Download Part 2. Crystal Dynamics

PHYS 353 Solid State Physics
Supplemental Problems
Part 2. Crystal Dynamics
2-1. Show that if you substitute the proposed wave solution of Eq. (2.8) into the Newton’s Second Law
expression of Eq. (2.7), you do get the dispersion relation of Eq. (2.9).
2-2. Consider the dispersion relation for the one-dimensional atomic chain stated in Eq. (2.9).
(a) Find an expression for the group velocity of a displacement wave traveling along the chain.
Recall that the group velocity is d/dk.
(b) Find an expression for the maximum angular frequency that the chain can support.
(c) What is the group velocity if this maximum frequency is the main frequency of the group?
Explain why this answer makes sense if you examine the graph of the dispersion relation.
(d) Find an expression for the phase velocity of a wave at this maximum frequency.
2-3. Let’s model copper (fcc lattice with a basis of one atom) as a one-dimensional chain and calculate
approximate values for different quantities.
Elastic modulus:
Density:
Atomic mass:
Lattice constant:
1.2x1011 N/m2
8.92 g/cm3
63.55 amu (1 amu = 1.66x10-27 kg)
3.6 Å
(a) Find the speed of sound in copper using Eq. (2.3).
(b) Recall that these sound waves have very long wavelengths compared to a. Show that in this
limit, the dispersion relation in Eq. (2.9) gives the speed to be a K / M .
(c) By equating your answers to (a) and (b), find a value for the spring constant K.
(d) With this spring constant, show that the maximum frequency that can be supported by the
chain is around 3x1012 Hz.
(e) If electromagnetic radiation with this frequency hits a copper object, then it should be
strongly absorbed since its energy can be strongly coupled into lattice vibrations. In what
part of the spectrum is light with this frequency?
2-4. Let’s model gallium arsenide (fcc lattice with a two-atom basis) as a one-dimensional chain with
two types of atoms and calculate approximate values for different quantities.
Elastic modulus:
Density:
Atomic masses:
Lattice constant:
7.55x1010 N/m2
5.32 g/cm3
Ga 69.72 amu As 74.92 amu (1 amu = 1.66x10-27 kg)
5.65 Å
(a) Find the speed of sound in GaAs using Eq. (2.3).
(b) These sound waves are near point O in the dispersion relation plot in Fig. 2.7. Using this
fact, find an approximate value for the spring constant for the chain.
(c) Now show that the maximum frequency supported by the chain is around 4x1012 Hz. This
vibration is near point A on the optical branch of the dispersion equation plot. What type of
light has this frequency?
2-5. Eq. (2.29) gives the heat capacity due to a single mode of lattice vibration.
(a) Show that this heat capacity does tend to kB in the “high temperature” limit of kBT >> .
(b) Notice that the condition kBT >>  is equivalent to requiring T >> . Evaluate  for a
frequency of 1012 Hz. (Recall that this frequency is near the maximum allowed frequency for a
1-D chain and that light with this frequency is in the infrared part of the spectrum.) Does the
“high temperature” limit mean that the temperatures are really that high?
2-6. The figure below shows the Raman scattering spectrum of a sample of GaInP. It was produced by
illuminating the sample with 0.15 W of focused 488-nm light from an Argon-ion laser. The
scattered light was collected and its wavelength measured by a spectrometer. The figure shows the
relative intensity of the collected light in arbitrary units versus the “Raman shift”. This shift is k
= ko – k where ko is the wave number of the incident light and k is the wave number of the scattered
light. There are three peaks in the spectrum at 330 cm-1, 360 cm-1, and 380 cm-1 due to a transverse
optical (TO) phonon, an InP-like longitudinal optical (LO) phonon, and a GaP-like LO phonon.
The lattice constant of GaInP is around 5.7 Å.
(a) Find the wavelengths of the light collected at the three peaks and then calculate the individual
shifts in wavelength experienced by the photons at the three peaks. Does the spectrometer need
good resolution to detect these shifts?
(b) Show that the wave numbers ko and k are much smaller than the value of /a. This
demonstrates that this Raman scattering is occurring near the center of the dispersion zone and
is a normal process, not an Umklapp process. Thus, it is correct to assume that the wave
number of the incident photon is equal to the sum of the wave numbers of the scattered photon
and the phonon, i.e. that k is simply the wave number of the phonon. (We don’t have to add or
subtract some multiple of 2/a.)
(c) Using conservation of energy, calculate the energies of the three phonons in electronvolts.
About how many times smaller are these phonon energies compared to the photon energies?