Pairing-induced speedup of nuclear spontaneous fission
Jhilam Sadhukhan, J. Dobaczewski, W. Nazarewicz, J. A. Sheikh, A. Baran
PPairing-induced speedup of nuclear spontaneous fission
Jhilam Sadhukhan,
1, 2, 3
J. Dobaczewski,
4, 5, 6
W. Nazarewicz,
7, 2, 4
J. A. Sheikh,
1, 2 and A. Baran Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA Physics Division, Oak Ridge National Laboratory,P. O. Box 2008, Oak Ridge, Tennessee 37831, USA Physics Group, Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, India Institute of Theoretical Physics, Faculty of Physics,University of Warsaw, Pasteura 5, PL-02-093 Warsaw, Poland Department of Physics, P.O. Box 35 (YFL), University of Jyv¨askyl¨a, FI-40014 Jyv¨askyl¨a, Finland Joint Institute of Nuclear Physics and Applications,P. O. Box 2008, Oak Ridge, Tennessee 37831, USA Department of Physics and Astronomy and NSCL/FRIB Laboratory,Michigan State University, East Lansing, Michigan 48824, USA Institute of Physics, University of M. Curie-Sk(cid:32)lodowska, ul. Radziszewskiego 10, 20-031 Lublin, Poland (Dated: May 25, 2018)
Background:
Collective inertia is strongly influenced at the level crossing at which quantum system changesdiabatically its microscopic configuration. Pairing correlations tend to make the large-amplitude nuclear collectivemotion more adiabatic by reducing the effect of those configuration changes. Competition between pairing andlevel crossing is thus expected to have a profound impact on spontaneous fission lifetimes.
Purpose:
To elucidate the role of nucleonic pairing on spontaneous fission, we study the dynamic fission trajec-tories of
Fm and
Pu using the state-of-the-art self-consistent framework.
Methods:
We employ the superfluid nuclear density functional theory with the Skyrme energy density functionalSkM ∗ and a density-dependent pairing interaction. Along with shape variables, proton and neutron pairing cor-relations are taken as collective coordinates. The collective inertia tensor is calculated within the nonperturbativecranking approximation. The fission paths are obtained by using the least action principle in a four-dimensionalcollective space of shape and pairing coordinates. Results:
Pairing correlations are enhanced along the minimum-action fission path. For the symmetric fission of
Fm, where the effect of triaxiality on the fission barrier is large, the geometry of fission pathway in the space ofshape degrees of freedom is weakly impacted by pairing. This is not the case for
Pu where pairing fluctuationsrestore the axial symmetry of the dynamic fission trajectory.
Conclusions:
The minimum-action fission path is strongly impacted by nucleonic pairing. In some cases, thedynamical coupling between shape and pairing degrees of freedom can lead to a dramatic departure from thestatic picture. Consequently, in the dynamical description of nuclear fission, particle-particle correlations shouldbe considered on the same footing as those associated with shape degrees of freedom.
PACS numbers: 24.75.+i, 25.85.Ca, 21.60.Jz, 21.30.Fe, 27.90.+b
Introduction — Nuclear fission is a fundamental phe-nomenon that is a splendid example of a large-amplitudecollective motion of a system in presence of many-bodytunneling. The corresponding equations involve poten-tial, dissipative, and inertial terms [1]. The individual-particle motion gives rise to shell effects that influencethe fission barriers and shapes on the way to fission, andalso strongly impact the inertia tensor through the cross-ings of single-particle levels and resulting configurationchanges [2–4]. The residual interaction between crossingconfigurations is strongly affected by nucleonic pairing:the larger pairing gap ∆ the more adiabatic is the collec-tive motion [5–9].The enhancement of pairing correlations along the fis-sion path was postulated in early Ref. [10] using simplephysical arguments. Since the collective inertia roughlydepends on pairing gap as ∆ − [5, 11–14], by choosing apathway with larger ∆, the fissioning nucleus can lowerthe collective action. This means that in searching for the least action trajectory the gap parameter should betreated as a dynamical variable. Indeed, macroscopic-microscopic studies [15–18] demonstrated that pairingfluctuations can significantly reduce the collective action;hence, affect predicted spontaneous fission lifetimes.Our long-term goal is to describe spontaneous fis-sion (SF) within the superfluid nuclear density func-tional theory by minimizing the collective action in many-dimensional collective space. The important milestonewas a recent paper [19], which demonstrated that pre-dicted SF pathways strongly depend on the choice of thecollective inertia and approximations involved in treatinglevel crossings. The main objective of the present work isto elucidate the role of nucleonic pairing on SF by study-ing the dynamic fission trajectories of Fm and
