P461 - Nuclei I1 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 I2 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 I3 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 p n n p 

P461 - Nuclei I4 Nuclear Sizes and Densities Use e + A  e + A scattering completely EM p e = 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, K L ) in strong interactions to give n,p distributions Find density ~same for all but the lowest A nucleii

P461 - Nuclei I5 Nuclear Densities can write density as an energy density Note Quark-Gluon Plasma occurs if

P461 - Nuclei I6 Nuclear Densities

P461 - Nuclei I7 Nuclear Densities

P461 - Nuclei I8 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 Z 2 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 I9 Semiempirical Mass Formula M(Z,A)=f0 + f1 +f2 + f3 + f4 + f5 f0 = m p Z + m n (A-Z) mass of constituents f1 = -a 1 A A ~ volume  binding energy/nucleon f2 = +a 2 A 2/3 surface area. If on surface, fewer neighbors and less binding energy f3 = +a 3 Z 2 /A 1/3 Coulomb repulsion ~ 1/r f4 = +a 4 (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) = a 5 A -.5 want to pair terms (up+down) so nuclear spin = 0 Binding energy from term f1-f5. Find the constants (a i ’s) by fitting the measured nuclei masses

P461 - Nuclei I10 Semiempirical Mass Formula the larger the binding energy E b, 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. A E b =  E/A volume surface Coulomb N/Z asymmetry Total

P461 - Nuclei I11 synthesis.com/webbook/33_segre/segre.html Number of neutrons number of protons most stable (valley)

P461 - Nuclei I12 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 U 235. U 236 likelier to be in an excited state.

P461 - Nuclei I13 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 I14 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 = E F + B Start filling up states in Fermi sea (separate for p/n) Scattering inhibited  1’ + 2’ as states 1’ and 2’ must be in unfilled states  nucleons are quasifree vs (ignore Coulomb) n p V B

P461 - Nuclei I15 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 I16 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 I17 Nuclei:Fermi motion solid lines are modified Fermi gas calculation (tails due to interactions) electron energy loss

P461 - Nuclei I18 n in C nucleus

P461 - Nuclei I19 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 I20 C Fe Physics Reports 1972 C.H. Llewellen-Smith Fermi gas “shell” model includes spin effects energy transfer 1-Suppression factor

P461 - Nuclei I21 Nuclei: Fermi Suppression and Pauli Exclusion important for neutrino energies less than 1 GeV. prevents accurate measurement of nuetrino energy in detector