Einstein Theory of Specific Heat of Solids

Failure of Classical Theory

Classical theory (Dulong–Petit law) predicts:

\[ C_V = 3Nk_B \]

This agrees at high temperature but fails at low temperature where experimental results show $C_V \to 0$ as $T \to 0$.

Assumptions of Einstein Model

A solid consists of N atoms.
Each atom behaves like a 3D quantum harmonic oscillator.
All oscillators vibrate with the same frequency $\nu$.
Energy levels are quantized:

Einstein model successfully introduced quantum ideas in solid state physics but was later improved by Debye theory.

Einstein theory of Specific Heat (1907):

In every crystal the atom is surrounded by a group of first nearest neighbours, second nearest neighbours etc. Every atom vibrates in the vicinity of it's Lattice point or equillibrium position in a force field exerted by all it's neigbours . Each atom vibrates independently and at a constant frquency. in three dimensional space .Each atom can be treated like a simple harmonic oscillator and the energy of nth ocillator is given by : $$ E_{n}=\left(n+\frac{1}{2}\right) h f \quad \text { Where } \mathrm{n}=0,1,2,3 \ldots . $$

Mean energy of the oscillator is :

As energy of every harmonic oscillator is quantised and energy of nth oscillator is : $$ U_{n}=\left(n+\frac{1}{2}\right) h f $$ Then the average energy of such oscillator is:

$$ \begin{aligned} \langle U\rangle&=\dfrac{\sum_{n=0}^{n=\infty} U_{n} e^{-\left(\dfrac{U_{n}}{k T}\right)}}{\sum_{n=0}^{n=\infty} e^{-\left(\frac{U_{n}}{k T}\right)}}\\ & =\frac{\sum_{0}^{\infty}\left(n+\frac{1}{2}\right) h f e^{-\frac{\left(n+\frac{1}{2}\right) h f}{k T}}}{\sum_{0}^{\infty} e^{-\frac{\left(n+\frac{1}{2}\right) h f}{k T}}} \\ &= h f \frac{\sum_{0}^{\infty} n e^{-\frac{n h f}{k T}} e^{-\frac{h f}{2 k T}}}{\sum_{0}^{\infty} e^{\frac{-n h f}{k T}} e^{-\frac{h f}{2 k T}}}+\frac{1}{2} h f \frac{\sum_{0}^{\infty} e^{-\frac{n h f}{k T}} e^{-\frac{h f}{2 k T}}}{\sum_{0}^{\infty} e^{\frac{-n h f}{k T}} e^{-\frac{h f}{2 k T}}} \\ &= h f \frac{\sum_{0}^{\infty} n e^{-\frac{n h f}{k T}}}{\sum_{0}^{\infty} e^{\frac{-n h f}{k T}}+\frac{1}{2} h f} \frac{\sum_{0}^{\infty} e^{-\frac{n h f}{k T}}}{\sum_{0}^{\infty} e^{\frac{-n h f}{k T}}}=h f\left(\frac{\sum_{0}^{\infty} n e^{-\frac{n h f}{k T}}}{\sum_{0}^{\infty} e^{\frac{-n h f}{k T}}}+\frac{1}{2}\right) \\ &= h f\left(\frac{0+e^{-x}+2 e^{-2 x}+3 e^{-3 x}+4 e^{-4 x}+\ldots . .}{1+e^{-x}+e^{-2 x}+e^{-3 x}+e^{-4 x}+\ldots . .} \frac{1}{2}\right) T a k i n g x=\frac{h f}{k T} \\ &=h f\left(-\frac{d}{d x} \ln \left(1+e^{-x}+e^{-2 x}+e^{-3 x}+e^{-4 x}+\ldots .\right)+\frac{1}{2}\right) \\ &=h f\left(-\frac{d}{d x} \ln \left(1-e^{-x}\right)^{-1}+\frac{1}{2}\right) \\ &=h f\left(\frac{d}{d x} \ln \left(1-e^{-x}\right)+\frac{1}{2}\right)=h f\left(\frac{e^{-x}}{1-e^{-x}}+\frac{1}{2}\right)=h f\left(\frac{1}{2}+\frac{1}{e^{x}-1}\right) \\ &=h f\left(\frac{1}{2}+\frac{1}{e^{h f / k T}-1}\right) \end{aligned} $$

