To be published in Chinese Physics C.

#  Production of super-heavy nuclei in cold fusion reactions

V. Yu<EMAIL_ADDRESS>I. Yu. Sedykh3 1 Institute for
Nuclear Research, Prospect Nauki 47, 03028 Kiev, Ukraine
2 Faculty of Physics, Taras Shevchenko National University of Kiev, Prospect
Glushkova 2, 03022 Kiev, Ukraine
3 Financial University under the Government of the Russian Federation,
Leningradsky Prospekt 49, 125993 Moscow, Russian Federation

###### Abstract

The model for the cold-fusion reactions related to the synthesis of super-
heavy nuclei in collisions of heavy projectile-nuclei with 208Pb target
nucleus is discussed. In the framework of this model the production of the
compound nucleus by two paths through, the di-nuclear system and the fusion
way, are taken into account simultaneously. The formation of the compound
nucleus in the framework of the di-nuclear system is related to the transfer
of nucleons from the light nucleus to the heavy one. The fusion way is linked
to the sequential evolution of the nuclear shape from the system of contacting
nuclei to the compound nucleus. It is shown that the compound nucleus is
mainly formed by the fusion way in the cold-fusion reactions. The landscape of
the potential energy related to the fusion path is discussed in detail. This
landscape for very heavy nucleus-nucleus systems has the intermediate state,
which is linked to the formation of both the compound nucleus and the quasi-
fission fragments. The decay of the intermediate state is taken into account
in the calculation of the compound nucleus production cross sections and the
quasi-fission cross sections. The values of the cold-fusion cross sections
obtained in the model are well agreed with the experimental data.

##  I introduction

Synthesis of Super-Heavy nuclei (SHN) is very interesting, exciting and
puzzling physical task as for experimentalists as for theoreticians. The
elements beyond Md with the number of protons $Z=102-118$ were synthesized by
the fusion of heavy nuclei h ; h-b ; h-c ; 102jinr ; 102jinr1 ; 104gsi ;
104gsi-a ; 104lbnl ; 106gsi ; 106lbnl ; 108gsi ; 108riken ; 110riken ; 110lbnl
; 112gsi ; 112riken ; 112riken1 ; folden ; m ; m1 ; m3 ; og ; og1 ; 118 ; 118a
; sh ; dp . The element Og with $Z=118$ is the heaviest element, which has
been synthesized to now by Oganessian et al. og ; 118 . Recently, the
experiments aimed at the synthesis of isotopes of element Z = 119 and 120 or
study the properties of related reactions have been performed 119 ; 120 ; 120a
; 120b ; 120c ; 120d ; 120e , but no decay chains consistent with the fusion-
evaporation reaction products were observed.

The cold-fusion reactions for SHN synthesis are the reactions between heavy-
ion projectiles with the mass/charge $A\geq 48/Z\geq 20$ and lead or bismuth
targets. This type of reaction has been proposed by Oganessian et al. cfr .
Using these reactions the SHN with charges Z=102$\div$113 have been
successfully synthesized in experiments h ; h-b ; h-c ; 102jinr ; 102jinr1 ;
104gsi ; 104gsi-a ; 104lbnl ; 106gsi ; 106lbnl ; 108gsi ; 108riken ; 110riken
; 110lbnl ; 112gsi ; 112riken ; 112riken1 ; folden ; m ; m1 ; m3 .

Many different models have been proposed for a description of the cross
section of the synthesis of the SHN in heavy-ion collisions, see, for example,
Refs. dp ; sw ; sw1 ; sw2 ; sw3 ; sww2 ; sww3 ; sww4 ; dh ; d-sym ; zg ; jap1
; jap4 ; shen1 ; kos2 ; kos3 ; boilley ; abe3 ; liang1 ; umar ; umar1 ; umar2
; torres2 ; torres ; torres1 ; v ; dns1 ; dns1c ; dns1aa ; dns1aa1 ; dns1aa2 ;
dns1aa4 ; dns2a ; adamian ; dns3 ; dns31 ; dns32 ; dns33 ; dns3a ; dns4b ;
dns4c ; dns4d ; dns5 ; dns444 ; dns4444 ; param ; param1c ; param1d ; param2c
; param3 ; param4a ; param4b ; param5 ; param6 ; param7  and papers cited
therein. The common feature of the models is that the cross section of the SHN
production is described as the product of the capture cross section, the
probability of compound nucleus formation, and the survival probability of the
compound nucleus. The capture process is related to the formation of the
system of touching nuclei. The probability of compound nucleus formation is
connected to the process of the evolution from the system of contacting nuclei
to the spherical or near spherical compound nucleus. The survival probability
of compound nucleus is linked to the competition between the fission and
neutron emission processes. When the excitation energy of a compound nucleus
drops down the residue nuclei emits alpha-particles and/or divided on two
fission fragments, which are detected in experiments. The chain of alpha-
particles and/or the fission fragments observed in the experiment are the
experimental signals of the successful synthesis of the SHN h ; h-b ; h-c ;
102jinr ; 102jinr1 ; 104gsi ; 104gsi-a ; 104lbnl ; 106gsi ; 106lbnl ; 108gsi ;
108riken ; 110riken ; 110lbnl ; 112gsi ; 112riken ; 112riken1 ; folden ; m ;
m1 ; m3 ; og ; og1 ; 118 ; 118a ; sh ; dp .

The differences between various models sw ; sw1 ; sw2 ; sw3 ; sww2 ; sww3 ;
sww4 ; dh ; d-sym ; zg ; jap1 ; jap4 ; shen1 ; kos2 ; kos3 ; boilley ; abe3 ;
liang1 ; umar ; umar1 ; umar2 ; torres1 ; torres2 ; torres ; v ; dns1 ; dns1c
; dns1aa ; dns1aa1 ; dns1aa2 ; dns1aa4 ; dns2a ; adamian ; dns3 ; dns31 ;
dns32 ; dns33 ; dns3a ; dns4b ; dns4c ; dns4d ; dns5 ; dns444 ; dns4444 ;
param ; param1c ; param1d ; param2c ; param3 ; param4a ; param4b ; param5 ;
param6 ; param7  at the description of the capture process are related to
using of the different nucleus-nucleus interaction potentials, consideration
of different shapes and mutual orientations of interacting nuclei, and
applications of various approximations at the evaluation of the capture cross
sections. The differences of various models at the description of the survival
probability are connected to applying of different statistical nuclear decay
models, various expressions for the energy level densities, different values
of the fission barrier of the SHN obtained in the frameworks of various
nuclear structure models, various dependencies of the fission barrier on the
excitation energy of the compound nucleus, different values of the neutron
binding energy taken from various nuclear mass models. Consequently, the
corresponding differences between the results obtained in the frameworks of
various approaches to the capture cross-section and the survival probability
of the compound nucleus are natural, reasonable and understandable. The
physical mechanisms of the capture cross-section and the survival probability
of the compound nucleus are the same in various models.

The process of the compound nucleus formation from the two touching nuclei
during the synthesis of SHN is most undefined up to now, because there are the
two alternative mechanisms of this process, the fusing sw ; sw1 ; sw2 ; sw3 ;
sww2 ; sww3 ; sww4 ; dh ; d-sym ; zg ; jap1 ; jap4 ; shen1 ; kos2 ; kos3 ;
boilley ; abe3 ; liang1 ; torres2 ; torres ; umar ; umar1 ; umar2  and Di-
Nuclear System (DNS) v ; dns1 ; dns1c ; dns1aa ; dns1aa1 ; dns1aa2 ; dns1aa4 ;
dns2a ; dns3 ; dns31 ; dns32 ; dns33 ; dns3a ; dns4b ; dns4c ; dns4d ; adamian
; dns5 ; dns444 ; dns444 , which are actively discussing.

The fusing approach to the compound nucleus formation is related to the smooth
subsequent shape evolution from the system of two touching nuclei to the
compound nucleus, see Refs. sw ; sw1 ; sw2 ; sw3 ; sww2 ; sww3 ; sww4 ; dh ;
d-sym ; zg ; jap1 ; jap4 ; shen1 ; kos2 ; kos3 ; boilley ; abe3 ; liang1 ;
umar ; umar1 ; umar2 ; torres ; torres2  and papers cited therein. The models
considered in Refs. sw ; sw1 ; sw2 ; sw3 ; sww2 ; sww3 ; sww4 ; dh ; d-sym ;
zg ; jap1 ; jap4 ; shen1 ; kos2 ; kos3 ; boilley ; abe3 ; liang1 ; umar ;
umar1 ; umar2 ; torres ; torres2  are based on different approaches to the
shape evolution and/or the shape parametrization of the nuclear system during
the compound nucleus formation. The compound nucleus formation is formed in a
competition to the quasi-fission and deep-inelastic processes in the framework
of the fusing approach.

The alternative mechanism of the compound nucleus formation from the two
colliding nuclei is proposed by Volkov in Ref. v . This mechanism is related
to the evolution of DNS formed by the two contacting nuclei after penetration
through the fusion barrier of the incident nuclei. In the framework of the DNS
model the compound nucleus is formed by the multi-nucleon transfer from the
light nucleus to the heavy one. The transfer of nucleons in the opposite
direction (from heavy nucleus to the light one) as well as the decay of the
DNS through the fusion barrier are the processes competing to the compound
nucleus formation. The both nuclei touch each other during these multi-nucleon
transfers. This DNS mechanism of the compound nucleus formation is applied to
the SHN production in Refs. v ; dns1 ; dns1c ; dns1aa ; dns1aa1 ; dns1aa2 ;
dns1aa4 ; dns2a ; dns3 ; dns31 ; dns32 ; dns33 ; dns3a ; dns4b ; dns4c ; dns4d
; adamian ; dns5 ; dns444 ; dns4444 , see also papers cited therein.

Note, the same experimental of values of the production cross section of SHN
are described by applying various approaches for the compound nucleus
formation. However, the fusion and DNS mechanisms of compound nucleus
formation are very different. It should be also noted that the probability of
the compound nucleus formation is also evaluated by various phenomenological
or semi-phenomenological expressions sw ; sw1 ; sw2 ; sww2 ; sww3 ; param ;
param1c ; param1d ; param2c ; param3 ; param4a ; param4b ; param5 ; param6 ;
param7 .

For the first time the DNS approach is successfully applied to a description
of the deep inelastic heavy-ion collisions v2 . In this case, two nuclei
collide at high energies and form a fast-rotating DNS. The fast rotation of
DNS prevents the fusion of two nuclei and stabilize the rotating DNS, because
the potential of rotating DNS shows the repulsion at small distances between
nuclei due to the contribution of the centrifugal force. Therefore, the multi-
nucleon exchange between the fast-rotating nuclei forming the DNS is naturally
taken place during the deep inelastic heavy-ion collisions. After rotation on
some angle the DNS decays on two excited nuclei, which have other nucleon
compositions and smaller values of the relative kinetic energies than the
incident nuclei v2 .

The cold fusion reactions are taking place at the collision energies near the
barrier of the nucleus-nucleus potential dn ; dnest . Due to this the possible
frequency of rotation of the DNS formed in the SHN formation reaction is much
smaller than the one in the case of the deep inelastic heavy-ion collisions.
Therefore, the stabilization of DNS due to rotation is impossible for heavy-
ion systems leading to the SHN. Another possibility of the of the formation of
the barrier at small distances between ions is a diabatic behaviour of nuclear
levels during fast collisions blnr . The diabatic shift of heavy-ion potential
energy occurs when the relative velocity of heavy ions is very high, therefore
nucleons occupy diabatic levels and cannot quickly relax in time to the
adiabatic ones. However, the relative velocity of ions is disappeared during
the penetration of the fusion barrier and the evolution of dinuclear system is
related to small relative velocities. The potential energy surfaces obtained
in the framework of various microscopic or semimicroscopic calculations have
not show the barrier for reactions related to the SHN synthesis zg ; jap1 ;
jap4 ; shen1 ; kos2 ; kos3 ; boilley ; abe3 ; umar ; umar1 ; umar2 .
Nevertheless, the DNS potential calculated in Refs. dns1 ; dns1c ; dns1aa ;
dns1aa1 ; dns1aa2 ; dns1aa4 ; dns2a ; adamian ; dns3 ; dns31 ; dns32 ; dns33 ;
dns3a ; dns4b ; dns4c ; dns4d ; dns5 ; dns444 ; dns4444  shows the strong
repulsion at small distances between nuclei, which stabilizes the DNS system
in this model. Such behaviour of the DNS potential in Refs. dns1 ; dns1c ;
dns1aa ; dns1aa1 ; dns1aa2 ; dns1aa4 ; dns2a ; adamian ; dns3 ; dns31 ; dns32
; dns33 ; dns3a ; dns4b ; dns4c ; dns4d ; dns5 ; dns444 ; dns4444  relates to
the calculation of the nucleus-nucleus potential in the frozen-density
approach. The frozen-density nucleus-nucleus potential is strongly repulsive
at small distances between nuclei, because the frozen nucleon densities of the
colliding nuclei overlap well at such distances and form a high-density region
dn ; dnest . Due to the high value of the incompressibility of nuclear matter
the potential energy of nucleus-nucleus system drastically rises, when the
nucleon density became higher than the equilibrium density of nuclear matter
dn ; dnest .

The density distribution of nucleons in colliding nuclei can relax during
collisions at small relative velocities of nuclei, which are taken place at
the collision energies close the barrier. Therefore, the high nucleon density
region with the density noticeably higher than the equilibrium density of
nuclear matter is not formed during heavy-ion collision leading to the SHN. As
a result, the realistic nucleus-nucleus potential does not show strong
repulsion at small distances between two nuclei. The behaviour of the
potential at such distances is related to the sequential evolution of the
shape of nuclear system. Therefore, the nuclei can fuse and the both
mechanisms of compound nucleus formation, the fusion and the DNS, should be
simultaneously taken into account in the calculation of the SHN production in
heavy-ion collisions. The both mechanisms of the compound nucleus formation
are shortly considered in Ref. torres1 . Below we discuss new model for SHN
formation, which includes the both mechanisms of the compound nucleus
formation and also the decay of primary-formed excited compound nucleus
related to the neutron evaporation in competition with fission. We propose new
shape parametrization for a description the fusion path, obtain the cross
section of SHN in cold fusion reactions, and compare the calculated cross
section values with the experimental data.

