CHEMISTRY OF SOLID STATE SYLLABUS UNIT III: CHEMISTRY OF SOLID STATE II: DIFFRACTION METHODS Band theory of solids- non-stoichiometry- point defects – linear defects- effects due to dislocations- electrical properties of solids-conductor, insulator, semiconductor-intrinsic-impurity semiconductors-optical properties-lasers and phosphors-elementary study of liquid crystals. Difference between point group and space group – screw axis – glide plane - symmetry elements –relationship between molecular symmetry and crystallographic symmetry – The Concept of reciprocal lattice – X-ray diffraction by single crystal – rotating crystal – powder diffraction. Neutron diffraction: Elementary treatment – comparison with X-ray diffraction. Electron diffraction- Basic principle. Crystal Growth methods: From melt and solution (hydrothermal, Page 1 of 67
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CHEMISTRY OF SOLID STATE
SYLLABUS
UNIT III: CHEMISTRY OF SOLID STATE II: DIFFRACTION METHODS
Band theory of solids- non-stoichiometry- point defects – linear defects- effects
due to dislocations-electrical properties of solids-conductor, insulator,
semiconductor-intrinsic-impurity semiconductors-optical properties-lasers and
phosphors-elementary study of liquid crystals.
Difference between point group and space group – screw axis – glide plane -
symmetry elements –relationship between molecular symmetry and
crystallographic symmetry – The Concept of reciprocal lattice – X-ray diffraction
by single crystal – rotating crystal – powder diffraction. Neutron diffraction:
Elementary treatment – comparison with X-ray diffraction. Electron diffraction-
Basic principle. Crystal Growth methods: From melt and solution (hydrothermal,
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UNIT III: CHEMISTRY OF SOLID STATE II: DIFFRACTION METHODS
1. BAND THEORY OF SOLIDS
According to band theory the energy spectrum of materials contains
conduction band and valence band. On the basis of distance between conduction
band and valence band, the materials are classified in to three categories.
1. Conductors:
If there is no energy gap between conduction band and valence band,
such materials are called conductors.
Examples: metals
2. Insulators:
Those materials in which the energy gap between conduction band and
valence band is very high , are called insulators.
3. Semiconductors:
If the gap between conduction band and valence band is very low , the
materials are called semiconductors.
Example: germanium and silicon
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2. NON-STOICHIOMETRY
Definition:
Compounds with non- integer values of atomic composition are called
non- stoichiometric compounds.
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Example : Ni 0.999O
Origin of non- stoichiometry
Impurities are the main reason
For example NaCl heated in Na vapour results Na 1.5 Cl
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Stoichiometric Defects
The compounds in which the number of positive and negative ions are exactly in
the ratios indicated by their chemical formulae are called stoichiometric
compounds. The defects do not disturb the stoichiometry (the ratio of numbers of
positive and negative ions) are called stoichiometric defects. These are of
following types,
(a) Interstitial defect: This type of defect is caused due to the presence of ions in
the normally vacant interstitial sites in the crystals.
(b) Schottky defect: This type of defect when equal number of cations and anions
are missing from their lattice sites so that the electrical neutrality is maintained.
This type of defect occurs in highly ionic compounds which have high co-
ordination number and cations and anions of similar sizes. e.g., NaCl, KCl, CsCl
and KBr etc.
(c) Frenkel defect: This type of defect arises when an ion is missing from its
lattice site and occupies an interstitial position. The crystal as a whole remains
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electrically neutral because the number of anions and cations remain same. Since
cations are usually smaller than anions, they occupy interstitial sites. This type of
defect occurs in the compounds which have low co-ordination number and cations
and anions of different sizes. e.g., ZnS, AgCl and AgI etc. Frenkel defect are not
found in pure alkali metal halides because the cations due to larger size cannot get
into the interstitial sites. In AgBr both Schottky and Frenkel defects occur
simultaneously.
CRYSTAL IMPERFECTIONS( CRYSTAL DEFECTS)
Any deviation in a crystal from a perfect periodic lattice structure is called
crystal defects. The three types of defects are
1. Point defects 2. Line defects( dislocations) 3. Surface defects(plane defects)
3. POINT DEFECTS
1. POINT DEFECTS
The deviation in a crystal,
from a perfect periodic lattice structure is localised in the vicinity of only few
atoms, it is called point defects. The different point defects are
1.1Vacancies
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1.2Interstial defects
1.3Frenkel defects (Vacancies and interstitial
1.4Schotky defects
1.5Substitutional defects
Stoiciometric defects:
1.3 FRENKEL DEFECTS (VACANCIY AND INTERSTITIAL DEFECTS):
When a missing atom, occupies the interstitial position, the defect caused is
known as Frenkel defects. This is most common in ionic crystals in which the
positive ions are smaller in size.
