A General Approach to Radius Ratios of Simple Ionic Crystals California State University. Los Angeies. CA 90032 Los Angeles Pierce College, Woodland Hllls. CA 91371
Gd' has suggested a general equation to calculate the radius ratios of simple ionic crystals. We present a simpler derivation of a different general equation that does not require the elaborate geometrical constructs to calculate the cosine of an unknown angle. The basic arrangement of the cation in relation to two identical anions in contact is shown as a cross section in the figure. In the figure the circle with the center 0 is the cross section of the cation, and the circles with centers A and B are cross sections of the identical anions. r- is the radius of the anion and r+ is the radius of the cation. Furthermore, AO=BO=r_+r+ AD=BD=r-
Radius of Spheres Fining In Holes of the Closest Packlng TVOBof Hole
C.N.
Y
rr
blgonal tetrahedral octahedral
3 4 6
60° 54- 44' 45O
0.15469r0.22478r0.41421r-
I t is ohvious that the angle AOB at the cation is 90°. Angle y takes the value of 45O, and r+ -=--
r_
A O D = B O D = % A O B = y, and LODA = LODB = 90°. Therefore,
1
sin 45O
1=----- I
0.70711
1= 0.41421
Case 3: Cubic, eightfold coordination of the cation as in CsC1. It is obvious that the anele AOB at the cation is anv one of the acute angles of intersection between the body diagonals of the cube at the corners of which lie the anions (the obtuse angles of intersection between the body d i o & of the to anions diaeonallv. O.D. D O S in ~ ~a face of the cube corres~ond cube with norontact and havea value of 109" 28'1.The angle AOB at the cation then is (180D 109" 28') = 70' 32'. Thus the angle y is 35' 16', which gives
-
'
1 r+ = -I=-r _ sin (35* 16') 0.51738
Hence,
Case 1: Tetrahedral coordination of the cation as in ZnS; coordination number 4. It is obvious that the angle AOB at the cation is necessarily 109' 28'. Therefore, the angle y is 54O 44'. Hence, 1 r+ = -1 =----- I r _ sin (54O 44') 0.81647
1= 0.22418
Case 2: Octahedral coordination of the cation as in NaC1; coordination number 6.
1 = 0.73196
Case 4: Tetradecahedral coordination as in the closest packing of equal spheres; coordination number 12. (Note that the coordination is not dodecahedral.) In this case the angle AOB is 60' and the angle y is 30".
No ionic compound in which the cation is tetradecahedrally coordinated is known. Known examples of this geometry are elements with hexagonal close or cubic close packing of equal spheres in which each sphere is coordinated to 12 other spheres. These packing8 create holes discussed below. The Holes The equation r+ 1 -=-r- sin y
is also applicable in relating the size of trigonal, tetrahedral, and octahedral holes in the closest packing of equal spheres. In this case r+ is the radius of the sohere fittine in the hole and r- is the radius of the equal sphe.res. y is h a f t h e angle of intersection at the center of the hole of the lines ioinine. the center of the hole and the centers of two adjacent equal spheres in contact. In the table we give the values of y and the value of r+ in terms of r- for the three cases mentioned above. Cross section showing arrangement d cation in rsiatlon to two idemlcal anions.
504
Journal of Chemical Education
