13th Lecture

Chemistry 2810 Lecture Notes Dr. M. Gerken Page77
5.2 Ionic structures
We are now ready to consider a few of the most common ionic crystal structure types. Several of the Group 1 and 2
halides have become common model systems for structural types that are used widely by ionic compounds of other elements,
as well as some quite covalent transition metal oxides or sulfides, and even for some of the metallic alloys. Ions are obtained
by the loss of electrons from or addition of electrons to a neutral atom.
Li (1s 2 2s 1 ) Li + (1s 2 )
F 1s 2 2s 2 2p 5 F – (1s 2 2s 2 2p 6 )
This results in dramatic changes in the sizes of the ions, as shown graphically below:
The radii changes involved for Li and F are as follows: Li is 1.52 Å, Li + 0.90 Å, F 0.72 Å, F – 1.19 Å. This translates into an
astounding 5-fold reduction in volume for lithium, and a 4-fold increase in volume for fluorine!
It is therefore not surprising that in most ionic structures the anions are considerably larger than the cations. This can be
verified readily from the graphic presentation of some common ionic radii arranged in periodic table format below. A few
exceptions to this general rule can be obtained if the largest cations are combined with the smallest anions (e.g. CsF).
Be 2+ Fe 2+
Li + Graphical Comparison of CN6 Radii of Cations and Anions N 3- O 2- F -
Fe 3+
Al 3+ Sb 5+
Na +
Mg 2+
S 2-
Cl-
Sc 3+ Ti 4+ V 5+ Cr 3+ Co 2+ Ni 2+ Cu + Zn 2+ Ga 3+
Ca 2+
K + Mn 2+ Se2- Br -
Rb + Sr 2+
Ag + Cd 2+ In 3+ Te 2- I -
Sn 4+
Cs + Ba 2+ Au 2+ Hg 2+ Tl 3+ Pb 4+
The definition of ionic radius is not without dispute, and in fact ion size
depends on the structures within which they occur. Obviously they also
depend on the oxidation state of the ion, as emphasized for the iron ions in the
chart above. Ionic radii are determined from X-ray crystallography. The
earliest determinations used simple geometric factors to calculate average
radii. More recently, accurate electron density maps have been used to obtain
more reliable estimates of absolute ion radii. Consider the plot of electron
density in a crystal of lithium fluoride shown at right. The actual minimum in
the electron density distribution is indicated at 0.92 Å. The original Pauling
radii puts Li at 0.60 Å (a value often presented in General Chemistry texts),
while the more accurate Shannon-Prewitt radius of Li with CN6 is 0.90 Å.
This is clearly a better estimate of the location where the cation ends and the
anion begins. But for such a shallow minimum, considerable variation - and
dispute - is expected to exist. A detailed list of the most up-to-date Shannon-
Prewitt ionic radii follows.

