In
condensed matter physics and
inorganic chemistry, the cation-anion radius ratio can be used to predict the
crystal structure of an
ionic compound based on the relative size of its atoms. It is defined as the ratio of the
ionic radius of the positively charged cation to the
ionic radius of the negatively charged anion in a cation-anion compound. Anions are larger than cations. Large sized anions occupy lattice sites, while small sized cations are found in voids.
In a given structure, the ratio of cation radius to anion radius is called the radius ratio. This is simply given by .
Ratio rule and stability
Critical Radius Ratio. This diagram is for octahedral interstices (coordination number six): 4 anions in the plane shown, 1 above the plane and 1 below. The stability limit is at rC/rA = 0.414
The radius ratio rule defines a critical radius ratio for different crystal structures, based on their
coordination geometry.[1] The idea is that the anions and cations can be treated as incompressible spheres, meaning the crystal structure can be seen as a kind of
unequal sphere packing. The allowed size of the cation for a given structure is determined by the critical radius ratio.[2] If the cation is too small, then it will attract the anions into each other and they will collide hence the compound will be unstable due to anion-anion repulsion; this occurs when the radius ratio drops below the critical radius ratio for that particular structure. At the stability limit the cation is touching all the anions and the anions are just touching at their edges. For radius ratios greater than the critical ratius ratio, the structure is expected to be
stable.
The rule is not obeyed for all compounds. By one estimate, the crystal structure can only be guessed about 2/3 of the time.[3] Errors in prediction are partly due to the fact that real chemical compounds are not purely ionic, they display some
covalent character.[1]
The table below gives the relation between critical radius ratio, , and
coordination number, , which may be obtained from a simple geometrical proof.[4]
^
abMichmerhuizen, Anna; Rose, Karine; Annankra, Wentiirim; Vander Griend, Douglas A. (2017-08-09). "Radius Ratio Rule Rescue". Journal of Chemical Education. 94 (10). American Chemical Society (ACS): 1480–1485.
doi:
10.1021/acs.jchemed.6b00970.
ISSN0021-9584.
^Nathan, Lawrence C. (1985). "Predictions of crystal structure based on radius ratio: How reliable are they?". Journal of Chemical Education. 62 (3). American Chemical Society (ACS): 215.
doi:
10.1021/ed062p215.
ISSN0021-9584.
^Toofan, Jahansooz (1994). "A Simple Expression between Critical Radius Ratio and Coordination Number". Journal of Chemical Education. 71 (2). American Chemical Society (ACS): 147.
doi:
10.1021/ed071p147.
ISSN0021-9584. (and Erratum 71(9): 749
doi:
10.1021/ed071p749), Following the erratum, equations should read and .
^
abJensen, William B. (2010-04-23). "The Origin of the Ionic-Radius Ratio Rules". Journal of Chemical Education. 87 (6). American Chemical Society (ACS): 587–588.
doi:
10.1021/ed100258f.
ISSN0021-9584.
^Goldschmidt, V. (1927). Geochemische Verteilungsgesetze der Elemente. VIII. Untersuchungen über Bau und Eigenschaften von Krystallen (in German). Oslo: Dybwad. pp. 14–17.
OCLC19831825.
^Pauling, Linus (1929). "The principles determining the structure of complex ionic crystals". Journal of the American Chemical Society. 51 (4). American Chemical Society (ACS): 1010–1026.
doi:
10.1021/ja01379a006.
ISSN0002-7863.