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 1 February 211, Published 25 May 211 Abstract: The fine structure constant α / c 1/137.36 and the blacbody radiation constant α R (a R /B )1/3 1/157.555 are two dimensionless constants, derived respectively from a discrete atomic spectra and a continuous radiation spectra and lined by an infinite prime product. The blacbody radiation constant governs large density matter where oscillating charges emit or absorb photons that obey the Bose-Einstein statistics. The new derivations of Planc s law, the Stefan-Boltzmann law, and Wein s displacement law are based on the fine structure constant and a simple 3D interface model. The blacbody radiation constant provides a new method to measure the fine structure constant and lins the fine structure constant to the Boltzmann constant. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Blacbody Radiation; Planc s Law; Fine Structure Constant; Boltzmann Constant PACS (21):..+a; 32.1.Fn; 5.3.-d 1. Introduction Planc and Einstein each noted respectively in 195 and 199 that /c h have the same order and dimension.[1, 2] This was before Sommerfeld s introduction of the fine structure constant α / c in 1916.[3] Therefore, the search for a mathematical relationship between /c h began with blacbody radiation.[] The Stefan-Boltzmann law states that the radiative flux density or irradiance is J σt [erg cm 2 s 1 ]incgs units. From the Planc law, the Stefan-Boltzmann constant σ 5.67() 1 5 [erg cm 2 K s 1 ]is XK6771@gmail.com
38 Electronic Journal of Theoretical Physics 8, No. 25 (21179 388 σ 2π x 3 dx e x cb (hc 2π5 15 2πΓ()ζ () c B (hc 2 π 5 cb 5! (hc c B (hc (1) The Stefan-Boltzmann law can be expressed as the volume energy density of a 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 B 5! (hc (2) In 191, Lewis and Adams noticed that the dimension of the radiation density constant divided by the th power Boltzmann constant a R /B is (energy length) 3, while is (energy length). However, they obtained an incorrect result equivalent to α 1 c/ 32π (π 5 /5!) 1/3 137.38.[5] In 1915, Allen rewrote it as α / c (15/π 2 ) 1/3 /(π) 2.[6] In CGS units, (.83227(12) 1 1 ) 2 [erg cm], a R 7.56576738 1 15 [erg cm 3 K ], and B (1.3865(2) 1 16 ) [erg K ]. We get the experimental dimensionless constant[7] ( ) 1/3 α R ar 1 B 157.558787.6369982 (3) This is the dimensionless blacbody radiation constant α R. 2 2. Relationship to the Fine Structure Constant The dimensionless blacbody radiation constant α R is on the same order of the fine structure constant α / c, and equal to α R.8697668 α () ( ) π 5 1/3 ( ) 1/3 ( ) Γ()ζ () π 2 1/3 α α α 2 π 5! π 2 15 Therefore, α R α, both α and α R are experimental results incapable of producing the α math formula. Physically, the fine structure constant α is obtained from the atomic discrete spectra, while the blacbody radiation constant α R is obtained from the thermal radiation of a 3D cavity in the continuous spectra. However, their relationship can be given by the Riemann zeta-function or by the modification of Euler s product formula 2 Not to be confused with the Stefan-Boltzmann constant σ or hc/ (blacbody radiation constant)
Electronic Journal of Theoretical Physics 8, No. 25 (21179 388 381 α 3 R α 3 π2 15 ζ () ζ (2) p 2 2 3 2 ( p 2 p 2 +1 5 2 (2 2 +1) (3 2 +1) (5 2 +1) (7 +1) 2 ) 7 2 (5) where the Euler product extends over all the prime numbers. In other words, the fine structure constant and the blacbody radiation constant can be lined by the prime numbers. The pattern of Planc spectra is given by f(x) x 3 /(e x ) where the photon hν is hidden in the argument x hν/ B T. The photon integral in (1) is equal to a dimensionless constant (Fig. 1) Figure 1 Photon integral is a dimensionless number Γ()ζ () π 15 6.93939 x 3 dx e x π Γ()ζ () 15 2 3 ( p p 2 3 2 3 (2 ) (3 ) (5 ) (7 ) p 5 ) 7 (6) where the Euler product extends over all the prime numbers. The photon distribution integral (6) yields a zeta-function that is lined to the Euler prime products. (5) and (6) clearly show how the fine structure constant α for the discrete spectra in () is converted to the blacbody radiation constant α R for the continuous spectra by multiplying a dimensionless constant (Fig. 2). (5) and (6) also indicate that this dimensionless constant can be expressed as an Euler infinite prime number product. Figur α and α R from the discrete and continuous spectra.
