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Beta decay of 252Cf on the way to scission from the exit point K. Pomorski1, B. Nerlo-Pomorska1 and P. Quentin2,3 1Katedra Fizyki Teoretycznej, Uniwersytet Marii Curie-Skl odowskiej, 20031 Lublin, Poland 2Université de Bordeaux, CENBG, UMR5797,33170 Gradignan, France 3CNRS, IN2P3, CENBG, UMR5797, 33170 Gradignan, France (Dated: January 13, 2015) Upon increasing significantly the nuclear elongation, the beta-decay energy grows. This paper − investigateswithin asimpleyetpartlymicroscopic approach,thetransition rateof theβ decay of the252Cfnucleusonthewaytoscissionfromtheexitpointforaspontaneousfissionprocess. Arather crude classical approximation is made for the corresponding damped collective motion assumed to be one dimensional. Given these assumptions, we only aim in this paper at providing the order − of magnitudes of such a phenomenon. At each deformation the energy available for β decay, is 5 determinedfrom suchadynamicaltreatment. Then,for agivenelongation, transition ratesfor the 1 allowed (Fermi) 0+ −→0+ beta decay are calculated from pair correlated wave functions obtained 0 withinamacroscopic-microscopic approach andthenintegratedoverthetimecorrespondingtothe 2 whole descent from exit to scission. The results are presented as a function of the damping factor n (inverseofthecharacteristicdampingtime)inuseinourclassicaldynamicalapproach. Forinstance, a in the case of a descent time from the exit to the scission points of about 10−20 second, one finds J a total rate of beta decay corresponding roughly to 20 events per year and per milligram of 252Cf. 2 The inclusion of pairing correlations does not affect much these results. 1 PACSnumbers: 21.10.DR,21.10.Ma,21.60.Jz,25.85,Ca ] h t - I. PURPOSE tio ρ = R12/R0 of the distance R12 between the mass l c centreofnascentfragmentsandtheradiusR0 ofthe cor- u responding spherical nucleus). One sees clearly that the Due to the decrease of the Coulomb energy and the n beta stability valley shifts from Californium almost to increase of the surface energy, upon strongly deform- [ ing a nucleus, it may happen that a nucleus which Nobelium when the nuclear shape varies from a sphere 1 is β−-stable at equilibrium deformation gets a positive to configurations close to the scission point where one v Q(β−) = (M(Z,A) M(Z +1,A) m )c2 at large de- has typically in such very heavy nuclei ρ 2.5. 1 − − e Thisresultcouldbealsoattainedupon≈consideringthe formation. This is exemplified on Fig. 1 within a Liq- 0 opposite behavior of proton and neutron separation en- uid Drop Model (LDM) approach [1], where the masses 7 ergies as could be seen from an extremely crude liquid 2 of A = 252 isobars (from Plutonium to Kurchatovium) drop approach. Indeed, let us consider a Liquid Drop 0 are plotted for various deformations (defined as the ra- . energy formula comprising merely volume, surface, di- 1 rectCoulomband volumesymmetry terms. For a spher- 0 icalshape, onegets forthe (positive)separationenergies 5 1 90 1.4 for protons (Sp) and neutrons (Sn) with a usual nota- sph. : A=252 tion and approximating the energy differences as differ- v entiablequantities(i.e. makingacontinuousapproxima- Xi eV] 80 1.6 tion for the variables Z or N): M ar s excess [ 70 12..80 SSnp ==SSp(n(00))++3131aacc ZZ22AA−−3443 −−2332aass AA−−3131 −2ac ZA−31(1) Mas 60 2.2 where Sp(0) and Sn(0) are deformation independent terms given e.g. for the protons by 2.4 50 ρ (N Z)(Z+3N) S(0) = a +a − (2) 94 96 98 100 102 104 p − v sym. A2 Z It appears clearly that upon deforming the nuclear FIG.1. MassexcessofPu-KunucleiwithA=252evaluated shape, both the surface and Coulomb deformation- withintheLSDmodel[1]. Eachcurvecorrespondstodifferent dependent terms present in Sn will contribute to its de- elongation of