Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1417–1420. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1475–1478. Original Russian Text Copyright c 2005 by Gangrsky, Zhemenik, Mishinsky, Penionzhkevich. NUCLEI Experiment
Independent Yields of Kr and Хе Fragments in the Photofission of 237Np and 243Am Odd Nuclei
Yu.P.Gangrsky*, V. I. Zhemenik, G. V. Mishinsky, and Yu. E. Penionzhkevich Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received August 5, 2004
Abstract—Results are presented that were obtained by measuring the independent yields of Kr (A = 89−93)andХе(A = 135−142) appearing as fragments in the photofission of 237Np and 243Am odd nuclei. The respective experiments were performed in a beam of bremsstrahlung photons from electrons accelerated to an energy of 25 MeV at the microtron of the Laboratory of Nuclear Reactions at the Joint Institute for Nuclear Research (JINR, Dubna). Use was made of the procedure involving the transportation of fragments emitted from the target by a gas flow along a capillary and the condensation of inert gases in a cryostat at liquid-nitrogen temperature. The identification of Kr and Хе appearing as fragments was performed by the gamma spectra of their daughter products. The mass-number distributions of the independent yields of Kr and Хе isotopes were obtained, along with those for the complementary fragments (Y and La in the fission of 237Np and Nb and Pr in the fission of 243Am). c 2005 Pleiades Publishing, Inc.
INTRODUCTION Хе inert gases—originating from the photofission of 237 243 Measurement of fragment yields and of their de- Np and Am odd nuclei and is a continuation 232 238 pendences on various features of fissile nuclei and of similar experiments performed with Th, U, fragments formed (nucleonic composition, excitation and 244Pu even–even nuclei and reported in [3, 4]. energy, angular momentum) is one of the lines of in- An analysis of the measured fragment-mass-number vestigation into the mechanism of the nuclear-fission dependences of the yields with allowance for known process. Measurement of the yields of primary frag- similar data on even nuclei would make it possible to ments (independent yields)—that is, those fragments explore the effect of an odd particle in a fissile nucleus that were formed upon the scission of a fissile nucleus on the formation of fragments. There are virtually and neutron emission, but which have not yet under- no data on the independent yields of fragments in gone beta decay—is of particular interest. Such mea- the photofission of nuclei featuring an odd number of surements furnish important information about the protons or neutrons (we can only indicate the study formation of the nucleonic composition of fragments reported in [5]). Moreover, the use of known data on in the process of their transition from the saddle to the the number of prompt fission neutrons would make scission point. it possible to deduce additional information about the However, data on this process are obviously in- yields of the complementary fragments as well—Nb sufficient and refer predominantly to the neutron- and Pr in the photofission of 243Am and Y and La in induced fission of U and Pu nuclei and to the spon- the fission of 237Np. taneous fission of 252Cf [1, 2]. It is of interest to extend the range of such investigations by including in them fission reactions caused by other bombarding EXPERIMENTAL PROCEDURE particles—for example, photons. Special features of photofission reactions (a fixed angular momentum In the present experiments, we employed the same that is introduced in a fissile nucleus by a photon and procedure as in the studies of our group that were the absence of Coulomb energy and binding energy) devoted to the photofission of even–even nuclei and make it possible to obtain new information about the which were reported in [3, 4]. Within this procedure, effect of external conditions on the formation of the one implements an efficient separation of Kr and Хе nucleonic composition of fragments. isotopes from other fission fragments by using the fact The present study is devoted to measuring the that the chemical properties of the former are strongly independent yields of fragments—isotopes of Kr and different from the chemical properties of the latter. Krypton and xenon nuclei originating as fission frag- *e-mail: [email protected] ments from a target irradiated with bremsstrahlung
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Table 1. Independent yields of Kr and Хе fragments after the scission of a fissile nucleus and neutron emission, but which did not have time to undergo beta 237Np (γ, f) 243Am (γ, f) decay. The measurements were performed in a room Frag- that was protected from microtron radiation—that is, ment −1 −1 Yrel, % Yabs, fiss. Yrel, % Yabs, fiss. under conditions of a low gamma-ray and neutron 89Kr 65(8) 0.0019(2) 81(3) 0.019(2) background. 91Kr 100 0.025(3)∗ 100 0.023(2)∗ For targets subjected to irradiation, we used 50-µg/cm2-thick layers made from Np and Am 92 Kr 44(7) 0.011(1) 42(2) 0.0095(9) oxides and deposited onto an aluminum substrate 93Kr 18(4) 0.0045(5) 20 µm thick. At these layer and substrate thick- 135 nesses, half of the fragments formed in a target were Xe 61(3) 0.023(3) 42(5) 0.015(2) emitted from it and were moderated in the gas. 137 ∗ ∗ Xe 100 0.038(4) 100 0.035(4) In order to measure the gamma spectra in ques- 138Xe 80(10) 0.030(3) 76(8) 0.027(3) tion,weemployedanHpGedetectorofvolume 100 cm3 and resolution 2.1 keV for the 1332-keV 139 31(4) 0.012(1) 35(3) 0.012(1) Xe gamma line of 60Со. The resulting spectra were saved 140Xe 8.0(9) 0.0030(3) 9.8(6) 0.0034(3) in the memory of a PC for a subsequent analysis with 141Xe 3.0(3) 0.0011(1) 6.0(6) 0.0021(2) theaidoftheAKTIVcode[7].Theyieldsofidentified Kr and Хе fragments were determined from the areas 142Xe 2.0(3) 0.00076(8) 3.4(7) 0.0012(1) of their gamma lines (or the gamma lines of their ∗ Estimated yield values. daughter products) in the spectra with allowance for the detection efficiency, the intensity per decay event, the irradiation time, the time of transportation along photons were moderated in a gas and were trans- the capillary, and the time of the measurements. ported by its flow along a Teflon capillary to a cryostat, The experiments were performed at an accelerated- where they were condensed on the walls of a 1-m- electron energy of 25 MeV,the corresponding average long copper pipe coiled into a spiral and placed in excitation energy of a fissile nucleus being 13 MeV (it a Dewar flask that was filled with liquid nitrogen. was determined from the shape of the bremsstrahlung All other fragments stopped in the gas were blocked spectrum [8] and the excitation function for the by a filter at the inlet of the capillary. Thus, only Kr photofission of 237Np and 243Am nuclei under the and Хе isotopes, whose half-lives were longer than assumption that this function is similar to the well- 0.2 s (the time of fragment transportation along the known dependence that was determined in [9] for capillary was about 0.5 s), and the products of their 238U). beta decay reached the copper pipe. For the carrier gas, we employed pure helium at a pressure of 2.5 atm in the chamber. This gas was characterized by the RESULTS OF THE MEASUREMENTS highest velocity of fragment transportation and was not activated by neutrons or bremsstrahlung from the By analyzing the measured gamma spectra, we microtron used. were able to identify four krypton isotopes and seven xenon isotopes. The ratios of the independent yields In identifying Kr and Хе isotopes and in determin- 91 137 ing their yields, we relied on the spectra of gamma of these isotopes to the yields of Kr and Хе, radiation that they emitted (for data on these spectra, respectively, are given in Table 1. In order to deter- see [6]). The identification of short-lived Kr and Хе mine these yields per fission event, we measured the isotopes (in the case of half-lives shorter than 1 min) cumulative yields of fragments characterized by the was based on the gamma spectra of their daughter mass numbers of A =91and A = 137 and estimated the fractions of Kr and Хе (fractional yields) in these products (Rb and Sr for Kr and Ва and La for Хе). isobars. The cumulative yields were obtained from These daughter products, which were formed directly 91 in the scission of a fissile nucleus, were blocked by the intensities of the gamma lines of the Sr and 137 the filter at the inlet of the capillary. Therefore, they Cs isotopes, which are at the end of the beta-decay did not contribute to the measured yields of Kr and chains for these A values. The results proved to be Хе fragments. Also, the filter blocked Se, Br, Te, and 4.2(2)% and 6.3(3)% of the number of fission events. I fragments, whose beta decay could produce Kr and The fractional yields of 91Kr and 137Хеfragments Хе isotopes being studied. Thus, one can see that, in were estimated by using the systematics of yields in order to measure gamma spectra, we separated only the neutron-induced fission of U and Pu nuclei [1] those Kr and Хе fragments that emerged immediately and were found to be 0.60(6) for 237Np and 0.55(6)
PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 INDEPENDENT YIELDS OF Kr AND Хе FRAGMENTS 1419
243 for Am. These values make it possible to deter- Y, fiss.–1 Xe mine the independent yields per fission event for 91Kr Kr and 137Хе and, from them, for all other Kr and Хе fragments. The yield values obtained in this way are 10–2 also given in Table 1, while their dependences on the fragment mass number are displayed in the figure. These dependences (isotope distributions of frag- ments) are usually approximated by the Gaussian 10–3 distribution (A − A¯)2 Y (A)=Kexp − , (1) 2σ2 ×10 ×10 10–4 where A¯ and σ are, respectively, the mean mass num- ber and the variance of the distribution, while K is 88 90 92 134~ ~ 136 138 140 142 a normalization factor. From the figure, it can be A seen that, in just the same way as in the photofission of even–even nuclei, the measured distributions are Mass-number (isotope) distributions of the independent well described by expression (1). However, an excess yields of Kr and Хе fragments in the reactions (closed of the measured yield values above their calculated circles) 237Np(γ, f)and(opencircles)243Am(γ, f). counterparts is observed for neutron-rich xenon frag- The points represent experimental data, while the dashed 141 142 curves correspond to the calculation by formula (1) whose ments ( Хеand Хе). This may be associated with parameters are set to the values in Table 2. the nuclear structure of these fragments.
The fitted values of the parameters of isotope dis- 237 243 tributions are given in Table 2. A comparison with the energies of the Np and Am nuclei undergoing known distributions for even–even nuclei shows that, fission, one can evaluate ν. The result is ν =4for 237 243 both for 237Np and for 243Am, the values of A¯ are well Np and ν =46for Am. These values of ν make ¯ below those for 232Th, 238U, and 244Pu [3], but that it possible to determine the mean mass numbers A of they are close to the value of A¯ for 235U fission induced Y, Nb, La, and Pr fragments. The results are given in by thermal neutrons [2]. This relationship between Table 2. the A¯ values reflects their dependence on the neutron The data obtained in the present study for the 237 243 excess in a fissile nucleus. The ratio N/A is 1.587 for photofission of Np and Am nuclei supplement 238U (similar values were found previously for 232Th the systematics of independent yields of fission frag- and 244Pu), 1.549 for 237Np, and 1.558 for 243Am. The ments. They give sufficient grounds to conclude that an odd particle in a fissile nucleus does not change σ 237 243 values of for Np and Am correspond to the significantly the isotope distribution of fragments in systematics of variances of isotope distributions in the relation to what was observed for even–even nuclei neutron-induced fission and photofission of nuclei. and that the distributions themselves are close to From the results obtained here for the isotope dis- those that were obtained in the fast-neutron-induced tributions of Kr and Хе, one can derive the analogous fission of the corresponding nuclei. distributions for the complementary fragments. For Kr fragments, these are La isotopes in 237Np fission Table 2. and Pr isotopes in 243Am fission. For Хе fragments, Parameters of the isotope distributions of photofission fragments these are, respectively, Y and Nb isotopes. For frag- ments characterized by identical independent yields, 237Np(γ, f) 243Am(γ, f) we have Element A¯ σ A¯ σ Al + Ah + ν = A0, (2) Kr 903(2) 1.2(1) 90.1(3) 1.4(2) where Al, Ah,andA0 are the mass numbers of, re- spectively, a light fragment, a heavy fragment, and the Y 96.3(3) 1.3(1) fissile nucleus being considered, while ν is the total Nb 101.5(2) 1.5(1) number of neutrons emitted from the two fragments. By employing the known numbers of neutrons Xe 136.7(2) 1.3(1) 136.9(1) 1.5(1) emitted in the neutron-induced fission of nuclei in La 142.7(3) 1.3(1) the reactions 236Np(n, f)and242Am(n, f) [10, 11] and taking into account corrections for the excitation Pr 148.3(5) 1.4(2)
PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1420 GANGRSKY et al. ACKNOWLEDGMENTS 5. D. De Frenne, H. Thierens, B. Proot, et al.,Phys. Rev. C 26, 1356 (1982). We are grateful to Yu.Ts. Oganessian and M.G. Itkis for their support of this study, A.G. Belov 6. E. Broune and R. B. Firenstone, Table of Radioac- tive Isotopes (Wiley, New York, 1986). for his assistance in irradiating the samples at the microtron, and G.V. Buklanov for manufacturing the 7. V. E. Zlokazov, Comput. Phys. Commun. 28,27 targets. (1982). 8. Ph. G. Kondev, A. P. Tonchev, Kh. G. Khristov, and V.E.Zhuchko,Nucl.Instrum.MethodsPhys.Res.B REFERENCES 71, 126 (1992). 1. A. C. Wahl, At. Data Nucl. Data Tables 39, 1 (1988). 9. J. T. Caldwell, E. L. Dowly, B. L. Berman, et al.,Phys. 2. T. R. England and B. E. Rides, Preprint No. La-Ur- Rev. C 21, 1215 (1980). 94-3106, LANL (Los Alamos National Laboratory, 10. F.Manero and V.A. Konshin, At. Energy Rev. 10, 657 Los Alamos, 1994). (1972). 3. Yu. P. Gangrsky, V. I. Zhemenik, N. Yu. Maslova, 11.Yu.P.Gangrsky,B.Dalkhsuren,andB.N.Markov, et al.,Yad.Fiz.66, 1251 (2003) [Phys. At. Nucl. 66, Nuclear-Fission Fragments (Energoatomizdat,´ 1211 (2003)]. Moscow, 1986) [in Russian]. 4.Yu.P.Gangrsky,V.I.Zhemenik,S.G.Zemlyanoı˘ , et al., Izv. Akad. Nauk, Ser. Fiz. 67, 1475 (2003). Translated by A. Isaakyan
PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1421–1432. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1479–1490. Original Russian Text Copyright c 2005 by Kadmensky, Rodionova. NUCLEI Theory
Subthreshold Photofission of Even–Even Nuclei S. G. Kadmensky* andL.V.Rodionova Voronezh State University, Universitetskaya pl. 1, Voronezh, 394693 Russia Received February 14, 2005
Abstract—Within quantum-mechanical fission theory, the angular distributions of fragments originating from the subthreshold photofission of the even–even nuclei 232Th, 234U, 236U, 238U, 238Pu, 240Pu, and 242Pu are analyzed for photon energies below 7 MeV. Special features of various fission channels are assessed under the assumption that the fission barrier has a two-humped shape. It is shown that the maximum value of the relative orbital angular momentum Lm of fission fragments can be found upon taking into account deviations from the predictions of A. Bohr’s formula for the angular distributions of fission frag- ments. The result is Lm ≈ 30.Theexistenceofan“isomeric shelf” for the angular distributions of fragments from 236Uand238Uphotofission in the low-energy region is confirmed. c 2005 Pleiades Publishing, Inc.
