Photon scattering off 52Cr: Two-phonon E1 strength at the N = 28 shell closure?

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 636 (1998) 139-155

Photon scattering off 52Cr: Two-phonon E1 strength at the N = 28 shell closure?* J. Enders a, E von Brentano b, J. Eberth h, R.-D. Herzberg b, N. Huxel a'l, H. Lenske c, p. von Neumann-Cosel a, N. Nicolay b, N. Pietralla b, H. Prade d, j. Reifd, A. Richter a, C. Schlegel a,2, R. Schwengner d, S. Skoda b'd, H.G. Thomas b, I. Wiedenh6ver b, G. Winter d'l, A. Z i l g e s a a lnstitutfu'r Kernphysik, Technische Universittit Darmstadt, D-64289 Darmstadt, Germany b Institut fiir Kernphysik, Universi~'t zu K61n, D-50937 K61n, Germany c lnstitutfilr Theoretische Physik, Justus-Liebig-Universitiit Gieflen, D-35392 Gieflen, Germany d lnstitut fiir Kern- und Hadronenphysik, Forschungszentrum Rossendo~ D-01314 Dresden, Germany

Received 12 March 1998; revised 15 April 1998; accepted 21 April 1998

Abstract Results of a natcr(y, "y) experiment at the S-DALINAC are reported for energies up to 7 MeV utilizing a Euroball Cluster detector. The excitation of a J = 1 state at 5544 keV is observed which is believed to belong to 52Cr because of the strength of the signal and the large natural abundance. By a variety of empirical arguments a quadrupole-octupole-coupled two-phonon 1character is suggested for this transition. Quasiparticle random-phase-approximation (QRPA) calculations are able to reproduce the energy and B(E1) transition strength remarkably well. The underlying microscopic structure is suggested to be more complex than a pure two-phonon picture. -1 Furthermore, the calculations indicate a pure neutron (2P3/22pl/2) magnetic dipole structure for an excitation experimentally seen at 5098 keV which would provide a direct measure of ground-state correlations in 52Cr. © 1998 Elsevier Science B.V. PACS: 21.60.Jz; 23.20.Lv; 25.20.Dc; 27.40.+z Keywords: NUCLEAR REACTIONS natCr(y, yt), E0 = 7.0 MeV; measured E~,, I~,. 5°,52,53Crdeduced levels, J, B(tr.~). 52Cr deduced two-phonon 1- state. QRPA calculations.

* Supported by the BMBF under contract Nos. 06 DR 6661 and 06 OK 862 I, and by the Deutsche Forschungsgemeinschaft under Ri 242/12-1 and Br 799/6-2. I Deceased. 2 Present address: GSI, D-64220 Darmstadt, Germany. 0375-9474 / 98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PH S 0 3 7 5 - 9 4 7 4 ( 9 8 ) 0 0 2 1 3 - 9

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J. Enders et al./Nuclear Physics A 636 (1998) 139-155

1. I n t r o d u c t i o n

Collective excitations into low-lying states of natural parity are a well-known feature of spherical nuclei, and they are commonly referred to as phonon excitations [ 1 ]. The coupling of two or even three of these modes leads to multiplets of states with energies close to the sum of the single-phonon excitations, as it is well established for the 12+ ® 2+;0 + , 2 +, 4 + } case. Evidence for states with a 13~-@ 3~-;0+, 2 + , 4 + , 6 +) structure is rather sparse (see e.g. Refs. [2-5] ), whereas the coupling of a quadrupole and an octupole phonon resulting in a quintuplet of states with J~ = 1-, 2 - , 3 - , 4 - , 5 - has been subject of theoretical [6-8] and recent experimental efforts (see e.g. Refs. [9,10] and references therein). Especially the J~ = 1- member of the multiplet has been studied extensively in nuclear resonance fluorescence (NRF) experiments in nuclei around the shell closures N = 82 [11-13], Z = 50 [14-16], and N = 50 [17]. Typical E1 transition strengths to the ground state of some 10 -3 Weisskopf units are reported for these nuclei, which is roughly one order of magnitude larger than average E1 strengths in this excitation energy region expected from empirical systematics [ 18]. The full signature for a two-phonon E1 excitation is the existence of collective decay transitions to the 2 + and 3 - single-phonon states. This was recently demonstrated for the N = 82 nuclei 144Sm and 142Nd in y-decay experiments following inelastic resonant proton scattering into isobaric analogue states [ 19,20]. Topics of current interest are the coupling of an unpaired nucleon to multi-phonon states [21,22] and the role of two- and three-phonon strength in the fine structure of E1 strength below the particle threshold as discussed in Refs. [23,24]. Furthermore, one is interested in a complete survey of two-phonon E1 excitations towards lighter nuclei. The aim is to study the degree of collectivity and possible anharmonicities of multi-phonon states in these nuclei when surface effects become more important and also to clarify whether a clear two-phonon structure forms at all. In this paper we report a search for a two-phonon E1 excitation at the N = 28 shell gap, where the nucleus 52Cr was chosen as an example. Here, the sum energy of the 2 + and the 3 - phonons with individual energies of 1434 and 4563 keV, respectively, is close to 6 MeV. Although the decay of a two-phonon state into single-phonon levels can hardly be observed in NRF due to the non-resonant background rising towards lower y-ray energies, the comparison of the g.s. transition width with results of heavy nuclei provides empirical evidence for a collective structure. The experimental findings on the dipole transitions are confronted with a QRPA calculation which should be able to account realistically for core polarization effects important for the appearance of sizable E1 strength at excitation energies well below the giant dipole resonance. In Section 2 the experiment at the superconducting Darmstadt electron linear accelerator S-DALINAC is briefly reviewed, followed in Section 3 by the results of the experiment and a comparison to empirical expectations for a quadrupole-octupole 1state. Section 4 presents a QRPA calculation of dipole transitions in 52Cr. Concluding remarks are given in Section 5.

J. Enders et al./Nuclear Physics A 636 (1998) 139-155

Scattering Target

J'

141

Orernstarget

,

.

_

i,-

B.or, Commoto,

K//.,/,~//.,.V,~,C/~ I " u "'/~ I

1 m

I

x

~ EUROBALL CLUSTER Detector Single Capsule Detector

Fig. 1. Experimental setup (schematically).

2. Experiment The experiment was performed at the S-DALINAC [25] utilizing an electron beam of 7 MeV and average beam currents of about 30 /zA. The setup is displayed in Fig. 1. The electron beam was stopped in an air-cooled tantalum bremstarget, the radiation was collimated and impinged on the scattering target located approximately 140 cm downstream of the converter target. The scattering target was sealed in a thin polyethylene foil and consisted of 3.14 g natural chromium powder (83.8% 52Cr) and 0.462 g natural boron powder for calibration purposes. The T-rays scattered off the target were detected with a Euroball Cluster module [26] built of seven individually encapsulated HPGe detectors [27] surrounded by a BGO suppression shield, which represents a very powerful instrument for NRF experiments at energies of several MeV [28-30]. The Cluster was placed under 130° relative to the incident beam direction, and a single HPGe detector with 60% efficiency under 90 ° was added for the determination of the spins of the excited levels by measuring angular correlations. The composite design of the Cluster detectors allows summing of coincident events of the HPGe crystals resulting from Compton scattering and pair production. Therefore, especially at high photon energies of several MeV the efficiency of the Cluster is considerably enhanced in comparison to individual detectors [ 31 ]. In order to reduce low-energy background radiation stemming e.g. from Compton or nuclear Thomson scattering in the target material, absorbers made of lead and copper with thicknesses of 15 mm each were located in front of the detectors. The efficiency of the detectors was determined with a S6Co source for energies up to 3500 keV. For higher photon energies the well-known transitions from liB were used in combination with

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J. Enders et al./Nuclear Physics A 636 (1998) 139-155

Monte Carlo simulations [32] which have been proven to agree well with experimental bremsstrahlung spectra [ 33,34]. Data were taken for 35 h with the Cluster and (because of a failure of an electronic module) for 11 h only with the single HPGe counter at 90 °, respectively. Thus, the determination of the transition multipole orders from the angular distributions was restricted to the strongest transitions. For the principal methods of the data analysis of ( y , y ' ) experiments see e,g. Refs. [10,35].

3. Results and discussion 3.1. Transitions in 5°'52"53Cr

The spectrum of the natcr(y, y') reaction obtained with the Euroball Cluster detector placed under 130 ° with respect to the incident beam is displayed in Fig. 2, and the numerical results are summarized in Table 1. For some transitions observed in the experiment discussed here, the transition strengths have been omitted in Table 1, since a large feeding from higher-lying states had to be assumed, although no decay branches could be observed due to the large non-resonant background at small energies in NRF experiments. Peaks without additional marks in Fig. 2 were assigned to background sources by the comparison of the spectrum with the results of other measurements performed under identical conditions. Transitions assigned to the calibration material liB and 12'13C caused by the polyethylene foil are marked with the symbol of the affiliating nuclide. The same is done for transitions assigned to 5°Cr and 53Cr. The nuclide 5°Cr was present in the target with its natural relative abundance of 4.3%. Here, only the two decay branches of the 1+ level at 3628 keV to the g.s. and first 2 + state (at 783 keV excitation energy) have been observed. This M1 excitation is well known from electron scattering experiments [36], and it exhibits features similar to the scissors mode [ 37 ] in heavier nuclei. For other deformed fp-shell nuclei, lowlying strong M1 excitations have been found, too [38]. Within the errors, the derived transition strength agrees with the electron scattering result of [ 36]. Some peaks visible in Fig. 2 have been identified with known transitions in 53Cr [ 39]. Taking E2/M1 mixing parameters from the literature [39], one obtains the transition strengths summarized in Table 1. Where no mixing parameters are given, transitions with pure multipole character are assumed. Those peaks in Fig. 2 which are labelled by their excitation energies are identified as g.s. transitions in 52Cr. Some excitations have already been known from the literature [40], and the 5098 keV excitation could be identified to belong to 52Cr due to a decay branch of 3664 keV to the 2 + state. An origin of previously unknown transitions from other chromium isotopes present in the target material cannot be excluded, but seems unlikely because of the much smaller abundances. The determination of angular distributions has only been possible for the strongest transitions. The dipole character of the 3628 keV excitation in 5°Cr could be confirmed.

