Physics Unit 4 Notes
1 sections · English
Open in GPAI
,Contents
1 Electromagnetic Induction and Alternating Current
,Electromagnetic Induction and Alternating Current
4.1 Free Electron Theory
Basis of free electron theory
developed by Drude and Lorentz in 1900
valence electrons involved in electrical conduction in metals and alloys
valence electrons become free in solids → move randomly like gas molecules
◦ thus, known as free electrons
neglecting electron-electron and electron-ion interaction, electrons undergo periodic collision with ions in the lattice
Success of Free Electron Theory
successfully explains Ohm's law
Electrical conductivity of metal:
◦ explains electrical conductivities of metals
◦ Electrical conductivity σ = neμ
▪ : electron density
n
▪ : charge on electron
e
▪ : mobility of electrons
μ
Thermal Conductivity:
◦ successfully explains thermal conductivities of metals at lower temperatures
◦ thermal conductivity of metals K =
KB n ν λ
: Boltzmann constant
2
▪ KB
Relation between electrical and thermal conductivity (Wiedemann-Franz Law):
◦ ratio of thermal conductivity ( ) and electrical conductivity ( ) of a metal is proportional to temperature ( )
K σ T
◦ K /(σT ) = 1.11 × 10
−8
W ⋅ Ω/K
2
◦ also known as Lorentz's number
Limitations of Free Electron Theory
conductivity of metals proportional to electron concentration (according to theory)
◦ ex) divalent atoms (cadmium, zinc) and trivalent atoms (aluminum) should have more electrical conductivity than monovalent
atoms (copper, silver) → theory fails
cannot explain classification of solids into conductors, semiconductors, and insulators
theoretical values of mean free path of electrons do not agree with experimental values
theoretical values of specific heat and electronic specific heat do not agree with experimental values
could not explain positive values of Hall effect in some metals (ex: zinc)
Wiedemann-Franz law deviates at low temperatures
4.2 Opening of band gap due to internal electron diffraction
to find allowed energies of electrons in solids, Schrödinger's wave equation is applied for an electron in a crystal lattice
Kronig and Penney Model
suggested a simplified model for atoms as a one-dimensional infinite row of rectangular potential wells separated by barriers
◦ each well has width and depth
b V0
◦ barriers have width
b
interatomic spacing → period of potential is
a (a + b)
, electron moves through a lattice of positive ions → experiences varying potentials
Schrödinger's equation in one dimension
2
d ψ 2m
+
(E − V )ψ = 0
2 2
dx ℏ
Boundary conditions and Schrodinger's equation
in region 0 < x < a , potential energy V = 0
2
d ψ 2mE
+
ψ = 0
2 2
dx ℏ
can be written as
2
d ψ
2
+ α ψ = 0 for 0 < x < a
2
dx
where α =
2mE
ℏ
2
Solution using Bloch Theorem
solution for periodic potential found using Bloch theorem:
maV0 b sin(αa)
( )
+ cos(αa) = cos(ka)
2
ℏ αa
provides allowed solutions to Schrödinger's equation
due to cosine term, right hand side varies between and +1 −1
◦ left hand side also allowed to vary between these two values
only certain values of are possible α
asα =
2mE
ℏ
, energy is restricted to lie within certain ranges
2
E
4.3 Band theory of solids
Energy vs wave number
relation between energy and wave number is parabolic
E k
motion of free electron interrupted at certain values of , shown by broken curve k
plot shows discontinuities in Energy of electron at k = ±
π
a
,±
2π
a
,±
3π
a
1 sections · English
Open in GPAI
,Contents
1 Electromagnetic Induction and Alternating Current
,Electromagnetic Induction and Alternating Current
4.1 Free Electron Theory
Basis of free electron theory
developed by Drude and Lorentz in 1900
valence electrons involved in electrical conduction in metals and alloys
valence electrons become free in solids → move randomly like gas molecules
◦ thus, known as free electrons
neglecting electron-electron and electron-ion interaction, electrons undergo periodic collision with ions in the lattice
Success of Free Electron Theory
successfully explains Ohm's law
Electrical conductivity of metal:
◦ explains electrical conductivities of metals
◦ Electrical conductivity σ = neμ
▪ : electron density
n
▪ : charge on electron
e
▪ : mobility of electrons
μ
Thermal Conductivity:
◦ successfully explains thermal conductivities of metals at lower temperatures
◦ thermal conductivity of metals K =
KB n ν λ
: Boltzmann constant
2
▪ KB
Relation between electrical and thermal conductivity (Wiedemann-Franz Law):
◦ ratio of thermal conductivity ( ) and electrical conductivity ( ) of a metal is proportional to temperature ( )
K σ T
◦ K /(σT ) = 1.11 × 10
−8
W ⋅ Ω/K
2
◦ also known as Lorentz's number
Limitations of Free Electron Theory
conductivity of metals proportional to electron concentration (according to theory)
◦ ex) divalent atoms (cadmium, zinc) and trivalent atoms (aluminum) should have more electrical conductivity than monovalent
atoms (copper, silver) → theory fails
cannot explain classification of solids into conductors, semiconductors, and insulators
theoretical values of mean free path of electrons do not agree with experimental values
theoretical values of specific heat and electronic specific heat do not agree with experimental values
could not explain positive values of Hall effect in some metals (ex: zinc)
Wiedemann-Franz law deviates at low temperatures
4.2 Opening of band gap due to internal electron diffraction
to find allowed energies of electrons in solids, Schrödinger's wave equation is applied for an electron in a crystal lattice
Kronig and Penney Model
suggested a simplified model for atoms as a one-dimensional infinite row of rectangular potential wells separated by barriers
◦ each well has width and depth
b V0
◦ barriers have width
b
interatomic spacing → period of potential is
a (a + b)
, electron moves through a lattice of positive ions → experiences varying potentials
Schrödinger's equation in one dimension
2
d ψ 2m
+
(E − V )ψ = 0
2 2
dx ℏ
Boundary conditions and Schrodinger's equation
in region 0 < x < a , potential energy V = 0
2
d ψ 2mE
+
ψ = 0
2 2
dx ℏ
can be written as
2
d ψ
2
+ α ψ = 0 for 0 < x < a
2
dx
where α =
2mE
ℏ
2
Solution using Bloch Theorem
solution for periodic potential found using Bloch theorem:
maV0 b sin(αa)
( )
+ cos(αa) = cos(ka)
2
ℏ αa
provides allowed solutions to Schrödinger's equation
due to cosine term, right hand side varies between and +1 −1
◦ left hand side also allowed to vary between these two values
only certain values of are possible α
asα =
2mE
ℏ
, energy is restricted to lie within certain ranges
2
E
4.3 Band theory of solids
Energy vs wave number
relation between energy and wave number is parabolic
E k
motion of free electron interrupted at certain values of , shown by broken curve k
plot shows discontinuities in Energy of electron at k = ±
π
a
,±
2π
a
,±
3π
a
