Chemistry 2810 Lecture Notes
5.2
Dr. R. T. Boeré
Page83
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 (1s22s1) à Li+ (1s2)
F 1s22s22p5 à F– (1s22s22p6)
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+
Graphical Comparison of CN6 Radii of Cations and Anions
Li+
N 3-
O2-
F-
S2-
Cl -
Se2-
Br -
Te2-
I-
Al 3+
Na +
Mg 2+
Fe 2+
Sc 3+ Ti4+ V5+ Cr3+
Fe 3+
Co 2+ Ni2+ Cu +
Zn2+
Ga 3+
K+
Ca 2+
Rb +
Sr2+
Ag+
Cd2+
In3+
Cs +
Ba 2+
Au2+
Hg 2+
Tl3+
Mn2+
Sn 4+ Sb5+
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.
83
Chemistry 2810 Lecture Notes
1
Dr. R. T. Boeré
18
H
+1 -0.24(1)
-0.04(2)
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)
Page84
He
Chemistry 2810 Ionic Radii (Å)
2
13
χ ≥ 1.50
Be
+2 0.31(3)
0.41(4)
0.59(6)
B
+3 0.15(3)
0.24(4)
0.41(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)
Al
3
Sc
+3 0.885(6)
1.010(8)
Y
+3 1.040(6)
1.159(8)
Lu
+3 1.001(6)
1.117(8)
4
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)
5
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)
6
7
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)
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)
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)
0.77(6)
0.72(6)
0.69(6)
0.52(4)
0.67(6)
Fe
Co
Ru
+3
+4
+5
+7
+8
0.82(6)
0.76(6)
0.705(6)
0.52(4)
0.52(8)
+4
+5
+6
+7
+8
0.77(6)
0.715(6)
0.685(6)
0.665(6)
0.53(4)
Re
+4
+5
+6
+7
9
+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)
Tc
+4 0.785(6)
+5 0.74(6)
+7 0.51(4)
0.70(6)
8
Os
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)
10
Ni
+2 0.69(4)
0.83(6)
+3 0.70(6L)
0.74(6H)
+4 0.62(6L)
Pd
11
12
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
Au
Ga
+3 0.61(4)
0.76(6)
Cd
+2 0.78(4SQ) +1 0.81(2)
+2 0.92(4)
1.00(6)
1.14(4)
1.09(6)
+3 0.90(6)
1.16(4SQ)
1.24(8)
+4 0.755(6)
1.29(6)
1.45(12)
1.42(8)
+2 0.93(4SQ)
1.08(6)
+3 0.81(4SQ)
0.89(6)
Pt
+3 0.53(4)
0.675(6)
In
+3 0.76(4)
0.940(6)
1.06(8)
Hg
+2 0.74(4SQ) +1 1.51(6)
+1 1.11(3)
0.94(6)
+3 0.82(4SQ)
1.33(6)
+4 0.765(6)
0.99(6)
+2 0.83(2)
+5 0.71(6)
+5 0.71(6)
1.10(4)
1.16(6)
1.28(8)
Tl
+1 1.64(6)
1.73(8)
1.84(12)
+3 0.89(4)
1.025(6)
1.12(8)
14
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)
Pb
15
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)
Sb
16
O
-2 1.21(2)
1.22(3)
1.24(4)
1.26(6)
1.28(8)
Bi
F
Cl
Te
Po
+4 1.08(6)
1.22(8)
+6 0.81(6)
Ar
-1 1.67(6)
+5 0.26(3PY)
+7 0.22(4)
0.41(6)
Se
-2 1.84(6)
+4 0.64(6)
+6 0.42(4)
0.56(6)
Ne
-1 1.145(2)
1.16(3)
1.17(4)
1.19(6)
+7 0.22(6)
S
-2 1.70(6)
+6 0.26(4)
0.43(6)
+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)
+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)
17
Br
-1
+3
+5
+7
Kr
1.82(6)
0.73(4SQ)
0.45(3PY)
0.39(4)
0.53(6)
I
Xe
-1 2.06(6)
+8 0.54(4)
+5 0.58(3PY)
0.62(6)
1.09(6)
+7 0.56(4)
0.67(6)
At
Rn
+7 0.76(6)
Lr
1f
2f
3f
4f
5f
6f
7f
8f
9f
10f
11f
12f
13f
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
+3 1.172(6)
1.30(8)
1.41(10)
1.50(12)
χ ≤ 1.50
Ac
+3 1.26(6)
+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)
+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)
+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)
+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)
+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)
+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)
+3 1.078(6)
1.193(8)
Cm
+3 1.11(6)
+4 0.99(6)
1.09(8)
+3 1.063(6)
1.180(8)
+4 0.90(6)
1.02(8)
+2 1.21(6)
1.33(8)
+3 1.052(6)
1.167(8)
Bk
+3 1.10(6)
+4 0.97(6)
1.07(8)
Cf
+3 1.09(6)
+4 0.961(6)
1.06(8)
+3 1.041(6)
1.155(8)
1.26(10)
Es
+3 1.03(6)
1.144(8)
Fm
+2 1.17(6)
+3 1.02(6)
1.134(8)
Md
14f
Yb
+2 1.16(6)
1.28(8)
+3 1.008(6)
1.125(8)
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
84
Chemistry 2810 Lecture Notes
Dr. R. T. Boeré
Page86
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.
