EJTP 8, No. 25 (2011) 379–388
Electronic Journal of Theoretical Physics
Dimensionless Constants and Blackbody Radiation Laws Ke Xiao∗ P.O. Box 961, Manhattan Beach, CA 90267, USA
Received 6 July 2010, Accepted 10 February 2011, Published 25 May 2011 Abstract: The fine structure constant α = e2 /c ≈ 1/137.036 and the blackbody radiation 4 )1/3 ≈ 1/157.555 are two dimensionless constants, derived respectively constant αR = e2 (aR /kB from a discrete atomic spectra and a continuous radiation spectra and linked by an infinite prime product. The blackbody radiation constant governs large density matter where oscillating charges emit or absorb photons that obey the Bose-Einstein statistics. The new derivations of Planck’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 blackbody radiation constant provides a new method to measure the fine structure constant and links the fine structure constant to the Boltzmann constant. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Blackbody Radiation; Planck’s Law; Fine Structure Constant; Boltzmann Constant PACS (2010): 44.40.+a; 32.10.Fn; 05.30.-d
1.
Introduction
Planck and Einstein each noted respectively in 1905 and 1909 that e2 /c ∼ h have the same order and dimension.[1, 2] This was before Sommerfeld’s introduction of the fine structure constant α = e2 /c in 1916.[3] Therefore, the search for a mathematical relationship between e2 /c ∼ h began with blackbody radiation.[4] The Stefan-Boltzmann law states that the radiative flux density or irradiance is J = σT 4 [erg · cm−2 · s−1 ] in CGS units. From the Planck law, the Stefan-Boltzmann constant σ = 5.670400(40) × 10−5 [erg · cm−2 K−4 s−1 ] is ∗
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ˆ∞ σ = 2π 0
4 ckB4 2π 5 ckB x3 dx · = ex − 1 (hc)3 15 (hc)3
= 2πΓ(4)ζ (4)
(1)
4 4 ckB 42 π 5 ckB = 5! (hc)3 (hc)3
The Stefan-Boltzmann law can be expressed as the volume energy density of a blackbody εT = aR T 4 [erg · cm−3 ], where the radiation density constant aR is linked to the StefanBoltzmann constant 4 4σ 43 π 5 kB (2) = aR = c 5! (hc)3 In 1914, Lewis and Adams noticed that the dimension of the radiation density con4 stant divided by the 4th power Boltzmann constant aR /kB is (energy × length)−3 , while e2 is (energy × length). However, they obtained an incorrect result equivalent 1/3 to α−1 = c/e2 = 32π (π 5 /5!) = 137.348.[5] In 1915, Allen rewrote it as α = e2 /c = (15/π 2 )1/3 /(4π)2 .[6] 2 In CGS units, e2 = (4.80320427(12) × 10−10 ) [erg·cm], aR = 7.56576738 × 10−15 4 4 [erg · cm−3 K−4 ], and kB = (1.3806504(24) × 10−16 ) [erg4 K−4 ]. We get the experimental dimensionless constant[7]
1/3
aR 1 αR = e = 4 kB 157.5548787 = 0.00634699482 2
(3)
This is the dimensionless blackbody radiation constant αR .2
2.
Relationship to the Fine Structure Constant
The dimensionless blackbody radiation constant αR is on the same order of the fine structure constant α= e2 /c, and equal to αR = 0.8697668 · α =
2 π
5
π 5!
1/3
α=
Γ(4)ζ (4) π2
1/3
α=
2
π 15
(4)
1/3
α
Therefore, αR = α, both α and αR 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 blackbody radiation constant αR 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/k (blackbody radiation constant)
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3 π2 αR p2 ζ (4) = = = α3 15 ζ (2) p2 + 1 p 22 32 52 72 = 2 ··· (2 + 1) (32 + 1) (52 + 1) (72 + 1)
(5)
where the Euler product extends over all the prime numbers. In other words, the fine structure constant and the blackbody radiation constant can be linked by the prime numbers. The pattern of Planck spectra is given by f (x) = x3 /(ex − 1) where the photon hν is hidden in the argument x = hν/kB T . The photon integral in (1) is equal to a dimensionless constant (Fig. 1)
Figure 1 Photon integral is a dimensionless number Γ(4)ζ (4) =
ˆ∞ 0
x3 dx p4 π4 = Γ(4)ζ (4) = = 2 · 3 · ex − 1 15 p4 − 1 p
= 2·3·
π4 15
= 6.4939394
(6)
34 54 74 24 ··· (24 − 1) (34 − 1) (54 − 1) (74 − 1)
where the Euler product extends over all the prime numbers. The photon distribution integral (6) yields a zeta-function that is linked to the Euler prime products. (5) and (6) clearly show how the fine structure constant α for the discrete spectra in (4) is converted to the blackbody radiation constant αR 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.
