Ionic radii 3.7.6



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3.7.6 Ionic radii

Empirical values for ionic radii are used for predicting interatomic distances and packing in crystals, even when the forces are far from being purely ionic. The ions are regarded as hard spheres, whose size is, to a first approximation, constant in all environments. To a second approximation, it has long been recognized that there is a dependence on coordination number, and, more recently, for transition-metal ions, a dependence also on spin state. Separate values are listed accordingly in the table.

The coordination number of an atom is the number of its first-nearest neighbours of opposite sign. It may sometimes be ambiguous when the neighbours are not all at exactly the same distance; judgement and experience, rather than formal rules, must then decide which of them are to be counted as first-nearest. Different decisions on this point will lead to small differences in estimates of the mean cation–anion distance within the polyhedron (the figure whose vertices are the first-nearest neighbour anions around the cation).

The spin state of a transition-metal ion refers to its spin angular momentum. Normally this depends on the ground state of the free atom, and is given by Hund's rule; this is the high-spin state. In a particular crystal structure, however, ions may be situated in strong ligand fields which reduce the spin angular momentum; this is the low-spin state².

In using the table, no a priori knowledge of the ionic character of the bonds, i.e. of the degree of polarization or covalent character, is needed. An estimate of the degree of polarization is given by the Pauling electrostatic valence, whose value is the valency of the cation divided by its coordination number Z. The radii tabulated for the different coordination numbers have been so derived as to allow for this, and will predict the mean cation–anion distance within a polyhedron to an accuracy, generally, of about ±3 pm. Individual distances may, however, show a rather larger scatter. The tabulated values also predict the shortest anion–anion polyhedron edges and packing distances for polyhedra where the Pauling valence is not greater than 1; for greater Pauling valences, the polyhedron edges are considerably shorter than the anion diameter.

The radii r tabulated are taken, with some simplification, from those listed by R. D. Shannon and C. T. Prewitt, Acta Cryst., 1969, B25, 925, with corrections ibid., 1970, B26, 1046. Methods of derivation and references to earlier work will be found there. For the sort of corrections to be made for more precise work, see, for example, W. H. Baur, Trans. Am. Cryst. Assoc., 1970, 6, 129.

Ionic radii

Z Cation	4	6	8	12	Z Cation		4	6	8	12	Z Cation	4	6	8	12
Z Cation	r/pm				Z Cation		r/pm				Z Cation	r/pm
Li⁺	59	74			Pb²⁺		94	118	129	149	Si⁴⁺	26	40
Na⁺		102	116								Ge⁴⁺	40	54
K⁺		138	151	160	B³⁺		12				Sn⁴⁺		69
Rb⁺		149	160	173	Al³⁺		39	53			Pb⁴⁺		78	94
Cs⁺		170	182	188	Ga³⁺		47	62
					In³⁺			79	92		Ti⁴⁺		60
Ag⁺		115	130		Tl³⁺			88	100		Zr⁴⁺		72	84
Tl⁺		150	160	176							Hf ⁴⁺		71	83
					Sc³⁺			73	87		Th⁴⁺		100	106
Be²⁺	27				Y³⁺			90	101		U⁴⁺		97	100
Mg²⁺	58	72			La³⁺			106	118	132
Ca²⁺		100	112	135							P⁵⁺	17	35
Sr²⁺		116	125	144	Cr³⁺			61			As⁵⁺	34	50
Ba²⁺		136	142	160	Mn³⁺	LS		58			Sb⁵⁺		61
					Mn³⁺	HS		65			Bi⁵⁺		74
Zn²⁺	60	75			Fe³⁺	LS		55			V⁵⁺	36	54
Cd²⁺	80	95	107	131	Fe³⁺	HS	49	65			Nb⁵⁺	32	64
					Co³⁺	LS		53			Ta⁵⁺		64	69
Mn²⁺ LS		67			Co³⁺	HS		61
Mn²⁺ HS		82									S⁶⁺	12
Fe²⁺ LS		61			Bi³⁺			102	111
Fe²⁺ HS	63	78									Cr⁶⁺	30	52
Co²⁺ LS		57			Pu³⁺			100			Mo⁶⁺	42	60
Co²⁺ HS		74									W⁶⁺	41	58
Ni²⁺		70									U⁶⁺	48	75
Cu²⁺		73
Z Anion	2	4	6		Z Anion		2	4	6
Z Anion	r/pm				Z Anion		r/pm
F⁻	128	131	133		O²⁻		135	138	140
Cl⁻		181^†			S²⁻			184^†
Br⁻		196^†
I^–		220^†

LS, low-spin state; HS, high-spin state. ^† Independent of coordination number.

M.L.McGlashan

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