nuclei ρ defined in the text. These values are crease. ThesetermsarealsopresentinS butaresupple- p reported on the r.h.s. side of the every second curve. The mented by a term which increases with the deformation. upper curve (labelled as ”sph.”) presents the mass excesses We thus conclude that the difference S S is increas- p n calculated for spherical nuclei. ing with deformation from about zero −for the ground 252Cf 2 10 Ground state 252Cf Scission point V] 5 VBexp= 4.8 MeV Q(β-) [MeV] e 0 0 M 0 -2 -2 [exp -5 turning point MLSD+δEmic M - -10 -4 -4 D λ S -15 252 n e [ MeV]sp -6 λp λn e [ MeV]sp -6 λp n E = MpL --2250 Cf scissio -8 -8 -30 1 1.5 2 2.5 -10 -10 ρ = R12 / R0 -12 p n -12 p n FIG. 3. Macroscopic (solid line) and macroscopic- microscopic (dashed line) fission barriers and the beta-decay FIG. 2. Proton (p) and neutron (n) single-particle levels in energyQβ (dottedline)of252Cfasafunctionofnuclearelon- theground-state (l.h.s.) and at thescission point (r.h.s.). gation. state of a β stable nucleus to a value which could be fission barrier down to the scissionpoint, along with the of the order of some MeV. Resorting now to state of the underlying LDM one. artmacroscopic-microscopiccalculations(usingtheLDM As a result we find that apart from the ground state model of Ref. [1] and the Yukawa folded potential of region, and in particular at large deformations (namely Ref. [2]) including pairing correlations using a seniority R /R > 1.5), the LDM energy curve represents a rea- pairing force, one gets an inversion of the relative posi- 12 0 sonable approximation of the results, where shell and tionofneutronandprotonFermilevelsofthe 252Cfbeta pairing effects are taken into account. However, for stable nucleus upon increasing the elongation, the lat- the calculated ground state one has definitely to make ter becoming larger than the former. This is illustrated a choice about its exact definition. One may rule out on Fig. 2 showing the neutron and protonsingle particle the LDM solution since shell and pairing effects increase (s.p.) spectra around the Fermi levels for ρ 0.9 and ≈ its height. As a goodapproximationof the macroscopic- ρ 2.8 corresponding respectively to the ground state ≈ microscopicexitpoint,ascanbecheckedonFig.3,isthe and the scission point solutions. exit point on the LDM fission barrier (on the side of the − We are therefore considering the following β decay descent towards fission) which is located at exactly 4.8 during the spontaneous fission process (beyond the exit MeV below the saddle point. This value corresponds to point): theexperimentalfissionbarrierheight(notingenpassant 252Cf 252 Es +e−+ν¯ (3) that our macroscopic-microscopic calculations do repro- 154 153 −→ duce it quite well). Welimitourselvestoallowedtransitions. Thespinand In what follows we will use also, instead of the ratio parity of the compound nucleus being conserved during ρ = R /R , a collective variable q obtained by a trans- 12 0 thefissionprocess(intheabsenceofparticleemission)we lationofρsuchthatq =0attheexitpoint. Wedefinein are considering thus a 0+ 0+ transition. This rules a somewhat arbitrary fashion a single scission point. It −→ out a Gamow-Teller transition and we are thus simply correspondstothepointonthe LDMfissionpath,where left with a Fermi transition. thenuclearliquiddropceasestobe consistingofonesin- gle piece (it is obtained here for ρ 2.9). It corresponds ≈ to a gain in energy of ∆E 28 MeV with respect to max II. APPROXIMATIONS ≈ theexitpointenergy(orthegroundstateone,sincethey are of course the same). − At this very preliminary stage of our study, we make On Fig. 3 we have also plotted the LDM Q(β ) value thefollowingapproximationspertainingtothestaticand as a function of ρ. It varies from -1.2 MeV at sphericity dynamical calculations