The equation has nontrivial solutions if the matrix
The classical equation of motion (11.1.28) is replaced by a system of equations in which the mass M m,μ of the μth atom in the mth primitive cell depends on the cell index m as well, and owing to the breakdown of translational symmetry force constants do not depend only on the separation of lattice points any more:įourier transformation with respect to time gives The previous method can now be generalized to three-dimensional lattices in which one atom is substituted by another atom of di erent mass, and thus force constants around the impurity are di erent from those in a regular crystal. It looks as if the impurity were resonating, therefore such states are called resonances.ġ1.5.2 Impurities in a Three-Dimensional Lattice Every atom of the crystal participates in these vibrations, however the displacement amplitude is much larger for the impurity than for distant atoms. Therefore somewhere around the middle of the spectrum frequencies are denser than elsewhere. They are shifted significantly, almost to the frequency just below.