Ab initio no-core properties of 7Li and 7Be with JISP16 and NNLO_opt interactions

0.72 MB·

by  Taihua Heng

Ab initio no-core properties of 7Li and 7Be with JISP16 and NNLO interactions
opt
Taihua Heng1,2, James P. Vary2 and Pieter Maris2∗
1School of Physics and Material Science, Anhui University, Hefei, China, 230601
2Department of Physics and Astronomy, Iowa State University, Ames, IA, United States, 50011
(Dated: February 2, 2016)
We investigate the properties of 7Li with the JISP16 and chiral NNLOopt nucleon-nucleon
interactions and 7Be with the JISP16 interaction in the ab initio no-core full configuration
(NCFC) approach. We calculate selected observables that include energy spectra, point proton
root-mean-square radii, electromagnetic moments and transitions. We compare our results with
experimental results, where available, as well as with results obtained using nucleon-nucleon plus
three-nucleon interactions. We obtain reasonable agreement between theory and experiment for
low-lying states that are dominated by p -shell configurations.
6
1
0 PACS numbers: 21.60.De, 27.20.+n, 21.10.Ky
2

n I. INTRODUCTION tion (NCFC) approach[21][22][23] which is an extension
a
of the NCSM approach. In the NCFC approach, the
J
An outstanding problem in nuclear physics is to direct solution of the nuclear many-body problem is
0
3 study the properties of atomic nuclei based on realistic obtained by diagonalization in a sufficiently large basis
interactions among the protons and neutrons. The space that converged binding energies are obtained-

] rapid development of ab initio methods for solving either directly or by simple extrapolation.
h
finite nuclei provides a theoretical foundation that can
t
- be numerically solved to high precision using realistic We select a traditional harmonic oscillator (HO)
l
c nucleon-nucleon (NN) and three-nucleon (NNN) basis so there are two basis space parameters, the
u interactions. A number of meson-exchange potentials, HO energy ~Ω and the many-body basis space cutoff
n sometimes combined with phenomenological terms to Nmax. Nmax is defined as the maximum number of
[
achieve high accuracy in fitting NN data (CD-Bonn total oscillator quanta allowed in the many-body basis

