Shell Model and Mean-Field Description of Band Termination

0.18 MB·English

by  M. Zalewski

Shell Model and Mean-Field Description of Band Termination
M. Zalewski,1 W. Satu la,1,2 W. Nazarewicz,3,4,1 G. Stoitcheva,5 and H. Zdun´czuk1
1 Institute of Theoretical Physics, University of Warsaw, ul. Hoz˙a 69, 00-681 Warsaw, Poland
2Joint Institute for Heavy Ion Research, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
3 Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996
4 Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
5Lawrence Livermore National Laboratory, P.O. Box 808, L-414, Livermore, California 94551
(Dated: February 9, 2008)
We study nuclear high-spin states undergoing the transition to the fully stretched configuration
with maximum angular momentum I within the space of valence nucleons. To this end, we
max
performasystematictheoreticalanalysisofnon-fully-stretchedI −2andI −1fn seniority
max max 7/2
isomers and d−1fn+1 intruder states in the A∼44 nuclei from the lower-fp shell. We employ
3/2 7/2
two theoretical approaches: (i) the density functional theory based on the cranked self-consistent
7 Skyrme-Hartree-Fock method, and (ii) the nuclear shell model in the full sdfp configuration space
0 allowing for 1p-1h cross-shell excitations. We emphasize the importance of restoration of broken
0 angular momentum symmetry inherently obscuring the mean-field treatment of high-spin states.
2 Overall good agreement with experimentaldata is obtained.

n
PACSnumbers: 21.10.Pc,21.10.Hw,21.60.Cs,21.60.Jz,23.20.Lv,27.40.+z
a
J

9
I. INTRODUCTION be applied to nuclei with several valence nucleons out-

sidethemagiccore. The effectiveHamiltonianisexactly
1
v The phenomenon of band termination is a splendid diagonalizedinasubspaceofmany-bodySlaterdetermi-
0 manifestation of a competition between nuclear single- nantsresultingincorrelatedwavefunctionsthatpreserve
2 particleandcollectivemotion. Atlowangularmomenta, angular momentum, parity, particle number, and - usu-
0 a rotational band is associated with a collective reorien- ally - isospin.
1
tation of a deformed nucleus in space, with many nu- For medium-mass and heavy nuclei containing many
0
7 cleons contributing coherently to the total spin. With valencenucleons,thetoolofchoiceisthenuclearDFT[5].
0 increasedrotationalvelocity,however,the Coriolisinter- Here, the nucleus is described in terms of one-body den-
/ actioncausesnucleonicpairstobreak,andthedecoupled sitiesandcurrentsrepresentingdistributionsofnucleonic
h
t nucleons align their individual angular momenta. It of- matter, spins, momentum, and kinetic energy, as well as
- ten happens that breaking relatively few nucleonic pairs their derivatives. The associated mean fields, obtained
l
c cangiverisetoanuclearstatewithafairlylargespin. In by means of the self-consistent Hartree-Fock (HF) pro-
u the languageofthe nuclearshellmodel,ofparticularim- cedure, are usually deformed, i.e., they spontaneously
n
portancearethe “seniorityisomers”or“fully-aligned(or break the symmetries of the underlying Hamiltonian. In
:
v stretched,oroptimal)states,”whichcarrythemaximum this way, many essential many-body correlations can be
Xi angularmomentumImax within the spaceof valence nu- incorporated into a single product state [3, 5, 6]. How-
cleons. ever, the price paid for the simple intrinsic picture is
r
a The transitionprocess fromcollective rotationat high high: the HF wave function is no longer an eigenstate
spins to a single-particlepicture atI is referredto as of symmetry operators;hence, the transformationto the
max
bandtermination[1]. Terminatingbandsarecommonin laboratory reference frame has to be carried out to re-
nuclei; they have been observed across the table of the store broken symmetries.
