# Three-body calculations of beta decay applied to 11Li

E. Garrido1, A.S. Jensen2, H.O.U. Fynbo2, K. Riisager2 1Instituto de
Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain
2Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C,
Denmark

###### Abstract

A novel practical few-body method is formulated to include isospin symmetry
for nuclear halo structures. The method is designed to describe beta decay,
where the basic concept of isospin symmetry facilitates a proper
understanding. Both isobaric analogue and anti-analogue states are treated. We
derive general and explicit formulas for three-body systems using
hyperspherical coordinates. The example of the beta decaying 11Li
(9Li+$n$+$n$) is chosen as a challenging application for numerical
calculations of practical interest. The detailed results are compared to
existing experimental data and good agreement is found at high excitation
energies, where the isobaric analogue and anti-analogue states are situated in
the daughter nucleus. An interpretation of the decay pattern at lower
excitation energies is suggested. Decays of the 9Li-core and the two halo-
neutrons are individually treated and combined to the daughter system with
almost unique isospin, which we predict to be broken by about $0.4\%$
probability. Properties of decay products are predicted as possible future
tests of this model.

##  I Introduction

Nuclear halo states are characterized by a significant decoupling of the halo
particle(s) from the remainder of the nucleus Jen05 ; Rii13 . Many accounts of
halo nuclei focus on the spatial decoupling of the halo and the core, but the
decoupling of dynamic degrees of freedom is of similar importance. This has
implications for many aspects of their structure and for the dynamics when
undergoing nuclear reactions. The aim of the current paper is to explore how
beta decay of halo nuclei is affected through the use of a few-body model that
enables treatment also of the continuum behaviour.

Most established halo states are ground states of light nuclei in the vicinity
of the neutron or proton driplines. Here beta decay energies are large, and
allowed transitions dominate the decay pattern, see Bla08 ; Pfu12  for general
overviews. Beta decays are a valuable probe of nuclear structure with
operators that for allowed transitions do not involve the spatial coordinates.
The decays of halo nuclei are reviewed in a recent paper Rii22 , and two
specific features are important for the current work. The first is that the
large spatial extension of the halo wave functions can give rise to decays
that are most efficiently described as proceeding directly to continuum
states, as done naturally in few-body models. The second is that the
decoupling between halo and core may lead to distinct patterns in the decay as
discussed in the next section. It should be recalled that the physics
structure is often seen best in transitions of large beta strength that only
rarely correspond to those with high branching ratio.

The nucleus 11Li is not only one of the best studied halo nuclei, but also has
a very diverse break-up pattern during the beta decay process Rii22 , and it
is therefore natural to take a close look at this decay. Furthermore, it has
been difficult to pin down the many decay branches of the nucleus, not only
experimentally, but also from the theoretical point of view. This is connected
to the fact that most of the beta strength in the 11Li decay leads to break-up
into continuum states. For example, with the shell model Suz97 ; Mad09a  or
the antisymmetrized molecular dynamics Kan10 , the calculations are often
restricted to the beta strength distribution in 11Be, and they do not attempt
to describe the subsequent particle emission or the coupling to the continuum.
The only exception is calculations Ohb95 ; Zhu95 ; Bay06  of the beta-delayed
deuteron branch that assume decays direct to the continuum.

In this work, we shall investigate the beta decay of the 11Li nucleus in its
ground state. The calculations will be based on a three-body framework (core
plus two halo nucleons), by means of the hyperspherical adiabatic expansion
method nie01 . Earlier the method was used to describe not only 11Li, but also
its mirror nucleus 11O, the two extremes of a $T=5/2$ isospin multiplet Gar20
. We now extended this two- and three-body general formulation to treat the
isospin symmetry fully and consistently for other less extreme members of the
multiplet. The extension to multiplet members in between, such as the Isobaric
Analog State (IAS) or the Anti-Analog State (AAS) in 11Be will be considered
in detail. We shall provide the necessary technical developments to be
implemented into the standard three-body method. The extended method also
allows to determine which asymptotic channels the decays proceed through.

After this introduction, we start in Section II by outlining isospin
structures and concepts in the beta decay process that are the basis of the
subsequent sections. The general numerical three-body method used in the
calculations, and in particular, the new implementation of the isospin degrees
of freedom will be described in Section III. In Section IV, we focus on the
particular case of 11Li, describing the isospin structure of the wave
functions, matrix elements, and energies. In Sections V and VI we continue
with the corresponding numerical calculations of respectively, three-body wave
functions and beta decay strength. A comparison to the experimentally
established decay scheme follows in Section VII. Finally, we conclude in
Section VIII with a brief outlook. Some technical details and supplementary
information are given in appendices.

##  II Conceptual background

We use the convention where the neutron has third isospin component
$-\frac{1}{2}$. The operators for allowed Fermi and Gamow-Teller $\beta^{\mp}$
decay then have the form $\hat{\cal O}_{F}^{\mp}=\sum_{i}t_{i}^{\pm}=T^{\pm}$
and $\hat{\cal O}_{GT}^{\mp}=\sum_{i}\bm{\sigma}_{i}t_{i}^{\pm}$; here
$\bm{t}_{i}$ and $\bm{\sigma}_{i}\hbar/2$ are, respectively, the isospin and
spin operators acting on nucleon $i$, and $\bm{T}$ is the total isospin
operator of the nucleus. The ladder operator $t_{i}^{+}$ is defined as
$t_{i}^{+}=t_{x_{i}}+it_{y_{i}}$, where $t_{x_{i}}$ and $t_{y_{i}}$ are the
Cartesian $x$\- and $y$-components of the vector operator $\bm{t}_{i}$.

###  II.1 Core and halo decays

Let us consider first the case where a halo state has one or more neutrons in
the halo component (the modifications to proton halos are then trivial) and
that the mass numbers of the core and halo components are $A_{c}$ and $A_{h}$
with total mass number $A=A_{c}+A_{h}$. The total isospin is split into core
and halo parts, $\bm{T}=\bm{t_{c}}+\bm{t_{h}}$, and we can assume that the
quantum numbers fulfill $t_{h}=A_{h}/2$, $T=t_{c}+t_{h}$, whose respective
third components are given by $T_{z}=-T$, $t_{c}^{z}=-t_{c}$,
$t_{h}^{z}=-t_{h}$.

Our basic approximation is taken from Nil00 ; Jon04 . It is applicable for
pronounced halo states where the wave function can be factorized into a core
and a halo part. An allowed beta-decay operator
$\hat{\mathcal{O}}_{\beta}^{A}$, either Fermi or Gamow-Teller, is a sum over
operators for each nucleon. Therefore, one formally can divide the sum into
two components, corresponding to core decay and halo decay:

$\hat{\cal O}_{\beta}^{A}|\mathrm{c+h}\rangle=\hat{\cal
O}_{\beta}^{A}\left(|\mathrm{c}\rangle|\mathrm{h}\rangle\right)=|\mathrm{c}\rangle\left(\hat{\cal
O}_{\beta}^{A_{h}}|\mathrm{h}\rangle\right)+\left(\hat{\cal
O}_{\beta}^{A_{c}}|\mathrm{c}\rangle\right)|\mathrm{h}\rangle,$ (1)

where $|\mathrm{c}\rangle$ and $|\mathrm{h}\rangle$ represent the (decoupled)
core and halo parts of the wave function.

As illustrated shortly, the two final components will in general not have an
individual well-defined isospin. Furthermore, the core-decay component often
contains several terms corresponding to different levels in its daughter
nucleus. Therefore, the right-hand side in general does not give eigenstates
in the final system, but may suggest a skeleton to be used in interpreting
patterns in the beta decay. Some experimental support for our approximation is
found in the decays of the halo nuclei 6He ($\alpha+2n$) and 14Be (10Be$+4n$)
Rii22 ; Nil00 ; Jon04 : in the former case $t_{c}=0$ and the decay is a pure
halo decay, in the latter case the main decay branch (to a low-lying $1^{+}$
level in 14B) is very similar to the decay of the 12Be core so this branch
looks like a “core decay”.

Our procedure will then be to calculate the initial wave function in a few-
body model that includes all halo degrees of freedom, and act on it with the
beta-decay operators according to Eq. (1). The resulting wave function then
must be projected on the final states. Our computational challenge is not to
find an accurate halo wave function, but to obtain a realistic description of
the final states. The main simplifying assumption is to assume few-body
structures in the final states also, and calculate these in a similar manner,
but of course with adjusted interactions. This cannot be expected to reproduce
all details of the final states, but may provide the main features of how the
beta strength is distributed in energy and thus could be a good approximation
for the components with large beta strength.

###  II.2 Isospin, IAS and AAS

Isospin is a good quantum number for most nuclear states and is particularly
relevant for beta decay, since the operator for allowed Fermi decay is simply
the isospin raising/lowering operator. We refer to Appendix A for a more
extensive discussion and give here the relevant wave functions for the general
(neutron-rich) halo system considered above. The notation reflects that we
shall mainly be interested in two-neutron halo nuclei where the isospins of
the halo nucleons, $t_{2}$ and $t_{3}$, couple to the halo isospin, $t_{h}$,
which in turn couples to the core isospin, $t_{c}$, to give the total isospin,
$T$, with projection $T_{z}$.

The isospin wave functions are then for the halo state, the isobaric analogue
state (IAS), and the corresponding anti-analogue state (AAS), given by:

$\displaystyle|\mathrm{halo}\rangle$ $\displaystyle=$
$\displaystyle|(t_{2},t_{3})t_{h},t_{c};T=t_{h}+t_{c},T_{z}=-t_{h}-t_{c}\rangle$
(2) $\displaystyle=$
$\displaystyle|t_{c},t_{c}^{z}=-t_{c}\rangle|t_{h},t_{h}^{z}=-t_{h}\rangle,$
$\displaystyle|\mathrm{IAS}\rangle$ $\displaystyle=$
$\displaystyle|(t_{2},t_{3})t_{h},t_{c};T=t_{h}+t_{c},T_{z}=-t_{h}-t_{c}+1\rangle$
(3) $\displaystyle=$
$\displaystyle\sqrt{\frac{t_{h}}{t_{c}+t_{h}}}|t_{c},t_{c}^{z}=-t_{c}\rangle|t_{h},t_{h}^{z}=-t_{h}+1\rangle$
$\displaystyle+$
$\displaystyle\sqrt{\frac{t_{c}}{t_{c}+t_{h}}}|t_{c},t_{c}^{z}=-t_{c}+1\rangle|t_{h},t_{h}^{z}=-t_{h}\rangle,$

and

$\displaystyle|\mathrm{AAS}\rangle$ $\displaystyle=$
$\displaystyle|(t_{2},t_{3})t_{h},t_{c};T=t_{h}+t_{c}-1,T_{z}=-t_{h}-t_{c}+1\rangle$
(4) $\displaystyle=$
$\displaystyle\sqrt{\frac{t_{c}}{t_{c}+t_{h}}}|t_{c},t_{c}^{z}=-t_{c}\rangle|t_{h},t_{h}^{z}=-t_{h}+1\rangle$
$\displaystyle-$
$\displaystyle\sqrt{\frac{t_{h}}{t_{c}+t_{h}}}|t_{c},t_{c}^{z}=-t_{c}+1\rangle|t_{h},t_{h}^{z}=-t_{h}\rangle,$

where $t_{c}^{z}$ and $t_{h}^{z}$ are the projections of $t_{c}$ and $t_{h}$,
respectively. If $t_{c}=0$ only the IAS exists and it is given by the first
term in Eq. (3). Note that whereas the IAS normally remains quite pure, the
AAS will often be close to states in the daughter nucleus with the same
(lower) isospin, spin and parity and may mix with those states.

##  III Numerical three-body method

In this work we assume that the nuclei involved in our investigation can be
described as a core surrounded by two halo nucleons. Therefore, together with
the three-body method itself, all the required internal two-body interactions
need to be specified. On top of that, the decays of the core and nucleon halo
have to be both considered.

###  III.1 Formulation

The total three-body wave function describing the system will be obtained by
means of the hyperspherical adiabatic expansion method detailed in nie01 .
According to this method the three-body wave functions are written as:

$\Psi(\bm{x},\bm{y})=\frac{1}{\rho^{5/2}}\sum_{n}f_{n}(\rho)\Phi_{n}(\rho,\Omega),$
(5)

where $\bm{x}$ and $\bm{y}$ are the usual Jacobi coordinates, from which one
can define the hyperradius $\rho$ and the five hyperangles (collected into
$\Omega$) nie01 . The angular functions $\Phi_{n}(\rho,\Omega)$ are the
eigenfunctions of the angular part of the Schrödinger, or Faddeev, equations,
with eigenvalues $\lambda_{n}(\rho)$. The radial functions, $f_{n}(\rho)$, are
obtained as the solution of a coupled set of differential equations in which
the angular eigenvalues, $\lambda_{n}(\rho)$, enter as effective potentials:

$V_{\mathrm{eff}}(\rho)=\frac{\hbar^{2}}{2m}\frac{\lambda_{n}(\rho)+\frac{15}{4}}{\rho^{2}},$
(6)

where $m$ is the arbitrary normalization mass used to construct the Jacobi
coordinates nie01 .

In the calculations, the angular eigenfunctions $\Phi_{n}$ are expanded in
terms of the basis set $\\{|{\cal Y}_{q}\rangle\\}$, where ${\cal Y}_{q}$
represents the coupling between the usual hyherspherical harmonics and the
spin terms, and where $q$ collects all the necessary, orbital and spin,
quantum numbers. The number of components, as well as the number of terms in
the expansion in Eq.(5), should of course be large enough to get a converged
three-body wave function.

The calculations, as described in Ref. nie01 , have now to be extended to make
explicit the isospin quantum numbers associated to the system under
investigation. As mentioned above, the isospin state of the halo system, the
IAS, and the AAS, are given by the isospin wave functions specified in
Eqs.(2), (3), and (4), respectively. The isospin quantum numbers characterize
each of the states to be computed. With this in mind, the simplest procedure
to incorporate the isospin state into the calculation is to proceed as
described in nie01 , but where the basis set used for the expansion of the
angular eigenfunctions, $\Phi_{n}$, is not just given by the $\\{|{\cal
Y}_{q}\rangle\\}$ terms formed after the coupling of the hyperspherical
harmonics and the spin functions, but by the set $\\{|{\cal
Y}_{q}\rangle|TT_{z}\rangle\\}$, where
$|TT_{z}\rangle\equiv|(t_{2}t_{3})t_{h},t_{c};TT_{z}\rangle$ is the isospin
function of the system to be computed.

In other words, following the procedure sketched above, the full three-body
wave function, $\Psi$, is in practice written as:

$\Psi=\Psi_{3b}|TT_{z}\rangle,$ (7)

where $\Psi_{3b}$ describes the relative motion between the three constituents
of the system.

###  III.2 Core energy

An important point to take into account is that the total hamiltonian is
actually the sum of the one describing the relative motion of the three
constituents of the system, plus the core hamiltonian, ${\cal
H}_{\mathrm{core}}$, which describes the core degrees of freedom. The
hamiltonian ${\cal H}_{\mathrm{core}}$ is such that:

${\cal
H}_{\mathrm{core}}|t_{c},t_{c}^{z}\rangle=\xi_{\mathrm{core}}|t_{c},t_{c}^{z}\rangle$
(8)

where $\xi_{\mathrm{core}}$ is the energy of the core.

