Shape coexistence in neutron-deficient Hg isotopes studied via lifetime measurements in $^{184,186}$Hg and two-state mixing calculations

0.74 MB·

by  L. P. Gaffney

Shape coexistence in neutron-deficient Hg isotopes studied via lifetime measurements
in 184,186Hg and two-state mixing calculations
L. P. Gaffney,1,2,∗ M. Hackstein,3,† R. D. Page,1,‡ T. Grahn,1,4 M. Scheck,1,5 P. A. Butler,1 P. F. Bertone,6
N. Bree,2 R. J. Carroll,1 M. P. Carpenter,6 C. J. Chiara,6,7 A. Dewald,3 F. Filmer,1 C. Fransen,3
M. Huyse,2 R. V. F. Janssens,6 D. T. Joss,1 R. Julin,4 F. G. Kondev,8 P. Nieminen,4 J. Pakarinen,1,4,9
S. V. Rigby,1 W. Rother,3 P. Van Duppen,2 H. V. Watkins,1 K. Wrzosek-Lipska,2 and S. Zhu6
1Oliver Lodge Laboratory, University of Liverpool, Liverpool L69 7ZE, United Kingdom
2KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium
3Institut fu¨r Kernphysik, Universita¨t zu Ko¨ln, Zu¨lpicher Str. 77, D-50937 K¨oln, Germany
4Department of Physics, University of Jyva¨skyl¨a, P.O. Box 35, FI-40014 Jyva¨skyl¨a, Finland
5Institut fu¨r Kernphysik, TU Darmstadt, D-64289 Darmstadt, Germany
6Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
7Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, USA
4 8Nuclear Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
1 9CERN-ISOLDE, CERN, CH-1211 Geneva 23, Switzerland
0
(Dated: February 12, 2014)
2
Theneutron-deficientmercuryisotopes,184,186Hg,werestudiedwiththeRecoilDistanceDoppler-
b
Shift(RDDS)methodusingtheGammaspherearrayandtheKo¨lnPlungerdevice. TheDifferential
e
F Decay Curve Method (DDCM) was employed to determine the lifetimes of the yrast states in
184,186Hg. An improvement on previously measured values of yrast states up to 8+ is presented as

1 wellasfirstvaluesforthe9 statein184Hgand10+ statein186Hg. B(E2)valuesarecalculatedand
3
1 compared to a two-state mixing model which utilizes the variable moment of inertia (VMI) model,
allowing for extraction of spin-dependent mixing strengths and amplitudes.
]
x
e
- I. INTRODUCTION Neutron Number, N
l 92 96 100 104 108 112 116 120 124
c
nu beNenucolfeiinetxehriebsittiinngnduicffleeraerntstrsuhcatpuerseaetvelorwsiennceergthyehdaivse- 3.0 12+ 12+ 8+
v2 [ cdtwoiuvesee,rnya1so8sf7oHcaigalataerndgdewj1ui8tm5hHpaginodbrtashemervametidceaicnnh-saiqsnougtaeorpeiden scshhhiaafrtpgemeberaae--- y [MeV] 22..05 6+ 8+ 10+8+ 64++ 10+ 6+
30 surements [1]. Calculations based on Strutinsky’s shell- Energ 1.5 4+ 6+ 2+
0 correction method [2] interpreted this result as a tran- on 0+ 4+ 2+
.5 sniotuionncefdropmrolaatwe-edaekfloyr-mdeefdorsmhaedpe.obFlautrethteor aisomtoopree sphrioft- xcitati 1.0
1 E
measurements reveal that the weakly-deformed oblate 0.5
0
character extends down to A = 182 in the even-mass
4 0+
Hg isotopes [3]. Calculations in these isotopes using 0.0
1
: theNilsson-Strutinskyapproach[4]predicttwodeformed 172 176 180 184 188 192 196 200 204
v
minima, where the lowest-energy minimum corresponds Mass Number, A
i
X toanoblateshape, β (cid:39)−0.15, andthesecondtoamore
deformed prolate shape with β (cid:39) 0.27. In shell-model FIG.1. (Coloronline)Levelenergysystematicsofeven-mass
r
a terms, these minima are associated with a proton zero- mercury isotopes. Red circles refer to the assumed intruder
particle-two-hole configuration, π(0p−2h), and a two- states while blue squares refer to the assumed oblate states,
proton excitation across the Z = 82 shell gap yielding a guidedbytheresultsofthecalculationsinthisworkandthat
ofRef.[5]. ThefigureisanupdatedversionofthatinRef.[6]
π(2p−4h) configuration, respectively.
using data taken from the NNDC database [7].