nuclides (see recent reviews [2, 3]). The nature of the The language in which the nucleus is pictured as a
termination process strongly depends on the size of the wave packet with anisotropic density/current distribu-
valence space. For instance, if only several valence nu- tionisparticularlyusefulforthedescriptionofrotational
cleonsarepresent,the staticnucleardeformationcannot motion within the cranking formalism. Here, the many-
develop and the collective effects have dynamic charac- body Hamiltonian Hˆ is replaced by a rotating Hamilto-
ter. In this case, Imax can be generated by breaking nian (Routhian), Hˆω = Hˆ −ωJˆz, where the rotational
veryfewnucleonicpairs,andthetransitiontothesingle- frequency ω is interpreted as a Lagrange multiplier de-
particle limit is rapid. On the other hand, for deformed termined from the angular momentum constraint. The
nuclei having many valence nucleons, the transition to resulting cranked HF method (CHF) can also be used
the non-collective regime is long and gradual, often in- to describethe so-callednon-collective rotations,i.e.,ro-
volving many intermediate stages. tations around symmetry axis [2, 7]. The non-collective
Theoretically, high-spin band terminations and fully CHF technique applied in this work has proven to be
aligned configurationsare often discussed within the nu- a very efficient and accurate way to generate stretched
clearshellmodel(SM) and/orthe self-consistentdensity solutions within a self-consistent framework.
functionaltheory(DFT).Thenuclearshellmodel[4]can In practice, most applications of nuclear DFT are
2
based on Skyrme energy density functionals optimized During the last decade, extensive SM calculations for
to various experimental data [5, 8]. Recently, such an thelightfp-shellnucleihavebeenconductedusingdiffer-
approachwasapplied[9,10]to fully-stretched,high-spin entSMinteractions[14,15,16,17,18]. Inmostcases,SM
states associated with the [fn ] and [d−1fn+1] reproduces well the excitation energies of normal-parity
7/2 Imax 3/2 7/2 Imax
SMconfigurations(ndenotesthenumberofvalencepar- andintruderstates,aswellastransitionrates. Ingeneral,
ticles outside the 40Ca core) of 20 Z N 24. the intruder d−1fn+1 states are predicted and observed
3/2 7/2
Thesestudieshavedemonstratedthat≤those≤fully-a≤ligned tobemorecollectivethanthefn structures,i.e.,theas-
7/2
states have a fairly simple SM structure, and, therefore, sociatedin-bandE2ratesaresignificantlygreaterwithin
they provide an excellent testing ground for both the intruderbands. TheSMresultsinthisworkareobtained
time-odd densities and fields that appear in the mean- using the code ANTOINE [19] in the sdfp configuration
field description and for SM effective interactions. In space limited to 1p-1h cross-shell excitation from the sd
particular, it was shown that the energy difference be- shell to the fp shell. In the fp-shell SM space we took
tween the excitation energies of the terminating states, the FPD6 interaction [20]. The remaining matrix ele-
E([d−3/12f7n/+21]Imax)−E([f7n/2]Imax),is asensitiveprobeof ments are those of Ref. [21]. As compared to the earlier
time-oddspincouplingsandthestrengthofthespin-orbit work [14], the mass scaling of the SM matrix elements
term in the Skyrme functional [9], and that with prop- was done here consistently, thus reducing the sd interac-
erly modifying functionals, the nuclear DFT provides a tion channel by 4%.
∼
descriptionofthedataforstretchedstatesthatisofsim-
ilar quality as the fully correlated SM [10].
The aim of this study is to understand the nature of
III. UNFAVORED-SIGNATURE TERMINATING
the[f7n/2]Imax−1 (Sec.III)and[d−3/12f7n/+21]Imax−1 (Sec.IV) f7n/2 STATES
configurationsin the A 44 mass region. These I 1
max
∼ −
states, usually referred to as unfavored-signature termi- Webeginwith[fn ] configurationsin20<Z
nating states, can be obtained from fully-stretched con- 7/2 Imax−1 ≤
N 24 nuclei. Figure 1 displays the energy difference
figurations by signature-changing particle-hole (p-h) ex- ≤
between the stretched I=I and the lowest I
citations. Consequently, their structure is sensitive to max max −
1 states. It is gratifying to see a qualitative, and – in
both shell structure and time-odd nuclear fields. We
most cases – quantitative agreement between SM and
demonstrate that non-collective cranking may lead to a
experimental data.
dramatic violation of rotational symmetry even for the
Within the na¨ıve non-collective cranking, the unfa-
cases when the nuclear shape is almost spherical. We
voredI 1statescanbe obtainedbyeitherinverting
also analyze in Sec. V the Imax − 2 weakly collective the signmaatxu−re of a single proton (π) or a single neutron
states in terminating normal-parity and intruder struc-
tures. The associated correlation energy, mainly associ- (ν). Theenergiesofthosestates,EI(mCHaxF−,π1) andEI(mCHaxF−,ν1),
ated with quadrupole effects, is calculated to be large, are displayed in Fig. 1 (top) with respect to the energy
around 2MeV. E(CHF) of the stretched configuration. It is seen that
Imax
agreement between CHF and experiment is rather poor.
