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PHYSICAL REVIEW C 79, 034307 (2009)

E0 transition strengths from X(5) to the rigid rotor

J. Bonnet, A. Krugmann, J. Beller, and N. PietrallaInstitut fur Kernphysik, Technische Universitat Darmstadt, Darmstadt, Germany

R. V. JolosJoint Institute for Nuclear Research, Dubna, Russia

(Received 18 August 2008; published 9 March 2009)

Relative and absolute E0 transition strengths [ρ2(E0)] on the transitional path between the X(5) solution andthe rigid rotor limit have been evaluated within the framework of the confined β soft (CBS) rotor model. RelativeE0 transition strengths between the β-vibrational band and the ground state band decrease with increasing angularmomentum for a given potential stiffness. Absolute E0 transition strengths drop with increasing potential stiffnesstoward zero in the rigid rotor limit. Interband E0–E2 correlation observables are examined to estimate ρ2(E0)as a function of the R4/2 ratio.

DOI: 10.1103/PhysRevC.79.034307 PACS number(s): 21.60.Ev

I. INTRODUCTION

As a function of particle number, heavy nuclei canundergo rapid quantum phase transitions with respect totheir deformation. These shape phase transitions have beenstudied experimentally and theoretically for many years [1,2].This discussion intensified when Iachello proposed analyticalsolutions of the geometrical Bohr Hamiltonian near thecritical points of various nuclear shape phase transitions[3–5]. The solutions labeled E(5) and X(5) have attracteda great deal of interest and initiated extensive research.Besides the characteristic excitation energy ratios R4/2 =E(4+

1 )/E(2+1 ) = 2.20 [for E(5)] and R4/2 = 2.90 [for X(5)]

or the evolution of the E2 transition rates as a function of spinalong the ground state band, the properties of the quadrupolecollective excited 0+ states, including the E2 decays, areconsidered as the identifying signatures of these models.Several experimental studies [6–9] have dealt with thesesignatures.

Another prominent decay mode, in particular for 0+ states,is the E0 transition. Von Brentano et al. [10] pointed out thatin the interacting boson model, the E0 transition rates to theground state increase at the shape phase transitional point andcontinue to have larger values up to the SU(3) or the O(6)dynamical symmetries.

E0 transition strengths have not yet been reported for theX(5) solution. In this article, we study E0 transition ratesin the confined β soft (CBS) rotor model [11]. This modelinterpolates between the X(5) solution and the rigid rotor limitas a function of one structural parameter. Using the CBS rotormodel, we have obtained relative E0 transition rates betweenbands with one, two, or three nodes in the β wave function.We will also study their evolution toward the rigid rotorlimit.

The next section defines the E0 transition rates. The CBSrotor model will be briefly introduced in Sec. III. Section IVreports on E0 transition rates in the X(5) limit and its evolutiontoward the rigid rotor limit as a function of other collectiveobservables such as the R4/2 ratio or B(E2) values. A summarywill be given in Sec. V.

II. E0 TRANSITIONS IN QUADRUPOLE COLLECTIVEMODELS

For this introduction, we partly follow the review by Woodet al. [12]. The E0 transition rate is defined by

1

τ (E0)= ρ2

if (�K + �Li+ · · · + �IP ), (1)

where

ρ2if (E0) = |〈�final|m(E0)|�initial〉|2

(eR2)2, (2)

with the E0 transition operator

m(E0) =∫

ρ(�r)r2d 3r (3)

obtained in the long wavelength approximation [1]. Subse-quently, we will consider prolate axially quadrupole-deformednuclei with the intrinsic surface parametrization

R(�) = R

(1 − α2

20

4π+ α20Y20(�)

), (4)

where we have used the constant-volume constraint. R =1.2A

13 fm is the nuclear radius, and α2µ are the intrinsic

quadrupole deformation parameters. In the homogeneous-charge density approximation

ρ(�r) = ρ0 = Ze4π3 R3

, (5)

one obtains

m(E0) = 3

5ZeR2

(1 + 5

4πα2

20 + 10

7π

√5

16πα3

20 + · · ·)

