The edge of a nucleus can be roughly modeled as a square potential barrier. An a

Question

The edge of a nucleus can be roughly modeled as a square potential barrier. An alpha particle in an unstable nucleus can be modeled as a particle with a specific energy, bouncing back and forth between these square potential barriers. Each time the alpha particle comes into contact with a potential barrier, there is some probability T that it will tunnel out. Since the tunneling probability is small, it may be approximated by

T=Ge^{-2kappa L},
where

G=16rac{E}{U_0}left(1-rac{E}{U_0} ight)hbox{ and }kappa=rac{sqrt{2m(U_0-E)}}{hbar} ; ,
and where E is the total energy of the alpha particle, U_0 is the potential energy of the barrier, m is the mass of the alpha particle, and hbar is Planck's constant divided by 2pi. This is a complicated expression, but the important part for this problem is simply that T is a constant if we assume that the barrier and alpha particle's energies are both fixed.

Consider a nucleus of radius r and an alpha particle with kinetic energy E (i.e., let the potential energy within the nucleus be zero) and mass m.


Assuming that the alpha particle moves along a diameter of the nucleus and that it moves at low enough speed that relativistic effects are negligible, what is the time tau between successive encounters between each edge of the nucleus and the alpha particle?
Express your answer in terms of E, r, and m.
tau =

Explanation / Answer

2r/(2E/m)^1/2 2r/sqrt2E/m 2r/sqrt2E/m

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