This is a very difficult question to answer. There are (at least) two reasons. First, we have detailed, numerically exact wave functions for stable, light nuclei only up to, just recently, $A=12$ (like $^{12}C$). The Argonne-Los Alamos-Urbana collaboration uses quantum Monte Carlo (QMC) techniques to evaluate the ground and excited states of *bound* nucleons (*ie.* nuclear states that are $L^2$ normalized). And the fact that the QMC methods look at only bound states indicates the second reason: we're really interested in the eigenstates of the nuclear system in the continuum -- that is, the *scattering* states. This is a much tougher problem than evaluating the energy of the bound states, whose nucleons range over effectively bounded regions, since we have to do an integral over and infinite region numerically; or be clever and figure out an equivalent, finite-region problem. (There's been some recent work in this direction by Bob Wiringa and Ken Nollett, building on earlier work by Kievsky and collaborators -- check the preprint archive for their recent work.) So although we know a little about the wave function for $^{12}C$, the $A=12$ scattering problem is something we're just starting to learn more about.

Before talking about an alternative to bound QMC for describing (ie., parametrizing, not solving) the scattering states, let me digress on the issue of the meaning of *ab initio* solutions of many-body quantum mechanical problems. Basically, even if you solve the problem exactly, unless you're very lucky (and smart) and identify a single (or very few) physical mechanism(s) (usually a collective phenomenon like GDR, pairing, etc.) that's particularly relevant for the experimental observation you're trying to describe, you're probably not going to have a great one-line, 'take-home' message that says, "The reason that $DD$ and $DT$ don't have as narrow a resonance as $p^{11}B$ is XXX." The answer to your question would require: 1) very good wave functions and 2) a concomitant study of the 2-,3-,...,?-body correlation functions in the $A=4$, $5$, and $12$ problems (with, of course, the right quantum numbers). Even then you might not identify an 'smoking gun' mechanism that says, "Here, look, that's why $A=4$ and $5$ don't show the narrow peak that $A=12$ has." But you might...

One way, alternative to QMC, that we have of studying/describing/parametrizing the reactions of light nuclei, that doesn't assume that the states are bound, is Wigner's $R$ matrix. (There are *ab initio* methods like the resonating group method and no-core shell model, too.) You can find a lot of literature through google scholar. But the basic idea is that one (artificially, if you will) separates the scattering problem into 'internal' and 'external' regions. The internal region is hard to solve -- all the dynamics of the interacting nucleons, when they're all close together (the "compound nucleus"), are at play. The external region is easy to solve: one ignores 'polarizing' (ie., non-Coulomb) forces (because they're small). The complicated hypersurface in the $3A-3$ dimensional space that separates the internal from the external is called the channel surface. We generally assume a sharp, simple form for this surface that roughly corresponds to (though is usually significantly less than) the distance between the target and projectile (or daughter particles) and parametrized by the "channel radii", $R_c$. (We only consider two-body channels -- a limitation of the method.)

Now, the wave functions are known in the external region (just appropriate sums of regular and irregular spherical Bessel functions modified by Coulomb phases if it's a charged channel reaction). Inside, however, we describe the system by Wigner's $R$ matrix:
\begin{align}
R_{c'c} &= \sum_{\lambda=1}^\infty \frac{\gamma_{\lambda c'}\gamma_{\lambda c}}{E_\lambda-E},
\end{align}
which you might recognize as the Green's function in the presence of some boundary conditions (Wigner's insight gave a particularly useful, simple condition) at the channel radii. The $R$ matrix is a meromorphic function of the energy, $E$ and depends on an infinite number of levels, $E_\lambda$ corresponding to the eigenstates of the Schr\"{o}dinger equation in the finite cavity (with given, Wigner-type BC's). The reduced widths (basically the fractional parentage coefficients of the bound, compound system as it "decays" to particular channels $c,c'$) are related (in a very complicated way) to the partial widths of the compound nucleus. In sum, the $R$ matrix makes an almost intractable problem a little easier.

So, what's my point? You can calculate or parametrize the $R$ matrix, then derive the $T$ (transition) (or $S$, scattering) matrix from it and find its poles near the physical region. This will tell you where the resonances are. And this procedure will give you insight into why a particular compound nucleus has resonance at a particular energy. If there is a "strong" (ie., small reduced width) $R$-matrix level at a particular energy, you can learn what the relevant (LS) quantum numbers of that level are.

The next step in the program is to ask: what type of 2-, 3-, ..., ?-body correlations/forces give rise to strong interaction in this LS channel-state? This, unfortunately, is a much more difficult question to answer and, incidentally, occupies a good part of my "free-time" as this is the type of research that I'm currently working on.

And I'm pretty sure that we have a ways to go to answer it.

This post imported from StackExchange Physics at 2014-07-13 04:40 (UCT), posted by SE-user MarkWayne