Chemistry 2810 Lecture Notes Dr. M. Gerken Page78
1 18
He
13 14 15 16 17
H
+1 -0.24(1)
-0.04(2) 2
Chemistry 2810 Ionic Radii (Å)
Li
+1 0.73(4)
0.90(6)
1.06(8)
Na
+1 1.13(4)
1.16(6)
1.30(8)
1.53(12)
K
+1 1.52(6)
1.65(8)
1.73(10)
1.78(12)
Rb
+1 1.66(6)
1.75(8)
1.80(10)
1.86(12)
1.97(14)
Cs
+1 1.81(6)
1.88(8)
1.95(10)
2.02(12)
Fr
+1 1.94(6)
Be
+2 0.31(3)
0.41(4)
0.59(6)
Mg
+2 0.71(4)
0.86(6)
1.03(8)
Ca
+2 1.14(6)
1.26(8)
1.37(10)
1.48(12)
Sr
+2 1.32(6)
1.40(8)
1.50(10)
1.58(12)
Ba
+2 1.49(6)
1.56(8)
1.66(10)
1.75(12)
Ra
+2 1.62(8)
1.84(12)
χ ≤ 1.50
Sc
+3 0.885(6)
1.010(8)
Y
+3 1.040(6)
1.159(8)
Lu
+3 1.001(6)
1.117(8)
χ ≥ 1.50
3 4 5 6 7 8 9 10 11 12
Lr
Ti
+2 1.00(6)
+3 0.81(6)
+4 0.56(4)
0.745(6)
0.88(8)
Zr
+4 0.73(4)
0.86(6)
0.98(8)
Hf
+4 0.72(4)
0.85(6)
0.97(8)
V
+2 0.93(6)
+3 0.78(6)
+4 0.72(6)
0.86(8)
+5 0.494(4)
0.68(6)
Nb
+3 0.86(6)
+4 0.82(6)
0.93(8)
+5 0.62(4)
0.78(6)
0.88(8L)
0.92(8H)
Ta
+3 0.86(6)
+4 0.82(6)
+5 0.78(6)
0.88(8)
Cr
+2 0.87(6L)
0.94(6H)
+3 0.76(6)
+4 0.55(4)
0.69(6)
+5 0.49(4)
0.71(8)
+6 0.44(4)
0.58(6)
Mo
+3 0.83(6)
+4 0.79(6)
+5 0.60(4)
0.75(6)
+6 0.55(4)
0.73(6)
W
+4 0.80(6)
+5 0.76(6)
+6 0.56(4)
0.74(6)
Mn
+2 0.81(6L)
0.96(6H)
1.07(8)
+3 0.72(6L)
0.79(6H)
+4 0.67(6)
+7 0.39(4)
0.60(6)
Tc
+4 0.785(6)
+5 0.74(6)
+7 0.51(4)
0.70(6)
Re
+4 0.77(6)
+5 0.72(6)
+6 0.69(6)
+7 0.52(4)
0.67(6)
Fe Co
+2 0.77(4H) +2 0.72(4H)
0.75(6L) 0.79(6L)
0.92(6H) 0.88(6H)
+3 0.63(4H) +3 0.67(6L)
0.69(6L) 0.75(6H)
0.785(6H)
Ru
+3 0.82(6)
+4 0.76(6)
+5 0.705(6)
+7 0.52(4)
+8 0.52(8)
Os
+4 0.77(6)
+5 0.715(6)
+6 0.685(6)
+7 0.665(6)
+8 0.53(4)
Rh
+3 0.805(6)
+4 0.74(6)
+5 0.69(6)
Ir
+3 0.82(6)
+4 0.765(6)
+5 0.71(6)
Ni
+2 0.69(4)
0.83(6)
+3 0.70(6L)
0.74(6H)
+4 0.62(6L)
Pd
+2 0.78(4SQ)
1.00(6)
+3 0.90(6)
+4 0.755(6)
Pt
+2 0.74(4SQ)
0.94(6)
+4 0.765(6)
+5 0.71(6)
Cu Zn
+1 0.60(2) +2 0.74(4)
1.74(4) 0.880(6)
1.91(6) 1.04(8)
+2 0.76(4SQ)
0.87(6)
+3 0.68(6L)
Ag
+1 0.81(2)
1.14(4)
1.16(4SQ)
1.29(6)
1.42(8)
+2 0.93(4SQ)
1.08(6)
+3 0.81(4SQ)
0.89(6)
Au
+1 1.51(6)
+3 0.82(4SQ)
0.99(6)
+5 0.71(6)
Cd
+2 0.92(4)
1.09(6)
1.24(8)
1.45(12)
Hg
+1 1.11(3)
1.33(6)
+2 0.83(2)
1.10(4)
1.16(6)
1.28(8)
B
+3 0.15(3)
0.24(4)
0.41(6)
Al
+3 0.53(4)
0.675(6)
Ga
+3 0.61(4)
0.76(6)
In
+3 0.76(4)
0.940(6)
1.06(8)
Tl
+1 1.64(6)
1.73(8)
1.84(12)
+3 0.89(4)
1.025(6)
1.12(8)
C
+4 0.06(3)
0.29(4)
0.30(6)
Si
+4 0.40(4)
0.540(6)
Ge
+2 0.87(6)
+4 0.53(4)
0.67(6)
Sn
+2 1.41(8)
+4 0.69(4)
0.83(6)
0.95(8)
N
-3 1.32(4)
+3 0.30(6)
+5 0.044(3)
0.27(6)
P
+3 0.58(6)
+5 0.31(4)
0.52(6)
As
+3 0.72(6)
+5 0.475(4)
0.60(6)
Pb Bi
