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Crystals. Crystal Structures. Atoms (and later ions) will be viewed as hard spheres. In the case of pure metals, the packing pattern often provides the greatest spatial efficiency (closest packing).

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crystal structures
Crystal Structures

Atoms (and later ions) will be viewed as hard spheres. In the case of pure metals, the packing pattern often provides the greatest spatial efficiency (closest packing).

Ionic crystals can often be viewed as a close-packed arrangement of the larger ion, with the smaller ion placed in the “holes” of the structure.

unit cells
Unit Cells

Crystals consist of repeating asymmetric units which may be atoms, ions or molecules. The space lattice is the pattern formed by the points that represent these repeating structural units.

unit cells4
Unit Cells

A unit cell of the crystal is an imaginary parallel-sided region from which the entire crystal can be built up.

Usually the smallest unit cell which exhibits the greatest symmetry is chosen. If repeated (translated) in 3 dimensions, the entire crystal is recreated.

close packing
Close Packing

Since metal atoms and ions lack directional bonding, they will often pack with greatest efficiency. In close or closest packing, each metal atom has 12 nearest neighbors.

The number of nearest neighbors is called the coordination number. Six atoms surround an atom in the same plane, and the central atom is then “capped” by 3 atoms on top, and 3 atoms below it.

close packing6
Close Packing

If the bottom “cap” and the top “cap” are directly above each other, in an ABA pattern, the arrangement has a hexagonal unit cell, or is said to be hexagonal close packed.

If the bottom and top “caps” are staggered, the unit cell that results is a face-centered cube. This arrangement is called cubic close packing.

close packing8
Close Packing

Either arrangement utilizes 74% of the available space, producing a dense arrangement of atoms. Small holes make up the other 26% of the unit cell.

holes in close packed crystals
Holes in Close Packed Crystals

There are two types of holes created by a close-packed arrangement. Octahedral holes lie within two staggered triangular planes of atoms.

holes in close packed crystals10
Holes in Close Packed Crystals

The coordination number of an atom occupying an octahedral hole is 6.

For n atoms in a close-packed structure, there are n octahedral holes.

octahedral holes
Octahedral Holes

The green atoms are in a cubic close-packed arrangement. The small orange spheres show the position of octahedral holes in the unit cell. Each hole has a coordination number of 6.

octahedral holes12
Octahedral Holes

The size of the octahedral hole = .414 r

where r is the radius of the cubic close-packed atom or ion.

holes in close packed crystals13
Holes in Close Packed Crystals

Tetrahedral holes are formed by a planar triangle of atoms, with a 4th atom covering the indentation in the center. The resulting hole has a coordination number of 4.

tetrahedral holes
Tetrahedral Holes

The orange spheres show atoms in a cubic close-packed arrangement. The small white spheres behind each corner indicate the location of the tetrahedral holes.

tetrahedral holes15
Tetrahedral Holes

For a close-packed crystal of n atoms, there are 2n tetrahedral holes.

The size of the tetrahedral holes = .225 r

where r is the radius of the close-packed atom or ion.

of atoms unit cell
# of Atoms/Unit Cell

For atoms in a cubic unit cell:

  • Atoms in corners are ⅛ within the cell
of atoms unit cell17
# of Atoms/Unit Cell

For atoms in a cubic unit cell:

  • Atoms on faces are ½ within the cell
of atoms unit cell18
# of Atoms/Unit Cell

A face-centered cubic unit cell contains a total of 4 atoms: 1 from the corners, and 3 from the faces.

of atoms unit cell19
# of Atoms/Unit Cell

For atoms in a cubic unit cell:

  • Atoms in corners are ⅛ within the cell
  • Atoms on faces are ½ within the cell
  • Atoms on edges are ¼ within the cell
other metallic crystal structures
Other Metallic Crystal Structures

Body-centered cubic unit cells have an atom in the center of the cube as well as one in each corner. The packing efficiency is 68%, and the coordination number = 8.

other metallic crystal structures21
Other Metallic Crystal Structures

Simple cubic (or primitive cubic) unit cells are relatively rare. The atoms occupy the corners of a cube. The coordination number is 6, and the packing efficiency is only 52.4%.


