Instanton-motivated study of spontaneous fission of odd-A nuclei
IInstanton - motivated study of spontaneous fission of odd-A nuclei
W. Brodzi´nski, J. Skalski
National Centre for Nuclear Research, Pasteura 7, PL-02-093 Warsaw, Poland (Dated: July 3, 2020)Using the idea of the instanton approach to quantum tunneling we try to obtain a method ofcalculating spontaneous fission rates for nuclei with the odd number of neutrons or protons. Thisproblem has its origin in the failure of the adiabatic cranking approximation which serves as thebasis in calculations of fission probabilities. Selfconsistent instanton equations, with and withoutpairing, are reviewed and then simplified to non-selfconsistent versions with phenomenological single-particle potential and seniority pairing interaction. Solutions of instanton-like equations withoutpairing and actions they produce are studied for the Woods-Saxon potential along realistic fissiontrajectories. Actions for unpaired particles are combined with cranking actions for even-even coresand fission hindrance for odd- A nuclei is studied in such a hybrid model. With the assumed equalmass parameters for neighbouring odd-A and even-even nuclei, the model shows that freezing the K π configuration leads to a large overestimate of the fission hindrance factors. Actions with adiabaticconfigurations mostly show not enough hindrance; instanton-like actions for blocked nucleons correctthis, but not sufficiently. I. INTRODUCTION
Nuclear fission is thought to be a collective process, classically envisioned in analogy to fragmentation of a liquiddrop. In reactions induced by neutrons and light or heavy ions, fission is one of many possible deexcitation channels ofa formed compound nucleus. On the other hand, spontaneous fission is a decay of the nuclear ground state (g.s.) whichexhibits its meta-stability and involves quantum tunneling through a potential barrier. In a theoretical approach,the fission barrier follows from a model of the shape-dependent nuclear energy. In practical terms, it is calculatedeither from a selfconsistent mean-field functional or a microscopic-macroscopic model, as a landscape formed by thelowest energies E ( q ) at fixed values of a few arbitrarily chosen coordinates q = ( q , ..., q i , ... ) (for simplicity assumenddimensionless) describing nuclear shape. The obscure part of the current approach relates to a) the likely insufficiencyof included coordinates and b) a description of tunneling dynamics, essentially shaped after the Gamow method, butwithout a clear understanding of mass parameters and conjugate momenta entering the formula for decay rate.The experimentally well established presence of pairing correlations in nuclei gives rationale for using cranking [1, 2]or adiabatic Time-Dependent Hartree-Fock(-Bogolyubov) - ATDHF(B) - approximation [3–5] in the description offission in even-even (e-e) nuclei. Indeed, as the lowest two-quasiparticle excitation in such nuclei has energy of atleast twice the pairing gap 2∆, which in heavy nuclei amounts to more than 1 MeV, one can, for collective velocities | ¯ h ˙ q | reasonably smaller than that, solve the time-dependent Schr¨odinger (or mean-field) equation to the first orderin ˙ q and obtain kinetic energy of shape changes: (cid:80) ij B q i q j ( q ) ˙ q i ˙ q j , with cranking (or ATDHFB) mass parameters B q i q j ( q ). Then one can apply the Jacobi variational principle to the imaginary under-the-barrier motion in order tofind the quasiclassical tunneling path q ( τ ) by minimizing action: S [ q ( τ )] = (cid:90) q fin q ini (cid:88) i p i dq i = (cid:90) q fin q ini (cid:113) B qq ( q ( τ ))[ E ( q ( τ )) − E ] dq. (1)Here, p i = (cid:80) j B q i q j ( q ) ˙ q j are the conjugate momenta; q (without index) is an effective coordinate along a path, usuallythe one of q i that controls elongation of the nucleus; B qq = (cid:80) kl B q k q l dq k dq dq l dq is the effective mass parametr along thefission path with respect to q . The Jacobi principle requires that a) q ini and q fin - the initial and final points of the paththrough a barrier - be fixed for all tunneling paths and b) on each trial path, E ( q ) − (cid:80) ij B q i q j ( q ) ˙ q i ˙ q j (the potentialminus kinetic energy) be constant and equal to E = E ( q ini ) = E ( q fin ), usually chosen as E g.s. + E zp - the g.s. energyaugmented by the zero-point energy of oscillations around the g.s. minimum in direction of fission, E zp = ¯ hω . Thespontaneous fission rate is given to the leading order by: ( ω π ) e − S min / ¯ h , with S min - the minimal action. By thefirst equality in (1), S equals the integral of twice the collective kinetic energy, B qq ˙ q , with (¯ h ˙ q ) = E ( q ) − E ] B qq , overthe time of passing the barrier. Estimating a posteriori collective velocities of the fictitious under-barrier motionfor heavy nuclei, with typical cranking mass parameter for the Woods-Saxon potential, B qq > ∼ h /MeV, and thefission barrier < ∼ h ˙ q < ∼ .
25 MeV, so the error of the cranking approximation might be believedmoderate.Situation changes rather dramatically for odd- Z or/and odd- N nuclei. For odd number of particles, their contri- a r X i v : . [ nu c l - t h ] J u l bution to the cranking mass parameter B q i q j , derived as if the adiabatic approximation were legitimate , reads: B q i q j = 2¯ h (cid:34) (cid:88) µ,ν (cid:54) = ν (cid:104) µ | ∂ ˆ h∂q i | ν (cid:105)(cid:104) ν | ∂ ˆ h∂q j | µ (cid:105) ( E µ + E ν ) ( u µ v ν + u ν v µ ) (2)+ 18 (cid:88) ν (cid:54) = ν (cid:16) ˜ (cid:15) ν ∂ ∆ ∂q i − ∆ ∂ ˜ (cid:15) ν ∂q i (cid:17) (cid:16) ˜ (cid:15) ν ∂ ∆ ∂q j − ∆ ∂ ˜ (cid:15) ν ∂q j (cid:17) E ν (cid:35) + 2¯ h (cid:88) ν (cid:54) = ν (cid:104) ν | ∂ ˆ h∂q i | ν (cid:105)(cid:104) ν | ∂ ˆ h∂q j | ν (cid:105) ( E ν − E ν ) ( u ν u ν − v ν v ν ) . Here, the odd nucleon occupies the orbital ν in the g.s.; ˆ h is the mean-field single - particle (s.p.) Hamiltonian, (cid:15) µ are its eigenenergies, ˜ (cid:15) ν = (cid:15) ν − λ , E µ = (cid:113) ˜ (cid:15) µ + ∆ , u and v are the usual BCS amplitudes. A common pairing gap∆ and Fermi energy λ were assumed for the g.s. and its two-quasiparticle excitations: those with the odd particle inthe state ν which give contribution in the square bracket that has the same form as the mass parameter for an e-enucleus, and those with the odd particle in the state ν (cid:54) = ν and the orbital ν paired, given by the last term of theformula. The latter becomes nearly singular, ∼ ( E ν − E ν ) − , at close avoided level crossings where E ν − E ν can be ofthe order of keV or less. This invalidates the very assumption underlying the cranking formula, except for ridiculouslysmall collective velocities. But there is still another deficiency: a departure from the symmetry preserved on a part ofthe fission trajectory often produces a negative contribution to the inertia parameter whose magnitude would dependon the proximity of the relevant crossing of levels of different symmetry classes. Although some calculations of fissionhalf-lives for odd nuclei with the cranking mass parameters (2) were done in the past, e.g. [6], the above-mentionedproblems make the precise minimization of action (1) for those nuclei both questionable and practically very difficult- a good illustration of near-singular cranking mass parameter [calculated with a formula more refined than (2)] inthe odd nucleus is provided in [7] (the middle panel of Fig. 4 there) [8].The well known experimental evidence, reviewed recently in [9], shows that the spontaneous fission rates of oddnuclei are three to five orders of magnitude smaller than those of their e-e neighbours. Although the explanationusually invokes the specialization energy - an increase in the fission barrier by the blocking of one level by a singlenucleon - a quantitative understanding is lacking at present. In particular, the combination of axial symmetry of thenuclear deformation and very different densities of s.p. levels with low- and high-Ω quantum numbers (Ω being theprojection of the s.p. angular momentum on the symmetry axis of a nucleus) could suggest a higher specializationenergy, and thus smaller fission rate, for configurations based on high-Ω orbitals, but the data [9] contradict this.While estimates of fission half-lives rely on the assumption of nearly adiabatic motion, doubtful for odd- A nuclei,the real-time solutions of Schr¨odinger-like dynamics are regular for any velocity profile ˙ q and any avoided crossings. Ingeneral, they lead to a population of levels above the Fermi energy. Analogous possibility must exist in the fictitiousimaginary-time motion, pertinent to quantum tunneling. In this light, a consideration of non-adiabatic tunneling -with fission paths formed at least in part by non-adiabatic configurations - presents itself as an interesting subject.Beyond-cranking effects could provide corrections to the standard cranking spontaneous fission rates in e-e nuclei andcan be crucial for spontaneous fission of odd- A nuclei and high- K isomersIn this paper, we present an attempt towards replacing the adiabatic cranking approximation by a scheme includingnon-adiabatic fission paths, motivated by the instanton method [10–14]. Instantons are solutions with the infiniteperiod to time-dependent mean-field equations in imaginary time τ = it , with the nuclear g.s. wave function as theboundary value. They arise from the saddle-point approximation to the path integral representation of the propagatorand give the leading contribution to spontaneous fission rate of the form: A inst exp( − S inst / ¯ h ). Here, S inst - instantonaction, is the counterpart of 2 S [ q ( τ )] in (1), while the prefactor A inst - the ratio of determinants including frequenciesof quadratic fluctuations around the instanton and the g.s. - for review see e.g. [15–17] - will not be considered it in thefollowing. The instanton with the smallest action (there can be more than one as the instanton equation determineslocal minima of action) gives fission half-life without the necessity of defining mass parameters . The resulting fissionpath involves all degrees of freedom of the mean-field state, not only shape parameters.The difficulty in solving for a selfconsistent instanton including pairing is beyond that of solving real-time TDHFBequations: the generically exponential τ -dependence of the HFB Z matrix [18], introducing components differing byorders of magnitude, has to be found from equations non-local in τ (see Sect. II C). Here, we treat the selfconsistenttheory as a motivation, and solve imaginary-time-dependent Schr¨odinger equation (iTDSE) with the phenomenologicalWoods-Saxon (W-S) potential to calculate action along various chosen paths. We use micro-macro energy for E ( q ).Since we reject cranking mass parameters for odd- A nuclei, we have to provide ˙ q without them. To this aim weuse cranking mass parameters of the neighbouring e-e nucleus. With this prescription, we can calculate manifestlybeyond-cranking actions and study their behaviour. Although we formulate equations with pairing, in the presentpaper we present iTDSE instanton-like solutions without it. To the best of our knowledge, such solutions and theiractions are discussed for the first time. Then, we combine instanton-like solutions for the odd nucleon with thecranking action with pairing for the e-e core in a hybrid model to study fission hindrance in odd- A nuclei. Within thismodel we calculate and compare fission half-lives obtained with and without constraining the Ω π (with π - parity)g.s. configuration.The presented approach cannot be as yet a basis for the systematic minimization of action over fission paths.Moreover, it differs from the instanton method by ignoring the anti-hermitean part of the imaginary-time mean-field.We think, however, that it presents some features of the instanton method and may be useful for developing either amore refined non-selfconsistent method or ways to implement the selfconsistent instanton treatment of spontaneousfission half-lives, including odd- A nuclei and high- K isomers.The paper is organized as follows: in sect. II we briefly describe the instanton formalism with and without pairing,specifying a simplification of each of them to a non-selfconsistent version with the phenomenological s.p. potential.To provide an illustration of imaginary-time solutions, in sect. III we discuss the two-level model, in particular thedependence of action on the interaction between levels and the collective velocity. Properties of solutions and actionsobtained from the iTDSE with the realistic W-S potential are described in sect. IV, including an example of the actioncalculation along the path through non-axial deformations. Sect. V contains a study of the fission hindrance in oddnuclei made within a hybrid model utilizing adiabatic cranking action for the e-e core and the iTDSE action withoutpairing for the odd nucleon. This approach is meant to mimic a model with pairing which we have not solved yet. Asa byproduct, we study the effect of freezing the configuration along the path of axially-symmetric deformations onthe fission rate. This is done under the assumption that the collective velocity along a given path in odd- A nucleus isas if it had the mass parameter of the e-e neighbour; stated otherwise, the difference in ˙ q between the odd- A nucleusand its e-e A − II. INSTANTON-MOTIVATED APPROACH
The instanton approach to nuclear fission was formulated in the mean-field setting in [11, 12, 19–21]. After reviewingthe selfconsistent formulation without pairing in Subsect. A, in Subsect. B, we formulate the non-selfconsistent versionwith the phenomenological nuclear potential, the solutions to which we present in this work. For completeness, as thepairing interaction is crucial to nuclear fission, we review also the selfconsistent equations with pairing in Subsect.C, and formulate the model with the phenomenological potential and the monopole pairing with the selfconsistentpairing gap in Subsect. D.
