Journal of Nuclear Sciences

Not for reproduction, distribution or commercial use. Provided for non-commercial research and education use. Volume 4, No. July 07 p-issn: 47-7736, e-issn:48-398 Ankara University Institute of Nuclear Sciences Journal of Nuclear Sciences Editor-in-Chief Haluk YÜCEL, Ph.D. sistant Editor-in-Chief George S. POLYMERIS, Ph.D. Editorial Board A. HEIDARI, Ph.D. Ayşe KAŞKAŞ, Ph.D. Birol ENGİN, Ph.D. Erkan İBİŞ, M.D. Gaye Ö. ÇAKAL, Ph.D. İskender Atilla REYHANCAN, Ph.D. İsmail BOZTOSUN, Ph.D. Mehmet TOMBAKOĞLU, Ph.D. M.Salem BADAWI, Ph.D. Mustafa KARADAĞ, Ph.D. Niyazi MERİÇ, Ph.D. Osman YILMAZ, Ph.D. Özlem BİRGÜL, Ph.D. Özlem KÜÇÜK, M.D. Slobodan JOVANOVIC, Ph.D. Turgay KARALI, Ph.D. Owner on behalf of Institute of Nuclear Sciences, Ankara University, Director Niyazi MERİÇ, Ph.D. http://dergipark.gov.tr/ankujns

Hosted by Ankara University Journal of Nuclear Sciences p-issn: 47-7736, e-issn:48-398 Journal homepage: http://dergipark.gov.tr/ankujns DOI: 0.50/nuclear_00000000 On the Role of Strong Coupling Constant and Nucleons in Understanding Nuclear Stability and Binding Energy U.V.S. Seshavatharam *, S. Lakshminarayana ORCID: 0000-000-695-6037,----- Honorary faculty, I-SERVE, Survey no-4, Hitech city, Hyderabad-84, Telangana, India Dept. of Nuclear Physics, Andhra University, Visakhapatnam-03,AP, India ABSTRACT Received.07.07; received in revised form.08.07; accepted 4.08.07 In this study, we dealt with beta stability line and nuclear binding energy in a simplified semi empirical approach by taking into account nucleon and electron rest masses and with reference to nuclear charge radius and strong coupling constant. In this context, we have considered ( α s ) ( e 8πϵ 0 R ) 4.86 MeV as a single nuclear binding energy coefficient or 0 potential. Keywords: Nucleon, Nuclear binding energy, Beta stability line, Strong coupling, Semi-empirical mass formula, Liquid drop model. Introduction In nuclear physics, the semi-empirical mass formula [- 3] is used to approximate the mass and various other properties of an atomic nucleus. its name suggests, it is based partly on theory and partly on empirical measurements. Although refinements have been made to the coefficients over the years, the structure of the formula remains the same today. The simple semi empirical mass formula (SEMF) constitutes of five different energy coefficients and 5 different energy terms. The major drawback of SEMF is that, it does not throw light on the implementation of strong interaction concepts and strong coupling constant [4,5] in understanding nuclear binding energy. In this study, we attempt: ) To propose a very simple and direct relation for understanding beta stability line [5, 6, 7] with reference to neutron, proton and electron rest masses, ) To propose a very simple semi empirical relation for fitting the nuclear binding energy with only one energy constant i.e. 4.86 MeV, which is assumed to be connected with nuclear charge radius R 0.5 fm and strong coupling constant α s 0.86, 3) To divide the nuclear binding energy scheme into 3 steps: a. To understand the binding energy at stable mass number of Z, b. To understand the binding energy below the stable mass number of Z, c. To understand the binding energy above the stable mass number of Z; 4) To report an attempt to fit and interrelate the SEMF binding energy coefficients.. Liquid Drop Model and the Semi Empirical Mass Formula According to the well-known liquid drop model: ) Atomic nucleus can be considered as a drop of incompressible fluid; *Corresponding author. E-mail address: seshavatharam.uvs@gmail.com, (Seshavatharam U.V.S) Journal of Nuclear Sciences, Vol. 4, No., July 07, 7-8 Copyright, Ankara University, Institute of Nuclear Sciences ISSN: 48-7736

