# Proton inelastic scattering reveals deformation in 8He

M. Holl  R. Kanungo  Z. H. Sun  G. Hagen  J.A. Lay  A. M. Moro  P. Navrátil
T. Papenbrock  M. Alcorta  D. Connolly  B. Davids  A. Diaz Varela  M. Gennari
G. Hackman  J. Henderson  S. Ishimoto  A. I. Kilic  R. Krücken  A. Lennarz  J.
Liang  J. Measures  W. Mittig  O. Paetkau  A. Psaltis  S. Quaglioni  J. S.
Randhawa  J. Smallcombe  I. J. Thompson  M. Vorabbi  M. Williams  Astronomy
and Physics Department, Saint Mary’s University, Halifax, Nova Scotia, B3H
3C3, Canada  TRIUMF, Vancouver, BC, V6T 2A3, Canada  Physics Division, Oak
Ridge National Laboratory, Oak Ridge, TN 37831, USA  Department of Physics and
Astronomy, University of Tennessee, Knoxville, TN 37996, USA  Departamento de
Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Apartado 1065,
E-41080 Sevilla, Spain  Instituto Interuniversitario Carlos I de Física
Teórica y Computacional (iC1), Apdo. 1065, E-41080 Sevilla, Spain  Department
of Physics, University of Guelph, Guelph, ON, N1G 2W1, Canada  High Energy
Accelerator Research Organization (KEK), Ibaraki 305-0801, Japan  McMaster
University, Hamilton, ON, L8S 481, Canada  Department of Physics, University
of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom  National
Superconducting Cyclotron Laboratory, Michigan State University, East Lansing,
MI 48824-1321, USA  Department of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824-1321, USA  Lawrence Livermore National
Laboratory, P.O. Box 808, L-414, Livermore, California 94551, USA  National
Nuclear Data Center, Bldg. 817, Brookhaven National Laboratory, Upton, New
York 11973-5000, USA  Department of Physics, University of York, York YO10
5DD, United Kingdom

###### Abstract

A measurement of proton inelastic scattering of 8He at $8.25A$ MeV at TRIUMF
shows a resonance at 3.54(6) MeV with a width of 0.89(11) MeV. The energy of
the state is in good agreement with coupled cluster and no-core shell model
with continuum calculations, with the latter successfully describing the
measured resonance width as well. Its differential cross section analyzed with
phenomenological collective excitation form factor and microscopic coupled
reaction channels framework consistently reveals a large deformation parameter
$\beta_{2}$ = 0.40(3), consistent with no-core shell model predictions of a
large neutron deformation. This deformed double-closed shell at the neutron
drip-line opens a new paradigm.

###### keywords:

††journal: Physics Letters B

Helium, the second most abundant element in the universe, has a closed shell
($Z$ = 2) of protons. The $N$ = 2 closed shell of neutrons makes 4He doubly-
magic. However, the conventionally expected doubly-magic heavier isotope,
10He, is unbound. The He chain terminates at the most neutron-rich nucleus,
8He, with $N/Z=3$. It has an interesting structure with four neutrons forming
a neutron-skin around a 4He core [1]. Despite being at the neutron drip-line
of the He isotopes it has a larger two-neutron separation energy than 6He [2].
This stronger binding suggests a possible closed sub-shell at $N$ = 6 which
would make 8He a doubly closed shell nucleus. Our knowledge thus far has shown
the handful of doubly closed-shell nuclei to be spherical. Here we investigate
if that holds true for 8He from its inelastic excitation that characterizes
nuclear deformation.

The measured charge radius of 8He is smaller than that of 6He [3]. This
decrease in charge radius compared to the preceding isotope is consistent with
other $N$ = 6 isotones [4] providing a tantalizing hint of a sub-shell gap in
He, Li and Be. This sub-shell feature has also been discussed in Li isotopes
in terms of spectroscopic studies and neutron separation energies [5]. The
matter radius of 8He is slightly larger than that of 6He while both are more
extended than 4He [6, 7, 8]. 6He exhibits a two-neutron halo, while four
neutrons form the neutron skin in 8He. Reactions of 6He and 8He on a Au target
[9] also demonstrate differences in transfer of neutron pairs in the two
nuclei, indicative of different configurations. In order to understand the
nature of the potential sub-shell gap at $N$ = 6 however, a precise knowledge
of the low-lying excited-state(s) in 8He is required - which thus far remains
elusive.