Below we present the new model for the description of the cross section of the
SHN synthesis in heavy-ion collisions, which takes into account simultaneously
both the fusion and DNS mechanisms of the compound nucleus formation. The
mechanisms of the nucleus-nucleus capture and the survival of the compound
nucleus applied in our model are close to tradition the ones, but we introduce
some new features at the consideration of the capture and survival stages of
SHN synthesis. The detail description of our model is presented in Sec. II.
The discussion of results obtained in our model for the cold fusion reactions
is given in Sec. III. Conclusions are drawn in Sec. IV.

##  II The model

The cold fusion reactions of the SHN synthesis are related to the collisions
of the projectiles nuclei 48Ca, 50Ti, 52,54Cr, 58Fe, 59Co, 64Ni, 65Cu, and
70Zn with the spherical target nuclei 208Pb and 209Bi h ; 102jinr ; 102jinr1 ;
104gsi ; 104gsi-a ; 104lbnl ; 106gsi ; 106lbnl ; 108gsi ; 108riken ; 110riken
; 110lbnl ; 112gsi ; 112riken ; 112riken1 ; folden ; m . The nuclei 48Ca,
50Ti, and 70Zn are spherical in the ground state, while the ground states of
52,54Cr, 58Fe, 59Co, 64Ni, and 65Cu are weakly-deformed msis . It is well-
known that the nucleus-nucleus interaction potential depends on the nucleon
density distributions and deformations of the interacting nuclei dp ; dn ;
dnest ; dpil ; dpil2 ; dms ; ds17 ; dps . The deformation of heavy nucleus
effects stronger on the nucleus-nucleus potential than the deformation of the
light nuclei in the case of asymmetric interacting system. Due to both the
very small deformations of projectile nuclei and the weak effect of the
deformation of the light nucleus on the interaction potential we may neglect
by the small deformations of the projectile nuclei in the case of the cold
fusion reactions. Therefore, we can consider that the spherical nuclei are
participating in the cold fusion reactions.

The cross section of the SHN synthesis in collisions of spherical nuclei with
the subsequent emission of $x$ neutrons from the formed compound nucleus in
competition with the fission is given as

$\displaystyle\sigma^{xn}(E)=\frac{\pi\hbar^{2}}{2\mu
E}\sum_{\ell}(2\ell+1)T(E,\ell)P(E,\ell)W^{xn}(E,\ell).$ (1)

Here $\mu$ and $E$ are the reduced mass and the collision energy of incident
nuclei in the center of the mass system, respectively. $T(E,\ell)$ is the
transmission coefficient through the fusion barrier formed by the Coulomb,
centrifugal, and nuclear parts of the nucleus-nucleus interaction, $P(E,\ell)$
is the probability of compound nucleus formation, and $W^{xn}(E,\ell)$ is the
survival probability of the compound nucleus related to the evaporation of $x$
neutrons in a competition with the fission. In a case of the cold fusion
reactions $x$ equals $1$ or $2$ and rarely $3$ or $4$. The next subsections
are devoted to detail description of our approaches to the calculations of
$T(E,\ell)$, $P(E,\ell)$, and $W^{xn}(E,\ell)$, respectively.

###  II.1 The transmission through the fusion barrier

The total potential between spherical nuclei with the numbers of proton
$Z_{1}$ and $Z_{2}$ is

$\displaystyle V_{\ell}(r)=\frac{Z_{1}Z_{2}e^{2}}{r}+V_{\rm N}^{\rm
sph}(r)+\frac{\hbar^{2}\ell(\ell+1)}{2\mu r^{2}}.$ (2)

Here $r$ is the distance between centers of mass of nuclei, $e$ is the charge
of proton, $V_{\rm N}^{\rm sph}(r)$ is the nuclear part of the nucleus-nucleus
potential, and $\ell$ is the value of orbital angular momentum in $\hbar$
units.

The total interaction potential energy of two spherical nuclei can be
approximated around the barrier by the parabola. The transmission coefficient
through a parabolic barrier hw  is known exactly and given by

$\displaystyle T_{\rm par}(E,B^{\rm
fus}_{\ell},\ell)=1/\left[1+\exp{\left(\frac{-2\pi(E-B^{\rm
fus}_{\ell})}{\hbar\omega_{\ell}}\right)}\right],$ (3)

where $B_{\ell}^{\rm fus}=B_{0}^{\rm fus}+\frac{\hbar^{2}\ell(\ell+1)}{2\mu
r_{\ell}^{2}}$ is the barrier height of the potential, $r_{\ell}$ is the
barrier radius, and
$\hbar\omega_{\ell}=\left.\left[-\frac{\hbar^{2}}{\mu}\frac{d^{2}V^{\rm
fus}_{\ell}(r)}{dr^{2}}\right]^{1/2}\right|_{r=r_{\ell}}$ is the curvature of
the barrier.

The fusion barrier distribution simulated the realistic multichannel coupling
is often taken into account at an evaluation of the sub-barrier heavy-ion
fusion cross sections dhrs ; motstef  and SHN formations zg ; dns4b ; dns4c ;
dns4d ; dns5 . In this case the total transmission coefficient is given as

$\displaystyle T(E,\ell)=\int_{B_{1}}^{B_{2}}dB\;T_{\rm
par}(E,B,\ell)\;f(B,B^{\rm fus}_{\ell}),$ (4)

where $f(B,B^{\rm
fus}_{\ell})=\frac{1}{g\sqrt{\pi}}\exp{\left[-\left(\frac{B-B^{\rm
fus}_{\ell}}{g}\right)^{2}\right]}$ is the barrier distribution function,
which usually approximates by the Gauss function zg ; dns4b ; dns4c ; dns4d .
The typical values of the barrier distribution width $g$ are several MeVs zg ;
dns4b ; dns4c ; dns4d .

In the case of sub-barrier energies $E\ll B^{\rm fus}_{\ell}$ Eq. (3) can be
approximated in the form $T_{\rm par}(E,B^{\rm
fus}_{\ell},\ell)\approx\exp{\left(\frac{2\pi(E-B^{\rm
fus}_{\ell})}{\hbar\omega_{\ell}}\right)}.$ Substituting this expression into
Eq. (4) and extend the limits of the integral to infinity we get
$T(E,\ell)\approx\exp{\left(\frac{2\pi(E-(B^{\rm
fus}_{\ell}-\Delta_{B}))}{\hbar\omega_{\ell}}\right)},$ where $\Delta_{B}=\pi
g^{2}/(2\hbar\omega_{\ell})$ is the the shift of the barrier value due to the
barrier distribution. Using this property, we approximate the transmission
coefficient through the distribution of the parabolic barriers for any values
of $E$ as

$\displaystyle T(E,\ell)\approx 1/\left[1+\exp{\left(\frac{-2\pi(E-(B^{\rm
fus}_{\ell}-\Delta_{B}))}{\hbar\omega_{\ell}}\right)}\right].$ (5)

It is obvious that the values of $T(E,\ell)$ obtained using this formula at
energies far from the barrier value are the same as the calculated with the
help of Eqs. (3)-(4). The values of $T(E,\ell)$ calculated by the approximate
formula (5) deviate from the exact values using Eqs. (3)-(4) at energies
around the barrier. This deviation decreases with a decreasing $g$. The
application of the parameter $\Delta_{B}$ is simpler than a numerical
integration of the barrier distribution (4) and leads to the same effect.

We should know the nucleus-nucleus potential for evaluating the transmission
coefficient using Eq. (5). The nuclear part of the nucleus-nucleus potential
consists of the macroscopic and the shell-correction contributions d2015

$\displaystyle V_{\rm N}^{\rm sph}(r)=V_{\rm macro}(r)+V_{\rm sh}(r).$ (6)

The macroscopic part $V_{\rm macro}(r)$ of the nuclear interaction of nuclei
is related to the macroscopic density distribution and the nucleon-nucleon
interactions of colliding nuclei. It is the Woods-Saxon form at $r>R_{\rm t}$
d2015

$\displaystyle V_{\rm macro}(r)=\frac{v_{1}C+v_{2}C^{1/2}}{1+\exp[(r-R_{\rm
t})/(d_{1}+d_{2}/C)]}.$ (7)

Here $v_{1}=-27.190$ MeV fm-1, $v_{2}=-0.93009$ MeV fm-1/2, $d_{1}=0.78122$
fm, $d_{2}=-0.20535$ fm2, $C=R_{1}R_{2}/R_{\rm t}$ is in fm, $R_{\rm
t}=R_{1}+R_{2}$, $R_{i}=1.2536A_{i}^{1/3}-0.80012A_{i}^{-1/3}-0.0021444/A_{i}$
is the radius of $i$-th nucleus in fm, $i=1,2$, and $A_{i}$ is the number of
nucleons in nucleus $i$.

The shell-correction contribution $V_{\rm sh}(r)$ to the potential is related
to the shell structure of nuclei, which is disturbed by the nucleon-nucleon
interactions of colliding nuclei. When the nuclei approach each other, the
energies of the single-particle nucleon levels of each nucleus are shifted and
split due to the interaction of nucleons belonging to different nuclei. This
changes the shell structures of both nuclei at small distances between them.
Therefore, the shell-correction contribution into the total nuclear
interaction of nuclei is introduced in Ref. d2015 . This representation of the
total nuclear potential energy of two nuclei is similar to the Strutinsky
shell-correction prescription for the nuclear binding energy s ; s1 ; s2 ; s3
. The shell-correction part of the potential at $r>R_{\rm t}$ is given as
d2015

$\displaystyle V_{\rm sh}(r)=[\delta E_{1}+\delta
E_{2}]\left[\frac{1}{1+\exp{\left(\frac{R_{\rm sh}-R}{d_{\rm
sh}}\right)}}-1\right],$ (8)

where $R_{\rm sh}=R_{\rm t}-0.26$ fm, $d_{\rm sh}=0.233$ fm, and

$\displaystyle\delta E_{i}=B^{\rm exp}_{i}-B^{\rm m}_{i}$ (9)

is the phenomenological shell correction for nucleus $i$.

$\displaystyle B^{\rm m}_{i}$ $\displaystyle=$
$\displaystyle-15.86864A_{i}+21.18164A^{2/3}_{i}-6.49923A^{1/3}_{i}$
$\displaystyle+$
$\displaystyle\left[\frac{N_{i}-Z_{i}}{A_{i}}\right]^{2}\left[26.37269A_{i}-23.80118A^{2/3}_{i}\right.$
$\displaystyle-$ $\displaystyle\left.8.62322A^{1/3}_{i}\right]$
$\displaystyle+$
$\displaystyle\frac{Z^{2}_{i}}{A^{1/3}_{i}}\left[0.78068-0.63678A^{-1/3}_{i}\right]+P_{p}+P_{n}$

is the macroscopical value of the binding energy in MeV founded in
phenomenological approach, $B_{\rm exp}$ is the binding energy of nucleus in
MeV obtained using the evaluated atomic masses be , $P_{p(n)}$ are the proton
(neutron) pairing terms, which equal to
$P_{p(n)}=5.62922(4.99342)A^{-1/3}_{i}$ in the case of odd $Z$ ($N$) and
$P_{p(n)}=0$ in the case of even $Z_{i}$ ($N_{i}$), $N_{i}$ is the number of
neutrons in nucleus $i$.

The parametrization of $V_{\rm N}^{\rm sph}(r_{\ell})$ from Ref. d2015  is
used in our model, because the barrier heights $B^{\rm fus}_{0}$ calculated
with the help of this parametrization agree well with the empirical values of
barrier heights for light, medium and heavy nucleus-nucleus systems d2015 ;
bass ; empbar1 ; empbar2a . The values of the barriers for the spherical
systems 48Ca, 48,50Ti, 52Cr, 54Cr, 56,58Fe, 64Ni, 70Zn + 208Pb leading to the
SHN obtained in our approach are presented in Table 1. These values of the
barriers well agree with the available values of the barriers derived from an
analysis of the experimental data for the quasi-elastic backscattering
mitsuoka , see Table 1. Note, this parametrization is also successfully used
for the description of the fragment mass distribution at the fission of
highly-excited nuclei dms ; ds17  and the ternary fission dps . So, the values
barrier heights obtained in our approach are reliable.

Using Eqs. (2), (5)-(10) we can evaluate the capture cross section

$\displaystyle\sigma_{\rm cap}(E)=\frac{\pi\hbar^{2}}{2\mu
E}\sum_{\ell}(2l+1)T(E,\ell).$ (11)

The capture cross section is related to the fusion barrier penetration. The
capture cross section coincides with the compound nucleus production cross
section in the case of collisions of light and medium nuclei, when the decay
of DNS on fragments and the quasi-fission process give negligible
contributions dhrs ; motstef ; den-subfus ; ds2019 ; empbar2a . In the case of
collisions of heavy nuclei, the capture cross section links to the formation
of the DNS.

Table 1:  The values of barrier heights between spherical nuclei $B^{\rm fus}_{\ell}$ for $\ell=0$, Q values of the compound nucleus formation $Q_{\rm CN}$ obtained using the evaluated atomic masses be , the excitation energies of the compound nucleus at the collision energies equal to the barrier heights $E_{\rm bar}^{*}=E+Q_{\rm CN}$, and the available values of the barrier heights $B_{\rm qebs}$ derived from an analysis of the experimental data for the quasi-elastic backscattering mitsuoka . All values are given in MeV. Collision systems | $B^{\rm fus}_{0}$ | $-Q_{\rm CN}$ | $E_{\rm bar}^{*}$ | $B_{\rm qebs}$
---|---|---|---|---
48Ca + 208Pb | 172.5 | 153.8 | 18.7 |
48Ti + 208Pb | 191.0 | 164.5 | 26.5 | 190.1
50Ti + 208Pb | 189.8 | 169.5 | 20.3 |
52Cr + 208Pb | 207.2 | 183.7 | 23.5 |
54Cr + 208Pb | 206.0 | 187.1 | 19.0 | 205.8
56Fe + 208Pb | 223.2 | 201.9 | 21.3 | 223.0
58Fe + 208Pb | 222.0 | 205.0 | 17.0 |
64Ni + 208Pb | 236.6 | 224.9 | 11.9 | 236.0
70Zn + 208Pb | 251.1 | 244.2 | 6.9 | 250.6

###  II.2 The probability of compound nucleus formation

####  II.2.1 Expression for the probability of compound nucleus formation

The height of the fusion barrier between incident spherical nuclei is high.
The collision energy of the two nuclei starts to dissipate just before the
barrier. After passing the incident fusion barrier the nuclei locate in the
capture well, which is close to the contact distance of nuclei. The nuclei
form the DNS in the capture well. The DNS formed in capture well is the
injection point for subsequent stages of the SHN formation and the DNS
evolution.

