The equipartition theorem assumed that each atom can be modeled as a classic harmonic oscillator. However, at low temperatures this led to a discrepancy in the heat capacity between the law of Dulong-Petit and the observed heat capacity. To explain the behaviour of $C$ at low temperatures, Einstein treated each atom as a _quantum_ harmonic oscilator, instead of a classic harmonic oscillator. Einstein also assumed that each atom oscillates with the same frequency $\omega_0$. To summarize, the Einstein model of a solid is characterised by the following two assumptions:
## The Einstein model
The equipartition theorem assumed that each atom can be modeled as a classic harmonic oscillator. However, at low temperatures this led to a discrepancy in the heat capacity between the law of Dulong-Petit and the observed heat capacity. To explain the behaviour of $C$ at low temperatures, Einstein treated each atom as a _quantum_ harmonic oscilator, instead of a classic one. Einstein also assumed that each atom oscillates with the same frequency $\omega_0$. To summarize, the Einstein model of a solid is characterised by the following two assumptions:
* The atoms are independent quantum harmonic oscillators
* Each atom has the same frequency $\omega_0$
For simplicity, we consider a 1D quantum harmonic oscillator (it turns out that the heat capacity in the 3D case is simply 3 times the heat capacity of the 1D case).
The energy levels and wavefunctions of a 1D quantum harmonic oscillator are shown below.
The energy spectrum and wavefunctions of a 1D quantum harmonic oscillator are shown below.
The energy spectrum of a quantum harmonic oscillator is given by
$$E_n=\left(n+\frac{1}{2}\right)\hbar\omega_0$$
where $\omega_0$ is the eigenfrequency of the oscillator, which is the same for each atom.
What is the thermal occupation and corresponding thermal energy stored in this oscillator? A harmonic oscillator is a bosonic mode, therefore its occupation number is described by the Bose-Einstein distribution:
What is the thermal occupation and corresponding thermal energy stored in this oscillator? A quantum harmonic oscillator is a bosonic mode, therefore its occupation number is described by the Bose-Einstein distribution:
Here we have substituted the Bose–Einstein distribution $n_{BE}(\omega,T)$ for the expected occupation number $\langle n \rangle$.
The left plot below shows the Bose-Einstein distribution as a function of the energy $\hbar omega$. We observe that low-energy states have a higher expected occupation number than high-energy states.
The right plot shows the expected value of the energy as a function of the temperature. We see that for high $T$, $\langle E \rangle$ is linear in $T$. For low $T$, there is insufficient energy to excite the atoms into an excited state and therefore all atoms occupy the ground state ($n = 0$). Hence, $\langle E \rangle$ converges towards a constant value of $\hbar omega_0/2$, which is the _zero-point energy_. Moreover, we see that the energy in the oscillator becomes approximately constant when $k_{\rm B}T\ll\hbar \omega_0$. This implies that the heat capacity becomes small when $k_{\rm B}T\ll\hbar \omega_0$ and goes to zero when $T\rightarrow0$.
Here we have substituted the Bose–Einstein distribution $n_{BE}(\omega,T)$ for the expected occupation number $\langle n \rangle$.
A plot of the Bose-Einstein distribution and the expected value of the energy are shown below.
The left plot below shows the Bose-Einstein distribution as a function of the energy $\hbar \omega$. We observe that low-energy states have a higher expected occupation number than high-energy states.
The right plot shows the expected value of the energy as a function of the temperature. We see that for high $T$, $\langle E \rangle$ is linear in $T$. For low $T$, there is insufficient energy to excite the atoms into an excited state and therefore all atoms occupy the ground state ($n = 0$). Hence, $\langle E \rangle$ converges towards a constant value of $\hbar \omega_0/2$, which is the _zero-point energy_. Moreover, we see that the energy in the oscillator becomes approximately constant when $k_{\rm B}T\ll\hbar \omega_0$. This implies that the heat capacity becomes small when $k_{\rm B}T\ll\hbar \omega_0$ and goes to zero when $T\rightarrow0$.
Having found an expression for $\langle E \rangle$ as a function of $T$, we can now calculate the heat capacity per atom $C$ explicitly. To do so, we need to differentiate $\langle E \rangle$ with respect to $T$.
$$
\begin{align}
C &= \frac{\partial\langle E \rangle}{\partial T}\\
@@ -301,7 +300,8 @@ C = k_b \left(\frac{T_E}{T}\right)^2\frac{e^{T_E/T}}{(e^{T_E/T} - 1)^2},
$$
where we have introduced the *Einstein temperature* $T_E \equiv \hbar \omega_0 / k_B$.
This is the characteristic temperature for which the thermal excitations "freeze out" in the harmonic oscillator. This means that there is not enough thermal energy to excite the harmonic oscillators into an excited state, which leaves them in the ground state. Consequently, it is also the temperature scale for which the heat capacity of an Einstein solid starts significantly decreasing. We also observe that the Einstein temperature depends on the eigenfrequency $\omega_0$. As different materials have a different eigenfrequency $\omega_0$, the Einstein temperature is a material-dependent parameter.
The Einstein temperature $T_E$ is the characteristic temperature for which the thermal excitations of the quantum harmonic oscilator start to "freeze out".
This means that there is not enough thermal energy to excite the harmonic oscillators into an excited state, which leaves them in the ground state. Consequently, it is also the temperature scale for which the heat capacity of an Einstein solid starts significantly decreasing. We also observe that the Einstein temperature depends on the eigenfrequency $\omega_0$. As different materials have a different eigenfrequency $\omega_0$, the Einstein temperature is a material-dependent parameter.
```python
# Defining variables
...
...
@@ -352,7 +352,7 @@ control = interact(einstein_temp_plot, T_E = float_slider)
The horizontal dashed line is the classical value, $k_{\rm B}$. The shaded area is the difference between the classical value $k_B$ and the value predicted by the Einstein model. Integrating over the shaded area yields $\frac{1}{2}\hbar\omega_0$, which is the zero-point energy of the oscillator, which cannot be extracted from the system. The vertical dashed line depicts the Einstein temperature $T_E$, at which the heat capacity $C \approx 0.92 k_B$.
The calculations above were done for a single harmonic oscillator. In order to obtain the heat capacity of a full material, we would have to multiply $C$ (or $\langle E \rangle$) by $3N$. The $N$ comes from the number of harmonic oscilators in the solid and the factor 3 originates from the dimensionality of the system.
??? 1D versus 3D harmonic oscillator
??? note "1D versus 3D harmonic oscillator"
One 3D (quantum) harmonic oscillator is equivalent to three independent 1D (quantum) harmonic oscillators. Therefore, we can simply multiply by a factor of 3.
