Fusion probability and survivability in estimates of heaviest nuclei production

Production of the heavy and heaviest nuclei (from Po to the region of superheavy elements close to Z=114 and N=184) in fusion-evaporation reactions induced by heavy ions has been considered in a systematic way within the framework of the barrier-passing model coupled with the statistical model (SM) of de-excitation of a compound nucleus (CN). Excitation functions for fission and evaporation residues (ER) measured in very asymmetric combinations can be described rather well. One can scale and fix macroscopic (liquid-drop) fission barriers for nuclei involved in the calculation of survivability with SM. In less asymmetric combinations, effects of fusion suppression caused by quasi-fission (QF) are starting to appear in the entrance channel of reactions. QF effects could be semi-empirically taken into account using fusion probabilities deduced as the ratio of measured ER cross sections to the ones obtained in the assumption of absence of the fusion suppression in corresponding reactions. SM parameters (fission barriers) obtained at the analysis of a very asymmetric combination leading to the production of (nearly) the same CN should be used for this evaluation.


Introduction
Complete fusion between an accelerated heavy ion and heavy target nucleus is defined as their amalgamation into a compound system inside the fission barrier.This process is unambiguously identified by the direct observation of evaporation residues (ER) produced after cooling of an excited compound nucleus (CN) by light particles (neutrons) evaporation.Complete fusion reactions have been successfully used in synthesis of the heaviest elements at the border of nuclear stability in experiments performed during the last 20 years [1,2].Such experiments are extremely challenging as the formation of heavy/superheavy ER is heavily suppressed not only by the CN-fission, but also by a non-equilibrium process called quasi-fission (QF) [3][4][5][6][7].Following the capture of a projectile and target nuclei, a system may re-separate prematurely, not forming a true CN.Such events represent the transition between deep-inelastic collisions (DIC) and complete fusion.In DIC the entrance channel mass-asymmetry is preserved, but there can be large dissipation of kinetic energy and angular momentum [8].CN formation, in contrast, is characterized by equilibration of all degrees of freedom, and hence completes loss of identity of the entrance channel.Intermediate EPJ Web of Conferences between DIC and CN fission, QF has full energy dissipation but incomplete drift toward the energetically favoured mass-symmetric configuration [5,7].
Intensive studies of the mechanisms of complete fusion in fission experiments [4,6,7] and in experiments on ER production [9][10][11][12] in reactions leading to heavy and very heavy composite systems, make it evident that the growing QF effect leads to a rapid decrease in the fusion probability and consequently to the drop of production cross sections for the heaviest elements as their atomic number is increased.Such dependence is clearly seen in the case of 'cold' fusion reactions [1].One of a qualitative explanation of the fusion suppression effect was proposed in the framework of the liquid drop (LD) model [11].It is suggested that the transition from the contact configuration to the CN configuration is determined by a presence of the conditional barrier along the mass-asymmetry coordinate [13] and the entrance-point position with respect to the top of the barrier, i.e., relatively to the Businaro-Gallone (BG) point [14].The Generalized Liquid-Drop Model (GLDM) [15] calculations of the potential energy surface (PES) in the (Z, N)-plane for the contact configuration of spherical nuclei give us an examination of the PES for the 216 Ra * CN in more detail (see Figure 1) than considered in [11].The calculations take into account proximity energy [16].The entrance points for asymmetric combinations (with Ar and lighter projectiles) lay well above the bottom of the valley (minima of the potential energy corresponding to the driving potential), on mountainsides of the PES.Shell corrections [17] strongly modulate the PES and a driving potential.In the framework of this approach fusion with 48 Ca and 86 Kr could be considered as the suppressed one, since their entrance points lay on the bottom of the valley, in front of the conditional barrier on a way along the mass-asymmetry coordinate to complete fusion as shown in Figure 1.In the 48 Ca+ 168 Er→ 216 Ra * experiment fusion suppression is observed and it corresponds to the fusion probability P fus ≈0.3 [12].The value deduced by comparing measured ER cross sections with those obtained in the 12 C+ 204 Pb→ 216 Ra * reaction [11,12].
