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34: Planck's Formula 
On the preceding page,
we studied about
the heat capacity of
a cavity
or a vacuum.
The result said that,
if we naively apply
the law of equipartition
of energy,
the total energy
of a vacuum (cavity)
becomes infinite.
This is completely
different from the reality.
Is the law of equipartition of energy inapplicable for the cavity radiation? The law is naturally derived from the classical theory consisting of Newtonian mechanics and Maxwellian electromagnetism. This seems to suggest that there must be something wrong in the classical theories. Then, where is the secret hidden? This is one of the biggest problems in physics at the end of the 19th century. 
[RayleighJeans's Formula]
As discussed on the preceding page, Maxwellian electromagnetism gives the number of the normal oscillations with the frequency between _{} and _{} per unit volume as If the constant energy kT is partitioned to all these normal oscillations according to the law of equipartition of energy, then the energy of the radiation (or light) with the frequencies between _{} and _{} per unit volume becomes This is the RayleighJeans formula proposed by J. W. S. Rayleigh (UK, 1842  1919) and J. H. Jeans (UK, 1877  1946). If we measure the frequencies and intensities of lights existing inside a hightemperature space like a blast furnace in an iron industry, we can obtain the spectrum (intensity distribution for each frequency of light). The experimental data are shown in the following figure. 
Experimental spectra of the cavity radiation
The positions of the hill tops of the solid curves on the horizontal axis denote the frequency of the brightest light. As the absolute temperature T of each curve becomes high, the frequency of the brightest light shifts gradually to higher. This means that the color of cavity changes from red to white, as the temperature becomes higher. The dashed curve shows the values of RayleighJeans's formula for T = 1646 K. 
For example,
when heating up
an iron block,
the color of the ion
is gray
at a low temperature.
It would be bright red
at about
_{}
and become white
and dazzling at
_{}
This feature
is shown in
the above figure.
In the above figure, the values of the RayleighJeans formula are shown by a dashed curve. It is well fit to experimental data at low frequencies, but becomes worse at high frequency. This appears to imply that the law of equipartition of energy is not valid for high frequencies. 
[Wien's Formula]
W. Wien (Germany, 1864  1928) extended a very ingenious discussion of generality to obtain a formula describing the intensity distribution in the cavity radiation (1896). Since Wien's idea is somewhat difficult to explain simply, we omit it and represent only the result here. According to Wien, the energy distribution of the cavity radiation for a unit volume is given by We cannot obtain the form of the function F(x) only from Wien's discussion. However the most important is that the functional value is determined only by the ratio of the frequency _{} to the temperature T . This is quite well fit to experiment. This law is called Wien's law (or Wien's displacement law). Let the wave length of the brightest light in the cavity radiation be _{}. Differentiating the above Wien's law by _{} we can easily obtain This is often called Wien's displacement law. Setting F(x) = k/x in Wien's law, we have the RayleighJeans formula. Wien took and obtained Wien's formula If the constant _{} is adjusted to an appropriate value, the values of Wien's formula are well fit to the experimental data at the highfrequency region. 
[Planck's Formula]
As discussed above, the RayleighJeans formula concerning the cavity radiation gives very good fit to experiment in the low frequency region. On the other hand, Wien's formula is very good in the high frequency region. Then, M.K.E.L. Planck (Germany, 1858  1947) proposed an interpolating formula which can unify these two formulae and brings about very good fit to experiment over all regions of frequencies. This is the famous Planck's formula which is one of the biggest discoveries at the end of the 19th century. Planck proposed that the function F(x) in the above mentioned Wien's law should be taken as Accordingly, the energy distribution in the cavity radiation becomes This is Planck's formula, and the constant k is Boltzmann's constant that is a universal constant, Another constant _{} is adjustable, which is usually written _{}, where h is called Planck's constant whose value is given as You can easily see in the following figure how well Planck's formula reproduces the experimental data. 
Comparison between
the experimental
spectrum of the
cavity radiation
and Planck's formula.
The small circles are experimental data and the solid curves denote the values of Planck's formula. Be careful the abscissa shows the wave length, not the frequency. 
[Comparison between
the Three Formulae]
As discussed above, RayleighJeans's formula shows a good fit to experiment in the lowfrequency region and Wien's formula is good in the highfrequency region. Planck's formula can reproduce the experimental energy distribution throughout all frequency regions. A comparison between these three formulae is given in the following figure. You can see from this figure that Planck's formula is just an interpolation of RayleighJeans's formula and Wien's formula. 
Comparison between
RayleighJeans's,
Wien's
and
Planck's formula

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