5.4 Using the Debye approximation for a onc-dimensional monatomic lattice with atomic spacing a and sound speed v. show that TCU hwp and 5.5 For Problem 5.4, derive the integral expressions for the thermal energy and phonon heat capacity. Show that n'k T Cy per unit length at low temperatures %3D a per unit length at high temperatures. 5.6 Solve the above problem for a two-dimensional crystal and show that at low temperatures Cy c T?.