Fine Structure Constant α 1/ and Blackbody Radiation Constant α R 1/

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1 Fine Structure Constant α / and Blacbody Radiation Constant α R / Ke Xiao Abstract The fine structure constant α e 2 / c / and the blacbody radiation constant α R e 2 (a R/ B) /3 / are lined by rime numbers. The blacbody radiation constant is a new method to measure the fine structure constant. It also lins the fine structure constant to the Boltzmann constant. Planc and Einstein noted resectively in 905 and 909 that e 2 /c h have the same order and dimension.[, 2] This was before the introduction of the fine structure constant α e 2 / c by Sommerfeld in 96.[3] Therefore, the search for a mathematical relationshi between e 2 /c h was started from the blacbody radiation. The Stefan-Boltzmann law states that the radiative flux density or irradiance is J σt [erg cm 2 s ] in CGS units or [W m 2 ] in SI units. From the Planc law, the Stefan-Boltzmann constant σ (0) 0 5 [erg cm 2 K s ] is ˆ x 3 dx σ 2π e x 0 c B (hc 2π5 5 2πΓ()ζ () c B (hc 2 π 5 cb 5! (hc c B (hc () The Stefan-Boltzmann law can be exressed as the volume energy density of the blacbody ε T a R T [erg cm 3 ], where the radiation density constant a R is lined to the Stefan-Boltzmann constant a R σ c 3 π 5 XK677@gmail.com Present Address: P.O. Box 96, Manhattan Beach, CA 90267, USA Acnowledgment: The Author thans Bernard Hsiao for discussion 5! B (hc (2)

2 In 9, Lewis and Adams noticed that the dimension of the radiation density constant divided by the th ower Boltzmann constant a R /B is (energy length) 3, while e 2 is (energy length). However, they obtained an incorrect result equivalent to α c/e 2 32π ( π 5 /5! ) / [] In 95, Allen rewrote it as α e 2 / c (5/π 2 ) /3 /(π) 2.[5] In CGS units, e 2 ( (2) 0 0) 2 [erg cm], ar [erg cm 3 K ], and B ( (2) 0 6) [erg K ]. We get the exerimental dimensionless constant[6] ( ) /3 α R e 2 ar B (3) This is the blacbody radiation constant α R, which is on the same order of the fine structure constant α e 2 / c α R 2 ( ) π 5 /3 ( ) π 2 /3 α α () π 5! 5 ( ) /3 Γ()ζ () α π 2 Therefore, α R α, both α and α R are exerimental results incaable of roducing the α math formula. Physically, the fine structure constant α is obtained from the atomic discrete sectra, while the blacbody radiation constant α R is obtained from the thermal radiation of a 3D cavity in the continuous sectra. However, their relationshi can be given by the Riemann zeta-function or by the modification of Euler s roduct formula (737) α 3 R α 3 π2 5 ζ () 2 2 n n 3 2 n n ) ( 2 ( ) (2 2 + ) (3 2 + ) (5 2 + ) (7 2 + ) ( ) where the Euler roduct extends over all the rime numbers. In other words, the fine structure constant and the blacbody radiation constant can be lined by the rime numbers. From (5), the Stefan-Boltzmann law as the volume energy density of the blacbody ε T is related to the fine structure constant α and the oscillating charge e 2 with the different resonating frequencies in a cavity (5) 2

3 and the radiative flux density is ε T a R T σ c T 3 π 5 5! ζ () e 2 B T B (hc T (6) e 2 B T J σt ζ () c e 2 BT c e 2 B T (7) and the total brightness of a blacbody is B J π ζ () c e 2 π BT c π e 2 B T (8) and the inner wall ressure of the blacbody cavity is P σ 3c T ζ () e 2 3 BT 3 e 2 B T (9) According to the Bose-Einstein model of hoton-gas,[7] the free energy of the thermodynamics is F P V σ 3c V T ζ () V e 2 3 BT V 3 e 2 B T (0) and the total radiation energy is E 3F 3P V σ c V T ζ () ( V e 2 B T αr V e 2 B T () where the hoton gas E 3P V is the same as the extreme relativistic electron gas, and the entroy is S F T 6σ 3c V T 3 ζ () e 2 V 3 BT 3 V 3 e 2 B T 3 (2) and the secific heat of the radiation is ( ) E C V 6σ T c V T 3 ζ () ( V e 2 B T 3 αr V e 2 B T 3 (3) V Landau assumed that the volume V in (0) (3) must be sufficiently large in order to change from discrete to continuous sectra. Exerimentally, solids or dense-gas have the continuous sectra, and hot low-density gas emits the discrete atomic sectra. The attern of Planc sectra is given by f(x) x 3 /(e x ) 3

