Nuclear masses and binding energies. The curve of binding energy per nucleon, B/A. b i. m nucl c 2 = m( A ZX N )c 2 Zm e c 2 +

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1 Nuclear masses and binding energies The mass energy of a certain nuclide is m nucl c 2 = m( A ZX N )c 2 Zm e c 2 + Z i=1 b i where m( A ZX N ), mass of the neutral atom ( A 1000 MeV/c 2 ) m e, electron mass (0.511 MeV/c 2 ) b i, electron binding energies ( at most 100 kev, neglect!) The difference between the nuclear mass and the mass of its constituents corresponds to the binding energy B of the nucleus: B = {Zm p + Nm n [m( A ZX N ) Zm e ]} c 2 = {Zm( 1 H) + Nm n m( A ZX N )} c 2 The curve of binding energy per nucleon, B/A Hyperphysics c CR Nave Georgia State University

2 Exercises ف مو 属 لم فم م ه م م مو م ف ى. فكى م ى مح م ك ص 238 Noting that for 238 U it is B/A 7.6 MeV and that for A = 119 it is B/A 8.5 MeV then we have that the energy locked into the final nuclei is MeV 119 = 2023 MeV whereas before fission it was 7.6 MeV 238 = 1809 MeV The difference MeV, is the energy that is released in the process (mostly as kinetic energy of the two produced nuclei) This amount of energy is 10 8 times larger than the energy released per atom when burning coal or oil. Teixeira-Dias PH Atomic and Nuclear Physics Royal Holloway Univ of London The curve of binding energy per nucleon, B/A Note that B/A is remarkably constant (within ±10% of 8 MeV) for all nuclei, with the sole exception of the very light nuclei; peaks at A 60. Therefore energy will be released by nuclear fission : breaking up heavy nuclei into lighter nuclei e.g., nuclear reactors, nuclear weapons nuclear fusion : assembling light nuclei into heavier nuclei e.g., thermonuclear H-bomb, future(?) fusion reactors, stellar nucleosynthesis up to the most tightly bound nucleus: 56 Fe (Q: How do the elements with Z > 56 come into being?)

3 The distribution of stable nuclei Z N=Z proton number neutron number, c Brokhaven National Lab Stable nuclei are indicated by black squares N The distribution of stable nuclei N Z symmetry ىم ك وهى ن ( Z A/٢) Z N > ى فى مل م فم ك ى Z ف increasing Coulomb repulsion ( Z 2 ) need more neutrons to hold nucleus together Number of A N Z stable nuclei Even Even 166 Even Odd Odd 8 Even Odd 57 Odd Odd Even 53 م فو ل م ىم ك م قف even Z ف ل ف even N nucleons of the same species can form strongly interacting pairs which make a rather large contribution to the binding energy.

4 The semi-empirical mass formula (SEMF) مى م فم ك ن ه ى مل م ك م ه م م ه ىل ىق ن م ك مو ىف م ➊ volume term: to lowest order B A, so we have a contribution +av A to the binding energy; recall R = R 0 A +1/3 so A R 3 note A dependence, and not A(A 1) A 2 as one would expect if each nucleon interacted with all others short range of strong nuclear force. ➋ surface term: nucleons at the surface of the nucleus have fewer neighbours so volume term overestimates their contribution to binding energy; surface of nucleus R 2 A 2/3, therefore as A 2/3 The semi-empirical mass formula (cont d) ➌ Coulomb term: each proton repels each other Z(Z 1); for a uniformly charged sphere 3 5 Z(Z 1)e 2 4πɛ 0 R = 3 5 e 2 Z(Z 1)A 1/3 4πɛ 0 R 0 and so we have a contribution to the binding energy ac Z(Z 1)A 1/3 ➍ symmetry term: Z A/2 (nuclear chart, light nuclei), decreasing with A: asym (A 2Z) 2 /A if Z = A/2, Bsym = 0; Z A/2 is disfavoured as it implies Bsym < 0;

5 The semi-empirical mass formula (cont d) ➎ pairing term: as seen earlier, nucleons of same species tend to pair (pp or nn), thus explaining why so many stable nuclei have even N and/or even Z, and why so few nuclei have odd Z and odd N. δ = ap A 3/4 for Z & N odd; 0 for A odd; +ap A 3/4 for Z & N even;