Is the potential barrier of a nucleus related to its binding energy and size?

[sunrah]

I am confused about spontaneous fission. My basic understanding is that like α-decay the tunnel effect is responsible. We have a potential barrier caused by the superposition of surface tension energy and coulomb potential through which an energised nuclear fragment can tunnel with a certain probability.

How does the potential barrier of the nucleus relate to its binding energy? I have read that in reality it is only very large nuclei that undergo spontaneous fission; these nuclei have have lower binding energies per nucleon. How does reduction in B/A effect size of potential barrier? Also in class we have derived the tunneling probability

T \propto e^{-2G}

where G is the Gamov funktion. I have read that G \propto m and that heavier particles/fragments have lower tunneling probability that is why α-decay is much more common than spontaneous fission of heavy nuclei. If so why are heavy nuclei more likely to fission spontaneously than lighter ones?

 

[Crazymechanic]

Not all of the answer you asked for but atleast part of for sure.

https://en.wikipedia.org/wiki/Nuclear_force

http://en.wikipedia.org/wiki/Nuclear_fission

 

[sunrah]

thanks but I have already read those pages enough :)

 

[anorlunda]

Here is a speculation. Of elements heavier than iron, the heavier the element the less binding energy per nucleon.

Of course elements lighter than iron can't spontaneously fission at all.

 

[Crazymechanic]

just to add to what anorlunda already said , heavy nuclei can fission spontaneously and deliberately because they are unstable to begin with , it has to do with the proton neutron number as it gets too big the nuclear force which holds everything together ( like glue) is having a hard time to hold the big structure together and it can break apart.
Small nuclei which have few protons and neutrons ( Hydrogen being the smallest one proton one electron) are very sable , few particles holding strongly together.

 

[sunrah]

thanks for the replies!

so these large nuclei are unstable to begin with because the binding energy per nucleon drops significantly whilst coulomb repulsion increases causing strong oscillations/deformation in the ground state?

also I couldn't find a definite answer to this: is the height of the potential barrier E_f above the ground state equal to the binding energy E_b ? are they one and the same? here is what I mean

http://www.kutl.kyushu-u.ac.jp/seminar/MicroWorld3_E/3Part3_E/3P33_E/fission_barrier_E.jpg

 

[Crazymechanic]

as for the first one the answer is yes.
as for the second i will let some more experienced people comment , but as far as I know you can tell how strongly something was held together by calculating how much force it took you to break it apart , now to my understanding this would also be the case to your question number too and I tend to think the answer should be yes.