Particle in a Box
0 < x < Lx ,
0 < y < Ly ,
0 < z < Lz
ψkx,ky ,kz (r
r) ∝ sin kxx sin ky y sin kz z
π
sin kxLx = 0 ⇒ kx = nx
Lx
�
ky
π/Lx
π/Ly
kx
�
nx = 1, 2, 3, · · ·
Lx Ly Lz
V
r
D(k)wavevectors =
= 3
π π π
π
V
r
D(k)states = (2S+1) 3
π
8.044 L19B1
ψrk (r
r) ∝ exp[irk · r]
Plane Wave
Periodic Boundary Conditions
ψrk (r
r + mxLxx̂ + my Ly ŷ + mz Lz ẑ) = ψrk (r
r)
2π
ik
(x+m
L
)
ik
x
x
x
x
x
e
=e
⇒ kx = nx
Lx
�
ky
�
nx = ±1, ±2, ±3, · · ·
Lx Ly Lz
V
r
D(k)wavevectors =
=
2π 2π 2π
(2π)3
2π/Lx
2π/Ly
kx
V
r
D(k)states = (2S+1)
(2π)3
8.044 L19B2
The # of wavevectors with |-k'| < k is the same in
both cases.
1
#wavevectors(k) =
8
4 3 V
4 3
V
πk
=
πk
3
3
π
3
(2π)3
n2k2
For free particles; E =
→ k(E) =
2m
2m √
E
2
n
3/2
4
V
V
2m
3/2
#wv(E) =
πk(E)3
=
E
3
(2π)3
6π 2 n2
d
V
2m 3/2 1/2
E
Dwv(E) = #wv(E) =
2
2
dE
4π
n
8.044 L19B3
For free particles
V
2m
Dstates(E) = (2S + 1) 2
4π
n2
�
�3/2
E1/2
D(ε)
ε
8.044 L19B4
Fermions: Non-interacting, free, spin 1/2, T = 0
kz
D( ε)
kF
ky
εF
ε
kx
Fermi sea
Fermi wave vector kF
Fermi surface
2 /2m
Fermi energy EF = n2kF
8.044 L19B5
4 3
8
V
3
N = 2 × Dwavevectors(k) × πkF = π
k
3
3 (2π)3 F
kF =
EF =
�
�1/3
2
3π (N/V )
2
n2 k F
2m
=
n2
⎛
2m
⎝
∝ (N/V )1/3
⎞2/3
2
3π N
V
⎠
∝ (N/V )2/3
8.044 L19B6
D(�) = a �1/2
N =
U =
� �
F
0
� �
F
0
D(�) d� =
�D(�) d� =
2 a �3/2
3 F
2 a �5/2
5 F
3N �
=5
F
∝ N (N/V )2/3
8.044 L19B7
Consequences: Motion at T = 0
p
P = h̄Pk
p
PF = h̄PkF
h̄ k
vF = m
F
Copper, one valence electron beyond a filled d shell
N/V = 8.45 × 1022 atoms-cm−3
kF = 1.36 × 108 cm−1
vF = 1.57 × 108 cm-s−1
pF = 1.43 × 10−19 g-cm-s−1
EF /kB = 81, 000K
8.044 L19B8
Consequences: P (T = 0)
U =3
5 N EF
EF ∝ (N/V )2/3
∂U
∂EF
2 (N/V )E
3
P =−
= −5N
= 5
F
∂V N,S
∂V
NN
'
�
�
�
�
EF
−2
3 V
∝ (N/V )5/3
�
∂P
5 (P/V ) at T = 0
= − 3
∂V N,T
�
8.044 L19B9
1
KT = −
V
∂V
31
3
1
=
=
∂P N,T
5P
2 (N/V )EF
For potassium
EF = 2.46 × 104K = 3.39 × 10−12 ergs
(N/V )conduction = 1.40 × 1022 cm−3
−12 cm3-ergs−1
KT = 1.40×10221.5
=
31.6
×
10
×3.39×10−12
The measued value is 31 × 10−12 !