Obviously, in order to obtain the three-body wave function, the
diagonalization of the full hamiltonian will require inclusion of the all the
matrix elements of ${\cal H}_{\mathrm{core}}$ between all the elements of the
basis set used for the expansion, $\\{|{\cal Y}_{q}\rangle|TT_{z}\rangle\\}$.
Making use of Eqs.(2), (3), and (4), we can easily see that these matrix
elements for the halo states, the IAS, and the AAS, are given by:

$\langle{\cal Y}_{q};\mathrm{halo}|{\cal H}_{\mathrm{core}}|{\cal
Y}_{q^{\prime}};\mathrm{halo}\rangle=\xi_{-t_{c}}\delta_{qq^{\prime}},$ (9)
$\displaystyle\langle{\cal Y}_{q};\mathrm{IAS}|{\cal H}_{\mathrm{core}}|{\cal
Y}_{q^{\prime}};\mathrm{IAS}\rangle=$ (10)
$\displaystyle\left(\frac{t_{h}}{t_{c}+t_{h}}\xi_{-t_{c}}+\frac{t_{c}}{t_{c}+t_{h}}\xi_{-t_{c}+1}\right)\delta_{qq^{\prime}},$

and

$\displaystyle\langle{\cal Y}_{q};\mathrm{AAS}|{\cal H}_{\mathrm{core}}|{\cal
Y}_{q^{\prime}};\mathrm{AAS}\rangle=$ (11)
$\displaystyle\left(\frac{t_{c}}{t_{c}+t_{h}}\xi_{-t_{c}}+\frac{t_{h}}{t_{c}+t_{h}}\xi_{-t_{c}+1}\right)\delta_{qq^{\prime}},$

where $\xi_{-t_{c}}$ and $\xi_{-t_{c}+1}$ are, respectively, the energies of
the core represented by the $|t_{c},t_{c}^{z}=-t_{c}\rangle$ and
$|t_{c},t_{c}^{z}=-t_{c}+1\rangle$ isospin states.

Usually, individual three-body energies are referred to the three-body
threshold, which amounts to taking $\xi_{\mathrm{core}}=0$. However, when
comparing the energy between different three-body systems, a unique zero
energy point has to be chosen, which implies that, when different cores are
involved, only the energy of one of them can be taken equal to zero, whereas
the energy of the other ones has to be referred to the first one.

As we can see, the effect of the core hamiltonian is a shift of the effective
potentials, different for the halo state, the IAS, and the AAS, and which is
dictated by the core energies. We have assumed that ${\cal H}_{\mathrm{core}}$
does not mix the $|t_{c},t_{c}^{z}=-t_{c}\rangle$ and
$|t_{c},t_{c}^{z}=-t_{c}+1\rangle$ core states.

###  III.3 Potential matrix elements

At this point it should be noticed that the isospin wave functions in Eqs.(2),
(3), and (4) explicitly show the three-body structures involved in each case,
with a core described by the isospin state $|t_{c},t_{c}^{z}\rangle$, and a
halo state described by $|t_{h},t_{h}^{z}\rangle$. In particular, the halo
state contains a single configuration with the core and the halo in the
$|t_{c},t_{c}^{z}=-t_{c}\rangle$ and the $|t_{h},t_{h}^{z}=-t_{h}\rangle$
isospin states, respectively. On the contrary, the IAS and AAS mix two
configurations, one with the core and the halo in the
$|t_{c},t_{c}^{z}=-t_{c}\rangle$ and $|t_{h},t_{h}^{z}=-t_{h}+1\rangle$
isospin states, and the other one with the core and the halo in the
$|t_{c},t_{c}^{z}=-t_{c}+1\rangle$ and $|t_{h},t_{h}^{z}=-t_{h}\rangle$
isospin states.

In any case, in order to perform the calculations, this decomposition of the
isospin states is not the most convenient one. This is because the core-
nucleon interactions will in general depend on the total core-nucleon isospin,
$t_{cN}$, resulting from the coupling between the core and nucleon isospins.
For this reason, it is preferable to rotate the isospin wave function
$|(t_{2},t_{3})t_{h},t_{c};T,T_{z}\rangle$, which is written in what we
usually call the first Jacobi set (or T-set), into the second or third Jacobi
sets (Y-sets), where the core and the nucleon isospins are first coupled into
$t_{cN}$, and then coupled to the isospin of the second halo nucleon in order
to produce the total isospin $T$ of the system. In this way, the dependence of
the isospin wave functions on $t_{cN}$ appears explicitly. In particular, when
rotating into the second Jacobi set we get:

$|(t_{2},t_{3})t_{h},t_{c};T,T_{z}\rangle=(-1)^{t_{c}+t_{2}+t_{3}}\sum_{t_{cN}}(-1)^{t_{cN}+t_{2}}\sqrt{2t_{cN}+1}\sqrt{2t_{h}+1}\left\\{\begin{array}[]{ccc}t_{3}&t_{2}&t_{h}\\\
T&t_{c}&t_{cN}\end{array}\right\\}|(t_{3},t_{c})t_{cN},t_{2};T,T_{z}\rangle,$
(12)

which makes evident that the isospin states given in Eqs.(2), (3), and (4),
can, in general, mix different values of the core-nucleon isospin, $t_{cN}$.
The curly brackets denote the six-$j$ symbol.

When solving the three-body problem, it is always necessary at some point to
compute the matrix elements of the two-body potentials between the different
terms of the basis set used in the expansion of the wave function, $\\{|{\cal
Y}_{q}\rangle|TT_{z}\rangle\\}$. Thanks to the expression in Eq.(12), and
assuming that the core-nucleon potential, $V_{cN}$, does not mix different
$t_{cN}$ values, we easily get for the potential matrix elements:

$\langle{\cal Y}_{q};T,T_{z}|V_{cN}|{\cal
Y}_{q^{\prime}};T,T_{z}\rangle=\sum_{t_{cN}}(2t_{cN}+1)(2t_{h}+1)\left\\{\begin{array}[]{ccc}t_{3}&t_{2}&t_{h}\\\
T&t_{c}&t_{cN}\end{array}\right\\}^{2}\langle{\cal
Y}_{q};(t_{3},t_{c})t_{cN},t_{2};T,T_{z}|V_{cN}^{(t_{cN})}|{\cal
Y}_{q^{\prime}};(t_{3},t_{c})t_{cN},t_{2};T,T_{z}\rangle,$ (13)

where we can see that the calculation of the different core-nucleon potential
matrix elements requires knowledge of the potentials for the different
possible values of $t_{cN}$, i.e., knowledge of all the possible
$V_{cN}^{(t_{cN})}$ potentials.

Furthermore, in order to fully determine the potential matrix elements, it is
also necessary to decouple $t_{3}$ and $t_{c}$, such that the values of the
third isospin components, $t_{3}^{z}$ and $t_{c}^{z}$, are made explicit. This
is required in order to know what precise core state and nucleon state are
actually interacting. After doing this we get:

$\displaystyle\langle{\cal
Y}_{q};(t_{3},t_{c})t_{cN},t_{2};T,T_{z}|V_{cN}^{(t_{cN})}|{\cal
Y}_{q^{\prime}};(t_{3},t_{c})t_{cN},t_{2};T,T_{z}\rangle=$ (14)
$\displaystyle\hskip 28.45274pt\sum_{t_{cN}^{z}t_{2}^{z}}\langle
t_{cN}t_{2};t_{cN}^{z}t_{2}^{z}|TT_{z}\rangle^{2}\sum_{t_{c}^{z}t_{3}^{z}}\langle
t_{3}t_{c};t_{3}^{z}t_{c}^{z}|t_{cN},t_{cN}^{z}\rangle^{2}\langle{\cal
Y}_{q};t_{c}t_{c}^{z};t_{3}t_{3}^{z}|V_{cN}^{(t_{cN})}|{\cal
Y}_{q^{\prime}};t_{c}t_{c}^{z};t_{3}t_{3}^{z}\rangle,$

which, after insertion into Eq.(13), gives the detail of the required
potential matrix elements. The matrix element, $\langle{\cal
Y}_{q};t_{c}t_{c}^{z};t_{3}t_{3}^{z}|V_{cN}^{(t_{cN})}|{\cal
Y}_{q^{\prime}};t_{c}t_{c}^{z};t_{3}t_{3}^{z}\rangle$, refers to the matrix
element between the basis elements ${\cal Y}_{q}$ and ${\cal Y}_{q^{\prime}}$,
when the core is in the $|t_{c},t_{c}^{z}\rangle$ isospin state, and the
nucleon in the $|t_{3},t_{3}^{z}\rangle$ state, coupled to the two-body
isospin $t_{cN}$, and interacting via the potential $V_{cN}^{(t_{cN})}$. We
have assumed here that the potential does not mix different isospin states.

Therefore, in order to describe a particular system, all the core-nucleon
potentials required to compute all the possible $\langle{\cal
Y}_{q};t_{c}t_{c}^{z};t_{3}t_{3}^{z}|V_{cN}^{(t_{cN})}|{\cal
Y}_{q^{\prime}};t_{c}t_{c}^{z};t_{3}t_{3}^{z}\rangle$ matrix elements need to
be specified. Once this is done, we can then compute the three-body wave
function, $\Psi_{3b}$, entering in Eq.(7) as described in Eq.(5), which
contains the isospin dependence of the potentials through the matrix elements
in Eqs.(13) and (14), as well as the dependence on the excitation energy of
the core through the matrix elements in Eqs.(9) to (11).

###  III.4 Beta decay strength

Let us now consider the beta decay of some initial state
$\Psi^{\mathrm{(in)}}=\Psi_{3b}^{\mathrm{(in)}}|T^{\prime},T^{\prime}_{z}\rangle$
by the action of the beta decay operator $\hat{\cal O}_{\beta}^{A}$, as given
in Eq.(1), and project on some final state
$\Psi^{\mathrm{(fin)}}=\Psi_{3b}^{\mathrm{(fin)}}|T,T_{z}\rangle$, where
$|T^{\prime},T^{\prime}_{z}\rangle$ and $|T,T_{z}\rangle$ describe the
respective initial and final isospin states.

For Fermi and Gamow-Teller decays the corresponding transition strengths are
given by:

$B=\frac{1}{2J_{i}+1}\left|\langle\Psi_{3b}^{\mathrm{(fin)}}||\langle
T,T_{z}|\hat{\cal
O}_{\beta}^{A}|T^{\prime},T^{\prime}_{z}\rangle||\Psi_{3b}^{\mathrm{(in)}}\rangle\right|^{2},$
(15)

where $\hat{\cal O}_{\beta}^{A}$ is either the Fermi, ${\cal
O}_{F}^{\mp}=\sum_{j}t_{j}^{\pm}$, or Gamow-Teller, ${\cal
O}_{GT}^{\mp}=\sum_{j}\bm{\sigma}_{j}t_{j}^{\pm}$, decay operators, and where
$J_{i}$ is the angular momentum quantum number of the initial state and $j$
runs over all the $A$ nucleons of the initial system.

As shown in Eq.(1), the Fermi and Gamow-Teller operators can be split in two
pieces, the first one involving the decay of the valence nucleons and the
second one involving the decay of the core nucleons. In this way, the reduced
matrix elements contained in Eq.(15) can be written as:

$\displaystyle\langle\Psi_{3b}^{\mathrm{(fin)}}||\langle T,T_{z}|\hat{\cal
O}_{\beta}^{A}|T^{\prime},T^{\prime}_{z}\rangle||\Psi_{3b}^{\mathrm{(in)}}\rangle=$
(16) $\displaystyle\langle\Psi_{3b}^{\mathrm{(fin)}}||\langle
T,T_{z}|\hat{\cal
O}_{\beta}^{A_{h}}|T^{\prime},T^{\prime}_{z}\rangle||\Psi_{3b}^{\mathrm{(in)}}\rangle$
$\displaystyle+\langle\Psi_{3b}^{\mathrm{(fin)}}||\langle T,T_{z}|\hat{\cal
O}_{\beta}^{A_{c}}|T^{\prime},T^{\prime}_{z}\rangle||\Psi_{3b}^{\mathrm{(in)}}\rangle,$

where

$\hat{\cal O}_{\beta}^{A_{h}}=\sum_{j=v_{1},v_{2}}t_{j}^{\pm}\mbox{ or
}\hat{\cal O}_{\beta}^{A_{h}}=\sum_{j=v_{1},v_{2}}\bm{\sigma}_{j}t_{j}^{\pm}$
(17)

describes the Fermi or Gamow-Teller decay of the halo nucleons, denoted by
$v_{1}$ and $v_{2}$, and

$\hat{\cal O}_{\beta}^{A_{c}}=\sum_{j\in\mathrm{core}}t_{j}^{\pm}\mbox{ or
}\hat{\cal
O}_{\beta}^{A_{c}}=\sum_{j\in\mathrm{core}}\bm{\sigma}_{j}t_{j}^{\pm}$ (18)

describes the Fermi or Gamow-Teller decay of the $A_{c}$ nucleons in the core.

From Eqs.(2), (3), and (4), it is simple to see that the core isospin matrix
element in the equation above becomes:

$\displaystyle\langle\mathrm{halo};T,T_{z}|\hat{\cal
O}_{\beta}^{A_{c}}|\mathrm{halo};T^{\prime},T^{\prime}_{z}\rangle=$ (19)
$\displaystyle\langle t_{c},t_{c}^{z}=-t_{c}|\hat{\cal
O}_{\beta}^{A_{c}}|t^{\prime}_{c},t_{c}^{\prime z}=-t^{\prime}_{c}\rangle,$
$\displaystyle\langle\mathrm{IAS};T,T_{z}|\hat{\cal
O}_{\beta}^{A_{c}}|\mathrm{halo};T^{\prime},T^{\prime}_{z}\rangle=$
$\displaystyle\sqrt{\frac{t_{c}}{t_{c}+t_{h}}}\langle
t_{c},t_{c}^{z}=-t_{c}+1|\hat{\cal
O}_{\beta}^{A_{c}}|t^{\prime}_{c},t_{c}^{\prime z}=-t^{\prime}_{c}\rangle,$

or

$\displaystyle\langle\mathrm{AAS};T,T_{z}|\hat{\cal
O}_{\beta}^{A_{c}}|\mathrm{halo};T^{\prime},T^{\prime}_{z}\rangle=$
$\displaystyle-\sqrt{\frac{t_{h}}{t_{c}+t_{h}}}\langle
t_{c},t_{c}^{z}=-t_{c}+1|\hat{\cal
O}_{\beta}^{A_{c}}|t^{\prime}_{c},t_{c}^{\prime z}=-t^{\prime}_{c}\rangle,$

which permit to write the reduced beta decay matrix element Eq.(16) as:

$\displaystyle\langle\Psi_{3b}^{\mathrm{(fin)}}||\langle T,T_{z}|\hat{\cal
O}_{\beta}^{A}|T^{\prime},T^{\prime}_{z}\rangle||\Psi_{3b}^{\mathrm{(in)}}\rangle=$
$\displaystyle\langle\Psi_{3b}^{\mathrm{(fin)}}||\langle T,T_{z}|\hat{\cal
O}_{\beta}^{A_{h}}|T^{\prime},T^{\prime}_{z}\rangle||\Psi_{3b}^{\mathrm{(in)}}\rangle$
$\displaystyle+f_{\mathrm{fin}}\langle\Psi_{3b}^{\mathrm{(fin)}}||\Psi_{3b}^{\mathrm{(in)}}\rangle\langle
t_{c},t_{c}^{z}|\hat{\cal O}_{\beta}^{A_{c}}|t^{\prime}_{c},t^{\prime
z}_{c}\rangle,$

where $f_{\mathrm{fin}}=1,\sqrt{t_{c}/(t_{c}+t_{h})}$, or
$-\sqrt{t_{h}/(t_{c}+t_{h})}$, depending on the character (halo state, IAS, or
AAS) of the final state, and where
$\langle\Psi_{3b}^{\mathrm{(fin)}}||\Psi_{3b}^{\mathrm{(in)}}\rangle$ is the
overlap of the initial and final three-body wave functions, excluding the
isospin part. The final matrix element describes the beta decay of the core,
whose initial and final isospin states are given by $|t^{\prime}_{c},t^{\prime
z}_{c}\rangle$ and $|t_{c},t^{z}_{c}\rangle$, respectively.

