Terminating states as a unique laboratory for testing nuclear energy density functional

0.2 MB·English

by  M. Zalewski

February 6, 2008 7:41 WSPC/INSTRUCTION FILE K˙V1
International JournalofModernPhysicsE
(cid:13)c WorldScientificPublishingCompany
7
0
0 TERMINATING STATES AS A UNIQUE LABORATORY FOR
2 TESTING NUCLEAR ENERGY DENSITY FUNCTIONAL

n
a
J
M.ZALEWSKI∗
0
Institute of Theoretical Physics, Universityof Warsaw,
3
ul. Hoz˙a 69, 00-681 Warsaw, Poland

2
v W.SATUL A†
3 Institute of Theoretical Physics, Universityof Warsaw,
8 ul. Hoz˙a 69, 00-681 Warsaw, Poland,
0
1
0 Received(receiveddate)
7 Revised(reviseddate)
0
/
h
Systematic calculations of favoredsignaturemaximum-spinImax andunfavored signa-
-t ture Imax−1 terminating states for [f7n/2] and [d−3/12f7n/+21] configurations (n denotes
cl number of valence particles) in A∼44 mass region are presented. Following the result
u of Zdun´czuk et al., Phys. Rev. C71 (2005) 024305 the calculations are performed us-
n ing Skyrme energy density functional with empirical Landau parameters and slightly
: reducedspin-orbitstrength.Theaimistoidentifyandphenomenologicallyrestorerota-
v tionalsymmetrybrokenbytheSkyrme-Hartree-Focksolutions.Inparticular,itisshown
Xi thatcorrelationenergyduetosymmetryrestorationisabsolutelycrucialinordertore-
ar plersosdiumcpeoerntearngtyfosprl[idtt−3i/n12gf7En/+(2I1m]caoxn)fi−gEur(aItmioanxs−. 1)in[f7n/2]configurationsbutisrelatively
1. Introduction
Structureofheavynucleimaybedescribedusingtwomajortheoreticalapproaches:
1 2
the nuclearshell-model (SM) andthe nucleardensity functionaltheory (DFT).
The main advantage of SM is proper treatment of many-body correlations among
limited number of valence particles. The resulting wave functions are eigenstates
to symmetry invariants of the effective Hamiltonian. In the language of DFT, nu-
cleus is treated as A-body composite object described by one-body densities and
3
currents. According to the Hohenberg-Kohn-Sham (HKS) theorem , this method
is formally exact. However, since the HKS theorem does not provide any method
of constructing such an exact functional standard procedures employed in nuclear
physics attempt to construct the functional in a systematic way guided by basic
∗[email protected]
†[email protected]
1
February 6, 2008 7:41 WSPC/INSTRUCTION FILE K˙V1
2 M. Zalewski, W. Satul a
4
symmetryrequirements .AparticularcaseofthenuclearEDFisthefunctionalin-
spiredbywellstudiedSkyrmemodel2,5,6 (S-EDF)which,intheisoscalar-isovector
t=0,1 representation, takes the following form:
Skyrme = d3r (TE)(r)+ (TO)(r) , (1)
E tX=0,1Z hHt Ht i
where
(TE)(r)=Cρρ2+C∆ρρ ∆ρ +Cτρ τ +CJJ2+C∇Jρ ∇ J , (2)
Ht t t t t t t t t t t t t · t
(TO)(r)=Css2+C∆ss ∆s +CTs T +Cjj2+C∇js (∇ j ). (3)
Ht t t t t t t t· t t t t t· × t
The functional is uniquely expressed as a bilinear form of time-even (te) ρ,τ,J
H
and time-odd (to) s,T,j local densities, currents, and by their derivatives. Exact
expressions linking auxiliary Skyrme parameters x ,t , i = 0,1,2,3 and W,α with
i i
2
coupling constants C can be found, for example, in Ref. .
