Systematics of binding energies and radii based on realistic twonucleon plus phenomenological threenucleon interactions
ABSTRACT We investigate the influence of phenomenological threenucleon interactions on the systematics of groundstate energies and charge radii throughout the whole nuclear mass range from 4He to 208Pb. The threenucleon interactions supplement unitarily transformed twobody interactions constructed within the Unitary Correlation Operator Method or the Similarity Renormalization Group approach. To be able to address heavy nuclei as well, we treat the manybody problem in HartreeFock plus manybody perturbation theory, which is sufficient to assess the systematics of energies and radii, and limit ourselves to regularized threebody contact interactions. We show that even with such a simplistic threenucleon interaction a simultaneous reproduction of the experimental groundstate energies and charge radii can be achieved, which is not possible with unitarily transformed twobody interactions alone. Comment: 10 pages, 10 figures

Article: Quasiparticle Random Phase Approximation with Interactions from the Similarity Renormalization Group
[Show abstract] [Hide abstract]
ABSTRACT: We have developed a fully consistent framework for calculations in the Quasiparticle Random Phase Approximation (QRPA) with $NN$ interactions from the Similarity Renormalization Group (SRG) and other unitary transformations of realistic interactions. The consistency of our calculations, which use the same Hamiltonian to determine the HartreeFockBogoliubov (HFB) ground states and the residual interaction for QRPA, guarantees an excellent decoupling of spurious strength, without the need for empirical corrections. While work is under way to include SRGevolved 3N interactions, we presently account for some 3N effects by means of a linearly densitydependent interaction, whose strength is adjusted to reproduce the charge radii of closedshell nuclei across the whole nuclear chart. As a first application, we perform a survey of the monopole, dipole, and quadrupole response of the calcium isotopic chain and of the underlying singleparticle spectra, focusing on how their properties depend on the SRG parameter $\lambda$. Unrealistic spinorbit splittings suggest that spinorbit terms from the 3N interaction are called for. Nevertheless, our general findings are comparable to results from phenomenological QRPA calculations using Skyrme or Gogny energy density functionals. Potentially interesting phenomena related to lowlying strength warrant more systematic investigations in the future.Physical Review C 04/2011; 83:064317. · 3.72 Impact Factor  SourceAvailable from: J. Wambach[Show abstract] [Hide abstract]
ABSTRACT: In 16O and 40Ca an isoscalar, lowenergy dipole transition (ISLED) exhausting approximately 4% of the isoscalar dipole (ISD) energyweighted sum rule is experimentally known, but conspicuously absent from recent theoretical investigations of ISD strength. The ISLED mode coincides with the socalled isospinforbidden E1 transition. We report that for N=Z nuclei up to 100Sn the fully selfconsistent RandomPhaseApproximation with finiterange forces, phenomenological and realistic, yields a collective ISLED mode, typically overestimating its excitation energy, but correctly describing its IS strength and electroexcitation form factor. The presence of E1 strength is solely due to the Coulomb interaction between the protons and the resulting isospinsymmetry breaking. The smallness of its value is related to the form of the transition density, due to translational invariance. The calculated values of E1 and ISD strength carried by the ISLED depend on the effective interaction used. Attention is drawn to the possibility that in NnotequalZ nuclei this distinct mode of IS surface vibration can develop as such or mix strongly with skin modes and thus influence the pygmy dipole strength as well as the ISD strength function. In general, theoretical models currently in use may be unfit to predict its precise position and strength, if at all its existence.European Physical Journal A 11/2010; 47(1). · 2.42 Impact Factor
Page 1
arXiv:1005.1599v1 [nuclth] 10 May 2010
Systematics of binding energies and radii based on realistic twonucleon
plus phenomenological threenucleon interactions
A. G¨ unther,1, ∗R. Roth,1, †H. Hergert,2and S. Reinhardt1
1Institut f¨ ur Kernphysik, Technische Universit¨ at Darmstadt, 64289 Darmstadt, Germany
2National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA
(Dated: May 11, 2010)
We investigate the influence of phenomenological threenucleon interactions on the systematics of ground
state energies and charge radii throughout the whole nuclear mass range from4He to208Pb. The threenucleon
interactions supplement unitarily transformed twobody interactions constructed within the Unitary Correlation
Operator Method or the Similarity Renormalization Group approach. To be able to address heavy nuclei as
well, we treat the manybody problem in HartreeFock plus manybody perturbation theory, which is sufficient
to assess the systematics of energies and radii, and limit ourselves to regularized threebody contact interactions.
