A Simple Expression between Critical Radius Ratio and Coordination Numbers Jahansooz Toofan Oregon State University, Co~allis,OR, 97331
For the structures of ionic compounds, the radius ratio is defined as the radius of positive ion to the radius of negative ion (small ion /large ion), ( I ) . It is easy to calculate this ratio by simple geometry (see the figure). We can show that: sin a = DCIBD
Where 2a = bond angle. The radius ratio R is defined as
where r is the ion radius. Also, it should be clarified that the coordination number (CN), in the crystallographic sense, implies the number of nearest neighbors associated with the species under consideration. In coordination compounds, the number of ligands attached to the metal ion is called its coordination number. I have found that, the relationship between a and certain highly symmetrical coordination groups with (CN = 3, 4,6, and 8),is
By substituting for a in eq 1,we get
, '' /
"J"
R = -[I2
-I
(3)
The above eqs 2 3 are helpful in remembering the order of these values in inorganic chemistly. The table gives the results of eqs 2-3 for common cwrdination numbers. Literature Cited 1. Nathan, L.C.J . Chrm. Edue. 1985.62. 215-218.
Critical Radius Ratio for Common Coordination Numbers
i CN
Asection of the two ions structure model, forthe limiting case of contact.
a (degree) Bond angle
R (critical)Arrangement
(degree)
3
60.00
120.00
0.1547 triangular
4
54.74
109.47
0.2247 tetrahedral
6
45.00
90.00
0.4142 octahedral
8
35.26
70.53
0.7320 cubical
Volume 71 Number 2 February 1994
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