In particular, the strong particle number variation in
II. THE MODELS the energy difference is not reproduced by CHF. This
discrepancy has its origin in the spontaneous violation
ThedetailsofDFTandSMframeworksappliedinthis of rotational invariance by the mean-field solutions, in
study follow Ref. [10] in which the stretched configura- spite of the fact that the underlying CHF states are al-
tionsinthe A 44massregionhavebeensuccessfullyex- most spherical in all the considered cases. Indeed, since
∼
plained. The CHF calculations are carriedout using the the cranking procedure only constrains an expectation
HFODD code of Ref. [11]. We employed the SLy4 [12] value of the angular momentum projection, CHF states
and SkO [13] Skyrme energy density functionals slightly contain components with different angular momentum.
modified along the prescriptionof Refs. [9]. Without go- In the case of unfavored Imax 1 states, two compo-
−
ing into detail, the modifications concern the coupling nents are expected to dominate: spurious reorientation
constants of s2 and s∆s terms giving rise to the time- mode Imax;Imax 1 and physical stretched configura-
· | − i
oddspinmeanfields. Thishasbeendonebyconstraining tion Imax 1;Imax 1 .
| − − i
the functionals to the empirical spin-isospin Landau pa- The appearance of spurious components in partially-
rameters. In addition, the strength of the spin-orbit in- aligned cranking configurations is well known [22]. A
teractionhasbeenreducedby5%fromtheoriginalSLy4 classic example is the cranking treatment of two iden-
and SkO values. As discussed in Ref. [10], the DFT re- tical nucleons in a spherical single-j shell. While the
sults for the T=0 states in N=Z nuclei have to be cor- stretched configurationwith J =M=2j 1 can be asso-
z
−
rected for the isospin breaking effects. Here we assume ciated with the stretched state having I =2j 1, the
max
−
that the isospin correction weakly depends on angular cranked J =2j 2 solution is simply a magnetic sub-
z
momentum;hence,ithasbeenneglectedwhendiscussing state of I (th−e I 1 state does not exist in a j2
max max
−
energy differences between high-spin states. configuration as a result of the Pauli principle). In or-
3
TABLEI:MaximumspinstatesIπ andsquaredunnormal-
max
ized expansion amplitudes for fn and d−1fn+1 configura-
7/2 3/2 7/2
tionsin A∼44nuclei. Inthelattercase only N6=Z nucleiare
-0.5 considered.
I a2 b2 I a2 b2 c2
-1.0 max max
42Ca 11− 12 7 3
-1.5 EXP 44Ca 13− 16 7 3
SM 42Sc 7+ 1 1
) -2.0 43Sc 19/2− 12 7 27/2+ 12 12 3
eV -2.5 44Sc 11+ 15 7 15− 15 12 3
M f n SLy4n nnn 45Sc 23/2− 16 7 31/2+ 16 12 3
) ( -3.0 7/2 SLy4p ppp 44Ti 12+ 1 1
-1 45Ti 27/2− 15 12 33/2+ 15 15 3
max 46Ti 14+ 16 12 17− 16 15 3
(I 46V 15+ 1 1
E -0.5 47V 31/2− 16 15 35/2+ 16 16 3
-)
ax-1.0
m
E(I -1.5 EXP
SM
where aˆ† (τ=π or ν) represents a particle in the f
-2.0 τm 7/2
shell carrying magnetic quantum number m. By assum-
SLy4A AAA
-2.5 SLy4B BBB ingthatthevalencecouplingscheme(1)canbeextended
tothe CHFcase,the I ;I 1 statecanberepre-
-3.0 | max max− i
sentedasauniquecombinationofCHFsolutions ν and
43Sc 45Sc 45Ti 46V
| i
π corresponding to the lowest neutron and proton p-h
42Sc 44Sc 44Ti 46Ti 47V | i
signature-changingexcitations.