J. Enders et al./Nuclear Physics A 636 (1998) 139-155

800

I

.

.

.

.

I

.

.

.

.

I

.

.

.

143 .

.

I

~Cr('y,7') 600

Eo =

7.0

0

130 °

=

MeV

400

200 o

0

3000

r,~

3500 .

¢-

.

.

.

I

4000 .

.

.

.

I

4500 .

.

.

.

I

200

0

0

100

0

5000

5500

Phofon Energy

6000

(keV)

6500

Fig. 2. Photon scattering spectrum of the natCr(T,y t) reaction between 2700 keV and 6600 keV observed with a Euroball Cluster detector placed under 130 ° relative to the incident beam. Ground state transitions assigned to 52Cr are labelled by their energies, transitions in 5°Cr and 53Cr are indicated separately like those of ]IB or 12,t3C. Other transitions correspond to inelastic transitions, escape peaks, or result from background radiation.

This transition exhibits an intensity ratio for the two detectors placed under 90 ° and 130 ° with respect to the incoming beam of W ( 130 ° ) / W ( 9 0 °) = 1.35 4- 0.17 in excellent agreement with the theoretical ratio of 1.36 for the 0 ---, 1 ~ 0 cascade. Furthermore, the angular distribution ratio of the 5544 keV transition in 52Cr is very close to the correlation of a dipole transition [W(130°)/W(90 °) = 1.89 4- 0.41], whereas for the 5098 keV transition neither a quadrupole [W(130°)/W(90 °) = 0.45] nor a dipole distribution was preferred [ W( 130 ° ) / W ( 9 0 °) = 0.94 4- 0.19] within the experimental errors. The 6136 keV transition shows an angular correlation of W(130°)/W(90 °) = 0.34 4- 0.14 in agreement with A = 2 and thus J~ = 2 +. Where the angular correlation could not be determined the spin assignments have been taken from the literature.

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J. Enders et al./Nuclear Physics A 636 (1998) 139-155

Table 1 Transitions observed in the natcr(~,'yt) reaction. Listed are excitation energies Ex, spin and parity jTr, branching ratio F ( J *r ---* 2+)IF, the multipole mixing parameter 8 (for transitions assigned to 53Cr) and the reduced transition strengths B(o-, A)T. If no branching is observed, F ( J ~r ~ 2+)/F = I is used for the determinationof the transition strengths. Levels marked with an asterisk are populated from higher-lying levels, therefore transition strengths have been omitted. Errors for the determinationof the excitationenergies are less than 1 keV. Ex

jr

(keY)

5°Cr

3627.8

1+

53Cr

1973.7 2320.9 3180.8 3262.8

(5/2)(3/2)(3/2)(5/2) + (5/2)(1/2)-

3617.0 52Cr

1434.1 3161.7 3739.6 3770.5 4800.1 4841.3 5098.4 5490.8 5544.4 5796.0 6136.7 d 6459.6 6493.8

2+ 2+ 1±,2 + 2+ 1±,2 + 1±,2 + 1± 2+ 1±,2 + 1± 1±,2 + 2+ 1±,2 + 2+

r(:--.2~) F

t~a

0.38(13)

B(M1)'[

B(E1)T

B(E2)T

(/z~)

(10 -3 e 2 fm 2)

(e 2 fm 4)

0.99(12) 0.48 -0.11

* * 0.06(1) b

85(19) b 0.98(20)

--0.25 0.19 --35

0.08(2) 0.10(1) 8(1) X 10-5

6.7(14) 3.8(5) 110(15)

~0.9 c 0.008(1)

0.09(1)

0.009(2) 0.011(2) 0.085(13)

0.10(2) 0.126(23) 0.94(14)

0.008(2) 0.19(4) 0.017(5)

0.09(3) 2.1(4) 0.19(5)

0.044(25)

0.49(28)

0.76(15)

0.42(15) 0.27(11)

15(2) 76(11) 10.5(20) 13.1(24) 71(12) 7.4(20) 14(4) 430(11) 29(16) 61(36)

a From Ref. [39]. b Mixing parameter unknown;assuming pure M1 and E2 transitions,respectively. e No branching to the g.s. observed. d Probably doublet with a backgound line.