+2 1.12(4PY) +3 1.17(6)
1.33(6) 1.31(8)
1.43(8) +5 0.90(6)
1.63(12)
+4 0.79(4)
0.915(6)
1.08(8)
O
-2 1.21(2)
1.22(3)
1.24(4)
1.26(6)
1.28(8)
S
-2 1.70(6)
+6 0.26(4)
0.43(6)
Se
-2 1.84(6)
+4 0.64(6)
+6 0.42(4)
0.56(6)
Sb Te
+3 0.90(4PY) -2 2.07(6)
0.90(6) 1.63(12)
+5 0.74(6) +4 0.80(4)
1.11(6)
+6 0.57(4)
0.70(6)
Po
+4 1.08(6)
1.22(8)
+6 0.81(6)
F
-1 1.145(2)
1.16(3)
1.17(4)
1.19(6)
+7 0.22(6)
Cl
-1 1.67(6)
+5 0.26(3PY)
+7 0.22(4)
0.41(6)
Br
-1 1.82(6)
+3 0.73(4SQ)
+5 0.45(3PY)
+7 0.39(4)
0.53(6)
I
-1 2.06(6)
+5 0.58(3PY)
1.09(6)
+7 0.56(4)
0.67(6)
At
+7 0.76(6)
1f 2f 3f 4f 5f 6f 7f 8f 9f 10f 11f 12f 13f 14f
La
+3 1.172(6)
1.30(8)
1.41(10)
1.50(12)
Ac
+3 1.26(6)
Ce
+3 1.15(6)
1.283(8)
1.39(10)
1.48(12)
+4 1.01(6)
1.11(8)
1.21(10)
1.28(12)
Th
+4 1.08(6)
1.19(8)
1.27(10)
1.35(12)
Pr
+3 1.13(6)
1.266(8)
+4 0.99(6)
1.10(8)
Pa
+3 1.18(6)
+4 1.04(6)
1.15(8)
+5 0.92(6)
1.05(8)
Nd
+2 1.43(8)
+3 1.123(6)
1.249(8)
1.41(12)
U
+3 1.165(6)
4+ 1.03(6)
1.14(8)
1.31(12)
+5 0.90(6)
+6 0.66(4)
0.87(6)
1.00(8)
Pm
+3 1.11(6)
1.233(8)
Np
+2 1.24(6)
+3 1.15(6)
+4 1.01(6)
1.12(8)
+5 0.89(6)
+6 0.86(6)
+7 0.85(6)
Sm
+2 1.41(8)
+3 1.098(6)
1.219(8)
1.38(12)
Pu
+3 1.14(6)
+4 1.00(6)
1.10(8)
+5 0.88(6)
+6 0.85(6)
Eu
+2 1.31(6)
1.39(8)
1.49(10)
+3 1.087(6)
1.206(8)
Am
+2 1.40(8)
+3 1.115(6)
1.23(8)
+4 0.99(6)
1.09(8)
Gd
+3 1.078(6)
1.193(8)
Cm
+3 1.11(6)
+4 0.99(6)
1.09(8)
Tb
+3 1.063(6)
1.180(8)
+4 0.90(6)
1.02(8)
Bk
+3 1.10(6)
+4 0.97(6)
1.07(8)
Dy
+2 1.21(6)
1.33(8)
+3 1.052(6)
1.167(8)
Cf
+3 1.09(6)
+4 0.961(6)
1.06(8)
Ho
+3 1.041(6)
1.155(8)
1.26(10)
Er
+3 1.03(6)
1.144(8)
Tm
+2 1.17(6)
+3 1.02(6)
1.134(8)
Yb
+2 1.16(6)
1.28(8)
+3 1.008(6)
1.125(8)
Es Fm Md No
+2 1.24(6)
NOTES: (4) = tetrahedral unless SQ; (6) = octahedral; (8) = sq. anti-prism; (>8) not defined; H = high spin; L = low spin; Source of data: R.D. Shannon Acta Cryst. (1976) A32, 751
Ne
Ar
Kr
Xe
+8 0.54(4)
0.62(6)
Rn

Chemistry 2810 Lecture Notes Dr. M. Gerken Page79
5.2.1. Common ionic crystal structures
The common ionic structures that we will consider in detail are depicted in the following picture. In each case, the smaller
circles represent the cations, while the larger circles represent the anions. This kind of open-lattice unit cell picture shows the
ions at 20% or less of their true radii. This gives the advantage that it is possible to see inside the cell and observe the relative
orientation of the ions and their chemical connectivity. It should be borne in mind that in reality in all ionic lattices the cations