as well as to the evaluation of to enter the instability region for ρ 1.5 which happens ≈ transition rates. coincidentally, to be very close to the exit point to reach a)Weconsideronlyasinglefissionpathassumedtobe 2.5 MeV around the scission point. ≈ the mostprobable one outof static macroscopic- micro- b)Inthepresentapproach,whichaimsonlyatprovid- scopic calculations of the above discussed type. Pairing ing orders of magnitude, we make the following approxi- is included, as it should, to define adequately this opti- mations for the collective dynamics: mal path. On Fig. 3, we have plotted the corresponding - As a function of the collective coordinate q, one may 3 approximatereasonably(seeFig.3)the potentialenergy III. EFFECTIVE Qβ VALUES AND AVAILABLE E (q)asalinearlydecreasingfunctionwhenqvariesfrom PHASE-SPACE p zero to q which the value of this parameter at scission. s One has thus Atagivenelongationqoratgiventimet(q),theparent nucleus has an excitation energy of E⋆(q). The effective E (q)= αq α 20 (0 q q ) (4) p − ≈ ≤ ≤ s Qβ(q)valueforatransitionfromaneutronsingleparticle stateitoaprotonsingleparticlestatef isthus givenby -Weapproximatethemassparameterm(q)asbeingcon- stantm(q)=µduringthe descentfromthe exitpointto Q(i,f)(q)=E⋆(q)+e (i) e (f))+0.78 MeV , (12) scission and equal to the reduced mass between the two β n − p final fragments (to assess the relevance of this approxi- wherethelastconstantoriginsfromthemassdifferenceof mation see Ref. [3]). neutron andproton plus the electron mass. Defining the -Weassumethatthedampingofthemotionresultsfrom neutron Fermi level λ and similarly for protons λ , one n p a simple friction force kq˙ with k>0. willconsideraneffectiveneutronFermilevelasλeff(q)as − n Then the equation of motion yields the following triv- ial solution (introducing of course a vanishing collective λeff(q)=λ +E⋆(q)+0.78 MeV . (13) n n velocity at the exit point) Possibletransitionsbetweenneutrons.p. statesandpro- q = α[t µ(1 e−kµt)] , (5) tons.p. states willrequire the followingconditions to be k − k − satisfied which could be numerically inverted to get the time t(q) K (i)=K (f) , as a function of the deformation q. λn e (f)p e (i) λeff(q) . (14) Then the collective kinetic energy will be given by p ≤ p ≤ n ≤ n EK(q)= 12 µkα22(1−e−kµt)2 . (6) IV. TRANSITIONS RATES The available energy is The nuclear matrix element for such a (i,f) Fermi transition is given with a usual notation by ∆E(q)= E (q)=αq , (7) p − from which one gets the excitation energy E⋆(q) as Mi,f =ZZZ φ⋆p(f)(~r)φn(i)(~r)d3r . (15) E⋆(q)= µα2[x g(x) 1g(x)2] , (8) A special note is to be made here, about the intrinsic k2 − − 2 paritybreaking. Letuscall Ψ theintrinsicwavefunction | i breakingtheleft-rightsymmetry,eitherfortheparentor where the dimensionless quantity x is defined by thedaughterstates. Oneshouldprojectapositiveparity kt state out of it, as x= (9) µ 1 Ψ(+) = (Ψ +Πˆ Ψ ) , (16) and with | i| √2 | i| | i g(x)=1 e−x . (10) where Πˆ is the parity operator. − The above nuclear matrix element will thus comprise c) To compute beta decay rates, the effect of pairing two parts corresponding to the two overlaps Ψ(i)Ψ(f) correlations will be taken into account. For the sake of and Ψ(i)Πˆ Ψ(f) . Howeveritisknownfromhparit|yproi- clarity we will first discuss our approach without them h | | i jection calculations of the fission barrier of heavy nuclei, andinaseparatesectionthemodificationsbroughtinby see Ref. [4], that somewhere before the second fission their inclusion. barrier and beyond, the intrinsic parity breaking defor- The conservation of the axial symmetry is assumed mation is so large that the overlap between the corre- along the path. Each s.p. state is thus defined in par- spondingwavefunctionandits parityimageis negligible. ticular by the usual