Potential Energy Surfaces of Actinide Nuclei from a Multi-dimensional Constraint Covariant Density Functional Theory: Barrier Heights and Saddle Point Shapes

1.5 MB·English

by  Bing-Nan Lu

Potential Energy Surfaces of Actinide Nuclei from a Multi-dimensional Constraint
Covariant Density Functional Theory: Barrier Heights and Saddle Point Shapes
Bing-Nan Lu,1 En-Guang Zhao,1,2,3 and Shan-Gui Zhou1,2,∗
1Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China
2Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China
3School of Physics, Peking University, Beijing 100871, China
(Dated: January 5, 2012)
Forthefirsttimethepotentialenergysurfacesofactinidenucleiinthe(β20,β22,β30)deformation
space areobtained from a multi-dimensional constrained covariant density functional theory. With
thisnewlydevelopedtheoryweareabletoexploretheimportanceofthetriaxialandoctupoleshapes
simultaneously along thewhole fission path. It is found that besides theoctupole deformation, the
2 triaxialityalsoplaysanimportantroleuponthesecondfissionbarriers. Theouterbarrieraswellas
1 the inner barrier are lowered by the triaxial deformation compared with axially symmetric results.
0 This lowering effect for the reflection asymmetric outer barrier is 0.5 ∼ 1 MeV, accounting for
2 10 ∼ 20% of the barrier height. With the inclusion of the triaxial deformation, a good agreement
n with thedata for the outerbarriers of actinide nuclei is achieved.
a
PACSnumbers: 21.60.Jz, 24.75.+i,25.85.-w,27.90.+b
J

4

Since the first interpretation of the nuclear fission by tocombinetheadvantagesoftheMMandself-consistent
]
h the barrier penetration [1], it has been a difficult task models,suchastheextendedThomas-Fermimethod[22].
-t to describe this phenomenon theoretically. For example, The double-humped fission barriers of actinide nuclei
l in order to study the fission problem, one should first can be used to benchmark the predictive power of the-
c
u have veryaccurate informationabout the fissionbarrier; oretical models [13, 23–25]. Various shape degrees of
n a 1 MeV difference in the fission barrier could result in freedom play important and different roles in the occur-
[ severalorders of magnitude difference in the fission half- rence and in determining the heights of the inner and

life. Particularly, to explore the island of stability of outer barriers. For examples, it has long been known
2
v superheavy nuclei (SHN) [2–4], it is more and more de- from MM model calculations that the inner fission bar-
9 sirable to have accurate predictions of fission barriers of rier is usually lowered when the triaxial deformation is
6 SHN [5–8]. allowed, while for the outer barrier the reflection asym-
7 Nowadays three types of models are used for calculat- metric (RA) shape is favored [26, 27]. Later on these
6 ing fission barriers. During a long period the majority points were also revealed in the non-relativistic [24] and
0. of these works is based on the macroscopic-microscopic relativistic [15, 28] density functional calculations, re-
1 (MM)models[5,9–11]. TheMMmodelsmakeuseofthe spectively. It is thus customary to consider only the tri-
1 Strutinskyshellcorrectionmethod,allowingfastcalcula- axial and reflection symmetric (RS) shapes for the in-
1 tions of multi-dimensional PES’s containing most of the ner barrier and axially symmetric and RA shapes for
:
v important shape degrees of freedom. Until now it is still the outer one [5, 29, 30]. It has been pointed out that
i animportantcandidate forlargescalefissionbarriercal- “there is no reason for a fissioning actinide nucleus not
X
culations basedonthe examinationofmulti-dimensional to penetrate all symmetry-breaking shapes on its way
ar PES’s[10]. Inrecentyears,the rapiddevelopmentofthe fromthe first(triaxial)tothe second(mass-asymmetric)
density functional theories (DFT) also makes it possible saddle” [31]. The non-axial octupole deformations are
tocalculatethefissionbarriersfullyself-consistently[12– considered in both the MM models [32] and the non-
15]. Therearemainlytworeasonsto applyDFT’s inthe relativistic Hartree-Focktheories [33]. However,a multi-
study of fission properties. First, many new functional dimensional structure of PES’s including both the triax-
forms and effective interactions are proposed with much ial and RA shape degrees of freedom has not been ex-
better performances for the excited state as well as the plored yet in the framework of covariant DFT. In this
groundstatecalculations[16–20]. Fissionbarriercalcula- paper, we will investigate the influence of the triaxiality
tionsarealsohelpfulfordevelopingtheseDFT’s. Second, and the octupole shape on the PES’s all the way from
in DFT’s much more shape degrees of freedom can be the ground state to the fission configuration when both
included self-consistently. For example, the symmetry- shapedegreesoffreedomareincludedsimultaneously. To
unrestrictedSkyrme-Hartree-Fock-Bogoliubovmodelhas this end, not only as many as self-consistent symmetries
been applied for the fission studies [21]. Besides these shouldbebroken,butalsomulti-dimensionalconstraints
two types of models, there exist also methods intending are needed [34].