A number of models using PES in calculations were developed to describe complete fusion of massive nuclei leading to production of heavy and heaviest nuclei.Among them there are calculations based on a di-nuclear system (DNS) concept (see, e.g., results of calculations in [18,19]).Other ones use a two-centre model approach [20].One should mention that these models use more sophisticated multi-dimensional PES with taking into account a distance between reactants, their static and dynamic deformations, a relative angular momentum, etc (see, e.g., [21]).In calculations, the resulting ER cross section can be parameterized as follows: where σ cap is a capture cross section as a function of the c.m. energy E and angular momentum L, and W sur is survivability of nuclei in the evaporation-fission process of the CN decay at the corresponding excitation energy and angular momentum L. Despite of variety of theoretical models they Fig. 1.Driving potential extracted from the PES obtained in the framework of GLDM [15,16] for the 216 Ra CN as a function of the mass asymmetry (solid and dashed lines corresponding to the calculations with and without shell corrections [17], respectively).Filled and open circles correspond to the positions of the entrance points on PES for some combinations, which are also calculated with and without shell corrections [17], respectively.
demonstrate rather good agreement of calculated ER cross sections with experimental ones.This agreement is shown, e.g., in the analysis [22] using a number of models that result in the calculated cross sections for the heaviest ER produced in 'cold' fusion reactions [1].At the same time, this analysis shows a large spread in the P fus values given by these models in calculations (a difference exceeds two orders of the value).To determine P fus strictly the authors [22] measured the capture cross section and extracted the CN-fission component with the decomposition of measured angular distributions for fission-like fragments produced in the 50 Ti+ 208 Pb→ 258 Rf * reaction.Such procedure is not completely model-independent one since it implies some constraints on the angular momenta and moments of inertia inherent in the CN-fission and QF processes [6].
In a contrast to the previous study with the decomposition of angular distributions integrated over a mass and energy of fission-like fragments [22], in our study of the 48 Ca+ 154 Sm→ 202 Pb * reaction the CN-fission component was extracted from measured mass-angular distributions with taking into account a total kinetic energy (TKE) of fission fragments (FF) [23].With these constraints, FF of the masses in the region of A CN /2±20 only had a symmetric angular distribution in the c.m. system.These FF may correspond to 'true' CN-fission.Therefore, the fraction of the QF component and the corresponding P fus values could be 'definitely' extracted [23].Nevertheless a comparison of the ER cross sections measured in this reaction [24] with those obtained in 16 O+ 186 W leading to the same 202 Pb * CN shows distinctly lower values of the fusion probability.This comparison (together with the calculated P fus values obtained in the framework of the DNS concept [25]) is shown in Figure 2. The question arises: why is there the difference in the P fus estimates obtained with the comparison of the ER data and from fission experiments?One should mention that in our experiments FF and ER were detected in the same runs following the same normalization procedure based on Rutherford scattering for the cross section estimates.At the same time, within our data processing procedure, we could i) underestimate quantity of QF events (using simple cutting and Gaussian fits) because of the interference of QF events with deepinelastic ones and ii) overestimate quantity of CN-fission events corresponding to a 'right' variance of the mass distributions and to the symmetric mass-angular distributions.The latter implies some equilibration and compactness of a system near the saddle point without passing over the fission barrier, i.e., without 'true' CN formation.So fission data may not give us adequate values of the fusion probability whereas the detection of ER is an unambiguous signature of CN formation, as mentioned above.

Method of analysis
One can propose two empirical approaches for the extraction of the fusion probability values from ER cross section data.In the first one, ER cross sections measured in the reaction under study are compared to those obtained in a very asymmetric combinations leading to the same CN.For the latter a lack of fusion suppression at energies above the Coulomb barrier is assumed, i.e., P fus =1.Bearing in mind that in the framework of a barrier passing (BP) model the ER cross section can be written as (2) where k is a wave number and T L is a transmission probability of the potential-barrier passing for L partial wave, which is about 1 at energies well above the barrier and survivability is the same for the same CN; one can came to a simple relation for the estimates of the fusion probability (3) where the terms with super-scripts asym correspond to the values measured and/or calculated for a very asymmetric combination for which P fus =1 is assumed.