A. Instantons of Hartree-Fock equations
A transition to imaginary time, t → − iτ , transforms TDHF equations for s.p. amplitudes ψ k ( t ) into imaginary-TDHF (iTDHF) equations for amplitudes φ k ( x, τ ) = ψ k ( x, − iτ ), with the complex-conjugate amplitudes ψ ∗ k ( t ) be-coming ψ ∗ k ( x, − iτ ) = φ ∗ k ( x, − τ ), so that the scalar products (cid:104) ψ k ( t ) | ψ l ( t ) (cid:105) transform to (cid:104) φ k ( − τ ) | φ l ( τ ) (cid:105) . Mean-fieldsolutions dominating the quasiclassical tunneling rate are periodic [11, 12], hence the iTDHF equations acquire theadditional terms ζ k φ k , with ζ k - Floquet exponents with the dimension of energy, which ensure periodicity:¯ h ∂φ k ( τ ) ∂τ = − (ˆ h ( τ ) − ζ k ) φ k ( τ ) . (3)The mean-field hamiltonian ˆ h ( τ ) = ˆ h [ φ ∗ ( − τ ) , φ ( τ )] is defined by: ˆ h ( τ ) φ k ( τ ) = δ H /δφ ∗ k ( − τ ), where H ( τ ) is the energyoverlap (cid:104) Φ( − τ ) | ˆ H | Φ( τ ) (cid:105) , playing the same role as energy in the usual TDHF, H ( τ ) = H [ φ ∗ ( − τ ) , φ ( τ )] = (cid:90) d x (cid:40) (cid:88) k occ ¯ h m ∇ φ ∗ k ( − τ ) ∇ φ k ( τ ) + V [ φ ∗ ( − τ ) , φ ( τ )] (cid:41) , (4)with | Φ( τ ) (cid:105) - the Slater determinant built of occupied orbitals { φ k ( τ ) } , and V - a two-body interaction energy densitycomposed as in the HF, but with φ k ( τ ) in place of ψ k ( t ), and φ ∗ k ( − τ ) in place of ψ ∗ k ( t ). The instanton solving (3) thatdescribes quantum tunneling, called bounce, has to fulfil specific consditions: amplitudes at the boundary are equalto static Hartree-Fock (HF) solutions at the metastable state (m.s.) minimum, φ k ( − T /
2) = φ k ( T /
2) = ψ HFk , withHF energy E m.s. , while the states φ k ( τ = 0) form a normalized Hartree-Fock state with the same energy E m.s. at theouter slope of the barrier, that corresponds to the exit point from the barrier q fin in Eq. (1). An infinite period T corresponds to a decay from the m.s. - evolution becomes infinitely slow close to the m.s. minimum. Hence, ∂φ k /∂τ become zero as τ → ±∞ , and Eq. (3) reduce there to the static HF equations. So, in the selfconsitent theory, theFloquet exponents are equal to s.p. energies at the m.s. state.Both, energy overlaps H ( τ ) and the mean-field Hamiltonian ˆ h ( τ ), depend on φ k ( τ ) and φ k ( − τ ), so Eq. (3) are nonlocal in τ and one cannot solve them as an initial value problem. Together with the periodicity condition, thismakes iTDHF equations a kind of a nonlinear boundary value problem in four dimensions.Eq. (3) conserve energy overlap H ( τ ), diagonal overlaps of solutions, and give the exponential τ -dependence totheir non-diagonal overlaps. As the HF solutions at the boundary are orthonormal, so remain the bounce solutions: (cid:104) φ i ( − τ ) | φ j ( τ ) (cid:105) = δ ij . (5)From ˆ H † = ˆ H , one has H ( − τ ) = H ∗ ( τ ), and the mean field hamiltonian ˆ h ( τ ) is in general not hermitean, butfulfils the condition: ˆ h ( − τ ) = ˆ h † ( τ ). It may be presented as a sum of its hermitean and antihermitean parts,ˆ h ( τ ) = ˆ h R ( τ ) + ˆ h A ( τ ), with: ˆ h R ( − τ ) = ˆ h R ( τ ) = ˆ h † R ( τ ) and ˆ h A ( − τ ) = − ˆ h A ( τ ) = ˆ h † A ( τ ); the τ -odd, antihermitianpart ˆ h A comes from τ -odd parts of densities building energy overlap H ( τ ). In tunneling, at least one τ -odd densityis provided by the current density j , in imaginary time given by: j ( τ ) = (cid:80) k [ φ k ( τ ) ∇ φ ∗ k ( − τ ) − φ ∗ k ( − τ ) ∇ φ k ( τ )] / j ( − τ ) = − j ∗ ( τ ). Decomposing amplitudes into τ -even and τ -odd parts, φ k ( τ ) = ϕ k ( τ ) − ξ k ( τ ), φ k ( − τ ) = ϕ k ( τ ) + ξ k ( τ ), one has: j = (cid:88) k occ [ (cid:60) ( ϕ ∗ k ∇ ξ k − ξ ∗ k ∇ ϕ k ) + i (cid:61) ( ξ ∗ k ∇ ξ k − ϕ ∗ k ∇ ϕ k )] . (6)One can see that, even if φ k are purely real, the τ -odd components ξ k in the first part of this expression generate the τ -odd antihermitean mean field ˆ h A . For small collective velocities, the τ -odd mean field ˆ h A is a direct analogy in theimaginary-time formalism of the Thouless-Valatin potential of the ATDHF method in real time [22].After finding iTDHF solutions one can calculate action. Since in the mean-field theory with a Slater determinantΨ( t ), (cid:104) Ψ( t ) | i ¯ h∂ t − ˆ H | Ψ( t ) (cid:105) plays a role of Lagrangian, action (cid:82) dt (cid:104) Ψ( t ) | i ¯ h∂ t | Ψ( t ) (cid:105) in the imaginary-time versionbecomes [11, 12]: S = ¯ h (cid:90) T/ − T/ dτ N (cid:88) i =1 (cid:10) φ i ( − τ ) (cid:12)(cid:12) ∂ τ φ i ( τ ) (cid:11) = (cid:90) T/ − T/ dτ N (cid:88) i =1 (cid:68) φ i ( − τ ) (cid:12)(cid:12) ζ i − ˆ h ( τ ) (cid:12)(cid:12) φ i ( τ ) (cid:69) , (7)where the summation runs over the occupied s.p. states.Contrary to the unfortunate and erroneous statement in [20] [in the paragraph containing the formula (14) there],repeated in [21] [after the formula (7) there], this expression is obviously composed of changes in φ i ( τ ) parallel to φ i ( − τ ). B. Non-selfconsistent instanton-motivated approach
In order to gain some idea about solutions of imaginary-time-dependent Schr¨odinger-like equations with instantonboundary conditions and resulting actions we replace the mean-field hamiltonian ˆ h [ φ ∗ ( − τ ) , φ ( τ )] by a simple one withthe phenomenological W-S s.p. potential. Releasing the selfconsistency makes these equations linear iTDSEs andremoves non-locality in τ , thus considerably simplifying solution. Certainly, we lose generality: the non-hermiteannature of the mean potential in tunneling is lost, we have to resort to the usual paramerization of nuclear shapes andhave to externally provide the collective velocity ˙ q ( τ ) which in the selfconsistent theory would follow from the energyconstraint H ( τ ) = E m.s. . However, we gain a possibility to study iTDSE solutions and their actions for manifestlynon-adiabatic imaginary-time motions along trial fission paths which in current treatments of fission are commonlyconsidered realistic. To have an approximate energy conservation we assume the effective collective velocity given by: B evenqq ( q ) ˙ q = 2( E ( q ) − E m.s. ) , (8)with: dτ = dq ˙ q ( τ ) . (9)Here, E ( q ) is the microscopic-macroscopic energy and B evenqq ( q ) is the adiabatic mass parameter along the fission pathof the even - even nucleus - the one in question or the nearest neighbour in case of the odd- A . The motivation willbe given in section V B. This whole procedure may be viewed as an attempt to simplify the selfconsistent theory toa micro-macro version.As a result, the phenomenological s.p. Hamiltonian ˆ h ( τ ) is:ˆ h ( q ( τ )) = − ¯ h m ∇ + V ( q ( τ )) , (10)where V is the phenomenological s.p. potential, including Coulomb repulsion for protons, depending on the collectivecoordinate q which itself depends on τ . In solving the equation (3) with the above s.p. hamiltonian along a given pathwe restrict to the subspace spanned by N adiabatic s.p. orbitals ψ µ ( q ). In this subspace, there are N bounce solutions φ i ( τ ), each of which tends to the s.p. orbital ψ i ( q min ) at the metastable minimum as T → ±∞ . By expanding thesesolutions onto adiabatic orbitals φ i ( τ ) = (cid:88) µ C µi ( τ ) ψ µ ( q ( τ )) , (11)we obtain the following set of equations for the square matrix of the coefficients C µi ( τ ):¯ h ∂C µi ∂τ + ˙ q (cid:88) ν (cid:104) ψ µ ( q ( τ )) | ∂ψ ν ∂q ( q ( τ )) (cid:105) C νi = [ ζ i − (cid:15) µ ( q ( τ ))] C µi . (12)Here, ζ i , i = 1 , ..., N , are the Floquet exponents in imaginary time, which for the selfconsistent instanton would beeqal to the s.p. energies at the metastable minimum, ζ i = (cid:15) i ( q min ). However, for a finite imaginary-time interval[ − T / , T / ζ i (cid:54) = (cid:15) i ( q min ), although they should tend to this limit when T → ∞ .The conservation of overlaps (cid:104) φ i ( − τ ) | φ j ( τ ) (cid:105) = δ ij leads to the condition on C µl ( τ ): N (cid:88) µ =1 C ∗ µi ( − τ ) C µj ( τ ) = δ ij . (13)This means that the matrix C µi ( τ ) has the inverse C + ( − τ ) and the adiabatic states can be expanded on (all N )bounce states: ψ µ ( q ( τ )) = N (cid:88) i =1 C ∗ µi ( − τ ) φ i ( τ ) = N (cid:88) i =1 C ∗ µi ( τ ) φ i ( − τ ) , (14)where in the second equality we assumed that q ( τ ) = q ( − τ ) which strictly holds for any real bounce observable: q ( τ ) = (cid:80) i occ (cid:104) φ i ( − τ ) | ˆ q | φ i ( τ ) (cid:105) = q ∗ ( − τ ). Then, the orthonormality of ψ µ , combined with the overlaps Eq. (13),produces the relation: N (cid:88) i =1 C µi ( τ ) C ∗ νi ( − τ ) = δ µν . (15)Thus, the quantity p µi ( τ ) = C ∗ µi ( − τ ) C µi ( τ ) may be considered as a quasi-occupation (it can be negative or complexin general case) of the adiabatic level µ in the bounce solution i , with (cid:80) µ p µi ( τ ) = 1, or as the quasi-occupation ofthe bounce state i in the adiabatic state µ , where (cid:80) i p µi ( τ ) = 1. The sums over the occupied states: (cid:80) i occ p µi ( τ )are diagonal elements ρ µµ ( τ ) of the density matrix ρ µν ( τ ) determined by the Slater states | Φ( τ ) (cid:105) .From (11) and (14) one obtains the relation: φ i ( − τ ) = N (cid:88) j =1 (cid:32) N (cid:88) µ C µi ( − τ ) C ∗ µj ( − τ ) (cid:33) φ j ( τ ) = N (cid:88) j (cid:0) C + ( − τ ) C ( − τ ) (cid:1) ji φ j ( τ ) , (16)where the matrix C + ( − τ ) C ( − τ ) is hermitean and positive. One can define: C + ( − τ ) C ( − τ ) = exp(2 ˆ S ( τ )) T , so thatˆ S ( τ ) is τ -odd and hermitean and: φ i ( − τ ) = exp( ˆ S ( τ )) ψ i ( τ ) , φ i ( τ ) = exp( − ˆ S ( τ )) ψ i ( τ ) , (17)where the states ψ i ( τ ) are τ -even and orthonormal, so they could be considered as some ”mean” TDHF orbitalsrelated to the bounce solutions φ i ( τ ) [20].Action is equal to the sum over the occupied iTDHF solutions: S = (cid:60) (cid:88) i occ (cid:90) T/ − T/ (cid:104) φ i ( − τ ) | ζ i − ˆ h | φ i ( τ ) (cid:105) = (cid:90) T/ − T/ (cid:88) i occ N (cid:88) µ =1 [ ζ i − (cid:15) µ ( q ( τ ))] C ∗ µi ( − τ ) C µi ( τ ) dτ, (18)so, using the quasi-occupations p µi , it can be written as: S = (cid:90) T/ − T/ (cid:88) i occ N (cid:88) µ =1 [ ζ i − (cid:15) µ ( q ( τ ))] p µi ( τ ) dτ. (19)From this, the sum of actions for all individual s.p. bounce states is the integral of a difference between two sums: ofall Floquet exponents and all adiabatic s.p. energies: (cid:80) N i =1 ( ζ i − (cid:15) i ). It can be shown that this integral vanishes [23],so the sum of all actions is zero.When the collective motion is nearly adiabatic, one recovers from this formalism action (1) with the cranking massparameter and, ususally not mentioned, related formula for the Floquet exponent - see Appendix A. C. Instantons with pairing interaction
