- #1
matteoargos
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- TL;DR Summary
- I am trying to reconstruct the historical sequence between famous Planck's formula of the black-body radiation with respect the ‘Rayleigh-Jeans’ formula and I conclude with a reflection on the small discrepancy between the ‘Rayleigh-Jeans’ original formula appeared in the paper dated 1905 compared to the formula that is usually given in today's physics textbooks.
The search for a mathematical formulation that would explain the shape of the blackbody radiation (which experimentally has that famous ‘bell-shaped’ as a function of wavelength, i.e. zero for wavelength values tending to zero as well as for those tending to infinity) was attempted for several years in the end of 19th century that by the tools of classical mechanics, which led to results that did not correspond to those of experience.
In the various physics textbooks I read that classical mechanics leads (for the blackbody radiation) to the famous Rayleigh-Jeans law (1.1) which generates the paradox of the ultraviolet catastrophe (diverging for wavelength values equal to zero).
Where
Also in these physics textbooks is written that "Planck solved the dilemma with the famous introduction of the quantum of energy , in particular that the atoms of the cavity walls behave like harmonic oscillators that can only take on discrete energy values ##E_n=nh\nu## where ##\nu=c/\lambda##, which led to his famous formula (1.2)
Actually, the exact formulation of (1.1) - or a close relative of it (*) - first appeared in J.H Jeans' “On the partition of Energy between Matter and Aether” ,issue 55 of The London Edinburg and Dublin Philosophical Magazine and Journal of Science, in the July 1905 [see (7) of the Figure 1].
FIGURE 2
It is true that Lord Rayleigh stated that blackbody radiation is proportional to ##1/\lambda^4## in the article entitled “Remarks upon the Law of Complete Radiation,” The London, Edinburgh, and Dublin Philosophical Magazine And Journal Of Science,” Vol. 50, Fifth Series July-December 1900 (see Figure 3), so before the Planck formula of December 1900, but the official formulation as we know it today by Rayleigh-Jeans was not officially published until 5 years later.
FIGURE 3
(*)
---------------
There is small a discrepancy between (1.1) and its original formulation (7) in in FIGURE 1 (highlighted in FIGURE 4 below)
Actually in the above-mentioned 1905 article, James Jeans written the formula
$$u(\lambda, T) = \frac{8 \pi R T}{\lambda^4}\tag{1.3}$$
through a constant ##R## of the value of ##9,3 \times 10^{-17}## ##erg/K##
One must be careful because J. Jeans when he wrote (1.3) did so according to the measurement system of the time which was in C.G.S (centimeters, grams, seconds). Whereas (1.1) is written in the M.K.S system (meters, kelvins, seconds).
So in reality to compare (1.1) with (1.3) one must convert between the two system measurements.
We can write the conversion formula
$$u_{\tiny m.k.s }(\lambda, T) = \frac{8 \pi R_{\tiny c.g.s } \times 10^{-7}(T{\tiny c.g.s }+273,15) \times 10^{8} }{\lambda^4}\tag{1.5}$$
which lead to
$$u_{\tiny m.k.s }(\lambda, T) = \frac{80 \pi R (T{\tiny m.k.s }) }{\lambda{\tiny m.k.s }^4}\tag{1.6}$$
Now we can compare (1.6) vs (1.1) and evaluate the differences by calculating,
for (1.1) the multiplying factor for ##T/\lambda^4## is
##2 \pi k_B c=2,600 \times10^{-14}##
for (1.6) he multiplying factor for ##T/\lambda^4## is
##80 \pi R=2,330 \times10^{-14}##
So (1.1) and (1.6) differs from a factor of ##1,116## .Quite similar but not exactly the same.
Probably the empirical experimental constant derived by Rayleigh was later replaced by more precise experimental data which lead to (1.1).
I honestly could not reconstruct where the (1.1) was first originally formulated
If anyone would like to help me to fill this gap I would be grateful.
