Ionic radius

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Ionic radius, rion, is the radius of a monatomic ion in an ionic crystal structure. Although neither atoms nor ions have sharp boundaries, they are treated as if they were hard spheres with radii such that the sum of ionic radii of the cation and anion gives the distance between the ions in a crystal lattice. Ionic radii are typically given in units of either picometers (pm) or angstroms (Å), with 1 Å = 100 pm. Typical values range from 31 pm (0.3 Å) to over 200 pm (2 Å).


The concept can be extended to solvated ions in liquid solutions taking into consideration the solvation shell.

Unit cell parameters (in pm, equal to two M–X bond lengths) for sodium and silver halides. All compounds crystallize in the NaCl structure.
Relative radii of atoms and ions. The neutral atoms are colored gray, cations red, and anions blue. Atomic & ionic radii.svg
Relative radii of atoms and ions. The neutral atoms are colored gray, cations red, and anions blue.

Ions may be larger or smaller than the neutral atom, depending on the ion's electric charge. When an atom loses an electron to form a cation, the other electrons are more attracted to the nucleus, and the radius of the ion gets smaller. Similarly, when an electron is added to an atom, forming an anion, the added electron increases the size of the electron cloud by interelectronic repulsion.

The ionic radius is not a fixed property of a given ion, but varies with coordination number, spin state and other parameters. Nevertheless, ionic radius values are sufficiently transferable to allow periodic trends to be recognized. As with other types of atomic radius, ionic radii increase on descending a group. Ionic size (for the same ion) also increases with increasing coordination number, and an ion in a high-spin state will be larger than the same ion in a low-spin state. In general, ionic radius decreases with increasing positive charge and increases with increasing negative charge.

An "anomalous" ionic radius in a crystal is often a sign of significant covalent character in the bonding. No bond is completely ionic, and some supposedly "ionic" compounds, especially of the transition metals, are particularly covalent in character. This is illustrated by the unit cell parameters for sodium and silver halides in the table. On the basis of the fluorides, one would say that Ag+ is larger than Na+, but on the basis of the chlorides and bromides the opposite appears to be true. [1] This is because the greater covalent character of the bonds in AgCl and AgBr reduces the bond length and hence the apparent ionic radius of Ag+, an effect which is not present in the halides of the more electropositive sodium, nor in silver fluoride in which the fluoride ion is relatively unpolarizable.


The distance between two ions in an ionic crystal can be determined by X-ray crystallography, which gives the lengths of the sides of the unit cell of a crystal. For example, the length of each edge of the unit cell of sodium chloride is found to be 564.02 pm. Each edge of the unit cell of sodium chloride may be considered to have the atoms arranged as Na+∙∙∙Cl∙∙∙Na+, so the edge is twice the Na-Cl separation. Therefore, the distance between the Na+ and Cl ions is half of 564.02 pm, which is 282.01 pm. However, although X-ray crystallography gives the distance between ions, it doesn't indicate where the boundary is between those ions, so it doesn't directly give ionic radii.

Front view of the unit cell of an LiI crystal, using Shannon's crystal data (Li = 90 pm; I = 206 pm). The iodide ions nearly touch (but don't quite), indicating that Lande's assumption is fairly good. LiI unit cell, front.png
Front view of the unit cell of an LiI crystal, using Shannon's crystal data (Li = 90 pm; I = 206 pm). The iodide ions nearly touch (but don't quite), indicating that Landé's assumption is fairly good.

Landé [2] estimated ionic radii by considering crystals in which the anion and cation have a large difference in size, such as LiI. The lithium ions are so much smaller than the iodide ions that the lithium fits into holes within the crystal lattice, allowing the iodide ions to touch. That is, the distance between two neighboring iodides in the crystal is assumed to be twice the radius of the iodide ion, which was deduced to be 214 pm. This value can be used to determine other radii. For example, the inter-ionic distance in RbI is 356 pm, giving 142 pm for the ionic radius of Rb+. In this way values for the radii of 8 ions were determined.

Wasastjerna estimated ionic radii by considering the relative volumes of ions as determined from electrical polarizability as determined by measurements of refractive index. [3] These results were extended by Victor Goldschmidt. [4] Both Wasastjerna and Goldschmidt used a value of 132 pm for the O2− ion.

Pauling used effective nuclear charge to proportion the distance between ions into anionic and a cationic radii. [5] His data gives the O2− ion a radius of 140 pm.

A major review of crystallographic data led to the publication of revised ionic radii by Shannon. [6] Shannon gives different radii for different coordination numbers, and for high and low spin states of the ions. To be consistent with Pauling's radii, Shannon has used a value of rion(O2−) = 140 pm; data using that value are referred to as "effective" ionic radii. However, Shannon also includes data based on rion(O2−) = 126 pm; data using that value are referred to as "crystal" ionic radii. Shannon states that "it is felt that crystal radii correspond more closely to the physical size of ions in a solid." [6] The two sets of data are listed in the two tables below.

