Re: nuclear reaction coordinates

["John S. Denker" <jsd@MONMOUTH.COM>]

Ludwik Kowalski wrote:
> My recollection is that in order to perform tunneling
> calculations, either for alpha particles, or for other
> nuclear chunks, one must know nuclear potential.

OK

> well
> parameters chosen for alphas gave wrong lambda
> for fission, and vice versa.

> Please share what you know.

The primary thing that I know is that I'm not an expert
in this area.  But, secondarily speaking, I'll share
what little I know.  Long ago in a galaxy far away I
thought about an atomic-physics problem (i.e. chemical
reactions) which might be reasonably analogous to this
sort of nuclear reactions.

I think the key concept here is _reaction coordinate_.
Starting in the 1850s theoretical physicists got really
good at performing change-of-coordinates transformations.
Rectangular -> cylindrical -> spherical coordinates are
elementary examples -- but it gets lots fancier than
that.  For the fission reaction, you might start by
imagining a reaction coordinate W such that for small
W the nucleus is oblate, for slightly larger W the
nucleus is prolate, for larger W it gets really proate
like a dumbbell, and for very large W it is just two
separated fragments.  You then write a potential V(W) as
a function of W.  (The real reaction coordinate must be
even more complicated than I have described.)

For alpha decay, the situation is probably much simpler.
You can (to a first approximation) write a potential V(r)
as a function of r, the distance of the alpha particle from
the rest of the nucleus.

For sure it would be unhelpful to speak of "the" potential
as a function of "the" coordinate.

> But there were other complications as well.  For
> instance, how often two
> protons and two neutrons happen to be bound into
> an alpha particle inside a nucleus?

In some sense, the answer is "pretty often".  An alpha
particle is a pretty favorable configuration of nucleons.

In another sense, it almost doesn't matter.  You can
treat nuclear matter as a collection of N individual
nucleons _or_ as a collection of roughly N/4 alpha
particles (with a few leftovers) by a more-or-less
purely mathematical change of basis.  The point is
that the quantum equations of motion explore every
physically-possible possibility _in parallel_ and if one
of the possibilities involves alpha particles, and tends
to go to completion, that's the one that will go to
completion.

This exploring-every-option-in-parallel behavior is
quite central to quantum mechanics.  Quantum computing
exploits this heavily.