The kinetic energy related to the relative motion of the colliding nuclei is
quickly dissipated at the initial collision stage, when the tails of nucleon
densities of the nuclei is starting to overlap. As a result, the kinetic
energy of the relative motion of nuclei transfers into the intrinsic energy of
the DNS v2 ; fl . Therefore, the subsequent stages of the SHN synthesis can be
considered in the framework of the statistical approach. In this approach the
probability of compound nucleus formation is linked to the ratio of the decay
widths of different processes. The widths considered in this subsection are
defined in the framework of the Bohr-Wheeler transition state statistical
approach bw .

The DNS formed in capture well may decay into different channels as, for
example, the spherical or deformed incident nuclei, new DNSs formed at the
transfer of nucleons between the incident nuclei, and formation of the
compound nucleus. Due to the transfer of nucleons between nuclei the DNS may
decay into the more symmetric nucleus-nucleus systems with subsequent decay on
the deformed nuclei or the more asymmetric nucleus-nucleus systems with
subsequent formation of the compound nucleus. The DNS may decay into the
compound nucleus by the smooth shape evolution too. The corresponding decay
branches of the DNS are linked to the respective decay widths and barriers.

The probability of specific decay process is related to the passing through
the barrier in competition to other processes. The compound nucleus is formed
by passing the barrier of the fusion way and the DNS barrier related to the
transfer of nucleons from the light nucleus to the heavy one. The passing
through other barriers are linked to the decay of the DNS on scattered nuclei.

Therefore, the probability of the compound nucleus formation from the DNS is
determined in our model as the ratio of the widths leading to the compound
nucleus to the total decay widths of the DNS, i.e.

$\displaystyle P(E,\ell)=\frac{\Gamma_{\rm CN}^{\rm DNS,f}(E,\ell)+\Gamma_{\rm
CN}^{\rm DNS,tr}(E,\ell)}{\Gamma^{\rm tot}_{\rm CN}(E,\ell)}.$ (12)

Here

$\displaystyle\Gamma^{\rm tot}_{\rm CN}(E,\ell)$ $\displaystyle=$
$\displaystyle\Gamma_{\rm CN}^{\rm DNS,f}(E,\ell)+\Gamma_{\rm CN}^{\rm
DNS,tr}(E,\ell)+\Gamma_{\rm DIC}^{\rm DNS}(E,\ell)$ (13) $\displaystyle+$
$\displaystyle\Gamma_{\rm sph}^{\rm DNS}(E,\ell)+\Gamma_{\rm def}^{\rm
DNS}(E,\ell)$

is the total decay widths of the DNS. $\Gamma_{\rm CN}^{\rm DNS,f}(E,\ell)$ is
the width related to the compound nucleus formation through the fusion path by
the smooth evolution of shape of nuclear system. $\Gamma_{\rm CN}^{\rm
DNS,tr}(E,\ell)$ is the width of the compound nucleus production through the
multi-nucleon transfer from the light to heavy nuclei. This is the DNS path of
compound nucleus formation used in the various versions of the DNS model dns1
; dns1c ; dns1aa ; dns1aa1 ; dns1aa2 ; dns1aa4 ; dns2a ; adamian ; dns3 ;
dns31 ; dns32 ; dns33 ; dns3a ; dns4b ; dns4c ; dns4d ; dns5 . $\Gamma_{\rm
DIC}^{\rm DNS}(E,\ell)$ is the width of the DNS decay into two nuclei with
nucleon transfer from heavy to light nuclei. During this process the DNS
decays into scattered nuclei, which are close to the incident nuclei. This
process is the deep-inelastic collisions (DIC). $\Gamma_{\rm sph}^{\rm
DNS}(E,\ell)$ and $\Gamma_{\rm def}^{\rm DNS}(E,\ell)$ are the width of the
DNS decay into incident spherical or deformed nuclei, respectively. The widths
$\Gamma_{\rm sph}^{\rm DNS}(E,\ell)$ and $\Gamma_{\rm def}^{\rm DNS}(E,\ell)$
are connected to the different quasi-elastic decay modes of the DNS.

Eq. (12) is written for the case of a direct coupling of the DNS to the
equilibrium shape of compound nucleus. In this case the potential energy
landscape has the valley, the bottom of which is directly connecting the
system of contacting nuclei and the equilibrium shape of compound nucleus,
see, for example, the landscapes presented in Fig. 1. If the final point of
the valley started from the DNS is related to the non-equilibrium shape of
compound nucleus and the equilibrium shape of compound nucleus locates in
another valley, see, for example, the landscapes presented in Fig. 2, then the
intermediate state should be introduced. The intermediate state can decay into
both the compound nucleus in equilibrium shape and to the quasi-fission
fragments. Such case of the compound nucleus formation will be considered in
Sec. 2.B.3.

According to Eqs. (12)-(13), the compound nucleus in our model can be formed
by both the fusion and DNS ways. If we put $\Gamma_{\rm CN}^{\rm
DNS,f}(E,\ell)=0$ and introduce $\Gamma_{\rm quasi-
fission}(E,\ell)=\Gamma_{\rm DIC}^{\rm DNS}(E,\ell)+\Gamma_{\rm sph}^{\rm
DNS}(E,\ell)+\Gamma_{\rm def}^{\rm DNS}(E,\ell)$ then our probability of
compound nucleus formation (12)-(13) equals to the one in Refs. v ; dns1 ;
dns1c ; adamian . Here we use the original name of the width $\Gamma_{\rm
quasi-fission}(E,\ell)$ used in Refs. v ; dns1 ; dns1c ; adamian . Note, in
our model the quasi-fission process is related to the decay of one-body
nuclear shape on the fission fragments bypassing the formation of the compound
nucleus with the equilibrium shape. (Recently, the authors of the DNS model
have used the master equation approach for evaluation of the probability of
compound nucleus formation dns2a ; dns3 ; dns31 ; dns32 ; dns33 ; dns3a ;
dns4b ; dns4c ; dns4d ; dns5 . However, the old v ; dns1 ; dns1c ; adamian
and new dns2a ; dns3 ; dns31 ; dns32 ; dns33 ; dns3a ; dns4b ; dns4c ; dns4d ;
dns5  approaches of the DNS model have the same mechanism of the compound
nucleus formation. In our model the probability of compound nucleus formation
is described by the ratio of the decay widths in Eqs. (12)-(13). This is very
convenient, because all decay processes are considered in the same approach.)

The shape and properties of the potential energy landscape related to the
fusion trajectory of compound nucleus formation is discussed in next
subsection.

####  II.2.2 The landscape of potential energy related to the fusion path from
the DNS to compound nucleus

The DNS is formed in the collision of two spherical nuclei in the case of cold
fusion reactions. The DNS after formation can evolve to the compound nucleus
or divide on two spherical or deformed nuclei. The division of the DNS on two
deformed fragments can be linked as to the quasi-fission as to the immediately
decay of the DNS. Therefore, the shapes of nucleus related to the analysis of
the trajectory from the DNS to the compound nucleus with possibility of the
quasi-fission should include the two spherical or deformed nuclei as well as
the spherical, well-deformed, and pre-ruptured one-body nuclear shapes. The
parametrization describing such different shapes should be simple as possible.
The axial-symmetric parametrization

$\displaystyle\rho=\left\\{\begin{array}[]{ll}0,&\textrm{if}\;z<-a,\\\
b\sqrt{1-(z/a)^{2}},&\textrm{if}\;-a\leq z\leq z_{0},\\\
c\sqrt{1-((z-R)/d)^{2}},&\textrm{if}\;z_{0}\leq z\leq R+d,\\\
0,&\textrm{if}\;z>R+d\end{array}\right.$ (18)

is satisfied the proposed conditions. Here $\rho$ and $z$ are the cylindrical
coordinates. The parametrization depends on the 6 parameters $a,b,c,d,z_{0}$,
and $R$.

The radius $\rho$ should be continued at the point $z_{0}$, therefore

$\displaystyle b\sqrt{1-(z_{0}/a)^{2}}=c\sqrt{1-((z_{0}-R)/d)^{2}}.$ (19)

This equation couples the two parameters of the parametrization, for example,
$z_{0}$ and $R$. The total volume of the nuclear system should be conserved
during the shape evolution of the nuclear system. As a result of these
constraints, the shape parametrization has 4 independent parameters.

We fix values $a=b=R_{1}$, where $R_{1}$ is the radius of the light incident
nucleus. Due to this fixing the shape parametrization (14) depends only on two
independent parameters. Note that two independent parameters are often used
for description of the compound nucleus formation at evolution of the one-body
form, see, for examples, Refs. shen1 ; boilley ; torres2 ; torres ; torres1 .

The two touching nuclei is described by Eq. (14) at $z_{0}=R_{1}$. During
fusion the nuclei get close and the value of $z_{0}$ is smoothly reduced. The
radius of the neck connecting nuclei is rising from 0 at $z_{0}=R_{1}$ up to
the radius of the light nucleus $R_{1}$ at $z_{0}=0$. After that the heavy
nucleus absorbs the light one and $z_{0}$ approaches to $-R_{1}$. The light
nucleus is fully absorbed by the heavy one at $z_{0}=-R_{1}$.

It is useful to consider two independent variables $z_{0}$ and $\beta_{2}$ for
specification of the shape of fusing nuclei. Parameter $\beta_{2}$ is coupled
to the ratio $d/c$ by the equation
$d/c=[1+\beta_{2}Y_{20}(\theta=0^{\circ})]/[1+\beta_{2}Y_{20}(\theta=90^{\circ})]$,
where $Y_{20}(\theta)$ is the spherical harmonic function vmk . Parameter
$\beta_{2}$ describes the quadrupole deformation of heavy nucleus in the case
of the touching nuclei at $z_{0}=R_{1}$ or the quadrupole deformation of the
one-body shape at $z_{0}=-R_{1}$. (Here we neglect the difference between the
equal-volume shapes of the axial-symmetric ellipsoid and the nucleus with the
surface radius $R(\theta)=R_{0}[1+\beta_{2}Y_{20}(\theta)]$. Such shapes are
very close to each other at small deformations.) Our shape parametrization at
$z_{0}=-R_{1}$ describes the fission process related to the evolution of the
quadrupole deformation too. Note that the quadrupole deformation is
successfully used for a discussion of the fission process by Bohr and Wheeler
bw .

Our two-parameters parametrization is useful for the simultaneous description
of various nuclear shapes related to the fusion of asymmetric nuclei, compound
nucleus formation, fission and quasi-fission. This parametrization has not
used anywhere, see, for example, Ref. hasse  devoted to the compilation of
various nuclear shapes used in literature.

For every value $z_{0}$ and $\beta_{2}$ we find parameters $R^{fit}$ and
$\beta_{L}^{fit}$, at which the parametrization

$\displaystyle
R(\theta)=R^{fit}[1+\sum_{L=2}^{9}\beta_{L}^{fit}Y_{L0}(\theta)]$ (20)

fits the shape described by Eq. (14). Using obtained values of parameters
$R^{fit}$ and $\beta_{L}^{fit}$ we calculate the shell correction energies as
the function of the parameters $z_{0}$ and $\beta_{2}$ by using the code
WSBETA wsbeta . This code uses a Woods-Saxon potential with a ’universal’
parameter set and the parametrization of the nuclear shape in the form
$R(\theta)\propto[1+\sum_{L=2}^{9}\beta_{L}^{fit}Y_{L0}(\theta)]$. The radius
parameter of the Woods-Saxon potential is fixed by the ’universal’ parameter
set wsbeta . Therefore, the parameters $z_{0}$ and $\beta_{2}$ are coupled to
the deformation parameters $\beta_{L}^{fit}$ only. The residual pairing
interaction is calculated by means of the Lipkin-Nogami method paring . The
macroscopic part of the deformation energy is evaluated using the Yukawa-plus-
exponential potential macro .

The dependencies of the potential energy landscape on the variables $z_{0}$
and $\beta_{2}$ for the cold-fusion systems 50Ti, 52,54Cr, 58Fe, 64Ni, 70Zn,
and 78Ge + 208Pb are presented in Figs. 1–2. The dependencies of the potential
energy of nuclei on $\beta_{2}$ at $z_{0}=-R_{1}$ presented in these Figs. are
typical for the fissioning nucleus. So, we see the ground-state well and the
fission barrier along the line $z_{0}=-R_{1}$ in these Figures.

There are many heavy and super-heavy nuclei with the two-hampered fission
barrier bs ; bl ; sob ; sob1 ; etfsi ; d-uhe ; mol ; snp ; pnsk ; pnsk-a ;
kowal ; kowal-a ; kjs ; afanas ; afanas-a ; agbemava ; warga . The values of
$\beta_{2}$ at the ground state of the compound nucleus, which we can see in
Figs. 1–2 at $z_{0}\approx-R_{1}$, are close to the corresponding ones
obtained in Ref. kjs . The values of the inner fission barrier evaluated from
Figs. 1-2 at $z_{0}\approx-R_{1}$ are close to the ones obtained in Refs.
kowal ; kowal-a ; kjs ; etfsi ; mol  in the framework of the shell-correction
approach. Note that Refs. sob ; sob1 ; mol ; kowal ; kowal-a ; kjs  are
devoted to the accurate calculations of the ground state and saddle point
properties of SHN using a rich set of the multipole deformations. Our
deformation space is limited by two independent variables, therefore the
agreement of the fission barrier values extracted from Figs. 1-2 with the
results obtained in the framework of other models is approximate.

Figure 1:  The potential energy landscape in depending on the variables
$z_{0}$ and $\beta_{2}$ for cold-fusion systems 50Ti, 52,54Cr, and 58Fe +
208Pb. The dashed lines are the trajectories of the compound nucleus
formation, which are drawn by eye.

Figure 2:  The potential energy landscape in depending on the variables
$z_{0}$ and $\beta_{2}$ for cold-fusion systems 64Ni, 70Zn, and 78Zn + 208Pb.
The dashed lines are the compound nucleus formation trajectories, which are
drawn by eye.