1 [1], Nijmegen [2], Argonne [3]), have been developed space above the minimum for that nucleus. We obtain
v that should be used together with modern NNN forces convergence in this two-dimensional parameter space
6 (Urbana [4][5], Illinois [6], Tucson-Melbourne [7][8][9]) (~Ω,Nmax), where convergence is defined as indepen-
5
to describe properties of many-body nuclear systems. denceofbothparameterswithinestimateduncertainties.
1
0 A very important step in the theory of inter-nucleon
0 interactions is the emergence of realistic NN and NNN In the present work, we investigate the properties of
. interactions tied to quantum chromodynamics (QCD) 7Li and 7Be in the ab initio NCFC approach with the
2
0 via Chiral Perturbation Theory (CPT) [10][11][12]. JISP16[24][25][26]andchiralNNLOopt [27]interactions.
The JISP16 NN interaction, proposed in Ref. [26], is
6
1 In addition, recent advances in the utilization of constructed in the J-matrix inverse scattering approach
: high-performance computing systems offer an oppor- [24][28]. It is known to provide an excellent description
v tunity for ab initio approaches to be at the forefront of np scattering data with χ2/datum = 1 [29]. The
i
X of nuclear structure explorations. Various microscopic interaction was fitted in Ref. [26] by means of phase-
r many-body methods have been developed for no-core equivalent transformations to a few binding energies of
a treatments of nuclei, including No-Core Shell Model nuclei with A ≤ 16, and it provides a good description
(NCSM) [13][14][15], the Green’s Function Monte Carlo of bindings and spectra of light nuclei without referring
(GFMC) approach [16][17][18], the Coupled-Cluster to three-nucleon forces [26][30][31][33][32].
(CC) method [19][20], etc. The name NCSM is due to
the fact that all the nucleons of the nucleus are active Chiral effective field theory (EFT) is a promising the-
andtreatedonanequalfooting andthus no inertcore is oretical approach to obtain a quantitative description of
assumed. The NCSM method was introduced as a finite the nuclear force from first principles [34]. Interactions
matrix truncation of the infinite matrix problem with a from chiral EFT employ symmetries and the pattern of
renormalized Hamiltonian specific to that truncation. spontaneous symmetry breaking of QCD [34][35]. More-
over, the interaction is parametrized in terms of low-
Here we adopt the ab initio No-Core Full Configura- energy constants (LECs) that are determined by fitting
experimentaldata. OuradoptedNNLO interactionat
opt
next-to-next-to-leading order (NNLO) is constructed by
the optimization tool POUNDerS (Practical Optimiza-
∗ [email protected], [email protected], tionUsing NoDerivativesfor Squares)inthe phase-shift
[email protected],[email protected] analysis[36][37]. Theoptimizationofthelow-energycon-
2
stants in the NN sector at NNLO yields a χ2/datum of the MFDn code, a hybrid MPI/OpenMP configuration
about one for laboratory scattering energies below 125 interaction code for ab initio nuclear structure calcula-
MeV.TheNNLO NN interactionisalsofittedtopro- tions [39][40][41].
opt
videverygoodagreementwithbindingenergiesandradii
forA=3and4nuclei. Somekeyaspectsofnuclearstruc- The properties of 7Li with JISP16 and NNLO and
opt
ture, such as excitation spectra, the position of the neu- 7Be with JISP16 are given in the next section. Results
trondriplineinoxygen,shell-closuresincalcium,andthe for 7Li with JISP16 up to and including the N = 14
max
neutronmatter equationof state atsubsaturationdensi- truncationarein agreementwith those inRef. [22]. The
ties,arereproducedbytheNNLOopt interactionwithout values of 7Li with the NNLOopt interaction, as well as
the addition of three-nucleon forces. the 7Li results with JISP16 at N = 16, are reported
max
hereforthefirsttime. Thepropertieswepresentfor7Be,
the mirror nuclei of 7Li, are more extensive than those
II. ab initio NO-CORE FULL CONFIGURATION presented in the related references. In order to assess
(NCFC) APPROACH convergence, we present results for even values of the
parameter N from 8 to 16, and for ~Ω over a range
max
from 10 MeV to 40 MeV.
WegiveabriefreviewoftheNCFC methodand,for
moredetails,wereferthe readerto Refs. [21][22][32][33].
TheHamiltonianfortheA-body systeminrelativecoor-
dinates is III. RESULTS
1 (p~ −~p )2
HA = X i j +XVNN(~ri−~rj)+XVC(~ri−~rj), The ground state energy is one of the primary
A 2m
i<j i<j i<j observables in nuclear physics. Our results for the
(1) ground state energies of 7Li with JISP16 and NNLO ,
opt
where V (V ) is the two-nucleon (Coulomb) interac- and of 7Be with JISP16, are plotted in Fig. 1 as a
NN C
tion. OurexpressionforH issomewhatschematicsince functionoftheHOenergyatourselectedvaluesofN .
A max
the NN interaction may, in general, be non-local. Fur- With increasing N , the minima of the U-shaped
max
thermore,inthepresentwork,weneglectNNN interac- curves are closer to each other and the shapes exhibit
tions. We solve the corresponding Schr¨odinger equation reduced dependence of ~Ω. We observe in Fig. 1 that
the energies with JISP16 are lower than those with
HAΨ(~r1,~r2,...,~rA)=EΨ(~r1,~r2,....~rA) (2) NNLOopt for 7Li at each value of Nmax. Moreover, the
ground state energies from the JISP16 interaction for
with numerical techniques formulated within a basis ex- both7Liand7Be showbetter convergencepatternsthan
pansion (configuration interaction) approach. In the that from NNLO . The lowest minima (points closest
opt
NCFC, the wave function Ψ(~r1,~r2,....~rA) is a superpo- to convergence) in the three panels are near ~Ω = 20
sition of Slater determinants Φi of single-particle HO MeV. We also indicate the first breakup thresholds in
states. Fig. 1 (3H+4He for 7Li and 3He+4He for 7Be). Note
that we use the experimental energies for the thresholds
Ψ(~r1,~r2,....~rA)=XciΦi(~r1,~r2,....~rA) (3) since both JISP16 and NNLO give very accurate
opt
i binding energies for nuclei with A ≤ 4. The energies
with JISP16 in panel 1 and panel 3 are lower than these
thresholds. The energy with NNLO is very close to
opt
For practical calculations, a finite basis space is the threshold where it appears near the tangent of the
specified by the Nmax truncation. We aim for results Nmax = 16 U-shaped line. We will next see that, upon
thatareconvergentasdeterminedbytheirindependence extrapolation, the ground state energies of all cases in
of the parameters ~Ω and Nmax as Nmax is increased. Fig. 1 lie below the lowest thresholds.
We present results for even values of N that cor-
max
respond to states with the same parity as the lowest
