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Nuclear Energy Slide Nuclear Forces • Opposite charges attract, like charges repel • Hydrogen has one proton and no neutrons • Everything else has more than one proton • Why don’t the protons repel each other? • If r is the spacing between two protons, the respective electrostatic repulsion force F between the two protons and the stored electrostatic energy U are 2 F e 2 4 0 r e 2 U 4 0 r What is r ? • It is difficult to determine nuclear size using scattering by charged particles because long range Coloumb force dominates (like trying to determine diameter of sun using orbit of a comet) • Need some head-on collisions • Rutherford shot alpha particles at aluminum and obtained some 1800 scattering • At instant of direction reversal, the alpha particle must be stationary and all kinetic energy is converted into potential energy • This gives an upper bound for nuclear radius Estimate upper bound on aluminum nucleus radius based on backscatter of 7.7 MeV alpha particle 2 Z Z Ale U 4 0 r U 7. 7 MeV Z 2 Z Al 13 19 Z Z e 2 2 13 1. 6 10 r Al 4. 9 10 15 m 12 6 4 0 U 4 8. 8 10 7. 7 10 Determining size of nucleus using neutron scattering • Much easier, since no Coulomb interaction (no electrostatic repulsion) • Simply use R 1 R0 nd • Neutrons either hit or miss (no grazing) • These give a nuclear radius scaling as 1/3 r r0A 15 where r0 1.37 10 m 1/3 Implications of r r0A • Volume of a nucleus scales as A • Densities of all nuclei are about the same • Nuclear mass density is Am 27 3 1u 1.66043 10 1. 5 1017 kg m 4 r 3 4 15 3 3 3 1.37 10 Or more picturesquely, the nuclear density is about 150 thousand metric tons per cubic mm like the mass of an ocean liner crammed into the head of a pin Nuclear force • Attracts protons and neutrons together • Balances proton electrostatic repulsion • Short range, dies out very quickly beyond nuclear radius • Protons and neutrons in nucleus described by quantum mechanics in manner similar to electrons in an atom • e.g. shell structure, Pauli exclusion principle hold • alpha particle is like a completed K shell (2 protons, 2 neutrons, one each spin up, spin down) and so is stable Nuclear force as a generic potential well Nucleon trapped In potential well L Nominal dimension Classical energy 2 2 E 1 mv 2 m v p 2 2m 2m Quantum mechanics: quantize momentum p k n where k n /nL Quantized energy for L 2r0 p 2 2 2 2 E n n 2m 2 8mr0 342 2 2 1. 05 10 n 8 1. 6 10 27 2 1. 37 10 15 2 1. 1 10 12 n 2 Joules 7n 2 MeV Extremely small size of nucleus implies quantized energy levels are in MeV !!! Coulomb Barrier • Potential well seen by protons and neutrons differs • Neutrons just see nuclear force (strong force) • Protons also see electrostatic repulsive force due to other protons • Nuclear force is short range, electrostatic force is long range Coulomb Barrier, cont’d Potential seen by a neutron Neutron outside well feels no force, can wander in Neutron trapped In potential well L Well is due to attractive nuclear force from all other neutrons, protons (short range) Coulomb Barrier, cont’d Proton outside well does not feel short range nuclear force Ridge (lip) due to repulsive but does feel repulsive electrostatic force is called Coulomb barrier force due to protons inside nucleus Proton trapped In potential well L Nominal dimension Well is due to attractive nuclear force from all other neutrons, protons (short range) Magnitude of nuclear energy • If nuclear force balances proton repulsion, then stored energy U per pair of protons must be of the order of the electrostatic energy of two protons separated by r0 2 e 13 U 1.78 10 J 1 MeV 4 0r0 • Any substantial change in configuration of protons, neutrons in nucleus will generally involve releasing or absorbing energies of the order of MeV Compare to energy of an electron orbiting in an atom • The distance is of the order of the Bohr radius so the order of magnitude of the energy for an electron orbiting an atom is e2 Uatom 30 eV 4 0rBohr • Chemical processes involve rearranging electron orbitals and so are an order of magnitude smaller, a few eV or less Cross-section can also be used to characterize rate of a process • e. g. can define a cross-section for fission, or a cross-section for neutron capture • Cross-section is usually given in “barns” where 1 barn =10-24 cm2 • “like hitting a barn wall” Reaction-rate cross section • Let R0 be incident number of particles per second that can instigate a process in target particles having density n in a sheet of thickness d • Let Rprocess be the rate of the process in question • Define the cross-section for the process as Rprocess 1 process R0 nd Number of protons and neutrons as function of atomic mass • Protons repel each other • Nuclear force is “glue” between protons and neutrons that holds the nucleus together • Use “glue” analogy because glue is a short range force Electrostatic potential energy In a sphere of charge, Gauss’s law shows that the charge behaves as if it is all at the center. The radial electric field of a sphere of radius r with Z charges is Ze Er 2 4 0r The repulsive radial force on each of these charges is Ze2 Fr 2 4 0r Here e 1. 