Ann. Physik 3 (1994) 107- 117 Annalen der Physik @ Johann Ambrosius Barth 1994 Cluster radioactivities of odd-mass nuclei D. N. Poenaru 'I* *, E. Hourani ', and W. Greiner' Institut de Physique Nuclkaire, F-91406 Orsay, France 'Institut fur Theoretische Physik der Universittit, Postfach 11 1932, D-60054 Frankfurt/M., Germany Received 22 November 1993, revised version 17 January 1994, accepted 3 February 1994 Abstract. The partial half-lives of the hypothetical even-even equivalent of an odd-mass nucleus for cluster transitions toward various excited states of the daughter, used as a reference to find the hindrance factor, can be calculated within analytical superasymmetric fission model, by taking into account the angular momentum of the emitted cluster. Detailed tables are presented for I4C radio22JAc;24Ne radioactivity of 233U,23'Pa, and 23Fdecay of 23'Pa, showing activity of 221Fr,221.223Ra, that, except for "*Ac, the existing experimental evidences, do not exclude (moderate) hindered transitions to the ground states of the daugther nuclei. Keywords: Cluster radioactivities; Transitions to excited states; Fission model. 1 Introduction A rich variety of nuclear decay modes have been discovered and intensively studied during the last decade [l]. Among them we would like to mention cluster (or '%?exotic") radioactivities - intermediate phenomena between fission and alpha decay, in which a parent nucleus AZ is splitted into two fragments: the emitted cluster *cZe and the daughter The spontaneously emitted light fragment 'tZe is a small nucleus (like '.4C, 24Ne, etc.) heavier than a particle, but lighter than the lightest fission fragment. The hadronic numbers are conserved: A = A,+Ad and Z = Z e + Z d . A growing interest for these new processes has been manifested since 1984, when the first experiment has been reported, confirming our predictions of 1980 that 14C should be the most probable emitted cluster from Ra isotopes with mass numbers A = 222 and 224. The related theoretical models [2,3] and experimental results [4,5] have been recently reviewed. The predicted halflives within analytical superasymmetric fission model (ASAFM) have been experimentally confirmed. Like for spontaneous fission or alpha decay, the basic explanation relies on the quantum mechanical tunnelling through a potential barrier. Theoretically any parent nucleus for which the released energy is a positive quantity, Q > 0, can be a cluster emit- * Permanent address: Prof. Dr. D. N. Poenaru, Institute of Atomic Physics, P.O.Box MG-6, RO-76900 Bucharest, Romania. 108 Ann. Physik 3 (1994) ter, but practically there is a severe selection imposed by the available techniques, requesting a short-enough halflife and a large enough branching ratio. We shall give below some examples of technical achievements. The decay constant L = In 2/T characterising the well known exponential law e - I r of variation in time, t, of the number of parent nuclei, can be expressed as a product of three (model dependent) quantities where v is the frequency of assaults on the barrier per second (related to the zero-point vibration energy E, = h v/2, with h the Planck constant), Po,is the preformation probability of the cluster at the nuclear surface, strongly dependent on the nuclear structure, and P, is the quantum penetrability of the potential barrier. Usually P, is calculated by using the semiclassical Wentzel-Kramers-Brillouin (WKB) approximation. Up to now there are successful measurements on 14C, "0, 23F 24-26Ne, 28*30Mg, and 32*34Si cluster decays of the following trans-francium nuclei: i2'Fr, 221-224*226Ra, 225Ac,228*230Th , 231P a, 232-234U7 236,238Pu, and (preliminary) 242Cm.The shorter halflife, T= 10" s, corresponds to the 14Cradioactivity of 222Raparent nucleus, leading to a double magic daughter 'O'Pb. There is a strong competition with 01 decay. The largest branching ratio with respect to a-decay, b = Ta/T is about 10-9.2 and has been observed in I4C radioactivity of 223Ra.With the high sensitivity of solid state track detectors, it was possible to measure a branching ratio