Ionic radius, rion, is the radius of an atom's ion in ionic crystals structure. Although neither atoms nor ions have sharp boundaries, they are sometimes 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&nbs;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.

## Trends

X- NaX AgX
F 464 492
Cl 564 555
Br 598 577
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.

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.

## Determination

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 a 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 in 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]
Number Name Symbol 3- 2- 1- 1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+
1 Hydrogen H -4 (2)
3 Lithium Li 90
4 Beryllium Be 59
5 Boron B 41
6 Carbon C 30
7 Nitrogen N 132 (4) 30 27
8 Oxygen O 126
9 Fluorine F 119 22
11 Sodium Na 116
12 Magnesium Mg 86
13 Aluminium Al 67.5
14 Silicon Si 54
15 Phosphorus P 58 52
16 Sulfur S 170 51 43
17 Chlorine Cl 167 26 (3py) 41
19 Potassium K 152
20 Calcium Ca 114
21 Scandium Sc 88.5
22 Titanium Ti 100 81 74.5
23 Vanadium V 93 78 72 68
24 Chromium ls Cr 87 75.5 69 63 58
24 Chromium hs Cr 94
25 Manganese ls Mn 81 72 67 47 (4) 39.5 (4) 60
25 Manganese hs Mn 97 78.5
26 Iron ls Fe 75 69 72.5 39 (4)
26 Iron hs Fe 92 78.5
27 Cobalt ls Co 79 68.5
27 Cobalt hs Co 88.5 75 67
28 Nickel hs Ni 83 70 62 ls
28 Nickel ls Ni 74
29 Copper Cu 91 87 68 ls
30 Zinc Zn 88
31 Gallium Ga 76
32 Germanium Ge 87 67
33 Arsenic As 72 60
34 Selenium Se 184 64 56
35 Bromine Br 182 73 (4sq) 45 (3py) 53
37 Rubidium Rb 166
38 Strontium Sr 132
39 Yttrium Y 104
40 Zirconium Zr 86
41 Niobium Nb 86 82 78
42 Molybdenum Mo 83 79 75 73
43 Technetium Tc 78.5 74 70
44 Ruthenium Ru 82 76 70.5 52 (4) 50 (4)
45 Rhodium Rh 80.5 74 69
46 Palladium Pd 73 (2) 100 90 75.5
47 Silver Ag 129 108 89
49 Indium In 94
50 Tin Sn 83
51 Antimony Sb 90 74
52 Tellurium Te 207 111 70
53 Iodine I 206 109 67
54 Xenon Xe 62
55 Caesium Cs 181
56 Barium Ba 149
57 Lanthanum La 117.2
58 Cerium Ce 115 101
59 Praseodymium Pr 113 99
60 Neodymium Nd 143 (8) 112.3
61 Promethium Pm 111
62 Samarium Sm 136 (7) 109.8
63 Europium Eu 131 108.7
65 Terbium Tb 106.3 90
66 Dysprosium Dy 121 105.2
67 Holmium Ho 104.1
68 Erbium Er 103
69 Thulium Tm 117 102
70 Ytterbium Yb 116 100.8
71 Lutetium Lu 100.1
72 Hafnium Hf 85
73 Tantalum Ta 86 82 78
74 Tungsten W 80 76 74
75 Rhenium Re 77 72 69 67
76 Osmium Os 77 71.5 68.5 66.5 53 (4)
77 Iridium Ir 82 76.5 71
78 Platinum Pt 94 76.5 71
79 Gold Au 151 99 71
80 Mercury Hg 133 116
81 Thallium Tl 164 102.5
83 Bismuth Bi 117 90
84 Polonium Po 108 81
85 Astatine At 76
87 Francium Fr 194
89 Actinium Ac 126
