1. Crystal Binding (Bonding) Continued: More on Ionic Bonding

2. Ionic Crystals Large Electronegativity Differences.
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.

3. Ionic Bonding Na (1s22s22p63s1) Na+ (1s22s22p6) + e-
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) Na+ (1s22s22p6) + e-
Cl (1s22s22p63s23p5) + e Cl- (1s22s22p63s23p6)

4. 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.

5. Melting Points of Ionic Materials Decrease with Increasing Interionic Distance9 17 35 53
Sodium with various anions 12 20 38 56
+2 cations
+1 cations 3 11 19 37

6. Mechanical Hardness of Ionic Materials
Decreases with Increasing Interionic Distance 4 12 20
38 56 22 21 12 11

7. 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.

8. 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.

9. rI = (½)d (d = bond length or nearest neighbor distance)
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 a specific 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 a
 Cu+ Ionic Radius
= (¼)(2a2)½ = 1.28 Å. 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.

10. For example, for atom X, this radius depends on:
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 (# nearest-neighbors) for X in that material.
3. The bond type.
4. The % of ionic or covalent character of that bond.

11. 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.

12. The Periodic Table & Atomic Radii

13. Atomic Radius vs. Atomic Number

14. Summary Ionization energy decreases Electronegativity decreases
Nuclear charge increases
Atomic radius increases
Shielding increases
Ionic size increases
Shielding is constant
Atomic Radius decreases
Ionization energy increases
Electronegativity increases
Nuclear charge increases

15. Atomic Radius
The overall trend in atomic radius looks like this.

16. 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)?

17. Positive Ions
Positive ions are always smaller than the neutral atom, due to their loss of outer shell electrons.

18. Negative Ions
Negative ions are always larger than the neutral atom due to the fact that they’ve gained electrons.

19. 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.

20. Ion size Trends in Columns
Ion size increases going down a column for both positive and negative ions

21. 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.

22. Charged Systems VI = Σn v(ri-rj+nL)
How can charged systems be handled? The Coulomb potential is long-ranged!
To do calculations, people often treat the Coulomb potential as if it were a short-ranged potential:
Cutoff the potential at r > (½)L. 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 22

23. 1-D Madelung Sum:
Consider a Model Ionic Lattice in 1 Dimension, as in the figure: R - + – … –
The value of α depends on whether it is defined in terms of the lattice constant R. Start on a negative ion, summing (left and right)
This sum is conditionally convergent! This means the order of the terms in the sum matters! 23

24. Model Ionic Lattice in 1 DimensionR - + – … –
The value of α depends on whether it 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! 24

25. Results for some Lattices:
3-D Madelung Sum
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 25 25

26. Long-Ranged Potentials
Why make the potential long ranged?
Consider a cubic lattice with +1 charges, and its Coulomb potential.
Approximate integral diverges!
Correct! Non-neutral system with infinite charge has infinite potential.
Consider a cubic lattice with charge neutrality, i.e. with ±1 charges.
Again need convergent lattice sum.
Energy is finite in charge neutral cell 26 26

27. 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-ranges if converges at rate < r–3.
In 2D, a potential is long-ranges if converges at rate < r–2.
In practice, we often use techniques for potentials that are not strictly long-ranged.
MOTIVATION for 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) 27 27

28. Total Lattice Energy = Ion-pair energy x Madelung constant M (or α)
Crystal Structure Madelung constant
NaCl
CsCl
Zinc blende
Wurtzite
Fluorite
Rutile 28

29. Origin of the Madelung Constant29

30. A Concentric Cube Calculation Does Converge:
The Madelung Series
does not converge:
A Concentric Cube Calculation Does Converge: 30

31. What decides which shape an ionic lattice takes?
Ionic Solids are 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. 31

32. Ionic charge has a huge effect on lattice energy.

33. Lattice Energy
Attempts to predict lattice energies are generally based on Coulomb’s law:
VAB = (Zae)(Zbe)
4πεorAB
Za and Zb = charge on cation and anion
e = charge of an electron (1.602 x 10-19C)
4πεo= permittivity of vacuum ( x 10-10J-1C2m-1)
rAB = distance between nuclei

34. Since ionic crystals involve more than 2 ions, the attractive and 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 and their relative distance from a given ion.
The constant is specific to the crystal type (unit cell), but independent of interionic distances or ionic charges.

35. Madelung Constants Structure Constant Crystal Madelung Fluorite 2.519
Cesium chloride
Fluorite
Rock salt (NaCl)
Sphalerite
Wurtzite

36. Estimating The Lattice Energy
Ec = NM(Z+)( Z-) e2
4πεor
where N is Avogadro’s number, and M is the
Madelung constant (sometimes represented by A)
This estimate is based on coulombic forces, and
assumes 100% ionic bonding. A further
modification, the Born-Mayer equation corrects for complex repulsion within the crystal.
Ec = NM(Z+)( Z-) e2 (1-ρ/r)
for simple compounds, ρ = 30pm

37. If we consider electrostatic and Born forces, we arrive at the Born-Mayer equation (evaluated at equilibrium internuclear separation, do)
r corrects for repulsions at short distances. Typically, a value of 30 pm is used for r.
This equation
will enable us
to predict lattice
energies (called
the calculated
lattice energy
Correction for
Born forces