Possibilities of direct production of superheavy nuclei with Z=112--118 in different evaporation channels
J. Hong, G. G. Adamian, N. V. Antonenko, P. Jachimowicz, M. Kowal
aa r X i v : . [ nu c l - t h ] J u l Possibilities of direct production of superheavy nuclei with Z = J. Hong,
Department of Physics and Institute of Physics and Applied Physics,Yonsei University, Seoul 03722, Korea
G. G. Adamian,
Joint Institute for Nuclear Research, Dubna 141980, Russia
N. V. Antonenko,
Joint Institute for Nuclear Research, Dubna 141980, RussiaTomsk Polytechnic University, 634050 Tomsk, Russia
P. Jachimowicz,
Institute of Physics, University of Zielona G´ora,Szafrana 4a, 65516 Zielona G´ora, Poland
M. Kowal ∗ National Centre for Nuclear Research,Pasteura 7, 02-093 Warsaw, Poland (Dated: July 17, 2020) bstract The production cross sections of heaviest isotopes of superheavy nuclei with charge numbers112–118 are predicted in the xn –, pxn –, and αxn –evaporation channels of the Ca-induced com-plete fusion reactions for future experiments. The estimates of synthesis capabilities are based on auniform and consistent set of input nuclear data. Nuclear masses, deformations, shell corrections,fission barriers and decay energies are calculated within the macroscopic-microscopic approach foreven-even, odd-Z and odd-N nuclei. For odd systems the blocking procedure is used. To find, theground states via minimisation and saddle points using Immersion Water flow technique, multidi-mensional deformation spaces, containing non-axiallity are used. As shown, current calculationsbased on a new set of mass and barriers, agree very well with experimentally known cross-sections,especially in the 3 n –evaporation channel. The dependencies of these predictions on the mass/fissionbarriers tables and fusion models are discussed. A way is shown to produce directly unknown super-heavy isotopes in the 1 n – or 2 n –evaporation channels. The synthesis of new superheavy isotopesunattainable in reactions with emission of neutrons is proposed in the promising channels withemission of protons ( σ pxn ≃ −
200 fb) and alphas ( σ αxn ≃ −
500 fb).
PACS numbers: 25.70.Hi, 24.10.-i, 24.60.-kKey words: Superheavy nuclei; Complete fusion reactions; Production of new isotopes; Superheavy nuclei; xn –, pxn –, and αxn –evaporation channels ∗ Electronic address: [email protected] . INTRODUCTION The production and spectroscopic study of superheavy nuclei (SHN) is currently one ofthe important topics in nuclear experiment and theory. Due to the extremely short lifetimesof SHN and the exceptionally low probabilities of their production the final cross-sectionsare extremely small. Although determine them is a huge challenge for today theory, onlyhaving reached this stage it will be be possible to make reliable predictions of probabilities forsynthesis even heavier, still non-existent SHN. The Ca-induced complete fusion reactionshave been successfully used to synthesize SHN with the charge numbers Z =112–118 in theneutron evaporation channels ( xn − evaporation channels) [1–11] and to approach to ”theisland of stability” of SHN predicted at Z =114–126 and neutron numbers N =172–184 bythe nuclear shell models [12–15]. The most of these SHN have been obtained in the 3 n − and 4 n − evaporation channels. Only in the reactions Ca+
Pu, Ca+
Am, and Ca +
Cm the evaporation residues have been detected in the 2 n -evaporation channel. The nuclei , Fl and
Ts have been also produced in the 5 n -evaporation channel. On the agendais to expand the region of SHN in the direction of the magic neutron number N = 184, thecenter of the predicted ”island of stability”. For this purpose, we should study both newexperimental possibilities and possible reaction channels. New isotopes of heaviest nucleiwith Z =112–117 can be synthesized in the Ca-induced actinide-based complete fusion-evaporation reactions with the emission of charged particle (proton ” p ” or alpha-particle” α ”) and neutron(s) from the compound nucleus (CN). Note that the possibility of theproduction of new heaviest isotopes of superheavy nuclei with charge numbers 113, 115, and117 in the proton evaporation channels with rather high efficiency was suggested