In the case, when the capture cross section for the reaction under study is known, the formulae (1) and (2) allow to estimate the fusion probability as follows (4) where the calculated ER cross section is based on the fitted capture cross section which is assumed to be equalled to the fusion cross section (P fus =1) and can be calculated as (5) A conventional barrier passing (BP) model that generally reproduces an experimental capture cross section as the barrier-passing one is used for the calculation of The calculated BP cross section can be associated with the fusion cross section, implying for the fusion probability P fus =1.The effect of coupling the entrance channel to other reaction channels is taken into account phenomenologically via the fluctuations of the radius-parameter r 0 .The radius-parameter fluctuations are generated with a Gaussian distribution around its average r 0 =1.12 fm with the barrier fluctuation parameter σ(r 0 ) [26].These fluctuations simulate coupling effects [27] at energies around the Coulomb barrier to a degree sufficient for the purpose of the present analysis.All the BP model parameters are fixed with the exception of the strength V 0 and the relative barrier fluctuation parameter σ(r 0 )/r 0 in the nuclear exponential potential [26].Variations of these parameters allow one to reproduce the experimental cross-section data for fission and ER production at sub-barrier energies in calculations.Transmission probabilities for the potential barrier passing are calculated using the WKB approximation.Note that for strongly fissile compound nuclei the ER cross sections at energies well above the nominal fusion barrier [28] are weakly sensitive to the form of the nuclear potential and are mainly determined by the survivability calculated.
The survivability is calculated in the framework of statistical model approximations.The Reisdorf's expression [29] is used to calculate the macroscopic level-density parameters and in fission and evaporation channels, respectively.Macroscopic fission barriers adjusted with the scaling factor k f at the rotating LD fission barriers [30].Therefore k f is the only fitting parameter in the expression determining a height of fission barrier: The empirical masses [31] are used to calculate shell corrections (as the difference between the empirical and LD masses [32]) as well as for the calculations of excitation and separation energies.The influence of the shell effects on the level density is included with an energy backshift in the lev-el-density parameter that is exponentially damped with increasing excitation energy.The same damping constant E D =18.5 MeV was used as in a number of the previous works [9,10,12,29].
The BP model and SM used in the present analysis are combined into the HIVAP code [26,29] used for the calculations.Usually, in the analysis of the most asymmetric combinations, excitation functions at high energies can be fitted with the variation of k f mainly.The lack of any fusion suppression (P fus =1) is evident if ER and fission excitation functions are fitted with the same parameter values of the nuclear potential and k f in the whole range of excitation energies.In less asymmetric combinations the absence of the fusion suppression is not evident.Repeating aforesaid, the fusion probability can be obtained with the same scaling for fission barriers and other parameters of SM, as obtained in the analysis of the most asymmetric combination leading to the same CN [12,33].

ER production in the vicinity of N=126 shell
In Figure 3 the reduced cross sections for ER produced in the 16 O+ 186 W and 48 Ca+ 154 Sm reactions are compared [24,33] with the aim to derive the fusion probability values.As we see, experimental values for the former exceed the ones for the latter by a factor of 1.5 at ≥70 MeV, where the effect of the fusion barriers [28] of both the reactions is negligible.The ER excitation function calculated for 16 O+ 186 W reproduces the measured cross sections very well, whereas the similar one calculated for 48 Ca+ 154 Sm with the same parameters of SM as used in 16 O+ 186 W, but with the barrierpassing cross sections fitted to the experimental values corresponding to σ cap =σ fis +σ ER , exceeds the measured ER cross sections by the same factor of 1.5 at ≥70 MeV.The excitation functions calculated for both the reactions are converged at ≥80 MeV that implies formation of indeed the same CN at these energies.The ER excitation function calculated for 48 Ca + 154 Sm with the normalisation to P fus =0.57allows one to reproduce the experimental data at ≥60 MeV.The empirical P fus values shown in Figure 2 have been derived from the ratio of the measured ER cross sections to the calculated ones corresponding to the barrier-passing cross sections fitted to the experimental values of σ cap =σ fis +σ ER and to the survivability obtained in the 16 O+ 186 W reaction.The latter is characterized by the scaling factor k f =0.85 at the LD fission barriers for nuclei produced in the 202 Pb * CN decay chains [33].
