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Semi-Empirical Mass Formula Applications – II Nucleon Separation Energies and Fission [Sec Dunlap]

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The Semi-Empirical Mass Formula Let us see how this equation can be applied to (i)Neutron Separation Energies (ii)Alpha Particle Decay Energies (iii)Fission

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Single neutron separation energies Fig 4.8 PULLING NEUTRONS OUT OF ODD-A NUCLIDES The arrows show the transitions from the odd A parabola to the even (A-1) parabolas for the two cases of (e,o) (o,o) breaking pairing on neutron side (o,e) (e,e) breaking no pairing bond

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Single neutron separation energies In an earlier lecture we found that the separation energy for a neutron was: This can be written in terms of mass of constituents and binding energies OddEven Apply the SEMF assuming B(A,Z) is continuous in A. A B 1 to o-o to e-e

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Single neutron separation energies Now apply the SEMF: This is an interesting result because it can give us an equation for the “neutron drip” line by putting S n =0

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Mass Parabolas Neutron number Proton number Z=N Neutron drip line

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Alpha Particle Decay Q We saw in a previous lecture that the Q-value (energy released) in -decay is: where From which:

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Alpha Particle Decay Q

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Energy released in Fission The diagram shows the Q (energy released) from the fission of 236 U as a function of the A of one of the fragments (as obtained from the SEMF). Note that maximum energy release is 210MeV/Fission for the nucleus splitting into equal fragments.

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Energy released in Fission This figure shows the prediction of the SEMF for the energy released in FISSION when two equal fragments are formed.

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Energy released in Fission

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The Fission Barrier The origin of the fission barrier can be seen by reversing the fission process. Two fission fragments approach with (1/r) potential – consider the fragments equal. When r decreases until the two fragments are nearly touching the nuclear attractive strong force takes over – the potential energy is less than that calculated by Coulomb law. Fission barrier

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Understanding the Fission Barrier Consider the stability of an Ellipsoidal Deformation, =eccentricity of ellipse How do B S and B C vary on deformation?

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Understanding the Fission Barrier SURFACE ENERGY The surface area increases on deformation and so does B S. The nucleus becomes LESS bound The mass energy increases with deformation – This produces a potential that seeks to keep =0, I.E. the nucleus in SPHERICAL condition Surface tenstion

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Understanding the Fission Barrier COULOMB ENERGY The Coulomb energy has the opposite tendency. On deformation the charge in the nucleus is less condensed – the electrostatic “blow apart” energy is less Nuclear deformation makes the nucleus MORE BOUND.

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Understanding the Fission Barrier

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The fission barrier on the SEMF To calculate the height of the fission barrier using the SEMF is fairly complex, but can be done as seen in this study – Fig12.3 Dunlap. The dotted lines show variations that are understood on the shell model. Note that the barrier is only small ~3MeV for A>250.

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The Fissionability The Fissionability parameter Z 2 /A as a function of A. Note that the fastest decaying man-made transuranics still have F<45

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The rate of spontaneous fission NOTE log of the decay rate (period) is approximately proportional to the fissionability Z 2 /A

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