E2 transitions and quadrupole moments in the E(5) symmetry

E2 transitions and quadrupole moments in the E„5…symmetry J. M. Arias Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Facultad de Fı ´sica, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain 共Received 1 December 2000; published 12 February 2001兲 E2 transitions and quadrupole moments are studied in the recently proposed E共5兲symmetry by using the intrinsic state formalism. It is shown that the values of these magnitudes can be obtained for the different bands to higher order in the boson number Nby projecting the intrinsic state on ␥ and ␤ variables. The formalism allows to find easily the dependence of those magnitudes on the structure parameter of the quadrupole operator, ␹ . DOI: 10.1103/PhysRevC.63.034308 PACS number共s兲: 21.60.Fw, 21.10.Ky, 21.10.Re I. INTRODUCTION Recently a new class of dynamic symmetry has been proposed by Iachello 关1兴. This symmetry is expected to be of use when analyzing systems undergoing phase transitions between traditional dynamic symmetries. In particular, the example presented in Ref. 关1兴considers the Bohr Hamiltonian 关2兴and discusses the case in which the potential is ␥ independent and in addition the ␤ dependence of the potential is modeled by a five-dimensional infinite well. This seems to be applicable in nuclear spectroscopy when nuclei are at the critical point in a transition from spherical to ␥ -unstable shape. The E(5) symmetry is discussed in 关1兴in connection with the interacting boson model 共IBM兲关3兴. Energy levels are given and transition probabilities for selected states are calculated by using the following quadrupole operator depending linearly on ␤ : T ␮ (E2)⫽t ␤ 冋 D ␮ 0 (2)共 ␪ i兲cos ␥ ⫹1 冑 2„D ␮ 2 (2)共 ␪ i兲 ⫹D ␮ ⫺2 (2) 共 ␪ i兲…sin ␥ 册 ,共1兲 where tis a scale factor. Experimental examples of this new class of symmetry have already been proposed 关4兴. Within the geometrical model the case of ␥ independent potential surface was discussed some time ago by Wilets and Jean 关5兴, while the equivalent situation within the IBM is known as the O共6兲limit and was discussed first in Ref. 关6兴. In both cases the energy surface has a definite equilibrium value for ␤ , being otherwise ␥ independent. On the other hand, the vibrational Bohr Hamiltonian 关2兴and the corresponding SU共5兲limit in IBM 关7兴provide with a situation in which the energy surface has equilibrium value ␤ ⫽0 and is ␥ independent too. As mentioned above, the newly proposed E共5兲symmetry seems to be appropriate when discussing potentials with flat behavior as a function of some coordinate, as it could be the case of the ␤ coordinate in a SU共5兲-O共6兲 transition. First, in Sec. II a brief review of the example of E共5兲 symmetry presented in Ref. 