For Fermi decay, the transition operator depends on the isospin only. It is
then possible to factorize the matrix element
$\langle\Psi_{3b}^{\mathrm{(fin)}}||\langle T,T_{z}|\hat{\cal
O}_{\beta}^{A_{h}}|T^{\prime},T^{\prime}_{z}\rangle||\Psi_{3b}^{\mathrm{(in)}}\rangle$
into one term involving only the coordinate and spin parts of the initial and
final three-body wave functions ($\Psi_{3b}$ in Eq.(7)), and another matrix
element involving only the isospin part. In other words, for Fermi decay we
can write:

$\displaystyle\langle\Psi_{3b}^{\mathrm{(fin)}}||\langle
T,T_{z}|\sum_{j=1}^{A}t_{j}^{+}|T,T^{\prime}_{z}\rangle||\Psi_{3b}^{\mathrm{(in)}}\rangle=\langle\Psi_{3b}^{\mathrm{(fin)}}||\Psi_{3b}^{\mathrm{(in)}}\rangle$
$\displaystyle\times\left(\langle
T,T_{z}|\sum_{j=v_{1},v_{2}}t_{j}^{+}|T^{\prime},T^{\prime}_{z}\rangle+f_{\mathrm{fin}}\langle
t_{c},t_{c}^{z}|\sum_{j\in\mathrm{core}}t_{j}^{+}|t^{\prime}_{c},t^{\prime
z}_{c}\rangle\right),$

which means that the Fermi decay matrix element is given just by the overlap
of the initial and final three-body wave functions (excluding the isospin
part) times an isospin dependent global factor.

###  III.5 Contribution from the continuum background

The matrix elements derived in the previous section correspond to the beta
decay transition from some halo initial state into some either halo, or IAS,
or AAS, final states, each of them characterized by specific isospin quantum
numbers. However, when computing the strength for a particular transition, it
is necessary to include the contribution from transitions into the continuum
states having the same quantum numbers as the final state, either halo state,
IAS, or AAS.

In this work the continuum states will be computed as discrete states after
imposing a box boundary condition. In other words, for a given set of quantum
numbers, the radial wave functions in Eq.(5) are obtained by solving the
radial part of the three-body equations in a box with a sufficiently maximum
large value for the hyperradius. To be specific, we shall use the finite-
difference method described for instance in Chapter 19 of Ref. pre97 . This
method permits to reduce the system of coupled differential equations into an
eigenvalue problem, such that the eigenfunctions are the radial solutions
evaluated at specific chosen values of the radial coordinate, and the
eigenvalues are the “discrete” continuum energies.

In this way, together with the precise three-body wave function for the final
state (halo state, IAS, or AAS), we get, for each of them, a family of
“discrete” continuum states with the same quantum numbers. Therefore, the
total transition strength to the states with a definite set of quantum numbers
will be given by the sum of all the strengths given in Eq.(15) for Fermi or
Gamow-Teller strengths, respectively. The sum runs of course over all the
computed discrete states.

At this point it is also interesting to consider the differential transition
strength, which permits to obtain the distribution of the strength as a
function of the final three-body energy. When dealing with discrete states,
and following Eq.(15), we can easily write the differential transition
strength for Fermi and Gamow-Teller transitions as:

$\displaystyle\frac{dB}{dE}=\frac{1}{2J_{i}+1}\times$
$\displaystyle\sum_{k}\delta(E-E_{k})\left|\langle\Psi_{3b}^{\mathrm{(fin)}}(E_{k})||\langle
T,T_{z}|\hat{\cal
O}_{\beta}^{A}|T^{\prime},T^{\prime}_{z}\rangle||\Psi_{3b}^{\mathrm{(in)}}\rangle\right|^{2},$

where $k$ runs over all the discrete final states, with energy $E_{k}$, and
wave function $\Psi_{3b}^{\mathrm{(fin)}}(E_{k})$. The integral over the
three-body energy, $E$, trivially provides the total strength, and the reduced
matrix elements for the Fermi and Gamow-Teller decays are given by Eq.(III.4),
which for Fermi decay can be simplified into Eq.(III.4).

In our calculations the $\delta$-function will be replaced by the normalized
Gaussian

$\delta(E-E_{k})\approx\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(E-E_{k})^{2}}{2\sigma^{2}}},$
(25)

where the Gaussian width, $\sigma$, is made as small as possible, but
producing a smooth strength function (otherwise Eq.(III.5) would become a
sequence of delta functions located at the energies of the different discrete
states). Typical $\sigma$ values range within the interval $\sigma=0.1$ MeV
and $\sigma=0.3$ MeV and therefore larger than the distance between the
discrete continuum states, which, in our calculations, will be in the range of
0.05 MeV to 0.1 MeV.

##  IV Application to 11Li beta decay

We shall here employ the extended three-body formalism described in the
previous section to investigate beta-decay of the 11Li ground state. After
specifying the isospin formalism in the context, we proceed to calculate
energies, potentials and wave functions.

###  IV.1 Specific information

In the following we denote the ground state wave function by 11Lig.s., which
decays into different 11Be states. The 11Lig.s. state will be constructed as a
9Li core in its ground state, 9Lig.s., and two halo neutrons.

For 11Be we shall consider as well the case of a two-neutron halo system
having 9Be in its ground state, 9Beg.s., as core. We shall denote this state
as 11Beg.s.. Together with it, we shall consider as well transitions into the
IAS of 11Lig.s., denoted as 11BeIAS, and into the AAS of 11Lig.s., denoted as
11BeAAS.

It is important to note again that beta decay does not change the spatial
structure of the system. Therefore, all the three 11Be states considered here,
11Beg.s., 11BeIAS, and 11BeAAS, are assumed to have the same partial wave
structure ($s$, $p$, and $d$ partial wave components) as 11Lig.s., and they
all have spin and parity $3/2^{-}$ (so, contrary to 11Lig.s., 11Beg.s. is not
the ground state of 11Be, and the subscript refers only to having 9Be in its
ground state as core).

It should be noted from the outset that our calculations will not at the
moment be able to describe all of the decay. The decay of the core, 9Lig.s.,
proceeds about half of the time to 9Beg.s., and the rest of the time to
excited states that are typically thought of as 8Be+n and 5He+$\alpha$
combinations before decaying into a final $\alpha\alpha$n continuum. Three-
body calculations of the resonances in 9Be are doable Alv10 , but extending
them to five-body calculations by adding two nucleons is beyond current
capabilities.

Since beta decay does not change the spatial wave function, the main strength
will, if we neglect the small change in Coulomb energy, go to states with
similar energy, and for the Fermi strength even the same energy. For 11Li this
implies that most strength should lie within about 5 MeV of the IAS as the
Gamow-Teller Giant Resonance, that includes most of the GT strength, is
roughly at the IAS energy in this region of the nuclear chart Sag93 . This is
an excitation energy where many channels are open and continuum degrees of
freedom are therefore important.

For the halo decay, the GT sum rule gives a total strength of 6. The 11Li halo
consists mainly of $s_{1/2}^{2}$ and $p_{1/2}^{2}$ configurations. In a halo
decay one of these neutrons is transformed into a proton; in most cases the
ensuing configuration will not differ much in energy (and could be treated in
few-body models), the main exception being for a final $p_{3/2}$ proton that
could combine with the 9Li core to produce states in 10Be. The remaining
$p_{1/2}$ neutron may then combine to form low-lying states in 11Be, a prime
example being the first excited $1/2^{-}$ state Suz94 . Beta transitions of
this type (moving a halo particle into the core) cannot easily be reproduced
in our few-body calculations.

For core decays, transitions that involve the 9BeIAS are straightforward to
treat theoretically, even though it is inaccessible in the 9Li decay. For all
other states fed in 9Be one has to treat them individually, find their
interactions with the halo neutrons and extract the relevant 9Li to 9Be∗
matrix elements either from experiment or from other theoretical models. We
shall attempt this for the 9Be ground state, but will not be able to treat
other states. The consequences of this will be discussed in section VII, as
will the extent to which the 11Li strength resembles the 9Li strength.

###  IV.2 The isospin wave functions

From Eq.(2) we can get the isospin wave functions for 11Lig.s. and 11Beg.s.,
which are given by:

$\displaystyle|\mbox{${}^{11}$Li}_{\mathrm{g.s.}}\rangle$ $\displaystyle=$
$\displaystyle|(t_{2},t_{3})t_{h}=1,t_{c}=\frac{3}{2};T=\frac{5}{2},T_{z}=-\frac{5}{2}\rangle$
(26) $\displaystyle=$
$\displaystyle|t_{c}=\frac{3}{2}t_{c}^{z}=-\frac{3}{2}\rangle|t_{h}=1,t_{h}^{z}=-1\rangle$
$\displaystyle\equiv$
$\displaystyle|^{9}\mbox{Li}_{\mathrm{g.s.}}+n+n\rangle,$

and

$\displaystyle|\mbox{${}^{11}$Be}_{\mathrm{g.s.}}\rangle$ $\displaystyle=$
$\displaystyle|(t_{2},t_{3})t_{h}=1,t_{c}=\frac{1}{2};T=\frac{3}{2},T_{z}=-\frac{3}{2}\rangle$
(27) $\displaystyle=$
$\displaystyle|t_{c}=\frac{1}{2}t_{c}^{z}=-\frac{1}{2}\rangle|t_{h}=1,t_{h}^{z}=-1\rangle$
$\displaystyle\equiv$
$\displaystyle|^{9}\mbox{Be}_{\mathrm{g.s.}}+n+n\rangle,$

where the core states $|t_{c}=\frac{3}{2},t_{c}^{z}=-\frac{3}{2}\rangle$ and
$|t_{c}=\frac{1}{2},t_{c}^{z}=-\frac{1}{2}\rangle$ correspond to 9Lig.s. and
9Beg.s., respectively. Therefore, 11Lig.s. and 11Beg.s. correspond to three-
body structures having either 9Lig.s. or 9Beg.s. as core, surrounded by two
neutrons (since $t_{h}^{z}=-1$, the halo is necessarily formed by two
neutrons).

Similarly, for 11BeIAS and 11BeAAS, making use of Eq.(3) and Eq.(4), we get:

$\displaystyle|\mbox{${}^{11}$Be}_{\mathrm{IAS}}\rangle$ $\displaystyle=$
$\displaystyle|(t_{2},t_{3})t_{h}=1,t_{c}=\frac{3}{2};T=\frac{5}{2},T_{z}=-\frac{3}{2}\rangle$
(28) $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{5}}|t_{c}=\frac{3}{2}t_{c}^{z}=-\frac{3}{2}\rangle|t_{h}=1,t_{h}^{z}=0\rangle+\sqrt{\frac{3}{5}}|t_{c}=\frac{3}{2}t_{c}^{z}=-\frac{1}{2}\rangle|t_{h}=1,t_{h}^{z}=-1\rangle$
$\displaystyle\equiv$
$\displaystyle\sqrt{\frac{2}{5}}|\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+n+p\rangle+\sqrt{\frac{3}{5}}|\mbox{${}^{9}$Be}_{\mathrm{IAS}}+n+n\rangle$

and

$\displaystyle|\mbox{${}^{11}$Be}_{\mathrm{AAS}}\rangle$ $\displaystyle=$
$\displaystyle|(t_{2},t_{3})t_{h}=1,t_{c}=\frac{3}{2};T=\frac{3}{2},T_{z}=-\frac{3}{2}\rangle$
(29) $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{5}}|t_{c}=\frac{3}{2}t_{c}^{z}=-\frac{3}{2}\rangle|t_{h}=1,t_{h}^{z}=0\rangle-\sqrt{\frac{2}{5}}|t_{c}=\frac{3}{2}t_{c}^{z}=-\frac{1}{2}\rangle|t_{h}=1,t_{h}^{z}=-1\rangle$
$\displaystyle\equiv$
$\displaystyle\sqrt{\frac{3}{5}}|\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+n+p\rangle-\sqrt{\frac{2}{5}}|\mbox{${}^{9}$Be}_{\mathrm{IAS}}+n+n\rangle,$

where, as before, $|t_{c}=\frac{3}{2},t_{c}^{z}=-\frac{3}{2}\rangle$
corresponds to 9Lig.s., but $|t_{c}=\frac{3}{2},t_{c}^{z}=-\frac{1}{2}\rangle$
corresponds to 9Be populating the IAS of 9Lig.s., i.e., 9BeIAS. Therefore,
both, 11BeIAS and 11BeAAS, mix two different three-body structures, one having
9Lig.s. as core, plus one neutron and one proton (as it corresponds to
$t_{h}^{z}=0$), and a second system having 9BeIAS as core, plus two neutrons.
As expected, the weights are such that the IAS and AAS described in Eqs.(28)
and (29) are orthogonal.

Note that for all the three-body systems considered here, 11Lig.s., 11Beg.s.,
11BeIAS, and 11BeAAS, the two halo nucleons couple to $t_{h}=1$. This means
that the isospin function is always symmetric under exchange of both nucleons,
and therefore the three-body wave function describing the relative motion
between the three constituents, $\Psi_{3b}$, has to be in all the cases
antisymmetric under the same exchange.

###  IV.3 Potential matrix elements

As mentioned, in order to obtain the three-body wave functions, it is always
necessary, at some point, to compute the matrix elements $\langle{\cal
Y}_{q};T,T_{z}|V_{cN}^{(t_{cN})}|{\cal Y}_{q^{\prime}};T,T_{z}\rangle$,
Eq.(13), which involve the different core-nucleon interactions and the basis
terms formed by the usual hyperspherical harmonics, ${\cal Y}_{q}$, and the
isospin function of the system, $|TT_{z}\rangle$. These matrix elements are,
in general, computed numerically, and, as shown in Eq.(14), they permit to
determine the two-body potentials needed in order to perform the calculation.

The particularization of the potential matrix elements for the systems
considered in this work, 11Lig.s., 11Beg.s., 11BeIAS, and 11BeAAS, can be made
by use of the isospin wave functions given in Eqs.(26) to (29). The details
are given in Appendix B.

In particular, from Eqs.(43) to (46) we get that, in order to obtain the
three-body wave functions for the 11Lig.s., 11Beg.s., 11BeIAS, and 11BeAAS
states, the following core-nucleon interactions need to be specified: $i)$ The
interaction between the 9Lig.s. core and the neutron (only $t_{cN}=2$ is
possible), $ii)$ the interaction between the 9Beg.s. core and the neutron
(only $t_{cN}=1$ is possible), $iii)$ the interaction between the 9Lig.s. core
and the proton for $t_{cN}=2$, $iv)$ the interaction between the 9BeIAS core
and the neutron for $t_{cN}=2$, $v)$ the interaction between the 9Lig.s. core
and the proton for $t_{cN}=1$, and $vi)$ the interaction between the 9BeIAS
core and the neutron for $t_{cN}=1$.

###  IV.4 Core energy

In section III.2 we described how the core hamiltonian, ${\cal
H}_{\mathrm{core}}$, gives rise to a shift of the effective three-body
potentials determining the energy of the different three-body states. This
shift is governed by the energy of the core, which, as we can see from
Eqs.(26) to (29), for the systems considered here can be either 9Lig.s.,
9Beg.s., or 9BeIAS, whose respective energies will be denoted by
$\xi_{\mbox{\scriptsize${}^{9}$Li${}_{\mathrm{g.s.}}$}}$,
$\xi_{\mbox{\scriptsize${}^{9}$Be${}_{\mathrm{g.s.}}$}}$, and
$\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{IAS}}}$.

The details of the calculation of these core matrix elements are given in
Appendix C, and, more specifically, the shift of the effective three-body
potentials is given by Eqs.(50) to (53).

###  IV.5 Relative energies

Figure 1: Relative energies between the different states involved in the
calculations. Data from Ref. til04 . To keep the figure clean, the separation
between the states is not always properly scaled.

In this work we are describing four different three-body systems, i.e.,
11Lig.s., 11Beg.s., 11BeIAS, and 11BeAAS. When computed individually, it is
common to refer the energy of the three-body states to three-body thresholds.
In other words, the zero-energy point is taken as the energy of the core for
each of them.