90 Thorium Th 108
91 Protactinium Pa 116 104 92
92 Uranium U 116.5 103 90 87
93 Neptunium Np 124 115 101 89 86 85
94 Plutonium Pu 114 100 88 85
95 Americium Am 140 (8) 111.5 99
96 Curium Cm 111 99
97 Berkelium Bk 110 97
98 Californium Cf 109 96.1
99 Einsteinium Es 92.8[7]
Effective ionic radii in pm of elements in 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]
Number Name Symbol 3- 2- 1- 1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+
1 Hydrogen H -18 (2)
3 Lithium Li 76
4 Beryllium Be 45
5 Boron B 27
6 Carbon C 16
7 Nitrogen N 146 (4) 16 13
8 Oxygen O 140
9 Fluorine F 133 8
11 Sodium Na 102
12 Magnesium Mg 72
13 Aluminium Al 53.5
14 Silicon Si 40
15 Phosphorus P 44 38
16 Sulfur S 184 37 29
17 Chlorine Cl 181 12 (3py) 27
19 Potassium K 138
20 Calcium Ca 100
21 Scandium Sc 74.5
22 Titanium Ti 86 67 60.5
23 Vanadium V 79 64 58 54
24 Chromium ls Cr 73 61.5 55 49 44
24 Chromium hs Cr 80
25 Manganese ls Mn 67 58 53 33 (4) 25.5 (4) 46
25 Manganese hs Mn 83 64.5
26 Iron ls Fe 61 55 58.5 25 (4)
26 Iron hs Fe 78 64.5
27 Cobalt ls Co 65 54.5
27 Cobalt hs Co 74.5 61 53 hs
28 Nickel ls Ni 69 56 48 ls
28 Nickel hs Ni 60
29 Copper Cu 77 73 54 ls
30 Zinc Zn 74
31 Gallium Ga 62
32 Germanium Ge 73 53
33 Arsenic As 58 46
34 Selenium Se 198 50 42
35 Bromine Br 196 59 (4sq) 31 (3py) 39
37 Rubidium Rb 152
38 Strontium Sr 118
39 Yttrium Y 90
40 Zirconium Zr 72
41 Niobium Nb 72 68 64
42 Molybdenum Mo 69 65 61 59
43 Technetium Tc 64.5 60 56
44 Ruthenium Ru 68 62 56.5 38 (4) 36 (4)
45 Rhodium Rh 66.5 60 55
46 Palladium Pd 59 (2) 86 76 61.5
47 Silver Ag 115 94 75
49 Indium In 80
50 Tin Sn 69
51 Antimony Sb 76 60
52 Tellurium Te 221 97 56
53 Iodine I 220 95 53
54 Xenon Xe 48
55 Caesium Cs 167
56 Barium Ba 135
57 Lanthanum La 103.2
58 Cerium Ce 101 87
59 Praseodymium Pr 99 85
60 Neodymium Nd 129 (8) 98.3
61 Promethium Pm 97
62 Samarium Sm 122 (7) 95.8
63 Europium Eu 117 94.7
65 Terbium Tb 92.3 76
66 Dysprosium Dy 107 91.2
67 Holmium Ho 90.1
68 Erbium Er 89
69 Thulium Tm 103 88
70 Ytterbium Yb 102 86.8
71 Lutetium Lu 86.1
72 Hafnium Hf 71
73 Tantalum Ta 72 68 64
74 Tungsten W 66 62 60
75 Rhenium Re 63 58 55 53
76 Osmium Os 63 57.5 54.5 52.5 39 (4)
77 Iridium Ir 68 62.5 57
78 Platinum Pt 80 62.5 57
79 Gold Au 137 85 57
80 Mercury Hg 119 102
81 Thallium Tl 150 88.5
83 Bismuth Bi 103 76
84 Polonium Po 94 67
85 Astatine At 62
87 Francium Fr 180
89 Actinium Ac 112
90 Thorium Th 94
91 Protactinium Pa 104 90 78
92 Uranium U 102.5 89 76 73
93 Neptunium Np 110 101 87 75 72 71
94 Plutonium Pu 100 86 74 71
95 Americium Am 126 (8) 97.5 85
96 Curium Cm 97 85
97 Berkelium Bk 96 83
98 Californium Cf 95 82.1
99 Einsteinium Es 83.5[7]

