Statistical Thermodynamics of
Defects in Solids
Defects in Solids
May have a perfect crystal at 0 K, but
otherwise defects will occur in the
structure
• Important influence on properties
• Electronic and thermal conduction
• Mechanical strength
• Diffusion
• Colour etc.
• Exploited in many applications
otherwise defects will occur in the
structure
• Important influence on properties
• Electronic and thermal conduction
• Mechanical strength
• Diffusion
• Colour etc.
• Exploited in many applications
Classification
1. Intrinsic vs extrinsic
• Intrinsic - integral to the pure crystal
• Extrinsic - foreign atoms.
2. Dimensionality
• Point defects at isolated atomic positions
• Extended
• Linear, planar, 3D
• Intrinsic - integral to the pure crystal
• Extrinsic - foreign atoms.
2. Dimensionality
• Point defects at isolated atomic positions
• Extended
• Linear, planar, 3D
3. Thermodynamic equilibrium
– Can represent a minimum in the free energy of
the crystal
– Independent of sample history
– Or can be metastable and change with the
history of the sample.
– Can represent a minimum in the free energy of
the crystal
– Independent of sample history
– Or can be metastable and change with the
history of the sample.
• Two categories
– Schottky
• Vacancies in the lattice
• For an ionic compound, consist of a combination of cation
and anion vacancies, to maintain charge neutrality
– Frenkel
• Interstitials and vacancies in the lattice
• Tend to be cation interstitials due to size
– Schottky
• Vacancies in the lattice
• For an ionic compound, consist of a combination of cation
and anion vacancies, to maintain charge neutrality
– Frenkel
• Interstitials and vacancies in the lattice
• Tend to be cation interstitials due to size
Why do they exist at
equilibrium?
equilibrium?
• Creation of a defect
normally costs
energy.
• But it also increases
the entropy of the
crystal!
• Defects increase in
concentration until
the free energy is a
minimum.
normally costs
energy.
• But it also increases
the entropy of the
crystal!
• Defects increase in
concentration until
the free energy is a
minimum.
• Thus need to be able to understand how the
enthalpy and the entropy vary with defect
concentration.
• If the defects are truly isolated from each
other, then the enthalpy should just be
proportional to the number of defects
• E.g. for ns Schottky pairs
∆H = ns
∆Hs
where ∆Hs
is the energy to create one defect pair
enthalpy and the entropy vary with defect
concentration.
• If the defects are truly isolated from each
other, then the enthalpy should just be
proportional to the number of defects
• E.g. for ns Schottky pairs
∆H = ns
∆Hs
where ∆Hs
is the energy to create one defect pair
The entropy
• Consider a 1:1 ionic crystal with N cation sites
N anion site, ns Schottky cation vacancies
and ns anion vacancies.
• The vacancies will be able to take up many
different possible positions in the crystal so
there will be a configurational entropy
associated with their distribution
∆S = k lnW = k lnWcWa
• What is the magnitude of the enthalpy
term?
• Normally about 60-600 kJ mol
-1
• So typical concentrations (ns/N) are:
term?
• Normally about 60-600 kJ mol
-1
• So typical concentrations (ns/N) are:
T/K | Conc. for enth. =300 kJmol -1 | Conc. for enth. =60 kJmol -1 |
300 | 6x10-27 | 5.7x10-6 |
1000 | 1.4x10-8 | 2.7x10-2 |
Duque Franky
C.I: 15.990.445
CRF
http://foord.chem.ox.ac.uk/teaching/3rd-yr-solidssurfaces-solids4.pdf
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