1. Exploring the decay of 16Be in a three-body model
Amy Lovell
Michigan State University and
National Superconducting Cyclotron Laboratory
In collaboration with:
Filomena Nunes (MSU/NSCL)
Ian Thompson (LLNL)
Thank you.
ECT* Workshop on Unbound Nuclei and Continuum States
October 21, 2016

2. Exotic systems are found in light nuclei
Much effort has gone into studying systems up to the dripline, but now we want to explore beyond it 1H 2H 7H 3H 6H 5H 4H
3He
4He
9He
5He
8He
7He
6He
6Li
7Li
12Li
8Li
11Li
10Li
9Li
3Li
4Li
5Li
9Be
10Be
15Be
11Be
14Be
13Be
12Be
7Be
8Be
5Be
10He
13Li
16Be
10B
11B
16B
12B
17B
14B
13B 8B 6B
15B
18B
19B
20B
21B
Borromean System
Neutron Halo
Proton Halo
Stable System
Neutron drip line
6Be 7B 9B
Exotic structures are found in light nuclei, especially near the driplines (as shown here for the neutron dripline). These include Borromean systems in which the three-body system is bound compared to each of the two-body systems, one and two proton halos.
I.J. Thompson and F.M. Nunes, Nuclear Reactions for Astrophysics, (Cambridge University Press, Cambridge, 2009)

3. Exotic decay modes across the nuclear chart
45Fe, 48Ni, 54Zn
26S, 30Ar, 34Ca
16Ne, 17Ne, 18Ne, 19Mg
2p decay from ground state
2p decay from excited states
Possible 2p decay candidates
These exotic systems can lead to exotic decay modes, not just in the light isotopes as we looked at before, but in the heavier isotopes as well. This chart lists examples of two proton decays at the limits of stability that have been observed from the ground state or excited states – as well as those that are predicted to be candidates for two proton decay.
6Be, 8B, 8C, 10C, 12O
R. Charity, Two-Proton Decay Experiments

4. Types of two-nucleon decays
Sequential Decay
Three-body Decay
Dinucleon Decay
These two proton (and also two neutron) decays can take place in a variety of ways, shown schematically here, along with a representation of the energies levels in the A, A-1, and A-2 systems. During sequential decay, one nucleon is emitted followed by the second nucleon. In this case, a state in the A-1 system is energetically assessable to decay through. For a three-body decay, the two nucleons are emitted simultaneously but are not correlated in any matter. The two nucleons are also emitted simultaneously in a dineutron or diproton decay, but they are strongly correlated and come out together. These last two are simultaneous decays, and the A-1 state is energetically inaccessible. A
A-2 A
A-2 A
A-1
A-2

5. Proposition of two-proton decay
Sequential 2p decay
Diproton decay
V.I. Goldansky, Nucl. Phys (1960)
Two-proton decay was first theorized in 1960 by Goldansky, and he looked at this spectrum of decay methods from sequential to diproton. So far, several cases of diproton decays have been observed, including those listed here.
Several cases have been observed so far, including:
J. Giovinazzo, et. al., PRL (2002)
M. Pfützner, et. al., Eur. Phys. J A (2002)
B. Blank, et. al., PRL (2005)
I. Mukha, et. al., PRL (2007)
C. Dossat, et. al., PRC (2005)
45Fe, 54Zn, 19Mg, 48Ni

6. First observation of diproton decay
Two independent observations of 45Fe at GANIL and GSI
J. Giovinazzo, et. al., PRL (2002)
M. Pfützner, et. al., Eur. Phys. J A (2002)
The first observation of diproton decay from the ground states was of 45Fe, in two independent observations at GANIL and GSI. Subsequent experiments were able to observe these decays with Time Projection Chambers, where you can see the 45Fe come in and then the two protons decaying simultaneously from it. Comparing observables, such as the opening angle between the two protons, with different structure models of the two valence protons. These correlations can tell us something of the internal structure of the system.
K. Miernik, et. al., PRL (2007)
Optical Time Projection Chamber, NSCL