6 1019 is the charge on a proton 12 and 0 8. 854 10 is the permittivity of vacuum The work done per charge on bringing all the charges in from infinity to surface of a sphere of radius r is r U Frdr per charge r 2 Ze dr 2 4 0r Ze2 r 4 0r Ze2 4 0r The work done on all the charges for doing this is Z2 e2 U 4 0r This is the electrostatic potential energy of Z protons grouped together on surface of a sphere of radius r. If charges are uniformly distributed in the volume of the sphere, then find (HW) that 3 Z2 e2 U 5 4 0r Protons on surface of a sphere Z2e 2 U 4 0r Protons filling up volume of a sphere 3 Z2e 2 U 5 4 0r 3 Z2 e2 U 5 4 0r The nuclear radius scaled as 1/3 r r0A and since Z A/2 the electrostatic potential energy of nuclei scales with A as 2 2 A/22 e2 U 3 Z e 3 A5/3 1/3 5 4 0r 5 4 0r 0A and so the electrostatic energy per nucleon scales as U A2/3 A Thus, proportionally more neutrons are required to hold together nuclei with large A General structure of nuclide chart • Average nuclear force between neutrons and protons is about twice as much as force between two neutrons or two protons • Implies binding energy ought to be largest if there are equal numbers of neutrons and protons • But, when there are lots of protons their mutual electrostatic repulsion becomes important, so need extra neutrons • Large stable nuclides are neutron rich Nuclide Chart Trends Proton rich Neutron rich Heavy elements are progressively more neutron rich Z=N Light nucleii have approximately the same number of neutrons as protons Heavy nuclei have about 1. 5 the number of neutrons as protons Breaking up a heavy nucleus results in neutron-rich light nuclei Thermal fission A low energy neutron becomes attached to a heavy nuclide having an odd value of A. The nuclide resonates like a jiggled water droplet. It develops a non-spherical shape. The long-range electrostatic repulsion force is only slightly reduced by the non-spherical shape. The short-range attractive nuclear force is greatly reduced by the non-spherical shape. The repulsive electrostatic force wins and the nucleus splits in two (fission). neutron Nucleus with odd A Nucleus with even A + + + + + + + + + + + + + + + + + + Vibrating nucleus + + + + + + + + +++++ +++++ +++++ +++++ +++++ +++++ Electrostatic repulsion +++++ +++++ +++++ +++++ +++++ +++++ Uranium fission example First neutron hits U-235 and excites it 235 236 92 U n 92 U where the asterisk means excited state (can vibrate) 144 89 Suppose the U-236 splits into 56 Ba, 36Kr and 3 neutrons Using 1/3 r r0A 15 where r0 1. 37 10 m the radii of the Barium and Krypton are 15 1/3 15 rBa 1. 37 10 144 7. 18 10 m 15 1/3 15 rKr 1. 37 10 89 6. 12 10 m If we consider the Barium and Krypton as two spheres just touching each other, the distance between the centers of the spheres is r rBa rKr and so the potential energy of the spheres is 2 ZBaZKre UBa,Kr 4 0r 56 36 1. 6 10 19 4 8. 8 1012 7. 18 1015 6. 12 1015 220 MeV This gets converted into kinetic energy as the two positive nuclides fly away from each other The numerical value of this estimate is about 25% more than the true value Shows that fission energy is mainly kinetic energy of fragments which comes from electrostatic repulsion +++++ +++++ +++++ +++++ +++++ +++++ Nuclear Fission: A slow neutron collides with U-235, excites it, and then it breaks up into fragments, including a few neutrons Light nucleii have approximately the same number of neutrons as protons Heavy nuclei have about 1. 5 the number of neutrons as protons Breaking up a heavy nucleus results in neutron-rich light nuclei Fission products will also have ~1.5 x number of neutrons as protons i.e., will lie below line of stability for their atomic mass number and so will be radioactive 235 Typical 235U fission U 235U -> 147La + 87Br +n Z=92, N=143 147La Z=57, N=90 ~8 excess neutrons compared to stable 87Br Z=35, N=52 ~7 excess neutrons compared to stable Quantifying fission process • Thus, if we shoot neutrons at a sheet of U-235 with rate R0 and measure the number of reactions per second, we can determine the fission cross-section Rfission 1 fission R0 nd Properties of cross-sections • Cross-sections can depend on velocity of incident particle • There can be different cross-sections for different competing processes • Comparing cross-sections tells what will happen • Cross-sections
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