Non-Stoichiometry & Solid Solution The document discusses various types of point defects that can occur in non-stoichiometric compounds including Schottky defects, Frenkel defects, and anti-site defects. It provides examples of intrinsic point defects in metal-deficient, metal-excess, oxygen-deficient, and oxygen-excess metal oxides. Solid solutions are formed when one or more minor components dissolve uniformly within the crystal lattice of a major component. Types of solid solutions include interstitial, substitutional, ordered, and disordered solutions. Experimental techniques like X-ray diffraction and density measurements can be used to study properties of non-stoichiometric compounds and solid solutions.

1. Non-Stoichiometry &
Solid Solution
Debabrata Bagchi
Ph. D. Student, NCU
JNCASR, Bangalore

2.  Schottky defects
• Cation and anion vacancies
 Anti-Schottky defects
• Cation and anion interstitials
• (not common)
 Frenkel defects
• Cation vacancies and interstitials
 Anti- or anion-Frenkel defects
• Anion vacancies and interstitials
Anti-site defects
• Cation and anion swap
• (not common)
Stoichiometric compounds: intrinsic point defect

3. • Examples for oxides:
• Metal deficient oxides, e.g. M1-xO
• Metal vacancies are majority point defects, compensated by electron holes
• Examples: Co1-xO, Ni1-xO, and Fe1-xO
• Metal excess oxides, e.g. M1+xO
• Metal interstitials are majority point defects, compensated by defect electrons
• Example: Cd1+xO
• Oxygen deficient oxides, e.g. MO2-y
• Oxygen vacancies are majority point defects, compensated by defect electrons
• Examples: ZrO2-y, CeO2-y
• Oxygen excess oxides, e.g. MO2+y
• Oxygen interstitials are majority point defects, compensated by electron holes
• Example: UO2+y
 Non-stoichiometric compounds have formulae that do not have simple integer ratios of atoms
they exhibit a range of composition and can be made by introducing impurities into a system,
Non-stoichiometric compounds

4. Non-Stoichiometric Defects
Anion vacancies Interstitial
anion/ cation Cation vacancies
F-centre NaCl in Na vapor

5. Defect Concentration

6. Effect of Non-Stoichiometry
Electrical
Kinetic (e.g.
diffusion)
Mechanical (e.g.
Strength, toughness,
hardness)
Magnetic
Optical

7. Effect of Non-Stoichiometry

8. Non stoichiometry in 1st row TM Oxides

9. Non-Stoichiometry in Wustite (FeO)
 Ferrous oxide is known as wustite it has the NaCl (rock salt) crystal
structure.
 Chemical analysis proved that it is non-stoichiometric and deficient in
iron
 Below 570°C, wustite disproportionate to α-iron and Fe3O4.
 Fe deficiency can be happened in two ways:
1. Fe vacancy (leads to Fe1-xO), or
2. Excess of oxygen in interstitial positions (giving FeO1+x).

10. Cation Vacancy or Interstitial Anion
Experimental Observations
a = 430.1 pm
d =5.728 kg m-3
Fe/O = 0.945
V = (430.1 pm)3 = 7.956 x 10-29 m3
4 formula units of FeO in a perfect unit cell (rock salt structure)
1 mole of FeO weighs (55.85 + 16.00) g = 0.07185 kg
4 mole weigh = 4 x 0.07185 kg
4 formula units weigh = 4 x 0.07185 kg / NA = 4.733 x 10-25 kg

11. Cation Vacancy
Experimental Observations
a = 430.1 pm
Fe/O = 0.945
V = (430.1 pm)3 = 7.956 x 10-29 m3
If Fe0.945O1
1 mole of Fe0.945O1 weighs ((55.85x0.945) + 16.00) g = 0.06877 kg
4 mole is 3.78Fe + 4 O weigh = 4 x 0.06877 kg
4 formula units weigh = 4 x 0. 06877 kg / NA = 4.567 x 10-25 kg
dcv= 4.567x10-25 /7.956x10-29 kg m-3 = 5.742 x 103 kg m-3