We now discuss how we extrapolate the results ob-
HO configuration (the N = 0 configuration) and are
max tained in finite basis spaces to the infinite basis space
called the ”natural” parity states.
andproduceanextrapolationuncertainty. Toobtainthe
extrapolated ground state energy E(∞), we use the ex-
We also need to address the issue of center of ponential form[21][22][23][32][33][42],
mass (COM) motion. In the above discussion, the
Hamiltonian is in the relative coordinate and not in E(N )=aExp(−cN )+E(∞) (4)
max max
the single-particle coordinates where the HO states
are specified. The methods of solution, including the We follow the method described as ”Extrapolation B”
methodto constrainthe COMmotion,arenotdescribed inRef. [21]. Under the assumptionthat the convergence
in this paper, because these details have been explained is exponential, we use three successive value of N
max
in Refs. [21][22][32][33]. We obtain our results with at each value of ~Ω. The numerical uncertainty is the
3
difference between the extrapolated results from two With increasing ~Ω, the patterns of the rms radii are
consecutive sets of N values. The extrapolated similarfor 7Be and7Li forthe two differentinteractions.
max
resultsusingthehighestavailablesetofN values(12, Theradiiof7Liand7Bedecreasemonotonicallywithin-
max
14 and 16) and their numerical uncertainties (depicted creasing~Ω and showa weaktrend towardsconvergence
as error bars) are shown in Fig. 1. These numerical as is frequently found for this long-range observable
uncertainties due to extrapolations, also defined in calculated in a HO basis. The value of the rms radius is
Ref. [21], are minimal and the extrapolations nearly a little larger with NNLO than that with JISP16 for
opt
independentofHOenergynear~Ω=20MeVforJISP16 7Li at the same ~Ω and N values, which may simply
max
and near 25 MeV for NNLO indicating favorable be a reflection of the lower binding energy of NNLO
opt opt
convergence estimates at those values of the HO energy. relative to JISP16.
We present the energies of the four lowest excited The magnetic dipole moments in units of µN are
states as a function of the HO energy with a series of depicted in Fig. 4 as a function of the HO energy for
NmaxvaluesinFig. 2andcomparewiththeexperimental 7Li and 7Be with the JISP16 andNNLOopt interactions.
excitation energies. The Y-axis denotes the excitation It is easy to see that the magnetic dipole moment for
relative to the ground state at the same HO energy and 7Be is negative and those for 7Li are about 3µN with
Nmax value. Ingeneral,manyoftheresultsshowreason- the two different interactions. With increasing Nmax,
able convergence in the ~Ω range of 15 - 25 MeV where the magnetic moments tend to decrease slightly in
thegroundstateenergiesareclosetotheirminima. With magnitude but converge to within about 1-2%. The
increasing ~Ω, the JISP16 and NNLO interactions converged value of 7Be appears to be a little below
opt
producesimilar~Ω-dependencepatternsfor7Beand7Li. −1.08µN. The converged magnetic dipole moment of
7Li with NNLO is close to 2.97µ while the result
opt N
We now focus on the upper two excited states which with JISP16 is a little less than 2.96µN.
have the same total angular momentum J = 5. In the
2
left and right panels, i.e. for 7Li and 7Be respectively Our results for the groundstate quadrupolemoments,
another important electromagnetic observable, are
with JISP16, the two states are nearly degenerate.
However, with the NNLO interaction for 7Li (central shown in Fig. 5. The patterns for the quadrupole
opt
moment results reflect the patterns seen above for the
panel),thereisabouta1MeVlargerenergygapbetween
the two states. In addition, both the J = 5 curves for rms radii since they are closely related observables. In
2
NNLO increase monotonically with increasing ~Ω particular, we again observe a very weak convergence
opt
pattern with increasing N . One may visually esti-
while the corresponding JISP16 curves display peaks max
and dips in the lower ~Ω range which become less mate that the converged results will be in the region of
-3.5 e·fm2 for 7Li and -5.5 e·fm2 for 7Be.
pronouncedwithincreasingN . Thesenon-monotonic
max
features in the case of JISP16 may be attributed, at
least in part, to the mixing of these two states as we Next, we present the results of two B(E2) transitions
−
discuss below. to the ground state (Jπ = 3 ) in Fig. 6 where the
2
−
transition from the Jπ = 1 state is represented by
2
Let us now turn our attention to observables other the top three panels and from the Jπ = 7− state
than the energies. The calculated point proton root- 2
by the bottom three panels. The left panels and the
mean-square (rms) radii are presented for the ground middle panels are for 7Li with JISP16 and NNLO
states of 7Li and 7Be as a function of HO energy for a respectively. The right panels are for 7Be with JISP1op6t.
range of N values in Fig. 3. We first note that these
max As may be expected, the patterns of these B(E2)
rms radii are approximately independent of N near
max transitions are also similar to those of point proton rms
~Ω = 12 MeV. This feature of the rms radii, sometimes
radii in Fig. 3. We again observe a very weak tendency
referred to as defining the ”interaction” or ”crossover”
for convergence reflecting the major role of the radius
point, has been noted in Refs. [23][43][44] and is −
operator in the B(E2). Comparing both Jπ = 1 and
sometimes taken as the estimate of the converged rms 2
−
radius. We adopt the practice of quoting the crossover 7 → ground state (signified by ”g.s.” in the label)
2
point of the rms radii in the current work since the B(E2) transitions for 7Li (with either the JISP16 and
robustextrapolationsof rms radiiwill require additional NNLOopt interactions), the results for the latter transi-
developments and/or larger basis spaces. Similarly, tion are about half those of the former transition. This
for other observables quoted below, we cite only an patternpersistsinthe7BetransitionsalsoseeninFig. 3.
estimate based on a visual inspection of the convergence
pattern without an associated uncertainty. Over time, The excitation energies, quadrupole moments and
we anticipate that further theoretical developments will B(E2)’s that we have presented and discussed for 7Be
provide reliable extrapolation tools that yield refined arekey observablesthat playa criticalrolein the identi-
convergedestimates from our results. fication and characterizationof its yrast rotationalband
[45–48]. Indeed,theirsystematicsalongwiththesystem-
4