FIGURE 5
In the various physics textbooks I read that classical mechanics leads (for the blackbody radiation) to the famous Rayleigh-Jeans law (1.1) which generates the paradox of the ultraviolet catastrophe (diverging for wavelength values equal to zero).
$$u_{\tiny RAYJEANS}(\lambda, T) = \frac{2 \pi k_B c T}{\lambda^4}\tag{1.1}$$
Where
- ##u(\lambda, T)## is the specific electromagnetic energy density of radiation at wavelenght ##\lambda## and temperature ##T##,
- ##\lambda## is the wavelenght of the radiation,
- ##k_B## is Boltzmann's constant, ##1.380649 \times 10^{-23}\text{ }Jk^{-1}##
- ##T## is the absolute temperature, in Kelvin
- ##c## is the speed of light = ##2,998\times 10^{-8}\text{ }ms^-1##
Also in these physics textbooks is written that "Planck solved the dilemma with the famous introduction of the quantum of energy , in particular that the atoms of the cavity walls behave like harmonic oscillators that can only take on discrete energy values ##E_n=nh\nu## where ##\nu=c/\lambda##, which led to his famous formula (1.2)
$$u_{\tiny PLANCK}(\lambda, T) = \frac{8 \pi h c}{\lambda^5} \cdot \frac{1}{e^{\frac{h c}{\lambda k_B T}} - 1} \tag{1.2}$$
where the constant h is mentioned for the first time. - ##h## is Planck's constant = ##6,626\times 10^{-34}\text{ }Js##
Actually, the exact formulation of (1.1) - or a close relative of it (*) - first appeared in J.H Jeans' “On the partition of Energy between Matter and Aether” ,issue 55 of The London Edinburg and Dublin Philosophical Magazine and Journal of Science, in the July 1905 [see (7) of the Figure 1].
FIGURE 1
Where the first version of (1.2) was written by Planck in the «Deutschen Physikalischen Gesellschaft» on 14th of December 1900 specifically in the paper named «Zur Theorie des Gesetzes der Energievereilung in Normalspectrum», (FIGURE 2) thus long before the final theoretical formulation of Rayleigh-Jeans.FIGURE 2
It is true that Lord Rayleigh stated that blackbody radiation is proportional to ##1/\lambda^4## in the article entitled “Remarks upon the Law of Complete Radiation,” The London, Edinburgh, and Dublin Philosophical Magazine And Journal Of Science,” Vol. 50, Fifth Series July-December 1900 (see Figure 3), so before the Planck formula of December 1900, but the official formulation as we know it today by Rayleigh-Jeans was not officially published until 5 years later.
FIGURE 3
(*)
---------------
There is small a discrepancy between (1.1) and its original formulation (7) in in FIGURE 1 (highlighted in FIGURE 4 below)
Actually in the above-mentioned 1905 article, James Jeans written the formula
$$u(\lambda, T) = \frac{8 \pi R T}{\lambda^4}\tag{1.3}$$
through a constant ##R## of the value of ##9,3 \times 10^{-17}## ##erg/K##
FIGURE 4
One must be careful because J. Jeans when he wrote (1.3) did so according to the measurement system of the time which was in C.G.S (centimeters, grams, seconds). Whereas (1.1) is written in the M.K.S system (meters, kelvins, seconds).
So in reality to compare (1.1) with (1.3) one must convert between the two system measurements.
We can write the conversion formula
$$u_{\tiny m.k.s }(\lambda, T) = \frac{8 \pi R_{\tiny c.g.s } \times 10^{-7}(T{\tiny c.g.s }+273,15) \times 10^{8} }{\lambda^4}\tag{1.5}$$
which lead to
$$u_{\tiny m.k.s }(\lambda, T) = \frac{80 \pi R (T{\tiny m.k.s }) }{\lambda{\tiny m.k.s }^4}\tag{1.6}$$
Now we can compare (1.6) vs (1.1) and evaluate the differences by calculating,
for (1.1) the multiplying factor for ##T/\lambda^4## is
##2 \pi k_B c=2,600 \times10^{-14}##
for (1.6) he multiplying factor for ##T/\lambda^4## is
##80 \pi R=2,330 \times10^{-14}##
So (1.1) and (1.6) differs from a factor of ##1,116## .Quite similar but not exactly the same.
I make the assumption that the constant R in (1.3) was an empirical constant derived from Rayleigh, which Jeans mentions for the 1st time in his "The Dynamical Theory of Gases, 1st edition, Cambridge at the University Press 1904", FIGURE 5
Probably the empirical experimental constant derived by Rayleigh was later replaced by more precise experimental data which lead to (1.1).
I honestly could not reconstruct where the (1.1) was first originally formulated
If anyone would like to help me to fill this gap I would be grateful.
FIGURE 5
