oops, let's change that to: (see comments)
\(\gamma = 40 \: eV/nm^2\),
Also, let's use the mass of a proton, (the mc^2 for that) and a =0.2 nm. How does that sound?
a) Plot the lattice vibration frequency as a function of k.
b) what is the frequency of the zone boundary, (\(k = \pi/a\)), lattice vibration mode?
c) what is the characteristic velocity associated with sound propagation in this crystal*?
d) do you need another parameter to answer some of these? what is a reasonable value for it?
e) extra credit. what is the energy, in meV, of a zone boundary acoustic phonon in this crystal?
* extra credit. How does this speed compare with the speed associated with electron dispersion near the K point in graphene?
do we need a reduced mass m to solve this? or maybe a value of mc^2?
ReplyDeleteawesome thanks
ReplyDeleteDon't we need a value for a to find the velocity?
ReplyDeletegood point. does a =0.2 nm seem reasonable?
Deletegood points. I am realizing reading this that I made the value of gamma too small by a lot. I think we should change gamma to about 40 eV/nm^2.
ReplyDeleteThat seems awfully large, but it is the same as 0.4 eV/angstrom^2, which maybe looks more reasonable. The distances associated with nuclei movement in lattice vibrations are typically pretty small, like about 1/10 or 1/100 angstrom I think.