In the presence of pairing interaction a proper mean-field formalism is the imaginary-time-dependent HFB (iTD-HFB) method. The Bogolyubov transformation from the fixed, independent of time creation operators a † µ to time-dependent quasiparticle creation operators α † i ( t ), after passing to imaginary time t → − iτ , can be written [20]: α † i ( τ ) = (cid:88) µ ( A µi ( τ ) a † µ + B µi ( τ ) a µ ) ,α i ( − τ ) = (cid:88) µ ( A ∗ µi ( − τ ) a µ + B ∗ µi ( − τ ) a † µ ) , (20)where amplitudes A µi ( t ) i B µi ( t ) became functions of τ , and their complex conjugate A ∗ µi ( t ) and B ∗ µi ( t ) depend nowon − τ . The unitarity of the Bogolyubov trnsformation in real time translates to the following condition in imaginarytime: (cid:18) A T ( τ ) , B T ( τ ) B † ( − τ ) , A † ( − τ ) (cid:19) − = (cid:18) A ∗ ( − τ ) , B ( τ ) B ∗ ( − τ ) , A ( τ ) (cid:19) . (21)The hamiltonian overlap (cid:104) Φ( τ ) | ˆ H | Φ( − τ ) (cid:105) can be expressed by the following contractions: (cid:104) Φ( τ ) | a + ν a µ | Φ( − τ ) (cid:105) = ρ µν ( τ ) = ( B ∗ ( − τ ) B T ( τ )) µν , (22) (cid:104) Φ( τ ) | a ν a µ | Φ( − τ ) (cid:105) = κ µν ( τ ) = ( B ∗ ( − τ ) A T ( τ )) µν , (cid:104) Φ( τ ) | a + ν a + µ | Φ( − τ ) (cid:105) = ˜ κ µν ( τ ) = ( A ∗ ( − τ ) B T ( τ )) µν , which, due to conditions (21), have the following properties when regarded as matrices: ρ ( − τ ) = ρ + ( τ ) , (23) κ T ( τ ) = − κ ( τ ) , ˜ κ ( τ ) = κ + ( − τ ) . Using those and proceeding as in the derivation of the TDHFB equations we arrive at imaginary-TDHFB (iTDHFB)equations written symbolically (where only the second index of the amplitudes is explicit):¯ h∂ τ (cid:18) A k ( τ ) B k ( τ ) (cid:19) + (cid:18) ˆ h ( τ ) − λ, ˆ∆( τ ) − ˆ∆ ∗ ( − τ ) , − (ˆ h ∗ ( − τ ) − λ ) (cid:19) (cid:18) A k ( τ ) B k ( τ ) (cid:19) = ζ k (cid:18) A k ( τ ) B k ( τ ) (cid:19) . (24)Here, for a given two-body interaction (cid:80) µνγδ v µνγδ a † µ a † ν a δ a γ , the self-consistent potential: Γ µν ( τ ) = (cid:80) γδ ( v µγνδ − v µγδν ) ρ δγ ( τ ) and the pairing potential: ∆ µν ( τ ) = (cid:80) γδ v µνγδ κ γδ ( τ ) have the properties: ˆΓ( − τ ) = ˆΓ + ( τ ), and ˆ∆ T ( τ ) = − ˆ∆( τ ). The same properties hold for the mean fields with additional rearrangement terms that follow from a densityfunctional. These ensure the property ˆ h ( − τ ) = ˆ h + ( τ ) of the mean-field Hamiltonian (ˆ t - kinetic energy) ˆ h ( τ ) = ˆ t +ˆΓ( τ ),and the same property, ˆ h ( − τ ) = ˆ h + ( τ ) of the total HFB mean-field Hamiltonian ˆ h ( τ ) given by the matrix in Eqs.(24).As a result of this, the equations (24) conserve both energy overlap (cid:104) Φ( τ ) | ˆ H | Φ( − τ ) (cid:105) and all relations (21). Theterms with constants ζ k on the r.h.s. fix the periodicity of solutions and these constants are equal to the quasi-particle energies at the HFB m.s. The bounce solution to Eqs.(24) has to be periodic and provide a path in thespace of imaginary-time quasiparticle vacua which connects the HFB m.s. | Φ( ± T / (cid:105) = | Ψ gs (cid:105) with some HFB state | Φ( τ = 0) (cid:105) at the same energy beyond the barrier.One has to emphasize that in Eq. (24) appears the Fermi energy λ (this term is missing in [20]). It does not haveto appear in an initial value problem, as TDHFB equations preserve the expectation value of the particle number T r ( ρ ), both in real [24] and in imaginary time. Here we look for a solution to the boundary value problem. Without λ , T r ( ρ ) would be incorrect at the boundary and one has to enforce its proper value. In particular, the solution hasto tend to the metastable HFB state | Φ( ± T / (cid:105) at the boundaries as τ → ± T /
2, and that fixes the value of λ .Eq. (24) have the property analogous to that of the HFB equations, that if ( A µi ( τ ) , B µi ( τ )) is a periodic solutionwith the Floquet exponent ζ i , then ( B ∗ µi ( − τ ) , A ∗ µi ( − τ )) is also a solution with the Floquet exponent − ζ i . So, itsuffices to find half of solutions. The proper state | Φ( τ ) (cid:105) should contain exactly one of each pair of two solutionswith ζ i and − ζ i which then corresponds to α i ( τ ). For ground states of e-e nuclei, it is natural to choose the solutionswith ζ i > α † i since in the limit τ → ± T / | Φ( − τ ) (cid:105) should be composed of solutions with ζ i which at τ → ± T / A µi ( τ ) and B µi ( τ ) correspond at τ → ± T / ζ i . Asthe boundary condition fixes the correspondence with the initial HFB state, the construction of matrices ρ and κ forodd nuclei is analogous to that in the HFB method [18]: one of the solutions ( A ( τ ), B ( τ )) with positive ζ i is replacedby ( B ∗ ( − τ ), A ∗ ( − τ )) with − ζ i .Decay rate is determined by instanton action which for a state | Φ( τ ) (cid:105) can be presented in terms of the amplitudes A and B [20]: S/ ¯ h = (cid:90) T/ − T/ dτ (cid:104) Φ( τ ) | ∂ τ Φ( − τ ) (cid:105) = 12 (cid:90) T/ − T/ dτ T r [ ∂ τ A † ( − τ ) A ( τ ) + ∂ τ B † ( − τ ) B ( τ )]= − (cid:90) T/ − T/ dτ T r [ A † ( − τ ) ∂ τ A ( τ ) + B † ( − τ ) ∂ τ B ( τ )] . (25)Substituting ∂ τ A µi ( τ ) and ∂ τ B µi ( τ ) from the iTDHFB equation (24) and using conditions (21) we obtain for theaction integrand: − (cid:88) i occ ζ i − (cid:88) µν (cid:0) ( h µν ( τ ) − λδ µν )(2 ρ νµ ( τ ) − δ µν ) + κ µν ( τ )∆ ∗ µν ( − τ ) + κ ∗ µν ( − τ )∆ µν ( τ ) (cid:1) . (26)One can cast the instanton method in a form analogous to the density matrix formalism. The matrix: R ( τ ) = (cid:18) ρ ( τ ) , κ ( τ ) − κ ∗ ( − τ ) , I − ρ ∗ ( − τ ) (cid:19) (27)satisfies the equation: ¯ h∂ τ R ( τ ) + [ˆ h ( τ ) , R ( τ )] = 0 , (28)which follows directly from (24,21). The matrix R has the property: R ( τ ) = R ( τ ), as a result of: ρ ( τ ) κ ( τ ) = κ ( τ ) ρ ∗ ( − τ ) and ρ ( τ ) − κ ( τ ) κ ∗ ( − τ ) = ρ ( τ ). However, being non-hermitean, it does not represent any real-time HFBdensity matrix. D. Phenomenological potential model with the selfconsistent pairing gap ∆( τ ) The above scheme can be simplified by replacing the mean-field ˆ h by the s.p. Hamiltonian with the W-S potentialand using the pairing interaction with the constant matrix element. The τ -dependent HFB transformation may bepresented as a composition: a + n → b + µ → α + i , where the first transformation diagonalizes the deformation-dependentW-S hamiltonian in the deformation-dependent basis ψ µ ( q ) = b + µ ( q ) | (cid:105) [note that now the independent of timeoperators a † carry the Latin indices n, m , not the Greek ones as in the preceding part of this section, which are nowreserved for eigenstates of the phenomenological ˆ h ( τ )]: b + µ ( q ) = (cid:88) n C nµ ( q ) a + n . (29)The second transformation is a genuine HFB one: α + i = (cid:88) µ (cid:0) A µi ( τ ) b + µ ( q ( τ )) + B µi ( τ ) b µ ( q ( τ )) (cid:1) . (30)We assume the pairing interaction with the constant matrix element G > µ ¯ µ . The only non-zero matrix elements of this interaction are: v µ ¯ µν ¯ ν = − G , and those related by the antisymmetry.Since the matrix C is q -dependent it must be differentiated in the iTDHFB equation (24), so that this equation inthe adiabatic basis becomes symbolically:¯ h∂ τ (cid:18) A i ( τ ) B i ( τ ) (cid:19) + (cid:18) ˆ (cid:15) ( q ) + ˆ D, ˆ∆( τ ) − ˆ∆ ∗ ( − τ ) , − ˆ (cid:15) ( q ) + ˆ D ∗ (cid:19) (cid:18) A i ( τ ) B i ( τ ) (cid:19) = ζ i (cid:18) A i ( τ ) B i ( τ ) (cid:19) . (31)Here, ˆ (cid:15) ( q ) is a diagonal matrix with elements ˆ (cid:15) µν ( q ) = δ µν ( (cid:15) µ ( q ) − λ ) ( (cid:15) µ are s.p. energies), ˆ D is the matrix ofadiabatic couplings, D µν ( τ ) = ¯ h (cid:104) µ | ∂ν∂τ (cid:105) = ¯ h ˙ q (cid:104) µ | ∂ν∂q (cid:105) , with (cid:104) µ | ∂ν∂τ (cid:105) = ˙ q ( τ ) (cid:80) n C ∗ nµ ( q ) ∂ q C nν ( q ), and only non-zeroelements of the matrix ˆ∆ are: ∆ µ ¯ µ ( τ ) = − ∆ ¯ µµ ( τ ) = − ∆( τ ), where:∆( τ ) = G (cid:88) µ> ¯ κ µ ¯ µ , (32)with ¯ κ the anomalous density in the adiabatic basis. The connection between density matrices ¯ ρ and ¯ κ in the adiabaticbasis, and ρ and κ (with indices m , n ) in the basis independent of time, reads: ρ ( τ ) = C ( q ( τ ))¯ ρ ( τ ) C † ( q ( τ )) , (33) κ ( τ ) = C ( q ( τ ))¯ κ ( τ ) C T ( q ( τ )) , where: δ µν (cid:15) µ ( q ) = (cid:16) C + ( q ( τ ))ˆ h ( q ( τ )) C ( q ( τ )) (cid:17) µν .Next, we intend to use further the Kramers degeneracy of s.p. states, already used in defining the pairing interaction.This is quite natural for e-e nuclei. In odd- A nuclei, the odd nucleon perturbs the mean field, breaking its invarianceunder time-reversal and the Kramers degeneracy; three new time-reversal-odd densities emerge in the mean-fieldtreatment [25]. However, we will neglect this effect here as if it would be small (see [26] for the effect of time-oddterms on the HF+BCS barrier). This means that also in odd- A nuclei we assume two groups of states, µ and ¯ µ , with (cid:15) µ = (cid:15) ¯ µ , D ¯ µ ¯ ν = D ∗ µν . There will be two sets of solutions, i and ¯ i , with ρ µ ¯ ν = ρ ¯ µν = κ µν = κ ¯ µ ¯ ν = 0, for which Eq.(31) separates into two independent sets with matrices: (cid:18) ˆ (cid:15) ( q ) + ˆ D, − ∆( τ ) · ˆ I − ∆ ∗ ( − τ ) · ˆ I, − ˆ (cid:15) ( q ) + ˆ D (cid:19) and : (cid:18) ˆ (cid:15) ( q ) + ˆ D ∗ , ∆( τ ) · ˆ I ∆ ∗ ( − τ ) · ˆ I, − ˆ (cid:15) ( q ) + ˆ D ∗ (cid:19) , (34)with ˆ I - the block unit matrix. Let the solutions with ζ i > A µi ( τ ) , B ¯ µi ( τ )), andfor the second set: ( A ¯ µ ¯ i ( τ ) , B µ ¯ i ( τ )). Then the solutions with ζ i < B ∗ ¯ µi ( − τ ) , A ∗ µi ( − τ )) - to the second set ofequations, and ( B ∗ µ ¯ i ( − τ ) , A ∗ ¯ µ ¯ i ( − τ )) - to the first one. If, additionally, ˆ D = ˆ D ∗ , which holds, for example, for a meanfield ˆ h with the axial symmetry or the one having the reflexion symmetry in three perpendicular planes (like for shapeswith deformations: β , γ , β , β = β − , β = β − , etc, cf Sec. IV), ∆ will also be real and then, the solutions ofthe second set of equations are: ( A ¯ µ ¯ i ( τ ) , B µ ¯ i ( τ )) = ( A µi ( τ ) , − B ¯ µi ( τ )). In such a case, both sets of equations producethe same sets of ζ i , one has: ¯ ρ ¯ µ ¯ ν = ¯ ρ µν , ¯ κ ¯ µν = − ¯ κ µ ¯ ν and it suffices to know the half of density matrices (in theadiabatic basis) which, from (28,34), fulfill the equations (cf e.g. [27] for comparison with the TDHFB):¯ h∂ τ ¯ ρ µν ( τ ) = ( (cid:15) ν ( q ) − (cid:15) µ ( q ))¯ ρ µν ( τ ) − ¯ κ µ ¯ ν ( τ )∆( − τ ) + ∆( τ )¯ κ µ ¯ ν ( − τ ) (35)+ [¯ ρ ( τ ) , ˆ D ] µν , ¯ h∂ τ ¯ κ µ ¯ ν ( τ ) = ∆( τ )( δ µν − ¯ ρ µν ( τ ) − ¯ ρ νµ ( τ )) − ( (cid:15) ν ( q ) + (cid:15) µ ( q ) − λ )¯ κ µ ¯ ν ( τ )+ [¯ κ ( τ ) , ˆ D ] µ ¯ ν . The Eq. (31) are a counterpart of (12) for instanton-like solutions with pairing. One should notice that, in spite ofusing a phenomenological potential in place of the selfconsistent one, we could not avoid nonlocality in time - thematrix in Eq. (31) depends on both ∆( τ ) and ∆( − τ ), and the function ∆( τ ) has to be selfconsistent - it should fulfilthe condition (32). In the process of iterative solution for ∆( τ ) its value at the current step would differ in generalfrom the value ∆ r ( τ ) resulting from the integration of the Eq. (31) in this step. Using the equation for densities onehas: ¯ h ∂ ∆ r ∂τ = G (cid:34) ( N r − N ) − (cid:88) µ> ( (cid:15) µ ( τ ) − λ ) κ ¯ µµ ( τ ) (cid:35) , (36)where N r = 2 (cid:80) µ> ρ µµ ( τ ) is the expectation value of the number of particles, not necessarily equal to the assumedone, and N - the number of included doubly degenerate levels. On the other hand, from these equations:¯ h ∂N r ∂τ = 2 G (∆ r ( τ )∆ ∗ ( − τ ) − ∆( τ )∆ ∗ r ( − τ )) . (37)One can see that the expectation value of the number of particles is constant for a selfconsistent solution with∆ r ( τ ) = ∆( τ ).Test solutions with a few adiabatic W-S levels indicate that the (rather long) iterative procedure applied to Eq.(31), equivalent to Eq. (35), leads to the exponential dependence of ∆( τ ), which is large on the interval [ − T / , , T / τ )∆( − τ ). This case is considerably more involved thanthe the equation with the W-S potential alone.Assuming that we have solutions to Eq. (31), one can write down action (25) for an e-e nucleus: S = (cid:90) T/ − T/ dτ (cid:40) − (cid:88) i> ζ i − (cid:88) µ> (cid:0) (2¯ ρ µµ ( τ ) − (cid:15) µ ( τ ) − λ ) + ∆( τ )¯ κ ∗ µ ¯ µ ( − τ ) + ¯ κ µ ¯ µ ( τ )∆ ∗ ( − τ ) (cid:1)(cid:41) (38)= (cid:90) T/ − T/ dτ (cid:40) − (cid:88) i> ζ i − (cid:88) µ> (2¯ ρ µµ ( τ ) − (cid:15) µ ( τ ) − λ ) + 2 ∆( τ )∆ ∗ ( − τ ) G (cid:41) , (39)where the summation runs over solutions i > µ >