for the firsttime in Ref. [16]. This extremely intriguing suggestion was tested in Refs. [17] and [18].One can also observe new isotopes in the 1 n − and 2 n − evaporation channels of the Ca-induced actinide-based complete fusion reactions. Using the predictions of superheavy nucleiproperties of Ref. [19], we have recently studied these possibilities in Refs. [17, 20]. Wehave revealed how rapidly the evaporation residue cross section decreases with decreasingbeam energy in the sub-barrier region.An interesting question is how the estimation of production cross sections change if we re-place the mass table containing the predictions of SHN properties. Taking other mass table,we should incorporate it in all steps of the calculation of the evaporation residue cross sec-3ions. As known, the evaporation residue cross sections depend on the capture cross section,fusion probability (formation of the CN), and survival probability (the survival with respectto fission). The last one seems to be the most sensitive to the SHN properties. However, thefusion probability also crucially depends on the change of the mass table because it affectsthe potential energy surface driving two colliding nuclei to the CN. The capture cross sectiondepends on the deformations predicted for the colliding nuclei. So, in the present article, em-ploying the mass table of Ref. [21–23] based on the microscopic-macroscopic method, we willpredict the excitation functions in the xn − , pxn − , and αxn − evaporation channels of the Ca-induced complete fusion reactions and, correspondingly, the maximum cross sectionsat the optimal energies of these channels.
II. MODEL
For the excited SHN, the emission of charged particles is suppressed by the high Coulombbarrier and competes with the neutron evaporation and fission. The evaporation residuecross section can be written in factorized form [17, 20, 24–38]: σ s ( E c . m . ) = X J =0 σ cap ( E c . m . , J ) P CN ( E c . m . , J ) W s ( E c . m . , J ) . (1)The evaporation residue cross section in the evaporation channel s depends on the par-tial capture cross section σ cap for the transition of the colliding nuclei over the entrance(Coulomb) barrier, the probability of CN formation P CN after the capture and the survivalprobability W s of the excited CN. The formation of CN is described within a version of thedinuclear system model [17, 20, 37, 38].In the first step of fusion reaction the projectile is captured by the target. In the calcu-lation of σ cap in Eq. (1), the orientation of the deformed actinide target nuclei is taken intoaccount [37]. The bombarding energy E c . m . at which the capture for all orientation becomespossible is defined by the Coulomb barrier at sphere-side orientation. At smaller E c . m . somepartial waves fall under the barrier. The position and height of the Coulomb barrier aremainly affected by the quadrupole deformation of actinide nucleus. The quadrupole defor-mation used were extracted in Ref. [39] from the measured quadrupole moments. So, theeffect of deformations of higher multipolarities is taken partially into consideration in ourcalculations. Because the uncertainty in quadrupole deformation affects the Coulomb bar-4ier stronger than the hexadecopole deformation, we consider only quadrupole deformationin our calculations.In the second step the formed dinuclear system (DNS) evolves to the CN in the massasymmetry coordinate η = ( A − A ) / ( A + A ) ( A and A are the mass numbers of theDNS nuclei) [17, 20, 24–28, 30–38]. Because the bombarding energy E c . m . of the projectileis usually higher than the Q value for the CN formation, the produced CN is excited.When successful, hot fusion creates a heavy nucleus in a highly excited state that rapidlyemits two, three or four fast neutrons, each removing a few MeV of energy from the system,whereby it cools down. At every stage of this emission the neutrons compete with thefission processes that lead to nucleus splitting. This means that the nucleus generatedthrough hot fusion must be resistant to nuclear fission and explain great importance of thefission barrier B f as the main parameter which protect nucleus against fission. In anotherwords, in the