The nucleus 8He has no bound excited states. Several experiments, with limited
statistics, report unbound states. An initial study using inelastic proton
scattering at 72$A$ MeV identified the first excited state to be $2^{+}$ at an
excitation energy of 3.57(12) MeV with a width of $\Gamma=0.50(35)$ MeV [10].
The excitation energy resolution however was $\sim$ 1.8 MeV FWHM, which
renders this determination of the resonance width ambiguous. In Ref.[11] this
data is explained using phenomenological density distributions with a
quadrupole deformation parameter of 0.3 in an eikonal model analysis. A
coupled channel analysis of the angular distribution with microscopic
potentials based on model transition densities show the data agrees with
phenomenological densities predicting neutron quadrupole transition matrix
elements ($M_{n}$) ranging from 3.65 - 5.0 fm2 [12]. It is discussed that the
(p,p’) scattering at lower energy will have stronger sensitivity to $M_{n}$.
In contrast, measurements performed at higher energies (227$A$ MeV) using both
Coulomb excitation [13] and fragmentation [14] reported a very narrow state,
considered to be possibly $2^{+}$, below 3 MeV, lower than that observed in
the (p,p′) experiment. This state overlaps with a very broad second excited
state which was conjectured to be a 1- excitation. The result from a
${}^{10}\mathrm{Be}(^{12}\mathrm{C},^{14}\mathrm{O})$ multi-nucleon transfer
reaction however finds the energy of the first excited state to be in
agreement with that from inelastic proton scattering. In addition, three
higher energy resonances at energies of 4.54(25) MeV, 6.03(10) MeV, and
7.16(4) MeV were reported [15, 16]. A similar energy and width of the first
excited state was also reported in studies of the
$t(^{6}\mathrm{He},p)^{8}\mathrm{He}$ reaction [17, 18]. However, these works
proposed that a significant contribution from a $1^{-}$-state close to the
two-neutron threshold, $S_{2n}$ = 2.13 MeV [19], better describes the data. A
recent measurement of the breakup of 8He at 82$A$ MeV [20] interpreted the
resonance spectrum with conclusions more in line with Ref. [14, 13].

To derive the quadrupole deformation parameter and to resolve the
inconsistencies regarding the dipole resonance and first 2+ state, this Letter
reports the first low-energy ($\sim$8.25$A$ MeV) measurement of proton
inelastic scattering with high statistics and high energy resolution. The
experiment was performed at the charged particle spectroscopy station IRIS at
TRIUMF in Canada [21]. The 8He nuclei, produced from the spallation of a SiC
target with a 500 MeV proton beam, were re-accelerated to an energy of 8.25$A$
MeV by the superconducting linear accelerator [22] and transported to the
ISAC-II experimental hall where the IRIS facility is located. The beam had an
average intensity of $\sim 10^{4}$ pps and a purity of
$80-90\leavevmode\nobreak\ \%$ at IRIS. The beam impurity was 8Li, identified
event-by-event from energy-loss measured using a low-pressure ionization
chamber, operated with isobutane gas at 19.5 Torr, at the entrance of the
experiment setup. Following this, the beam impinged on a $\leavevmode\nobreak\
100\leavevmode\nobreak\ \mu$m solid H2 target formed on a
$4.5\leavevmode\nobreak\ \mu$m Ag backing foil cooled to $\sim
4\leavevmode\nobreak\ $K. The target cell was surrounded by a copper heat
shield cooled to $\sim 28\leavevmode\nobreak\ $K.

Figure 1:  $\Delta E-E$ identification plots from (a) Telescope 1 detecting
$p,d,t$ and 6,8He and (b) Telescope 2 detecting 6,8He ions.

The target-like reaction products, protons ($p$), deuterons ($d$) and tritons
($t$), as well as helium nuclei were detected and identified using an array of
100$\leavevmode\nobreak\ \mu$m thick segmented silicon strip detectors and a
12 mm thick CsI(Tl) array behind it. This $\Delta E-E$ telescope (Telescope 1)
was placed 12.5 cm downstream of the target, covering laboratory angles of
21-46∘. The top panel of Fig. 1 shows the identification plot using this
telescope showing the $p$, $d$ and $t$ loci clearly separated. A second
$\Delta E-E$ telescope (Telescope 2), consisting of $60\leavevmode\nobreak\
\mu$m and 1 mm annular double-sided silicon strip detectors, was used to
detect the beam-like He and Li nuclei. Telescope 2, placed 18 cm from the
target, covered scattering angles of $3-10^{\circ}$. The identification plot
of the beam-like He nuclei in coincidence with proton detection by Telescope 1
is shown in the bottom panel of Fig. 1. The 6,8He events are clearly
distinguished.

