Anomalous behavior of the first excited 0$^{+}$ state in $N \approx Z$ nuclei

0.17 MB·English

by  K. Kaneko

Anomalous behavior of the first excited 0+ state in N ≈ Z nuclei
K. Kaneko1, R. F. Casten2, M. Hasegawa3, T. Mizusaki4,
Jing-ye Zhang2,5, E. A. McCutchan2, N. V. Zamfir2, R. Kru¨cken6
1Department of Physics, Kyushu Sangyo University, Fukuoka 813-8503, Japan
2WNSL, Yale University, New Haven, Connecticut 06520-8124, USA
3Laboratory of Physics, Fukuoka Dental College, Fukuoka 814-0193, Japan
4Institute of Natural Sciences, Senshu University, Kawasaki, Kanagawa, 214-8580, Japan
5Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
6Physik Department E12, Technische Universi¨at Mu¨nchen, 85748 Garching, Germany
(Dated: February 9, 2008)
A study of the energies of the first excited 0+ states in all even-even Z ≥ 8 nuclei reveals an
anomalous behavior in some nuclei with N = Z, Z ± 2. We analyze these irregularities in the
framework of the shell model. It is shown that proton-neutron correlations play an important role
5 in this phenomenon.
0
0 PACSnumbers: 21.10.Re,21.60.Fw,27.70.+q
2

n
30
A topic of currentinterest and varying interpretations
a
J is the nature of excited Jπ = 0+ states in nuclei. Tra- Experimental
0 ditionally, the lowest excited 0+ states in even-even de- 25 Calculated
2 formednucleihavebeeninterpretedas“β”vibrations[1].
In recent years, various papers [2, 3, 4, 5, 6] have dis-
20
1 cussedthelowestK =0+excitationasacollectiveexcita-
v
tion built on the γ vibration. The observationof numer-
48 ous excited 0+ states in 158Gd[7] prompted severaltheo- R0/2 15
reticalinterpretations. Theseincludetheirdescriptionas
0 12,10
1 two-phononoctupole in character[8] or as quasi-particle 10 24,24 26,26 10,12
0 excitationsbasedontheprojectedshellmodel[9]. Inlight 24,26 12,12
5 nuclei, descriptions in terms of pairingvibrations,multi- 5 10,10
0 particle - multi-hole intruder states and isobaric analog
/
h states have been discussed for decades. Nevertheless, to
0
t date, a complete understanding of the origin of excited 0.0 0.5 1.0 1.5 2.0 2.5
-
l 0+ states remains elusive.
c E(2+) (MeV)
The purpose of this Rapid Communication is twofold.
u 1
n First, we will show a striking, and heretofore unrecog-
v: nized,anomalyin0+2 energiesthatoccursincertain(but FIG.1: Theenergy ratio R0/2 ≡E(0+2)/E(2+1) as afunction
i not all) light nuclei with N = Z, N = Z ± 2. Secondly, of the energy of the first excited 2+1 state for all even-even
X we will present shell model calculations that show the nuclei with Z ≥ 8. The points above the main trajectory
r significant role of the T = 0 part of the residual proton- are labelled by their (Z,N) values. The solid squares denote
a neutron interaction in these anomalies. the experimental values and the open circles denote the cal-
With the high current interest in N ∼ Z nuclei and culated values for some of these nuclei (see text). The three
the likelihood that new examples of such nuclei will be unlabelled,calculatedpointsthatlieabovethetrajectorynear
E(2+) ∼ 1.0 and 1.3 MeV correspond to44,46,48Ti.