(6)

from the integration of Eq. (3) within the limits given byEq. (4). Tabulations of the “electronic” factors �i in Eq. (1)are given by Bell et al. [13]. To lowest order in α2, the E0operator is in general obtained as [12]

T (E0) = 3

5ZeR2

(1 + 5

4π�µ|α2µ|2

). (7)

0556-2813/2009/79(3)/034307(7) 034307-1 ©2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevC.79.034307

BONNET, KRUGMANN, BELLER, PIETRALLA, AND JOLOS PHYSICAL REVIEW C 79, 034307 (2009)

The constant term in Eq. (6) cannot induce transitionsbetween orthogonal states. Therefore, to lowest order in thedeformation parameter, nuclear E0 transitions originate foraxially symmetric quadrupole deformation from the part

T (E0)tr = 3

4πZeR2 β2 (8)

due to∑

µ |α2µ|2 = |α2,0|2 = β2 for axial symmetry with γ =0. Data on E0 transitions have been reviewed by Wood et al.[12]. The E0 transition operator is similar in structure to theβ-dependent part of the E2 transition operator

T (E2)K=0 = 3

4πZeR2β , (9)

again in the axially symmetric case with γ = 0. To proceed,we have to specify the nuclear wave functions in Eq. (2).

III. CBS ROTOR MODEL

In this article, we study the evolution of E0 transitionstrengths for axially symmetric quadrupole deformation fromIachello’s X(5) solution [4] toward the rigid rotor limit withinthe confined β soft rotor model. The CBS rotor modelrepresents an approximate analytical solution to the BohrHamiltonian [14]

H = − h2

2B

[1

β4

∂

∂ββ4 ∂

∂β+ 1

β2 sin 3γ

∂

∂γsin 3γ

∂

∂γ

− 1

4β2

∑k

Q2k

sin2(γ − 2

3πk)]

+ V (β, γ ) (10)

in the quadrupole shape parameters β and γ . Assuming aseparable potential V (β, γ ) = u(β) + v(γ ), the wave equationapproximately separates into

�(φ, θ, ψ, β, γ ) = ξL(β) ηK (γ )DLM,K (θi), (11)

where DLM,K denotes the Wigner functions with θi being the

Euler angles for the orientation of the intrinsic system. ηK

denotes the appropriate wave function in γ . For sufficientlyaxially symmetric prolate nuclei, one might consider a steepharmonic oscillator in γ [4]. ξL(β) describes the part of thewave function depending on the deformation variable β. Theapproximate separation of variables [4] leads to the differentialequation

− h2

2B

[1

β4

∂

∂ββ4 ∂

∂β− 1

3β2L(L + 1) + u(β)

]ξL(β)

= E ξL(β) (12)

for ξL(β). It can be considered as the “radial” equation in thespace of quadrupole deformation parameters. It contains theangular momentum dependence through the centrifugal term.The CBS rotor model assumes for prolate axially symmetricnuclei an infinite square well potential u(β), with boundaries atβM > βm > 0. For this potential, the differential equation (12)is analytically solvable. The ratio rβ = βm/βM parametrizesthe width of this potential, that is, the stiffness of the nucleus inthe β degree of freedom. For rβ = 0, the X(5) limit is obtained

with large fluctuations in β. The rigid rotor limit withoutfluctuations in β corresponds to rβ → 1. The quantizationcondition of the CBS rotor model is

Qrβ

ν(L)(z) = Jν(L)(z)Yν(L)(rβz) − Jν(L)(rβz)Yν(L)(z) = 0, (13)

with Jν and Yν being Bessel functions of first and secondkind of irrational order ν =

√[L(L + 1) − K2]/3 + 9/4. For

a given structural parameter rβ and any spin value L thesth zero of Eq. (13) is denoted by z

rβ

L,s . The full solution ofEq. (12) with the aforementioned choice of the CBS square-well potential is then given as

ξL,s(β) = cL,s β− 32

[Jν

(zrβ

L,s

β

βM

)