where hoton hν is hidden in x hν/ B T. The hoton integral in () is equal to a dimensionless constant (Fig. ) Figure : Photon integral is a dimensionless number Γ()ζ () π 5 6.

4 where hoton hν is hidden in x hν/ B T. The hoton integral in () is equal to a dimensionless constant (Fig. ) Figure : Photon integral is a dimensionless number Γ()ζ () π ˆ 0 x 3 dx e x Γ()ζ () π ( ) (2 ) (3 ) (5 ) (7 ) (7 ) ( ) ( ) >5 () where the Euler roduct extends over all the rime numbers. The hoton distribution integral () yields a zeta-function that is lined to the Euler rime roducts, therefore, there is no hν in (6) (3). () shows clearly how the fine structure constant α for the discrete sectra in () is converted to the blacbody radiation constant α R for the continuous sectra by multilying a dimensionless constant. (5) and () indicate that this dimensionless constant can be exressed as the Euler infinite rime number roduct. In (6) (3), the oscillators of the thermal electrons α/e 2 / c or α R /e 2 lay a critical role in the electromagnetic couling on a 3D surface (Fig. 2). Figure 2: Blacbody radiation is related to α and e 2 in a 3D cavity.

5 This 3D box (or shere) model does not necessarily have solid walls, the lasma gas layer of a star can have the same effect. This lins the quantum theory to the classical theory of blacbody radiation with or without using the Planc constant a R e 2 B ζ () e 2 B 5 e 2 B ζ () ( 2π B (5) hc The Planc constant h with the revolutionary concet of energy quanta is a bridge between classical hysics and quantum hysics. Einstein s roosal of the light quanta hν in 905 was based on the Planc constant, however, Planc always had reservations due to the continuous sectra of blacbody radiation and the wave-article duality. In 95, Einstein said that, All these fifty years of conscious brooding have brought me no nearer to the answer to the question, What are light quanta? [7] In QED, the hoton is treated as a gauge boson, and the erturbation theory involves the finite ower series in α. The discretecontinuous sectra is bridged by the Bose-Einstein distribution, and the rime sequences lin the fine structure constant α to the blacbody radiation constant α R. The blacbody radiation constant is a new method to measure the fine structure constant. It also lins the fine structure constant to the Boltzmann constant. References [] M. Planc, letter to P. Ehrenfest, Rijsmuseum Leiden, Ehrenfest collection (accession 96), July (905) [2] A. Einstein, Phys. Zeit., 0, 92 (909) [3] A. Sommerfeld, Annalen der Physi 5(7), -9 (96) [] G. N. Lewis, E. Q. Adams, Phys. Rev (9) [5] H. S. Allen, Proc. Phys. Soc (95) [6] T. H. Boyer, Foundations of Physics, 37, 7, 999 (2007) [7] L. D. Landau, E. M. Lifshit, Statistical Physics, Vol. 5 (3rd ed.), 83 (980) [8] A. Einstein, letter to Michael Besso, Dec. 2 (95); The Born-Einstein Letters Max Born, translated by Irene Born, Macmillan (97) 5

Dimensionless Constants and Blackbody Radiation Laws

EJTP 8, No. 25 (21179 388 Electronic Journal of Theoretical Physics Dimensionless Constants and Blacbody Radiation Laws Ke Xiao P.O. Box 961, Manhattan Beach, CA 9267, USA Received 6 July 21, Accepted