(a) Classical
Dulong and Petit (1819)
Cv=3Nk, Correct at high temperature
(b) Einstein
Based on Planck's quantum hypothesis (1901)
Quantised energy, Showed exponential dependence of Cv
(c) Debye
Showed complete dependence (1912)
The classical model for specific heats considered the atoms as being simple harmonic oscillators vibrating about a mean position in the lattice. Each atom could be simulated by three simple harmonic oscillators (SHOs) vibrating in mutually perpendicular directions.
For a classical SHO:
Average kinetic energy = ½ kT
Average potential energy = ½ kT
Total average energy per oscillator = kT
Total average energy per atom = 3kT
For N atoms the total average energy = U = 3NkT
The specific heat capacity is
Cv=3*R
Classical treatment - Dulong and Petit - the specific heat capacity of a given number of atoms of a given solid is independent of T and is the same for all solids.
So far, the treatment of the vibrational behaviour of materials has been entirely classical. For a harmonic solid, the vibrational excitations are the collective, independent normal modes, having frequencies w determined by the dispersion relationship w(k) with the allowed values of k set by the boundary conditions. In the classical limit, the energy of a given mode with frequency w, determined by the wave amplitude, can take any value.
· MB energy distribution: as T is raised F(Ehigh) increases!
· The energy of the atomic vibrations becomes greater as we go from low to high T
Einstein produced a theory of heat capacity based upon Planck's quantum hypothesis. He assumed that each atom of the solid vibrates about its equilibrium position with an angular frequency w. Each atom has the same frequency and vibrates independently of other atoms. The quantum mechanical result, treating each normal mode as an independent harmonic oscillator with frequency w, is that the energy is quantised and can only take values characterised by the quantum number n(k,p) for a particular branch p. A vibrational state of the whole crystal is thus specified by giving the excitation numbers n(k,p) for each of the 3N normal modes. Instead of describing the vibrational state of a crystal in terms of this number, it is more convenient and convenntional to say, equivalently, that there are n(k,p) phonons (i.e. particle like entities representing the quantised elastic waves).
· Einstein replaced the classical SHO with a Quantum SHO: energy does not increase continuously but in discrete steps: 1D SHO,
n=0,1,2…..· Note that the quantum mechanical expression for the energy implies that the vibrational energy of a solid is non zero even when there are no phonons present: the residual energy of a given mode, 0.5*h*w , is the zero point energy
· choose zero energy at 0.5 h * w
· take
The probability of occupation of this level is:
· choose zero energy at 0.5 h * w
· take
The probability of occupation of this level is:
The total energy of the solid becomes,
Taking account of the zero point energy and using the above result, the mean energy is therefore,
This energy may be considered either as the time-averaged energy for a particular atom, or it can be thought of as the average energy of all the atoms in the assembly at any instant in time.Einstein model conlusion!
· successfully predicts that C falls with decreasing T
· however, exponential decrease is not observed; if low frequencies are present, then will be small, much smaller than kT even at low temperatures; C will remain at 3kT to much lower frequencies and the fall off is not as dramatic as predicted by the Einstein model
· assumption of 'an average' single frequency w is too simplistic
· need a spread of frequencies - a frequency spectrum!
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BIBLIOGRAFIA:
xbeams.chem.yale.edu/~batista/vaa/node28.html
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