Consider the phonon dispersion relation for a 10cm long GaAs
crystal, shown in the figure below.
(a) How can you obtain the phonon speed from the dispersion
relation? What is the speed of the phonons with the shortest
wavelength? (2 marks) (b) Which is more effective at carrying
energy across a solid: long wavelength or short wave- length
phonons? Why? (2 marks) (c) What is the wavelength of the
fundamental mode of this crystal? (1 mark) (d) In the long
wavelength limit, how many optical phonons would be created if 8.08
_10_21 J of heat was transferred to the crystal? (2 marks) (e)
Does the long wavelength limit correspond to low or high sample
temperatures? Explain. (1 mark) (f) In the long wavelenth limit,
would this sample of GaAs be transparent to light having a
frequency of 5.1 _1013 rad/s? How about 3.2 _1013 rad/s? (1 mark)
5.1 x 1013 rad/s – – – – – – – – – – – – – – – – 3.2 x 1013 rad/s o(4) _____________ == __E______________ 0 E/G Wavevector, k
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