+ Jν

(rβz

rβ

L,s

)Yν

(rβz

rβ

L,s

)Yν

(zrβ

L,s

β

βM

)]. (14)

For convenience, we occasionally use the relative deformationparameter b = β

βM∈ [rβ, 1]. The normalization constant cL,s

of the wave function (14) is given by the normalizationcondition

1 =∫ βM

βm

β4[ξL,s(β)]2 dβ. (15)

The eigenvalues of Eq. (12) are obtained as

EL,s = h2

2Bβ2M

(zrβ

L,s

)2. (16)

The CBS rotor model describes well the evolution of low-energy 0+ bands [11], ground bands of strongly deformednuclei [15], and the dependence of relative moments of inertiaas a function of spin in deformed transitional nuclei [16].

IV. E0 TRANSITION RATES

Electromagnetic transition rates can be calculated from thewave functions given above. For E0 transitions, we obtain withEqs. (2) and (8)

ρ2if (E0) =

(3Z

4π

)2

|〈ψf |β2|ψi〉|2 (17)

=(

3

4π

)2

Z2β4M |〈ξf |b2|ξi〉|2 (18)

between states that only differ in the β-dependent part of thewave function, i.e., 〈Df |Di〉 = 〈ηf |ηi〉 = 1.

As a function of the choice of the potential, i.e., as afunction of the nuclear deformation (βM ) and the nuclearstiffness against centrifugal stretching (rβ), we calculate theE0 transition rates as

ρ2if (E0) =

(3

4π

)2

Z2β4M

[∫ 1

rβ

ξ ∗f (b)b2ξi(b)b4db

]2

(19)

= qE0(βM )[m2,rβ

(Ji → Jf )]2

, (20)

034307-2

E0 TRANSITION STRENGTHS FROM X(5) TO THE . . . PHYSICAL REVIEW C 79, 034307 (2009)

FIG. 1. Partial level schemes for low-energy K = 0 bands obtained with the CBS rotor model with relative E0 transition strengthsρ2(E0, Ji → Jf )/ρ2(E0, 0+

2 → 0+1 ). Left: for X(5) [rβ = 0], R4/2 = 2.90. Right: for rβ = 0.2, R4/2 = 3.10.

with the matrix element

mk,rβ(Ji → Jf ) =

∫ 1

rβ

ξi(b) bkξf (b) b 4db (21)

of kth order in the reduced deformation variable b, which onlydepends on the nuclear stiffness parameter rβ . The size ofthe maximum deformation βM appears in the scaling constantqE0(βM ). The expression for B(E2) transition strengths forK = 0 transitions follows in this notation as

B(E2; Ji → Jf ) = qE2(βM )(C

Jf 0Ji020

)2[m1,rβ

(Ji → Jf )]2

,

(22)

with the scaling constant

qE2 (βM ) =(

3

4π

)2

β2MZ2e2R4. (23)

CJf 0Ji020 denotes a Clebsch-Gordan coefficient. Deviations from

the homogeneous-charge density approximation might betaken into account to lowest order by an effective chargeZe → qEλZe, with qEλ ≈ 1 for electric transitions withmultipolarity λ.

A. Stiffness-dependence of relative E0 transition strengths

First we derive the E0 transition strengths in the X(5)solution, i.e., for the structural parameter rβ = 0. The width ofthe potential well, βM , is considered as the sole free parameter,

and thus we normalize the E0 transition rates relative tothe 0+

s=2 → 0+s=1E0 transition. The left-hand side of Fig. 1

shows these relative E0 transition rates for rβ = 0, which isX(5). Some of the values are also tabulated in Table I. Wenote that the ρ2(E0) values are similar in size for all E0transitions that have s = 1, while s > 1 E0 transitions aresuppressed by an order of magnitude. The right-hand side ofFig. 1 shows the relative E0 transition rates normalized againto the 0+

s=2 → 0+s=1 transition for the CBS model parameter

rβ = 0.2, which corresponds to an intermediate situation withan excitation energy ratio R4/2 = 3.10. Again, we observe anapproximate selection rule for E0 transitions between bandswith s > 1.