3.2. Possible candidates f o r a quadrupole-octupole excitation in 52Cr

In Fig. 2 one recognizes two strong transitions depopulating states below 6 MeV: an excitation at 5098 keV and another one at 5544 keV which might be candidates for a two-phonon 1 - configuration. A n excitation at 5546 keV (average taken from Ref. [40] ) was previously observed in proton scattering [41,42] and in the (3He, a ) reaction [43]. The angular m o m e n t u m transfer L = 1 found in the (3He, ol) experiment suggests positive parity, but the assignment of other angular m o m e n t u m transfers (and thus negative parity) cannot be excluded. In addition, the excitation energy in that experiment was determined only with a precision of about 4-20 keV. Thus, it is not clear whether it should correspond to the 5544 keV level identified in the (% "/I) measurement. Decay branches other than to the ground state have not been observed for the level

J. Enders et aL /Nuclear Physics A 636 (1998) 139-155

145

at 5544 keV, neither to the quadrupole nor to the octupole phonon state. Within the assumption of a pure two-phonon structure, the E2 decay to the collective 3 - state at 4563 keV is expected to be rather strong [comparable to B(E2;2 + ---, 0+)], but the signal could not be identified in NRF due to the large non-resonant background rising towards lower energies. The partial width of the E3 decay to the 2 + level annihilating the octupole phonon is small due to the high multipole order, and E1 admixtures are not expected to be large. The experimental upper limit of the branching ratio can be estimated to be F ( 1 - ~ 2 + ) / 1 " <. 0.022. Therefore, the detection limit of the present experiment is of the same order of magnitude as for the experimental findings in the N = 82 nuclei [ 19,20,44]. From shell-model results no strong M1 excitations below about 7 MeV are expected in the N = 28 isotones [45], but there might exist certain configurations due to groundstate correlations which allow for M1 transitions at lower excitation energies also. Highresolution proton scattering experiments which could be sensitive to spin M1 strength show no signs up to 6 MeV [46]. The state observed at 5098 keV excitation energy exhibits a sizeable branch to the quadrupole vibration. Since a branching of this amount is not expected for the decay of the two-phonon state, we most probably can exclude this level as a candidate for the quadrupole-octupole vibration. The QRPA results presented below suggest an alternative explanation. Therefore the strong dipole transition depopulating the level at 5544 keV represents the best candidate for a quadrupole-octupole 1- state. It is strongly excited, and decay branches other than to the g.s. have not been observed. Furthermore, the excitation energy deviates less than 10% from the sum energy of the single-phonon excitations, a fact that is also found in heavy nuclei. 3.3. Empirical arguments f o r the two-phonon nature

In this subsection we want to argue that the 5544 keV dipole transition depopulates a level which dominantly contains the aforementioned two-phonon structure. Starting from the result for the g.s. transition strength we will discuss three different approaches: First, the E1 g.s. transition strength is expressed in terms of the single-phonon excitation strengths. Within this picture one finds that the strength scales with the collectivity of the one-phonon states over a wide mass range. The second approach takes the admixture of the isovector giant dipole resonance (GDR) to the low-lying El state into account and neglects the decay of the unperturbed two-phonon structure. Starting from the observed E1 strength one can deduce a mixing matrix element within first order perturbation theory which is comparable to the findings in heavy nuclei. The third argument is based on the mixing between one particle-one hole ( l p l h ) states representing the main components of the GDR and two particle-two hole (2p2h) states, the latter contributing dominantly to two-phonon excitations [47]. Of course, the structure of the proposed two-phonon E1 excitation is more complex than assumed within these three approaches: Even for the N = 82 nuclei, where the quadrupole-octupole picture is well established, detailed microscopic calculations suggest a delicate interplay of one- and two-phonon

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J. Enders et al./Nuclear Physics A 636 (1998) 139-155

