The Planck Distribution
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We saw that the Rayleigh Jeans law fails to explain the black body spectrum at higher frequencies. Planck in trying to explain this discrepancy noticed that the average total energy for the modes was kT at low frequencies but must fall to zero at high frequencies to describe the experimental curve. This required a high frequency cut off for the oscillators in violation of the equipartition theorem which assumes that the oscillators may have any energy from 0 to infinity with equal a priori probability resulting in an average total energy per mode independent of frequency.
Planck departed from this hypothesis and postulated that the oscillators have
discrete energies which are equally spaced that is E=0, ΔE. 2 ΔE..... and ΔE=hν was proportional to the frequency and the proportionality constant h was refered to as the Planck's constant.
As ΔE=hν the energy of the oscillators E=nhν with n=0,1,2,3.....
An average energy computed from this discrete values using the Boltzmann
distribution as in the Classical Equipartition Theorem ( using summations over discrete values instead of integrals) yielded the Plancks distribution formula for the average energy
Eav=[hν/ e(hν/kT) -1]
It is obvious from the formula that the desired high frequency cut off for the oscillators is obtained as E goes to zero with large values of frequencies and E
goes to (kT) ( expanding the exponential to first order) for small values of frequency.
Multiplying this by the Jeans number of modes in an infinitessimal frequency interval we obtain the energy density of the cavity radiation to be
ρ(ν)dν=(8πν2/c3 [hν/ e(hν/kT) -1] dν
This is the Planck Black body Spectrum and completely matches the experimental black body curve ( upto a proportionality constant).
http://www.youtube.com/watch?v=cW4vmr3hb2o&feature=related
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We saw that the Rayleigh Jeans law fails to explain the black body spectrum at higher frequencies. Planck in trying to explain this discrepancy noticed that the average total energy for the modes was kT at low frequencies but must fall to zero at high frequencies to describe the experimental curve. This required a high frequency cut off for the oscillators in violation of the equipartition theorem which assumes that the oscillators may have any energy from 0 to infinity with equal a priori probability resulting in an average total energy per mode independent of frequency.
Planck departed from this hypothesis and postulated that the oscillators have
discrete energies which are equally spaced that is E=0, ΔE. 2 ΔE..... and ΔE=hν was proportional to the frequency and the proportionality constant h was refered to as the Planck's constant.
As ΔE=hν the energy of the oscillators E=nhν with n=0,1,2,3.....
An average energy computed from this discrete values using the Boltzmann
distribution as in the Classical Equipartition Theorem ( using summations over discrete values instead of integrals) yielded the Plancks distribution formula for the average energy
Eav=[hν/ e(hν/kT) -1]
It is obvious from the formula that the desired high frequency cut off for the oscillators is obtained as E goes to zero with large values of frequencies and E
goes to (kT) ( expanding the exponential to first order) for small values of frequency.
Multiplying this by the Jeans number of modes in an infinitessimal frequency interval we obtain the energy density of the cavity radiation to be
ρ(ν)dν=(8πν2/c3 [hν/ e(hν/kT) -1] dν
This is the Planck Black body Spectrum and completely matches the experimental black body curve ( upto a proportionality constant).
http://www.youtube.com/watch?v=cW4vmr3hb2o&feature=related
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