u/Radlib123
What is Black-body radiation? It's basically radiation that all objects radiate, all matter radiates. When you increase the temperature of any body, it starts to radiate more electromagnetic waves in all of its frequencies.
What formula determines the intensity spectrum of black body radiation? Meaning the formula, describing the intensity of each frequency of radiation a black body radiates, at different temperatures of the matter.
It is described by the Plank's law.
What is the explanation for that formula? Planck had one explanation of it, but Einstein disagreed with Planck's explanation, and provided his own.
Einstein explained it, that black body radiation comes from two sources - stimulated emission, and spontaneous emission. Spontaneous emission being the main bulk of black body radiation, while stimulated emission causing additional intensity at longer wavelength, lower frequency portion of the spectrum.
What is the formula of intensity spectrum, from each of those contributions? I couldn't find the exact formula for the stimulated emission contribution to the black body radiation, but i did find the exact formula for the contribution of spontaneous emission (which is the main contributor to black body radiation).
It is Wien's law, or Wien approximation:
https://en.wikipedia.org/wiki/Wien_approximation
How did he come up with that equation? You can read about it here:
Turns out he simply refined the ideas of another earlier physicist, Wladimir Michelson, to come up with his formula. Specifically, the ideas of Michelson from this paper:
https://hal.science/jpa-00238766v1/document (written in French)
Here is translation of it to English:
https://archive.org/details/michelson-1887-translation
His main ideas were:
Each particle in matter oscillates in a spherical shell of fixed radius, thus emitting EM waves. The frequency of this oscillation was proportional to the velocity of that particle:
f ∝ v, (equation 4 in Michelson's paper, with period turned into frequency)
The intensity of this radiation was proportional to:
- Number of particles emitting radiation of same frequency, thus having the same velocity. More particles of same velocity, means higher intensity of radiation at the corresponding frequency.
- some function of surrounding temperature. Higher the temperature of the body, higher intensity of radiation for all frequencies. Higher is the intensity of radiation from each particle.
- some function of particle's velocity, thus its emitting frequency (since it's proportional to velocity). Higher the velocity of the particle (thus higher is its frequency), higher is the intensity of the radiation that particle emits.
He was able to derive a formula, that is similar to present day Planck's law and WIen's law, except with much less accuracy.
Wien refined his ideas, in the following manner:
He dropped the assumption of radiation being result of particle oscillating in a spherical shell, being the mechanical cause of radiation. And only assumed, that frequency of emitted radiation from each particle, was proportional to velocity.
He kept the assumptions of Michelson, of the radiation's intensity being proportional to the number of particles of same velocity, some function of temperature, and some function of that frequency.
He then applied restrictions on the possible formula, based on other empirical measurements of the black body radiation spectrum, and came up with his formula:
This formula accurately describes the spontaneous emission contribution to the black body radiation (but doesn't include the contribution of the stimulated emission).
So, what did Wien fix in Michelson's assumptions?
He implicitly assumed, that each particle of given velocity, emits radiation of frequency:
f=(mv^2)/(2h).
Difference from Michelson's assumption, is that now radiation scales as proportional to velocity squared, rather than just velocity:
f ∝ v^2
He kept the same assumption, of intensity of radiation being proportional to the number of particles that have the same velocity. But he implicitly ignores it for deriving the formula, as the black body radiation formula doesn't exactly deal with absolute intensities, but rather the relative ones. The main goal of that formula is to get the curvature right.
He implicitly assumed, that intensity of radiation was proportional to temperature with power of 1.5. Determining the undetermined function of Michelson. More precisely:
(kT/h)^1.5, where k is the Boltzmann constant, h is Planck's constant.
And he implicitly assumed, that intensity of radiation was proportional to frequency with power of 2.5. Determining the undetermined function of Michelson:
f^2.5
When he applied those assumptions, to the Maxwell-Boltzmann distribution of kinetic energies, in matter, he derived his black body radiation formula.
Let's verify it.
Step 1. Formula of Maxwell-Boltzmann distribution of energies (formula 9 in the "Distribution for the energy" section):
https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution#Distribution_for_the_energy
Step 2. Transform the horizontal axis into frequency. f=(mv^2)/(2h), thus hf=(1/2)mv^2, thus E=hf. Substitute E = hf, dE = h df:
Step 3. Apply Michelson's temperature intensity scaling assumption, (kT/h)^1.5, which cancels with the term from Maxwell distribution:
Step 4. Apply Michelson's frequency intensity scaling assumption, f^2.5. Which merges with the frequency term from previous formula: √f · f^2.5 = f^3
Step 5. Multiply by some constant of proportionality A:
To get Wien's law, we need to assume that A = (√π)h)/c^2:
And we derive Wien's law! Which is a real, accurate formula for describing the spontaneous emission contribution to black body radiation.