) Nuclear fluid is made of protons and neutrons, which are held together by the strong nuclear force. Mathematical analysis of the theory delivers an equation which attempts to predict the binding energy of a nucleus in terms of the numbers of protons and neutrons it contains. This equation includes five terms on its right hand side. These correspond to the cohesive binding of all the nucleons by the strong nuclear force, the electrostatic mutual repulsion of the protons, a surface energy term, an asymmetry term which is derivable from the protons and neutrons occupying independent quantum momentum states, and a pairing term which is partly derivable from the protons and neutrons occupying independent quantum spin states. The coefficients are calculated by fitting to experimentally measured masses of nuclei. Their values can vary depending on how they are fitted to the data. In the following formulae, let A be the total number of nucleons, Z the number of protons, and N the number of neutrons. The mass of an atomic nucleus is given by: m Zm Nm B c p n () where m p and m n represent the rest mass of a proton and a neutron, respectively, and B is the binding energy of the nucleus. The semi-empirical mass formula states that the binding energy will take the following form, /3 Z( Z ) ( A Z) ap B av A as A ac a /3 a () A A A Here a υ is volume energy coefficient, a s is the surface energy coefficient, a c is the coulomb energy coefficient, a a is the asymmetry energy coefficient and a p is the pairing energy coefficient. If we consider the sum of the volume energy, surface energy, coulomb energy, asymmetry energy and pairing energy, then the picture of a nucleus as a drop of incompressible liquid roughly accounts for the observed variation of binding energy of the nucleus. given in Table, the currently adopted coefficients in unit of energy (MeV) are used in semi empirical mass formula [3]. Table. Adopted SEMF binding energy coefficients a v a s a c MeV MeV MeV MeV MeV 5.78 8.34 0.7 3..0 5.58 6.6 0.689.0 0.08 a a a p 3. Estimation of Stable Mass with Proton By maximizing B(A, Z) with respect to Z, it can be found the number of protons Z of the stable nucleus of atomic weight A as, A Z ac aa A 0.4A AZ A 00 This is roughly A/ for light nuclei, but for heavy nuclei there is an even better agreement with nature. By substituting the above value of Z back into B, one obtains the binding energy as a function of the atomic weight, B(A). The maximization of B(A)/A with respect to A would give the nucleus which is most strongly bound or most stable. In this context, we would like to suggest that, independent of SEMF concepts, nuclear beta stability line can be understood within neutron, proton and electron rest masses. It is also possible to show that [4]: mn mp c.9333mev exp exp 4 mc e 0.50998946MeV Based on this observation and without considering the binding energy coefficients, beta stability line can be understood with the following empirical relations. These relations can be compared with the computational relations pertaining to isotopic shift and drip lines proposed in reference [3], N s = 0.96805Z + 0.00658803Z. Let, /3 mn mp c ln 4.53045 mc e k 0.00633 4 Z Z Z 0.00633Z Z kz 4 Z Ns Z Z 0.00633Z Z kz 4 Z Z 0.00633Z kz 4. (3) (4) (5) With even-odd corrections much better correlations can be observed. For light and medium atomic nuclides, there is some mismatch, which can be attributed to shell structure and needs for further study. indicated in 8

Table the stable nucleon number can be estimated with its corresponding proton number by fitting. Table. The fit results for the stable mass numbers versus proton number Proton Z Estimated Stable Mass, A s A s with Even Odd Correction 44.80 45 9 63.3 63 37 8.67 83 47 07.99 07/09 53 3.79 3/5 60 4.80 4 69 68.5 67/69 79 97.5 97 83 09.6 09/ 9 37.60 38 00 63.33 6 303.43 30 8 34.7 34 4. Characteristic Nuclear Binding Potential and Nuclear Binding Energy Coefficients In this analysis, it is assumed that with reference to nuclear charge radius, R 0.5 fm, and the strong coupling constant, α s 0.86 by the following relation. mp 4 av B0 5.9 MeV; as av B0 8.34 MeV m e m 4 p ac B0 0.745 MeV; aa as B0 3.0 MeV me ap aa as B0.6 MeV; 3 as av B0 ; aa as B0 ; aa av B0 ; 5. Proposed Method of Estimating Nuclear Binding Energy for Z= to 83 Starting from Z= to 83, Step : Close to beta stability line, nuclear binding energy can be expressed with the following semi empirical relation. B where, A Z A s s x 4.86 MeV Z A s 6 Z Z to 9, x and for 30 Z 83, x.0 30 (7) (8) B 0 e 4.85655 MeV 4.86 MeV s 8 0R0 (6) B A A Z A s s xy 4.86 MeV Z A s (9) With this binding energy potential, nuclear binding energy can be fitted with the following two-term semi empirical relation. It should be noted that, with further research and analysis, qualitatively a simplified and unified method can be developed. With this energy potential, energy coefficients of the SEMF can be fitted in the following way. Step : Below and above the stable mass numbers, where, 3 A Z A A, 6 s y Z Z to 9, x Z 30 and for 30 Z 83, x.0 A Z A, y Z 9