1. INTRODUCTION of the qualitative physical assumption that fission fragments are emitted along or against the direc- The discovery [1] of a sizable anisotropy in the tion of the fissile-nucleus-symmetry axis. Within the angular distributions of fragments originating from 232 238 quantum-mechanical theory of binary nuclear fis- the subthreshold photofission of Thand Unu- sion [9], it is shown that this assumption is at odds clei served as a basis for introducing the concept with the quantum-mechanical uncertainty relation of A. Bohr’s transition fission states [2, 3] (fission between the orbital angular momentum and the par- channels) that are formed at the saddle points of the ticle emission angle. Therefore, A. Bohr’s formula deformation potential and which are associated with can only be valid approximately under the condition “cold” intrinsic states of a fissile nucleus. The analysis that, in the asymptotic region of a fissile nucleus, of the angular distributions of photofission fragments where primary fission fragments have already been that was performed in the review article of Ostapenko formed, there arise rather high orbital angular mo- et al. [4] on the basis of a vast body of experimental menta of these fragments, L Lm. A virtually uni- information and under the assumption [5] that the versal mechanism of pumping of the orbital angular deformation potential of a fissile nucleus has a two- momenta and spins of fission fragments was justified humped shape confirmed that transition fission states in [10]. This mechanism is due to a strong non- associated both with the inner and with the outer sphericity of the fragment-interaction potential in the fission barrier play a decisive role in the formation of vicinity of the scission point of a fissile nucleus and these distributions. leads to a value of Lm ≈ 30 for the maximum rela- For the angular distributions of fragments from tive orbital angular momentum of fission fragments. 236 238 Uand Uphotofission in the low-energy region, So large a value of Lm is consistent with the limit- an interesting phenomenon was discovered in [6, 7] ing estimate obtained in [10] for the maximum value that was associated with the emergence of an “iso- of the total spin of primary fragments, F = J1 + J2, meric shelf,” which was determined by the radiative on the basis of experimental data on the multiplic- population of shape-isomer states [8] in the second ity of prompt neutrons and on the multiplicity and well of the deformation potential of a fissile nucleus. multipolarity of prompt photons emitted by primary fragments, the condition that the maximum spins of We note that the results obtained previously from || an analysis of angular distributions of photofission these fragments are parallel, (J1)max (J2)max,being fragments—for example, in [4]—were based on the satisfied. use of A. Bohr’s [2, 3] formula for the angular dis- On the basis of quantum-mechanical fission the- tributions of fragments originating from the fission ory [9], we analyzed in [11] the angular distribu- channel where an axisymmetric fissile nucleus has tion of fragments from 238Uphotofission induced by aspinJ whose projection onto the symmetry axis bremsstrahlung photons of endpoint energy Eγ = of this nucleus is K. This formula is a realization 5.2 MeV, at which the experimental parameters of this distribution were determined in [12] to a fairly *e-mail: [email protected] high degree of precision. For the maximum relative
1063-7788/05/6809-1421$26.00 c 2005 Pleiades Publishing, Inc. 1422 KADMENSKY, RODIONOVA orbital angular momentum in question, we found the М1 photons. At the same time, one can disregard + range 25
PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SUBTHRESHOLD PHOTOFISSION OF EVEN–EVEN NUCLEI 1423 undergo a uniform statistical mixing owing to the where the quantities a , b ,andc are defined as 0 0 0 dynamical enhancement of the Coriolis interaction JK JK effect [13]. A similar idea is also often used [3, 4] for a0 = αJKP ,b0 = βJKP , (7) compound states in the second well, but it has yet to JK JK JK be rigorously justified. c0 = γJKP . The normalized (to unity) angular distribution JK T JK(θ) of the fragments of photofission in the JK The values of the coefficients αJK, βJK,andγJK for exit fission channel is independent of the spin projec- →∞ ± the cases of Lm (A. Bohr’s limit), Lm =30,and tion M = 1 and is usually calculated by A. Bohr’s L =20are given in Table 1. formula [3] m In analyzing the experimental angular distribu- JK 2J +1 | J |2 | J |2 tions of photofission fragments, one usually deter- T (θ)= [ D1K (ω) + D1−K (ω) ]β=θ, ◦ 8π mines the asymmetry Wγf(θ) ≡ σγf(θ)/σγf(90 ) of (3) these distributions with respect to the angle θ =90◦. where DJ (ω) is a generalized spherical harmonic This asymmetry can be represented by formula (6) MK upon the replacement of the quantities a , b ,andc that depends on the Euler angles ω ≡ (α, β, γ) deter- 0 0 0 by the quantities mining the orientation of the symmetry axes of the a b c fissile nucleus with respect to the axes in the lab- a = 0 ,b= 0 ,c= 0 , (8) oratory frame. Within quantum-mechanical fission a0 + b0 a0 + b0 a0 + b0 JK theory [9–11], the quantity T (θ) takesadifferent respectively; obviously, we have a + b =1. form, JK 2J +1 | J |2 T (θ)= dω[ D1K (ω) (4) 3. ANALYSIS OF THE ANGULAR 16π2 DISTRIBUTIONS OF FRAGMENTS | J |2 ORIGINATING FROM THE SUBTHRESHOLD + D1−K (ω) ]F (Lm,θ). PHOTOFISSION OF NUCLEI Here, F (L ,θ) is the normalized (to unity) angular m Let us investigate the photofission of 232Th, 234U, distribution of photofission fragments in the intrinsic 236 238 238 240 242 coordinate system of the fissile nucleus, U, U, Pu, Pu, and Pu even–even nu- clei that is induced by photons of energy in the range F (Lm,θ)=b(Lm) (5) E ≤ 7 MeV. We select only those results of the ex- γ 2 perimental studies reported in [12, 14–18] for which Lm − L 1+ππ1π2( 1) relative root-mean-square deviations of the measured × YL0(ξ )YL0(1) , 2 quantities a and c from a and c are less than 0.5. L=0 For the ensuing analysis, we employ the quanti- −1 JK JK 10 Lm − L 2 ties G = P /P , which possess the following (2L +1) 1+ππ1π2( 1) JK 10 11 11 10 b(Lm)= , properties: G > 0, G =1, G =Γ /Γ ,and 4π 2 f f L=0 G22 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1424 KADMENSKY, RODIONOVA Table 1. Values of the coefficients αJK, βJK,andγJK for Lm →∞, Lm =30,andLm =20 Lm α10 α11 α20 α22 β10 β11 β20 β22 γ20 γ22 ∞ 0 0.75 0 0 0.75 −0.375 0 0.625 0.9375 0.15625 30 0.024 0.74 0.060 0.030 0.714 −0.357 −0.022 0.592 0.854 0.142 20 0.04 0.73 0.087 0.044 0.697 −0.349 −0.033 0.576 0.815 0.136 11 20 − c(α11 + β11)(α22 − a(α22 + β22))], For all cases studied here, the G and G val- ues calculated with allowance for the experimental d = c[(α11 + β11)α20 − α11(α20 + β20)] (11) errors in determining a and c appear to be positive definite in A. Bohr’s limit (Lm →∞). As to finite − a(α11 + β11)γ20 + α11γ20. values of Lm, the situation for them proves to be By substituting, into these formulas, the experimental more complicated, as might have been expected. One values of the quantities a and c, one can calculate G11 can single out cases where the values of G11 and 20 22 20 and G versus the quantities Lm and G ,whichare G for Lm =30and Lm =20provetobepositive considered as parameters. Since the quantities G11 definite and rather close to the values of G11 and 20 and G20 are positive by definition, a comparison of the G , respectively, in A. Bohr’s limit. These cases calculated values of G11 and G20 with the conditions arise at rather large values of experimentally deter- G11 > 0 and G20 > 0 enables one to determine the mined a and c and correspond to the situation where A. Bohr’s formula (3) for the angular distributions of possible minimal values of Lm. fragments is highly accurate. Table 2 gives the G11 It was indicated above that, by using experimental and G20 values calculated for cases where A. Bohr’s data on the multiplicity and multipolarity of prompt formula (3) is no longer correct, so that there arise photons emitted by primary fission fragments, the significant distinctions between the absolute values maximum value of the relative orbital angular mo- G11 G20 L L mentum of fission fragments was estimated in [10] of and for m =30and m =20and their ≈ counterparts in A. Bohr’s limit. From Table 2, one as Lm 30. In order to test this estimate, we will 11 20 can see that, at Lm =30, the values of G and G consider the cases of Lm =30, Lm =20,andLm → ∞ (A. Bohr’s limit) in our ensuing calculations. provetobepositivedefinite in the majority of cases studied here, but the current experimental accuracy in 22 20 The ratio G /G is equal to the fission-width determining a and c is insufficient for the root-mean- 22 20 11 ratio Γf /Γf , which is much less than unity if the square deviations of all G and, in some cases (for axisymmetric shape the nucleus is conserved in the example, for 232Th), of G20 from their mean values to process of subthreshold fission. This is explained by be much less than unity. At the same time, it turns out thefactthatthe2+2 transition fission state, which that, in eight of the 78 cases presented in Table 2 (at 232 is associated with the γ vibrations of a nucleus, has energies in the range 5.9 ≤ Eγ ≤ 6.4 MeV in Th, a rather high excitation energy (0.7 MeV [19]) with at energies in the range 5.75 ≤ E ≤ 5.97 in 234U, + γ respect to the 2 0 transition fission state. Therefore, 236 22 22 and at an energy of Eγ =5.75 MeV in U), positive the quantity G was varied in the interval 0 ≤ G ≤ 11 20 values of G can only be obtained by shifting the ex- 0.2G , where the upper boundary was chosen to perimental values of a toward greater values by more 22 exceed, by a good margin, the possible values of G . than two standard deviations in determining a.We For all cases investigated here, these variations of the note in passing that, in some cases, the most recent quantity G22 lead to changes of not more than 25% measurements of a lead to positive definite values in the absolute values of G11 and G20 calculated by of G11, in contrast to measurements of a that were formulas (9)–(11). Therefore, one can disregard, with performed previously. By way of example, we indicate a high degree of precision, the effect of G22 on G11 that such a situation arises for 232Thphotofission 20 22 and G and employ a value of zero for G . This at Eγ =6.4 MeV, in which case the results of the conclusion, along with the above conclusion that the more recent measurements of a that were performed effect of the 2+1 fission channel is negligible, actually in [15] differ significantly from those presented earlier validates the procedure extensively used earlier [4] in [12]. Therefore, the above situation where there that describes the angular distributions of fragments appear negative values of G11 is likely to stem from from the subthreshold photofission of nuclei without an insufficient experimental accuracy in determining taking into account the 2+1 and 2+2 fission channels. a in the region of its smallest values ( a ≤0.014 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SUBTHRESHOLD PHOTOFISSION OF EVEN–EVEN NUCLEI 1425 11 20 Table 2. Values of G and G for Lm →∞, Lm =30,andLm =20 L =30 L =20 L = ∞ Eγ , c a m m m MeV