Einstein's Specific Heat Theory
Summary
Einstein's specific heat theory addresses the deviation from classical predictions of specific heat in solids at low temperatures. Classical theory, based on the Dulong-Petit law, suggested that specific heat capacity is constant, but experimental data showed a decrease at low temperatures. Einstein proposed a quantum mechanical model to explain this behavior.
Einstein's Model
Einstein treated each atom in a solid as an independent quantum harmonic oscillator. The frequency of these oscillators is considered to be the same for all atoms.
Derivation of Specific Heat
1. Quantum Harmonic Oscillator
The energy levels \( E_n \) of a quantum harmonic oscillator are given by:
\( E_n = \left(n + \frac{1}{2}\right)h\nu \)
where \( n \) is a non-negative integer, \( h \) is Planck's constant, and \( \nu \) is the frequency of vibration.
2. Partition Function
The partition function \( Z \) for a single oscillator is:
\( Z = \sum_{n=0}^{\infty} e^{-\frac{E_n}{k_B T}} \)
where \( k_B \) is Boltzmann's constant and \( T \) is the temperature.
Substituting \( E_n \):
\( Z = \sum_{n=0}^{\infty} e^{-\frac{\left(n + \frac{1}{2}\right)h\nu}{k_B T}} \)
This is a geometric series:
\( Z = e^{-\frac{h\nu}{2k_B T}} \sum_{n=0}^{\infty} \left(e^{-\frac{h\nu}{k_B T}}\right)^n \)
\( Z = \frac{e^{-\frac{h\nu}{2k_B T}}}{1 - e^{-\frac{h\nu}{k_B T}}} \)
3. Average Energy
The average energy \( \langle E \rangle \) of a single oscillator is:
\( \langle E \rangle = -\frac{\partial \ln Z}{\partial \beta} \)
where \( \beta = \frac{1}{k_B T} \):
\( \langle E \rangle = \frac{h\nu}{e^{\frac{h\nu}{k_B T}} - 1} \)
4. Specific Heat
The heat capacity \( C_V \) is obtained by differentiating the average energy with respect to temperature:
\( C_V = \frac{\partial \langle E \rangle}{\partial T} \)
\( C_V = \frac{\partial}{\partial T} \left[ \frac{h\nu}{e^{\frac{h\nu}{k_B T}} - 1} \right] \)
After differentiation, the specific heat capacity of a solid with \( N \) oscillators is:
\( C_V = 3Nk_B \left( \frac{\theta_E}{T} \right)^2 \frac{e^{\frac{\theta_E}{T}}}{\left( e^{\frac{\theta_E}{T}} - 1 \right)^2} \)
where \( \theta_E = \frac{h\nu}{k_B} \) is the Einstein temperature.