studiedingreaterdetailwithexoticbeamexperimentsin 1
upcomingyears,thediscoveryofnewphenomenainsuch
nuclei takes on heightened interest. Looking over the
entirenuclearchart,thetrendoffirstexcited0+ energies 22,24Mg,48,50Cr and52Fe with Z and/orN = 10,12,24,
exhibits an interesting behavior. To compare nuclei over and 26. An important aspect of the anomaly is that not
awiderangeofstructures,wenormalizetheenergyofthe all nuclei with N = Z, Z ± 2 exhibit it. Specifically,
0+2 state by the energy of the 2+1 state, defining R0/2 ≡ nuclei with N or Z magic or with N or Z = 14, 16, 18
E(0+)/E(2+). ThisR ratioisplottedasafunctionof or 22 lie within the main trajectory. The phenomenon
2 1 0/2
theenergyofthefirst2+ stateinFig. 1,foralleven-even occurswhen,inthemostsimpleviewofshellfilling,there
1
nuclei with Z ≥ 8. For the majority of nuclei, the R0/2 is an open 1d5/2 shell or an open 1f7/2 shell, except for
ratios follow a compact trajectory. There are, however, N,Z = 22. Any interpretationof this phenomenon must
obvious deviations from this trajectory: 7 points which account not only for its existence, but also its locus and
clearly stand out above the main trajectory. must explain why it is not universal in N = Z and N =
Interestingly,thedeviationsinFig. 1compriseasetof Z ± 2 light nuclei.
nuclei with N = Z and N = Z ± 2, specifically 20,22Ne, Inspection of the individual excitation energies E(0+)
2
2

6 8
a) Ne b) Ti c) Cr
10,12 12,10 26,26 7
5 24,26 V) 6 Mg Ti (0+3) Fe
e
M 5
10,10 (
4 24,24 12,12 gy 4 0+2
R0/2 3 Ener 23 0+2 0+2 2+
1 2+ 2+ 1
1 1
0
2 6 d) Ne e) Ti f) Cr
Mg Ti (0+) Fe
5 3
1
4
2
0 R0/ 3
1.6 2.0 2.4 2.8 3.2 2
R 1
4/2
0
0 2 4 6 0 2 4 6 0 2 4 6
FIG.2: TheenergyratioR asafunctionofR forsdand
0/2 4/2 N - Z N - Z N - Z
fpshelleven-evennucleiwith8≤Z ≤30. Thosenucleiwith
anomalous R valuesare labelled by their (Z,N) values.
0/2 FIG. 3: Excitation energies of the first excited 0+ and 2+
states and as a function of N - Z for the (a) Ne and Mg (b)
and E(2+) shows that the ratio R is large for these Ti and (c) Cr and Fe isotopes. Ratio R0/2 ≡ E(0+2)/E(2+1)
1 0/2 as a function of N - Z for the (d)Ne and Mg (e) Ti and (f)
nuclei due to the combination of a large value of E(0+2) Cr and Fe isotopes. The solid and open symbols denote the
and a small value of E(2+1) compared with neighbors in experimental and calculated values, respectively. The cross
thecorrespondingmassregion. Sincesmall2+ excitation symbols in (b) and (e) correspond to the second excited 0+
1
energies are in general associated with soft or deformed state in theTi isotopes.