G11 G20 G11 G20 G11 G20 238Pu 5.25 [14] 1.412 ± 0.139 0.408 ± 0.103 0.368 ± 0.169 1.540 ± 0.049 0.284 ± 0.173 1.603 ± 0.050 0.523 ± 0.163 1.416 ± 0.048 5.5 [14] 1.513 ± 0.112 0.330 ± 0.063 0.240 ± 0.096 1.571 ± 0.056 0.153 ± 0.099 1.635 ± 0.057 0.399 ± 0.090 1.448 ± 0.053 5.75 [14] 0.654 ± 0.055 0.414 ± 0.037 0.431 ± 0.062 0.711 ± 0.043 0.381 ± 0.065 0.738 ± 0.044 0.523 ± 0.059 0.659 ± 0.04 6.0 [14] 0.370 ± 0.018 0.526 ± 0.011 0.644 ± 0.022 0.434 ± 0.018 0.607 ± 0.023 0.450 ± 0.018 0.714 ± 0.020 0.402 ± 0.017 240Pu 5.45 [14] 1.147 ± 0.07 0.102 ± 0.044 −0.009 ± 0.052 1.038 ± 0.039 −0.075 ± 0.054 1.077 ± 0.04 0.109 ± 0.049 0.966 ± 0.037 5.7 [14] 0.710 ± 0.052 0.222 ± 0.034 0.160 ± 0.046 0.685 ± 0.037 0.11 ± 0.047 0.711 ± 0.038 0.251 ± 0.043 0.638 ± 0.035 5.95 [14] 0.331 ± 0.013 0.533 ± 0.010 0.660 ± 0.020 0.390 ± 0.013 0.625 ± 0.02 0.405 ± 0.013 0.727 ± 0.019 0.361 ± 0.012 242Pu 5.0 [14] 3.702 ± 0.424 0.532 ± 0.308 0.485 ± 0.593 4.646 ± 0.530 0.275 ± 0.583 4.892 ± 0.570 0.850 ± 0.598 4.119 ± 0.402 5.25 [14] 0.965 ± 0.032 0.448 ± 0.053 0.461 ± 0.089 1.078 ± 0.003 0.396 ± 0.090 1.122 ± 0.003 0.580 ± 0.088 0.995 ± 0.001 5.35 [14] 1.018 ± 0.069 0.418 ± 0.046 0.408 ± 0.078 1.114 ± 0.042 0.341 ± 0.080 1.159 ± 0.044 0.531 ± 0.074 1.028 ± 0.040 5.5 [14] 0.734 ± 0.034 0.310 ± 0.022 0.272 ± 0.033 0.748 ± 0.025 0.219 ± 0.034 0.776 ± 0.026 0.367 ± 0.031 0.695 ± 0.023 5.75 [14] 0.422 ± 0.012 0.488 ± 0.008 0.572 ± 0.015 0.482 ± 0.011 0.532 ± 0.015 0.500 ± 0.011 0.646 ± 0.014 0.447 ± 0.010 232Th 5.90 [14] 0.084 ± 0.014 0.010 ± 0.005 −0.028 ± 0.006 0.072 ± 0.012 −0.054 ± 0.006 0.074 ± 0.012 0.01 ± 0.005 0.068 ± 0.011 5.95 [14] 0.074 ± 0.010 0.014 ± 0.004 −0.024 ± 0.005 0.063 ± 0.008 −0.049 ± 0.005 0.066 ± 0.009 0.014 ± 0.004 0.06 ± 0.008 6.2 [14] 0.079 ± 0.010 0.012 ± 0.003 −0.026 ± 0.004 0.068 ± 0.008 −0.051 ± 0.004 0.07 ± 0.009 0.012 ± 0.003 0.064 ± 0.008 6.4 [12] 0.028 ± 0.006 0.021 ± 0.003 −0.014 ± 0.003 0.024 ± 0.005 −0.037 ± 0.004 0.025 ± 0.005 0.021 ± 0.003 0.023 ± 0.005 6.4 [15] 0.060 ± 0.040 0.060 ± 0.030 0.025 ± 0.034 0.052 ± 0.034 0 ± 0.036 0.054 ± 0.035 0.062 ± 0.032 0.049 ± 0.032 6.4 [15] 0.028 ± 0.006 0.044 ± 0.002 0.01 ± 0.002 0.024 ± 0.005 −0.014 ± 0.003 0.025 ± 0.005 0.045 ± 0.002 0.023 ± 0.005 6.5 [14] 0.022 ± 0.014 0.022 ± 0.005 −0.012 ± 0.006 0.019 ± 0.012 −0.035 ± 0.006 0.02 ± 0.012 0.022 ± 0.005 0.018 ± 0.011 6.7 [14] 0.009 ± 0.008 0.023 ± 0.002 −0.01 ± 0.003 0.008 ± 0.007 −0.033 ± 0.003 0.008 ± 0.007 0.023 ± 0.002 0.007 ± 0.006 6.9 [14] 0.020 ± 0.022 0.032 ± 0.007 −0.002 ± 0.009 0.017 ± 0.019 −0.025 ± 0.009 0.018 ± 0.02 0.033 ± 0.007 0.016 ± 0.018 7.0 [14] 0.038 ± 0.012 0.036 ± 0.004 0.001 ± 0.005 0.033 ± 0.01 −0.023 ± 0.005 0.034 ± 0.011 0.037 ± 0.004 0.031 ± 0.01 234U 5.6 [18] 0.195 ± 0.018 0.032 ± 0.008 −0.014 ± 0.009 0.169 ± 0.015 −0.044 ± 0.01 0.175 ± 0.015 0.033 ± 0.008 0.158 ± 0.014 5.75 [16] 0.152 ± 0.012 0.025 ± 0.004 −0.018 ± 0.005 0.131 ± 0.01 −0.046 ± 0.005 0.136 ± 0.01 0.025 ± 0.004 0.123 ± 0.009 5.97 [16] 0.158 ± 0.012 0.028 ± 0.004 −0.016 ± 0.005 0.136 ± 0.01 −0.044 ± 0.005 0.141 ± 0.01 0.028 ± 0.004 0.128 ± 0.009 6.05 [16] 0.110 ± 0.014 0.041 ± 0.005 0.001 ± 0.006 0.096 ± 0.012 −0.025 ± 0.007 0.099 ± 0.012 0.042 ± 0.005 0.09 ± 0.011 6.17 [16] 0.158 ± 0.012 0.050 ± 0.004 0.007 ± 0.005 0.138 ± 0.01 −0.021 ± 0.005 0.143 ± 0.011 0.051 ± 0.004 0.13 ± 0.01 6.2 [16] 0.183 ± 0.017 0.086 ± 0.007 0.043 ± 0.009 0.163 ± 0.015 0.014 ± 0.009 0.169 ± 0.015 0.09 ± 0.008 0.153 ± 0.014 6.3 [16] 0.089 ± 0.018 0.084 ± 0.008 0.047 ± 0.01 0.079 ± 0.016 0.022 ± 0.011 0.082 ± 0.016 0.088 ± 0.009 0.074 ± 0.015 6.35 [16] 0.083 ± 0.015 0.133 ± 0.006 0.102 ± 0.008 0.076 ± 0.013 0.076 ± 0.008 0.078 ± 0.014 0.142 ± 0.007 0.071 ± 0.013 6.4 [16] 0.020 ± 0.016 0.133 ± 0.008 0.106 ± 0.01 0.018 ± 0.015 0.083 ± 0.011 0.019 ± 0.015 0.143 ± 0.009 0.017 ± 0.014 6.45 [16] 0.087 ± 0.013 0.163 ± 0.005 0.136 ± 0.007 0.081 ± 0.012 0.11 ± 0.007 0.084 ± 0.012 0.177 ± 0.006 0.076 ± 0.011 6.5 [16] 0.046 ± 0.019 0.157 ± 0.009 0.132 ± 0.012 0.042 ± 0.017 0.108 ± 0.013 0.044 ± 0.018 0.17 ± 0.011 0.04 ± 0.016 6.6 [16] 0.044 ± 0.018 0.185 ± 0.009 0.165 ± 0.012 0.041 ± 0.017 0.141 ± 0.013 0.043 ± 0.017 0.204 ± 0.011 0.039 ± 0.016 6.7 [16] 0.042 ± 0.018 0.238 ± 0.009 0.231 ± 0.013 0.041 ± 0.017 0.207 ± 0.014 0.042 ± 0.018 0.27 ± 0.012 0.038 ± 0.016 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1426 KADMENSKY, RODIONOVA Table 2. (Contd.) Lm =30 Lm =20 Lm = ∞ Eγ , c a MeV G11 G20 G11 G20 G11 G20 236U 4.3 [15] 1.71 ± 0.35 0.72 ± 0.25 1.064 ± 0.665 2.369 ± 0.004 0.957 ± 0.677 2.481 ± 0.012 1.249 ± 0.625 2.133 ± 0.021 5.0 [15] 0.250 ± 0.15 0.200 ± 0.10 0.175 ± 0.133 0.23 ± 0.129 0.143 ± 0.139 0.238 ± 0.134 0.229 ± 0.124 0.215 ± 0.121 5.1 [15] 0.187 ± 0.048 0.04 ± 0.02 −0.005 ± 0.024 0.162 ± 0.040 −0.034 ± 0.0254 0.167 ± 0.042 0.041 ± 0.021 0.152 ± 0.028 5.25 [15] 0.12 ± 0.04 0.06 ± 0.03 0.020 ± 0.034 0.105 ± 0.034 −0.006 ± 0.036 0.105 ± 0.035 0.062 ± 0.032 0.099 ± 0.032 5.5 [15] 0.479 ± 0.016 0.048 ± 0.006 −0.018 ± 0.007 0.419 ± 0.013 −0.058 ± 0.008 0.434 ± 0.013 0.049 ± 0.006 0.393 ± 0.012 5.5 [15] 0.500 ± 0.13 0.150 ± 0.07 0.094 ± 0.089 0.459 ± 0.103 0.052 ± 0.094 0.475 ± 0.106 0.165 ± 0.082 0.429 ± 0.096 5.6 [15] 0.503 ± 0.028 0.040 ± 0.012 −0.027 ± 0.014 0.438 ± 0.022 −0.069 ± 0.015 0.454 ± 0.023 0.041 ± 0.013 0.411 ± 0.020 5.75 [15] 0.452 ± 0.015 0.027 ± 0.005 −0.037 ± 0.006 0.391 ± 0.012 −0.076 ± 0.007 0.405 ± 0.012 0.027 ± 0.005 0.367 ± 0.011 6.0 [15] 0.166 ± 0.013 0.042 ± 0.005 −0.002 ± 0.006 0.144 ± 0.011 −0.03 ± 0.006 0.149 ± 0.011 0.043 ± 0.005 0.136 ± 0.01 6.25 [15] 0.110 ± 0.013 0.069 ± 0.005 0.03 ± 0.006 0.097 ± 0.011 0.004 ± 0.007 0.1 ± 0.012 0.071 ± 0.005 0.091 ± 0.011 6.5 [15] 0.080 ± 0.014 0.112 ± 0.006 0.078 ± 0.008 0.072 ± 0.012 0.053 ± 0.008 0.075 ± 0.013 0.119 ± 0.007 0.068 ± 0.012 6.75 [15] 0.051 ± 0.015 0.199 ± 0.008 0.181 ± 0.011 0.048 ± 0.014 0.157 ± 0.011 0.05 ± 0.014 0.221 ± 0.01 0.045 ± 0.013 238U 4.85 [12] 1.380 ± 0.134 0.231 ± 0.079 0.124 ± 0.107 1.346 ± 0.069 0.045 ± 0.11 1.399 ± 0.072 0.266 ± 0.101 1.245 ± 0.066 4.93 [12] 1.350 ± 0.088 0.171 ± 0.053 0.052 ± 0.067 1.272 ± 0.045 −0.023 ± 0.069 1.322 ± 0.047 0.189 ± 0.063 1.18 ± 0.043 5.0 [12] 1.139 ± 0.031 0.135 ± 0.020 0.026 ± 0.024 1.051 ± 0.017 −0.04 ± 0.025 1.091 ± 0.018 0.145 ± 0.023 0.977 ± 0.016 5.13 [12] 1.026 ± 0.054 0.162 ± 0.033 0.065 ± 0.042 0.96 ± 0.033 0.003 ± 0.043 0.996 ± 0.034 0.177 ± 0.039 0.893 ± 0.031 5.2 [14] 0.910 ± 0.080 0.100 ± 0.035 0.006 ± 0.043 0.821 ± 0.057 −0.051 ± 0.046 0.851 ± 0.059 0.106 ± 0.039 0.765 ± 0.053 5.2 [12] 0.907 ± 0.045 0.090 ± 0.024 −0.005 ± 0.029 0.814 ± 0.03 −0.062 ± 0.03 0.844 ± 0.031 0.095 ± 0.026 0.759 ± 0.028 5.4 [12] 0.306 ± 0.026 0.050 ± 0.012 −0.004 ± 0.014 0.267 ± 0.021 −0.037 ± 0.015 0.277 ± 0.022 0.051 ± 0.013 0.251 ± 0.02 5.45 [14] 0.155 ± 0.021 0.038 ± 0.009 −0.005 ± 0.011 0.134 ± 0.018 −0.033 ± 0.011 0.139 ± 0.018 0.039 ± 0.009 0.126 ± 0.017 5.5 [14] 0.039 ± 0.014 0.078 ± 0.005 0.045 ± 0.006 0.035 ± 0.012 0.021 ± 0.007 0.036 ± 0.013 0.081 ± 0.005 0.032 ± 0.012 5.5 [12] 0.161 ± 0.007 0.040 ± 0.004 −0.004 ± 0.005 0.14 ± 0.006 −0.032 ± 0.005 0.145 ± 0.006 0.041 ± 0.004 0.131 ± 0.005 5.5 [18] 0.172 ± 0.033 0.049 ± 0.013 0.005 ± 0.016 0.15 ± 0.028 −0.024 ± 0.017 0.155 ± 0.029 0.050 ± 0.014 0.141 ± 0.026 5.6 [12] 0.074 ± 0.012 0.054 ± 0.005 0.017 ± 0.006 0.065 ± 0.01 −0.008 ± 0.006 0.067 ± 0.011 0.056 ± 0.005 0.061 ± 0.01 5.65 [14] 0.040 ± 0.010 0.034 ± 0.005 −0.001 ± 0.006 0.035 ± 0.009 −0.025 ± 0.006 0.036 ± 0.009 0.035 ± 0.005 0.033 ± 0.008 5.65 [12] 0.085 ± 0.011 0.034 ± 0.006 −0.004 ± 0.007 0.074 ± 0.009 −0.03 ± 0.007 0.076 ± 0.01 0.035 ± 0.006 0.069 ± 0.009 5.7 [12] 0.041 ± 0.009 0.054 ± 0.004 0.019 ± 0.005 0.036 ± 0.008 −0.005 ± 0.005 0.037 ± 0.008 0.056 ± 0.004 0.034 ± 0.007 5.75 [18] 0.043 ± 0.014 0.041 ± 0.005 0.006 ± 0.006 0.037 ± 0.012 −0.018 ± 0.007 0.039 ± 0.012 0.042 ± 0.005 0.035 ± 0.011 5.8 [12] 0.052 ± 0.014 0.068 ± 0.007 0.033 ± 0.008 0.046 ± 0.012 0.009 ± 0.009 0.047 ± 0.013 0.07 ± 0.008 0.043 ± 0.011 5.9 [12] 0.058 ± 0.007 0.054 ± 0.004 0.018 ± 0.005 0.051 ± 0.006 −0.006 ± 0.005 0.052 ± 0.006 0.056 ± 0.004 0.048 ± 0.006 5.97 [18] 0.128 ± 0.013 0.079 ± 0.005 0.039 ± 0.006 0.113 ± 0.011 0.012 ± 0.007 0.117 ± 0.012 0.082 ± 0.005 0.107 ± 0.011 6.0 [12] 0.048 ± 0.012 0.099 ± 0.006 0.066 ± 0.007 0.043 ± 0.011 0.042 ± 0.008 0.044 ± 0.011 0.104 ± 0.007 0.04 ± 0.01 6.05 [18] 0.125 ± 0.016 0.087 ± 0.007 0.048 ± 0.009 0.111 ± 0.014 0.021 ± 0.009 0.115 ± 0.014 0.091 ± 0.008 0.105 ± 0.013 6.17 [18] 0.123 ± 0.012 0.090 ± 0.005 0.051 ± 0.006 0.11 ± 0.01 0.024 ± 0.007 0.114 ± 0.011 0.094 ± 0.005 0.103 ± 0.010 6.2 [12] 0.034 ± 0.038 0.128 ± 0.019 0.1 ± 0.024 0.031 ± 0.034 0.076 ± 0.026 0.032 ± 0.035 0.137 ± 0.022 0.029 ± 0.032 6.2 [18] 0.138 ± 0.016 0.086 ± 0.007 0.046 ± 0.009 0.123 ± 0.014 0.019 ± 0.009 0.127 ± 0.014 0.090 ± 0.008 0.115 ± 0.013 6.3 [18] 0.038 ± 0.018 0.110 ± 0.008 0.079 ± 0.01 0.034 ± 0.016 0.056 ± 0.011 0.035 ± 0.017 0.116 ± 0.009 0.032 ± 0.015 6.35 [18] 0.032 ± 0.013 0.103 ± 0.005 0.072 ± 0.006 0.029 ± 0.012 0.049 ± 0.007 0.03 ± 0.012 0.109 ± 0.006 0.066 ± 0.009 6.4 [14] 0.034 ± 0.008 0.127 ± 0.004 0.098 ± 0.005 0.031 ± 0.007 0.075 ± 0.005 0.032 ± 0.007 0.136 ± 0.005 0.029 ± 0.007 6.4 [18] 0.038 ± 0.020 0.114 ± 0.009 0.084 ± 0.011 0.034 ± 0.018 0.06 ± 0.012 0.035 ± 0.019 0.121 ± 0.010 0.032 ± 0.014 6.45 [18] 0.078 ± 0.011 0.111 ± 0.004 0.077 ± 0.005 0.07 ± 0.01 0.052 ± 0.006 0.073 ± 0.01 0.118 ± 0.004 0.066 ± 0.009 6.5 [18] 0.052 ± 0.018 0.123 ± 0.009 0.093 ± 0.011 0.047 ± 0.016 0.068 ± 0.012 0.049 ± 0.017 0.131 ± 0.010 0.044 ± 0.015 6.6 [18] 0.043 ± 0.018 0.126 ± 0.008 0.097 ± 0.01 0.039 ± 0.016 0.073 ± 0.011 0.04 ± 0.017 0.135 ± 0.009 0.037 ± 0.015 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SUBTHRESHOLD PHOTOFISSION OF EVEN–EVEN NUCLEI 1427 G11 G20 102 1 (‡) 10 (b) 101 100 100 10–1 10–1 10–2 10–2 4 5 6 7 4 5 6 7 Eγ, MeV 11 20 236 11 Fig. 1. Energy dependence of (a) G and (b) G for U (open circles) in the limit Lm →∞and of G (closed circles) at 11 Lm =30.Inthisfigure and those that follow, the values of G for Lm →∞were multiplied by a factor of 10. for 232Th, a ≤0.028 for 234U, and a ≤0.027 for For all cases presented in Table 2, the values of 236 11 U) and may serve as a basis for performing more G for Lm =30prove to be significantly smaller than accurate investigations. From an analysis of our re- the values of G11 that were found in A. Bohr’s limit. sults for G11 that are presented in Table 2 for the This circumstance, which is illustrated in Figs. 1–7, nuclei and the energies Eγ investigated here, one can results in the need for revisiting some parameters of draw the conclusion that the maximum possible value the model of doorway states [20, 21], which was used of L =30is by and large compatible with the entire in [4] to calculate cross sections and angular distribu- m tions of fragments from the subthreshold photofission body of available experimental data. This conclusion →∞ is in accord with the results presented in [10, 11]. of nuclei in A. Bohr’s approximation (Lm ). As to the quantities G20, one can see from Ta- The situation changes radically at Lm =20.From ble 2 that their values at Lm =30are virtually co- Table 2, it can be seen that, for 25 of the 78 cases incident with those in A. Bohr’s limit. In Figs. 1–7, ≤ ≤ studied here (at energies in the range 5.9 Eγ we therefore present the values of G20 for A. Bohr’s 232 7 MeV in Th, at energies in the range 5.75 ≤ Eγ ≤ limit, which tend to increase with decreasing energy 6.17 MeV in 234U, at energies in the range 5.5 ≤ Eγ and, in some cases, reach a plateau region at 236 E Eγ ≤ 6 MeV in U, and at energies in the range extremely low values of the energy γ , in accordance 238 11 with the expected properties of the fission widths Γ10 5.2 ≤ Eγ ≤ 5.65 MeV in U), positive values of G f 20 can be obtained only upon shifting the values found and Γf in the subbarrier region [4]. Here, one can for a experimentally toward greater values by more see resonance structures in the behavior of G20 in than two standard deviations in the definition of a. 236U, 238U, and 232Th nuclei. The origin of these It follows that, for the experimental-data array being structures was discussed in [4] and is associated with considered, the value of Lm =20can be realized only the appearance of resonance states of the deformation with a probability that is negligible in relation to the fission mode in the second well of the deformation probability of realization of the value of Lm =30. potential. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1428 KADMENSKY, RODIONOVA G11 G20 2 10 (‡) 101 (b) 101 100 100 10–1 10–1 10–2 10–2 4 5 6 7 5 6 7 Eγ, MeV 11 20 238 Fig. 2. Energy dependence of (a) G and (b) G for U (open circles and triangles) in the limit Lm →∞and (closed circles and triangles) Lm =30. Presented here are experimental data from (open and closed circles) [12, 14] and (open and closed triangles) [15, 17, 18]. G11 G20 101 100 (‡) (b) 100 10–1 10–1 10–2 10–2 5 6 7 5 6 7 Eγ, MeV Fig. 3. As in Fig. 1, but for 234U. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SUBTHRESHOLD PHOTOFISSION OF EVEN–EVEN NUCLEI 1429 G11 G20 0 100 10 (‡) (b) 10–1 10–1 10–2 10–2 5 6 7 5 6 7 Eγ, MeV Fig. 4. As in Fig. 1, but for 232Th. G11 G20 102 1 (a) 10 (b) 101 100 100 10–1 10–1 5 6 7 5 6 7 Eγ, MeV Fig. 5. As in Fig. 1, but for 238Pu. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1430 KADMENSKY, RODIONOVA 11 G 20 2 G 10 (a) 101 (b) 101 100 100 10–1 10–1 10–2 5 6 7 5 6 7 Eγ, MeV Fig. 6. As in Fig. 1, but for 240Pu. G11 G20 102 1 (a) 10 (b) 101 100 100 10–1 5 6 7 5 6 7 Eγ, MeV Fig. 7. As in Fig. 1, but for 242Pu. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SUBTHRESHOLD PHOTOFISSION OF EVEN–EVEN NUCLEI 1431 4. “ISOMERIC SHELF” IN THE ANGULAR were employed to deduce information about the val- DISTRIBUTIONS OF FRAGMENTS ues of G11 and G20. At the same time, the experi- FROM THE SUBTHRESHOLD mental values of a that were used in our calculations PHOTOFISSION OF NUCLEI actually correspond to the quantities a¯ introduced above, which take into account the isomeric-shelf- The “isomeric shelf” phenomenon discovered ex- is 11 perimentally in analyzing the total cross sections for induced correction a0 . As a result, the quantity G 236 238 the subthreshold photofission of 238U [6, 7] and 236U for the Uand Unuclei(seeFigs.1,2)first [7,15] nuclei consists in the following: as one goes decreases with decreasing Eγ and attains a minimum, 11 over from the photon endpoint energies in the re- whereupon there occurs a sharp increase in G in gion E >Eis to those in the range E ≤ Eis,the the region of the lowest energies Eγ . This behavior γ γ 11 rate of the decrease in the photofission cross section of G differs from the behavior expected in the case with decreasing Eγ abruptly becomes much lower. where the prompt-fission mechanism is dominant [4] 11 Concurrently, the isotropic component of the angular and where G first decreases with decreasing Eγ distribution of photofission fragments grows in the and then reaches the region of constant values. It is is is interesting to note that, as can be seen from Figs. 1 region Eγ ≤ E , becoming dominant at Eγ ≈ E . 0 and 2, the regions where the isomeric shelf manifests An explanation of this phenomenon was proposed 236 238 in [22, 4] on the basis of introducing the mechanism itself for Uand U prove to be close in A. Bohr’s →∞ of delayed fission. Finding its way to the second well limit (Lm )andatLm =30.FromFigs.3– 234 232 238 240 in the process of fission, a fissile nucleus can undergo 7, one can see that, for U, Th, Pu, Pu, fission either via fission without any change in energy and 242Pu nuclei, the behavior of the quantity G11 and without delay (prompt fission) or via radiative in all regions of the energy Eγ that were studied decay accompanied by the population of a shape- here has a character that corresponds to the prompt- isomer state and followed by fission from this state— fission mechanism and which does not feature the that is, with a delay characterized by the lifetime of isomeric-shelf effect. This observation is in line with the nucleus in the second well (delayed fission). Since the conclusions drawn in [4] from an analysis of the the spin of a shape isomer is zero for an even–even subthreshold photofission of 234Uand232Thnucleiin nucleus, the angular distribution of fragments from the region of extremely low values of the energy Eγ . delayed fission is isotropic. Therefore, the quantity a0, which appears in the definition (6) of the angular 5. CONCLUSION distribution of photofission fragments and which has Within quantum-mechanical fission theory, we the form (7), is replaced, upon taking into account have analyzed the angular distributions of fragments is is delayed fission, by a¯0 =(a0 + a0 ),wherea0 grows originating from the subthreshold photofission of is seven even–even nuclei. The results of this analysis with decreasing Eγ from negligible values (a0 a0) is is is have confirmed the conclusion drawn in [10, 11] for Eγ >E to a0 b0 a0 at Eγ = E0 .Inorder that the maximum value Lm of the relative orbital to observe the isomeric-shelf effect in the angular angular momentum L of fission fragments cannot distributions of nuclear-photofission fragments, one differ significantly from a limiting value of Lm ≈ 30, usually studies [4] the experimental values of the ratio which was obtained in [10, 11] from an analysis of the b0/a¯0, which can be represented in the form b0/a¯0 = multiplicity of prompt neutrons and the multiplicity −1 ¯b/a¯ =(1− a¯)/a¯ =(¯a) − 1 and which can be ex- and multipolarity of prompt photons emitted by pri- − pressed in terms of (¯a) 1 [the quantities a¯ and ¯b are mary fission fragments. This result can be considered given by formulas (8), where a¯0 must be substituted as an additional substantiation of the mechanism of −1 −1 for a0]. For prompt fission, (¯a) = a first increases pumping of high spins and orbital angular momenta monotonically with decreasing Eγ and then reaches of primary fission fragments that was proposed in [10] values that remain constant to the smallest observ- and which was associated with a strong nonsphericity able energies Eγ [4]. But if one takes into account of the potential of interaction of these fragments. delayed fission associated with the population of a shape isomer, the behavior of (¯a)−1 in response to a ACKNOWLEDGMENTS decrease in Eγ is the following: upon reaching a max- We are grateful to V.M.Vakhtel’, G.A. Petrov, and is V.E. Bunakov for stimulating discussions. imum at Eγ ≈ E , it begins decreasing and attains a ≈ is This work was supported by the Russian Foun- minimum in the region around Eγ E0 . This effect dation for Basic Research (project no. 03-02-17469), was studied in detail by Ostapenko et al. [4]. the Dynasty Foundation for support of noncommer- In the present study, formulas (1), (2), (7), and (8), cial programs, and the International Center for Fun- which are based on the prompt-fission mechanism, damental Physics in Moscow. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1432 KADMENSKY, RODIONOVA REFERENCES 13. S. G. Kadmenskiı˘ ,V.P.Markushev,andV.I.Furman, 1. E. J. Winhold, P.T. Demos, and I. Halpern, Phys. Rev. Yad. Fiz. 35, 300 (1982) [Sov. J. Nucl. Phys. 35, 166 87, 1139 (1952). (1982)]. 2. A. Bohr, in Proceedings of the International Con- 14.N.S.Rabotnov,G.N.Smirenkin,A.S.Soldatov, ference on the Peaceful Uses of Atomic Energy, et al.,Yad.Fiz.11, 508 (1970) [Sov. J. Nucl. Phys. New York, 1956, Vol. 2, p. 175. 11, 285 (1970)]. 3. A. Bohr and B. Mottelson, Nuclear Structure (Ben- jamin, New York, 1969, 1974; Moscow, 1971, 1977), 15.V.E.Zhuchko,Yu.B.Ostapenko,G.N.Smirenkin, Vols. 1, 2. et al.,Yad.Fiz.30, 634 (1979) [Sov. J. Nucl. 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Fiz. 65, 1424 (2002) [Phys. At. Nucl. 65, 1390 (2002)]. 20. P. D. Goldstone and P. Paul, Phys. Rev. C 18, 1733 10. S. G. Kadmensky, Yad. Fiz. 67, 167 (2004) [Phys. At. (1978). Nucl. 67, 170 (2004)]. 21. U. Goerlach, D. Habs, M. Just, et al.,Z.Phys.A287, 11. S. G. Kadmensky and L. V. Rodionova, Yad. Fiz. 66, 171 (1978). 1259 (2003) [Phys. At. Nucl. 66, 1219 (2003)]. 12. A. V. Ignatyuk, N. S. Rabotnov, et al.,Zh.Eksp.´ 22. C. D. Bowman, Phys. Rev. C 12, 856 (1975). Teor. Fiz. 61, 1284 (1971) [Sov. Phys. JETP 34, 684 (1971)]. Translatedby A. Isaakyan PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1433–1442. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1491–1500. Original Russian Text Copyright c 2005 by Kadmensky, Rodionova. NUCLEI Theory Angular Distributions of Fragments Originating from the Spontaneous Fission of Oriented Nuclei and Problem of the Conservation of the Spin Projection onto the Symmetry Axis of a Fissile Nucleus S. G. Kadmensky* andL.V.Rodionova Voronezh State University, Universitetskaya pl. 1, Voronezh, 394693 Russia Received April 19, 2004; in final form, September 7, 2004 Abstract—The concept of transition fission states, which was successfullyused to describe the angular distributions of fragments for the spontaneous and low-energyinduced fission of axisymmetric nuclei, proves to be correct if the spin projection onto the symmetry axis of a fissile nucleus is an integral of the motion for the external region from the descent of the fissile nucleus from the external fission barrier to the scission point. Upon heating a fissile nucleus in this region to temperatures of T ≈ 1 MeV (this is predicted bymanytheoretical models of the fission process), the Coriolis interaction uniformlymixes the possible projections of the fissile-nucleus spin for the case of low spin values, this leading to the loss of memory about transition fission states in the asymptotic region where the angular distributions of fragments are formed. Within quantum-mechanical fission theory, which takes into account deviations from A. Bohr’s formula, the angular distributions of fragments are calculated for spontaneously fissile nuclei aligned byan external magnetic field at ultralow temperatures, and it is shown that an analysis of experimental angular distributions of fragments would make it possible to solve the problem of spin-projection conservation for fissile nuclei in the external region. c 2005 Pleiades Publishing, Inc. 1. INTRODUCTION estimate of lm obtained with allowance for the mech- anism of pumping of high spins and relative orbital On the basis of the observation that there is a angular momenta of fragments (this mechanism was sizable anisotropyin the angular distributions of frag- proposed in [12]). ments originating from the subthreshold photofis- The angular distributions of fragments originating