0 9 38 47 56 65 74 83 9 Binding energy (MeV) Seshavatharam and Lakshminarayana/Journal of Nuclear Sciences Vol 4 () 7-8 6. Alternative Expression for ymmetry Energy Term at Beta Stability Line With reference to the proposed beta stability line coefficient, k ( 4π ) 0.006333, close to the beta stability line, with trial error, we noticed that: For light and medium atoms, A s Z 3 A s k A N s s (0) 8. Binding Energy Data Fitting given in Table 3 the nuclear binding energy was estimated at stable mass numbers. Fig. shows nuclear binding energy at stable atomic nuclides. The plotted solid curve between the binding energy and the proton number indicates the actual or SEMF binding energy (MeV) and dotted red curve indicates estimated binding energy (MeV). For heavy atoms, A Z 3 s s s k A N () 7. Proposed Method of Estimating Nuclear Binding Energy for Z > 83 Based on the above observation, for Z > 83, binding energy close to beta stability line can be expressed with the following relation. 000 750 500 50 000 750 500 50 0 B x k A N where x.0 3 A s s s Z 4.86 MeV A s 3 BA xy k Ns 4.86 MeV Z Below and above the stable mass number, where, () (3) Proton number Fig. Nuclear binding energy at stable atomic nuclides seen in Table 4, the isotopic binding energies are calculated for Z=0, 30, 40, 50, 60, 70, 80 and 90. The column- represents the proton number, column- represents the stable estimated mass number, column-3 represents the neutron number, column-4 represents the nucleon number, column-5 represents the binding energy calculated with SEMF, column-6 represents the binding energy calculated with proposed relations and column 7 represents the %error with respect to SEMF. x.0, A Z A Z A, y and A, y Z Z 3 0

Table 3. Estimated nuclear binding energy at stable mass numbers A s from 4 to 63 Proton Z Stable Mass A s A s with Even Odd Correction Estimated Binding Energy (MeV) Actual [8] or *SEMF [] Binding Energy (MeV) 4 4 4.8 8.96 3 6 7 40. 39.44 4 8 8 55.6 58.08 5 0 7.4 76.05 6 89. 9.6 7 4 4 06.8 04.659 8 6 6 4.8 7.69 9 9 9 43.3 47.80 0 6.0 67.406 3 3 8. 86.564 5 4 00. 98.57 3 7 7 0.0 4.95 4 9 8 39.7 36.537 5 3 3 59.9 6.97 6 34 34 80.6 9.839 7 36 35 300.7 98. 8 38 38 3.8 37.343 9 40 39 34.4 333.74 0 43 4 363.8 36.896 45 45 385.5 387.848 47 46 406.5 398.93 3 49 49 48.5 *46.34 4 5 5 450.8 456.349 5 54 55 473. 48.075 6 56 56 494.7 49.58 7 59 59 57.5 57.33 8 6 60 539. 56.846 9 63 63 56. 55.385 30 66 66 585.4 578.36 3 68 67 607.5 *58. 3 70 70 64. 60.5 33 73 73 644. *634.34 34 75 74 663.0 64.89 35 78 77 683.0 *667.66 36 80 80 703.0 695.434 37 83 83 73. *70.46 38 85 84 74.8 78.906 39 88 87 76.8 *755.05 40 90 90 78.9 783.893 4 93 93 80.0 805.765 4 95 94 80.7 84.56

