Journal of Theoretics Volume 5-6, Dec 2003/Jan 2004

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1 Journal o Theoretics Volume 5-6, Dec 003/Jan 00 A New Physical odel or the Atomic ass Calculation Dezso Sarkadi hunatom@eudoramail.com Research Centre o Fundamental Physics Address: H-7030 Paks, Kishegyi 16. Hungary Abstract According to generally accepted physical model, the synthesis o the heavy elements may happen in the supernova explosions at a very high temperature. As a consequence o nuclear usion, the supernova stars emit a very strong electromagnetic (E) radiation dominantly in orm o gamma rays and X-rays. The intensive E radiation drastically decreases the masses o the exploding stars, directly causing mass deects o the nuclei. The general description o the black body E radiation is based on the Planck s amous radiation theory, which supposes the existence o independent quantum oscillators inside the black body. In this paper, it is supposed that in exploding supernova stars, the E radiating oscillators can be identiied with the nascent heavy elements loosing their speciic yields o own masses in the radiation process. The inal binding energy o the nuclei is determined additionally by the strong neutrino radiation, which also ollows the axwell- Boltzmann distribution in extremely high temperature. Extending to very high temperatures o the Planck s E radiation law or discrete radiation energies, a very simple ormula has been obtained or the description o the measured neutral atomic masses. Keywords: atomic mass, Planck s radiation law, the origin o the elements, nuclei binding energy, nuclear synthesis model. 1. Introduction Historically the theoretical determination o the neutral atom masses dates back to the year 1935, when C.F. von Weizsäcker [1] published his amous liquid drop model or the calculation o the nuclei masses. The model provides a general overview o masses and related stability o nuclei, and assu the nucleus behaves in a gross collective manner, similar to a charged drop o liquid. The semi-empirical mass ormula based on this phenomenological model successully was applied mainly in the earlier period o the nuclear physics. From the simple drop model one can easily calculate approximately the mass o a neutral atoms by adding the Z number electron masses to the mass o X(A,Z) nuclei. However, the liquid drop model does not give answers or many important questions related to the structure and orces inside the nuclei. The long-standing goal o nuclear physics has been to understand how the structure o nuclei arises rom the interactions between the nucleons. It is known that nucleons are composed o quarks, but nuclear interactions have not yet been successully derived rom the undamental interactions between the quarks. The standard way o modern calculation or light nuclei is based mainly on the non-relativistic quantum mechanics. In the world, many realistic phenomenological models o two- and three-nucleon interactions have been developed by its to nucleon-nucleon (NN) scattering data and the properties mainly o H, 3 H, and He. The used non-relativistic Hamiltonian usually contains two-body and three-body potentials. Dierent types o approximation method are now

2 available in the literature to solve the ew-body problems; nevertheless, the itting o the experimental data is remained necessary to date. In this paper, we had no intention to investigate any urther the nuclear structure and orces avoiding the diicult theoretical treatments and calculations. We only have ocused to the birth o the nuclei in stars and given a very simple physical model or it. It is widely known that the main part o the elements o periodic table is synthesized in the stars. The synthesis o the heavy elements may happen only in supernova star explosions in very high temperature. In consequence o the nuclear usion, the supernova stars and o course, the ordinary stars emit a very strong electromagnetic (E) radiation dominantly in orm o gamma and X-rays. In addition, the E radiation is combined with strong neutrino radiation, which also ollows the axwell-boltzmann distribution in the case o extremely high temperatures. The intense energy radiation continuously decreases the masses o the stars directly, thus causing the mass deects o the nascent nuclei and the strong binding o nuclei. The individual nuclei represent quantitized black body oscillators; their requencies are determined by their mass numbers. From this simple physical model, one can conclude that the binding energy curve o the nuclei is in immediate connection to Planck s radiation law in the high temperature region. In our paper, we have itted Planck s radiation law to the binding energy curve o the nuclei, supposing that the radiation requency o an arbitrary nucleon is proportional to the root-square o its mass number.. Extension o Planck s Radiation Law According to the Planck s radiation law the energy density o the E radiation in unction o the radiation requency is: 3 8 h dε = π d, (1) 3 h / kt c e 1 where T is the absolute temperature, c is the speed o light, k is the Boltzmann constant, h is the Planck s constant, and is the radiation requency. Our model or the explanation o the origin o the elements requires discrete radiation requencies o the stars depending on the mass numbers o the nuclei. In the classical electrodynamics, the radiation energy density o a simple dipole antenna is proportional to the requency s ourth power: ε = cst.. () From the analogy the discrete energy emitted by the individual nuclei at T absolute temperature must be: Erad ( A, Z) = cst. ; = (A,Z). h / kt e 1 (3) The binding energy o the nuclei X(A,Z) is equal to the negative value o the emitted energy. We can suppose that expression (3) is a natural generalization o Planck s radiation law or discrete radiation requencies. The most important task was to determine the mathematical relation between the radiation requency and the arbitrary nucleon signed X(A,Z). Our original goal was to calculate the neutral atomic mass values; thereore we have supposed that the radiation requency mainly depends on its mass, i.e. on the mass number A. We have thought that the Z dependence may be very small in the requency expression and it can take account later as a small correction.

