ELSETIER Physics Letters B 393 ( 1997) 7- I2

6 February 1997

PHYSICS LETTERS B

Electric dipole transitions near the 2 = 50 and N = 82 closed shells

K. Heyde, C. De Coster ’ Vakgroep Subatomaire en Stralingsjjvica, Institute for Theoretical Physics. Proeftuinstraat 86, B-9000 Genf, Belgium

Received 16 October 1996; revised manuscript received 29 November 1996 Editor: C. Mahaux

Abstract

We point out that El transition rates at the Z = 50 (and N = 82) closed sheik, describing the decay of the two-phonon quadmpole-octupole (2+ @ 3-; 1-j states, of the correct magnitude can be obtained. The mechanism of including lp-lh admixtures, at the tail of the giant dipole resonance (GDR), into the otherwise collective two-phonon states is at the origin of the observed El rates. At the same time, we show how a two-body El operator can be constructed that directly deexcites the two-phonon 1 - state. We give an estimate of the effective electric charge needed in this two-body El operator.

PACS: 21.6O.C~; 21.60 Ev; 23.20.-g Keywords: Electric dipole transitions; Particle-hole excitations; Two-phonon states

Recently, rather strong El ground-state transitions from J” = I- states have been observed in spher-

ical nuclei, in particular at and very near to closed shells [ 11. It has been proposed that these electric

dipole excitations arise by coupling the low-lying one-

octupole (3- ) and one-quadrupole (2+) excitations [2-41. The J” = I- member of this ]2+ @ 3-; JM)

multiplet has been observed by now in quite a number of N = 82 nuclei and adjacent odd-mass N = 83 nuclei [ 5,6] and the energy of the J” = I- state very closely

resembles the summed energy of E,( 2+) f E,( 3-)

[ 7-171. This near-equality even holds when approach- ing the region of deformed nuclei.

In i.

Similar J” = 1 - excitations have recently been observed in the Z = 50 closed proton shell nuclei ’ ‘6 I 24Sn with B ( El ) transition rates of the order of

ii,

I Postdoctoral fellow of the Belgian National Fund for Scientific Research (NFWO) .

In

4-6 x 10-‘e2 fm2, very close to the ones in the N = 82

region [ 181. This similarity points towards a common

origin i.e. the presence of admixtures of the tail of the giant-dipole resonance (GDR) into the two-phonon collective state [ 191.

Collective models have considered as the El tran- sition operator for deexciting these \2+@ 3-; 1 -) ex-

citations, a two-body operator [ 20-231. the present brief report we point out that: a qualitative understanding of the El decay of the two-phonon collective quadrupole-octupole

states can be obtained using lp-1 h admixtures into these collective configurations, and, a two-body El transition operator can be con- structed explicitly using perturbation theory, giv- ing at the same time an estimate of the effective charge of this two-body El operator. order to understand the decay (or excita-

0370-2693/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(96)0 16 I l-5

8 K. Heyde, C. De Caster/Physics Letters B 393 (1997) 7-12

tion) properties of the particular l- two-phonon )2+ @3-; I-) configuration, we start from a schematic

model incorporating 1 p- 1 h I- (one-phonon) admix- tures into the 12+ @ 3-; I -) two-phonon state. Using the fact that the two-phonon state can be depicted

microscopically as a 4 qp state, no direct ground state decay will occur through the standard one-body El operator unless one considers ground state correlations

in the theoretical description. The El decay should proceed solely via the particular lp-lh admixtures.