The coupling constants of the S-EDF (or Skyrme force (SF)) are adjusted to
globalpropertiesofnuclearmatterandtoselectedexperimentalobservablesinorder
to account for basic nuclear structure properties. Since there exist a multitude of
observables and no real consensus about which of these are to be selected in a
unique manner, there exist a multitude of different SF parameterizations. One of
the basic problems in adjusting the force parameters is related to the fact that the
single-particle (SP) states, that are so crucial for high accuracy calculations, are
in generalcoupled to collective motion and therefore difficult to determine. In this
context, any datasetrepresentingSP motionis aninvaluable source of information
that can be used for a rigorous test and subsequent fine-tuning of the parameters.
Our long standing experience tells us that terminating states which are
maximum-spin states within given SP configuration are one of the best examples
of unperturbed SP motion 7,8. This conclusion was recently confirmed by a com-
9
parativestudy betweenstateoftheartSMandSkyrme-Hartree-Fockcalculation
performed for maximum spin I states terminating within fn and d−1fn+1
max 7/2 3/2 7/2
configurations in N>Z, A 44 nuclei. Indeed, for these cases the energy difference
∼
∆E =E[d−1fn+1] E[fn ] , (4)
3/2 7/2 Imax − 7/2 Imax
calculated using Skyrme-Hartree-Fock (SHF) approach with modified spin-fields
10
and spin-orbit (SO) strength follows very closely the results of SM calculations.
This study revealed simultaneously that in N=Z nuclei the extreme SP picture
breaks down due to spontaneous breaking of isobaric symmetry.
The aim of this work is to explore the structural simplicity of maximum-spin
I (favored signature) as well as for I 1 (unfavored signature) terminating
max max
−
states in order to further constrain parameters of the S-LEDF as well as in order
to identify, quantitatively evaluate and subsequently restore in a phenomenologi-
cal way brokensymmetries inherently obscuring the SHF treatment. In particular,
we will show that the energy difference of Eq. (4) between I configurations
max
February 6, 2008 7:41 WSPC/INSTRUCTION FILE K˙V1
Terminating statesas a unique laboratory for testingnuclear energy density functional 3
p−1dn+1 and dn in A 16 mass regionand between I configurationsf−1gn+1
1/2 5/2 5/2 ∼ max 5/2 9/2
and gn in A 80 mass regionfollows closelythe trend found in A 44mass region
9/2 ∼ ∼
10
in Refs. with respect to the isoscalar effective mass scaled isoscalar strength of
SOtermandthatthisresultnicelycorrelateswithsimpleNilssonmodelpredictions
concerning N=Z=8 and N=Z=40 magic gaps. In the second part we will demon-
strate that unfavored signature I 1 terminating states in A 44 mass region
max
− ∼
manifestly violaterotationalinvariance.However,unlike in mostother caseswhere
the spontaneous breaking of rotationalsymmetry (SSB) leads to the occurrence of
deformation what allows to incorporate substantial fraction of many-body correla-
8
tionsintoasingledeformedSlaterdeterminant,seeRef. andrefs.quotedtherein,
the effect discussed here occurs at quasi-spherical shape. It appears that there
is a fundamental difference between these two situations. In deformed nuclei the
SSB mechanism works constructively. The violated symmetry can be subsequently
approximately restored within independent particle model based on cranking ap-
proximation what leads naturally to emergence of collective rotational states. In
thecaseofI 1terminatingstatestheSSBmechanismworksdestructively and
max
−
oneneedtogobeyondmean-fieldandperformconfigurationmixingcalculationsto
restore the symmetry. However,since the number of participating (dominant) con-
figurationsisverylimitedsuchconfigurationmixingcanbe performedanalytically,
as shown in detail in Sect. 3.
2. Isoscalar spin-orbit strength
10
In Refs. it was shown that the set of terminating states in N=Z, A 44 mass
6 ∼
regionprovides reliable constraints ontime-odd fields and the strengthof isoscalar
SO interaction of the S-EDF. In particular, it was demonstrated that constrain-
ing coupling constants of time odd spin-fields to the empirical spin-isospin Landau
parameters leads to unification of predictions for ∆E of Eq. (4) for such different
11 12 13 14
Skyrme parameterizations like SLy4 and SLy5 , SkO , SIII , and SkXc
with mean deviationbetween calculated and experimentalvalues ∆E 500keV i.e.
