3. Crystal vibrations
3. Crystal vibrations
3.1. Linear chain - dispersion relation
3. Crystal vibrations
75
77
3.1. Linear chain with two atoms per unit cell
M1, sn1
M2, sn2
a
(n-1)-th unit cell
n-th unit cell
optical
branch
(n+1)-th unit cell
acoustical
branch
3.1. Linear chain
3. Crystal vibrations
3.1. Linear chain – atomic displacements
3. Crystal vibrations
76
78
Dispersion relation for the
diatomic linear chain.
Displacement of atoms in a diatomic
linear chain for q ~ 0 and q = π/a.
Note: to understand this figure most of us
will need at least 10 minutes!
source: Th. Fauster
3.2. Lattice vibrations (3 dim.)
3. Crystal vibrations
3.2. Lattice vibrations (3 dim.)
3. Crystal vibrations
79
81
Example: (Si, Ge, diamond)
fcc lattice with basis of 2 atoms (a = 2)
lattice planes
6 branches:
1x LA + 2x TA
1x LO + 2x TO
q
Longitudinal mode
Transversal mode
L
Data points stem from inelastic neutron diffraction
3.2. Lattice vibrations (3 dim.)
3. Crystal vibrations
3.3. Thermal properties
80
3. Crystal vibrations
82
Bose Verteilung
3.3. Debye approximation
3. Crystal vibrations
3. Crystal vibrations
3.3. Debye specific heat
83
85
T3 – law at low temperatures
Specific heat of solid argon
ΘD = 92 K
Debye wave-vector
Debye temperature (material property)
3.3. Debye specific heat
3. Crystal vibrations
0K
2K
3.4. Density of states
3. Crystal vibrations
86
84
Example: Si
Good approximation for small and large T
regions with low dispersion ω(q) ↔ high density of states
3.4. Debye approximation
3. Crystal vibrations
3. Crystal vibrations
3.4. Debye approximation
87
3.4. Density of states
3. Crystal vibrations
87
3. Crystal vibrations
3.5. Anharmonic Effects
86
88
Thermal Expansion
a (Å)
Example: Si
Solid Argon
Silicon
T (K)
γ negative
regions with low dispersion ω(q) ↔ high density of states
3. Crystal vibrations
3.5. Anharmonic Effects
89
No umklapp:
Heat conduction by phonons
NaF
Umklapp: