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RAPID COMMUNICATIONS

PHYSICAL REVIEW C 76, 011301(R) (2007)

Contrasting behavior in the rotational structure of the Tz = 1/2 nuclei 73 Kr and 75 Rb: A possible fingerprint of T = 0 neutron-proton pairing correlations Ramon A. Wyss,1,* Paul John Davies,2,† Wojciech Satuła,3,1,‡ and Robert Wadsworth2,§ 1

KTH (Royal Institute of Technology), AlbaNova University Center, S-106 91 Stockholm, Sweden 2 Department of Physics, University of York, Heslington, York Y010 5DD, United Kingdom 3 Institute of Theoretical Physics, University of Warsaw, ul. Ho˙za 69, PL-00 681 Warsaw, Poland (Received 2 March 2007; published 2 July 2007)

Three different rotational bands have been observed to high spin in 75 Rb using Gammasphere. The structures are similar, but not identical to those found in the neighboring Tz = 1/2 nucleus 73 Kr. Conventional total Routhian surface (TRS) calculations with T = 1 pairing are able to reproduce the rotational bands in 75 Rb with extreme accuracy, but they completely fail in reproducing the spectra and decay pattern of the negative parity bands in 73 Kr. Simple qualitative arguments are put forward to show that the decay pattern observed in the negative parity bands in 73 Kr can be accounted for by means of T = 0 pair correlations. To further corroborate this scenario, deformation, and pairing self-consistent TRS calculations, including schematic T = 1 and T = 0 pairing, are performed for the first time indicating the onset of dynamical T = 0 pair-correlations at high angular momenta. DOI: 10.1103/PhysRevC.76.011301

PACS number(s): 21.30.Fe, 21.60.Jz, 27.50.+e

There is ample evidence, such as the odd-even mass differences and moments of inertia being approximatively one-third of the rigid body value, that the atomic nucleus can be described by a superfluid condensate of proton-proton (pp) and neutron-neutron (nn) Cooper pairs in the weak coupling limit suitable for the BCS description, see [1,2] and references quoted therein. In the spherical and isobaric symmetry limits the like-particle partners forming Cooper pairs move in time reversed orbits with total angular momentum, spin, and isospin (LST) = (001), i.e., they form an isovector pairing phase. The only bound nuclear two-body system is the deuteron, consisting of a neutron-proton (np) pair with (LST) = (010). A profound question in nuclear structure is whether there exists an isoscalar T = 0 pair condensate in heavy atomic nuclei and whether it will mix with the isovector T = 1 condensate, rendering a single ground-state (g.s.) wave function comprising two different Cooper condensates [3,4]. Note, such a mixing leads to α-like (or quartet-like) correlations which are difficult to incorporate within the conventional BCS-type approach due to the different properties of isovector and isoscalar Cooper pairs under the time-reversal transformation [5]. A major obstacle in the study of T = 0 pairing correlations is related to the signature of this pairing mode. One possible fingerprint of the T = 0 pairing is the mass excess along the N = Z line, known also as the Wigner energy (WE), see [6–8]. The nuclear shell-model (SM) supports the leading role of an isoscalar interaction in building up the WE [6,9]. However, in terms of pair-content, the SM predicts the WE to have a rather complex structure which is not dominated by deuteron-like J = 1, T = 0 pairs [6,10,11]. The mean-field (MF), which naturally separates particle-hole and pairing channels, faces problems related to corrections beyond the

*

[email protected] [email protected] ‡ [email protected] § [email protected] †

0556-2813/2007/76(1)/011301(5)

011301-1

MF, caused by the isobaric-symmetry violation [12] and the symmetry energy [13,14] in particular. The signal for T = 0 pairing is therefore rather elusive, see also the discussion in Refs. [15,16]. One simple reason is of course that even if the size of T = 0 pairing correlations are of the same magnitude as the T = 1 correlations, the static observables such as quasiparticle excitation energies, etc., depend on the squareroot of the sum of the two pair gaps [4,17], i.e., on = |T =1 |2 + |T =0 |2 . Hence, an expected enhancement in the effective gap felt by individual quasiparticles, if any, sinks in the typical uncertainties of nuclear models. By the same token, the effect on the moments of inertia (MoI) is not expected to be as pronounced as for the T = 1 channel alone, the reason being that the T = 0 condensate influences the MoI in isospace rather than in real space [18,19]. Still, signatures like correlations in terminating states and shifts in band crossing frequencies may occur, but these signs are controversial. Moreover, standard calculations involving only isovector pair-correlations provide reliable predictions even for heavy N ∼ Z nuclei, see, e.g., [20–22]. In this Rapid Communication we provide evidence for a new dynamical effect that occurs in the high-spin spectrum of 73 36 Kr37 [23], which indicates the presence of T = 0 pair correlations. We will start our discussion based upon standard total Routhian surface (TRS) [24] calculations, arguing that the predicted configurations contradict the observed decay pattern. Next, we will bring forward simple qualitative arguments that the T = 0 np configuration mixing can serve as a natural explanation of the observed phenomena. The arguments will be supported by pairing and deformation self-consistent TRS calculations that include schematic T = 1 and T = 0 paircorrelations treated on the same footing within the LipkinNogami prescription. Details of the method and specification of the pairing Hamiltonian can be found in Refs. [4,25]. In the second part of this paper we will discuss the results of a recent Gammasphere plus Microball experiment where the high-spin spectrum of the Tz = 1/2 nucleus 75 37 Rb38 has been significantly extended from previous work [26]. In spite ©2007 The American Physical Society