third step of the reaction the CN loses its excitation energy mainly by theemission of particles and γ − quanta [40–47]. In the de-excitation of CN, the particle emissioncompetes with the fission which is the most probable process besides fission in normalnuclei. In this paper, we describe the production of nuclei in the evaporation channels withemission of charged particle (proton or α -particle) and neutrons as in Refs. [17, 20, 38].The emissions of γ , deutron, triton, and clusters heavier than alpha-particle are expectedto be negligible to contribute to the total width of the CN decay. The de-excitation of theCN is treated with the statistical model using the level densities from the Fermi-gas model.The neutron separation energies B n , Q − values for proton and alpha-particle emissions, thenuclear mass excesses of SHN, and the fission barriers for the nuclei considered are taken fromthe microscopic-macroscopic model [21–23]. Recently, within this approach (with parametersadjusted to heavy nuclei [48]), it was possible to reasonably reproduce data on ground statemasses; first, second and third [22, 49–54] fission barriers in actinides nuclei for which someemipirical/experimental data are available.Within the microscopic-macroscopic method, the energy of deformed nucleus is calculatedas a sum of two parts: the macroscopic one being a smooth function of Z , N and deformation,and the fluctuating microscopic one that is based on some phenomenological single-particlepotential. The deformed Woods-Saxon potential model used here is defined in terms of thenuclear surface. Mononuclear shapes can be parameterized via nuclear radius expansion inspherical harmonics Y lm ( ϑ, ϕ ). We admit shapes defined by the following equation of the5 ABLE I: The theoretical barriers V i and energy thresholds B i = V i − Q i in the evaporationchannels with emission of proton and alpha-particle ( i = p, α ). The Q p,α − values are calculatedwith the microscopic-macroscopic models [19] and [21].Reaction V p V α B p [21] B α [21] B p [19] B α [19](MeV) (MeV) (MeV) (MeV) (MeV) (MeV) Ca+
Pu 12.6 25.1 16.7 15.0 17.1 16.6 Ca+
Pu 12.6 25.1 16.1 15.7 17.2 16.9 Ca+
Am 12.7 25.3 14.5 14.1 14.1 15.7 Ca+
Cm 12.8 25.5 15.5 14.7 15.5 14.6 Ca+
Cm 12.7 25.5 16.1 14.6 15.9 14.4 Ca+
Bk 12.8 25.6 14.0 14.1 14.2 13.9 Ca+
Cf 12.9 25.9 15.0 13.6 14.8 13.8 Ca+
Cf 12.9 25.9 15.7 14.0 15.1 13.3TABLE II: Calculated Q α -values [21] and experimental Q exp.α -values [3] for the indicated SHN.SHN Q α Q exp.α SHN Q α Q exp.α (MeV) (MeV) (MeV) (MeV) Og 12.09 11.81
Fl 10.32 10.07
Ts 11.25 11.12
Fl 10.44 10.16
Ts 11.45 11.36
Fl 10.80 10.37
Lv 10.80 10.68
Nh 9.68 9.89
Lv 10.92 10.77
Nh 10.27 10.33
Lv 10.89 10.89
Nh 10.34 10.30
Lv 11.14 10.99
Nh 10.71 10.78
Ms 10.11 10.42
Nh 11.56 11.85
Ms 10.56 10.69
Cn 9.35 9.32
Ms 10.54 10.70
Cn 9.91 9.67
Ms 10.61 10.74
Cn 10.78 10.46
Fl 9.93 9.97
Cn 11.66 11.62
140 150 160 170 180 190 200 210 163 166 169 172 175 178 181 Cn Fl Lv Og (a) M ( M e V ) N
140 150 160 170 180 190 200 210 163 166 169 172 175 178 181 Nh Mc Ts (b) M ( M e V ) N
163 166 169 172 175 178 181 Cn Fl Lv Og (c) Q α ( M e V ) N
163 166 169 172 175 178 181 Nh Mc Ts (d) Q α ( M e V ) N FIG. 1: Nuclear masses: M (top panels) and Q α -values (bottom panels) for even- Z (left hand-sidepanels): − Cn, − Fl, − Lv, − Og and odd- Z (right hand-side panels): − Nh, − Mc, − Ts. Experimental data for Q α are taken from [3]. nuclear surface: R ( ϑ, ϕ ) = c ( { β } ) R { ∞ X λ =1 + λ X µ = − λ β λµ Y λµ ( ϑ, ϕ ) } , (2)where c ( { β } ) is the volume-fixing factor and R is the radius of a spherical nucleus.The n p = 450 lowest proton levels and n n = 550 lowest neutron levels from N max =19 lowest shell of the deformed oscillator are taken into account in the diagonalizationprocedure. We have determined the single–particle spectra for every investigated nucleus.The Strutinsky smoothing was performed with the 6-th order polynomial and the smoothingparameter equal to 1 . ~ ω . For the systems with odd proton or neutron (or both), we use thestandard blocking method. Considered configurations consist of an odd particle occupyingone of the levels close to the Fermi level and the rest of particles forming paired BCS state onremaining levels. The ground states were found by minimizing over configurations (blockingparticles on levels from the 10-th below to 10-th above the Fermi level) and deformations.7 Cn Fl Lv Og (a) B f ( M e V ) N Nh Mc Ts (b) B f ( M e V ) N