12. Interstitial Anion
Experimental Observations
a = 430.1 pm
Fe/O = 0.945 O/Fe = 1/0.945 = 1.058
V = (430.1 pm)3 = 7.956 x 10-29 m3
If Fe1O1.058
1 mole of Fe1O1.058 weighs (55.85 + (16.00x1.058)) g = 0.07277 kg
4 mole is 4Fe + 4.232O weigh = 4 x 0.07277 kg
4 formula units weigh = 4 x 0.07277 kg / NA = 4.832x 10-25 kg
dia= 4.832x10-25 /7.956x10-29 kg m-3 = 6.076 x 103 kg m-3

13. Cation Vacancy or Interstitial Anion
 Experimental density d =5.728 kg m-3
 If Fe0.945O1 dcv = 5.742 x 103 kg m-3
 If Fe1O1.058 dia= 6.076 x 103 kg m-3
 Thus, FeO has cation vacancy
 Non-stoichiometric compounds are found to exist over a range of composition. It is possible to
determine whether the non-stoichiometry is accommodated by vacancy or interstitial defects using
density measurements.

14. Electronic Defects in FeO

15. Koch-Cohen Cluster
• NaCl type structure with 4 interstitial
Fe3+ ions in tetrahedral voids, 13
immediately surrounding octahedral Fe2+
sites must be vacant.
• This is referred to as sueperstructure or
superlattice
Fe2+ Oh site vacant
Fe3+ Td
interstitial

16. Uranium Dioxide

17. Vegard's law
 For most of the non-stoichiometric compounds, their unit cell size varies
smoothly with composition but the symmetry is unchanged. This is known as
Vegard’s Law.
Experimental and theoretical densities (103 kg m-3) for FeO

18. This law simply states that when you combine elements to form an alloy, the lattice
constant will follow a linear trend with the element concentrations, provided that there
is no phase change and lattice parameters do not differ by more than 5%.
Vegard's law continued…
Mathematical expression for Vegard’s law for a binary system A-B is:
 where X is the mole fraction of component B and a = lattice parameters of pure
components

19. Application of Vegard's law
 Lattice constant increases with increase in Cr substitution for Al in CuAlO2.

20. Application of Non-stoichiometric compound
 Oxidation catalysis: Reactions of hydrocarbon with oxygen, a conversion that is catalysed by metal oxides.
 Here transfer of "lattice" oxygen to the hydrocarbon substrate, a step that temporarily generates a vacancy (or
defect). Such catalysts rely on the ability of the metal oxide to form phases that are not stoichiometric
 Ion conduction: The defect sites provide pathways for
atoms and ions to migrate through the otherwise dense
ensemble of atoms that form the crystals.
 Oxygen sensors and solid state batteries are two
applications that rely on oxide vacancies.
 Superconductivity: Many superconductors are non-stoichiometric. YxBa2Cu3O7−x. arguably
the most notable high temperature superconductor, is a non-stoichiometric solid with the
formula YxBa2Cu3O7−x. The critical temperature of the superconductor depends on the exact
value of x.

21. Solid Solution
 Solid Solution is a solid mixture containing one or more minor components (solute)
uniformly distributed within the crystal lattice (matrix) of the major component (solvent).
 Such a mixture is considered a solution rather than a compound when the crystal structure of the solvent
remains unchanged by addition of the solutes, and when the mixture remains in a single homogeneous phase.
Figure: This binary phase diagram shows
two solid solutions.
Solid solution formation usually causes increase of electrical resistance and mechanical
strength and decrease of plasticity of the alloy.

22. Types of Solid Solution
Interstitial
solid solution
Substitutional
solid solution
Ordered
solid solution
Disordered
solid solution

23. Solute atoms are much smaller than solvent atoms (size of the solute is less than
40% that of solvent), so they occupy interstitial position in solvent lattice.
Carbon ,nitrogen ,hydrogen ,oxygen, lithium, sodium and boron are the element
which commonly form interstitial solid solution. Steel : C atoms solute in Fe.
Solvent Atoms
Solute Atoms
Interstitial Solid Solution

24. Substitutional Solid Solution
Solute atoms sizes are roughly similar to solvent atoms.
Due to similar size solute atoms occupy vacant site in solvent atoms.
Cu and Zn, Cu and Ni, are the example of substitutional solid solution.
Solvent Atoms
Solute Atoms

