1. Phonons and lattice vibration
Exercises 2
Phonons and lattice vibration

2. What is the magnitude of the force required to stretch a 20 cm-long spring, with a spring constant of 100 N/m, to a length of 21 cm? 
The spring changes from a length of 20 cm to 21 cm, hence it stretches by 1 cm or |Δx  | = 1 cm = 0.01 m.  | F | = k | Δx  | = 100 N / m × 0.01 m = 1 N
2. If the unit cell has 3 atoms, thus how many phonon modes are present
3N modes. Therefore 9 modes.
3. If the velocity of sound in a solid is taken to be 3 x 103m/s and interatomic distance as 5 x m, calculate the value of cutoff frequency assuming a linear lattice.
Velocity of sound in a solid (v) is 3 x 103m/s
Interatomic distance (a) is 5 x m
Velocity and frequency related equation:
Apply the known values in the equation.
Critical frequency (w) = 3× 1012 HZ

3. 4. Below is a displacement-time graph of a wave
4. Below is a displacement-time graph of a wave. (Displacement in metres is on the y-axis and time in seconds is on the x-axis.)
Determine: (a) the amplitude of the wave (b) the period of the wave (c) the frequency of the wave
2.2 m
4 s
0.25
5. A wave on a rope is shown below. (a) What is the wavelength of this wave? (b) If the frequency of this wave is 4 Hz, what is its wave speed? 3m
12m/s

4. 6. If the velocity of sound in a solid is of the order 10 3 m/s, compare the frequency of the sound wave λ = 20 Å for (a) a monoatomic system and (b) acoustic waves and optical waves in a diatomic system containing two identical atoms (M=m) per unit cell of interatomic spacing 2.2 Å.
Givenvalue:
Velocity of sound in a solid = 103 m/s
Sound wave λ = 20 x m
Interatomic spacing (a) = 2.2 ×10-10 m
In case of homogenous line, the frequency :

5. a. Acoustic wave in a diatomic ( identical M=m) lattice, the frequency varies from
b. Optical wave, the frequency varies from :

6. 7. The unit cell parameter of NaCl is 5
7. The unit cell parameter of NaCl is 5.65Å and the modulus of elasticity along [100] direction is 6 x 1010 N/m2. Estimate the wavelength at which an electromagnetic radiation is strongly reflected by the crystal. At. Wt. of Na = 23 and of Cl = 37.
Given value:
Unit cell parameter (a) = 5.65 x m
Modulus of elasticity (Y) = 6 x 1010 N/m2
The maximum frequency of radiation strongly reflected by the NaCl crystal will be given by:
Let us assume that an extension along [100] direction will have negligible effect on vertical springs. Therefore, we can write:

7. The frequency expression becomes
where M and m be the atomic weight of the crystal, and considered the atomic mass unit.
Atomic weight of chlorine (M) = 37
Atomic weight of sodium (m) = 23
Atomic mass unit = 1.67 × Kg
Hence the wavelength at which this radiation is strongly reflected is
where ω = 5.13×1013 rad/sec, c is the velocity of light then,