Many metals exhibit different crystal structures with changes in pressure and temperature. Typically, denser forms occur at higher pressures.

Higher temperatures often cause close-packed structures to become body-center cubic structures due to atomic vibrations.

atomic radii of metals
Atomic Radii of Metals

Metallic radii are defined as half the internuclear distance as determined by X-ray crystallography. However, this distance varies with coordination number of the atom; increasing with increasing coordination number.

atomic radii of metals24
Atomic Radii of Metals

Goldschmidt radii correct all metallic radii for a coordination number of 12.

Coord #Relative radius

12 1.000

8 0.97

6 0.96

4 0.88


Alloys are solid solutions of metals. They are usually prepared by mixing molten components. They may be homogeneous, with a uniform distribution, or occur in a fixed ratio, as in a compound with a specific internal structure.

substitutional alloys
Substitutional Alloys

Substitutional alloys have a structure in which sites of the solvent metal are occupied by solute metal atoms. An example is brass, an alloy of zinc and copper.

substitutional alloys27
Substitutional Alloys

These alloys may form if:

1. The atomic radii of the two metals are within 15% if each other.

2. The unit cells of the pure metals are the same.

3. The electropositive nature of the metals is similar (to prevent a redox reaction).

interstitial alloys
Interstitial Alloys

Interstitial alloys are solid solutions in which the solute atoms occupy holes (interstices) within the solvent metal structure. An example is steel, an alloy of iron and carbon.

interstitial alloys29
Interstitial Alloys

These alloys often have a non-metallic solute that will fit in the small holes of the metal lattice. Carbon and boron are often used as solutes. They can be dissolved in a simple whole number ratio (Fe3C) to form a true compound, or randomly distributed to form solid solutions.

intermetallic compounds
Intermetallic Compounds

Some mixtures of metals form alloys with definite structures that may be unrelated to the structures of each of the individual metals. The metals have similar electronegativities, and molten mixtures are cooled to form compounds such as brass (CuZn), MgZn2, Cu3Au, and Na5Zn2.

ionic compounds
Ionic Compounds

Since anions are often larger than cations, ionic structures are often viewed as a close-packed array of anions with cations added, and sometimes distorting the close-packed arrangement.

common crystal types
Common Crystal Types

1. The Rock Salt (NaCl) structure-

Can be viewed as a face-centered cubic array of the anions, with the cations in all of the octahedral holes, or

common crystal types33
Common Crystal Types

1. The Rock Salt (NaCl) structure-

A face-centered cubic array of the cations with anions in all of the octahedral holes.

common crystal types34
Common Crystal Types

1. The Rock Salt (NaCl) structure-

The coordination number is 6 for both ions.

common crystal types35
Common Crystal Types

2. The CsCl structure-

Chloride ions occupy the corners of a cube, with a cesium ion in the center (called a cubic hole) or vice versa. Both ions have a coordination number of 8, with the two ions fairly similar in size.

common crystal types36
Common Crystal Types

3. The Zinc-blende or Sphalerite structure-

Anions (S2-) ions are in a face-centered cubic arrangement, with cations (Zn2+) in half of the tetrahedral holes.

common crystal types37
Common Crystal Types

4. The Fluorite (CaF2) and Antifluorite structures

A face-centered cubic arrangement of Ca2+ ions with F- ions in all of the tetrahedral holes.

common crystal types38
Common Crystal Types

4. The Fluorite (CaF2) and Antifluorite structures

The antifluorite structure reverses the positions of the cations and anions. An example is K2O.

ionic radii
Ionic Radii

Ionic radii are difficult to determine, as x-ray data only shows the position of the nuclei, and not the electrons.

