Crystal Binding (Bonding)
Continued: Quantitative Models of Ionic Bonding
Ionic Crystals
• As we’ve already said, these consist of atoms with
Large Electronegativity Differences.
Most naturally occurring minerals are ionic crystals.
Further, many of these minerals are oxides.
• As a first approximation, these oxides can be thought of as an
array of oxygen atoms in a close packed arrangement, with
metallic ions fitting into interstitial sites between the oxygens.
• Most of these crystals are not very useful to physicists.
• But, some Geoscience friends are experts on these kinds of crystals.
In ionic crystals, the ions are
in close packed arrangements
to maximize the attractions &
to minimize the repulsions
between the ions.
Ionic Bonding
• This bonding occurs between atoms with Large
Electronegativity Differences. This means that the two atoms
are usually far removed from each other in the periodic table.
• This also means that they can easily exchange electrons & stabilize
their outer electron shells (which become more inert gas-like).
• So, electronically neutral bonds between cations (positive ions) &
anions (negative ions) are created. Example: NaCl
Na (1s22s22p63s1)
Cl (1s22s22p63s23p5) + e-
Na+ (1s22s22p6) + eCl- (1s22s22p63s23p6)
Physical Properties of Ionic Materials
In general, Ionic Materials:
• Are harder than metals or molecular solids, but
softer than covalently bonded materials.
• Have greater densities than metals or molecular solids,
but are less dense than covalently bonded solids.
• Have low electrical & thermal conductivities in
comparison with most other materials.
• These & other physical properties are caused by
the ionic bond strength, which is related to
1. The spacing between the ions.
2. The ionic charge.
Melting Points of Ionic Materials
Decrease with Increasing Interionic Distance
9
17
35
53
Sodium with various anions
12
+2 cations
20
38
56
+1 cations
3
11
19
37
Mechanical Hardness of Ionic Materials
4
Decreases with Increasing
Interionic Distance
12
20
38 56
22
21
12
11
Atomic & Ionic Radii
In Quantum Mechanics, the concepts
“Atomic Radius” & “Ionic Radius”
ARE NOT WELL-DEFINED!
• Electrons are waves & are spread out over the atomic volume
with no rigid “boundary” that is the “atomic radius”.
• However, sometimes in crystalline solids, these
concepts can be useful to obtain a qualitative
(sometimes even close to quantitative!) understanding
of interatomic (inter-ionic) distances (bond
lengths), as well as certain other material properties.
• How these radii are defined is certainly not rigorous &
varies from one bond type to another, sometimes from one
material to another & sometimes from one application to another.
• The Atomic Radius of a neutral atom can be crudely
thought of as the Mean Quantum Mechanical Radius
<r> of the orbital of the outer valence electron for that atom.
• Similarly, the Ionic Radius of an ion can be crudely
thought of as the Mean Quantum Mechanical Radius
<r> of the orbital of the outer valence electron for that ion.
• In quantum mechanics, <r> could be calculated using the
wavefunction (solution to Schrödinger’s Equation) of
the relevant valence electron.
However, this quantum mechanical
calculation is almost never done. Instead,
these radii are most often treated as parameters
which are obtained from crystal structure data.
• For example, a means to find an Ionic Radius for a metallic
ion is to obtain it from data on the crystalline solid for that
metal. In this case, the Ionic Radius would be half the ion-ion
bond length:
rI = (½)d (d = bond length or nearest neighbor distance)
• As an example, take the copper ion Cu+. The lattice structure of
Cu metal is FCC. See figure. X-ray data gives d100 = a = 3.61
Å. From the figure, nearest neighbor distance = (½)(2a2)½
 Cu+ Ionic Radius
= (¼)(2a2)½ = 1.28 Å.
a
a
Note:
The Cu+ ionic radius is different
than the radius of the Cu++ ion. It is
also different in the covalently
bonded material CuO2, etc., etc.
• The Atomic (or Ionic) Radius of a given
atom (or ion) can be different, depending on
the material of interest.
• For example, for atom X, this radius depends on:
1. The crystal structure of the material that X is in.
2. The coordination number (# nearestneighbors) for X in that material.
3. The bond type.
4. The % of ionic or covalent character of that
bond.
5. .....
Variation of Atomic Radii with
Position in the Periodic Table
It increases from top to bottom down a column.
• Why? Going down a column, the energies of valence
electrons increases, so their binding energy with the
nucleus gets weaker moving down the column. They are
not bound as tightly to the nucleus as the electrons in the
filled shells because they are screened or shielded (
pushed away) by other electrons in inner levels.
It decreases from left to right in a row.
•Why? The number of protons in the nucleus
increases to the right. This pulls electrons closer to
the nucleus going from left to right.
The Periodic Table & Atomic Radii
Atomic Radius vs. Atomic Number
Atomic Radius (pm)
250
K
200
Na
Li
150
Mg
Al Si
Be
100
Ca
P S Cl
B C N
O F
Ar
Ne
50
H
0
0
He
2
4
6
8
10
12
Element
14
16
18
20
Nuclear charge increases
Shielding increases
Atomic radius increases
Ionic size increases
Ionization energy decreases
Electronegativity decreases
Summary
Shielding is constant
Atomic Radius decreases
Ionization energy increases
Electronegativity increases
Nuclear charge increases
Atomic Radius
• The overall trend in atomic radius looks like this.
Ionic Radii
• Metallic Elements easily lose electrons.
• Non-Metals more readily gain electrons.
How does losing or gaining
an electron effect the radius
of the atom (ion)?
Positive Ions
• Positive ions are always smaller than
the neutral atom, due to their loss of
outer shell electrons.
Negative Ions
• Negative ions are always larger than
the neutral atom due to the fact that
they’ve gained electrons.
Ion size Trends in Rows.
• Going from left to right there is a
decrease in size of positive ions.
• Starting with group 5, there is sharp
increase followed by a decrease in
the size of the anion as you move
from left to right.
Ion size Trends in Columns
•Ion size increases going
down a column for both
positive and negative ions
Covalent Radii
• Obviously, applies to atoms in a covalently
bonded material.
In a pure elemental solid, the
Covalent Radius is simply half of
the bond length.
• Once that is found for a given atom, then
the covalent radius of that atom is assumed
to be the same in any material in which that
atom participates in a covalent bond.
Charged Systems
• How are charged systems handled? The Coulomb
potential is long-ranged!
To do calculations, people often treat the
Coulomb potential as if it were a shortranged potential:
• Cutoff the potential at r > (½)L. L is empirically
determined. Problem:
– The effect of the discontinuity never disappears:
(1/r) (r2) gets bigger as r gets bigger!
• An Image Potential solves this:
VI = Σn v(ri-rj+nL)
– But this summation diverges!
22
Model 1-D Madelung Sum:
• Consider a Model Ionic Lattice in 1
Dimension, as in the figure:
…
-–
R
+
-–
–-
+