Figure 2 α and αR from the discrete and continuous spectra.
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3.
Photon-Gas Model
From (5), the Stefan-Boltzmann law written as the volume energy density of a blackbody εT is related to the fine structure constant α and the oscillating charge e2 with T4 different resonating frequencies in a cavity εT = aR T 4 = 4σ c
3
ζ (4) α εT = ζ (2) e2
4 4 kB T
3
αR = e2
4 4 kB T
(7)
and the radiative flux density is J = σT 4
ζ (4) α J= ζ (2) e2
3
c 4 4 c αR k T = 4 B 4 e2
3
4 4 kB T
(8)
and the total brightness of a blackbody is B = J/π
ζ (4) α B= ζ (2) e2
3
c 4 4 c kB T = 4π 4π
αR e2
3
ζ (4) α ζ (2) e2
3
1 4 4 1 αR k T = 3 B 3 e2
(9)
4σ 4 T 3c
and the inner wall pressure of the blackbody cavity is P = P =
4 4 kB T
3
4 4 kB T
(10)
According to the Bose-Einstein model of photon-gas,[8] the free energy in the thermodynamics is F = −P V = − 4σ V T4 3c
ζ (4) α F =− ζ (2) e2
3
3
V
4 4 kB T
=V
αR e2
4σ V c
and the total radiation energy is E = −3F = 3P V = ζ (4) α E= ζ (2) e2
V 4 4 V kB T = − 3 3
αR e2
3
4 4 kB T
(11)
T4
3
4 4 kB T
(12)
where the photon gas E = 3P V is the same as the extreme relativistic electron gas, and the entropy is S = − ∂F = 16σ V T3 ∂T 3c
S=
ζ (4) α ζ (2) e2
3
4V 4 3 4V k T = 3 B 3
and the specific heat of the radiation is CV =
CV =
ζ (4) α ζ (2) e2
3
∂E ∂T V
αR e2
16σ V c
=
4 3 4V kB T = 4V
3
αR e2
kB4 T 3
(13)
T3
3
4 3 kB T
(14)
We have
ζ (4) α N kB T = ζ (2) e2
3
V 4 4 V k T = 3 B 3
αR e2
3
4 4 kB T
From P V = N kB T , the total number of photons in blackbody radiation is
(15)
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ζ (4) α 3 V 4 4 V αR 3 4 4 N= kB T (16) k T = ζ (2) e2 3 B 3 e2 Landau assumed that the volume V in (11)∼(16) must be sufficiently large in order to change from discrete to continuous spectra.[7] Planck’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) = x3 /(ex − 1) where x = hν/kB T , therefore, there is no hν in (7)∼(16). In (7)∼(16), the charged oscillators α/e2 = 1/c or αR /e2 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 Blackbody radiation is related to α and e2 in a 3D cavity.
4.
Planck’s Law and the Stefan-Boltzmann law
Planck’s original formula is a experimental fitting result. There are many derivations to explain the blackbody radiation law, including Planck in 1901, Einstein in 1917, Bose in 1924, Pauli in 1955.[9, 10, 11] We are not reinventing the blackbody radiation law, but instead pointing out that the surface charge is Planck’s oscillator, and it is related to the fine structure constant. Planck’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, Planck’s law in terms of the spectral energy density in [erg · cm−3 · sr−1 · Hz−1 ] can be rewritten as
u(ν, T ) =
1 8πhν 3 3 hν/k T −1 B c e
(17)
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2
3
1 1 hν = h· α 2 π e ehν/kB T − 1 2 3 1 α ε3 u(ε, T ) = h · π e2 eε/kB T − 1 where 1/π is related to a solid angle in fractions of the sphere [1sr = 1/4π fractional area]; and the interaction ratio of photon hν and charge e2 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 ∞
ne−nhν/kB T 1 = hν/k T −hν/k T B B e −1 n=0 e
n=0
∞
(18)
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 1 + (hν/me c2 )(1 − cos θ)
(19)
Fig. 3 shows that the scattering angle θ varies during each photon-electron interaction involving the fine structure constant. For εT = aR T 4 , using dν = (1/h)dε ˆ∞ εT = 0
2
u(ε, T )dε 1 = h π
Γ(4)ζ (4) α = π2 e2
3
α e2
3
(kB T )
αR (kB T )4 = e2
3
4
ˆ∞ 0
x3 dx ex − 1
(20)
(kB T )4
This links the quantum theory to the classical theory of blackbody radiation with or without using the Planck constant
ζ (4) α 3 4 kB = aR = ζ (2) e2 3 α 4 9 4 = · kB · · 2 e 5 10
αR 3 4 ζ (4) kB = 2 e ζ (2) 25 49 121 169 · · · 26 50 122 170
2π 3 4 kB hc 289 361 · · ··· 290 362
(21)
Planck’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 = caR /4, therefore, we multiply c/4 on u(ν, T ) in (17) c 1 2πhν 3 · u(ν, T ) = 2 hν/k BT − 1 4 c e 3 1 2 1 hν α 2 = (hc) · 2π e ehν/kB T − 1 2 3 1 α ε3 I(ε, T ) = (hc) · 2π e2 eε/kB T − 1
I(ν, T ) =
For J = σT 4 ,
(22)
Electronic Journal of Theoretical Physics 8, No. 25 (2011) 379–388
ˆ∞ J = 0
2
I(ε, T )dε c = h 4
=
c Γ(4)ζ (4) α 4 π2 e2
1 π
3
α e2
385
3
(kB T )
c αR (kB T )4 = 4 e2
4
3
ˆ∞ 0
x3 dx ex − 1
(23)
(kB T )4
and the Stefan-Boltzmann constant is
σ=
c ζ (4) α 4 ζ (2) e2
3
4 kB =
c αR 4 e2
3
4 kB
(24)
The Planck 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 1905 was based on the Planck 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 link the fine structure constant α to the blackbody radiation constant αR .