Most systems assign a radius to the oxide ion (often 1.26Å), and the radius of the cation is determined relative to this assigned value.

ionic radii40
Ionic Radii

Like metallic radii, ionic radii seem to vary with coordination number. As the coordination number increases, the apparent ionic radius increases.

ionic radii41
Ionic Radii

1. Ionic radii increase as you go down a group.

2. Radii of ions of similar charge decrease across a period.

3. If an ion can be found in many environments, its radius increases with higher coordination number.

4. For cations, the greater the charge, the smaller the ion (assuming the same coordination #).

5. For atoms near each other on the periodic table, cations are generally smaller than anions.

predicting crystal structures
Predicting Crystal Structures

General “rules” have been developed, based on unit cell geometry, to predict crystal structures using ionic radii.

Radius ratios, usually expressed as the (radius of the cation)/(radius of the anion) are used.

predicting crystal structures43
Predicting Crystal Structures

General “rules” have been developed, based on unit cell geometry, to predict crystal structures using ionic radii.

Radius ratios, usually expressed as the (radius of the cation)/(radius of the anion) are used. This assumes that the cation is smaller than the anion.

predicting crystal structures44
Predicting Crystal Structures


8 ≥0.70 quite reliable

6 0.4 -0.7 moderately reliable

4 0.2 –0.4 unreliable

3 0.10 -0.20 unreliable

energetics of ionic bonds
Energetics of Ionic Bonds

The lattice energy is a measure of the strength of ionic bonds within a specific crystal structure. It is usually defined as the energy change when a mole of a crystalline solid is formed from its gaseous ions.

M+(g) + X-(g)  MX(s)

lattice energy
Lattice Energy

M+(g) + X-(g)  MX(s) ∆E = Lattice Energy

Lattice energies cannot be measured directly, so they are obtained using Hess’ Law. They will vary greatly with ionic charge, and, to a lesser degree, with ionic size.


1/2 bond energy of Cl2

Electron Affinity of Cl

Ionization energy of K

Lattice Energy of KCl

∆Hsub of K}

∆Hf of KCl

lattice energy49
Lattice Energy

Attempts to predict lattice energies are generally based on coulomb’s law:

VAB = (Zae)(Zbe)


Za and Zb = charge on cation and anion

e= charge of an electron (1.602 x 10-19C)

4πεo=permittivity of vacuum (1.1127 x 10-10J-1C2m-1)

rAB = distance between nuclei

lattice energy50
Lattice Energy

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
The Madelung Constant

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.

madelung constants
Madelung Constants

Crystal StructureMadelung Constant

Cesium chloride 1.763

Fluorite 2.519

Rock salt (NaCl) 1.748

Sphalerite 1.638

Wurtzite 1.641

estimating lattice energy
Estimating Lattice Energy

Ec = NM(Z+)(Z-)e2


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.

estimating lattice energy54
Estimating Lattice Energy

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

solubility of ionic crystals
Solubility of Ionic Crystals

The dissolving of ionic compounds in water may be viewed in terms of lattice energy and the solvation of the gaseous ions.

MX(s)  M+(g) + X-(g) Lattice energy

M+(g) + H2O(l)  M+(aq) Solvation

X-(g) + H2O(l)  X-(aq) Solvation

MX(s) ) + H2O(l)  M+(aq) + X-(aq) ΔHsoln

solubility of ionic crystals56
Solubility of Ionic Crystals

Factors such as ionic size and charge, hardness or softness of the ions, crystal structure and electron configuration of the ions all play a role in the solubility of ionic solids. The entropy of solvation will also play a role in solubility.

ionic size
Ionic Size

Smaller ions have a stronger coulombic attraction for each other and also for water. They also have less room to accommodate the waters of hydration.

Larger ions have weaker electrostatic attraction for each other and also for water. They also have accommodate more waters of hydration.

ionic size58
Ionic Size

The overall result of these factors result in low solubility of salts containing two large ions (soft-soft) or two small ions (hard-hard).

For salts containing two small ions, especially with the same magnitude of charge, the greater lattice energy dominates, and cannot be easily overcome by the hydration energy of the ions.

ionic size60
Ionic Size

For two large ions, the hydration energies are considerably lower, so the lattice energy dominates the process and results in a positive value for the enthalpy of hydration.

effect of entropy
Effect of Entropy

All ionic crystals will have an increase in entropy upon dissolution. This increase in entropy will increase the solubility of salts that have an endothermic enthalpy of solution.