However, in order to describe the decay of some system (11Lig.s. in our case)
into some other systems (three different 11Be states in our calculations) it
is necessary to refer all the energies to a single zero-energy value. In our
calculations we have chosen the zero-energy point as the 11Lig.s. threshold,
which implies that $\xi_{\mbox{\scriptsize${}^{9}$Li${}_{\mathrm{g.s.}}$}}=0$
in Eq.(47). After this choice, as one can find in til04 , we have that
$\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{g.s.}}}=-13.6067$ MeV and
$\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{IAS}}}=0.7855$ MeV.

The relative energies between the different systems involved in our
calculations are sketched in Fig. 1 (data from Ref. til04 ). The values of
$\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{g.s.}}}$ and
$\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{IAS}}}$ given above amount to an
excitation energy of 14.3922 MeV of 9BeIAS with respect to 9Beg.s., which in
turn is 6.8122 MeV above the 10Be ground state energy. This leads to an energy
threshold of 21.2044 MeV for the 9BeIAS+neutron state above the 10Be ground
state. Also, the experimental excitation energy of 11BeIAS with respect to
11Beg.s., 21.16 MeV, gives rise to an excitation energy of only 0.24 MeV with
respect to the 11Lig.s. threshold.

##  V Three-body calculations

In order to perform the three-body calculations, the key quantities are the
two-body interactions between the three constituents of the systems. We now
describe how these interactions are constructed, and discuss the three-body
wave functions obtained after the subsequent calculations.

###  V.1 Two-body potentials

The systems involved in this work are 11Lig.s., 11Beg.s., 11BeIAS, or 11BeAAS.
The required core-nucleon interactions are the six interactions mentioned in
section IV.3, which, together with the nucleon-nucleon potential, constitute
all the interactions needed for the calculations.

For the nucleon-nucleon interaction we use the potential given in Eq.(5) of
Ref. gar04 . This is a simple potential employing Gaussian shapes for the
central, spin-spin, spin-orbit, and tensor terms, and whose parameters
(strength and range) are adjusted to reproduce the experimental $s$\- and
$p$-wave nucleon-nucleon scattering lengths and effective ranges.

| $t_{cN}=2$ | $t_{cN}=1$ |  $n-^{9}$Beg.s.
---|---|---|---
Pots. | $i),iii),iv)$ | $v),vi)$ | $ii)$
$S_{c}^{(s)}$ | $-5.4$ | $-23.61$ | $-11.47$
$S_{c}^{(p)}$ | $260.75$ | 246.70 | $-1.89$
$S_{c}^{(d)}$ | $260$ | 246.70 | $-44.22$
$S_{ss}^{(s)}$ | $-4.5$ | $-1.83$ | $0.59$
$S_{ss}^{(p)}$ | $1.0$ | $-3.35$ | $-4.0$
$S_{ss}^{(d)}$ | $-9.0$ | $-3.35$ | $-0.88$
$S_{so}^{(p)}$ | 300 | 300 | 30.0
$S_{so}^{(d)}$ | $-300$ | $-300$ | $-10.0$
Table 1: Strengths, in MeV, of the nuclear Gaussian potentials
($V^{(\ell)}(r)=S^{(\ell)}e^{-r^{2}/b^{2}}$) for the six potentials, labeled
as in section IV.3, entering in this work. The range of the Gaussians is
$b=3.0$ fm for potential $ii)$ and $b=2.55$ fm for the other five potentials.

For all the necessary core-nucleon interactions we take an $\ell$-dependent
potential with the form:

$V_{cN}^{(\ell)}(r)=V_{c}^{(\ell)}(r)+V_{ss}^{(\ell)}(r)\bm{s}_{c}\cdot\bm{j}_{N}+V_{so}^{(\ell)}(r)\bm{\ell}\cdot\bm{s}_{N},$
(30)

with central, spin-spin, and spin-orbit terms (plus Coulomb interaction when
needed). In the expression above $\bm{s}_{c}$ and $\bm{s}_{N}$ are the spins
of the core and the nucleon, respectively, $\bm{\ell}$ is the relative orbital
angular momentum between them, and $\bm{j}_{N}$ results from the coupling of
$\bm{\ell}$ and $\bm{s}_{N}$. This choice of the spin-spin and spin-orbit
operators is particularly useful to preserve the shell-model quantum numbers
for the nucleon state, making easier the implementation of the Pauli principle
gar03 . In all the cases the potential functions $V_{c}^{(\ell)}$,
$V_{ss}^{(\ell)}$, and $V_{so}^{(\ell)}$ are taken to be Gaussians whose
parameters are adjusted to reproduce the available experimental information
about the core-nucleon system.

The connection to an explicit isospin dependence is described in Appendix A,
where the expression of the potential can be found for use if the isospin
formalism is preferred.

Below we describe how the different potentials, $i)$, $ii$), $iii)$, $iv)$,
$v)$, and $vi)$, as labeled in section IV.3, are selected:

$i)$ and $iii)$ For the neutron-9Lig.s. interaction (potential $i)$), which
enters in the matrix elements (43), (45), and (46), we use the interaction
specified in Ref. Gar20  to describe the 11Li ground state. The given Gaussian
potentials are shallow enough to avoid binding of a halo nucleon into a Pauli
forbidden state. In particular, since the $p_{3/2}$-shell is fully occupied by
the neutrons in the core, the $p$-wave spin-orbit potential is used to push
the $p_{3/2}$-states away. As shown in Eq.(35), the neutron-9Lig.s.
interaction corresponds fully to a core-neutron isospin $t_{cN}=2$. The
precise parameters of the potential for $s$, $p$, and $d$ waves are given in
the second column of Table 1. The same interaction is used for the
proton-9Lig.s. potential with $t_{cN}=2$ (potential $iii)$), entering in the
matrix elements (45) and (46), simply by adding the corresponding core-proton
Coulomb repulsion.

$ii)$ For the neutron-9Beg.s. interaction the potential is fitted to reproduce
the known 10Be properties til04 . The 9Be core has spin and parity
$J^{\pi}=\frac{3}{2}^{-}$, which, when coupled to an $s_{1/2}$-neutron, gives
rise to a $1^{-}$ and a $2^{-}$ state. Experimentally, 10Be is known to have
such a 1- and $2^{-}$ states, both bound with respect to the 9Be-neutron
threshold by 0.8523 MeV and 0.5489 MeV, respectively. These two states are
then used to construct the $s$-wave potential. In the same way the $2^{+}$
state, bound by 0.8538 MeV, is used to obtain the $p$-wave potential.
Experimentally, there is no low-lying $1^{+}$ state, and the spin-spin
interaction is then used to push away this state. Also, since the partial wave
structure of the system is the same as the one in 11Lig.s., the spin-orbit
potential is employed to move away the Pauli forbidden $p_{3/2}$ states.
Finally, the $4^{-}$ and $3^{-}$ states, with excitation energies 9.27 MeV and
10.15 MeV, respectively, are used to estimate the potential parameter for
$d$-waves. This interaction enters in the matrix element (44) only, and it
corresponds to $t_{cN}=1$. The parameters of the neutron-9Beg.s. potential are
given in the fourth column of Table 1.

$iv)$ For the neutron-9BeIAS interaction and $t_{cN}=2$ we take into account
that 9BeIAS is the IAS of the 9Lig.s.. The members of a given isobaric
multiplet preserve the nuclear structure, in such a way that the energy
difference between them is provided almost exclusively by the different
strength of the Coulomb repulsion. For this reason, when $t_{cN}=2$, we use
for the neutron-9BeIAS interaction, which enters in the matrix elements (45)
and (46), the same nuclear potential as for the neutron-9Lig.s. interaction,
which is given in the second column of Table 1.

$v)$ and $vi)$ For the proton-9Lig.s. and neutron-9BeIAS potentials with
$t_{cN}=1$, since, as mentioned above, the 9Lig.s. and 9BeIAS cores belong to
the same isospin multiplet, we take in both cases the same nuclear potential.
The Coulomb repulsion is the only difference between both interactions.

For the case of the 9BeIAS+neutron system, the $t_{cN}=1$ states are the Anti-
Analog States of the 10Li low-lying states, whose energy is about 20 MeV above
the 10Be ground state, as shown in Fig. 1. Similar pairs of known IAS-AAS
partners suggest that the Anti-Analog States lie a few MeV below the Isobaric
Analog States. Therefore, the 9BeIAS+neutron ($t_{cN}=1$) states are expected
to be located about 16 to 18 MeV above the 10Be ground state. Good candidates
are therefore the experimentally known 10Be states with excitation energies
17.12 MeV, 17.79 MeV, and 18.55 MeV til04 . This is actually supported by the
fact that, for the 9BeIAS+proton (10B) system, which except for the Coulomb
repulsion is equivalent to the 9BeIAS+neutron system, there is a known series
of isospin 1 states at an energy of about 18 MeV above the 10Be ground state,
among which a $2^{-}$, a $2^{+}$, and a $1^{-}$ state have been observed.

For this reason we have assumed that the first and third of the mentioned 10Be
states, at 17.12 MeV and 18.55 MeV, are $2^{-}$ and $1^{-}$ states arising
from the coupling of an $s_{1/2}$ neutron and the core
($J^{\pi}=\frac{3}{2}^{-}$), and the second state, at 17.79 MeV, is assumed to
be a $2^{+}$ state coming from the coupling of a $p_{1/2}$ neutron and the
core. As in the neutron-9Lig.s. interaction, the $p_{3/2}$ neutron states are
pushed away. The precise parameters are given in the third column of Table 1.
For the $d$-waves we use the same potential as for the $p$-waves but changing
the sign in the spin-orbit potential (as systematically happens for the other
potentials). At this point it is important to keep in mind the relative
energies between all the systems involved in our calculations, which are shown
in Fig. 1. The $2^{-}$, $2^{+}$, and $1^{-}$ states in 10Be used to determine
the $t_{cN}=1$ part of the interaction, with excitation energies all around 18
MeV above the 10Be ground state, are actually below the 9BeIAS-neutron
threshold, and therefore they are bound with respect to this threshold by
$4.08$ MeV, $3.41$ MeV, and $2.65$ MeV, respectively.

An additional simplification comes from treating the 9BeIAS core as a single
particle; 9BeIAS is far from being bound, although it is quite narrow (less
than one keV). One could envisage describing the 9BeIAS+neutron system as a
four-body ($\alpha+\alpha+n+n$) system, and consequently the corresponding
11Be states as a five-body system. In any case this is out of our reach, as it
would make the calculations significantly more complex.

###  V.2 Three-body wave functions

Using the interactions as described above, we have made the three-body
calculations for 11Lig.s., 11Beg.s., 11BeIAS, and 11BeAAS. Among them the more
tricky ones are the calculations of 11BeIAS and 11BeAAS, since, as shown in
Eqs.(45) and (46), they both combine different core-nucleon potentials, and,
in the case of 11BeAAS, also different values of $t_{cN}$.

Figure 2: (a) The four lowest effective potentials for the IAS (solid red) and
AAS (dashed green) in 11Be.The inset shows a zoom of the large distance
behavior. (b) Radial wave functions for the four lowest adiabatic terms for
11BeIAS (solid red), 11BeAAS (dashed green), 11Lig.s. (dotted blue), and
11Beg.s. (dot-dashed brown).

In Fig. 2a we show for these two systems the four lowest computed effective
adiabatic potentials as defined in Eq.(6). The solid-red and dashed-green
curves correspond, respectively, to the 11BeIAS and the 11BeAAS systems. We
can immediately see that the potentials for the AAS are clearly deeper than
the ones for the IAS. This is expected, since in the AAS case the interaction
contains the $t_{cN}=1$ part of the potential, which, as explained above,
binds by several MeV the lowest $2^{-}$, $1^{-}$, and $2^{+}$ states in the
9BeIAS-neutron system with respect to the 9BeIAS-neutron threshold. In fact,
the lowest AAS potentials go asymptotically to the energies of the bound
9BeIAS-neutron states, although weighted with the geometric factors involved
in the matrix element in Eq.(42).

The higher potentials in the AAS case, as well as the ones for the IAS, go
asymptotically to the value determined by the energy shift imposed by the core
hamiltonian in Eqs.(52) and (53), i.e.,
$\frac{3}{5}\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{IAS}}}$ and
$\frac{2}{5}\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{IAS}}}$ for the IAS and
AAS, respectively, where
$\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{IAS}}}=0.7855$ MeV (see Fig. 1).
This is seen in the inset in Fig. 2a, which shows the large distance part of
the potentials.

As shown in Fig. 1, the experimental excitation energy of 11BeIAS is 21.16
MeV, which is 0.24 MeV above the 9Lig.s.+$n$+$n$ threshold that we take as our
zero-energy point. This value of the IAS energy, 0.24 MeV, is below the
asymptotic energy limit,
$\frac{3}{5}\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{IAS}}}=0.47$ MeV, of
the potentials shown in the inner panel in Fig. 2a, which implies that the
computed radial wave function will decay exponentially at large distances, as
a regular bound state. The same of course happens for the AAS, which, as one
can anticipate from Fig. 2a, is going to be clearly more bound than the IAS.

In the calculation for 11Lig.s. and 11BeIAS it has been necessary to introduce
a small attractive effective three-body force in such a way that the
respective experimental energies, $-0.369$ MeV and 0.24 MeV with respect to
the energy threshold shown in Fig. 1, are well reproduced. For 11BeAAS the
same effective three-body force as for the IAS has been used, and a binding
energy of $-5.53$ MeV with respect to our zero energy point has been obtained.
This amounts to an excitation energy of this 11Be state of 15.39 MeV, which is
consistent with observations in other IAS/AAS partners. Also, the computed
energy difference between the IAS and AAS is equal to 5.77 MeV, consistent as
well with the upper limit of 7 MeV estimated in Appendix A.

Finally, for 11Beg.s. the three-body calculation without any three-body
potential gives rise to a binding energy of $-18.03$ MeV with respect to our
chosen zero-energy value. This means, see Fig. 1, that the computed 11Beg.s.
wave function has an excitation energy of 2.89 MeV with respect to the ground
state of 11Be. Since this excitation energy is in the vicinity of the
experimentally known 3/2- states in 11Be with excitation energy 2.69 MeV and
3.97 MeV, we then have not included any three-body potential for 11Beg.s..

| Exc. energy |  9Lig.s.+$n$+$n$
---|---|---
11Lig.s. | 0 | $-0.37$
11Beg.s. | 2.89 | $-18.03$
11BeIAS | 21.16 | 0.24
11BeAAS | 15.39 | $-5.53$
Table 2: Computed excitation energies (second column) and energies relative to
the 9Lig.s.+$n$+$n$ threshold (third column) for 11Lig.s., 11Beg.s., 11BeIAS,
and 11BeAAS. All energies are given in MeV.

The computed energies for 11Lig.s., 11Beg.s., 11BeIAS, and 11BeAAS are
summarized in Table 2, where the second column gives the excitation energies
and the third one the same energies but relative to the 9Lig.s.+$n$+$n$
threshold used in this work. All the energies are given in MeV.

In Fig. 2b we plot the radial wave functions associated to the four lowest
adiabatic terms, Eq.(5), for each of the systems computed in this work, i.e.,
11BeIAS (solid red), 11BeAAS (dashed green), 11Lig.s. (dotted blue), and
11Beg.s. (dot-dashed brown). As we can see, in all the cases the first two
terms provide almost the full wave function. The wave functions for 11BeIAS
and 11Lig.s. are very similar. This is due to the fact that in both cases only
the $t_{CN}=2$ part of the core-nucleon potential enters. Since the cores in
both systems belong to the same isospin multiplet the nuclear interaction is
therefore the same for the two of them. The only difference comes from the
Coulomb repulsion present in the 9Lig.s.+$n$+$p$ structure entering in the IAS
case, see Eq.(28).

##  VI Beta decay strength

The energy distribution for a beta decay transition is given by Eq.(III.5).
The initial state is 11Lig.s., whose isospin wave function is given by
Eq.(26). For the final states we consider either 11Beg.s., Eq.(27), 11BeIAS,
Eq.(28), or 11BeAAS, Eq.(29).