as low as 10-'6.25for Mg emission from 238Pu.The longer partial s, is that of 231Panucleus against 23Femishalf-life determined up to now, of = sion. Usually only one kind of emitted cluster (the most probable) could be experimentally observed, for the next one the emission rate being smaller by some orders of magnitude. Nevertheiess, there are few exceptions: both F and Ne emitted by 231Pa;Ne and Mg from 234U,and Mg+Si from 238Pu. The main difficulty of such an experiment comes from necessity to select few rare events from an enormous background of alpha particles. Either solid state track detectors (which are not sensitive to alpha particles), or magnetic spectrometers (in which alpha particles are deflected by a strong magnetic field), have been used to overcome this difficulty. The superconducting spectrometer SOLENO, at I.P.N. Orsay has been employed since 1984 to detect and identify the 14Cclusters spontaneously emitted from 222i223*224*226Ra parent nuclei. Moreover, its good energy resolution has been exploited in 1989 to discover 161 a 'ifine structure" in the kinetic energy spectrum of 14C emitted by 223Ra.Cluster emission leading to excited states of the final fragments have been considered for the first time [7] in 1986. When the fine structure of 14C radioactivity of 223Rahas been discovered it was shown that the transition toward the first excited state of the daughter nucleus is stronger than that to the ground state. In other words, like in spontaneous fission of odd-mass nuclei [8], or in fine structure of a-decay, one has [9] a hindered and a favoured transition, respectively. Spontaneous fission rates of odd-A nuclei are slower than that of e - e nuclei, indicating a higher fission barrier, due to shell effects. In both kind of theories the physical explanation relies on the single-particle spectra of neutrons or/and protons. If the uncoupled nucleon is left in the same state both in parent and heavy fragment, the transition is favoured. Otherwise the difference in structure leads to a large hindrance D. N. Poenaru et al., Cluster radioactivities of odd-mass nuclei 109 H = T eX P/Te- , where TeXPis the measured partial half-life for a given transition, and Te-.eis the corresponding quantity for a hypothetical even-even equivalent, estimated either from a systematics (log T versus Q-"* for example) or from a model. The released energy during the process, Q, is given by the difference between the initial (parent) mass M a n d the final one (the sum of fragment masses M e f M d ) expressed in units of energy (that is - multiplied by the light velocity c squared): Q = [M- (Me+ M d ) ]c2. A transition is favoured if H = 1, and it is hindered if H%-1. Particularly interesting are the transitions in which at least one of the three partners possess an odd number of nucleons, because of the spectroscopic informations one can get. Unlike in a-decay, where the initial and final states of the parent and daughter are not so far one from the other, in cluster radioactivities of odd-rnass nuclides, one has a unique possibility to study a transition from a well deformed parent nucleus with complex configuration mixing, to a spherical nucleus with a pure shell model wave functions. In this way, one can get direct spectroscopic information on spherical components of deformed states. In 1986, after taking properly into account the even-odd effect, we have improved our earlier predictions. The purpose of the present paper is to learn about some possible hindered transitions by analyzing the still existing discrepancies. Before displaying the results we shall compare two different kinds of systematics of experimental data of even-even nuclei, which could be used to determine Te-e from Eq. (2). 