3 Our working hypothesis was that the classical oscillator energy is proportional to the square o its requency at constant amplitude which is shown as E osc cst. =, () and thereore the binding energy o a nucleon must be nearly proportional to the mass o the nucleon: ( = cst. A. (5) 3. The new mass-ormula or the neutral atoms Neglecting the dependence on Z o the neutral atoms, the mass calculation is based on the next simple expression ( = A E ( c, (6) 0 rad / where 0 is a phenomenological constant and E rad is the emitted energy rom the star related to the nucleon with mass number A. The detailed orm o (6) gives where we have supposed by (5): ( = A0 c1 ; (7a) h / kt e 1 = ( = c A. (7b) Replacing new variables or the unknown constants c 1 and c, we have 1 λf ( = 0 A F = F ; η 1 A, (8) where λ, η and 0 are the new itting parameters or the experimentally determined atomic mass values. Here we introduced a new variable F proportioning to the radiation requency.. Numerical results In the numerical procedure we have observed, that the parameter η precisely equal to an important mass ratio: m Electron mass λ = =, (9) Proton mass and thereore the mass ormula will be: ( = m F A. (10) η 1 0 F For the variable F deined in (8), we have ound a better expression F = A, (11)

4 which means that the binding energy o the deuteron is very small, so its radiation requency is approximately zero compared to heavier nuclei. Equations (10) and (11) yield excellent agreement with the average trend o the measured masses or all stable atoms except those o very small A. For this reason we have tried to decrease the values F or the small A values. Ater some investigation, we have obtained a very successul solution: F = A( A 6). (11) Finally, the new mass ormula can be written: m A 6 ( = A 0 1 A η 1, (1) which involves only two its: 0 and η. The introduced new mass ormula represented by the (1) was itted to near 000 measured neutral atomic masses as obtained rom the publication o G.Audi and A.H.Wapstra.[] The results o the itting procedure are: 0 = a. u. η = (11) The accuracy o the new ormula was determined by the relative standard deviation 1 calc - σ = , (1) = N which yields excellent results. Figure 1 shows the relative errors o the itted atomic masses. 5.E-03.E-03 3.E-03 calc.e-03 1.E-03 0.E E-03 A -.E-03-3.E-03 -.E-03-5.E-03 Figure 1. Relative deviations between calculated and measured atomic masses

5 5. Conclusions Based on our new successul atomic mass ormula we have concluded that in extreme high temperatures, the nuclear synthesis can be physically described exclusively ollowing Planck s radiation law. In the Planck model it is supposed that the black body oscillators are independent to each other and have axwell-boltzmann energy distribution. Already in the earlier nuclear physics, there were some experiences showing that all the nuclei inside the atom are weekly bound. This experimental act was also proved theoretically here in the present work. The accuracy o the introduced double-parameterized ormula is comparative with the accuracy o the ive-parameterized liquid drop model established by von Weizsäcker. Nevertheless, our model has a serious chance to improve its accuracy taking account additionally the Z-dependence o the atomic masses or more parameters (i.e., nucleon-spin, parity, etc.). Reerences 1. C.F. von Weizsäcker Z. Phys. 96, p.31, (1935).. G.Audi, A.H.Wapstra Nuclear Physics A595, Vol., p.09-80, (1995). Acknowledgements The author would like to acknowledge the Hungarian Research Centre o Fundamental Physics or the support o the present work realisation. The author is also very grateul to colleagues o RFP or providing interest and encouragement in developing o the new atomic mass model. Received arch 10, 003 Journal Home Page Journal o Theoretics, Inc. 003