Fermi Energy of Proton and the SEMF Energy Coefficients

Fermi energy of proton and the SEMF energy coeﬃcients

U. V. S. Seshavatharam, Honorary faculty, I-SERVE Alakapuri, Hyderabad-35, AP, India. E-mail: [email protected]

Prof. S. Lakshminarayana, Dept. of Nuclear Physics, Andhra University Visakhapatnam-03, AP, India. E-mail: [email protected]

Abstract: Considering the Avogadro number as a scal- Assumption-2 ing factor an attempt is made to understand the origin The key conceptual link that connects the gravitational of the strong, weak and electromagnetic interactions in and non-gravitational forces is - the classical force limit a uniﬁed approach. Nuclear characteristic size, proton size, nucleon rest masses and magnetic moments were c4 F =∼ =∼ 1.21026 × 1044 newton (1) ﬁtted. It is noticed that, nuclear binding energy can be C G understood with the ‘molar electron mass’ and ‘fermi en- It can be considered as the upper limit of the string ten- ergy’ concepts. sion. In its inverse form it appears in Einstein’s theory of gravitation as 8πG . It has multiple applications in Black Keywords: Avogadro number; nuclear strong force; nu- c4 hole physics and Planck scale physics [4]. It has to be clear weak force; weak coupling angle; fermi energy; nu- measured either from the experiments or from the cosmic clear binding energy constants; and astronomical observations.

1 Introduction Assumption-3

The subject of unification is very interesting and very Ratio of ‘classical force limit = FC ’ and ‘weak force mag- 2 complicated. By implementing the Avogadro number N nitude = FW , ’ is N where N is a large number close as a scaling factor in unification program, one can probe to the Avogadro number. the constructional secrets of elementary particles [1,2]. FC ∼ 2 ∼ Upper limit of classical force The Planck’s quantum theory of light, thermodynam- = N = (2) FW nuclear weak force magnitude ics of stars, black holes and cosmology totally depends ∼ c4 ∼ upon the famous Boltzmann constant kB which in turn Thus the proposed weak force magnitude is FW = N 2G = depends on the Avogadro number [3]. From this it can 3.33715×10−4 newton and can be considered as the char- be suggested that, Avogadro number is more fundamen- acteristic nuclear weak string tension. It can be mea- tal and characteristic than the Boltzmann constant and sured in the particle accelerators. indirectly plays a crucial role in the formulation of the quantum theory of radiation. In this connection it is Assumption-4 noticed that, ‘molar electron mass’ and ‘fermi energy’ concepts play a very interesting role in fitting the semi The strong force magnitude can be defined as follows. empirical mass formula energy coefficients. r F S =∼ 2π ln N 2 =∼ 4π ln (N) (3) FW ∼ 2 Key assumptions in unification Thus FS = 157.9944 newton. Assumption-1 2.1 The lepton-quark mass generator Nucleon behaves as if it constitutes molar electron mass. In the earlier published papers [1,2] it was defined that Molar electron mass (N.me) plays a crucial role in nu- r clear and particle physics. 4π N 2Gm2 X =∼ 0 e =∼ 295.0606338 (4) E e2

1 U. V. S. Seshavatharam & S. Lakshminarayana: Fermi energy of proton and the SEMF energy coefficients. 2 where N is the Avogadro number, G is the gravitational Proton rest mass is close to constant and m is the rest mass of electron. It can e F 1 be called as the lepton-quark-nucleon mass generator. S 2 ∼ 2 ∼ + XE − 2 · EW = mpc = 938.18 MeV (13) It plays a very interesting role in nuclear and particle FW α physics. Using this number leptons, quarks and nucleon s 2 rest masses can be fitted [1]. It can be expressed as e FW where E =∼ =∼ 1.731843735×10−3 MeV (14) W 4π s 0 4π G (Nm )2 X =∼ 0 e =∼ 295.0606338 (5) Neutron and proton mass difference is close to E e2 r FS 2 ∼ 2 2 ∼ Weak coupling angle was defined as + XE · EW = mnc − mpc = 1.2966 MeV (15) FW m c2 1 u ∼ ∼ ∼ where m and m are the neutron and proton rest masses 2 = sin θW = = 0.464433353 (6) n p mdc XEα respectively [3]. where mu is the rest mass of up quark, md is the rest mass of down quark and sin θW is the weak coupling 3 Fermi energy of proton and the nuclear angle. In the modified SUSY, the fermion and boson binding energy mass ratio Ψ can be fitted in the following way. 2 2 ∼ 3.1 Size of proton Ψ ln 1 + sin θW = 1 (7) ∼ It is noticed that, Thus Ψ = 2.262706. If mf is the mass of fermion and mb is the mass of its corresponding boson then s 2 ∼ e 2G (N.me) ∼ mf Rp = 2 · 2 = 0.90566 fm (16) m =∼ (8) 4πε0Gmp c b Ψ With this idea super symmetry can be observed in the This obtained magnitude can be compared with the rms strong interactions [1] and can also be observed in the charge radius of proton 0.8768(69) fm [3]. Here the error electroweak interactions [2]. is 3.28 %. This may be an accidental coincidence also.