0, and the last equality holds for the selfconsistentsolution. For an odd nucleus, one has to exchange in densities (23) one amplitude with positive ζ by the other onewith − ζ .In the limit of no pairing, the positive Floquet exponents of decoupled Eq. (31) are: ζ NPi − λ for amplitudes A of empty states, and λ − ζ NPi for amplitudes B of occupied states, where ζ NPi are Floquet exponents of solutions to(12). Density ¯ ρ µµ , composed of amplitudes of occupied states, expressed in terms of quasi-occupations p µi of Sec.II B, is: (cid:80) i> ,ζ NPi <λ p µi . For solutions i > ρ µµ − (cid:80) ζ NPi <λ p µi − (cid:80) ζ NPi >λ p µi (since (cid:80) i> p µi = 1).Hence, the sum in the integrand (38) is equal to the difference (cid:80) ζ NPi <λ − (cid:80) ζ NPi >λ of the following expressions:( ζ NPi − λ ) − (cid:80) µ> p µi ( (cid:15) µ − λ ). The terms with λ vanish after summation as a consequence of: (cid:80) µ> p µi = 1; one isthus left with the difference of sums of actions without pairing for solutions i >
0: (below) − (above) the Fermi level.We know from Sec. II B that those sums add to zero; therefore the result is 2 × the sum of actions for i > i and ¯ i ) occupied states. III. TWO - LEVEL MODEL
It turns out that a main difficulty in integrating Eq. (12) are avoided crossings with a minuscule interlevel interaction- see Sec. IV C. Here we study a dependence of bounce-like action for such a crossing on the collective velocity and0level slopes in a simple model with two s.p. levels - a kind of analogy with the Landau - Zener problem [28–30]. TheHamiltonian is: ˆ h ( q ( τ )) = (cid:18) E ( q ( τ )) VV ∗ E ( q ( τ )) (cid:19) , (40)where q ( τ ) is a time-dependent parameter, e.g. some nuclear deformation. We assume: V = V ∗ , E , = ± E ( q − q ),so that diagonal elements are linear in q and cross at q . The states: | χ (cid:105) = (1 , T , | χ (cid:105) = (0 , T we call diabatic ;the basis: | ψ (cid:105) = (cid:18) cos θ sin θ (cid:19) , | ψ (cid:105) = (cid:18) − sin θ cos θ (cid:19) , (41)in which ˆ h is diagonal with eigenvalues: (cid:15) , = ∓ (cid:112) ( E − E ) + 4 V (42)we call adiabatic . Here, tan θ = VE − E . So, for q < q , θ → | ψ , (cid:105) → | χ , (cid:105) .At the pseudo-crossing q , θ = − π/ q approach their minimal distance (cid:15) − (cid:15) = 2 V . For q > q , θ → − π and | ψ (cid:105) → −| χ (cid:105) (note the change of sign), | ψ (cid:105) → | χ (cid:105) , so after passing the pseudo-crossing the adiabatic states exchange theircharacteristics. The coupling of adiabatic states in the iTDSE is: (cid:28) ψ (cid:12)(cid:12)(cid:12)(cid:12) dψ dq (cid:29) = − dθdq = 12 EVE ( q − q ) + V = 12 α ( q − q ) + α , (43)where we introduced α = V /E . It has the Lorentz shape with a maximum at q and the width and height regulatedby α . In the limit V →
0, i.e., α →
0, the coupling element tends to the Dirac δ -function.To define the model we have to specify q ( τ ) and the resulting collective velocity ˙ q ( τ ). In the following we use theansatz: q ( τ ) = q fin − q ini cosh(Γ τ ) + q ini , (44)where q ini , q fin are the initial and final collective deformation (e.g. the entrance and exit from the barrier). Sodefined q ( τ ) has an impulse shape, typical for instanton, which means that the motion takes place in a finite timeinterval around τ = 0, while in the asymptotic region, τ → ±∞ , q ( τ ) → q ini with vanishingly small ˙ q . The equationreads: ¯ h ˙ c = − (cid:15) c − ¯ h ˙ q (cid:104) ψ | ∂ q ψ (cid:105) c , (45)¯ h ˙ c = − (cid:15) c + ¯ h ˙ q (cid:104) ψ | ∂ q ψ (cid:105) c . After using definitions of the model and introducing dimensionless time parameter z = τ | E | ¯ h the following form ofiTDSE is obtained: ddz ˜ c = (cid:112) ( q − q ) + α ˜ c + 12 β tanh( βz ) ( q − q ini ) α ( q − q ) + α ˜ c , (46) ddz ˜ c = − (cid:112) ( q − q ) + α ˜ c − β tanh( βz ) ( q − q ini ) α ( q − q ) + α ˜ c , where ˜ c i ( z ) = c i ( τ ) and β = ¯ h Γ / | E | . The following parameters were fixed: q ini = 0 . q fin = 0 . q = 0 . c k ( z ) and action depend on two parameters: α and β : S = S ( α, β ). Pertinentto difficulties of realistic calculations are the non-obvious changes in S for small α and β - see Sec. IV C. Accordingly,other parameters were set as follows: Γ = 0 . × s − (the maximal possible velocity was | ˙ q max | ≈ . × s − ), E = 5 , , ... MeV defined values of β , and V covered a range of exponentially small values. Solutions wereobtained by the method described in Appendix B, but for small α Eq. (45) was solved in the diabatic basis.In Fig. 1 the calculated action is displayed as a function of the parameter α at fixed values of β . The parameter α is proportional to V - the strength of interaction between levels. The extremal cases are when V is very large orvery small. In the first case, levels are repelling each other and transitions between the adiabatic levels are reduced1 S / - h log ( α ) β =15.20 1/ β =30.401/ β =45.60 1/ β =60.79 1/ β =75.99 1/ β =91.19 1/ β =106.38 1/ β =121.58 1/ β =136.78 FIG. 1: Action S ( α ) for various parameters 1 /β . -0.5 0 0.5 1 1.5 2 2.5-3 -2.5 -2 -1.5 -1 -0.5 0 p ( τ ) τ [10 -21 s]1/ β =30.40 log ( α )=-2.43log ( α )=-3.17log ( α )=-3.66log ( α )=-4.39log ( α )=-5.13log ( α )=-6.11 p ( τ ) τ [10 -21 s]1/ β =30.40 log ( α )=-2.43log ( α )=-3.17log ( α )=-3.66log ( α )=-4.39log ( α )=-5.13log ( α )=-6.11 FIG. 2:
Left panel:
Pseudo-occupation of the lower adiabatic level for solutions with various α at fixed 1 /β = 30 .
40. Thecorresponding S ( α ) is shown in Fig. 1. The pseudocrossing occurs at τ c ≈ − . Right panel :
The same in greater detail,close to τ c . - one can expect a small action (note that the adiabatic limit of small β/α = ¯ h ˙ q/V is not covered in Fig. 1). When V →
0, the transitions between diabatic levels cease, and action tends to zero again. A larger action can be expectedfor intermediate values of α and there has to be at least one maximum of S . Calculated values of S ( α ) in Fig. 1 showa maximum at some α max , while for smaller and larger values of α , respectively, action rises from, and falls down tozero. In the covered range of α , one can observe an approximate scaling: S (log α, β (cid:48) ) ∼ ( β/β (cid:48) ) S (( β (cid:48) /β ) log α, β ).For an illustration of non-adiabatic transitions, in Fig. 2 we show the pseudo-occupation p ( τ ) defined in Sect.II B [after the formula (15)]. It is displayed for the same α values which were used to calculate S ( α ) in Fig. 1, for1 /β = 30 .
40. It can be seen that for α greater than α max (log ( α max ) ≈ − . p is concentratedin the lower adiabatic state; a transition to the upper adiabatic state takes place only around the pseudo-crossing,while behind it the system returns to the lower state, i.e. p ( τ = 0) = 1. This behaviour changes when we approachthe maximum of action - for log ( α ) = − .
39 - the system behind the crossing remains partially excited to the upperadiabatic level (0 < p ( τ = 0) < α < α max , behind the pseudo-crossing the system occupiesexclusively the upper adiabatic level, till the end of the barrier ( p ( τ = 0) = 0; p ( τ = 0) = 1). In such a case wehave a continuation of the diabatic state.In Fig. 3 is shown a plot of action as a function of 1 /β at the fixed α , which corresponds to the fixed matrix element (cid:104) ψ | ∂ q ψ (cid:105) . One can see its jump-like character: for small 1 /β action is close to zero, over a short interval of 1 /β it2 FIG. 3: Action S (1 /β ) for various values of α .FIG. 4: Left panel:
Pseudo-occupation of the lower adiabatic level for solutions with various 1 /β at fixed log ( α ) = − . S (1 /β ) is shown in Fig. 3. The pseudo-crossing occurs at τ c ≈ − . Right panel:
The same ingreater detail, close to τ c . rises rapidly to a maximal value and then it decreases very slowly. The jump is more sharp and larger for smallervalues of α , which correspond to a sharper pseudo-crossing between the adiabatic levels. As 1 /β ∼ / Γ ∼ / ˙ q max , thegreater the velocity, the stronger the coupling between the adiabatic levels, so for sufficiently large ˙ q (small 1 /β ) onecan expect a diabatic continuation (transition to an upper adiabatic level) when passing through the pseudo-crossing,which means a small action. One should notice that action vanishing in the limit of very large ˙ q is an artificial propertyof the model with a finite number of states - after reaching the highest one the system cannot excite anymore.For smaller ˙ q , after passing through the pseudo-crossing, pseudo-occupations of both adiabatic states becomecomparable - action becomes sizable. For still smaller ˙ q , the pseudo-occupation p of the upper adiabatic state isnon-zero only around the pseudo-crossing, and action does not change much. This also can be seen in Fig. 4 wherethe pseudo-occupation of the lower adiabatic state is shown for the lower iTDSE solution at the fixed value of α . Thediabatic behaviour - a sharp fall of p from 1 to 0 at the pseudo-crossing (red and black lines) - gives way to anintermediate situation - 0 < p < p = 1except the close neighbourhood of the pseudocrossing (all other lines). One can notice from Fig. 3 that a smaller α means a larger domain of diabatic behaviour in 1 /β , i.e. as α decreases, the interval of a diabatic - to - adiabatic3transition shifts towards smaller collective velocities (larger 1 /β ).Presented solutions determine whether the evolution is diabatic, intermediate or adiabatic. Since values of α pertinent to nuclear potential with nonaxial deformation can be as small as ∼ − - 10 − , cf Sec. IV C, this simplemodel demonstrates a possibility of large variation in action for a fixed α , resulting from the dependence on thecollective velocity ˙ q at the crossing. As Fig. 1 suggests, even for very small V one can get sizabele action. In arealistic case, with many interacting levels, it is difficult to predict the effect of one pseudo-crossing on the value ofaction without solving for the instanton-like solution.Independent of the above results, we have checked that in the adiabatic limit of small ˙ q/V = β/α , the two-levelmodel produces action which tends to the value given by the formula (A12) with the cranking mass parameter , see[31]. IV. INSTANTON-LIKE SOLUTIONS WITH THE WOODS-SAXON POTENTIAL
From this point on, we shall consider instanton-like iTDSE solutions related to the realistic s.p. Woods-Saxonpotential within the microscopic-macroscopic framework briefly described below.Deformation enters the s.p. potential via a definition of the nuclear surface by [32]: R ( θ, ϕ ) = c ( { β } ) R { (cid:88) λ> β λ Y λ ( θ, ϕ ) + (cid:88) λ> ,µ> ,even β λµc Y cλµ ( θ, ϕ ) } , (47)where c ( { β } ) is the volume-fixing factor. The real-valued spherical harmonics Y cλµ , with even µ >
0, are defined interms of the usual ones as: Y cλµ = ( Y λµ + Y λ − µ ) / √
2. Here we restrict shapes to reflection-symmetric ones and allowonly for the quadrupole non-axiality β . The n p = 450 lowest proton levels and n n = 550 lowest neutron levels from N max = 19 lowest major shells of the deformed harmonic oscillator were taken into account in the diagonalizationprocedure. Eigenenergies are used to calculate the shell- and pairing corrections. The macroscopic part of energyis calculated by using the Yukawa plus exponential model [33]. All parameters used here, of the s.p. potential, thepairing strength and the macroscopic energy, are equal to those used previously in the calculations of masses [34, 35]and fission barriers [36–39] of heaviest nuclei, whose results are in reasonable agreement with data. In particular, wetook the ”universal set” of potential parameters and the pairing strengths G n = (17 . − . · I ) /A for neutrons, G p = (13 .