interstial
Fe 2+ O 2-Fe 2+ O 2-Fe 2+ O 2-
Fe 2+ vacancy
Fe 2+ O2-O 2-Fe 2+ O 2-
Fe 2+ O 2-Fe 2+ O 2-Fe 2+ O 2-
Number of Frenkel defects in a crystal can be calculated by the formula
N = √N N i√e−EKT
Where N total number of atoms and Ni number of interstitial positions
Derivation:
Let the energy required to displace an atom, from its proper position to an
interstitial position be E1. If there are N atoms and Ni interstitial positions , then
the number of ways in which ‘n’ Frenkel defects can be formed is given by
W = N !
n ! (N−n )!× ¿ !
n ! (¿−n )!
The change in Helmholtz free energy by the creation of ‘n’ Frenkel defects is
∆A = nE – TS
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= nE – T [ kln W]
= nE – T k ln [ N !
n ! (N−n )!× ¿ !
n ! (¿−n )! ]
= nE – T k [ ln N! + ln Ni! - 2 ln n! – ln( N-n)! – ln( Ni – n) !]
Using Sterling ‘s approximation ln N! = N ln N - N we get
∆A = nE – T k{[ N ln N – N] +[Ni ln Ni – Ni] - 2 [ nln n – n] –[ (N – n) ln( N-
n)] –
(N – n) ] – [( Ni – n) ln( Ni – n) - ( Ni – n) ] }
= nE – T k{[ N ln N – N +Ni ln Ni - Ni - 2 nln n +2 n – (N – n) ln( N-n)+
(N – n) – ( Ni – n) ln( Ni – n) + ( Ni – n) }
= nE – T k{[ N ln N – N +Ni ln Ni - Ni - 2 nln n +2 n – (N – n) ln( N-n)+
(N – n) – ( Ni – n) ln( Ni – n) + ( Ni – n) }
= nE – T k{ N ln N +Ni ln Ni - 2 nln n – (N – n) ln( N-n)– ( Ni – n) ln( Ni – n)
}
Differentiating with respect to ‘n’ at constant temperature,
¿ ) T = E - T k { -2 [n (1n) + ln n] - [ ( N-n) ( −1
N−n¿+ ln (N-n) (0-1)]
- [ ( Ni – n) × ( −1
(¿ – n)¿ + ln ( Ni – n) ( 0-1)}
= E - T k { -2 -2 ln n] - [ -1 - ln (N-n) ] -[- 1- ln ( Ni – n) }
= E - T k { -2 -2 ln n] + 1 + ln (N-n) + 1+ ln ( Ni – n) }
= E - T k { -2 ln n + ln (N-n) + ln ( Ni – n) }
= E - T k { ln (1n2 ) + ln [ (N-n) × ( Ni – n) ] }
= E - T k { ln(N−n)×(¿ – n)
n2 }
When equilibrium is attained, the Helmholtz free energy is constant and its first
derivative is equal to zero. i. e ¿ ) T = 0
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∴0 = E - T k { ln(N−n)×(¿ – n)
n2 }
E = T k { ln(N−n)×(¿ – n)
n2 }
ET k= { ln
(N−n)×(¿ – n)n2 }
When N >>n, N- n ≈ N similarly, When Ni >>n, Ni- n ≈ Ni
Therefore the above equation becomes, ET k= { ln (N )×(¿)n2 }
Taking exponential on both sides,
eET k = N∋ ¿
n2 ¿
n2 = NNie−ET k
∴ n = (NNie−ET k ) ½
This is the expression for the number of ways of forming the defects
SCHOTKY DEFECTS
When a positive as well as negative ions of a crystal are missing, the defect
is known as Schotky defects.
In Schotky defect the displaced atom migrates in successive steps
eventually settles at the surface. Since the number of missing positive ions and
negative ions is same, the crystal remains as neutral
Na + Cl-
Na + Cl- Na + Cl- Na + Cl–
Na + Cl- Na + Na + Cl-
Na + Cl- Cl- Na + Cl-
Na + Cl- Na + Cl- Na + Cl-
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Number of Schotky l defects in a crystal can be calculated by the formula
n = N×e−E2KT
Where N total number of atoms .
Derivation:
Suppose a crystal contains N atoms and ‘n’Schotky defects are produced
by removing ‘n’ cations and ‘n’ anions from the crystal. Let the energy required
to displace an atom, from its proper position to an interstitial position be E1. The
number of ways in which ‘n’ schotky defects can be formed is given by
W = N !
n ! (N−n )!× N !
n ! (N−n )!
