Formula page

Formula page
1.
LATTICES
Reciprocal lattice G is defined by eiG·R = 1 with R the lattice vector.
Pd
ai is the primitive direct lattice vector R = i ai ni , d is the dimension, ni ∈ Z.
bi is the primitive reciprocal lattice vector ai · bj = 2πδij , or
b1 = 2π
a2 × a3
a1 · (a2 × a3 )
b2 = 2π
a3 × a1
a1 · (a2 × a3 )
b3 = 2π
a1 × a2
a1 · (a2 × a3 )
The primitive cell volume V = |a1 · (a2 × a3 )|.
Three dimensional Bravais Lattices:
2.
DIFFRACTION
Bragg condition |G| = 2 |k| sin φ/2, with k = 2π
λ the incident wave and λ the wavelength.
P
iG·di
The structure factor SG =
, with di the basis vectors, fi (G) the atomic form factor
i fi e
R
2
1
iG·r
fi (G) = − e r ρi (r)e
. The intensity is I ∼ |SG | .
2
3.
COHESIVE ENERGY
Lennard Jones potential
σ 12
σ 6 u = 2 A12
+ A6
r
r
X 1
An =
n
|R|
R6=0
2
Coulomb energy u = − ed α , α =
(−1)j
j |rj /d| .
P
4.
PHONONS
The Harmonic energy with potential φ(r)
U Harmonic =
1 X
2
X
[uµ (R1 ) − uµ (R2 )]
R1 ,R2 µ,ν=x,y,z
∂ 2 φ(r)
[uν (R1 ) − uν (R2 )]
∂rµ ∂rν
∂U
The equations of motions are mü(R) = − ∂u(R)
, with periodic boundary conditions we seek solution of the form
ik·R−iωt
u=e
, the polarization vector.
In k space the dynamical matrix D̈(k)u = M ω 2 u. For oscillation in d dimensions of p atoms in a unit cell, there are
d acoustic branches and (p − 1)d optical branches.
The specific heat
Z
dk
∂ X
~ωs (k)
cV =
∂T s F BZ (2π)3 eβ~ωs (k) − 1
Specific heat in low temperature, cV =
Lattice energy is U =
R
ω
2π 2 kB
5
kB T 3
,
~c
with 1/c3 the angular average of the velocity speed 1/c3s (k).
g(ω)f (ω)~ω, with the density of states g(ω) =
∂n
∂ω
,
d3 k
(2π)3
= g(ω)dω.
In an alternative form
g(ω) =
XZ
s
X
d2 k
δ(ω − ωk ) =
3
(2π)
s
Z
d2 k δ(k − k(ω))
(2π)3 |∇ωs (k)|
where the integral is over the FBZ.
Debye model replaces all branches with the acoustic branches, each branch has linear dispersion, ω = ck, in addition
the integral over FBZ is replaced by an integral over a sphere (circle in two dimensions) of radius kD . Debye
temperature is ΘD = ~ωD /kB = ~ckD /kB .
Einstein model replaces the frequency of each optical branch by ωE , a constant in k.
3
5.
ELECTRONS
The Drude conductivity is σ = ne2 τ /m, with n the number of electron per unit volume, and τ the mean free time.
The equation of motion for the momentum in Drude theory is
p
dp
p
= −e E +
×H −
dt
mc
τ
2
The cyclotron frequency is ωc = eH
mc , the plasma frequency is ωp =
conductivity tensor σ is defined by j = σE.
4πne2
m
the current density is j =
ne
m p,
and the
The occupation functions are
f (ω) =
 1
 eβ~ω +1

Fermi-Dirac
1
Bose-Einstein
eβ~ω −1
Fermi surface (k) = F . Sommerfeld expansions are
Z
∞
Z
µ
H(x)dx +
H(x)f (x)dx =
−∞
−∞
π2
7π 4
(kB T )2 H 0 (µ) +
(kB T )4 H 000 (µ) + . . .
6
360
0
2
π g (F )
(kB T )2
6 g(F )
π2
g(F )(kB T )2
u = u0 +
6
µ = F −
The specific heat of electron gas in low temperature is cv =
g() = ∂n()
∂ .
∂u
∂T
=
π2 2
3 kB T g(F ),
where the density of states is
Bloch theorem ψ(r + R) = eik·R ψ(r), electron velocity in energy En (k): vn (k) = ~1 ∇k En (k).
Effective mass tensor
1 ∂ 2 (k)
1
=
m∗i,j
~2 ∂ki ∂kj
i, j = x, y, z
dk
Therefore, m∗ dv
dt = ~ dt
Electron k in a periodic potential
0k−G ck−G +
X
UG0 −G ck−G0 = k−G ck−G
G0
where UG =
1
vp.u.c
R
p.u.c
U (r)e−iG·r dr are the Fourier components of the potential, p.u.c = primitive unit cell. the
ck−G are the Fourier components of the wave function and 0k−G =
For a weak potential, near a couple of degenerate states
1
= (0k + 0k−G ) ±
2
0k − 0k−G
2
2
+ |Uk |
1/2
~
2m (k
− G)2 is the free electron energy.
4
Far from degenerate states
2
k = 0k +
X
G
|UG |
0k − 0k−G
Tight Binding model
(k) = Es − β −
X
γ(R) cos kR
where Es is the energy of atomic s-level, and
Z
2
β=−
dr∆U (r) |φ(r)|
Z
γ(R) = − drφ∗ (r)∆U (r)φ(r − R)
6.
CONSTANTS, INTEGRALS, MISC
Constants:
1.054 · 10−34 [J s]
1.38 · 10−23 [J/K]
6.022 · 1023 [1/mole]
9.11 · 10−31 [kg]
1.6 · 10−19 [C]
=
=
=
=
=
~
kB
NA
me
e
Integrals:
∞
Z
0
1
Z
Z
0
A
lim
A→0
Z
0
∞
0
∞
Z
0
Z
Z
1
0
A
lim
A→0
x4 e x
4π 2
dx =
2
− 1)
15
(ex
0
x4 e x
dx ≈ 0.32
(ex − 1)2
x4 e x
A3
dx
=
(ex − 1)2
3
x3 e x
dx = 6ζ(3) ≈ 7.2
(ex − 1)2
x4 e x
7π 4
dx
=
(ex + 1)2
30
x4 e x
dx ≈ 0.042
+ 1)2
(ex
x4 e x
A5
dx
=
(ex + 1)2
20
Ellipse equation and its area:
x 2
a
+
y 2
b
=1 ,
A = πab