Crystal ionic radii in pm of elements as a function of ionic charge and spin
(ls = low spin, hs = high spin).
Ions are 6-coordinate unless indicated differently in parentheses
(e.g. 146 (4) for 4-coordinate N3−). [6]
1 Hydrogen H208−4 (2)
3 Lithium Li90
4 Beryllium Be59
5 Boron B41
6 Carbon C30
7 Nitrogen N132 (4)3027
8 Oxygen O126
9 Fluorine F11922
11 Sodium Na116
12 Magnesium Mg86
13 Aluminium Al67.5
14 Silicon Si54
15 Phosphorus P5852
16 Sulfur S1705143
17 Chlorine Cl18126 (3py)41
19 Potassium K152
20 Calcium Ca114
21 Scandium Sc88.5
22 Titanium Ti1008174.5
23 Vanadium V93787268
24 Chromium lsCr8775.5696358
24 Chromium hsCr94
25 Manganese lsMn81726747 (4)39.5 (4)60
25 Manganese hsMn9778.5
26 Iron lsFe756972.539 (4)
26 Iron hsFe9278.5
27 Cobalt lsCo7968.5
27 Cobalt hsCo88.57567
28 Nickel lsNi837062
28 Nickel hsNi74
29 Copper Cu918768 ls
30 Zinc Zn88
31 Gallium Ga76
32 Germanium Ge8767
33 Arsenic As7260
34 Selenium Se1846456
35 Bromine Br18273 (4sq)45 (3py)53
37 Rubidium Rb166
38 Strontium Sr132
39 Yttrium Y104
40 Zirconium Zr86
41 Niobium Nb868278
42 Molybdenum Mo83797573
43 Technetium Tc78.57470
44 Ruthenium Ru827670.552 (4)50 (4)
45 Rhodium Rh80.57469
46 Palladium Pd73 (2)1009075.5
47 Silver Ag12910889
48 Cadmium Cd109
49 Indium In94
50 Tin Sn83
51 Antimony Sb9074
52 Tellurium Te20711170
53 Iodine I20610967
54 Xenon Xe62
55 Caesium Cs167
56 Barium Ba149
57 Lanthanum La117.2
58 Cerium Ce115101
59 Praseodymium Pr11399
60 Neodymium Nd143 (8)112.3
61 Promethium Pm111
62 Samarium Sm136 (7)109.8
63 Europium Eu131108.7
64 Gadolinium Gd107.8
65 Terbium Tb106.390
66 Dysprosium Dy121105.2
67 Holmium Ho104.1
68 Erbium Er103
69 Thulium Tm117102
70 Ytterbium Yb116100.8
71 Lutetium Lu100.1
72 Hafnium Hf85
73 Tantalum Ta868278
74 Tungsten W807674
75 Rhenium Re77726967
76 Osmium Os7771.568.566.553 (4)
77 Iridium Ir8276.571
78 Platinum Pt9476.571
79 Gold Au1519971
80 Mercury Hg133116
81 Thallium Tl164102.5
82 Lead Pb13391.5
83 Bismuth Bi11790
84 Polonium Po10881
85 Astatine At76
87 Francium Fr194
88 Radium Ra162 (8)
89 Actinium Ac126
90 Thorium Th108
91 Protactinium Pa11610492
92 Uranium U116.51039087
93 Neptunium Np124115101898685
94 Plutonium Pu1141008885
95 Americium Am140 (8)111.599
96 Curium Cm11199
97 Berkelium Bk11097
98 Californium Cf10996.1
99 Einsteinium Es92.8 [7]
Effective ionic radii in pm of elements as a function of ionic charge and spin
(ls = low spin, hs = high spin).
Ions are 6-coordinate unless indicated differently in parentheses
(e.g. 146 (4) for 4-coordinate N3−). [6]
1 Hydrogen H139.9−18 (2)
3 Lithium Li76
4 Beryllium Be45
5 Boron B27
6 Carbon C16
7 Nitrogen N146 (4)1613
8 Oxygen O140
9 Fluorine F1338
11 Sodium Na102
12 Magnesium Mg72
13 Aluminium Al53.5
14 Silicon Si40
15 Phosphorus P212 [8] 4438
16 Sulfur S1843729
17 Chlorine Cl18112 (3py)27
19 Potassium K138
20 Calcium Ca100
21 Scandium Sc74.5
22 Titanium Ti866760.5
23 Vanadium V79645854
24 Chromium lsCr7361.5554944
24 Chromium hsCr80
25 Manganese lsMn67585333 (4)25.5 (4)46
25 Manganese hsMn8364.5
26 Iron lsFe615558.525 (4)
26 Iron hsFe7864.5
27 Cobalt lsCo6554.5
27 Cobalt hsCo74.56153
28 Nickel lsNi695648
28 Nickel hsNi60
29 Copper Cu777354 ls
30 Zinc Zn74
31 Gallium Ga62
32 Germanium Ge7353
33 Arsenic As5846
34 Selenium Se1985042
35 Bromine Br19659 (4sq)31 (3py)39
37 Rubidium Rb152
38 Strontium Sr118
39 Yttrium Y90
40 Zirconium Zr72
41 Niobium Nb726864
42 Molybdenum Mo69656159
43 Technetium Tc64.56056
44 Ruthenium Ru686256.538 (4)36 (4)
45 Rhodium Rh66.56055
46 Palladium Pd59 (2)867661.5
47 Silver Ag1159475
48 Cadmium Cd95
49 Indium In80
50 Tin Sn118 [9] 69
51 Antimony Sb7660
52 Tellurium Te2219756
53 Iodine I2209553
54 Xenon Xe48
55 Caesium Cs167
56 Barium Ba135
57 Lanthanum La103.2
58 Cerium Ce10187
59 Praseodymium Pr9985
60 Neodymium Nd129 (8)98.3
61 Promethium Pm97
62 Samarium Sm122 (7)95.8
63 Europium Eu11794.7
64 Gadolinium Gd93.5
65 Terbium Tb92.376
66 Dysprosium Dy10791.2
67 Holmium Ho90.1
68 Erbium Er89
69 Thulium Tm10388
70 Ytterbium Yb10286.8
71 Lutetium Lu86.1
72 Hafnium Hf71
73 Tantalum Ta726864
74 Tungsten W666260
75 Rhenium Re63585553
76 Osmium Os6357.554.552.539 (4)
77 Iridium Ir6862.557
78 Platinum Pt8062.557
79 Gold Au1378557
80 Mercury Hg119102
81 Thallium Tl15088.5
82 Lead Pb11977.5
83 Bismuth Bi10376
84 Polonium Po223 [10] 9467
85 Astatine At62
87 Francium Fr180
88 Radium Ra148 (8)
89 Actinium Ac106.5 (6)
122.0 (9) [11]
90 Thorium Th94
91 Protactinium Pa1049078
92 Uranium U102.5897673
93 Neptunium Np11010187757271
94 Plutonium Pu100867471
95 Americium Am126 (8)97.585
96 Curium Cm9785
97 Berkelium Bk9683
98 Californium Cf9582.1
99 Einsteinium Es83.5 [7]

