Trial functions of the form
are used to find solutions for the system.
is a vector displacement for the mass in unit cell
is the position of the unit cell
are the unit cell vectors;
is a vector in reciprocal space. The vectors
together form an eigenvector of the equations of motion of the system for this assumed solution, where
are the eigenvalues of this system. For this two-atom basis, there are four such
point, each with a different characteristic motion. A control is available to pick between these characteristic angular frequencies.
Free parameters available as controls in this Demonstration include the spring constants
, the masses
and the primitive unit cell lattice vectors
Changing any of these free parameters requires an update selection ("U" in the upper right corner) to make them take effect.
Three continuously updated controls are available:
. These can be varied for each selection of the set of free parameters, after update.
Selecting run for the time control will animate the lattice vibrations.
General theory describing oscillations around the lattice equilibrium points can be found in Chapters 21 and 22 of .
 N. W. Ashcroft and N. D. Mermin, Solid State Physics,
New York: Holt, Rinehart and Winston, 1976.