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Homework Help: Quantum Tunneling Minimum Energy

  1. May 18, 2010 #1
    Good day, everyone!
    Lately I faced the necessity of solving a problem from a field I know literally nothing about. So I just made an online research but without success. Any help (hints, good sources, relevant equations) would be greatly appreciated!

    1. The problem statement, all variables and given/known data

    Gold nucleus (Au Z=79 A=197) of radius R is bombarded by an alpha particle (He z=2 A=4) of radius r. Find the minimum energy of the alpha particle required to penetrate inside the nucleus.

    2. Relevant equations

    Would be great to know.

    3. The attempt at a solution

    (the following most likely has nothing to do with a solution and is not to be read :) )

    A nucleus and a particle are repulsed by the Coulomb force when colliding, so the problem refers to overcoming the Coulomb barrier. Its approximate height for an arbitrary nucleus is

    [itex] B = \frac{Zz}{A^{\frac{1}{3}}} [/itex]

    What I am missing here is a "minimum energy required to penetrate". In quantum mechanics, the particle with energy far lower than a barrier can still penetrate, isn't it just a matter of possibility? At what point does the penetration become impossible?

    Thus, we can find the possibility of overcoming our barrier (which is its transparency coefficient [itex]D[/itex]). Given the formula for a rectangular barrier

    [itex] D = \exp{\left(-\frac{2}{\hbar} d \sqrt{2m(U-E)}\right)} [/itex]

    where [itex]d[/itex] is a barrier width, [itex]U[/itex] is a barrier height and [itex]E[/itex] is a particle energy, we can get one for an arbitrary barrier by breaking it into thin rectangular stripes and integrating over them:

    [itex] D = \exp{\left(-\frac{2}{\hbar} \int_{x_1}^{x_2} dx \, \sqrt{2m(U-E)} \right)} [/itex]

    Now let's apply this formula to our case. The potential energy of a particle on a distance [itex]r \geq R[/itex] is defined by the energy of a Coulomb interaction (I just stumbled across this formula and am not sure where it comes from):

    [itex] U(r) = E_{C} \frac{R}{r} [/itex]
    [itex] E_0 = m_\alpha c^2[/itex] is a rest energy of an alpha particle

    The barrier boundaries are [itex]R[/itex] and [itex]r_\alpha[/itex], where [itex]r_\alpha = R\frac{E_C}{E_\alpha}[/itex] is a distance where the energy of a particle becomes equal to the energy of a Coulomb repulsion.

    So finally,

    [itex] D = \exp{\left(-\frac{2}{\hbar c} \sqrt{2 E_0} \int_{R}^{r_\alpha} dr \, \sqrt{E_{C} \frac{R}{r} - E_\alpha} \right)} [/itex]

    I'm actually clueless how this can help.
    And I'm also not sure how to make use of an alpha particle size.

    Thanks in advance!
  2. jcsd
  3. May 19, 2010 #2
    Ok, admittedly I stopped reading after the first equation, as I think the rest may be irrelevant (what level of study are you at?).

    I'd see this as a coulomb repulsion problem. Your statement: 'In quantum mechanics, the particle with energy far lower than a barrier can still penetrate, isn't it just a matter of possibility?' I assume is talking about quantum tunnelling, not really applicable in this case. Also QM is a game of probabilities, so try not to go down that route.
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