Soft-sphere model

Soft-sphere ionic radii (in pm) of some ions
Cation, MRMAnion, XRX

For many compounds, the model of ions as hard spheres does not reproduce the distance between ions, , to the accuracy with which it can be measured in crystals. One approach to improving the calculated accuracy is to model ions as "soft spheres" that overlap in the crystal. Because the ions overlap, their separation in the crystal will be less than the sum of their soft-sphere radii. [12]

The relation between soft-sphere ionic radii, and , and , is given by


where is an exponent that varies with the type of crystal structure. In the hard-sphere model, would be 1, giving .

Comparison between observed and calculated ion separations (in pm)
MXObservedSoft-sphere model

In the soft-sphere model, has a value between 1 and 2. For example, for crystals of group 1 halides with the sodium chloride structure, a value of 1.6667 gives good agreement with experiment. Some soft-sphere ionic radii are in the table. These radii are larger than the crystal radii given above (Li+, 90 pm; Cl, 167 pm). Inter-ionic separations calculated with these radii give remarkably good agreement with experimental values. Some data are given in the table. Curiously, no theoretical justification for the equation containing has been given.

Non-spherical ions

The concept of ionic radii is based on the assumption of a spherical ion shape. However, from a group-theoretical point of view the assumption is only justified for ions that reside on high-symmetry crystal lattice sites like Na and Cl in halite or Zn and S in sphalerite. A clear distinction can be made, when the point symmetry group of the respective lattice site is considered, [13] which are the cubic groups Oh and Td in NaCl and ZnS. For ions on lower-symmetry sites significant deviations of their electron density from a spherical shape may occur. This holds in particular for ions on lattice sites of polar symmetry, which are the crystallographic point groups C1, C1h, Cn or Cnv, n = 2, 3, 4 or 6. [14] A thorough analysis of the bonding geometry was recently carried out for pyrite-type compounds, where monovalent chalcogen ions reside on C3 lattice sites. It was found that chalcogen ions have to be modeled by ellipsoidal charge distributions with different radii along the symmetry axis and perpendicular to it. [15]

See also

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