nd Jogj Interntionl Physics Conference Enhncing Network nd Collortion Developing Reserch nd Eduction in Physics nd Nucler Energy Septemer 6-9, 007, Yogykrt-Indonesi Creting A New Plnck s Formul of Spectrl Density of Blck-ody Rdition y Mens of AF(A) Digrm Ali Yunus Rohedi 1* 1 Deprtment of Physics, Fculty of Mthemtics nd Nturl Sciences, ITS Keputih, Sukolilo, Sury 60111 Astrct This pper reports derivtion of new Plnck s formul of spectrl density of lck-ody rdition, tht ws originted y modeling the interpoltion formul of Plnck s lw of otining the men of energy of lck-ody cvity in nd order of Bernoulli eqution. The new Plnck s formul is creted y mens AF(A) digrm of solving rctngent differentil eqution fter trnsforming the Bernoulli eqution into the rctngent differentil eqution The New Plnck s formul not only contins the terms of the photon energy nd the energy difference etween two sttes of the motion of hrmonic oscilltor ( ), ut lso contins oth terms of the minimum energy of hrmonics oscilltor ( / ) nd the phse differences ( π / ) s representing the intermodes-orthogonlity, hence it cn nswer why the explntion of lck-ody rdition hs een ssocited with the hrmonic oscilltors. Keywords : Interpoltion formul of Plnck s lw, Bernoulli eqution, AF(A) digrm, * Corresponding uthor. E-mil ddress: rohedi@physics.its.c.id I. Introduction The er of developing the modern scientific ws originted y presence the Plnck s lw of lck-ody rdition s representtion of the light sources in therml equilirium. Plnck not only could complete the Ryleigh-Jens nd Wien s lws, oth of rdition lws previously tht of ech only pproprites to the experimentl for the rnge of long wvelength nd short wvelength respectively, ut he lso creted new constnt h clled s Plnck s constnt tht not known previously in clssicl physics [1]. Bsed on his constnt, Plnck postulted the discretittion of electromgnetic energy in pcket of energy clled s photon, where for every ngulr frequency ( ω ), the energy per photon is E =, which ws justified lter y Einstein through fotoelectric effect []. Plnck s lw for lck-ody rdition lso ecme the primry se of derivtion Einstein coefficients of spontneous nd stimulted of emission rtes for generting the light sources such s mser nd lser []. More thn one century since pulished t the first time, the Plnck s formul for the spectrl density of lck ody rdition ( ρ ) is of form ρ =, where ω nd c re / π c e respectively the ngulr frequency nd the velocity h of light in vcuum, T is temperture nd h =, π while k is the Boltzmnn constnt [],[4],[5] It is well-known tht the Plnck s formul is ω in multipliction form of U nd tht re π c respectively represent the men energy per unit volume in modes frequency nd the numer of modes per intervl of ngulr frequency ( d ω ) etween the ngulr frequency ω nd ω + d ω. Until now there is no lterntive form for the Plnck s formul, especilly to nswer why the explntion of lck-ody rdition hs een lwys ssocited to the motion of hrmonic oscilltors, lthough the quntity ssocites to oth the photon energy nd the energy differences etween two sttes of hrmonic oscilltor. This pper explores the formultion of the spectrl rdition of lck-ody rdition y converting the originl Plnck s interpoltion formul of the rte equtions of 1/T vries respect to the U i.e, d 1 k = into Bernoulli eqution du T U du + h ωu = U, where β = 1, tht yields the usul Plnck s formul [5]. In this pper, creting the new Plnck s formul is performed y trnsforming the Bernoulli eqution into the rctngent differentil eqution. We give 0