The mixing coefficients a and b introduced in Eq. (1)
for 43Sc can be calculated in a similar manner for any
nucleus. They are displayed in Table I. Assuming that
FIG. 1: The energy difference between I and I −1 contributions from other shells are small, the physical
max max
states in f7n/2 configurations in A∼44 nuclei. Dots and cir- state |Imax−1;Imax−1i can be represented by the or-
clesmarkexperimentaldataandtheSMresults,respectively. thogonal combination:
Top: CHF-SLy4 results for unfavored signature terminating
states corresponding to neutron (filled triangles) and proton I 1;I 1 = b ν +a π . (2)
max max
| − − i − | i | i
(open triangles) signature inversion. Bottom: CHF-SLy4 re-
sults including the angular momentum correction calculated This 2 2 configurationmixing can be dealt with using
×
usingprescription A (filledtriangles) and B(open triangles). twoslightlydifferentmethods. Inthefirstvariant(called
A in the following), one requires that the spurious solu-
tionisdegeneratewithrespectto theCHFoptimalstate
der to remove spurious components, angular momentum having energy E(CHF). This assumption leads to a sim-
Imax
needstoberestored. SincethenumberofM-schemecon- ple expression for the energy of the unfavored-signature
figurations around I is very limited, in our cranking terminating state
max
analysis we resort to an approximate projection scheme.
Inthefollowing,theadoptedangularmomentumrestora- EI(mCHaxF−,A1) =EI(mCHaxF−,π1)+EI(mCHaxF−,ν1)−EI(mCHaxF). (3)
tion procedure is discussed for the case of 43Sc .
In the second variant (called B), the mixing coefficients
Due to a near-sphericity of the CHF solutions, single-
are taken directly from Table I and the corresponding
particle (sp) states can be labeled using the angular mo-
energy can be written as
mentum projection m. The optimal state I ;I
max max
| i
can be viewed as a CHF vacuum around which p-h ex- a2E(CHF,π) b2E(CHF,ν)
citations are built. The I ;I 1 spurious state E(CHF,B) = Imax−1 − Imax−1 . (4)
| max max − i Imax−1 a2 b2
can be obtained by acting on the vacuum with the low- −
ering operator Iˆ−. For 43Sc, which in a SM picture has Note that this methodcannotbe applied to N=Z nuclei
oneprotonandtwoneutronsinaf7/2 shell,Imπax=19/2− where a2=b2. Moreover, method B does not guarantee
and the (unnormalized) spurious state can be written as that the energy of the spurious state is degenerate with
the optimal state.
|Imax;Imax−1i=2√3aˆ†ν3/2aˆν5/2|Imax;Imaxi+ staTtehseobentaeirngeyddiniffvearreinacnetss AbeatwndeeBn aIrmeadxispalnadyedIminaxt−h1e
√7aˆ†π5/2aˆπ7/2|Imax;Imaxi≡a|νi+b|πi, (1) lower panel of Fig. 1. It is seen that the the effect of
4
symmetry restoration is large. In particular, the ener- The intruder case represents a 3 3 mixing problem.
