3.
Debye solid in one dimension.
Consider a one-dimensional crystalline solid consisting of $N$ atoms; treat this according to the Debye model. The
frequencies of the longitudinal elastic waves in the crystal are given by $\omega_n = \pi c_s n/L$, where $c_s$ is the velocity
of sound, $L$ is the length of the crystal, and $n$ can take integer values $n = 1, 2, 3, \dots n_{max}$ so that the available
frequencies are bounded from above by $\omega_{max} = \pi c_s n_{max}/L$.
(a) How many modes are there in the Debye model in one dimension?
(b) Hence obtain the expression for the Debye temperature $T_D$ in terms of $N$ and other relevant parameters. If
$N = 3.5 \times 10^8$, $L = 10$ cm, and $c_s = 5000$ m/s, what is the Debye temperature in kelvins?