The dependence of the potential energy of two contacting nuclei on the value
of quadrupole deformation of heavy nucleus is presented in Figs. 2–3 at
$z_{0}=R_{1}$. We see the potential energy has a local minimum at small values
of $\beta_{2}\approx 0$ at $z_{0}=R_{1}$, which is related to the spherical
ground-state shape of 208Pb. Note that the similarity of the shapes described
by the parametrizations (14) and (16) is worsened at $z_{0}\rightarrow R_{1}$.
Nevertheless, the shape described by Eq. (16) is close to the shape of two
contacting nuclei.

The landscape of potential energy is strongly changed near $z_{0}=R_{1}$ for
the system 78Ge + 208Pb. As a result, the contour lines are difficultly
separated, therefore we present the potential landscape for this system for
$z_{0}\leq 4.4$ fm, see Fig. 2.

The trajectory of the compound nucleus formation, which is drawn by eye in
Figs. 1–2, connects the point of two contacting spherical nuclei at
$z_{0}=R_{1},\beta_{2}=0$ and the point of the ground state of the compound
nucleus at $z_{0}\approx-R_{1}$ and $\beta_{2}$ from the range
$0\leq\beta_{2}\leq 0.25$. These trajectories for the cold-fusion systems
50Ti, 52,54Cr, 58Fe + 208Pb are located in the bottom of the valley leading to
the compound nucleus formation, see Fig. 1. Similar valleys are also obtained
for the cluster emission from heavy nuclei with the daughter nucleus near
208Pb in Ref. warda18 . The fusion path leading to the compound nucleus
formation is also studied in Ref. moller ; moller-a . The dependencies of the
potential energies on the elongation along the fusion paths presented in Refs.
moller ; moller-a  for reactions 50Ti and 70Zn + 208Pb look like the ones in
Figs. 1 and 2 for the these systems. Unfortunately, the direct comparison of
the potentials along the fusion trajectories obtained in these different
approaches is not possible due to using various shape parametrizations.

The high ridge separates the fusion valley and the quasi-fission area in Fig.
2. This ridge merges smoothly to the inner fission barrier at $z_{0}=-R_{1}$.
The true quasi-fission (or fast fission) process in heavy-ion reactions is
related to the fission of nuclear system before reaching the ground-state
shape of the compound nucleus, while the true fission is started from the
ground-state shape of the compound nucleus. The large difference between the
energies at the bottom of the fusion valley and at the ridge leads to a
statistical suppression of the quasi-fission process in comparison to the
compound nucleus formation for systems 50Ti, 52,54Cr, 58Fe + 208Pb. Therefore,
the probabilities of compound nucleus formation are well determined by Eqs.
(12)-(13) for these systems.

We see the saddle points at the point $z_{0}\approx 2-4$ fm and
$\beta_{2}\approx 0.1-0.25$ on the fusion path for systems 64Ni, 70Zn, and
78Ge + 208Pb in Fig. 2. The heights of these saddle points define the barriers
for the transition from the DNS to the compound nucleus along the fusion
trajectories for these systems. In contrast to this, such saddle points absent
for the fusion paths for reactions 50Ti, 52,54Cr, and 58Fe + 208Pb in Fig. 1.
Therefore, the barriers related to the formation of the compound nucleus along
the fusion trajectory for reactions 50Ti, 52,54Cr, and 58Fe + 208Pb are
defined as the highest values of the potential energies of spherical or near-
spherical nuclei at the contact point, see Fig. 1. This is because the
excitation energies of these systems in the contact point should be above or
equal these potential energies for the successful formation of the compound
nuclei.

The trajectories of the compound nucleus formation for reactions 64Ni, 70Zn,
and 78Ge + 208Pb have the saddle points near the point $z_{0}\approx 4$ fm and
$\beta_{2}\approx 0.1-0.2$, which are linked to the decay of the slightly
overlapped nuclei (or the DNS) on two fragments, see Fig. 2. The decay of DNS
on two fragments is also described by the widths $\Gamma_{\rm DIC}^{\rm
DNS}(E,\ell)$, $\Gamma_{\rm sph}^{\rm DNS}(E,\ell)$ and $\Gamma_{\rm def}^{\rm
DNS}(E,\ell)$. The width $\Gamma_{\rm def}^{\rm DNS}(E,\ell)$ is connected to
the lowest value of the barrier, which is taken place for the two-body
systems. The height of the saddle point near the point $z_{0}\approx 4$ fm and
$\beta_{2}\approx 0.1-0.2$ is higher the height of the barrier related to the
width $\Gamma_{\rm def}^{\rm DNS}(E,\ell)$. Therefore, we may neglect by the
influence of this saddle point on both the formation of the compound nucleus
and the decay of DNS on fragments.

The ridge, which separates the compound nucleus formation valley and the
quasi-fission valley, merges to the outer fission barrier near $z_{0}\approx-
R_{1}$ for the cold-fusion systems 70Zn and 78Ge + 208Pb, see Fig. 2. The
compound nucleus formation valley is merged with the potential energy well
between the inner and outer fission barrier near $z_{0}\approx-R_{1}$.
Therefore, the intermediate state is formed in this well. The compound nucleus
is formed at the decay of the intermediate state through the inner fission
barrier. The quasi-fission fragments can be appeared at the decay of the
intermediate state through the outer fission barrier. The quasi-fission
fragments are different from the ones produced in the decay of the DNS on two
fragments and related to the DIC or quasi-elastic processes. This is because
the yield of the DIC or quasi-elastic fragments is concentrated around the
incident nuclei, while the yield of the quasi-fission fragments is similar to
the one for the compound nucleus fission.

The probabilities of compound nucleus formation for the systems 70Zn, and 78Ge
+ 208Pb are not described by Eqs. (12)-(13), because we should take into
account the decay branches of the intermediate state. We consider the
probability of compound nucleus formation in such case in next subsection.

####  II.2.3 Expression for the probability of compound nucleus formation in
the case of intermediate state

The formation of compound nucleus in the case of the intermediate state is
occurred by two steps. The first step is related to the formation of the
intermediate state from the DNS, while the second one is linked to the decay
of the intermediate state into the compound nucleus with the equilibrium
shape. The intermediate state may decay into the compound nucleus, the quasi-
fission fragments, and back to the DNS.

The consideration of the two-step process in the model is similar to the
discussion of the sequential stages of the SHN formation used in Eq. (1). The
intermediate state has taken place on the fusion way of the compound nucleus
formation and has not any influence on the DNS path of the compound nucleus
formation. As a result, the probability of compound nucleus formation in this
case is determined as

$\displaystyle P(E,\ell)=\frac{\Gamma_{\rm CN}^{\rm DNS,f}(E,\ell)P_{\rm
is}(E,\ell)+\Gamma_{\rm CN}^{\rm DNS}(E,\ell)}{\Gamma^{\rm tot}_{\rm
CN}(E,\ell)},$ (21)

where

$\displaystyle P_{\rm is}(E,\ell)=\frac{\Gamma^{\rm is}_{\rm
CN}(E,\ell)}{\Gamma^{\rm is}_{\rm tot}(E,\ell)}$ (22)

is the decay probability of the intermediate state into the compound nucleus.
The widths presented in Eq. (17) have been discussed already, see Eqs.
(12)-(13). Now we consider the widths appeared in Eq. (18).

$\displaystyle\Gamma^{\rm is}_{\rm tot}(E,\ell)=\Gamma^{\rm is}_{\rm
CN}(E,\ell)+\Gamma_{\rm qf}^{\rm is}(E,\ell)+\Gamma_{\rm DNS}^{\rm
is}(E,\ell)$ (23)

is the total decay width of the intermediate state, $\Gamma^{\rm is}_{\rm
CN}(E,\ell)$, $\Gamma_{\rm fiss}(E,\ell)$, and $\Gamma_{\rm DNS}^{\rm
is}(E,\ell)$ are the decay widths of the intermediate state to the compound
nucleus, the quasi-fission fragments, and the DNS, respectively. The width
$\Gamma_{\rm qf}(E,\ell)$ is described the true quasi-fission process, which
is related to the fission of the one-body nuclear system bypassing the
formation of the compound nucleus with equilibrium shape.

The probability of compound nucleus formation decreases due to the decay of
the intermediate state to the quasi-fission fragments or back to the DNS,
because $P_{\rm is}(E,\ell)\leq 1$. Eq. (17) coincides with Eq. (12), when
$P_{\rm is}(E,\ell)=1$.

The cross sections of the compound nucleus formation and the true quasi-
fission are related to the corresponding decay branches of the intermediate
state. Therefore, these cross sections can be defined, respectively, as

$\displaystyle\sigma_{\rm CN}(E)=\frac{\pi\hbar^{2}}{2\mu
E}\sum_{\ell}(2l+1)T(E,\ell)P(E,\ell),$ (24) $\displaystyle\sigma_{\rm
qf}(E)=\frac{\pi\hbar^{2}}{2\mu E}\sum_{\ell}(2l+1)T(E,\ell)P_{\rm
qf}(E,\ell).$ (25)

Here

$\displaystyle P_{\rm qf}(E,\ell)=\frac{\Gamma_{\rm CN}^{\rm
DNS,f}(E,\ell)}{\Gamma^{\rm tot}_{\rm CN}(E,\ell)}\times\frac{\Gamma_{\rm
qf}^{\rm is}(E,\ell)}{\Gamma^{\rm is}_{\rm tot}(E,\ell))}\;\;$ (26)

is the probability of the quasi-fission decay. The first factor in Eq. (22) is
the probability of intermediate state formation, while the second one is the
decay probability of the intermediate state to the quasi-fission fragments.

The number of the successive intermediate states $k$ may be more than one in
the case of the very complex potential energy landscape. In such case the
probability of compound nucleus formation is also determined by Eq. (17) in
which the probability $P_{\rm is}(E,\ell)$ is substituted by the product of
the decay probabilities of $k$ successive intermediate states $P_{\rm
is\;1}(E,\ell)\cdot P_{\rm is\;2}(E,\ell)\cdot...\cdot P_{{\rm
is}\;k}(E,\ell)$. Eqs. (21)-(22) for the quasi-fission cross section should be
also modified by respective way, because the quasi-fission fragments can be
emitted at the decay of any intermediate state. The total number of
corresponding parameters, which is needed for the reaction description, rises
with the number of the intermediate states. Therefore, we consider in our
model only the one intermediate state for reactions 70Zn and 78Ge + 208Pb.

The main decay channel of the superheavy compound nucleus is fission,
consequently, the values of compound nucleus production cross sections are
very close to the compound nucleus fission cross sections. The probabilities
of formation of the compound nucleus fission fragments (12), (13) or quasi-
fission fragments (22) are very small for the heavy cold-fusion systems.
Therefore, the probabilities of these processes are strongly lower than the
probability of the DIC fragments formed at the DNS decay, because it is
necessary to form the compound nucleus or the intermediate state additionally
to the DNS formation. This is strongly correlated to the experimental yields
of near-symmetric fission or quasi-fission fragments and very asymmetric DIC
or quasi-elastic fragments for various reactions 120e ; itkis ; itkis-a ;
itkis-c .

####  II.2.4 The decay widths

Eqs. (12)-(13), (17)-(19), and (22) include of the two types of the widths.
The widths $\Gamma_{\rm CN}^{\rm DNS,f}(E,\ell)$, $\Gamma^{\rm is}_{\rm
CN}(E,\ell)$, $\Gamma_{\rm qf}^{\rm is}(E,\ell)$, $\Gamma_{\rm DNS}^{\rm
is}(E,\ell)$ are related to the one-body shape of nucleus, while the widths
$\Gamma_{\rm CN}^{\rm DNStr}(E,\ell)$, $\Gamma_{\rm DIC}^{\rm DNS}(E,\ell)$,
$\Gamma_{\rm sph}^{\rm DNS}(E,\ell)$, $\Gamma_{\rm def}^{\rm DNS}(E,\ell)$ are
linked to the two-body nuclear systems.

The widths linked to the various one-body shapes are determined as

$\displaystyle\Gamma_{\rm one-body}(E,\ell)=\frac{1}{\rho_{\rm
in}(E)}\int_{0}^{E-B}d\varepsilon\;\rho_{A,\ell}(\varepsilon).$ (27)

Here $\rho_{\rm in}(E)$ is the energy level density of the nuclear system in
the initial state, $\rho_{A,\ell}(\varepsilon)$ is the energy level density of
the nuclear system in the final state, $B$ is the height of the saddle point
taken place on the way from the initial state to the final one, $\varepsilon$
is the excitation energy. The corresponding values of $E$, $\ell$ and $B$
should be applied at the calculation of the widths $\Gamma_{\rm CN}^{\rm
DNS,f}(E,\ell)$, $\Gamma^{\rm is}_{\rm CN}(E,\ell)$, $\Gamma_{\rm qf}^{\rm
is}(E,\ell)$, $\Gamma_{\rm DNS}^{\rm is}(E,\ell)$.

We use the back-shifted Fermi gas energy level density of nucleus with the
excitation energy $\varepsilon$, $A$ nucleons and the angular momentum $J$
ripl3 , which is written as

$\displaystyle\rho_{A,J}(U)$ $\displaystyle=$
$\displaystyle\frac{(2J+1)}{4\sqrt{2\pi}\sigma_{J}^{3}}\exp{\left\\{-[(J+1/2)/\sigma_{J}]^{2}/2\right\\}}$
(28) $\displaystyle\times$ $\displaystyle\frac{\sqrt{\pi}}{12(a_{\rm
dens}U^{5})^{1/4}}\exp{\left[2\sqrt{a_{\rm dens}U}\right]}.$

Here

$\displaystyle U=\varepsilon-\delta=a_{\rm dens}T^{2}$ (29)

is the back-shifted excitation energy, which is connected with the temperature
$T$,

$\displaystyle\delta=12nA^{-1/2}+0.173015$ (30)

is the energy shift with $n=-1,0$ and $1$ for odd-odd, odd-A, and even-even
nuclei, respectively, $\sigma_{J}^{2}=(0.83A^{0.26})^{2}$ is the spin cut-off
parameter. The level density parameter depends on the excitation energy of the
nucleus ist  and equals

$\displaystyle a_{\rm dens}=a_{\rm inf}[1+(\delta_{\rm
shell}/U)(1-\exp{(-\gamma U)})],$ (31)

where

$\displaystyle a_{\rm inf}=0.0722396A+0.195267A^{2/3}$ (32)

is the asymptotic level density parameter, $\gamma=0.410289/A^{1/3}$ is the
damping parameter, $\delta_{\rm shell}$ is the phenomenological shell
correction ripl3 . The value of phenomenological shell correction is
determined as the difference $\delta_{\rm shell}=M_{\rm exp}-M_{\rm ld}$ ripl3
, where $M_{\rm exp}$ is the experimental value of the nuclear mass taken from
Ref. be  and $M_{\rm ld}$ is the liquid drop component of the mass formula ms
. All parameter values used for the evaluation of the energy level density are
taken from Ref. ripl3  without any changes. (Note the phenomenological shell
corrections $\delta_{\rm shell}$ and $\delta E_{i}$, see Eq. (9), are the same
physical sense. However, they are obtained using the different mass formulas
for the calculation the liquid-drop contribution d2015 ; ripl3 ; ms . The
values of parameters $\delta_{\rm shell}$ and $\delta E_{i}$ are,
respectively, linked to the values of other parameters of the energy level
density and the nuclear part of the interaction potential. Therefore, we use
for calculations of $\delta_{\rm shell}$ and $\delta E_{i}$ different
expressions.) The values of shell corrections are very important for the
properties of SHN d-shell , therefore the influence of shell correction on the
level density should be taken into account.