The H2 target thickness was measured from the energy difference without and
with the H2 target using the downstream telescope. This was done from the
peaks of the energy distributions of both 8He and 8Li nuclei scattered off the
silver foil. In addition, a silicon surface barrier detector was
intermittently inserted into the beam at 0∘ located at the extreme downstream
end of the setup as another measurement of the target thickness. Measurements
with a warm target cell without hydrogen were used to estimate the background
from fusion-evaporation reactions coming from the silver foil. The detection
efficiency and acceptance of the telescopes were determined from simulations
of the experiment in which the energies and momenta of the particles were
generated according to phase space decays and which included the experimental
resolution of the detectors.

Figure 2:  (a) Measured excitation energy spectrum of 8He. The red / blue
histogram shows the measured background spectrum from the Ag foil / the
spectrum with H2 target after Ag foil background subtraction. (b) The
background subtracted non-elastic excitation spectrum with coincident
detection of protons and 6He. The red shaded band shows the non-resonant
background from reactions (A) and (B). The individual non-resonant components
are shown by the green and magenta curves for channels (A) and (B),
respectively. (c) The observed resonance after subtraction of the non-resonant
background. The simulated resonance spectra (see text ) for different
$L^{rel}$ decay possibilities are shown by the red / blue dotted/green dashed
curves labeled in the legend.

The excitation energy of 8He was reconstructed from the measured energies and
angles of the detected protons using the missing mass technique. The
excitation spectrum is shown in Fig. 2(a). The ground state can be clearly
seen and has negligible background from reactions in the Ag foil. Above the
neutron threshold the excitation spectrum has a strong contribution from non-
resonant reactions together with resonant excitations. Looking only at events
in which 6He was detected in coincidence with scattered protons, the non-
resonant background can be caused by two reactions,
$(\mathrm{A})\>p+{}^{8}\mathrm{He}\rightarrow p+{}^{6}\mathrm{He}+n+n$ and
$(\mathrm{B})\>p+{}^{8}\mathrm{He}\rightarrow p+{}^{7}\mathrm{He}+n$.

The non-resonant reaction kinematics were simulated considering isotropic
emission of the reaction products in the center of mass frame. The simulation
includes detector geometrical acceptance and resolution effects. The resulting
energy of the protons from these non-resonant channels was used to construct
the excitation energy spectrum of 8He in the identical process of missing mass
technique as adopted for the (p,p’) inelastic scattering reaction channel. The
measured non-elastic spectrum (Fig. 2(b)) was fitted with a sum of non-
resonant channels (A) and (B) with their amplitudes as free fit parameters and
a simulated resonance with Voigt function profile where the resonance energy,
the width and amplitude were free parameters in the fit. The resonance width
contains decay angular momentum energy dependence.

The blue curve in Fig. 2(b) shows the best obtained fit. The red hatched area
denotes the contribution with uncertainty by the non-resonant background. The
overall total strength of the non-resonant contributions from reactions (A)
(green curve, Fig. 2(b)) and (B) (magenta curve, Fig. 2(b)) was determined by
the resulting best fit parameters considering non-resonant and resonant
contributions to the total spectrum. The hatched band indicates the
uncertainty. This leads to the non-resonant phase space describing the high
energy end of the spectrum. We have not assumed any high excitation energy
resonance since there is no clear resonance peak observed in this region.
Theoretical predictions of, the 1+ excited state energy by continuum shell
model [23] and the no core shell model discussed below is $\sim$ 6 MeV which
is at the limit of our detection. The differential cross sections for the
total spectrum and the derived non-resonant backgrounds can be found in
Fig.1(Sup) of the Supplementary Material.