###  VI.1 Overlap between initial and final wave functions.

One of the ingredients that enter in both, Fermi and Gamow-Teller, transition
strengths is the overlap between the initial and final three-body wave
functions (excluding the isospin part), $\Psi_{3b}^{\mathrm{(in)}}$ and
$\Psi_{3b}^{\mathrm{(fin)}}$. In order to investigate this overlap, we
introduce the overlap function:

$F_{\mathrm{ov}}(\rho)=\int\rho^{5}\Psi_{3b}^{\mathrm{(fin)}}(\rho,\Omega)\Psi_{3b}^{\mathrm{(in)}}(\rho,\Omega)d\Omega,$
(31)

where the integral is over the five hyperangles only. With this definition we
have that:

$\int d\rho
F_{\mathrm{ov}}(\rho)=\langle\Psi_{3b}^{\mathrm{(fin)}}|\Psi_{3b}^{\mathrm{(in)}}\rangle=\frac{1}{\sqrt{2J_{i}+1}}\langle\Psi_{3b}^{\mathrm{(fin)}}||\Psi_{3b}^{\mathrm{(in)}}\rangle.$
(32)

Figure 3: $F_{\mathrm{ov}}(\rho)$ function in Eq.(31) for the overlap between
the 11Lig.s. wave function and the one corresponding to 11BeIAS (solid),
11BeAAS (dashed), and 11Beg.s. (dotted).

In Fig. 3 we show the overlap function in Eq.(31) where
$\Psi_{3b}^{\mathrm{(in)}}$ is the wave function of 11Lig.s. and
$\Psi_{3b}^{\mathrm{(fin)}}$ is the wave function of either 11BeIAS (solid
curve), 11BeAAS (dashed curve), or 11Beg.s. (dotted curve). Due to the
similarity between the 11Lig.s. and 11BeIAS wave functions, dotted and solid
curves in Fig. 2b, the integral over $\rho$, Eq.(32), of the corresponding
overlap function (solid curve in Fig. 3) is 0.99, very close to 1. As we can
also anticipate from Fig. 2b, the cases of 11BeAAS and 11Beg.s. are different.
These wave functions are clearly narrower than the one of 11Lig.s., and, as a
consequence, the integral over $\rho$ of the overlap function (dashed and
dotted curves in Fig. 3) will be significantly smaller, 0.72 and 0.82 for
11BeAAS and11Lig.s., respectively.

Figure 4: Overlap strength, as given in Eq.(33), as a function of the three-
body energy, for transitions into states, bound and continuum, with the same
quantum numbers as 11BeIAS (solid-red), 11BeAAS (dashed-green), and 11Beg.s.
(dotted-blue). The thin curves show the contribution from core-nucleon
$s$-waves only.

At this point it is interesting to introduce the effect of the continuum
background in the overlap functions by means of the differential distribution:

$\frac{dB_{\mathrm{ov}}}{dE}=\frac{1}{2J_{i}+1}\sum_{k}\delta(E-E_{k})\left|\langle\Psi_{3b}^{\mathrm{(fin)}}(E_{k})||\Psi_{3b}^{\mathrm{(in)}}\rangle\right|^{2},$
(33)

which is analogous to the differential transition strength introduced in
Eq.(III.5), and where $k$ runs over all the discretized continuum states.

The result from Eq.(33) is shown in Fig. 4 for transitions into states, bound
and continuum, with the same quantum numbers, as 11BeIAS (solid-red), 11BeAAS
(dashed-green), and 11Beg.s. (dotted-blue). The thin curves in the figure give
the contribution when only the relative core-nucleon $s$-waves are considered.
The three main peaks are produced by the transitions into the specific
11BeIAS, 11BeAAS, and 11Beg.s. states, which provide most of the contribution
to $B_{\mathrm{ov}}$. The remainder is the contribution from the continuum
states. We can see there is a non-negligible contribution from the continuum
states for transitions into the states with the 11BeAAS and 11Beg.s. quantum
numbers. For the 11BeIAS, the $s$-wave contribution (thin-solid) has a bump at
about 2 MeV, but this bump disappears from the total due to the interference
with the other partial waves, in such a way that the contribution from the
continuum states is negligible.

|  11BeIAS |  11BeAAS |  11Beg.s.
---|---|---|---
$B_{\mathrm{ov}}$ (total) | 0.99 | 0.87 | 0.90
$B_{\mathrm{ov}}$ (no cont.) | $0.98(99\%)$ | 0.52 (60%) | 0.67 (74%)
$B_{\mathrm{ov}}$ (continuum) | $0.01(1\%)$ | 0.35 (40%) | 0.23 (26%)
$B_{\mathrm{ov}}$ ($s$-waves) | 0.63 (64%) | 0.56 (64%) | 0.59 (66%)
$B_{\mathrm{ov}}$ ($p$-waves) | 0.30 (30%) | 0.28 (32%) | 0.23 (26%)
$B_{\mathrm{ov}}$ ($d$-waves) | 0.00 (0%) | 0.00 (0%) | 0.00 (0%)
Table 3: Integrated overlap function, $B_{\mathrm{ov}}$, from Eq.(33), for the
states with same quantum numbers as 11BeIAS, 11BeAAS, and 11Beg.s.. The
following two rows give the contributions to the total from the non-continuum
states and the continuum background, respectively. The last three rows give
the contribution to the total when only relative core-nucleon $s$-waves,
$p$-waves, and $d$-waves are included. The values within parenthesis give the
percentage over the total given by each contribution.

The integral over $E$ of the different curves in Fig. 4 gives the total value,
$B_{\mathrm{ov}}$, of the overlap, which is 0.99, 0.87, and 0.90 for states
with the same quantum numbers as 11BeIAS, 11BeAAS, and 11Beg.s., respectively.
The total contribution from the specific 11BeIAS, 11BeAAS, and 11Beg.s. states
is given by the square of Eq.(32), which, as obtained after the analysis of
Fig. 3, is equal to 0.98, 0.52, and 0.67 for the three cases. In other words,
the contribution from the continuum background is very small ($\sim 0.01$) for
the IAS case, but quite relevant, 0.35 and 0.23, for the other two cases.
These results are summarized in the first three rows in Table 3.

In the table, in the last three rows, we also give the contribution to the
total when only the relative core-nucleon $s$-waves, $p$-waves, and $d$-waves
are included. As we can see, the contribution to the overlaps from the
$d$-waves is negligible in all the three cases. Also, the weight of $s$ and
$p$ partial waves is similar in the three cases too, especially if we consider
the percentage over the total given by each of them, which is indicated in the
table within parenthesis. Note that the sum of the $s$-, $p$-, and
$d$-contributions does not provide the total value of $B_{\mathrm{ov}}$. The
missing contribution, which ranges from 4% to 8% depending on the case, is the
contribution from the $s$-, $p$-, and $d$-interferences when squaring the
overlap in Eq.(32), which, by construction, are omitted when the calculation
is limited to $s$-, $p$-, or $d$-waves.

Note as well that, although the value of the integral over $E$ of Eq.(33) is
independent of $\sigma$, see Eq.(25), the height of the peaks of the curves in
Fig. 4 is very sensitive to its value. In this calculation we have taken
$\sigma=0.2$ MeV, but a value slightly smaller can produce curves with clearly
higher peaks, but that one could still consider smooth. Therefore, whereas the
curves in Fig. 4 can be seen as an estimate of the differential overlap, the
integral under the curves, the integrated strengths, Table 3, which are the
relevant quantities in order to obtain the total transition strengths, are
well established results of the calculations.

###  VI.2 Fermi decay.

For Fermi transitions, the beta decay matrix element is given by Eq.(III.4),
which simply is the overlap
$\langle\Psi_{3b}^{\mathrm{(fin)}}||\Psi_{3b}^{\mathrm{(in)}}\rangle$ as given
in Eq.(32), multiplied by a global isospin dependent factor. Therefore, as
seen from Eq.(III.5), the corresponding differential transition strength will
be the overlap strength defined in Eq.(33), multiplied by the isospin global
factor squared. In other words, the differential Fermi transition strength for
the different transitions considered in this work, is given by the curves in
Fig. 4, but, each of them, has to be multiplied by the corresponding isospin
factor squared.

As we can see in Eq.(III.4), the isospin factor contains three ingredients,
$f_{\mathrm{fin}}$, the halo isospin matrix element $\langle
T,T_{z}|\sum_{j=v_{1},v_{2}}t_{j}^{+}|T^{\prime},T^{\prime}_{z}\rangle$, and
the core isospin matrix element $\langle t_{c},t_{c}^{z}|\sum_{j\in\mbox{
\scriptsize${}^{9}$Li${}_{\mathrm{g.s.}}$}}t_{j}^{+}|t^{\prime}_{c}=\frac{3}{2},t^{\prime
z}_{c}=-\frac{3}{2}\rangle$.

Daughter | $\langle\sum_{j=v_{1},v_{2}}t_{j}^{+}\rangle$ | $f_{\mathrm{fin}}$ | $\langle\sum_{j\in\mbox{ \scriptsize${}^{9}$Li${}_{\mathrm{g.s.}}$}}t_{j}^{+}\rangle$ | Total
---|---|---|---|---
11Beg.s. | 0 | 1 | 0 | 0
11BeAAS | $\sqrt{\frac{6}{5}}$ | $-\sqrt{\frac{2}{5}}$ | $\sqrt{3}$ | 0
11BeIAS | $\frac{2}{\sqrt{5}}$ | $\sqrt{\frac{3}{5}}$ | $\sqrt{3}$ | $\sqrt{5}$
Table 4: For decays into 11Beg.s., 11BeAAS, and 11BeIAS, we give the values of
the halo isospin matrix element (second column), $f_{\mathrm{fin}}$ factor
(third column), and core isospin matrix element (fourth column), entering into
the total isospin term (fifth column) in Eq.(III.4).

For the decay of 11Lig.s. into 11Beg.s., due to the different initial and
final core isospins, $\frac{3}{2}$ in the case of 9Lig.s. and $\frac{1}{2}$
for 9Beg.s., both the halo isospin matrix element and the core isospin matrix
element (which in this case corresponds to $t_{c}=\frac{1}{2}$ and
$t_{c}^{z}=-\frac{1}{2}$) are equal to zero. Therefore the isospin factor is
in this case equal to zero, and there is no Fermi transition between 11Lig.s.
and 11Beg.s.. These values are collected into the second row of Table 4.

For transitions from 11Lig.s. into 11BeAAS, as shown in Eq.(III.4) for
$t_{c}=3/2$ and $t_{h}=1$, we have that $f_{\mathrm{fin}}=-\sqrt{2/5}$. In
this case the halo isospin matrix element takes the value $\sqrt{6/5}$, and
the core isospin matrix element, with $t_{c}=\frac{3}{2}$ and
$t_{c}^{z}=-\frac{1}{2}$, is equal to $\sqrt{3}$. When these values are
combined to get the isospin factor in Eq.(III.4), we obtain that its value is
equal to zero (third row of Table 4). The Fermi transition between 11Lig.s.
and 11BeAAS is forbidden as it connects states with different $T$.

Finally, for transitions from 11Lig.s. into 11BeIAS, Eq.(III.4) for
$t_{c}=3/2$ and $t_{h}=1$ results into $f_{\mathrm{fin}}=\sqrt{3/5}$. In this
case the halo isospin matrix element is equal to $2/\sqrt{5}$, and, as for
transitions into the AAS, the core isospin matrix element is $\sqrt{3}$. All
this results in a global isospin term in Eq.(III.4) equal to $\sqrt{5}$
(again, these values are collected in Table 4, last row). Therefore, a Fermi
transition between 11Lig.s. and 11BeIAS is possible, and the corresponding
differential transition strength is given by the solid (red) curve in Fig. 4,
but multiplied by a factor of 5. As a consequence, as $B_{\mathrm{ov}}=0.99$
in this case (see Table 3), the total Fermi strength is also approximately
equal to 5 for this transition.

As mentioned, the products of a beta decay process have, in general, a non
well-defined isospin. This quantum number is not a fundamental symmetry, since
the long-range Coulomb interaction breaks the symmetry. Note that isospin
breaking has been found to be small in earlier explicit calculations for halo
states Suz91 ; Han93 .

Therefore, the daughter state of the 11Lig.s. nucleus could actually mix both,
the 11BeIAS and the 11BeAAS states. In order to estimate the isospin admixture
of these states we closely follow the procedure described in Han93  (more
precisely, Eqs.(7) and (8) of that reference). The matrix element between the
spatial wave functions of the IAS and AAS of the Coulomb potential,
$Z_{c}e^{2}/r$ (where $Z_{c}$ is the 9Li charge, $e^{2}=1.44$ MeV$\times$fm,
and $r$ is the core-proton distance) is equal to 1.31 MeV. For the Coulomb
displacement of the core, $\Delta E_{c}$ in Han93 , we use the energy shift
between 9Lig.s. and 9BeIAS, which, as shown in Fig. 1, is equal to 0.7855 MeV.
All these values, together with the overlap of the spatial IAS and AAS wave
functions, computed to be 0.74, and the core and halo isospins, $t_{c}$ and
$t_{h}$, leads to the expectation value $\langle
H_{\mathrm{coupl}}\rangle=0.35$ MeV, which by means of Eq.(8) in Han93 , and
making use of the computed energy difference between the IAS and AAS, 5.77
MeV, leads to an isospin mixing $\alpha^{2}=\sin^{2}\theta=3.7\times 10^{-3}$.

###  VI.3 Gamow-Teller decay.

The Gamow-Teller decay matrix element is given by Eq.(III.4), and it contains
the summation of two terms: the matrix element of the Gamow-Teller operator
involving the valence nucleons, and a term proportional to the same overlap,
$\langle\Psi_{3b}^{\mathrm{(fin)}}||\Psi_{3b}^{\mathrm{(in)}}\rangle$, as the
one needed for the Fermi decay, and which comes from the Gamow-Teller decay of
the core. The mixing of these two terms is dictated by the factor

$f_{\mathrm{mix}}=f_{\mathrm{fin}}\langle t_{c},t_{c}^{z}|\sum_{j\in\mbox{
\scriptsize${}^{9}$Li}}\bm{\sigma}_{j}t_{j}^{+}|t^{\prime}_{c}=\frac{3}{2},t^{\prime
z}_{c}=-\frac{3}{2}\rangle,$ (34)

which includes the Gamow-Teller matrix element describing the decay of the
9Lig.s. core into a core with isospin quantum numbers $t_{c}$ and $t_{c}^{z}$.

The first of the two terms mentioned above, the matrix element describing the
Gamow-Teller decay of the valence nucleons, is zero for decays into 11Beg.s..
This is due to the different core isospin in the initial
($t^{\prime}_{c}=\frac{3}{2}$) and final ($t_{c}=\frac{1}{2}$) states. For
decays into 11BeIAS and 11BeAAS this is not true anymore, because, as seen in
Eqs.(28) and (29), the wave functions for these two states have a component
corresponding to a 9Lig.s. core plus a neutron and a proton. However, the
numerical calculation of this matrix element for these two transitions gives
rise to a very small number, about $10^{-3}$ in both cases, which permits to
predict that the contribution to the Gamow-Teller strength from the decay of
one of the halo neutrons is going to be, in any case, very small. This is due
to the fact that this contribution is given only by the components in the
initial and final wave functions with relative nucleon-nucleon $p$-waves and
nucleon-nucleon spin equal to 1, which have a very minor weight in the wave
functions.

Figure 5: Gamow-Teller strength, Eq.(III.5), for decay of 11Lig.s. into
states, bound and continuum, with the same quantum numbers as 11BeIAS (solid
curves), 11BeAAS (dashed curves), and 11Beg.s. (dot-dashed curves), as a
function of the energy of the final state. The mixing factor,
$f_{\mathrm{mix}}$, is equal to 0.55, $-0.45$, and 0.32, for each of the three
cases, respectively. The hardly visible thick curves in the low-right corner
show the very small contribution from the decay of the halo neutrons.