As we have pointed, the widths $\Gamma_{\rm CN}^{\rm DNS,tr}(E,\ell)$,
$\Gamma_{\rm DIC}^{\rm DNS}(E,\ell)$, $\Gamma_{\rm sph}^{\rm DNS}(E,\ell)$,
$\Gamma_{\rm def}^{\rm DNS}(E,\ell)$ are related to the DNS, which are
consisted of two nuclei with various shapes and nucleon compositions. The
width of the DNS built by the nuclei with numbers of nucleons $A_{1}$ and
$A_{2}=A-A_{1}$, correspondingly, is written as

$\displaystyle\Gamma_{\rm DNS}(E,\ell)=\frac{1}{\rho_{\rm
in}(E)}\int_{0}^{E-B_{\ell}}d\varepsilon\;\rho_{A_{1},A_{2}}(\varepsilon,\ell),$
(33)

where

$\displaystyle\rho_{A_{1},A_{2}}(\varepsilon,\ell)=\int_{0}^{\varepsilon}d\varepsilon^{\prime}\;\rho_{A_{1},0}(\varepsilon^{\prime})\;\rho_{A_{2},0}(\varepsilon-\varepsilon^{\prime}).$
(34)

Here $\rho_{\rm in}(E)$ is the energy level density of the nuclear system in
the initial state. $\rho_{A_{1},A_{2}}(\varepsilon,\ell)$ is the energy level
density of the DNS, $\rho_{A_{1},\ell}(\varepsilon)$ and
$\rho_{A_{2},\ell}(\varepsilon)$ are the energy level density of the nuclei
with $A_{1}$ and $A_{2}$ nucleon in the final state, $B_{\ell}$ is the height
of the saddle point taken place on the way from the initial state to the final
one. We neglect by the transfer of the orbital moment of the DNS system into
the orbital momenta of nuclei for the sake of simplicity.

The probabilities of the compound nucleus formation $P(E,\ell)$ and the decay
of the intermediate state $P_{\rm is}(E,\ell)$ depend on the ratio of the
decay widths into a specific states to the total decay width of the initial
state. Therefore, these probabilities are independent on $\rho_{\rm in}(E)$.

We should define the barrier heights for the calculation of various decay
widths and the probabilities. Let us consider the barriers for corresponding
widths in details.

####  II.2.5 The barrier heights for different processes

The width $\Gamma_{\rm sph}^{\rm DNS}(E,\ell)$ depends on the barrier $B^{\rm
fus}_{\ell}$. The value of $B^{\rm fus}_{\ell}$ was obtained at the
calculation of the transmission probability $T(E,\ell)$, see Eq. (5). This
barrier of the total potential energy of the spherical incident nuclei can be
found using Eqs. (2), (6)-(10). Substituting value of $B^{\rm fus}_{\ell}$ to
Eq. (29) we obtain $\Gamma_{\rm sph}^{\rm DNS}(E,\ell)$.

The width $\Gamma_{\rm def}^{\rm DNS}(E,\ell)$ connects to the barrier $B^{\rm
fus}_{\ell,{\rm def}}$. This barrier is determined as the minimal values of
the barrier of the total potential energy of deformed nuclei. The nucleon
compositions of these deformed nuclei are the same as in the incident channel.
The total potential energy of deformed nuclei is calculated in the framework
of the approach developed in Refs. dms ; ds17 ; dps . Now we improve it by
taking into account the realistic surface stiffness of interacting nuclei.

As shown in Refs. dp ; dpil ; dpil2 ; dpfus-a ; dm ; dm-a ; dm-b  the axially-
symmetric nuclei, which is elongated along the line connected the mass
centers, have the lowest value of the barrier height. Therefore, we consider
that the DNS decays preferably by such mutual orientation of the axially-
symmetric nuclei. The other nucleus-nucleus configurations have higher values
of the barrier. Consequently, such configurations have the lower values of the
thermal excitation energy of the DNS and the smaller values of the statistical
yield. As a result, such configurations we may be neglected.

The total potential energy of interacting deformed nuclei $V_{\rm
DNS}(r,\ell,\\{\beta_{L1}\\},\\{\beta_{L2}\\})$ consists of the nuclear
$V_{\rm N}(R,\\{\beta_{L1}\\},\\{\beta_{L2}\\})$, Coulomb $V_{\rm
C}(R,\\{\beta_{L1}\\},\\{\beta_{L2}\\})$, and centrifugal
$V_{\ell}(R,\\{\beta_{L1}\\},\\{\beta_{L2}\\})$ energies as well as the
deformation energies $E_{{\rm def}i}(\\{\beta_{Li}\\})$ of each nucleus. So,
the total potential energy equals to

$\displaystyle V_{\rm DNS}(r,\ell,\\{\beta_{L1}\\},\\{\beta_{L2}\\})=V_{\rm
N}(r,\\{\beta_{L1}\\},\\{\beta_{L2}\\})$ $\displaystyle+V_{\rm
C}(r,\\{\beta_{L1}\\},\\{\beta_{L2}\\})+V_{\ell}(r,\\{\beta_{L1}\\},\\{\beta_{L2}\\})$
$\displaystyle+E_{{\rm def}1}(\\{\beta_{L1}\\})+E_{{\rm
def}2}(\\{\beta_{L2}\\}),$ (35)

where $\\{\beta_{Li}\\}=\beta_{0i},\beta_{1i},\beta_{2i}$,
$\beta_{3i},\beta_{4i}$ is the set of the surface multipole deformation
parameters of nucleus $i$, $i=1,2$. These deformation parameters are related
to the surface radius of deformed nucleus

$\displaystyle
R_{i}(\theta)=R_{0i}\left[1+\sum_{L}\beta_{Li}Y_{L0}(\theta)\right],$ (36)

where $R_{0i}$ is the radius of spherical nucleus $i$ and $Y_{L0}(\theta)$ is
the spherical harmonic functions vmk . The parameters $\beta_{0i}$ and
$\beta_{1i}$ provide the volume conservation and non-movement of the position
of the mass center for nucleus $i$. The values of the deformation parameters
$\\{\beta_{L1}\\},\\{\beta_{L2}\\}$ are determined by the condition of the
minima of the total interaction potential energy of these nuclei $V_{\rm
DNS}(r,\ell,\\{\beta_{L1}\\},\\{\beta_{L2}\\})$ at given $r$. Note the
contributions of higher multipole deformations $\beta_{L\geq 5}$ into the
value of $V_{\rm DNS}(r,\ell,\\{\beta_{L1}\\},\\{\beta_{L2}\\})$ are
negligible.

According to the proximity theorem derjaguin ; prox , the nuclear part of the
interaction potential between deformed nuclei can be approximated as dms ;
ds17

$\displaystyle V_{\rm N}(r,\\{\beta_{L1}\\},\\{\beta_{L2}\\})\approx
S(\\{\beta_{L1}\\},\\{\beta_{L2}\\})$ $\displaystyle\times V_{\rm N}^{\rm
sph}(d(r,\\{\beta_{L1}\\},\\{\beta_{L2}\\})+R_{01}+R_{02})).$ (37)

Here

$\displaystyle
S(\\{\beta_{L1}\\},\\{\beta_{L2}\\})=\frac{\frac{R_{1}(\pi/2)^{2}R_{2}(\pi/2)^{2}}{R_{1}(\pi/2)^{2}R_{2}(0)+R_{2}(\pi/2)^{2}R_{1}(0)}}{\frac{R_{01}R_{02}}{R_{01}+R_{02}}}$
(38)

is the factor related to the modification of the strength of nuclear
interaction of the deformed nuclei induced by the surface deformations, which
is derived in Ref. dms ,

$\displaystyle d(r,\\{\beta_{L1}\\},\\{\beta_{L2}\\})=r-R_{1}(0)-R_{2}(0)$
(39)

is the smallest distance between the surfaces of deformed nuclei, which
coincides to the distance between the surfaces of spherical nuclei. The
potential $V_{\rm N}^{\rm sph}$ determines the nuclear part of the interaction
between spherical nuclei, see Eqs. (6)-(10).

The expression of the Coulomb interaction of the two deformed arbitrary-
oriented axial-symmetric nuclei is obtained by an expansion on the deformation
parameters in Ref. dpil . The accuracy of this expression is very high. The
values of the Coulomb interaction of the two deformed arbitrary-oriented
axial-symmetric nuclei evaluated by using the expression from Ref. dpil  and
by the numerical calculations are very good agreed to each other iam . Taking
into account the considered orientation of axial-symmetric nuclei at the
searching the value of the lowest barrier height we rewrite the expression
from Ref. dpil  in a simple form

$\displaystyle V_{\rm C}(r)$ $\displaystyle=$
$\displaystyle\frac{Z_{1}Z_{2}e^{2}}{r}\\{1$ (40) $\displaystyle+$
$\displaystyle\sum_{L\geq
1}\left[f_{L1}(r,R_{01})\beta_{L1}+f_{L1}(r,R_{02})\beta_{L2}\right]$
$\displaystyle+$ $\displaystyle
f_{2}(r,R_{01})\beta_{21}^{2}+f_{2}(r,R_{02})\beta_{22}^{2}$ $\displaystyle+$
$\displaystyle\left.f_{3}(r,R_{01},R_{02})\beta_{21}\beta_{22}\right\\},$

where

$\displaystyle f_{L1}(r,R_{0i})=\frac{3R_{0i}^{L}}{2\sqrt{\pi(2L+1)}r^{L}},$
(41) $\displaystyle f_{2}(r,R_{0i})=\frac{3R_{0i}^{2}}{7\pi
r^{2}}+\frac{9R_{0i}^{4}}{14\pi r^{4}},$ (42) $\displaystyle
f_{3}(r,R_{01},R_{02})=\frac{27R_{01}^{2}R_{02}^{2}}{10\pi r^{4}}.$ (43)

This expression is taken into account linear and quadratic terms on the
quadrupole deformation parameters and linear terms of high-multipolarity
deformation parameters. The volume correction, which is appeared in the second
order of the quadrupole deformation parameter and important for heavy systems,
is taken into account in this expression.

The nuclei formed DNS after penetration the fusion barrier are excited.
Therefore, the momentum of inertia of the DNS can be well approximated in the
framework of the solid-state model. The centrifugal potential energy of DNS
nuclei is

$\displaystyle
V_{\ell}(r,\\{\beta_{L1}\\},\\{\beta_{L2}\\})=\frac{\hbar^{2}\ell(\ell+1)}{2(\mu
r^{2}+J_{1}+J_{2})},$ (44)

where

$\displaystyle J_{i}=(2/5)m_{n}R_{0i}^{2}A_{i}(1+\sqrt{5/(16\pi)}\beta_{2i})$
(45)

is the momentum of inertia of nucleus $i$, and $m_{n}$ is the mass of nucleon.
Here we take into account only quadrupole deformation, because the
contribution of higher multipolarities into the momentum of inertia is
negligible.

The incident nuclei participated in the cold-fusion reactions have the
spherical equilibrium shape. The nuclei involved to the DNS evolution are
deforming due to interaction between them. The deformation energy of the
nucleus induced by a deviation from the spherical shape consists of the
surface and Coulomb contributions. This energy in the liquid-drop
approximation bormot  is given as

$\displaystyle E_{{\rm def}i}^{\rm ld}(\\{\beta_{Li}\\})=\sum_{L=2}^{4}C^{\rm
ld}_{LA_{i}Z_{i}}\frac{\beta_{Li}^{2}}{2},$ (46)

where

$\displaystyle C^{\rm ld}_{LA_{i}Z_{i}}=\frac{(L-1)(L+2)b_{\rm
surf}A^{2/3}_{i}}{4\pi}-\frac{3(L-1)e^{2}Z^{2}_{i}}{2\pi(2L+1)R_{0i}}$ (47)

is the surface stiffness coefficient obtained in the liquid-drop
approximation, and $b_{\rm surf}$ is the surface coefficient of the mass
formula msis .