2 Two kinds of systematics The main quantities experimentally determined are the partial halflife, and the kinetic energy of the emitted cluster Ek = QAd/A. This equation is a direct consequence of the "cold" character of this decay mode - the total kinetic energy of the two fragments practically exhausts the released energy Q, which is shared between the two fragments. Intuitively one can imagine that the preformation probability is a decreasing function of the cluster mass number A,; e.g. the chance to find 14 nucleons close together would be smaller than that of only 4 of them. One may assume an empirical linear dependence of the logarithm of this quantity in the range of emitted clusters already measured: log Po, = (Ae-l)logP:u 3 (3) where the preformation probability of a particle P:" can be determined by fit with experimental data. On this basis we found [lo] a single universal curve log T =f(log P,), for each kind of cluster radioactivity of even-even parent nuclei. This has been done by making a further assumption that the frequency of assaults v , which enters into the decay constant relationship as was shown above, is independent on the emitted cluster and the daughter = constant. From a fit of experimental data we obtained nucleus v(Ae,Ze,Ad,Zd) s-', leading to Pzu = 0.0160694 and v = 110 Ann. Physik 3 (1994) log T = -log P, -22.169 +0.598 ( A ,- 1) (4) which is a straight line for a given A,, with a slope equal to unity. For all cases of practical importance the daughter nucleus has a spherical shape (a doubly magic 208Pbor one of its neighbours). When the cluster is preformed, by neglecting in the first approximation its deformation, we can assume a touching point configuration of two spherical nuclei with a separation distance R = R, between centers expressed in fm, given by the Myers-Swiatecki liquid drop model: R, = 1.2249(Ai’3+A:/3). For any combination of fragments AeZe,AdZd one can calculate easily the WKB penetrability of the external (Coulomb) part of the potential barrier E ( R ) = Z,Zde2/R, between the classical turning points R, and R b . The electron charge e = 1.43998 MeV-fm and Rb is defined by E(R,) = Q. The nuclear inertia B ( R ) at the touching point equals the reduced mass p = mpA = mAdAe/A, where rn is the nucleon mass. P, = exp (-K,) ; K, = 2 1v2p[E(R)-Q]dR Rb R, in which K, is the action integral. By expressing the length in units of fm we get where r = Rt/Rb,Rb = 1.43998z d z e / Q . The advantage of such a handy relationship is evident. The up to now 14 even-even half-life measurements are well reproduced (within a ratio 3.86, or rms = 0.587 orders of magnitude). A similar result is obtained with the “hand-pocket” formula presented in Ref. 3 . Another argument of its universality is illustrated in Fig. 1. Instead of having different lines for various parent nuclei, like in the “classical” systematics log T = f ( Q - ’ l 2 ) ,one get practically only one line when we plot log T =f(log P,). We had to use calculated (ASAFM) values for the half-lives in both kind of systematics presented in Fig. 1, but in principle a linear least-square method through experimental points (if available), would give similar results. None of these two systematics can take into account the angular momentum of the emitted cluster, which can play a role for odd-mass nuclei. Fig. 1 Comparison of a “classical” systematics (left hand side) with universal curve-like systematics (right hand side), calculated within ASAFM for 14C radioactivity of even-even Ra isotopes (full line) and of Pb isotopes (dashed line). D. N. Poenaru et al., Cluster radioactivities of odd-mass nuclei 111 3 Analytical superasymmetric fission model The above mentioned universal curve has been recently developed and it is certainly of limited use in comparison with the more general ASAFM. Since 1980, in order to make a systematic search for new cluster decay modes we had to take into account a large number of combinations parent-emitted nuclei (at least 2000x250 = 5 . lo’). The corresponding large amount of computations can be performed in a reasonable time by using an analytical relationship for the halflife. Such a formula has been obtained on the basis of the Myers-Swiatecki liquid drop model adjusted with a phenomenological shell correction in the spirit of Strutinsky method. Within ASAFM we can easily simulate the even-even assumption. We also can take into account the allowed angular