PC2352 Examples 8

19.

Exam question, 1995. The diatomic molecules of a certain gas (each with moment of inertia $I$) have rotational energy levels given by

$$E_l = \frac{\hbar ^2}{2I} l(l+1) ; l=0, 1, 2, \ldots$$

with degeneracy $g_l = (2l + 1)$.

Show that the one-particle rotational partition function is given by

$$Z_1 (T) = 1 + 3e^{- \hbar ^2 /Ik_{\scriptscriptstyle B}T}$$

in the low temperature limit ( $T \ll \hbar ^2 /k_{\scriptscriptstyle B}I$) and by

$$Z_1 (T) \approx 2Ik_{\scriptscriptstyle B}T/\hbar ^2$$

in the high-temperature limit ( $T \gg \hbar ^2 /k_{\scriptscriptstyle B}I$).

[In the second case the summation over $l$ may be replaced by an integral (Why?) which may then be evaluated by the substitution $x=l(l+1)$].

Calculate the rotational contribution to the internal energy of one mole of $N_2$ at $20^0$C ( $I = 1.42 \times 10^{-46}$kgm$^2$).

For the last part, you have to change a sum over $l$ to an integral, and make the substitution $x=l(l+1)$, with ${\rm d}x=(2l+1) {\rm d}l$. Luckily, the factor $(2l+1)$ is just the degeneracy factor, so the integrand is a simple exponential. See here.

20.

Exam question, 1994, slightly altered. Show that the partition function for a simple harmonic oscillator of frequency $\omega$ is

$$Z_1={1\over 2 \sinh({\textstyle \frac 1 2}\beta\hbar\omega)}.$$

(You will need the expression for the sum of a geometric series from your maths formula sheet. $\beta=1/(k_{\scriptscriptstyle B}T)$.)

According to Einstein's theory of specific heats due to lattice vibrations of a crystal, a solid of $N$ atoms behaves like $3N$ simple harmonic oscillators of frequency $\omega_{\scriptscriptstyle E}$. Derive expressions for the internal energy $E$ and the Helmholtz free energy $F$ of the crystal as predicted by Einstein's theory. Show that as $T\to 0$ the internal energy is just the zero-point energy of all the oscillators.

The thermal expansion of the crystal can be explained if $\omega_{\scriptscriptstyle E}$ varies with volume as $V^{-\gamma}$ where $\gamma$ is a constant. Show that in this case the pressure exerted by the lattice vibrations is

$$P={\gamma E\over V}.$$

Experimentally the low-temperature heat capacity of a crystal is observed to be proportional to $T^3$. Does the Einstein theory reproduce this result?

Remember the energy levels of a harmonic oscillator are $(n+{\textstyle \frac 1 2})\hbar \omega$, where ${\textstyle \frac 1 2}\hbar \omega$ is the zero-point motion. See here for a single oscillator; the crystal is just like $3N$ distinguishable oscillators with $Z=Z_1^{3N}$. The pressure can be obtained from $\left\langle P \right\rangle =({\partial F\over \partial V})_{\lower0.4ex\hbox{${\scriptscriptstyle T}$} }$ and $\left\langle F \right\rangle =-k_{\scriptscriptstyle B}T \ln Z$. The model does not reproduce the observed low temperature behaviour.