To calculate the potential energy surfaces and fission
barriers, in this work we use the covariant density func-
∗ [email protected] tional theory (CDFT) [17, 19, 20, 35, 36]. By breaking
2

not only the axial [37, 38] but also the reflection sym-
CimnDgetFrfioTeus’sr[i3fno9r]w,mhwsi:cehdtthehveeelmfoupensecodtniomneuxallcthic-aadnnimgbeeenoosrnioepnooaiflnttch-ocenofsuotplrlaloiiwnngt- 108 240Pu PC-SPK1 AS & RS
) A
nucleon interactions combined with the non-linear or V 6 RS
e
density-dependent couplings [40]. If not specified, the M
functionalformofthe point-couplingnucleoninteraction E ( 4
with non-linear self energy terms and the parameter set 2 Bif
PC-PK1 [41] are used in this work. o
Fortheparametrizationofthenuclearshape,weadopt 0 Bf
the conventional ansatz in mean field calculations,
0.0 0.5 1.0 1.5
20
4π
βλµ = 3ARλhQλµi (1) FIG. 1. (Color online) Potential energy curvesof 240Pu with
various self-consistent symmetries imposed. The solid black
where Qλµ is the mass multipole operators. When the curverepresentsthecalculatedfissionpathwithV4symmetry
axial and reflection symmetries are broken, the nuclear imposed,thereddashedcurvethatwithaxialsymmetry(AS)
imposed, thegreen dotted curvethat with reflection symme-
shape is invariant under the reversion of x and y axes.
try (RS) imposed, the violet dot-dashed line that with both
In other words, the intrinsic symmetry group is V and
4 symmetries(AS&RS)imposed. Theempiricalinner(outer)
all shape degrees of freedom β with µ even, includ-
λµ barrier height Bemp is denoted by the grey square (circle).
ing the triaxial (µ 6= 0) and octupole (λ = 3) deforma-
The energy is normalized with respect to the binding energy
tions, are possible. Irrespective with the self-consistent
of the ground state. The parameter set used is PC-PK1.
symmetries, the single particle wave functions and vari-
ous densities are expanded on an axially deformed har-
monic oscillator basis [38, 42]. In order to get a fast
morethan2MeVandresultsina better agreementwith
convergence of the results against the basis size, in the
the empirical datum; the RA shape is favored beyond
elongated direction more states are included in the ba-
the fission isomer and lowers very much the outer fis-
sis. FollowingWardaetal.[43],thebasisistruncatedas:
sion barrier. Besides these features, we observe for the
nz/Qz+(2n⊥+|m|)/Q⊥ ≤Ncut. Herenz,n⊥,andmare
first time that the outer barrier is also considerably low-
quantum numbers characterizingeach state in the basis,
ered by about 1 MeV when the triaxial deformation is
Qz = MAX(1,bz/b0), Q⊥ = MAX(1,b⊥/b0), and b0, bz,
allowed. Again, a better reproduction of the empirical
and b⊥ are the oscillator lengthes. The calculated bind-
barrierheightcanbe seenfor the outer barrier. We note
ing energy of 240Pu at β = 1.3 varies only about 130
20
that this feature can only be found when the axial and
keV and 20 keV when N increases from 16 to 18 and
cut
reflection symmetries are simultaneously broken.