382 Electronic Journal of Theoretical Physics 8, No. 25 (21179 388 3. Photon-Gas Model From (5), the Stefan-Boltzmann law written as the volume energy density of a blacbody ε T is related to the fine structure constant α and the oscillating charge with different resonating frequencies in a cavity ε T a R T σt c ε T ζ () ( ) α 3 ζ (2) e BT 2 and the radiative flux density is J σt J ζ () ( ) α 3 c ζ (2) BT c and the total brightness of a blacbody is B J/π B ζ () ( ) α 3 c ζ (2) π BT c π BT (7) and the inner wall pressure of the blacbody cavity is P σ 3c T P ζ () ( ) α 3 1 ζ (2) 3 BT 1 3 BT (8) BT (9) BT (1) According to the Bose-Einstein model of photon-gas,[8] the free energy in the thermodynamics is F PV σ 3c VT F ζ () ( ) α 3 V ζ (2) 3 BT V 3 and the total radiation energy is E 3F 3PV σ c VT E ζ () ( ) α 3 V ζ (2) e BT V 2 BT (11) BT (12) where the photon gas E 3PV is the same as the extreme relativistic electron gas, and the entropy is S F 16σ T 3c VT3 S ζ () ( ) α 3 V ζ (2) 3 BT 3 V 3 and the specific heat of the radiation is C V ( E T We have C V ζ () ( ) α 3 V ζ (2) e BT 3 V 2 N B T ζ () ( ) α 3 V ζ (2) 3 BT V 3 ) 16σ V c BT 3 (13) VT3 BT 3 (1) BT (15) From PV N B T, the total number of photons in blacbody radiation is
Electronic Journal of Theoretical Physics 8, No. 25 (21179 388 383 N ζ () ( ) α 3 V ζ (2) 3 BT V 3 BT (16) Landau assumed that the volume V in (11) (16) must be sufficiently large in order to change from discrete to continuous spectra.[7] Planc s law is violated at microscopic length scales. Experimentally, solids or dense-gas have the continuous spectra, and hot low-density gas emits the discrete atomic spectra. The photon hν is hidden in f(x) x 3 /(e x ) where x hν/ B T, therefore, there is no hν in (7) (16). In (7) (16), the charged oscillators α/ 1/ c or α R / play a critical role in the electromagnetic coupling on a 3D surface (Fig. 3). Therefore, the traditional 3D box (or sphere) model is not necessarily composed of solid walls; the plasma gas photosphere layer of a star can have the same effect. Figure 3 Blacbody radiation is related to α and in a 3D cavity.. Planc s Law and the Stefan-Boltzmann law Planc s original formula is a experimental fitting result. There are many derivations to explain the blacbody radiation law, including Planc in 191, Einstein in 1917, Bose in 192, Pauli in 1955.[9, 1, 11] We are not reinventing the blacbody radiation law, but instead pointing out that the surface charge is Planc s oscillator, and it is related to the fine structure constant. Planc s radiation law is derived as the result of the 3D interface interaction between photons and charged particles. Using the 3D surface charge model in Fig. 3 and the energy quanta ε hν, Planc s law in terms of the spectral energy density in [erg cm 3 sr 1 Hz 1 ] can be rewritten as u(ν, T ) 8πhν3 c 3 1 e hν/ BT (17)
38 Electronic Journal of Theoretical Physics 8, No. 25 (21179 388 ( ) ( 1 2 h α hν 1 π e hν/ BT ( ) 1 2 ( ) α 3 ε 3 u(ε, T )h π e ε/ BT where 1/π is related to a solid angle in fractions of the sphere [1sr 1/π fractional area]; and the interaction ratio of photon hν and charge is regulated by fine structure constant α, the cubic term involving the closed 3D cavity wall; Under the statistic thermodynamic equilibrium, Bose-Einstein distribution can be derived as n ne nhν/ BT n e hν/ BT 1 e hν/ BT The frequency of photons in ε hν is constantly shifted into a continuous spectra through photon-electron scattering, such as the Compton effect ν ν (19) 1+(hν/m e c 2 )(1 cos θ) Fig. 3 shows that the scattering angle θ varies during each photon-electron interaction involving the fine structure constant. For ε T a R T, using dν (1/h)dε ε T u(ε, T )dε h