40 + 44 . · I ) /A for protons ( I = ( N − Z ) /A ), as adjusted in [34]. As always within this model, N neutronand Z proton s.p. levels have been included when solving BCS equations.First we discuss the iTDSE solutions for axially-symmetric nuclear shapes composed of multipoles with even λ . Inthis case the τ -evolution of groups of states with different Ω π are indepedendent of each other. As an example we take8 neutron Ω π = 1 / + states in the W-S potential for Mt along the axially symmetric fission path shown on theenergy map in Fig. 5. The map was obtained from the four-dimensional (4D) calculation by minimizing energy of thelowest odd proton and neutron configuration over β , β at each β , β , i.e. without keeping the K π configurationof the g.s. Then, to assure a continuity of the path, β and β were chosen continuous and close to those of theminimization, with energy changed by no more than 200-300 keV. Collective velocity was calculated from Eq. (8) bytaking the effective (i.e. tangent to the path) cranking mass parameter of the e-e ( Z − N −
1) nucleus
Hs. Theadiabatic neutron levels in the basis for solving iTDSE were chosen so, that in the g.s. the lower four are occupied(the fourth one singly) and the upper four are empty. In Fig. 6, they are shown along β which, here and in thefollowing, will play a role of the effective collective coordinate q along fission paths.The method which we used for solving the iTDSE in this and all other cases reported here is described in AppendixB. We find solutions for a finite period T in a finite adiabatic basis and for each of them we calculate action. A naturalquestion then is what would be the limiting values of S i for occupied states when T → ∞ and the dimension of thebasis N → ∞ . We tried to answer this by finding actions for increased periods, and by incresing dimension of theadiabatic basis and inspecting the quasi-occupation coefficients. Results of such tests showed that with moderatelylong periods and rather small bases one can obtain reasonably stable action values for occupied states - see AppendixC.For the discussed eight levels in
Mt, the iTDSE solutions were obtained with the period T = 30 × − s.The amplitudes C µi ( τ ) of solutions have exponential τ -dependence, reach very large values in the interval [ − T / , , T / p µi of adiabaticstates for selected solutions. This also makes sense from the point of view of action (19) which is built of thesequantities. In Fig. 7, quasi-occupations p µi are shown for two solutions, φ and φ . It can be seen that at τ = ± T / p µi ∼ = δ µi , with minuscule admixtures which should vanish completely for T = ∞ . During imaginary-time evolution,4 p µi are concentrated on the corresponding adiabatic states ψ µ = i , except around the pseudo-crossings where a partialexcitation to the nearest-neighbour state occurs. Until a pseudo-crossing is isolated (there is no other pseudo-crossingnearby) excitations to other states are negligible. If successive pseudo-crossings follow one after another, the quasi-occupations of other adiabatic levels are possible, as seen for the solution φ which locally becomes a combination of ψ and ψ , and then of ψ and ψ - see Fig. 7.Next we discuss some properties of iTDSE solutions which seem relevant for their physical interpretation andapplications. A. Rise of action with the collective velocity ˙ q With cranking mass parameters fixed along a path, the collective velocity of tunneling is proportional to (cid:112) E ( q ) − E , where E ( q ) − E is a plot of the fission barrier (reduced by E zp ). In a selfconsistent instanton cal-culation, the increase in barrier height also relates to an increase in the magnitude of ˙ q necessary to increase thedifference between | Φ( τ ) (cid:105) and | Φ( − τ ) (cid:105) in order to keep their energy overlap H ( τ ) constant. On the other hand, inour non-selfconsistent treatment, ˙ β , i.e. our ˙ q ( τ ), is simply an assumed functional parameter of the solution to Eq.(12). However, having in mind its implied physical relation to the barrier height, we tested the action dependence on | ˙ β | . The collective velocity for Mt determined from (8) with the cranking mass parameter from the neighbouringe-e nucleus (Z=108, N=168) along the path depicted in Fig. 5 is shown in Fig. 8. This profile was then scaled bythe factors 1.3 and 1.6. The action calculated for all occupied neutron states of positive parity for three collectivevelocities is given in Table I. One can see that action indeed increases with | ˙ q | , as the expected relation with thebarrier height would suggest. Detailed outcome is dependent on the s.p. level scheme, in particular, pseudocrossingsclose to the Fermi level. In Eq. (12), the coupling terms causing non-adiabatic transitions are ˙ q (cid:104) ψ i | ∂ q ψ j (cid:105) , so the maininfluence on S have regions in q where a large | ˙ q | occurs at a sharp pseudocrossing. FIG. 5: Energy surface of
Mt; a chosen trajectory coloured in red. -12-10-8-6-4-2 0 2 4 0.3 0.4 0.5 0.6 0.7 ε [ M e V ] β FIG. 6: Neutron levels Ω π = 1 / + around the Fermi level of Mt along the trajectory shown in Fig. 5.Collective velocity S tot = (cid:80) Ω + S Ω + [¯ h ]˙ q . q . q S tot for neutron states of positive parity in Mt as a function of scaled collective velocity. The profile ˙ q corresponds to the formula (8) for the path in Fig. 5. B. Integrand of action vs mass parameters
One can ask whether it would be possible to define a mass parameter B ( q ) from the τ - even action integrand inEq. (19) by: (cid:88) i,occ N (cid:88) µ =1 [ ζ i − (cid:15) µ ( q ( τ ))] p µi ( τ ) = B qq ( q ) ˙ q . (48)In Fig. 9 are shown contributions to the integrand of action from s.p. bounce-like states and their sum for even andodd number of particles (19). Calculations were done for the same Ω π = 1 / + neutron states in Mt, for a pathshown in Fig. 5. It can be seen that while integrands of single iTDSE solutions sometimes show a rather complicatedpattern, their sum is much more regular. This comes from a cancellation of excitations among solutions correspondingto occupied levels and only excitations to levels above the Fermi level count. There is no drastic difference betweenthe even- and odd-particle-number case - it is just a contribution from one singly occupied instanton-like solution,which may be both positive or negative in general. This is in contrast to the cranking approach, where for the odd- A case, mass parameter (2) and the action integrand (1) would show large peaks at pseudocrossings of the unpairedlevel.As seen in Fig. 9, the integrand (48) becomes negative around the endpoints τ → ± T /
2, so it cannot define anymass parameter. This follows from differences between the Floquet exponents ζ i and s.p. energies (cid:15) i at the g.s.minimum, which, as stated in Sect. II B, is the artefact of using T < ∞ in practical calculation. The same difficultywill probably remain in the selfconsitent calculations.However, even for a positive integrand of action there would be a more general impediment to deriving the massparameter. The beyond-cranking treatment means that the integrand of action depends on all even powers of ˙ q . Thus,for a given path, B qq of (48) would be dependent on | ˙ q | . On the other hand, since a solution along the prescribedpath depends on it, two different paths tangent at a common point q (what would imply eqal effective cranking massparameters at q ) would have generally different integrands of action at q .6 -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-15 -10 -5 0 5 10 15 τ [10 -21 s] c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ ) -1.5-1-0.5 0 0.5 1 1.5 2 2.5-15 -10 -5 0 5 10 15 τ [10 -21 s] c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ ) FIG. 7: Pseudo-occupations of the adiabatic states for instanton - like iTDSE solutions; upper panel: for φ , lower panel: for φ . Colours correspond to the levels of Fig. 6. C. Calculations along nonaxial path for neutron states in Mt A solution of iTDSE equations for nonaxial shapes turns out to be more difficult than in the case of axial defor-mations considered hitherto. The W-S spectrum along a nonaxial fission path shows many sharp pseudocrossingsbetween levels of the same parity, some with interaction as small as V ∼ − − − MeV (see Fig. 10). Althoughfor V → V ≈
0, may lead to large errors in calculated action. On the other hand, many pseudo-crossings with a veryweak interaction, leading to extremely high peaks in the matrix elements which couple involved adiabatic states, arethe obstacle in solving iTDSE. The encountered problem and its (rather cumbersome) solution are described below.Calculations were performed along the chosen nonaxial path for
Mt, see Fig. 10, for N = 32 neutron states ofpositive parity. In the first version, we used the data from the W-S code along the path with a variable step, notshorter than ∆ β = 10 − . In the second version, the minimal step was smaller, ∆ β = 10 − . Finally, in the thirdversion, we used the procedure described in the Appendix D, with the minimal step ∆ β = 10 − , and the analyticmodel (D2) adjusted to those peaks for which the minimal stepsize still did not cover their range with a sufficientprecision. Actions calculated for occupied instanton levels and their sum are given in Table II. It can be seen that7 dq / d τ β FIG. 8: Collective velocity ˙ q in units od 10 s − calculated from (8) for Mt along the path shown in Fig. 5. -2 0 2 4 6 8 10 12 -10 -5 0 5 10 A c t i on i n t eg r and [ s - ] τ [10 -21 s] -2 0 2 4 6 8 10 -10 -5 0 5 10 τ [10 -21 s] integrand - state 1integrand - state 2integrand - state 3integrand - state 4 FIG. 9:
Left:
The total action integrand in units 10 s − - the sum of individual contributions - for six (in black) and seven(in red) neutrons - taken from [31]. Right:
Contributions to the integrand of action from individual s.p. solutions. actions for some individual levels in the first and second versions of the calculation differ widely - this means thatthe step ∆ β = 10 − is not sufficient. This is consistent with an insufficient density of points for a description ofparticular pseudocrossings, as revealed by the inspection of related coupling matrix elements. In spite of this, thetotal action is similar in two versions of calculation. This is yet another sign that action depends on pseudo - crossingsclose to the Fermi levels - the details of crossings far above or below the Fermi energy (between both occupied or bothunoccupied levels) do not have effect on total action.In the third version of the calculation, the highest peaks in the coupling matrix elements were replaced by thepeaks modelled analytically (D2). Actions obtained within this method (in the third column in Table II), both forindividual solutions and the total, are close to those of the second version. This is probably related to the fact thatdifficult couplings that were modelled occur at such q , where ˙ q ≈
0, so that they were suppresed in the instantonequations (12). In general, however, the procedure of peaks modelling seems indispensable for obtaining sufficientlyexact actions if the instanton equations are to be solved in the adiabatic basis (in particular, when a very largenonadiabatic coupling occurs close to the Fermi energy).We also checked the dependence of action on the dimension N of the adiabatic basis. We changed N from 14 to32, always keeping the Fermi level at N / -10-9-8-7-6-5-4-3 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ε [ M e V ] β FIG. 10:
Left:
Energy landscape for
Mt in β − β , minimized over β , β , β with a chosen fission path (marked in red). Right:
Display of 14 positive-parity neutron levels around the Fermi energy along the fission path; the 7-th level from below isthe last occupied. Nr ∆ β = 10 − ∆ β = 10 − ∆ β = 10 − plus fit1 3.2143 3.2057 3.19362 0.9453 8.0320 8.05553 3.2931 6.9294 6.91184 3.2790 -8.7864 -8.78675 -0.0346 2.1493 2.16846 -1.7771 -2.3285 -2.35317 0.9953 1.1126 1.11298 8.8511 9.1817 9.14589 4.1217 -1.3617 -1.445510 5.5588 9.6487 9.829911 -2.9214 -2.3793 -2.381712 -4.5752 -4.5158 -4.566013 -0.4160 -0.3668 -0.378814 6.7950 6.4864 6.484815 6.6443 6.4057 6.403316 2.8743 2.8123 2.8128 S tot / ¯ h 36.8479 36.2254 36.2069 TABLE II: Actions for separate s.p. solutions occupied at the g.s. and their sum - the total action for a nonaxial path; firstcolumn: calculations with the minimal step ∆ β = 10 − ; second column: calculations with the minimal step ∆ β = 10 − ; third column: calculations with the minimal step ∆ β = 10 − augmented with the modelling of the highest peaks in thenonadiabatic couplings by the formula (D2). Action obtained for the trajectory along nonaxial shapes was compared to the one along the axially symmetric path(shown in Fig. 5) in Table IV. In both cases the same neutron levels with positive parity were included. It can beseen that action along the shorter, axially symmetric path is smaller in spite of the fact that the barrier is lower by ∼ q .It has to be emphasized that the last result cannot be treated as general - it merely shows that the instantonmethod applied to reasonably chosen paths can lead to situations similar as in calculations with the cranking massparameters. Deciding whether axial or nonaxial path prevails would require a minimization procedure not defined9 N S tot = (cid:80) N / i =1 S i [¯ h ]16 27.031320 35.828924 35.970528 36.118732 36.2069TABLE III: Action (in ¯ h ) for neutrons of positive parity along the nonaxial path for various numbers N of included adiabaticstates. Path B f [MeV] S tot / ¯ h axial 8.4 21.35nonaxial 6.5 36.21TABLE IV: Fission barrier heights B f and actions S tot (in ¯ h ) for neutrons of positive parity in Mt along the axial (Fig. 5)and nonaxial (Fig. 10) fission paths. here.
V. FISSION HINDRANCE IN ODD NUCLEI - A STUDY
Usually, the spontaneous fission hindrance factors HF for odd nuclei are defined as T osf /T eesf , where T osf is thespontaneous fission half-life of an odd nucleus and T eesf is a geometric mean of the fission half-lives of its e-e neighbours[9]. Experimental facts are that 1) most of HF values lie between 10 to 10 , 2) they do not display any strongdependence on the K (= Ω) quantum number of the g.s. configuration [9].Here, we will use HF calculated as: HF = T osf T esf , (49)where T osf i T esf are fission half-lives of an odd-A nucleus and its A − HF s can be converted into relations between actions for odd- A ande-e neighbours by using the W KB -motivated formula for spontaneous fission half-lives:log ( T sf [ s ]) = − .
54 + 0 . S ¯ h − log (cid:18) E zp . M eV (cid:19) . (50)Here, S is the minimal action chosen among all possible fission paths, and E zp is the zero-point energy (in MeV)of vibration along the fission direction around the m.s. Assuming a universal value of E zp , which is surely anapproximation, one obtains: log ( HF ) ≈ . S odd − S even ¯ h . (51)Calculations were performend for selected superheavy nuclei with known half-lives and, in some cases, known g.s.spin and parities, indicating possible configurations. A similar calculations for actinide nuclei would be much moreinvolved in view of their much longer and more complex barriers. A. Instanton-like action without pairing for
Rf, Rf By solving iTDSE for a given path and collective velocity profile ˙ q ( τ ) one can calculate action for both even andodd nuclei, neglecting pairing. Such results would correspond to a scenario originally put forward by Hill and Wheeler0 Nucleus ( K π ) Rf (1 / + ) Rf (11 / − ) RfAction [¯ h ] S inst S cr ( ˙ q P ) S inst S cr ( ˙ q P ) S inst S cr ( ˙ q P )Neutrons (+) 27.29 86.40 31.23 68.41 19.52 32.11Neutrons ( − ) 73.71 1378.97 82.06 1539.65 65.53 1172.65Protons (+) 46.19 9530.25 50.46 9754.98 46.09 9393.87Protons ( − ) 15.34 21.76 19.11 46.94 12.87 16.39 Sum 162.53 11017.38 182.86 11409.98 144.01 10615.02
TABLE V: Actions (in ¯ h ) for Rf and both configurations in
Rf obtained with collective velocities ˙ q P (see text) alongpaths shown in Fig. 11: instanton-like S inst and with the cranking mass parameter without pairing - S cr ( ˙ q P ). Contributionsfrom neutrons and protons of each parity (indicated in parentheses) are given separately. [40]. Without pairing they cannot be realistic, but allow to notice a few things, among them how much fission wouldbe hindered without pair correlations.We choose the odd nucleus Rf as the example. Its I π = 1 / + g.s., which well corresponds to the K π = Ω π = 1 / + configuration in the W-S model, has a known spontaneous fission half-life of T oddsf = 423 s [9]. Also known is theexperimental lower limit of T oddsf >
490 s [9] for the half-life of the excited I π = 11 / − state, corresponding to the K π = 11 / − configuration in our micro-macro model. The experimental spontaneous fission half-life for the e-eneighbour Rf is T evensf = 6 . HF = 6 . × (for K π = 1 / + configuration) and HF > . × (for K π = 11 / − ).The tunneling path was chosen as follows. First, micro-macro energy landscapes of two nuclei were calculatedby using mass-symmetric axial deformations: for each β − β energy was minimized over β , β , with the steps∆ β = 0 .