momenta, I, determined from the spin (1)and parity (n)conservation: The half-life T = [(hIn 2)/(2E,)]exp K (8) of a parent nucleus A 2 against the split into a cluster AeZe and a daughter A d z d is calculated by using the WKB approximation, according to which the action integral is given by with B ( R ) = p - the reduced mass defined above, and E ( R ) replaced by [E(R)-E,,,]-Q. In fact the action integral contains two terms K = K,,+K,, the former corresponding to the overlapping stage of the fragments (the limits of the integral R,,R,), and the latter - to separated fragments (limits R,,Rb). E,,, is a phenomenological correction energy similar to the Strutinsky shell correction, also taking into account the fact that Myers-Swiatecki’s liquid drop model overestimates fission barrier heights, and the effective inertia in the overlapping region is different from the reduced mass. In order to avoid a lengthy numerical computation of the deformation energy of two overlapping spheres from a separation distance R = R, to the touching point R = R,, and to obtain an analytical relationship for K O , , we are using a parabolic approximation of the potential barrier: E ( R ) = Q+(Ej-Q)[(R-Ri)/(R,-Rj)J2 ; R S R , (10) where the interaction energy at the top of the barrier, in the presence of a nonnegligible angular momentum, I A , is given by Coulomb and centrifugal terms: Ei = E , + E , = e 2 Z e Z d / R , + h 2 1 ( 1 + 1 ) / ( 2 p R ~.) (11) The touching point separation distance R, = Re +Rd represents the distance between the centers of the two tangent spheres. Rj = rOAj”(j = O,e,d; ro = 1.2249 fm) are the 112 AM. Physik 3 (1994) radii of parent, emitted and daughter nuclei. Initially, one has an internal touching point, when the separation distance is Rj = Ro -R e . During the deformation process from one parent nucleus to two final fragments we presume that the radius of the small sphere, representing the emitted cluster, remains constant R2 = Re, and that of the daughter decreases from R t = Ro to R , = R d , allowing to conserve the total volume a condition imposed by the incompressibility of the nuclear matter. After substitution into the above integral we obtain a closed formula where the parameter b 2 = (Ecor+E*)/EEappears as a consequence of the following expression of the first turning point (defined by E(R,) = Q ) : R, = Rj + (Rt -Rj) [(E,,, +E *)/E:]‘I2 (1 3) in which E* is the excitation energy concentrated in the separation degree of freedom, and E t = Ej- Q is the barrier height before correction. For separated spherical fragments in the Coulomb-plus-centrifugal potential E ( R ) = EcR,/R+E,Rj/R2 ; R > R t (14) it is not necessary to make any approximation; we get K, = 0.4392[(Q+EC,,+E*)A,A,j/A]‘/2RbJ, (15 ) with in which r = Rt/Rb and c = rEc/(Q+E,,, + E * ) . When the angular momentum I = 0, one has c = 1 and the above equations take a simpler form. The external turning point (defined as E ( R b )= Q ) is given by In order to reduce the number of fitting parameters, we took E, = E,,, though it is evident that, owing to the exponential dependence, any small variation of E,,, induces a large change of T, and thus plays a more important role compared to the preexponential factor variation due to E, (see Eq. (8)). Both shell and pairing effects are included in E,,, = ai(Ae)Q(i = 1,2,,3,4 for even-even, odd-even, even-odd, and odd-odd parent nuclei). Pairing effects are clearly seen: for a given cluster radioactivity we have four values of the coefficients ai, the largest for even-even parent and the smallest for the D. N. Poenaru et al., Cluster radioactivities of odd-mass nuclei 113 odd-odd one [14]. The shell effects for every cluster radioactivity is implicitly contained in the correction energy due to its proportionality with the Q value, which is maximum when the daughter nucleus has a magic number of neutrons and protons. Another contribution of shell effects, which is always present, comes from the fact that we subtract from the deformation energy (with corrections) the true value of the released energy Q, obtained as a difference of experimentally determined atomic masses. 