The second term in the Gamow-Teller strength comes from beta decay of the
core, whose beta decay transition matrix element enters in the parameter given
in Eq.(34). From Eqs.(III.4) and (III.4) we see that for both, the decay into
11BeIAS and 11BeAAS, the core decay matrix element describes the decay 9Lig.s.
into the same core state, in particular the core state with
$t_{c}=\frac{3}{2}$ and $t_{c}^{z}=-\frac{1}{2}$, which corresponds to 9BeIAS.
In the same way, for a transition into 11Beg.s, we see from Eq.(19) that
$t_{c}=\frac{1}{2}$ and $t_{c}^{z}=-\frac{1}{2}$, and the core matrix element
in Eq.(34) refers to the decay of 9Lig.s. into 9Beg.s..

Appropriate values for the core Gamow-Teller matrix elements may be estimated
from shell-model calculations for the decay of 9Lig.s., as shown in Ref. Suz97
. In this work the values of 0.5 and 0.1 are given for the Gamow-Teller
strength for decays of 9Lig.s. into 9BeIAS and 9Beg.s., respectively. The
square root of these values gives then the transition matrix element entering
in Eq.(34) for each case, which give rise to $f_{\mathrm{mix}}=0.55$, $-0.45$,
and 0.32 for decays into 11BeIAS, 11BeAAS, and 11Beg.s., respectively. To get
these numbers one has to recall that $f_{\mathrm{fin}}$ is equal to
$\sqrt{3/5}$, $-\sqrt{2/5}$, and 1 for each of the transitions mentioned above
(third column in Table 4).

|  11BeIAS |  11BeAAS |  11Beg.s.
---|---|---|---
Fermi | 5.0 | 0 | 0
Gamow-Teller | 0.30 | 0.18 | 0.09
GT (no cont.) | $0.29$ | 0.10 | 0.07
GT (continuum) | $<0.01$ | 0.08 | 0.02
$f_{\mathrm{mix}}$ | 0.55 | $-0.45$ | 0.32
Table 5: Integrated strengths for Fermi and Gamow-Teller decay of 11Lig.s.
into 11BeIAS, 11BeAAS, and 11Beg.s.. For the case of Gamow-Teller decay we
split the total strength into the contributions from the non-continuum states
and the continuum background. The last row gives the$f_{\mathrm{mix}}$ values,
Eq.(34), used to compute the Gamow-Teller strengths.

Using the $f_{\mathrm{mix}}$ values given above, we obtain the computed Gamow-
Teller strengths, Eq.(III.5), shown in Fig. 5 for decay of 11Lig.s. into
states, bound and continuum, with the same quantum numbers as 11BeIAS (solid
curves), 11BeAAS (dashed curves), and 11Beg.s. (dot-dashed curves). The thick
curves in the low-right corner of the figure show the tiny contribution to the
strength from the beta decay of the halo neutrons (which is exactly zero for
the case of decay into 11Beg.s.). This contribution is hardly visible, and the
consequence is that the curves in Fig. 5 are simply the ones in Fig. 4 scaled
by the factor $f_{\mathrm{mix}}^{2}$. As a consequence, the integrals under
the curves is nothing but the product of $f_{\mathrm{min}}^{2}$ and the values
of $B_{\mathrm{ov}}$ given in Table 3. In this way we get the total Gamow-
Teller strengths 0.30, 0.18, and 0.09 for transitions into 11BeIAS, 11BeAAS,
and 11Beg.s., respectively, from which 0.29, 0.10, and 0.07 correspond to
transitions into the non-continuum IAS, AAS, and 11Beg.s., and the rest is
given by decay into the continuum. These results have been collected into the
lower part of Table 5, where the last row gives the $f_{\mathrm{mix}}$ values
used, and where, for completeness, we also give the strengths for Fermi decay.

##  VII Comparison to data

A brief overview of the current experimental state of knowledge on the beta
decay of 11Li and its core 9Li is given in Appendix D and Fig. 7. Here we
emphasize those aspects that are relevant to compare with our calculations.

###  VII.1 Three-body energies

Our calculations should be most reliable for the $3/2^{-}$ IAS and AAS. The
IAS energy has been adjusted to the experimental position at 21.16 MeV as
determined in a charge-reaction experiment Ter97 , but neither the total
strength nor the isospin purity could be accurately extracted in the
experiment. The AAS position is then predicted to be at 15.39 MeV (see Table
2), which fits naturally with the experimentally established $3/2^{-}$ level
at 16.3 MeV.

###  VII.2 The IAS and AAS strengths

The Fermi strength of 5 to the IAS is significantly larger than the predicted
Gamow-Teller strength of 0.29 and will dominate the transition. The transition
is not energetically allowed in beta decay and it has not been possible to
extract an accurate strength from the existing charge-reaction experiment. The
predicted GT beta strength to the AAS is 0.10 and thereby close to the so far
identified experimental strength of 0.12. We note that not all experimental
strength may have been identified yet and that the Fermi strength is
negligible. The predicted isospin impurity of the IAS, $3.7\times 10^{-3}$, is
large for such a light nucleus, but still too small to be easily measured. The
magnitude is, as is the mismatch of 0.02 in the spatial overlap 11Li to IAS of
0.98, increased due to the halo structure.

Figure 6: Lowest effective potential for the 9Lig.s.+$n$+$p$ system with
$t_{h}=1$, and turning points, $\rho_{1}$ and $\rho_{2}$, for the energy of
the 11BeIAS state. The WKB transmission coefficient through the barrier, $T$,
is obtained as indicated in the figure.

###  VII.3 Decay branches

As shown in Eq.(28), the IAS contains two different three-body components, the
9BeIAS+$n$+$n$ and the 9Lig.s.+$n$+$p$ structures, both with $t_{h}=1$. As
seen in Fig. 1, the IAS energy is below the 9BeIAS+$n$+$n$ threshold by 0.55
MeV, but above the 9Lig.s.+$n$+$p$ threshold by 1.02 MeV. Therefore, whereas
the IAS can not decay through the 9BeIAS+$n$+$n$ channel, it actually can do
it through the 9Lig.s.+$n$+$p$ channel. An estimate of the width for this
decay can be obtained by means of the WKB approximation, as described in Ref.
gar04b . In Fig. 6 we show the lowest effective potential for the
9Lig.s.+$n$+$p$ ($t_{h}=1$) system, obtained with potential $i)$, given in
Table 1, and indicate the turning points, $\rho_{1}$ and $\rho_{2}$,
corresponding to the IAS excitation energy. The transmission coefficient
through the barrier, $T$, is obtained as indicated in the figure. The width,
$\Gamma$, is then estimated as $\Gamma/\hbar=fT$, where the knocking rate,
$f$, is obtained as $f=v/\rho_{1}$, and the velocity $v$ is extracted after
equating the kinetic energy, $mv^{2}/2$, and the energy difference between $E$
and the maximum depth of the effective potential. Following this procedure we
have obtained a WKB estimate for the width of the IAS of about 0.5 MeV, which
agrees nicely with the experimental width, Ref. Ter97 , of 0.49(7) MeV.

Figure 7: Comparison of the beta decays of 9Li and 11Li. A selection of states
fed in the decays is shown along with important thresholds for particle
emission.

We do not predict strong particle decay branches from our AAS state, but as
remarked earlier it is likely to mix with close-lying states of the same spin-
parity. We therefore expect its decay pattern to be very similar to that of
the 18 MeV level, an expectation that agrees with the current experimental
knowledge Mad09a .

Moving now to the 11Beg.s. calculation, they are based on a shell-model
strength for the 9Be ground state transition that is a factor 4 above the
experimental value. Our above predictions of a Gamow-Teller strength of 0.07
to a level at 2.89 MeV excitation energy in 11Be and a strength of 0.02
situated around the 2$n$-threshold (see Table 5) should therefore be scaled
down and are then below the experimental strength of 0.075 to the $3/2^{-}$
levels at 2.69 MeV and 3.97 MeV. We shall return below to the question of what
the origin of the strength to the low-lying states in 11Be is. Still, we note
that Fig. 5 predicts strength to the states just below the one neutron
threshold in 10Be as well as strength going directly to the two-neutron
continuum. The barely bound states are experimentally known to be fed and have
been proposed to have halo structure causing them to be preferentially fed in
the 11Li decay, see Mat09  and references therein. Here again the experimental
strength points to the presence of other contributions.

###  VII.4 Core decay

Model calculations for core decays into excited 9Be states would require five-
body calculations and are not currently doable. The following observations are
therefore of a more speculative nature: Both decaying nuclei, 9Li and 11Li
have large Q-values with low particle separation energies in the daughter
nuclei and final states that in many cases can be considered composed of alpha
particles and neutrons. The observable beta-strength will be concentrated at
high excitation energy and therefore clearly in the continuum. The spatial
extension of the halo neutrons will tend to have larger overlap with final
nucleons at low energy whereas the excitation energy residing in the core will
be higher. The (final state) interactions between nucleons and alpha particles
are strong at relative energies up to a few MeV, so it is not surprising that
final states in the 9Li decay tend to cluster into intermediate 8Be+n and
5He+$\alpha$ states, see e.g. figure 1 in Pre05 . A similar clustering has
been found Mad08  for 11Li decays into two charged fragments where final state
4He+7He structures have been clearly seen and there is evidence also for a
6He(2+)+5He channel; both configurations appear by adding two neutrons to one
fragment in the $\alpha$+5He channel. The different final channels may be
difficult to distinguish at low total energies, so the first few MeV above the
two-neutron threshold in 11Be could be challenging experimentally as well as
numerically.

The first excited state in 9Be fed in beta decay is the narrow (0.78(13) keV)
$5/2^{-}$ level at 2.43 MeV where the $B$(GT) is 0.046(5) til04 . This could
correspond to the $5/2^{-}$ 10.6 MeV level in 11Be (see Fig. 7) to which the
branching ratio seems to be around 2% (adding the values in Mat09 ; Mad08 ,
the branching ratio from Hir05  is higher, but is inconsistent with the data
of Mad08 ) giving a $B$(GT) around 0.1.

The second state fed in 9Be is the very broad (1.08(11) MeV) $1/2^{-}$ level
at 2.78 MeV. It is the only $1/2^{-}$ level fed in the 9Li decay and has a
strong $2\alpha+$n clustering. A 11Li beta decay with this “core component”
would naturally lead to $2\alpha+3$n final configurations. The higher-lying
states fed in 9Be are also broad and would also be expected to lead to high-
lying multi-body final states. As a rough indicator, the partial half life for
all excited states in 9Be — all leading to $2\alpha+$n— is around 350 ms,
which would correspond to a 11Li branching ratio of 2.5% that is close to
$P_{3n}=1.9(2)$%. This lends support to the suggested parallel between the 9Li
and 11Li decays.

The main fact speaking against this suggested parallel is that the spin-parity
of the strongly fed 11.8 MeV state in 9Be is $5/2^{-}$, whereas the
corresponding strongly fed 18 MeV state in 11Be is reported as $3/2^{-}$, see
figure 7. Both spin determinations come from angular distributions assuming
the decay channels could be clearly separated. If this discrepancy is
confirmed when a better decay scheme is established, a core decay scenario
would be excluded.

###  VII.5 Decay into $t_{h}=0$ components

Finally, let us consider the additional halo decay components. The IAS and AAS
correspond to $t_{h}=1$ final state components. Decay into the $t_{h}=0$
components ($T=\frac{3}{2}$, $T_{z}=-\frac{3}{2}$) will include core-deuteron
structures. These states can be populated after Gamow-Teller beta decay of
11Lig.s., and the corresponding strength will be given by the first term in
Eq.(III.4) only, i.e., the daughter nucleus is populated by decay of one of
the halo neutrons in 11Lig.s..

Figure 8: Gamow-Teller strength, Eq.(III.5), for decay of 11Lig.s. into
$t_{h}=0$ states in 11Be with total angular momentum and parity
$J^{\pi}=\frac{3}{2}^{-}$ (solid), $J^{\pi}=\frac{1}{2}^{-}$ (dashed), and
$J^{\pi}=\frac{5}{2}^{-}$ (dotted). The sum of the three terms is shown by the
brown (thin-solid) curve. The strength is given as a function of the relative
9Lig.s.-deuteron energy. The experimental data are from Rii22 ; Raa08 .

In Fig. 8 we plot the Gamow-Teller strength for decay of 11Lig.s. into
$t_{h}=0$ states in 11Be with total angular momentum and parity
$J^{\pi}=3/2^{-}$ (solid), $J^{\pi}=1/2^{-}$ (dashed), and $J^{\pi}=5/2^{-}$
(dotted). In the case of $J^{\pi}=5/2^{-}$ there is a 11Be state about 0.8 MeV
below the 9Lig.s.+deuteron threshold, which is responsible for the large peak
observed in the dotted curve. For the other two cases, $J^{\pi}=1/2^{-}$ and
$J^{\pi}=3/2^{-}$, all the computed states are found above the deuteron
threshold. The sum of the three terms is shown by the thin-solid curve. The
total integrated Gamow-Teller strengths are 1.21, 0.65, and 1.88 for the
$3/2^{-}$, $1/2^{-}$, and $5/2^{-}$ cases, respectively. There is at the
moment no experimental evidence for the $5/2^{-}$ level below the threshold,
but decays directly to the deuteron continuum have been seen, the latest
experiment Rii22 ; Raa08  observing deuterons in the energy range 0.2 MeV to
1.6 MeV above the deuteron threshold with a total experimental $B$(GT) of 0.7
Rii22 . These experimental data are shown by the dots in the figure. The
predicted integrated strengths in this range are 0.52, 0.25, and 0.11 for the
$3/2^{-}$, $1/2^{-}$, and $5/2^{-}$ cases, respectively. The sum of 0.88 is
slightly above the experimental value and the theoretical strength increases
with energy whereas the experimental is close to constant.

###  VII.6 Decays outside the model

What remains is the decays that are outside our model space, namely the final
states containing proton $p_{3/2}$ states fed by decay of a $p_{1/2}$ neutron.
The $\bm{\sigma}$ operator favours the $p_{1/2}$ to $p_{3/2}$ transition to
$p_{1/2}$ to $p_{1/2}$. For a simple estimate, the existing proton in 11Li
will block one out of the four $p_{3/2}$ substates and the two protons will
couple to $0^{+}$ or $2^{+}$, these states in 10Be can then couple with the
remaining $p_{1/2}$ neutron to give $1/2^{-}$ or $3/2^{-}$ and $5/2^{-}$
states, respectively. The former could explain the feeding to the first
excited 11Be state Suz94 , the latter will contribute to the excited states up
to around 9 MeV.

We have calculated the one-particle wave function overlap between a 11Li
$p_{1/2}$ neutron and a well-bound proton or the halo neutron in the first
excited 11Be state to be 0.75 and 0.96, respectively. The $(p_{1/2})^{2}$
fraction in 11Li is 0.35, the $p_{1/2}$ to $p_{3/2}$ fraction is 8/9, adding
the Pauli blocking, the Gamow-Teller quenching and the wave function overlap
all in all reduces the total halo strength of 6 to around 0.9. The sum of the
currently assigned experimental strength to this region is around 0.6, which
is in fair agreement.

##  VIII Conclusion

We have extended the theoretical description of three-body systems to include
isospin explicitly, thereby allowing a treatment of beta-decay processes. The
framework reproduces selected parts of the decay well, in particular the decay
to the isobaric analogue state. It is also expected to reproduce the main
features of the decay to the anti-analogue state that, however, can mix with
surrounding levels. A distinct advantage is the ability to treat decays (both
weak and strong) to final continuum states thereby giving a full description
of the decay process. This is important for decays of halo nuclei. We have
applied the framework to the decay of 11Li that is a challenging case due to
the non-zero spin of the core, 9Li, and to the many relevant final-state
channels.