∼
at the level of 10%. Let us stress that our high-spin estimate of Landau pa-
∼
rameters is fairly consistent with the data extracted based on completely different
experimentalinput like giantresonances,beta decays,or moments ofinertia 15,16.
The remaining mean discrepancy ∆E ∆E ∆E between theory and
exp th
≡ −
experiment can be further reduced by slightly reducing SO strength. Indeed, the
quantity ∆E of Eq. (4) is governed by the size of N = Z = 20 magic gap ∆e
20
which,inturn,stronglydependsontheSOstrength.Itcanbenicelydemonstrated
17
within the spherical Nilsson Hamiltonian where:
∆e =~ω (1 6κ 2κµ). (5)
20 0
− −
Forlightnucleiµ 0i.e.flatbottomandsurfaceeffectdonotplayimportantrole.
∼
The width of the potential well ~ω determines global energy scale in low energy
0
nuclear physics. It is rather well constrained by data and even small variations
February 6, 2008 7:41 WSPC/INSTRUCTION FILE K˙V1
4 M. Zalewski, W. Satul a
can spoil in general good agreement between theory and experiment, especially in
heavy nuclei. Hence, the SO term κ plays indeed a dominant role in magnitude of
the magic gap ∆e .
20
2.0
SkM*
W*
0
W
0
1.5
MSk1
]
V
e SkP
M
1.0
[

E SkXc
Sly.. Sly..
DDDD 0.5
SkO
SIII
0
120 140 160 180
W ; W* [MeV fm5]
0 0
Fig. 1. Mean value ∆E as a function of the isoscalar strength W0∗ (filled diamonds) and W0
(open circles) for different parameterizations of the SF. See text for further discussion. Taken
fromRef.10.
10
In Refs. it was shown, that ∆E in A 44 nicely correlates with isoscalar SO
∼
strength.However,duetonon-localmomentumdependentterms,suchacorrelation
cannotbedoneatthelevelofbareSOstrengthW ,seeblackdotsinFig.2,butmust
0
be performedat the levelof the so called asymptoticallyequivalent representation:
m
φ˜(~r)= φ (~r). (6)
i rm∗(~r) i
In this representation free particles in the infinity r acquire not effective m
→∞ ∗
but bare mass m while the SO potential takes the form:
m∗(~r) 1 1
V (q,r) W ρ′(r) W ρ′(r) ~l~s. (7)
LS ≈ m (cid:26) 0r 0 ± 1r 1 (cid:27)
Note, that the true SO strengthfelt by a nucleonis W∗ m∗W and not W . The
correlation between the effective-mass-scaled isoscalar0s≡trenmgth0is W∗ and 0∆E is
0
now evident, see diamonds in Fig.2. All forces giving similar ∆E including SLy4,
Sly5,SIII, SkO andSkXc havealsosimilar W∗ ( 135 10MeVfm5). For SkP 18,
MSk1 19 or SkM* 20 which give unacceptabl0y l∼arge ∆±E, W∗ is also considerably
0
larger.
∗
Tofurtherinvestigatecorrelationbetween∆E andtheSOstrengthW wehave
0
performed similar calculations for terminating states ∆E = E([p−1dn+1] )
1/2 5/2 Imax −
February 6, 2008 7:41 WSPC/INSTRUCTION FILE K˙V1
Terminating statesas a unique laboratory for testingnuclear energy density functional 5
0.5
SLy5 SLy5
0.0 SkO
SLy4 SkXc 0.0 SLy4
V]-0.2 SkO V]-0.5 SIII SkXc
Me-0.4 SIII SkP Me MSk1
E[ref-0.6 E[ref-1.0 SkP
E--0.8 [d ]-1[f ]n+1 MSk1 E--1.5
(cid:0) 3/2 7/2 (cid:0)
-1.0 [f5/2]-1[g9/2]n+1 SkM* -2.0 [p1/2]-1[d5/2]n+1
SkM*
-2.5
130 140 150 160 170 130 140 150 160 170
W0* [MeV fm5] W* [MeV fm5]
0
Fig. 2. Left figure shows mean value of ∆E−Eref in A∼44 nuclei in comparison to analogical
quantity calculated for ∆E = E([f−1gn+1])−E([gn ]) configurations in A∼80 nuclei. Right
5/2 9/2 9/2
figure shows ∆E−Eref calculated for∆E =E([p−1/12dn5/+21])−E([dn5/2])configurations in A∼16
nuclei.Seetextforfurtherdetails.
