Constraints on Nuclear Fusion Reactions

1
Constraints on Nuclear Fusion Reactions

• Binary reactions dominate
• Energy Conservation (Energy release)
• Charge conservation (d2H nucleus)
• pp ? deEnergy ??
• Angular momentum conservation
• Proton, electron spin (1/2)h/(2p)
• Deuteron spin h/(2p)
• Neutrino predicted by E.Fermi
• pp ? de ??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 A5 nuclei
3
pp chains in Sun
31
69
99.7
(pp I)
0.3
(pp II)
(pp III)
4
pp chainsHHHH?4He2e?2e2?

• 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 gt 0?? h/p
• If Kinetic energy E-U lt 0
• Wave is exponentially damped exp(-?r)?
  U-E1/2.

5
Billiard Ball model of pp collisions

• Proton density ?, Temperature T
• Velocity distribution (Maxwell-Boltzman)
• P(vT) exp(-mv2/(2KT)) / 2pkT3/2.
• ltmv2/2gt ltmvx2/2gt ltmvy2/2gt ltmvz2/2gt 3kT/2
• vrms kT/m1/2.
• Collision rate per unit volume R.
• ?? (10-15m)2.
• Rate R ?2??v ?2? kT/m1/2.

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/mc21/2
• Surface kT 0.5 eV. vrms 7000 m/s
• Central kT 1KeV. vrms 3 ?105 m/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/(4p?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