关1兴is given. In Sec. III the formalism used is developed, including the intrinsic state description and the projection onto the laboratory frame. Results are presented in Sec. IV. Finally, Sec. V is devoted to a summary. II. THE E„5…SYMMETRY Consider the Bohr Hamiltonian H⫽⫺ ប2 2B 冋 1 ␤ 4 ⳵ ⳵ ␤␤ 4 ⳵ ⳵ ␤ ⫹1 ␤ 2sin3 ␥ ⳵ ⳵ ␥ sin3 ␥ ⳵ ⳵ ␥ ⫺1 4 ␤ 2兺 ␬ Q ␬ 2 sin2共 ␥ ⫺1 3 ␲ ␬ 兲 册 ⫹V共 ␤ , ␥ 兲,共2兲 where ␤ , ␥ are the shape variables and the Q ␬ ’s are the components of the angular momentum written in terms of Euler angles. In cases in which the potential depends only on ␤ , V( ␤ , ␥ )⫽U( ␤ ), the wave function can be factorized as ⌿共 ␤ , ␥ , ␪ i兲⫽f共 ␤ 兲⌽共 ␥ , ␪ i兲,共3兲 where ␪ istands for the three Euler angles, and the Schro ¨- dinger equation can be split into two equations, 冋 ⫺1 sin3 ␥ ⳵ ⳵ ␥ sin3 ␥ ⳵ ⳵ ␥ ⫹1 4兺 ␬ Q ␬ 2 sin2共 ␥ ⫺2 3 ␲ ␬ 兲 册 ⌽共 ␥ , ␪ i兲 ⫽ ␶ 共 ␶ ⫹3兲⌽共 ␥ , ␪ i兲; ␶ ⫽0,1,2,..., 共4兲 and 冋 ⫺ប2 2B 冉 1 ␤ 4 ⳵ ⳵ ␤␤ 4 ⳵ ⳵ ␤ ⫺ ␶ 共 ␶ ⫹3兲 ␤ 2 冊 ⫹U共 ␤ 兲 册 f共 ␤ 兲⫽Ef共 ␤ 兲. 共5兲 If U( ␤ ) can be modeled as a five dimensional infinite well, the problem is exactly solvable and the corresponding symmetry is called E共5兲. The solutions of the Schro ¨dinger equations in ␤ and ( ␥ , ␪ i) with the appropriate boundary conditions are known 关1兴. The wave functions on ␤ are f ␰ , ␶ 共 ␤ 兲⫽C ␰ , ␶ ␤ ⫺3/2J ␶ ⫹3/2 冉 x ␰ , ␶ ␤ w ␤ 冊 ,共6兲 PHYSICAL REVIEW C, VOLUME 63, 034308 0556-2813/2001/63共3兲/034308共5兲/$15.00 ©2001 The American Physical Society63 034308-1 where ␶ is the label associated to the O共5兲algebra, ␰ is a label that enumerates the zeros of the relevant Bessel function, x ␰ , ␶ is the ␰ th zero of the Bessel function J ␶ ⫹3/2(x), C ␰ , ␶ are normalization constants, and ␤ wis the range of the potential in the ␤ variable. The solutions of the ( ␥ , ␪ i) part were studied in Ref. 关5兴 and tabulated in Ref. 关12兴, where ⌽ ␶ ,L,M( ␥ , ␪ i) are written in terms of Dfunctions as ⌽ ␶ ,L,M共 ␥ , ␪ i兲⫽兺 ␳ g ␶ ,L, ␳ 共 ␥ 兲DM ␳ (L)共 ␪ i兲.共7兲 The intrinsic functions g ␶ ,L, ␳ ( ␥ ) are explicitly given in 关12兴 and only even values of ␳ appear in the sum. The E共5兲states are labeled by 兩 N; ␰ ␶ LM 典 .Nis the boson number, ␰ is a label related to the solution of the Schro ¨dinger equation in the ␤ variable as mentioned above, ␶ is the label associated to the O共5兲algebra, Lis the total angular momentum, and Mits projection on one axis. III. THE FORMALISM In this work it is shown that the values of the E2 transitions and quadrupole moments in the E共5兲dynamic symmetry can be obtained by using the intrinsic state formalism in IBM 关8–10兴and projecting on the appropriate variables. The starting point is that the E2 transition operator in IBM is written as T ␮ (E2)⫽qQ ␮ (sd),共8兲 where qis a scale factor and Q(sd)is the IBM quadrupole operator, Q ␮ (sd)⫽共s†d ˜ ⫹d†s兲 ␮ (2)⫹ ␹ 共d†d ˜ 兲 ␮ (2) .