+
()
Madelung const.

j i
R
Rpij
–-
+
…
(rij  Rpij )
• The value of α is defined in terms of the lattice constant
R. Start on a negative ion, summing (left & right)
• This sum is conditionally convergent! This
means the order of the terms in the sum matters!
Model Ionic Lattice in 1 Dimension
…
–-
+
R
-–
–-
+

–-
+
()
Madelung const.

j i
R
Rpij
+
…
(rij  Rpij )
The value of α is defined in terms of the lattice constant R. Start
on a negative ion, summing (left & right)
This sum is conditionally convergent! This means the order of the
terms in the sum matters!
1 1 1 1

()
   j i
 2      ...
1 2 3 4

pij
Since
2
3
4


x
x
x
ln(1 x)   x     ...
2
3 4


  2 ln 2
• Since ionic crystals involve more than 2 ions,
the attractive & repulsive forces between
neighboring ions, next nearest neighbors, etc.,
must be considered.
• The Madelung Constant is derived for each
type of ionic crystal structure. It is the sum of
a series of numbers representing the number of
nearest neighbors & their relative distance from
a given ion.
• The Madelung Constant is specific to the
crystal type (unit cell), but independent of
interionic distances or ionic charges.
3-D Madelung Sums
• In 3D this series presents greater difficulty.
The series will not converge unless successive
terms in the series are arranged so that + and
terms nearly cancel.
Powerful methods were developed by Ewald (Ann. Physik 64, 253 (1921)
Evjen (Phys. Rev. 39, 675 (1932) and Frank (Phil. Mag. 41, 1287 (1950) .
Results for some Lattices:
Velectrostatic ~ α/R
Lattice
NaCl
CsCl
ZnS
α = 1.747565
α = 1.762675
α = 1.6381
26
Long-Ranged Potentials
• Why not make the potential long ranged? To answer this, consider
a cubic lattice with only +1 charges, & its Coulomb potential.

1
2 
V (ri )  
  dr 4 r
r
L  0 | ri  L |
0
• The approximate integral diverges!
Correct! A charged system with infinite
charge has infinite potential.
• Consider instead a cubic lattice with
charge neutrality, i.e. with ±1 charges.
Vcell
1
 
2 i j
Zi Z j
| r  r
L
i
j
L|
• Again, we need a convergent lattice sum.
– Energy is finite in a charge neutral cell
What is a Long-Ranged Potential?
• A potential is long-ranged if the real-space lattice
sum does not (naively) converge.
– In 3d, a potential is
long-ranged if it converges at rate < r–3.
– In 2d, a potential is
long-ranged if converges at rate < r–2.
– In practice, we often use techniques for potentials that are
not strictly long-ranged.
• MOTIVATION for this bothersome math: Most
interesting systems contain charge:
– Any atomic system at the level of electrons and nuclei.
– Any system with charged defects (e.g., Frenkel defects)
– Any system with dissolved ions (e.g. biological cases)
– Any system with partial charges (e.g. chemical systems)
In general, Total Lattice Energy =
Ion-pair energy 
Madelung constant M (α)

Z Z M 1
U o  1389
1  
ro  n 
-
Crystal Structure
NaCl
CsCl
Zinc blende
Wurtzite
Fluorite
Rutile
Madelung Constant
1.748
1.763
1.638
1.641
2.519
2.408
Origin of the Madelung Constant
The Madelung Series does not converge:
However, a concentric cube calculation
does converge:
What decides the shape an ionic lattice takes?
• Ionic Solids can be thought of as anion structures
with cations filling cavities (holes) between the anions.
• Bonding is strongest with the most cation-anion
interactions that do not crowd the anions into each other
(which is the same as leaving gaps between anions and the cation.)
At the “stability limit” the cation is touching all the
anions and the anions are just touching at their edges.
Beyond this stability limit the compound will be stable.
Ionic charge has
a huge effect on
the lattice energy.
Madelung Constants
Crystal
Structure
Cesium Chloride
Fluorite
Rock Salt (NaCl)
Sphalerite
Wurtzite
Madelung
Constant
1.763
2.519
1.748
1.638
1.641