sion of axisymmetric deformed even–even nuclei [1], 236 234 A. Bohr [2, 3] introduced the concept of transition from the fission of Uand U nuclei formed upon fission states that are formed at the saddle points of the capture of resonance and low-energyneutrons of energyin the range 0.4 1063-7788/05/6809-1433$26.00 c 2005 Pleiades Publishing, Inc. 1434 KADMENSKY, RODIONOVA between the fission amplitudes for different compound a rather high statistical accuracyof experiments that states of the fissile nucleus in the first well of the studythe angular distributions in question. deformation potential, which have different energies The objective of the present studyis to explore the and, in general, different spins, manifests itself in problem of conservation of the spin projection K for cross sections and in the angular distribution of frag- a fissile axisymmetric nucleus in the fission process ments for nuclear fission induced byresonance and and to analyze, within quantum-mechanical fission low-energyneutrons. Such an interference does not theory[9 –12], the potential of experiments that would contribute to cross sections and angular distributions studythe angular distributions of fragments for the of fragments for photofission, since the character- spontaneous fission of oriented nuclei with the aim of istic energyinterval of averaging photo fission cross solving this problem. sections is verywide ( ≥100 keV) in the experiments reported in [8]. The veryfact that there are distinctions between 2. PROBLEM OF CONSERVATION the results of descriptions of photofission, on one OF THE FISSILE-NUCLEUS-SPIN PROJECTION IN THE FISSION PROCESS hand, and fission induced byresonance and low- energyneutrons, on the other hand, demonstrates Let us consider the case where the shape of the typical difficulties that arise in extracting reliable in- deformed fissile nucleus in the first well of its deforma- formation about fission physics in analyzing the fis- tion potential possesses axial symmetry. If we elim- sion of nuclei from highlyexcited compound states of inate the effect of weak interactions, the spin J and a fissile nucleus that possess a complicated structure. the parity π of fissile-nucleus states will be integrals In this connection, it would be of paramount im- of the motion at all stages of the fission process. As to portance to implement the project [16] of studying the projection K of the fissile-nucleus spin onto the the angular distributions of fragments for the spon- symmetry axis of the fissile nucleus, it can change in taneous fission of nuclei aligned byan external mag- the fission process because of the effect of two main netic field at ultralow temperatures. For anisotropyin factors. The first factor is associated with the possibil- these angular distributions to be observed, it is neces- itythat, in the fission process, the nucleus undergoing sarythat the ground-state spin J of the fissile nucleus fission maydevelop triaxial deformations speci fied by γ satisfythe condition J>1/2. This means that the a nonzero parameter , which lead to the mixing of states characterized bydi fferent values of K. This nuclei being studied must be odd or odd–odd, which possibilitywas discussed in [17]. The second factor creates difficulties for planned experiments since the is associated with the effect of Coriolis interaction, branching fractions of spontaneous fission for odd which, as was shown in [18, 19], undergoes a dynam- and especiallyodd –odd nuclei are small, falling well ical enhancement in the case of rather high excitation below the analogous branching fractions for even– energies E∗ of the fissile nucleus and thermalization even spontaneously fissile nuclei. of this energyover a large number of multiquasipar- Investigation of the spontaneous fission of axisym- ticle states (compound states) and leads to a uniform metric deformed nuclei is advantageous in that, for statistical mixing of the projections K of the spins J the ground states of these nuclei, the projection K of of these states. the spin J always takes the value K = J. It follows For actinide nuclei, it is generallyaccepted that, as that, within the concept of transition fission states [2, collective deformation coordinates of the nucleus un- 3], spontaneous fission is determined exclusivelyby dergo changes in the process of its fission, the nuclear the onlydeep-subbarrier transition fission state of shape retains axial symmetry. Therefore, we will dis- π π quantum numbers J K = J J. This circumstance regard the effect of the first factor on the conservation simplifies substantiallythe analysisof the angular of the spin projection K in the fission process. As to distributions of fragments for the spontaneous fission the second factor, which is associated with Coriolis of oriented nuclei in relation to the analysis of the interaction, its inclusion seems mandatory. angular distributions of fragments for fission from In calculating fission widths and the angular dis- compound states of fissile nuclei (see above). tributions of fragments for induced nuclear fission, On the basis of quantum-mechanical fission the- use is usuallymade [3, 11] of the concept that, at ory[9], we demonstrated in [10] the deviations of the initial stage of the fission process—this stage the angular distributions of fragments for the spon- involves the formation of highlyexcited compound taneous fission of nuclei oriented to the maximum states of a fissile nucleus at the first well of the de- possible degree byan external magnetic field from formation potential—the projections K of the spins the predictions of A. Bohr’s formula, this making it J of these compound states are distributed uniformly possible to determine the maximum value lm of rela- at rather low values of J. As a rule, this concept is tive orbital angular momenta of fission fragments at also applied [3, 11] to the compound states of the PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 ANGULAR DISTRIBUTIONS OF FRAGMENTS 1435 fissile nucleus in the second well of the deformation mixing of all values of the projections K of the fissile- potential. However, there is presentlyno consistent nucleus spin J at rather low J and to the loss of validation of this concept for the case of practical memoryabout the spin projection in the transition interest where the excitation energies of compound fission states of the nucleus. states in the second well are much lower than the excitation energies of the analogous states in the first This conclusion contradicts the fact that there well. appear sizable anisotropies in the angular distribu- In describing spontaneous and low-energyin- tions of fragments for the subthreshold fission of nu- duced fission of nuclei, use is extensivelymade of the clei [8, 10, 11] and for the fission of aligned nuclei concept of transition fission states [2, 3], which are that is induced bylow-energyand resonance neu- formed at the saddle points of the deformation poten- trons [12, 13]; it is also at odds with the observation tial of a fissile nucleus and which are characterized of P -odd [28] and P -even [29] asymmetries in the bya spin J,aparityπ, and a spin projection K onto angular distributions of fragments originating from the symmetry axis of the fissile nucleus. For the states nuclear fission induced bypolarized cold neutrons, as being considered, the spin projection K takesasingle well as with the discoveryof signi ficant fluctuations value for the reason that, since the internal state of in the fission widths of neutron resonances [30]. By the fissile nucleus is cold, Coriolis mixing in K does employing the formula obtained for the angular dis- not playanysigni ficant role at the saddle points. tributions of fragments originating from subthreshold For the concept of transition fission states to be nuclear fission [3, 8, 10, 11] and fragments originating operative and for these states to be crucial for the for- from the fission of aligned nuclei that is induced by mation of fission widths and the asymptotic behavior low-energyand resonance neutrons [13, 14], as well of the fissile-nucleus wave function in the region of al- astheformulasforthecoefficients of P -odd and readyformed primary fission fragments, the following P -even asymmetries in the angular distributions of condition of paramount importance must be satisfied: fragments in nuclear fission induced bycold polarized the spin projections K associated with transition fis- neutrons [31–33], it can be proven rigorouslythat the sion states at the outer saddle point of the deformation anisotropies and the coefficients of P -odd and P -even potential must remain integrals of the motion over asymmetries in the angular distributions of fission the entire external region of the fissile system from fragments vanish in the case of a uniform distribution the descent of the nucleus from this saddle point to of all projections K of the spins J of fissile-nucleus the point of its scission into primary fission fragments states in the vicinityof the scission point. and a subsequent divergence of these fragments. This means that the fissile nucleus must remain quite cold Upon considering that, at nuclear eхcitation ener- throughout this region in order that the Coriolis in- gies in the range E∗ < 2 MeV,the Coriolis interaction teraction not come into playand not mix, with equal effect on the mixing of the projections K of the spins J probabilities, all possible projections K of the fissile- of excited states of an A ≈ 250 axisymmetric nucleus nucleus spin. is weak at rather low values of J [18], the use of The majorityof modern fission models [20–27], formula (1) makes it possible to obtain a qualitative which describe successfullythe mass, charge, and estimate of maximum possible fissile-nucleus tem- energydistributions of fragments originating from the peratures Tm at which the projection K of the fissile- spontaneous and induced fission of various nuclei, in- nucleus spin J can still be considered as an integral volve the concept that, because of dissipation effects, of the motion, Tm ≤ 0.25 MeV. This estimate con- a considerable part of the difference of the potential tradicts the result obtained within the fission models energies of a fissile nucleus at the outer saddle point that were proposed in [20–27] and in which, as was and at the point of its scission into fragments goes indicated above, the temperature T of a fissile nucleus into heating the fissile nucleus, whose state near the in the vicinityof its scission point is about 1 MeV. It scission point is characterized bya temperature of is of importance to develop new fission models that T ≈ 1 MeV.According to [3], the relation between the ∗ would be able to describe basic features of the fission nuclear excitation energy E and the nuclear temper- process, such as the mass, charge, and energydistri- ature T at rather high T is butions of fission fragments, and which, at the same ∗ E = aT 2, (1) time, would be compatible with the experimentally corroborated concept of transition fission states [2, 3]. where a =(A/8) MeV−1 [3]. For an A ≈ 250 fissile nucleus, this yields the estimate E∗ ≈ 30 MeV.It was An investigation of the angular distributions of indicated above that, upon the thermalization of so fragments originating from the spontaneous fission high an excitation energy, Coriolis interaction in an of aligned nuclei mayfurnish additional arguments in axisymmetric nucleus leads to [18, 19] a complete favor of the above concept. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1436 KADMENSKY, RODIONOVA 3. ANGULAR DISTRIBUTIONS fragments in the bodyframe of the fissile nucleus. OF FRAGMENTS FOR THE FISSION Specifically, we have OF ALIGNED NUCLEI lm √ Within quantum-mechanical fission theory[9 – F (lm,θ)=b(lm) Yl0(θ ) 2l +1Rl , (6) 12], the normalized (to unity) angular distribution l=0 dP Jπ(θ)/dΩ of fragments originating from the spon- −1/2 taneous fission of axisymmetric nuclei aligned by an lm 2 external magnetic field at ultralow temperatures can b(lm)= (2l +1)Rl , (7) be represented in the form l=0 Jπ JπK JπK l dP (θ) (2 − δ )Γ dP (θ) where Rl =1+ππ1π2(−1) , lm is the maximum = K,0 , (2) dΩ ΓJπ dΩ value of the relative orbital angular momentum l of K≥0 fission fragments, and Yl0(θ ) is a spherical harmonic. JπK As follows from specific calculations, the function where dP (θ)/dΩ is the angular distribution of fragments for fission from the JπK state of the fissile F (lm,θ) at high values of lm is virtuallyindependent nucleus, of the parity π of the fissile nucleus and the parities π1 and π2 of fission fragments—that is, on the parity JπK dP (θ) J Jπ of the relative orbital angular momentum l of fission = aM TMK(θ); (3) →∞ dΩ fragments. For lm , the angular distribution M 2 F (lm,θ) reduces to the sum of delta functions of Ω ≡ (θ,ϕ) is the solid angle that determines the di- the form rection of light-fission-fragment emission in the lab- 1 [δ(ξ − 1) + δ(ξ +1)], oratoryframe; ΓJπK is the partial fission width for 4π the transition of the parent nucleus to the asymptotic where ξ =cosθ , so that the angular distribution (4) state of spin J,parityπ, and spin projection K (M) assumes the form predicted byA. Bohr’s formula [2, 3], onto the symmetry axis of the nucleus (z axis in the laboratoryframe); and ˜Jπ 2J +1 J TMK(θ)= BMK(ω) , (8) Jπ JπK 8π β=θ Γ = (2 − δK,0)Γ K>0 which is usuallyused [3, 8, 13, 14] in calculating the is the total fission width of the parent nucleus, the fac- angular distributions of fission fragments. tor (2 − δK,0) taking into account double degeneracy The mechanism of pumping of relative orbital an- of K>0 states of an axisymmetric nucleus. In (3), gular momenta and spins of fission fragments due to aJ is a parameter that characterizes the relative pop- a stronglynonspherical character of the potential of M interaction between primary fission fragments in the ulation of the JπM sublevels of the fissile nucleus, J J Jπ vicinityof the point where the fissile nucleus under- M=−J aM =1,andTMK(θ) is the normalized (to goes scission into fragments was proposed in [12]. unity) angular distribution of fragments for fission This mechanism leads to a value of lm ≈ 30 for the from the JπMK states of the fissile nucleus. For this maximum relative orbital angular momentum of fis- angular distribution, we have [9, 10] sion fragments. This value of lm was confirmed in [10, 2J +1 11], where the angular distributions of fragments were T Jπ (θ)= dωBJ (ω)F 2(l ,θ), (4) MK 16π2 MK m analyzed for the subthreshold photofission of a set of even–even nuclei and will therefore be used below. BJ (ω)=|DJ (ω)|2 + |DJ (ω)|2, (5) Byusing the multiplication theorem for spheri- MK MK M−K cal harmonics and the formula for transforming the where θ is the angle between the direction of light- spherical harmonic Yl0(θ ) from the bodyframe to fission-fragment emission and the symmetry axis the laboratoryframe, we can represent the angular J F 2(l ,θ) of the fissile nucleus and DMK(ω) is a generalized distribution m in the form spherical harmonic, which depends on the Euler 2 2 [b(lm)] (2l + 1)(2l +1) angles ω ≡ (α, β, γ) determining the orientation of F (lm,θ)= √ (9) 4π (2L +1) the symmetry axes of the fissile nucleus with respect Lk ll ∗ to the axes of the laboratoryframe. The function ×| L0 |2 L Cll00 YLk(θ)Dk0 (ω)RlRl , F (lm,θ) in (4) coincides with the amplitude of the L0 normalized (to unity) angular distribution of fission where Cll00 is a Clebsch–Gordan coefficient. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 ANGULAR DISTRIBUTIONS OF FRAGMENTS 1437 In turn, the use of the multiplication theorem for can recast the angular distribution dP JπK(θ)/dΩ (3) generalized spherical harmonics makes it possible to into the form J JπK represent the quantity BMK(ω) in (5) as dP (θ) 1 2J +1 = + (15) J JL L dΩ 4π 4π BMK(ω)=2 BMKD00(ω), (10) × J JL L=0,2,4... aM BMKDL(lm)PL(θ). M L=2,4,... − BJL =(−1)M K CL0 CL0 (11) MK JJM−M JJK−K For a further analysis of the angular distributions (15), =(−1)2J−M−K CL0 CL0 . J JJM−M JJK−K it is convenient to employthe quantity fL that is Performing integration in (4) with respect to ω and referred to as the nuclear-orientation parameter and which is given by[34] taking into account the explicit expressions (10) BJ (ω) F 2(l ,θ) −1/2 2 and (9) for MK and m , respectively, we J (2J + L +1)! (L!) −L J Jπ fL = J GL, can recast the angular distribution TMK(θ) into the (2L + 1)(2J − L)! (2L)! form (16) 2J Jπ 2J +1 JL where TMK(θ)= DL(lm)BMKPL(θ), 4π J J − J−M L0 L=0,2,4... GL = aM ( 1) CJJM−M (17) (12) M where PL(θ) is a Legendre polynomial and the coeffi- is the Fano statistical tensor. If a fissile nucleus is cient DL(lm) is given bythe expression not oriented because of an equiprobable population of all JπM sublevels of this nucleus at different values of − DL(lm) (13) J 1 M, in which case we have aM =(2J +1) ,theFano lm statistical tensor GJ and the nuclear-orientation 2 (2l + 1)(2l +1) L0 2 L =[b(lm)] RlRl (C ) , J J 2L +1 ll 00 parameter fL become, respectively, GL =(2J + ll −1/2 J 1) δL,0 and fL = δL,0 byvirtue of the following whence it follows that DL(lm) ≤ 1 and D0(lm)=1 properties of Clebsch–Gordan coefficients: for anyvalues of lm. L0 00 C − C − (18) If, for the generalized spherical harmonic DL (ω), JJM M JJ MM 00 M which appears in the definition of BJ (ω),one MK L0 J−M 1 L | = C − (−1) √ = δL,0. employs the relation D00(ω) β=θ = PL(θ), then the JJM M 2J +1 ˜Jπ M angular distribution TMK(θ) (8) in A. Bohr’s limit (lm →∞) can be represented in the form In this case, all of the L =0 terms vanish in the angu- 2J lar distribution (15), which becomes purelyisotropic, 2J +1 ˜Jπ JL is TMK(θ)= BMKPL(θ). (14) dP JπK(θ) 1 4π = . L=0,2,4... dΩ 4π Formula (14) can also be obtained from (12) upon With the aid of formulas (11), (12), (16), and (17), considering that, in A. Bohr’s limit, the quantity the angular distribution (15) can in general be rep- l D l →∞ DL( m) in (13) reduces to L( m )=1for any resented in the form values of L, whence it can be seen that, at finite values JπK JπK is of lm, the angular distribution in (12) differs from dP (θ) dP (θ) = W JπK(θ), (19) A. Bohr’s angular distribution in (14) onlybecause dΩ dΩ of the deviation of the coefficients DL(lm) from unity. where At lm =30,thecoefficients DL(lm) are independent of the parity π of the fissile nucleus or the parities JπK J W (θ)=1+ DL(lm)f AL(J, K)PL(θ) π and π of fission fragments, taking the values L 1 2 L=2,4,... of D2(30) = 0.95, D4(30) = 0.91, D6(30) = 0.87, (20) D8(30) = 0.82, ..., which decrease with increas- ing L. is the reduced angular distribution of fission frag- Employing formula (12) and considering that, at ments with JL −1 − J−K L0 L =0, the quantity BMK is equal to (2J +1) ,we AL(J, K)=( 1) CJJK−K (21) PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1438 KADMENSKY, RODIONOVA J Orientation parameters fL and coefficients AL(J, K) for J =7/2 and J =9/2 and various values of K versus L J =7/2 J =9/2 L A (J, K) A (J, K) J L J L fL fL K =7/2 K =5/2 K =3/2 K =1/2 K =9/2 K =7/2 K =5/2 K =3/2 K =1/2 2 0.57 4.08 0.58 −1.75 −2.92 0.59 4.6 1.53 −0.76 −2.30 −3.07 4 0.08 7.96 −14.78 −3.41 −10.23 0.11 10.75 −13.14 −10.16 1.79 10.75 6 0.003 10.21 −51.06 91.91 −51.06 0.008 17.30 −63.43 57.67 34.60 −46.13 8 0.0002 20.85 −145.96 417.04 −583.86 291.92 (2J +1)JL(2L)! (2L + 1)(2J − L)! 1/2 unityif the distribution of all projections K of the × . (L!)2 (2J + L +1)! fissile-nucleus spin J is equiprobable in the asymp- totic region, in which case the partial fission widths For a complete alignment of a spontaneously fissile ΓJπK in (24) are independent of K. This result is nucleus in an external magnetic field at ultralow tem- associated with the propertyof Clebsch –Gordan co- J J peratures, in which case aM becomes aM = δM,±J , efficients that arises from (18) upon the substitution depending on the sign of the gyromagnetic ratio for a of the index K for the index M and which leads to J fissile nucleus, the Fano tensor GL (17) assumes the the vanishing of all L =0terms in (24). This means form that the discoveryof a sizable anisotropyin experi- Jπ 2 1/2 mental reduced angular distributions W (θ) would J L0 (2L + 1)(2J!) GL = CJJJ−J = , indicate that, in the external region introduced above, (2J − L)!(2J + L +1)! the fissile nucleus being studied is not heated to sig- while the nuclear-orientation parameter reduces nificant temperatures. It would also be of interest to to [34] obtain a piece of evidence in support of an exact spin- projection conservation for a spontaneously fissile nu- 2 (L!) − (2J)! f J = J L . (22) cleus in the external region, in which case the sum L (2L)! (2J − L)! over K in (24) would be dominated bythe K = J term. The total angular distribution dP Jπ(θ)/dΩ of fragments for the spontaneous fission of oriented nuclei is obtained bysubstituting (19) and (20) 4. ANGULAR DISTRIBUTIONS into (2), OF FRAGMENTS FOR THE SPONTANEOUS FISSION OF COMPLETELY ORIENTED dP Jπ(θ) dP JπK(θ) is NUCLEI 255Es AND 257Fm = W Jπ(θ), (23) dΩ dΩ A further analysis will be performed by consid- Jπ ering spontaneous fission from the ground states of where the reduced angular distribution W (θ) is 255 257 JπK the Es and Fm odd nuclei that are completely expressed in terms of the distribution W (θ) (20) oriented in an external magnetic field at ultralow tem- as peratures and which were chosen for investigation in JπK (2 − δK,0)Γ the project formulated in [16]. Although there have W Jπ(θ)= W JπK(θ). (24) Jπ so far been no reliable measurements of the spins of ≥ Γ K 0 these nuclei, one can conclude from the systematics One can then see that, in just the same wayas in the of spins of neighboring odd isotopes of Es and Fm case of angular distributions of fragments for the sub- that the spins J of the 255Es and 257Fm nuclei are 7/2 threshold fission of nuclei [8, 10, 11] and fragments and 9/2, respectively. Since the angular distributions originating from the fission of aligned nuclei that is of fragments in the spontaneous fission of aligned induced bylow-energyand resonance neutrons [13, nuclei are sensitive to the spins J of these nuclei, one 14], the reduced angular distribution W Jπ(θ) of fis- can obtain direct information about the spins J from sion fragments becomes fullyisotropic and equal to an analysis of these distributions, thereby confirming PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 ANGULAR DISTRIBUTIONS OF FRAGMENTS 1439 π WJ K(θ) 1.6 L = 0, 2 255Es J = 7/2, K = 3/2 L = 0, 2, 4, 6 1.2 0.8 0.4 0 30 60 90 120 150 180 L = 0, 2, 4 θ, deg –0.4 Fig. 