SLy5 [f ]-1[g ]n+1
5/2 9/2
V] 0.5 SLy4 SIII [p1/2]-1[g9/2]n+1
SkO
e SkXc
M
0.0
SkP
[
f
e
r
E -0.5 MSk1
-
E
∆
SkM*
-1.0
130 140 150 160 170
W* [MeV fm5]
0
Fig. 3. Comparison of ∆E − Eref for two possible particle-hole excitations [f5−/12g9n/+21] and
[p−1gn+1]throughthemagicgap40.
1/2 9/2
E([dn ] ) in A 16 mass region and for two different terminating configura-
5/2 Imax ∼
tions ∆E = E([f−1gn+1] ) E([gn ] ) and ∆E = E([p−1gn+1] )
5/2 9/2 Imax − 9/2 Imax 1/2 9/2 Imax −
E([gn ] )inA 80massregion.The resultsaredepictedinFigs.2and3.Since
9/2 Imax ∼
thereisnodataforterminatingstatesinthesenucleiwecomparepurelytheoretical
trends by showing mean value ∆E E (averaging over different N=Z nuclei)
ref
− 6
where E denotes mean value of ∆E calculated using SkO, SIII and Sly4 forces
ref
February 6, 2008 7:41 WSPC/INSTRUCTION FILE K˙V1
6 M. Zalewski, W. Satul a
i.e. those which perform best in A 44 region.
Note, that dependence of ∆E ∼E on W∗ for [p−1dn+1] and [f−1gn+1] con-
− ref 0 1/2 5/2 5/2 9/2
figurations is very similar to each other and to A 44 case, as shown in Fig. 2. For
[p−1gn+1] states one observes completely differen∼t pattern, see Fig. 3. The reason
1/2 9/2
forthatbecomes clearwhenone looksatthe hierarchyofdifferentcontributionsto
the corresponding energy gaps emerging within Nilsson model:
∆e(d5/2−p1/2) =~ω (1 4κ), (8)
N=Z=8 0 −
∆e(f7/2−d3/2) =~ω (1 6κ 2κµ), (9)
N=Z=20 0 − −
∆e(g9/2−f5/2) =~ω (1 8κ 3κµ), (10)
N=Z=40 0 − −
∆e(g9/2−p1/2) =~ω (1 6κ 13κµ). (11)
N=Z=40 0 − −
The value of µ vary from zero in light nuclei where the surface and flat bottom
effects are small to the pseudo-spin limit µ 1/2 in heavy-nuclei. In (8), (9) and
∼
(10) contribution to ∆e from flat bottom and surface effect is small in comparison
with contribution from SO interaction. The ratio of these two terms (assuming
µ=1/2) for (9) and (10) cases is equal:
µ 3µ
0.17, (12)
3 ≈ 8 ≈
and for (8) it is equal zero.In case of ph excitation from p to g sub-shell this
1/2 9/2
ratio is:
13µ
1, (13)
6 ≈
and one may anticipate qualitative change in physical scenario what indeed takes
place as shown in Fig. 3.
3. Imax −1 states
Unfavored signature [f7n/2]Imax−1 terminating states can be obtained within mean-
field theory by changing either the signature of valence neutron or the signature
of valence proton. These two independent Slater determinants will be labeled ν
| i
and π , respectively. Energy difference, between these states and I solution
max
| i
calculated using modified SkO force is shown in the left hand part of Fig. 4. Note
thattheSHFvaluesareincompletedisagreementbothwithexperimentaldataand
21
SM results. Similar result holds for SLy4 force, see Ref. .
