4. A one dimensional material has a density of vibrational states g(s-')=A. {a} The total number of vibrational modes is 1N in 11:) where N is the number of atoms in the material. Find the constant A in terms of N and a Debye frequency (b}Derive an expression for the heat eapaeit'},r ofa 1D material to} What are the hightemperature and lowtemperature limits of the 1D heat capacity"