nuclei, we investigate the connection between R and
0/2
deformationinFig. 2,focusingontheregion,8≥Z≥30,
where the anomalies are located. As a measure of the state in N ≈ Z nuclei, we perform full shell model
structure, we use the energy ratio R4/2 = E(4+1)/E(2+1), calculations for a wide range of nuclei. For the sd
whereR4/2 =2.0forsphericalnuclei,3.33foraxiallyde- shell, with model space (1d5/2,2s1/2,1d3/2), we adopt
formednuclei. Asthelow2+ energysuggests,eachofthe the Wildenthal interaction[22] to perform calculations
1
anomalousnuclei have R > 2.4. However,from Fig.2, for 20−26Ne, 22−30Mg, 28−32Si, 32−34S, 36Ar. We note
4/2
onecanalsoseethatthereareseveralothernucleiwhich that, in terms of energies, the results obtained for the
have similar structure but do not exhibit an anomalous sd shell are identical to those obtained in shell model
R . This observation reiterates that the anomaly re- calculations by Brown[23]. For the fp shell, with model
0/2
sults from a combination of a low 2+1 excitation energy space (1f7/2,2p3/2,1f5/2,2p1/2), we make use of the KB3
and a high 0+ excitation energy. interaction[24] for calculations on 44−50Ti, 48−54Cr and
The remain2derof this paper willfocus onthe originof 52−58Fe. Due to the enormous size of the configura-
the highlying 0+ statesin these nucleiusing shellmodel tion space for 54−58Fe, we employed an extrapolation
2
calculations. Since the abnormalities in R occur for method[25] for the shell model calculations of these nu-
0/2
nuclei with N ≈ Z, an appropriate aspect to consider is clei. The calculated values of R0/2 are indicated by the
p−ninteractions,whicharestrongforN =Z nucleiand open circles in Fig. 1 and follow very well the exper-
playasignificantroleindeterminingtheirstructure[10]. imental trend. In particular, the large deviations from
Numerous studies ofp−ncorrelationsincorporatingthe the simple trajectory for the seven nuclei near N ≈ Z
competition between T = 0 andT = 1 components have are well reproduced. The calculations for the remaining
investigated the energy spectrum as well as the binding nuclei lie within the main trajectory, with the exception
energyofprotonrichnuclei. The largep−n correlations of 3 points corresponding to 44−48Ti. This peculiarity
atN ≈Z giverisetoseveralinterestingphenomenasuch associated with the Ti nuclei will be discussed below.
assingularitiesinthep−ninteractionenergy[11,12,13], Acomparisonwithexperimental0+ and2+energiesas
2 1
alpha-like correlations[14], the Wigner energy[15], and a function of N - Z for Ne and Mg is given in Fig. 3(a)
degenerate T = 0 and T = 1 lowest states in odd-odd, andfor the CrandFe isotopesinFig. 3(c). Includedare
N = Z nuclei[16, 17, 18, 19, 20, 21]. We can expect all seven nuclei previously highlighted, except for 22Mg
that large p−n correlations may also contribute to the which should be identical to 22Ne assuming isospin in-
properties of the 0+ state in N ≈ Z nuclei. variance. For 20,22Ne and 48,50Cr, i.e., with N = Z, Z
2
In order to analyze the above behavior of the 0+ + 2, the 0+ energies are large while the 2+ energies are
2 2 1
3

small. As neutron number increases, the 0+2 energy de- (a) d ‚V Ú
creases and the 2+1 energy increases. In the Mg and Fe 8 Jp = 0+ d ‚VT=0 Ú
isotopes, the 0+2 energy is large only for 24Mg and 52Fe ) 6 d ‚VT=1Ú
(eNxpe=rimZe)natanldtrtehnednsdoefctrheeas0e+s wanitdh2n+euentreorngieesxcaerses.repTrhoe- MeV 4 int
ducedwellbythecalculation2s. Figur1es3(d)and(f)show ÚV ( 2
‚
the corresponding experimental and calculated R ra- 0
0/2 d
tios. As expected from the discussion of the 0+2 and 2+1 -2 Ne
energies, the ratios for 20,22Ne and 48,50Cr are large and
-4
almosttwotimesthoseoftheheavierNeandCrisotopes. (b)
In the Mg and Fe isotopes, however, the ratios for 26Mg 8
and 54Fe (N = Z + 2, with N = 14 and Z = 28) are 6 Jp = 0+
small while those of 24Mg and 52Fe (N = Z) are large. V) 4
e
Again, the experimental behavior is reproduced by the M
( 2
calculations. ÚV
While the anomaly for N=Z and N=Z ± 2 nuclei is ‚ 0
d
widespread in the open 1d and 1f shell nuclei, it is -2 Mg
5/2 7/2
conspicuously absent experimentally in the Ti isotopes. -4
In contrast, in these same nuclei, the 0+ state and R 0 2 4 6
2 0/2
N - Z
ratio is actually very low in energy as seen in Figs 3(b)
and (e). This result is not found in the calculations.