Our description of both the IAS and AAS, where we identify the latter as the
16.3 MeV resonance, in 11Be is consistent with all existing experimental data.
We adjust the position of the IAS and then predict the position of the AAS to
be close to the experimental one. The decay width of the IAS matches the
experiment, and the decay pattern of the AAS is close to that of the nearby 18
MeV level, as could be expected. We note that the different potentials for
core nucleon systems with $t_{cN}=2$ and $t_{cN}=1$ have been fitted
independently, but that the strengths of the central potentials for different
angular momenta are consistent with the difference being due to a common
isospin-isospin nucleon-nucleon potential, as discussed in Appendix A. Our
predicted isospin mixing for the IAS of $3.7\times 10^{-3}$ is very large for
a light nucleus and significantly larger than expected for more bound nuclei.

The core internal degrees of freedom are only taken crudely into account
through the matrix elements for decay of 9Li into 9Be states, and we
furthermore only calculate explicitly results for decays into the 9Be ground
state. Nevertheless, a consistent overall interpretation of the beta decay of
11Li emerges as follows: the core beta decay explains much of the structure
from around the two-neutron threshold up to the Q-value. The decay to the
ground state of 9Be has low intensity (actually, the sum of $B$(GT) values to
the lowest 9 MeV in 9Be is of order 0.1, only) and will give a small
contribution to the 11Li decays to the lowest 9 MeV in 11Be that has a summed
$B$(GT) of order 0.6. These decays get a larger contribution from the decay of
the $p_{1/2}^{2}$ halo component where a produced $p_{3/2}$ proton couples
with the core to make intermediate 10Be $0^{+}$ and $2^{+}$ states. Apart from
the transition to the AAS mentioned above and the transition to the 18 MeV
state that is assumed to correspond to the 9Li transition to the 11.8 MeV
state (note however the need for a reinvestigation of this particular branch),
the other high beta strength transition at high excitation is the beta-delayed
deuteron branch that is fairly well reproduced as a halo beta decay.

The large Qβ value for 11Li is explained by the fact that adding two neutrons
to 9Li produces a barely bound nucleus, whereas adding two neutrons to 9Be
gives a significant energy gain, see Fig. 7. The gain comes mainly from the
first neutron that fills the $p_{3/2}$ neutron orbit, whereas the second
neutron again produces barely bound states. The above interpretation of where
core and halo decays proceed to nicely fit this observation.

There are several clear consequences of this interpretation that can be tested
experimentally. In general, the spectator halo neutrons from core decays
should give low-energy final state neutrons, this is consistent with the
results from the two-neutron coincidence spectrum in Del19  where the majority
of the coincident neutrons lie below 2 MeV. A specific requirement is that the
11.8 MeV state in 9Li must have spin $3/2^{-}$ rather than the current
experimental value of $5/2^{-}$.

There is a need for improved experimental data on both Li-decays. The current
discrepancies for the decay scheme of 11Li must be resolved and more detailed
data on decays feeding the two- and three-neutron unbound continuum would be
very valuable, although challenging to obtain as neutron detection is
difficult. For 9Li it is important to look closer at the strength residing
above 10 MeV, both in order to resolve the puzzle of the spin of the 11.8 MeV
level and in order to determine whether other levels contribute as predicted
by several models. A renewed look at and comparison with the mirror decay of
9C may be useful.

The beta decay of 11Li is challenging for several reasons. It would be
interesting to test the methods developed here to decays of other dripline
nuclei that can be described with few-body methods.

###### Acknowledgements.

This work has been partially supported by the Independent Research Fund
Denmark (9040-00076B), by grant No.PGC2018-093636-B-I00, funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”.

##  Appendix A Isospin and interactions

The basic nucleon-nucleon interaction contains isospin-isospin terms, e.g.
from one-pion exchange, and as Lane lan62  pointed out this will for a system
with one nucleon outside a core give an interaction term
$(U/A_{c})\bm{t}\cdot\bm{t}_{c}$ where the constant $U$ in mid-mass nuclei is
of order 100 MeV. Robson rob65  showed how this naturally leads to isobaric
analogue states of high purity, and also noted that the isospin mixing mainly
arises from the region external to the nucleus. These argument can be
generalized to our case. The isospin-isospin interaction for a two-neutron
halo will have the form
$(U/A)(\bm{t}_{2}\cdot\bm{t}_{c}+\bm{t}_{3}\cdot\bm{t}_{c}+\bm{t}_{2}\cdot\bm{t}_{3})=(U/A)(\bm{t}_{h}\cdot\bm{t}_{c}+\bm{t}_{2}\cdot\bm{t}_{3})$.
Here the first term can be rewritten as
$\bm{t}_{h}\cdot\bm{t}_{c}=1/2(t_{h}^{+}t_{c}^{-}+t_{h}^{-}t_{c}^{+})+t_{h}^{z}t_{c}^{z}$.
It is straightforward to verify that the explicit charge-exchange terms will
transform the two components of the wave functions in Eqs.(3) and (4) into
each other and thereby, in the absence of isospin-breaking terms in the
Hamiltonian, enforce the structure of the IAS and the AAS.

The AAS is established in many light nuclei, e.g. in 12C, where it is situated
at 12.71 MeV somewhat below the IAS at 15.11 MeV. It is known to play a role
in both gamma and beta decays, see e.g. the contributions in wil69 , and often
gives the main contribution to isospin purity violation zel17 .

The spatial and spin parts of the wave functions should have the same
structure for the IAS and the AAS (the spatial extent can differ, as shown in
Fig. 2b), and a substantial part of the difference in their energies will
therefore come from the isospin-isospin interaction. For a two-neutron halo
the $\bm{t}_{2}\cdot\bm{t}_{3}$ term gives the same contribution in the two
states, so the relevant quantity is the expectation value of
$\bm{t}_{h}\cdot\bm{t}_{c}$. This differs by $t_{h}+t_{c}$ between the AAS and
the IAS. However, the effective value of $U$ will be less than for bulk
nuclear matter: we are in light nuclei and furthermore are dealing with
loosely bound structures, so since the nucleon-nucleon isospin-isospin
interaction is of short range its average value will fall below a simple $1/A$
scaling. The value of
$E_{\mathrm{IAS}}-E_{\mathrm{AAS}}=(U_{\mathrm{eff}}/A_{c})(t_{h}+t_{c})$ may
be estimated from 12C. Scaling with mass number and isospin this would give a
difference of about 7 MeV between the IAS and the AAS of 11Li, a value that
should be taken as an upper limit since the effect of the halo extension is
not considered.

In our two-body potentials in Section V.1 we have not employed any isospin
dependent potential, but rather deduced potentials for each isospin sector. It
is therefore interesting that the difference in the strengths for $t_{cN}=2$
and $t_{cN}=1$ in Table 1 is of similar order-of-magnitude for the central
potential, consistent with the isospin dependence being effectively included.

##  Appendix B Potential matrix elements for 11Lig.s., 11Beg.s., 11BeIAS, and
11BeAAS

Here we specify how the matrix elements of the core-nucleon potentials in
between the different terms of the basis set $\\{|{\cal
Y}_{q}\rangle|TT_{z}\rangle\\}$ are computed for the particular cases of
11Lig.s., 11Beg.s., 11BeIAS, and 11BeAAS.

From Eq.(12) we immediately get that after rotation from the first to the
second Jacobi set, the isospin wave functions for the 11Lig.s., 11Beg.s.,
11BeIAS, and 11BeAAS states is given by:

$|\mbox{${}^{11}$Li}_{\mathrm{g.s.}}\rangle=|(t_{2},t_{3})t_{h}=1,t_{c}=\frac{3}{2};T=\frac{5}{2},T_{z}=-\frac{5}{2}\rangle=|(t_{3},t_{c})t_{cN}=2,t_{2}=\frac{1}{2};T=\frac{5}{2},T_{z}=-\frac{5}{2}\rangle,$
(35)
$|\mbox{${}^{11}$Be}_{\mathrm{g.s.}}\rangle=|(t_{2},t_{3})t_{h}=1,t_{c}=\frac{1}{2};T=\frac{3}{2},T_{z}=-\frac{3}{2}\rangle=|(t_{3},t_{c})t_{cN}=1,t_{2}=\frac{1}{2};T=\frac{3}{2},T_{z}=-\frac{3}{2}\rangle,$
(36)
$|\mbox{${}^{11}$Be}_{\mathrm{IAS}}\rangle=|(t_{2},t_{3})t_{h}=1,t_{c}=\frac{3}{2};T=\frac{5}{2},T_{z}=-\frac{3}{2}\rangle=|(t_{3},t_{c})t_{cN}=2,t_{2}=\frac{1}{2};T=\frac{5}{2},T_{z}=-\frac{3}{2}\rangle,$
(37) $\displaystyle|\mbox{${}^{11}$Be}_{\mathrm{AAS}}\rangle$ $\displaystyle=$
$\displaystyle|(t_{2},t_{3})t_{h}=1,t_{c}=\frac{3}{2};T=\frac{3}{2},T_{z}=-\frac{3}{2}\rangle$
(38) $\displaystyle=$
$\displaystyle\sqrt{\frac{5}{8}}|(t_{3},t_{c})t_{cN}=1,t_{2}=\frac{1}{2};T=\frac{3}{2},T_{z}=-\frac{3}{2}\rangle-\sqrt{\frac{3}{8}}|(t_{3},t_{c})t_{cN}=2,t_{2}=\frac{1}{2};T=\frac{3}{2},T_{z}=-\frac{3}{2}\rangle,$

where we can see that 11Lig.s. and 11BeIAS can only hold $t_{cN}=2$ values
(otherwise the total isospin $T=\frac{5}{2}$ could not be reached) and the
11Beg.s. can only have $t_{cN}=1$ (otherwise the total isospin $T=\frac{3}{2}$
could not be reached). On the contrary, 11BeAAS mixes two different $t_{cN}$
values, $t_{cN}=1$ and $t_{cN}=2$.

From the expressions above, and making use of Eq.(13), we can split the matrix
elements corresponding to the core-nucleon interaction into the different
$t_{cN}$ parts:

$\langle{\cal Y}_{q}\mbox{${}^{11}$Li}_{\mathrm{g.s.}}|V_{cN}|{\cal
Y}_{q^{\prime}}\mbox{${}^{11}$Li}_{\mathrm{g.s.}}\rangle=\langle{\cal
Y}_{q};(t_{3},t_{c})t_{cN}=2,t_{2}=\frac{1}{2};\frac{5}{2},-\frac{5}{2}|V_{cN}^{(t_{cN}=2)}|{\cal
Y}_{q^{\prime}};(t_{3},t_{c})t_{cN}=2,t_{2}=\frac{1}{2};\frac{5}{2},-\frac{5}{2}\rangle,$
(39) $\langle{\cal Y}_{q}\mbox{${}^{11}$Be}_{\mathrm{g.s.}}|V_{cN}|{\cal
Y}_{q^{\prime}}\mbox{${}^{11}$Be}_{\mathrm{g.s.}}\rangle=\langle{\cal
Y}_{q};(t_{3},t_{c})t_{cN}=1,t_{2}=\frac{1}{2};\frac{3}{2},-\frac{3}{2}|V_{cN}^{(t_{cN}=1)}|{\cal
Y}_{q^{\prime}};(t_{3},t_{c})t_{cN}=1,t_{2}=\frac{1}{2};\frac{3}{2},-\frac{3}{2}\rangle,$
(40) $\langle{\cal Y}_{q}\mbox{${}^{11}$Be}_{\mathrm{IAS}}|V_{cN}|{\cal
Y}_{q^{\prime}}\mbox{${}^{11}$Be}_{\mathrm{IAS}}\rangle=\langle{\cal
Y}_{q};(t_{3},t_{c})t_{cN}=2,t_{2}=\frac{1}{2};\frac{5}{2},-\frac{3}{2}|V_{cN}^{(t_{cN}=2)}|{\cal
Y}_{q^{\prime}};(t_{3},t_{c})t_{cN}=2,t_{2}=\frac{1}{2};\frac{5}{2},-\frac{3}{2}\rangle,$
(41)

and

$\displaystyle\langle{\cal
Y}_{q}\mbox{${}^{11}$Be}_{\mathrm{AAS}}|V_{cN}|{\cal
Y}_{q^{\prime}}\mbox{${}^{11}$Be}_{\mathrm{AAS}}\rangle$ $\displaystyle=$
$\displaystyle\frac{5}{8}\langle{\cal
Y}_{q};(t_{3},t_{c})t_{cN}=1,t_{2}=\frac{1}{2};\frac{3}{2},-\frac{3}{2}|V_{cN}^{(t_{cN}=1)}|{\cal
Y}_{q^{\prime}};(t_{3},t_{c})t_{cN}=1,t_{2}=\frac{1}{2};\frac{3}{2},-\frac{3}{2}\rangle$
(42) $\displaystyle\\!\\!\\!\\!\\!+$
$\displaystyle\\!\\!\\!\\!\\!\frac{3}{8}\langle{\cal
Y}_{q};(t_{3},t_{c})t_{cN}=2,t_{2}=\frac{1}{2};\frac{3}{2},-\frac{3}{2}|V_{cN}^{(t_{cN}=2)}|{\cal
Y}_{q^{\prime}};(t_{3},t_{c})t_{cN}=2,t_{2}=\frac{1}{2};\frac{3}{2},-\frac{3}{2}\rangle$

where we have assumed that the potential does not mix different values of the
core-nucleon isospin, $t_{cN}$.

Finally, from Eq.(14) we can find the specific core-nucleon states, and
therefore the specific core-nucleon interactions, involved in the different
three-body systems under consideration:

$\langle{\cal Y}_{q};\mbox{${}^{11}$Li}_{\mathrm{g.s.}}|V_{cN}|{\cal
Y}_{q^{\prime}};\mbox{${}^{11}$Li}_{\mathrm{g.s.}}\rangle=\langle{\cal
Y}_{q};\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+n|V_{cN}^{(t_{cN}=2)}|{\cal
Y}_{q^{\prime}};\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+n\rangle,$ (43)
$\langle{\cal Y}_{q};\mbox{${}^{11}$Be}_{\mathrm{g.s.}}|V_{cN}|{\cal
Y}_{q^{\prime}};\mbox{${}^{11}$Be}_{\mathrm{g.s.}}\rangle=\langle{\cal
Y}_{q};\mbox{${}^{9}$Be${}_{\mathrm{g.s.}}$}+n|V_{cN}^{(t_{cN}=1)}|{\cal
Y}_{q^{\prime}};\mbox{${}^{9}$Be${}_{\mathrm{g.s.}}$}+n\rangle,$ (44)
$\displaystyle\langle{\cal
Y}_{q}\mbox{${}^{11}$Be}_{\mathrm{IAS}}|V_{cN}|{\cal
Y}_{q^{\prime}}\mbox{${}^{11}$Be}_{\mathrm{IAS}}\rangle=\frac{1}{5}\langle{\cal
Y}_{q};\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+n|V_{cN}^{(t_{cN}=2)}|{\cal
Y}_{q^{\prime}};\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+n\rangle$ (45)
$\displaystyle+\frac{1}{5}\langle{\cal
Y}_{q};\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+p|V_{cN}^{(t_{cN}=2)}|{\cal
Y}_{q^{\prime}};\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+p\rangle+\frac{3}{5}\langle{\cal
Y}_{q};\mbox{${}^{9}$Be${}_{\mathrm{IAS}}$}+n|V_{cN}^{(t_{cN}=2)}|{\cal
Y}_{q^{\prime}};\mbox{${}^{9}$Be${}_{\mathrm{IAS}}$}+n\rangle,$
$\displaystyle\langle{\cal
Y}_{q}\mbox{${}^{11}$Be}_{\mathrm{AAS}}|V_{cN}|{\cal
Y}_{q^{\prime}}\mbox{${}^{11}$Be}_{\mathrm{AAS}}\rangle=\frac{5}{32}\langle{\cal
Y}_{q};\mbox{${}^{9}$Be${}_{\mathrm{IAS}}$}+n|V_{cN}^{(t_{cN}=1)}|{\cal
Y}_{q^{\prime}};\mbox{${}^{9}$Be${}_{\mathrm{IAS}}$}+n\rangle$ (46)
$\displaystyle+\frac{9}{160}\langle{\cal
Y}_{q};\mbox{${}^{9}$Be${}_{\mathrm{IAS}}$}+n|V_{cN}^{(t_{cN}=2)}|{\cal
Y}_{q^{\prime}};\mbox{${}^{9}$Be${}_{\mathrm{IAS}}$}+n\rangle+\frac{15}{32}\langle{\cal
Y}_{q};\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+p|V_{cN}^{(t_{cN}=1)}|{\cal
Y}_{q^{\prime}};\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+p\rangle$
$\displaystyle+\frac{3}{160}\langle{\cal
Y}_{q};\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+p|V_{cN}^{(t_{cN}=2)}|{\cal
Y}_{q^{\prime}};\mbox{${}^{9}$Li${}^{\mathrm{g.s.}}$}+p\rangle+\frac{3}{10}\langle{\cal
Y}_{q};\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+n|V_{cN}^{(t_{cN}=2)}|{\cal
Y}_{q^{\prime}};\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}+n\rangle.$

Therefore, the required interactions are: $i)$ The interaction between the
9Lig.s. core and the neutron (only $t_{cN}=2$ is possible), $ii)$ the
interaction between the 9Beg.s. core and the neutron (only $t_{cN}=1$ is
possible), $iii)$ the interaction between the 9Lig.s. core and the proton for
$t_{cN}=2$, $iv)$ the interaction between the 9BeIAS core and the neutron for
$t_{cN}=2$, $v)$ the interaction between the 9Lig.s. core and the proton for
$t_{cN}=1$, and $vi)$ the interaction between the 9BeIAS core and the neutron
for $t_{cN}=1$.