B. Angular momentum dependence ofrelative E0 transition strengths

Next we compare relative E0 transition strengths fordifferent width parameters rβ and different angular momentaJ for transitions between states with s = 2(β band) ands = 1 (ground state band). The results are shown in Fig. 2.Here, the E0 transitions are again normalized to the 0+

s=2 →0+

s=1 transition for every width parameter rβ , separately. Forincreasing angular momentum, the model predicts a decreasingE0 transition rate. For X(5), the 10+

s=2 → 10+s=1E0 transition

rate is reduced by 38% relative to the corresponding 0+s=2 →

0+s=1 transition. Notice also, that for rising potential stiffness,

the relative E0 transition strengths rapidly approach a constant

034307-3


0 2 4 6 8 10

1.0

0.8

0.6

0.4

0.2

ρ2 (E0)

J+ 2

J+ 1/ρ

2 (E0)

0+ 2 0

+ 1

J

RR

X(5)

rβ =0.5...0.90.40.30.20.10

0.0

FIG. 2. (Color online) Relative E0 transition strengths for variousnuclear stiffnesses with rβ ranging from 0 to 0.9 in steps of 0.1.

value as a function of spin, which is the typical behavior of therigid rotor limit.

To understand this dependence of the E0 transition rateson angular momentum and potential stiffness, we considerthe E0 matrix elements of Eq. (21) in more detail. Figure 3illustrates the wave functions for states of angular momentumJ = 0 (left) and J = 10 (right) for two different stiffnessparameters, rβ = 0 (top) and rβ = 0.5 (bottom). All functionsof the relative deformation parameter b contributing to theE0 transition matrix elements are plotted. The matrix elementrepresents the overlap integral of the initial and final wavefunctions, weighted by the square of the relative variable,b2. Because of the orthogonality of the wave functions, theoverlaps of the initial and final wave functions are equal inabsolute size and opposite in sign to the left and to the rightof the node of that wave function with s = 2. Consequently,the partial E0 matrix elements on either side of that node

have opposite signs while the respective overlap integrals areweighted by b2.

The lengths of the arrows in Fig. 3 denote the absolutevalue of the E0 matrix element on the corresponding side ofthat node. The difference between these absolute values givesthe total matrix element. The positions of the arrows indicatethe root mean square of b weighted by the wave function’soverlap on either side of that node.

In the case of small potential stiffness (rβ = 0, top), thewave functions for 0+ states and 10+ states differ substantiallydue to centrifugal stretching. Therefore, the absolute valuesfor the partial E0 matrix elements between the node and thepotential boundaries (length of the arrows) strongly depend onthe wave function’s angular momentum. As a consequence, thetotal E0 matrix element (the difference between the length ofthe two arrows) varies substantially with angular momentumfor the case of low potential stiffness.

In the case of high stiffness (rβ = 0.5, bottom) the wavefunctions of the 0+ and 10+ states do not differ much anymore,because centrifugal stretching is strongly suppressed in the stiffpotential. As a consequence, the total E0 matrix element (thedifference between the length of the two arrows) is almostconstant for wave functions with any moderate spin quantumnumber. The spin dependence of the E0 transition rates isnegligible in this case, as seen in Fig. 2 for the lines labeledrβ = 0.5 · · · 0.9.

C. Evolution of absolute E0 rates

Now we focus on absolute transition strengths. Figure 4shows E0 transition strengths for different angular momentadepending on the width parameter rβ . All transition strengthsare normalized to the transition 0+

2 → 0+1 for X(5), i.e., we

plot the quantity

ρ2(E0, Jf → Ji ; rβ)

ρ2(E0, 0+2 → 0+

1 ; rβ = 0)=

[m2,rβ

(Ji → Jf )

m2,0(0+2 → 0+

1 )

]2

FIG. 3. (Color online) Behavior ofE0 transition strengths between wavefunctions for J = 0+ and J = 10+ forthe transition s = 2 → s = 1 for differentvalues of rβ . Upper: rβ = 0 [X(5)]. Lower:rβ = 0.5.

034307-4


1.00.80.60.40.20.0

1.0

0.8

0.6

0.4

0.2

8+

6+

10+

4+

2+

ρ2 (E0

) J+ 2 J

+ 1

/ρ2 (E

0) 0

,X(5

)

rβ

0+

0.0

FIG. 4. (Color online) E0 transition rates from s = 2 to s = 1between states with angular momenta J = 0–10 as a function of rβ .The theoretical values are normalized to the 0+

2 → 0+1 E0 transition

strength in X(5), ρ20,X(5) ≡ ρ2(E0, 0+

2 → 0+1 ; rβ = 0).

for a fixed deformation βM . For rβ approaching unity, all E0transition strengths drop to zero. The evolution of this decreaseas a function of rβ depends on angular momentum, as discussedabove. E0 transition strengths from 0+ states monotonicallydecrease with increasing rβ . E0 transition strengths betweenstates with larger angular momenta are significantly effectedonly above a certain value of rβ , which depends on angularmomentum. This behavior originates in the phenomenon ofcentrifugal stretching for states at higher spins, which causesfor higher rotational frequencies a lower sensitivity to thepotential at small values of β [11].

For higher stiffness (rβ >∼ 0.5) the decrease of E0 transitionrates as a function of rβ is entirely dictated by the narrowing ofthe square-well potential. Independent of low enough spin, theE0 transition rates decrease almost linearly with increasingrβ .

Table I provides an overview over the values of E0transition strengths for the width parameters of rβ = 0, 0.2,and 0.4. To present some global values, the E0 transitionstrengths in Table I are divided by the appropriate powersof the nuclear charge Z and the scale of its deformation, βM .

A comparison of E0 transition strengths, predicted by theCBS rotor model, to experimental values is given in Table II.

TABLE II. Comparison of experimental values for ρ2(E0) for152Sm [17] with the CBS rotor model with stiffness parameter rβ =0.14. The second line contains absolute predictions (see text). Thevalues given in the third line are normalized to the E0 strength of the0+

2 → 0+1 transition in 152Sm.

Transition 0+2 → 0+

1 0+3 → 0+

1 0+3 → 0+

2

ρ2(E0) experiment 0.058(6) 0.0007(4) 0.022(9)ρ2(E0) CBS from E2 0.3041 0.0114 0.3534ρ2(E0) CBS renormalized 0.058 0.0022 0.0674

Data on 152Sm, for which the strengths of the E0 transitionrates between the 0+

1 , 0+2 , and the 0+

3 states are known, areused as examples and were taken from Ref. [17]. The CBSvalues were calculated using the stiffness parameter rβ = 0.14obtained in Ref. [11] for a good reproduction of the levelscheme of 152Sm.

Reproduction of the 3.47(7) B(E2, 0+1 → 2+

1 ) = 3.47 ±7e2 b2 results in a width parameter βM = 0.46 and an averagedeformation of 〈β〉 = 0.29. This parameter-free absoluteprediction of the 0+

2 → 0+1 E0 transition strength for 152Sm

coincides with the data within a factor of about 5 when noeffective charges are considered, qE0/qE2 ≡ 1.

Data and theory, furthermore, agree on the approximateselection rule s = 1, which suppresses the 0+

3 → 0+1 E0

transition by one order of magnitude. Normalizing the theoryto the 0+

2 → 0+1 transition in the spirit of using an effective E0

charge yields a good description for the E0 decay rates of the0+

3 state. This is quite satisfactory.

V. INTERBAND E2-E0 CORRELATION OBSERVABLES

Finally we discuss observables that relate E0 and E2transition strengths between the β band and the ground stateband. The Z-independent quantity X was introduced in the1960s by Rasmussen [18]. He defined the ratio of the E0transition strength ρ2(E0, 0+

2 → 0+1 ) and B(E2, 0+

2 → 2+1 )

value between the β-band and the ground state band, i.e.,

XRasm = ρ2(0+2 → 0+

1 )e2R4

B(E2, 0+2 → 2+

1 ), (24)

TABLE I. E0 transition strengths in convenient units, see Eq. (18), for rβ = 0, 0.2, and 0.4.

