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 wave vector at the zone boundary. 

(b) Sketch the crystals 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 T_c, and at T = T_c. Add to the graph the same information for phonons in the new structure at T well below T_c.