Interestingly, if we transform the temperature into most probable velocity (the biggest number of particles having the same velocity in a given body), v_mp:
And transform that, into frequency f_mp, using the earlier formula f=(mv^2)/(2h), we learn that Michelson's assumption of (kT/h)^1.5, transforms into f_mp^1.5:
(kT/h)^1.5=f_mp^1.5
We also can express Maxwell-Boltzmann distribution, that we previously derived for frequency being the horizontal axis, using f_mp:
If you multiply it by:
A * f^2.5 * f_mp^1.5
We get Wien's law, expressed using f_mp!
Connection to De Broglie wavelength, Schrodinger equation, matter waves, photoelectric effect
The formula, of how the velocity of the particle correlates to the frequency of radiation it emits, implicitly present in Wien's paper, is very interesting.
f=mv^2/2h. Because re-arranged, it is hf=(1/2)mv^2
If we assumed, that a traveling particle, left the trail making oscillation of this frequency along the way, it would create a wave line behind the traveling particle. And the wavelength of this spatial pattern, would be twice the De Broglie wavelength, which is the wavelength of a matter wave that accompanies all particles, according to the Schrodinger equation.
This wave like pattern is stationary (since we see it as a static trail left by the particle), thus from particle's perspective, this wave is traveling in a direction opposite to the particle's velocity, and equal to particle's velocity:
v_trail=v (v is particle's velocity)
v_tail=λ_trail * f (the standard wave formula, of how wavelength, frequency and velocity of the wave relate)
Substituting:
v= λ_trail * (mv^2/2h)
λ_trail = 2h/mv
Which is double the De Broglie wavelength:
λ_db=h/mv
If we assume, just like in the Schrodinger equation, the velocity of this matter wave is half the particle's velocity, then the rate at which the particle passes its own matter wave is f=mv^2/2h. Exactly the same formula, that Wien was implicitly using, when deriving his law of black body radiation.
v_wave=v/2 (from Schrodinger, and v_trail is no longer just a trial, as it moves, thus i renamed it v_wave)
v_wave=λ_wave * f
(v/2)= λ_wave * (mv^2/2h)
λ_wave = h/mv = λ_db
Thus, Wien has implicitly been using the matter wave formula, for his derivation of the black body radiation law.
De Broglie himself, also assumed that each particle in motion, had an inherent frequency in them, same assumption as Wien. But his formula for this particle's frequency, was different:
f=mc^2/h.
You can see how it is similar to our formula:
f=mv^2/2h
De Broglie's frequency formula forced him to imagine the matter wave's velocity as superluminal. While Wien's implicit frequency formula, for each particle in motion, led to assumption that the velocity of the matter wave is half the particle's velocity. Exactly the same assumption as in Schrodinger equation. In Schrodinger, the frequency at which the particle passes its own matter wave, is that exact same frequency.
hf=(1/2)mv^2 is also the formula for the photoelectric effect, assuming that the work function is 0.
Note on the history of the Black Body radiation law.
Here is the full history of the black body radiation law.
In 1887, Wladimir Michelson was the first person to come up with the approximate formula, describing the black body radiation spectrum.
In 1896, Wilhelm Wien refined Michelson's ideas, deriving the accurate formula of the spontaneous emission contribution to the Black Body radiation.
In 1900, the inaccuracy of Wien's law for the low frequency, long wavelength portion of black body radiation spectrum was noticed, and Lord Rayleigh derived a formula, that accurately predicted just the long wavelength/low frequency part of the black body radiation formula.
The same year, Planck created the modern Black Body radiation law by simply extrapolating Rayleigh's formula and Wien's law together, deriving an empirical formula of black body radiation. He initially derived it only empirically, without a theory to back it up. Only couple months later, did he present a theory, to explain his formula that he derived empirically earlier.
In 1905 and years following, Einstein disagreed with Planck's explanation of the black body radiation law. He instead wrote, that black body radiation was made from contributions of stimulated emission, and spontaneous emission. And Wien's law correctly calculated the spontaneous emission contribution to the black body radiation.
Also, the "ultraviolet catastrophe" of 19th century did not exist. Its a historical misinformation widely accepted by the modern scientific community. The ultraviolet catastrophe only stems from Rayleigh-Jeans law, which only accurately described the long wavelength/low frequency portion of the black body radiation, and never claimed otherwise. And prior to that law, Wien's law was used for describing the black body radiation spectrum, which described the whole spectrum approximately accurately.
You can read the history of it here: https://en.wikipedia.org/wiki/Planck%27s_law#History
u/Radlib123 — 14 days ago