4 Hindrance factors In the following we shall present the results of calculations within ASAFM, concerning cluster emission from the ground state of some odd mass number parent nuclei toward ground state and first excited states of the corresponding daughters. As can be seen from the Table 1, the spin and parity of the (initial) ground state of 223Rais JP = 3/2+. In the final states of 2wPb one has J7f = 9/2+ (gs), 11/2+ (first excited state, at 0.779 MeV), and 1512- (second excited state, at 1.423 MeV). Consequently the allowed angular momenta are 4 and 6 units of li in the first two cases, and 7 and 9 in the last one. The favoured transitions are explained within the present version of ASAFM (one should allow an accuracy of + I order of magnitude of any theoretical calculations), but the hindered one needs a larger action integral, K. Such an increase can be obtained Table 1 Hindrance factors, H = TeXP/Tee and log Tee for I4C transitions from 223Ra(gs 3/2+) to various excited states of 209Pbcalculated from a systematics (S) log T = j ’ ( Q - ” 2 ) of 3 even-even neighbours (linear least-sq.) and1 within analptical superasymmetric fission model (ASA). T is given in seconds. ~~ E* keV ~ J? ~ Q log Texp I log Tfe log T f A HS NASA 0 4 6 0 4 6 0 7 9 0 2 4 0 2 0 13.21 13.36 13.60 13.86 14.90 15.14 15.40 16.22 16.89 17.29 16.52 16.59 16.76 17.51 17.58 18.47 18.54 18.71 18.58 18.65 640.3 456.4 264.9 145.8 2.5 1.4 0.8 2.4a MeV 0 9/2+ 31.852 16.02 778 11/2+ 31.073 15.29 1422 15/2- 30.429 16.60a 1567 5/2+ 30.285 2032 1/2+ 29.820 2470 7/2+ 29.382 2520 3/2+ a preliminary. 29.332 4 0 2 14.79 16.13 16.44 17.45 18.41 18.53 3.2 2.9a 0.5a 0.2a 114 Ann. Physik 3 (1994) Table 2 log Teeand H = Texp/Tee for 14C transitions from Z2‘Ra(gs??) to various excited states of 207Pbcalculated from a systematics (S) log T =f(Q-’’*)of 3 even-even neighbours (linear least-sq.) and within analytical superasymmetric fission model (ASA). T is given in seconds. log TexP= 13.39. E* keV J3 Q I 0 3 0 t 0 HS HASA 17.4 8.9 6.5 0.7 0.7 0.2 0.2 MeV 0 1/2- 32.394 570 512- 32.824 898 312- 3 1.496 12.15 13.27 13.93 1 12.44 12.58 13.53 13.55 14.17 14.19 1.3 0.3 Table3 log Teeand H = Texp/T,,for “Ne transitions from 233U(gs 5/2 *) to various excited states of ‘”b calculated from a systematics (S)log T = f ( Q - ” 2 ) of 3 even-even neighbours (linear leastsq.) and within analytical superasymmetric fission model (ASA). T is given in seconds. log Texp= 24.84. E* keV J7 Q 1 log T:e log T”,”” HS HASA 23.51 23.55 23.65 23.80 24.76 24.89 25.05 25.25 25.81 26.01 26.19 26.43 26.04 26.08 26.18 44.0 21.3 19.4 15.6 11.0 MeV 0 9/2+ 60.503 0 2 4 6 23.20 778 11/2+ 59.724 24.44 1422 1512- 59.080 0 4 6 8 0 25.49 5 1567 5/2+ 58.937 7 9 0 2 4 25.73 2.5 0.2 1.2 0.9 0.6 0.4 0.1 0.1 0.0 3.1 0.0 0.1 0.1 0.0 with a larger potential barrier, due to the so-called “specializationenergy” [I 11, in a similar way to what has been done for spontaneous fission of odd-mass nuclei 112, 131. This energy arises from the conservation of spin and parity of the odd particle during the fission. With an increase of deformation (distance between centers within the singleparticle two center shell model) the odd nucleon may not be transferred on a low energy at a level crossing, if in this way it cannot conserve the spin and parity. The corresponding barrier becomes higher and wider, compared to that of the even-even neighbour. The examples given in [I 1- 131 are showing how the barrier height and width become larger due to this effect. If we introduce the correction, this would produce a larger area of the barrier E ( R )-E,,, integrated over the range of separation distances from the first to the second turning point, (Ra,Rb). We need a difference of about 15.76 MeV * fm, in order to get an hindrance factor of 400. D. N. Poenaru et al., Cluster radioactivities of odd-mass nuclei 115 Table 4 log Teeand H = Tcxp/Tee for I4C transitions from 221Fr(gs 5 / 2 - ) to various excited states of 207Tl calculated within analytical superasymmetric fission model (ASA). T is given in