More severe fusion suppression is observed in reactions leading to more fissile Th * compound nuclei.An example of data analysis is shown in Figure 4 for the reactions leading to the 220 Th * CN.The data obtained in the very asymmetric 16 O+ 204 Pb reaction can be reproduced with the corresponding scaling of the LD fission barriers.For the less asymmetric combination with 40 Ar and for the nearly symmetric one with 124 Sn, ER cross sections calculated with these barriers at energies well above the fusion barrier, exceed the experimental ones by a factor of 4 and 8, respectively.At sub-barrier energies, we obtain even stronger fusion suppression using barrier-passing cross sections corresponding to the capture cross sections fitted.For the nearly symmetric 124 Sn+ 96 Zr combination, one had to use the extra-extra-push (EEP) formalism with the parameters obtained in [7], giving the additional ΔB=17.1 MeV to the potential barrier.This additive to the fusion barrier together with the normalisation of calculations with the P fus =0.14 value allow one to reproduce the ER cross sections.
More neutron-deficient Th compound nuclei can be only produced in reactions with massive nuclei.Their production in a very asymmetric combination, which is similar to the 16 O+ 204 Pb one, is impossible.Therefore, a simple extrapolation with the proportionality of the k f values, corresponding to the data fits with and without fusion suppression for similar reactions leading to the 220 Th * CN could be used.This extrapolation gives us considerably higher magnitudes of macroscopic fission barriers than one can get with a straight fit of calculations to the measured ER cross sections.For example, in the case of reactions with the 32 S [38, 39], 40 Ar [36] and 60 Ni [39] projectiles leading to the 214,216 Th * compound nuclei, straightforward fits give us k f =0.56 comparing to the k f =0.614 value obtained with the extrapolation.Using this scaling one can obtain fusion probability values (with some dependence on the extrapolated k f values) and reproduce ER cross sections measured in the investigated reactions leading to the neutron-deficient Th compound nuclei.Some examples of fusion probabilities obtained with this empirical approach for reactions leading to the production of Th compound nuclei are shown in Figure 5.In the case of the 214 Th * CN one can compare the present data with the fusion probabilities extracted from the fission data [38] and calculated with the DNS model [40].It should be stressed that the resulting fusion probability values at energies well above the fusion barrier are noticeably smaller than unity in contrast to the previous studies for similar reactions [9,41].This is the result of the normalization of experimental ER cross sections to the calculation with fission barriers obtained in the analysis of very asymmetric reactions.To shed light on the effect of static deformations of fusioning nuclei, one should consider relative positions of the fusion barriers corresponding to 'tip' and 'side' collisions for projectile and target nuclei (see Figure 5).These positions (relatively to the Bass barrier [28]) were calculated according to [21] using the nuclear proximity potential and tabulated deformation parameters [17] for projectile and target nuclei.As one can see from the figure, no evident correlations are observed between the barrier spans corresponding to their minimal and maximal magnitudes and slopes of fusion probability curves in the vicinity of the nominal fusion barrier [28].We see an expected growth of the fusion probability in going from 'tip' to 'side' configurations due to the reduction in the corresponding distance between fusioning nuclei.
One should mention that the similar analysis of reactions leading to Po compound nuclei shows the fusion probabilities approaching the unity at energies well above the fusion barrier [43].In this study we also extracted the macroscopic component of fission barriers in a wide region of neutron numbers, which were compared with predictions of different models and tabulated values.In the same way the macroscopic components of fission barriers for the heavier nuclei from Rn to Th were obtained, but in this case only very asymmetric combinations were considered in the SM analysis.These data are shown in Figure 6 together with the predictions of different models [30,[44][45][46][47] and tabulated values [48] for the macroscopic barriers (bottom panels).Shell corrections calculated in [17] and those obtained in the framework of the present analysis (see Section 2) could be also compared (upper panels).As one can see, the extracted macroscopic fission barriers for neutron-deficient nuclei are lower than any model calculations, with the exception of the predictions given by the LSD model [47].At the same time, the values for relatively neutron-rich Ra and Th nuclei approach to the tabulated ones [48], as observed for Po nuclei [43].

