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Alpha emission paradox says that an alpha particle with energy ~4 MeV is able to "come out" of the nucleus of U-238 but an alpha particle of energy ~9 MeV is unable to penetrate the Coulomb barrier of the nucleus. My question is, if the charge on the nucleus is positive then the barrier which the 4 MeV alpha particle has to overcome is the nuclear force and not the Coulomb repulsion barrier of the nucleus as the nucleus anyways would repel the alpha particle making it more possible to come out. So why in Gamow's theory we take the barrier as the Coulomb barrier for the 4 MeV alpha particle?

Edit: The potential barrier used to solve the problem is that of Coulomb potential and not of nuclear potential. Coulomb potential barrier makes sense if we are considering the outer alpha particle but when we consider the inner one, Coulomb repulsive potential just makes it easier for the inner one to come out. So why we are considering Coulomb potential barrier for the inner alpha particle?

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The potential barrier keeping an alpha inside a nucleus is on the order of 25 MeV (the work required to bring it from infinity into the range of the strong nuclear force), while the alpha's inside are zipping around with 4 MeV (up to 9 MeV) and simply cannot get over the hump. However; they do strike it $10^{21}$ times per second, and each time have a small probability of tunneling out. If the half life is million to billions of years, that's a lot of strikes, but finally: the alpha tunnels with 4-9MeV.

An external alpha particle with 9 MeV has no hope of getting near the 25 MeV peak, and, although it can tunnel, at what density will it strike the nucleus at $10^{21}$ Hz...in which case it could still take billions of years to tunnel?

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    $\begingroup$ No I meant, the potential barrier used to solve the problem is that of Coulomb potential and not of nuclear potential. Coulomb potential barrier makes sense if we are considering the outer alpha particle but when we consider the inner one, Coulomb repulsive potential just makes it easier for the inner one to come out. So why we are considering Coulomb potential barrier for the inner alpha particle? $\endgroup$ – Korra Apr 2 at 5:01
  • $\begingroup$ The Coulomb potential is 0 at infinity and rises to 25 MeV (-ish) at the point a pion reaches out and grabs it (QHD scale). Likewise, the inner alpha has 25 MeV of glue pulling it in until that fades and the + charge sends it out. The latter never happens, as quantum tunneling makes low energy decay possible. $\endgroup$ – JEB Apr 2 at 18:31
  • $\begingroup$ I still didn't understand, sorry. When the force acting on the inner alpha is nuclear and outer alpha is Coulomb, how can we have the potential barrier to be Coulomb potential? Shouldn't both the nuclear and Coulomb potentials be considered? $\endgroup$ – Korra Apr 3 at 8:14

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