Puin a four-dimensional collective space. In addition to twoquadrupole moments defining the elongation and triaxi-ality of nuclear shape we consider the strengths of neu-tron and proton pairing fluctuations. Since in our model a r X i v : . [ nu c l - t h ] O c t the effect of triaxiality on the fission barrier is larger for Fm ( ∼ Pu ( ∼ Theoretical framework — To calculate the SF half-life,we closely follow the formalism described in Ref. [19].In the semi-classical approximation, the SF half-life canbe written as [23, 24] T / = ln 2 / ( nP ), where n is thenumber of assaults on the fission barrier per unit time(here we adopt the standard value of n = 10 . s − ) and P = 1 / (1 + e S ) is the penetration probability expressedin terms of the fission action integral, S ( L ) = (cid:90) s out s in (cid:126) (cid:112) M eff ( s ) ( V ( s ) − E ) ds, (1)calculated along the fission path L ( s ). The effec-tive inertia M eff ( s ) [19, 23–25] is obtained from themulti-dimensional nonperturbative cranking inertia ten-sor M C . E is the collective ground state energy, and ds is the element of length along L ( s ). To compute the po-tential energy V , we subtract the vibrational zero-pointenergy E ZPE from the Hartree-Fock-Bogoliubov (HFB)energy E HFB obtained self-consistently from the con-strained HFB equations for the Routhian:ˆ H (cid:48) = ˆ H HFB − (cid:88) µ =0 , λ µ ˆ Q µ − (cid:88) τ = n,p (cid:16) λ τ ˆ N τ − λ τ ∆ ˆ N τ (cid:17) , (2)where ˆ H HFB , ˆ Q µ , and ˆ N τ represent the HFB hamilto-nian, axial ( µ = 0) and nonaxial ( µ = 2) componentsof the mass quadrupole moment operator, and neutron( τ = n ) and proton ( τ = p ) particle-number opera-tors, respectively. The particle-number dispersion terms∆ ˆ N τ = ˆ N τ − (cid:104) ˆ N τ (cid:105) , controlled by the Lagrange mul-tipliers λ τ , determine dynamic pairing correlations ofthe system [26, 27]. That is, λ τ = 0 corresponds tothe static HFB pairing. Dynamic pairing fluctuationsstronger than those obtained within the static solutionare described by λ τ >
0. The overall magnitude of pair-ing correlations (static + dynamic) can be related to theaverage pairing gap ∆ τ [28, 29]. In this study, λ τ areused as two independent dynamical coordinates to scanover a wide range of pairing correlations (or ∆ τ ).The one-dimensional path L ( s ) is defined in the multi-dimensional collective space by specifying the collectivevariables { X i } ≡ { Q , Q , λ n + λ p , λ n − λ p } as func-tions of path’s length s . Furthermore, to render collectivecoordinates dimensionless, we define x i = X i /δq i . In thisway, ds becomes dimensionless as well. In this work, wetake δq = δq = 1 b and δq = δq = 0 .
01. The dy-namical coordinates x and x control respectively theisoscalar and isovector pairing fluctuations.As a continuation of our previous study [19], we firstconsider the SF of Fm, which is predicted to un-dergo a symmetric split into two doubly magic
Sn fragments [30]. Therefore, the crucial shape degreesof freedom in this case are elongation and triaxiality;they are represented by quadrupole moments Q and Q defined as in Table 5 of Ref. [31]. To computethe total energy E HFB and inertia tensor M C , we em-ployed the symmetry-unrestricted HFB solver HFODD(v2.49t) [32]. To be consistent with the previous work[19], we use the Skyrme energy density functional SkM ∗ [33] in the particle-hole channel.The particle-particle interaction is approximated bythe density-dependent mixed pairing force [34]. The zero-point energy E ZPE is estimated by using the Gaussianoverlap approximation [16, 35, 36]. To obtain the ex-pression for E ZPE , we neglected the derivatives of thepairing fields with respect to λ τ ; we checked, how-ever, that the topology of fission path