×
gies corrected for the angular momentum mixing follow Two different analytical approximate projection tech-
fairlyaccuratelyexperimentandSM.Moreover,methods niques have been developed [23]. In the first method
A and B give fairly similar results, although method B (called A), we assume real mixing matrix elements and
is slightly closer to the data. Similar results were also require the eigenvector (a,b,c) of Eq. (5) and Table I
obtained in the CHF-SkO variant. to correspond to zero energy mode relative to the CHF
energy E(CHF) of the optimal solution. By introducing
Imax
e =E(CHF,α) E(CHF),whereα=ν (or1),π (or2),and
IV. UNFAVORED-SIGNATURE TERMINATING α Imax−1 − Imax
d−1fn+1 STATES πˆ (or 3),the energiesofphysicalsolutions relativeto the
3/2 7/2 CHF optimal state E(CHF) are:
Imax
Wenowconsiderthe[d−1fn+1] intruderconfig-
3/2 7/2 Imax−1 1
urations. Letusrecallthatinthiscasethe[d−1fn+1] λ = e ( e )2 4Z , (6)
3/2 7/2 Imax ± 2 i± i − 
configurations are uniquely defined only for N = Z nu- i s i
6 X X
clei. Indeed, as discussed in Ref. [10], the CHF solutions  
where
for N=Z systems violate isobaric symmetry and can no
longerserveasreferencestates. Hence,weshalllimitour Z = (e e V 2) (7)
i j ij
considerations to 20 Z <N 24 nuclei. Moreover,in −| |
thefollowing,wewill≤considero≤nlyconfigurationsinvolv- Xi<j
ing 1p-1h proton excitation across the Z=20 gap. The and
latter assumption concerns the N Z=1 nuclei in which
the neutron cross-shell excitations−can also give rise to V = −e1a2−e2b2+e3c2,
12 2ab
aligned I 1 states. It is worth mentioning that the
max− e a2+e b2 e c2
1p-1h neutron excitations in N Z=1 nuclei are higher V = − 1 2 − 3 , (8)
thanthelowest1p-1hprotonexc−itations. Moreover,they 13 2ac
donotmixatthelevelofCHF,leadingtoasevereisospin V = e1a2−e2b2−e3c2.
symmetry violation. Restoration of the isospin symme- 23 2bc
try for these cases is, however, beyond the scope of this
In the second method (called B), we admit complex
work.
mixingamplitudes. Inthiscasewesetthespuriousmode
Under these assumptions, within the non-collective
to zero but do not require the correspondingeigenvector
cranked CHF picture, there are three obvious ways of
to be equal (a,b,c). This procedure leads to exactly the
changing the signature quantum number of an intruder
same set of formulas (6)-(7) as above but with V =
state. Namely, one can invert the signature of a single αβ
f proton(π ), asinglef neutron(ν ), oraproton √eαeβ (see Ref. [23] for further details).
7/2 | i 7/2 | i The SM and CHF results corresponding to the lowest
d hole (π¯ ). In this case, SM yields two independent
(I3m/2ax −1)|i (ii=1,2) low-lying solutions. In CHF, both b(Iemtwaxee−n1S)M1 staantdeseaxrpeersihmoewnnt iins aFgiga.in2.exTcehlelenagt,reeexmceepntt
these solutions are polluted by the presence of spurious
for the T=1/2 nuclei 43Sc and 47V where theory overes-
I ;I 1 components.
| mSainxcemthaxe−CHiFsolutionsforthealignedintruderstates timatesthe energydifference betweenImax andImax−1
intruder states. In order to understand this apparent
are nearly spherical, we can employ the same technique
discrepancy, Fig. 2 also shows the SM results for ex-
as proposed earlier for the fn stretched states to ap-
7/2 cited (I 1) states in T=1/2 nuclei. It turns out
max 2
proximately restore the rotational invariance. The only that for47V−the calculated33/2+ levelhas a neutronin-
difference is that in addition to active f7/2 particles, one truder character, while it is the1 second 33/2+ state in
has to consider active d protonholes (denoted as π¯ in 2
3/2 which the proton intruder configuration dominates. For
bthyeafcotlilnogwiwnigt)h. IAˆsounsutahle,tIhe |Im,Iax,ImCaxH−F1viaicsuoubmta.inFeodr 45Ti, the calculated 31/2+1,2 states have a mixed proton-
− max max neutron intruder character with a slight preference for a
| i
the representative case of 42Ca one obtains: protonconfiguration,andthesameistrueforthe25/2+
1,2
states in 43Sc. Clearly, the energetics of (I 1) and
|Imax,Imax−1i = √12aˆ†ν3/2aˆν5/2|Imax,Imaxi (Imax 1)2 states in T=1/2 nuclei strongmlyaxde−pen1ds on
−
+ √7aˆ† aˆ I ,I the mixing between proton and neutron intruder states,
π5/2 π7/2| max maxi i.e, the isospin dependence of the cross-shell sd fp
+ √3aˆ†π¯1/2aˆπ¯3/2|Imax,Imaxi interaction. ↔
aν +bπ +cπ¯ . (5) The CHF calculations displayed in Fig. 2 were carried
≡ | i | i | i out for the SLy4 (top) and SkO (bottom) Skyrme pa-
Similar calculations can be carried out for all N = Z rameterizations. It is interesting to see that predictions
nuclei. The resulting mixing coefficients are collecte6d in for d−1fn+1 intruder states strongly depend on the en-
3/2 7/2
Table I. ergy density functional. The mean empirical value of
5
jectionmethodBtoCHF-SLy4yieldsanalmostconstant
offsetof∆E ∆E 0.450keV.Indeed,byshift-
EXP SLy4
− ∼
d- 1 fn+1 ing the theoretical results up by 0.450keV, one repro-
SM 3/2 7/2 ∼
ducessurprisinglywelltheempiricalisotopicandisotonic
1.5 dependence,includingthecasesof43Scand47V.Itisim-
portanttopointoutthatbothmethodsAandBarefree
from adjustable parameters. In both cases we have sim-
1.0
EXP plifiedthepicturebylimitingthesizeoftheconfiguration
*
B space to three states, and in both cases we have forced
) * *
V A the spurious mode to have zero energy.