共9兲 The parameter ␹ is its structure constant. The operators d ˜ m ⫽(⫺1)md⫺mare introduced so as to have tensors with the appropriate properties under spatial rotations. The basic idea of the intrinsic frame formalism is to consider that the pure quadrupole states are globally described by a boson condensate of the form 兩 g 典 ⫽1 冑 N!共⌫g †兲N 兩 0 典 ,共10兲 where the basic boson is given by ⌫g †⫽1 冑 1⫹ ␤ 2 冋 s†⫹ ␤ cos ␥ d0 †⫹1 冑 2 ␤ sin ␥ 共d2 †⫹d⫺2 †兲 册 , 共11兲 which depends on the ␤ and ␥ shape variables. The equilibrium values for ␤ and ␥ are obtained by minimizing the energy surface in the boson condensate 共10兲. In the SU共5兲 and O共6兲limits of IBM this minimization leads to definite values of ␤ ( ␤ 0⫽0 and ␤ 0⫽1, respectively兲and the energy surface is ␥ independent in both cases. At the critical point in the phase transition from SU共5兲to O共6兲the energy surface is expected to be rather flat in the ␤ variable and the symmetry E共5兲seems to be appropriate. When calculating E2 transitions and moments in the traditional 关SU共5兲or O共6兲兴 IBM limits, the ␤ variable is fixed in Eqs. 共10兲and 共11兲to its equilibrium value and integration on ␥ has to be performed. In the E共5兲case, since the behavior of the energy surface is flat in both ␤ and ␥ variables, integration on both variables has to be done. Electromagnetic E2 transition rates and quadrupole moments are evaluated by taking matrix elements of the quadrupole operator 共8兲and 共9兲. These matrix elements in the boson condensate 兩 g 典 , 具 g 兩 Qm (sd) 兩 g 典 ⬅Qm (sd)( ␤ , ␥ ), have already been calculated 关11兴, Q0 (sd)共 ␤ , ␥ 兲⫽N 1⫹ ␤ 2 冋 2 ␤ cos ␥ ⫺ 冑 2 7 ␹ ␤ 2cos2 ␥ 册 , 共12兲 Q2 (sd)共 ␤ , ␥ 兲⫽Q⫺2 (sd)共 ␤ , ␥ 兲 ⫽1 冑 2 N 1⫹ ␤ 2 冋 2 ␤ sin ␥ ⫹ 冑 2 7 ␹ ␤ 2sin2 ␥ 册 . 共13兲 The matrix elements of Q(sd)not specified are zero. In the O共6兲limit the IBM E2 transition operator is usually defined with ␹ ⫽0 since in that case it is a generator of O共6兲and definite selection rules appear. This approximation has proved to be good for studying nuclei at the O共6兲limit, but for transitional SU共5兲-O共6兲nuclei at the critical point the quadrupole operator could depend on ␹ . Thus, in the following the general form in Eqs. 共12兲and 共13兲will be kept. With the help of Eqs. 共6兲and 共7兲, states in the laboratory can be obtained from the boson condensate 共10兲and 共11兲as 兩 N; ␰ ␶ LM 典 ⫽f ␰ , ␶ 共 ␤ 兲⌽ ␶ ,L,M共 ␥ , ␪ i兲 兩 g 典 .