Γ()ζ () π 2 ( ) 1 2 ( ) α 3 (B T ) π ( α (B T ) (B T ) x 3 dx e x This lins the quantum theory to the classical theory of blacbody radiation with or without using the Planc constant a R ζ () ζ (2) ( α B B ζ () ζ (2) (18) (2) ( ) 2π 3 B (21) hc ( ) α 3 B 5 9 1 25 26 9 5 121 122 169 17 289 29 361 362 Planc s law in terms of the spectral radiative intensity or the spectral radiance in [erg s 1 cm 2 sr 1 Hz 1 ] has a electromeganetic radiation with lightspeed c and σ ca R /, therefore, we multiply c/ onu(ν, T ) in (17) For J σt, I(ν, T ) c 1 u(ν, T )2πhν3 c 2 e hν/ BT ( ) ( 1 2 (hc) α hν 1 2π e hν/ BT ( ) 1 2 ( ) α 3 ε 3 I(ε, T )(hc) 2π e ε/ BT (22)
Electronic Journal of Theoretical Physics 8, No. 25 (21179 388 385 J I(ε, T )dε h ( c )( ) 1 2 ( ) α 3 (B T ) π ( ) ( ) c Γ()ζ () α 3 (B T ) c π 2 x 3 dx e x (B T ) (23) and the Stefan-Boltzmann constant is σ c ( ) ζ () α 3 ζ (2) B c B (2) The Planc constant h with the revolutionary concept of energy quanta is a bridge between classical physics and quantum physics. Einstein s proposal of the light quanta hν in 195 was based on the Planc constant. In QED, the photon is treated as a gauge boson, and the perturbation theory involves the finite power series in α. The discretecontinuous spectra is bridged by the Bose-Einstein distribution, and the prime sequences lin the fine structure constant α to the blacbody radiation constant α R. 5. Wien s Displacement Law Wien s frequency displacement law is ν max b ν T (25) where b ν 5.878933(1) 1 1 [Hz K] in CODATA-26. It has the numerical solution from e x (x 3) + 3 (26) where x ν hν/ B T 3+W ( 3e 3 ) 2.8213937212278893. Lambert W-function W (x) ( n) n 1 W (x) x n (27) n1 n! is a convergent series for x < 1/e, and i.e., ν max [3 + W ( 3e 3 )] B h b ν [3+W ( 3e 3 )] α 2π Wien s wavelength displacement law is T (28) c B e (29) λ max b λ T (3)
386 Electronic Journal of Theoretical Physics 8, No. 25 (21179 388 Figure Numerical solution for e x (x N)+N. where b λ 2.8977685(51) 1 3 [m K] in CODATA-26. It has a numerical solution from e x (x 5) + 5 (31) where x λ hc/λ max B T 5+W ( 5e 5 ).965112317276337, and ( ) 1 hc b λ (32) 5+W ( 5e 5 ) B It is noticed that λ has a dimension of V 1/3 B /a R c B /σ hc/ B. We can get another good approximate for b λ in (32) ( ) 1/3 ( cb 1/3 e e b λ 3σ 3) 2 (33) B α R ( ) 2 1/3 c ( ) 5 1/3 hc π 2 B 2π 5 where b λ 2.89772 1 3 [m K] with a uncertainty u r 1.6 1 5 to CODATA-26. From x λ /x ν 1.75978586382117887, we can get B b ν c x ν c [5 + W ( 5e 5 )] b λ x λ b λ [3 + W ( 3e 3 )] 5.87932 1 1 [Hz K] (3) From Wien s displacement law, the Boltzmann constant can be directly lined to the fine structure constant b ν h B [3 + W ( 3e 3 )] b ν 2π e [3 + W ( 3e 3 )]αc 2π e b λ [5 + W ( 5e 5 )]α (35)
Electronic Journal of Theoretical Physics 8, No. 25 (21179 388 387 The blacbody radiation constant is a new method to measure the fine structure constant. It lins the fine structure constant to the Boltzmann constant. Acnowledgment The Author thans Bernard Hsiao for discussion. References [1] M. Planc, letter to P. Ehrenfest, Rijsmuseum Leiden, Ehrenfest collection (accession 196), July (195) [2] A. Einstein, Phys. Zeit., 1, 192 (199) [3] A. Sommerfeld, Annalen der Physi 51 (17), 1-9 (1916) [] M. Buchanan, Nature Physics, 6, 833 (21) [5] G.N. Lewis, E.Q. Adams, Phys. Rev. 3 92 12 (191) [6] H.S. Allen, Proc. Phys. Soc. 27 25 31(1915) [7] T.H. Boyer, Foundations of Physics, 37, 7, 999 (27) [8] L.D. Landau, E.M. Lifshitz, Statistical Physics, Vol. 5 (3rd ed.), 183 (198) [9] A. Einstein, Phys. Zs. 18, 121 (1917) [1] S.N. Bose, Z. Phys. 26, 178-181 (192) [11] W. Pauli, Statistical Mechanics (Vol. of Pauli Lectures on Physics). MIT Press, Cambridge, MA, 173 pp. (1973)