7. Decay models probe internal structure
Diproton Decay
Three-body Decay
2% p2
24% p2
The decay mode of 45Fe (diproton or three-body) changes based on the underlying composition of the system
The decay mode itself also probes the internal structure (as I’m sure Leonid will talk about). As Leonid Grigorenko has explored (and again, probably discussed), models for different amounts of p^2 give rise to different decay modes being dominant. For example, these plots in terms of the distance between the core and 2p system and the distance between the two protons show that very little p^2 gives rise to a diproton decay being dominant (two protons stay close together as they travel away from the core), which increasing the p^2 component gives rise to a three body decay (three bodies travel away from each other). While a lot of studies have investigated two-proton decays, a lesser amount of work, both experimentally and theoretically, has focused on two neutron decays. This is what I’m going to focus on today.
L.V. Grigorenko and M.V. Zhukov, PRC (2003)

8. 16Be: first observation of dineutron decay
The first observation of dineutron decay was for 16Be in a 2012 experiment at the NSCL. They compared observables, such as the energy of the two neutrons, the opening angle between the two neutrons, and the opening angle between the 14Be core and one of the neutrons, to models for sequential decay (blue), three-body decay (red), and dineutron decay (black) and found that their data was best described by the dineutron decay.
All of these modes can coexist in a full three-body model
A. Spyrou, et. al., PRL (2012)

9. 16Be is an ideal case for dineutron decay1H 2H 7H 3H 6H 5H 4H
3He
4He
9He
5He
8He
7He
6He
6Li
7Li
12Li
8Li
11Li
10Li
9Li
3Li
4Li
5Li
9Be
10Be
15Be
11Be
14Be
13Be
12Be
6Be
7Be
8Be
5Be
10He
13Li
16Be
10B
11B
16B
12B
17B
14B
13B 7B 8B 9B 6B
15B
18B
19B
20B
21B
1.35 MeV
575 keV
14Be
15Be
16Be
1.8 MeV
16Be is an ideal case for dineutron decay. In the previous experiment, the ground state energy was measured to be 1.35 MeV while the ground state energy of 15Be was measured to be 1.8 MeV, which would be energetically inaccessible, depending on the width of that state – which seems to support this proposition.
A. Spyrou, et. al., PRL (2012)
J. Snyder, et. al., PRC (R) (2013)

10. Solving the three-body problem
Jacobi and Hyperspherical Coordinates x
T-basis y
16-body problem
3-body problem x y
Now we can exactly model the degrees of freedom relevant to the decay
We are interested in taking the 16-body problem of 16Be and reducing it to a three-body problem of 14Be core plus two neutrons. This allows us to exactly model the degrees of freedom that are relevant for the decay that we are interested in. We can use Jacobi coordinates to model our system, where x is the distance between two of the bodies and y is the distance between the third body and the center of mass of the first two. With two identical neutrons, this gives us two separate coordinate systems, however, because of the symmetry between the two neutrons, we typically work in the T-system. For reasons that will become apparent in a minute, we work in hyperspherical coordinates. These are defined in terms of rho which gives the overall size of the system and is invariant between the two bases, and theta which describes the relative magnitude of x and y.
Y-basis

11. Hyperspherical harmonics basis expansion
The wave function is now separable
We have an eigenfunction of the angular part of the Hamiltonian
Have a coupled-channel hyper-radial equation to solve for each value of K
We can write the SE in hyperspherical coordinates. What’s nice about these coordinates is that the wave function is now separable into a part that depends on rho and an angular part which is an eigenfunction of the angular part of the Hamiltonian. Then all we are left to solve is a radial equation for each value of the eigenvalue K. Because we’re in the continuum, this has to be done using scattering boundary conditions.
Coupling potential produces strong off-diagonal elements
Using scattering boundary conditions:
I.J. Thompson and F.M. Nunes, Nuclear Reactions for Astrophysics, (Cambridge University Press, Cambridge, 2009)

12. Hyperspherical R-matrix method
Create an orthogonal basis, w, by solving the uncoupled equations in a box – fix the logarithmic derivatives at the boundary
Solve the full coupled problem (in a box) by constructing a basis, g, from a linear combination of the w’s
We solve our radial equation using the hyperpherical R-matrix method. We first create an otherognal basis, w, by solving the uncoupled problem in a box of length a and fixing the logarithmic derivatives at the boundary. To solve the full coupled problem – still in the box – a basis, g, is constructed from a linear combinations of the w’s (as shown here). The full radial solution is constructed as a combination of these g’s, where the A coefficients contain information about the asymptotics.
Construct the full radial part of the wave function through a linear combination of the g’s
I.J. Thompson and F.M. Nunes, Nuclear Reactions for Astrophysics, (Cambridge University Press, Cambridge, 2009)

