A simple model based on the group SO(5) suggests that both the like-particle and neutron-proton components of isovector pairing correlations in odd-A nuclei are Pauli blocked. The same effect emerges from Monte Carlo Shell-model calculations of proton-rich nuclei in the full fp shell. There are small differences between the two models in their representation of the effects of an odd nucleon on the competition between like-particle and neutron-proton pairing, but they can be understood and reduced by using a two-level version of the SO(5) model. On the other hand, in odd-odd nuclei with N not equal to Z SO(5) disagrees more severely with the shell model because it incorrectly predicts ground-state isospins. The shell model calculations for any fp-shell nuclei can be extended to finite temperature, where they show a decrease in blocking

Engel, J

Langanke, K

Vogel, P

Physics Letters B

arXiv

J. Engel

K. Langanke

P. Vogel

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Isovector pairing in odd-A proton-rich nuclei

A simple model based on the group SO(5) suggests that both the like-particle and neutron-proton components of isovector pairing correlations in odd-A nuclei are Pauli blocked. The same effect emerges from Monte Carlo Shell-model calculations of proton rich nuclei in the full fp shell. There are small differences between the two models in their representation of the effects of an odd nucleon on the competition between like-particle and neutron-proton pairing, but they can be understood and reduced by using a two-level version of the SO(5) model. On the other hand, in odd-odd nuclei with N≠Z, SO(5) disagrees more severely with the shell model because it incorrectly predicts ground-state isospins. The shell model calculations for any fp-shell nuclei can be extended to finite temperature, where they show a decrease in blocking

Engel, J.

Langanke, K.

Vogel, P.