nd Jogj Interntionl Physics Conference Enhncing Network nd Collortion Developing Reserch nd Eduction in Physics nd Nucler Energy Septemer 6-9, 007, Yogykrt-Indonesi the generl solution of the rctngent differentil eqution in modultion function, where its mplitude (A) modulted in the phse function F(A), tht cn e simplified y pplying AF(A) digrm [6],[7]. II. Governing the corresponding differentil eqution of the Plnck s Lw According to the explntion in oth of modern physics [] nd sttisticl physics [5], the men energy of cvity used in formulting the Ryleigh-Jens s Lw for the rnge of shortfrequency ( << ) derived clssiclly using the thermodynmic concept is in from: U =, (1) while for the Wien s Lw tht holds for the rnge of high-frequency ( h ω >> ) derived using the sttisticl Boltzmnn involving the Mxwell distriution otined in the from: / U = h ωe () Plnck thought, it ws impossile to crete formultion of the density of spectrl rdition of lck-ody rdition tht cover ll of the rnge of wvelength or frequency y using thermodynmic concept. Plnck originted his formultion y writing the U formuls of Ryleigh-Jens nd Wien s lws in the rte equtions of 1/T vries respect to the U. The corresponding rte eqution for Ryleigh-Jens lw ws otined from Eqs.(1) in the form: d 1 k = () du T U while from Eqs.() for the Wien lw otined in the form: d 1 k =, (4) du T U Plnck then interpolted oth of the rte equtions into the following form: d 1 k = (5) du T U We cll Eqs.(5) s the interpoltion formul of Plnck s lw. One wy of creting the usul formul of the spectrl density of lck-ody rdition (ρ ) cn e performed y trnsforming Eqs.(5) into the following integrl : 1 du = k + C (6) T U where C is the integrting constnt. Next, solve the ove integrl using indefinite integrl [8] 1 tht ville on the tle of integrls, x p + qe y pplying the oundry vlue T for U, or y using RohediSmrt formul for Bernoulli eqution [9]. 1 But ctully, y defining β =, the interpoltion formul of Plnck s lw in Eqs.(5) cn e represented in form of Bernoulli differentil eqution : du + h ωu = U (7) We think, due to the Bernoulli integrl s integrl form of the Bernoulli eqution ws not ville on the tle of integrls ecomes the reson why Plnck did not grft Bernoulli s nme s once reference when pulishing his lck-ody rdition s lw. II. Creting AF(A) digrm for solving n rctngent differentil eqution The first order nonliner differentil eqution clled n rctngent differentil eqution is in the form [7]: dy = y + (8) with the initil vlue y0 t x0 for ritrry nd coefficients. The common wy of solving Eqs.(8) is y trnsforming it into rctngent du. This purpose only cn e done y mking u +1 the chnge vriles : u = y, (9) hence the corresponding rctngent to the Eqs.(8) is of form : du = C + (10) u + 1 where C is the integrting constnt. Next, y using the definition of rctngent to the left sides of Eqs.(10), thus we find : tn y = ( x x0 ) + C (11) nd y pplying its initil vlue, we find = tn y C 0. Sustituting this C into
nd Jogj Interntionl Physics Conference Enhncing Network nd Collortion Developing Reserch nd Eduction in Physics nd Nucler Energy Septemer 6-9, 007, Yogykrt-Indonesi tn ( y) = jtnh ( jy) (14) tnh ( y) = jtn ( jy) (14) ( ) = ( ) + y x tn x x tn y 0 0 (1) nd lso the reltion etween tngent nd hyperolic-tngent function Eqs.(11) gives the generl solution of rctngent differentil eqution in the form : Due to Eqs.(1) forms modultion function, in which the mplitude term (in this cse A = ) lso modulted`in the tngent function of the phse function, the solving rctngent differentil eqution cn e simplified using AF(A) digrm [7]. Creting the AF(A) digrm is performed y writing Eqs.(8) in form : Solution y dy = ( y) + y0 ( x) = tn ( x x ) + tn 0 Digrm AF(A) / On utilizing the ove AF(A) digrm, y cts s the solution, / eside cts s mplitude lso involving (modulted) in the phse function, oth of constnt nd ct s multiplier constnt for ( x x 0 ) of independent vrile nd strting point t the phse function, while oth the initil vlue y0 nd / involving in the initil phse. For oth of nd re positive constnts, =, therefore the solution of Eqs.(8) for this cse is in the form : ( ) ( ) (1) y x = tn x x + 0 tn y0 The nlyticl solution of Eqs.(8) for ritrry of comintion vlues of nd cn e found using Eqs.(1) fter involving the following de Moivre reltion relted rctn to hyperolicrctn [8] : tn tnh ( x) jtnh( jx) ( x) jtn( jx) = (14c) = (14d) III. Solving the corresponding rctngent differentil eqution of interpoltion formul of Plnck s lw Solving of Eqs.(7) tht corresponds to the interpoltion formul of Plnck s lw y using AF(A) digrm cn e performed y writing the Bernoulli differentil eqution in form : du 1 1 = U + h ω + j (15) Creting the solution y AF(A) digrm yields : 1 U + 1 1 1 0 U+ = j tn j β β ( ) + 0 tn (16) 1 j Sustituting β = 1/ nd y using the reltion in Eqs.