angular momentum projection K We are just left here with the Ψ(i)Ψ(f) overlap as in quantum number. h | i the non-parity breaking case. Yet, in this case one can- We will assume that there is no polarization i.e. that not, of course, apply a selection rule on the quantum the s.p. wavefunctions in the 252Cf or 252Es mean fields number π which is not conserved but only on the angu- arethesameandfurthermorethatonehas,withatrans- lar momentum projection K which we have assumed to parentnotation,thefollowingrelationbetweenthebind- be a good quantum number. ing energies of the parent and daughter nuclei (Koop- The associated transition rate is [5] mans approximation) M 2 m5c4 E(252Es)=E(252Cf)+e e . (11) R =G2 | i,f| e f(Z =99,Qi,f) , (17) p− n i,f 2π3 ~7 β 4 −21 which after inserting of the week interaction coupling t [10 s] constant (G) and other constants values takes the fol- 0 5 10 15 20 lowing form: 1.4 1 2 4 R =1.105 10−4 M 2 f(Z =99,Qi,f) . (18) 1.2 i,f | i,f| β 1 A rough estimate of the function f as a function of the k/µ=6 MeV//h (i,f) 0.8 effective Q value Q (q) is given for Z =99 by β q 0.6 log (f)=3.5 log (Q(i,f)(q))+3 , (19) 0.4 10 10 β 0.2 where Q(i,f)(q) is expressed in MeV. β 0 0 5 10 15 20 25 30 t [ /h/MeV ] V. INCLUDING PAIR CORRELATED NUCLEAR STATES FIG.4. Collectiveelongationq=(R12−R1ex2it)/R0asfunction oftime(eq.5)fordifferentvaluesofthenucleardampingk/µ. Whenpairingcorrelationsareincludedinboththepar- ent and daughter nuclear states (assuming again no polarization effects, i.e. taking the s.p. wavefunctions, oc- onthe retainedvalueforthe frictionparameterk. Inthe cupation probabilities and quasi-particle energies as ob- followingwewilluseourestimatesofPdecay asafunction tained in the parent nucleus) one has just to multiply of the time tsc to evaluate the number of β-decays from the overlaps M by the BCS factor u v while the a sample of 252Cf in a given time period. i,f p(f) n(i) effective Qi,f(q) becomes β VII. RESULTS Q(i,f)(q) =E⋆(q) [Eqp(i)+Eqp(f)]+(λ λ ) β − n p n− p +0.78 MeV , (20) t [10−21 s] where Enqp(i) and Epqp(f) are the initial neutron and the 0 5 10 15 20 final proton quasiparticle energies. 6 In this case,one may consider all transitions from sin- 1 gle neutron to single proton states which satisfy the fol- 5 x lowing conditions beyond the angular momentum com- ponent matching rule (K (i)=K (f)): V ] 4 n p e -theprotonstateshouldnotbefullyoccupied(i.e. lying M 3 below the valence space), [ K 2 - the neutron state should not be fully unoccupied (i.e. E k/µ=2 MeV//h lying above the valence space), 1 x - the final state energy should be lower or equal to the 4 x initial state energy, i.e. Qi,f(q) 0 . 0 6 x β ≥ 0 5 10 15 20 25 30 t [ /h/MeV ] VI. TOTAL TRANSITION RATES FIG.5. Pre-scission kineticenergyfor a few valuesof there- At a given time (or at a given elongation) the total ducedfrictionasfunctionoftime. Thetimeatwhichscission configuration is reached fora givenk/µ valueismarked bya rate of decay will be obtainedby summing all individual cross. rates R to get R(q). For the whole descent from exit i,f to scission, one will get the number of decay for a single fission, i.e. the probability of decay, by integrating Let us first use the results of our simple dynamical R(q) over the whole time of descent from the exit to the calculations to describe the collective motion from the scission points exit to the scission points. Our results will be presented for a few values of the reduced friction parameter k/µ, qs k −kt −1 which inverse is the damping time (5). The damping Pdecay =Z R(q) α (1−e µ) dq . (21) parameters k/µ is given in units MeV/~ 1.5 1021s−1. 