ by substituting into equation 6.6 values of rx and the measured internuclear Li—X distances. 

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EXAMPLE 1.11 Sample exercise Deciding the relative sizes of ions... 

Empirical Crystal Radii.—A set of crystal radii, derived empirically by Wasastjerna16 with the aid of mole refraction values, is at present generally accepted17 as giving satisfactorily the sizes of ions in crystals. Wasastjerna s radii are given in Table IV. 

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. 

Clearly, it is the size of ions that is decisive in ion-pair formation. Moreover, the coulombic interactions can extend even to more distant neighbours. The Blander equation is then, of course, no longer applicable. 

We expect forces in ionic compounds to increase as sizes of ions become smaller and as ionic charges become greater. As the forces between ions become stronger, a higher temperature is required to melt the crystal. In the series of compounds NaF, NaCl, NaBr, and Nal, the anions are progressively larger, and thus the ionic forces become weaker. We... 

Linus Pauling, "The Sizes of Ions and the Structure of Ionic Crystals," JACS 49 (1927) 766, quoted in Servos (1990 283). 

In the next few years, in addition to his duties at Caltech, Pauling lectured regularly at Berkeley, glad to talk with Lewis and other physical chemists there about his work. Lewis wrote his former student, Joseph Mayer, that Uhlenbeck and Sommerfeld both had been at Berkeley in spring 1929, "but the best of all by far was Pauling.. . . He gave a course three hours a week in quantum mechanics, and one of one hour a week on the size of ions and other similar problems." 59... 

Pauling L. (1927a). The sizes of ions and the structure of ionic crystals. Jour. Amer. Chem. 

In further developments (9), the sizes of ions are considered as concentration-dependent parameters, and new expressions are derived for the activity coefficients. 

TABLE 11 Possible Substitution Studies in La2Cu04 According to the Ionic Sizes of Ions in Different Coordination Sites. 

However, just because it is possible to form a particular ion, does not mean that this ion will always exist whatever the circumstances. In many structures, we find that the bonding is not purely ionic but possesses some degree of covalency the electrons are shared between the two bonding atoms and not merely transferred from one to the other. This is particularly true for the elements in the centre of the Periodic Table. This point is taken up in Section 1.6.4 where we discuss the size of ions and the limitations of the concept of ions as hard spheres. 

Shannon and Prewitt using data from almost a thousand crystal structure determinations and based on conventional values of 126 pm and 119 pm for the radii of the and F ions, respectively. These values differ by a constant factor of 14 pm from traditional values but it is generally accepted that they correspond more closely to the actual physical sizes of ions in a crystal. A selection of this data is shown in Table 1.9. 

Several important trends in the sizes of ions can be noted from the data in Table 1.9 ... 

The Born equations (2.37) and (2.43) are widely used, but suffer from lack of accurate information about the real sizes of ions in aqueous solutions. In the derivations used above, ionic radii have been used, but these give numerical answers that are exaggerated, in particular for those of cations. The values of the absolute enthalpies of hydration of the ions of Table 2.3 are given in Table 2.8, based on the conventional values with that of the proton taken to be — I 110 kJ mol-1. 

Without further data, it should thus be possible to make some predictions regarding sizes of ions, and for this purpose we will confine our attention to ions with noble-gas structures. In any period the radii of the ions must decrease with increasing positive charge, as in the following series ... 

In general the results of recent investigations support the thesis that the forces operative in ionic crystals are those, described above, that underlie the Bom equation for the crystal energy and we may feel justified in investigating the further consequences of this postulate. The question of the sizes of ions is studied from this point of view in the following section. 

THE SIZES OF IONS UNIVALENT RADII AND CRYSTAL RADII ... 

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