Spectroscopy of the even-mass mercury isotopes re-
veals a systematic trend of the intruding 0+ band head
2
(shown in Fig. 1) which minimizes in energy near the
for A ≤ 186, and the 2+ levels become close enough in
neutron mid-shell at N = 104. The excited states built
upon these configurations become yrast above Iπ = 4+ energy to mix strongly.
Large E0 components in 2+ →2+ transitions indicate
2 1
a large degree of mixing. An attempt to understand the
mixing between the bands was made by measuring this
∗ Correspondingauthor(184Hg): Liam.Gaff[email protected] E0 component in-beam in 180Hg [8] and 186Hg [9]. The
† Correspondingauthor(186Hg): [email protected] conversion coefficient of this transition can also be mea-
‡ Correspondingauthor(calc.): [email protected] sured following β decay, and there is an effort to provide
2
15 16 (a)
(a) 15 16 (b) 356.7 1000 6+→ 4+ 2+→ 0+ 8+→ 6+
13 409.3 646.0 381.8 13 611.0 14 680000 9-→ 7- sh us sh 11-→ 9- sh 50µm 10+s→h 8+
sh
445.2 14 11 sh
11 452.4 10 255.8 581.6 400
400.3 397.7 603.9 10 12 200
1010.7
9 12
948.9 480.8 754.9 542.0 0
329.1
7 548.8 551.3 8 1244.3 10 2000 (b)
5 273.5 1037.9 10 477.6 V]
571.8 708.8 6 488.9 ke 1500 400µm
6 489.2 8 1/
1126.0 8 s [ 1000
4 462.6 1193.8 418.3 6 4 597.6 424.2 6 Count 500
356.8
340.1 4
719.8 4 675.1 186.4 2 0
119.4 2 402.7
2 287.0 167.6 0 2 215.6 0 2500 (c)
534.4 2000µm
366.8 405.3 2000
0 0
1500
FIG. 2. Partial level schemes of (a) 184Hg and (b) 186Hg 1000
showingstatesofinterest. DataaretakenfromRefs.[20,21].
500
0
moreexperimentaldatainthisregion[10]. Mixingofthe 300 350 400 450 500
0+ statescanbequantifiedbycomparisonofρ(E0)2 val- Energy [keV]
ues [11, 12] which have been experimentally determined
in180Hg[13], 184Hg[14], and188Hg[15]. Asdescribedin FIG. 3. Gamma-ray spectra from the Gammasphere detec-
tors at θ ≈ 53◦, gated on the shifted (sh) component of the
a recent review on the topic [16], the microscopic shell-
4+ → 2+ transition in 184Hg at a target-to-stopper distance
model approach and the theoretical mean-field approach 1 1
of(a)50µm,(b)400µm,and(c)2000µm. Transitionswhich
canbothsuccessfullyreproducetheobservedmixing. An feed the 4+ state only have their shifted (sh) component in
analysis of α-decay hindrance factors [13, 17] indicates a coincidenc1e, while one also observes the unshifted (us) com-
smallerprolatecontributiontotheground-stateband0+ ponent of the 2+ →0+ depopulating transition.
1 1
state.
To determine the magnitude and type of deformation
of the two bands, their mixing strength, and to test the
suppressed, high-purity Ge detectors [22] arranged into
picture of shape coexistence, more precise data are re- 17 rings of constant polar angle, θ, with respect to the
quired on the absolute transition strengths between the
beam. For this experiment, 100 detectors split into 16
excitedstatesofthenucleiinthismassregion. Lifetimes
rings were in use. The K¨oln plunger device was installed
ofexcitedstatesin180,182Hghavebeenrecentlymeasured
at the target position to allow for the Recoil Distance
by this collaboration [18, 19]. To extend this knowledge
Doppler-Shift (RDDS) lifetime measurements. The dis-
andaddressthesemissingdata,lifetimemeasurementsof tance of the 11 mg/cm2-thick Au stopper foil was varied
excited states in 184,186Hg have been performed. Partial withrespecttothe0.6mg/cm2 thickSmtargetwithina
level schemes of the nuclei studied in this work are given range of 2–2000 µm and data were taken at 12 distances
in Fig. 2. for 184Hg (10 for 186Hg).
For the analysis of the data, γγ-coincidence matrices
werebuiltusingGSSORT[23]andtheROOTframework[24].
II. EXPERIMENTAL DETAILS
The data from different HPGe detectors of Gammas-
phere were grouped by rings at similar angles. In the
Excited states in 184Hg and 186Hg were popu- analysis of 184Hg, the lifetimes are obtained by taking
lated in the heavy-ion induced, fusion-evaporation reac- the weighted average of the lifetimes for different rings.
tions148Sm(40Ar,4n)184Hgand150Sm(40Ar,4n)186Hg,at Here,itwaspossibletoobtainthevelocityindependently
beam energies of 200 MeV and 195 MeV. The average for each ring, utilizing the measured Doppler shift of the
recoil velocity was v/c=1.94% and 1.90%, respectively. peaks. In the case of 186Hg, where measurement times
The primary beam was provided by the ATLAS facil- wereshorter, thelifetimeswerederiveddirectlyfromthe
ityattheArgonneNationalLaboratoryanddeliveredto weighted average of the intensities whilst the velocity is
the target position inside the Gammasphere spectrom- obtained by weighting the velocities from the individual
eter. The latter nominally consists of 110 Compton- γγ-coincidencespectraforallcombinationsofringpairs.
3
2
Typicalγγ-coincidencespectraobtainedduringtheex- ⌧(6+)=8.82(24)ps
perimentarepresentedinFig.3,inwhichitispossibleto 9.6 (a)
see the variation of intensity of the fully Doppler-shifted
9.2
component of the depopulating transition, I , with in-
sh s]
creasing distance. At the smallest distances the flight [p 8.8
times of the nuclei are very short leading to few γ rays ⌧
8.4
emitted in-flight. Consequently, when gating in the γ-γ
matrix, a low number of events are observed, an effect 8.0
present in Fig. 3 (a).
Toaccountforthevaryingmeasurementtimeatdiffer- 6000 Ish
ent target-to-stopper distances, the intensities had to be
5000
normalized. For184Hg,thedistanceswerenormalizedby (b)
4000
employing a gate on the total intensity of the 340-keV,
6+ → 4+ transition and averaging the intensities of the 3000
1 1
287-keV, 4+ → 2+ and 367-keV, 2+ → 0+ transitions 2000
1 1 1 1
at all angles. For 186Hg, all possible γγ coincidences for 1000
theyrasttransitionsuptothe10+ state, inspectrawith 0
1
θ < 90◦, were used for the normalization. The use of 2000 Ius
thecoincidencedifferentialdecaycurvemethod(DDCM) (c)
eradicates the influence of recoil de-orientation [25]. 1500
In the analysis of 184Hg, lifetimes were obtained us-
1000
ing the prescriptive coincidence DDCM [26, 27]. While
here we explain only the features of the method that
500
are required for this analysis, the full details are con-
tainedinRefs.[26,27],towhichthereaderisreferredfor 0
more information. The coincidence technique allows for 10 100 1000
the elimination of systematic errors usually introduced Distance[µm]
in RDDS-singles measurements by the unknown feeding
history from states that lie higher in energy. The decay FIG. 4. (a) Lifetime values, τ, extracted at each distance in
curves,asafunctionofdistance,x,forthisanalysiswere the sensitive region, for the 340-keV, 6+1 → 4+1 transition in
184Hg,measuredinthedetectorsatθ≈53◦. Thedashedand
constructed using gates on the shifted component of the
dotted lines show the weighted average and the associated
yrast transition directly feeding the state of interest, en-
uncertainty, respectively. (b) Normalized intensity curve for
suring a significant simplification whereby the lifetime is
the shifted component, I , fitted (solid line) with a function
determinedusingonlytheratiooftheunshifted(I )and sh
us consistingofcontinuous,piecewisesecond-orderpolynomials.