13. Two- and three-body potentials
1s1/2, 1p3/2, 1p1/2 are assumed to be full and projected out with a supersymmetric transformation
14Be-n interaction to reproduce 15Be
Structure of 15Be
1d5/2
1d3/2
2s1/2
Neutron-neutron interaction
The last piece that we need to talk about before we can solve our problem are the two- and three-body potentials that will be used. We include a 14Be-n interaction that reproduces 15Be. Typically, this would come from experimental studies, however, only one state has been measured in 15Be, so we rely on shell model calculations performed by Alex Brown, shown here. In order to match the experimental result, the states that we’ve included in our calculation are the shell model states lowered by 1 MeV. We include a neutron-neutron interaction (the GPT interaction). As well as a three-body interaction which is used to reproduce the experimental resonance energy – and takes into account the degrees of freedom that are missing in the model.
Three-body interaction reproduces the experimental resonance energy, takes into account the degrees of freedom missing in the model
Shell model (WBP)
This work
Experimental
B.A. Brown, private communication (2013)
E.K. Warburton and B.A. Brown, PRC (1992)
D. Gogny, P. Pires, and R. De Tourreil, Phys. Lett. 32B 7 (1970)
J. Snyder, et. al., PRC (R) (2013)

14. Convergence of the basis expansion
Also need to converge:
Number of Jacobi polynomials for hyperangular discretization
Number of hyperradial basis states, for the R-matrix calculation
Size of the R-matrix box (depends on the number of hyperradial basis states)
Because we are using a basis expansion, we need to make sure that we have included enough terms in our expansion. Here, I show the three-body energy as a function of the max value of K that was included in the calculation. We can see that it converges around K=28. There are a list of other observables that we need to make sure converge, such as the number of Jacobi polynomials for the hyperangular discretization, number of hyperradial basis states for the R-matrix calculation, and the size of the R-matrix box, which dependon the number of hyperradial basis states.

15. Single channel resonance
After converging the system, we can look at single channel phase shifts to get an idea of the three-body resonance energy and single channel width.
K=0, L=0, lx= 0, ly=0
With this method:
Eres=1.858 MeV
Γres=0.20 MeV
E≈1.858 MeV
Experimental:
Eres=1.35 MeV
Γres=0.8 MeV
After convergence, we can look at single channel phase shifts to get an idea of the three-body resonance energy and single channel width. (With a small three-body interaction to help with convergence.) With this three-body potential, we have a resonance energy of 1.65 MeV and a width of approximately 0.2 MeV, compared to the experimental resonance energy of 1.35 MeV and width of We can also check that the resonance energy extracted from this single phase shift is indeed the resonance energy of the full system by calculating the three-body elastic scattering cross section. This peak gives the resonance energy which is in good agreement with the resonance energy extracted from the K=0 phase shift.
The three-body elastic cross section reinforces the location of the resonance by including all channels.

16. Reproducing the experimental resonance energy
K=0, L=0, lx=0, ly=0
Eres (exp) = 1.35 MeV
Total elastic cross section reinforces the resonance position across all channels
V3B = 0.0 MeV
Eres = MeV
Γres=0.20 MeV
E≈1.38 MeV
V3B = MeV
Eres = MeV
Γres=0.17 MeV
E≈1.90 MeV
A mentioned before, we include a three-body interaction to reproduce the experimental resonance energy. Show the K=0 phase shifts for the system without a three-body interaction and with a three-body interaction (WS) with depth of 7.19 MeV which reproduces the experimental resonance energy. We can again look at the elastic scattering cross section to make sure that our single channel phase shift does indeed capture the resonance position across all of the channels.

17. Three-body configuration
A dineutron decay of 16Be was seen experimentally
Eres = MeV
E (off-resonance) = MeV
A dineutron was seen experimentally. We can look at the density distribution calculated from our wave function at the resonance energy to look at the configuration. Here we show the density distribution as function of the distance between the core and the COM of the two neutrons, as well as the distance between the nn. From this, we can see that we have a very strong dineutron configuration, and to a lesser extent, a small helicopter configuration. We can also compare this to a density distribution of the system off-resonance, to make sure this is a feature of the resonance and not the system. This dineutron is very strong, and we can explore what in the internal structure of 16Be is giving rise to this strong component.

