Comment on "New modes of halo excitations in the 6He nucleus"

2 Pages·0.08 MB·English

by  D. V. Fedorov

Comment on ”New modes of halo excitations in the 6He nucleus”
D. V. Fedorov, A. Cobis, and A. S. Jensen
Institute of Physics and Astronomy, University of Aarhus, 8000 Aarhus C, Denmark
6
Wetrytoexplainthedifferencesinthe6Hedipolestrength
functionin[1]and[2]. Weperformthefullbasiscalculationof
4
thestrengthfunctionwiththesamerenormalized interaction e
u
as in [1] and show that the size of the basis, needed for con- al
verged calculations of the 6He continuum spectrum, is much nv 2
e
larger thanthatfor thediscretespectrum. Therenormalized g
interaction of [1] therefore cannot be used for thecontinuum ei
8 spectrum calculations with the same basis as for the ground ular 0 KKma=x=199
9 state. g max
n K =99
9 a -2 asymmpatxotics
1

n In a recent paper [1] the continuum properties of
a 6He were discussed within a three body nnα model. -4
0 10 20 30 40
J The strength functions were computed using the hyper- hyperradius (fm)
9 spherical harmonics approach. The authors argued that FIG.1. The lowest angular eigenvalue for 6He (Jπ =1−)
1 the main features of the continuum spectrum of the calculated with differentKmax. Alsoshown is theeigenvalue