1. Reduced angular distributions W JπK(θ) (20)inthecaseof255Es for various L values included in their calculation. π WJ K(θ) 257Fm 1.5 J = 9/2, K = 5/2 1.0 L = 0, 2, 4 L = 0, 2 L = 0, 2, 4, 6 0.5 L = 0, 2, 4, 6, 8 0 30 60 90 120 150 180 θ, deg –0.5 Fig. 2. As in Fig. 1, but for 257Fm. (or disproving) the spin values proposed above for the in (20). The calculations reveal that, in describing nuclei in question. angular distributions of fragments for fission from the From the table, which presents the values calcu- ground state of a J = K fissile nucleus, one can dis- lated byformulas (22) and (21) for the coe fficients regard the contributions of the L =6terms for 255Es J 257 fL and AL(J, K) at J =7/2 and J =9/2,onecan and the contributions of the L =8terms for Fm. J In describing the angular distributions W JπK(θ) for see that the quantities fL decrease with increasing K PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1440 KADMENSKY, RODIONOVA π WJ K(θ) 255Es 4 J = 7/2 K = 7/2 3 K = 1/2 2 K = 5/2 1 K = 3/2 0 30 60 90 120 150 180 θ, deg Fig. 3. Reduced angular distributions W JπK(θ) (20) in the case of 255Es for various values of K. The solid and dashed curves represent the results of the calculations at, respectively,lm =30and lm = ∞ (A. Bohr’s formula). π WJ K(θ) 257Fm J = 9/2 5 K = 9/2 4 3 K = 1/2 2 K = 3/2 K = 7/2 1 K = 5/2 0 30 60 90 120 150 180 θ, deg Fig. 4. As in Fig. 3, but for 257Fm. PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 ANGULAR DISTRIBUTIONS OF FRAGMENTS 1441 see that, at J =7/2 and K =3/2, the shape of the ACKNOWLEDGMENTS angular distribution changes markedlyin the vicinity ◦ WearegratefultoV.E.Bunakov,G.A.Petrov,and of the angle θ =0 upon taking into account the L = W.I. Furman for stimulating discussions. 6 term. Figure 2 shows that, at J =9/2 and K =5/2, the inclusion of the L =8term is necessarysince, This work was supported bythe program “Uni- versities of Russia” (grant no. 02.01.011), the Dy- otherwise, the distribution W JπK(θ) takes negative ◦ ◦ nastyfoundation for support of noncommercial pro- values in the vicinities of the angles 0 and 180 . grams, and the International Center for Fundamental As can be seen from Figs. 3 and 4, the shape of Physics in Moscow. the reduced angular distributions W JπK(θ) for J = 7/2 and J =9/2 changes stronglywith K—from the REFERENCES distributions at low K, which have a maximum at θ =90◦, to the distributions at K = J,whichhavea 1. E. J. Winhold, P.T. Demos, and I. Halpern, Phys. Rev. minimum at θ =90◦. At the same time, a compar- 87, 1139 (1952). ison of the data in Figs. 3 and 4 shows that, in the 2. A. 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Nauk 25. V.A. Rubchenya and S. G. Yavshits, Yad. Fiz. 40, 649 136, 3 (1982) [Sov. Phys. Usp. 25, 1 (1982)]. (1984) [Sov. J. Nucl. Phys. 40, 416 (1984)]. 32. S. G. Kadmensky, Yad. Fiz. 66, 1739 (2003) [Phys. 26. W. Greiner, J. P. Park, and W. Schield, Nuclear Molecules (World Sci., Singapore, 1995). At. Nucl. 66, 1691 (2003)]. 27. T. M. Shneideman et al., Phys. Rev. C 65, 064302 33. S. G. Kadmensky, Yad. Fiz. 67, 258 (2004) [Phys. At. (2002). Nucl. 67, 241 (2004)]. 28. G. V. Danilyan et al.,Pis’maZh.Eksp.´ Teor. Fiz. 26, 68 (1977) [JETP Lett. 26, 186 (1977)]. 34. Alpha-, Beta-, andGamma-ray Spectroscopy ,Ed. byK. Siegbahn (North-Holland, Amsterdam, 1968; 29. A.K.Petukhov,G.A.Petrov,et al.,Pis’maZh.Eksp.´ Teor. Fiz. 30, 324 (1979) [JETP Lett. 30, 439 (1979)]. Atomizdat, Moscow, 1969), Vol. 3. 30. I. Halpern, Nuclear Fission (Fizmatgiz, Moscow, 1962). Translatedby A. Isaakyan PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1443–1452. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1501–1510. Original Russian Text Copyright c 2005 by Kuklin, Adamian, Antonenko. NUCLEI Theory Spectroscopic Factors and Barrier Penetrabilities in Cluster Radioactivity S. N. Kuklin1), 2), G.G.Adamian1), 3)*, and N. V. Antonenko1) Received December 2, 2004 Abstract—The cold cluster decay model is presented in the framework of a dinuclear system concept. Spectroscopic factors are extracted from barrier penetrabilities and measured half-lives. The deformation of the light cluster and residual nucleus is shown to affect the nucleus–nucleus potential and decay characteristics. Half-lives are predicted for neutron-deficient actinides and intermediate-mass nuclei. The connection between spontaneous fission and cluster radioactivity is discussed. c 2005 Pleiades Publish- ing, Inc. 1. INTRODUCTION description of cluster decay is similar to the classi- cal description of the nuclear fission process. Models Spontaneousbinarynucleardecayispossibleif based on this description are called adiabatic or fission this process is exothermic, i.e., occurs with positive models [4–7]. At the same time, one can assume that Q>0 energy release . This condition is satisfied for a light cluster is formed on the surface of the parent many decay modes of numerous nuclei in the second nucleus. The decay of such a system consisting of half of the Periodic Table. Only α decay (decay super- two daughter nuclei does not require a significant symmetric in product masses) and/or spontaneous adjustment of the system to the final decay channel. fission (symmetric or asymmetric decay) is primarily In this nonadiabatic approach [1, 2, 8, 9], α decay is realized. The yields of carbon, fluorine, oxygen, neon, well described. Thus, in adiabatic models, the shape magnesium, and silicon isotopes are measured only 208 of a fissioning nucleus is assumed to change con- for the case where one of daughter nuclei is Pb tinuously and smoothly with time, and the process is or nuclei close to it in narrow ranges 90 ≤ Z ≤ 93 ≤ ≤ described dynamically beginning with the parent nu- and 206 A 212 of the charge and mass numbers, cleus and ending with the prescission configuration. α respectively. Light clusters, as well as the particle, In nonadiabatic models, the probability of the almost are in the ground state, and the heavy residual nucleus instantaneous formation of a two-cluster system and is in the ground state or in one of low-lying states 223 → 14 209 the penetrability of the potential barrier are calculated (e.g., the fine structure of the Ra C + Pb separately. decay [1, 2]). This phenomenon is called cluster ra- dioactivity and has many common features with α The model discussed in this work is based on decay and cold fission in which the decay fragments the assumption that the ground nuclear state can are almost unexcited [1, 2]. Experiments point to be represented as a superposition of cluster states the decisive role of structure effects such as the fine and a mononucleus [10]. Cluster states are described structure and exclusion of transitions from odd parent as dinuclear systems (DNSs) [11]. A DNS decays nuclei [1–3]. if nuclei overcome a potential barrier preventing the The process of cold nuclear decay can be repre- instantaneous decay of the DNS. There are many sented in the form of the collective flow of nucleons methods for calculating a nucleus–nucleus poten- of the initial nucleus to the daughter nuclei, when tial (e.g., the proximity potential, folding potential, nuclei have time to occupy levels with the minimum double-folding potential, optical potential, etc.) [1, 2, energy. This process is accompanied with the consid- 12]. In addition, different models involve different pa- erable rearrangement of the parent nucleus. Such a rameters for calculating the interaction potential. The form of the potential is determined primarily by the 1) Joint Institute for Nuclear Research, Dubna, Moscow spectroscopic factor and barrier penetrability. We use oblast, 141980 Russia. a nucleus–nucleus potential based on the double- 2)Mari State University, Ioshkar Ola, Russia. 3)Institute of Nuclear Physics, Academy of Sciences of folding procedure. The aim of this work is to extract Uzbekistan, Tashkent, 702132 Uzbekistan. the spectroscopic factors of clusters from the available *e-mail: adamian@thsun1.jinr.ru experimental half-lives and calculated penetrabilities 1063-7788/05/6809-1443$26.00 c 2005 Pleiades Publishing, Inc. 1444 KUKLIN et al. U, MeV resonances in the classically allowed surface of the potential energy for a particular parent nucleus. In 130 the case under consideration, this frequency is related to the frequency of zero modes in the variable η near 110 |η| =1. The allowed region arises because the poten- tial energy of the α configuration is comparable or 90 even lower (particularly for neutron-deficient nuclei) | | R0 R1 than the potential energy of the mononucleus ( η = 70 1) [10]. The angular frequency has a specificvaluefor 10 14 18 R, fm aspecific parent nucleus, but we take the mean value ω =1.2 28 206 0 MeV [10] in calculations for all decays Fig. 1. Interaction potential between the Mg and Hg under consideration. nuclei in the decay of 234U. Calculations were performed for two values of the quadrupole nuclear deformation: The probability of tunneling through a potential (solid curve) βx = βf =0and (dashed curve) βx =0.49, barrier in R, i.e., its penetrability P ,iscalculatedby βf =0.05. The horizontal straight line shows the abso- the formula lute value of Q for the given decay. P =1/(1 + exp G), (2) where of potential barriers and to predict the half-lives of R1 neutron-deficient nuclei. 2 − G = 2m[U(R) Q]dR. 2. MODEL R0 The R0 and R1 values are given in Fig. 1 for the Cluster decay can be described as the evolution 28 206 of the system along the mass-asymmetry coordinates interaction potential between the Mg and Hg nuclei, m = m A A /(A + A ) is the reduced mass η =(Af − Ax)/(Af + Ax) (Af and Ax are the mass 0 x f f x numbers of the heavy and light clusters, respectively) of the system (m0 is the nucleon mass), and Q is the ≈ and along the distance R between the centers of decay energy. The contact configuration at R = R0 mass of clusters. With low probability, a strongly Rx(1 + 5/(4π)βx)+Rf (1 + 5/(4π)βf )+0.5 fm asymmetric DNS exists in the ground nuclear state. 