Global study of the spectroscopic properties of the first 2+ state in even-even nuclei

Abstract

We discuss the systematics of the 2+ excitation energy and the transition probability from this 2+ to the ground state for most of the even-even nuclei, from O16 up to the actinides, for which data are available. To that aim we calculate their correlated J=0 ground state and J=2 first excited state by means of the angular-momentum and particle-number projected generator coordinate method, using the axial mass quadrupole moment as the generator coordinate and self-consistent mean-field states only restricted by axial, parity, and time-reversal symmetries. The calculation, which is an extension of a previous systematic calculation of correlations in the ground state, is performed within the framework of a non-relativistic self-consistent mean-field model using the same Skyrme interaction SLy4 and a density-dependent pairing force to generate the mean-field configurations and mix them.

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arXiv:nucl-th/0611089v1 24 Nov 2006
Global study of the spectroscopic properties of the first 2+state in even-even nuclei
B. Sabbey,1M. Bender,2,3G. F. Bertsch,1and P.-H. Heenen4
1Department of Physics and Institute for Nuclear Theory,
Box 351560, University of Washington, Seattle, WA 98195
2Dapnia/SPhN, CEA Saclay, F-91191 Gif sur Yvette Cedex, France
3Centre d’Etudes Nucl´ eaires de Bordeaux Gradignan, BP120, F-33175 Gradignan Cedex, France
4Service de Physique Nucl´ eaire Th´ eorique, Universit´ e Libre de Bruxelles, CP 229, B-1050 Brussels, Belgium
(Dated: November 22 2006)
We discuss the systematics of the 2+excitation energy and the transition probability from this 2+
to the ground state for most of the even-even nuclei, from16O up to the actinides, for which data are
available. To that aim we calculate their correlated J = 0 ground state and J = 2 first excited state
by means of the angular-momentum and particle-number projected generator coordinate method,
using the axial mass quadrupole moment as the generator coordinate and self-consistent mean-field
states only restricted by axial, parity, and time-reversal symmetries. The calculation, which is
an extension of a previous systematic calculation of correlations in the ground state, is performed
within the framework of a non-relativistic self-consistent mean-field model using the same Skyrme
interaction SLy4 and a density-dependent pairing force to generate the mean-field configurations
and mix them. To separate the effects due to angular-momentum projection and those related to
configuration mixing, the comparison with the experimental data is performed for the full calcula-
tion and also by taking a single configuration for each angular momentum, chosen to minimize the
projected energy. The theoretical energies span more than 2 orders of magnitude, ranging below
100 keV in deformed actinide nuclei to a few MeV in doubly-magic nuclei. Both approaches sys-
tematically overpredict the experiment excitation energy, by an average factor of about 1.5. The
dispersion around the average is significantly better in the configuration mixing approach compared
to the single-configuration results, showing the improvement brought by the inclusion of a dispersion
on the quadrupole moment in the collective wave function. Both methods do much better for the
quadrupole properties; using the configuration mixing approach the mean error on the experimental
B(E2) values is only 50%. We discuss possible improvements of the theory that could be made by
introducing other constraining fields.
I.INTRODUCTION
Self-consistent mean-field methods (SCMF) are the
only computationally tractable methods which can be
applied to medium and heavy nuclei and have a well-
justified foundation in many-body theory [1]. Recently
there has been considerable progresses in using these
methods to compute nuclear mass tables [2].
the appealing features of the SCMF is that the proper-
ties of all nuclei are derived from a fixed energy density
functional that depends on a small number of universal
parameters, and can be used for the entire chart of nu-
clei. The Skyrme family of functionals which is used in
the present study, depends on about 10 parameters for
the particle-hole interaction with 2-3 extra parameters
for the pairing interaction.
In a previous study [3, 4], we have used an extended
SCMF theory to calculate the ground state binding en-
ergies of the about 600 even-even nuclei whose masses
are known experimentally.
sions were introduced to treat correlation effects going
beyond a mean field approach.
tion of the SCMF wave functions to restore symmetries
broken by the mean field: particle numbers and angu-
lar momentum.The second is a mixing of projected
mean-field states corresponding to different intrinsic axial
quadrupole deformations. These calculations were per-
formed with the same energy functional as for the de-
One of
In particular, two exten-
The first is a projec-
termination of the mean-field configurations, so they do
not require to introduce new parameters. Our main aim
was to determine the effect of correlations on masses.