5.
Wien’s Displacement Law Wien’s frequency displacement law is νmax = bν · T
(25)
where bν = 5.878933(10) × 1010 [Hz · K] in CODATA-2006. It has the numerical solution from ex (x − 3) + 3 = 0
(26)
where xν = hν/kB T = 3 + W0 (−3e−3 ) 2.8214393721220788934. Lambert W-function W0 (x) W0 (x) =
∞
(−n)n−1 n=1
n!
xn
(27)
is a convergent series for |x| < 1/e, and νmax =
[3 + W0 (−3e−3 )]kB ·T h
(28)
α c kB 2π e2 ke
(29)
i.e., bν = [3 + W0 (−3e−3 )] Wien’s wavelength displacement law is λmax =
bλ T
(30)
386
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Figure 4 Numerical solution for ex (x − N ) + N = 0. where bλ = 2.8977685(51) × 10−3 [m · K] in CODATA-2006. It has a numerical solution from ex (x − 5) + 5 = 0
(31)
where xλ = hc/λmax kB T = 5 + W0 (−5e−5 ) 4.9651142317442763037, and 1 bλ = 5 + W0 (−5e−5 )
hc kB
(32)
It is noticed that λ has a dimension of V 1/3 = kB /aR = ckB /σ = hc/kB . We can get another good approximate for bλ in (32)
1/3
ckB 4 1/3 ke e2 = bλ 3σ 3 kB αR 1/3 20 5 1/3 hc c · = = π2 kB 2π 5 kB
(33)
where bλ = 2.897720 × 10−3 [m · K] with a uncertainty ur = 1.6 × 10−5 to CODATA-2006. From xλ /xν = 1.75978058603821178870, we can get c · xν c · [5 + W0 (−5e−5 )] = b λ xλ bλ [3 + W0 (−3e−3 )] = 5.8790302 × 1010 [Hz · K]
bν
(34)
From Wien’s displacement law, the Boltzmann constant can be directly linked to the fine structure constant bν h [3 + W0 (−3e−3 )] bν 2πke e2 = [3 + W0 (−3e−3 )]αc 2πke e2 = bλ [5 + W0 (−5e−5 )]α
kB =
(35)
Electronic Journal of Theoretical Physics 8, No. 25 (2011) 379–388
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The blackbody radiation constant is a new method to measure the fine structure constant. It links the fine structure constant to the Boltzmann constant.
Acknowledgment The Author thanks Bernard Hsiao for discussion.
References [1] M. Planck, letter to P. Ehrenfest, Rijksmuseum Leiden, Ehrenfest collection (accession 1964) , July (1905) [2] A. Einstein, Phys. Zeit. , 10 , 192 (1909) [3] A. Sommerfeld, Annalen der Physik 51 (17), 1-94 (1916) [4] M. Buchanan, Nature Physics , 6 , 833 (2010) [5] G.N. Lewis, E.Q. Adams, Phys. Rev. 3 92–102 (1914) [6] H.S. Allen, Proc. Phys. Soc. 27 425–31(1915) [7] T.H. Boyer, Foundations of Physics , 37 , 7, 999 (2007) [8] L.D. Landau, E.M. Lifshitz, Statistical Physics , Vol. 5 (3rd ed.), 183 (1980) [9] A. Einstein, Phys. Zs. 18 , 121 (1917) [10] S.N. Bose, Z. Phys. 26 , 178-181 (1924) [11] W. Pauli, Statistical Mechanics (Vol. 4 of Pauli Lectures on Physics). MIT Press, Cambridge, MA, 173 pp. (1973)
