Crystal Binding (Bonding) Part IVQuantitative Models of Ionic Bonding

Ionic Crystals• As we’ve already said, these consist of atoms with

Large Electronegativity DifferencesMost naturally occurring minerals are ionic crystals.Further, many of these minerals are oxides.

• As we’ve already said, a first approximation is that 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.

Ionic Bonding

•This bonding occurs between atoms with Large Electronegativity Differences. This means that the 2 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: NaClNa (1s22s22p63s1) Na+ (1s22s22p6) + e-

Cl (1s22s22p63s23p5) + e- Cl- (1s22s22p63s23p6)

Ionic Bonding

Example: NaCll

Na (1s22s22p63s1) Na+ (1s22s22p6) + e-

Cl (1s22s22p63s23p5)+e- Cl (1s22s22p63s23p6)

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.

Physical Properties of Ionic Materials

Melting Points of Ionic Materials Decrease with Increasing Interionic Distance

+1 cations

+2 cations

Sodium with various anions

12

20

38

56

3

11

19

37

9

17

35

53

4

12

20

38 56

22

21

12

11

Mechanical Hardness of Ionic MaterialsDecreases with Increasing Interionic Distance

Atomic & Ionic RadiiIn 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.

In Quantum Mech, “Atomic Radius” & “Ionic Radius” Aren’t Well-Defined!

• Sometimes, 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, one 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. X-ray data gives d100 = a = 3.61 Å.

rI = (½)d(d = bond length or nearest neighbor distance)

• 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, near neighbor distance = (½)(2a2)½

a

a

Cu+ Ionic Radius = (¼)(2a2)½ = 1.28 Å.

Note: The Cu+ ionic radius is different than the radius of the Cu++ ion. It is also different in the covalently bonded material CuO2.

•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. 5. .....

Variation of the Atomic Radius withPosition in the Periodic Table

1. It increases from top to bottom down a column.(This is what I called a “Chemical Trend” earlier)

• Why? Going down a column, the energies of valence electrons increase, 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.

2. 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

SummaryShielding is constantAtomic Radius decreasesIonization energy increasesElectronegativity increasesNuclear charge increases

Nuc

lear

cha

rge

incr

ease

sSh

ield

ing

incr

ease

sA

tom

ic ra

dius

incr

ease

sIo

nic

size

incr

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sIo

niza

tion

ener

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decr

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sEl

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dec

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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 one or more 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, this 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 (like ions) handled?• In the early days of solid state physics, this was

a difficult problem. • Why? Because the Coulomb potential is long-

ranged! That is, in principle, the Coulomb potential due to a given (+ or -) ion extends all the way out to r

To do calculations, people often treat the Coulomb potential as if it were a short-

ranged potential:

27

Approximation:The Coulomb potential is short-ranged:

• In practice, 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!

Model 1-D Madelung Sum:

Madelung const. R

()Rpij

ji (rij Rpij )

•The value of α is defined in terms of the lattice constant

R. Start on a negative ion, summing (left & right)

R- + -– + -– + -– +… …

•This sum is conditionally convergent! Thismeans the order of the terms in the sum matters!

•Consider a Model Ionic Lattice in 1Dimension, as in the figure.

–

Madelung const. R

()Rpij

ji (rij Rpij )

The value of α is defined in terms of the lattice constant R. Start on a negative ion, summing (left & right)

()pij

ji 2 11

12

13

14

...

R- + -– + -– + -– +… …

Since ln(1 x) x x2

2

x3

3

x4

4 ...

2 ln 2

Model Ionic Lattice in 1 Dimension–

This sum is conditionally convergent! This means the order of the terms in the sum matters!

• Since ionic crystals involve more than 2 ions, the attractive & repulsive forces betweenneighboring 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.

31

3-D Madelung Sums•In 3D this series presents greater difficulty.

The series will not converge unless successiveterms 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

LatticeNaCl α = 1.747565 CsCl α = 1.762675

ZnS α = 1.6381

Long-Ranged Potentials• Why not make the potential long ranged? To answer this, consider

a cubic lattice with only +1 charges, & its Coulomb potential.

V (ri ) 1

| ri L |L0 dr 4r2

r0

• 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 12 ij

ZiZ j

| ri rj L |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.

What is a Long-Ranged Potential?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)

nrU

oo

11MZZ 1389 -

Crystal Structure Madelung ConstantNaCl 1.748CsCl 1.763Zinc blende 1.638Wurtzite 1.641Fluorite 2.519Rutile 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.)

What decides the shape an ionic lattice takes?• 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 ConstantsMadelung Constants CrystalCrystal Madelung Madelung StructureStructure ConstantConstantCesium ChlorideCesium Chloride 1.7631.763FluoriteFluorite 2.5192.519Rock Salt (NaCl)Rock Salt (NaCl) 1.7481.748SphaleriteSphalerite 1.6381.638WurtziteWurtzite 1.6411.641