In particular, the error on experimental masses in mi-
croscopically based methods presents arches between the
magic numbers. The correlations added in Refs. [3, 4]
clearly reduce the amplitude of the arches in the mass
residuals, but do not remove them completely. For the
parameterization we have used, however, the arches are
much more pronounced along isotopic chains than along
isotonic chains, which suggests that their appearance is
not only related to missing correlations, but also to defi-
ciencies of the currently used effective interactions. There
is a clear improvement when looking at mass differences
between neighboring nuclei around magic ones, in partic-
ular when crossing proton shells. Similarly, the system-
atics of charge radii is also improved, particularly in the
transitional region between spherical and well-deformed
nuclei. Altogether, this study clearly confirms the im-
portance of incorporating some beyond mean-field corre-
lations explicitly in the method and not heuristically in
the energy density functional.
In this work, we extend our previous study to two new
observables: the excitation energy of the first 2+state
and the B(E2) value for the transition between the first
2+and the ground state.
A simple approximation that can be systematically ap-
plied is to start from a set of constrained SCMF config-

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urations corresponding to different axial quadrupole mo-
ments and to project them on angular momentum. Then,
one searches the projected configuration that has the low-
est energy for each angular momentum.
this procedure minimization after projection (MAP). A
more sophisticated procedure requires an additional step:
for each J-value, the total energy is further minimized
by mixing projected SCMF configurations correspond-
ing to different deformations. The mixing of constrained
SCMF configurations is called the Generator Coordinate
Method (GCM), and is based on the solution of the Hill-
Wheeler (HW) equation. We shall label configuration
mixing calculations by HW.. Although numerically de-
manding, this approach has nowadays been used to study
the detailed structure of low-energy collective excitation
spectra of nuclei up to the actinide region using non-
relativistic Skyrme [5, 6, 7, 8, 9] and Gogny [10, 11, 12]
interactions as well as relativistic Lagrangians [13, 14].
Here, however, as in [3, 4], we aim at something differ-
ent: the calculation of a few very specific properties of
the lowest 0+and 2+states for several hundred nuclei.
For this purpose, the numerical procedure necessary for a
detailed study by the GCM is too costly to be applied on
such a large scale with present computational resources.
Our bias on lowest collective 0+and 2+states, however,
permits to set-up a tailor-made numerical scheme that
reduces the effort considerably. For the angular momen-
tum projection, we will generalize the topological Gaus-
sian overlap approximation [15, 16] (GOA) used in our
previous work. The GCM calculations will also be re-
duced in size by truncations of the configuration space.
We will call
II.CALCULATIONAL DETAILS
As mentioned in the introduction, calculations are per-
formed along the lines of Ref. [4]. We will briefly sum-
marize the most important points and give details only
for the necessary extensions to calculate 2+states and
matrix elements of the quadrupole operator. The calcu-
lations reported here go beyond mean field in three re-
spects: (i) projections on good particle numbers; (ii) pro-
jection on angular momentum J = 0 and 2; and (iii) mix-
ing of states with different intrinsic deformation. All the
results presented in this paper include particle-number
projection and we drop explicit reference to particle num-
ber from the notation.
A.DFT Calculations
We use the code ev8 (see Refs. [17, 18]) to solve
the SCMF equations for an energy functional based on
the Skyrme interaction. The single-particle orbitals are
discretized on a three-dimensional Lagrange mesh cor-
responding to a cubic box in coordinate space.
code imposes time-reversal symmetry on the many-body
state, assuming pairs of conjugated states linked by
The
time-reversal and having the same occupation number,
which limits the description to even-even nuclei, and non-
rotating states. The only other restriction on the wave
function is that the Slater determinant of the orbits is
invariant with respect to parity and axial rotations. For
this work we take the SLy4 Skyrme parameterization [19],
the same as we used in our previous global study. For
the pairing interaction we choose an energy functional
that corresponds to a density-dependent zero-range pair-
ing force, with cutoffs at 5 MeV above and below the
Fermi energy, as described in [20]. As in earlier projected
GCM studies, the pairing strength is taken to be −1000
MeV fm3for both protons and neutrons.
To avoid a breakdown of pairing correlations for small
level densities around the Fermi surface, we enforce the
presence of pairing correlations using the Lipkin-Nogami
(LN) prescription as described in Ref. [21]. However, we
emphasize that the LN prescription is only used to gener-
ate wave functions of the BCS form; physical properties
are calculated with the code promesse [22], which per-
forms projections on proton and neutron particle num-
bers and provides the matrix elements needed for angular
momentum projection.
Mean-field states with different mass quadrupole mo-
ments are generated by adding a constraint to the mean-
field equations to force the intrinsic axial quadrupole mo-
ment q to have a specific value. Higher-order even ax-
ial multipole moments are automatically optimized for
a given mass quadrupole moment. A typical calculation
for a nucleus involves the construction of about 20 SCMF
wave functions that span a range of deformations suffi-
cient to describe the ground state.