43 98 97 840.7 *836.6 44 00 00 860.9 86.98 45 03 03 88.0 884.63 46 05 04 899.6 89.8 47 08 07 99.7 95.63 48 0 939.8 940.646 49 3 3 960.0 963.094 50 6 6 980. 988.684 5 8 7 998.7 *99.78 5 0 08.8 07.8 53 4 3 039.0 *038.77 54 6 6 059. 063.909 55 9 9 079.3 *085.08 56 3 3 099.5 0.038 57 35 35 9.7 *3 58 37 36 38. 38.79 59 40 39 58.3 *60.56 60 43 4 78.5 85.4 6 46 45 98.7 *03.86 6 48 48 8.9 5.39 63 5 5 39. 44.4 64 54 54 59.3 66.67 65 57 57 79.5 *87.38 66 60 60 99.8 309.455 67 6 6 38. *34.3 68 65 64 338.4 336.447 69 68 67 358.6 *356.45 70 7 70 378.8 378.3 7 74 73 399.0 *397.78 7 77 76 49.3 48.80 73 80 79 439.5 *438.48 74 83 8 459.7 459.335 75 86 85 479.9 478.34 76 89 86 498.3 484.807 77 9 9 50.4 58.088 78 95 94 540.6 539.577 79 98 97 560.9 559.386 80 0 00 58. 58.8 8 04 03 60.4 600.87 8 07 06 6.6 6.35 83 0 09 64.8 640.3 84 3 646.6 *655.76 85 6 5 666.0 *669. 86 9 8 685.3 *685.75 87 704.6 *70.68 88 5 4 73.8 *79.3

89 8 7 743.0 *735.53 90 3 30 76. *753.97 9 34 33 78. *770.78 9 38 38 803.7 80.69 93 4 4 8.7 *87.3 94 44 44 84.6 *835.45 95 47 47 860.5 *85.73 96 50 50 879.3 *868.97 97 54 53 898.0 *884.38 98 57 56 96.8 *90.6 99 60 59 935.4 *97.39 00 63 6 954.0 *935. Table 4. Estimated isotopic nuclear binding energy of Z=0 to 90 Proton Estimated Stable Mass Neutron Nucleon SEMF Binding Energy in MeV Proposed Binding Energy in MeV %Difference 0 4 0 40 339.7 34.4-0.50 0 4 4 350. 35.7-0.74 0 4 4 363. 363.8-0.7 0 4 3 43 37.6 37.0-0.0 0 4 4 44 38.7 380.0 0.7 0 4 5 45 389.3 387.8 0.38 0 4 6 46 398.7 395.5 0.80 0 4 7 47 403.8 403.0 0.9 0 4 8 48 4.8 40.4 0.33 0 4 9 49 45.5 47.7-0.53 0 4 30 50 4. 44.8-0.65 0 4 3 5 44.7 43.8 -.68 0 4 3 5 430. 438.8 -.99 0 4 33 53 43.7 445.6-3. 30 66 30 60 509.5 58.4 -.75 30 66 3 6 5. 59.9 -.68 30 66 3 6 535 54. -.6 30 66 33 63 545. 55.4 -.3 30 66 34 64 557.8 563.5 -.0 30 66 35 65 566.7 574.5 -.38 30 66 36 66 577.9 585.4 -.30 30 66 37 67 585.7 593.5 -.33 30 66 38 68 595.8 60.4-0.95 30 66 39 69 60.4 609.3 -.5 3

30 66 40 70 6.5 67. -0.9 30 66 4 7 67. 64.7 -. 30 66 4 7 65.3 63.3 -. 30 66 43 73 630. 639.8 -.54 30 66 44 74 637. 647. -.57 30 66 45 75 64. 654.5 -.07 30 66 46 76 647.6 66.7 -.8 40 90 40 80 666. 673.8 -.4 40 90 4 8 678.6 685.0-0.94 40 90 4 8 693. 696. -0.43 40 90 43 83 704.3 707. -0.40 40 90 44 84 77.7 78.0-0.05 40 90 45 85 78 78.9-0. 40 90 46 86 740.4 739.6 0. 40 90 47 87 749.7 750.3-0.08 40 90 48 88 76. 760.9 0.04 40 90 49 89 769.6 77.4-0.4 40 90 50 90 780. 78.9-0. 40 90 5 9 787.8 789.7-0.4 40 90 5 9 797.6 797.4 0.03 40 90 53 93 804.4 805.0-0.08 40 90 54 94 83.4 8.6 0.0 40 90 55 95 89.5 80. -0.07 40 90 56 96 87.8 87.5 0.04 40 90 57 97 833.3 834.8-0.8 40 90 58 98 840.8 84. -0.6 40 90 59 99 845.7 849.4-0.43 50 6 56 06 887.6 878.4.03 50 6 57 07 898. 888.8.03 50 6 58 08 90.4 899..3 50 6 59 09 90. 909.5.5 50 6 60 0 93.8 99.8.9 50 6 6 940.7 930.0.4 50 6 6 95.6 940.. 50 6 63 3 960.0 950..0 50 6 64 4 970. 960..03 50 6 65 5 977.9 970. 0.79 50 6 66 6 987.5 980. 0.75 50 6 67 7 994.6 987.5 0.7 50 6 68 8 003.6 994.8 0.87 50 6 69 9 00. 00. 0.79 50 6 70 0 08.5 009.4 0.90 50 6 7 04.4 06.5 0.77 4