is hardly sensi-tive to the detailed structure of E ZPE . The inertia ten-sor M C was obtained from the nonperturbative crankingapproximation to Adiabatic Time Dependent HFB as de-scribed in Refs. [19, 37]. The density-matrix derivativeswith respect to collective coordinates used to compute M C [37], were obtained by using finite differences withsteps δq i . Finally, to obtain the minimum action path-ways we adopted two independent algorithms to ensurethe robustness of the result: the dynamical program-ing method (DPM) [23] and the Ritz method [24]. Inall cases considered, both approaches give consistent an-swers. Results — In the first step, to assess the relative im-portance of isoscalar and isovector pairing degrees offreedom, the minimum-action path was calculated inthe three-dimensional space of coordinates x , x , and x . To this end, we adopted a 90 × ×
31 mesh with21 ≤ x ≤ − ≤ x ≤
50, and − ≤ x ≤
15. Co-ordinate x was fixed according to the two-dimensionaldynamical path of Ref. [19]. The contour maps of V inthe x – x plane for x = 0 and x – x plane for x = 0are displayed in Fig. 1 (left).Since the individual components of the full (three-dimensional) inertia tensor are difficult to interpret, fol-lowing [19] in Fig. 1 (right) we show the cubic-root-determinant of inertia tensor |M C | / . It can be seenthat at large values of x , the peaks in |M C | / due tolevel crossings disappear and, moreover, the magnitudeof inertia generally decreases with x . This is consis-tent with general expectations for the effect of paring oncollective inertia. On the other hand, variations in x have little effect on |M C | / and V . This result is con-firmed by computing the minimum action path in the( x , x , x ) space: the fissioning system prefers to main-tain large proton and neutron pairing gaps and, at thesame time, x ≈
0. Consequently, in the SF, this degreeof freedom seems to play less important role.In the previous work, we have shown that the min-imum action path breaks the axial symmetry to avoidlevel crossings and minimize the level density of single-
50 70 90 (d) (c) -8081624 (a)
FmV
50 70 90-12-60612 | M C | (b) x ( x = ) x ( x = ) x FIG. 1. (Color online) Contour maps of V (left, in MeV) and |M C | / (right, in (cid:126) MeV − /1000), calculated for Fm inthe x – x plane for x = 0 (top) and x – x plane for x = 0(bottom). The energies are plotted relatively to the ground-state value. (c)
30 50 70 90 (d) | M C | (a) V Fm
30 50 70 9001020 (b) x x ( x = ) x ( x = ) FIG. 2. (Color online) Similar as in Fig. 1 except for the x – x plane for x = x = 0 (top) and x – x plane for x = x = 0 (bottom). particle states around the Fermi level. In contrast, pair-ing correlations grow with single-particle level density.Consequently, pairing is expected to impact V and M eff in a different way. Namely, as pairing (or x ) increases,the potential energy is expected to grow – as one de-parts from the self-consistent value – while the collectiveinertia is reduced. The interplay between these two op-posing tendencies determines the least-action trajectory.To evaluate how the fission path is modified due to pair- ing, in the next step we minimize the collective action inthe ( x , x , x ) space. Here we assume x = 0 and adoptthe value of E = 1 MeV to be consistent with Ref. [19].Figure 2 displays the resulting contour maps of V and |M C | / . The upper panels correspond to the situationdiscussed in Ref. [19], in which dynamical pairing is disre-garded ( x = 0). As seen in Fig. 2(a), triaxial coordinate x reduces the fission barrier height by slightly more than4 MeV. The fluctuations seen in |M C | / in Fig. 2(c) re-flect crossings of single-particle levels at the Fermi level.The results shown in the lower panels correspond to theaxial shape ( x = 0); they are again consistent with thegeneral dependence of potential energy and collective in-ertia on pairing correlations.
123 4 5 4 374 41 3 7 7 515913 Fm s t a t i c
2D (a)
40 60 80 10001020 (b)3D x x x FIG. 3. (Color online) Projections of the three-dimensional(3D, solid line) dynamic SF path for