Carolina Digital Repository

arXiv:nucl-th/9803018v1  5 Mar 1998Isovector pairing in odd-A proton-rich nucleiJonathan EngelDept. of Physics and Astronomy, CB3255, University of North Carolina, Chapel Hill, NC 27599Karlheinz LangankeInst. for Phys. and Astr. and Cen. for Theor. Astrophys., Univ. of Aarhus, DK-8000 Aarhus C, DenmarkPetr VogelPhysics Department, California Institute of Technology, Pasadena, California 91125August 8, 2018AbstractA simple model based on the group SO(5) suggests that both the like-particle and neutron-proton components of isovector pairing correlations inodd-A nuclei are Pauli blocked. The same effect emerges from Monte CarloShell-model calculations of proton-rich nuclei in the full fp shell. There aresmall differences between the two models in their representation of the effectsof an odd nucleon on the competition between like-particle and neutron-protonpairing, but they can be understood and reduced by using a two-level versionof the SO(5) model. On the other hand, in odd-odd nuclei with N 6= Z, SO(5)disagrees more severely with the shell model because it incorrectly predictsground-state isospins. The shell model calculations for any fp-shell nuclei canbe extended to finite temperature, where they show a decrease in blocking.1Proton-rich nuclei play an important role in explosive nucleosynthesis and are increas-ingly accessible to experiment. For this reason the subject of neutron-proton pairing hasexperienced a revival. Even-A nuclei have received most of the attention so far, but odd-Anuclei, in which pairing is affected by the odd nucleon, are equally important. In a previ-ous paper[1] we analyzed the competition between neutron-proton (np) and like-particle(pp and nn) pairing in even-even fp-shell nuclei and in odd-odd N = Z nuclei, arguingthat a simple model based on the group SO(5) captured the essentials of full shell-modelcalculations, despite deformation, spin-orbit splitting, and other physics that the simplemodel omits. Here we turn to odd-A nuclei, discussing the same competition when anextra nucleon is present and pairing correlations are blocked. We also touch on odd-oddnuclei, of which only those with N = Z were treated in Ref. [1]. Though our focus is onground states, we briefly discuss the pairing competition at finite temperature as well.We begin with the predictions of the simple model, a full description of which is in Ref.[1]. Briefly, the isovector angular-momentum-zero nn, pp, and np pair creation operators,the corresponding annihilation operators, the three isospin generators, and the numberoperator form the algebra SO(5). The Hamiltonian consists of three equally weightedpairing terms and has the ground-state expectation valueE = −G2[n(Ω−n− 64)− ν(Ω−ν − 64) + t(t+ 1)− T (T + 1)], (1)where G is a the pairing-force strength constant, n is the number of nucleons, Ω is halfthe number of (degenerate) single-particle levels, T is the isospin, and ν and t are thethe seniority and “reduced isospin”, which take the values ν = 0, t = 0 in even-A groundstates and ν = 1, t = 1/2 in odd-A ground states. When ν = 0 Eq. (1) becomesE = −G2[n(Ω−n− 64)− T (T + 1)]. (2)When ν = 1 we haveE = −G2[n(Ω−n− 64)− (Ω +12)− T (T + 1)]. (3)These equations imply a blocking in odd-A nuclei very similar to what would be presentwithout np pairing; in fact in the ordinary like-particle seniority model the pairing energiesalong any even-Z isotope chain, in which there are no odd-odd nuclei, differ only by asmall constant from Eqs. (2) and ( 3). This prediction is confirmed by shell-model MonteCarlo calculations, which we describe shortly. Since the np pairing replaces some of thelike-particle pairing when an isospin-symmetric interaction is used, it would seem thatboth pairing modes are blocked in odd-A nuclei, though pp pairing is apparently almostunaffected by an odd neutron.This can be seen explicitly by examining the competition among nn, pp, and np pairs,something we discussed in even-A nuclei in Ref. [1]. Defining pair “number operators” asin that paper, (for a brief but exhaustive discussion of these operators see Ref. [2]), weuse techniques described in Ref. [3] to obtain for odd-A ground states:〈Npp〉 =(Tc + 1)(N − Tc)(1 −N−Tc−32Ω)2Tc + 3〈Nnp〉 =(N − Tc)(1 −N−Tc−22Ω)2Tc + 3〈Nnn〉 =(Tc + 1)(N + Tc)(1 −N+Tc+12Ω) + (Tc +NΩ)2Tc + 3(4)if N ≡ (n− 1)/2 (the total number of pairs) and Tc ≡ T − 1/2 (the core isospin) are botheven or both odd, and〈Npp〉 =Tc (N − Tc)(1−N−Tc−12Ω)2Tc + 1〈Nnp〉 =(N − Tc + 1)(1 −N−Tc−12Ω)2Tc + 1〈Nnn〉 =Tc (N + Tc)(1−N+Tc−32Ω) + (Tc − 1)(1−N+TcΩ)2Tc + 1(5)if N and Tc ≡ T+1/2 are both even or both odd (the relation between Tc and T is differentin the two sets of expressions). Combining these results with those for even-A nuclei fromRef. [1], we plot on the left-hand side of Figure 1 the numbers of each kind of pair (scaledby 0.5; we discuss this factor shortly) as neutron number increases in the Cr isotopes. Asthe figure shows, 〈Nnn〉 oscillates