0 ≈ As seen in Fig. 4, the collective descent from q =0 to This probability, and similarly the pre-scission kinetic q 1.4takesfrom 3.510−21sforalargedumpingtime energy E0 = E (q ), are of course strongly dependent (k≈/µ=1intheabo≈vediscussedunits)to 1810−21sfor K K s ≈ 5 asmalldampingtime(k/µ=6)). Inallconsideredcases VIII. CONCLUSIONS the collective motion is completely damped at scission as exemplified on Fig. 5. The pre-scission kinetic energy This paper aimed at pointing out the strong depen- EK valuesrangefromacouplehundredkeVforthelarge dence of the beta-decay stability as a function of the dampingcase(k/µ=6)to 5MeVforthelowdamping nuclear deformation. From state of the art macroscopic- ≈ case (k/µ=1). microscopic calculations it has been claimed that a very Theresultingnumberofbetadecaysfora1mgsample heavynucleuslike252Cf,whilebeta-stableaswell-known of 252Cf, is given in Fig. 6, as a function of the descent in its ground state becomes instable near the fission exit time t from the exit to the scission points (or equiva- point. Throughaverysimpledescriptionofthecollective sc lently as a function of the damping parameter k/µ). It dynamics, the transition rates of the Fermi beta decay ranges from 6 for the low damping case (k/µ = 1) to during the spontaneousfissionprocessof this nucleus up ≈ 46for the largedampingcase(k/µ=6). The rangeof tothescissionpoint,havebeencalculated. Theydepend ≈ consideredvaluesofthedampingparameterk/µistaken of courseon the amount of damping ofthe fissioncollec- following estimates made in Ref. [6] (see Fig. 10 of this tivemodewhichregulatestheexcitationenergyavailable Reference). at each time, for such a decay. In so far as these rates could prove to be experimentally reachable, they would provideamuchneededsourceofinformationsonthepre- k/µ [MeV//h] scissiondynamics,ase.g. theaveragepre-scissionkinetic 0 1 2 3 4 5 6 50 energy or the value of the reduced friction parameter. Taking or not into account pairing correlationsis not an 40 252 important factor. Cf As a result a relatively weak number of beta decay 1 mg sample 30 are expected from our results during the descent from y / β the exit to the scission points for standard values of the N 20 dampingparameter. Thecapacityofexistingexperimen- taldevicestoassessthisconclusionfortractableamounts 10 of 252Cf is questionable. Improving these so far very without pairing crude estimates by a better treatment of the static and with pairing 0 dynamic parts of these calculations as well as exploring 5 10 15 20 this phenomenon for other nuclei (involving possibly the -21 tsc [10 s] inclusion of Gamow-Teller decays as well) will be under- taken. FIG. 6. Number of beta decays per year for a 1 mg sample of252Cfasafunction ofthedescenttimetsc from theexitto ACKNOWLEDGEMENTS thescission points. ThisworkwassupportedinpartbythePolishNational The previous figures are given for an evaluationwhich Science Centre under Grant No. 2013/11/B/ST2/04087. does not take into account pairing correlations effects. Oneoftheauthor(P.Q.)gratefullyacknowledgesthesup- As seen of Fig. 6, including these correlations increases portofthe Polish-FrenchcooperationagreementCOPIN slightly the above given numbers (only by 2% for 08-131 and the warm welcome extended to him during ≈ k/µ = 6), this correction being an increasing function a visit to the Department of Theoretical Physics at the of the damping parameter k/µ. Maria Curie Skl odowska University. [1] K. Pomorski and J. Dudek, Phys. Rev. C 67, 044316 [4] T.V.Hao, P.Quentin,andL.Bonneau, Phys.Rev.C86, (2003). 064307 (2012). [2] K.T.R.DaviesandJ.R.Nix,Phys.Rev.C14,1977(1976). [5] C.S. Wu and S.A. Moszkowski, /it Beta Decay, Inter- [3] J. Randrup, S.E.Larsson, P. Moller, S.G. Nilsson, K. Po- science monographs and texts in physics and astronomy, morski, and A.Sobiczewski, Phys. Rev.C 13, 229 (1976). v. 16, Interscience Publishers, 1966 [6] E. Strumberger, K. Dietrich, K. Pomorski, Nucl. Phys. A529, 522 (1991).