Albert Einstein (1879-1955)
“ Imagination is more
important than knowledge.
For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and
understand. ”
Einstein
Einstein
Stimulated Emission Stimulated Emission
( ) ( )
i ij j ji ji
N B u ν = N ⎡ ⎣ A + B u ν ⎤ ⎦
At equilibrium
Stimulated Emission Stimulated Emission
( ) / /
3 3
/
( ) ( )
( ) /
1
8 /
( )
1
j i B B
B
i ij j ji ji
ji ji
i ij
j ji
E E k T h k T
i j
ij h k T
ji
N B u N A B u A B
u N B
N B
N e e
N
h c
u B
B e
ν
ν
ν ν
ν
ν π ν
−
⎡ ⎤
= ⎣ + ⎦
= ⎛ ⎞⎛ ⎞
⎜ ⎟⎜ ⎟ −
⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
= =
= ⎛ ⎞
⎜ ⎟ −
⎜ ⎟
⎝ ⎠
Boltzmann factors
Planck’s distribution
Emission’s A, B Coefficients Emission’s A, B Coefficients
3 3
8
ij ji
ji ji
B B B
A A h B
c π ν
= ≡
≡ =
Stimulated emission should exist!
( )
0
0
3 3
internal energy of 1kilomole of soild : Avogadro's number
3 3
B
V
V
V
E N k T RT
N c dE
dT d RT
c R
dT
= =
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
= =
Specific Heat of Solids Specific Heat of Solids
Dulong-Petit law
( )
/
0
2 /
/ 2
1 3
3
1
B
B
B
h k T
h k T
V h k T
V B
h e
E N
dE h e
c R
dT k T e
ν
ν ν
ε ν
ε
ν
= −
=
⎛ ⎞
⎛ ⎞
= ⎜ ⎟ = ⎜ ⎟
⎝ ⎠ ⎝ ⎠ −
Einstein’s Theory of Specific Heat of Solids
Einstein’s Theory of Specific Heat of Solids
Coupled harmonic oscillators
Debye’s Theory of Specific Heat of Solids Debye’s Theory of Specific Heat of Solids
Phonons
Fermi-Dirac Distribution
Fermi-Dirac Distribution
( )
( )
3/ 2 3
/
0 0 /
( )
8 2
( ) 1
( ) 1
F B
F B
E E k T
E E k T
g E dE D EdE D m
h
D EdE n E dE
e
N EdE
n E dE D
V e
π
−
∞ ∞
−
=
=
= +
= =
∫ ∫ +
Density of States in 3D Structure (Metal)
Density of States in 3D Structure (Metal)
3/ 2 0
2 / 3 2
At 0
2 3 (0) 3
2 8
E
FF
F
T K
N D EdE DE V
h N
E m π V
=
= =
⎛ ⎞
= ⎜ ⎝ ⎟ ⎠
∫
Fermi Energy in Metal
Fermi Energy in Metal
표
9.2
페르미 에너지eV