We show that even with such a simplistic threenucleon interaction a simultaneous reproduction of the experi
mental groundstate energies and charge radii can be achieved, which is not possible with unitarily transformed
twobody interactions alone.
PACS numbers: 21.30.Fe,21.45.Ff,21.60.Jz
I.INTRODUCTION
Nuclearstructuretheoryisapproachinganeraofsystematic
manybody calculations using nuclear Hamiltonians based on
Quantum Chromodynamics (QCD). An important step along
these lines is the formulation of nuclear interactions within
chiral effective field theory [1–3], leading to a consistent hier
archy of two, three and manynucleon interactions starting
from the relevant degrees of freedom and symmetries for the
lowenergy nuclear structure regime. The use of these two,
three and manynucleoninteractions in nuclear structure cal
culations is a formidable task.
Inadditiontofewbodycalculationsthemostpromisingnu
clear structure calculations using the chiral two plus three
nucleon interaction consistently have been performed in the
nocore shell model (NCSM) for mid pshell nuclei [4]. An
immense numerical effort is needed to compute and man
age the threebody matrix elements in these calculations,
which limits the range of applicability of these calculations
at present. Recently, the use of consistent two plus three
nucleon interactions resulting from a Similarity Renormaliza
tion Group evolution of the chiral two plus threenucleon in
teraction was demonstrated also in the context of the NCSM
[5]. This approach, a unitary transformation of the chiral
Hamiltonianaimingat a prediagonalizationthat improvesthe
convergence properties of NCSM substantially, holds great
potential also for the use in other manybody schemes and
will play a significant role in the future. However, the com
putational effort for including those two plus threenucleon
interactions into manybody calculations, be it exact or ap
proximate, is still the limiting factor for many applications.
In this paper we follow a more pragmatic route to explore
the impact of threebody forces in connection with unitarily
transformed twonucleon interactions. We start from the Ar
∗Electronic address: anneke.guenther@physik.tudarmstadt.de
†Electronic address: robert.roth@physik.tudarmstadt.de
gonne V18 highprecision twonucleon potential [6], which
is still widely used although it does not have the same sys
tematic link to QCD like the chiral effective field theory in
teractions and is consideredphenomenologicalin this respect.
We then use the Similarity Renormalization Group [7–11] as
well as the Unitary Correlation Operator Method [10, 12–14]
to construct a transformedtwonucleoninteraction, which has
a much better convergence behavior and allow us to use sim
plified manybody schemes. At this level neither genuine nor
induced threenucleon interactions are included. From vari
ous applications of these unitarily transformed twonucleon
interactions we know that there are characteristic deviations
of basic nuclear observables from the experimental systemat
ics that might be connected to threebody interactions. For
example, unitarily transformed twobody interactions which
yield a realistic systematics for binding energies tend to un
derestimate the charge radii [10, 15]. Here we study to what
extend these systematic deviations can be cured by including
a threebodyinteraction. Note that we are not aiming at a pre
cision description of individual nuclei but rather the complete
systematics from light nuclei,4He, to heavy nuclei,208Pb.
To facilitate calculations for the full mass range from4He
to208Pb we have to simplify the approach compared to the
consistent. The first simplification consists in the use of a
phenomenologicalthreebodyinteraction,whichallowsforan
efficient computation of matrix elements but violates the con
sistency discussed above. The second simplification consists
in the use of HartreeFock plus manybody perturbation the
ory for the approximate solution of the manybody problem.
Despite ofthese simplifications, we will obtainvaluableinfor
mation on the interplay between realistic twobody and phe
nomenological threebody interactions and on how well the
experimental systematics of groundstate energies and charge
radii can be reproduced. Furthermore, these studies prepare
thegroundforcalculationswithconsistentlytransformedtwo
plus threenucleon interactions.
