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Summary

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English: This shows how Planck's model predicts that oscillators with "photon" constrained-energies ε ≡ hν = hc/λ greater than kT get significantly frozen out of the black body mix, thus solving the ultraviolet catastrophe to which the Rayleigh-Jeans model gave birth. This freeze-out may be explained if (for whatever reason) "quantum oscillators" exist only in packets with energy/frequency & momentum/spatial-frequency linked by the constant h = E/ν = p/g = 2πL.


The dot-dashed lines show the (additive) fractional-contribution of one, two, three, and four photon excitations to the average energy per oscillator at frequency ν. As you can see, when hν>>kT the single-photon fraction dominates as the average energy-per-oscillator drops to zero i.e. finding one of them in the mix is unlikely and finding more than one is much much less likely. This compensates for the fact that with increasing frequency (also decreasing wavelength) the number of oscillators per unit frequency (or wavelength) interval is going through the roof.
 
This plot was created with Gnuplot by v.
Date
Source

https://systemreboot.net/post/black-body-energy-as-a-sum-of-photon-oscillator-energies

The probability of an oscillator having photons is given by

The energy contribution of an oscillator with photons is given by

The average black body energy is the sum of contributions from all oscillators.

Author Arun Isaac
gnuplot source
InfoField
click to expand
#! /usr/bin/env gnuplot

# This work is in the public domain.
#
# It was completed by Arun I in 2016.
#
# Though you are not legally obliged to do so, I would appreciate it
# if you credit me for this work by not removing this notice, and
# hopefully linking to my blog post at
# https://systemreboot.net/post/black-body-energy-as-a-sum-of-photon-oscillator-energies

# Base filename for output files
filename='blackbody-energies'

# Set terminal and output file
set term epslatex color colortext standalone
set output sprintf('%s.tex', filename)

# Axes' properties, title and key
set xrange [-0.01:5]
set yrange [0:1.05]
set mxtics 5
set ytics 0.2
set mytics 4
set xlabel '$ h\nu/kT $'
set ylabel 'Average oscillator energy (in units of $ kT $)'
set title "Planck's solution as a freeze-out of\nphoton oscillator-energies as $ h\\nu/kT \\to \\infty $"
unset key

# Line styles
set style line 1 lc 'blue' dt 4
set style line 2 lc 'red' dt 4
set style line 3 lc 'dark-yellow' dt 4
set style line 4 lc 'forest-green' dt 4

# Labels
set arrow from graph 0, first 1 to graph 1, first 1 nohead lc 'black' dt 6
set label '$ E_{\textrm{\tiny avg}}^{\textrm{\tiny Rayleigh}} = kT $' at graph 1, first 1 right offset graph -0.05, -0.05
set label '\footnotesize $ h\nu $' at 1.4, 0.15 tc ls 1
set label '\footnotesize $ 2h\nu $' at 1, 0.33 tc ls 2
set label '\footnotesize $ 3h\nu $' at 0.8, 0.45 tc ls 3
set label '\footnotesize $ 4h\nu $' at 0.65, 0.53 tc ls 4
set label '$ E_{\textrm{\tiny avg}}^{\textrm{\tiny Planck}} $' at 0.6, 0.8

# Expressions to plot
En(x,n) = n*x*exp(-n*x)*(1-exp(-x))
yN(x,N) = sum [n=1:N] En(x,n)

plot x/(exp(x) - 1) with lines lc 'black', \
     for [N=1:4] yN(x,N) with lines ls N

# Convert tex -> pdf -> svg
set output
system(sprintf('pdflatex %s.tex && pdf2svg %s.pdf %s.svg', filename, filename, filename))

Licensing

w:en:Creative Commons
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This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
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Under the following conditions:
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31 October 2016

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current22:58, 31 October 2016Thumbnail for version as of 22:58, 31 October 2016450 × 315 (109 KB)ArunisaacUser created page with UploadWizard

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