The document discusses the Einstein and Debye models of heat capacity in solids. The Einstein model from 1907 treats each atom as an independent harmonic oscillator vibrating at the same frequency. In contrast, the Debye model from 1912 treats the solid as a continuous elastic body with a range of vibration frequencies, improving on Einstein's assumptions. Both models aim to explain how heat capacity decreases with temperature and approach zero at low temperatures, with the Debye model providing a more accurate theoretical framework through its treatment of phonons and coupled oscillators across different frequencies.

1. Einstein
and Debye
model of
heat
capacity

2. Contents
Einstein Model of Heat Capacity
Debye Heat Model of Heat Capacity
Comparison

3. Einstein Model
• The Einstein model is a theoretical model used to explain the heat capacity
of solids at low temperatures. It was proposed by Albert Einstein in 1907,
based on his understanding of the quantization of energy in solids.
• Dulong-petit value for high temperature is correct.
• But it failed to explain that why in the low temperature range the specific
heat of all solid is found to zero.

4. Einstein Model
“According to Einstein model solids are considered as a collection of
atoms or molecules that vibrate around their equilibrium positions.”
• Einstein made several key assumptions in his model:
 All atoms vibrate with same frequency
 Every atom have independent S.H.O
 Discrete energy spectrum
 Energy levels of vibrating atoms or molecules are equally spaced

5. Mathematical Expression
Einstein model Equation:
 Cᴠ = nR(Θᴱ/T)² * e^(Θᴱ/T) / (e^(Θᴱ/T) - 1)²
Where:
Cᴠ is the molar heat capacity at constant volume.
n is the number of atoms or molecules per mole.
R is the gas constant.
Θᴱ is the characteristic Einstein temperature of the solid.
T is the temperature.

7. Debye Heat Model
The Debye heat model is a theoretical framework used to describe the heat
capacity of solids. It was proposed by Peter Debye and is based on the idea
that the atoms in a solid vibrate and can be treated as harmonic oscillators.
Debye Model Improvement:
 Lattice vibrating at low frequencies at low temperatures.
Corresponding wavelengths much longer than atomic spacing.
 Assuming crystal as a continuous elastic body.
Treating lattice vibration as elastic vibration.
Applying the concept of phonons.
g(v) = 4πV v²

8. Mathematical Expression
An elastic wave can be de-coupled into 2 transverse and 1 longitudinal waves:
Vtrans : velocity of the transverse wave
Vlong : velocity of the longitudinal wave
Define average velocity as

9. Mathematical Expression
Maximum frequency vD such that
Debye frequency

10. Mathematical Expression
The Debye heat capacity equation is given by:
 Cv = 9Nk(dT/θ_D)^3∫(0 to θ_D/T) (x^4e^x / (e^x - 1)^2)dx
where:
Cv is the heat capacity at constant volume
N is the number of atoms per unit volume
k is Boltzmann's constant
T is the temperature
θ_D is the Debye temperature, which represents the characteristic
vibrational energy of the solid

11. Comparison Between Einstein and
Debye’s Theory of Heat
Similarities:
• The solid is made up of a lattice of atoms.
• The atoms vibrate about their equilibrium position.
• The vibrations can be treated as a system of harmonic oscillators.
• The energy of each oscillator is quantized.
• The total energy of the solid is the sum of the energies of all the oscillators.
• Constant value of specific heat at high temperature.

12. Differences:
Einstein theory of specific heat:
•All atoms in the solid vibrate at the same frequency.
•The number of oscillators is proportional to the number of atoms in the solid
•The oscillators are independent of each other.
•The energy of each oscillator is quantized in the same way.
•The total energy of the solid is the sum of the energies of all the oscillators.
Debye theory of specific heat:
•The solid has a continuous range of vibrational frequencies.
• The number of oscillators is proportional to the volume of the solid.
• The oscillators are coupled to each other.
• The energy of each oscillator is quantized in a unique way based on its frequency.
• The total energy of the solid is the integral of the energies of all the oscillators.

14. Thank you