from 18 to 20. This means a good convergence and such
a truncation scheme with N = 16 ensures a 0.2 MeV How the PES of 240Pu becomes unstable against the
cut
accuracy for the deformation range we are interested in. triaxial distortion can be seen much more clear in Fig. 2
In the present work, N =16 (20) is used in the triax- in which we show 2-d PES’s from calculations without
cut
ial (axial) calculations. More details of the convergence and with the triaxial deformation. When the triaxial
studywillbegiveninRef.[40]. TheBCSapproachisim- deformation is allowed, the binding energy of 240Pu as-
plemented in our model to take into account the pairing sumes its lowest possible value at each (β20,β30) point.
effect. Since it has been found that the BCS calculation Atsomepointswegetnon-zeroβ22 values. Thatis,non-
witha constantpairinggapcannotprovideanadequate axial solutions are favoredat these points than the axial
descriptionofthefissionbarriers[14],weuseadeltaforce ones. The triaxial deformation appears mainly in two
for the pairing interaction with a smooth cutoff [41, 44]. regions in Fig. 2. One region starts from the first saddle
point and extends roughly along the direction of the β
30
We performed one- (1-d), two- (2-d), and three- axis up to a very asymmetric shape with β30 ∼ 1.0. In
dimensional (3-d) constrained calculations for the ac- this region the values of β22 are about 0.06 ∼ 0.12 cor-
◦
tinide nucleus 240Pu. In Fig. 1 we show the 1-d poten- responding to γ ∼ 10 . The energy, especially the inner
tial energy curves (PEC) from an oblate shape with β barrierheight, is loweredby about 2 MeV. The other re-
20
about −0.2 to the fission configuration with β20 beyond gion is around the outer barrier and the β2◦2 values are
2.0 which are obtained from calculations with different about 0.02 ∼ 0.03 corresponding to γ ∼2 . About 1
self-consistent symmetries imposed: the axial (AS) or MeV is gained for the binding energy at the second sad-
triaxial (TS) symmetries combined with reflection sym- dle point due to the triaxiality. In other regions, e.g., in
metric (RS) or asymmetric cases. The importance of the ground state and fission isomer valleys, only axially
the triaxial deformation on the inner barrier and that of symmetric solutions are obtained.
the octupole deformation on the outer barrier stressed Next we examine the full 3-d PES of 240Pu obtained
by earlier studies [15, 24, 28] are clearly seen here: The from the newly developed multi-dimensional constraint
triaxial deformation reduces the inner barrier height by CDFT. For simplicity, in Fig. 3 are shown only five typ-
3
ical sections of the 3-d PES of 240Pu in the (β ,β )
22 30
1.0 (a) w/o plane calculated at β20= 0.3 (around the ground state),
240 0.6(aroundthefirstsaddlepoint),0.9(aroundthefission
Pu
isomer),1.3(aroundthesecondsaddlepoint)and1.6(be-
0.5 P C-PK1 yond the outer barrier), respectively. Many conclusions
30 can be drawn by examining these 3-d PES’s. First, the
ground state and the fission isomer are both axially and
0.0
30 V) reflectionsymmetricaswhatisshowninthe1-dPECand
20 e the 2-d PES. But with the 3-d PES one can investigate
1.0 (b) E (M the stability of 240Pu against the β22 and β30 deforma-
tions. One finds that the stiffness of the fission isomer is
10
0.5 much larger than that of the ground state against both
the β and β deformations. Second, while around the
22 30
0 inner barrier the shape of 240Pu is triaxial and reflec-
0.0 tionsymmetric,the secondsaddlepointwhichiscloseto
w/
β = 1.3 appears as both triaxial and reflection asym-
20
0.0 0.5 1.0 1.5 2.0
metric shape. Third, the triaxialdistortionappears only
20 on the top of the fission barriers.
FIG. 2. (Color online) Potential energy surfaces of 240Pu in It has been pointed out that one may obtain spurious
the(β20,β30)planefromcalculations(a)withoutand(b)with saddle points if only a small number of shape degrees
the triaxial deformation included. The energy is normalized of freedom are constrained, see, e.g., Ref. [5]. That is,
with respect to the binding energy of the ground state. The the calculated fission path may jump from one valley
numbers in (b) show the values of β22 at these points. The to another and results in discontinuities in the lower-
fissionpathisrepresentedbyadash-dottedline. Theground
dimensional PES’s; in some cases, a continuous path
state and fission isomer are denoted by full and open circles.
may even cross a higher saddle point. Although the
The first and second saddle points are denoted by full and
spurious saddle points may not be excluded completely,
open triangles. Thecontour interval is 1 MeV.
most of them can be avoided if (1) the obtained fission
path keeps to be continuous in the energy as well as the
most important shape degrees of freedom and (2) the
results are examined by higher-dimensionalcalculations.