Heaviest nuclei production
In contrast to the description of Po to Th nuclei production, the same SM analysis shows that Fm excitation functions measured in very asymmetric reactions can be reproduced with the scaled LD fission barriers which are about 20% higher than the original ones [51].Applying these LD barriers to the production of Fm isotopes in the 'cold' fusion reaction, one can obtain the averaged fusion probability value about 14% allowing to reproduce the xn evaporation cross sections measured in the 40 Ar+ 208 Pb reaction [52].This value is comparable with P fus =(32±17)% derived from the fission experiment carried out at a higher energy [53].In the case of the Fm production in the more symmetric Fig. 6.Macroscopic fission barriers derived with the SM analysis of the ER and fission cross sections measured in very asymmetric reactions leading to the Rn * , Fr * , Ra * and Th * compound nuclei (different symbols with error bars correspond to the observed CN evaporation chains).Lines correspond to the LD [30], empirical [44], finite-range [45] (FR), Thomas-Fermi [46] (TF) and Lublin-Strasbourg drop [47] (LSD) model calculations performed for these nuclei (bottom panels).Refs to the experimental data obtained in reactions leading to the Rn * , Fr * , Ra * and Th * compound nuclei can be found in [49], [50], [12,33] and [34,49], respectively.Shell corrections obtained in the present SM analysis are compared with the ones tabulated for the finite range droplet (FRD) and finite range liquid drop (FRLD) models [17].The region of near-spherical nuclei corresponding to the deformation parameter |β 2 |≤0.15 [17] is indicated (upper panels).06001-p.8suppression than in the previous case.The 50 Ti+ 198 Pt data [52] can be described with the 1.5% value of the fusion probability together with the additional ΔB=11.3MeV energy to the fusion barrier, which is introduced according to the EEP formalism, similar to the way it was done in the case of 124 Sn+ 96 Zr (see above).Resulting values of the fusion probability for both reactions are shown in Figure 7 that demonstrates essentially different dependences of P fus on energy for 'cold' fusion reactions induced on spherical and deformed target nuclei.
The SM analysis of the No isotopes production in the 12,13 C+ 244,246,248 Cm reactions [54] also shows that the LD fission barriers should be 20% higher than the nominal ones in to describe excitation functions for the production of these nuclei.Using these macroscopic fission barriers for the analysis of the ER and fission cross sections obtained in the 48 Ca+ 208 Pb reaction (see [55] and [56], respectively, and Refs therein), one can come to the similar result as obtained for 40 Ar+ 208 Pb.Namely, the averaged value of P fus ≈18% can be derived with the macroscopic fission barriers obtained in the SM analysis of the xn excitation functions measured for the C+Cm reactions.This value is significantly lower than the P fus values extracted from the fission data [56] and obtained with the theory [57] but is much higher than it follows from the DNS model calculation [58], as shown in Figure 7.The P fus values obtained in the present analysis of 'cold' fusion reactions slightly differ from those estimated earlier in the preliminary study [51].One should also mention that the alternative independent fits to both sets of the data give us significantly smaller LD fission barriers corresponding to k f =0.9 and 0.8 for 40 Ar+ 208 Pb and 48 Ca+ 208 Pb, respectively, [51], i.e., the values that contradict the result of the very asymmetric reactions analysis, which yields k f =1.2.[53,56] and obtained with theoretical models [57,58] are also shown.See the text for details.