The major source of disagreement is related to spontaneous violation of rota-
tional symmetry by these quasi-spherical SHF solutions. Indeed, simple m-scheme
countingshowsthatbothI andI 1representationsaresingle-foldedwithin
max max
−
the fn Hilbert space irrespective of number of valence particles n. The SHF solu-
7/2
tions are eigenstates of an angular moment projection with K =I 1. Hence,
max
−
they are mixtures of ”spurious” I ,I 1 and physical I 1,I 1
max max max max
| − i | − − i
states:
I ,I 1 =aν +bπ , (14)
max max
| − i | i | i
February 6, 2008 7:41 WSPC/INSTRUCTION FILE K˙V1
Terminating statesas a unique laboratory for testingnuclear energy density functional 7
I 1,I 1 = bν +aπ . (15)
max max
| − − i − | i | i
The symmetry restorationis therefore limited to a simple 2 2 mixing problem:
×
e V a a
1 =λ . (16)
(cid:18)V e (cid:19)(cid:18)b(cid:19) (cid:18)b(cid:19)
2
where e E(ν ) E(I ) and e E(π ) E(I ). The problem can be
1 max 2 max
≡ | i − ≡ | i −
solvedanalyticallyintwodifferent ways.The firstmethod (A) determinesV based
on strict requirement that ”spurious” solution should be placed at zero energy
λ1 0.ThiscorrespondstoV =√e1e2 andλ2 =e1+e2.Thisisenergyofphysical
≡
I 1staterelativetoenergyofI state.The resultsofthis procedureforthe
max max
−
SkO-SHF calculations are shown in the right hand side of Fig. 4 (black triangles).
-0.5 -0.5
V] V]
Me-1.0 Me-1.0
1) [-1.5 1) [-1.5
I-max-2.0 EXP I-max-2.0 EXP
E( SM E( SM
)-max-2.5 SSkkOOnp nnnppp )-max-2.5 SSkkOOAB BBAABA
I I
E(-3.0 E(-3.0
43Sc 45Sc 45Ti 46V 43Sc 45Sc 45Ti 46V
42Sc 44Sc 44Ti 46Ti 47V 42Sc 44Sc 44Ti 46Ti 47V
Fig. 4. Energy difference between Imax and Imax−1 states for f7n/2 configuration. Left panel
shows the SkO-SHF solutions |πi and |νi which break rotational symmetry. Right panel show
theeffectofsymmetryrestorationaccordingtothemethodsAandB,respectively.Inbothcases
resultsarecomparedwithexperimental data(dots)andtheSMcalculations(circles).
The method B uses values of a and b coefficients calculated by acting on
Imax,Imax state with Iˆ− operator, see Ref. 21 for details. For example, in 43Sc
| i
we obtain:
Iˆ− Imax,Imax = Imax,Imax 1 =2√3π +√7ν . (17)
| i | − i | i | i
Knowing a and b one may find from (16) the value of interaction V as well as
energies of ”spurious” and physical solutions λ and λ :
1 2
r(e e )
1 2
V = − , (18)
1 r2
−
e r2e
1 2
λ = − , (19)
1 1 r2
−
e r2e
2 1
λ = − , (20)
2 1 r2
−
February 6, 2008 7:41 WSPC/INSTRUCTION FILE K˙V1
8 M. Zalewski, W. Satul a
where r b/a. Physical solution λ is shown the right hand side panel of Fig. 4.
2
≡
Note, that in this case the energy, λ , of ”spurious” state is not equal zero, but
1
for SkO force it doesn’t exceed 0.1 MeV (with exception of 45Ti). Note also that
±
method B doesn’t work for N = Z nuclei where r = 1. It is clearly seen from
Fig 4 that our simple symmetry restoring schemes provide very accurate results.
More detailed discussion concerning this issue including results for Sly4 force can
21
be found in Ref. .
