Classical Planck Spectrum for Relative Thermal Radiation, Classical Zero-Point Radiation, and Scale Parameter
aa r X i v : . [ phy s i c s . c l a ss - ph ] J u l Classical Planck Spectrum for Relative ThermalRadiation, Classical Zero-Point Radiation, and ScaleParameter
J. Tapia ,H. Gonz´alez , and R. Rubiano Universidad Surcolombiana, Physical of Program, Neiva, A.A 385, Colombia.
Received: date / Revised version: date
Abstract
In this work we obtain Planck’s blackbody spectrum from the thermal scalar radiationcontained in a resonant cavity of volume V in the context of classical mechanics, whichprovides the classical zero-point electromagnetic radiation in terms of a scale parameterthat depends on geometric properties of the enclosure and electrical magnitudes. Thescale parameter of the classical zero-point electromagnetic radiation is associated to ex-perimental measurements of Casimir forces, but we show that theoretically its value forknown radiant cavities that approach a blackbody has a numerical value of the order ofPlanck’s constant.
Keywords—
Planck’s blackbody spectrum, scalar radiation, resonant cavity, classicalmechanics, zero-point electromagnetic radiation, scale parameter.
The contributions of Rayleigh-Jeans [1, 2] and Planck [3] to elucidate the nature of theradiation emitted by hot bodies was elaborated before the year 1900, a time in whichthe special theory of relativity had not been formulated [4]. The foregoing indicates thatthis classical radiation law, and its quantum version in terms of Planck’s constant doesnot contain space-time geometry explicitly, nor the Special Theory of Relativity [5]. TheZero-point radiation is included in quantum field theory [6–10]; however, this zero-pointenergy also emerges in classic relativistic electrodynamics and is determinant for explain-ing Casimir forces, Van der Waals forces, specific heats, and other phenomena apparentlybelonging to quantum mechanics [11–13].There are derivations of blackbody thermal radiation spectrum that lead to Planck’slaw, with the classical zero-point energy, and that do not take into account the assumptionsabout quantization of energy [14–17]. The foregoing would indicate that radiation emittedby hot bodies can also arise in a classical context, with the incorporation of the SpecialTheory of Relativity in the formulation of classical electrodynamics and the classical rel-ativistic field theory. In reference to Planck’s constant part of the scientific communitydiscards its use as a universal constant, it can appear in diverse contexts [18–21], and inthis classical approximation, a parameter of the order of Planck’s constant emerges, Inthis article, we incorporate to the Planck radiation law derived from classical relativistic lectrodynamics and with zero-point radiation a scale parameter associated with the geo-metrical and electromagnetic characteristics of the thermal radiation contained in a finiteenclosure of volume V. This scale parameter is associated with experimental measures ofCasimir forces, in such a way that the structure of Planck’s law differs from that generallypresented in literature in terms of Planck’s constant ~ .The organization of this work is distributed as follows: In the first section we presenta synthesis of Rayleigh-Jeans and Planck radiation spectra. The second section containsthe thermodynamics of normal radiation modes and the derivation of the Planck spectrumusing classical physics, including classical zero-point radiation In the third section, whichis the central part of our work, we present a version of the Planck spectrum for thermalradiation involving classical zero-point radiation, and the inclusion of scale parameter ofzero-point energy. In the fourth section, we illustrate some specific examples of physicalsystems we apply the obtained Planck’s radiation law to and determine the scale parametervalue for each of them. In the last section we present an analysis of our results and theirrespective conclusions. A blackbody absorbs and emits radiation perfectly, that is, there is no dominant frequencyrange. Therefore, the intensity of the radiation emitted is related to the