We can also evaluate the realistic deformation energy of a nucleus at small
surface deformations in the framework of the shell correction method s ; s1 ;
s2 ; s3  and approximate the dependence of this energy on the deformation
parameters by

$\displaystyle E_{{\rm def}i}^{\rm sc}(\\{\beta_{Li}\\})=\sum_{L=2}^{4}C^{\rm
sc}_{LA_{i}Z_{i}}\frac{\beta_{Li}^{2}}{2}.$ (48)

Here $C^{\rm sc}_{LA_{i}Z_{i}}$ is the total surface stiffness coefficient
obtained with the shell correction method. Using the shell-correction method
we can split both the deformation energy and the stiffness coefficient on the
shell-correction and liquid-drop parts

$\displaystyle E_{{\rm def}i}^{\rm sc}(\\{\beta_{Li}\\})=E_{{\rm def}i}^{\rm
shell}(\\{\beta_{Li}\\})+E_{{\rm def}i}^{\rm ld}(\\{\beta_{Li}\\})$
$\displaystyle=\sum_{L=2}^{4}[C^{\rm shell}_{LA_{i}Z_{i}}+C^{\rm
ld}_{LA_{i}Z_{i}}]\frac{\beta_{Li}^{2}}{2}$
$\displaystyle=\sum_{L=2}^{4}\left[\left(\frac{C^{\rm
sc}_{LA_{i}Z_{i}}}{C^{\rm ld}_{LA_{i}Z_{i}}}-1\right)+1\right]C^{\rm
ld}_{LA_{i}Z_{i}}\frac{\beta_{Li}^{2}}{2}.$ (49)

The deformation energy of a nucleus at small surface deformations can be
obtained in the harmonic oscillator model too bormot ; wong68 . In this model
the deformation energy of a nucleus is described as

$\displaystyle E_{{\rm def}i}^{\rm ho}(\\{\beta_{Li}\\})=\sum_{L=2}^{4}C^{\rm
ho}_{LA_{i}Z_{i}}\frac{\beta_{Li}^{2}}{2}.$ (50)

Here $C^{\rm ho}_{LA_{i}Z_{i}}$ is the surface stiffness coefficient in the
harmonic oscillator model, which is connected to the energy ${\cal
E}_{LA_{i}Z_{i}}$ and the total zero-point amplitude $\beta_{LA_{i}Z_{i}}^{0}$
of the surface oscillations (or the transition probability for exciting the
surface oscillations $B(E,0\rightarrow L)$) bormot ; wong68

$\displaystyle C^{\rm ho}_{LA_{i}Z_{i}}$ $\displaystyle=$
$\displaystyle\frac{(2L+1){\cal
E}_{LA_{i}Z_{i}}}{2(\beta_{LA_{i}Z_{i}}^{0})^{2}}$ (51) $\displaystyle=$
$\displaystyle\left(\frac{3ZeR^{L}}{4\pi}\right)^{2}\frac{(2L+1){\cal
E}_{LA_{i}Z_{i}}}{2B(E,0\rightarrow L)}.$

The known experimental values of ${\cal E}_{LA_{i}Z_{i}}$,
$\beta_{LA_{i}Z_{i}}^{0}$, and/or $B(E,0\rightarrow L)$ for nuclei are
tabulated for $L=2$ and 3 in Refs. be2 ; be3 . Note the coupling of the
incident channel with the low-energy surface vibration channels is often taken
into account in the framework of the harmonic oscillator approach at
description of various heavy-ion reactions fl ; dhrs ; motstef ; den-subfus .
The characteristics of heavy-ion reactions depend strongly on the properties
of the surface vibrations.

The harmonic oscillator $C^{\rm ho}_{LA_{i}Z_{i}}$ and shell-correction
$C^{\rm sc}_{LA_{i}Z_{i}}$ values of the surface stiffness parameters should
be close to each other. Therefore, we rewrite Eq. (45) in the form

$\displaystyle E_{{\rm def}i}^{\rm
sc}(\\{\beta_{Li}\\})=\sum_{L=2}^{4}\left[\left(\frac{C^{\rm
ho}_{LA_{i}Z_{i}}}{C^{\rm ld}_{LA_{i}Z_{i}}}-1\right)+1\right]C^{\rm
ld}_{LA_{i}Z_{i}}\frac{\beta_{Li}^{2}}{2}.\;\;\;$ (52)

This expression for deformation energy is useful for further application,
because using experimental values of ${\cal E}_{LA_{i}Z_{i}}$,
$\beta^{0}_{LA_{i}Z_{i}}$ we find the values of the ratio $C^{\rm
ho}_{LA_{i}Z_{i}}/C^{\rm ld}_{LA_{i}Z_{i}}$, see in Table 2. We put $C^{\rm
ho}_{LA_{i}Z_{i}}/C^{\rm ld}_{LA_{i}Z_{i}}=1$, if the experimental data for
evaluation of $C^{\rm ho}_{LA_{i}Z_{i}}$ are unknown.

The values of the ratio $C^{\rm ho}_{LA_{i}Z_{i}}/C^{\rm ld}_{LA_{i}Z_{i}}$
for $L=2,3$ presented in Table 2 have an irregular behaviour from one nucleus
to another, see also in Refs. bormot ; wong68 . This ratio very strongly
deviates from 1 near the magic nuclei. Nucleus 208Pb is very stiff for the
surface quadrupole and octupole distortions, because $C^{\rm
ho}_{LA_{i}Z_{i}}/C^{\rm ld}_{LA_{i}Z_{i}}\gg 1$ for $L=2,3$. In contrast to
this, nuclei 58Fe and 78Ge are soft for the surface quadrupole distortions,
because $C^{\rm ho}_{LA_{i}Z_{i}}/C^{\rm ld}_{LA_{i}Z_{i}}\ll 1$ for $L=2$.
These nuclei are well deformed during the DNS decay.

Table 2: The ratio $C^{\rm ho}_{LA_{i}Z_{i}}/C^{\rm ld}_{LA_{i}Z_{i}}$ obtained using the experimental properties of the low-energy surface vibrational states with multiplicities $L=2$ be2  and $L=3$ be3 . We put $C^{\rm ho}_{LA_{i}Z_{i}}/C^{\rm ld}_{LA_{i}Z_{i}}=1$ in the case of unknown the experimental properties of the low-energy surface vibrational states for some multiplicity and nucleus.  Nucleus | $C^{\rm ho}_{LA_{i}Z_{i}}/C^{\rm ld}_{LA_{i}Z_{i}}$
---|---
| $L=2$ | $L=3$ | $L=4$
50Ti | 2.03 | 1.46 | 1
52Cr | 12 | 3.15 | 1
54Cr | 0.45 | 1 | 1
58Fe | 0.36 | 2.0 | 1
64Ni | 1.46 | 1.36 | 1
70Zn | 0.54 | 0.56 | 1
78Ge | 0.33 | 1 | 1
208Pb | 44.9 | 2.2 | 1

The typical values of excitation energy of DNS, formed by incident nuclei in
cold-fusion reactions with 1–3 evaporated neutrons are 15–40 MeV. The
amplitudes of shell correction energy at such excitation energy are
approximately reduced 2–4 times ds ; ach ; ach-b ; bq ; dah ; lpc ; snp ; pnsk
; pnsk-a ; zp . We expect a similar effect for the value of the stiffness
parameter, which should approach to the hydrodynamical one at high excitation
energies.

Moreover, the single-particle spectra of nuclei near the contact point became
more homogeneous due to a level splitting and shifting induced by the nucleus-
nucleus interaction. This leads to a reduction of the amplitudes of the shell
correction energies in interacting nuclei, see also Eq. (8). Consequently, the
values of realistic surface stiffness coefficient of nuclei should approach to
the liquid-drop one at small distances between them due to the nucleus-nucleus
interaction.

Taking into account the excitation energy and nucleus-nucleus interaction
effects on the shell correction energies we modify Eq. (48) as

$\displaystyle E_{{\rm
def}i}(\\{\beta_{Li}\\})=\sum_{L=2}^{4}\left[\left(\frac{C^{\rm
ho}_{LA_{i}Z_{i}}}{C^{\rm ld}_{LA_{i}Z_{i}}}-1\right)k_{LA_{i}Z_{i}}+1\right]$
$\displaystyle\times\frac{C^{\rm ld}_{LA_{i}Z_{i}}\beta_{Li}^{2}}{2}.$ (53)

Here $k_{LA_{i}Z_{i}}\approx 0.1$ is the parameter, which is described the
attenuation of the shell-correction effect on the surface stiffness
coefficient of the incident nuclei formed the DNS in the cold-fusion
reactions. If $C^{\rm ho}_{LA_{i}Z_{i}}=C^{\rm ld}_{LA_{i}Z_{i}}$ then the
deformation energy is determined by the liquid-drop properties and independent
on $k_{LA_{i}Z_{i}}$. Note, the deformation energy is only defined by the
liquid drop properties in the framework of various versions of the DNS model
of the SHN production dns1 ; dns1c ; dns1aa ; dns1aa1 ; dns1aa2 ; dns1aa4 ;
dns2a ; adamian ; dns3 ; dns31 ; dns32 ; dns33 ; dns3a ; dns4b ; dns4c ; dns4d
; dns5 .

The double-magic target nucleus 208Pb and magic or close to magic projectile
nuclei are involved in the incident channel of the cold-fusion reactions.
Therefore, we should take into account the realistic surface stiffness of
nuclei at calculation of the width $\Gamma_{\rm def}^{\rm DNS}(E,\ell)$. The
width $\Gamma_{\rm def}^{\rm DNS}(E,\ell)$ is linked to $B^{\rm
DNS}_{\ell,{\rm def}}$, which is calculated with the help of Eqs. (31)-(41),
(43), (47), (49). The value of $B^{\rm DNS}_{\ell,{\rm def}}$ rises with a
rising $C^{\rm ho}_{2A_{i}82}/C^{\rm ld}_{2A_{i}82}$, because the barrier has
taken place at smaller values of the deformation parameter of nuclei. The
using very stiff nuclei in the cold fusion reaction leads to a higher value of
$B^{\rm DNS}_{\ell,{\rm def}}$ and, as a result, a smaller value of
$\Gamma_{\rm def}^{\rm DNS}(E,\ell)$. This leads to the increasing of the
probability of compound nucleus formation $P(E,\ell)$ described by Eq. (12).
In the contrary, fusion reactions between soft nuclei have a smaller value of
$B^{\rm DNS}_{\ell,{\rm def}}$ and, as a result, higher value of $\Gamma_{\rm
def}^{\rm DNS}(E,\ell)$ and smaller value of $P(E,\ell)$.

The correlation between the surface stiffness of incident nuclei and the
production cross sections is clearly observed experimentally. For example, the
values of $C^{\rm ho}_{2A_{i}82}/C^{\rm ld}_{2A_{i}82}$ for nuclei
208,206,204Pb obtained using data from Ref. be2  are, respectively, close to
45, 25, 17. The values of cross-section maxima for reactions
208,206,204Pb(48Ca,$2n)^{254,252,250}$No are $3\cdot 10^{6},4\cdot
10^{5},7\cdot 10^{3}$ b 102jinr ; 102jinr1 , correspondingly. So, we clearly
see that reactions with more stiff target nuclei have higher values of the SHN
production cross section. (Note, the other effects may also contribute into
the cross-section values.) The surface stiffness effect may be also
significant for the synthesis of SHN with $Z>118$ in hot fusion reactions,
when the stiff projectile 48Ca is substituted by more soft the one as 50Ti or
similar.

Let us consider the other barriers related corresponding decay widths used in
our model. The barrier $B_{\ell,{\rm DIC}}^{\rm DNS}=B_{0,\ell,{\rm DIC}}^{\rm
DNS}+Q_{\rm tr}$ is defined as the barrier between deformed contacting nuclei
formed after the nucleon transfer from the heavy nucleus to the light one,
where $Q_{\rm tr}$ is the transfer reaction Q-value evaluated with the help of
an atomic mass table be . This barrier is taken place at evolution of initial
DNS system to the more symmetric one. The DNS after passing the barrier
$B_{\ell,{\rm DIC}}^{\rm DNS}$ can decay in the two scattered nuclei with new
nucleon composition or the nucleon exchange between nuclei can continue
further. The interaction potential energy of touching nuclei after nucleon
exchange $B_{0,\ell,{\rm DIC}}^{\rm DNS}$ is calculated the similar way as
$B^{\rm DNS}_{\ell,{\rm def}}$. The barrier $B_{\ell,{\rm DIC}}^{\rm DNS}$ is
the minimal value of among the barriers related to various nucleon transfer
paths from the incident DNS to the more symmetric the one. Substituting the
obtained value of the barrier into Eq. (29) we can find the width $\Gamma_{\rm
DIC}^{\rm DNS}(E,\ell)$.

The compound nucleus formation using the DNS path is related to the barrier
$B_{\ell,{\rm CN}}^{\rm DNS,tr}$, which takes place at nucleons transfer from
the light nucleus to the heavy one. The values of $B_{\ell,{\rm CN}}^{\rm
DNS,tr}$ for every system formed at along various multi-nucleon transfer paths
is evaluated similarly to $B_{\ell,{\rm DIC}}^{\rm DNS}$. The surface
deformations of both nuclei are also taken into account. The barrier
$B_{\ell,{\rm CN}}^{\rm DNS,tr}$ is the minimal value among the barriers
related to various paths from the DNS formed by incident nuclei to the
compound nucleus. The width $\Gamma_{\rm CN}^{\rm DNS,tr}(E,\ell)$ is
calculated substituting the value of $B_{\ell,{\rm CN}}^{\rm DNS,tr}$ into Eq.
(29).

The width $\Gamma_{\rm CN}^{\rm DNS,f}(E,\ell)$ is related the barrier

$\displaystyle B_{\ell,{\rm CN}}^{\rm DNS,f}=B_{0,{\rm CN}}^{\rm
DNS,f}+\frac{\hbar^{2}\ell(\ell+1)}{2J_{\rm CN}^{\rm fus}}+Q_{\rm CN}.$ (54)

Here $B_{0,{\rm CN}}^{\rm DNS,f}$ the height of a corresponding saddle point
evaluated relatively the ground state of the compound nucleus using the
potential energy surface presented in Figs. 1–2, $J_{\rm CN}^{\rm fus}$ is the
momentum of inertia of compound nucleus at the saddle point, and $Q_{\rm CN}$
is the Q-value of the compound nucleus formation evaluated with the help of an
atomic mass table be . The width $\Gamma_{\rm CN}^{\rm DNS,f}(E,\ell)$ is
obtained using Eq. (23) and the value $B_{\ell,{\rm CN}}^{\rm DNS,f}$.

The widths related to the intermediate state can be found similar way as the
width $\Gamma_{\rm CN}^{\rm DNS,f}(E,\ell)$.

After obtaining the values of all widths we can determine the probability of
compound nucleus formation. Now we can determine the survival probability of
compound nucleus.