0.5 SM’
e
M
SLy4
(
)
1 0
-
x
a
m
(I SM V. FAVORITE SIGNATURE TERMINATING
E
I −2 STATES
- 1.5 max
)
x
a
m
I In the standard picture of band termination, the
(
E1.0
* EXP Imax−2 configurations contain some quadrupole collec-
tivity. That is, these states have small quadrupole de-
* *B
formations that give rise to a small collective rotational
0.5
A
component in the wave function. One can therefore ex-
SkO
pectthattheenergydifferencebetweenI andI 2
42Ca 43Sc 45Sc 46Ti max max
−
0 statesshouldmainly dependontime-evennuclearmulti-
44Ca 44Sc 45Ti 47V polefieldsassociatedwiththenucleardeformability. The
energyofthelastquadrupoletransitionwithinthetermi-
nating sequence,∆E E(I ) E(I 2), is shown
2 max max
inFig.3forbothfn (≡top)andd−−1fn+1(b−ottom)struc-
7/2 3/2 7/2
tures. The agreementbetweenSMandexperimentis ex-
FIG. 2: Theenergy difference between Imax and (Imax−1)1 cellent. In 42Ca, 43,44Sc, and 45Ti, the B(E2) rates for
states in d−1fn+1 configurations in A∼44 nuclei. Dots and
3/2 7/2 the I (I 2) transitionsaresignificantlylarger
max max 2
circles mark experimental data and the SM results, respec- → −
than those for the yrast I (I 2) transitions.
tively. The asterisk symbols (SM’) show the SM results for max→ max − 1
In these cases, the quadrupole strength is fragmented
the(Imax−1)2 states in T=1/2 Sc,Ti, and V isotopes. The
and the band structures cannot be easily identified. It
CHFresultswiththeSLy4andSkOfunctionalsareshownin
the upper and lower panel, respectively. The lowest cranked is interesting to see that ∆E2 for the f7n/2 configurations
solution (|π¯i) is marked by squares while open (filled) trian- is significantly smaller (and sometimes even negative) as
gles mark theresults obtained within theapproximate angu- comparedtothed−1fn+1 values. Thisisconsistentwith
3/2 7/2
larmomentumprojectionA(B)aremarkedwithopen(filled)
the fact that the quadrupole polarizationin the intruder
triangles.
states is significantly greaterthan in the fn structures.
