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Bound States and Tunneling

Bound States and Tunneling . State A will be bound with infinite lifetime. State B is bound but can decay to B->B’+X (unbound) with lifetime which depends on barrier height and thickness. Also reaction B’+X->B->A can be analyzed using tunneling

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Bound States and Tunneling

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  1. Bound States and Tunneling • State A will be bound with infinite lifetime. State B is bound but can decay to B->B’+X (unbound) with lifetime which depends on barrier height and thickness. • Also reaction B’+X->B->A can be analyzed using tunneling • tunneling is ~probability for wavefunction to be outside well B’ V B A E=0 0 0 P460 - barriers

  2. Alpha Decay • Example: Th90 -> Ra88 + alpha • Kinetic energy of the alpha = mass difference • have V(r) be Coulomb repulsion outside of nucleus. But attractive (strong) force inside the nucleus. Model alpha decay as alpha particle “trapped” in nucleus which then tunnels its way through the Coulomb barrier • super quick - assume square potential • more accurate - 1/r and integrate P460 - barriers

  3. Alpha Decay A B C • Do better. Use tunneling probability for each dx from square well. Then integrate • as V(r) is known, integral can be calculated. See E+R and Griffiths 8.2) 4pZ=large number P460 - barriers

  4. Alpha Decay • T is the transmission probability per “incident” alpha • f=no. of alphas “striking” the barrier (inside the nucleus) per second = v/2R, If v=0.1c f=1021 Hz Depends strongly on alpha kinetic energy P460 - barriers

  5. Fusion in the Sun • “Classically” two particles have to get within about 1 F for strong interaction reactions to occur • in p-p collisions, need to overcome Coulomb repulsion. If assume classical distance of closest approach Reaction occurs at lower temperature as proton wavefunctions can overlap-same as tunneling P460 - barriers

  6. Fusion in the Sun • Proton can “tunnel” and so the wavefunctions have non-zero probability of overlapping even if DCA>>size of proton To determine energy or fusion reactions need to convolute with Boltzman distribution for a given T P460 - barriers

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