###  II.3 The survival probability of compound nucleus

The survival probability of the compound nucleus formed in the cold-fusion
reaction relates to the competition between the evaporation of $x$ neutrons
and the fission. It can be approximated by expression

$\displaystyle W^{xn}(E,\ell)=P_{xn}(E_{{\rm
CN}\ell}^{*})\frac{\Gamma_{1n}(E^{*}_{1},\ell)}{\Gamma_{1n}(E^{*}_{1n},\ell)+\Gamma_{\rm
f}(E^{*}_{1},\ell)}$
$\displaystyle\times\frac{\Gamma_{2n}^{A-1}(E^{*}_{2},\ell)}{\Gamma_{2n}^{A-1}(E^{*}_{2},\ell)+\Gamma_{\rm
f}^{A-1}(E^{*}_{2},\ell)}\times...$
$\displaystyle\times\frac{\Gamma_{xn}^{A-x+1}(E^{*}_{x},\ell)}{\Gamma_{xn}^{A-x+1}(E^{*}_{x},\ell)+\Gamma_{\rm
f}^{A-x+1}(E^{*}_{x},\ell)}.$ (55)

Here $P_{xn}(E^{*})$ is the realization probability of the $xn$-evaporation
channel jack , $E^{*}_{{\rm CN}\ell}=E-Q_{\rm
CN}-\hbar^{2}\ell(\ell+1)/(2J_{\rm gs})$, $J_{\rm gs}$ is the ground-state
momentum of inertia. $E^{*}_{1}=E-Q_{\rm CN}$ is the excitation energy of the
compound nucleus formed in the heavy-ion fusion reaction.
$\Gamma_{yn}^{A-y+1}(E^{*}_{y},\ell)$ and $\Gamma_{\rm
f}^{A-y+1}(E^{*}_{y},\ell))$ are, respectively, the width of neutron emission
and the fission width of the compound nucleus formed after emission of $(y-1)$
neutrons. $E^{*}_{y}=E^{*}_{y-1}-B_{n,y-1}-2T_{y-1}$ is the excitation energy
before evaporation of the $y$-th neutron, where $B_{n,y-1}$ is the separation
energy of the $(y-1)$-th neutron. $T_{y-1}$ is the temperature of the compound
nucleus after evaporation of $(y-1)$ neutrons and obtained from
$E^{*}_{y-1}=a_{\rm dens}T_{y-1}^{2}$, where $a_{\rm dens}$ is defined by Eqs.
(27)-(28).

The width of neutron emission from the nucleus with $A$ nucleons is given as
vh

$\displaystyle\Gamma_{n}(E^{*},\ell)=\frac{g_{n}m_{n}R_{n}^{2}}{\pi\hbar^{2}\rho_{A,\ell}(E^{*})}\int_{0}^{E^{*}-B_{n}}d\varepsilon\;\varepsilon$
$\displaystyle\times\rho_{A-1,\ell}(E^{*}-B_{n}-\varepsilon,\ell),\;\;$ (56)

where $B_{n}$ is the neutron separation energy from the nucleus,
$\rho_{A,\ell}(E^{*})$ and $\rho_{A-1,\ell}(E^{*})$ are, correspondingly, the
energy level densities of the compound nuclei before and after neutron
emission, $g_{n}$ is the neutron intrinsic spin degeneracy, and $R_{n}$ is the
radius of the neutron-nucleus interaction.

The fission width of the nucleus depends on the fission barrier height, which
is consisted of the liquid-drop and shell-correction contribution in the
Strutinsky shell correction prescription s ; s1 ; s2 ; s3 . The excitation
energy of a compound nucleus formed in cold-fusion reactions $E^{*}$ belongs
to the range 10 to 25 MeV, therefore $T\lesssim 1$ MeV. The liquid-drop part
of the fission barrier weakly depends on temperature at $T\lesssim 2$ MeV ds ;
pvb ; bgh ; gsb . As a result, the temperature dependence of the shell
correction contribution ds ; ach ; ach-b ; bq ; lpc ; dh ; dp ; dah  induces
the temperature dependence of the fission barrier of SHN.

The exponential reduction of the fission barrier of SHN with thermal
excitation energy is obtained in the framework of the finite-temperature self-
consistent Hartree-Fock+BCS model with Skyrme force in Refs. pnsk ; pnsk-a ;
snp ; zp . The exponential dependence of the fission barrier height is also
used in Refs. dh ; dns1 ; dns1c ; dns1aa ; dns1aa1 ; dns1aa2 ; dns1aa4 ; dns2a
; dns3 ; dns31 ; dns32 ; dns33 ; dns3a ; dns4b ; dns4c ; dns4d ; param3 ; ds ;
ds2019 . We also consider the exponentially decreasing of the fission barrier
with the excitation energy and define the fission barrier of excited rotating
nuclei as

$\displaystyle B_{\rm f}(\varepsilon,\ell)=B_{\rm f}^{\rm ld}+B_{\rm f}^{\rm
sh}e^{-\gamma_{D}\varepsilon}+\frac{\hbar^{2}\ell(\ell+1)}{2}\left[\frac{1}{J_{\rm
s}}-\frac{1}{J_{\rm gs}}\right].\;\;$ (57)

Here $B_{\rm f}^{\rm ld}$ and $B_{\rm f}^{\rm sh}(\varepsilon)$ are the
liquid-drop and shell-correction contribution of the fission barrier,
$\gamma_{D}$ is the damping parameter dp ; ds ; ds2019 ; dh ; snp ; pnsk ;
pnsk-a . The last line of equation describes the rotational contribution to
the barrier. $J_{\rm gs(s)}=m_{n}(2/5)R_{0}^{2}A(1+\sqrt{5/(16\pi)}\beta_{\rm
gs(s)})$ and $\beta_{\rm gs(s)}$ are, respectively, the ground-state (fission
saddle-point) momentum of inertia and quadrupole deformations of the compound
nucleus.

The dependence of the fission barrier of SHN on the excitation energy should
be taken into account in the evaluation of the survival probability. The Bohr-
Wheeler expression for the fission width bw ; vh  is obtained in the
transition state approach with the fission barrier independent of the
excitation energy. This expression is not consistent with the barrier
dependent on the excitation energy ssww ; ssww-a .

The number of states over barrier increases with the thermal reduction the
fission barrier height. Taking into account both the dependence of the fission
barrier on the excitation energy and the rising of the number of states over
the energy-dependent fission barrier we derived new expression for the fission
width in the form ds

$\displaystyle\Gamma_{\rm
f}(E^{*},\ell)=\frac{2}{2\pi\rho_{A,\ell}(E^{*})}\int_{0}^{\varepsilon_{\rm
max}}d\varepsilon\frac{\rho_{A,\ell}(\varepsilon)}{N_{\rm tot}}N_{\rm
saddle}(\varepsilon).$ (58)

Here the ratio $\rho(\varepsilon)/N_{\rm tot}$ is the probability to find the
fissioning nucleus with the intrinsic thermal excitation energy $\varepsilon$
in the fission transition state, $N_{\rm tot}=\int_{0}^{\varepsilon_{\rm
max}}d\varepsilon\rho_{A,\ell}(\varepsilon)$ is the total number of states
available for fission in the case of the energy-dependent fission barrier,
$N_{\rm saddle}(\varepsilon)=\int^{E-B_{\rm
f}(\varepsilon)}_{\varepsilon}de\rho_{A,\ell}(e)$ is the number of states
available for the fission at $\varepsilon$ and the barrier value $B_{\rm
f}(\varepsilon)$. $\varepsilon_{\rm max}$ is the maximum value of the
intrinsic thermal excitation energy of nucleus at the saddle point, which is
determined as the solution of the equation

$\displaystyle\varepsilon_{\rm max}+B_{\rm f}(\varepsilon_{\rm
max},\ell)=E^{*}.$ (59)

This equation is related to the energy conservation law, i.e. the sum of
thermal $\varepsilon_{\rm max}$ and potential $B_{\rm f}(\varepsilon_{\rm
max},\ell)$ energies at the saddle point equals to the total excitation energy
$E^{*}$.

The difference between the Bohr-Wheeler fission width bw  and $\Gamma_{\rm
fiss}(E^{*},\ell)$ is discussed in Refs. ds2019 ; ds . Note, $\Gamma_{\rm
f}(E^{*},\ell)$ is equal to the Bohr-Wheeler fission width in the case of the
energy-independent fission barrier ds .

##  III Discussion

In this section we compare the theoretical values of the SHN production cross
sections obtained in the framework of our model for various cold fusion
reactions with the experimental ones. At the beginning we like to point some
important experimental features of the cross-section data of the SHN
production.

The experimental cross sections for reaction 208Pb(64Ni,$n)^{271}$Ds are
measured in Gesellschaft fur Schwerionenforschung (GSI) h ; 108gsi , Lawrence
Berkeley National Laboratory (LBNL) 110lbnl ; folden , and Institute of
Physical and Chemical Research (RIKEN) 110riken . Unfortunately, the collision
energies and the values of the cross section in the maxima obtained in
different laboratories for this reaction are different. The difference between
the lowest h ; 108gsi  and highest 110riken  collision energies of the maxima
of the cross sections for reaction 208Pb(64Ni,$n)^{271}$Ds obtained in
different experiments is close to 5 MeV (see also Fig. 7 and Ref. folden ).
This value of energy difference is very large, because the cold fusion
reaction is usually taken place at sub-barrier collision energies. The values
of the sub-barrier fusion cross section may be changed on several orders at
the variation of the collision energy of 5 MeV. Such changes of the sub-
barrier fusion cross section on collision energy are typical for light and
medium heavy-ion systems, see, for example, Refs. dp ; fl ; dpil2 ; dhrs ;
motstef ; den-subfus . Similar dependence of the capture process should be in
the case of cold fusion reactions too.

The targets used in experiments for the SHN synthesis are thick. The reaction
of SHN production may take place when the projectile is just entering into the
target or just before the projectile escaping the target. The beam loss energy
in the target is close to 3–4 MeV folden . As a rule, the collision energies
in the middle of the target or the laboratory beam energies are pointed in the
experimental works. In these cases, the reaction may take place at energies on
approximately $\pm 2$ MeV or $4$ MeV lower from the experimentally pointed
ones.

The maximal values of the cold-fusion cross section measured in different
laboratories are strongly varied too. For example, the maximal values of the
cross section for reaction 208Pb(50Ti,$n)^{257}$Rf presented in Refs. 104gsi ;
104gsi-a ; 104lbnl  differ about three times, (see also Fig. 3). Note that the
number of reaction events is relatively high for this reaction. Due to this
the statistical errors are low in both experiments. In contrast to this, 1 or
2 reaction events are only obtained in experiments for the production of very
heavy SHN in cold-fusion reactions. Therefore, the experimental errors of the
cross section are very high, (see, for example, Fig. 8).

Due to these reasons, the exact values of the heavy-ion collision energies at
which the SHN form in experiments and the cross-section values have not been
well-defined now. Consequently, this is not a sense to describe the cross-
section of the SHN production precisely. Therefore, we use a ”soft” criterion
of the agreement between the experimental data and the theoretical
calculations. This criterion assumes that we describe both the maximal value
of the cross section for some experimental measurement with 50% accuracy and
the energy position of the maximum of the cross section with a precision of
several MeVs. Moreover, we see the more physical sense in the smaller changes
of the fitting parameters for the nearest reactions, than in better agreement
with the data. Below we will point, when the fitting parameters drastically
changed for the nearest reactions and discuss the reason of these changes.

###  III.1 Reaction 208Pb(50Ti,$xn)^{258-x}$Rf

Figure 3:  A comparison of our theoretical calculations of the cross sections
for reactions 208Pb(50Ti,$xn)^{258-x}$Rf with $x=1,2$ and 3 with available
experimental data. The cross sections for reactions 208Pb(50Ti,$xn)^{258-x}$Rf
with $x=1$ and 2 are measured in Refs. 104gsi ; 104gsi-a  (GSI) and 104lbnl
(LBNL) and the cross sections for reaction 208Pb(50Ti,$3n)^{255}$Rf are
obtained in Ref. 104gsi ; 104gsi-a  (GSI).

The values of the cross sections for reactions 208Pb(50Ti,$xn)^{258-x}$Rf with
$x=1$ and 2 are measured in GSI 104gsi ; 104gsi-a  and LBNL 104lbnl . The
experimental cross sections for reaction 208Pb(50Ti,$3n)^{255}$Rf are only
obtained in GSI 104gsi ; 104gsi-a . We calculate the cross sections for these
reactions in the framework of our model and compare the obtained values with
the experimental data in Fig. 3. Our results agree well with the LBNL data
104lbnl , but the GSI data shift to the lower collision energies relatively
both the LBNL data and our results, see Fig. 3. Taking into account the ”soft”
criterium we conclude that the experimental cross sections for reactions
208Pb(50Ti,$xn)^{258-x}$Rf with $x=1\div 3$ are well described in the
framework of our model.

The parameters of the model used in the calculation of reactions
208Pb(50Ti,$xn)^{258-x}$Rf are presented in Tables 2–4. Note, these values of
parameters are not unique. It is also possible to describe the data by
choosing slightly other values of parameters.

The starting values of the fission barrier $B_{\rm f}(0,0)$ and the compound
nucleus formation barrier $B_{\rm 0,CN}^{\rm DNSf}$ are extracted from Fig. 1.
However, the final values of the parameters presented in Tables 3–4 are chosen
by the fine fitting of the experimental data. The values of these barriers
given in Tables are close to the ones extracted in Fig. 1. The ground state
$\beta_{\rm gs}$ and saddle point $\beta_{\rm sp}$ deformations of fissioning
nuclei are taken from Fig. 1.