In considering the lowest-lying lp-lh l- state with a large El single-particle matrix element 1241, which occurs at E, M 7.0 MeV for the 2 = 50 Sn nuclei

and is the proton configuration llg$lh,1/2(7.r); l-),

perturbation theory results in the particular l- wave

function:

(I-)=12+@3-;I-)

+ jlg$hll,2(7r); 1-1v12+ @3--; 1-j

AE

x Ilg,l:lh,,,?-(“); 1-j. (1)

A realistic estimate of the unperturbed energy differ- ence AE between the pure 2+@ 3- two phonon con- figuration at E,t z 3.5 MeV and the lg,,, -’ lh11/2(r)

Ip-lh configuration at E, z 7.0 MeV is AE M 3.5

MeV. We also need an estimate of the coupling matrix element (V). This needs the calculation of a 2p-2h-

lp-1 h coupling matrix element. The detailed evalua- tion of the coupling matrix element connecting a lp and a 2p- 1 h configuration (or similarly of a lp-1 h to a 2p-2h configuration) has been given by Kuo and Brown and is denoted by a L-matrix element. Full de- tails are discussed in Ref. [ 251. Applications in the Ni-Zr region [26] lead to typical coupling matrix el-

ements for the above configurations of 0.2-0.3 MeV. In evaluating the coupling matrix element in Eq. ( 1 ), one should in principle decompose the 2 phonon state

as a linear combination of 4qp configurations and so, a sum over various L-matrix elements will result. In the Sn mass region, only the 3- level can be consid-

ered to have a rather strong collective character. In or- der to estimate the difference in considering a single term in the coupling matrix element in Eq. (1) and considering a collective 3- state, we could obtain a macroscopic estimate since the 3- phonon is quite col- lective and the L-matrix element coupling vertex may

______)( El

A P h

Fig. 1. The polarization El charge diagrams, modifying the pure

single-particle charge, resulting from coupling the single-particle

motion to the collective dipole (GDR) mode (denoted by the

wavy line).

be approximated by a typical particle-core coupling matrix element. Studies in the Sb, Sn nuclei result in typical values of 0.3 MeV [ 271, consistent with the microscopic studies of Kuo and Brown. The present comparison indicates that our method of obtaining the

coupling matrix element in Eq. ( 1) can be considered to be correct. Combining these results we obtain the wave function

11-j 2 12+@3-;1-) +0.1~lg~]21~,l,2(?T); 1-j.

(2)

The subsequent El decay to the O+ ground state (act- ing via the Ip-1 h admixture) leads to the matrix ele- ment

(I-IIT(El>llO+) = 0.1(lg~/~11~(E~)l1~~,,/2). (3)

In studying the El decay to the O+ ground state, one should in a similar way as before, include particle- hole ground-state correlations. To lowest order, this

results into 2p-2h O+ admixtures [ 281. In the light of the very large energy denominator for such processes

(AE(2p-2h) 2 8 MeV in the even-even Sn nuclei), those admixtures will be very small. Even more, these 2p-2h correlations can only be reached, using the one- body E 1 operator, through the 1 p- 1 h admixtures in the l- wave function of EQ. (2), which are themselves quite small. So, we have used the approximate value given by Eq. (3) in the further discussion.

K. Heyde, C. De Caster/ Physics Letters B 393 (1997) 7-12

Fig. 2. On the left hand side, we indicate the most important unperturbed proton lp-lh I- configurations (and the sp. El strength in units fin*) that play a tale at the Z = 50 closed shell (A = 116 124). Fig. 6.16 from Ref. [ 24 I has been modified conform with a closed proton shell at 2 = 50 and filled 1g7i2 and 2ds,, neutron shells at N = 64. The proton (rr) and neutron (v) Ip-I h I- excitations are presented separately. On the right hand side, results for a more realistic calculation of the I- strength resulting from a full diagonalization in the

2qp model space for 116Sn are shown ( 1311). The energy positon of the [2+ @J 3-, l-) two-phonon state is shown too, in both cases.

