P460 - Quan. Stats. III1 Nuclei Protons and neutrons in nuclei separately fill their energy levels: 1s, 1p, 1d, 2s, 2p, 2d, 3s…………… (we’ll see in 461 their.
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P460 - Quan. Stats. III1 Nuclei Protons and neutrons in nuclei separately fill their energy levels: 1s, 1p, 1d, 2s, 2p, 2d, 3s…………… (we’ll see in 461 their ordering and split by total J) often easier to analyze as 2 Fermi gases of (mostly) non-interacting particles density ~1/fm 3 slightly higher for neutrons large A 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
P460 - Quan. Stats. III2 Nuclei gives Fermi momentum ~same for all except H 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.
P460 - Quan. Stats. III3 Nuclei: Fermi 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. In SN1987, most observed events were from antineutrinos (or off electrons) even though (I think) 1000 times more neutrinos. Detectors were water…..
P460 - Quan. Stats. III4 Fermi Gases in Stars Equilibrium: balance between gravitational pressure and “gas” (either normal or degenerate) pressure total gravitational Energy: density varies in normal stars (in Sun: average is 1 g/cm 3 but at r=0 is 100 g/cm 3 ). More of a constant in white dwarves or neutron stars will have either “normal” gas pressure of P=nkT (P=n ) or pressure due to degenerate particles. Normal depends on T, degenerate (mostly) doesn’t n = particle density in this case
P460 - Quan. Stats. III5 Degenerate Fermi Gas Pressure Start with p = n non-relativistic relativistic P depends ONLY on density Pressure decreases if, for a given density, particles become relativistic
P460 - Quan. Stats. III6 Older Sun-like Stars Density of core increases as H-->He. He inert (no fusion yet). Core contracts electrons become degenerate. 4 e per He nuclei. Electrons have longer wavelength than He electrons move to higher energy due to Pauli exclusion/degeneracy. No longer in thermal equilibrium with p, He nuclei pressure becomes dominated by electrons. No longer depends on T allows T of p,He to increase rapidly without “normal” increase in pressure and change in star’s equilibrium. Onset of 3He->C fusion and Red Giant phase (helium flash when T = 100,000,000 K)
P460 - Quan. Stats. III7 White Dwarves Leftover cores of Red Giants made (usually) from C + O nuclei and degenerate electrons cores of very massive stars are Fe nuclei plus degenerate electrons and have similar properties gravitational pressure balanced by electrons’ pressure which increases as radius decreases ---> radius depends on Mass of star Determine approximate Fermi Energy. Assume electron density = 0.5(p+n) density electrons are in this range and often not completely relativistic or non-relativistic---> need to use the correct E 2 = p 2 + m 2 relationship
P460 - Quan. Stats. III8 White Dwarves + Collapse If the electron energy is > about 1.4 MeV can have: any electrons > E T “disappear”. The electron energy distribution depends on T (average E) the “lost” electrons cause the pressure from the degenerate electrons to decrease the energy of the neutrinos is also lost as they escape ---> “cools” the star as the mass increases, radius decreases, and number of electrons above threshold increases #e’s E F E T
P460 - Quan. Stats. III9 White Dwarves+Supernovas another process - photodisentegration - also abosrbs energy “cooling” star. Similar energy loss as e+p combination At some point the not very stable equilibrium between gravity and (mostly) electron pressure doesn’t hold White Dwarf collapses and some fraction (20-50% ??) of the protons convert to neutrons during the collapse gives Supernovas
P460 - Quan. Stats. III10 Neutron Stars-approx. numbers Supernovas can produce neutron stars - radius ~ 10 km - mass about that of Sun. always < 3 mass Sun - relative n:p:e ~ 99:1:1 gravity supported by degenerate neutrons plug into non-relativistic formula for Fermi Energy -----> 140 MeV (as mass =940 MeV, non-rel OK) look at wavelength can determine radius vs mass (like WD) can collapse into black hole
P460 - Quan. Stats. III11 Neutron Stars 3 separate Fermi gases: n:p:e p+n are in the same potential well due to strong nuclear force assume independent and that p/n = 0.01 (depends on star’s mass) need to use relativistic for electrons but not independent as p n plus reactions with virtual particles free neutrons decay. But in a neutron star they can only do so if there is an available unfilled electron state. So suppresses decay
P460 - Quan. Stats. III12 Neutron Stars Will end up with an equilibrium between n-p-e which can best be seen by matching up the Fermi energy of the neutrons with the e-p system neutrons with E > E F can then decay to p-e-nu (which raises electron density and its Fermi energy thus the balance) need to include rest mass energies. Also density of electrons is equal to that of protons can then solve for p/n ratio (we’ll skip algebra) gives for typical neutron star: