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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 [2(E0)] on the transitional path between the X(5) solution andthe rigid rotor limit have been evaluated within the framework of the confined soft (CBS) rotor model. RelativeE0 transition strengths between the -vibrational band and the ground state band decrease with increasing angularmomentum for a given potential stiffness. AbsoluteE0 transition strengths drop with increasing potential stiffnesstoward zero in the rigid rotor limit. Interband E0E2 correlation observables are examined to estimate 2(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[35]. 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 [69] 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 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 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 onE0 transition rates in the X(5) limit and its evolutiontoward the rigid rotor limit as a function of other collectiveobservables such as theR4/2 ratio orB(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

(E0)= 2if (K +Li + +IP ), (1)

where

2if (E0) =|final|m(E0)|initial|2

(eR2)2, (2)

with the E0 transition operator

m(E0) =

(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(

1 220

4+ 20Y20()

), (4)

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

13 fm is the nuclear radius, and 2 are the intrinsic

quadrupole deformation parameters. In the homogeneous-charge density approximation

(r) = 0 = Ze43 R

3, (5)

one obtains

m(E0) = 35ZeR2

(1 + 5

4220 +

10

7

5

16320 +

)(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 2, the E0operator is in general obtained as [12]

T (E0) = 35ZeR2

(1 + 5

4|2|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 = 34

ZeR2 2 (8)

due to

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

T (E2)K=0 = 34

ZeR2 , (9)

again in the axially symmetric case with = 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 fromIachellos X(5) solution [4] toward the rigid rotor limit withinthe confined soft rotor model. The CBS rotor modelrepresents an approximate analytical solution to the BohrHamiltonian [14]

H = h2

2B

[1

4

4

+ 1

2 sin 3

sin 3

142

k

Q2k

sin2( 23k

)]+ V (, ) (10)

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

(, , , , ) = L() K ( )DLM,K (i), (11)where DLM,K denotes the Wigner functions with i being theEuler angles for the orientation of the intrinsic system. Kdenotes the appropriate wave function in . For sufficientlyaxially symmetric prolate nuclei, one might consider a steepharmonic oscillator in [4]. L() describes the part of thewave function depending on the deformation variable . Theapproximate separation of variables [4] leads to the differentialequation

h2

2B

[1

4

4

1

32L(L+ 1) + u()

]L()

= E L() (12)for L(). 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(), with boundaries atM > m > 0. For this potential, the differential equation (12)is analytically solvable. The ratio r = m/M parametrizesthe width of this potential, that is, the stiffness of the nucleus inthe degree of freedom. For r = 0, the X(5) limit is obtained

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

Qr(L)(z) = J(L)(z)Y(L)(rz) J(L)(rz)Y(L)(z) = 0, (13)

with J and Y being Bessel functions of first and secondkind of irrational order =

[L(L+ 1) K2]/3 + 9/4. For

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

rL,s . The full solution of

Eq. (12) with the aforementioned choice of the CBS square-well potential is then given as

L,s() = cL,s 32[J

(zrL,s

M

)

+ J(rz

rL,s

)Y

(rz

rL,s

)Y(zrL,s

M

)]. (14)

For convenience, we occasionally use the relative deformationparameter b =

M [r, 1]. The normalization constant cL,s

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

1 = Mm

4[L,s()]2 d. (15)

The eigenvalues of Eq. (12) are obtained as

EL,s = h2

2B2M

(zrL,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. ForE0 transitions, we obtain withEqs. (2) and (8)

2if (E0) =(

3Z

4

)2|f |2|i|2 (17)

=(

3

4

)2Z24M |f |b2|i|2 (18)

between states that only differ in the -dependent part of thewave function, i.e., Df |Di = f |i = 1.

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

2if (E0) =(

3

4

)2Z24M

[ 1r

f (b)b2i(b)b

4db

]2(19)

= qE0(M )[m2,r (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 strengths2(E0, Ji Jf )/2(E0, 0+2 0+1 ). Left: for X(5) [r = 0], R4/2 = 2.90. Right: for r = 0.2, R4/2 = 3.10.

with the matrix element

mk,r (Ji Jf ) = 1r

i(b) bkf (b) b

4db (21)

of kth order in the reduced deformation variable b, which onlydepends on the nuclear stiffness parameter r . The size ofthe maximum deformation M appears in the scaling constantqE0(M ). The expression for B(E2) transition strengths for

K = 0 transitions follows in this notation asB(E2; Ji Jf ) = qE2(M )

(C

Jf 0Ji020

)2[m1,r (Ji Jf )

]2,

(22)

with the scaling constant

qE2 (M ) =(

3

4

)22MZ

2e2R4. (23)

CJf 0Ji020

denotes a Clebsch-Gordan coefficient. Deviations fromthe homogeneous-charge density approximation might betaken into account to lowest order by an effective chargeZe qEZe, with qE 1 for electric transitions withmultipolarity .

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 r = 0. The width ofthe potential well, M , 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. 1shows these relative E0 transition rates for r = 0, which isX(5). Some of the values are also tabulated in Table I. Wenote that the 2(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 parameterr = 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 r and different angular momentaJ for transitions between states with s = 2( 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 r , separately. Forincreasing angular momentum, the model predicts a decreasingE0 transition rate. For X(5), the 10+s=2 10+s=1E0 transitionrate is reduced by 38% relative to the corresponding 0+s=2 0+s=1 transition. Notice also, that for rising potential stiffness,the relativeE0 transition strengths rapidly approach a constant

034307-3

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

0 2 4 6 8 10

1.0

0.8

0.6

0.4

0.2

2(E

0)J+ 2

J+ 1

/2 (

E0)

0+ 2 0

+ 1

J

RR

X(5)

r =0.5...0.90.40.30.20.10

0.0

FIG. 2. (Color online) RelativeE0 transition strengths for variousnuclear stiffnesses with r 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, r = 0 (top) and r = 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 functionsoverlap on either side of that node.

In the case of small potential stiffness (r = 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 functions 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 (r = 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 labeledr = 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 r . All transition strengthsare normalized to the transition 0+2 0+1 for X(5), i.e., weplot the quantity

2(E0, Jf Ji ; r)2(E0, 0+2 0+1 ; r = 0)

=[m2,r (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 r . Upper: r = 0 [X(5)]. Lower:r = 0.5.

034307-4

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

1.00.80.60.40.20.0

1.0

0.8

0.6

0.4

0.2

8+

6+

10+

4+

2+

2(E

0) J

+ 2 J

+ 1

/2 (

E0

) 0,X

(5)

r

0+

0.0

FIG. 4. (Color online) E0 transition rates from s = 2 to s = 1between states with angular momenta J = 010 as a function of r .The theoretical values are normalized to the 0+2 0+1 E0 transitionstrength in X(5), 20,X(5) 2(E0, 0+2 0+1 ; r = 0).

for a fixed deformation M . For r approaching unity, all E0transition strengths drop to zero. The evolution of this decreaseas a function of r depends on angular momentum, as discussedabove. E0 transition strengths from 0+ states monotonicallydecrease with increasing r . E0 transition strengths b...