Eigenvalues wrt Crystal Oscillations

[Jon Fram]

Getting back to James' original question about the the importance of eigenstuff
in crystal oscillation...

Solid state physicists model crystals as repeating systems of harmonic
OSCILLATORS. There are 14 lattice types in three dimensions, and each type has
three basis vectors. For example, a simple cubic bravais lattice has three
mutually perpendicular basis vectors of equal magnitude. The postion of every
molecule (lattice point) in the crystal is then just a linear combination of
the basis vectors with respect to an origin. Even through the interaction force
is more complicated, physicists model the interaction between the molecules in
the crystals with springs connected between the the nearest neighbors of each
molecule. The equation of motion dictated by the spring forces is stuffed into
a matrix whose determinant yields EIGENVALUES. After a lot of messy math and
subtle physics, this microscopic model is used to explain macroscopic
properties like heat capacity, reflectivity, conductance, cool resonances... 

The harmonic OSCILLATOR -> EIGENVALUE method is amazingly powerful (considering
its simplicity). It is used extensively in every major field of physics I can
think of. 

The real question is why is he named Eigenvalue? What is so magically
fundamental about him? I dunno.

By the way James, why did you ask about "crystal oscillations" with respct to
eigenvalues in the first place?

For a more comprehensive and comprehensible explanation of eigenvalues and
crystal oscillations see Introduction to Solid State Physics by Charles Kittel. 

FRAM