V

:3 t~

2

i¢) IM V

O4 I.l.I tv~

V V

ua

0

S2Cr ~Sr Ir'Sn =4Sn BeBa ~°Ce W=Nd I

I

I

I

I

I

I

Nuclide Fig. 3. Ratio between two-phonon E! and the product of one-phonon E2 and E3 transition strengths in closed-shell nuclei. All values are expressedin single-particleunits. The ratio is normalizedto the mean value of all data shown. The main contributionsto the error bars are from the experimentalB(E3) values. components [ 8,14,24 ]. In a first step the E1 strengths of transitions of this type in nuclei at different shell closures are compared. One typically finds transition strengths of the order of some 10 -3 Weisskopf units. Using a collective approach, the E1 operator can basically be written (see e.g. Ref. [48] ) as a product of quadrupole- and octupole-phonon creation and annihilation operators b~, b~ T(E1) ,-~ [(b~ + b2)(b~ + b3)]j=,

(1)

From this one would expect a proportionality of the two-phonon E1 strength to the products of the B(E2) and B(E3) strengths from the single phonons to the ground state B ( E 1 ; 0 + ~ 1 - ) ,-~ 1(2+ ® 3 - ; 1-11b~b~ll0~>l 2 ,

(2)

B ( E 1 ; 0 + ~ 1 - ) ~ B(E2) × B ( E 3 ) ,

(3)

whereas the E1 decay to the single-phonon 2 + state is not allowed within this purely bosonic picture. Therefore the ratio of the two-phonon E1 strength and the product of the E2 and E3 one-phonon strengths should be approximately constant for all spherical nuclei if one removes the mass dependence, i.e. single-particle units are used. Alternatively, one could make use of quadrupole and octupole deformation parameters f12 and/33. Fig. 3 displays the results obtained for the nuclei 88Sr [ 17], 116,124Sn [ 14], 138Ba, 14°Ce [ 13] and 142Nd [ 12] in comparison to 52Cr, where the 5544 keV state has been assumed to be of two-phonon structure. The B(E2) and B(E3) strengths have been taken from Refs. [49,50], respectively, and it should be noted that the main contributions to the errors are from the B(E3) values. In addition, the data have been normalized to their mean value. One recognizes a typical scattering of a factor of about two which is a reasonable agreement considering the uncertainties of the experimental data and the simplicity of the ansatz.

J. Enders et al./Nuclear Physics A 636 (1998) 139-155

147

The second empirical argument for a two-phonon structure is based on the assumption that the rather large g.s. E1 transition strengths to [2+@ 3-; 1-) states can be interpreted as mixing of the GDR into the low-lying levels [51]. Such an analysis for a large body of data in deformed nuclei has been discussed in Ref. [52]. From perturbation theory one gets (Hc) [GDR) I1-) = [2+ ® 3-) + [El- - EODRI x

(4)

where the Coulomb matrix element (Hc) = (GDR]IT(E1)II 2+ ® 3-) and IGDR) stands schematically for the wave function of the GDR. Assuming that the direct decay of two-phonon states to the g.s. is much smaller than the contribution from the GDR

(2 + ® 3-11T(E1)II0~-) ~ 0, (GDRI ITCE1)110~+)

(5)

one finds

? 8(E1,0~ ~ 1-) (Hc) "~ ]El- -- E~DRI B(E1,0 + ~ GDR) "

(6)

A quantity related to the mixing matrix element and approximately independent from the particular nucleus is the so-called spreading width F 1 defined by F J.=

2rr

(Hc)2 IE1- - EaDRI

(7)

(see e.g. Ref. [53] ). For the strongest low-lying electric dipole transitions in heavier nuclei one finds typical spreading widths of 10 to 100 keV [52]. Taking the 5544 keV state as the two-phonon vibration in 52Cr and with the GDR photoabsorption cross sections [54] of 730(60) MeVmb at a mean excitation energy of 20.4 MeV and 240(50) MeVmb at 23.0 MeV for the (7,p) and (9',n) channels, respectively, the mixing matrix element and spreading width amount to (Hc) = 212(30) keV

and

F l= 18.4(52) keV,

in good agreement to typical values presented in the systematics of Refs. [ 52,53,55 ]. In a similar approach - also starting from first order perturbation theory - one can take the strongest one particle-one hole ( l p l h ) component of the GDR to estimate the admixture to the two-phonon E1 strength as suggested in [47]. For 52Cr this would e.g. be the 7 r ( l f ~ l g 9 / 2 ) configuration. In analogy to Eq. (4) one can write the wave function of the 1- state as I1-) = [ 2 + ® 3 - ) + ( l p l h l V I 2 + @ 3 - ; I - ) IE1- - glplh]

× Ilplh; 1 - ) .