seconds. log Texp= 14.53. E* keV J? 0 Q I log T$tA HASA MeV 1/2+ 31.295 0 3 13.60 13.74 8.5 6.2 35 1 3/2' 30.944 0 1 3 2.0 1.6 1.2 1340 11/2- 29.955 0 4 6 8 14.29 14.32 14.44 16.32 16.57 16.83 17.19 1670 (5/2)+ 29.625 0 1 3 5 17.03 17.05 17.17 17.39 Table 5 log Teeand H = Texp/Tee for I4C transitions from 2 2 5 A(gs ~ (3/2-)) to various excited states of 211Bicalculated within analytical superasymmetric fission model (ASA). T is given in seconds. log Texp= 17.16. E* keV J? Q I log T$A HASA MeV 0 (9/2)- 30.479 0 4 6 17.00 17.24 17.50 1.4 0.8 0.5 405 (7/2)- 30.074 0 2 4 17.86 17.94 18-10 0.2 0.2 0.1 766 29.818 0 18.42 832 ? (9/2)- 29.752 0 4 6 18.56 18.80 19.06 95 1 ? 29.633 0 18.83 We have performed similar calculations for other odd-mass parent nuclei (see Tables 2-7). The influence of the angular momentum is not very large, as can be seen by comparing the results obtained for I = 0 and for the allowed values of 1. The spin and parity of ''Ra in its ground state (see Table 2) and of some excited states of 2"Bi (Table 5 ) are not known. The Tfe values obtained from the systematics of even-even neighbours (when the measurements are available, like in Tables 1- 3), are not essentially different from the calculated ones T$A within ASAFM. 116 Ann. Physik 3 (1994) Table 6 log Teeand €2 = TCXp/Tee for 24Netransitions from 23'Pa(gs 3/2-) to various excited states of 207Tlcalculated within analytical superasymmetric fission model (ASA). T is given in seconds. log TexP= 22.89. ~ J? E* keV 0 1/2+ Q 1 log Tf:A HASA 0 22.37 22.39 22.93 22.94 23.01 24.51 24.72 24.90 3.3 3.2 0.9 0.9 0.8 MeV 60.421 1 351 3/2+ 60.070 0 59.081 3 0 1 1340 11/2- 5 7 Table 7 log Teeand H = TexP/Te, for 23Ftransitions from 231Pa(gs 3/2-) to various excited states of 2osPb calculated within analytical superasymmetric fission model (ASA). T is given in seconds. log TexP= 26.02. J? E* keV Q log T$"" I ~ O+ 0 HASA MeV 51.844 0 1 0 2 4 2615 3- 49.229 3198 5- 48.646 0 4 6 ~~ 24.93 24.94 29.74 29.79 29.89 30.87 31.03 31.19 ~~ 12.4 12.0 We got a large hindrance factor (H = 44) for "Ne decay of 233U(gs 5/2+) to the gs (912') of 209Pb;the best fit with TeXP = 1024.84 s is obtained for the transition to the = 1024.89. first excited state (11/2+ at 778 keV) with 1 = 4, TAsAFM Other hindered transitions could be: 23Fdecay of 231Pa(gs 3/2-) to the O+ gs of 208Pb ( H = 12); I4C decay of "'Fr (gs 5/2-) to the 1/2+ gs of 207Tl(H = 8.5); I4C decay of 221Rato the 1/2- gs of 207Pb(H= 9), and 24Ne decay of 23'Pa to the gs of 207~1. The I4C transition from '"Ac (gs(3/2-)) to the (9/2-) gs of "'Bi seems to be not hindered. An experimental search for the fine structure in this decay mode will be performed soon. References 11J 121 [3] [4] D. N. Poenaru, W. Greiner (eds) Nuclear Decay Modes, to be published D.N. Poenaru, W. Greiner, Chapter 4 in Ref. 1 R. Blendowske, T. Fliessbach, H. Walker, Chapter 5 in Ref. 1 E. Hourani, Chapter 6 in Ref. 1 D. N. Poenaru et al., Cluster radioactivities of odd-mass nuclei 117 R. Bonetti, A. Guglielmetti, Chapter 7 in Ref. I and contribution at the 2nd International Conference on Atomic and Nuclear Clusters, Santorini, Greece, 1993; R. Bonetti, C. Chiesa, A. Guglielmetti, R. Matheoud, C. Migliorino, A. L. Pasinetti, H. L. Ram, Nucl. Phys. A562 (1993) 32 L. 3rillard, A. G. Elayi, E. Hourani, M. Hussonnois, J. F. Le Du, L. H. Rosier, L. Stab, C. R. Acad. Sci. 309 (1989) 1105 M. Greiner, W. Scheid, J. Phys. G. 12 (1986) L285 D.C. Hoffman, T. M. Hamilton, M. R. Lane, Chapter 8 in Ref. 1 D.N. Poenaru, W. Greiner, E. Hourani, M. Hussonnois, Z. Phys. D, in print D.N. Poenaru, W. Greiner, Phys. Scripta 44 (1991) 427 J.A. Wheeler, in Niels Bohr and the Development of Physics, W. Pauli, L. Rosenfeld, V. Weisskopf (eds), Pergamon Press, London 1955, p. 163 J. Randrup, S. E. Larsson, P. Mbller, S . G. Nilsson, K. Pomorski, A. Sobiczewski, Phys. Rev. C 13 (1 976) 229 Z. Lojewski, A. Baran, Z. Phys. A322 (1985) 695 D.N. Poenaru, D. Schnabel, W. Greiner, D. Mazilu, R. Gherghescu, Atomic Data Nucl. Data Tables 48 (1991) 231

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