-30 -30
7 7
Li JISP16 Li NNLO 7
opt Be JISP16
Nmax=8
Nmax=10
-32 Nmax=12 -32
Nmax=14
J=1.5
V) Nmax=16
e Threshold
M-34 Exp. -34
( Extrap.

y
g
r
e
n
E-36 -36
-38 Exp. -38
Exp.
Exp.
-40 -40
10 15 20 25 30 35 10 15 20 25 30 35 10 15 20 25 30 35 40
FIG. 1. (Color online) Theenergy of theground state (J=23) for 7Be and 7Li with theJISP16 and NNLOopt interactions as a
function of HO energy. In this figure and the following figures, for 7Li and 7Be, the Nmax value ranges from 8 up to 16. The
increment of Nmax is 2. Extrapolated ground state energies are shown in purplewith uncertainties depicted as vertical bars.
atics ofthe other members of the same bandobtainedin whichwerediscussedaboveinconnectionwiththebehav-
NCFC calculations with JISP16 are essential to demon- iors of the excitation energies, are also apparent in the
stratingemergenceofcollectiverotationalmotionin7Be. B(M1) transitions of Fig. 7. The low and the high ~Ω
−
We present three B(M1) transitions as a function of regions of the 7Li transitions from these 5 states with
2
the HO energyin Fig. 7 for 7Liand7Be witha sequence the JISP16 interaction, for example, appear to support
ofN values. The three top graphsdisplaythe B(M1) the discussionsofmixingthatwerestimulatedbythe re-
max
− −
transitions from the Jπ = 1 state to the Jπ = 3 sults for the excitation energies. This mixing is again
2 2
ground state. The three middle graphs and the three seen to decrease with increasing Nmax. In addition, one
bottom graphs are from the Jπ = 5− state and the may now interpret the results for the B(M1)s from the
− 21 two 5− states in 7Be as suggesting mixing that also de-
Jπ = 5 state to the ground state, respectively. The 2
22 creases with increasing N .
− max
subscript 1 (2) on the 5 signifies the lower (upper) of
2
−
the two states with Jπ = 5 . In order to better examine the nuclear structure and
2
the relationship between the different states, we present
the total magnetic moment and the contributions to
It is noteworthy that the top three graphs have
the same convergence pattern for 7Li and 7Be with the total angular momentum from the orbital motions
of the proton and neutron as well as the contributions
the JISP16 and NNLO interactions. Considering
opt
from intrinsic spin in Fig. 8. We follow the procedures
the greatly expanded scales used for these B(M1)
presented in Ref. [32] and define these contributions
results, one observes that good convergence is actually
though matrix elements of the projections of these
attained in all cases shown in Fig. 7. In particular
individual contributions on the state’s total angular
the convergence at the highest N shown is good
max
over a fairly large range in ~Ω from about 15 MeV to momentum, i.e. by matrix elements of the terms on the
right-hand side of the following equation:
about 35 MeV. These B(M1)’s as well as the magnetic
dipolemomentscontinuetobeamongthebestconverged
oftheelectromagneticobservablesinNCFCcalculations. 1
J = (<J~·L~ >+<J~·L~ >+<J~·S~ >+<J~·S~ >).
p n p n
J +1
−
Featuressuggestiveofthemixingofthetwo 5 states, (5)
2
5