(14) gives : 1 1 1 1 U0 U( T) = tnh + tnh 0 / (17) Tking the initil vlue T =, or β 0 = 0 nd U 0 =, then Eqs.(17) reduces to the new formul of Plnck s Lw for lck-ody rdition 1 h π U( T) = tnh ω j, ( 18) here, we hve used tnh 1 ( ) = jπ /. By the new formul of U, then the density of spectrl rdition (ρ ) of Plnck s Lw is of the form : 1 1 π ρ = tnh j (19) π c Let compre with the usul form of Plnck Lw ρ = (0) / π c e The plotting of oth Plnck s formul for severl tempertures re shown in Fig.1 elow,
nd Jogj Interntionl Physics Conference Enhncing Network nd Collortion Developing Reserch nd Eduction in Physics nd Nucler Energy Septemer 6-9, 007, Yogykrt-Indonesi Hence the corresponding of ngulr frequency is` ω mx =,814 / h. Fig.1. The density of spectrl rdition of oth lck-ody rdition As pprer in the Fig.1 ove, the new formul of Plnck s lw gives the equl density of spectrl rdition to the usul grph. Furthermore, the corresponding of ngulr frequency to the mximum of density of spectrl rdition of lck-ody rdition ws dρ commonly determined from ( ) 0 d ω mx ω = y ωmx defining xmx = h. For the new Plnck s formul, the vlues of x mx is determined from 1 π 1 1 π tnh xmx j + xmxsech xmx j = 0 (1) 6 while for the usul Plnck s formul ws otined from [5] : x ( x ) e mx mx = 0 () Both of the ove nonliner eqution for determinting the vlue of x mx only cn e solved y numericl methods, while the result given in Fig., Fig. Plotting the grphicl of otining x mx prmeter As shown in Fig., oth formul of Plnck s rdition give the sme vlue of x mx =, 814. IV. Discussion We hve shown tht representtion of interpoltion formul of the rte equtions of 1/T vries respect to the U.into Bernoulli eqution in Eqs.(7) trnsformed into rctngent differentil eqution yields the new formul of U in Eqs.(18), nd hence cn crete the new formul of the spectrl density of lck-ody rdition in Eqs.(19). The new Plnck s formul of lck-ody rdition not only contin the terms of the photon energy nd the energy difference etween two sttes of the motion of hrmonic oscilltor, ut lso contins oth terms of the minimum energy of hrmonics oscilltor ( h ω/ ) nd the phse differences ( π / ) s representing the intermodesorthogonlity. Hence. this new formul cn lso nswer why the explntion of lck-ody rdition hs een ssocited with hrmonic oscilltors. V. Conclusion Creting the new Plnck s formul of lck-ody rdition y solving the corresponding rctngent differentil eqution of interpoltion formul of Plnck s lw using AF(A) digrm` hs een presented. The new Plnck s formul of lck-ody rdition cn nswer why the explntion of lck-ody rdition hs een ssocited with hrmonic oscilltors. References : 1. B. F. Bymn, M. Hmmermesh, A Review Undergrdute Physics, p.199, John Wiley & Sons, Cnd, 1986.. A. Beiser, Konsep Fisik Modern, pp:40-46, Erlngg, Jkrt, 198.. J. Wilson, J.F.B. Hwkes, Optoelectronics : An Introduction, pp:176-8, Prentice Hll Interntionl, 198. 4. D.J.Griffiths, Introduction to Quntum Mechnics, pp:16-17, Prentice Hll Interntionl, 1984. 5. F.Mndll, Sttisticl Physics, pp: 50-6, John Wiley & Sons, Ltd, 1971. 6. A.Y. Rohedi, Anlyticl Solution of the Rictti Differentil Eqution for High Frequency Derived y Using Stle
nd Jogj Interntionl Physics Conference Enhncing Network nd Collortion Developing Reserch nd Eduction in Physics nd Nucler Energy Septemer 6-9, 007, Yogykrt-Indonesi Modultion Technique, Presented on Internrtionl Conference of Mthemtics nd Nturl Sciences, Poster Edition, Fculty of Nturl Sciences, ITB, Nopemer, Bndung, 005. 7. A.Y. Rohedi, Anlytic solution of Nonliner Schrödinger Eqution y Mens of A New Approch Presented on Interntionl Symposium of Modern Optics nd Its Applictions, Physics Deprtment ITB, Bndung, 007 8. M.R. Spiegel, Mthemticl Hndook of Formuls nd Tles, Schum s Outline Series, McGRAW-Hill Book Compny, 1968. 9. A.Y. Rohedi, Applying of Modultion Scheme of Solving Bernoulli Differentil Eqution, Journl of Physics nd Its pplictions, Vol, No.1, pp 1-6, Deprtment of Physics, ITS, Sury, 007. 4