7/2
The mean p-h ∆J = 2 excitation energy calculated
z
−
in HF-SLy4 lies 2MeV below the data. By performing
∆E E(I ) E (I 1), averaged over all nu- ∼
≡ max − 1 max − self-consistent cranking, one arrives at a slightly collec-
clei considered, is ∆E 0.990keV. For the lowest
EXP ≈ tive solutionthat, onthe average,is not too far fromex-
CHF solutions corresponding to π¯-type configurations
periment. Fortheweaklycollective,normal-paritystruc-
(markedbysquaresinthe figures),the calculationsyield
tures, the CHF theory poorly reproduces the particle-
∆E 0.330keV and ∆E 0.730keV. The cal-
SLy4 ≈ SkO ≈ number dependence. The situation is significantly im-
culated corrections are of similar size for both Skyrme
proved in the more collective intruder states where the
functionals, and in all cases they improve the agreement
mean-field theory (both in CHF-SLy4 and CHF-SkO
with the data. Within method A, the CHF curves are
variants) performs much better. Here, the energy gain
shifted up, on the average, by δ∆E 0.090keV
SLy4 ≈ duetodeformationcanbeaslargeas 2MeV.Ofcourse,
and δ∆E 0.140keV. The corrections obtained in ∼
SkO a big part of the remaining discrepancy between CHF
≈
variant B are δ∆E 0.210keV and δ∆E
SLy4 SkO results and experiment is due to angular momentum vi-
≈ ≈
0.270keV.
olation in the I 2 states. Unfortunately, for those
max
−
In variant B, the average splitting calculated in CHF- states, the approximate methods of restoring rotational
SkO is fairly close to the experimental value, ∆E invariancediscussedintheprevioussectionsdonotapply
SkO
≈
∆E 1MeV. However, clear discrepancies in the becauseofmanystatesinvolved(i.e.,pronouncedconfig-
EXP
≈
isotopicandisotonicdependenceareseen. Applyingpro- uration mixing).
6
VI. CONCLUSIONS
Inthis workweperformedatheoreticalanalysisofop-
SM
2 timal Imax states and non-fully-stretched Imax 2 and
−
I 1statesintheA 44nucleifromthelower-fpshell.
max
− ∼
Overall, the level of agreement between SM results and
1 SLy4 experimentaldataforfn seniorityisomersandd−1fn+1
7/2 3/2 7/2
intruderstatesisexcellent. WehaveshownthatCHFso-
)
V
EXP lutionsforunfavored-signatureterminatingstatesareaf-
e 0
M fectedbydramaticviolationofrotationalsymmetryeven
) ( iftheshapeofnucleiunderconsiderationisalmostspher-
-2 -1 fn SLy4, p-h ical. Approximatemethodsofrestoringrotationalinvari-
x 7/2
ma anceinImax 1configurationshavebeenproposed. The
I −
( energy corrections due to angular momentum projection
-E d3-1/2f7n/+21 canbesignificant,andtheirinclusionimproveagreement
) x 3 with the data. Finally, we investigated the weakly col-
a
m lective I 2members ofterminating structures. The
I max
( −
E 2 * correlationenergyinthesestates,mainlyofaquadrupole
SkO
* * nature, is fairly large. While for nearly-spherical high-
* SLy4
1 spinf7n/2 statesthe CHFmethodgivesonlyaroughesti-
mate of the energy splitting between I 2 and I
max max
−
states, the agreement is more quantitative for intruder
0
SLy4, p-h configurations.
-1
42Ca 42Sc 44Sc 44Ti 46Ti 47V
44Ca 43Sc 45Sc 45Ti 46V
Acknowledgments
FIG.3: Thetransition energy between Imax and (Imax−2)1
states in fn (top) and d−1fn+1 (bottom) configurations in
7/2 3/2 7/2 This work was supported in part by the U.S.
A∼44 nuclei. Dots and circles mark experimental data and
Department of Energy under Contract Nos. DE-
the SM results, respectively. The intruder(Imax−2)2 states
in42Ca,43,44Sc,and45Tiareshownbyasterisks. Inthosenu- FG02-96ER40963 (University of Tennessee), DE-AC05-
00OR22725withUT-Battelle,LLC(OakRidgeNational
clei,theB(E2)ratesfortheImax→(Imax−2)2transitionsare
significantlylargerthanthosefortheyrastImax→(Imax−2)1 Laboratory), DE-FG05-87ER40361 (Joint Institute for
transitions. The CHF-SLy4 results are marked by a short- Heavy Ion Research), W-7405-Eng-48with University of
dashed line. The corresponding average particle-hole excita- California (Lawrence Livermore National Laboratory);
tion energy is shown by a dotted line. For comparison, the by the Polish Committee for Scientific Research (KBN)
CHF-SkO results (dash-dotted line) are given in the bottom under contract No. 1 P03B 059 27; and by the Founda-
panel. tion for Polish Science (FNP).
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