amount of energyin the body in thermal equilibrium. The blackbody radiation theory development historyis very interesting because it gave rise to the discovery of quantum theory [22]. Thefirst experimental studies established that the emissivity of a blackbody is a function offrequency and temperature. A measure of emissivity may be the quantity ρ ( v, T ) whichis the density of radiated energy per volume unit and per frequency unit at an absolutetemperature T and at a frequency v . The first theoretical studies used Maxwell’s equationsfor calculating the electromagnetic modes and for determining ρ ( v, T ) density of modes.For example, Wilhelm Wien in 1896 used a simple model to derive the expression: ρ ( v, T ) = av exp (cid:2) − bv (cid:3) (1)Where a, b were constant. However, the above equation fails in the range of exper-imental data low frequencies. In 1900 Lord Rayleigh published a model based on themodes of electromagnetic waves of a cavity. Each mode had a particular frequency andcan give or receive energy in a continuous way. The total number of modes per volume, N ( v ), is [23]: dN ( v ) dV = 8 πv c (2)Rayleigh assigned an energy k B / k B / k B = 1 . K/J is Boltzmann constant. In thisway, the density of electromagnetic energy by frequency ρ ( v, T ) becomes: ρ ( v, T ) = dN ( v ) dV k B T = 8 πv c k B T (3)The last equation is known as the Rayleigh-Jeans distribution of blackbody radiationand fails dramatically in the ultraviolet part of the spectrum (historically referred toas the ”ultraviolet catastrophe”) [24]. It has been suggested that Planck discovered hisradiation formula on the afternoon of October 7th, 1900. Planck had taken into accountsome additional experimental data from Heinrich Reubens and Ferdinand Kurlbaum, aswell as Wien’s formula, and he deduced an expression that fit all the experimental dataavailable. His formula was the one now known as the blackbody radiation formula givenby: ( v, T ) = 8 πv c hv exp h hvk B T i − h = 6 , × − Joule.seconds is known as Planck’s constant. The aboveexpression is reduced to the Wien formula for high frequencies (i.e., hv/k B T ≫
1) andRayleigh-Jeans formula for low frequencies (i.e., hv/k B T ≪ A small harmonic oscillator reaches equilibrium with thermal radiation at the same av-erage energy as a normal mode of radiation at the same frequency as the oscillator [25].Alternatively, we can think that a radiation mode behaves like a harmonic oscillator.The normal mode of radiation thermodynamics has two thermodynamic variables Tand ω , like a harmonic oscillator, and takes a particularly simple form [25]. The averageenergy for the normal mode is related to the canonical potential function φ ( ω/T ) by meansof the equation: U ( ω, T ) = − ωφ ′ ( ω/T ) (5)The first law of thermodynamics is: dQ = dU + dW (6)The change of dS entropy is related to temperature T by: dS = dQT (7)From the laws of thermodynamics, it is inferred that the energy for each normal modeis expressed by: U = ωf (cid:16) ωT (cid:17) (8) f (cid:0) ωT (cid:1) is an unknown function. The above equation is known as Wien’s theorem. [14]Helmholtz free energy is related to U ( ω, T ) energy by: F ( ω, T ) = U ( ω, T ) − T S (9) U ( ω, T ) = F ( ω, T ) + T S (10)This random radiation that exists at temperature T = 0 is a classical zero-point elec-tromagnetic radiation. Thus, to account for Casimir forces observed experimentally, and or each normal mode, the average energy becomes: U ( ω,
0) = F ( ω,
0) = ~ ω U ( ω, T ) = ~ ω exp h ~ ωk B T i − ~ ω Consider the electromagnetic radiation contained in an enclosure of volume V in the freespace. Assuming that the contributions of electric and magnetic parts to energy densityare equal, in these circumstances in SI units: u = 12 ε E + 12 µ B = ε E (13)With the equation of motion for electromagnetic field given by:1 c ∂ φ∂t − ∂ φ∂x − ∂ φ∂y − ∂ φ∂z = 0 (14)Where Lagrangian density is inferred from: L = ε " c (cid:18) ∂φ∂t (cid:19) − (cid:18) ∂φ∂x (cid:19) − (cid:18) ∂φ∂y (cid:19) − (cid:18) ∂φ∂z (cid:19) (15)And consequently the energy: U = ε Z " c (cid:18) ∂φ∂t (cid:19) − (cid:18) ∂φ∂x (cid:19) − (cid:18) ∂φ∂y (cid:19) − (cid:18) ∂φ∂z (cid:19) dV (16)Random thermal movements can be expressed in terms of the oscillations of the nor-mal modes with random phases. The field expressed in electromagnetic modes is given y the expansion term [26]. φ ( ct, x, y, z ) = ∞ X m = −∞ ∞ X n = −∞ ∞ X l = −∞ g (cid:16) c~k (cid:17) ( V ) / cos h ~k.