(8)

The matrix element (V) is now a measure of the coupling between lplh and 2p2h subspaces [56] with a typical value V ~ 300 keV in the fp-shell region [57]. The

148

J. Enders et al./Nuclear Physics A 636 (1998) 139-155

unperturbed 7r( lf~121g9/2); 1- state is found at an energy of 9.5 MeV using a harmonic oscillator potential. The polarization of the single-particle E1 strength by the collective dipole mode necessitates the introduction of an effective charge [ 1] eeff(E1) = - ~1 ( rz

N --~ Z)(I+x)

'

(9)

where rz = -t-1/2 depending on the particle type (proton or neutron). For the exchange term X = - 0 . 7 can be taken [ 1]. The result is B(E1)T= 2.5 × 10 -3 e2fm2 in excellent agreement with the experiment. However, the good correspondence might be partly fortuitous, because other GDR l p l h components and the interaction between them are neglected here. Thus, the simplified approach represents an order-of-magnitude guess only demonstrating that the experimentally observed large B(E1) strength is compatible with a two-phonon picture.

4. QRPA description of core polarization in S2Cr The low excitation energy of the 21+ state in 52Cr indicates that this semi-magic nucleus is easily polarizable and exhibits collective features. The wave functions may contain statically deformed components but vibrational excitations must be equally important as indicated by a strong excitation of the collective 3- level. Low-lying positive parity excitations into which a collective 3- state at about Ex ~ 4.5 MeV is immersed are typical spectroscopic features in this mass region. Representative examples are the neutron-rich Ca-nuclides and Ti-isotopes [58-61]. Already from the spectra of these nuclei it has to be expected that anharmonicities play an important role in the 1f2pshell region. The nucleus 52Cr is an especially interesting case allowing a study at the N = 28 shell closure. For a theoretical description, the softness of 1f2p-shell nuclei is a peculiar feature because in such systems dynamical correlations and multi-particle configuration mixing are strongly enhanced. The latter aspect is properly accounted for by full l f 2 p shellmodel calculations which, however, give no access to the coupling to high-lying states from the giant resonance region and beyond. Obviously, they also do not account for the possibility of intruder states from the next higher lg2d3s-shells which are another typical experimental finding in the N = 20-28 shell region. A special role is played by lg9/2 states for which a sizable amount of strength is observed experimentally in the lf2p-shell energy region, see e.g. Refs. [58,59]. Here, we are mainly interested in a realistic description of the electromagnetic response rather than in the full many-body structure of S2Cr. For that purpose, the quasiparticle RPA (QRPA) is well suited because it allows correlations to be described over a broad energy range. Inherent to the QRPA is the assumption that response functions of one-body operators like the electromagnetic field are mainly determined by the 1-particle-l-hole ( l p l h ) or two-quasiparticle (2qp) excitations, respectively, of a

J. Enders et aL /Nuclear Physics A 636 (1998) 139-155

149

ground state which already includes correlations by itself [62]. Extensions of RPA and QRPA to higher order configurations have been developed [63] and are still under active consideration [ 64]. In the present context it is of particular interest that response functions [65,66] and particle and hole strength functions [58,60,61] in even and odd mass lf2p-shell nuclei, respectively, are well described by RPA and QRPA methods. Dynamical correlations from the coupling to higher order configurations lead to energy shifts and a redistribution of the single-particle strengths over a certain energy interval. As an important consequence, the transition strengths of e.g. electromagnetic multipole operators are quenched and shifted in energy with respect to simple independent particle estimates. Theoretically, these effects are taken into account by first describing dynamical core polarizations (DCP) microscopically for single-particle and hole configurations relative to the 52Cr mean-field ground state. Similar as in Refs. [59-61] one-quasiparticle strength functions are obtained by coupling single quasiparticle states a t to QRPA excitations ,(2 t of the 52Cr core. The correlated wave functions lil t) are defined by a superposition of one (lqp) and three (3qp) quasi-particle configurations with respect to a mean-field ground state denoted 10)

Znan(J ) + Z n

Zkc[a~(jk) ®

(lO)

kc

The spin and parity j,r of the full state is defined by the "leading" lqp parts given by a superposition of lqp states with energies enid. The 3qp components introduce spreading and energy shifts. Their selection is only constrained by angular momentum coupling and parity. The DCP states are then used as a refined basis in extended QRPA calculations. The correlated 2qp states R t ( j , f ) "~ _ flj/3j, t t include already higher order configurations and correlations because of the coupling to the core excitations. In order to minimize double counting, only interactions between the 2qp components of the correlated states are taken into account. Theoretically, this amounts to solve the QRPA with vertex and propagator renormalizations due to the dynamical self-energies of the DCP basis.