Theypredictthe anomalyin44,46Tiandevenin48Ti(N FIG. 4: Correlation energy differences, hVinti and the corre-
sponding T = 0 and T = 1 components for the (a) Ne and
=Z +4). Arecentneweffectiveinteraction,GXPF1[26],
(b) Mg isotopes.
yieldssimilarresultsfortheTiisotopes. Theexplanation
of this lies in the well-known presence of low lying 0+
intruderstatesinTiarisingfrom2p-2hprotonand2p-2h
To examine the contributions of the T = 0 and T = 1
neutronexcitationsacrosstheZ,N =20shellgap,which
components inthe shellmodelinteraction,we cancalcu-
leads to a collective, low-lying 0+ state. This state is
2 latetheir expectationvaluesinthe eigenstatesofthe full
clearlybeyondthespaceofthepresentcalculations. The
Hamiltonian, Eq. (1). Since the excitation energy de-
reasonthat such a mode is not found in Fe, for example,
pends strongly upon the competition between the corre-
is simply that the excitation of 2 protons from below Z
lationenergies ofthe excited state andthe groundstate,
= 20 in Fe (Z = 26) would lead to a filled f proton
7/2 we define the following correlation energy difference:
shell with the consequent decrease,rather than increase,
inthenumberofvalenceprotons. Indeed,inothernuclei
with a filled 1f shell, (i.e., 54Fe) the anomaly is also
7/2 δhOi=hOiex−hOigr, (3)
not found. The 0+ states in the Ti isotopes (crosses
3
in Fig. 3) likely correspond to an fp shell excitation where hOi and hOi denote the expectation values of
gr ex
and their energies are reproducedreasonablywell by the theoperatorOforthegroundandfirstexcited0+ states,
calculations (open diamonds). respectively,andOisaphysicaloperatorsuchastheshell
Whilewehaveshownthatshellmodelcalculationscan model interaction, V or the T = 0, 1 interactions.
int
reproduce the anomalous behavior of R0/2, a more de- Figure 4 shows the correlation energy difference of 0+
tailed analysis of the calculations is required to under- states as a function of N - Z in the Ne and Mg isotopes,
stand how and why such good agreement is obtained. including the shell model interaction energy, hV i, and
int
We first consider the following Hamiltonian: its decomposition into isoscalar, hVT=0i, and isovector,
hVT=1i, parts. From Fig. 4 we see a correspondence be-
tween the ratio R in Fig. 3(d) and the correlation
0/2
H =H +V , (1) energy differences, δhV i. For each of the Ne and Mg
sp int int
nuclei, with the exception of 20Ne, the T =1 component
where H is the single particle Hamiltonian and V is is largerthan the T = 0 component. Inthe Mg isotopes,
sp int
the realistic shell model interaction, such as the Wilden- Fig. 4(b), the overall evolution of the correlationenergy
thal interaction as used in the following analysis. To difference, δhV i, follows very closely the behavior of
int
examine the roles of the T = 0 and T = 1 correlations the T = 0 component. Both the R0/2 value and corre-
in the 0+ state, we separate the shell model interaction lation energy difference are large in 24Mg and decrease
2
Vint into two parts corresponding to the isoscalar,VT=0, suddenly in 26Mg. This is attributed to the T = 0 com-
and isovector, VT=1, interactions as follows: ponent, which displays a singular negative value for N
- Z = 2, resulting in a sudden decrease in δhV i. For
int
the Ne isotopes, shown in Fig. 4(a), the T = 1 compo-
Vint =VT=0+VT=1 (2) nentremainsrelativelyconstant(∼3.5MeV) for N = 10
4
- 14 dropping to ∼ 1 MeV for N = 16. Because of the large for 20Ne and 24Mg and decrease smoothly with in-
nearly constant behavior of the T = 1 component, the creasing proton number. The behavior of the 0+ states
2
correlation energy differences in the Ne isotopes follow in Fig. 5(a) is similar to that of the correlation energy
the trend of the T = 0 component, decreasing smoothly differences, δhV i, in Fig. 5(b), with the exception of
int
with increasing neutron number. This behavior results 36Ar. A similar behavior as described above is observed
in large correlation energy differences in the Ne isotopes in the N = Z + 2 nuclei. For these nuclei, only 22Ne
with N = Z and N = Z + 2, which will generate high displays a large R ratio and with increasing proton
0/2
excitation energies for the 0+ state in these nuclei com- number, the ratio decreases rapidly.
