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A crystal consists of atoms which are arranged in a periodic manner and are bound togather by strong binding forces.
“In the classical theory it is consumed that each atom of a crystal acts as a three dimensional harmonic oscillator and all the atoms vibrate independent of one other.” Further a system of N vibrating atoms or N independent three dimensional harmonic oscillators is equivalent to a system of 3N identical and independent onedimensional Harmonic oscillators.
Assuming that the distribution of oscillators in energy obeys the MaxwellBoltzmann distribution law the average energy of each harmonic oscillator is given by
According to Equipartition Theory (K.E) = 1/2 k_{B}T
According to Hooks Law (K.E) = (P.E) = 1/2(k_{B}T)
But E =K .E+ P.E
⇒ (E) = 1/2(k_{B}T) + 1/2(k_{B}T) = k_{B}T
Thus the total vibration energy of the crystal containing N identical atoms or 3N onedimensional Harmonic oscillator become
E = 3 N < E >
E = 3Nk_{B}T
Using the definition of specific heat of C_{V} = (∂E/∂T)_{V}
The above result so that the molar specific heat of all the solids is constant and is independent of temperature and frequency this is called Dulog and Petit’s law
Einstein, in 1911, attempted to resolve the discrepancies of the classical theory of specific heat by applying the Planck’s quantum theory. Einstein retained all the assumptions of the classical theory as such except replacing the classical harmonic oscillator by quantum
Harmonic oscillator i.e.
The salient features of the Einstein’s theory are listed as:
To calculate the average energy of an oscillator, we replace integration by summation in expression for the mB distribution of energy and obtain.
The expression for the internal energy of the crystal become
Then, (let Einstein temperature
CaseI:
High temperature behavior
C_{V} = 3Nk_{B}= 3R
which is the Dulong and Petits law as obtain from classical theory.
CaseII:
Low temperature behavior
Thus, for T << θ_{E} the heat capacity is proportional to e^{−θ}_{E}^{/ T} which is the dominating factor. But e rimentally it is found to vary as T^{3} for most of the solid.
Thus Einstein theory failed to explain actual variation of specific heat.
In this model, the vibrational motion of the crystal as a whole was considered to be equivalent to the vibrational motion of system of complete harmonic oscillator which can propagate a range of frequency rather than a single frequency. Debye proposed that crystal can propagate elastic waves of wave lengths ranging from low frequency of (sound wave) to high frequencies corresponding to infrared absorption. This means that a crystal can be a number of modes of vibration. The number of vibration modes per unit frequency range is called density of modes z (v ) .
Thus, the number of possible modes of vibration is
In general, the elastic waves propagating in solid are of two types, transverse waves and longitudinal wave
For the total number of vibrational modes with frequencies ranging from zero to v_{D} . We can write
We can associate a Harmonic oscillator of the same frequency with each vibrational mode. Thus, the vibrational energy of the crystal is given by
(ii) where
Using equation (i) and (ii)
Putting
Let Debye temperature is
For more general
d = 1for 1D, d = 2for 2D, d = 3for 3D
We will discuss for 3 dimension
CaseI: High temperature x >> T >> θ_{D} e^{x} − 1 ≈ x
⇒ E = 3Nk_{B}T
⇒ C_{V}= 3R
Thus, at high temperatures, the Debye’s theory also obeys the Dulong and Petit law as obeyed by classical theory and the Einstein theory.
CaseII: Low temperature
For T << θ_{D} ,x_{m} = θ_{D}/T → ∞
⇒ CV ∝ T^{3}
Thus, at very low temperature, the specific heat is proportional to T^{3}. This is called the debye T^{3} law.
Note: The heat capacity dependency on temperature in 3D, 2D & 1D as follows
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