Distinction in the assumptions between Einstein and Debye models of specific heat of solid
Einstein and Debye models are two theoretical models used to describe the specific heat of solids at low temperatures. They were proposed by Albert Einstein and Peter Debye, respectively, and both models are based on different assumptions about the behaviour of atoms or molecules in a solid lattice.
Einstein Model:
Independent Harmonic Oscillators: In the Einstein model, it is assumed that each atom or molecule in the solid behaves as an independent three-dimensional harmonic oscillator. This means that the atoms vibrate back and forth around their equilibrium positions like tiny springs.
Quantization of Vibrational Modes: Einstein assumed that the vibrational energy levels of the harmonic oscillators are quantized, just like in quantum mechanics. Each oscillator can only have specific discrete energy levels.
Constant Einstein Temperature: The key assumption of the Einstein model is that all the atoms in the solid have the same characteristic vibrational frequency. This frequency is represented by the Einstein temperature (θ_E), which is the same for all substances.
Debye Model:
Lattice Vibration: In the Debye model, the assumption is that the atoms or molecules in the solid are not independent harmonic oscillators but are instead part of a lattice where they interact with each other through forces.
Continuum of Vibrational Modes: The Debye model treats the lattice vibrations as continuous rather than discrete energy levels. It assumes that the vibrational energy of the lattice has a range of allowed values, forming a continuum of vibrational modes.
Debye Frequency and Temperature: The Debye model introduces the concept of the Debye frequency (ω_D), which represents the highest vibrational frequency in the lattice. The Debye temperature (θ_D) is defined as θ_D = ħω_D / k_B, where ħ is the reduced Planck's constant and k_B is Boltzmann's constant. Unlike the Einstein model, the Debye model allows different substances to have different Debye temperatures based on their lattice structures.
Temperature Dependence of Specific Heat:
Einstein Model: The specific heat predicted by the Einstein model follows a step-like behaviour, where it increases with temperature until it reaches the characteristic Einstein temperature (θ_E), after which it remains constant.
Debye Model: The specific heat predicted by the Debye model increases linearly with temperature at low temperatures and approaches a constant value at high temperatures. It provides a more accurate description of specific heat behaviour at low temperatures than the Einstein model.
The number of independent models of vibration per unit volume in between the frequency range 𝛾 and 𝛾 + d𝛾 = 4πV / v³ 𝛾² dr
Since the solid is assumed to be elastic body, both longitudinal and transverse wave are present. For each, longitudinal vibration, there are two transverse vibrations. So the total number of independent modes of vibration in volume V containing N molecules can be written as
The range of frequency from 0 to Vm indicates that there are limited number of modes of vibration.
Since energy associated with each is
Now, energy associated with the solid having frequency range 𝛾 and 𝛾 + d𝛾 is given by
Hence total energy associated with the frequency 0 to 𝛾m is
𝜃_D is called Debye's characteristic temperature
Case I : At high temperature , equation 2 becomes
which shows that at high temperature, Debye's specific heat agrees with classical result
Case II: At low temperature,
Using 2 neglecting 1 in 2nd term
Cv ∝ T³
Which is Debye T³ Law
The specific heat behaviour of materials is a complex subject and depends on the crystal structure, phonon modes, and other factors. Different materials may exhibit different specific heat behaviours, and the "Cv ∝ T³" relationship is specific to certain materials at low temperatures and does not apply universally to all substances. Debye's Law sheds light on the lattice vibrations and thermal properties of materials, opening up avenues for research and practical applications in various scientific and technological fields.
In summary, the main distinctions between the Einstein and Debye models lie in their assumptions about the vibrational behaviour of atoms or molecules in a solid lattice and the temperature dependence of specific heat. While the Einstein model assumes independent harmonic oscillators with a constant Einstein temperature, the Debye model considers lattice vibrations with a continuum of vibrational modes and introduces the concept of the Debye temperature, which varies for different substances. The Debye model provides a more accurate description of specific heat at low temperatures and has broader applicability to various solids.
This note is a part of the Physics Repository.