On Wien’s peaks
DOI:
https://doi.org/10.31349/RevMexFis.21.010501Keywords:
Thermal radiation, Planck’s Law, Jacobian’s transformation, Wien´s peakAbstract
Most Modern Physics Books contain a chapter on the Laws of Black Body radiation when introducing the principles of Quantum Mechanics. These laws govern many phenomena that we encounter in daily life, technological developments, and scientific research. For that, this old subject is still of great importance, and even now some issues require our attention. This work addresses one of these topics. We describe why it has no sense to think that the wavelength at which Planck’s black-body spectral radiance distribution plotted as a function of the wavelength has its maximum value, must be the same that the wavelength calculated from the peak value obtained when the distribution is plotted as a function of another variable, such as the energy of the photons. We will show how the issue lies in using the correct form to calculate this wavelength from measured quantities.
References
Physics Textbook Review Committee, Quibbles, misunderstandings, and egregious mistakes, Phys. Teach. 37 (1999) 297, https://doi.org/10.1119/1.880292 -. See “Modern Physics” Item 2 on p. 304.
M. Planck, Über das Gesetz der Energieverteilung im Normalspectrum, Ann. d. Phys. 4 (1901) 553, https://doi.org/10.1002/andp.19013090310
B. H. Soffer, D. K. Lynch, Some paradoxes, errors, and resolutions concerning the spectral optimization of human vision, Am. J. Phys. 67 (1999) 946, DOI:10.1119/1.19170
J. Mooney,.P. Kambhampati, Get the Basics Right: Jacobian Conversion of Wavelength and Energy Scales for Quantitative Analysis of Emission Spectra, J. Phys. Chem. Lett. 4 (2013) 3316, https://doi.org/10.1021/jz401508t
J. M. Overduin, Eyesight and the solar Wien peak, Am. J. Phys. 71 (2003) 216, DOI:10.1119/1.1528917
M. A. Heald, Where is the Wien peak?, Am. J. Phys. 71 (2003) 1322, https://doi.org/10.1119/1.1604387
A. Hobson, Solar blackbody spectrum and the eye’s sensitivity, Am. J. Phys. 71 (2003) 295, https://doi.org/10.1119/1.1545762
S. M. Stewart, Wien peaks and the Lambert W function, Rev. Bras. Ens. Fis. 33 (2011) 3308, https://doi.org/10.1590/S1806-11172011000300008
A.L. Xavier Jr and S. Celaschi, Black body radiation as a function of frequency and wavelength: an experimentally oriented approach, Rev. Bras. Ens. Fis. 34 (2012) 2304, https://doi.org/10.1590/S1806-11172012000200007
J. M. Marr and F. P. Wilkin, A better presentation of Planck's radiation law, Am. J. Phys. 80 (2012) 399, http://dx.doi.org/10.1119/1.3696974
S. M. Stewart, Spectral peaks and Wien’s displacement law, J. Thermophys. Heat Transfer 26 (2012) 689.
C. Bohren, Spectral distribution and the color of blackbodies, Am. J. Phys. 83 (2015) 485, https://doi.org/10.2514/1.T3789
E. Marín and A. Calderón, Conversion of Wavelength and Energy Scales and the Analysis of Optical Emission Spectra, J. Phys. Chem. Lett. 13 (2022) 8376, https://doi.org/10.1021/acs.jpclett.2c02621
J. Mooney,.P. Kambhampati. Correction to “Get the Basics Right: Jacobian Conversion of Wavelength and Energy Scales for Quantitative Analysis of Emission Spectra,” J. Phys. Chem. Lett. 5 (2014) 3497, https://doi.org/10.1021/jz502066v
One trivial example of a narrow distribution function is the Dirac Delta, δ(λ − λ0), which is centered at λ=λ0. For this function, the average wavelength is hλi = R λδ(λ − λ0)dλ/ R δ(λ − λ0)dλ = λ0 = R (hc/E)δ(E − E0)dE/ R δ(E − E0)dE = hch1/Ei = hc/E0 = hc/hEi. It is obvious to see that, for δ(λ − λ0), λmax = hc/Emax is also fulfilled.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Ernesto Marin, Antonio Calderón
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors retain copyright and grant the Revista Mexicana de Física E right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