The excitation spectrum after subtraction of the non-resonant background is
shown in Fig. 2(c). In the configuration of the 8He(2+) state only the
component with core 6He${}_{gs;0^{+}}$ decays. The possible decay branches can
be one neutron emission to the 7He+n threshold and two-neutron emission to the
6Hegs+$nn$ threshold. For 8He(2+) = 6Hegs(0+)+$nn$, since the combined
intrinsic spin of neutron-neutron ($nn$) cluster $S_{nn}$ = 0, the $nn$
orbital angular momentum $L_{nn}$ = 0, 2 leads to possible 6He-$nn$ relative
angular momentum $L^{rel}_{2}$ = 2, 0 for two-neutron decay to 6Hegs. The
relative angular momentum for one-neutron decay to 7He+n is $L^{rel}_{1}$ =1.
The spectrum extends below the 7He+n threshold signifying decay to 6Hegs+$nn$
to be present. We analyzed the spectrum with a Voigt function with an energy
dependent width ($\Gamma(E)=\Gamma_{0}\sqrt{(E/E_{r})}$) [24] of the Breit-
Wigner resonance profile. $E_{r}$ is the resonance energy. This energy
dependence corresponds to $L^{rel}_{2}$ = 0 (Fig. 2(c) red curve). The
resultant reduced chisquare from the fit is 1.43. We also performed a fit of
the data considering a single resonance state to decay by sum of $L^{rel}_{1}$
= 1 and $L^{rel}_{2}$ = 0, resulting in reduced chisquare value of 1.83 (Fig.
2(c) blue dotted curve). The similarity of the two fits suggests that the
effects of detector acceptance and resolution probably masks a clear
distinction. The sum of $L^{rel}_{1}$ = 1 and $L^{rel}_{2}$ = 2 (Fig. 2(c)
green dashed curve) does not explain the data having a reduced chisquare value
of 6.9. The narrower width for the $L^{rel}_{2}$ =2 curve is due to its
smaller penetrability.

The determined position and intrinsic width of the resonance from the red
curve (Fig. 2(c)) is $E^{*}=3.53(4)$ MeV and $\Gamma=0.89(11)$ MeV,
respectively. The resonance peak from the blue dotted curve (Fig. 2(c)) is
3.56(4) MeV which is in agreement with that from the red curve. The average
resonance energy derived from the two fits is 3.54(6) MeV. The excitation
energy resolution was 0.15 MeV ($\sigma$) at an excitation energy of 3.5 MeV
as determined from simulations which were consistent with the elastic
scattering peak width. The excitation energy is in agreement with the previous
measurements using inelastic scattering and transfer reactions. However, this
high resolution measurement defines precisely the resonance width which agrees
only with the upper uncertainty end of that reported in Ref. [10]. Including
an additional resonance in the fit does not improve the description of the
data. The angular distribution is not consistent with a dipole excitation and
hence does not align with the conclusion from the breakup experiments [14, 13,
20]. The (p,p’) and (d,d’) reactions can populate low-lying dipole resonance
states as seen in Refs.[25, 26]. This suggests that the breakup reactions
likely exhibit strong non-resonant dipole transitions to the continuum, as in
11Be [27] and 8B [28] breakup.

We mention here that a fit to the full spectrum with the sum of non-resonant
background channels and two separate resonance states described by Voigt
functions with $L^{rel}_{1}$ = 1 and $L^{rel}_{2}$ = 0 whose peak positions
and widths are free parameters in the fit results in a reduced chisquare value
of 2.76. The spectrum fit and the angular distributions resulting from the two
different resonance peaks are included in the Supplementary Material
(Fig.2(Sup)) to show that neither of them are consistent with a dipole
excitation.

Figure 3:  Differential cross sections in the center-of-mass frame for elastic
(a) open/closed symbols are from detection of 8He/$p$ and resonant inelastic
scattering (b). The curves show CC and DWBA calculations. The red solid / blue
dashed curves are with optical potential Set 1/Set 2 and with $L$=2 excitation
and rotational model in (b). The pink dashed-dotted curve shows CC calculation
for $L$=2 excitation with Set 2 and vibrational model. The green solid/dashed
curve in (b) is for $L$=1 excitation with potential Set1/Set2.