0.8 We have carefully checked the full 3-d PES and found
20 = 0.3 20 = 0.6
0.4 that the fission path enters and exits the triaxial con-
figuration rather smoothly, which tells that no sudden
0.0
jump is found and the 1-d (with the β deformation
20
-0.4 constrained and β ,β deformations imposed) and 2-
22 30
d (with β ,β deformations constrained and the β
0.8 20 30 22
20 = 0.9 20 = 1.3 deformation imposed) calculations of the fission barriers
0.4 may be well justified for 240Pu. It is clear that the con-
30 0.0 tinuity of the fission path found in a lower-dimensional
constraint calculation is a necessary but not sufficient
-0.4
condition for locating the correct saddle point. In order

0.8 tohaveastrictlydefiniteconclusion,onecertainlyshould
20 = 1.6 -0.1 0. 0 0.1 0.2 carry out multi-dimensional constraint calculations with
0.4
12 V) even higher-multipolarity deformations included.
0.0 8 Me
4 E ( For the RS calculations, the triaxiality also lowers the
-0.4 fission path by a few MeV beyond the second saddle
0
point. ThispointisillustratedbythedottedlineinFig.1
-0.2 -0.1 0.0 0.1 0.2 andthelocalminimawithβ =0.0intheβ =1.6sub-
30 20
22 figures of Fig. 3. However, it is relatively unimportant,
because the RA fission is still the most favoredone even
FIG.3. (Coloronline)Sectionsofthethree-dimensionalPES
when triaxiality is included.
of240Puinthe(β22,β30)planecalculatedatβ20=0.3(around
Guided by the features found in the 1-d, 2-d, and 3-d
the ground state), 0.6 (around the first saddle point), 0.9
(around the fission isomer), 1.3 (around the second saddle PES’s of 240Pu, the fission barrier heights are extracted
point) and 1.6 (beyond the outer barrier), respectively. The for even-even actinide nuclei whose empirical values are
energyisnormalizedwithrespecttothebindingenergyofthe recommendedin RIPL-3 (see Table XI in [45]). The em-
groundstate. Thecontourintervalis0.5MeV.Localminima phasis is put on the influence of the triaxialdeformation
are denoted by crosses. on the two fission barriers.
Asithasbeenshownpreviously,aroundtheinnerbar-
4

248
10 230-232Th 232-238U 238-244Pu 242-248Cm 6 (a) Cm
empirical barrier
PC-PK1
eV) 8 V) 4
M e
i B (f6 E (M 2
0
path I
4 -2 path II

1.0
8 230-232Th 232-238U 238-244Pu 242-248Cm 12.0
(b)
9.0
V) 6 V)
o B (Mef4 w/o 300.5 5.0 E (Me
1.0
w/
2 exp PC-PK1 0.0
-3.0

1.0 1.5 2.0
Z, A
20
FIG.4. (Coloronline) Theinner(Bi)andouter(Bo)barrier
f f FIG. 5. (Color online) (a) One-dimensional potential energy
heights of even-even actinide nuclei. The axial (triaxial) re-
sultsaredenotedbyopen(full)symbols. Theempiricalvalues curveE ∼β20 and (b)two-dimensional potentialenergy sur-
are taken from Ref. [45] and represented bygrey squares. faceE ∼(β20,β30)of248Cmtheouterbarrierregionwiththe