163 166 169 172 175 178 181 Cn Fl Lv Og (c) B n ( M e V ) N
163 166 169 172 175 178 181 Nh Mc Ts (d) B n ( M e V ) N -4-3-2-1 0 1 163 166 169 172 175 178 181 Cn Fl Lv Og (e) B f − B n ( M e V ) N -4-3-2-1 0 1 163 166 169 172 175 178 181 Nh Mc Ts (f) B f − B n ( M e V ) N FIG. 2: Fission barriers: B f (top panels), neutron separation energies: B n (middle panels) andtheir differences: B f − B n (bottom panels) for even- Z (left-hand side panels): − Cn; − Fl; − Lv; − Og and odd- Z (right-hand side panels): − Nh; − Mc; − Ts. Exper-imental data for barriers, marked by crosses, are taken from [72].
For nuclear ground states it was possible to confine analysis to axially-symmetric shapes.More details can be found in Ref. [22]. The simplest extension of the WS model to oddnuclei required three new constants which may be interpreted as the mean pairing energiesfor even-odd, odd-even and odd-odd nuclei [22]. They were fixed by a fit to the masses with Z ≥
82 and
N >
126 via minimizing the rms deviation in particular groups of nuclei what8s rather standard procedure [55, 56]. The experimental nuclear masses of heavy nuclei weretaken from [57]. The obtained rms deviation in masses for 252 nuclei is about 400 keV withblocking scenario [22] used here. Similar rms error is obtaimed for 204 Q α values. For 88measured Q α values in SHN, the quantities outside the region of the fit, we obtained therms deviation of about 250 keV [22].To estimate the survival probability, the fission barriers from adiabatic scenario, i.e. thesmallest possible ones, are taken [23]. The main problem in a search for saddle points is that,since they are neither minima nor maxima, one has to know energy on a multidimensionalgrid of deformations (the often used and much simpler method of minimization with imposedconstraints may produce invalid results) [50, 58–60]. To find saddles on a grid we used theImaginary Water Flow technique. This conceptually simple and at the same time veryefficient (from a numerical point of view) method was widely used and discussed [50, 58, 61–64]. Based on this and our previous results showing that triaxial saddles are abundantin SHN [54], we consider that quadrupole triaxial shapes have to be included for the firstbarriers with which we are dealing with the nuclei considered here. So, the saddle pointsare searched in the five dimensional deformation space spanned by: β , β , β , β , β - defined in Eq. (2), using the Imaginary Water Flow technique. All details regarding themethodology of searching for the right saddles with the exact specification of the deformationspaces used, can be found in Ref. [23]. Finally, we want to emphasize, that recently we havesystematically determined inner and outer fission barrier heights for 75 actinides, withinthe range from actinium to californium, including odd- A and odd-odd systems, for whichexperimental estimates were accessible [53]. A statistical comparison of our inner and outerfission barrier heights with available experimental estimates gives the average discrepancyand the rms deviation not larger than 0.82 MeV and 0.94 MeV, respectively. This allowsus to have some confidence in the macroscopic-microscopic model used here. Significantdifferences in the fission barriers obtained in various modern nuclear models were noticed inRef. [65]. A broad discussion of the problems arising from this can be found in Refs. [23, 66].Owing to the dependence of the shell effects on nuclear excitation, the value of shellcorrection effectively depends on the excitation energy with the damping parameter E d = 25MeV. In comparison to Refs. [18, 67, 68], which are based for even-even nuclei on thesame mass table, we employ the equivalent method to calculate the survival probability[17, 20, 38] taking into account the shell effect damping in the potential energy surface and9symptotic level-density parameter a . However, we would like to emphasise that for oddnuclei in Refs. [18, 67, 68] the