Differential cross sections in the center-of-mass system for elastic
scattering are shown in Fig. 3(a) and for the excited state at 3.54 MeV in
Fig. 3(b). The cross sections were obtained from the background subtracted
spectra where the non-resonant contribution for inelastic scattering is
subtracted as well. The excited state cross section is obtained over the
entire excitation range with counts in the resonance profile extracted from
the data as described above. The elastic scattering cross section was also
obtained from detection of the scattered 8He only (Fig. 3(a) open symbols). In
order to determine the $p+^{8}$He optical potential parameters, these two
angular distributions were fitted simultaneously with coupled-channel (CC) and
one-step distorted wave Born approximation calculations (DWBA) using the code
SFRESCO [29] to obtain the best-fit solution. The DWBA calculations with
collective form factor use the rotor model which in first order is same for
vibrational model. The CC calculations with rotor model and vibrational model
give slightly different fits ((Fig. 3). The deformation length $\delta$ was
included in the fit to describe the inelastic scattering data containing the
imprint of deformation. The inelastic scattering angular distribution is
explained by a quadrupole transition, $L=2$, to the first excited state.

We derived two sets of optical potential parameters (Table 1) that describe
the data with DWBA (Set 1) and CC (Set 2) calculations for the entire angular
range of elastic scattering. The fits to the data using them are shown in Fig.
3. It is noteworthy that the sets require large deformation lengths
$\delta_{2}^{ex}$ = 1.24 - 1.40 fm within the adopted reaction model to
explain the inelastic scattering data within 2$\sigma$ lower uncertainty for
$\theta_{cm}<$ 70∘. The $\delta_{2}^{ex}$ derived in this framework is found
to be consistent with microscopic reaction model calculations presented below.
Considering the measured matter radius of 8He [30] they correspond to a large
quadrupole deformation parameter of $\beta_{2}$ = 0.40(3) showing that 8He has
a deformed sub-shell gap at $N$ = 6.

The collective vibrational model form factor cannot distinguish between static
and dynamic deformation. However, the no-core shell model (NCSM) calculations
reported below shows large neutron deformation in the 2+ state of 8He. The
large neutron quadrupole moment for the 2+ state predicted by the NCSM
calculations suggest 8He as a nucleus with a significant intrinsic deformation
in contrast to a spherical (vibrational) picture, for which the 2+
reorientation term would vanish. The microscopic transition density obtained
in the no-core shell model leads to a quadrupole deformation length consistent
with that derived from the collective form factor as discussed below.

Table 1: Optical potential parameters for ${}^{8}\mathrm{He}+p$, determined from a simultaneous fit to the elastic and inelastic scattering data. The depth, radius, diffuseness parameters for the real potential are $V,r,a$ and for the surface imaginary potential are $V_{s},r_{s},a_{s}$, respectively. The degrees of freedom ($dof$) were 30. |  |  $V$ [MeV] |  $r$ [fm] |  $a$ [fm] |  $W_{s}$ [MeV] |  $r_{s}$ [fm] |  $a_{s}$ [fm] |  $\delta$ [fm] | $\chi^{2}/dof$
---|---|---|---|---|---|---|---|---|---
Set 1 | DWBA | 46.3 | 1.65 | 0.35 | 22.8 | 1.77 | 0.27 | 1.40 | 1.60
Set 2 | CC - Rotor Model | 50.5 | 1.51 | 0.33 | 20.2 | 1.79 | 0.19 | 1.24 | 1.36
Set 2 | CC - Vibrational Model | 50.5 | 1.51 | 0.33 | 20.2 | 1.79 | 0.19 | 1.32 | 1.60

The $\beta_{2}$ values for heavier $N$ = 6 isotones, are 1.14(6) for 10Be and
0.582(24) for 12C [31] which suggests that deformation persists from stable
nuclei to the neutron-rich region. The excitation energies of the 2+ states
for the $N$ = 6 isotones are similar, 4.44 MeV in 12C, 3.37 MeV in 10Be and
3.54 MeV in 8He. Therefore, the extent of the sub-shell gap at $N$ = 6 may be
similar along the isotonic chain but it becomes prominent towards the neutron-
drip line due to the disappearance of the strong shell closure at $N$ = 8. In
14O however, the 2+ state lies at a much higher excitation energy of 6.6 MeV,
reflecting the presence of protons in the filled 1$p_{1/2}$ orbital, causing a
wider gap at $N$ = 6 due to the attractive proton-neutron tensor force. The
question remains open regarding the extent of deformation in heavier nuclei
with closed neutron sub-shells, such as neutron-rich Ca isotopes, where some
theoretical predictions suggest the drip-line extending to 72,74Ca [32, 33].

A small part of the resonance spectrum extends below the 7He+n threshold
indicating that decay to the 6Hegs+2n threshold is important. The measured
angular distribution is not supportive of a low-energy dipole resonance ($L$ =
1, Fig. 3(b)). Further studies may aid in a complete understanding of this
feature.

