3. Crystal vibrations 3. Crystal vibrations 3.1. Linear chain - - - PowerPoint PPT Presentation

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Mar 04, 2023 293 likes •364 views

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 M 1 , s n1 M 2 , s n2 a optical (n-1)-th unit cell (n+1)-th unit cell n-th unit

SLIDE 1

75

a

3. Crystal vibrations
3. Crystal vibrations

3.1. Linear chain with two atoms per unit cell

n-th unit cell (n+1)-th unit cell (n-1)-th unit cell M1, sn1 M2, sn2

76

3.1. Linear chain

3. Crystal vibrations

77

3.1. Linear chain - dispersion relation

3. Crystal vibrations

acoustical branch

ptical

branch

78

3.1. Linear chain – atomic displacements

3. Crystal vibrations

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! Dispersion relation for the diatomic linear chain.

source: Th. Fauster

SLIDE 2

79

3.2. Lattice vibrations (3 dim.)

3. Crystal vibrations

q

Longitudinal mode Transversal mode lattice planes

80

3.2. Lattice vibrations (3 dim.)

3. Crystal vibrations

81

3.2. Lattice vibrations (3 dim.)

3. Crystal vibrations

Example: (Si, Ge, diamond) fcc lattice with basis of 2 atoms (a = 2) 6 branches: 1x LA + 2x TA 1x LO + 2x TO

Data points stem from inelastic neutron diffraction

82

3.3. Thermal properties

3. Crystal vibrations

Bose Verteilung

SLIDE 3

83

3.3. Debye approximation

3. Crystal vibrations

Debye wave-vector Debye temperature (material property)

84

3.3. Debye specific heat

3. Crystal vibrations

Good approximation for small and large T

85

3.3. Debye specific heat

3. Crystal vibrations

Specific heat of solid argon ΘD = 92 K

T3 – law at low temperatures

2 K 0 K

86

3.4. Density of states

3. Crystal vibrations

regions with low dispersion ω(q) ↔ high density of states

Example: Si

SLIDE 4

87

3.4. Debye approximation

3. Crystal vibrations

86

3.4. Density of states

3. Crystal vibrations

regions with low dispersion ω(q) ↔ high density of states

Example: Si

87

3.4. Debye approximation

3. Crystal vibrations

88

3.5. Anharmonic Effects

3. Crystal vibrations

Thermal Expansion

a (Å) T (K) Silicon Solid Argon γ negative

75

a

3.1. Linear chain with two atoms per unit cell

n-th unit cell (n+1)-th unit cell (n-1)-th unit cell M1, sn1 M2, sn2

76

3.1. Linear chain

77

3.1. Linear chain - dispersion relation

acoustical branch

branch

78

3.1. Linear chain – atomic displacements

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! Dispersion relation for the diatomic linear chain.

source: Th. Fauster

79

3.2. Lattice vibrations (3 dim.)

q

Longitudinal mode Transversal mode lattice planes

80

3.2. Lattice vibrations (3 dim.)

81

3.2. Lattice vibrations (3 dim.)

Example: (Si, Ge, diamond) fcc lattice with basis of 2 atoms (a = 2) 6 branches: 1x LA + 2x TA 1x LO + 2x TO

Data points stem from inelastic neutron diffraction

82

3.3. Thermal properties

Bose Verteilung

83

3.3. Debye approximation

Debye wave-vector Debye temperature (material property)

84

3.3. Debye specific heat

Good approximation for small and large T

85

3.3. Debye specific heat

Specific heat of solid argon ΘD = 92 K

T3 – law at low temperatures

2 K 0 K

86

3.4. Density of states

regions with low dispersion ω(q) ↔ high density of states

Example: Si

87

3.4. Debye approximation

86

3.4. Density of states

regions with low dispersion ω(q) ↔ high density of states

Example: Si

87

3.4. Debye approximation

88

3.5. Anharmonic Effects

Thermal Expansion

a (Å) T (K) Silicon Solid Argon γ negative

89

3.5. Anharmonic Effects

Heat conduction by phonons

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