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A Study of Nuclear Binding Energy of Magic Number Nuclei and Energy Splitting Considering Independent Particle Shell Model

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International Journal of Advanced Scientific Research and Management, Volume 3 Issue 2, Feb 2018. www.ijasrm.com ISSN 2455-6378 A study of Nuclear Binding Energy of Magic Number Nuclei and Energy Splitting considering Independent particle shell model Dipesh Chanda Department of Physics, Siliguri College, Siliguri,West Bengal, India. Pin 734001 Abstract according to the configuration 1s-2s-2p-3s-3p-4s- It is well established about the higher nuclear 3d-4p-5s-4d-5p-6s. binding energies of atoms having magic number Similarly nuclear binding energies are high nuclei compared to their nearest neighbouring compared to their adjacent nuclei when proton nuclei and as a whole. In the present case there will number and or neutron numbers are 2, 4, 8, 20, 50, be a comparative study about the nuclear binding 82, 126. The nuclei having the number of protons energies of highly stable nucleus and the nuclei and or neutrons are called magic nuclei. These having one more or less nucleons using nuclear nuclei are not only highly stable but show addition liquid drop model and independent particle shell some characteristics, which are greater relative model simultaneously. In addition to that, there abundances in nature, greater number of stable would be a try to explain the reason of higher isotopes, the separation energy of one proton or one nuclear binding energies of magic number nuclei neutron very large, probability of capturing a by considering additional s l interactions of neutron is much lower, the energy of first excited nucleons. This s l interaction are considered to states of nuclei is high, the alpha disintegration be much stronger and opposite in sign to the energy is small etc. These unexpected behaviors coupling of electrons in the atom. The can be explained by introducing a nuclear model, interaction is very strong coupling of spin and experimental results also reflects these.. orbital motion of the nucleons in the nucleus of the To explain the properties of the nucleus, various magic number nuclei with negative sign is required nuclear models are proposed. Among these, liquid to confirm energy splitting. drop model is one of the most suitable one to calculate the exact value of nuclear binding energy. Keywords: Magic number nuclei, stable nucleus, In the model the atomic nucleus is assumed to be a independent particle model, coupling small water droplet composed of molecules. The properties like shape, density independent of its volume, short range force, etc. of a nucleus are 1. Introduction analogous to the microscopic depiction of a water drop. According the model, when binding energy From the study of atoms, it is found that when of the nucleus are calculated, different factors like number of electrons in an atom are 2, 10, 18, 36, volume energy/effect, surface energy/effect, 54, 86 show high chemical stability, ionization coulomb energy/effect, asymmetric energy/effect potentials are also high and do not interact with and pairing energy/effect, of a nucleus are assumed other atoms. These are called atomic magic to be the only factors effected the actual binding numbers and chemical stability of these atoms can energy of the nucleus, as a result weizsacker be explained on the basis of electronic closed shells assumed a formula that is called semi empirical and sub-shells according Pauli’s exclusion mass formula. Out of the five factors volume effect principle. He(2), Ne(10), Ar(18), Kr(36), Xe(54), increases the binding energy , whereas the next are the examples of atomic magic numbers. three factors decreases the binging energy and the Electrons are filled in shells and sub-shells last factors increases the value for even- even N and Z, decreases when both N and Z are odd. 106 International Journal of Advanced Scientific Research and Management, Vol. Issue , 2018. www.ijasrm.com ISSN 2455-6378 From the study of quadrupol moments of the 931x[Zmp (A Z)mn M] MeV------------- (2) ground state of magic number nuclei, shows zero Where A=mass number and M= isotopic mass of value i.e. these indicates their spherical shape having closed structure. the nucleus. mp , mn , mass of protons and The single particle nuclear shell model proposed neutrons, are used standard values as from the basic as was found in atoms, where m p = 1.007825u, mn =1.008665u electrons moves independently in a central orbit Using the above formula nuclear binding energy of round the nucleus due to the central Coulomb force magic nuclei as well as for their neighbours can be created by the nucleus. In the model it assumed that calculated. the