nnα system can be obtained using only a few hyper- obtainedfromtheasymptoticlargedistanceexpansionofthe
1
spherical harmonics with the hyper-spherical quantum Faddeev equations [4].
v
3 number K ≤ 6, if one simultaneously renormalizes the
3 nα interaction by introducing additional attraction. All lengths are in fm and energies in MeV. Due to
0 However the dipole strength function in [1] differs sig- the overattractionthis interactionfails to reproducethe
1 nificantly from our recent calculation [2], where we used experimental nα phase shifts.
0 the same three-body model. The purpose of this com- For the nn interaction we use a simple Gaussian [7],
8
ment is to explain these differences. similar to the the one used in [5]. The potential repro-
9
We show that the continuum state calculations of the ducestheexperimentaleffectiverangeins-waveandscat-
/
h nnαsystemdemand amuchlargerbasis,K ≤100,com- tering lengths in s- and p-waves,
t
- pared to the bound state. In such case the simple pre-
cl scriptiontorenormalizethe nαinteractionaimingatthe Vˆnn(r)=exp(−r2/1.82)× (2)
u bound state might not be adequate for the continuum. (2.92+45.22s1s2−12.08l·(s1+s2)+26.85Sˆ12),
n The method we use is, in principle, similar to that
v: of [1] – the angular part of the wave function is ex- wheres1,s2arethespinsoftheneutrons,Sˆ12istheusual
i panded in terms of the hyper-spherical harmonics and tensor operator.
X
then the resulting system of coupled hyper-radial equa- InFig.1weillustratetheconvergenceofthelowestan-
ar tions is solved. However our approach has the follow- gular eigenvalue using, as examples, calculations with
ing modifications [3]: i) We use Faddeev rather than the
K = 9, K = 19, and K = 99. Obviously the
max max max
Schr¨odingerequation;ii)Priortosolvingthehyper-radial
basis limited to K < 9, used in [1], is by far not con-
equations we calculate the eigenvalues λn, as functions verged. It is only the curve for Kmax ≈ 100 that finally
ofthe hyper-radiusρ,ofthe angularpartofthe Faddeev
converged towards the large-distance asymptotics. This
operator. The quantities λn(ρ)ρ−2 then serve as effec- longtailofthe eigenvaluearises,inthe presentcase,due
tive potentials in the radial equations; iii) We use an-
to large scattering length in the nn subsystem [4]. The
alytic large-distance solutions of the Faddeev equations scale of convergence towards the asymptotic behavior is
for short range potentials [4] which greatly enhances the
then around 20 fm.
method and allows fully convergedcalculations. The convergence problems for the lowest effective po-
The nα interaction, originally fit to the scattering
tential is illustrated in Fig. 2. The converged curve has
data, was somewhat modified in [1,5,6] in order to com- the minimum value of about 0.15 MeV and the low (0.3
pensate for the limited basis,
MeV)barrierextendingintotheasymptoticregionbegin-
Vˆnα(r)=exp(−r2/2.352)× (1) ning atabout25fm. Incontrastthe curvefor Kmax =9
has minimum and maximum values at 0.7 MeV and 1.4
(50Pˆ −48.3Pˆ −23.0Pˆ −11.71l·s).
s p d MeV, respectively. A more realistic potential calculated
Here Pˆ is a projection operator on the corresponding with a correct nα interaction [2], is accidentally close to
l
orbital momentum state l and s is the neutron spin. the curve for Kmax =19.
1
2 We have to note, however, that due to the additional
attractionintroducedinthenαinteractionin[1]thecon-
V) vergeddipolestrengthfunctionhereisexcessivelyshifted
e
M 1.5 Kmax=9 towardslowerenergiesandthereforeisnotrealistic. The
( K =19
al Kmax=99 realistic dipole strength function [2] is close to the curve
nti asymmpatxotics Kmax = 19. In this case the limited basis is approx-
ote 1 imately compensated by the over-attractive potential.
e p Note that for the 0+ bound state calculations the cor-
v responding compensatory basis was K = 6 [1]. This
u max
ct 0.5 showsthatthe continuumspectrumdemandsmorecare-
e
eff ful and accurate calculations.
In conclusion, in order to explain the differences
0 between the dipole strength functions from [1] and [2]
0 10 20 30 40
we have performed a fully converged calculation of the
FIG.2. Thelowesteffehcytipveerpraodteiunst i(aflm()λ1(ρ)+15/4)ρ−2× Jπ = 1− continuum spectrum of 6He using the same
¯h2/(2m) for 6He (Jπ = 1−) calculated with different Kmax renormalized nα interaction as in [1]. We have shown
that the number of basis functions needed for converged
(m is the nucleon mass).
calculations of the 6He continuum spectrum is much
larger than that for the discrete spectrum. We conclude
Note that that the K < 9 basis seems to be sufficient
forthe0+groundstatewiththerenormalizedinteraction thatthetechniqueof[1]torenormalizethenαinteraction
to compensate for the limited basis becomes ambiguous
[1]. Thisshowsthattherenormalizationtechniqueneeds
whenappliedtothecontinuumspectrum. Thereasonfor
different bases for the ground state and the continuum
the differences is therefore that this renormalized inter-
state calculations.
actionisnotappropriateforcontinuumcalculationswith
The dipole strength function was calculated in [1]
the same basis size as for the ground state.
with a basis size corresponding to K ≤ 6. In Fig. 3 we
showtheresultsofourcalculationsofthedipolestrength
function for different K . Our K =9 curve agrees
max max
quite well with [1]. Increasing K causes the main
max
peaktoshifttowardslowerenergieswiththesecondpeak
formingandalsomovingtowardslowerenergies. There-
sult for K = 9 is rather smooth and deviates rela-
max [1] B.V.Danilin, T.Rogde, S.N.Ershov, H.Heiberg-Andersen,
tively little from the corresponding result obtained with
− J.S.Vaagen, I.J.Thompson and M.V.Zhukov, Phys. Rev.
plane waves for the 1 continuum wave function. This
C 55, R577 (1997).
leads to a presumably misleading impression that there
[2] A.Cobis,D.V.Fedorov,andA.S.Jensen,Phys.Rev.Lett.
−
is little structure in the 1 continuum. The other two 79, 2411 (1997).
curves, K =19 and K =99, on the contrary show
max max [3] D.V. Fedorov, A.S. Jensen, and K. Riisager, Phys. Rev.
visible concentration of the strength at small energies. C 50, 2372 (1994).
[4] A.S.Jensen,E.Garido,andD.V.Fedorov,Few-bodySys-
tems 22, 193 (1997)
K =9 [5] B.V. Danilin, M.V. Zhukov, A.A. Korsheninnikov, and
max
Kmax=19 L.V Chulkov,Sov.J. Nucl. Phys. 53, 45 (1991).
1) 0.4 Kmax=99 [6] B.V.Danilin,M.V.Zhukov,Phys.Lett.B302,129(1993).
-V Ref.[1] (arb. un.)
e [7] E. Garrido, D.V. Fedorov, and A.S. Jensen, Phys. Rev.
M C 55, 1327 (1997)
2 0.3
m
2 f
e 0.2
(
)
E
(
S 0.1
0
0 1 2 3 4 5
E (MeV)
FIG. 3. The dipole strength function S(E) = P ×
n
|hnEkM(E1)kgsi|2 as a function of thethree-body energy E
for the transition from the ground state (0+) to the contin-
uum state (1−) of 6He calculated with different Kmax. The
curvefrom [1] is given in arbitrary units.
2