共14兲 Thus, the matrix elements of the quadrupole operator are given by 具 N; ␰ ␶ LM 兩 Q ␮ (sd)共lab兲 兩 N; ␰ ⬘ ␶ ⬘L⬘M⬘ 典 ⫽ 具 N; ␰ ␶ LM 兩 兺 mQm (sd)共int兲D ␮ m (2)共 ␪ i兲 兩 N; ␰ ⬘ ␶ ⬘L⬘M⬘ 典 ⫽ 冕 d⍀ 冕 ␤ 4d ␤ 冕 兩 sin3 ␥ 兩 d ␥ f ␰ , ␶ *共 ␤ 兲⌽ ␶ ,L,M *共 ␥ , ␪ i兲 ⫻兺 m 具 g 兩 Qm (sd)共int兲 兩 g 典 D ␮ m (2)共 ␪ i兲f ␰ ⬘, ␶ ⬘共 ␤ 兲 ⫻⌽ ␶ ⬘,L⬘,M⬘共 ␥ , ␪ i兲,共15兲 where the IBM quadrupole operator in the laboratory has been transformed to the intrinsic frame by using Dfunctions. The intrinsic matrix elements 具 g 兩 Qm (sd)(int) 兩 g 典 are those given in Eqs. 共12兲and 共13兲. J. M. ARIAS PHYSICAL REVIEW C 63 034308 034308-2 IV. RESULTS Within the scheme presented in the preceding section, the calculation of matrix elements of the quadrupole operator implies integration on the variables ␤ and ␥ , as well as on the Euler angles ␪ i, in addition to the relevant matrix elements in the intrinsic frame, Eqs. 共12兲and 共13兲. Thus, the quadrupole matrix elements are given by 具 N; ␰ ␶ LM 兩 Q ␮ (sd)共lab兲 兩 N; ␰ ⬘ ␶ ⬘L⬘M⬘ 典 ⫽8 ␲ 2 2L⫹1 具 2 ␮ L⬘M⬘ 兩 LM 典 兺 m ␳␳ ⬘ 具 2mL⬘ ␳ ⬘ 兩 L ␳ 典 ⫻ 冕 兩 sin3 ␥ 兩 d ␥ g ␶ ,L, ␳ 共 ␥ 兲 ⫻ 冋 冕 ␤ 4d ␤ f ␰ , ␶ 共 ␤ 兲Qm (sd)共 ␤ , ␥ 兲f ␰ ⬘, ␶ ⬘共 ␤ 兲 册 g ␶ ⬘,L⬘, ␳ ⬘共 ␥ 兲, 共16兲 where the integration on the Euler angles has already been done. Only even values of m, ␳ , and ␳ ⬘appear in the sum. With this expression it is straightforward to calculate E2 transition and moments since the integrals in ␤ and ␥ can be easily evaluated. The quadrupole moments of the different states ( ␰ , ␶ ,L) are given by 关qis the scale factor in Eq. 共8兲兴 Q共 ␰ , ␶ ,L兲 ⫽ 冑 16 ␲ 5 具 N; ␰ ␶ LM⫽L 兩 qQ0 (sd)共lab兲 兩 N; ␰ ␶ LM⫽L 典 . 共17兲 The corresponding quadrupole moments for some selected states are Q共 ␰ ⫽1, ␶ ⫽1,L⫽2兲⫽0.1425N ␹ q,共18兲 Q共 ␰ ⫽1, ␶ ⫽2,L⫽2兲⫽⫺0.0514N ␹ q,共19兲 Q共 ␰ ⫽1, ␶ ⫽2,L⫽4兲⫽0.2400N ␹ q,共20兲 Q共 ␰ ⫽1, ␶ ⫽3,L⫽6兲⫽0.3112N ␹ q,共21兲 Q共 ␰ ⫽1, ␶ ⫽3,L⫽4兲⫽0.0751N ␹ q,共22兲 Q共 ␰ ⫽2, ␶ ⫽1,L⫽2兲⫽0.1202N ␹ q,共23兲 Q共 ␰ ⫽2, ␶ ⫽2,L⫽2兲⫽⫺0.0428N ␹ q,共24兲 Q共 ␰ ⫽2, ␶ ⫽2,L⫽4兲⫽0.1998N ␹ q.共25兲 The calculation of E2 transition probabilities can be done straightforward from B共E2; ␰ , ␶ ,L→ ␰ ⬘, ␶ ⬘,L⬘兲 ⫽1 2L⫹1 兩 具 N; ␰ ␶ L 兩兩 qQ(sd)共lab兲 兩兩 N; ␰ ⬘ ␶ ⬘L⬘ 典 兩 2. 共26兲 The corresponding B(E2) transition rates for some selected transitions are B共E2; ␰ ⫽1, ␶ ⫽1,L⫽2→ ␰ ⫽1, ␶ ⫽0,L⫽0兲⫽0.1459N2q2, 共27兲 B共E2; ␰ ⫽1, ␶ ⫽2,L⫽2→ ␰ ⫽1, ␶ ⫽0,L⫽0兲⫽0.0044N2 ␹ 2q2, 共28兲 B共E2; ␰ ⫽1, ␶ ⫽2,L⫽2→ ␰ ⫽1, ␶ ⫽1,L⫽2兲⫽0.2282N2q2, 共29兲 B共E2; ␰ ⫽1, ␶ ⫽2,L⫽4→ ␰ ⫽1, ␶ ⫽1,L⫽2兲⫽0.2282N2q2, 共30兲 B共E2; ␰ ⫽1, ␶ ⫽3,L⫽6→ ␰ ⫽1, ␶ ⫽2,L⫽4兲⫽0.2806N2q2, 共31兲 B共E2; ␰ ⫽2, ␶ ⫽0,L⫽0→ ␰ ⫽1, ␶ ⫽1,L⫽2兲⫽0.0710N2q2, 共32兲 B共E2; ␰ ⫽2, ␶ ⫽0,L⫽0→ ␰ ⫽1, ␶ ⫽2,L⫽2兲⫽0.0082q2N2 ␹ 2, 共33兲 B共E2; ␰ ⫽1, ␶ ⫽3,L⫽0→ ␰ ⫽1, ␶ ⫽1,L⫽2兲⫽0.0090N2 ␹ 2q2, 共34兲 B共E2; ␰ ⫽1, ␶ ⫽3,L⫽0→ ␰ ⫽1, ␶ ⫽2,L⫽2兲⫽0.2806N2q2. 共35兲 TABLE I. Comparison of some B(E2) ratios in 134Ba with the E共5兲symmetry. Experimental data are from Ref. 