sharply while the other pair-numbers stagger less. Thereason is that when N is even as well as Z, nn and pp pairing (particularly the formerfor N > Z) are enhanced at the expense of np pairing. Adding a neutron to make Nodd blocks the nn pairing, thereby reducing 〈Nnn〉, but also blocks np pairing to a degreeso that 〈Nnp〉 cannot take advantage of the reduced coherence in the nn condensate. Alittle surprisingly, perhaps, 〈Npp〉 is not able to take any advantage of the slight drop innp pairing; it doesn’t increase until N becomes even again and 〈Nnp〉 is reduced moresignificantly by the increased strength of the nn condenstate.This picture augments that described in Ref. [1] and it is natural to ask how muchit has to do with reality. Since the strengths of the individual pairing modes and eventhe total isovector pairing strength are hard to extract from the limited data available,we again turn to a large-scale shell model Monte Carlo (SMMC) calculation in the full fpshell to see if the physics plays out in the same way. Unfortunately in odd-A and odd-odd (excepting N = Z) nuclei the notorious sign-problem[4] inherent in SMMC studieswith realistic interactions cannot be circumvented at low temperatures (T ≤ 0.8 MeV)by “g-extrapolation”[5]. We in large part avoid the problem, however, by using “pairingplus multipole-multipole” interaction of the kind used in Ref. [6]. In even-even nucleithis interaction has been shown to do a good job with essential features of the spectrum,including isovector pairing correlations, which have been checked against the predictionsof the realistic KB3 interaction. We fix the temperature at T = 0.4 MeV, which shouldbe sufficient to cool a nucleus to near its ground state. Although a mild sign problemstill affects the nuclei considered here (the sign for 49Cr is 0.35 ± 0.01 at T = 0.4 MeV)SMMC calculations can be performed without g-extrapolation. The only consequenceof the residual sign problem is that statistical uncertainties for odd-A nuclei are slightlyworse than for even-even nuclei, in which there is no sign-problem at all.Figure 1: The quantities Ω〈Npp〉 (solid line), Ω〈Nnn〉 (dashed line), and Ω〈Nnp〉 (dottedline) for the Cr isotopes. On the left are the SO(5) results described in the text (withΩ = 10, half the number of single-particle levels in the fp shell) scaled by a factor 0.5. Onthe right are the results of the Shell Model Monte Carlo calculation with Gpair = 20/AMeV and the quadrupole coupling constant χ = 134A−11/3 MeV/fm4.The right-hand sides of Figure 1 contains the SMMC results for the same quantitiesdiscussed in SO(5), and is clearly similar to the left-hand side. The minor differencesbetween the two panels, most apparent in 〈Nnn〉 for large N − Z, are most likely due topresence of several nondegenerate levels in the fp shell. We conclude, as before, that thepresence of physics beyond SO(5) reduces the strength of each the three pairing modes byabout a factor of two, but does not drastically alter the balance of power among them.Where exactly does the factor of two come from? The shell-model contains manyphysical effects not included in SO(5), but we focus here on the role of spin-orbit splitting,which we can mock up in a two-level version of the SO(5) model. (For the formalism,applicable to seniority-zero states, see Ref. [7, 8].) Although the matrices one must diag-onalize are no longer tiny, they are still small. To mimic the separation of the f7/2 levelfrom the rest of the fp shell, we take our two levels to have Ω1 = 4 and Ω2 = 6 and splitthem by ǫ = 10G, a number close to the real ratio of spin-orbit splitting to pairing-forcestrength. In Figure 2 we plot the numbers of the 3 kinds of pairs together with the SMMCpredictions, now just for even-A Cr isotopes (i.e. for every other point in Figure 1). Thesplitting weakens the coherence of the pairs so that the numbers now agree quite wellwith the shell-model calculations without the factor of 2 scaling, except for 〈Nnn〉 at largeN −Z. The disagreement there clearly reflects the residual splitting between the other fplevels; it doesn’t appear until there are more neutrons than can be accomodated in the f7/2level. All this leads us to attribute the differences in scale between simple SO(5) and fullshell-model calculations to spin-orbit splitting, or more generally to the nondegeneracy ofthe the single particle states, which can also reflect deformation.0510152048 50 52 54 56A051015pp and npnnContribution to Pairing EnergyFigure 2: Comparison of the two-level model and shell model. In the upper panel arethe quantities Ω〈Npp〉 and Ω〈Nnp〉 and in the lower one Ω〈Nnn〉. The two level model isdescribed in the text; the results are connected by full lines. The SMMC results, with thesmall error bars as indicated, are connected by the dashed lines.We turn now to the Mn isotopes, which have Z = 25 and are odd-odd as well asodd-even. Figure 3 shows SO(5) and SMMC results for the pair numbers. The figure issplit into 