2.0
exp SLy4
V] SM corrected SLy4
e exp
M
1.5 SM
[
)
1
-
x
a
m1.0
I
(
E
–
)
x0.5
a
m 4 SkO
I 5
E( 5k corrected SkO
e
0 V
42Ca44Ca43Sc44Sc45Sc45Ti46Ti47V 42Ca44Ca 43Sc 44Sc 45Sc45Ti 46Ti 47V
Fig.5.Energydifferencebetween Imax andthelowestImax−1terminatingstatesford−3/12f7n/+21
configuration.TrianglesandgreydotsdenoteSMresultsandexperimentaldatarespectively.Black
dotsrepresentbareSHFsolutions.CirclesrepresentsymmetryrestoredSHFresultsaccordingto
the method A. Left (right) panel illustrates the SLy4-SHF (SkO-SHF) results, respectively. In
order to facilitate comparison of isotopic and isotonic dependence of the Sly4 results we include
additional curve showing symmetry restored SLy4-SHF solution shifted arbitrarily by 455keV.
The method B, which is not shown for the reasons of clarity, gives larger correction than the
methodAby∼100keVforbothSly4andSkOforces.
For d−1fn+1 configuration there are three possibilities of creating I 1
3/2 7/2 max −
states at the mean-field level. One may change the signature of either proton or
neutron in f sub-shell. These states will be labeled as π and ν , respectively.
7/2
| i | i
Alternatively, one may change the signature of the proton in d sub-shell. This
3/2
Slaterdeterminantwillbedenotedas π¯ .SimpleM-schemecountingmethodshows
| i
that our Hilbert space contains two-foldI 1 representation.Hence, to restore
max
−
brokensymmetrywehavetodealherewith3 3eigen-problemhavingtwophysical
×
andone”spurious” I ,I 1 solutions.ThemostgeneralmixingHamiltonian
max max
| − i
matrix can in this case be written as:
e V V∗ a a
1 12 13
V1∗2 e2 V23b=λb, (21)
V V∗ e c c
 13 23 2    
where e E[i] E[I ], and i= π , ν , π¯ . Two different methods of restoring
i max
≡ − | i | i | i
symmetry (called A and B) are proposed below. In both methods the energy of
February 6, 2008 7:41 WSPC/INSTRUCTION FILE K˙V1
Terminating statesas a unique laboratory for testingnuclear energy density functional 9
”spurious” solution is rigorously set to zero. The method A assumes additionally
real interaction and precise knowledge of mixing coefficients a, b and c. The values
of a, b and c are calculated again by using simple angular momentum algebra,
in particular by applying Iˆ− to Imax,Imax reference state, see 21 for details.
| i
Method B admits complex interaction and sets no further constraints on a, b and
c coefficients.
In both cases the problem can be solved analytically. Method A gives:
e a2 e b2+e c2
1 2 3
V = − − ,
12
2ab
e a2+e b2 e c2
1 2 3
V = − − , (22)
13
2ac
e a2 e b2 e c2
1 2 3
V = − − ,
23
2bc
while in method B we obtain:
V =√e e , V =√e e , V =√e e , (23)
12 1 2 13 1 3 23 2 3
if e ,e ,e are of the same sign or:
1 2 3
V =i√ e e , V = i√ e e , V =√e e , (24)
12 1 2 13 1 3 23 2 3
− − −
for e <0 and e ,e > 0. Other possibilities are analogous. Knowing the values of
1 2 3
interaction one may find the energies of physical I 1 states. They are equal:
max
−
1
λ± = Σ √∆ (25)
2(cid:16) ± (cid:17)
where
∆=Σ2 4Z,
−
Σ =e +e +e ,
1 2 3
Z =e e +e e +e e V 2 V 2 V 2.
