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Constraints on Nuclear Fusion Reactions

Constraints on Nuclear Fusion Reactions. Binary reactions dominate Energy Conservation (Energy release) Charge conservation (d= 2 H nucleus) pp  d+e + +Energy +?? Angular momentum conservation Proton, electron spin = (1/2)[h/(2π)] Deuteron spin = h/(2π) Neutrino predicted by E.Fermi

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Constraints on Nuclear Fusion Reactions

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  1. Constraints on Nuclear Fusion Reactions • Binary reactions dominate • Energy Conservation (Energy release) • Charge conservation (d=2H nucleus) • pp  d+e++Energy +?? • Angular momentum conservation • Proton, electron spin = (1/2)[h/(2π)] • Deuteron spin = h/(2π) • Neutrino predicted by E.Fermi • pp  d+e++ +Energy • Detected in 1950s by Reines and Cowen next to a nuclear reactors • Weak Interactions (lepton number conservation) • n  p + e+ anti-. Half Life 12,33 year • Weak interactions are Weak  Reaction Rates are slow

  2. Light Nuclei, http://www.nndc.bnl.gov/chart/ No stable A=5 nuclei

  3. pp chains in Sun 31% 69% 99.7% (pp I) 0.3% (pp II) (pp III)

  4. pp chainsH+H+H+H4He+2e+2e++2 First Step • pp  D + e+ +  + MeV • Weak interaction process • D=(pn) = nucleus of 2H. • Neutron half life = 10.23 min • Suppressed by pp Coulomb repulsion • Quantum Tunneling required • Protons are waves • If Kinetic energy = E-U > 0= h/p • If Kinetic energy = E-U < 0 • Wave is exponentially damped exp(-r) = [U-E]1/2.

  5. Billiard Ball model of pp collisions • Proton density , Temperature T • Velocity distribution (Maxwell-Boltzman) • P(v;T) = exp(-mv2/(2KT)) / [2πkT]3/2. • <mv2/2>= <mvx2/2> +<mvy2/2> +<mvz2/2> =3kT/2 • vrms ≈ [kT/m]1/2. • Collision rate per unit volume: R. • ≈ (10-15m)2. • Rate R ≈ 2v|≈ 2 [kT/m]1/2. Volume = vt

  6. Ballistic Burn Time of Sun • Number density of sun = /MH. • Average = 8 1029 /m3. • Central = 1032 /m3. • Temperature, rms Velocity vrms ≈ [kT/mc2]1/2 • Surface kT = 0.5 eV. vrms ≈ 7000 m/s • Central kT = 1KeV. vrms ≈ 3 105m/s • Proton Mass = 0.9 GeV • Time for all protons to collide / R = /[|v|] • Surface 1/[|v|] ≈ 10-4 s • Center ≈ 30 ns • Crucial role of Coulomb Repulsion and Weak interactions to regulate nuclear fusion rate in sun

  7. pp fusion slowed by coulomb repulsion • Kinetic energy required to bring two protons to within 1 fm • e2/(4π0 r) = 1.4 MeV • Center of sun kT ≈ 1 KeV • Maxwell Boltzmann distribution • P(v) = exp{-[mv2/(2kT)]} • P[-(1.4MeV)/(1 KeV)] ≈ exp(-1000) • Classical pp Collision rate in sun ≈ 0 • No pp fusion, even worse for heavier elements • Quantum barrier penetration

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