After a brief reminder of the basic concepts of the Unitary
CorrelationOperatorMethodandthe Similarity Renormaliza
tion Groupwe introducethe phenomenologicalthreebodyin
Page 2
2
teraction and calculate the matrix elements in the harmonic
oscillator basis in the second section. In the third section, we
discuss the inclusion of the threebody interaction in Hartree
Fock and manybody perturbation theory and discuss the sys
tematics of groundstate energies and charge rmsradii across
the whole nuclear mass range and its dependence on the two
and threenucleon interaction.
II. FORMALISM
A.Unitary Correlation Operator Method and Similarity
Renormalization Group
The Unitary Correlation Operator Method and the Similar
ity RenormalizationGroupprovidetwo conceptuallydifferent
but physically related approaches for the construction of soft
phaseshift equivalent interactions.
The Similarity Renormalization Group (SRG) [7–11] aims
at the prediagonalizationof the Hamiltonian for a given basis
by means of a unitary transformation implemented through
the renormalizationgroupflow equation:
dHα
dα
= [ηα,Hα] ,
(1)
where α is the flow parameter and Hαthe evolved Hamilto
nian, with H0 = H being the initial or ‘bare’ Hamiltonian.
The antihermitian generator ηα defines the specifics of the
flow evolution, e.g. the representation with respect to which
the Hamiltonian should become diagonal or blockdiagonal.
Various choicesfor this generatorhavebeen investigated[11],
we restrict ourselves to the simple generator [7, 9]
ηα= [Tint,Hα] (2)
with Tint= T − Tcmbeing the intrinsic kinetic energy, which
leads to a prediagonalizationof the Hamiltonian with respect
to the eigenbasis of the kinetic energyor momentumoperator.
Once the generator is fixed, the Hamiltonian and all operators
of interest can be evolved easily using a matrix representation
of the flow equation (1).
In Abodyspace the evolutiongenerates up to Abodyoper
atorseveniftheinitialHamiltoniancontainsonlyuptotwoor
threebody operators. For reasons of practicability one has to
truncate the evolution at some low particle number—typically
this is done by solving the evolution equation in a matrix rep
resentation in two or threebody space. For the moment we
restrict ourselves to transformations in twobody space, i.e.,
we will discard any induced threebody interactions.
The aim of the Unitary Correlation Operator Method
(UCOM) [10, 12–14, 16] is to explicitly treat shortrange cor
relations inducedby the nuclearinteractionvia a static unitary
transformation. This transformationcan either be used to cor
relate the manybody states or to similarity transform opera
tors of interest, e.g. the Hamiltonian
˜ H = C†HC ,
(3)
using the correlation operator C. The dominant shortrange
correlations are induced by the strong shortrange repulsion
and the tensor part of the nuclear interaction. Therefore the
correlation operator is written as a product of two unitary op
erators, Crfor the central correlations and CΩfor the tensor
correlations. We choose an explicit form of the correlation
operators:
C = CΩCr= exp
?
− i
?
j<k
gΩ,jk
?
exp
?
− i
?
j<k
gr,jk
?
(4)
with the following ansatz for hermitian generators grand gΩ:
gr=1
gΩ=3
2[qrs(r) + s(r)qr] ,
2[(σ1· r)(σ2· qΩ) + (σ1· qΩ)(σ2· r)] ,
(5)
where qr=1
The strengths and radial dependencies of the two transforma
tions are governed by the correlation functions s(r) and ϑ(r)
for the central and tensor correlations, respectively. One can
obtain these functions via an energy minimization in the two
body system [16]. Recently, we have also employed the SRG
as tool for the determination of the UCOM correlation func
tions s(r) and ϑ(r) as discussed in Refs. [9, 10]. Here, we will
use these SRGoptimized UCOM correlation functions only.
Though the SRG and UCOMtransformations have a dif
ferent formal background, they address the same physics of
shortrange correlations. A first connection becomes clear at
the level of the generators [8]—the SRG generator (2) in two
body space at α = 0 reveals the same operator structures that
appear in the UCOMgenerators (5). At the level of matrix
elements, both the SRG and UCOMtransformations lead to
a suppression of the offdiagonal momentumspace matrix el
ements and an enhancement of the lowmomentum matrix el
ements as discussed in detail in Ref. [10].
