As a first step towards something grander (I hope), I'm trying to
understand the dispersion for the monatomic simple cubic case.

Starting with a simple cubic crystal with 2 angstrom spacings I
construct a 4x4x4 chunk of crystal with a single atom type.  I place
springs between atoms that are within 3 angstroms, and applying
periodic boundary conditions each atom has 26 neighbors.  I construct
the dynamic matrix (64*3=192 dimensions) from the periodic spring
hessian for wavevectors(q) in a given direction, which is then used to
construct the dispersion relation (192 frequencies) as a function of
q.  From my understanding, this chunk of crystal should have the
effect of reducing the Brillouin zone by a factor of 4 (+- pi/4a) and
displace of the acoustic branches upwards to look like optical
branches (with no displacement at the boundaries).  To explore this, I
computed the dispersion for the dynamic matrix corresponding to one
atom in the crystal (3x3 matrix).  I find that indeed, the hunk of
crystal has the same acoustic modes as the single atom (displaced as
expected) which is nice, but the thing I can't  nail down conceptually
is all the extra stuff (from the 192 frequencies).  Should they (192)
all degenerate to the acoustic branches?  Am I missing something?

here is a link to the figure:

http://kandinsky.chem.wisc.edu/~riccardi/monatomic-sc-dispersion.html

Thank you for your time!

Demian