2
pared to isotopes with larger N. Thus, the behavior of
correlation energy differences, which strongly affects the
ratio R , is attributed to the combined effects of T=1
0/2
and T=0 components. The former contributes the main
amplitude, while the latter correlates with variations in
R .
0/2
From the above discussion, we might expect large val-
uesoftheratioR forallnucleiwithN =Z. However,
0/2
as previously mentioned, not all nuclei with N = Z, Z
± 2 areanomalous. The phenomenonoccurs when there
is an open 1d shell or an open 1f shell, except for
5/2 7/2
N,Z = 22. We now focus on the N = Z sd-shell nuclei
and address why only 20Ne and 24Mg exhibit large R
0/2
ratios. Figure 5(a) illustrates the excitation energies of
the 0+ and 2+ states and the corresponding R ratios.
2 1 0/2
Again,theexcitationenergiesarewellreproducedbythe In conclusion, we have investigated the first excited
shell model calculations. Compared with neighboring N 0+ states in light even-even nuclei. It is found that an
= Z nuclei, 20Ne and 24Mg exhibit large excitation en- anomalyoccursintheratioR ≡E(0+)/E(2+)insome
ergies of the 0+2 states, small excitation energies of the even-even N ≈ Z nuclei. Th0e/2anomaly2 results1 from the
2+1 states and therefore large R0/2 ratios. The correla- combinationoflargeE(0+2)energiesandsmallE(2+1)en-
tion energy differences, along with their T = 0 and T = ergies. Thelatterareassociatedwithasoftnesstodefor-
1 components, are given in Fig. 5(b). The former are mation in these nuclei. Shell model calculations repro-
duce well the observed behavior. Concentrating on the
Ne andMg isotopes, the anomaliesof the 0+ states were
8 8 2
(a) 2+ analyzed using the correlation energy differences and a
1
0+ decomposition of the shell model interaction into T = 0
6 R2 6 and T =1 components. From this analysis, it is shown
) 0/2
V that the anomalousR values can be attributed to the
E (Me 4 40/2 R jcooimntpeoffneecnttscoofnthtreibTut=e0s0/2taontdhTe m=a1incoammppolniteundtes:wthhieleTt=h1e
2 2 T=0 component correlates with the variation of R0/2.

0 0
(b) d ‚V Ú
T=0
8 d ‚V Ú
T=1
) 6 d ‚V Ú
V int
e
M 4
(
ÚV 2
‚

0
d
-2
-4
20Ne 24Mg 28S i 32S 36Ar
FIG. 5: (a) Excitation energies of the 0+ and 2+ states and
2 1
the corresponding R ratios for the N = Z sd-shell nu-
0/2
clei. The solid and open symbols denote the experimental We are grateful to B. A. Brown, R. Broglia, Larry
and calculated values, respectively. (b) Correlation energy Zamick and Igal Talmi for inspiring discussions. This
differences hVinti and the corresponding T = 0 and T = 1 workwassupportedbyU.S.DOEGrantNos. DE-FG02-
components for the N = Z sd-shell nuclei.
91ER-40609and DE-FG05-96ER-40983.
5
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