WYSS, DAVIES, SATUŁA, AND WADSWORTH

2.5


(-,-)

73Kr

(+,+)

Eω [MeV]

2.0 1.5 1.0 0.5 0.0 -0.5 0.5

1.0

1.5

0.5

hω [MeV]

1.0

1.5

FIG. 1. Calculated (white and grey) and experimental (black symbols) Routhians E ω versus rotational frequency. Left panel represents positive parity, positive signature band. Right part depicts negative parity, negative signature band. White triangles represent the 1qp configuration at low spins and the continuation after the band crossing. Grey triangles represent the 3qp band. See text for discussion. 75 of expected similarities, the spectra of 73 36 Kr37 [23] and 37 Rb38 show rather contrasting behavior. In particular, it appears that the data in 75 Rb unlike in the case of 73 Kr can be reproduced within the conventional TRS formalism invoking only T = 1 pp and nn pairing. We start the discussion with the conventional interpretation of rotational bands in 73 Kr based on the standard TRS model calculations [24] including only T = 1 pairing. It has already been shown that this model has been able to describe band-heads and alignment processes in this mass region rather well [20,27]. Comparing the calculated Routhians, see Fig. 1, and spin Ix , see Fig. 2, with the experimental data reveals that the positive parity band, built upon the νg9/2 1qp state, is well reproduced by the calculations, whereas the negative parity band, built upon the ν( fp) state, agrees only up to the band crossing. However, the high-spin part can be made to agree with the calculations provided that the negative parity band changes configuration in the band crossing region in the following way ν( fp) → νg9/2 ⊗πg9/2 ⊗ π ( fp). This configuration change implies that the neutron

30

73Kr

73Kr

(+,+)

(-,-)

25

Ix

20 3qp

5qp

15 10

1qp

5 0.5

1.0

1.5

0.5

hω [MeV]

1.0

1.5

FIG. 2. Similar to Fig. 1 but for angular momentum Ix versus frequency.

1qp configuration changes from ν( fp) to νg9/2 and at the same time a proton 2qp configuration is formed. The total parity is preserved, since the proton 2qp configuration has negative parity. This 3qp configuration not only reproduces the alignment pattern at high-spin in our paired TRS calculations based on a Woods-Saxon potential but is also energetically favored, see Fig. 1. A similar conclusion was reached in Nilsson-Strutinsky single-particle calculations performed in Ref. [23]. Note also, that at low spin this 3qp configuration is almost 2 MeV above the 1qp ν( fp) configuration and can therefore be safely discounted in the low spin regime. For the 1qp configuration, both proton and neutron g9/2 qp’s align, forming effectively a 5qp configuration after the band crossing. Similarly, in the 3qp band, the nonblocked g9/2 proton and neutron (see Fig. 2) align smoothly with increasing spins. Let us now discuss the physical consequences of the inherent configurations in 73 Kr. Within the conventional isovector pairing scenario the 1qp and 3qp wave functions, corresponding respectively to the low- and highspin parts of the negative parity bands, can be written † † as αˆ ν( fp) |BCS and αˆ †νg9/2 αˆ †πg9/2 αˆ π( fp) |BCS where |BCS = |BCSν |BCSπ denotes the local BCS (or HFB) separable vacuum state. The major consequence of the difference between the 1qp and 3qp configurations is that it inhibits electric quadrupole transitions between the bands. Indeed, the electric quadrupole transition operator O(E2) is a onebody operator. Hence, it cannot simultaneously change both proton and neutron wave functions in the transition matrix † element, thus BCS|αˆ ν( fp) O(E2)αˆ †νg9/2 αˆ †πg9/2 αˆ π( fp) |BCS ≈ 0. Consequently, the E2-decay between these bands is expected to be strongly hindered. At the same time, one would expect the 3qp configuration to decay via E1 transitions into the neutron νg9/2 band. Indeed, in this case the transition matrix element is † BCS|αˆ νg9/2 O(E1)αˆ †νg9/2 αˆ †πg9/2 αˆ π( fp) |BCS and hence only the proton part of the wave function changes. This kind of decay is in fact a standard decay mode which is observed in a wide range of nuclei, where two (three) qp configurations decay via E1 transitions into zero (one) qp configurations. The above simple implications, resulting from the assumed configuration assignments, are in complete contradiction to the data, where one sees (i) collective quadrupole transitions between the negative parity 1qp and 3qp bands that extend over the entire range of spins and (ii) the absence of any electric dipole transitions connecting the positive and negative parity bands. This means that either the assumed 3qp configuration is wrong and one should find an alternative assignment, or one should seek an additional source of configuration mixing that would eventually dissolve the distinctive features of the 1qp and 3qp negative parity configurations, thus enabling substantial mixing to occur. As discussed below, the latter scenario can be naturally achieved by invoking T = 0 pairing. Within a scenario involving T = 0 pairing, the valence particles of 73 Kr, with respect to the closed-shell nucleus 56 Ni, can be schematically visualised in terms of np pairs occupying the same Nilsson orbitals originating from the g9/2 and f5/2 − p3/2 subshells, respectively, as depicted in Fig. 3. The 1qp configuration corresponds to the case where the