4.1. QRPA and DCP results In the numerical calculations the 52Cr ground state was obtained from a Hartree-FockBogoliubov (HFB) mean-field calculation with microscopic pairing. Parameters were adjusted such that the experimental binding and neutron and proton separation energies of 52Cr [67] were reproduced. Because of the N = 28 shell closure we neglect pairing for neutrons. The lf7/2 proton valence particles are scattered by about 20% mainly into the 2p orbits. In the QRPA calculations unperturbed 2qp states up to 100 MeV and with total spin J ~< 4 were taken into account. The single-particle continuum was discretized by imposing a box boundary condition at RB = 30 fm giving a sufficiently dense spectrum for states with orbital momenta up to g = 6. A density dependent residual

J. Enders et al./Nuclear Physics A 636 (1998) 139-155

150

'

1.5

~E v

1.0

to

S2Cr protons

0.5

o

E

,- 0.0 ~

E I,_

m

1.5

S~Cr neutrons

1.0 N

0.5 Q_

~:o 0"00

4

8

12

Excitation energy (MeV) Fig. 4. Proton and neutron total g9/2 single-particle strength functions in 52Cr from the QRPA results. The arrows indicate the energetic position of the corresponding single-particle states in the mean-field calculation.

interaction was obtained from a G-matrix adjusted to nuclear matter and the ground state properties of finite nuclei [68]. The DCP states were constructed by including QRPA states up to 25 MeV, i.e. well beyond the giant resonance region. The DCP states were then reinserted into QRPA as described above. For 2qp configurations above 25 MeV core polarization was neglected. In principle, full self-consistency could be obtained by iteration of the whole cycle of calculations but the exponential growth in the number of states clearly inhibits such an approach. Spurious admixtures to dipole excitations were projected out by orthogonalization of the full 1- spectrum to a purely spurious isoscalar dipole state exhausting the full isoscalar sum rule strength. Similar to the observations made for the Ca and Ti region [58,60,61 ], core polarization leads to a significant amount of low-energy proton and neutron 9/2 + strength also in Cr. This is demonstrated in Fig. 4 where the corresponding particle strength functions in 52Cr are displayed for excitation energies up to 12 MeV. The position of the unperturbed single-particle strength is indicated by arrows. Clearly, the coupling to core degrees of freedom induces strong fragmentation and a shift of a significant part of the strengths to lower energies. However, one should note that the centroid energy still agrees within a few hundred keV with the mean-field prediction. Strong 9/2 + components lowered by about 2 MeV for the neutron case and even about 5 MeV for protons are visible in Fig. 4. These low-energy parts of the lg9/2 single-particle strength are responsible for the appearance of a j~r = 1- state which can be appreciably excited in the photon scattering experiment. The results of Fig. 4 also illustrate the main difference of a DCP description to a simple independent particle approach. In the latter case the spectral densities are assumed as sharp energy distributions of 8 function-type at the mean-field

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Table 2 Experimental excitation energies and electromagnetic g.s. transition strengths of low-lying states in 52Cr compared to QRPA predictions Experiment J~"

Ex (keV)

B(Eh.)T (e 2 fm 2a)

2+ 311+

1434 4563 5544 c 5098 d

660(30) a 6520(340) b 2.1(4) × 10 - 3 c

QRPA B(MI)T (/z 2)

Ex (keV)

B(EA)T (e 2 fm 2~) 733 6208 2.6 x 10 - 3

0.085(13) d

1818 5147 5328 4766

B(M1)T (/tL2 )

0,039

a From Ref. [491. b From Ref. [50]. c Present experiment. d Present experiment, assuming j r = 1+ (see text).

eigenvalues. Core polarization also contributes to the other members of the lg2d3s-shell and the i f and 2p states but the shifts and depletions are smaller. The 9/2 + configurations are responding most sensitively to core polarization mainly because of two reasons. Energetically, the unperturbed lg9/2 mean-field level is located in a region where the core states already have high level densities. Furthermore, the high spin and positive parity of the orbit provides a large amount of core coupling configurations for E1 excitations. As a result, the gap between the lf2p-shell and the next higher shell is effectively lowered and the lg9/2 orbits overlap, while in a pure mean-field picture they are well separated in energy. The redistribution of strength within the lf2p-shell does not only affect the lowenergy positive parity states but also the collective 3 - state. Only if core polarization is included the B(EA) strengths of the collective transitions to the one-phonon 2 + and 3 - states are obtained simultaneously and consistently with the same set of parameters. As seen from Table 2 the energies and transition probabilities agree satisfactorily well with the experimental observations. The excitation energies are met within a couple of 100 keV which has to be considered as very reasonable in view of the restrictions set by the QRPA and DCP configuration spaces. Core polarization contributes to the B(EA) values mainly like a rescaling factor of the individual 2qp matrix elements. Hence, the calculations lead to state-dependent effective charges which are obtained theoretically from core polarization. The lowering especially of the 9/2 + strength has important consequences for negative parity states in 52Cr. In combination with the 7 / 2 - hole configurations excited states with a large variety of angular momenta can be formed, including in particular also 1- states. With core polarization we find indeed a (non-spurious) 1~- state at Ex = 5328 keV. The theoretical energy position and also the transition strength, B(E1)T = 2.58 x 10 -3 e 2 fm 2, are in remarkable agreement with the measured values. The (1 f ~ lg9/2) configuration accounts for about 50% of the wave function of the 1~- state. The 2p3/2 ® 3 - and