and the anions ought to be in contact. The anions may also contact the other anions, but more normally they are somewhat
pulled apart. An anion-anion contact is electrostatically repulsive, while the cation-anions contacts are attractive. Some
alternate graphical representations are provided later on for some of these structures. Note that some of these will be
stereoviews. These can simulate a 3-dimensional look. To see the perspective in such structures, one relaxes the eyes and
allows then to cross, such that the stereo image forms mid-way between the two flat images. Alternatively, separate the two
views from the other eye by holding a piece of white cardboard between the images, extending from the surface of the paper to
the observers eyes.

Chemistry 2810 Lecture Notes Dr. M. Gerken Page80
a) NaCl lattice
AB stoichiometry Cation : anion ratio is 1:1
Each Na ion and Cl ion has six nearest neighbours, so that the
coordination number, CN = 6.
Na-Cl-Na angles are 180° and 90°.
Coordinates: Cl – (0, 0, 0) (½,½, 0) (½, 0, ½) (0, ½, ½)
Na + (0, ½, ½) (½, 0, ½) (½, ½, 0) (½, ½, ½)
Consider the "sliced" view at the right. Note the strong similarity to the
FCC metallic lattice. In fact, the Cl – ions are arranged just the same as the
metals in FCC. This leaves the larger O h holes for the Na + to occupy, which
leads to a single sodium ion at the center of the unit cell, and 12 in total
along the centres of each unit cell edge. These edge atoms are on ¼ within the volume of the cubic cell. The next image is
known as a stereoview, and gives 3-D insight into the structure. Note that in this graphic, the positions of the sodium and
chloride ions have been reversed, and this serves to emphasize the important fact that the NaCl structure is interchangeable
between cation and anion.
b) CsCl lattice
AB stoichiometry
Eight nearest neighbours. CN = 8.
Angles 180°, 70.5° & 109.5°
Coordinates: Cs + at (½, ½, ½)
Cl – at (0, 0, 0)
Here again there is a "sliced" view that allows us to accurately count the unit
cell contents. The eight chloride ions at the corners each contribute only 1/8 th of
their volume to the cell, while the central cesium ion is completely within the cell.
It is a common mistake to call this structure "body-centred cubic" - this is false; in
fact the lattice type is primitive cubic, with the central cesium ion occupying the
cubic hole There is one net CsCl formula per unit cell.

Chemistry 2810 Lecture Notes Dr. M. Gerken Page81
c) Zinc blende (cubic ZnS), Sphalerite
AB stoichiometry
CN = 4 for cation and for anion.
