December 06. Lecture 12. 1. Lecture 12: Lattice vibrations. Quantised Lattice
vibrations: Diatomic systems in 1-D and in Phonons in 3-D. 6 Aims: ?
Lecture 12: Lattice vibrations Quantised Lattice vibrations: Diatomic systems in 1-D and in Phonons in 3-D Aims: Model systems (continued): Lattice with a basis: Phonons in a diatomic chain origin of optical and acoustic modes
Phonons as quantised vibrations Real, 3-D crystals: Examples of phonon dispersion: Rare gas solids Alkali halides
December 06
Lecture 12
1
Diatomic lattice Technically a lattice with a basis mmAA
mmBB
proceeding as before. Equations of motion are:
m Au2 n = α (u2 n +1 + u2 n −1 − 2u2 n )
mB u2 n +1 = α (u2 n + 2 + u2 n − 2u2 n +1 ) Trial solutions:
u2 n = U1 exp{ i(2nqa − ω t )}
u2 n +1 = U 2 exp{ i(( 2n + 1) qa − ω t )} substituting gives
(m
2 ω − 2α )U1 + (2α cos qa )U 2 = 0 A
(2α cos qa )U1 + (mB ω 2 − 2α )U 2 = 0
homogeneous equations require determinant to be zero giving a quadratic equation for ω2.
ω =
α
2
m A mB
{(m December 06
[(mA + mB ) ±
+ mB ) − 4m A mB sin qa 2
A
2
Lecture 12
}
12
Two Two solutions solutions for for each each qq
] 2
Acoustic and Optic modes Solutions q
0: Optic mode (higher frequency)
ω=
α 2(m A + mB ) m A mB
=
2α
Effective Effective mass mass µµ
µ
Acoustic mode (lower frequency)
ω2 ≈
α
m A mB
[(mA + mB ) −
(mA + mB ) 1 − 4 mA mB 2 (qa )2 (m A + m B )
12
]
2αa 2 ω ≈ q m A + mB ω= ω=√(2α √(2α/m /mBB))
ω= ω=√(2α √(2α/m /mAA))
December 06
Periodic: Periodic: all all distinguishable distinguishable modes modes lie lie in in |q|< |q|