The effective charge eeff( El) to be used in the evaluation of the El reduced matrix element is modi- fied from the bare El charge e by taking into account modifications resulting from coupling the single- particle motion to the collective dipole (GDR) mode (see Fig. 1) . This results into an effective El charge, largely reduced by a polarization (screening) charge [ 24,29,30] (with x = -0.7)

eeff(E1) +(,-!+..), (4)

or eeff(E1) cv +0.17e (protons) and eeff(El) N- -0.13 (neutrons) for the above Sn nuclei. Using in the Sn region as the strongest lp-lh El transition, the I tg$Wl,~; 1-j ---+ 10’) transition and the cor- respondmg reduced El matrix element (according to Eqs. 6.304a,b, pg. 469 of Ref. [ 241; see also the left- hand part of Fig. 2)) Fiq. (3) leads to the result

B(El;O’-+ 1-)

g (0.~)21~~ff(~~)12~~~~,211~(~~)11~~,,,*~2

2 1 x lo-* e* fm*

A similar analysis for the N = 82 nucleus 14’Ce with the 11 h;lj2 I it3/2; I-) configurations as the one giving the largest El matrix element, a B (El ) value of 6 x 10e3 e2 fm2 is obtained.

In the present schematic model we have considered that particular lp-lh component only that represents the largest direct El transition rate. Starting from this unperturbed energy, however, many more lp- 1 h exci- tations occur, albeit with a much smaller El strength as compared with the lg-’ Ih, 1,~ (TV) configuration (see Fig. 2). In more reali% calculations all unperturbed proton and neutron lp-1 h states need to be consid- ered, also including the residual interaction within this complete Ip- 1 h space in order to address the question of fragmentation of the particular two-phonon state 12+ @ 3-; I-). The diagonalization in the lp-lh I- space results into the formation of the El giant dipole resonance (GDR) at 15 17 MeV in the Sn nuclei pulling the unpertured lp-lh El strength into that res- onant state. B( El) results similar to the above more simple approach are obtained [ 18,311 (see Fig. 2).

A different approach, using perturbation theory, now leads to the possibility of constructing an explicit

10 K. Heyde, C. De Cosrer/ Physics Lerrers B 393 (1997) 7-12

two-body El operator. The effect of lowest-order cor- rections to the excited dipole states Ii), keeping the

ground-state 1 f) essentially unmodified gives

(5)

with If) denoting the O+ ground state, Ii) the ini- tial two-phonon ]2+ @ 3-; l-) state and In) the inter-

mediate lp-1 h configurations. We denote by 13, In the corrected states with If) = If). Now, the direct (fl IZ’( El) I Ii) contribution vanishes and we can so de- fine an effective El operator through the relation.

(flIT”ff(E1)(I~ = ~(fltW-l)ll~)~. (6) n ”

In the explicit situation of El decay from the two- phonon ]2+ @ 3-; 1-j state into the O+ ground state

via lp-lh admixtures, Eq. (6) reads

(O+((Tff(El)((2+@3-;l-)

= ~(O+IJT(El)]]ph-t; l-)

ph

x (ph-‘; I-(V(2+ @3-; I-)

AE,h,l- (7)

Here, the sum over all (ph- t ; I- ) states can be carried out (general quantum mechanics gives l= C I,yJ (xii) if we consider an average energy de- nominator within the given lp-I h 1 - configuration

space i.e. AE = Cph,, _ AEph,t -/N with N the dimen-

sion of the latter .P = I- space. Using an average to the actual lp-1 h energies, as can be noticed from Fig. 2 (left-hand-part) where Eph,,- varies between 7.5 MeV and 12 MeV, is a rather good approxima- tion. So, one obtains the effective El operator as T( El) 2 This operator can now be rewritten into a form that explicitly expresses the two-body El char- acter. Using second-quantized formalism, one obtains the expression

1

T(EI)& = ; c c (~lI~(E1)lI~)(~~lVlrs) CT?- w3,YS

1 x =a~a,a~a$asa, .

AE (8)

In evaluating matrix elements of the composite oper- ator of Eq. (8) between two-phonon (2p-2h configu- rations) states and the ground-state (vacuum) refer-

ence state, using Wick’s theorem, one can lower the rank of the operator (by contracting r and (Y (or /3)

or u with 6 (or y) ) . This then leads to the following simplified form

1 g_

2 C[C ((allT(El)ll~)(~pIVlr~)

@YS rz

+ (~PlVl~~)(~ll~(El)lly)) =&Za$aa,. 1 (9)