CNR*11
06001-p.9 The same LD fission barriers scaling can be obtained for the relatively neutron-rich Rf isotopes produced in very asymmetric combinations with 15 N and 16,18 O (see [51] and Refs therein).Applying this result to the data obtained in less asymmetric combinations with 22 Ne and 26 Mg leading to the same 264 Rf * CN, we get some overestimates for the xn evaporation cross sections in calculations.At the same time, the 26 Mg+ 238 U ER cross sections [59] can be better described using the fusion probability extracted from the fission data [7].The analysis of ratios of the CN-fission to capture cross sections obtained in the 238 U induced reaction on heavier targets [7] shows the decrease in the fusion probability with Z of a target nucleus without any dependence on the energy.The last statement is not fulfilled in the case of the 48 Ca+ 238 U fission data [7,60] for which decrease in the fusion probability is observed even at energies above the nominal fusion barrier [28].In further study one can use the systematics of the P fus values derived from the fission data [6,7] corresponding to energies above the nominal fusion barrier [28].The P fus values demonstrate an exponential dependence on the Coulomb factor as shown in Figure 8.The energy dependence of P fus at barrier and sub-barrier energies reveal the data obtained with ER cross sections measured in the 26 Mg, 30 Si+ 238 U reactions [59,61].These data are shown in Figure 9.They are derived from the ratios of the Σσ xn measured in the evaporation reactions to the calculated ones obtained with the capture cross sections measured in the same reactions.Asymptotic values of P fus corresponding to the fusion probability at energies well above the fusion barrier are estimated with the Fermi-function fits to the data.As one can see in Figure 8 these values do not contradict to the ones derived from fission data.Similar fits were used to estimate the asymptotic values of P fus for the 48 Ca + 238 U and 36 S+ 238 U data obtained in the fission experiments [7,60,63].The value extracted from the data [7] is included into the systematics shown in Figure 8. Similar values extracted from the data [60,63] obtained recently are not included into the systematics, but they are in agreement with it.
The application of this systematics to the analysis of the ER data obtained in the 34 S + 238 U reaction [64] leading to the 272 Hs * CN shows a good agreement of calculations with the meas-Fig.8. Fusion probabilities derived from the ratios of the CN-fission cross sections to those for capture, which obtained in [6,7] for deformed nuclei at energies above the Bass barrier [28] (spheres with error bars).Asymptotic values of P fus derived from Fermi-function fits to the 26 Mg, 30 Si + 238 U data and values corresponding to very asymmetric reactions leading to the 264 Rf * CN (see Refs in [51]) are shown by filled triangles with error bars.The P fus values obtained with the diffusion model [62] for reactions leading to the 274 Hs * CN are also shown by squares.Fig. 9. Fusion probabilities for the 26 Mg, 30 Si + 238 U reactions, which are derived as the ratio of the measured Σσ xn cross sections [59,61] to those calculated as described in the text (symbol with error bars).Fermi-function fits to the data are shown by solid lines.See details in the text.ured ER cross sections, which is achieved using the LD fission barriers without any scaling.At the same time, the description of the data with zero LD fission barriers and fusion probability equalled to unity seems to be also acceptable.The analysis of the Hs isotopes production in the 26 Mg+ 248 Cm and 48 Ca+ 226 Ra reactions [65,66] leading to the same 274 Hs * CN may clarify this situation.Thus, options with the macroscopic barriers equalled to the LD ones and to zero ones do not fit both the data, whereas the option with the half of LD barriers and with the fusion probabilities corresponding to the systematics shown in Figure 8 satisfies both the data as shown in Figure 10.The only exception is the 3n cross section data obtained in the 26 Mg + 248 Cm reaction at deep sub-barrier energies [65].To describe them one has to use a very large value of the fusion barrier fluctuation parameter σ(r 0 )/r 0 (see Section 2) which is not needed in the description of the heaviest nuclei production in the 26 Mg and 30 Si induced reactions on 238 U [59,61].One should mention in this connection that the isotope 271 Hs being the product of the 3n evaporation channel was not observed in our 48 Ca+ 226 Ra experiment [66].Thus, the unexpected enhancement of the 3n evaporation channel in the 26 Mg+ 248 Cm reaction needs further confirmations.
