Effective crystal radius

Table 1.11 Effective crystal radii (CR) and ionic radii (IR) of Shannon (1976). CN = coordination number SP = spin sp = square planar py = pyramidal EIS = high spin LS = low spin data in A. 

Table 4.6 Data from Shannon, R.D. (1976) Revised effective crystal radii and systematic studies of interatomic distances in halides and chalcogenides, Acta. Cryst. A32, 751.

Effective crystal radius, cm Number average particle size, cm Weight average particle size, cm Intracrystalline diffusivity, cm /sec... 

The radii found in this way are given in Table II, the univalent radii being also included, in parentheses. The dependence of the univalent crystal radius and the crystal radius on the atomic number is shown graphically in Fig. 3.14 The effect of the valence in causing the crystal radius to deviate from the regular dependence on the atomic number shown by the univalent crystal radius is clearly evident. 

The effect of deformation is shown in the sulfide, selenide and telluride of lead. Especially interesting is the decrease in the crystal radius in the series Mn++, Fe++, Co++, Ni++ there must then come an increase when the shell is completed, at Zn++, with the radius 0.74 A. 

The values of hj for different ions are between 0 and 15 (see Table 7.2). As a rule it is found that the solvation number will be larger the smaller the true (crystal) radius of the ion. Hence, the overall (effective) sizes of different hydrated ions tend to become similar. This is why different ions in solution have similar values of mobilities or diffusion coefficients. The solvation numbers of cations (which are relatively small) are usually higher than those of anions. Yet for large cations, of the type of N(C4H9)4, the hydration number is zero. 

The conductivities of melts, in contrast to those of aqueous solutions, increase with decreasing crystal radius of the anions and cations, since the leveling effect of the solvation sheaths is absent and ion jumps are easier when the radius is small. In melts constituting mixtures of two salts, positive or negative deviations from additivity are often observed for the values of conductivity (and also for many other properties). These deviations arise for two reasons a change in hole size and the formation of new types of mixed ionic aggregates. 

The effects of the anions (i.e., their specific adsorbabilities) increase in the order F < Cr < Br < I . This trend is due to the fact that the solvation energy decreases with increasing crystal radius as one goes from F to I , and the transfer of the ions to the inner Helmholtz plane is facilitated accordingly. The opposite picture is seen for surface-active cations (e.g., [N(C4H5)4]+) the descending branch of the ECC is depressed, and the PZC shifts in the positive direction. 

Micropore mass transfer resistance of zeoUte crystals is quantified in units of time by r /Dc, where is the crystal radius and Dc is the intracrystalline diffusivity. In addition to micropore resistance, zeolitic catalysts may offer another type of resistance to mass transfer, that is resistance related to transport through the surface barrier at the outer layer of the zeoHte crystal. Finally, there is at least one additional resistance due to mass transfer, this time in mesopores and macropores Rp/Dp. Here Rp is the radius of the catalyst pellet and Dp is the effective mesopore and macropore diffusivity in the catalyst pellet [18]. 

The solvation energy described by the Born equation is essentially electrostatic in nature. Born equations 8.116 and 8.120 are in fact similar to the Born-Lande equation (1.67) used to define the electrostatic potential in a crystal (see section 1.12.1). In hght of this analogy, the effective electrostatic radius of an ion in solution r j assumes the same significance as the equilibrium distance in the Born-Lande equation. We may thus expect a close analogy between the crystal radius of an ion and the effective electrostatic radius of the same ion in solution. 

The crystal radius thus has local validity in reference to a given crystal structure. This fact gives rise to a certain amount of confusion in current nomenclature, and what it is commonly referred to as crystal radius in the various tabulations is in fact a mean value, independent of the type of structure (see section 1.11.1). The crystal radius in the sense of Tosi (1964) is commonly defined as effective distribution radius (EDR). The example given in figure 1.7B shows radial electron density distribution curves for Mg, Ni, Co, Fe, and Mn on the M1 site in olivine (orthorhombic orthosilicate) and the corresponding EDR radii located by Fujino et al. (1981) on the electron density minima. 

Fia. 13-7.—The function F(p) showing the effect of radius ratio on equilibrium interionic distance of crystals with the sodium chloride arrangement. 

ACTINIDE CONTRACTION. An effect analogous to the Lanthanide contraction, which lias been found in certain elements of the Actinide series. Those elements from thorium (atomic number 90) to curium (atomic number 96) exhibit a decreasing molecular volume in certain compounds, such as those which the actinide tetrafluoiides form with alkali metal fluorides, plotted in Eig. 1. The effect here is due to the decreasing crystal radius of the tetrapositive actinide ions as the atomic number increases. Note that in the Actinides the tetravalent ions are compared instead of the trivalent ones as in the case of the Lanthanides, in which the trivalent state is by far the most common. 

Figure 19 shows sample isotherms and interface shapes predicted by the QSSM for calculations with decreasing melt volume in the crucible, as occurs in the batchwise process. Because the crystal pull rate and the heater temperature are maintained at constant values for this sequence, the crystal radius varies with the varying heat transfer in the system. Two effects are noticeable. First, decreasing the volume exposes the hot crucible wall to the crystal. The crucible wall heats the crystal and causes the decrease in... 

Figure 4.5 Effect of buffer cation on mobility, (a) Electrophoretic mobility (dansylalanine) versus reciprocal crystal radius (b) electroosmotic mobility (mesityl oxide) versus reciprocal crystal radius. Capillary 50 cm X 75 p,m I.D. fused silica. (Reprinted from Ref. 16 with permission.)...

An empirical set of effective ionic radii in oxides and fluorides, taking into account the electronic spin state and coordination of both the cation and anion, have been calculated (114). For six-coordinate Bk(III), the radii values are 0.096 nm, based on a six-coordinate oxide ion radius of 0.140 nm, and 0.110 nm, based on a six-coordinate fluoride ion radius of 0.119 nm. For eight-coordinate Bk(IV), the corresponding values are 0.093 and 0.107 nm, based on the same anion radii (114). Other self-consistent sets of trivalent and tetravalent lanthanide and actinide ionic radii, based on isomorphous series of oxides (145, 157) and fluorides (148, 157), have been published. Based on a crystal radius for Cf(III), the ionic radius of isoelectronic Bk(II) was calculated to be 0.114 nm (158). It is important to note, however, that meaningful comparisons of ionic radii can be made only if the values compared are calculated in like fashion from the same type of compound, both with respect to composition and crystal structure. 

In the case of nitriloacetate complexes, the changes in enthalpy have been explained in terms of the consequences of lanthanide contraction (i) increasingly exothermic complexation with decreasing crystal radius and (ii) decreasing exothermic complexation with decreasing hydration of the cation [22]. In the case of dipicolinates and diglycolates these effects become small as the coordination sphere loses water molecules. Thus AH3 and A S3 vary more regularly than A Hi and A Si. 

In the semiconductors of greater polarity, the dielectric constants are smaller and the effective masses larger, and the same evaluation leads to 0.07 eV in zinc selcnidc, for example many of the impurity states can be occupied at room temperature. As the energy of the impurity states becomes deeper, the effective Bohr radius becomes smaller and the use of the effective mass approximation becomes suspect the error leads to an underestimation of the binding energy. Thus, in semiconductors of greatest polarity- and certainly in ionic crystals— impurity states can become very important and arc then best understood in atomic terms. We will return to this topic in Chapter 14, in the discussion of ionic crystals. 

Type of radius Coordination Radius (pm) Crystal radius (pm) Effective ionic radius (pm)... 

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