The change in Helmholtz free energy by the creation of ‘n’ Frenkel defects is
∆A = nE – TS
= nE – T [ kln W]
= nE – T k ln [ N !
n ! (N−n )! ] 2
= nE – 2T k [ ln N! - ln n! – ln( N-n)! ]
Using Sterling ‘s approximation ln N! = N ln N - N we get
∆A = nE– 2T k{[ N ln N – N] - [ nln n – n] –[ (N – n) ln( N-n)] – (N – n) ]
= nE – 2T k{[ N ln N – N - nln n + n – (N – n) ln( N-n)+ (N – n)
= nE – 2T k{[ N ln N – N - nln n + n – (N – n) ln( N-n)+ (N – n) }
= nE – 2T k{ N ln N - nln n – (N – n) ln( N-n) }
Differentiating with respect to ‘n’ at constant temperature,
¿ ) T = E - 2T k { - [n (1n) + ln n] - [ ( N-n) ( −1
N−n¿+ ln (N-n) (0-1)]
= E - 2T k { -1 - ln n] - [ -1 - ln (N-n) ]
= E - 2 T k { -1 - ln n] + 1 + ln (N-n)
= E - 2T k { - ln n + ln (N-n) }
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= E - 2T k { ln(N−n)n }
When equilibrium is attained, the Helmholtz free energy is constant and its first
derivative is equal to zero. i. e ¿ ) T = 0
∴0 = E - 2T k ln(N−n)n
E = 2T k ln(N−n)n
ET k = 2 ln(N−n)
n
When N >>n, N- n ≈ N
Therefore the above equation becomes, E2T k = ln(N )
n
Taking exponential on both sides,
eE
2T k = Nn
∴ n = N ×e−E2T k
This is the expression for the number of ways of forming the defects
4. linear defects
5. effects due to dislocations
6. electrical properties of solids
7. CONDUCTOR, INSULATOR, SEMICONDUCTOR
1. Conductors:
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If there is no energy gap between conduction band and valence band,
such materials are called conductors.
Examples: metals
2. Insulators:
Those materials in which the energy gap between conduction band and
valence band is very high , are called insulators.
3. Semiconductors:
If the gap between conduction band and valence band is very low , the
materials are called semiconductors.
Example: germanium and silicon
8.INTRINSIC SEMICONDUCTORS:
A semi conductor which is pure and contains no impurity is known as intrinsic
semiconductor.
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9. IMPURITY SEMICONDUCTORS
Extrinsic Semiconductors:
A semiconducting material in which, the charge carriers originate from
impurity atoms added to the material, is called extrinsic semiconductor or
impurity semiconductor.
Theses are divided in to two types.
1 n- type semi conductor:
Pentavalent elements such as P, As, Sb , have five electrons in their
outermost orbits. When any one such impurity is added to the intrinsic semi
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conductor, four electrons are engaged in covalent bonding with four
neighbouring semi conductor atoms and the fifth electron is free.
Free electron
2 p- type semi conductor:
Trivalent elements such as Al, Ga or In have three electrons in their outer
most orbits. When such impurity is added to the intrinsic semi conductor, all the
three electrons are engaged in covalent bonding with three neighbouring semi
conductor atoms and creating a hole ( vacant electron site) on the semiconductor
atom.
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1. optical properties
2. lasers and phosphors
ELEMENTARY STUDY OF LIQUID CRYSTALS.
Solids yield a viscous cloudy liquids at a temperature known as transition
point. If the temperature is increased beyond the transition point, the cloudiness
disappear at the temperature called melting point
Between transition point and melting point the cloudy liquid shows double
refaction. This state is called mesomorphic state. And the compounds in this state
are called liquid crystals.
SMECTIC TYPE CRYSTALS WITH EXAMPLES
1.The word "smectic" originates from the Latin word having soap-like properties
2.There are two phases in smectic type. They are named as smectic A and smectic
C
3. The smectic A phase has molecules organized into layers.
4. In the smectic C phase , the molecules are tilted inside the layers.
5. The layers can slide over one another .
Example:
Smectic phase Transition
temperature
Melting
temperature
Ethyl – p- azoxy benzoate 114K 121 K
Ethyl – p- azoxy cinnamate 140K 249 K
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NEMATIC TYPE CRYSTALS WITH EXAMPLES
. 1.The word nematic comes from the Greek which means "thread".
2. In a nematic phase, organic molecules have no positional order,
3. The molecules are free to flow
4. Nematics are uniaxial:
5. Nematics have fluidity similar to that of ordinary liquids
6. They can be easily aligned by an external magnetic or electric field.
Example:
Nematic phase Transition
temperature
Melting
temperature
p- azoxy anizole 390K 410 K
p- azoxy phenetole 410K 440K
CHOLESTERIC TYPE CRYSTALS WITH EXAMPLES
They exhibit the unique property that they reflect circularly polarized light