05 and ∆ β = 0 . K π were kept constrained at K π = 1 / + and K π = 11 / − for the g.s. and the excited state, respectively. This means a continuation, possibly non-adiabatic, ofthe state Ω π occupied by the odd neutron at the energy minimum. A similar calculation, but without blocking, wasperformed for Rf. It can be seen from the maps in Fig. 11 that keeping the configuration in the odd nucleus leadsto a substantial increase and elongation of the barrier, especially for the excited configuration K π = 11 / − . Takinginto account the experience from action minimization calculations, the fission path was chosen piecewise straight andclose as possible to the minimal energy, in order to keep the path short and the barrier low (the path is also piecewisestraight in β , β ). It is depicted in red in Fig. 11Instanton-like action S inst was calculated by solving iTDSE with the collective velocity: ˙ q P = (cid:112) E ( q ) − E m.s. ) /B P ( q ), where E ( q ) − E m.s. is deformation energy with respect to the m.s. for each nu-cleus/configuration (i.e. with E zp set to zero), and B P ( q ) is the cranking mass parameter of Rf, both includingpairing and calculated along the chosen paths. So, strictly speaking, ˙ q P derives from the paired system, but iTDSEis solved for the system without pairing. For comparison, along the same paths we calculated actions: S cr ( ˙ q P ) = (cid:90) T/ − T/ dτ B NP ( q ) ˙ q P , (52)with the same ˙ q P ( τ ) and the cranking mass parameter B NP ( q ) without pairing for each nucleus (i.e. also for the oddone). The mass parameter B NP includes large peaks due to close avoided level crossings which should considerablyincrease action relative to S inst . We can calculate action S cr ( ˙ q P ) accurately thanks to the large number of points- few thousands per path. Both actions are given in Table V. We also calculated cranking action without pairing S crank , i.e. twice the expression of Eq. (1) with the integrand (cid:112) B NP ( q )( E ( q ) − E m.s. ), i.e. with the mass parameter B NP ( q ) and collective velocity ˙ q NP = (cid:112) E ( q ) − E m.s. ) /B NP ( q ).As might be expected, S cr ( ˙ q P ) hugely overestimates S inst - nearly by two orders of magnitude (Tab. V), mainlybecause of pseudo - crossings of s.p. levels close to the Fermi energy. Locally, around them, B NP >> B P , and thisresults in large local contributions to action S cr ( ˙ q P ). The local bumps in B NP , capriciously dependent on detailsof avoided level crossings, explain vastly different contributions to S cr ( ˙ q P ) from different groups of levels: ∼
90% of S cr ( ˙ q P ) comes from protons of positive parity, while the contributions from protons of negative parity in Rf and1/2 + state in Rf are similar as those to S inst (Tab. V). Using ˙ q NP , which differs from ˙ q P mainly in that it is muchsmaller at pseudo-crossings, largely reduces action: one obtains S crank = 199 .
28 ¯ h for Rf and 222 .
48 ¯ h for Rf( K π = 1 / + ), results larger than, but much closer to instanton-like action S inst .From (50), after assuming E zp = 0 . S = 42 .
24 ¯ h for Rf and1
FIG. 11:
Upper panel: energy landscapes for
Rf minimized over β , β with fixed K π = 1 / + (left) and K π = 11 / − (right) configuration. Lower panel: energy landscape for the neighbouring
Rf. Chosen fission paths marked in red. For e-e
Rf also a second path (marked in blue) was considered (see text). Note different ranges of β in maps. S = 53 .
34 ¯ h for the g.s. of Rf - these doubled actions should be compared to values from Tab. V. Thus, calculated S inst are ≈ . Rf S inst ( ˙ q P ) = 167 ¯ h , larger by 23 ¯ h than for the not very different red one. Apparently, in the absence of pairing, the details of pseudo - crossings havelarge influence on action. This shows that action minimization without pairing might be very difficult and would bedirecting into paths with more gentle crossings.The difference between instanton-like actions S odd and S even comes from: 1) a collective contribution - from thedifferences in deformation energy of the e-e and odd- A nuclei, which in turn comes from: a) different collectivevelocities and b) different lengths of the path; 2) a contribution to action from the odd nucleon [41].Note that in the instanton method without pairing, the odd - even effect in fission half-lives comes exclusively fromdifferent heights and lengths of the fission barriers. If not for these, action for odd- A would lie between those ofneighbouring A − A + 1 e-e species, as it is a sum of individual s.p. instanton-like actions, Eq. (19).For two configurations in Rf we have from Tab. V: ∆ S odd − even = 18 .
52 ¯ h for K π = 1 / + , and ∆ S odd − even =38 .
85 ¯ h for K π = 11 / − . This large difference of 20 .
33 ¯ h can be traced to a larger ˙ q P for the second configuration, and2could be predicted from their very different barriers in Fig. 11. This well illustrates the trend towards higher barriersin calculations with a fixed- K configurations, and those with higher K values in particular. Such K -dependence isabsent in experimental half-lives (see Fig. 17 in [9]).We note that for the relative quantities, ∆ S odd − even /S odd , for the g.s. of Rf and
Rf we obtain from (50),again assuming the same E zp , the ratio 0.114 vs. the experimental value 0.21. However, the minimization of action,not attempted here, could change this ratio. B. Calculations assuming collective mass parameter and an odd - particle contribution
Without having solved Eq. (31) with pairing, we will use unpaired iTDSE solutions to study odd-even fissionhindrance by adopting a hybrid model which incorporates both pairing and the odd particle contribution to action.We assume the following scheme. Action for an e-e nucleus is taken from Eq. (1) with both energy and the crankingmass parameter including pairing. For an odd- A nucleus we assume: S odd = S crank + 12 S insts.p. , (53)where S crank is the cranking action (1) of the e-e core, calculated with the micro-macro barrier for the odd- A nucleus, E odd ( q ) − E , where E = E m.s. + E zp , and the cranking mass parameter with pairing B evenP ( q ) of the neighbouringe-e A − S insts.p. is the contribution to action from the unpaired nucleon. It can be calculated as actionof the instanton-like solution corresponding to the unpaired Ω π state (i.e. the one blocked in the m.s.) with thecollective velocity ˙ q P = (cid:112) E odd ( q ) − E m.s. ) /B evenP ( q ), or as the difference in actions for occupied Ω π states betweenthe odd- A and e-e A − S insts.p. give very similar values; we will give those by thesecond method. The factor 1 / S inst corresponds to twice action of Eq. (1).The rationale behind the choice of the mass parameter and, consequently, of the collective velocity ˙ q P , is theassumed collectivity of quantum tunneling in spontaneous fission. We reject the cranking mass parameter for odd- A ,Eq. (2), as it leads to huge differences between collective velocities ˙ q at the neighbouring q points in an odd- A nucleus,and between A and A − q point. Outside regions where pseudo-crossings of the odd level takeplace, the cranking mass parameters for A and A − A is consistent with assuming its magnitude similar as in the even- A − q . Certainly, similar does not mean equal. However, lack of arguments for any definite ratio singlesout the made choice as the simplest one. It means that the difference in actions for A and A − A and e-e A − S insts.p. is the remaining difference between actions for odd- A and e-e A − A nucleus is calculated conserving the configuration Ω π of the g.s. or releasing this requirement and taking the minimalenergy among various configurations at each deformation. We performed calculations within our model in both waysin order to compare results.Included deformation parameters and the choice of fission paths were as discussed in the previous subsection. Weselected nuclei Z = 103 −
112 for which their, and their even- A − Db and its e-e neighbour
Rf. The g.s. configurationof
Db is K π = 9 / + . Both surfaces for Db, adiabatic (minimized over configurations) and constrained on the K π value, are given together with chosen fission paths. It can be seen that the fission barriers are double-humped,with a smaller second hump. A similar picture holds for other considered nuclei. A clear difference between adiabaticand K π - conserved surfaces can be observed for K = 9 / Db - one can notice higher and longer second barrier.For smaller K , like e.g. the K π = 1 / + configuration in Sg (not shown here), this difference is smaller. A largedifference in barriers for high- K configuration was also seen for Rf in Fig. 11.At this point one has to note that our calculations do not include nonaxial deformations, β , etc, which lowerthe first barrier, neither do they account for mass asymmetric deformations lowering the second barrier. Calcula-tions which include nonaxiality indicate that a path through the nonaxial saddle, lower by 1-2 MeV, has a sub-stantially greater length which moderates or even compensates the effect of the lower saddle. On the other hand,the mass asymmetry is lowering the second barrier and the path incorporating it is not much longer (in terms of ds = (cid:113)(cid:80) λµ ( dβ λµ /dβ ) dβ ) than the one considered here because the mass-asymmetric exit from the barrieroccurs for smaller β - thus the effect of β λ with odd λ is likely to decrease the action.3 FIG. 12:
Upper panel: energy landscapes for
Db, minimized over β , β with the kept g.s. configuration K π = 9 / + (left)and adiabatic (right). Lower panel: energy landscape for
Rf. Chosen fission path marked in red. Note different range of β in maps. It turns out that with realistic values of E zp around 0.5 - 1 MeV we obtain too large actions and half-lives for e-enuclei as compared to the experimental values. The reason lies in a too limited choice of nuclear shapes and in arelatively small strength of the pairing interaction, dictated by the local mass fit [34]. Indeed, we have checked for Rf, that with the pairing strengths and E zp = 0 . A nuclei we decided to artificially change zero-vibration energy E zp so that the mean - square deviation of fission half-lives in e-e nuclei from experimental values is minimal. Thishappens for E zp = 2 .