pairing was treated in different way compared to nuclear inputdata used here. Namely, the predictions of the Fusion-by-Diffusion model [18, 67, 68] forthe synthesis cross sections of 114120 elements were based on the macroscopic-microscopicproperties calculated within the quasiparticle method in pairing channel. The ground statesand consequently fission barrier heights for other nuclei were calculated separately by addingthe energy of the odd particle occupying a single-particle state. This quasiparticle energy E qp in the superconducting state takes a simple form: E qp = p ( ε qp − λ ) + ∆ , where ε qp isthe energy of the odd nucleon in the quasiparticle state, λ is the Fermi energy and ∆ is thepairing gap energy. In this scenario of fission barriers calculation the energy E qp was addedat every grid point as well as at every minimisation step in the gradient procedure usedfor the ground states. So, the calculations of masses and B f have been performed withoutblocking of any state in the calculations within the Fusion-by-Diffusion model [18, 67, 68]but with using the BSC-quasiparticle method.In the DNS model used here the damping parameter should be larger than in Refs. [18, 67,68]. With the expression a n = a = A/
10 MeV − for the asymptotic level-density parameterfor neutron ( A is the mass number of the CN) we obtain almost the same values as thoseused in Ref. [69] and found microscopically in Ref. [70]. The level-density parameters forfission, proton-emission, and α − emission channels are taken as a f = 0 . a , a p = 0 . a , and a α = 1 . a , respectively. The ratio between a and a f is closed to that found in Ref. [70].Here, we set these parameters for all reactions considered. Because the shell corrections atthe ground state are larger with the mass table [19], in Refs. [17, 20] the larger values of a f = 1 . a were used. Other parameters in Refs. [17, 20] were set the same as in this paper.So, taking other mass table for the properties of SHN, we change only the ratio a f /a .For the calculation of the Coulomb barrier, we use the expression V j = ( Z − z j ) z j e r j [( A − m j ) / + m / j ] , (3)where z j ( m j ) are the charge (mass) numbers of the charged particle (proton or α -particle)and r j is a constant. The charge Z (mass A ) number corresponds to the CN. There aredifferent theoretical estimations of r j [40, 47]. In the case of α − emission, r α varies from1.3 to 1.78 fm. We obtain r α from the energy of the DNS formed by the daughter nucleusand α -particle. We calculate the Coulomb barrier in the interaction potential between the10 -particle and the daughter nucleus [71], and find the value of r α from Eq. (3). For differentnuclei considered, we obtained r α = 1 .
57 fm using this method. Thus, in the calculationsof V α we set r α = 1 .
57 fm for nuclei considered. The parameter r p for the Coulomb barrierfor proton emission is taken as r p =1.7 fm from Refs. [35, 47]. With these values of r α and r p Eq. (3) results in V α and V p which are about 2.5 and 1.5 MeV (Table II), respectively,larger than those used in Refs. [18, 67, 68]. As shown in Refs. [18, 67, 68], the increase of V α and V p by 4 MeV leads to about one order of magnitude smaller σ s in the αxn and pxn evaporation channels. So, the difference of our r α and r p from those in Refs. [18, 67, 68]could create 2–4 times difference in the values of σ s . In Refs. [17, 20], the same values of V α and V p were used as in this paper. As seen in Table II, the values of energy thresholds forprotons and alpha-particles obtained with the mass table [19] deviate within 2.5 MeV fromthose calculated with the mass table [21].As found, the values of σ s near the maxima of excitation functions are almost insensitiveto the reasonable variations of the parameters used, but far from the maxima they changeup to one order of magnitude. Therefore, the results obtained in this paper have quite asmall uncertainty near the maxima of excitation functions which are important to get themaximum yield of certain nucleus in the experiments. We estimate the uncertainty of ourcalculations of σ s within a factor of 2–4. Our model was well tested in Ref. [38] for manyreactions in which the excitation functions of transfermium nuclei produced in the chargedparticle evaporation channels have been measured. III. CALCULATED RESULTS