axial symmetry imposed in the calculation. In both figures,
the energy is normalized with respect to the binding energy
of the ground state. The fission path I/II is represented by
full/dash dot lines and the corresponding saddle point is de-
rier an actinide nucleus assumes triaxial and reflection
noted by up/down triangles. The fission isomer is denoted
symmetric shapes. Thus in order to obtain the inner fis-
by open circle. In (a), the empirical outer barrier height is
sionbarrierheightwecansafelymakeaone-dimensional
depicted by dash-dotted line. In (b), the contour interval is
constraint calculation with the triaxial deformation al-
0.5 MeV.
lowedandthe reflectionsymmetry imposed. InFig. 4(a)
we present the calculated inner barrier heights Bi and
f
compare them with the empirical values. It is seen that
with different octupole deformations. In consequence,
the triaxiality lowers the inner barrier heights of these one often observes in the 1-d PEC two or more fission
actinide nuclei by 1 ∼ 4 MeV as what has been shown paths. This happens in 244,246Pu and 244,246,248Cm in
in Ref. [15]. In general the agreement of our calculation the present study and we present a typical example in
results with the empirical ones is very good with excep- Fig. 5 for 248Cm. In this figure, one finds that there are
tionsinthetwothoriumisotopesand238U.For230Thand two fission paths both in the 1-d E ∼ β curve and in
20
232Th,thecalculatedinnerbarrierheightsaresmallerby the 2-d E ∼ (β ,β ) PES. One path denoted by “I”
20 30
about 2 or 1 MeV than the empirical values depending favors shapes with larger octupole deformations and the
onwhetherthe triaxialdeformationis allowedornot. In other denoted by “II” favors less RA shapes. In such
thesetwonuclei,theouterbarrierishigherthantheinner nuclei, it is not safe to perform a 1-d constraint calcula-
one. This may result in some uncertainties when deter- tion in order to get Bo. Thus we first assume the axial
f
mining empirically the height of the inner barrier which symmetry and make a 2-d calculation in the (β ,β )
20 30
is not the primary one [12]. Similar results for 230Th planefromwhichwecanapproximatelyidentify thelow-
and232ThwereobtainedfromtheSkyrme-Hartree-Fock- est fission path βlowest(β ) and the location of the sec-
30 20
Bogoliubovmodel[12]andverysmallinnerbarrierheight ond saddle point. Then along this fission path, we per-
was gotfor 232Th in Ref. [15]. For 238U, Bfi fromthe ax- form a 1-d β20-constraint calculation with the triaxial
ial calculation agrees with the empirical value very well. and octupole deformations allowed. At each point with
The triaxiality reduces the barrier height by about 1.5 β , the initial deformations are taken as βini. = 0 and
20 22
MeV, thus bringing a discrepancy which was similar to βini. = βlowest(β ). In this 1-d PEC, we can locate the
30 30 20
the result in Ref. [15]. second saddle point and extract the outer barrier height
To obtain the outer fission barrier height Bo, the sit- for each nucleus.
f
uation becomes more complicated because more shape InthelowerpanelofFig.4weshowtheresultsofouter
degreesoffreedomhaveimportantinfluences aroundthe barrierheightsBo andcomparethemwithempiricalval-
f
outer fission barrier. For example, the inclusion of the ues. For most of the nuclei investigated here, the triaxi-
reflection asymmetric shape makes it possible to have in alitylowerstheouterbarrierby0.5∼1MeV,accounting
the (β ,β ) plane two or more competing fission paths forabout10 ∼ 20%ofthe barrierheight. One finds that
20 30
5
our calculation with the triaxiality agrees well with the densityfunctionaltheoryisdevelopedwhichallowsusto
empirical values and the only exception is 248Cm. From studythe importanceofthe triaxialandoctupole shapes
the calculation with the axial symmetry imposed, the simultaneously along the whole fission path. The one-
outerbarrierheightof248Cmisalreadysmallerthanthe dimensional PEC E ∼ β , two-dimensional PES E ∼
20
empirical value. The reason for this discrepancy may be (β ,β ),andthree-dimensionalPESE ∼(β ,β ,β )
20 30 20 22 30
relatedtothattherearetwopossiblefissionpathsbeyond ofactinide nucleiare shownand studied indetails. Both
the first barrier, as seen in Fig. 5. For the path I with the triaxiality and the reflection asymmetry plays cru-
a lower saddle point from which we get the outer fission cial roles at and around the second saddle point. The
barrierheight,thebarrierisverywideandforthepathII outer barrier as well as the inner barrier are lowered by
with a higher saddle point, the barrier is relatively nar- the triaxial deformation compared with axially symmet-
row. Therefore the empirical value of the outer fission ric results. For most of the nuclei investigated here, the
barrierheightmaynotbe easilyextractedforthefollow- triaxiality lowers the outer barrier by 0.5 ∼ 1 MeV, ac-
ing two reasons: (i) There must be a strong competition counting for about 10 ∼ 20% of the barrier height. The
between the two fission paths; (ii) When the empirical calculated results of the outer barrier heights agree well
valueoftheouterbarrierheightisevaluated,itisusually with the empirical values.
