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5/24/2018 Phon Script
1/12
SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
Lattice vibrations
5/24/2018 Phon Script
2/12
SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
Lattice vibrations
The motion of atoms in a linear chain is coupled, giving rise topropagating waves
The frequency of oscillation depends on the wavelength (i.e.
the wave vector) of the propagating wave
For an infinite chain, the possible frequency of oscillations is a
continuous
For a finite chain of quantum oscillator, only a discrete set offrequencies is possible
Each propagating wave with a certain frequency and hence acertain group velocity is called a phononphonon
The frequency of atomic vibrations in a phonon depends on the
phonon wave vector k: This defines the dispersion relation.
Phonon wave vectors for a 1D chain of length L are n2/L,
where n is integer. Number of phonons is
All phonon wave vectors lie between /a and /a. Therefore
the number of phonons in a 1D chain is 2/a / 2/L = L/a
Each phonon can be treated itself as a quantum oscillation. Forlow temperatures every atom can be approximated by an
harmonic oscillator, the energy of the oscillation is
RealReal spacespace: For a: For a chainchain withwith 1313 atomsatoms
separatedseparated byby a;a; lengthlengthLL isis12 a12 a
ReciprocalReciprocal spacespace:: LargestLargest frecuencyfrecuency appearsappearsatat thethe largestlargest kk valuesvalues/a and/a and /a/a
DiscreteDiscrete wavewave vectorvector valuesvalues nn22/L= n/L= n/6a/6a
In total 12In total 12 modesmodes betweenbetween/a and/a and /a/a
5/24/2018 Phon Script
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
Lattice vibrations
22aa
PropagatingPropagating wavewave
withwith infiniteinfinite
wavelengthwavelength ((allall
atomsatoms inin phasephase))
PropagatingPropagating
wavewave withwith
wavelengthwavelength 2a2a
((allall atomsatoms inin
antiphaseantiphase))
ItIt isis onlyonly neededneeded toto knowknow thethe dispersiondispersion relationrelation inin thethe rangerange betweenbetween andand--
aa
aa
5/24/2018 Phon Script
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
Two atoms per unit cell
equations of motion
two linear equations, two unknowns
(system of homogeneous linear equations)
ansatz
5/24/2018 Phon Script
5/12
SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
Two atoms per unit cell
this has only a solution when
coefficient matrix
two solutions for every value of k
5/24/2018 Phon Script
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
Two atoms per unit cell
5/24/2018 Phon Script
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
M1 und M2 schwingen in Phase
Akustische und optische Gitterschwingungen
M1 und M2 schwingen gegenphasig
5/24/2018 Phon Script
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
Periodic boundary conditions
Max Born and Theodore von Karman (1912)chain with N atoms:
longest wavelength for wave solutions
This restricts the possible k values
So there are N possible different vibrations (m=0....N-1)
5/24/2018 Phon Script
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
Finite chain with 10 unit cells and one atom per unit cell
N atoms give N so-called normal modes of vibration.
For long but finite chains, the points are very dense.
5/24/2018 Phon Script
10/12
SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
Long atomic chain: quantum model
The excitations of these oscillators are called phonons.
Strong analogy with photons: both bosonic excitations Both described by quantum mechanical harmonic oscillators Wave-particle duality
5/24/2018 Phon Script
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
Phonons in 3D crystals: Aluminium
Results from inelastic x-ray scattering / neutron scattering.
5/24/2018 Phon Script
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Lattice vibrations
Phonons in 3D crystals: diamond
Results from inelastic x-ray scattering / neutron scattering.
Acoustic and optical branches present.