B.Beyond mean field
Formally, eigenstates |JMq? of the angular momentum
operatorsˆJ2andˆJzwith eigenvalues J(J +1) and M are
obtained by application of the operator
ˆPJ
MK=2J + 1
16π2
?4π
0
dα
?π
0
dβ sin(β)
?2π
0
dγ D∗J
MKˆR, (1)
on the SCMF states |q?. The rotation operatorˆR and the
Wigner function DJ
α,β,γ.
In a further step, we consider the variational configu-
ration mixing in the framework of the Generator Coor-
dinate Method. Starting from the ansatz
MKboth depend on the Euler angles
|JMk? =
?
q
fJk(q)|JMq?,(2)
for the superposition of projected SCMF states, where
k labels the states for given J and M, the variation of
the energy ?JMk|ˆH|JMk?/?JMk|JMk? leads to the dis-
cretized Hill-Wheeler-Griffin equation
?
q′
?HJ(q,q′) − EJ,kIJ(q,q′)?fJ,k(q′) = 0(3)

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that determines the weights fJk(q) of the SCMF states
in the projected collective states, and the energy EJ,kof
the collective states.
We have to calculate diagonal and off-diagonal matrix
elements of the norm and Hamiltonian kernels. For the
sake of simple notations, we use a Hamiltonian operator
in all formal expressions, although there is no Hamil-
tonian corresponding to an effective energy functional.
In practice, as it is common procedure [11], we replace
the local densities and currents entering the mean-field
energy functional with the corresponding transition den-
sities.
The axial symmetry of the mean-field states allows to
simplify the 3-dimensional integral over Euler angles to
a one-dimensional integral:
IJ(q,q′) = ?JMq|JMq′?
=
1
NJqNJq′
?1
0
dcos(β)dJ
00(β)?q|ˆRβ|q′?
(4)
HJ(q,q′) = ?JMq|ˆH|JMq′?
=
1
NJqNJq′
?1
0
dcos(β)dJ
00(β)?q|ˆRβˆH|q′?
(5)
with normalization factors
N2
Jq= (2J + 1)
?1
0
dcos(β) dJ
00(β)?q|ˆRβ|q?.(6)
For the calculation of B(E2) values and spectroscopic
quadrupole moments we also have to evaluate matrix el-
ements of the quadrupole operator, which is outlined in
appendix A. A more detailed discussion of the method
can be found in Refs. [4, 9] and references given therein.
1.topGOA overlaps and Hamiltonian matrix elements
In Ref. [23], we found that for the description of the
properties of the ground state the J = 0 projected over-
laps can be computed with a sufficient accuracy for our
purpose with a 2- or 3-point approximation to the inte-
gral using an extension of the gaussian overlap approxi-
mation called the topGOA [24]. There the rotated over-
laps are parameterized by
?q|ˆRβ|q′?t=



?q|q′?e−c(q,q′)F(β)
or
?q|q′?e−c(q,q′)F(β)−d(q,q′)F2(β)
(7)
where F(β) = sin2(β) and the subscript t specifies the
topGOA approximation. This form satisfies the require-
ment of the GOA that F(β) → β2for small β as well
as the topological requirement that F(π − β) = F(β).
Projected matrix elements of the Hamiltonian are also
needed; these were calculated assuming the functional
form
?q|ˆRβˆH|q′?t2 = ?q|q′?e−c(q,q′)F(β)[h0(q,q′) − h2(q,q′)F(β)]
?q|ˆRβˆH|q′?t3 = ?q|q′?e−c(q,q′)F(β)−d(q,q′)F2(β)[h0(q,q′) − h2(q,q′)F(β) − h4(q,q′)F2(β)](8)
for the 2- and the 3-point approximations respectively.
In general, the 2-point approximation is adequate for
heavy nuclei and large deformations, but the 3-point ap-
proximation is necessary to describe light nuclei.
take points at β equal to zero, and to a value where
?q|ˆRβ|q′? ≈ 0.5 for the 2-point approximation. A third
point is added at β = π/2 for the 3-point approximation.
This is important for matrix elements between oblate and
prolate configurations, which have their maximum value
at π/2,
In Fig. 1, we show an example of an energy curve deter-
mined by this procedure for38Ar. Angular momentum
projection changes the quadrupole moment from the in-
trinsic one to the one observable in the laboratory frame,
which now depends on angular momentum. Most no-
tably, it is zero for J = 0 states independent of the
deformation of the intrinsic configuration. As a conse-
quence, it is more convenient and intuitive to use the
intrinsic quadrupole moment of the SCMF states to la-
We
bel the projected states. The marked points correspond
to the q values of the SCMF configurations that were
previously calculated. They are connected with lines to
distinguish the J = 0 and J = 2 curves.