assumed that the second barrier is in an anti-parabolic Helpful discussions with Jie Meng, P. Ring, D. Vrete-
shape with a fixed and smaller width [45]. nar, Xi-Zhen Wu, and Zhen-Hua Zhang are acknowl-
edged. This work has been supported by NSFC (Grant
We also examined the parameter dependency of our
Nos. 10875157, 10975100, 10979066, 11175252, and
results. Theloweringeffectofthetriaxialityontheouter
11120101005),MOST (973 Project 2007CB815000),and
fissionbarrierisalsoobservedwhenparametersetsother
CAS (Grant Nos. KJCX2-EW-N01 and KJCX2-YW-
than PC-PK1 are used.
N32). The computation of this work was supported by
Insummary,amulti-dimensionalconstrainedcovariant Supercomputing Center, CNIC of CAS.
[1] N.Bohr and J. A.Wheeler, Phys.Rev. 56, 426 (1939). [11] A.SobiczewskiandK.Pomorski,Prog.Part.Nucl.Phys.
[2] S. Hofmann and G. Mu¨nzenberg, Rev. Mod. Phys. 72, 58, 292 (2007).
733 (2000). [12] M. Samyn,S.Goriely, andJ. M.Pearson, Phys.Rev.C
[3] K. Morita, K. Morimoto, D. Kaji, T. Akiyama, S.- 72, 044316 (2005).
i. Goto, H. Haba, E. Ideguchi, R. Kanungo, K. Ka- [13] T.Burvenich,M.Bender,J.A.Maruhn, andP.-G.Rein-
tori, H. Koura, H. Kudo, T. Ohnishi, A. Ozawa, hard, Phys. Rev.C 69, 014307 (2004).
T. Suda, K. Sueki, H.-S. Xu, T. Yamaguchi, A. Yoneda, [14] S. Karatzikos, A. Afanasjev, G. Lalazissis, and P. Ring,
A.Yoshida, andY.-L.Zhao,J.Phys.Soc.Jpn.73,2593 Phys. Lett. B 689, 72 (2010).
(2004). [15] H. Abusara, A. V. Afanasjev, and P. Ring, Phys. Rev.
[4] Y.Oganessian,J.Phys.G:Nucl.Phys.34,R165(2007); C 82, 044303 (2010).
Y. T. Oganessian, F. S. Abdullin, P. D. Bailey, D. E. [16] M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev.
Benker, M. E. Bennett, S. N. Dmitriev, J. G. Ezold, Mod. Phys.75, 121 (2003).
J. H. Hamilton, R. A. Henderson, M. G. Itkis, Y. V. [17] D. Vretenar, A. Afanasjev, G. Lalazissis, and P. Ring,
Lobanov, A. N. Mezentsev, K. J. Moody, S. L. Nelson, Phys. Rep.409, 101 (2005).
A.N.Polyakov,C. E. Porter, A.V.Ramayya,F. D.Ri- [18] L. Guo, J. A. Maruhn, and P.-G. Reinhard, Phys. Rev.
ley, J. B. Roberto, M. A. Ryabinin, K. P. Rykaczewski, C 76, 034317 (2007).
R. N. Sagaidak, D. A. Shaughnessy, I. V. Shirokovsky, [19] J.Meng, H.Toki,S.G.Zhou,S.Q.Zhang,W.H.Long,
M.A.Stoyer,V.G.Subbotin,R.Sudowe,A.M.Sukhov, and L. S.Geng, Prog. Part. Nucl. Phys.57, 470 (2006).
Y. S. Tsyganov, V. K. Utyonkov, A. A. Voinov, G. K. [20] T. Niksic, D. Vretenar, and P. Ring, Prog. Part. Nucl.
Vostokin, andP.A.Wilk,Phys.Rev.Lett.104,142502 Phys. 66, 519 (2011).