## Soft-sphere model

Soft-sphere ionic radii (in pm) of some ions
Cation, M RM Anion, X RX
Li+ 109.4 Cl- 218.1
Na+ 149.7 Br- 237.2

For many compounds, the model of ions as hard spheres does not reproduce the distance between ions, ${\displaystyle {d_{mx}}}$, 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.[8]

The relation between soft-sphere ionic radii, ${\displaystyle {r_{m}}}$ and ${\displaystyle {r_{x}}}$, and ${\displaystyle {d_{mx}}}$, is given by

${\displaystyle {d_{mx}}^{k}={r_{m}}^{k}+{r_{x}}^{k}}$,

where ${\displaystyle k}$ is an exponent that varies with the type of crystal structure. In the hard-sphere model, ${\displaystyle k}$ would be 1, giving ${\displaystyle {d_{mx}}={r_{m}}+{r_{x}}}$.

Comparison between observed and calculated ion separations (in pm)
MX Observed Soft-sphere model
LiCl 257.0 257.2
LiBr 275.1 274.4
NaCl 282.0 281.9
NaBr 298.7 298.2

In the soft-sphere model, ${\displaystyle k}$ 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 ${\displaystyle k}$ 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,[9] 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.[10] 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.[11]

## References

1. ^ On the basis of conventional ionic radii, Ag+ (129 pm) is indeed larger than Na+ (116 pm)
2. ^ Landé, A. (1920). "Über die Größe der Atome". Zeitschrift für Physik. 1 (3): 191-197. Bibcode:1920ZPhy....1..191L. doi:10.1007/BF01329165. Retrieved 2011.
3. ^ Wasastjerna, J. A. (1923). "On the radii of ions". Comm. Phys.-Math., Soc. Sci. Fenn. 1 (38): 1-25.
4. ^ Goldschmidt, V. M. (1926). Geochemische Verteilungsgesetze der Elemente. Skrifter Norske Videnskaps--Akad. Oslo, (I) Mat. Natur. This is an 8 volume set of books by Goldschmidt.
5. ^ Pauling, L. (1960). The Nature of the Chemical Bond (3rd Edn.). Ithaca, NY: Cornell University Press.
6. ^ a b c d R. D. Shannon (1976). "Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides". Acta Crystallogr A. 32 (5): 751-767. Bibcode:1976AcCrA..32..751S. doi:10.1107/S0567739476001551.
7. ^ a b R. G. Haire, R. D. Baybarz: "Identification and Analysis of Einsteinium Sesquioxide by Electron Diffraction", in: Journal of Inorganic and Nuclear Chemistry, 1973, 35 (2), S. 489-496; doi:10.1016/0022-1902(73)80561-5.
8. ^ Lang, Peter F.; Smith, Barry C. (2010). "Ionic radii for Group 1 and Group 2 halide, hydride, fluoride, oxide, sulfide, selenide and telluride crystals". Dalton Transactions. 39 (33): 7786-7791. doi:10.1039/C0DT00401D. PMID 20664858.
9. ^ H. Bethe (1929). "Termaufspaltung in Kristallen". Annalen der Physik. 3 (2): 133-208. Bibcode:1929AnP...395..133B. doi:10.1002/andp.19293950202.
10. ^ M. Birkholz (1995). "Crystal-field induced dipoles in heteropolar crystals - I. concept". Z. Phys. B. 96 (3): 325-332. Bibcode:1995ZPhyB..96..325B. CiteSeerX 10.1.1.424.5632. doi:10.1007/BF01313054.
11. ^ M. Birkholz (2014). "Modeling the Shape of Ions in Pyrite-Type Crystals". Crystals. 4 (3): 390-403. doi:10.3390/cryst4030390.