the time derivative of the fully-shifted (I ) components
sh (c) Normalized intensity curve for the unshifted component,
of the depopulating transition. I . Thecurvein(c)isproportionaltothederivative ofthat
us
in(1b2)00andIbusothcurvesarefittedsimultaneously. Accordingto
τ(x)= Ius(x) , (1) Eq1.010,0theratioofthese⌧(g4iv+1e)s=th5e.6l(i2fe0t)impseofthestate,shown
ddtIsh(x) in (a8)0.0
600 (a)
Typical fits of continuously-connected second-order
polynomials, performed with Napatau [28, 29], are illus- 400
III. RESULTS
trated in Fig. 4. The lifetime is determined at every 200
distance and should sit at a constant value. Deviations 0
Th8.e0 final weighted averages of the mean lifetimes
from this behavior indicate systematic effects, which can
of all states studied are shown in Table I along with
be identified easily with this method. The weighted av- 6.0
the transition strengths, B(E2), and absolute transi-
erage, τav, is taken of the points inside of the sensitivity tiops]na4.l0quadrupole moments, |Q |, of the depopulating
region, i.e. where the derivative of the decay curve is [ t
largest. yr⌧as2t.0transitions. Transition quadrupole mome(nbt)s, Qt,
are related to the B(E2) values assuming a rotating
staLtiefewtiemreessimofiltahrleyydreatsetrmstianteedswoifth18t6hHegexucpepttoiotnheof1t0h+1e quad0r.0up0olede1f0o0rmed2n0u0cleus30u0singt4h0e0rota5t0io0nalm60o0del:
Distance[µm]
a4+1ndst2a+1te.→In0t+1histrcaanses,ittiohnesneraenrddeoruedbleittopfrtohbele4m+1at→ic2t+1o B(E2;I →I(cid:48))= 165π(cid:104)I020|I(cid:48)0(cid:105)2Q2t, (2)
use the simple “gate from above” method. Instead, the
method of “gating from below” [30] was used. The cor- where (cid:104)I020|I(cid:48)0(cid:105) is a Clebsch-Gordan coefficient and
responding τ plot for the 4+ state is found in Fig. 5. I(cid:48) =I−2.
1
The intensities involving the 6+ and 4+ analysis were In184Hg,thesevenindependentmeasurementsofeach
1 1
corrected for a contamination from the 15− →13− tran- ofthelifetimesoftheeven-spinyraststatesarepresented
2 2
sition, using a gate on the 13− →11− transition. in Fig. 6, as a function of the “ring” angle at which they
2 2
2
⌧(6+)=8.82(24)ps
9.6 (a)
9.2
s]
[p 8.8
⌧
8.4
8.0
I
6000 sh
5000
(b)
4000
3000
2000
1000
0
I
2000 us
(c)
1500
1000
500
0
10 100 1000
Distance[µm]
4
1200 Ius
1000 ⌧(4+1)=5.6(20)ps 10 24++ →→ 02++ 6+ → 4+ 108++ →→ 68++
800
9
600 (a)
400 8
200 b]
e 7
8.00 | [𝑄𝑡
| 6
6.0
ps]4.0 5
[
⌧2.0 (b) 4
0.00 100 200 300 400 500 600 3
180 182 184 186
Distance[µm]
Mass number, A
FIG.5. (a)Thenormalizedintensityoftheunshiftedcompo-
FIG. 7. (Color online) Experimental |Q | values, extracted
nent of the 4+ → 2+ depopulating transition in 186Hg. The t
1 1 from measured lifetimes, for yrast transitions in mercury nu-
fittedlineisasinFig.4. (b)Lifetimevaluesextractedateach
clei as a function of mass number, A. Data for A=180,182
distance in the sensitive region. The solid and dashed lines
aretakenfromRef.[18]andthe10+ →8+ valueforA=184
show the weighted average and the associated uncertainty.
is taken from Ref [33]. Some markers are slightly offset from
integer A values to maintain clarity.