25.  Hume Rothery studied a number of alloy systems and
formulated condition that favour extensive substitutional
solid solubility.
Conditions of Hume Rothery’s rule
1) The size difference between solute and solvent atoms must be less
than 15%.
2) The solubility of a metal with higher valence in a solvent of
lower valence is more compared to the reverse situation e.g.
Zn is much more soluble in Cu than Cu in Zn.
3) The crystal structures of metals must be same.
4) The electronegativity difference between the metals must be small.
Hume Rothery’s Rule

26. Ordered Substitutional Solid Solution
 If the atoms of the solute occupy certain
preferred sites in the lattice of the solvent,
an ordered solid solution is formed. It may
occur only at certain fixed ratios of the
solute and solvent atoms.
 In Cu – Au system, Cu atoms occupying
the face-centred sites and Au atoms
occupying the corner sites of the FCC unit
cell.
Solvent Atoms
Solute Atoms

27.  If the atoms of the solute are present
randomly in the lattice of the solute, it is
known as disordered solid solution.
 Most of the solid solutions are disordered
solid solutions
Disordered Substitutional Solid Solution
Solute Atoms
Solvent Atoms

28. X-ray powder diffraction, XRD :
1. Fingerprint method: Determination of the crystalline phases that are present (the detection limit of phases in a
mixture is usually of the order of 2–3 wt%)
2. Measure the XRD pattern of solid solutions accurately and obtain their unit cell dimensions, which may undergo
a small contraction or expansion as composition varies. The calibration graph of unit cell dimensions against composition can be
used to determine the composition of solid solutions in a particular sample.
Usually, a unit cell expands if a small ion is replaced by a larger ion and vice versa, contracts if a smaller ion is substituted into
the structure. From Bragg’s law and the d-spacing formulae, the whole pattern shifts to lower values of 2q with increasing unit
cell parameters..
According to Vegard’s law, unit cell parameters should change linearly with solid solution composition.
3. Third, using Rietveld refinement of the powder XRD patterns of solid solutions, and in particular the intensities of the XRD
reflections, it is possible to gain detailed crystallographic information such as the sites occupied by atoms and the location of
vacancies and interstitials.
Experimental methods for studying solid solutions

29. The mechanism of solid solution formation may be inferred by a combination of density and
unit cell volume measurements for a range of compositions.
The key parameter is the mass of the average unit cell contents and whether it increases or
decreases on solid solution formation.
 Density data for cubic CaO-stabilised zirconia solid solutions for samples quenched from 1600 ◦C
Density measurements

30.  Many materials undergo abrupt changes in structure or property on heating and, if the
material forms a solid solution, the temperature of the change usually varies with
composition.
The changes can often be studied by differential thermal analysis/differential scanning
calorimetry (DTA/DSC) since most phase transitions have an appreciable enthalpy of
transition.
 Effect of dopants on the ferroelectric Curie temperature of
BaTiO3 showing the effect of isovalent substitution of Sr,
Ca and Pb for Ba, isovalent substitution of Zr for Ti and
aliovalent substitution of Ca for Ti.
Changes in other properties – thermal activity and DTA/DSC

31.  Solid solution strengthening is a type of alloying that can be used to improve the strength
of a pure metal.
 Why strengthening of metal is required?
 As pure metal are inherently weak due to presence of dislocation.
 So, We can enhance the mechanical properties by eliminating dislocation.
 So, by introducing some mechanism that prohibits the mobility of dislocations, that are
called Strengthening mechanism.
Methods For Strengthening
Of Metal
Strain hardening
Grain boundary
strengthening
Precipitation
hardening
Solid solution
strengthening
Solid solution strengthening

32. Factors affecting Solid Solution Strengthening
Difference in size between
solute and solvent atoms
Amount of solute
added
Nature of distortion
produced by solute atoms
size difference increases
the intensity of stress field
around solute atom
resistance to dislocation
is increases
strength of metal
increases.
A large concentration means more
frequent obstacles to dislocation.
The strength increases in
proportion of C ½
Spherical distortion produced
by Substitutional solute atoms
Non spherical distortion produced
by interstitial solute atoms.

33.  Why is Steel so strong?
 Smaller carbon atoms fill some of the small spaces available
between the iron atoms and form Interstitial Solid Solution.
 Usually materials deform by the movement of dislocations. The carbon interstitials make steel stronger by fully
or partially blocking the movement of dislocations.
Fe
C
Application of Solid Solution