The J = 0 curve has two very shallow minima at
deformations q ≈ −100 and +100 fm2.
curve has a pronounced oblate minimum at −125 fm2
and a shallow secondary minimum at +175 fm2. For the
MAP calculation, we next estimate the quadrupole mo-
ment at the minimum by interpolating between the cal-
culated points. We then redo the calculations at the esti-
mated minimum to find the MAP energy and quadrupole
properties.For
q0= −96 fm2with and energy E0= −332.32 MeV. The
corresponding quantities for J = 2 are q2 = −120 fm2
and E2 = −328.33 MeV. The MAP excitation energy
is the difference, E20 ≡ E2− E0 = 3.9 MeV. This is
80 % higher than the experimental excitation energy of
2.17 MeV. This is a rather extreme case, in that the 2+
The J = 2
38Ar, the minimum for J = 0 is at

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FIG. 1: Energy landscapes for J = 0 and J = 2 angular-
momentum projected states in38Ar. The open circles show
the q values of the calculated configurations, with lines drawn
to guide the eye. Solid circles are ones included in the mixed
configuration calculations for the global survey.
of38Ar is very likely better described as a broken-pair
two-quasiparticle state than as a field-induced deformed
state. We will return to this point later.
C.Matrix elements of the quadrupole operator
The calculation of the quadrupole moments of pro-
jected states requires the calculation of all components
of the quadrupole tensor. Q2±1and Q2±2are of course
exactly zero for axial mean-field states with the z axis
as symmetry axis as chosen here, but they have non-
vanishing transition matrix elements between a rotated
and an unrotated state.
The detailed expressions for the quadrupole operator
and its projected matrix elements can be found in ap-
pendix A. For axial states, as used here, only the matrix
elements ofˆQ20and the real part of the matrix elements
ofˆQ21 andˆQ22 need to be calculated, which simplifies
the computational task.
To calculate matrix elements of the quadrupole oper-
atorˆQ2µ, some modifications of the GOA parameteriza-
tion are necessary since the functional behavior of the
operator depends on its azimuthal angular momentum
µ. In particular, for the matrix element of Q2±1, the
form F(β) = sin2(β) used in the polynomial expansion
of Eq. (8) is not topologically correct. We therefore de-
fine a topGOA by taking d2
µ0(β) for the argument of the
polynomial expansion
ℜ??q|ˆRβˆQ2µ|q′??
= ?q|ˆRβ|q′?t
?
a0+ a2d2
µ0(β) + a4
?d2
µ0(β)?2?
, (9)
where the coefficients of the polynomial ai depend on
q,q′. As with the other matrix elements, it is important
to include the point at β = π/2 when q and q′have
opposite signs.
There is an additional complication compared to the
norm and Hamiltonian kernels. While for these scalar
operators the kernels (7) and (8) are invariant under
exchange of |q? and |q′?, this is not the case for the
quadrupole operator, where ?q|ˆRβˆQ2µ|q′? is not equal to
?q′|ˆRβˆQ2µ|q?. To avoid the explicit calculation of both,
we express the latter matrix elements as a weighted sum
of the former using angular-momentum algebra and the
symmetry properties of the SCMF states [22]. A separate
topGOA is then set up to calculate the projected ma-
trix element from the ?q′|ˆRβˆQ2µ|q?. It has to be noted,
however, that the difference plays a role only for transi-
tion matrix elements between states with different angu-
lar momentum. As we are interested here in 2+→ 0+
transitions only, the topGOA for matrix elements with
exchanged arguments is needed for µ = 0 only, Eq. (A5).
An example for the fits of the integrand is shown in
Fig. 2 for52Cr at a deformation of q = q′= 150 fm2.
Starting from the bottom panel, the three panels show
the rotated overlap matrix element ?q|ˆRβˆQ2µ|q? for µ =
0, 1 and 2 respectively. The open circles are the points
used to evaluate the integral by a 12-point Gaussian
quadrature, as it would be used in a calculation for the
complete low-energy spectroscopy of this nucleus. The
solid circles are the points used for the topGOA fit, the
resulting curves being indicated by lines. The agreement
is excellent; the error associated with the topGOA is typ-
ically less than 1 % for the µ = 0 matrix element. This
is the only one needed to calculate the B(E2) value of
the 2+→ 0+transition (see appendix A). The middle
panel shows the integrand for the Q21operator. In effect,
only the middle point can be used for the fit because the
integrand vanishes at β = 0 and π. Nevertheless, this
approximation works rather well. It is less accurate for
some non-diagonal matrix elements, particularly for ma-
trix elements connecting configurations with very differ-
ent deformations which are needed to describe soft nuclei.