4 panels to separate the odd-A isotopes from the even-A (odd-odd) isotopes.The agreement between (scaled) SO(5) and the SMMC is good in the upper panels , butnoticeably less so in lower panels. The problem is that the model predicts the wrongisospin for the ground states in odd-odd isotopes with N−Z 6= 0. The ground states in allthese nuclei have T = Tz, while the model doesn’t even contain such states; they appearonly when a pair is broken and, if you believe the model, should lie much higher in energy.Figure 3: The quantities Ω〈Npp〉 (solid line), Ω〈Nnn〉 (dashed line), and Ω〈Nnp〉 (dottedline) in the odd-A (top) and odd-odd (bottom) isotopes of Mn. The left side contains theSO(5) results, the right the SMMC results.In the real world, however, the breaking of a pair is offset by the shell closure at N = 28and by the quadrupole-quadrupole force, for the following reason: When N ≥ 28 (A ≥ 53here) the lowest-lying states in the pure single-particle model have T = Tz because the filledneutron shell prohibits the operator τ+ from giving anything but zero. Higher T statescorrespond to particle-hole excitations, which have too much single-particle excitationenergy to be pulled all the way down when the pairing interaction is added. A similarargument in the Nilsson scheme implies that T = Tz in the deformed nucleus52Mn aswell. Our version of SO(5) includes only fully paired states even in the two-level model,and so overlooks the effects of single-particle splitting and deformation. As a result, itsdescription of odd-odd nuclei with N 6= Z is lacking. 11Interestingly, the SMMC with our chosen strength of the pairing plus quadrupole interaction also failsWe turn finally to the temperature dependence of pairing correlations in proton-richnuclei (related work on nuclei with N = Z nuclei appears in Refs. [6, 9]). As an examplewe have chosen 49Cr, an odd-A nucleus with N = Z + 1. We again use the SMMCwith the Hamiltonian and model space described above to calculate pair-numbers, butnow for temperatures spanning the range T = 0.4 − 2 MeV. Figure 4 shows the results.The blocking effect is clearly visible in the pair correlations: at low temperatures theneutron-neutron correlations are smaller than both the proton-neutron and proton-protoncorrelations, despite the presence of five neutrons to only four protons. As the temperatureincreases both the pairing strengths and the blocking effect decrease, so that for T > 1.2MeV the neutron-neutron pairing correlations are the largest and the neutron-protonthe weakest, just as if there were no blocking. It appears that although both pairingcorrelations and Pauli blocking decrease with temperature, the latter disappear first.0.0 0.5 1.0 1.5 2.0T [MeV]0.02.04.06.08.0Contribution to Pairing Energy49CrPROTON−PROTONNEUTRON−NEUTRONPROTON−NEUTRONFigure 4: The quantities Ω〈Npp〉 (circles), Ω〈Nnn〉 (squares), and Ω〈Nnp〉 (diamonds) inthe odd-A nucleus 49Cr as a function of temperature calculated in the SMMC.One can imagine using functional methods to study temperature-dependence in SO(5)as well, provided some higher seniority states are included. We defer that exercise, how-ever, and instead repeat in closing that although SO(5) predicts the wrong isospin forto predict the correct ground-state isospin in 52Mn. The problem is clearly in the quadrupole-quadrupoleforce, which is a little too weak. Strengthening it by about 10% pulls the T = Tz = 1 state below theT = 2 state, as it is in reality.odd-odd nuclei with N 6= Z, it makes the competition among the three kinds of isovectorpairs in all other isotopes very easy to understand. Most of the effects that escape themodel are recovered by a simple modification — two sets of degenerate levels instead ofone — even when Pauli blocking, which affects all three modes and the competition amongthem, is at play.We were supported in part by the U.S. Department of Energy under grants DE-FG05-94ER40827 and DE-FG03-88ER-40397, by the U.S. National Science Foundation undergrants PHY94-12818 and PHY94-20470, and by the Danish Research Council.References[1] J. Engel, K. Langanke, and P. Vogel, Phys. Lett. B389, 211 (1996).[2] J. Dobes, Phys.. Lett. B413, 239 (1997).[3] K.T. Hecht, The Vector Coherent State Method and Its Application to Problems ofHigher Symmetries (Springer-Verlag, Berlin Heidelberg, 1987).[4] S.E. Koonin, D.J. Dean and K. Langanke, Physics Reports 278, 1 (1997).[5] Y. Alhassid, D.J. Dean, S.E. Koonin, G. Lang and W.E. Ormand, Phys. Rev. Lett.72, 613 (1994).[6] K. Langanke, P. Vogel and D.-C. Zheng, Nucl. Phys. A626 (1997) 735.[7] G. G. Dussel, E. Maqueda, R. P. J. Perazzo, and J. A. Evans, Nucl. Phys. A450, 164(1987).[8] O. Civitarese, M. Reboiro, and P. Vogel, Phys. Rev. C56, 1840 (1997).[9] K. Langanke, D.J. Dean, P.B. Radha and S.E. Koonin, Nucl. Phys. A613 (1997) 253.

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Isovector Pairing in Odd-A Proton-Rich Nuclei

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