Soft modes and lattice transformations. Sketch a monatomic linear lattice of lattice constant a. (a) Add to each of six atoms a vector to indicate the direction of the displacement at a given time caused by a longitudinal phonon with wavevector at the zone boundary. (b) Sketch the crystal structure that results if this zone boundary phonon becomes unstable (w → 0) as the crystal is cooled through Tc. (c) Sketch on one graph the essential aspects of the longitudinal phonon dispersion relation for the monatomic lattice at T well above Tcand at T = Tc. Add to the graph the same information for phonons in the new structure at T well below Tc.

Soft modes and lattice transformations. Sketch a monatomic linear lattice of lattice constant a. (a) Add to each of six atoms a vector to indicate the direction of the displacement at a given time caused by a longitudinal phonon with wavevector at the zone boundary. (b) Sketch the crystal structure that results if this zone boundary phonon becomes unstable (w → 0) as the crystal is cooled through Tc. (c) Sketch on one graph the essential aspects of the longitudinal phonon dispersion relation for the monatomic lattice at T well above Tcand at T = Tc. Add to the graph the same information for phonons in the new structure at T well below Tc.