1 2 1 3 2 3 12 13 23
−| | −| | −| |
The results for the lowest I 1 states are shown in Fig. 3. One can see, that
max
−
now the effect of symmetry restoration is relatively small as compared to clearly
dominant mean-field splitting. There is also an interesting difference between Sly4
andSkOforces.FortheSly4forcetheisotonicandisotopicdependenceofE(I )
max
−
E(I 1)isverywellreproducedbutthereisaconstantoffsetbetweentheoretical
max
−
results and the data. This offset, after symmetry restoration, equals to 455keV
∼
for the method A and to 385keV for the method B. The opposite is true in the
∼
SkOcase.Thisforcedoesnotreproducedetailsofisotopicandisotonicdependence,
but reproduces quite well mean value of E(I ) E(I 1).
max max
− −
4. Summary
We have shown that terminating states in A 44 mass region are excellent play-
∼
ground for testing and constraining various aspects of the nuclear EDF. In par-
ticular, unification of otherwise quite random coupling constants connected with
February 6, 2008 7:41 WSPC/INSTRUCTION FILE K˙V1
10 M. Zalewski, W. Satul a
spin-fields to experimental data and slight reduction of SO strength allow to reach
very good agreement with experimental data for maximum-spin I terminating
max
states.Our study revealsthat the SHF method cannotbe directly applied to unfa-
voredsignatureI 1states.ForthesestatestheSHFsolutionsmanifestlybreak
max
−
rotationalinvarianceat almostsphericalshape. After identificationof the underly-
ing SSB mechanism we propose analytical symmetry restoration schemes for both
[f7n/2]Imax−1 and [d−3/12f7n/+21]Imax−1 configurations. It is shown that for [f7n/2]Imax−1
the configuration mixing is absolutely necessary in order to reproduce empirical
E(Imax)−E(Imax −1) splitting. For [d−3/12f7n/+21]Imax−1 configuration on the other
handthe splitting E(I ) E(I 1)isdominatedbymean-fielditselfandthe
max max
− −
symmetry restoration effect is relatively less important.
5. Acknowledgments
This work was supported in part the Polish Committee for Scientific Research
(KBN)undercontractNo.1P03B05927andbytheFoundationforPolishScience
(FNP).
References
1. E. Caurier et al.,Rev. Mod. Phys. 77 (2005) 427.
2. M. Bender, P.-H. Heenen,and P.-G. Reinhard, Rev.Mod. Phys. 75 (2003) 121.
3. P.HohenbergandW.Kohn,Phys.Rev.136, B864 (1964); W.KohnandL.J. Sham,
Phys. Rev.140, A1133 (1965); W. Kohn, Rev.Mod. Phys. 71, 1253 (1998).
4. E. Perlin´ska et al.,Phys.Rev. C 69, 014316 (2004).
5. T.H.R. Skyrme,Phil. Mag. 1 (1956) 1043; Nucl. Phys.9 (1959) 615.
6. D. Vautherinand D.M. Brink, Phys.Rev. C5 (1972) 626.
7. A.V. Afanasjev et al., Phys.Rep.322 (1999) 1.
8. W. Satul a and R. Wyss, Rep.Prog. Phys. 68 (2005) 131.
9. G. Stoitchevaet al.,Phys.Rev. C73 (2006) 061304(R).
10. H. Zdun´czuk, W. Satul a, and R. Wyss, Phys. Rev. C71 (2005) 024305; Int. J. Mod.
Phys. E14 (2005) 451; W. Satul a, R. Wyss, and H. Zdun´czuk, Eur. Phys. J. A 25,
s01, (2005) 551. .
11. E. Chabanat, et al.,Nucl. Phys.A627 (1997) 710; A635 (1998) 231.
12. P.-G. Reinhard et al., Phys.Rev.C60 (1999) 014316.
13. M. Beiner et al.,Nucl. Phys.A238 (1975) 29.
14. B.A. Brown, Phys. Rev. C58 (1998) 220.
15. F. Osterfeld, Rev.Mod. Phys. 64 (1992) 491.
16. M. Bender et al., Phys.Rev.C65 (2002) 054322.
17. S.G. Nilsson, Mat. Fys. Medd.Dan. Vid. Selsk. 29 (1955).
18. J. Dobaczewski, H.Flocard, and J. Treiner, Nucl. Phys. A422 (1984) 103.
19. F. Tondeur et al.,Phys.Rev. C62(2000) 024308.
20. J. Bartel et al., Nucl.Phys. A386 (1982) 79.
21. M.Zalewskietal.,submittedtoPhys.Rev.C(2006),LANLe-printnucl-th/0701020.