In the following, we employ both transformations to gener
ate oneparameterfamilies of phaseshiftequivalenttwobody
interactions starting from a specific initial NNinteraction,
the Argonne V18 (AV18) in our case.
transformation the flow parameter α directly spans this fam
ily of twobody interactions. We will study two versions of
the SRGtransformation, one where the flow equations are
solved for all partial waves and one where only the partial
waves containing relative Swaves, i.e. the1S0and the cou
pled3S1−3D1partial waves, are transformed. The latter is
motivated by the fact that shortrange correlations affect the
Swave channels most, because for all higher orbital angular
momenta the relative wave functions are suppressed by the
centrifugal barrier at short distances. We use the label ‘SRG’
for the fully transformed interactions and ‘SSRG’ for the S
waveonly transformations. For the UCOMtransformation
we use correlation functions determined from SRGevolved
twobody wave functions as discussed in Refs. [9, 10], thus
the flow parameter α also spans a family of different UCOM
transformed interactions. Note that the standard formulation
of UCOM only uses different transformationsfor the different
(S,T)channels. We thus use the SRGevolved wave func
tions for the lowest partial waves for each (S,T)channel to
define the correlation functions, leading to a transformed in
teraction labelled ‘UCOM(SRG)’. Analogously to the SSRG
2(r
r·q+q·r
r), qΩ= q−r
r·qr, and q =1
2[p1−p2].
For the SRG
Page 3
3
00.04 0.080.12
0.16
0.2
α [fm4]
30
29
28
27
26
25
24
.
E(4He) [MeV]
FIG. 1:
the flow parameter α obtained from a converged nocore shell
model calculation using the UCOM(SRG)transformed (•), the S
UCOM(SRG)transformed (?), the SRGtransformed (?), or the S
SRGtransformed (?) AV18 potential. The horizontal lines indicate
the experimental binding energy (
energy for the bare AV18 twobody interaction (
(color online) Binding energy of
4He as function of
) and the exact ground state
) [17].
transformation,we canalso use an SwaveonlyUCOM trans
formation, denoted ‘SUCOM(SRG)’, which acts only in the
1S0and the coupled3S1−3D1partial waves.
So far, we have assumed that both transformationsare eval
uated in twobody space, leading to a transformed interaction
containing twobody terms only. A consistent firstprinciples
treatment requires the transformation to be performed in A
body space, leading to a hierarchy of induced interactions up
to the Abody level, as mentioned earlier. The most advanced
attempts along these lines use the full SRGevolution at the
threebody level to construct a consistently transformed two
plus threenucleon interaction [5]. The use of those two plus
threebody interactions in manybody calculations is very de
manding and presently limited to rather small model spaces.
Therefore, we follow a more pragmatic path in this work.
We evaluate the unitary transformations at the twobody level
and mimic the threebody contributions (genuine plus in
duced) through a simple phenomenological threebody inter
action. By using a simplified threenucleon (3N) interaction,
e.g., a regularized contact or a Gaussian interaction, the cal
culation of the threebody matrix elements becomes formally
and computationally much less demanding. This allows us to
study the impact of 3N interactions on various nuclear struc
ture observables for nuclei and model spaces beyond the do
main accessible with realistic 3N interactions. Furthermore,
we can developand benchmarkapproximatetreatments of the
threebody contributions and establish the technical frame
worktoinclude3Ninteractionintodifferentmanybodymeth
ods.
The parameters of the phenomenological 3N interactions
will be adjusted depending on the flow parameter α of
the transformed twonucleon (NN) interaction. For a wide
range of α parameters the transformed twobody interaction
alone produces an overbinding compared to the experimen
tal ground state energy. This is illustrated in Fig. 1 for the
groundstate energy of4He as function of α obtained in con
vergednocoreshellmodelcalculationsfortheUCOM(SRG)
, the SUCOM(SRG), the SRG, and the SSRGtransformed
AV18 interaction. Thus the additional phenomenological in
teraction, which mimics the net effect of the genuine and the
induced 3N interaction, has to be repulsive in order to lead
to a4He binding energy consistent with experiment. Note
that the phenomenological threebody forces that are used in
connection with the bare AV18 interaction are generally at
tractive. Thus the induced 3N interaction resulting from the
unitary transformation of the NN interaction alone has to be
repulsiveand sufficiently strong to create an overall repulsive
threebody contribution.