10 10
9 9
8 8
) J=2.5
V
e J=2.5
7 J=2.5 7
M
( J=2.5 J=2.5 J=2.5
y 7
6g Li NNLO 6
opt
r
e
5n 5
E
n J=3.5 J=3.5 Nmax=6 J=3.5
4 4
o N =8
max
i
t N =10
a 7 max
3t Li JISP16 N =12 3
ci max 7
x Nmax=14 Be JISP16
2E 2
N =16
max
Exp.
J=0.5 J=0.5
1 1
J=0.5
0 0
10 15 20 25 30 35 4010 15 20 25 30 35 4010 15 20 25 30 35 40
FIG. 2. (Color online) The energies of the four lowest excited states (Jπ = 21−,72−,25−,52−,) with JISP16 and NNLOopt for
7Be and 7Li as a function of the HO energy. The corresponding experimental values are shown as a horizontal solid line. The
Y-axisindicates theenergy gap between the excited and ground state.
From top to bottom, the three graphs are for 7Li increasingN , the level crossingas a function of~Ω is
max
with JISP16, 7Li with NNLO and 7Be with JISP16 disappearing in the ~Ω range of 10 to 40 MeV.
opt
respectively. One may compare the top panel in Fig.
8 for 7Li with JISP16 to corresponding results in Ref. We also include the magnetic moment in Fig. 8
[32] where the results were shown through Nmax = 14. for each state (purple symbols and lines) which is a
From the left to the right in each panel, the individual sum over the weighted contributions from the other
− − − − −
frames are for the states Jπ = 3 ,1 ,7 ,5 and 5 results depicted. Thus, the magnetic moment shows a
2 2 2 21 22
successively. In the frames for the first three states, similar convergence pattern to those of its individual
we see that the four components are well converged as contributions.
they become nearly independent of the values of ~Ω and
−
Nmax with increasing Nmax. Let us examine the results for the 7 state in Fig. 8
2
in some detail. In all cases, the intrinsic spin (green)
In the two panels with the JISP16 interaction results provides almost no contribution to the total angular
in Fig. 8, the two states with J = 5 exhibit trends momentum. For 7Li the neutron orbital motion (blue)
2
suggesting there is crossing and mixing as a function of dominates while for 7Be the proton orbital motion
~Ω at lower values of N . We have elected to display (red) dominates as may be expected on the basis of
max
a larger range of results in Fig. 8 for these two states isopsin symmetry (an approximate symmetry since we
thanpresentedinRef. [32]inordertomapoutthis level include Coulomb and NNLO is charge-dependent).
opt
crossing as a function of N and ~Ω. Clearly with Due to the relative weightings of these contributions,
max
6