50 6 7 03.3 03.7 0.83 50 6 73 3 037.8 030.8 0.68 50 6 74 4 045. 037.8 0.70 50 6 75 5 050. 044.8 0.5 50 6 76 6 056.9 05.7 0.49 60 4 73 33 099 090.6 0.76 60 4 74 34 0. 00.5 0.87 60 4 75 35 9 0.4 0.77 60 4 76 36 9.7 0.3 0.83 60 4 77 37 38 30. 0.69 60 4 78 38 48 39.9 0.7 60 4 79 39 55.8 49.6 0.54 60 4 80 40 65.4 59.3 0.53 60 4 8 4 7.7 68.9 0.3 60 4 8 4 8.7 78.5 0.7 60 4 83 43 88.5 85.7 0.4 60 4 84 44 97. 9.8 0.36 60 4 85 45 03.5 99.9 0.30 60 4 86 46.6 06.9 0.39 60 4 87 47 7.5 3.9 0.30 60 4 88 48 5. 0.9 0.35 60 4 89 49 30.7 7.8 0.4 60 4 90 50 37.9 34.7 0.6 60 4 9 5 43 4.5 0. 60 4 9 5 49.9 48.3 0.3 60 4 93 53 54.6 55. -0.04 70 70 9 6 303.9 94.8 0.70 70 70 9 6 33.9 304. 0.73 70 70 93 63 3.8 33.7 0.6 70 70 94 64 33.3 33. 0.6 70 70 95 65 338.8 33.4 0.47 70 70 96 66 348.0 34.8 0.46 70 70 97 67 355.0 35. 0.9 70 70 98 68 363.8 360.4 0.5 70 70 99 69 370.5 369.6 0.07 70 70 00 70 378.8 378.8 0.00 70 70 0 7 385. 385.7-0.04 70 70 0 7 393.0 39.5 0.03 70 70 03 73 399.0 399.3-0.0 70 70 04 74 406.6 406. 0.04 70 70 05 75 4. 4.8-0.05 70 70 06 76 49.4 49.6-0.0 70 70 07 77 44.6 46. -0. 5

70 70 08 78 43.5 43.9-0.0 70 70 09 79 436.4 439.5-0. 70 70 0 80 443.0 446. -0. 70 70 8 447.5 45.7-0.36 80 00 9 509.8 50.0 0.58 80 00 9 58.7 50.0 0.57 80 00 3 93 55.6 59.0 0.43 80 00 4 94 534. 57.9 0.40 80 00 5 95 540.8 536.9 0.6 80 00 6 96 549.0 545.8 0. 80 00 7 97 555.3 554.6 0.04 80 00 8 98 563. 563.5-0.0 80 00 9 99 569. 57.3-0.0 80 00 0 00 576.7 58. -0.8 80 00 0 58.3 587.7-0.34 80 00 0 589.6 594. -0.9 80 00 3 03 595.0 600.7-0.36 80 00 4 04 60.9 607. -0.33 80 00 5 05 607.0 63.7-0.4 80 00 6 06 63.6 60. -0.4 80 00 7 07 68.5 66.6-0.50 80 00 8 08 64.8 633.0-0.50 80 00 9 09 69.4 639.3-0.6 80 00 30 0 635.4 645.7-0.63 80 00 3 639.7 65.0-0.75 90 30 3 699.8 685.7 0.83 90 30 3 707.9 694.3 0.80 90 30 33 3 74. 70.9 0.66 90 30 34 4 7. 7.4 0.6 90 30 35 5 78. 79.9 0.48 90 30 36 6 735.6 78.4 0.4 90 30 37 7 74.4 736.8 0.6 90 30 38 8 748.7 745.3 0.0 90 30 39 9 754. 753.7 0.03 90 30 40 30 76. 76. -0.05 90 30 4 3 766.5 768.4-0. 90 30 4 3 773. 774.6-0.08 90 30 43 33 778.3 780.9-0.5 90 30 44 34 784.8 787. -0.3 90 30 45 35 789.5 793.3-0. 90 30 46 36 795.8 799.5-0.0 90 30 47 37 800.3 805.6-0.30 90 30 48 38 806.3 8.7-0.30 6