8.044 L19B10
Magnetic Susceptibility
ε
εF
� = Hẑ
H
k,↑
= (k) + µeH
k,↓
= (k) − µeH
µe H
D ( ε)
D ( ε)
8.044 L19B11

N↓ − N ↑ =




N
D(EF )   N
D(EF ) 
+ µe H
− µe H
−
2
2
2
2
= µeHD(EF )
⇒
M = µ2
e
HD(EF ) ≡ χH
χ = µ2
e
D(EF )
This expression holds as long as kT « EF , so χ is
temperature independent in this region. This is not
Curie law behavior. It is called Pauli paramagnetism.
8.044 L19B12
Temperature Dependence of < n�k,m >
s
< n�k,m >= f (E) only, as in the Canonical Ensemble
s
<n>
~ kT
1
T=0
T <<
εF
εF/ k B
ε
8.044 L19B13
Estimate CV
Classical Quantum
# electrons
N
# electrons influenced N
N
∼ N × kT
�F
Δ�
3 kT
2
ΔU
3 N kT
2
∼N
CV
3N k
2
∼ 2N k kT
�
Exact result: CV =
∼ kT
(kT )2
�F
F
π 2 N k kT
�F
2
8.044 L19B14
N =
� ∞
0
< n(�, µ(T ), T ) > D(�) d�
This expression implicitly determines µ = µ(T ).
U =
� ∞
0
< n(�, µ(T ), T ) > � D(�) d�
To determine CV from this expression one must
take into account the temperature dependence of
µ in addition to the explicit dependence of < n >
on T .
8.044 L19B15
<n>
~ kT
1
T=0
T <<
εF
εF/kB
ε
[< n > −1/2]] = − [[< n > −1/2
]]
at E = µ + δ
at E = µ − δ
[
8.044 L19B16
D(ε)
1
ε
µ
εF
Because D(�) is an increasing function of �, µ must
decrease with increasing temperature.
8.044 L19B17
Elementary Excitations Excitations out of the ground
state in interacting many-body systems.
For a 3D Coulomb gas of electrons
Plasma oscillations (plasmons)
ε(k)
Collective modes: H.O.s
Quasi-particles
(‘dressed’ electrons)
=
2k2
2m∗
k
8.044 L19B18
Other collective modes
Phonons, lattice vibrations in solids
Spin waves, in Ferromagnets
Ripplons, waves on surfaces
Other quasi-particles
Polarons, (electron+lattice distortion)
in ionic materials
8.044 L19B19
Some possible electronic densities of states in solids
D(ε)
D(ε)
metal
εF ε
D(ε)
εF
ε
wide
D(ε)
intrinsic semiconductor
ε
insulator
doped semiconductor
εF
ε
narrow
8.044 L19B20
Finding µ(T ) in an intrinsic semiconductor
1
<n>
D(ε)
0
µ
εG
ε
εF
Assume the energy gap is � kB T .
8.044 L19B21
1
-5
-4
-3
-2
-1
0
1
2
3
4
5
Comparison of 1/(ex + 1) with e−x when x > 0 and
1 − ex when x < 0.
8.044 L19B21a
< ne > → e−(E−µ)/kB T = e−(EG−µ)/kB T e−(E−EG)/kB T
�
�
2me 3/2
V
1/2
Dstates,e(E) =
(E
−
E
)
G
2π 2 n2
Ne =
� ∞
EG
< ne > D(E) dE
�
�
V
2me 3/2 −(EG−µ)/kB T � ∞ √ −δ/kB T
=
e
dδ
δe
2
2
0
2π
n
V 2mekB T
=
4
πn2
�
�3/2
e−(EG−µ)/kB T
8.044 L19B22
< nh > = 1− < ne > → e−(µ−E)/kB T
V
2mh 3/2
1/2
Dstates,h(E) =
(−E)
2π 2 n2
Nh =
0
−∞
< nh > D(E) dE
V
2mh 3/2 −µ/kB T ∞ √ −δ/kB T
=
e
δe
dδ
2
2
0
2π
n
V 2mhkB T 3/2 −µ/kB T
=
e
2
4
πn
8.044 L19B23
N h = Ne
3/2 −µ/kB T
e
mh
3/2 −(�G−µ)/kB T
e
= me
(mh/me)3/2 = e−�G/kB T e2µ/kB T
(3/2)kB T ln(mh/me) = −�G + 2µ
µ = �G/2 − (3/4)kB T ln(me/mh)
8.044 L19B24
MIT OpenCourseWare
http://ocw.mit.edu
8.044 Statistical Physics I
Spring 2013
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