1. Phonons I: Crystal vibrations
one dimensional vibration
one dimensional vibration for crystals with basis
three dimensional vibration
quantum theory of vibration
M.C. Chang
Dept of Phys

2. One dimensional vibration
consider only the longitudinal motion
consider only the NN coupling
analyzed by Newton’s law of motion (classical)
dispersion relation (色散關係)

3.  Dispersion curve k /a -/a
(redundant)
The waves with wave numbers k and k+2/a describe the same atomic displacement
Therefore, we can restrict k to within the first BZ [-/a,/a]

4. Displacement of the n-th atom
Pattern of vibration:
k = 0時, exp(ikXn)=1.
所有原子同步振動，此時其恢復力接近於零。
k = /a時,exp(ikXn) = (-1)n.
相鄰的原子反向振動，形成駐波。此時恢復力最大。
(Similar to Bragg reflection)
Velocity of wave:
長波的頻率為 = (Ma/2)k。
這時候相速度/k與群速度d/dk的值相同。
在k = /a時由於振動波為駐波，所以群速度為零。

5. More on the group velocity (Stokes, Rayleigh, 1876/77)
The total energy E associated with a wave of amplitude f(x,t) is
Each harmonic wave has its own phase velocity k/, which is different from the velocity of the energy flow when the medium is dispersive (i.e. -k relation is not linear)
If F(k) is narrow (if not, more complicated), then we can approximate
Therefore, the waveform f(x,t), and hence the energy flow, moves at the group velocity!

6. How many different vibrations with different k’s?
For travelling waves, use periodic boundary condition
PBC: u0(t) = uN(t)
Consider a short 1-dim lattice with only 8 atoms
The value of k, like x, is discretized
k=2/Na  0 as N 
Each k describes a normal mode of vibration (i.e. a vibration with a specific frequency)

7. Crystal vibrations
one dimensional vibration
one dimensional vibration for crystals with basis
three dimensional vibration
quantum theory of vibration

8. Vibration of a crystal with 2 atoms in a unit cell

9. a d c b Two branches of dispersion curves (assume M2 > M1)
光頻支 d b c
聲頻支
Patterns of vibration
similar
See a very nice demo at

10. Crystal vibrations
one dimensional vibration
one dimensional vibration for crystals with basis
three dimensional vibration
quantum theory of vibration

11. Three dimensional vibration Along a given direction of propagation, there are one longitudinal wave and 2 transverse waves, each may have different velocities
Sodium (bcc)

12. Crystal with atom basis
FCC lattice with 2-atom basis
cm-1
Rules of thumb
一塊三維晶體如果每個單位晶胞有p種不同原子,則其聲子譜會有
3條acoustic branches, 3(p1)條optical branches。
一塊三維晶體如果有N個單位晶胞，則每個分支會有N個normal modes。
因此總振動模式數為 3pN (= total DOF of this system)。

13. Crystal vibrations
one dimensional vibration
one dimensional vibration for crystals with basis
three dimensional vibration
quantum theory of vibration

14. Quantization of a1-dim simple harmonic oscillator (DOF=1)
Review:
Quantization of a1-dim simple harmonic oscillator (DOF=1)
Classically, it oscillates with a single freq =(/M)1/2
The energy  can be continuously changed.
After quantization, the energy becomes discrete
is the energy of an energy quantum
The number n of energy quanta depends on the amplitude of the
oscillator.

15. Quantization of a 1-dim vibrating lattice (DOF=N)
Classically, for a given k, it vibrates with a single frequency (k).
The amplitude ( and hence energy ) can be continuously changed.
After quantization, the vibrational energy of the lattice becomes discrete (see Appendix C)
For a given k (or wave length), the energy quantum is
and the number of energy quanta (called phonons) is nk.
There are no interaction between phonons, so the vibrating lattice can
be treated as a “free” phonon gas.
Total vibrational energy E of the lattice = summation of phonon energies.

16. The total vibrational energy of the crystal is
k=2m/L, (L=Na, m=1…N). If we write nk as nm, then the microscopic vibrational state of the whole crystal (state of the phonon gas) is characterized by
At 0o K, there are no phonons being excited. The hotter the crystal, the more the number of phonons.
In 1-dim, if there are two atoms in a unit cell, then there are two types of phonons with the following energies:
The total vibrational energy of the crystal is
In general, for a crystal with more branches of dispersion curves, just add in more summations.

17. uniform translation of the crystal
A k-mode phonon acts as if it has momentum k in the scattering process (for a math proof, see Ashcroft and Mermin, App. M)
Recoil momentum of the crystal
Elastic scattering of photon: k’ = k + G (chap 2)
Inelastic scattering of photon: k’ = k  kphonon + G (Raman scattering)
However, the physical momentum of a vibrating crystal with wave vector k is zero
uniform translation of the crystal
no center-of-mass motion
Therefore, we call k a crystal momentum (of the phonon), in order not to be confused with the usual physical momentum.