86
Chemistry 2810 Lecture Notes
a)
Dr. R. T. Boeré
Page87
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 Oh 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 1/8th 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 cental cesium ion being
intimately related to the anion at the origin. The two make a pair. There is one net CsCl formula per unit cell.
c)
Zinc blende (cubic ZnS)
AB stoichiometry
CN = 4 for cation and for anion.
87
Chemistry 2810 Lecture Notes
Dr. R. T. Boeré
Page88
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. 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.
Coordinates:
Zn2+ : (¼, ¼, ¼)
(¼, ¾, ¾)
(¾, ¼, ¾)
(0, 0, 0)
Face atoms:
(½ , ½, 0)
(½, 0, ½)
(0, ½, ½)
Interior atoms
The local geometry is almost identical to that of zinc blende, but the overall symmetry of the lattice is hexagonal, not
cubic.
e)
Fluorite (CaF 2)
AB2 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°.
88
Chemistry 2810 Lecture Notes
f)
Antifluorite (e.g. K 2O)
Dr. R. T. Boeré
Page89
(same pictures as fluorite)
A2B 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 (TiO2)
AB2 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 Td and Oh 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:
- NaCl: ccp array of Cl-; Na+ in all the Oh holes.
[Note it is a special property of the ccp array alone, that the Oh 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 S2-; Zn2+ in every second Td
-
Wurtzite: hcp array of S2-; Zn2+ in every second Td hole.
-
Fluorite (CaF2): ccp array of Ca2+ (i.e. this is an anti-lattice), F- in all Td holes.
-
Antifluorite: the reverse of CaF2, where a 2- anion is ccp; 1+ cations in Td holes
-
Rutile: cannot be so described (distorted close packing)
hole.
89
Chemistry 2810 Lecture Notes
-
Dr. R. T. Boeré
Page90
CsCl: is not a close packed array.
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 speculation, 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
Just touching
-
-
e
e
-
e
+
−
-
e
-
e
Interpenetrating
-
e
Repulsive forces
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.
Cl
-
Cl
-
x = a√ 3
2
The radius of the cation is determined as follows:
+
−
r + r = d Cs− Cl
a 3
=
2
Cs
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.
-
x
r = a/2
+
-
+
r + r = a√ 3
2
•
+
Cl
-
Cl
-
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 anionanion contact, occurs without altering the lattice type.
(see next page)
90
Chemistry 2810 Lecture Notes
Dr. R. T. Boeré
Stoichiometry
1:1
r+/r_
0.00-0.155
0.225-0.414
0.414-0.732
0.732-1.00
1:2
0.225-0.414
0.414-0.732
0.732-1.00
Page91
Lattice type
C.N.
3
4
6
8
C.N. of smaller ion
4
6
8
No examples
Wurtzite and zinc blende
NaCl
CsCl
b-quartz (not dealt with previously)
Rutile
Fluorite (note r(Ca2+) / r(F-)
Summing it up in words: where 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 S2-, the ZnS structures are
favoured. Note these are only rules of thumb.
SAMPLE PROBLEMS
Predict the coordination numbers of the following ionic solids:
MgF2
use CN=6 radii r+/r- = .72/1.31 = 0.55
Predict rutile structure; but if use CN=4 radius of Mg2+, 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.
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 mixedmetal fluorides and for salts of the oxoanions like perchlorate. Radii for the latter are given in Table 4.5, p. 129 of the text.
LiCl
91