Figure 4:  (a) Comparison of the observed 2+ excited state in 8He to ab initio
predictions. The red bands show results using the NNLOsat interaction with the
EOM-CCSDT-3 method for the 2${}_{1}^{+}$ and 2${}_{2}^{+}$ states. The pink
bands show results for the 2${}_{1}^{+}$ and 2${}_{2}^{+}$ states with the
NNLOopt interaction using the EOM-CCSDT-3 and SMCC methods. The blue band
shows the result for the 2${}_{1}^{+}$ state with the NN-N4LO interaction in a
NCSMC calculation, no 2${}_{2}^{+}$ state is found within this excitation
energy range. The cyan / orange bands show results with the SMCC / EOM-CCSDT-3
methods using the NN-N4LO interaction. (b) The $2^{+}_{1}$ excitation energy
dependence of 8He on the basis size for the NCSM and NCSMC calculations with
the NN-N4LO interaction. Extrapolated values and the data are shown on the
right. The vertical bars represent resonance widths obtained in the NCSMC
calculations and in the experiment.

For theoretical descriptions of the 2+ states in 8He, we employed two many-
body approaches, namely coupled-cluster theory and the no-core shell model
(with continuum) using several chiral interactions. The measured energy of the
resonance (2${}_{1}^{+}$ state) is compared to the ab initio calculations in
Fig. 4.

We used two different coupled-cluster methods [34]. First, for the chiral
interaction NNLOsat [35] we employ the equation-of-motion technique with up to
three-particle–three-hole (3p-3h) excitations (so called EOM-CCSDT-3) [36] of
the closed-shell 8He reference state, and find the 2${}_{1}^{+}$ state at an
excitation energy of 3.5 to 3.8 MeV. The range reflects model-space
uncertainties. The Hartree-Fock basis is built from model spaces consisting of
11 to 15 oscillator shells with frequencies between 12 and 16 MeV. We find
that 8He is bound by almost 3 MeV with respect to 4He, in agreement with data
($S_{4n}\sim 3.11$ MeV). Second, we employ the chiral nucleon-nucleon (NN)
interaction NNLOopt [37]. For this interaction 8He is not bound and about 1.5
MeV above the 4He ground-state energy. We again used the EOM-CCSDT-3 approach
and also the shell-model coupled-cluster method (SMCC) [38]. This method
employs a 4He core and constructs a valence-space Hamiltonian in the
$0p_{3/2}$, $0p_{1/2}$, and $1s_{1/2}$ shells based on computations of the
$A=5,6$ body problems in 5 to 13 oscillator shells (and frequencies of 12 to
22 MeV). In the valence space four-neutron correlations are treated exactly.
The EOM-CCSDT-3 and SMCC methods yield an excited $2_{1}^{+}$ state at about
3.2 and 3.8 MeV, respectively, and we take this range as a systematic
uncertainty. Both predictions agree with the data within the theoretical
uncertainty band shown in Fig. 4(a) with and without the three-nucleon force.

We also applied the no-core shell model (NCSM) [39] to calculate properties of
8He. In the NCSM, the many-body wave function is expanded over a basis of
antisymmetric $A$-nucleon harmonic oscillator (HO) states. The basis contains
up to $N_{\rm max}$ HO excitations above the lowest possible Pauli
configuration and depends on an additional parameter $\Omega$, the frequency
of the HO well. We employed the same Hamiltonian as in our recent
investigation of 9He [40]. The NN interaction, denoted here as NN-N4LO, is
from the fifth order chiral expansion (N4LO) of Ref. [41] and was renormalized
by the SRG approach [42] with an evolution parameter $\lambda_{\rm SRG}{=}2.4$
fm-1. The three- and higher-body SRG induced terms were not included. We
performed calculations up to $N_{\rm max}{=}12$ with $\hbar\Omega{=}20$ MeV.
We find 8He bound by about 2 MeV with respect to 4He. The NCSM calculations
yield a large quadrupole neutron moment $Q_{n}=6.15\leavevmode\nobreak\ e$fm2
and a small proton quadrupole moment, $Q_{p}=0.60\leavevmode\nobreak\ e$fm2
for the 2${}^{+}_{1}$ state. For 12C we predict
$Q_{n}{\approx}Q_{p}{\sim}6\leavevmode\nobreak\ e$fm2. Thus, the neutron
deformation in 8He is similar to that in 12C and qualitatively consistent with
the experimental observations discussed above. In the Variational Monte Carlo
framework $Q_{p}$ of 8C(2+) is 5.6 $e$fm2 which can reflect the $Q_{n}$ of
8He(2+) considering charge symmetry [43].