Transition 103 ρ2(E0)Z2β4

MJf → Ji

s = 2 → s = 1 s = 3 → s = 1 s = 3 → s = 2

X(5) rβ = 0.2 rβ = 0.4 X(5) rβ = 0.2 rβ = 0.4 X(5) rβ = 0.2 rβ = 0.4

0+ 1.81 1.70 1.31 0.092 0.048 0.013 2.10 1.98 1.522+ 1.73 1.68 1.30 0.093 0.059 0.015 2.05 1.96 1.524+ 1.60 1.59 1.30 0.092 0.076 0.019 1.95 1.92 1.526+ 1.45 1.45 1.28 0.088 0.084 0.027 1.84 1.83 1.518+ 1.31 1.31 1.23 0.083 0.083 0.037 1.72 1.72 1.49

10+ 1.19 1.20 1.16 0.078 0.078 0.047 1.61 1.61 1.46

034307-5


within the rigid rotor model and obtained

XRasm = 4β2RR, (25)

where βRR denotes the deformation of the nucleus in the rigidrotor limit. This result has led to a wide-spread misconceptionabout the evolution of E0 transition rates in deformed nuclei.Equation (25) shows the increase of the E0/E2 ratio betweenthe ground state band and β band as a function of thenuclear deformation. It does not express an increase of theE0 transition strength to the β band for a sequence of nucleiapproaching the rigid rotor limit, i.e., with increasing R4/2

value (→3.33).In contrast, our previous results (see Figs. 3 and 4 and

Table I) show that the E0 strengths decrease with increasingstiffness of the nuclear quadrupole deformation. That is, fora typical deformation β <∼ 0.5, we must expect the followingconsequence: the better the nucleus resembles the rigid rotor(R4/2 ≈ 3.33), the larger is its stiffness, and the smaller arethe E0 and E2 transitions between the β and the ground statebands.

In the CBS model, we can trace the evolution of X asa function of nuclear stiffness. Therefore we consider thequantity

XCBS = ρ2(0+2 → 0+

1 )e2R4

B(E2, 0+2 → 2+

1 )(26)

=⟨�0+

1

∣∣b2∣∣�0+

2

⟩2β2

M⟨�2+

1

∣∣b∣∣�0+2

⟩2 (27)

=[

m2,rβ(0+

2 → 0+1 )

m1,rβ(0+

2 → 2+1 )

]2

β2M. (28)

As plotted in Fig. 5, the ratio m2,rβ(0+

2 → 0+1 )2/m1,rβ

(0+2 →

2+1 )2 converges to 4 in agreement with the limit obtained by

Rasmussen in the rigid rotor model. Data on X exist for thenuclides 152Sm and 156Gd. They are given in Table III andincluded in Fig. 5. The predicted increase of X as a function ofnuclear stiffness is obvious. However, the CBS overpredicts thedata by a factor of about 2. The ratio X/β2

M includes the scalingfactor β2

M and therefore is not determined completely by data.It is customary to extract β2

M from the B(E2, 2+1 → 0+

1 ) ina consistent, albeit model-dependent, way. The deviationsbetween theory and data for X/β2

M seen in Fig. 5 may thenbe attributed to the usage of data on both interband (X) andintraband [β2

M ∝ B(E2, 2+1 → 0+

1 )] transitions.

TABLE III. Comparison of experimental and theoretical correla-tion observables.

Xexp/β2M XCBS/β

2M Yexp YCBS

152Sm 0.304(25) 0.80 10.4(17) 21156Gd 0.921(36) 1.28 1.4+1.6

−0.5 × 102 62

XRasm

SmGd

0.2 0.4 0.6 0.8 1.0r

1

2

3

4

X/ M2

FIG. 5. Ratio XCBS/β2M plotted as a function of the stiffness

parameter rβ . A value of X/β2M = 4 has been derived by Rasmussen

[18] in the rigid rotor limit. The data points show the experimentalvalues for 152Sm [17] (rβ = 0.14) and 156Gd [19,20](rβ = 0.31).