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∆ [MeV]

CONTRASTING BEHAVIOR IN THE ROTATIONAL . . .

1.5 1.0 0.5 0 30

∆T=1 ∆T=0 73Kr

25 20

Ix

theory

15 10 5

FIG. 3. Schematic figure illustrating the 1qp (upper, conf I) and 3qp (lower, conf II) configurations discussed extensively in the text and their connection via np-pair scattering.

odd-neutron occupies the ( fp) state, whilst the 3qp configuration corresponds to the case where the odd-neutron occupies the g9/2 orbit. As seen from Fig. 3 the difference between these two configurations lies in one single np pair that scatters from a g9/2 orbit into an ( fp) orbit. Since the T = 0 Cooper pairs can scatter freely from g9/2 into the ( fp) and vice versa, preserving the parity, these two configurations are in fact merging. In other words, what in the T = 1 scenario appears as two distinctly different configurations, becomes a single configuration if mixing via the T = 0 pairing interaction is allowed. To shed more light on this scenario, we performed TRS calculations, including schematic T = 0,1 pairing correlations. In the isovector mode we admit pair-correlations in signature-reversed Nilsson orbits, the so called α α˜ mode, whereas the T = 0 mode couples particles in the same Nilsson orbits forming the so-called αα mode. The latter is very important at higher spins, which are of particular interest for our investigation. The strength of the T = 1 pairing GT =1 was increased by 10% compared to the standard average gap method value in order to mimic the somewhat increased pairing correlations in signature reversed orbits. The strength of the T = 0 pairing was chosen to be undercritical at low spins with GT =0 = 0.9GT =1 . The calculations are performed on a grid in deformation space and the energy is minimized at each frequency with respect to deformation parameters. At each grid point and frequency the pairing equations are solved self-consistently using Lipking-Nogami approximate particle number projection. For more details of the model, see Ref. [25]. The TRS calculation for 73 Kr is presented in Fig. 4. The low-spin regime does not differ in principle from the standard extended TRS calculations with T = 1 pairing only. However, after the alignment, the spins are in nice agreement with the data. This is achieved with only one single configuration, not two different ones. Clearly, in this case one expects no hindrance of the B(E2)’s, as the configuration is changing smoothly in the presence of T = 0 pairing within the same band structure. Apparently, calculations invoking T = 0 pairing are able to explain qualitatively both the decay pattern of the bands as well as their excitation energy. From Fig. 4, we see that at low spins only T = 1 pairing is present. After the first alignment, at h ¯ ω ≈ 0.6 MeV, there is a

exp

0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

hω [MeV]

FIG. 4. Calculated and experimental values of angular momentum Ix versus rotational frequency for negative parity bands in 73 Kr. Calculations involve both T = 0 and T = 1 pairing and were done for a single 1qp configuration. Upper part shows the changes in self-consistent T =1 and T =0 pair gaps taking place along the collective path.