Let us consider a choice of fission barriers in calculations for the heaviest nuclei.As shown in numerous studies as well as in the present analysis, fission barriers are the most indefinite input parameter determining the survivability of the heaviest nuclei in calculations.The simple scaling expressed by Eq. ( 6) was usually used in a number of previous studies [10,12,26,33,43,[49][50][51]55].The necessity of such scaling stems from the fact that model calculations demonstrate very different values of the macroscopic component of fission barriers as shown in Figure 6.The same is at the end of nuclear stability for the heaviest nuclei.At the same time, this component disappears in going from Hs to Cn nuclei according to the model calculations [30,[44][45][46][47].So fission barriers for the elements heavier than Cn are determined by the shell corrections only or by the difference between the empirical [31] and LD (macroscopic) masses [32].The comparison of the calculated Finite Range Droplet (FRD), Finite Range Liquid Drop (FRLD) [17], Thomas-Fermi (TF) [67] and Hartree-Fock-Bogoliubov (HFB-2) [68] masses, with the empirical ones (AWT) [31] for isotopes of the elements with Z=110 to 118 shows that the FRLD masses are the best ones from the point of view of their agreement with the empirical ones, whereas the FRD masses announced as the best ones in [17] are systematically shifted to the lower (M-A)-values and seem to be the worst ones.The values of TF and HFB-2 masses are somewhere between the predictions of the FRD and FRLD models.A quantitative analysis confirms this statement.The sums over the number of isotopes n tabulated in [31] for root mean square deviations of the calculated masses from the empirical ones have the smallest values for the FRLD masses [17] for the elements with Z=110-118, as one can see in Figure 11.Thus, one can recommend using the FRLD masses for further calculations.
In Figure 12 the measured cross sections for isotopes of the elements of Cn and 114 produced in fusion-evaporation reactions induced by 48 Ca [69] are shown and compared with the excitation functions calculated using the FRLD masses and the systematics of fusion probability presented in Figure 8.The EEP formalism (see above) was also applied to reproduce the measured cross sections.Similar comparison was also made for the production of isotopes of the elements of 116 and 118 in the reactions induced by 48 Ca [70].These comparisons show that production of the heaviest nuclei close to the region of superheavy elements could be quite satisfactory reproduced in the framework of the SM calculations using the FRLD masses for the estimates of fission barrier heights (B f =ΔW gs ) for nuclei involved in an evaporation-fission cascade.These input quantities determine mainly the calculated survivability for the heaviest nuclei produced in fusionevaporation reactions.
Another point is the fusion suppression that imposes a constraint on the CN formation in the entrance reaction channel.This constraint could be reproduced in calculations by the corresponding function of the fusion probability.The asymptotic values of this functions (fusion probabilities at energies well above the fusion barrier) could be determined with the empirical systematics shown in Figure 8.Besides this limitation the width of the fusion barrier distribution σ(r 0 )/r 0 (see Section 2) should be significantly decreased to reproduce the data shown in Figure 12.In fact, σ(r 0 )/r 0 is the only parameter varied within (0-1.5)% to describe the sub-barrier 2n and 3n evaporation cross sections.These values are much lower than the ones that are inherent in reactions with massive and deformed reactants leading to the nuclei near the N=126 shell and show σ(r 0 )/r 0 =(4-5)%, as indicated in Figure 4 and in Refs.[9,10,12,33,36,43,[49][50][51].Fusion suppression at barrier and sub-barrier energies could be also reproduced in the framework of the extra-extra-push formalism using parameter values obtained in [7].In Figure 12 this variant of calculations demonstrates a similar agreement with the experimental data.This set of parameter values give a corresponding addition to the fusion barrier of reactions ΔB indicated in the figure.Such increase in the height of the fusion barrier is accompanied by some increase in the width Fig. 11.The sum of root mean square deviations of the calculated masses from empirical ones [31] for the FRD, FRLD [17], TF [67] and HFB-2 [68] masses for isotopes of elements of Z=110-118.See designations in the text.
of the fusion barrier distribution corresponding to σ(r 0 )/r 0 =(1.5-3)%, which should be used to describe the data (see Figure 12).

Summary and conclusion
Being critical and correlating values in the estimates of evaporation residue cross sections, fusion probability and survivability in fusion-evaporation reactions leading to heavy and heaviest nuclei were considered using the potential barrier-passing model for capture of nuclei and statistical model (SM) for the compound nucleus de-excitation.
The survivability of heavy nuclei produced in the vicinity of a 126 neutron shell in very asymmetric projectile-target combinations can be reproduced with reduced values of the liquid-drop (LD) component of fission barriers in the framework of SM.These barriers can be used for the empirical estimates of the fusion probability for more symmetric reactions.
