Chap' 3 CRYSTAL BINDING

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Chap. 3 CRYSTAL BINDING
3.1 General Introduction
Electromagnetic interaction is major force
lt Repulsive force gt Pauli Exclusion Principle
(Exchange type interaction) One of the most
important rules in quantum chemistry is the Pauli
Exclusion Principle. It states that every
electron in an atom must have a unique set of
quantum numbers no two electrons in the same
atom can have the same quantum numbers. Short
range become repulsive when two atoms begin to
overlap with each other.
ltAttractive forcegt

• Vander Waals interaction
• Electrostatic Coulomb interaction
• Covalent bonding sharing of electrons
  (locally)
• Metallic sharing electrons (globally)

r
2

• Electronic wave function

Conductor metal
Insulator Molecular crystal

• Covalent bonding bond in which one or more
  pairs of electrons are shared by two atoms.

Ground state
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• Crystal Classification by cohesion

Ionic crystal
Molecular crystal
Covalent crystal
Metal crystal
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3.2 Molecular Crystal (Van der Waals bond)
Noble Gases ( Ar, Ne, Xe ,.) Spherical
distribution of charges
Almost no overlap between the nearest atoms
Vander walls-London interaction
Physical origin charge fluctuations in atoms
due to the zero point motion
? induced dipole moments give attractive
interaction
Induced dipole moment
Fluctuating dipole moment
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Brief Introduction on Electromagnetism
Electric potential,
fluctuating dipole moment
Induced dipole field,
a polarizability
Dipole Energy in Electric Field
? Principal attractive interaction in crystals of
inert gas and organic crystals
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Total potential energy of two atoms separated by
R
Pauli exclusion principle When the charge
distribution s of two atoms overlap there is a
tendency for electrons from atom B to occupy in
part states of atom A already occupied by
electrons of atom A.
Promote electrons to unoccupied high energy
states of the atoms.
Short range repulsive interaction
Lennard Jones Potential (6-12)
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• Equilibrium Lattice Constant

Equilibrium configuration is the lowest total
energy configuration (at zero temperature)
Total Energy of a N-atom system
For FCC structure
For hcp structure
there are 12 nearest neighbors
Equilibrium Distance
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• Cohesive Energy

Total energy of free atoms Total energy of a
crystal
Zero Point Fluctuations
Atoms can be considered as harmonic oscillators
Ground state energy in quantum mechanics ½ hw
K coupling constant
M mass
He does not crystallize due to zero point motion
at ambient pressure
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• Equilibrium Bulk modulus

The bulk modulus B
For a FCC lattice, the volume per particle is
given by
The equilibrium separation is that which
minimizes the energy per particle u gt the first
derivative vanishes
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3.3 Ionic Crystals (Ionic Bond)
One or more atoms lose electrons and other atoms
gain them in order to produce a noble gas
electron configuration, the bond thus formed is
called an ionic bond.
http//mychemistrypage.future.easyspace.com/Genera
l/bonding/ionic.html
Na Cl
Na Cl-
Ionize (charge transfer)
Strong electrostatic interaction between the
positive ions and the negative ions provide
cohesive energy (I VII) Alkali-Halide
Li F Li
F-
Li Li e 1st ionization
energy 5.39 eV
 ?E 1.99eV Energy should be supplied.
Fe F- Electron affinity 3.40
eV
But, the crystal gains attractive Coulomb energy
between two charged ions. (long range force)
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One dimension
Three dimensions
NaCl structure 1.747565 CsCl structure
1.762675 ZnS structure 1.6381
Coulumb force is long range
The energy per molecule unit of a crystal sodium
chloride is 6.4eV lower than the Energy of
separated neutral atoms.
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3.4 Covalent Crystals
Pauli exclusion principle Wave functions for
two identical fermions must be antisymmetric.
Total wave function
spin space ? antisymmetric
Hydrogen Molecule
http//hyperphysics.phy-astr.gsu.edu/hbase/molecul
e/hmol.htmlc1
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Delocalize -gt lower kinetic energy, raise
potential energy
Highly direction bonding
Hybridization gives covalent bond.
S2P2 ?SP3

• Characteristics
• - Non-closed packed structure
• - Quite strong bond (large cohesive energy)
• - Spatial charge distribution charge density is
  concentrated along the direction joining two atoms

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3.5 Metallic Crystal
Metal High electrical conductivity ? Free
electrons
e
e
e
e
e
e
delocalize

• Positive ions in a nearly uniform conduction
  electron sea
• -gt reduction of kinetic energy of valence
  electrons gives cohesive energy
•  
• Simple metal weakly binding
• Transition metal involve d-electron

In BCC crystal
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Observed values in Alkali metals 2-6
Positive ions are not point but occupy finite
volume (excluded volume increases) ? Kinetic
energy will be greater ? Potential energy will
be less negative The equilibrium radius will
increase with increasing ion core radius
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3.6 Hydrogen bond
Hydrogen has one electron and it should form a
covalent bond with only one atom Under certain
condition atom of hydrogen is attracted by rather
strong forces to two atoms
Hydrogen bond- A type of bond formed when the
partially positive hydrogen atom of a polar
covalent bond in one molecule is attracted to the
partially negative atom of a polar covalent bond
in another.
Largely ionic in character forms with most
electronegative atoms, F,O,N

• Ion core of a hydrogen atom is a bare proton 105
  smaller tan any other ion core
• Hydrogen is but one electron shy of the stable
  helium (two electrons)
• The ionization potential of atomic hydrogen is
  unusually high
• (H 13.59eV, Li 5.39eV)

Extreme case of hydrogen bonding
Hydrogen bonded water pentamer