drop in the T = 1 pairing gap, as expected from the breaking of a qp pair and the standard anti-coriolis effect. One also observes the onset of T = 0 pairing. The value of T =0 ∼ 0.5 MeV is relatively stable up to the highest observed spins pointing towards a dynamical rather than static scenario. The importance of this outcome is not the mere size of the gap, but the mechanism responsible for the mixing. In nuclei along the N = Z line, it appears that configurations which are well separated by the standard T = 1 interaction may become mixed due to the T = 0 pairing interaction matrix elements. Note also, that the possibility of a pairing phase transition from predominantly T = 1 to T = 0 at high angular momentum was pointed out some time ago in schematic calculations of 24 Mg [28]. Moving now to the second issue in this paper, we report new results on 75 Rb, which was populated via the 40 (40 Ca,αp) reaction at a beam energy of 165 MeV. The extended decay scheme is shown in Fig. 5. At low spins the known [26] positive parity band is interpreted as a one quasiparticle (qp) structure built upon the Nilsson orbital originating from the πg9/2 subshell, whereas the two negative parity bands are built upon the Nilsson orbitals originating from the pseudospin doublet πp3/2 − πf5/2 , denoted as π ( fp). Since 73 Kr has the same number of neutrons (N = 37) as the number of protons (Z = 37) in 75 Rb one might expect a very similar deexcitation pattern to occur in both nuclei. In particular, one may expect that the negative parity band associated with the 1qp proton occupying the π ( fp) configuration at low spins to change into π ( fp) → πg9/2 ⊗ νg9/2 ⊗ ν( fp) configuration in the crossing region. Then the arguments used in 73 Kr can be essentially repeated but with neutrons and protons interchanged in the configurations. The detailed TRS calculations show that this is not the case. The conventional TRS calculations are able to reproduce both the Routhians, see Fig. 6, and the aligned angular momentum,

011301-3


WYSS, DAVIES, SATUŁA, AND WADSWORTH


(73/2 )

−,+/-) (−

75Rb

2.0

(+,+)

1.5

Eω [MeV]

3896 (69/2 )

1.0 0.5 0.0

3552

α = −1/2 α = +1/2

-0.5 (65/2 ) (65/2 )

3142

0.5

3255 (61/2 )

2836 2960 (57/2 )

2519 2567 (53/2 )

2238 (49/2 )

2233 (49/2 )

1991 1924

(45/2 )

(43/2 )

1976 (45/2 )

1789 1769

(41/2 )

(39/2 )

1766 (41/2 )

1646 1656

1617

(37/2 )

(35/2 )

(37/2 )

1560 1555

(33/2 )

1482 (33/2 )

(31/2 )

1468 1437

(29/2 )

1352 (29/2 )

(27/2 )

1347

1299

(25/2 )

(23/2 )

0.5

hω [MeV]

1.0

1.5

Ix for 75 Rb, see Fig. 7, without any obvious need to invoke supplementary correlations. Although the calculations overestimate the relative band head energy between the positive and negative parity band by ∼ 300 keV they follow the relative excitation energy pattern with extreme accuracy. In particular, the positive parity band is reproduced within ±100 keV. Also the signature splitting between the negative parity bands that develops after the band crossing (compare white and black triangles) is nicely reproduced by the calculations. The negative parity 3qp configuration πg9/2 ⊗νg9/2 ⊗ν( fp), which is the analog to 73 Kr 3qp configuration νg9/2 ⊗ πg9/2 ⊗ π ( fp) is predicted to be the lowest among the negative parity configurations in the frequency range between 0.8 MeV h ¯ ω 1.5 MeV. Note, however, that this configuration shows clear signs of smooth termination at lower frequencies than the conventional 1qp band. This is in clear contrast to the case of 73 Kr as shown in Fig. 1 and may explain why this structure is not fed in the 75 Rb experiment. In this paper we have discussed the newly observed high spin level structure in the Tz = 1/2 nucleus 75 Rb and the equivalent structures in 73 Kr. It is demonstrated that in 75 Rb

(57/2 )

(47/2 )

1.5

FIG. 6. Calculated (white and grey) and experimental (black symbols) Routhians versus rotational frequency for positive (circles) and negative (triangles) parity bands in 75 Rb. Grey triangles represent the 3qp band. See text for discussion.

(61/2 )

(53/2 )

1.0

1227

(25/2 )

1201 (21/2 )

1141

1085 (21/2 )

(19/2 )

1041 922

(17/2 )

965

(17/2 ) (15/2 )

(13/2 )

(11/2 )

317

(7/2 )

354 (3/2 )

733 (661) 748

(13/2 )

773

570

(15/2 )

862

669

(11/2 )

515

495 68 (7/2 ) 253 (5/2 ) 232 164 463 234 378 210 340 (38) (3/2 ) (5/2 ) 144 (9/2 )

(9/2 )

FIG. 5. Level scheme for 75 Rb deduced from the present work.