1/3 (Rx,f = r0x,f A are the radii of the light and heavy Such a DNS decays due to tunneling through the x,f clusters, and β are the parameters of quadrupole barrier of the nucleus–nucleus potential. In the first x,f approximation, the process can be divided into two nuclear deformation) corresponds to the position of stages: the formation of the DNS can be treated as the local minimum of the nucleus–nucleus poten- evolution along η, and the decay of the DNS, as tial U. It is assumed that the DNS is axisymmetric, evolution along R. The probability S of the formation because the potential interaction energy for all other of the DNS can be found using the ground state orientations of the DNS nuclei is higher. We note that of the Schrodinger¨ equation in the mass asymmetry only quadrupole nuclear deformation is taken into variable. The probability P of tunneling in R can be account. considered in the WKB approximation. The potential U(R) can be represented as the sum In order to calculate the characteristics of cluster of three terms [12]: decay, we use formulas similar to the formulas of U(R)=UCoul(R)+UN (R)+UL(R), (3) α-decay theory [1]. The half-life of a nucleus with 2 respect to cluster decay is given by the expression where UCoul(R), UN (R),andUL(R)= L(L + 1)/(2(R)) are the Coulomb, nuclear, and centrifu- tx = /Γx, gal potentials, respectively, where (R) is the orbital where moment of inertia. Calculation of UN (R) is most ω difficult. Let us take it in the form of the double- Γ = 0 SP folding potential: x π is the decay width. The half-life can be expressed in UN (R)= ρx(rx)ρf (R − rf )F (rf − rx)drxdrf . terms of tx as π ln 2 The nucleon–nucleon forces T = t ln 2 = . (1) 1/2 x F (r − r ) ω0SP f x In the expression, the energy ω0 related to the angu- ρ0(rx) ρ0(rx) = C0 Fin + Fex 1 − δ(rf − rx), lar frequency is the distance between the neighboring ρ00 ρ00 PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SPECTROSCOPIC FACTORS 1445 Table 1. Calculated penetrabilities P and spectroscopic factors S, as well as measured half-lives T1/2; quadrupole- deformation parameters βx and βf of the cluster and residual nucleus, respectively [15]; and energy release Q in decay (the calculations were performed with the parameters a0x =0.55 fm, a0f =0.56 fm, r0x =1.15 fm, and r0f =1.16 fm) A → Ax + Af βx βf Q,MeV T1/2,s P S 221Fr → 14C + 207Tl −0.36 0.05 31.29 2.9 × 1014 8.2 × 10−30 5.2 × 10−7 221Ra → 14C + 207Pb −0.36 0.03 32.29 1.5 × 1013 8.9 × 10−28 9.1 × 10−8 222Ra → 14C + 208Pb −0.36 0.00 33.05 1.7 × 1011 1.9 × 10−26 3.8 × 10−7 223Ra → 14C + 209Pb −0.36 0.02 31.85 7.6 × 1015 6.0 × 10−29 2.9 × 10−9 224Ra → 14C + 210Pb −0.36 0.02 30.54 7.3 × 1015 3.2 × 10−31 5.3 × 10−7 226Ra → 14C + 212Pb −0.36 0.02 28.21 1.7 × 1021 1.0 × 10−36 6.8 × 10−7 225Ac → 14C + 211Bi −0.36 0.02 30.48 1.5 × 1017 9.3 × 10−33 8.6 × 10−9 228Th → 20O + 208Pb 0.26 0.00 44.72 5.4 × 1020 7.7 × 10−29 2.9 × 10−14 230Th → 24Ne + 206Hg 0.45 0.05 57.78 4.1 × 1024 3.4 × 10−26 5.8 × 10−21 231Pa → 23F + 208Pb 0.45 0.00 51.87 9.5 × 1025 3.9 × 10−27 3.3 × 10−21 231Pa → 24Ne + 207Tl 0.45 0.05 60.24 7.9 × 1022 8.0 × 10−23 1.9 × 10−22 232U → 24Ne + 208Pb 0.45 0.00 62.31 2.5 × 1020 2.3 × 10−22 2.1 × 10−20 233U → 24Ne + 209Pb 0.45 0.02 60.50 6.9 × 1024 1.5 × 10−24 1.1 × 10−22 233U → 25Ne + 208Pb 0.50 0.00 60.75 6.9 × 1024 3.9 × 10−24 4.4 × 10−23 234U → 24Ne + 210Pb 0.45 0.02 58.84 8.5 × 1025 8.3 × 10−27 1.7 × 10−21 234U → 26Ne + 208Pb 0.50 0.00 59.47 8.5 × 1025 6.5 × 10−26 2.1 × 10−22 234U → 28Mg + 206Hg 0.49 0.05 74.13 5.4 × 1025 6.3 × 10−22 3.5 × 10−26 235U → 24Ne + 211Pb 0.45 0.02 57.36 2.8 × 1027 3.6 × 10−29 1.2 × 10−20 235U → 25Ne + 210Pb 0.50 0.02 57.83 2.8 × 1027 6.2 × 10−28 6.5 × 10−22 235U → 26Ne + 209Pb 0.50 0.02 58.11 2.8 × 1027 1.1 × 10−27 1.5 × 10−21 236U → 30Mg + 206Hg 0.43 0.05 72.51 3.8 × 1027 7.7 × 10−25 4.0 × 10−25 236Pu → 28Mg + 208Pb 0.49 0.00 79.67 3.5 × 1021 8.6 × 10−19 4.2 × 10−25 238Pu → 28Mg + 210Pb 0.59 0.02 75.93 4.7 × 1025 2.0 × 10−21 1.3 × 10−26 238Pu → 30Mg + 208Pb 0.43 0.00 77.03 4.7 × 1025 9.9 × 10−23 2.5 × 10−24 238Pu → 32Si + 206Hg 0.22 0.05 91.21 1.9 × 1025 6.3 × 10−23 9.5 × 10−25 242Cm → 34Si + 208Pb 0.18 0.00 96.53 1.4 × 1023 1.5 × 10−21 1.5 × 10−23 A − 2Z A − 2Z −3 x x f f where ρ00 =0.17 fm and a0x,f are the nuclear dif- Fin,ex = ζin,ex + ζin,ex , Ax Af fusivity parameters. depend on the nuclear density, because ρ0(rx)= − ρx(rx)+ρf (R rf ). The constants ζin =0.09, ζex = The Coulomb potential UCoul is calculated by the − 3 2.59, ζin =0.42, ζex =0.54,andC0 = 300 MeV fm formula [14] are calculated by fitting the measured nuclear charac- 2 2 teristics [13]. The spatial axisymmetric nucleon den- e ZxZf 3 e ZxZf sity has the form of the Woods–Saxon distribution: UCoul(R)= + 3 R 5 R ρ00 2 ρ (r)= , × R βiY20(θi) x,f 1+exp(|r − R |/a ) 0i x,f 0x,f i=x,f PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1446 KUKLIN et al. Table 2. Calculated penetrabilities P and spectroscopic factors S for spherical clusters and residual nuclei, as well as measured half-lives T1/2 and energy release Q in decay (the calculations were performed with the parameters a0x =0.55 fm, a0f =0.56 fm, r0x =1.15 fm, and r0f =1.16 fm) A → Ax + Af Q,MeV T1/2,s P S 221Fr → 14C + 207Tl 31.29 2.9 × 1014 1.8 × 10−27 2.3 × 10−9 221Ra → 14C + 207Pb 32.29 1.5 × 1013 1.6 × 10−26 5.0 × 10−9 222Ra → 14C + 208Pb 33.05 1.7 × 1011 6.6 × 10−25 1.1 × 10−8 223Ra → 14C + 209Pb 31.85 7.6 × 1015 1.9 × 10−27 8.4 × 10−11 224Ra → 14C + 210Pb 30.54 7.3 × 1015 1.2 × 10−29 1.4 × 10−8 226Ra → 14C + 212Pb 28.21 1.7 × 1021 1.2 × 10−34 5.8 × 10−9 225Ac → 14C + 211Bi 30.48 1.5 × 1017 5.8 × 10−33 1.4 × 10−8 228Th → 20O + 208Pb 44.72 5.4 × 1020 1.5 × 10−31 1.5 × 10−11 230Th → 24Ne + 206Hg 57.78 4.1 × 1024 6.0 × 10−32 4.8 × 10−15 231Pa → 23F + 208Pb 51.87 9.5 × 1025 2.1 × 10−32 6.0 × 10−16 231Pa → 24Ne + 207Tl 60.24 7.9 × 1022 1.1 × 10−29 1.4 × 10−15 232U → 24Ne + 208Pb 62.31 2.5 × 1020 7.7 × 10−28 6.3 × 10−15 233U → 24Ne + 209Pb 60.50 6.9 × 1024 1.7 × 10−30 1.0 × 10−16 233U → 25Ne + 208Pb 60.75 6.9 × 1024 2.6 × 10−30 6.7 × 10−17 234U → 24Ne + 210Pb 58.84 8.5 × 1025 9.9 × 10−33 1.4 × 10−15 234U → 26Ne + 208Pb 59.47 8.5 × 1025 2.3 × 10−32 6.0 × 10−16 234U → 28Mg + 206Hg 74.13 5.4 × 1025 6.2 × 10−30 3.5 × 10−18 235U → 24Ne + 211Pb 57.36 2.8 × 1027 3.6 × 10−35 1.2 × 10−14 235U → 25Ne + 210Pb 57.83 2.8 × 1027 9.4 × 10−35 4.3 × 10−15 235U → 26Ne + 209Pb 58.11 2.8 × 1027 5.5 × 10−34 3.2 × 10−15 236U → 30Mg + 206Hg 72.51 3.8 × 1027 2.2 × 10−32 1.4 × 10−17 236Pu → 28Mg + 208Pb 79.67 3.5 × 1021 1.1 × 10−25 3.3 × 10−18 238Pu → 28Mg + 210Pb 75.93 4.7 × 1025 3.8 × 10−30 6.9 × 10−18 238Pu → 30Mg + 208Pb 77.03 4.7 × 1025 4.2 × 10−29 6.0 × 10−18 238Pu → 32Si + 206Hg 91.21 1.9 × 1025 1.2 × 10−27 5.0 × 10−20 242Cm → 34Si + 208Pb 96.53 1.4 × 1023 1.5 × 10−24 1.5 × 10−20 2 12 e ZxZf 3. RESULTS OF THE CALCULATIONS + [R β Y (θ )]2. 35 R3 0i i 20 i i=x,f Figure 1 shows the nucleus–nucleus potential for the DNS 28Mg + 206Hg at two different values of In terms of the measured half-lives T1/2 and cal- nuclear deformation. The inclusion of quadrupole de- culated penetrabilities P , the probabilities S of the formation changes the position and height of the formation of various clusters (spectroscopic factors) barrier, as well as the depth of the local potential are expressed as minimum corresponding to the DNS at the contact π ln 2 point; i.e., R0 and the area under the curve U(R) − Q S = . (4) change. As a result, penetrability P increases for βx > ω0T1/2P 0 and decreases for βx < 0 by one to eight orders Numerical results are given in the next section. of magnitude as compared to the values calculated PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SPECTROSCOPIC FACTORS 1447 for spherical nuclei (Tables 1 and 2). Penetrability 96 for identical clusters emitted from neighboring nuclei (a) always increases with charge asymmetry. 92 , MeV 88 If an additional degree of freedom—the angle θi between the emission direction of one of the nu- 84 clei (i = x, f) and the axial symmetry axis—is intro- duced, the nucleus–nucleus potential is a function PU 30 (b) of this collective variable. A change in the nucleus– nucleus potential leads to changes in the barrier pen- –log 26 etrability P and the potential energy Uc = Bx + Bf + U (B and B are the binding energies of the nuclei) 22 x f 0 π/6 π/3 π/2 of the contact configuration. As θi for deformed nuclei θ x increases, Р decreases and Uc increases. These de- pendences become sharper for large deformations and Fig. 2. (а) Contact-configuration potential U and (b)the large products ZxZf .Asanexample,P and U for the sign-reversed logarithm of the barrier penetrability P for 208 30 208 30 DNS Pb + Mg are shown in Fig. 2 as functions the dinuclear system Pb + Mg vs. the angle θx of the deviation of the light-nucleus axis of symmetry from the of θx.Bydefinition, the state with minimum potential energy is most probable for the contact system (it axis passing through the centers of masses of the nuclei. has the largest spectroscopic factor). In this case, it corresponds to the cluster configuration with θi =0. and by seven and eight orders of magnitude for 24Ne The S value for θi = π/6 is approximately 1/30 of the and 28Mg, respectively. value for θ =0. The barrier penetrability is maximal i According to Tables 1 and 2, the S value for 28Mg for θ =0. Therefore, the probability SP of cluster i is 17 orders of magnitude smaller than the value decay is also maximal, and the main contribution to for 14C. The spectroscopic factors for carbon ra- averaging of SP over θi comes from the vicinity of 222,224,226 θi ≈ 0. dioactivity of the even–even isotopes Ra as Analysis of the decays of odd nuclei requires calculated including deformation are close to the re- spective values calculated disregarding deformation. the inclusion of DNS states with nonzero orbital 223 angular momenta L. These states are effectively The S value in the odd isotope Ra is two orders of magnitude smaller than the values for the neighbor- taken into account by the centrifugal potential UL in nucleus–nucleus potential (3). Possible L values ing even–even isotopes. This relation indicates that are determined by the conservation laws for parity structure effects are noticeable for a nuclear decay and angular momentum. In particular, for the decay mechanism and expresses the known suppression of − the emission of an even particle from an odd nucleus 235U(7/2 ) → 26Ne(0+)+209Pb(9/2+), L can be as compared to the neighboring even–even nuclei. equalto1,3,5,and7.Figure3showsthedepen- This suppression explains strong deviation of the dences of the barrier penetrability on the possible widths measured for the decays 221Ra → 14C + 207Pb orbital angular momenta for the decays 235U → 26Ne, and 235U → 24,25,26Ne + 209,210,211Pb from known 223 → 14 229 → 21 θ Ra C, and Th Oat =0in compar- dependences. ison with P at L =0. It is seen that, as the orbital angular momentum increases, the penetrability de- The spectroscopic factors presented in Table 1 234 28 206 creases slightly due to an increase in the moment of (except for S for the decay U → Mg + Hg) inertia of the system. In the decay considered above, are one to four orders of magnitude smaller than the L does not exceed 10. All calculations are performed values for the phenomenological potential described with the inclusion of the angular momentum. in [16]. Our S values for 14C are two orders of magni- Tables 1 and 2 present spectroscopic factors S tude larger than the shell-model results [1], and our S 24 28 30 calculated by Eq. (4) using experimental data with, values for Ne, Mg, and Mg are two, five, and one respectively, the inclusion and exclusion of the static order of magnitude lower, respectively, than the cor- quadrupole deformation of the light cluster and resid- responding shell-model results [1]. The spectroscopic ual nucleus. The deformation parameters of the nu- factors obtained for 24Ne and 28Mg in [5] are three and clei in the ground state are taken from experimen- four orders of magnitude larger than the respective tal systematics [15], where only the absolute values spectroscopic factors in Table 1. The spectroscopic of the quadrupole deformation parameters were pre- factors obtained for 14С in [5] agree well with our sented. It is remarkable that, as the deformation of the calculations. light cluster is varied, S for a given channel changes Figures 4 and 5 show the spectroscopic factor strongly by more than one order of magnitude for 14C S extracted from experimental data as a function of PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1448 KUKLIN et al. –logP 31 29