Fm on the x – x planefor x = x = 0 (a) and x – x plane for x = x = 0 (b),calculated using the DMP technique. The dash-dotted lineshows for comparison the two-dimensional (2D) path com-puted without pairing fluctuations. The static SF path cor-responding to the minimized collective potential [19] is alsoplotted (dotted line). Symbols on the paths denote the pathlengths in units of 10. Potentials V of Fig. 2 are drawn as abackground reference. In Figs. 3(a) and 3(b), we show projections of the min-imum action path onto the x – x and x – x planes, re-spectively. The two-dimensional (2D) fission path calcu-lated without pairing fluctuations ( x = x = 0) and thestatic SF path corresponding to the valley of the mini-mized collective potential are also shown for comparison.Evidently, the triaxiality along the fission path 3D is re-duced at the expense of enhanced pairing. Nevertheless,owing to the reduced action S , the calculated SF half-lifeof Fm in the 3D variant is decreased by as much asthree decades.Figure 4 summarizes our results for
Fm. Namely, itshows V , M C eff , S , and ∆ τ along the fission paths cal-culated with dynamical (3D) and static (2D) pairing.Compared to the 2D path, the 3D path is shorter andit favors lower collective inertia at a cost of higher po-tential energy, both being the result of enhanced pairingcorrelations. It is interesting to notice that the collec-tive potentials V in 2D and 3D are fairly different, andthey both deviate from the static result that is usuallyinterpreted in terms of a fission barrier, or a saddle point. Fm (a)static (b)2D3D M C e ff V (d) p ( D ) n ( D ) p (3D)n (3D) Path length L (c)3D2D S FIG. 4. (Color online) Potential V (in MeV) (a), effectiveinertia M C eff (in (cid:126) MeV − /1000) (b), action S (c), and av-erage pairing gaps ∆ n and ∆ p (in MeV) (d) plotted alongthe 2D (static pairing, dotted line) and 3D (dynamic pairing,solid line) paths. The static fission barrier is displayed forcomparison in panel (a). While the least-action pathways in
Fm are not thatfar from the static SF path, this is not the case for
Pu,where the energy gain on the first barrier resulting fromtriaxiality is around 2 MeV, that is, significantly less thanin
Fm. To illustrate the impact of pairing fluctuationson the SF of
Pu, we consider the least-action collec-tive path between its ground state and superdeformed fis-sion isomer. In this region of collective space, reflection-asymmetric degrees of freedom are less important; hence,the 3D space of ( x , x , x ) is adequate.As seen in Fig. 5, in the region of the first saddle in Pu s t a ti c (a)
40 60 8002040 (b) x x x FIG. 5. (Color online) Similar as in Fig. 3 but for
Pu.The static SF path is marked by the dotted line. the static calculations, the impact of dynamics on theleast-action pathway for
Pu is dramatic. Comparedto static-pairing calculation, in the 2D calculations theeffect of triaxiality is significantly reduced, and in 3Dcalculations the axial symmetry of the system is fullyrestored.
Conclusions — In this study, we extended the self-consistent least-action approach to the SF by consideringcollective coordinates associated with pairing. Our ap-proach takes into account essential ingredients impactingthe SF dynamics [22]: (i) spontaneous breaking of meanfield symmetries; (ii) diabatic configuration changes dueto level crossings; (iii) reduction of nuclear inertia bypairing; and (iv) dynamical fluctuations governed by theleast-action principle.We demonstrated that the SF pathways and lifetimesare significantly influenced by the nonperturbative collec-tive inertia and dynamical fluctuations in shape and pair-ing degrees of freedom. While the reduction of the collec-tive action by pairing fluctuations has been pointed outin earlier works [10, 13, 15–17] and also very recently in aself-consistent approach [38], our work shows that pairingdynamics can profoundly impact penetration probability,that is, effective fission barriers, by restoring symmetriesspontaneously broken in a static approach.Our calculations for
Fm and
Pu show that thedynamical coupling between shape and pairing degreesof freedom can lead to a dramatic departure from thestandard static picture based on saddle points obtainedin static mean-field calculations. In particular, for