Angles 109.5° . Tetrahedra set on one edge.
S: (0, 0, 0) (½ , ½, 0) (½, 0, ½) (0, ½, ½)
Zn: (¼, ¼, ¼) (¼, ¼, ¾) (¾, ¼, ¾) ( ¾ , ¾, ¼)
This structure is closely related to the diamond structure, but the arrangement of the anions is FCC. Thus every second
tetrahedral hole in the lattice is occupied by the cations. There is a close geometrical affinity to the diamond structure, and in
fact it is like diamond with every second carbon atom replaced by a sulfur atom.
d) Wurtzite (hexagonal ZnS)
AB stoichiometry
CN = 4 for both ions
Angles 109.5°. Tetrahedra sitting on a face.
The local geometry is almost identical to that of zinc blende, but the overall symmetry of the lattice is hexagonal, not
cubic. The anion packing is hcc with the cations occupying half of the tetrahedral holes.
e) Fluorite (CaF 2 )
AB 2 stoichiometry
CN = 8 for cation; CN = 4 for anion.
Ca-F-Ca angles are 109.5° ; F-Ca-F angles are 180, 109.5 and 70.5°.
In this structure, Ca 2+ is actually larger than the F - anion. Therefore, we consider the packing of the cation being fcc with
anions occupying all tetrahedral holes.

Chemistry 2810 Lecture Notes Dr. M. Gerken Page82
f) Antifluorite (e.g. K 2 O) (same pictures as fluorite)
A 2 B stoichiometry
CN = 4 for cation; CN = 8 for anion.
Ca-F-Ca angles are 109.5°; F-Ca-F angles are 180°, 109.5° and 70.5°.
O-K-O angles are 109.5°; K-O-K angles are 180°, 109.5° and 70.5°.
g) Rutile (TiO 2 )
AB 2 stoichiometry.
CN = 6 for cation; CN = 3 for anion.
Ti-O-Ti angles are 120°; O-Ti-O angles are 180° and 90°.
In applying the close packed model, we imagine just the larger of the ions forming a close-packed lattice, in which the
anions exist as spheres just touching each other (for antilattices, the cations.) These anions are held in place, however, not by a
metallic bond, but by electrostatic attraction for cations, which occupy the T d and O h holes.
Now, having said this, considerable expansion of the "close-packed" structure may occur. I.e. the cations may be larger
than the basic size of the holes, and the anions may be in the same position as in a close-packed structure, but no longer
touching. Later we will consider what kind of size criteria cause alterations from the close-packed structures.
Not all ionic compounds can be understood by this model. Those which can include the following:
AB type:
- NaCl: ccp array of Cl - ; Na + in all the O h holes.
[Note it is a special property of the ccp array alone, that the O h holes describe a ccp array of their own. Thus NaCl
is often spoken of as two interpenetrating ccp arrays, one of Cl - , the other of Na + , displaced from each other by
a/2.]
- Zinc blende: ccp array of S 2- ; Zn 2+ in every second T d hole.
- Wurtzite: hcp array of S 2- ; Zn 2+ in every second T d hole.
- CsCl: is not a close packed array.
-
AB 2 type:
- Fluorite (CaF 2 ): ccp array of Ca 2+ (i.e. this is an anti-lattice), F - in all T d holes.

Chemistry 2810 Lecture Notes Dr. M. Gerken Page83
- Antifluorite: the reverse of CaF 2 , where a 2- anion is ccp; 1+ cations in T d holes
- Rutile: cannot be described in such a way (distorted close packing)
Remember that this is only one possible way of describing ionic solids. An alternate approach is to categorize each
separately by their main structural features, as is done below.
It is not easy to explain why certain ionic solids adopt certain structures! Whole books have been written on this subject
(see the reading list at the end of chapter 4). We will restrict ourselves to just one such rationalization, specifically that the
choice between various types of lattices is chiefly governed by relative ion sizes. Go back to our story of the sodium and
fluoride ions approaching each other. Presumably they will stop when they reach each other.