45
IV. DISCUSSION
40 𝜏(2+) = 35.7(15)
35
A. Yrast states in 184Hg and 186Hg
30
s]
p
e, [Lifetim𝜏 122505 𝜏(4+) = 30.2(10) 𝜏𝜏𝜏𝜏((((2468++++)))) sispoiTtnorapynerssaistwtiohnsetaraleteq1su8aa0drre≤upgioAvleen≤mion1m8F6e.ingt.sT,h7|Qefto2|,r+ftoshrteatthmeeserescvhueornwy-
no strong variation in |Q | with mass number, whereas
t
10 𝜏(6+) = 8.7(4) the collectivity of the 4+ states reduces with increasing
mass number. This can be compared to the energy level
5 𝜏(8+) = 3.19(14)
systematics in Fig. 1 where the energy of the intruder
0 states reaches a minimum at A=182.
0 20 40 60 80 100 120 140 160 180
Polar angle in Gammasphere [deg]
FIG. 6. (Color online) Lifetimes of yrast states in 184Hg as B. The 9 state in 184Hg
3
a function of angle. The weighted average and 1σ uncer-
tainty, calculated from the weighted standard deviation, for
each state are denoted by the solid and dashed lines, respec- After the even-spin, positive-parity yrast band in
tively. Please note, the τ(4+) values are offset by 5◦ on the 184Hg, the most populated band is the odd-spin rota-
x-axistomaintainclarityandsomeindividualerrorbarsmay tional band built upon the I = 5 state, observed at
be smaller than the marker size.
1.848MeV,whichbecomesyrastataround4MeV.Anal-
ogous bands have been observed in the neighboring iso-
topes, specifically 178Hg [34] and 180Hg [35] and their
structure discussed in terms of octupole correlations.
were determined in Gammasphere. Lifetime measurements of states in this band in 182Hg
have been performed and a consistency in the structure
It is worth noting that the new lifetime for 4+ state of these states has been extended to 184Hg using the en-
1
in 186Hg is smaller than the previously measured value ergy displacement of states differing by 3h¯ [19]. The
fromRef.[31]. Thediscrepancyinthefirstmeasurement quadrupolemomentmeasuredhereforthe9 →7 tran-
3 3
is likely due to complications that the authors encoun- sition in 184Hg, |Q | = 5.6(7) eb, is similar to that of
t
tered in resolving the doublet and subsequent assump- the even-spin yrast band, |Q | (cid:39) 7.7 eb, although it is
t
tions which were made. smaller than those measured in the lighter isotopes.
5
TABLE I. Properties of the states investigated in this study. The uncertainties presented on τ represent the 1σ statistical
av
error and include an additional systematic uncertainty which accounts for the choice of the fitting function and relativistic
effects [27], typically ≤ 3%. Gamma-ray energies (E ) and branching fractions (b.f.) of the depopulating γ-ray transitions
γ
(corrected for internal conversion), as well as spin and parity (Iπ) values, are taken from Refs. [20, 21]. In cases where only
one depopulating transition is observed b.f. is assumed to be equal to unity. τ values from Refs. [31–33] are shown for
prev
comparison.