In the above expression g, /? annihilate the h, h’ (hole) configurations and y, S annihilate the p, p’ (parti-

cle) configurations in the particular 2p-2h two-phonon

]2+ @ 3-; l-) state. This process of replacing the ac-

tion of the two-body interaction followed by the ac- tion of the El operator by means of a two-body El transition operator, as described above, is depicted di-

agrammatically in Fig. 3. Here, (al ]T( El) I la) is the standard, allowed El one-body matrix element with

effective charge e and thus, the total strength of the two-body effective El charge is modified according

to the expression

/-“((El) = e. (cr~lVlyS)& (10)

Using the similar values for the average two-body matrix element (v) and D as before in discussing

the Sn nuclei, an effective value eeff( El) N 0.1 e is obtained but now as the strength of a two-body El operator. This two-body El operator, annihilat- ing 2p-2h fermion configurations, in a single process

very closely relates to the El decay process in which two phonon configurations 2+@ 3- are annihilated. The effective charge derived in the present discus- sion ( eeff( El ) ) gives rise to El matrix elements of similar magnitude as the ones derived within a col- lective approach where a phenomenological two-body El transition operator has been introduced [ 20-231. A detailed study, however, on how to connect two- body boson operators to their underlying microscopic

K. Heyde, C. De Caster/Physics Letters B 393 f 19971 7-12

------- ----x

AA P h P’ h

Fig. 3. Illustration, in diagrammatic form, of the theoretical perturbation method that allows for the introduction of a two-body El transition

operator from the subsequent action of the two-body interaction and the one-body El operator. Contraction on the particle line (p”) and

the hole line (h”) is shown by the open circle on that particular line that is summed over (the label cy in Eq. (9) ).

4 quasi-particle structure, is outside the scope of this

short note.

So, as a conclusion, we have pointed out that i.

ii.

Ip-1 h admixtures into the two-phonon 12+ 63 3-; l-) collective state, with strong individual El single-particle matrix elements into the O+

ground state, can account for B (El ) values with

magnitude of 0.5-l x 10e2e2 fm2. Even though the derivation hinges on a simplified description, a more realistic diagonalization in the lp-1 h l- space with El strength mainly concentrated in the GDR, results into quite similar B( El) val- ues deexciting the two-phonon state. Using perturbation theory, we were able to de- rive a form of the El operator which has the

structure of a two-body transition operator. This

then closely relates to the El decay process that becomes possible by a two-phonon transition op- erator. An effective charge for the fermion two- body El transition operator is derived.

The authors are grateful to many participants of the

ECT* workshop on ‘Spin and isospin excitations in nuclear structure’, in March-April 1995. Discussions with I. Hamamoto, U. Kneissl, F! von Brentano, A. Richter and B. Mottelson are particularly appreciated. They also like to thank the NFWO/IIKW for financial support during our stay at the ECT in Trento.

12 K. Heyde, C. De Caster/Physics Lerrers B 393 (1997) 7-12

References

[ I] U. Kneissl, H.H. Pitz and A. Zilges, Progr. Part. Nucl. Phys. 37 ( 1996) 349.

]2] PO. Lipas, Nucl. Phys. 82 ( 1966) 91. [3] A. Raduta, A. Sandulescu and PO. Lipas, Nucl. Phys. A

149 (1970) Il. [4] P Vogel and L. Kocbach, Nucl. Phys. A 176 (1971) 33. [5] R.-D. Herzberg, A. Zilges, A.M. Ores, P von Brentano, U.

Kneissl, J. Margraf, H.H. Pitz and C. Wesselborg, Phys. Rev, C 51 (1995) 1226.

]6] A. Zilges, R.-D. Herzberg, P von Brentano, E DBnau, R.D. Heil, R.V. Jolos, U. Kneissl, J. Margraf, H.H. Pitz and C. Wesselborg, Phys. Rev. L&t. 70 (1993) 2880.

[7] F.R. Metzger, Phys. Rev. C 14 (1976) 543. [8] F.R. Metzger, Phys. Rev. C 18 (1978) 2138. [9] ER. Metzger, Phys. Rev. C 18 (1978) 1603.