03 MeV. The fission half-lives of e-e nuclei obtained with the adjusted E zp , which will serve asthe reference for the calculation of fission hindrance factors in odd- A nuclei, are given in Table VI. They are mostlyof the same order of magnitude as the experimental ones, except in Sg and
Cn. The effect of higher E zp cancelsthe contribution to action from the second barrier for Z = 102 − S crank of (53) obtained in two ways for odd nuclei: S confcrank - by keeping the fixedconfiguration, and S adcrank - by using adiabatic occupation of the odd nucleon. Differences between these actions,4 Nucleus S crank / ¯ h T expsf [s] T calcsf [s] No 21.60 1.2E-03 4.1E-03
Rf 18.46 2.3E-05 7.8E-06
Rf 21.91 6.4E-03 7.6E-03
Rf 22.97 2.2E-02 6.4E-02
Sg 21.92 2.6E-03 7.7E-03
Sg 23.62 7.0E-03 2.4E-01
Cn 18.82 9.1E-04 1.6E-05TABLE VI: Calculated actions (in ¯ h ) and calculated vs experimental fission half-lives (in seconds) for e-e nuclei after adjustingzero-point energy E zp to minimize the root-mean-square error.Nucleus K π S confcrank / ¯ h T cranksf [s] S adcrank / ¯ h ∆ S crank / ¯ h Lr 7/2- 33.32 6.2E+07 23.44 9.88
Rf 9/2- 56.06 3.5E+27 25.31 30.75
Rf 1/2+ 34.32 4.6E+08 22.58 11.74
Rf (m) 11/2- 48.89 2.1E+21 22.58 26.31
Db 9/2+ 40.79 1.9E+14 26.65 14.14
Sg 1/2+ 32.44 1.1E+07 23.23 9.21
Sg 3/2+ 30.75 3.6E+05 25.30 5.45
Cn 5/2+ 24.52 1.4E+00 21.56 2.96TABLE VII: For odd nuclei and their K π configurations shown in columns 1 and 2 are given cranking actions (1) calculatedwith the mass parameters of the e-e neighbour: for a fixed K π configuration S confcrank (col. 3), for adiabatic configuration S adcrank (col. 5), their difference ∆ S crank (col. 6), all in ¯ h ; half-lives T cranksf (in s) resulting from S confcrank are given in col. 4. The zeropoint energy E zp was adjusted to experimental fission half-lives of e-e nuclei. S confcrank − S adcrank , are greater than 9 ¯ h , except for Sg and
Cn. As we have checked, they remain large for a widechoice of adopted E zp values between 0.5 and 2 MeV. As for e-e nuclei, paths on the adiabatic surfaces effectivelydo not show the second barrier. With the preserved K π configuration, the contribution of the second barrier toaction is substantial and strongly dependent on the magnitude of K . Fission half-lives calculated with keeping the K π configuration, also given in Table VII, vastly overestimate the experimental values (see col. 3 of Table VIII forcomparison), except in Cn, with the largest discrepancy for large K . Therefore, we do not include odd-particleactions S insts.p. for them.Results pertaining to half-lives of odd- A nuclei and fission hindrance factors obtained with the adiabatic blockingare given in Table VIII and shown in Fig. 13. Here we include results obtained with S adcrank alone and with the addedodd-particle contribution S insts.p. . Obtained half-lives are much closer to the experimental ones than those for fixedconfigurations, but with no clear hindrance, i.e. HF s are mostly underestimated (with two exceptions - Rf and
Db). The modification of the half-life introduced by adding instanton-like action for the odd nucleon S insts.p. (53),shown in Tab. VIII, moves the calculated HF s closer to the experimental values, but the effect is still too small.Odd-even fission hindrance factors calculated assuming the same collective mass parameter in e-e and odd- A neigh-bours suggest the following conclusions:1. Keeping configuration K π of the fissioning states leads to the odd-even fission HF s larger by orders of magnitudethan in experiment.2. Keeping the lowest configuration leads mostly to (with two exceptions) too small hindrance factors.3. Instanton-like correction for the odd nucleon added to adiabatic cranking result S adcrank (53) acts in the rightdirection but is too small. As a result, the obtained HF s are on average smaller than the experimental valuesof 10 - 10 ; they are also more scattered than the latter.One can note that these conclusions concerning diffrences in T sf of odd- A and e-e closest neighbours do not seemto be much influenced by the lack of the action minimization: adiabatic energy landscapes of odd- A nuclei and theire-e neighbours are very similar, S adcrank are relatively smooth and the chosen paths are typical of realistic calculations.5 Nucleus data Adiabatic blocking A X I π T expsf [s] HF exp S insts.p. / ¯ h T crsf [s] T cr + instsf [s] HF crcalc HF cr + instcalc Lr 7/2- 27.4 2.3E+04 1.02 0.16 0.45 3.9E+01 1.1E+02
Rf 9/2- 3.15 1.4E+05 -1.37 6.83 1.73 8.8E+05 2.2E+05
Rf 1/2+ 423 6.6E+04 2.43 0.03 0.33 3.9E+00 4.34E+01
Rf (m) 11/2- > > Db 9/2+ 5.6 2.5E+02 0.04 99.6 103.6 1.56E+03 1.62E+03
Sg 1/2+ 8 3.1E+03 1.85 0.11 0.68 1.43E+01 8.83E+01
Sg 3/2+ 31 4.4E+03 0.61 6.7 12.32 2.79E+01 5.13E+01
Cn 5/2+ (*) 24 2.6E+04 2.76 0.0038 0.06 2.38E+02 3.75E+03TABLE VIII: For seven odd- A nuclei listed in the first column are given: configurations I π (experimental or from systematics),experimental spontaneous fission half-lives T expsf (after [9]) and fission hindrance factors HF exp according to (49), and calculatedquantities (for the g.s. or m.s. configurations K π = I π ): the odd nucleon instanton contribution to action S insts.p. , fission half-lives and HF s following from the adiabatic actions S adcrank for the e-e core (given in Tab. VII) and the same augmented with S insts.p. , S adcrank + S insts.p. . Half-lives are given in seconds, actions in units of ¯ h . Asterisk for Cn signals that the given T expsf isthe smaller of two conflicting experimental values and spin/parity is derived from our W-S spectrum. The symbol (m) denotesthe excited configuration. Lr Rf Rf Rf (m) Db Sg Sg Cn FIG. 13: Logarithms of fission hindrance factors, log HF , defined by Eq. (49): experimental (blue circles) vs. calculated with(red squares) and without (green triangles) the odd-particle instanton contribution for nuclei specified at the bottom of thepanel; an arrow for Rf(m) signifies that only the lower bound for HF is experimentally known; for further details - see text. VI. SUMMARY AND CONCLUSIONS
As the cranking or ATDHF(B) approximation commonly used in calculating spontaneous fission half-lives is incor-rect for odd- A nuclei and K -isomers, in the present paper we tried to include nonadiabatic, beyond-cranking effects inthe description of quantum tunneling. A treatment that avoids the adiabatic assumption is provided by the methodof instantons. For atomic nuclei, it takes a form of iTDHFB equations non-local in time, with specific boundarycondition, which seem unsolvable at present. This motivated us to simplify these equations to iTDSE and studyactions for resulting instanton-like solutions which relate to fission half-lives. The rationale for taking an intermediatestep before the full instanton theory is also related to the question of the energy overlaps (4): they are crucial in theselfconsistent theory, but their proper treatment is unknown for the majority of energy functionals presently used.The instanton equations of the selfconsistent theory were simplified to iTDSE version with the phenomenologicalpotential in the case without pairing, and to iTDHFB equations with the fixed potential and selfconsistent pairinggap for the seniority pairing interaction. The iTDSEs were solved for the phenomenological Woods-Saxon potential6in a number of cases. Since we do not want to relay on the cranking mass parameters for odd- A nuclei, we had toassume the collective velocity. We used for this purpose the cranking mass parameter of the neighbouring e-e nucleus- a plausible, but not unique assumption.The method of obtaining iTDSE solutions and actions was demonstrated for axially symmetric potential. It wasfound that actions may be reliably calculated using reasonably long periods and relatively small bases of adiabaticlevels, lying close to Fermi energy. Compared to the cranking approximation for odd- A nuclei, close avoided levelcrossings have milder influence on instanton-like actions. For collective velocities typical of e-e actinide or superheavynuclei, the quasi-occupations which characterize nonadiabatic excitations in iTDSE solutions are changing mostlyin the vicinity of pseudo-crossings. Instanton-like action rises with the (uniformly) rising collective velocity and thelength of the fission path can balance the lower barrier in the competition between trajectories.The case of triaxial potential turned out to be more demanding as a result of many very weakly-interacting pseudo-crossings. The solution of iTDSE in the adiabatic basis becomes difficult and an effective way of solution remains tobe found. One has to mention that the difficulty caused by many nearly-crossing levels may be less acute when oneincludes the antihermitean part of the mean field. This would make the eigenvalues of the mean-field ˆ h complex andinstanton solutions less susceptible to such crossings.In the study of odd-even fission hindrance factors we made use of iTDSE solutions without pairing by combiningthem with the cranking actions for the e-e cores. The premise of this study was that effective mass parameterspertinent to spontaneous fission are the same (or very similar) in neighbouring e-e and odd- A nuclei. The clear resultobtained under this proviso is that actions calculated for the fixed K π configurations along axially symmetric pathshugely overestimate values from experiment. The actions calculated with adiabatic energy landscapes are mostly tooclose to those of e-e neighbours. Since adiabatic energy landscapes of odd-A nuclei include the effect of the pairinggap decrease due to blocking, one may say that this effect alone is insufficient, while the additional effect of preserving K quantum number is unrealistically large. The instanton-like contributions from the odd nucleon, when added tothe e-e core actions obtained with adiabatic landscapes, are (in most cases) too small to provide for the observedhindrance factors. One could say that actions for odd-A nuclei seem to be closer to the scenario with unconstrainedconfigurations what would suggest changes in K in tunneling, possibly related to nonaxial or more exotic deformationsalong the fission paths.In the near future we plan to study the simplified iTDHFB actions including pairing of Sec. II C in order to see howthe above conclusions about fission hindrance factors change. In particular, it seems interesting whether one couldreproduce their relatively small experimental scatter of merely 2 orders of magnitude. We would also like to see if onecan effectively use the solution method for iTDSE studied here in the solution of the selfconsistent problem. It wouldbe also interesting to improve the presented micro-macro instanton-like procedure. This, however, would probablyrequire some non-selfconsistent version of the antihermitean part of the imaginary-time mean-field. Acknowledgments
The authors would like to thank Micha(cid:32)l Kowal for many inspiring discussions and suggestions, and Piotr Jachimowiczfor providing energy landscapes including effects of the axial- and reflection-asymmetry on fission saddles.
Appendix A: Cranking expressions for action & Floquet exponents
The cranking approximation in solving the real-time Schr¨odinger equation: i ¯ h∂ t ψ ( t ) = ˆ h ( q ) ψ ( t ), where q = q ( t ) isa collective coordinate, follows from expanding ψ ( t ) onto adiabatic states ψ µ ( q ) (11), substituting: C µ ( t ) = c µ ( t ) exp (cid:18) − i ¯ h (cid:90) t (cid:15) µ ( t (cid:48) )) dt (cid:48) (cid:19) , (A1)and solving equations for c µ ( t ): ∂ t c µ = − ˙ q (cid:88) ν (cid:104) ψ µ | ∂ q ψ ν (cid:105) c ν exp (cid:18) i ¯ h (cid:90) t ( (cid:15) µ − (cid:15) ν ) dt (cid:48) (cid:19) , (A2)to the leading order in ˙ q , assuming that the amplitude of the adiabatic ground-state dominates others: | c | ≈ | c µ | << µ >
0. For µ >
0, one can integrate (A2) under the assumption that the exponential gives the leading t -dependence: c µ ≈ i ¯ h ˙ q (cid:104) ψ µ | ∂ q ψ (cid:105) (cid:15) µ − (cid:15) c exp (cid:18) i ¯ h (cid:90) t ( (cid:15) µ − (cid:15) ) dt (cid:48) (cid:19) , (A3)7so the wave function in the cranking approximation is: ψ ( t ) = c exp (cid:18) − i ¯ h (cid:90) t (cid:15) dt (cid:48) (cid:19) (cid:32) ψ + i ¯ h ˙ q (cid:88) µ> (cid:104) ψ µ | ∂ q ψ (cid:105) (cid:15) µ − (cid:15) ψ µ (cid:33) . (A4)This form of integration, different from the usual one for an initial value problem, allows to obtain mass parameter(see below) as a function solely of the coordinate q . Other possible integrals of (A2) imply dissipation of collectivemotion, see e.g. [46] or the recent [47]. From (A4), the initial assumption | c µ | << ¯ h ˙ q(cid:15) µ − (cid:15) (cid:104) ψ µ | ∂ q ψ (cid:105) << does not hold in a vicinity of a sharp (avoided) level crossing, except for minuscule ˙ q .Substituting c µ of (A3) into Eq. (A2) for c one obtains: ∂ t c ≈ i ¯ h (cid:32) i ¯ h (cid:104) ψ | ∂ t ψ (cid:105) + (¯ h ˙ q ) (cid:88) µ> | (cid:104) ψ µ | ∂ q ψ (cid:105) | (cid:15) µ − (cid:15) (cid:33) c , (A5)where the expression in the parenthesis is real, so c evolves as a pure phase: c ≈ exp (cid:40) i ¯ h (cid:90) t (cid:32) i ¯ h (cid:104) ψ | ∂ t ψ (cid:105) + (¯ h ˙ q ) (cid:88) µ> | (cid:104) ψ µ | ∂ q ψ (cid:105) | (cid:15) µ − (cid:15) (cid:33) dt (cid:48) (cid:41) , (A6)with the first term in the exponent being the topological (Berry’s) phase [48]. Usually, the coeficient c is modified toassure normalization of ψ ( t ), (cid:80) µ | c µ | = 1, which introduces corrections quadratic in ˙ q to | c | , but does not changeits phase. As a result, the expectation value of ˆ h , (cid:104) ψ ( t ) | ˆ h ( q ) | ψ ( t ) (cid:105) ≈ (cid:15) ( q ) + ˙ q B qq ( q ), where: B qq ( q ) = 2¯ h (cid:88) µ> |(cid:104) ψ µ | ∂ q ψ (cid:105)| (cid:15) µ − (cid:15) (A7)is the cranking mass parameter.For a periodic hamiltonian with a period T , ˆ h ( t + T ) = ˆ h ( t ), the cranking wave function ψ ( t ) is quasiperiodic, witha phase augmented by − iζT / ¯ h after each period, where by Eq. (A4,A6), if topological phase gives no contribution, ζ = 1 T (cid:90) T [ (cid:15) ( q ) −
12 ˙ q B qq ( q )] dt. (A8)Thus, one can present ψ ( t ) as: ˜ ψ ( t ) exp( − iζt/ ¯ h ), where ˜ ψ ( t ) is periodic with the period T , and ζ is called the Floquetexponent. The function ˜ ψ ( t ) satisfies (in the cranking approximation) the equation: ( i ¯ h∂ t − ˆ h ( q )) ˜ ψ = − ζ ˜ ψ . Calculatingaction, (cid:82) T dt (cid:104) ˜ ψ | i ¯ h∂ t ˜ ψ (cid:105) , one thus obtains (cid:82) T dt ( (cid:15) + ˙ q B qq ( q ) − ζ ), which from (A8) equals (cid:82) T dtB qq ( q ) ˙ q . Thisaction may be used to quantize energy of collective modes, see e.g. [49].The analogous solution to the equation in imaginary time τ = it , ¯ h∂ τ φ + ˆ h ( q ) φ = 0, with − T / < t < T / q ( − τ ) = − ˙ q ( τ ), is: φ ( τ ) = c exp (cid:18) − h (cid:90) τ (cid:15) dτ (cid:48) (cid:19) (cid:32) ψ − ¯ h ˙ q (cid:88) µ> (cid:104) ψ µ | ∂ q ψ (cid:105) (cid:15) µ − (cid:15) ψ µ (cid:33) , (A9)where: c ≈ exp (cid:26) − h (cid:90) τ (cid:18) ¯ h (cid:104) ψ | ∂ τ ψ (cid:105) + 12 ˙ q B qq ( q ) (cid:19) dτ (cid:48) (cid:27) , (A10)although, due to the exponential character of solutions, the range of validity of the cranking approximation is probablymuch smaller than in the real-time. The corrections to c quadratic in ˙ q which ensure the condition (cid:104) φ ( − τ ) | φ ( τ ) (cid:105) = 1modify the τ -even part of c , but not its time-odd exponent. In this approximation, (cid:104) φ ( − τ ) | ˆ h ( q ) | φ ( τ ) (cid:105) ≈ (cid:15) ( q ) − ˙ q B qq ( q ). For a periodic hamiltonian, as the one with q ( τ ) describing a bounce solution, this wave functioncan be presented as φ ( τ ) = ˜ φ ( τ ) exp( − ζτ / ¯ h ), where ˜ φ ( τ ) is periodic; the Floquet exponent here is ζ = 1 T (cid:90) T/ − T/ [ (cid:15) ( q ) + 12 ˙ q B qq ( q )] dτ. (A11)8The periodic function ˜ φ satisfies the equation: ¯ h∂ τ ˜ φ = ( ζ − ˆ h ( q )) ˜ φ . Action defined for it by: S = (cid:82) T/ − T/ dτ (cid:104) ˜ φ ( − τ ) | ¯ h∂ τ ˜ φ ( τ ) (cid:105) , can be written by using the previous relations as: S = (cid:90) T/ − T/ dτ ( ζ − (cid:15) + 12 ˙ q B qq ( q )) = (cid:90) T/ − T/ dτ B qq ( q ) ˙ q , (A12)consistent with the cranking formula (1). Appendix B: Methods applied to obtain non-selfconsistent bounce solutions