As the $2^{+}_{1}$ state is unbound, its excitation energy convergence is slow
in NCSM as seen in Fig. 4(b). To improve the theoretical description, we
applied the no-core shell model with continuum (NCSMC) [44, 45, 46].
Optimally, three-body cluster NCSMC [47] with 6He+n+n or even five-body 4He+4n
continuum should be used. That is, however, beyond our current technical
capabilities. As the 7He ground state resonance is rather narrow (150 keV), it
is reasonable to use as the simplest alternative the 7He(gs)+n cluster to
extend the 8He NCSM basis. This results in a greatly improved convergence of
the $2^{+}_{1}$ excitation energy, with the extrapolated value of 3.58(6) MeV
(Fig. 4(b)). In addition, we calculate the $2^{+}_{1}$ width to be 750(50)
keV. Overall, with this interaction we obtain an excellent agreement with the
present experimental measurement. We note that the only other resonance we
find in the calculation below 6 MeV in 8He excitation energy is a broad
$1^{+}$ state. In particular, we do not see any evidence for a $1^{-}$
resonance in this energy range. Both the NCSM and NCSMC calculations were
performed using the HO frequency of $\hbar\Omega\leavevmode\nobreak\
{=}\leavevmode\nobreak\ 20$ MeV determined as optimal for the 8He ground state
with the NN-N4LO interaction as illustrated in Fig. 2 in Ref. [40]. Compared
to that paper, the present NCSM calculations were extended to $N_{\rm
max}\leavevmode\nobreak\ {=}\leavevmode\nobreak\ 12$ using the importance
truncation [48, 49]. The calculated $N_{\rm max}\leavevmode\nobreak\
{=}\leavevmode\nobreak\ 12$ ground-state energy, -28.2 MeV, is in line with
the extrapolation shown in Fig. 4 of Ref. [40].

We also benchmarked the coupled-cluster computations with the NCSM using the
NN-N4LO potential. For the ground-state energy, the extrapolated NCSM result
[40] is $E=-30.23(30)$ MeV, while SMCC and CCSDT-3 yield $-30.3$ and
$-29.0(5)$ MeV, respectively. For the 2${}^{+}_{1}$ state, we find excitation
energies of $3.8(2)$ and $3.1(2)$ MeV for SMCC and EOM-CCSDT-3, respectively.
The SMCC results agree with the NCSM, while CCSDT-3 and EOM-CCSDT-3 are less
accurate. We note that the EOM-CCSDT-3 calculations yield two nearby $2^{+}$
states at $3.1(2)$ and $3.8(2)$ MeV. The first state carries only about 40% of
$1p-1h$ amplitudes from the reference state, while the second state exhibits
about 70% of $1p-1h$ amplitudes. Thus, the EOM computations are not converged
with respect to wave function correlations, and 8He is not a closed-shell
nucleus for the employed potential.

Figure 5:  The measured differential cross sections for (a) 8He(p,p) (b)
8He(p,p’)8He(2+) and (c) 8He(p,d)7He shown by the symbols. The black dashed /
red solid curves are CC / CRC calculations with NCSM densities (see text). The
blue dotted / dashed-dotted curve shows CC / CRC calculation including an
R-matrix pole.