nucleons are moving independently in a combined force of central and non-central force field. For central field, it is explained as a short- Calculation of Separation energy of neutron Sn range potential. Various forms of potential had and proton S p : used for the calculation of nuclear energy level, Separation energy of a neutron or a proton can be like square well potential and harmonic oscillator treated as the amount of energy required to get free well etc. Three dimensional Schrodinger equation one neutron or a proton from the nucleus. i.e. for harmonic oscillator can be solved using minimum energy required for the separation of one spherical polar coordinates. Hence, achieved eigen neutron or one proton from a nucleus. These can value and eigen vectors. In the eigen vectors two be estimated by the equations parts, one is radial function and other spherical S [m M (Z, A1) M (Z, A)]931 MeV. harmonics. For the non central part, MM.Mayer n n and H.Jensen separately assumed the existence of And nuclear interaction (nuclear spin-orbit S p [mp M(Z 1, A1) M(Z, A)931 MeV coupling interaction). Nuclear spin-orbit coupling -------------- (3) where, mn and m p are the masses interaction is like that the atomic cases but much of neutron and proton, M (Z, A1),M (Z 1, A1) stronger and opposite in sign to the interaction of electrons. From this special property and M (X , A) are the masses of nuclei of one of coupling, energy splitting of shells are created in neutron short, one proton short and the mass of nuclear magic numbers, which are also verified original nuclei. experimentally. Estimation of energy of a shell and splitting 2. Theory ands Calculationl energy: Atomic nucleus is considered as a liquid drop so For the explanation of higher stability of magic binding energies of the highly stable nucleus i.e. nuclei, MM.Mayer and H.Jensen separately magic nuclei and their adjacent nuclei can be assumed the existence of nuclear interaction estimated using semi empirical mass formula. The (nuclear spin-orbit). The interaction is stronger and formula used to calculate binding energy is, opposite sign compared to spin orbit coupling 2 / 3 1/ 3 interactions in atoms. The strong interaction of spin B.E avAbsA ccZ(Z 1) / A --------- and orbital motion of the nucleons in the nucleus in d (A 2Z)2 / A / A3/ 4 the nucleus having opposite sign, is the result of a energy splitting in agreement with the experimental (1), nuclear magic numbers. Spin orbit interaction term where the successive terms are volume is considered to be added to the central potential energy/effect, surface energy, coulomb energy, which is non central and may written in the form asymmetry energy and pairing energy. A and Z are V f (r)(s.l ) ----- (4), where f (r) is the the mass no. and atomic no. of the nucleus and av, ls bs, cc, da, and ð are constants, their values are 15.8, potential function, s and l are spin and orbital 17.8, 0.71, 23.7 and 34 respectively. When A and Z angular momentum vectors, both combine to form both are even ð is positive, it is zero, when any one total angular momentum j . is odd and negative when both are odd. Finally binding energy and binding energy per nucleon for After simplification of cosine law magic nuclei and their neighbours can be j( j 1) l(l 1) s(s 1) (s.l ) and estimated. 2 1 1 Calculation of nuclear binding energy from the jmax l and jmin l 2 2 mass defect: Nuclear binding energy = 107 International Journal of Advanced Scientific Research and Management, Vol. Issue , 2018. www.ijasrm.com ISSN 2455-6378 l (l 1) and Easto Ev are estimated out. Than total binding i.e. (s.l ) j and (s.l ) j min ---- max 2 2 are calculated and binding energy per nucleon are (5) estimated. Next once again binding energy of each Ultimately using equation (5), the spin orbit energy magic number nuclei are calculated using mass splitting of two levels defect, For this calculation equation (2) is utilized. Ultimately binding energy per nucleon is 1 Vsl f (r)(2l 1) ----- (6) determined. 2 For the estimation of separation energies of The equation (6) confirms that the state with neutron and proton equation (3) are used. After that 1 1 j l lies below the state j l . some nuclei are chosen which have one more 2 2 nucleon compared to magic nuclei. Following Hence the nuclear orbit in each shells splits and above mention process and by using equations according energy states these are arrange as 1s1/2 (1), (2) and (3), the different ratios of energy terms, 1p3/2,1p1/2,1d5/2,1d3/2,2s1/2,1f7/2, binding energy, binding energy per nucleon, (two 1f5/2,2p3/2,2p1/2,1g9/2,1g7/2 and so on. different ways) and separation energies of neutron Solving the potential function f (r) ( in the form of and proton are estimated.
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