Atomic radius determining from crystal structure

Theoretically, the radius of an ion extends from the nucleus to the outermost orbital occupied by electrons. The very nature of the angular wave function of an electron, which approaches zero asymptotically with increasing distance from the nucleus, indicates that an atom or ion has no definite size. Electron density maps compiled in X-ray determinations of crystal structures rarely show zero contours along a metal-anion bond. 

SAMPLE PROBLEM 12.4 Determining Atomic Radius from Crystal Structure... 

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. 

As is seen from the behaviour of the more sophisticated Heine-Abarenkov pseudopotential in Fig. 5.12, the first node q0 in aluminium lies just to the left of (2 / ) / and g = (2n/a)2, the magnitude of the reciprocal lattice vectors that determine the band gaps at L and X respectively. This explains both the positive value and the smallness of the Fourier component of the potential, which we deduced from the observed band gap in eqn (5.45). Taking the equilibrium lattice constant of aluminium to be a = 7.7 au and reading off from Fig. 5.12 that q0 at 0.8(4 / ), we find from eqn (5.57) that the Ashcroft empty core radius for aluminium is Re = 1.2 au. Thus, the ion core occupies only 6% of the bulk atomic volume. Nevertheless, we will find that its strong repulsive influence has a marked effect not only on the equilibrium bond length but also on the crystal structure adopted. 

Physical Properties. The absorption of x-rays by iodine has been studied and the iodine crystal structure determined (12,13). Iodine crystallizes in the orthorhombic system and has a unit cell of eight atoms arranged as a symmetrical bipyramid. The cell constants at 18°C (14) are given in Table 1, along with other physical properties. From the interatomic distances of many iodine compounds, the calculated effective radius of the covalendy bound iodine atom is 184 pm (15). 

One property of a transition metal ion that is particularly sensitive to crystal field interactions is the ionic radius and its influence on interatomic distances in a crystal structure. Within a row of elements in the periodic table in which cations possess completely filled or efficiently screened inner orbitals, there should be a decrease of interatomic distances with increasing atomic number for cations possessing the same valence. The ionic radii of trivalent cations of the lanthanide series for example, plotted in fig. 6.1, show a relatively smooth contraction from lanthanum to lutecium. Such a trend is determined by the... 

In (CgHg ) P-Au-Mn(CO)g there is an approximately linear bond system P-Au-Mn, This is the normal stereochemical form that would be expected for an Au(I) atom forming two bonds. Considerable difficultly arose in the structure determination. The crystal is non-centrosymmetric and, as a further complication, contains two crystallographically different molecules with a pseudosymmetry of the heavy atom positions higher than that of the space-group. In this situation the common methods of crystal structure refinement are severely hampered. Nevertheless all atoms have been located. The distance Au-P may be used to get, by difference, a covalent radius applicable to the gold atom and hence, from the Au-Mn distance, a radius for Mn is found, A more suitable compound is the variant... 

The atomic radius reported in the table is difficult to define, as there is no precise outer boundary of an atom. Its value is obtained from determinations of the atomic distances in a metal by X-ray diffraction methods. The covalent radius is one-half of the distance between the nuclei of two identical atoms joined by a single covalent bond. The ionic radius is calculated from the distance between the nuclei of atoms joined by a ionic bond. The atomic radius of a metal in a metallic structure is usually much greater than the ionic radius of an ion of the same element in a salt crystal. 

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