Properties of Nuclei

1. Properties of Nuclei • Z protons and N neutrons held together with a short-ranged force  gives binding energy • P and n made from quarks. Most of the mass due to the strong interactions binding them together. Recent JLAB results show masses inside nucleus might be slightly smaller than free particles • P and n are about 1 Fermi in size and the strong force doesn’t compress. Size ~ range of strong force  all nuclei have the same density and higher A nuclei are bigger (unlike atoms) P461 - Nuclei I

2. Protons vs Neutron • neutron slightly heavier than proton and so it decays. No reason “why” just observation • quark content: n = udd and p = uud (plus g, qqbar) Mass up and down quarks 5-10 Mev • three generations of quarks. Only top quark ever observed as “bare” quark. Somehow up quark seems to be slightly lighter than down quark P461 - Nuclei I

3. Nuclei Force • Strong force binds together nucleons • Strong force nominally carried by gluons. But internucleons carried by pions (quark-antiquark bound states) as effective range too large for gluons • Each p/n surrounded by virtual pions. Strong force identical p-p, p-n, n-n (except for symmetry/Pauli exclusion effects) • Range of 1 F due to pion mass n p n p p P461 - Nuclei I

4. Nuclear Sizes and Densities • Use e + A  e + A scattering completely EM • pe = 1000 MeV/c  wavelength = 1.2 F now JLAB, in 60s/70s SLAC up to 20 GeV( mapped out quarks) • Measurement of angular dependence of cross section gives charge distribution (Fourier transform) • Can also scatter neutral particles (n, KL) in strong interactions to give n,p distributions • Find density ~same for all but the lowest A nucleii P461 - Nuclei I

5. Nuclear Densities • can write density as an energy density • Note Quark-Gluon Plasma occurs if P461 - Nuclei I

6. Nuclear Densities P461 - Nuclei I

7. Nuclear Densities P461 - Nuclei I

8. P461 Model of Nuclei • “billiard ball” or “liquid drop” • Adjacent nucleons have force between them but not “permanent” (like a liquid). Gives total attractive energy proportional to A (the volume) – a surface term (liquid drop) • Repulsive electromagnetic force between protons grows as Z2 • Gives semi-empirical mass formula whose terms can be found by fitting observed masses • Pauli exclusion as spin ½  two (interacting) Fermi gases which can be used to model energy and momentum density of states • Potential well is mostly spherically symmetric so quantum states with J/L/S have good quantum numbers. The radial part is different than H but partially solvable  shell model of valence states and nuclear spins P461 - Nuclei I

9. Semiempirical Mass Formula • M(Z,A)=f0 + f1 +f2 + f3 + f4 + f5 • f0 = mp Z + mn (A-Z) mass of constituents • f1 = -a1A A ~ volume  binding energy/nucleon • f2 = +a2A2/3 surface area. If on surface, fewer neighbors and less binding energy • f3 = +a3Z2/A1/3 Coulomb repulsion ~ 1/r • f4 = +a4(Z-A/2)2/2 ad hoc term. Fermi gas gives equal filling of n, p levels • f5 = -f(A) Z, N both even = 0 Z even, N odd or Z odd, N even = +f(A) Z., N both odd f(A) = a5A-.5 want to pair terms (up+down) so nuclear spin = 0 • Binding energy from term f1-f5. Find the constants (ai’s) by fitting the measured nuclei masses P461 - Nuclei I

10. Semiempirical Mass Formula volume surface Coulomb Eb= DE/A • the larger the binding energy Eb, the greater the stability. Iron is the most stable • can fit for terms • good for making quick calculations; understanding a small region of the nuclides. Total N/Z asymmetry A P461 - Nuclei I

11. www.meta-synthesis.com/webbook/33_segre/segre.html most stable (valley) number of protons Number of neutrons P461 - Nuclei I

12. Semiempirical Mass • the “f5” term is a paring term. For nuclei near U there is about a 0.7 MeV difference between having both n and p paired up (even A), odd A (and so one unpaired), and another 0.7 MeV for neither n or p being paired spin (even A) • so ~5.9 MeV from binding of extra n plus 0.7 MeV from magnetic coupling • easier for neutron capture to cause a fission in U235. U236 likelier to be in an excited state. P461 - Nuclei I

13. Fermi Gas Model • p,n spin ½ form two Fermi gases of indistinguishable particles  p n through beta decays (like neutron stars) and p/n ratio due to matching Fermi energy • In finite 3D well with radius of nucleus. Familiar: • Fermi energy from density and N/A=0.6 • Slightly lower proton density but shifted due to electromagnetic repulsion P461 - Nuclei I

14. Fermi Gas Model II • V = depth of well = F(A) ~ 50 MeV • Fermi energy same for all nuclei as density = constant • Binding energy B = energy to remove p/n from top of well ~ 7-10 MeV V = EF + B • Start filling up states in Fermi sea (separate for p/n) • Scattering inhibited 1 + 2  1’ + 2’ as states 1’ and 2’ must be in unfilled states  nucleons are quasifree B vs (ignore Coulomb) V n p n p P461 - Nuclei I

15. Nuclei • If ignore Coulomb repulsion, as n<->p through beta decay, lowest energy will have N=Z (gives (N-Z) term in mass formula) • proton shifted higher due to Coulomb repulsion. Both p,n fill to top with p<->n coupled by Weak interactions so both at ~same level (Fermi energy for p impacted by n) n p P461 - Nuclei I

16. Nuclei: Fermi motion • if p,n were motionless, then the energy thresholds for some neutrino interactions are: • but Fermi momentum allows reactions to occur at lower neutrino energy. dN/dp p P461 - Nuclei I

17. Nuclei:Fermi motion electron energy loss solid lines are modified Fermi gas calculation (tails due to interactions) P461 - Nuclei I

18. n in C nucleus P461 - Nuclei I

19. Nuclei:Pauli Suppression • But also have filled energy levels and need to give enough energy to p/n so that there is an unfilled state available. Simplest to say “above” Fermi Energy • similar effect in solids. Superconductivity mostly involves electrons at the “top” of the Fermi well • at low energy transfers (<40 MeV) only some p/n will be able to change states. Those at “top” of well. • Gives different cross section off free protons than off of bound protons. Suppression at low energy transfers if target is Carbon, Oxygen, Iron... • In SN1987, most observed events were from antineutrinos (or off electrons) even though (I think) 1000 times more neutrinos. Detectors were water….. P461 - Nuclei I

20. Physics Reports 1972 C.H. Llewellen-Smith C 1-Suppression factor Fermi gas “shell” model includes spin effects Fe energy transfer P461 - Nuclei I

21. Nuclei: Fermi Suppression and Pauli Exclusion • important for neutrino energies less than 1 GeV. prevents accurate measurement of nuetrino energy in detector P461 - Nuclei I