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Wien's Displacement Law for Thermal Radiation


Willy Wien

Willy Wien was an important participant in the search for a formula for the radiation emitted by a body as a function of temperature and wavelength. He was born in East Prussia to aristocratic parents who gave him the name Wilhelm Carl Werner Otto Fritz Franz Wien. He later became known more simply as Willy.

Josef Stefan had conjectured in 1879 that the total emission of radiant energy from a body at absolute temperature T is proportional to the fourth power of T. In 1884 Ludwig Boltzmann provided a rigorous proof for Stefan's conjecture.

Boltzmann's proof was thermodynamic and involved treating the radiation in a cavity as a gas. Furthermore this radiation gas was subjected to work involving reflection from a moving piston. Willy Wien recognized that the moving piston would subject the reflecting radiation to the Doppler Effect, thus changing its wavelength and frequency. Stefan and Boltzmann were not concerned with the wavelength (and frequency) distribution of radiation, but. Wien was. He saw that an extension of Boltzmann's thermodynamic analysis would yield information about the distribution of thermal radiation.

Wien was looking for the energy per unit wavelength interval, Eλ, and its equivalent, Bν, the energy per unit frquency interval. He found on the basis of his extension of Boltzmann's analysis that Eλ and Bν must have the forms

Eλ = F(λT)/λ5
Bν = ν3G(ν/T)

where F( ) and G( ) are unknown functions. Later It was discovered by Max Planck that the proper form for ν/T is hν/(kT), where h is a constant that subsequently became known as Planck's constant and k is Boltzmann's constant.

The Relationship between the Wavelength
for the Peak in Energy Emission
and the Absolute Temperature

Consider finding the wavelength, λmax, at which Eλ is a maximum. This involves finding the value of λ such that

dEλ/λ = 0
(d/dλ)[F(λT)/λ5] = F'(λT)T/λ5 −5F(λT)/λ6

When this expression is set equal to zero and the equation multiplied through by λ6 the result is

F'(λT)λT −5F(λT) = 0

Let λT be denoted as z. Then the above equation reduces to
F'(z)z = 5F(z)

There is presumably only one value of z satisfying this equation, say z*.

Thus it turns out that this implies that

λmaxT = z*

The value of z* is approximately 0.294 cm-K°. Thus λmax is inversely proportional to absolute temperature. This was confirmed by empirical measurement and became known as Wien's Displacement Law because the peak in the curve is displaced (changed) by changes in temperature.

Wavelength and frequency are related by

ν = c/λ

where c is the speed of light in space. Thus the frequency at which Bν is a maximum is directly proportional to T.

Wien went on to make further assumptions and on the basis of those assumptions concluded that the energy emission function must be of the following form

Eλ = c1exp(c2/(λT))/λ5

where c1 and c2 are constants.

This form did fit reasonably well the data available at the time. That data however only involved temperature up to 4000°K and wavelengths down to the values for violet light. When empirical measurements were obtained for wavelengths much greater than those of red light the formula failed. The formula that had been accepted as right was now obviously wrong. Willy Wien's reputation in physics suffered from this failure.


Reference:

Max Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill Book Co., New York, 1966.


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