As the crystal contianing N atoms has 3 N harmonic oscillators . The total internal energy of the crystal becomes.

$$ \begin{aligned} U_{\text {total }}& =3 N\langle U\rangle\\ &=3 N\left(\frac{1}{2} h f+\frac{h f}{e^{h f / k T}-1}\right) \\ \Rightarrow U_{\text {total }}&=3 N\left(\frac{1}{2} h f+\frac{h f}{e^{h f / k T}-1}\right) \end{aligned} $$

Then specific heat of the crystal at constant volume is :

$$ \begin{aligned} C_{V}&=\frac{d U_{\text {total }}}{d T}\\ &=0+3 N h f \frac{d}{d T}\left(\frac{1}{e^{h f / k T}-1}\right) \\ & =3 N h f \frac{d}{d T}\left(e^{h f / k T}-1\right)^{-1} \\ & =-(3 N h f)\left(e^{h f / k T}-1\right)^{-2}\left(-e^{h f / k T} \frac{h f}{k T^{2}}\right) \\ & =3 N k\left(\frac{h f}{k T}\right)^{2}\left(\frac{1}{e^{h f / k T}-1}\right)^{2} e^{h f / k T} \\ & =3 N k\left(\frac{\theta_{E}}{T}\right)^{2}\left(\frac{1}{e^{\theta_{E} / T}-1}\right)^{2} e^{\theta_{E} / T} \text { where } h f=k \theta_{E}, \theta_{E} \text { is Einstein Temperature } \end{aligned} $$

Special Case :

(1) When temperature is very high $ T \gg \theta_E $

\[ \begin{align} C_V &= 3Nk\left(\frac{\theta_E}{T}\right)^2 \left( \frac{1}{1 + \frac{\theta_E}{T} + \frac{1}{2}\left(\frac{\theta_E}{T}\right)^2 + \cdots - 1} \right)^2 \left(1 + \frac{\theta_E}{T} + \frac{1}{2}\left(\frac{\theta_E}{T}\right)^2 + \cdots \right) \\ &= 3Nk\left(\frac{\theta_E}{T}\right)^2 \left( \frac{1}{\frac{\theta_E}{T}} \right)^2 \left(1 + \frac{\theta_E}{T} \right) \\ &= 3Nk\left(1 + \frac{\theta_E}{T} \right) \\ &= 3Nk + 3Nk \cdot \frac{\theta_E}{T} \end{align} \]

Hence, when $ T \to \infty $, $ \frac{\theta_E}{T} \to 0 $,

\[ C_V = 3Nk = 3R \]

(2) When temperature is very low $ T \ll \theta_E $

\[ \begin{align} C_V &= 3Nk\left(\frac{\theta_E}{T}\right)^2 \left( \frac{1}{e^{\theta_E/T}} \right)^2 e^{\theta_E/T} \quad \text{(since } e^{\theta_E/T} - 1 \approx e^{\theta_E/T} \text{)} \\ &= 3Nk\left(\frac{\theta_E}{T}\right)^2 e^{-\theta_E/T} \\ &= 3R\left(\frac{\theta_E}{T}\right)^2 e^{-\theta_E/T} \end{align} \]

As temperature $ T \to 0 $, the exponential term $ e^{-\theta_E/T} \to 0 $, hence the specific heat $ C_V \to 0 $.

Merits

First quantum theory of solids.
Explains decrease of specific heat at low T.
Introduced concept of Einstein temperature.

Limitation of Einstein's Theory :

Einstein 's theory fails at low temperature as at low temperature $C_{V} \propto T^{3}$ instead of $C_{V} \alpha e^{-\left(\frac{\theta_{E}}{T}\right)}$
Einstein's assumption that each atomic oscillator vibrates independently at same frequency like other is wrong. In reality as per Debye Model The oscillators are coupled together and vibrate with wide range of frewuencies .
All atoms assumed same frequency.
Predicts exponential fall instead of $T^3$ law.
Cannot fully explain low-temperature behavior.