@@ -49,7 +49,7 @@ _(based on chapter 2.2 of the book)_
## Deficiency of the Einstein model
In the previous lecture, we observed that the Einstein model explained the heat capacity of solids quite well.
However, we can see that something goes wrong if we compare the heat capacity predicted by the Einstein model to the that of silver[^1]:
However, we can see that something goes wrong if we compare the heat capacity predicted by the Einstein model to the measured heat capacity of silver[^1]:
```python
pyplot.rcParams['axes.titlepad']=20
...
...
@@ -80,21 +80,21 @@ ax.set_title(r'Heat capacity of silver and a fit of the Einstein model')
ax.legend();
```
The low-temperature heat capacity of silver is underestimated by the Einstein model.
We see that the low-temperature heat capacity of silver is underestimated by the Einstein model.
This underestimation is not obvious at first, but as we will see, this subtle difference is due to a profound physical phenomenon.
??? question "How does $C$ predicted by the Einstein model behave at low $T$?"
??? question "How does $C$ predicted by the Einstein model scale with temperature at low $T$?"
When $T → 0$, $T_E/T → \infty$. Therefore neglecting $1$ in the denominator we get $C \propto \left(\frac{T_E}{T}\right)^2e^{-T_E/T}$, and the heat capacity should be exponentially small.
## Debye model
## The Debye model
The key simplification of the Einstein model is to consider the atoms as independent quantum harmonic oscillators.
Instead of independent oscillators, Peter Debye considered the collective motion of atoms as sound waves.
Instead of considering each atom as an independent harmonic oscillator, Peter Debye considered sound waves - the collective motion of atoms - as harmonic oscillators.
> ### Sound waves
>
> A sound wave is a collective motion of atoms through a solid. The displacement $\mathbf{\delta r}$ of an atom at position $\mathbf{r}$ and time $t$ is discribed by
> A sound wave is a collective motion of atoms through a solid. The displacement $\mathbf{\delta r}$ of an atom at position $\mathbf{r}$ and time $t$ is described by
@@ -110,20 +110,14 @@ Instead of independent oscillators, Peter Debye considered the collective motion
>
> The space containing all possible values of $\mathbf{k}$ is called the _$k$-space_ (also named the _reciprocal space_).
Despite having a position-dependent phase, each sound waves is an independent harmonic oscillator.
The quantum mechanical excitations of this harmonic oscillator motion are called *phonons*—the particles of sound.
Phonons are bosons and therefore their statistics is described by the Bose-Einstein distribution $n_B(\hbar \omega(\mathbf{k}))$.
Debye used the description of phonons to model the heat capacity of solids.
The frequency of these phonons depends on its wavevector $\mathbf{k}$ through the _dispersion relation_
The sound waves are characterized by their frequency, wavevector, and polarization. The frequency of the sound wave is determined by its wavevector $\mathbf{k}$ through the _dispersion relation_
$$
\omega = v_s|\mathbf{k}|,
$$
where $v_s$ is the _sound velocity_ of a material.
To summarize, instead of having $3N$ oscillators with the same frequency $\omega_0$, we now have $3N$ possible phonon modes with frequencies depending on $\textbf{k}$ through the dispersion relation $\omega(\mathbf{k}) = v_s|\mathbf{k}|$.
The expected value of the total energy (which we, for simplicity, from now on will denote as the total energy) is given by the sum over the energy of all possible phonon modes characterized by a wavevector $\mathbf{k}$:
where $v_s$ is the _sound velocity_ of a material. In the Debye model, each wave is considered as an independent quantum harmonic oscillators.
As in the previous lecture, the quantum mechanical excitations of a harmonic oscillator are called *phonons*, and the expected number of phonons in the oscillator at temperature $T$ is given by the Bose-Einstein distribution $n_B(\beta \hbar \omega(\mathbf{k}))$. So, instead of having $3N$ oscillators with the same frequency $\omega_0$ as in the Einstein model, we now have $3N$ oscillators (the vibrational modes) with frequencies depending on $\textbf{k}$ through the dispersion relation $\omega(\mathbf{k}) = v_s|\mathbf{k}|$. Apart from this crucial difference with the Einstein model, the calculation of the expectation value of the energy stored in the oscillators proceeds in the same way:
The expected value of the total energy stored in the oscillators (which, from now on, we will simply denote as the total energy) is given by the sum of the energy stored in all the individual oscillators. These oscillators are characterized by their wavevector $\mathbf{k}$:
\begin{align}
\langle E \rangle &= 3 \sum_\mathbf{k} \left(\frac{1}{2}\hbar\omega(\mathbf{k})+\hbar \omega(\mathbf{k}) n_{BE}(\hbar \omega(\mathbf{k}))\right)\\