Iπ (¯h) E (keV) b.f. τ (ps) τ (ps) B(E2)↓ (W.u.) |Q | (eb)
γ av prev t
184Hg 2+ 366.8 1 35.7(15) 30(7) 52(2) 4.04(8)
1
4+ 287.0 0.959(4) 30.2(10) 32.8(34) 191(6) 6.46(11)
1
6+ 340.1 1 8.7(4) 8.1(31) 308(15) 7.81(19)
1
8+ 418.3 1 3.19(14) 2.9+1.1 309(13) 7.65(17)
1 −1.6
9 329.1 0.65(16) 12.1(8) − 169(40) 5.6(7)
3
186Hg 2+ 405.3 1 24(3) 26(4) 47(6) 3.9(2)
1
4+ 402.7 0.93(2) 5.6(20) 13(4) 200(70) 6.6(12)
1
6+ 356.8 1 9.1(4) 7(3) 231(10) 6.82(15)
1
8+ 424.2 1 4.5(3) ≈4 202(14) 6.2(2)
1
10+ 488.9 1 1.9(2) − 238(25) 6.7(4)
1
C. Two-state mixing calculations calculation will be discussed.
Low-lying levels in 184,186Hg have also been inter-
preted assuming the mixing of spherical and deformed
Inordertoshedlightonthepropertiesofthecoexisting
states [39], while an alpha-plus-rotor model has been
structures in these light mercury isotopes, phenomeno-
usedtoextractspin-dependentinteractionstrengthsand
logical two-band mixing calculations have been carried
mixing amplitudes in 182,184,186Hg [5]. The latter study
out using the assumption of a spin-independent interac-
predicted a contribution of the more-strongly deformed
tion between two rotational structures. In the calcula-
structuretothe2+statein182Hgof76%,comparingvery
tions, the variable moment of inertia (VMI) model [36] 1
well to the value of 71% obtained in this work. We note
was used to fit known level energies of rotational bands
here that this contribution drops to only 2.3% for the
built upon the first two 0+ states, up to and including
same state in 188Hg. The corresponding isotones in the
Iπ =10+ and 4+ for yrast and non-yrast bands, respec-
platinumnucleiwerealsointerpretedrecentlyusingtwo-
tively. Employing the method of Lane et al. [37], one
band mixing calculations and qualitatively similar con-
can derive the wave-function amplitudes of the two con-
clusions were drawn regarding a strong degree of mixing
figurations from the mixing strengths. These are shown
for the low-spin states [38, 40].
in Table II along with the mixing strength, V, for each
It is possible to determine the transitional quadrupole
isotope and the band-head energies of the two configura-
moment of the unperturbed I →I−2 transitions in the
tions.
normal (n) and intruder bands (i), Q , using an aver-
I
Similar mixing calculations have previously been per- age of the moment of inertia of the two states, J(cid:48), such
formed for 180,182,184Hg [38]. The present results for I
that [36]
180,182Hg differ due to the inclusion of non-yrast states
identified in recent studies [8, 10], which provide addi- (cid:113) (cid:112)
tional constraints on the calculations. The calculation QI =k JI(cid:48) =k (JI +JI−2)/2, (3)
for 184Hg presented in Table II is essentially identical
to that in Ref. [38]. It places the I = 2 member of where J is the moment of inertia for a pure state with
I
the intruder band just 5 keV above the corresponding spin I, extracted from the fit. An evaluated value of the
state in the normal band before mixing. This leads to constant, fitted to data in the neutron-deficient A=170
almost complete mixing between the states, producing a region,k =45(2)ebkeV−1/2 [41],wasusedinthisstudy.
