Environmental Engineering Reference
In-Depth Information
thermodynamically favored at low temperature, may become important at high
temperatures.
5.4.1 Neutral and Charged Defects and Electroneutrality
Conditions
In a crystal of ionic compounds (oxides, halides, sulfides, etc.) the atoms are
charged, i.e., the compound crystal is made up of cations and anions. Point defects
may be neutral or charged. Neutral defects are designated by a cross-sign as
superscript, e.g., V
O
,V
x
M
,M
i
,O
i
, etc., which signifies zero charge. It is generally
more convenient to write defect formation reaction in a crystal by considering
the charge on the defect relative to the perfect crystal. The relative charge is
termed as the ''effective charge'' on the defect. As examples, singly and doubly
ionized oxygen vacancies are represented by V
•
O
and V
•
O
, respectively. A doubly
charged interstitial cation is written as M
•
i
. Similarly, singly and triply ionized
metal vacancies are represented by V
M
, respectively. Correspondingly,
an interstitial oxygen ion will have effective negative charge, e.g., O
′
M
, and V
i
. Also,
substitutionally dissolved foreign ions, which have valence different from that
of the host lattice ions (i.e., aliovalent foreign ion), will also show an effective
charge represented by F
•
M
or F
′
O
. In the study of band structure for
ionic solids or semiconductor compounds, the energy levels of V
O
,M
i
, and F
M
(F having higher valence than M) are nearer to the conduction band of the host
lattice and hence these centers act as donors, i.e., they donate electrons to the
conduction band and themselves become positively charged. Similarly, the en-
ergy levels of V
M
,O
i
, and F
M
(F having lower valence than M) will be nearer
to the valence band of the host crystal. These centers act as acceptors taking up
electrons from the valence band, thereby becoming negatively charged and in-
jecting positive holes in the valence band. Quasi-free electrons (excess electrons)
in the conduction band and positive holes in the valence band will exhibit effec-
tive negative and positive charges, respectively. They are represented in defect
reactions as e
′
M
or F
•
O
or F
and h
•
, respectively.
It should also be kept in mind that a crystal as a whole should be electrically
neutral, i.e., the sum of all effective positive charges must equal the sum of all
effective negative charges. This principle of electroneutrality condition serves as
the basic concept in formulating and treating defect equilibria and for evaluating
defect concentration in crystals.
′
5.4.2 Stoichiometry and Nonstoichiometry
Stoichiometric compounds also referred to as daltonides, follow the law of con-
stant proportion, i.e., in the case of oxides, a fixed proportion between metal and
oxygen is maintained. Schottky and Wagner [9] for the first time established