(2010). [21] A. Staszczak, A. Baran, J. Dobaczewski, and
[5] P. M¨oller, A. J. Sierk, T. Ichikawa, A. Iwamoto, W. Nazarewicz, Phys. Rev.C 80, 014309 (2009).
R. Bengtsson, H. Uhrenholt, and S. Aberg, Phys. Rev. [22] A.Mamdouh,J.M.Pearson,M.Rayet, andF.Tondeur,
C 79, 064304 (2009). Nucl. Phys. A 644, 389 (1998).
[6] J. C. Pei, W. Nazarewicz, J. A. Sheikh, and A.K. Ker- [23] M. Mirea and L.Tassan-Got, Cent. Eur.J. Phys.9, 116
man, Phys.Rev.Lett. 102, 192501 (2009). (2011).
[7] C.-J. Xia, B.-X. Sun, E.-G. Zhao, and S.-G. Zhou, Sci. [24] M. Girod and B. Grammaticos, Phys. Rev. C 27, 2317
China-Phys. Mech. Astron. 54 (Suppl. 1), 109 (2011), (1983).
arXiv: 1101.2725 [nucl-th]. [25] V. Blum, J. Maruhn, P.-G. Reinhard, and W. Greiner,
[8] Z.-H.LiuandJ.-D.Bao,Phys.Rev.C84,031602(2011). Phys. Lett. B 323, 262 (1994).
[9] P.M¨oller, D.G.Madland,A.J.Sierk, andA.Iwamoto, [26] V. V.Pashkevich, Nucl. Phys.A 133, 400 (1969).
Nature409, 785 (2001). [27] P.M¨ollerandS.G.Nilsson,Phys.Lett.B31,283(1970).
[10] P.M¨oller,A.J.Sierk, andA.Iwamoto,Phys.Rev.Lett. [28] K.Rutz,J.A.Maruhn,P.G.Reinhard, andW.Greiner,
92, 072501 (2004). Nucl. Phys. A 590, 680 (1995).
6
[29] J.L.EgidoandL.M.Robledo,Phys.Rev.Lett.85,1198 [39] L.-S.Geng,J.Meng, andH.Toki,Chin.Phys.Lett.24,
(2000). 1865 (2007).
[30] L.Bonneau,P.Quentin, andD.Samsoen,Eur.Phys.J. [40] B.-N. Lu et al.,in preparation.
A 21, 391 (2004). [41] P. W. Zhao, Z. P. Li, J. M. Yao, and J. Meng, Phys.
[31] J. Skalski, Phys.Rev.C 43, 140 (1991). Rev. C 82, 054319 (2010).
[32] P.Jachimowicz, M.Kowal, andJ.Skalski,Phys.Rev.C [42] P. Ring,Y.K.Gambhir, and G. A.Lalazissis, Comput.
83, 054302 (2011). Phys. Commun. 105, 77 (1997).
[33] J. Skalski, Phys.Rev.C 76, 044603 (2007). [43] M.Warda,J.L.Egido,L.M.Robledo, andK.Pomorski,
[34] P.RingandP.Schuck,TheNuclearMany-Body Problem Phys. Rev.C 66, 014310 (2002).
(Springer,1980). [44] M.Bender,K.Rutz,P.-G.Reinhard, andJ.A.Maruhn,
[35] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 Eur. Phys. J. A 8, 59 (2000).
(1986). [45] R. Capote, M. Herman, P. Oblozinsky, P. Young,
[36] P.Ring, Prog. Part. Nucl.Phys. 37, 193 (1996). S. Goriely, T. Belgya, A. Ignatyuk, A. Koning, S. Hi-
[37] J. Meng, J. Peng, S. Q. Zhang, and S.-G. Zhou, Phys. laire, V. Plujko, M. Avrigeanu, O. Bersillon, M. Chad-
Rev.C 73, 037303 (2006). wick, T.Fukahori,Z.Ge,Y.Han,S.Kailas, J.Kopecky,
[38] B.-N.Lu,E.-G.Zhao, andS.-G.Zhou,Phys.Rev.C84, V. Maslov, G. Reffo, M. Sin, E. Soukhovitskii, and
014328 (2011), arXiv:1104.4638v1 [nucl-th]. P. Talou, Special Issue on Nuclear Reaction Data, Nucl.
Data Sheets 110, 3107 (2009).