In Fig. 1, our results for nuclear masses (top panels) and calculated from them Q α -values(bottom panels) for SHN considered in this article are shown. As one can see, the availablealpha-decay energy measurements are perfectly reproduced. Only in the case of Cn and Nhnuclei with smaller number of neutrons our results slightly overestimate the experimentaldata. The exact values in some of the most important cases here are summarized in theTable II. Let us emphasize that only ground-state-to-ground-state alpha transition werecalculated. Apparent Q α values taking the parent ground-state configuration in odd andodd-odd systems as the final state in daughter were not considered. This may be the reasonfor the overestimation in a few cases, as especially in odd nuclei the decay may occur to11 -2 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV) (a)4n3n2n1n5n Ca+
U 10 -2 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV)4n3n2n1n5n (b) Ca+
Np10 -2 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV) (c)4n3n2n1n5n Ca+
Am 10 -4 -3 -2 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV) (d)4n3n2n1n5n Ca+ Bk FIG. 3: The measured (symbols) and calculated (lines) excitation functions for xn − evaporationchannels ( x = 1 −
5) of the indicated complete fusion reactions. The mass table of Ref. [21] is usedin the calculations. The black triangles at energy axis indicate the excitation energy E ∗ CN of theCN at bombarding energy corresponding to the Coulomb barrier for the sphere-side orientation.The blue diamonds, green squares, red circles, and gray pentagons represent the experimental data[3] with error bars for 2 n − , 3 n − , 4 n − , and 5 n − evaporation channels, respectively. The verticallines with arrow indicate the upper limits of evaporation residue cross sections. excited states of the daughter nucleus, which shortens the alpha transition lines.In Fig. 2 we provide a comparison of our calculated fission barriers with the availableexperimental estimates based on the observed evaporation residue production probabilities[72]. As in the case of Fig. 1, the isotopes: Fl, Cn, Lv and Og are shown on the left-hand sidewhile Nh, Mc, Ts on the right-hand side. As seen in Fig. 2, our calculated fission barriers B f are in quite good agreement with the experimental (empirical) estimates [72] marked12 -2 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV) (a)4n3n2n1n5n Ca+
Pu 10 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV) (b)4n3n2n1n5n Ca+
Pu10 -2 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV) (c)4n3n2n1n5n Ca+
Pu 10 -3 -2 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV) (d)4n3n2n1n5n Ca+ Pu FIG. 4: The same as in Fig. 3, but for other indicated complete fusion reactions. by crosses. For completeness, we show the neutron separation energies B n in the middlepart of Fig. 2. The strong staggering is clearly visible. As mentioned before, the height ofthis threshold in relation to the fission barrier is crucial for determining which process winsthe competition in nuclear decay, fission or neutron emission. In Fig. 2 (bottom panels),the differences between these two key parameters controlling survival probability are alsoshown.With the nuclear properties from Refs. [21–23] the calculated excitation functions for xn evaporation channels are presented in Figs. 3-5 for the complete fusion reactions Ca+ U, Np,
Am,
Bk, , , , Pu, , Cm, , Cf. In Ref. [20] and here, thesame model is used to calculate the evaporation residue cross sections. So, the comparisonwith the results of Ref. [20] reflects the difference in the predicted properties of SHN. Incomparison to Ref. [20] the bombarding energies corresponding to the Coulomb barriers13 -2 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV) (a)4n3n2n1n5n Ca+
Cm 10 -4 -3 -2 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV) (b)4n3n2n1n5n Ca+
Cm10 -2 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV) (c)4n3n2n1n5n Ca+
Cf 10 -3 -2 -1