first-excited2+ statewhichcomprises51%ofthenormal Combining knowledge of the wave-function amplitudes
configuration and 49% of the intruder configuration. An withtheintrinsicquadrupolemomentsofthepurestates,
alternative calculation in which the order of these states Q , it is possible to extract the B(E2;I →I−2) values
I
is reversed was also performed and yielded very similar ofthemixedstates[37]. Therelativesignoftheintrinsic
parameters to those presented in Table II. In this sce- quadrupole moments of the two configurations must be
nario the unmixed states are nearly degenerate, so the assumed to be positive or negative and was found to be
degree of mixing is even greater. The resulting B(E2) best reproduce the data when positive. This feature has
values(presentedlaterinFig.8)arenotsignificantlydif- beennotedinpreviouscalculations[37,42]andisatodds
ferent,soinwhatfollowsonlytheresultsfromtheformer with what is expected from the rotational model when
6
500
TABLE II. Wave-function amplitudes of the normal configu-
ration, αI, at each spin, I, and spin-independent interaction 400
strengths between members of the normal (n) band and in-
truder (i) band, V = |(cid:104)I |V |I (cid:105)| (equal for all values of I), 300
n I i
calculated using the model described in the text. The ex-
tracted unperturbed band-head energies are denoted by En 200
0
and E0i for the normal and intruder bands, respectively. 100 180Hg
Nucleus I [h¯] |αI| 0
0 0.9799 500
V =82.1 keV
2 0.7722
400
4 0.1571
180Hg En =16.7 keV
0 300
6 0.0817
8 0.0546 200
E0i =403.3 keV 10 0.0408 W.u.] 100 182Hg
0 0.9606 2) [
V =89.4 keV -(cid:2) 0
2 0.5382 (cid:1)
182Hg E0n =25.9 keV 46 00..11708013 B(2;(cid:1)(cid:2)(cid:3) 450000
Ei =309.0 keV 8 0.0697 300
0
10 0.0534
200
V =84.7 keV 0 0.9725 100 184Hg
2 0.7172
4 0.2014 0
184Hg En =20.4 keV
0
6 0.1025
400
8 0.0679
Ei =353.4 keV
0 10 0.0506 300
0 0.9915 200
V =66.3 keV
2 0.9506
100 186Hg
4 0.2604
186Hg En =8.9 keV
0
6 0.0978 0
8 0.0587 2 4 6 8 10
E0i =505.4 keV (cid:2)(cid:3)[(cid:1)]
10 0.0416
0 0.9954
V =76.3 keV FIG. 8. (Color online) Experimental B(E2) values measured
2 0.9884 in this work (points) plotted as a function of spin and com-
4 0.8933 pared to those extracted from the mixing calculations (solid
188Hg En =7.8 keV
0 black line, dashed line represents the uncertainty in the con-
6 0.2467
stant, k). For reference, the intra-band B(E2) values cal-
8 0.1095
Ei =791.8 keV culated for the pure unperturbed normal (red) and intruder
0
10 0.0691 (blue) bands are also shown. Data for 180,182Hg are taken
from Ref. [18] and the point at I = 10 h¯ in 184Hg is taken
from Ref. [33].
two bands with the same K quantum number have an
oppositesignofdeformation,i.e. anoblateandaprolate B(E2;2+ →0+)valuesaresimilarineachisotope, while
1 1
band. the B(E2;4+ →2+) varies and, in each case, is not con-
1 1
The results of the calculations for the yrast sequences sistent with a transition within either of the pure bands.
arecomparedinFig.8withthoseextractedfromthelife- The low B(E2;2+ → 0+) value is interpreted as being
1 1
times measured in this work. Good agreement is found due to a transition between the weakly-deformed oblate
forthemajorityofyrasttransitions,evenforcaseswhere 0+ ground state and a more strongly-deformed prolate
the states are strongly mixed. The observed discrep- 2+ state[18]. However,theadmixtureofthenormaland
ancies for I ≥ 6 in 186Hg may indicate a breakdown intruderconfigurationsforthe0+ and2+ statesisunique
of the simple two-band picture as other structures be- ineachisotopeandasimilarB(E2)valueisnotnecessar-
gin to influence the yrast states [43, 44]. For the nu- ilyindicativeofsimilarstructuresacrossthemassrange.
clei presented in Fig. 8, the experimental and calculated InFig.9,calculatedB(E2)valuesareplottedasafunc-
7
-200 0 200 400 ∆𝐸2 [keV] tinct differences in deformation for the assigned normal
350
182 184 𝐼 = 2+ and intruder bands could be shown. The lifetime of the
300 180 186 𝐼 = 4+ 4+ statein186Hgwasfoundtobeshorterthantheprevi-
1
𝐼 = 6+ ously measured value. Lifetimes of the 9 state in 184Hg
u.] 250 and the 10+ state in 186Hg have been m3easured for the
.W
-2) [𝐼 200 fivirosutstimmeea.suTrehmeeontthse,rwlhifielteimthees uanrececrotanisnisttyenctouwldithbeprree--
→ 150 duced significantly. These more precise lifetime values,
𝐼
2; with those of Ref. [18], have been a vital input to the
(𝐵𝐸 100 analysisofCoulombexcitationexperimentsattheREX-
ISOLDE facility [45, 46].