B.ThreeBody Contact Interaction
The simplest choice for a phenomenological3N interaction
is a spinisopinindependentcontact interaction
V3N= C3Nδ(3)(x1− x2) δ(3)(x1− x3) (6)
with variable strength C3N. Despite its simplicity it allows us
to study the impact of a 3N interaction on bulk observables
like groundstateenergies or chargeradii. Obviouslythis sim
plistic choice offers substantial computational advantages.
For evaluating the matrix elements of a realistic 3N in
teraction for the use in configurationspace HartreeFock or
nocore shell model type calculations one typically adopts
a twostep procedure: First the matrix elements are eval
uated in a Jacobicoordinate basis for the relative motion
in the threenucleon system. Then, through a sequence of
TalmiMoshinski transformations and angular momentum re
couplings, the matrix elements are transformed into the m
scheme to perform the manybody calculation. Both steps
are nontrivial and computationally demanding, thus limiting
the modelspace sizes for which those matrix elements can be
handled.
In contrast, the matrix elements of the contact interac
tion can be directly evaluated in the mscheme in a straight
forward manner. We first consider the matrix elements of the
3N contact interaction with respect to the spatial part of three
particle product states in the harmonic oscillator basis
?n1l1ml1,n2l2ml2,n3l3ml3V3Nn4l4ml4,n5l5ml5,n6l6ml6? . (7)
The spin and isospin quantum numbers and the antisym
metrization will be included subsequently. We can insert a
unitoperatorinpositionrepresentationusingcartesiancoordi
nates and directly evaluate the Kroneckerdeltas. This leaves
us with a single integration over a singleparticle coordinate,
which can be rewritten in spherical coordinates. Introducing
the position representation of the harmonic oscillator single
particle states, φnlml(x) = Rnl(x)Ylml(Ω), with radial wave
Page 4
4
functions Rnl(x) and spherical harmonics Ylml(Ω), we obtain:
?n1l1ml1,n2l2ml2,n3l3ml3V3Nn4l4ml4,n5l5ml5,n6l6ml6?
= C3N
dxx2Rn1l1(x)Rn2l2(x)Rn3l3(x)
?
× Rn4l4(x)Rn5l5(x)Rn6l6(x)
dΩ Y∗
×
?
l1ml1(Ω)Y∗
× Yl4ml4(Ω)Yl5ml5(Ω)Yl6ml6(Ω) .
l2ml2(Ω)Y∗
l3ml3(Ω)
(8)
The integraloverthe six radial wave functionsRnl(x) has to be
calculatednumericallywhiletheintegraloverthesixspherical
harmonics Ylml(Ω) can be evaluated analytically. The product
of three spherical harmonics can be reduced to one spherical
harmonic and the integral over the remaining two spherical
harmonics can be solved analytically, leading to
?
dΩY∗
l1ml1(Ω)Y∗
1
16π2ˆl1ˆl2ˆl3ˆl4ˆl5ˆl6
l2ml2(Ω)Y∗
l3ml3(Ω)Yl4ml4(Ω)Yl5ml5(Ω)Yl6ml6(Ω)
1
2L2+ 1
=
?
L1L2L3
ML1ML2ML3
?L1 l3
ML1
c
L3
ML3
c
× c
× c
× c
?l1l2
?l1
?l4
0 0
???L1
0
?
c
0 0
?
?
???L2
ML1ml3
?L3
0
?
c
?l4l5
???
???
0 0
L2
ML2
L2
ML2
???L3
?
?
0
?
c
?L3 l6
0 0
???L2
0
?
l2
ml1ml2
???
???
L1
?L1
ML3ml6
l3
l5
ml4ml5
l6
(9)
withˆl =
coefficients.
We precompute and store those angular integrals as well as
theradialintegralsin(8). Theinclusionofthespinandisospin
quantum numbers, the coupling of the singleparticle orbital
angular momenta and the spins, and the antisymmetrization
are then done on the flight during the manybody calculation.
This makes calculations in large model spaces feasible.