10 20 30 40 50 60 σ x n ( pb ) E CN* (MeV) (d)4n3n2n1n5n Ca+ Cf FIG. 5: The same as in Fig. 3, but for other indicated complete fusion reactions. for the sphere-side orientations lead to 2–3 MeV smaller E CN . As a result, the maxima ofexcitation functions σ xn ( E CN ) are slightly shifted to higher E CN in Figs. 3-5. If in Ref. [20] σ n > σ n in the maxima of excitation functions for the reactions Ca+ , Pu, , Cm,the present calculations with the data of Ref. [21] result in σ n < σ n and larger σ n /σ n .Though in the reactions Ca+ U, Np,
Am,
Bk the maximum production cross sec-tions are expected in the 3 n evaporation channel independently on the mass table, in Fig. 3the ratios σ n /σ n are about 2 times smaller than those in Ref. [20]. The mass table [21]leads to close maxima of σ n and σ n , relatively large σ n and smaller σ n in most reactions.The maximum cross sections in the 2 n -evaporation channel were found to be within factorof 10 smaller than the cross sections at the maxima of excitation functions of the 3 n − or4 n − evaporation channels. The cross sections in 1 n evaporation channel could be of interestfor the experimental study if they are larger than 5 fb. Thus, employing reactions in the14 -3 -2 -1
20 30 40 50 60 σ α x n ( pb ) E CN* (MeV) (a) α α α α Ca+
Am 10 -5 -4 -3 -2 -1
20 30 40 50 60 σ α x n ( pb ) E CN* (MeV) (b) α α α α Ca+ Bk -5 -4 -3 -2
20 30 40 50 60 σ p x n ( pb ) E CN* (MeV) (c)p3np2np1np Ca+ Am -7 -6 -5 -4 -3
20 30 40 50 60 σ p x n ( pb ) E CN* (MeV) (d)p3np2np1np Ca+ Bk FIG. 6: The same as in Fig. 3, but for αxn − evaporation channels ( x = 0 −
3) of the indicatedcomplete fusion reactions. n − and 2 n − evaporation channels, one can directly produce heaviest isotopes closer to thecenter of ”the island of stability”: , Cn, , Nh,
Lv,
Ts, and − Og. Manyof them were only produced as daughters in the α − decay chains. The isotopes Ts, and − Og are presently unknown.The comparison of the results in Figs. 3-5 with those in Refs. [67, 68] based on the samemass table allows us to stress the difference of the fusion models used. In spite of thedifference of the fusion models, the predicted values of cross sections are rather close formost reactions. While σ n > σ n for the Ca+
Cf reaction in Ref. [67], in Fig. 5 we obtain σ n < σ n . For the Ca+
Bk, we obtain smaller ratio σ n /σ n and larger σ n /σ n thanthose in Ref. [68].The calculated excitation functions for the channels with evaporation of charged particle15 -3 -2 -1
30 40 50 60 70 σ α x n ( pb ) E CN* (MeV) (a) α α α α Ca+
Pu 10 -4 -3 -2 -1
30 40 50 60 70 σ α x n ( pb ) E CN* (MeV) (b) α α α α Ca+
Pu10 -4 -3 -2 -1
20 30 40 50 60 σ p x n ( pb ) E CN* (MeV) (c)p3np2np1np Ca+
Pu 10 -3 -2 -1
30 40 50 60 σ p x n ( pb ) E CN* (MeV) (d)p3np2np1n Ca+ Pu FIG. 7: The same as in Fig. 3, but for αxn − and pxn − evaporation channels ( x = 0 −
3) of theindicated complete fusion reactions. are presented in Figs. 6-9. While in Ref. [17] σ α n > σ α n and σ p n > σ p n with mass table[19] in most reactions, we obtain rather close cross sections in α n and α n , p n and p n evaporation channels due to slightly smaller neutron separation energies in the mass table[21–23]. Because the same mass table is used (with a reservation regarding the differenttreatment of odd particles) in Ref. [18], the predicted cross sections are similar there tothose in Figs. 6-9. However, stronger dependence of fusion probability P CN on energy leadsto relative increase of the α n and p n evaporation channels in Figs. 6-9. The relativelysmaller yields in the α n and p n evaporation channels are due to the same reason.The production cross sections of almost all of these SHN in the xn -evaporation channelsare comparable or even larger than those in the charged particle evaporation channels. Theproduction cross sections of heaviest isotopes − Nh, − Mc, and − Ts ( , Cn,16 -3 -2 -1
20 30 40 50 60 σ α x n ( pb ) E CN* (MeV) (a) α α α α Ca+
Cm 10 -4 -3 -2 -1
20 30 40 50 60 σ α x n ( pb ) E CN* (MeV) (b) α α α α Ca+ Cm -4 -3 -2 -1
20 30 40 50 60 σ p x n ( pb ) E CN* (MeV) (c)p3np2np1np Ca+
Cm 10 -4 -3 -2 -1
20 30 40 50 60 σ p x n ( pb ) E CN* (MeV) (d)p3np2np1n Ca+ Cm FIG. 8: The same as in Fig. 3, but for αxn − and pxn − evaporation channels ( x = 0 −
3) of theindicated complete fusion reactions.