18. Removing the neutron-neutron interaction
Eres = 3.14 MeV
E≈1.38 MeV
E≈3.14 MeV
We can remove the nn interaction to see what effect that has. Looking at the three-body elastic cross section, this is a comparison between the resonance that we were just discussing (black) and the calculation without nn interaction (green). There is a slight bump around 3.14 MeV, which is much less pronounced than the previous resonance. Looking at the density distribution for this state, we see a more even split between dineutron and helicopter, with some three-body component as well.

19. Historical importance of low lying s-waves
10Li was predicted to have a p1/2 neutron ground state (shell model)
This did not give rise to a bound 11Li
An s1/2 neutron ground state was needed
Experimental evidence was found for an s-wave neutron ground state
Solid – total velocity spectrum
Dot-dashed – s-wave contributions (a=-30 fm)
Dashed – p-wave with Edecay=538 kev
There had also been some conversation about the possibility of a low-lying s-wave being present in 15Be that was not observed experimentally. Historically, low-lying s-waves have been important, for example in 10Li, where the ground state was predicted to have a p1/2 neutron, but this did not give rise to a bound state in 11Li which had been observed. It was found that an s1/2 neutron ground state was needed, and then experimental evidence was found for this state.
M. Thoennessen, et. al., PRC (1999)

20. Model for a low lying s-wave in 16Be
We can include a low lying 2s1/2 state (0.48 MeV)
The ground state of 16Be is bound
We can include a low-lying s-wave in 15Be (at 0.48 meV) to see what, if any, effect this has on our model. This is again the K=0 phase shift for the full model we discussed earlier (black) compared to the model with the low s-wave (red). It seems like we have a broad state up here around 2 MeV, but if we look at our three-body elastic scattering cross section, we do not see this state appear. Furthermore, it actually turns out that this leads to the ground state of 16Be being bound. Also, as we saw before, this state is not necessary to reproduce the experimental results.

21. Low lying d3/2 state
Kuchera, et. al., PRC (2015)
A repulsive three-body potential would be required to reproduce the experimental resonance energy
Looked for the 1d3/2 state in 15Be using the 3n+12Be channel, less than 11% strength at 2.69 MeV
Eres = 1.06 MeV
V3B = MeV
Eres = MeV
Recently, there was a proposal that there was a small chance of finding the 1d3/2 state in 15Be at 2.69 MeV (compared to where shell model has it around 6 or 7 MeV). We can move this state and see what that does to our 16Be. If we include this state – again showing the K=0 phase shift compared to our full three-body model, we see that it reduces the resonance energy, below the experimental resonance energy (even without a three-body interaction). This would require a repulsive three-body interaction to reproduce the experimental resonance energy, which is highly atypical. Still, if we calculate the density distribution, we still see a strong dineutron configuration. However, the need for a repulsive three-body interaction makes this highly unlikely.
V3B = 0.0 MeV
Eres = 1.06 MeV

22. Summary
Using hyperspherical coordinates and the R-matrix method, we constructed a three-body model for 16Be.
The density distribution favors a dineutron configuration over a helicopter configuration, consistent with experimental results.
The cause of the strong dineutron was investigated by changing the two- body interactions. The nn interaction seems to be the main source of this feature.
The dineutron is robust under changes to the structure of 15Be. However, changes to the level orderings do not reproduce other experimental results (ground state energy).
We’re done!

23. Outlook
Work is ongoing to compute experimental observables (two-body energy, opening angles, etc.) to directly compare to data, by making a connection between the full three-body wave function and the input information for the complex experimental simulation.
Better level schemes need to be developed in order to create more accurate input potentials. An MSU/NSCL group is working on implementing the Gamov basis for the shell model in the p and sd shells.
There are other three-body methods on the market. It is necessary to benchmark our results against these other methods to make sure they are robust, along with benchmarking against full 16-body calculations.

24. Acknowlegements Filomena Nunes and few-body group (MSU/NSCL)
Ian Thompson (LLNL)
Artemis Spyrou, Michael Thoennessen, Simin Wang, and Anthony Kuchera (MSU/NSCL)
iCER and HPCC (MSU)
This work is supported by the Stewardship Science Graduate Fellowship program, which is provided under grant number DE-NA
Let’s thank some people