For applications beyond the meanfield level a regulariza
tion of the contact interaction is inevitable. However, the
regularization should preserve the simplicity of the matrix
element calculation,which rules out momentumspacecutoffs
and such. Hence, we introduce an energy cutoff parameter
e3N, which defines an upper bound for total oscillator energy
ofthethreeparticlestate, (2n1+l1)+(2n2+l2)+(2n3+l3) ≤ e3N.
The implementation of this cutoff is trivial and it preserves all
computational advantages of the contact interaction.
√2l + 1 and c
?
l1
l2
ml1ml2
???L
ML
?
being ClebschGordan
III. MANYBODY CALCULATIONS
We adopt the 3N contact interaction together unitarily
transformed NN interactions for the study of the systematics
of nuclear groundstate energies and charge radii throughout
the whole mass range from4He to208Pb using HartreeFock
and manybodyperturbation theory.
A.HartreeFock Approximation
WehaveemployedtheHartreeFock(HF)approximationas
a first indicator for the gross systematics of binding energies
andchargeradiiobtainedwith unitarilytransformedtwobody
interactions in Refs. [10, 15] already. In order to assess the
impact of 3N contact interactions we extend our HF frame
work in a first step.
All calculations are based on the translationally invariant
Hamiltonian
Hint= Tint+ VNN+ V3N= H(2)
int+ V3N
(10)
with VNNbeing the UCOM or SRGtransformed NN inter
action and Tint = T − Tcmthe intrinsic kinetic energy. This
Hamiltonian includes all charge dependent and electromag
netic terms of the transformed AV18 potential as well as the
phenomenologicalthreebody force.
The HF equations are formulated in a harmonic oscilla
tor basis representation, i.e., the singleparticle states are ex
panded in the harmonic oscillator states:
νljmmt? =
?
n
C(νljmt)
n
nljmmt? ,
(11)
where nljmmt? denotes the harmonic oscillator eigenstates
with radial quantum number n, orbital angular momentum l,
total angular momentum j with projection m, and isospin pro
jection quantum number mt. Since we only consider closed
shell nuclei in the following, the expansion coefficients are
independent of m. The HF equations can now be written as
?
¯ n
h(ljmt)
n¯ n
C(νljmt)
¯ n
= ε(νljmt)C(νljmt)
n
(12)
with the singleparticle energies ε(νljmt). The matrix elements
of the singleparticle HF Hamiltonian
h(ljmt)
n¯ n
=
?
l′j′m′
t
?
n′¯ n′
?nljmt,n′l′j′m′
tH(2)
int¯ nljmt, ¯ n′l′j′m′
t? ̺(l′j′m′
t)
¯ n′n′
+1
2
?
l′j′m′
l′′j′′m′′
t
t
?
n′n′′
¯ n′¯ n′′
?nljmt,n′l′j′m′
t,n′′l′′j′′m′′
t×
× V3N¯ nljmt, ¯ n′l′j′m′
t, ¯ n′′l′′j′′m′′
t? ̺(l′j′m′
t)
¯ n′n′
̺(l′′j′′m′′
¯ n′′n′′
t)
(13)
are obtained by contractions of the antisymmetrized matrix
elements of the twobodypart of the Hamiltonian H(2)
threebody interaction V3Nwith the onebody density matrix
given by
intand the
̺(ljmt)
¯ nn
=
?
ν
O(νljmt)C(νljmt)∗
¯ n
C(νljmt)
n
(14)
with O(νljmt)being the numberof occupiedmagneticsublevels
which is 2j + 1 for closedshell nuclei.
In the following the HF approach is applied to selected
closedshell nuclei from4He to208Pb. The HF equations are
solved iteratively until full selfconsistency is reached. The
Page 5
5
14
12
10
8
6
4
.
E/A[MeV]
2
3
4
5
.
rch[fm]
4He
16O
24O
34Si
40Ca
48Ca
48Ni
56Ni
60Ni
78Ni
88Sr
90Zr
100Sn
114Sn
132Sn
146Gd
208Pb
FIG. 2: (color online) Groundstate energies per nucleon and charge
radii of selected closedshell nuclei resulting from HF calculations
based on pure twobody interactions for emax = 10: UCOM(SRG)
with α = 0.16fm4(•), SUCOM(SRG) with α = 0.16fm4(?), SRG
with α = 0.10fm4(?), SSRG with α = 0.10fm4(?). The bars
indicate the experimental values [18, 19].
model space is truncated at a given major oscillator quantum
number e = 2n+l ≤ emax, where emax= 10 is sufficient to ob
tain convergedgroundstateenergies and radii at the HF level.