Nh, , Fl, , Mc, and
Lv) in the pxn -channels ( αxn -channels) of the Ca-inducedfusion reactions were predicted: about 10–200 fb in the pxn − evaporation channels (about50–500 fb in the αxn − evaporation channels). IV. SUMMARY
For the Ca-induced actinide-based complete fusion reactions, the excitation functionsfor the production of the SHN with charge numbers 112–118 were calculated in xn − , αxn -,and pxn -evaporation channels using the predictions of SHN properties from Ref. [21–23].As it turns out, in modeling of reactions leading to the SHN, the use of a consistenti.e., coming from one source, set of nuclear data input plays a fairly important role. In17 -4 -3 -2 -1
20 30 40 50 60 σ α x n ( pb ) E CN* (MeV) (a) α α α α Ca+
Cf 10 -4 -3 -2 -1
20 30 40 50 60 σ α x n ( pb ) E CN* (MeV) (b) α α α α Ca+ Cf -3 -2 -1
30 40 50 60 σ p x n ( pb ) E CN* (MeV) (c)p3np2np1n Ca+
Cf 10 -4 -3 -2
30 40 50 60 σ p x n ( pb ) E CN* (MeV) (d)p3np2np1n Ca+ Cf FIG. 9: The same as in Fig. 3, but for αxn − and pxn − evaporation channels ( x = 0 −
3) of theindicated complete fusion reactions. the presented article, all used nuclear properties of the ground states and saddle pointswere calculated within the multidimensional macroscopic-microscopic approach with block-ing technique for odd nuclei.Obtained agreement of cross-sections for the reactions in 3 n channel is excellent. Excita-tion functions are only slightly shifted towards higher energies compared to the experimentwhen four neutrons are emitted in the cascade. Only for the reactions; Ca+
Pu and Ca+
Pu, the resulting cross-sections are underestimated - but less than one order ofmagnitude.The use of the charged particle evaporation channels allows us to increase the massnumber of heaviest isotopes of nuclei with Z = 113, 115, and 117 (112 and 114) up to 5, 3,and 1 (1 and 1) units, respectively, with respect to the xn evaporation channels. In addition,18n the nuclei produced the electron capture can occur by adding one more neutron in thedaughter nuclei. The proton evaporation channels seem to be more effective to approach N = 184 than the alpha emission channels. One can produce more neutron-rich isotopesin the reactions with even − Z targets than in the reactions with odd − Z ones. The pxn − and αxn − evaporation channels allows us to obtain an access to those isotopes which areunreachable in the xn − evaporation channels due to the lack of proper projectile-targetcombination. Thus, employing reactions suggested, one can produce the heaviest isotopescloser to the center of the island of stability. The pxn − and αxn − evaporation channels canbe only distinguished by different α − decay chains of the evaporation residues because theexcitation functions of these channels overlap with those from xn − evaporation channels.Our present results were compared with those obtained with the same fusion model andother mass table and with completely other fusion model [18, 67, 68] for which nuclear prop-erties where calculated using the same macroscopic-microscopic model but with quasiparticlemethod for pairing. Absolute values of cross sections are rather close. However, the ratioof the cross sections in the maxima of excitation functions is sensitive to the mass table.For example, σ p n > σ p n with the mass table [19], while σ p n ≈ σ p n in the calculationspresented. If the same mass table is used with different fusion model, the fusion probabilitycreates the difference in the cross sections obtained. For example, the ratios σ n /σ n and σ α n /σ α n are sensitive to the increase rate of P CN with excitation energy and, thus, to thefusion model. Acknowledgments
G.G.A. and N.V.A. acknowledge the partial supports from the Alexander von Humboldt-Stiftung (Bonn). This work was partly supported by RFBR (17-52-12015, 20-02-00176),DFG (Le439/6-1), and Tomsk Polytechnic University Competitiveness Enhancement Pro-gram grant. M.K. was co-financed by the National Science Centre under Contract No.UMO-2013/08/M/ST2/00257 (LEA-COPIGAL). [1] Yu.Ts. Oganessian, J. Phys. G , R165 (2007).
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