The differential cross section are also analyzed in terms of coupled-channels
(CC) and coupled-reaction-channels (CRC) calculations using whenever possible
structure inputs from the aforementioned NCSM calculations. In the CC
calculations, only the the elastic and inelastic channels were considered. The
p+8He(g.s.) and p+8He(2+) diagonal potentials as well as the quadrupole
coupling between these two channels were computed by a single folding
procedure, convoluting the JLMb interaction of Ref. [50] with the NCSM matter
and transition densities. The result of this calculation, shown by the black
dashed curve in Fig.5, does not describe well the shape of the data. In the
CRC calculations, in addition to the couplings considered in the CC
calculations, we included the coupling to the d+7He(g.s.) and p+n+7He(g.s.)
channels, the latter accounting for the deuteron continuum. The required
$\langle{\rm{}^{7}He}|{\rm{}^{8}He(g.s.)}\rangle$ overlap function was
approximated by a single-particle wavefunction calculated in a Woods-Saxon
potential with the depth adjusted to give the experimental separation energy
and the geometry adjusted to reproduce the NCSM overlap in the interior. The
same Woods-Saxon geometry was adopted for the
$\langle{\rm{}^{7}He}|{\rm{}^{8}He(2^{+})}\rangle$ overlap. The results of
these calculations are given by the red solid lines in Fig.5. The agreement
with the inelastic and transfer angular distributions is rather satisfactory,
but not for the elastic scattering scattering. The coupling between the
$p$+8He(2+) channel and the d + 7He channel was found to be very important in
reproducing the shape of the inelastic scattering data. It is interesting to
note that the deformation parameter derived from the NCSM quadrupole
transition density that explains the data is found to be $\delta_{2}$ = 1.39
fm, in good agreement with that derived from the fit to the experimental
inelastic cross sections using the collective model with a deformed Woods-
Saxon potential discussed above.

The incomplete description of the elastic scattering data (Fig.5) with CC and
CRC could be due to the effect of compound nucleus resonance(s) in 9Li.
Although a detailed investigation of this effect is beyond the scope of the
current work, to highlight the possible effect we have performed CC and CRC
calculations including an R-matrix pole representing the effect of a compound-
nucleus resonance with a $J^{\pi}=5/2^{+}$ and with the energy and reduced
width amplitudes adjusted to reproduce in the best possible way the measured
elastic and inelastic data. As a result, we obtain a formal energy of
$E_{\mathrm{c.m.}}\simeq 5.5$ MeV ($E_{x}=19.4$ MeV with respect to the
9Li(g.s.)). The resultant elastic and inelastic distributions are given by the
blue dotted curve for CC + R matrix pole and blue dashed-dotted curve for CRC
+ R matrix pole in Fig.5. It is seen that the inclusion of such resonance can
result in a significantly improved agreement with the data.

In summary, a measurement of proton inelastic scattering of 8He at 8.25$A$ MeV
affirms the first excited state to be an unbound 2+ state at an excitation
energy of 3.54(6) MeV with a width of 0.89(11) MeV (FWHM). Analysis of the
measured angular distribution yields a quadrupole deformation parameter of
$\beta_{2}$ = 0.40(3). The deformation length is consistent with calculations
in a no-core shell model framework. Microscopic CRC calculations with NCSM
densities explain the inelastic scattering yielding a deformation length 1.39
fm, providing further support for the large deformation. Ab initio
calculations in a coupled cluster framework and NCSMC find a 2${}^{+}_{1}$
excitation energy in good agreement with the data. The resonance width
predicted by the NCSMC is also consistent with the data. The high-quality
data, signalling deformation at the $N$ = 6 drip-line, open exciting prospects
for further investigations.

The authors thank the TRIUMF ISAC beam delivery team and the cyclotron
operations team. The support from NSERC, Canada Foundation for Innovation and
Nova Scotia Research and Innovation Trust is gratefully acknowledged. TRIUMF
receives funding via a contribution through the National Research Council
Canada. The support from RCNP for the target is gratefully acknowledged. It
was partly supported by the grant-in-aid program of the Japanese government
under the contract number 23224008 and 14J03935. The use of the S3 detector
provided by C.Y. Wu of LLNL is much appreciated. Computing support from
Westgrid and Compute Canada is gratefully acknowledged. This work was
supported by the Office of Nuclear Physics, US Department of Energy, under
grants DE-FG02-96ER40963 and DE-SC0018223, and at Lawrence Livermore National
Laboratory under Contract DE-AC52-07NA27344, as well as by the Field Work
Proposal ERKBP72 at Oak Ridge National Laboratory (ORNL). Computer time was
provided by the Innovative and Novel Computational Impact on Theory and
Experiment (INCITE) programme. This research used resources of the Oak Ridge
Leadership Computing Facility located at Oak Ridge National Laboratory, which
is supported by the Office of Science of the Department of Energy under
contract no. DE-AC05-00OR22725. J.A.L. and A.M.M. are supported by the Spanish
Ministerio de Ciencia, Innovación y Universidades (including FEDER funds)
under project FIS2017-88410-P and by the European Union’s Horizon 2020
research and innovation program under Grant Agreement No. 654002. Discussions
with R.B. Wiringa is gratefully acknowledged.

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