2.2 Applications of the strong force 3.2 The fermi energy of proton magnitude in nuclear physics Fermi energy of proton can be expressed in the following The characteristic nuclear size is way. Let the number density of protons in the nucleus is s −1 e2 4π 3 ∼ 44 −3 ∼ ∼ np = R = 3.213778 × 10 m (17) R0 = = 1.2084 fm (9) 3 p 4πε0FS where R =∼ 0.90566 fm. If each quantum state con- If (αX )−1 ∼ sin θ , magnetic moment of electron can p E = W stitutes 2 protons, number density of quantum states is be expressed as np 2 . So the Fermi energy of a proton is close to s 2 ∼ 1 e ∼ −24 ¯h2 n 2/3 µn = sin θW · ec · = 9.274 × 10 J/tesla 2 p ∼ 2 4πε0FW EF = 3π = 58.6974 MeV (18) 2mp 2 (10) Similarly magnetic moment of proton is close to Therefore, the average energy of a proton is given by s 1 e2 3 ∼ ∼ ∼ −26 Eav = EF = 35.218 MeV (19) µp = sin θW · ec · = 1.348 × 10 J/tesla 5 2 4πε0FS (11) 3.3 To fit the semi empirical mass formula If proton and neutron are the the two quantum states energy coefficients of the nucleon, by considering the radius of proton Rp, magnetic moment of neutron can be fitted as The semi-empirical mass formula (SEMF) is used to ap- 1 proximate the mass and various other properties of an µ =∼ sin θ · ec · R =∼ 9.782 × 10−27 J/tesla (12) n 2 W p atomic nucleus [5]. As the name suggests, it is based U. V. S. Seshavatharam & S. Lakshminarayana: Fermi energy of proton and the SEMF energy coefficients. 3

partly on theory and partly on empirical measurements. Z A Cal.BE in MeV Mes.BE in MeV %Error The theory is based on the liquid drop model proposed 26 56 491.53 492.254 +0.148 by George Gamow and was first formulated in 1935 by 28 62 546.10 545.259 -0.153 German physicist Carl Friedrich von Weizscker. Based 34 84 728.19 727.341 -0.119 on the ‘least square fit’, volume energy coefficient is 50 118 1006.60 1004.950 -0.164 av = 15.78 MeV, surface energy coefficient is as = 18.34 60 142 1182.5 1185.145 0.222 MeV, coulombic energy coefficient is ac = 0.71 MeV, 79 197 1554.27 1559.40 0.329 asymmetric energy coefficient is aa = 23.21 MeV and 82 208 1625.1 1636.44 0.693 pairing energy coefficient is ap = 12 MeV. The semi em- 92 238 1803.40 1801.693 -0.094 pirical mass formula is 2 2 Z (Z − 1) (A − 2Z) 1 ∼ 3 BE = Aav −A as − 1 ac − aa ± √ ap Tab. 1: SEMF binding energy with the proposed A 3 A A (20) energy constants In a unified approach it is noticed that, the energy coefficients are having strong inter-relation with the nucleon Acknowledgements rest masses and the fermi energy of proton. The interesting observations can be expressed in the following way. The first author is indebted to professor K. V. Krishna ∼ Murthy, Chairman, Institute of Scientific Research on ap + aa = Eav (21) Vedas (I-SERVE), Hyderabad, India and Shri K. V. R. ∼ av + as + ac = Eav (22) S. Murthy, former scientist IICT (CSIR) Govt. of India, Asymmetry energy coefficientt is close to Director, Research and Development, I-SERVE, for their 2 valuable guidance and great support in developing this a =∼ E =∼ 23.479 MeV (23) a 3 av subject. Pairing energy coefficient is close to 1 References a =∼ E =∼ 11.739 MeV (24) p 3 av [1] U. V. S. Seshavatharam and S. Lakshminarayana. Volume energy coefficient is close to Super Symmetry in Strong and Weak interactions. ∼ ∼ av = sin θW · Eav = 16.3564 MeV (25) Int. J. Mod. Phys. E, Vol.19, No.2, (2010), p.263. Coulombic energy coefficient is close to [2] U. V. S. Seshavatharam and S. Lakshminarayana. 3 SUSY and strong nuclear gravity in (120-160) GeV a =∼ (m − m ) c2 =∼ 0.776 MeV (26) c 5 n p mass range. Hadronic journal, Vol-34, No 3, 2011 Surface energy coefficient is close to June, p277-300 ∼ ∼ as = Eav − av + ac = 19.639 MeV (27) [3] P. J. Mohr and B.N. Taylor. CODATA Recom- mended Values of the Fundamental Physical Con- In table-1 considering the magic numbers, within the stants.2007. Http://physics.nist.gov/constants. range of (Z = 26; A = 56) to (Z = 92; A = 238) nuclear binding energy is calculated and compared with the mea- [4] U. V. S. Seshavatharam. Physics of rotating and sured binding energy [6]. If this procedure is found to expanding black hole universe. Progress in Physics, be true and valid then with a suitable fitting procedure Vol-2, April, 2010. Page 7-14. qualitatively and quantitatively magnitudes of the proposed SEMF binding energy coefficients can be refined. [5] P. Roy Chowdhury et al. Modified Bethe-Weizsacker mass formula with isotonic shift and new driplines. Mod. Phys. Lett. A20 (2005) 1605-1618 Conclusion [6] G. Audi and A.H. Wapstra. The 1993 atomic mass evolution.(I) Atomic mass table. Nuclear physics, A At one go a unified theory can not be developed. Search- 565, 1993, p1-65. ing, collecting, sorting and compiling the cosmic code is an essential part of unification. In this attempt the above observations can be given a chance. Further research and analysis may reveal the mystery of the strong, weak and electromagnetic interactions in a unified way.