Production of heavy nuclei from Fm to Rf in very asymmetric 'hot' fusion reactions can be reproduced with the 20% increase in the LD component of fission barriers with the use of a similar SM approach.The corresponding estimates of the fusion probability for more symmetric 'cold' fusion reactions differ from those predicted by theories and obtained in fission experiments.
Production of the heaviest nuclei with Z>104 in 'hot' fusion reactions induced by massive projectiles can be reproduced with the fusion probabilities obtained from fission data and with some reduction in the LD component of fission barriers (for Hs nuclei).Data analysis implies a disappearance of the macroscopic (LD) component for nuclei with Z>108.

EPJ Web of Conferences
Estimates of atomic masses and shell corrections are mainly responsible for the description of the heaviest nuclei production in reactions induced by 48 Ca and heavier projectiles, bearing in mind an absence of the macroscopic component in the fission barriers for these nuclei.

Fig. 3 .
Fig. 3. Comparison of the reduced ER cross sections obtained for the 16 O+ 186 W and 48 Ca+ 154 Sm reactions leading to the same 202 Pb * CN [24, 33].Experimental values and calculations are shown by different symbols and lines, correspondingly.See the text for further details.

Fig. 4 .
Fig.4.Fission (capture) and ER cross sections obtained in the16 O+ 204 Pb[34],40 Ar+ 180 Hf[36,36] and 124 Sn+ 96 Zr[9,37] reactions leading to the same 220 Th * CN (symbols).Calculations without any fusion suppression and with fission barriers obtained from the16 O+ 204 Pb data fit are shown by dashed lines for the40 Ar and 124 Sn induced reactions, whereas similar calculations with the adjusted P fus values are shown by solid lines for ER cross sections.Alternative fits corresponding to P fus =1 and smaller fission barriers (k f values) are shown by dash-dotted lines.See the text for further details.

Fig. 5 .
Fig. 5. Fusion probabilities obtained as the ratio of the experimental ER cross sections Σσ xn obtained in [36] ( 40 Ar), [39] ( 32 S, 60,64 Ni), [42] ( 70 Zn) and [9] ( 124 Sn) to the calculated ones for reactions leading to the 220,218,214 Th * compound nuclei (symbols with error bars).Solid lines correspond to the calculations based on the fit of barrier-passing cross sections to the capture ones measured in [36] ( 40 Ar+ 180,178 Hf), [37] ( 124 Sn+ 96 Zr) and [38, 39] ( 32 S+ 182 W), whereas dashed lines reproduce the calculations based on the assumed capture cross sections.For the latter the same parameter values are used as obtained for similar reactions at the fit of the calculated barrier-passing cross section.

Fig. 7 .
Fig. 7. Fusion probabilities derived as the ratio of the experimental ER cross sections Σσ xn obtained in the 40 Ar+ 208 Pb, 50 Ti+ 198 Pt and 48 Ca+ 208 Pb reactions [52, 55 and Refs therein] to the calculated ones (symbols with error bars).Solid line corresponds to the averaged P fus value for 40 Ar+ 208 Pb and 48 Ca+ 208 Pb, whereas a dashed line in the left panel corresponds to the calculation fitted to the measured 50 Ti+ 198 Pt ER cross sections.P fus values extracted from fission experiments[53,56] and obtained with theoretical models[57,58] are also shown.See the text for details.

Fig. 10 .
Fig. 10.ER cross sections obtained in the 26 Mg+ 248 Cm [65] and 48 Ca+ 226 Ra [66] reactions leading to the same 274 Hs * CN (symbols).Calculations with fusion suppression (the P fus values are taken from the systematics presented in Figure 8) and without it along with a different scaling for the LD fission barriers are shown by different lines for both the reactions.See the text for details.

Fig. 12 .
Fig. 12.The xn evaporation cross sections obtained in the 48 Ca+ 238 U and 48 Ca+ 242,244 Pu [69] reactions leading to the 286 Cn * and 290,292 114 compound nuclei (symbols).Calculations with the P fus values taken from the systematics presented in Figure8and with the EEP formalism[7] are shown by dashed and solid lines, respectively.In both variants of calculations the FRLD masses[17] were used for the estimates of fission barrier heights.See the text for details.