90 30 49 39 80.6 87.9-0.40 90 30 50 40 86.4 83.9-0.4 90 30 5 4 80.5 830.0-0.5 9. Discussion From the above proposed relations and the estimated results in tables to 4 it is possible to infer that: ) Nucleons and electrons play a crucial role in understanding nuclear stability. ) Strong coupling constant plays an important role in understanding the nuclear binding energy. 3) Nuclear binding energy can also be estimated with use of a single energy coefficient. 4) Z=30 seems to play an interesting role and we are working on understanding the physical significance of ( Z ) 6. 30 5) Z=53 is estimated to be stable A s=3 and its estimated binding energy is 039 MeV. Actually, it is stable at A s=7. From relation (9), binding energy of 53I 7 is ( 7 53 ) 039.0 068.7 MeV 3 53. Actual binding energy of 53I 7 is 07.57 MeV. 6) The authors are working in this line to deal with the nuclear binding energies and masses involving more realistic quantum physics, different shell corrections and odd-even phenomena [9-4]. 0. Conclusion Understanding and estimating nuclear binding energy with strong interaction concepts still stands as a really challenging task. So far no such model is available in physics literature. Even though some % difference is persisting in the proposed binding energy expressions, qualitatively they are very simple to follow. An interesting point to be noted is that, the number of energy coefficients can be minimized and their existence can be linked with strong interaction concepts. We are confident to say that, with further research, background physics of the proposed relations can be understood and thereby a clear and simple model can be developed. Acknowledgements Author Seshavatharam is indebted to professors shri M. Nagaphani Sarma, Chairman, shri K.V. Krishna Murthy, founder Chairman, Institute of Scientific Research in Vedas (I-SERVE), Hyderabad, India and Shri K.V.R.S. Murthy, former scientist IICT (CSIR), Govt. of India, Director, Research and Development, I- SERVE, for their valuable guidance and great support in developing this subject. Conflicts of Interest The authors have no conflict of interest. References [] Von C.F. Weizsäcker, On the theory of nuclear masses Journal of Physics (96) 43-458 (935). [] W. D. Myers and Swiatecki. W. J. Table of Nuclear Masses according to the Thomas-Fermi Model. (from nsdssd.lbl.gov), (994). [3] P. R. Chowdhury, C. Samanta and D. N. Basu, Modified Bethe-Weizsacker mass formula with isotonic shift and new driplines. Modern Physics Letters A 0. 605-68 (005). [4] C. Patrignani et al. (Particle Data Group), Chin. Phys. C (40) 0000 (06) and 07 update. [5] U. V. S. Seshavatharam and S. Lakshminarayana, Proceedings of the DAE-BRNS Symp. on Nucl. Phys. 60 8-9 (05). [6] U. V. S. Seshavatharam and S. Lakshminarayana, On the Ratio of Nuclear Binding Energy & Protons Kinetic Energy. Prespacetime Journal 6 (3) 47-55 (05). [7] U. V. S. Seshavatharam and S. Lakshminarayana, Consideration on Nuclear Binding Energy Formula. Prespacetime Journal 6 () 58-75 (05). [8] N. Ghahramany, S. Gharaati and M. Ghanaatian, A new approach to nuclear binding energy in integrated nuclear model. T.8, No (65) 69-8 (0). [9] J.G. Hirsch and J. Mendoza-Temis, Microscopic mass estimations. J. Phys. G: Nucl. Part. Phys. 37 06409 (00). 7

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