50 Rotational bands built upon the first two Iπ = 0+
states have been considered in terms of a two-state mix-
0
0 200 400 600 800 ing model and the mixing amplitudes of the two con-
Pure-band energy difference, ∆𝐸 = 𝐸i - 𝐸n [keV] figurations extracted as a function of spin. It is ob-
0 0 0
served that, while the ground state remains composed
FIG. 9. (Color online) Representative B(E2;I+ →(I−2)+) ofpredominantlyoneconfiguration,namelytheassumed
1 1
valuesasafunctionoftheenergyseparationofthepureband- weakly-deformed normal structure, in all of the even-
heads, ∆E0. The curves for I =2,4,6 are calculated assum- mass mercury isotopes considered (180 ≤ A ≤ 188), the
ing the mixing strength and moments of inertia are equal to first excited 2+ state changes dramatically in its com-
that extracted for 184Hg. As a guide, the energy difference
position. This is in contrast to a na¨ıve interpretation
between the pure 2+ states, ∆E2, under this assumption is emanating from the systematics of the 2+ level energies
alsoshown. Thedashedverticallinesshowtheextracted∆E
0 and B(E2;2+ → 0+) values, which are both strikingly
values for each of the four isotopes.
similar across the mass range.
tion of the energy difference between the pure, unmixed
ACKNOWLEDGMENTS
0+ band heads. The parameters used in the calculations
are those of 184Hg, but since they are not too dissimilar
The authors would like to thank the operators of the
for all of the isotopes, the curves can be considered rep-
ATLAS facility at ANL for providing the beams. This
resentative of the mercury isotopes around the neutron
work was supported by the United Kingdom Science
mid-shell. The four isotopes compared have ∆E values
0
and Technology Facilities Council, by the German DFG,
intherangeof250–550keV,markedbytheverticallines
grantNo. DE1516/1-1,bytheAcademyofFinland,con-
in Fig. 9. One clearly observes that above 200 keV the
B(E2;2+ →0+)valuesremainconstant,eventhoughthe tract number 131665, by the German BMBF, grant No.
square o1f the m1 ixing amplitude of the I = 2 states, α2, 06 KY 7153, by US Department of Energy, Office of Nu-
2 clearPhysics,underContractNo. DE-AC02-06CH11357
drops from 0.80 at 200 keV down to 0.015 at 1 MeV. In
contrast,theB(E2;4+ →2+)valuesvarysignificantlyin as well as Grant No. DE-FG02-94ER40834, by FWO-
1 1 Vlaanderen (Belgium), by GOA/2010/010 (BOF KU
the range of interest and are much more sensitive to α ,
2
while the B(E2;6+ →4+) values become more sensitive Leuven), by the Interuniversity Attraction Poles Pro-
1 1 gramme initiated by the Belgian Science Policy Office
at larger band-head energy differences.
(BriX network P7/12), by the European Commission
within the Seventh Framework Programme through I3-
ENSAR (contract no. RII3-CT-2010-262010) and by a
V. SUMMARY AND CONCLUSIONS
Marie Curie Intra-European Fellowship of the European
Community’s7thFrameworkProgrammeundercontract
Lifetimes of excited states have been measured by em- number (PIEF-GA-2008-219175). L.P.G. acknowledges
ployingtheRDDStechnique. YraststatesuptoIπ =8+ support from FWO-Vlaanderen (Belgium) as an FWO
in 184Hg and Iπ =10+ in 186Hg have been studied. Dis- Pegasus Marie Curie Fellow.
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