Sem 2 PT2 Flashcards
Equation of motion og a 1D harmonic monatomic chain
πΆ(π’π+1 β π’π) β πΆ(π’π β π’πβ1) = βπΆ(2π’π β (π’πβ1 + π’π+1)) = ππ^2π’π/ππ‘2
Travelling wave solution
π’π(π‘) = π’0 exp[π(ππ₯ β ππ‘)]
Dispersion relation of monotomic 1D chain of atoms
π(π) = 2[β(πΆ/π)]|sin (ππ/2)|
Long wavelength limit for monotomic 1D chans
π£π = π£π = π(πΆ/π)^1/2
Short wavelength limit for monotomic 1D chans
For larger k (smaller Ξ»), Ο(k) is non-linear and for k = Β±Ο/a, the group velocity dΟ/dk is zero. For these
values the solution is a standing wave in which adjacent atoms move in anti-phase. So waves with k =
Β±Ο/a, i.e. Ξ» = 2a cannot propagate.
Wave vector outside first Brillouin zone
πβ² = π + 2ππ/π
Each value of k outside of the first Brillouin zone corresponds to the
same motion of the atoms as that of a k value inside the first Brillouin zone.
Coupled equation of motion of daitomic chain of atoms.
βπΆ (2π’π β (π’πβ1/2 + π’π+1/2)) = π1 *π^2π’π/ππ‘^2
and
βπΆ (2π’π+1/2 β (π’π + π’π+1)) = (π^2)π2π’π+1/2/
ππ‘2
equation of motion solutions of daitomic chain of atoms.
π’π = π΄ exp[π(πππ β ππ‘)]
Acoustic mode
The behaviour of Ο(k) for this root,
π = β(2πΆ/π1)
adjacent atoms move almost in phase
Optical mode
sqaure of acoustic mode
djacent atoms move almost in anti-phase
Low velcoity
Allowed K-space in 1D
K- states are uniformly spaced along k-axis with allowed wavelengths L/j
One allowed k-state per 2pi/L
The number of allowed k-states is equal to the number of atoms in the chain. There is one longitudinal
and two transverse modes for each k-state so the total number of modes is 3N.
Allowed k-space in 3D
K- states are uniformly spaced with one state per volume (2pi/L)^3
What is a phonon
Quantum energy of a lattice vibration
What is crystal momentum
When phonon behave as they have momentum hbar k
What are the allowed modes
k-states within the first Brillioun Zone