Pu,pairing fluctuations restore the axial symmetry aroundthe fission barrier, which in the static approach is brokenspontaneously. The examples presented in this work, inparticular in Figs. 4 and 5, illustrate how limited is thenotion of fission barrier.The future improvements, aiming at systematic com-parison with experiment, will include: the full AdiabaticTime Dependent HFB treatment of collective inertia,adding reflection asymmetric collective coordinates, andemploying energy density functionals optimized for fis-sion [39]. The work along all these lines is in progress.Discussions with G.F. Bertsch, K. Mazurek, andN. Schunck are gratefully acknowledged. This studywas initiated during the Program INT-13-3 “Quanti-tative Large Amplitude Shape Dynamics: fission andheavy ion fusion” at the National Institute for Nu-clear Theory in Seattle. This material is based uponwork supported by the U.S. Department of Energy, Of-fice of Science, Office of Nuclear Physics under AwardNumbers No. DE-FG02-96ER40963 (University of Ten-nessee) and No. de-sc0008499 (NUCLEI SciDAC Col-laboration); by the NNSA’s Stewardship Science Aca-demic Alliances Program under Award No. DE-FG52-09NA29461 (the Stewardship Science Academic Alliancesprogram); by the Academy of Finland and Univer-sity of Jyv¨askyl¨a within the FIDIPRO programme; andby the Polish National Science Center under ContractNos. 2012/07/B/ST2/03907. An award of computer timewas provided by the National Institute for Computa-tional Sciences (NICS) and the Innovative and NovelComputational Impact on Theory and Experiment (IN-CITE) program using resources of the OLCF facility. [1] W. Swiatecki and S. Bjørnholm, Phys. Rep. , 325 (1972).[2] D. L. Hill and J. A. Wheeler, Phys. Rev. , 1102 (1953).[3] L. Wilets, Phys. Rev. , 372 (1959).[4] R. W. Hasse, Rep.Prog. Phys. , 1027 (1978).[5] M. Brack, J. Damgaard, A. S. Jensen, H. C. Pauli, V. M.Strutinsky, and C. Y. Wong, Rev. Mod. Phys. , 320(1972).[6] G. Sch¨utte and L. Wilets, Nucl. Phys. A , 21 (1975);Z. Phys. A , 313 (1978).[7] V. Strutinsky, Z. Phys. A , 99 (1977).[8] W. Nazarewicz, Nucl. Phys. A , 489 (1993).[9] T. Nakatsukasa and N. R. Walet, Phys. Rev. C , 1192(1998).[10] L. G. Moretto and R. P. Babinet, Phys. Lett. B , 147(1974). [11] M. Urin and D. Zaretsky, Nucl. Phys. , 101 (1966).[12] T. Ledergerber and H.-C. Pauli, Nucl. Phys. A (1973).[13] Y. A. Lazarev, Phys. Scr. , 255 (1987).[14] K. Pomorski, Int. J. Mod. Phys. E , 237 (2007).[15] A. Staszczak, A. Baran, K. Pomorski, and K. B¨oning,Phys. Lett. B , 227 (1985).[16] A. Staszczak, S. Pi(cid:32)lat, and K. Pomorski, Nucl. Phys. A , 589 (1989).[17] Z. (cid:32)Lojewski and A. Staszczak, Nucl. Phys. A , 134(1999).[18] M. Mirea and R. C. Bobulescu, J. Phys. G , 055106(2010).[19] J. Sadhukhan, K. Mazurek, A. Baran, J. Dobaczewski,W. Nazarewicz, and J. A. Sheikh, Phys. Rev. C ,064314 (2013).[20] A. Staszczak, A. Baran, and W. Nazarewicz, Int. J. Mod.Phys. E , 552 (2011).[21] J. A. Sheikh, W. Nazarewicz, and J. C. Pei, Phys. Rev.C , 011302 (2009).[22] J. Negele, Nucl. Phys. A , 371 (1989).[23] A. Baran, K. Pomorski, A. Lukasiak, and A. So-biczewski, Nucl. Phys. A , 83 (1981).[24] A. Baran, Phys. Lett. B , 8 (1978).[25] A. Baran, Z. (cid:32)Lojewski, K. Sieja, and M. Kowal, Phys.Rev. C , 044310 (2005).[26] N. L. Vaquero, T. R. Rodr´ıguez, and J. L. Egido, Phys.Lett. B , 520 (2011).[27] N. L. Vaquero, J. L. Egido, and T. R. Rodr´ıguez, Phys.Rev. C , 064311 (2013).[28] J. Dobaczewski, H. Flocard, and J. Treiner, Nucl. Phys.A (1984).[29] J. Dobaczewski, W. Nazarewicz, and T. R. Werner,Phys. Scr. , 15 (1995).[30] A. Staszczak, A. Baran, J. Dobaczewski, andW. Nazarewicz, Phys. Rev. C , 014309 (2009).[31] J. Dobaczewski and P. Olbratowski, Comput. Phys. Com-mun. , 158 (2004).[32] N. Schunck, J. Dobaczewski, J. McDonnell, W. Satu(cid:32)la,J. Sheikh, A. Staszczak, M. Stoitsov, and P. Toivanen,Comput. Phys. Commun. , 166 (2012).[33] J. Bartel, P. Quentin, M. Brack, C. Guet, and H.-B.H˚akansson, Nucl. Phys. A , 79 (1982).[34] J. Dobaczewski, W. Nazarewicz, and M. Stoitsov, Eur.Phys. J. A , 21 (2002).[35] A. Baran, A. Staszczak, J. Dobaczewski, andW. Nazarewicz, Int. J. Mod. Phys. E , 443 (2007).[36] A. Staszczak, A. Baran, and W. Nazarewicz, Phys. Rev.C , 024320 (2013).[37] A. Baran, J. A. Sheikh, J. Dobaczewski, W. Nazarewicz,and A. Staszczak, Phys. Rev. C , 054321 (2011).[38] S. A. Giuliani, L. M. Robledo, and R. Rodriguez-Guzman, arXiv:1408.6940 (2014).[39] M. Kortelainen, J. McDonnell, W. Nazarewicz, P.-G.Reinhard, J. Sarich, N. Schunck, M. V. Stoitsov, andS. M. Wild, Phys. Rev. C85