FIG. 7. Calculated (white symbols) and experimental (black symbols) Routhians versus rotational frequency for for positive (left panel) and negative parity positive signature (right panel) bands in 75 Rb. Grey triangles represent the 3qp band. See text for discussion. 011301-4


CONTRASTING BEHAVIOR IN THE ROTATIONAL . . .


conventional calculations invoking only T = 1 pairing are able to reproduce the observed rotational bands with extreme accuracy, thus leaving little room for additional correlations. We argue, however, that in the case of 73 Kr it is difficult (if not impossible) to account for the observed structures by standard calculations involving T = 1 pairing only. The data are then put into the context of T = 0 pairing correlations. In the presence of T = 0 pairing, an entirely different scenario of the band structure emerges; one that accounts quantitatively in a beautiful manner for the observed spectrum and decay properties. A search for evidence of this T = 0 pairing channel has so far been rather elusive. A direct measure of T = 0 pair transfer is difficult to achieve due to the different structure of the T = 0 ground state in even-even and odd-odd nuclei [5]. The present data, however, may be viewed as an indication of the T = 0 pair transfer, not between different nuclei, but between two different band structures, rendering a strong

mixing of what usually is characterized as one and three qp bands. Apparently, the fundamental excitations in nuclei close to the N = Z line have different properties to those in other regions of the nuclear chart. The present data set allows us to probe for the first time the T = 0 channel in rotational nuclei. It is a sensitive and important data set for further development of an effective interaction that fully takes into account the isospin degree of freedom, which was established long ago but until now not fully taken into account. Deformation and pairing self-consistent TRS calculations including schematic T = 1 and T = 0 pair-correlations suggest an enhanced but rather dynamical T = 0 scenario.

[1] D. J. Dean and M. Hjorth-Jensen, Rev. Mod. Phys. 75, 607 (2003). [2] W. Satuła, Phys. Scr. T125, 82 (2006). [3] A. L. Goodman, Adv. Nucl. Phys. 11, 263 (1979). [4] W. Satuła and R. Wyss, Phys. Lett. B393, 1 (1997). [5] S. Głowacz, W. Satuła, and R. A. Wyss, Eur. Phys. J. A 19, 33 (2004). [6] W. Satuła et al., Phys. Lett. B407, 103 (1997). [7] J. Engel, K. Langanke, and P. Vogel, Phys. Lett. B389, 211 (1996). [8] G. Röpke et al., Phys. Rev. C 61, 024306 (2000). [9] D. S. Brenner et al., Phys. Lett. B243, 1 (1990). [10] A. Poves and G. Mart´ınez-Pinedo, Phys. Lett. B430, 203 (1998). [11] J. A. Sheikh and R. Wyss, Phys. Rev. C 62, 51302(R) (2000). [12] K. Neerg˚ard, Phys. Lett. B537, 287 (2002); B572, 159 (2003). [13] W. Satuła and R. Wyss, Phys. Lett. B572, 152 (2003).

[14] W. Satuła, R. Wyss, and M. Rafalski, Phys. Rev. C 74, 011301R (2006). [15] A. O. Macchiavelli et al., Phys. Rev. 61, 041303 (2000). [16] S. M. Fischer, C. J. Lister, and D. P. Balamuth, Phys. Rev. C 67, 064318 (2003). [17] A. L. Goodman, Phys. Rev. C 60, 014311 (1999). [18] W. Satuła and R. Wyss, Phys. Rev. Lett. 86, 4488 (2001). [19] W. Satuła and R. Wyss, Phys. Rev. Lett. 87, 052504 (2001). [20] D. Rudolph et al., Phys. Rev. C 56, 98 (1997). [21] R. Wyss and W. Satuła, Acta Phys. Pol. B 32, 2457 (2001). [22] A. V. Afanasjev and S. Frauendorf, Phys. Rev. C 71, 064318 (2005). [23] N. S. Kelsall et al., Phys. Rev. C 65, 044331 (2002). [24] W. Satuła and R. Wyss, Phys. Rev. C 50, 2888 (1994). [25] W. Satuła and R. Wyss, Nucl. Phys. A676, 120 (2000). [26] C. J. Gross et al., Phys. Rev. C 56, R591 (1997). [27] F. Lerma et al., Phys. Rev. Lett. 83, 5447 (1999). [28] K. Mühlhans, E. M. Müller, K. Neerg˚ard, and U. Mosel, Phys. Lett. B105, 329 (1981).

This work has been supported by the Göran Gustafsson Foundation, the Swedish Science Research Council (VR), the U.K. EPSRC, the Swedish Institute (SI), and the Polish Committee for Scientific Research (KBN).

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