2.6 2.6
Nmax=6
2.5 Nmax=8 2.5
m) Nmax=10
(f2.4 Nmax=12 7 2.4
us Nmax=14 Be JISP16
adi2.3 Nmax=16 2.3
R
S 2.2 7Li JISP16 7Li NNLO 2.2
M opt
R2.1 2.1

n
o
ot2.0 2.0
r
P
t 1.9 1.9
n
i
o
P1.8 1.8

1.7 1.7
1.6 1.6
10 15 20 25 30 35 4010 15 20 25 30 35 4010 15 20 25 30 35 40
FIG.3. (Coloronline)Pointprotonroot-mean-square(rms)radius(infm)ofthegroundstateasafunctionofHOenergywith
JISP16and NNLOopt for asequenceofNmax values. Theleft andright graphsarewith JISP16, andthemiddlegraph iswith
NNLOopt.
the magnetic moment shows an apparent slower trend results. Nevertheless,the results with AV18+IL7appear
towards convergence. For example, if we take the case to be more consistent with experiment. The values in
−
of 7Li with NNLOopt, we see that the 72 state has a the fourth column for NNLOopt show somewhat larger
negative contribution from the neutron intrinsic spin differences with experiment than those with JISP16 and
that, due to the large weighting of the neutron intrinsic AV18+IL7possiblydue,primarily,toslowerconvergence
magneticmoment,providesaslowerconvergencepattern of observables in 7Li with this interaction.
− −
than observed for the 3 and 1 states. The source of
2 2
this weaker convergence is the weaker convergence of
−
the neutron spin contribution (green) in the 7 state IV. SUMMARY
2
compared to its contribution in the two lower-lying
states - though it may be challenging to see this detail
We havecalculatedthe properties of7Li and7Be with
in the central frame of the middle panel in Fig. 8.
the JISP16andNNLO interactions in the no-corefull
opt
configuration(NCFC)approach. Wepresentresultswith
Finally, the properties of 7Li with the JISP16 and the many-body truncation parameter N up through
max
NNLO interactions and 7Be with JISP16 are listed in 16 and for natural parity (N is even). We obtained
opt max
Table.1. The results are compared to the experimental the energies of the ground and excited states, point pro-
valuesaswellasthosefromAV18+IL7[33][50]andchiral ton rms radii, magnetic and quadrupole moments, E2
NN+NNN[51]. For 7Li, the results with different inter- and M1 transitions. To our knowledge, this is the first
actions are listed in columns 2-6, and those for 7Be are time that properties of7Liwith NNLO are presented.
opt
incolumns7-9. Inthepresentwork,wedonotquotethe The theoretical results and experimental data are com-
uncertaintywhichhasbeenexplainedanddiscussedinre- paredinTable. 1. Takingintoconsiderationtheattained
latedreferences. However,theuncertaintiesestimatedin degreeofconvergence,wefindtheresultswithJISP16or
Ref. [51]remainasvalidestimatesforourcurrentuncer- AV18+IL7 are in better agreement with the experimen-
tainties(see,forexample,theextrapolationuncertainties taldatathanthosewithNNLO . Inordertoaidinthe
opt
depicted in Fig. 1) so that we can assert the following: diagnosticsofindividualstatesandtobetter understand
the theoretical results with JISP16 and NNLO inter- thegenerallygoodconvergenceofmagneticmoments,we
opt
actionareinreasonableagreementwiththeexperimental decomposethe magnetic momentcontributionsintopro-
7
ton and neutron orbital and spin components. This aids no. 11205004), by the US Department of Energy under
us, for example, to observe level crossing in the J = 5 Grants No. DESC0008485 (SciDAC/NUCLEI) and No.
2
statesfoundwiththeJISP16interactioninboth7Liand DE-FG02-87ER40371,bytheUSNationalScienceFoun-
7Be. Interestingly, we did not observe this level crossing dation under Grant No. 0904782. Computational re-
in 7Li with NNLO . sources were provided by the National Energy Research
opt
Supercomputer Center (NERSC), which is supported by
the Office of Science of the U.S. Department of Energy
under Contract No. DE-AC02-05CH11231. Computa-
V. ACKNOWLEDGMENTS
tionalresourceswerealsoprovidedbytheArgonneLead-
ership Computing Facility (ALCF) (US DOE Contracts
We acknowledge useful discussions with Andrey Shi- No. DE-AC02-05CH11231 and DE-AC02-06CH11357)
rokov and Mark Caprio. This work was supported by and under an INCITE award (US DOE Office of Ad-
theNationalNaturalScienceFoundationofChina(grant vanced Scientific Computing Research).
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9