~r − kct − θ (cid:16) ~k (cid:17)i (17)Where g (cid:16) c~k (cid:17) is called amplitude or distribution function by mode. Thus, the energyof thermal radiation in a cavity can be expressed as a sum over the energies of the normalmodes of oscillation. Within classical physics, thermal radiation is treated as classicalelectromagnetic radiation defining: g (cid:16) c~k (cid:17) = δ h (cid:16) c~k (cid:17) (18)With δ as a constant factor extracted from g (cid:16) c~k (cid:17) , using eqs. (17) and (18) in eq.(16), electromagnetic energy U is Each oscillation mode will have an energy given by: U = ε δ ∞ X m = −∞ ∞ X n = −∞ ∞ X l = −∞ h h (cid:16) c~k (cid:17)i k cos h ~k.~r − kct − θ (cid:16) ~k (cid:17)i (19)Each oscillation mode will have an energy given by: U k = ε δ k h h (cid:16) c~k (cid:17)i (20)For eq. (18) to be dimensionally correct, and given that the term k h h (cid:16) c~k (cid:17)i hasunits of volume, δ must have electric field dimensions; U k = ε E k h h (cid:16) c~k (cid:17)i (21)Thus In eq. (22) has been taken δ = E . Being E the electric field averagesquare in the proximity of the radiant enclosure. It is possible to adapt the units in the k h h (cid:16) c~k (cid:17)i coefficient by extracting a constant factor having units of volume per time.Assuming that the constant factor is: α = L c (22)Where L is a factor with length dimensions and c is the speed of light in free space: h h (cid:16) c~k (cid:17)i = α h f (cid:16) c~k (cid:17)i (23) hus, energy by mode acquires the form: U k = ε E L c k h f (cid:16) c~k (cid:17)i = βk h f (cid:16) c~k (cid:17)i = k h F (cid:16) c~k (cid:17)i (24)The parameter β has been inserted in the function h F (cid:16) c~k (cid:17)i , so that: h F (cid:16) c~k (cid:17)i = β h f (cid:16) c~k (cid:17)i (25)We have found that parameter β has units of J.S, and is defined by: β = ε E L c (26) β can be determined numerically by knowing the linear dimension L of the enclosureand the value of E which is the average value of the electric field amplitude, which cor-responds to different modes of oscillation. The parameter β fixed, in this approximation,the classical zero-point energy. U = 12 βω (27)From U k in eq. (24), energy per oscillation mode, we can determine the values ofthe criterion of smooth monotonic behavior of the canonical partition function φ ( ω/T )at high and low temperature, to find Planck’s radiation spectrum with classic zero-pointenergy [17]. For small T there is the classic zero-point energy 1 / βω , from eq. (5); U = − ωφ ′ (cid:16) ωT (cid:17) = 12 βω (28)Taking into account that ω/T = z : − φ ′ ( z ) → β z → ∞ (29)As follows: − φ ( z ) → β z ; z → ∞ (30) or large values of T, Rayleigh-Jeans limit, eq. (5) gives: U ∞ = − ωφ ′∞ ( z ) = k B T (31)When, − φ ′∞ ( z ) → k B z ; z → − φ ∞ ( z ) → k B ln z ; z → k B is Boltzmann constant Eqs. (30) and (33) give extreme behaviors of thecanonical potential function φ ( z ). In general, the canonical potential function φ ( z ), likeits derivatives, must preserve its characteristic smooth monotonicity throughout the range(0 , ∞ ); the quotient: y = U U ∞ = βωk B T = βω k B T = β k B z (34)Is the argument of the smooth monotonic function M ( y ) , with extreme values givenby eqs (29) and (32), which determine energy U ( ω, T ). U ( ω, T ) = − ωφ (cid:16) ωT (cid:17) = − ωM ( y ) (35)In reference [17] the author demonstrated that an appropriate function that maintainsthe smooth monotony of φ ( z ) and its derivatives, the function of interpolation betweenthese two limits, in the entire range of variation of ω/T = z is given by the equation: φ ( z ) = − β ln (sinh z ) (36)Thus, φ ′ (cid:16) ωT (cid:17) = M ( y ) = − β (cid:18) βω k B T (cid:19) (37)Adapting this function to eq. (35): U ( ω, T ) = U coth (cid:18) βω k B T (cid:19) = 12 βω coth (cid:18) βω k B T (cid:19) (38) ( ω, T ) = βω exp h βωk