A cation and anion will approach till they gently nudge; closer than this will set up repulsion between the outer electrons in
each electron:
Attractive forces
+
−
Approaching
Attractive forces
e -
e -
+
−
e -
e -
e - e -
+
e - e - −
e - e -
Repulsive forces
Just touching
Interpenetrating
In the lattice, we want to maximize cations and anions just touching, for this is the greatest Coulombic attraction. But we
want to avoid anion-anion touching, since these are purely repulsive interactions. So the ideal packing of an ionic crystal is ion
pairs touching, but cations well separated.
This is where the idea of coordination number comes in, because what the most favourable interactions will be depends on
relative ion sizes. The radius ratio expresses the relative size of the cation and the anion. For each type of lattice, we can
calculate the ideal radius ratio for perfect packing using solid geometry. This is the minimum radius ratio that this type of
structure will tolerate; any thing less and the anions will remain touching, but the ion-pairs are no longer touching. When this
occurs, the lattice will switch to one with lower coordination number.
Consider the CsCl structure: (See the figures above, especially the stereodiagrams). We will now map out the CsCl
diagonal plane. That is a plane which stretches from one box-edge of the cube to the diagonally opposed box edge. This plane
will have sides with length of the box edge (two sides) and box face-diagonal (other two sides).
The radius of the anions is just r - = a/2.
The radius of the cation is determined as follows:
Cl - Cl -
x = a√ 3
2
+ −
a 3
r + r = dCs−Cl
=
2
Solving both equations gives r + /r _ = 0.73. (Note that for
fluorite and other "reversed" lattices, the radius ratio is r - /r + .)
Try such calculations yourself for NaCl and zinc blende.
Cl -
x
Cs +
•
Cl -
r - = a/2
r - + r + = a√ 3
2
r + = a√ 3 - a/2
2
= a ( √ 3 -1 )
2
r + /r - = 0.732
a
In practice, the range of radius ratio values which still lead to the same structure are given in the following table. The first
number in each range is the limiting ratio. Drop below this number, and the lattice should jump back to the structure with
lower coordination number. The upper number is the limit of the range in which lattice expansion, or loss of anion-anion
contact, occurs without altering the lattice type.

Chemistry 2810 Lecture Notes Dr. M. Gerken Page84
Stoichiometry r + /r _ Lattice type
1:1 C.N.
0.00-0.155 3 No examples
0.225-0.414 4 Wurtzite and zinc blende
0.414-0.732 6 NaCl
0.732-1.00 8 CsCl
1:2 C.N. of smaller ion
0.225-0.414 4 b-quartz (not dealt with previously)
0.414-0.732 6 Rutile
0.732-1.00 8 Fluorite (note r(Ca 2+ ) / r(F - )
Summing it up in words: when the cations are large, many anions can surround the cation; this leads to the CsCl structure.
For smaller cations, the NaCl lattice is observed, and for the largest anions, such as S 2- , the ZnS structures are favoured. Note
these are only rules of thumb.
SAMPLE PROBLEMS
Predict the coordination numbers of the following ionic solids:
MgF 2 use CN=6 radii r + /r - = .72/1.31 = 0.55
Predict rutile structure; but if use CN=4 radius of Mg 2+ , get 0.37; this suggests the b-quartz
structure; in fact is Rutile
KBr use CN=6 radii r + /r - = 1.52/1.82 = 0.835
Predict CsCl; in fact is NaCl.
LiCl use CN=6 radii r + /r - = 0.90/1.67 = 0.455
Predict NaCl, correctly
In fact, the radius ratio rule is least reliable for simple ionic halides and oxides, and most reliable for complex mixed-metal
fluorides and for salts of the oxoanions like perchlorate. Radii for the latter are given in Table 4.5, p. 129 of the text.