The exponential behaviour of solutions to Eq. (12) and the presence of many different exponents pose problemswhich require special care in the numerical treatment. In this section we address these difficulties and discuss methodsapplied to obtain instanton-like solutions in this work.Let us first notice, that the set of equations (12) without the ζ -term:¯ h ∂C µi ∂τ = − (cid:15) µ ( q ( τ )) C µi − ˙ q N (cid:88) ν (cid:104) ψ µ ( q ( τ )) | ∂ψ ν ∂q ( q ( τ )) (cid:105) C νi (B1)is of the form: ˙C i = A ( τ ) C i , where the matrix A ( τ ) is periodic: A ( − T /
2) = A ( T / C i is the column - vectorof coefficients C µi ( τ ) of the i -th solution. Therefore, according to the Floquet theorem, the linearly independentsolutions can be written as: C i ( τ ) = P i ( τ ) e − ζ i τ/ ¯ h , (B2)where P i ( τ ) is a periodic function with the period T while ζ i are determined by the eigenvalues e − ζ i T/ ¯ h of themonodromy matrix, M = G ( T / , − T / G ( τ , τ ) designating resolvent of (B1), propagating solutions from τ to some other time τ . Putting (B2) into (B1) we obtain equation for the unknown periodic functions: ˙P i = ( I ζ i − A ( τ )) P i ( τ ) , (B3)with the boundary condition: P ki ( − T /
2) = P ki ( T /
2) = v ki , where v ki is the k -th component of the i -th eigenvectorof M . The equation above is identical to Eq. (12), therefore P i ( τ ) are the sought bounce solutions with Floquetexponents ζ i and boundary values given by the eigenvalues and eigenvectors of the monodromy matrix. Theseconsiderations lead to the following scheme of solving iTDSE with the instanton - like boundary conditions, whichwas used in the present work:1. Calculate the monodromy matrix M of (B1) by a step-by-step forward integration along short intervals of τ inthe range τ ∈ (cid:104)− T / , T / (cid:105) , with the identity matrix as the initial condition;2. Perform the eigendecomposition of M ;3. Taking the consecutive eigenvectors as initial values and their corresponding eigenvalues as Floquet exponents,integrate numerically Eq. (B3) (at the final point τ = T /
2, according to the periodic boundary condition, oneshould recover the initial values). In this way one obtains N linearly independent bounce solutions.In this work, Eq. (12) and (B1) were treated as if the matrix A ( τ ) were piecewise constant on each integrationinterval. One step of integration of Eq. (B1) consists in calculating the exponential of a constant matrix and itsaction on the vector of coefficients of the previous step: C ( τ i +1 ) = exp ( A · ( τ i +1 − τ i )) C ( τ i ) = G ( τ i +1 , τ i ) C ( τ i ) . (B4)The resolvent matrix is obtained by a successive multiplication of the one-step exponentials.The chief difficulty in applying the above procedure comes from the exponential behaviour of solutions. We canwrite them in the form with the explicit exponential factor (which is an analogue of the phase factor in real-timequantum mechanics) as: C µi ( τ ) = c µi ( τ ) e − h (cid:82) τ − T/ (cid:15) µ ( q ( τ (cid:48) )) dτ (cid:48) . (B5)9This dependence, combined with the presence of markedly different adiabatic energies (cid:15) µ ( q ), leads to the exponentiallydivergent numerical scales. During the evolution, the coefficient associated with the lowest state will be amplifiedrelative to all others. Therefore, a simple numerical multiplication of successive one-step exponentials involves amixing of elements of different orders of magnitude, which results in the loss of accuracy (due to a finite numericalprecision). One needs a way of separating different scales at each matrix multiplication. In our work we adopt thesingular value decomposition (SVD) approach, described in [50]. The procedure consists of the following steps:1. SVD decomposition of the propagation matrix in the first step of integration: G ( τ , − T /
2) = U Σ V , where U i V are orthogonal matrices, and Σ is a diagonal matrix with singular values, which contain informationon magnitude scales present in the problem.2. For the successive integration steps one performs the following operations:(a) Calculation of the propagation matrix over a short interval ( τ i − , τ i ): G ( τ i , τ i − ) = exp( A · ( τ i − τ i − )),(b) Multiplication of matrices in order given by the brackets in the expression: [ G ( τ i , τ i − ) U i − ] Σ i − = S i ,(c) Performing the SVD decomposition of the matrix S i : S i = U i Σ i (cid:101) V i ,(d) Multiplication of the V matrices: S i V i − = U i Σ i ( (cid:101) V i V i − ) = U i Σ i V i – this leads to the SVD form of thepropagation matrix G ( τ i , − T /
2) with separated numerical scales stored in the diagonal elements (singularvalues) of the matrix Σ i .3. Performing steps ( i = 2 , . . . , N ) described above along the range of integration ( − T / ,
0) one obtains the SVDform of the propagation matrix: G (0 , − T /
2) = U N Σ N V N .The monodromy matrix has the form: M = G ( T / , − T /
2) = G ( T / , G (0 , − T / A ( τ ) = A † ( − τ ), fulfilled by the matrix of Eq. (B1), G ( T / ,
0) = G † (0 , − T / M = G † (0 , − T / G (0 , − T / M = V † N Σ † N Σ N V N , and the products: σ ∗ i σ i , with σ i the i -th singular value of Σ N , are equal to the eigenvalues e − ζ i T/ ¯ h of the monodromy matrix. It is thus sufficient tointegrate Eq. (B1) over a half of period, i.e. in the range ( − T / , ζ j > ζ (where ζ –the lowest ζ ). From Eq. (3) and its counterpart for φ ∗ i ( − τ ) one obtains: (cid:104) φ i ( − τ ) | φ j ( τ ) (cid:105) = (cid:104) φ i ( − τ ) | φ j ( τ ) (cid:105) e h ( ζ j − ζ i )( τ − τ ) . (B6)This means that if at some τ the overlap (cid:104) φ i ( − τ ) | φ j ( τ ) (cid:105) (cid:54) = 0 (which is inevitable due to a limited numericalprecision), the evolution causes its exponential rise and spoils φ j solution by increasing admixtures of φ i with lower ζ i to it. To eliminate this effect, the orthogonalisation of φ j with respect to all solutions with ζ i < ζ j was performedafter each integration step.The accuracy of the applied method of solution was tested by comparing the results with the ones of the algorithmwith a finer imaginary time-step (and thus more densly calculated adiabatic Woods-Saxon energies and wave functions)and by running the code in quadrupole precision. The other tests, of more physical significance, are described inAppendix C. Appendix C: Stability of solutions with respect to period and the size of the adiabatic basis
The stability of iTDSE solutions, in particular their actions, with respect to the assumed period T and basisdimension N was checked on a few examples. Here we give results obtained for the Ω π = 1 / + neutron levels in Mt, discussed in Section IV A.
1. Stability of action with respect to the period
The values of actions S i and Floquet exponents ζ i of solutions φ i change with increasing period T . As the instanton-like solution would correspond to T = ∞ , it is of relevance that S i and ζ i should stabilize above some T . It is indeedthe case: actions S i , shown in Tab. IX, change not more than ∼
3% except the very small ones, whose contribution isnegligible anyway. The convergence of the Floquet exponents to the eigenenergies at the initial (and final) state canbe well approximated by the formula: ζ i ( T ) = A i + B i /T with constant A i and B i , and in calculations the relation ζ i ( ∞ ) = A i ≈ (cid:15) i , although not axact, is approximated reasonably well - see Tab. X.0 Nr T=20 T=25 T=30 T=35 T=40 T=451 0.2893 0.2953 0.2970 0.2976 0.2978 0.29832 0.6306 0.6368 0.6399 0.6399 0.6401 0.64023 1.5633 1.5813 1.5854 1.5870 1.5874 1.58754 -0.0210 -0.0093 -0.0051 -0.0038 -0.0034 -0.0033TABLE IX: Action values (in ¯ h ) calculated for the four lowest iTDSE solutions for various assumed periods T (in 10 − s).Nr T=20 T=25 T=30 T=35 T=40 T=45 ζ T →∞ (cid:15) g . s -9.044 -8.990 -8.059 -8.061 -5.588 -5.600 -4.089 -4.037 TABLE X: Floquet exponents ζ i [MeV] for the four lowest instanton-like iTDSE solutions, for increasing values of the period T [10 − s], and the limiting value ζ i ( T → ∞ ) [MeV], estimated from the formula in the text, vs s.p. energies (cid:15) i [MeV] at theg.s. deformation.
2. Stability of action with respect to the dimension N of the adiabatic basis We also tested the change of the total action S tot (19) with increasing number of adiabatic basis states N included symetrically below and above the Fermi level . Intuition would suggest that the main contribution to action shouldcome from states lying close to the Fermi level. For trajectory depicted in Fig. 5, action values for increasing N arepresented in Tab. XI. One can see that for larger N changes in action become negligible.For the case of N = 14 basis states, in the upper panel of Fig. 14, we show quasi-occupations of adiabatic statesabove the Fermi energy, (cid:15) > (cid:15) F , in the lowest iTDSE solution φ . It can be seen that excitations to adiabatic statesabove the Fermi level are marginal and nearly do not contribute to action. In the lower panel of Fig. 14, are shownquasi-occupations of the same adiabatic states in the highest occupied instanton-like state φ . It can be seen thattransitions occur mainly to the adiabatic states closest in energy. These results indicate that adiabatic states in awide enough energetic window around the Fermi level suffice to calculate instanton - like action. Appendix D: Treating sharp pseudocrossings along nonaxial fission paths
Sharp pseudocrossings in the s.p. spectrum for nonaxial shapes generate very narrow (in q ) and large peaks in thematrix elements of the adiabatic coupling; an example is shown in Fig. 15. Those present an obvious impediment toan effective solution of iTDSE.A rapid change of adiabatic states with q at sharp pseudocrossings suggests the unsuitability of the adiabatic basis.In chemistry, there were many trials in such situations to find a suitable quasi-diabatic basis with smaller and regularcoupling between crossing states [52–54]. The diabatic basis, like {| χ i (cid:105)} in the two-level model (sect. III), might seem N S tot = (cid:80) N / i =1 S i [¯ h ]8 2.517210 2.538812 2.565714 2.5779TABLE XI: Total action S tot for the lower half of the iTDSE solutions (i.e. occupied instanton-like states) as a function of thenumber N of adiabatic basis states included in calculations. -0.5-0.4-0.3-0.2-0.1 0 0.1-10 -5 0 5 10 τ State 7 c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ )c (- τ )*c ( τ ) FIG. 14: Quasi-occupations of seven upper adiabatic states for the lowest ( upper panel ) and the seventh (i.e. last occupied; lower panel ) instanton solution for N = 14. Note ∼ a good candidate. It is related to the adiabatic basis via θ angle, being a function of α = V /E and q − q , where q is the crossing point. One can locally fit these parameters to each crossing and define a new basis by means of theangle θ , while leaving all not crossing levels unchanged. This is an approximation, so the resulting basis is not strictlydiabatic (with (cid:104) χ i | ∂ q χ j (cid:105) = 0), but quasi-diabatic ( (cid:104) χ i | ∂ q χ j (cid:105) << (cid:104) φ i | ∂ q φ j (cid:105) ). One can show that in the general case ofmany levels and many deformations q i a strictly diabatic basis does not exist [51].The calculations have shown that the quasi-diabatic basis found by this procedure does not bring any advantage incomparison with the adiabatic one: the density of points necessary to probe the neighbourhood of a crossing in orderto ensure an approximately correct action value is the same for both bases (very dense mesh is needed in both cases).An alternative solution would be solving instanton equations using a large basis, smoothly changing with defor-mation (like that of the harmonic oscillator), without resorting to the adiabatic basis. Then the problem of sharpcrossings would be avoided, however, not without a cost: large basis would be needed that probably would lead tothe necessity of using quadruple precision and more time-consuming calculations.We kept the adiabatic basis. In order to integrate Eq. (12) we used a changing step in β for calculating inputdata, i.e. energies and adiabatic couplings along the path. The step ∆ β was diminished when a change in any of thecouplings was above 10% of its preceding value. It was necessary to impose the minimal step value, ∆ β = 10 − (with2 β as the parameter of the path). Such a probing was dense enough for a nearly exact integration for most of thepeaks. However there were a few narrow and high peaks which were still not well rendered. In those cases, the shapeof such peak was modelled by the formula (43) (with parameters α and q ) using the least squares fit to the calculatedpoints. Next, for each such modelled crossing, a 2 × G ( τ fin , τ ini ) for the two crossing levels wasintegrated [defined by the Eq. (B4)], where τ ini , τ fin means the beginning and end of the peak. The integration ofa model peak is simple due to its analytic formula which makes many Woods-Saxon calculations unnecessary. Thenthe propagation matrix (cid:101) G ( τ fin , τ ini ) for all N levels is calculated as follows: propagation of the N − G calculated for the fittedmodel. Denoting the index of the lower crossing level i , one can schematically write the matrix (cid:101) G : . . . i i + 1 . . . N (cid:101) G (cid:101) G . . . . . . (cid:101) G N (cid:101) G (cid:101) G . . . . . . (cid:101) G N ... ... ... . . . ... ... . . . ... i . . . G i i G i i +1 . . . i + 1 0 0 . . . G i +1 i G i +1 i +1 . . . N (cid:101) G N (cid:101) G N . . . . . . (cid:101) G N N (D1)Thus, we neglect the cross terms, setting: (cid:101) G αl = (cid:101) G lα = 0, where α (cid:54) = i, i + 1 and l = i, i + 1. It means we treat thecrossing of two levels as isolated: the evolution of c i , c i +1 is dominated by the coupling between them, (cid:104) φ i | ∂ q φ i +1 (cid:105) ,while the effect of other states c α (cid:54) = i,i +1 on crossing levels and the effect of the pair on those other states can beneglected in the vicinity of crossing.This procedure was tested in few cases in which the vicinity of the crossing could be probed dense enough forthe solution without any fit to be exact. Then the solutions for smaller density of calculated points but with themodelled adiabatic coupling in the vicinity of crossing was compared to the exact one. It turned out that for thedesired accuracy the model for the coupling should include independent parameters for the height and half-width: (cid:28) φ (cid:12)(cid:12)(cid:12)(cid:12) dφ dq (cid:29) = 12 α ( q − q ) + σ . 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