关14兴. B共E2;41,2 ⫹→21,1 ⫹兲 B共E2;21,1 ⫹→01,0 ⫹兲 B共E2;02,0 ⫹→21,1 ⫹兲 B共E2;21,1 ⫹→01,0 ⫹兲 B共E2;02,0 ⫹→21,2 ⫹兲 B共E2;02,0 ⫹→21,1 ⫹兲 B共E2;01,3 ⫹→21,1 ⫹兲 B共E2;01,3 ⫹→21,2 ⫹兲 E共5兲a1.68 0.86 0 0 E共5兲b1.56 0.49 0.12 0.032 Expt. 1.56(18) 0.42(12) 0.18(8) 0.037(3) aE共5兲as calculated in Ref. 关1兴with operator 共1兲. bE共5兲from this work with the IBM quadrupole operator 共8兲and 共9兲and ␹ ⫽1. E2 TRANSITIONS AND QUADRUPOLE MOMENTS IN . . . PHYSICAL REVIEW C 63 034308 034308-3 It has been checked that these results converge to known results in simpler situations. On the one hand, the O共6兲values 关13兴are obtained if the integral in ␤ , between brackets in Eq. 共16兲, is substituted by Qm (sd)(1, ␥ )( ␤ ⫽1). On the other hand, these results reduce to those presented in Ref. 关1兴, where the E2 operator 共1兲is used, if ␹ is taken as zero in the IBM quadrupole operator 共8兲and 共9兲and the normalization factor 2/(1⫹ ␤ 2) in Eqs. 共12兲and 共13兲is substituted by 1. With these changes the IBM quadrupole operator used here reduces to the E2 transition operator used in 关1兴. 134Ba has been proposed 关4兴as a first evidence in nuclear physics of the E共5兲symmetry. In Table I some important E2 branching ratios for this nucleus are compared with the results obtained in the E共5兲symmetry. Experimental data are taken from Ref. 关14兴. The notation used for denoting the states is L ␰ , ␶ ␲ . Two kinds of E共5兲results are shown. The results E共5兲labeled with 共a兲are those taken from Ref. 关1兴. Due to the form of the transition quadrupole operator used there, Eq. 共1兲, transitions B(E2;02,0 ⫹→21,2 ⫹) and B(E2;01,3 ⫹ →21,1 ⫹) are forbidden. The results E共5兲labeled with 共b兲are those obtained in this work in which the E2 transition operator is Eqs. 共8兲and 共9兲with ␹ ⫽1. It is observed in Table I that the formalism presented here allows for even a better description of the 134Ba when comparing to the newly proposed E共5兲symmetry. This improvement comes from the two differences the IBM quadrupole operator, Eqs. 共8兲,共9兲 and Eqs. 共12兲,共13兲, introduces with respect to the usually used quadrupole operator 共1兲. On the one hand, the inclusion of the term depending on ␤ 2( ␹ ) in Eqs. 共8兲and 共9兲is crucial to describe the ratios B(E2;02,0 ⫹→21,2 ⫹)/B(E2;02,0 ⫹→21,1 ⫹) and B(E2;01,3 ⫹→21,1 ⫹)/B(E2;01,3 ⫹→21,2 ⫹). On the other hand, the normalization factor 2/(1⫹ ␤ 2) in Eqs. 共12兲and 共13兲 improves the description of the ratios B(E2;41,2 ⫹ →21,1 ⫹)/B(E2;21,1 ⫹→01,0 ⫹) and B(E2;02,0 ⫹→21,1 ⫹)/B(E2;21,1 ⫹ →01,0 ⫹). Therefore, the IBM transition operator 共8兲and 共9兲, with intrinsic matrix elements 共12兲and 共13兲, seems to provide a better description of the experimental data than the operator 共1兲. One important point as a signature of E共5兲symmetry in comparison with the O共6兲case is the transition B(E2;02,0 ⫹ →21,1 ⫹). This is forbidden in the O共6兲limit even if one considers the general form of the quadrupole operator including the ␹ term, while it gives the correct ratio B(E2;02,0 ⫹ →21,1 ⫹)/B(E2;21,1 ⫹→01,0 ⫹) in the E共5兲limit with ␹ ⫽1. In Fig. 1 the observed B(E2) transition rates in 134Ba 关14兴are compared with the results obtained in this work assuming ␹ ⫽1. Units are given in W.u. For the decay from the state 0⫹at 1.761 MeV to the states 22 ⫹and 21 ⫹only the branching ratio is known 27/1. This branching ratio is nicely reproduced in the calculation. V. SUMMARY In this paper it has been presented how to use the intrinsic state formalism to evaluate electromagnetic transition rates and quadrupole moments in the recently proposed E共5兲symmetry. It has been shown that dealing with ␤ and ␥ dependent objects in the intrinsic frame can be done easily. The same technique can be used to calculate expectation values of other observables. In addition, it has been shown that the IBM E2 operator provides a better description of the data than the operator 共1兲used in Ref. 关1兴. ACKNOWLEDGMENTS This work was supported in part by the Spanish DGICYT under Project No. PB98-1111. I acknowledge continuous collaboration with C.E. Alonso and A. Vitturi. 关1兴F. Iachello, Phys. Rev. Lett. 85, 3580 共2000兲. 关2兴A. Bohr and B. Mottelson, Nuclear Structure, Vol. II 共Benjamin, Reading, MA, 1975兲. 关3兴F. Iachello and A. Arima, The Interacting Boson Model 共Cambridge University Press, Cambridge, 1987兲. 关4兴R. F. Casten and N. V. Zamfir, Phys. Rev. Lett. 85, 3584 共2000兲. 关5兴L. Wilets and M. Jean, Phys. Rev. 102, 788 共1956兲. FIG. 1. Level schemes and B(E2) values 共in W.u.兲for 134Ba in the E共5兲symmetry with ␹ ⫽1共left兲and corresponding experimental data 关14兴共right兲. The scale factor qin Eq. 共8兲has been adjusted to match the experimental B(E2;21,1 ⫹ →01,0 ⫹) value. For the decay from the state 0⫹at 1.761 MeV to the states 22 ⫹and 21 ⫹only the branching ratio (27/1) is known. J. M. ARIAS PHYSICAL REVIEW C 63 034308 034308-4 关6兴A. Arima and F. Iachello, Ann. Phys. 共N.Y.兲123, 468 共1979兲. 关7兴A. Arima and F. Iachello, Ann. Phys. 共N.Y.兲99, 253 共1976兲. 关8兴J. N. Ginocchio and M. W. Kirson, Nucl. Phys. A350,31 共1980兲. 关9兴A. E. L. Dieperink, O. Scholten, and F. Iachello, Phys. Rev. Lett. 44, 1747 共1980兲. 关10兴A. Bohr and B. Mottelson, Phys. Scr. 22, 468 共1980兲. 关11兴C. E. Alonso, J. M. Arias, F. Iachello, and A. Vitturi, Nucl. Phys. A539,59共1992兲. 关12兴D. Be `s, Nucl. Phys. 10, 373 共1959兲. 关13兴C. E. Alonso, M. Lozano, C. H. Dasso, and A. Vitturi, Phys. Lett. B 212,1共1988兲. 关14兴Yu. V. Sergeenkov, Nucl. Data Sheets 71, 557 共1994兲. E2 TRANSITIONS AND QUADRUPOLE MOMENTS IN . . . PHYSICAL REVIEW C 63 034308 034308-5