Vibrations and the Einstein solid

A vibration has a ladder of equal rungs and its own hot-or-cold temperature

Intuition

A vibrating bond is like a spring that can only be stretched in whole steps. The rungs are equally spaced. A cold spring sits on the bottom rung. A hot spring has enough energy to visit many rungs. How hot is “hot” depends on how big one step is.

The harmonic oscillator has equally spaced levels $E_n = h\nu\left(n + \tfrac{1}{2}\right)$, or $E_n = h\nu n$ if we ignore the zero-point energy. The characteristic temperature for vibration is

$\Theta_{\text{vib}} = \frac{h\nu}{k}$

Typical values from the lecture: Ar(s) 40 K, Pb(s) 60 K, SO$_2$(g) stretch 2000 K, H$_2$O(g) stretch 5000 K. So a bond with a big $\Theta$ stays “cold” even at room temperature, and soft lattices like lead and argon are “hot” there.

The oscillator’s q can be summed exactly

Unlike translation, the vibrational sum can be done exactly. Write $x = e^{-\Theta_{\text{vib}}/T}$:

$q_{\text{vib}} = \sum_{n=0}^{\infty}e^{-\beta nh\nu} = 1 + x + x^2 + x^3 + \cdots$

This is a geometric series. For $|x| < 1$, $\sum x^n = 1/(1-x)$, so

$q_{\text{vib}} = \frac{1}{1 - e^{-\Theta_{\text{vib}}/T}}$

At low $T$, $x \to 0$ and $q \to 1$: only the bottom rung is used. At high $T$, $x \to 1$ and $q$ grows.

Try it: worked example

For an oscillator with Theta = 200 K, what are p_0 and p_1 at 100 K?

Show answer and steps

Answerq = 1.157, p_0 = 0.87, p_1 = 0.12, matching the slide inset (about 0.87 and 0.12).

1. $\Theta/T = 200/100 = 2$, so $x = e^{-2} = 0.135$.
2. $q_{\text{vib}} = \dfrac{1}{1 - 0.135} = 1.157$.
3. $p_0 = \dfrac{1}{q} = 0.87$.
4. $p_1 = \dfrac{x}{q} = \dfrac{0.135}{1.157} = 0.12$.
5. These match the 100 K inset on slide 20 ($p_0 \approx 0.87$, $p_1 \approx 0.12$). At 250 K the inset shows $p_0 \approx 0.55$, $p_1 \approx 0.25$, $p_2 \approx 0.1$. At 1000 K the populations spread over states 0 to 7.

Energy and heat capacity climb as the rungs open up

Use $E = NkT^2\,d\ln q/dT$ with this $q$. The energy above the zero-point energy is

$E = \frac{Nk\Theta_{\text{vib}}}{\exp(\Theta_{\text{vib}}/T) - 1}$

The heat capacity is its slope:

$C_V = \left(\frac{dE}{dT}\right)_V = Nk\left(\frac{\Theta_{\text{vib}}}{T}\right)^2\frac{\exp(\Theta_{\text{vib}}/T)}{\left(\exp(\Theta_{\text{vib}}/T)-1\right)^2}$

It rises with $T$ as molecules can absorb more energy, and approaches $R$ per mole of vibrations: $C_V^{\text{vib}} \to R$. The energy curve on the slide is near zero below about 50 K and then rises roughly linearly. This one-dimensional oscillator describes the vibrational heat capacity of diatomic molecules (O$_2$, N$_2$, CO, Cl$_2$) well. A diatomic’s heat capacity rises from $5R/2$ (translation plus rotation) towards $7R/2$ (translation, rotation and vibration).

Pitfall

Heat capacity is not zero or constant: it depends on whether $T$ is above or below $\Theta_{\text{vib}}$. Below, the vibration is frozen out and contributes almost nothing. Above, it contributes its full share.

An Einstein solid is a lattice of independent oscillators

Intuition

Picture a crystal as a jelly-like grid of balls on springs. Each ball jiggles on its own, in three directions. Einstein’s trick was to pretend every ball jiggles independently and at the same speed.

In simple solids (Ar(s), metals) vibration is the only possible motion. Atoms in a lattice vibrate in all three directions, so a lattice of $N$ atoms has $3N$ vibrations and $3N$ sets of energy levels. Einstein proposed his model to explain how the heat capacity of solids changes with temperature. It has two assumptions:

• Each atom is an independent 3D quantum harmonic oscillator.
• All atoms oscillate with the same frequency.

Each atom vibrates in 3 directions, so it contributes 3 times the heat capacity of a single oscillator. At high $T$ the molar heat capacity approaches

$3R = 24.9\ \text{J K}^{-1}\,\text{mol}^{-1}$

the Dulong-Petit law. The model fits the data well. When $T$ is scaled by $\Theta_{\text{vib}}$ the curves for different solids fall onto one (the slide shows C, Cu and Al). The differences between materials come from bond strengths, that is the vibrational frequencies $\nu$.

Optional aside

The Einstein model fails at very low temperatures, where interactions become important and give collective motion called phonons. A more accurate model is due to Debye. The lecture mentions this as an aside, with no formula.

Exam detail

Lecture 17 numbers: $\Theta_{\text{vib}}$ of 40 K (Ar), 60 K (Pb), 2000 K (SO$_2$ stretch) and 5000 K (H$_2$O stretch). The oscillator with $\Theta = 200$ K has $p_0 \approx 0.87$ at 100 K. $3R = 24.9$ J K$^{-1}$ mol$^{-1}$. Diatomic heat capacity runs from $5R/2$ towards $7R/2$.

Recall

What are the Einstein model's two assumptions and its high-temperature result?

Each atom is an independent 3D quantum harmonic oscillator, and all atoms share one frequency. At high $T$ the molar heat capacity approaches $3R = 24.9$ J K$^{-1}$ mol$^{-1}$.