3.06 3.06 -1.04
Nmax=6
Nmax=8
3.04 Nmax=10 3.04 -1.06
Nmax=12
Nmax=14
3.02 Nmax=16 3.02 -1.08
)
N 7
1( 7Li JISP16 Li NNLOopt
M3.00 3.00 -1.10
7
2.98 2.98 -1.12 Be JISP16
2.96 2.96 -1.14
2.94 2.94 -1.16
10 15 20 25 30 35 40 10 15 20 25 30 35 40 10 15 20 25 30 35 40
FIG.4. (Coloronline)Magnetic dipolemomentofthegroundstateshown asafunctionoftheHOenergyfor7Beand7Liwith
theJISP16 and NNLOopt interactions.

0.0 0.0
-0.5 -0.5
2 ) -1.0 -1.0
m
f -1.5 -1.5
e
(
t -2.0 -2.0
n
e
m -2.5 -2.5
o
M -3.0 Nmax=6 -3.0

le -3.5 Nmax=8 -3.5
o
p -4.0 Nmax=10 -4.0
u
r Nmax=12
d -4.5 -4.5
a N =14
max
u
Q -5.0 N =16 -5.0
max
-5.5 -5.5
7 7 7
Li JISP16 Li NNLO Be JISP16
-6.0 opt -6.0
10 15 20 25 30 35 10 15 20 25 30 35 10 15 20 25 30 35 40
FIG.5. (Coloronline)QuadrupolemomentofthegroundstateasafunctionoftheHOenergywithasequenceofNmax values
for 7Be and 7Li.
10

20 20 40
Nmax=6
)
4 fm16 7Li JISP16 Nmax=8 16 7Li NNLOopt 35 7Be JISP16
2 e Nmax=10 30
(
g.s.) 12 NNmmaaxx==1124 12 25
Nmax=16 20

2 8 8
/ 15
1
;
2
E 10
( 4 4
B
5

100 100 200

)
4
m
2 (ef 8 7Li JISP16 8 7Li NNLOopt 16 7Be JISP16

)
.
s
g. 6 6 12

2
7/ 4 4 8

;
2
E
B( 2 2 4
0 0 0
10 15 20 25 30 35 40 10 15 20 25 30 35 40 10 15 20 25 30 35 40
FIG. 6. (Color online) B(E2) to the ground state (Jπ = 3−) from the states Jπ = 1− (top) and Jπ = 7− (bottom) as a
2 2 2
function of the HOenergy with a sequenceof Nmax values. The abbreviation ”g.s.” signifies theground state.