B T i − βω U , and scale parameter β . The classic zero-point energy is given by: U = 12 βω (40)Where the scale parameter is: β = ε c (cid:2) E L (cid:3) (41)The scale parameter depends on geometric properties, linear dimension L of the en-closure, and electromagnetic properties, permittivity of the free space ε , speed of light c,and the square of the amplitude of the electric field average value. Scale parameter sets zero-point classical electromagnetic energy and has J.S dimensionsin the international system of units: β = ε c (cid:2) E L (cid:3) ∼ = 2 . × − C sN m (cid:16)(cid:2) E L (cid:3) (cid:17) (42)With the linear dimension L in meters and E in V /m or in
N/C . Taking into accountthat the parameter β is related to experimental measurements of Casimir forces and thatits order of magnitude is similar to ~ , an estimation of the linear dimension of some radi-ant systems reported in the literature [27] is made to observe if they are consistent withtheir real values. Table 1 shows some of the values reported for E and the estimate madefor the linear dimension L using eq. (41) where β/ . × − J.s is the experimentalvalue reported by Casimir Effect measurements [28] in the table 2.Using eq. (41) and the experimental value of β/ We have found the blackbody radiation spectrum in connection with zero-point energyusing only classical physics, given by eq. (38) and introducing a scale parameter, whichis defined in eq. (40) consequently, we have shown that the theoretical numerical value E .Radiant System Values of E (cid:0) NC (cid:1) (a) Theoretical Value L ( m )( b )Field just beneath a highpower line 0.5 0 . × − Field of a 100 W bulb at 1m distance 50 34 . × − Electric fields in laserpulses 10 . × − Electric fields in U ions 10 . × − Maximum practical electricfield in vacuum, limited byelectron pair production 1 . × . × − Maximum electric field innature 1 . × . × − Note: ( a ) Data taken from the text: C. Schiller , Motion Mountain The adventure of Physics VolumeI. Light, Charges and Brains (Creative Commons, Munich, Germany, 2011).Note: ( b ) Data taken from the text: C. Schiller , Motion Mountain; The adventure of Physics VolumeIII Light, Charges and Brains, 30th ed. (Creative Commons, Munich, Germany, 2018).
Table 2: Determination of the linear dimension L average values from equation (41) which setsthe experimental value of the scale parameter β .Radiant System Experimental Value L (m) Correspondence accordingto the value of L (c)Field just beneath a highpower line 0 . × − Unicellular protozoon ofthe amoeba genusField of a 100 W bulb at 1m distance 30 × − Diameter of a human hairElectric fields in laserpulses 79 × − Sodium atomic radioElectric fields in U ions 2 . × − Atomic nuclei sizeMaximum practical electricfield in vacuum, limited byelectron pair production 2 . × − Compton wavelength inelectronsMaximum electric field innature 1 . × − Planck length
Note: ( c ) Data taken from the text: C. Schiller , Motion Mountain The adventure of Physics VolumeI. Light, Charges and Brains (Creative Commons, Munich, Germany, 2011). epends on the geometrical properties of the resonant box and the electrical propertiesaccording to our approximation.Therefore, the scale parameter obtained for zero-point radiation energy was foundwith arguments of the classical theory and does not imply the quantization of energy;however, its numerical value is of the order of the Planck constant, since the typicalelectric fields generated by various oscillating physical systems have values consistentwith the theoretical estimation sizes of the radiant system (see column 2 of Table 1).In conclusion, the scale parameter found is related to the experimental measurementsof Casimir forces; In addition, the above arguments show that the Planck constant canappear in quantum mechanics or in classical mechanics, just as the zero-point of energycan be classical or quantum. Acknowledgement
We thank Professor Ricardo Gait´an for reading the manuscript and making importantsuggestions. We appreciate the total support of Surcolombiana University through theresearch project 2334.
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