The top panel shows the matrix element for the operator
Q22. Here there are effectively two points to determine
the topGOA fit.
To test this approximation further, we have compared
the topGOA quadrupole matrix elements with the matrix
elements obtained by a full integration for a variety of
nuclei and deformations. The result is shown in Fig. 3.
One can conclude that Eq. (9) is of sufficient accuracy
for our purpose.
D.Configuration mixing
As mentioned above, we typically compute about Nq=
20 SCMF configurations to construct the energy land-
scape. For many nuclei, however, only about half that
number can be kept in the configuration mixing calcula-
tion due to ill-conditioned norm matrices when the space
is overcomplete. Nevertheless, the full configuration mix-
ing calculation requires to compute about 50 projected
matrix elements, which is beyond our computational re-

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FIG. 2: The matrix element ?q|ˆRβˆQ2µ|q? for the nucleus52Cr
at an intrinsic deformation of q = 150 fm2. The open circles
show the points used for evaluating the integral by a 12-point
Gaussian quadrature. The solid circles show the points used
for the topGOA, and the resulting fit. The three panels show
the results for µ = 0,1 and 2 going up from the bottom panel.
FIG. 3: Comparison of the topGOA and full calculation for
Q2µ for various nuclei and deformations.
sources for a study of several hundred nuclei. In Ref. [23],
a GOA was developed for a coordinate corresponding to
the deformation q, permitting calculations to be made to
the needed accuracy only using the diagonal and subdi-
agonal elements of the configuration mixing matrix, i.e.
about 2Nqprojected matrix elements. Unfortunately, the
demands on the approximation are more severe when cal-
culating quadrupole matrix elements between states of
different angular momentum. The matrix element can
change sign, depending on the deformations. For matrix
elements connecting different manifolds of states (0+and
2+), there is no diagonal element to anchor the GOA.
Lacking a reliable GOA to determine the off-diagonal
quadrupole matrix elements, we took another approach
to simplify the configuration mixing calculation.
number of configurations has been reduced for each nu-
cleus to a number small enough to make a global cal-
culation feasible but large enough to have a sufficient
accuracy on the energy of the lowest 0+and 2+states.
Since we have to deal with energy curves of very different
topologies, some care must be taken into the selection of
points. The procedure that we have followed is explained
in appendix B. The number of selected configurations
varies from 3 to 10, but is most often equal to 6. We
have therefore labeled this approximation HW-6.
The points selected for38Ar are shown in Fig. 1 as
black circles. In this case, the single-configuration MAP
energy of the ground state is E0(MAP) = −332.25 MeV,
as quoted in the last section. The gain in energy from
configuration mixing with the large set of configurations
(11 in this case) is E0(HW) − E0(MAP) = −332.69 +
332.25 = −0.44 MeV. This is to be compared with
−332.61+332.25 = −0.36 MeV for the HW6 space. The
error, 0.08 MeV, is within our targeted limit of accuracy.
For the J = 2 projected wave functions, the energy gain
by the HW treatment is −0.68 MeV and the error of the
6-configuration truncation is 0.07 MeV with the same
sign as in the ground state energy. With our present
computer resources, we were able to test the HW6 trun-
cation for about 100 nuclei. The 2+excitation energies
computed both ways are compared in Fig. 4. The approx-
imation reproduces the energies to an r.m.s. accuracy of
better than 0.1 MeV. The worst cases are two nuclei with
coexisting minima at low excitation energy that are sep-
arated by very low barriers,188Pb and190Pb, visible as
points off the line at the bottom left-hand corner of the
graph.
The same comparison for the reduced matrix element
of the 2+→ 0+transition is shown in Fig. 5. The agree-
ment is very good except for the light Pb isotopes,182Pb,
188Pb, and190Pb. Among the nuclides, the light Pb iso-
topes are rather singular and we shall examine188Pb in
more detail in the next section. Overall, the accuracy of
the HW-6 approximation is more than adequate for the
present global survey.
The
III.SOME EXAMPLES
In this section we shall examine the results for a sam-
ple of nuclei with energy maps of different topologies: a
light doubly-magic system,40Ca; a heavy doubly-magic
system,208Pb; a transitional nucleus near magicity,38Ar;
a soft nucleus exhibiting triple shape coexistence,188Pb;
and a well-deformed heavy nucleus,240Pu. We first ex-