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1f7/2®3- couplings contribute to 25% and 18%, respectively, to the lowest 9/2+-particle configuration visible in Fig. 4. Combining these configurations with the lqp-components of the core excited states allows one to identify a 2 + ® 3 - structure as anticipated in the qualitative discussion. However, it should be noted that this does not completely agree with a pure two-phonon picture. In the core correlated parts of the ( l fT/21g9/2) -l coupling the 2qp structure of the 21+ QRPA-phonon is distributed over many components. Microscopically, phonons are characterized by the coherence of a large number of lqp configurations. The coherent nature of phonon wave functions is essential for the collectivity seen in transitions. This property is also of major importance for the intrinsic stability of phonons against perturbations typical for heavy nuclei. However, coherence is lost to some extent in the two-phonon parts of the 1- wave function. E.g., comparing the B(E2; 1- ~ 31) transition to the one-phonon annihilation B(E2; 2 + ~ 01+) suggested in Refs. [ 19,20] as a test of the two-phonon character, one finds a reduction of the B(E2) strength of about two. This provides a rough measure of the 2 + @ 3- content of the 1- state. The 1+ state found theoretically at Ex = 4766 keV (see Table 2) is predicted to be given almost exclusively by a (2p~/122pl/2) neutron configuration. It is tentatively identified with the experimental level at 5098 keV, since no other dipole state with sizable g.s. excitation strength is suggested by the QRPA results in the experimentally investigated energy range. The transition is dominated by the orbital components of the M1 operator. The appearance of this state reflects mainly the pairing ground state correlations in 52Cr because these are responsible for a partial occupation of the 2p3/2 level in contrast to independent particle model expectations. Core polarization also affects the B(M1) values because it acts like a rescaling of the electromagnetic vertex, similar to the renormalizations in electric transitions. For magnetic transitions, DCP introduces state-dependent effective magnetic moments. Most of the M1 strength is contained in the transition to a state predicted at Ex = 8.82 MeV with B(M1)T= 10.38 ~ which represents the major part of the spin-flip resonance experimentally observed in highresolution electron scattering [69].

5. Concluding remarks To summarize, we have performed an NRF experiment on the semi-magic 52Cr nucleus from which we conclude on the existence of two strong dipole excitations below 6 MeV. From empirical comparison to N = 82 nuclei and from a microscopic QRPA calculation we have found arguments for a strong low-energy (with respect to the GDR) E1 excitation in the N = 28 nucleus 52Cr with a dominant quadrupole-octupole twophonon structure. Although no direct proof of the two-phonon character can been given (e.g. by measuring the decay branches to the single-phonon states), it fits well into the systematics of comparable states in heavier nuclei. Detailed QRPA calculations are capable of reproducing the experimental finding remarkably well, if the spurious isoscalar parts of the E1 strength are properly removed. The theoretical results are only partially

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compatible with a two-phonon picture. The necessity of refinements may already be expected from the non-negligible anharmonicity of the excitation energy. Microscopically, the most important effect is a drastic lowering of the proton and neutron lg9/2 configurations due to core polarization effects. The 1- state is then interpreted to arise predominantly from the coupling to the 1f7/2 hole configuration. However, core-excited components from the low-lying 3 - phonon are also non-negligible. In addition, a magnetic dipole excitation with a neutron (2P3/~2pl/2) character induced by ground-state correlations is suggested by the calculations. Its observation would provide a direct measure of the depletion of the neutron lf7/2 configuration due to g.s. correlations. A candidate for this 1+ state might be the level at 5098 keV, but because neither the angular correlation nor the parity has been observed, additional experiments are needed to verify this. Further studies in neighbouring nuclei are necessary to establish a clear picture to what extent 2 + ® 3 - two-phonon structures exist in the vicinity of the N = 28 shell closure and how they are built up. The high counting efficiency of composite HPGe detectors like the Euroball Cluster should also allow us to proceed in the search for two-phonon El states towards higher energies and thus even lighter nuclei. Work along these lines is planned.

Acknowledgements The authors are grateful to K. Heyde, M. Grinberg, V.Yu. Ponomarev, and A. Poves for valuable discussions concerning the structure of dipole transitions in fp-shell nuclei. Thanks are due to H.-D. Gr~if and his coworkers for the operation of the S-DALINAC, to M.K. Kabadiyski for his help with the data acquisition, and to T. Servene, H. Schnare and L. K~iubler for their support in the setup of the experiment. This work was carried out in the scope of the Euroball project.

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