Quantum Tunneling Minimum Energy

[fiontie]

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!

Homework Statement

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.

Homework Equations

Would be great to know.

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

B = \frac{Zz}{A^{\frac{1}{3}}}

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 D). Given the formula for a rectangular barrier

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

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

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

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

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

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

So finally,

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)}

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!

 

[ben133]

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.