##  Appendix C ${\cal H}_{\mathrm{core}}$ matrix elements for 11Lig.s.,
11Beg.s., 11BeIAS, and 11BeAAS

Following Eq.(8), we have that the effect of the core hamiltonian, ${\cal
H}_{\mathrm{core}}$, on the different core states involved in the three-body
systems considered in this work, can be expressed as:

${\cal
H}_{\mathrm{core}}|t_{c}=\frac{3}{2},t_{c}^{z}=-\frac{3}{2}\rangle={\cal
H}_{\mathrm{core}}|\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}\rangle=\xi_{\mbox{\scriptsize${}^{9}$Li${}_{\mathrm{g.s.}}$}}|\mbox{${}^{9}$Li${}_{\mathrm{g.s.}}$}\rangle,$
(47) ${\cal
H}_{\mathrm{core}}|t_{c}=\frac{1}{2},t_{c}^{z}=-\frac{1}{2}\rangle={\cal
H}_{\mathrm{core}}|\mbox{${}^{9}$Be${}_{\mathrm{g.s.}}$}\rangle=\xi_{\mbox{\scriptsize${}^{9}$Be${}_{\mathrm{g.s.}}$}}|\mbox{${}^{9}$Be${}_{\mathrm{g.s.}}$}\rangle,$
(48) ${\cal
H}_{\mathrm{core}}|t_{c}=\frac{3}{2},t_{c}^{z}=-\frac{1}{2}\rangle=|\mbox{${}^{9}$Be}_{\mathrm{IAS}}\rangle=\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{IAS}}}|\mbox{${}^{9}$Be}_{\mathrm{IAS}}\rangle,$
(49)

where $\xi_{\mbox{\scriptsize${}^{9}$Li${}_{\mathrm{g.s.}}$}}$,
$\xi_{\mbox{\scriptsize${}^{9}$Be${}_{\mathrm{g.s.}}$}}$, and
$\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{IAS}}}$ are the energies of the
9Lig.s., 9Beg.s., and 9BeIAS cores, respectively.

Making now use of Eqs.(26) to (29), we can then easily get the matrix elements
of ${\cal H}_{\mathrm{core}}$ between the different terms of the basis set:

$\langle{\cal Y}_{q}\mbox{${}^{11}$Li}_{\mathrm{g.s.}}|{\cal
H}_{\mathrm{core}}|{\cal
Y}_{q^{\prime}}\mbox{${}^{11}$Li}_{\mathrm{g.s.}}\rangle=\xi_{\mbox{\scriptsize${}^{9}$Li${}_{\mathrm{g.s.}}$}}\delta_{qq^{\prime}},$
(50) $\langle{\cal Y}_{q}\mbox{${}^{11}$Be}_{\mathrm{g.s.}}|{\cal
H}_{\mathrm{core}}|{\cal
Y}_{q^{\prime}}\mbox{${}^{11}$Be}_{\mathrm{g.s.}}\rangle=\xi_{\mbox{\scriptsize${}^{9}$Be${}_{\mathrm{g.s.}}$}}\delta_{qq^{\prime}},$
(51) $\langle{\cal Y}_{q}\mbox{${}^{11}$Be}_{\mathrm{IAS}}|{\cal
H}_{\mathrm{core}}|{\cal
Y}_{q^{\prime}}\mbox{${}^{11}$Be}_{\mathrm{IAS}}\rangle=\left(\frac{2}{5}\xi_{\mbox{\scriptsize${}^{9}$Li${}_{\mathrm{g.s.}}$}}+\frac{3}{5}\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{IAS}}}\right)\delta_{qq^{\prime}},$
(52) $\langle{\cal Y}_{q}\mbox{${}^{11}$Be}_{\mathrm{AAS}}|{\cal
H}_{\mathrm{core}}|{\cal
Y}_{q^{\prime}}\mbox{${}^{11}$Be}_{\mathrm{AAS}}\rangle=\left(\frac{3}{5}\xi_{\mbox{\scriptsize${}^{9}$Li${}_{\mathrm{g.s.}}$}}+\frac{2}{5}\xi_{\mbox{\scriptsize${}^{9}$Be}_{\mathrm{IAS}}}\right)\delta_{qq^{\prime}},$
(53)

which are consistent with the general expressions given in Eqs.(9), (10), and
(11). We have assumed that ${\cal H}_{\mathrm{core}}$ does not mix the
$\mbox{${}^{9}$Li}_{\mathrm{g.s.}}$ and $\mbox{${}^{9}$Be}_{\mathrm{IAS}}$
wave functions.

##  Appendix D Decay data for 9Li and 11Li

Both decays involve broad (and partially overlapping) resonances in the
daughter nuclei so that branching ratios and beta strength may not always be
uniquely attributed to a given level. A discussion is given in Rii14  that
suggests to consider $B$(GT) as a function of the final-state energy when
comparing theory and experiment. For narrow levels the usual relation between
$B$(GT) and the $ft$-value is
$ft=6144(4)/(B(\mbox{F})+(g_{A}/g_{V})^{2}B(\mbox{GT}))$ where the conversion
factor is taken from superallowed Fermi transitions Har20  and neutron decay
gives $g_{A}/g_{V}=-1.2754(13)$ pdg20 . The empirical quenching factor for
Gamow-Teller strength was for light nuclei estimated in Cho93  to be 0.845 and
0.833 for mass 9 and 11 nuclei, giving effective values for
$(g_{A}/g_{V})^{2}$ of 1.16 and 1.13 that we shall use when extracting
experimental $B$(GT) values.

###  D.1 The 9Li decay

A detailed overview of the current knowledge on the 9Li decay can be found
from til04 ; Pre05 . It has historically been difficult to unravel the decays
via excited 9Be states, see figure 7, since the obvious decay channels,
through $\alpha$ \+ 5He and through n + 8Be, lead to the same final continuum
and can overlap in energy. The currently established branches go to the
$3/2^{-}$ ground state and excited states at 2.43 MeV ($5/2^{-}$), 2.8 MeV
($1/2^{-}$), 5.0 MeV ($3/2^{-}$), 7.9 MeV ($5/2^{-}$) and 11.8 MeV
($5/2^{-}$). The main Gamow Teller strength goes to the 11.8 MeV state, but
has been difficult to determine accurately; different approaches give numbers
in the range 5 to 8. The ground state transition has a strength of 0.026(1)
and the states in between have strengths of similar order of magnitude. For
the five lowest states this is all very consistent with earlier theoretical
treatments Mik88 ; Cho93 ; Suz97 ; Kan10  that all predict several strong
transitions in the energy range 10–13 MeV where only one has been observed.

###  D.2 The 11Li decay

A good overview of the existing data on the 11Li decay can be found in the
review tunl11 , the only more recent experimental result being the first hint
at the energy distribution for two-neutron coincidences Del19 . Most
experiments have focused on the low-energy range, roughly up to around the
two-neutron threshold in 11Be, and the overall features of the decay are known
here. However, the detailed interpretation differs between the most recent
experiments Mat09 ; Hir05  and there is still only few data from the higher-
energy region, so the distribution of beta strength is still uncertain. Figure
7 displays some of the levels that currently are believed to play a role in
the decay.

Around 93% of the decay goes to neutron unbound states, the bound state
feeding goes to the first excited $1/2^{-}$ halo state in 11Be. The
measurement of Hirayama et al. Hir05  detected neutrons as well as gamma-rays
and beta polarization. The extracted beta-asymmetry is for most neutron
energies consistent with 11Be spins of $3/2^{-}$ or $5/2^{-}$ (or a mixture of
both), it is only in very small regions at high neutron energy that $1/2^{-}$
is indicated with perhaps an indication of a contribution also a low neutron
energies. Their suggested level scheme differs in many details from the one
extracted from the latest analysis Mat09  of $\beta$-$n$-$\gamma$ decays, but
agree on including essentially the same $3/2^{-}$ and $5/2^{-}$ states up to
10.6 MeV. In terms of branching ratios they indicate that around 58% of the
decays go to states below the two-neutron threshold, and around 32% to states
above this up to the 10.6 MeV state. The total assigned probability for beta-
delayed one-neutron emission does not exceed 70%, which is considerably below
the tabulated value of $P_{1n}=86.3(9)$%.

Only one $\beta$-$n$-$\gamma$ experiment Fyn04  reported neutrons from states
at higher energies, the reported intensity of 6% does not include neutron
emission to the ground state nor the branching of around 1% to n-$\alpha$-6He.
The multi-neutron emission probabilities of $P_{2n}=4.1(4)$% and
$P_{3n}=1.9(2)$% combined with the small fractions of order $10^{-4}$ going
via $\beta$d and $\beta$t shows that a sizeable strength (and a non-negligible
branching) will go to the multi-particle continuum. The most recent
measurements on branches with two charged particles Mad08 ; Mad09a  point to a
significant role played by three levels at 10.6 MeV, 16.3 MeV and just above
18 MeV. The latter level has also been invoked in $\beta$t and $\beta$n
branches, whereas the $\beta$d decay is normally assumed to proceed directly
to the continuum. However, the overall experimental picture leaves many
questions, in particular above the two-neutron threshold.

## References

  * (1) A.S. Jensen, K. Riisager, D.V. Fedorov, E. Garrido, Rev. Mod. Phys. 76 (2004) 215.
  * (2) K. Riisager, Phys. Scr. T 152 (2013) 014001.
  * (3) B. Blank and M.J.G. Borge, Prog. Part. Nucl. Phys. 60 (2008) 403.
  * (4) M. Pfützner, M. Karny, L.V. Grigorenko, K. Riisager, Rev. Mod. Phys. 84 (2012) 567.
  * (5) K. Riisager, Beta Decay of Halo Nuclei. In: Tanihata, I., Toki, H., Kajino, T. (eds) Handbook of Nuclear Physics. Springer, Singapore, 2022.
  * (6) T. Suzuki and T. Otsuka, Phys. Rev. C 56 (1997) 847.
  * (7) M. Madurga et al., Phys. Lett. B 677 (2009) 255.
  * (8) Y. Kanada-En’yo, Phys. Rev. C 81 (2010) 034321.
  * (9) Y. Ohbayasi and Y. Suzuki, Phys. Lett. B 346 (1995) 223.
  * (10) M.V. Zhukov, B.V. Danilin, L.V. Grigorenko, and J.S. Vaagen, Phys. Rev. C 52 (1995) 2461.
  * (11) D. Baye, E.M. Tursunov and P. Descouvemont, Phys. Rev. C 74 (2006) 064302.
  * (12) E. Nielsen, D.V. Fedorov, A.S. Jensen, and E. Garrido, Phys. Rep. 347, 373 (2001).
  * (13) E. Garrido and A.S. Jensen, Phys. Rev. C 101 (2020) 034003.
  * (14) T. Nilsson, G. Nyman, K. Riisager, Hyperfine Interactions 129 (2000) 67.
  * (15) W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Cambridge University Press (1997) Chapter 19.
  * (16) B. Jonson, Phys. Reports 389 (2004) 1.
  * (17) R. Álvarez-Rodríguez, A.S. Jensen, E. Garrido and D.V. Fedorov, Phys. Rev. C 82 (2010) 034001.
  * (18) H. Sagawa, I. Hamamoto and M. Ishihara, Phys. Lett. B 303 (1993) 215.
  * (19) T. Suzuki and T. Otsuka, Phys. Rev. C 50 (1994) R555.
  * (20) D.R. Tilley, J.H. Kelley, J.L. Godwin, D.J. Millener, J. Purcell, C.G. Sheu, and H.R. Weller, Nucl. Phys. A 745 (2004) 155.
  * (21) E. Garrido, D.V. Fedorov, and A.S. Jensen, Phys. Rev. C 69 (2004) 024002.
  * (22) E. Garrido, D.V. Fedorov, and A.S. Jensen, Phys. Rev. C 68 (2003) 014002.
  * (23) Y. Suzuki and K. Yabana, Phys. Lett. B 272 (1991) 173.
  * (24) P.G. Hansen, A.S. Jensen and K. Riisager, Nucl. Phys. A 560 (1993) 85.
  * (25) T. Teranishi et al., Phys. Lett. B 407 (1997) 110.
  * (26) E. Garrido, D.V. Fedorov, A.S. Jensen, Nucl. Phys. A 733 (2004) 85.
  * (27) C.M. Mattoon, F. Sarazin, C. Andreoiu, A.N. Andreyev, R.A.E. Austin, G.C. Ball et al., Phys. Rev. C 80 (2009) 034318.
  * (28) Y. Prezado et al., Phys. Lett. B 618 (2005) 43.
  * (29) M. Madurga et al., Nucl. Phys. A 810 (2008) 1.
  * (30) Y. Hirayama et al., Phys. Lett. B 611 (2005) 239.
  * (31) R. Raabe, A. Andreyev, M.J.G. Borge, L. Buchmann, P. Capel, H.O.U. Fynbo et al., Phys. Rev. Lett. 101 (2008) 212501.
  * (32) F. Delaunay et al., Il Nuovo Cimento 42 C (2019) 98.
  * (33) A.M. Lane, Nucl. Phys. 35 (1962) 676.
  * (34) D. Robson, Phys. Rev. 137 (1965) B535.
  * (35) D.H. Wilkinson (ed.), Isospin in Nuclear Physics, North-Holland Pub. Co, Amsterdam (1969).
  * (36) V. Zelevinsky and A. Volya, Physics of Atomic Nuclei, Wiley-VCH Verlag GmbH & Co (2017), p. 40.
  * (37) K. Riisager, Nucl. Phys. A 925 (2014) 112; ibid 298.
  * (38) J.C. Hardy and I.S. Towner, Phys. Rev. C 102, 045501 (2020).
  * (39) P.A. Zyla et al., Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * (40) W.T. Chou, E.K. Warburton, and B.A. Brown, Phys. Rev. C 47 (1993) 163.
  * (41) D. Mikolas et al., Phys. Rev. C 37 (1988) 766.
  * (42) J.H. Kelley et al., Nucl. Phys. A 880 (2012) 88.
  * (43) H.O.U.Fynbo et al., Nucl. Phys. A 736 (2004) 39.