To fully avoid this dependence on the scaling factor βM

and, thus, any model dependence, we define the quantity

Y = ρ2(0+2 → 0+

1 ) (e2R4Z)2(

34π

)2

B(E2, 0+2 → 2+

1 )2

=⟨�0+

1

∣∣b2∣∣�0+

2

⟩2⟨�2+

1

∣∣b∣∣�0+2

⟩4 (29)

as the ratio between two appropriate powers of interbandtransition strengths. This observable theoretically dependssolely on the stiffness parameter rβ . Thanks to the monotonicrelation between rβ and the R4/2 ratio [11], the CBS modeluniquely relates the observable Y to the R4/2 ratio, which itselfis a fundamental, easily obtainable characteristic observable.Figure 6 shows plots of Y as a function of rβ (left) and R4/2

(right). Y increases monotonically with the stiffness parameterrβ and with the R4/2 ratio, too. Y diverges in the rigid rotorlimit.

Experimental data points for 152Sm and 156Gd have beenincluded in Figs. 5 and 6 as examples. Their values aretabulated in Table III. They were obtained by evaluatingthe quantities X and Y as functions of the easily obtainablebranching ratio �E0/�E2 for the decays of the 0+

2 state to the0+

1 and 2+1 members of the ground state band and the lifetime

τ0+2

of the 0+2 state. We obtained

Xexp = �E0

�E2

cE2(Eγ,E2/MeV)5e2R4

h( ∑

i �i

) , (30)

and

Yexp = Xexp

[e2R4Z2

(3

4π

)2]

τ0+2

h

(1 + α + �E0

�E2

), (31)

Sm

Gd

0.2 0.4 0.6 0.8 1.0rΒ

10

100

1000

104Y

Sm

Gd

3.0 3.1 3.2 3.3R4/2

10

100

1000

Y

FIG. 6. Ratio Y over rβ and R4/2. The data points representthe experimental values of Y for 152Sm (R4/2 ≈ 3.00) [17] and for156Gd (R4/2 ≈ 3.23) [19,20].

034307-6


where α represents the conversion coefficient of the 0+2 →

2+1 E2 transition, τ0+

2is the lifetime of the 0+

2 state and cE2 =8.064 (meV/e2 b2) is a numerical conversion factor. The dataon the stiffness observable Y for 152Sm and 156Gd support theexpected interband E2–E0 correlation on a quantitative level.

VI. SUMMARY

The analytical wave functions in the deformation coordinateβ of the confined β soft rotor model have been used toevaluate E0 transition strengths between low-energy K = 0bands in axially symmetric nuclei. As a function of the onlystructural parameter within the CBS rotor model, i.e., rβ ,which interpolates from X(5) to the rigid rotor, absolute E0transition strengths decrease with increasing rβ . The relativeE0 strengths for a given value of the structural parameterrβ decrease with increasing angular momentum. The Z-

independent quantity X ∝ ρ2(E0; 0+2 → 0+

1 )/B(E2; 0+2 →

2+1 ) has been traced between X(5) and the rigid rotor. It

reaches the value 4β2M at the rigid rotor limit, as previously

derived by Rasmussen. A new observable Y ∝ ρ2(E0; 0+2 →

0+1 )/B(E2; 0+

2 → 2+1 )2 has been proposed, which is indepen-

dent of the absolute nuclear deformation and solely depends onnuclear stiffness. Available data are in satisfactory agreementwith the CBS model.

ACKNOWLEDGMENTS

We thank F. Iachello, K. Heyde, P. von Brentano,G. Rainovski, and O. Moller for stimulating discussions.Support by the DFG under Grant No. SFB 634 and bythe Helmholtz International Center for FAIR is gratefullyacknowledged.

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