The oscillator parameteris chosen for each nucleus separately
such that the experimental charge radius is reproduced by a
shellmodel Slater determinant built from harmonic oscillator
singleparticle states.
As a first illustration of the behavior of unitarily trans
formed twobody interactions Fig. 2 summarizes the ground
state energies per nucleon and the charge radii obtained at
the HF level for nuclei up to208Pb. We adopt four differ
ent twobody interactions—UCOM(SRG), SUCOM(SRG),
SRG, and SSRG—with flow parameters relevant for the later
calculations including the 3N contact interaction. We observe
that the general trend of the binding energies and charge radii
is similar for the UCOM(SRG), the SUCOM(SRG), and the
SSRG interactions. All three interactions produce binding
energies that are within 2 MeV per nucleon of the experimen
tal values for the whole mass range. By includingcorrelations
beyond HF, e.g., through manybody perturbation theory, all
interactions would lead to an overbindingcompared to exper
iment. At the same time the charge radii are underestimated
for all but the lightest isotopes. Those systematic deviations
can be remedied by a repulsive 3N interaction, as it will be
included in the next step.
The SRGtransformed interaction exhibits a vastly differ
ent behavior. The binding energies per nucleon increase
rapidly with mass number, leading to an completely unphysi
cal overbinding already at the HF level for intermediate and
heavy nuclei. At the same time the charge radii are even
smaller than the ones obtained with the other transformed
interactions. Those strong systematic deviations have to be
compensated by the 3N interaction that is generated from the
8
6
4
.
E/A[MeV]
2
3
4
5
.
rch[fm]
4He
16O
24O
34Si
40Ca
48Ca
48Ni
56Ni
60Ni
78Ni
88Sr
90Zr
100Sn
114Sn
132Sn
146Gd
208Pb
FIG. 3: (color online) Groundstate energies per nucleon and charge
radii of selected closedshell nuclei resulting from HF calculations
for the pure twobody interaction SUCOM(SRG) for emax= 10 and
different flow parameters: α = 0.04fm4(•), α = 0.12fm4(?), α =
0.16fm4(?). The bars indicate the experimental values [18, 19].
initial NN potential in the course of the SRGevolution. Be
cause of the mere size of the threebody corrections needed
one cannot expect a simple phenomenological interaction to
beadequatetocapturethemainphysicscontainedinthethree
body contributions. Therefore, we will not consider the fully
SRGtransformed interactions in the following.
Before including the 3N contact interaction explicitly, we
analyze the dependence of the HF results obtained with the
transformed twobody interactions on the flow parameter α.
In Fig. 3 the binding energies and charge radii for the S
UCOM(SRG) interactions with α = 0.04fm4, 0.12fm4, and
0.16fm4are shown. For the smallest flow parameter α =
0.04fm4the groundstate energies reproduce the systemat
ics of the experimental values up to a constant shift. The
missing binding energy can be explained by beyondHF cor
relations that can be recovered, e.g., by perturbation theory.
This flow parameter would be used for calculations based on
the pure NN interaction, as they are discussed in detail in
Refs. [9, 10, 15].
When increasing the flow parameter entering into the con
struction of the SUCOM(SRG) interaction to α = 0.12fm4
or 0.16fm4the groundstate energy at the HF level decreases
substantially. For most nuclei the binding energy per nucleon
morethandoubleswhengoingfromα = 0.04fm4to 0.16fm4.
For heavier nuclei the increase is larger, thus leading to a tilt
of the groundstate energy systematics with respect to the ex
perimental behavior. Unlike the energies, the charge radii ex
hibit a very weak αdependence as shown in the lower panel
of Fig. 3. For all flow parameters considered here, the radii
are somewhat underestimated. The situation is very similar
for the UCOM(SRG) and the SSRG interactions.
This general phenomenology of groundstate energies and
charge radii obtained from unitarily transformed interactions
at larger flow parameters illustrates that the purely repulsive