2014-10-17
due date: 2014-10-23
10 a.m.
Solid State Physics (MS613)
Homework #3
1. Neutrons of energy 0.02 eV are scattered at an angle of 10o from solid helium with emission of a
phonon. Estimate the energy loss of the neutrons. What is the time of flight over a 10 m path of
unscattered and scattered neutrons? (Hint: speed of lattice vibration  speed of sound = 300 ms-1).
2. We may make a model of the stretching vibrations of a polyethylene chain ㅡCH=CHㅡCH=CHㅡ
…by considering a linear chain of identical masses M with alternating force constants K 1 and K2.
Show that the characteristic frequencies of such a chain are given by
1/ 2
 
 
2 1
4 K1 K 2 sin
ka  
K1  K 2  
2
2

 
1  1 
 
M
K1  K 2 2  
 
 
 
where a is the repeat distance of the chain. By obtaining values for ω as k→0 and k→±π/a, sketch the
dispersion curves for the optical and acoustic branches of the phonon spectrum.
(Hint: the relative lengths of single and double bonds are irrelevant)
3. By treating the atoms as simple harmonic oscillators and assuming that the average thermal energy
of an atom at temperature T is kBT, show that the amplitude of the atomic vibrations (xmax) is given by
x max
 2k T 
 B 
 k 
1/ 2
where, k is force constant. Given that the actual value of k for copper is about 100 N/m and the atomic
spacing is 0.256 nm, estimate the amplitude of vibration of the atoms at 300K as a percentage of the
equilibrium separation.
4. Carry the expansion the equation (4.24) one step further and calculate the time average of the scattered
amplitude. Rewrite the expansion
1
1
)2
1 − ⟨(𝐾 ∙ 𝑢𝑛 )2 ⟩𝑡 ≈ 𝑒 2⟨(𝐾∙𝑢𝑛 ⟩𝑡
2
By equating the time average with the ensemble average, calculate the scattered intensity for a primitive lattice.
In the prefactor exp(-<(K∙un)2>t the quantity W=½<(K∙un)2> is known as the Debye-Waller factor. For
a harmonic lattice the result is correct even for arbitrarily large values of
<(K∙un)2>. The proof, however, is
not straightforward. Calculate the temperature dependence of the Debye-Waller factor assuming that all atoms
vibrate as independent oscillators of a frequency ħω by using 2 equations below.
1
M𝜔2 ⟨𝑆 2 ⟩𝑡 = (𝑛 + ) ħ𝜔
2
⟨𝑛⟩ 𝑇 =
1
𝑒 ħ𝜔/λT − 1
(Text below is extracted from the textbook in case not having textbook currently.)