[IO] H.H. Pitz, U.E.P. Berg, R.D. Heil, U. Kneissl, R. Stock, C. Wesselborg and F! von Brentano, Nucl. Phys. A 509 ( 1990) 587.

[ 1 I] R.A. Gatenby, J.R. Vanjoy, E.M. Baum, E.L. Johnson, SW. Yates, T. Belgya, B. Fazekas, A. Veres and G. Molnar, Phys. Rev. C 41 (1990) R414.

[ 121 R.A. Gatenby, E.L. Johnson, E.M. Baum, S.W. Yates, D. Wang, J.R. Vanjoy, M.T. McEllistrem, T. Belgya, B. Fazekas and G. Molnar, Nucl. Phys. A 560 (1993) 633.

[ 131 J. Rico, B. Rubio, J.L. Tain, A. Gadea, J. Bea, L.M. Garcia-Raffi. 0. Tengblad, P Kleinheinz, R. Menegazzo, R. Wirowski, P von Brentano, G. Siems and J. Blomqvist, Z. Pbys. A 345 (1993) 245.

[ 141 R.-D. Herzbetg, I. Bauske, P von Brentano, Th. Eckert, R. Fischer, W. Geiger, U. Kneissl, J. Margraf, H. Maser, N. Pietralla, H.H. Pitz and A. Zilges, Nucl. Phys. A 592 ( 1995) 211.

[ 151 E. Miiller-Zanotti, R. Hertenberger, H. Kader, D. Hofer, G. Graw, Gh. Cata-Dan& G. Lazzari and PI? Bortignon, Phys. Rev. C 47 ( 1993) 2524.

Ll6J T. Belgya, R.A. Gatenby, E.M. Baum, E.L. Johnson, D.P. DiPrete, S.W. Yates, B. Fazekas and G. Molttar, Phys. Rev. C 52 (1995) R2314.

] 171 R.-D. Herzberg et al., Phys. Lett. B, accepted for publication. I 181 K. Govaea, L. Govor, E. Jacobs, D. De Frenne, W.

Mondelaers, K. Persyn, M.L. Yoneama, U. Kneissl, J. Matgraf, H.H. Pitz, K. Huber, S. Lindenstruth, R. Stock, K. Heyde, A. Vdovin and V.Yu. Ponomarev, Phys. Lett. B 335 (1994) 113.

[ 191 A. Zilges, P von Brentano and A. Richter, Z. Phys. A 341 (1992) 489.

[20] P von Brentano, N.V. Zamfir and A. Zilges, Phys. Lett. B 278 (1992) 221.

[ 2 I] N.V. Zamhr and P von Brentano, Phys. Lea. B 289 ( 1992) 24.

[22] A.F. Barfield, P von Brentano, A. Dcwald, K.O. Zell, N.V. Zamfir, D. Bucurescu, M. Ivagcu and 0. Scholten, Z. Phys. A 332 (1989) 29.

[ 231 P von Brentano, A. Zilges, R.D. Heil, R.-D. Herzbetg, U. Kneissl, H.H. Pitz and C. Wesselborg, Nucl. Phys. A 557 ( 1993) 593c.

1241 A. Bohr and B.R. Mottelson, Nuclear Structure, Vol. 2 (Benjamin, New York, 1975).

[25] T.T.S. Kuo and G.E. Brown, Nucl. Phys. 85 (1966) 40. [26] T.T.S. Kuo and G.E. Brown, Nucl. Phys. A 114 ( 1968) 241. (271 K. Heyde, P van Isacker, M. Waroquier, J.L. Wood and R.A.

Meyer, Phys. Reports 102 ( 1983) 291. ]28] M. Waroquier, J. Bloch, G. Wenes and K. Heyde, Phys. Rev.

C 28 (1983) 1791. [29] 1. Hamamoto, J. Holler and X.Z. Zhang, Phys. Len B 226

(1989) 17. [ 301 I. Hamamoto, private communication at the ECT* workshop

on Spin- and isospin excitations in nuclear structure ( 1995). [ 3 I] A. Vdovin, private communication.