Crystals

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Crystals
2
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.

3
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.

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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.

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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.

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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.

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Close Packing
8
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.

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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.

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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.

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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.

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Octahedral Holes

• The size of the octahedral hole .414 r
• where r is the radius of the cubic close-packed
  atom or ion.

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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.

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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.

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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.

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of Atoms/Unit Cell

• For atoms in a cubic unit cell
• Atoms in corners are ? within the cell

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of Atoms/Unit Cell

• For atoms in a cubic unit cell
• Atoms on faces are ½ within the cell

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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.

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of Atoms/Unit Cell

• For atoms in a cubic unit cell
• Atoms in corners are ? within the cell
• Atoms on faces are ½ within the cell
• Atoms on edges are ¼ within the cell

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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.

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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.

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Polymorphism

• 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.

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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.

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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

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Alloys

• 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.

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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.

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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).

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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.

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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.

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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.

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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.

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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

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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.

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Common Crystal Types

• 1. The Rock Salt (NaCl) structure-
• The coordination number is 6 for both ions.

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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.

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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.

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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.

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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.

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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.

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Ionic Radii

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

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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.

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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.

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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.

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Predicting Crystal Structures

• CN r/r- accuracy
• 8 0.70 quite reliable
• 6 0.4 -0.7 moderately reliable
• 4 0.2 0.4 unreliable
• 3 0.10 -0.20 unreliable

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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)

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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.

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1/2 bond energy of Cl2
Electron Affinity of Cl
Ionization energy of K
Lattice Energy of KCl
?Hsub of K
?Hf of KCl
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Ionic charge has a huge effect on lattice energy.
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Lattice Energy

• Attempts to predict lattice energies are
  generally based on coulombs law
• VAB (Zae)(Zbe)
• 4peorAB
• Za and Zb charge on cation and anion
• e charge of an electron (1.602 x 10-19C)
• 4peopermittivity of vacuum (1.1127 x
  10-10J-1C2m-1)
• rAB distance between nuclei

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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.

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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.

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Madelung Constants

• Crystal Structure Madelung Constant
• Cesium chloride 1.763
• Fluorite 2.519
• Rock salt (NaCl) 1.748
• Sphalerite 1.638
• Wurtzite 1.641

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Estimating Lattice Energy

• Ec NM(Z)( Z-) e2
• 4peor
• where N is Avogadros number, and
• M is the Madelung constant (sometimes
  represented by A)
• This estimate is based on coulombic forces, and
  assumes 100 ionic bonding.

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Estimating Lattice Energy

• A further modification, the Born-Mayer equation
  corrects for complex repulsion within the
  crystal.
• Ec NM(Z)( Z-) e2 (1-?/r)
• 4peor
• for simple compounds, ?30pm

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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) ?Hsoln

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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.

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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.

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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.

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Ionic Size
60
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.

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Ionic Size
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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.