Table 3: The liquid-drop $B_{\rm f}^{\rm ld}$ and shell $B_{\rm f}^{\rm sh}$ contributions of the fission barriers $B_{\rm f}^{0}=B_{\rm f}^{\rm ld}+B_{\rm f}^{\rm sh}$ of nuclei, the ground state $\beta_{\rm gs}$ and saddle point $\beta_{\rm sp}$ deformations of fissioning nuclei, and the damping parameter of the fission barrier $\gamma_{D}$. The fission barrier values $B_{\rm f}$ obtained in Refs. etfsi ; mol  are also presented. The values of barriers are given relatively the ground-state energy of the compound nucleus in MeV. The values of $\gamma_{D}$ are presented in MeV-1. Nucleus | $B_{\rm f}^{\rm ld}$ | $B_{\rm f}^{\rm sh}$ | $B_{\rm f}^{0}$ | $B_{\rm f}^{[127]}$ | $B_{\rm f}^{[135]}$ | $\beta_{\rm gs}$ | $\beta_{\rm sp}$ | $\gamma_{D}$
---|---|---|---|---|---|---|---|---
258Rf | 0.5 | 6.3 | 6.8 | 5.0 | 5.65 | 0.2 | 0.4 | 0.105
257Rf | 0.5 | 6.1 | 6.6 | 5.6 | 6.02 | 0.2 | 0.4 | 0.105
256Rf | 0.5 | 6.5 | 7.0 | 5.3 | 6.26 | 0.2 | 0.4 | 0.105
262Sg | 0.4 | 3.7 | 4.1 | 4.3 | 5.91 | 0.2 | 0.4 | 0.07
261Sg | 0.4 | 3.3 | 3.7 | 4.7 | 5.88 | 0.2 | 0.4 | 0.07
260Sg | 0.4 | 4.8 | 5.2 | 4.6 | 5.84 | 0.2 | 0.4 | 0.11
259Sg | 0.3 | 3.0 | 3.3 | 4.9 | 5.82 | 0.2 | 0.4 | 0.11
266Hs | 0.5 | 5.7 | 6.3 | 3.5 | 6.26 | 0.2 | 0.4 | 0.10
265Hs | 0.5 | 4.2 | 4.7 | 3.5 | 6.26 | 0.2 | 0.4 | 0.10
272Ds | 0.4 | 3.7 | 4.1 | 2.2 | 7.31 | 0.2 | 0.4 | 0.04
271Ds | 0.3 | 3.5 | 3.8 | 2.2 | 6.92 | 0.2 | 0.4 | 0.04
278Cn | 0.2 | 2.3 | 2.5 | 1.9 | 5.99 | 0.0 | 0.3 | 0.04
277Cn | 0.3 | 2.6 | 2.9 | 2.0 | 6.36 | 0.0 | 0.3 | 0.04
286Fl | 0.4 | 4.2 | 4.6 | 4.1 | 9.00 | 0.0 | 0.3 | 0.05
285Fl | 0.4 | 4.0 | 4.4 | 2.7 | 8.82 | 0.0 | 0.3 | 0.05

The experimental information on the fission barrier height in SHN is very
poor. The values of the fission barrier obtained in various models sob ; sob1
; etfsi ; d-uhe ; mol ; snp ; pnsk ; pnsk-a ; kowal ; kowal-a ; kjs ; afanas ;
afanas-a ; agbemava ; warga  for SHN are very different. The difference
reaches more than 100% in some cases agbemava . The reasons of uncertainties
in the predictions of the fission barrier heights in SHN have been discussed
in Ref. agbemava . Therefore, it is difficult to prefer any model for the
barrier calculation. Nevertheless, we present the fission barrier values
$B_{\rm f}$ obtained in Refs. etfsi ; mol  in Table 3 for a comparison. The
values of barriers $B_{\rm f}(0,0)=B_{\rm f}^{\rm ld}+B_{\rm f}^{\rm sh}$
obtained with the help of Fig. 1 and after the fine fitting of the data belong
to the range of barrier values found in other approaches, see Table 3. The
values of the liquid-drop $B_{\rm f}^{\rm ld}$ and shell $B_{\rm f}^{\rm sh}$
contributions into the fission barriers of nuclei and the ground state
$\beta_{\rm gs}$ and saddle point $\beta_{\rm sp}$ deformations of fissioning
nuclei are near the corresponding values from Refs. kjs . The values of the
liquid-drop $B_{\rm f}^{\rm ld}$ contribution of the fission barrier are close
to 10% of the shell contribution $B_{\rm f}^{\rm sh}$.

Table 4: The values of barrier of the fusion trajectory at the formation of the compound nucleus from the DNS $B_{\rm 0,CN}^{\rm DNS,f}$ and the parameter $\Delta_{B}$. The values of $\Delta_{B}$ and $B_{\rm 0,CN}^{\rm DNS,f}$ are given in MeV. The values of barriers $B_{\rm 0,CN}^{\rm DNS,f}$ is given relatively the ground state energy of the compound nucleus. The last column shows the laboratory, in which the experimental data are obtained for the reaction. Reaction | $\Delta_{B}$ | $B_{\rm 0,CN}^{\rm DNS,f}$ | Exp. lab.
---|---|---|---
208Pb+50Ti | 5.0 | 12.5 | GSI, LBNL
208Pb+52Cr | 4.0 | 14.7 | LBNL
208Pb+54Cr | 11.0 | 11.1 | GSI
208Pb+58Fe | 7.0 | 10.6 | GSI, RIKEN
208Pb+64Ni | 6.5 | 6.2 | GSI,LBNL,RIKEN
208Pb+70Zn | 4.0 | 2.7 | GSI, RIKEN
208Pb+78Ge | 4.0 | 4.9 |

The values of damping parameter of the fission barrier $\gamma_{D}$ depend on
the numbers of protons and neutrons in nucleus and belong to the range from
$\approx 0.1$ to $0.03$ MeV-1 snp ; pnsk ; pnsk-a . The values of $\gamma_{D}$
presented in Table 3 belong to this range. The influence of this parameter on
the evaporation characteristics are widely discussed, see, for details, Refs.
dp ; ds ; ds2019 ; dh ; snp ; pnsk ; pnsk-a ; dns1 ; dns1c ; dns2a ; dns3 ;
dns31 ; dns32 ; dns33 ; dns3a ; dns4b ; dns4c ; dns4d ; zg ; param3 . Note
that parameters $\gamma_{D}$ and $\gamma$ in Eq. (27) are different, because
they are obtained by the fitting of different physical quantities.

As has pointed earlier, the accurate values of the collision energies of
reaction leading to the SHN are not known due the thick target and the
differences of the experimental data obtained in various laboratories.
Therefore, it is reasonable to fit the maximum of the theoretical cross
section using the parameter $\Delta_{B}$, because the position of the maximum
depends strongly on it. We remind, that the parameter $\Delta_{B}$ is linked
to the width of the fusion barrier distribution $g$, see Eqs. (4)-(5) and
related text. The value of $g$ is not known experimentally. The value of
$\Delta_{B}$ obtained from the fit of the data 104lbnl  is given in Table 4.
The value of $\Delta_{B}$ presented in Table corresponds to the value of the
width of barrier distribution $g=3.3$ MeV, which belongs to the range of
typical values of $g=2-4$ MeV used in other models zg ; dns4b ; dns4c ; dns4d
; dns5 . Note, the GSI data can be fitted by using larger values of
$\Delta_{B}$.

The position of the barrier $B_{\rm 0,CN}^{\rm DNS,f}$ corresponds to the
contacting near-spherical incident nuclei formed the DNS, see Fig. 1. Due to
this the momentum of inertia related to this barrier, see Eq. (50), is
calculated as $J_{\rm CN}^{\rm fus}\approx\mu(R_{1}+R_{2})^{2}$.

###  III.2 Reaction 208Pb(52Cr,$xn)^{260-x}$Sg

Figure 4:  A comparison of our theoretical calculations of the cross sections
for reactions 208Pb(52Cr,$xn)^{260-x}$Rf with $x=1$ and 2 with the
experimental data measured in Ref. 106lbnl  (LBNL).

The values of the cross sections for reactions 208Pb(52Cr,$xn)^{260-x}$Sg with
$x=1$ and 2 are measured in LBNL 106lbnl . The results calculated in the
framework of our model agree well with the experimental data, see Fig. 4. The
parameters of the model used in the calculation of these reactions are
presented in Tables 2–4. The values of parameters are chosen similar way as
the ones for reactions 208Pb(50Ti,$xn)^{258-x}$Rf. The parameter values for
reactions 208Pb(52Cr,$xn)^{260-x}$Sg are close to the corresponding ones for
reaction 208Pb(50Ti,$xn)^{258-x}$Rf. However, we use a smaller value of the
dissipation parameter $\gamma_{D}$, see Table 3, which is responsible for
attenuation of the shell contribution of the fission barrier. Recall, that the
value of $\gamma_{D}$ depends strongly on the numbers of protons and neutrons
in nucleus snp ; pnsk ; pnsk-a .

###  III.3 Reaction 208Pb(54Cr,$xn)^{262-x}$Sg

The cross sections for reactions 208Pb(54Cr,$xn)^{262-x}$Sg with $x=1$ and 2
are measured in GSI 106gsi . We calculate the cross sections for these
reactions in the framework of our model. Our model describes well the
experimental data, see Fig. 5.

The parameters of the model used in calculation of reactions
208Pb(54Cr,$xn)^{262-x}$Sg are given in Tables 2–4. The parameters presented
in Table 4 are similar to the corresponding ones for reactions
208Pb(50Ti,$xn)^{258-x}$Rf and 208Pb(52Cr,$xn)^{260-x}$Sg.

Figure 5:  A comparison of our theoretical calculations of the cross sections
for reactions 208Pb(54Cr,$xn)^{262-x}$Rf with $x=1$ and 2 with the
experimental data measured in Ref. 106gsi  (GSI).

We remind that the maxima of the SHN cross sections for reactions measured in
GSI is taking place at smaller collision energies than the ones measured in
other laboratories. Therefore, for the sake of the data description, we use
the largest value of parameter $\Delta_{B}$ for this reaction in comparison to
other reactions, see Table 4. The value of the barrier $B_{\rm 0,CN}^{\rm
DNS,f}$ for 208Pb(52Cr,$xn)^{260-x}$Sg reaction is higher than the one for the
reaction 208Pb(54Cr,$xn)^{262-x}$Sg, see Table 4. The excitation energy of the
compound nucleus at the maximum of the SHN cross section reduces, when the
maximum is shifted to smaller collision energies. Therefore, we change the
parameters related to the survival probability of the compound nucleus. The
values of fission barriers for these nuclei presented in Table 3 are close to
the ones obtained in Refs. etfsi ; mol .

###  III.4 Reaction 208Pb(58Fe,$xn)^{266-x}$Hs

The values of the cross sections for reaction 208Pb(58Fe,$1n)^{265}$Hs are
measured in GSI h ; 108gsi  and for reactions 208Pb(58Fe,$xn)^{266-x}$Hs with
$x=1$ and 2 are obtained in RIKEN 108riken . The cross sections for these
reactions calculated in the framework of our model agree well with the
experimental data, see Fig. 6.

The difference in the collision energy values of the cross section maxima
measured in GSI and RIKEN for reaction 208Pb(58Fe,$1n)^{265}$Hs is close to 4
MeV, see Fig. 6. It is reasonable to make the maximum of the cross section in
our model for reaction 208Pb(58Fe,$1n)^{265}$Hs at the collision energy
between the ones obtained in GSI and RIKEN. Due to this the value of
$\Delta_{B}$ for this reaction lies between the ones for reactions with 52Cr
and 54Cr projectiles, see Table 4. The width of the cross section peaks are
close to the experimental ones.

Figure 6:  A comparison of our theoretical calculations of the cross sections
for reactions 208Pb(58Fe,$xn)^{266-x}$Hs with $x=1$ and 2 with available
experimental data. The cross sections for reaction 208Pb(58Fe,$1n)^{265}$Hs
are measured in Refs. h ; 108gsi  (GSI) and for reactions
208Pb(58Fe,$xn)^{266-x}$Hs $x=1$ and 2 are obtained in Refs. 108riken
(RIKEN).

The other parameters of the model used in the calculation of reactions
208Pb(58Fe,$xn)^{266-x}$Hs are presented in Tables 2–4. The values of
parameters are chosen similar way as for other reactions.

Our value of the fission barrier for 266Hs is very close to the one obtained
in Ref. mol  and larger the one presented in Ref. etfsi . Our value of the
fission barriers for 265Hs lies between the ones obtained in Refs. etfsi ; mol
.

###  III.5 Reaction 208Pb(64Ni,$xn)^{272-x}$Ds

The cross sections for reaction 208Pb(64Ni,$1n)^{271}$Ds are measured in GSI h
; 108gsi , LBNL 110lbnl , and RIKEN 110riken . We calculate the cross sections
for reactions 208Pb(64Ni,$xn)^{272-x}$Ds with $x=1$ and $2$ in the framework
of our model. The calculated values agree well with the available experimental
data, see Fig. 7.

The parameters of the model used in the calculation of reactions
208Pb(64Ni,$1n)^{271}$Ds are given in Tables 2–4. We select the value of
parameter $\Delta_{B}$ so that the theoretical peak of the cross sections lies
close to the RIKEN data. This value of parameter $\Delta_{B}$ is close to the
ones obtained for a description of the RIKEN data with other projectiles, see
Table 4. The value of the barrier of the fusion trajectory from the DNS to the
compound nucleus $B_{\rm 0CN}^{\rm DNS,f}$ is estimated using Fig. 2. We see
that the values $B_{\rm 0,CN}^{\rm DNSf}$ decrease with rising to projectile
mass, see Figs. 1–2 and Table 4. The smallest value of the dissipation
parameter $\gamma_{D}$ is used for this reaction, see Table 3. The values of
other parameters are chosen similar way as before.

The cross-section points of this reaction measured in GSI, see Fig. 7, locate
at smaller collision energies, than the ones measured in RIKEN and the results
obtained in our model. The difference in the position of the cross-section
maxima for this reaction measured in different laboratories is close to 5 MeV.

Figure 7:  A comparison of our theoretical calculations of the cross sections
for reactions 208Pb(64Ni,$1n)^{271}$Ds with available experimental data. The
cross sections for this reaction are measured in Refs. h ; 108gsi  (GSI),
110riken  (RIKEN) and 110lbnl  (LBNL). The theoretical calculation of the
cross section for reaction 208Pb(64Ni,$2n)^{270}$Ds is also presented.

The maximum of the cross section for the reaction with 2n emission
208Pb(64Ni,$2n)^{270}$Ds is located at high beam energy, which is not used in
any experiments. The calculated value of cross section in the maximum for
reaction 208Pb(64Ni,$2n)^{270}$Ds is strongly lower than the one for reaction
208Pb(64Ni,$1n)^{271}$Ds, see Fig. 7.

###  III.6 Reaction 208Pb(70Zn,$xn)^{278-x}$Cn

The cross sections for reaction 208Pb(70Zn,$1n)^{277}$Cn are measured in GSI h
; 112gsi  and RIKEN 112riken ; 112riken1 . We calculate the cross sections for
this reaction in the framework of our model. We describe well the data from
both laboratories in our model by the corresponding parameter choice, see Fig.
8. The values of parameters used in our calculations of reactions
208Pb(70Zn,$xn)^{277-x}$Cn with x=1 and 2 are presented in Tables 2–4.

The value of barrier $B^{\rm DNS,tr}_{0,{\rm CN}}$ for this system obtained in
our approach is 269.3 MeV, see Table 5. This value is close to the one
evaluated in Ref. dns3a  20.98 MeV - $Q_{\rm CN}$ = 20.98 MeV + 244.2 MeV =
265.18 MeV. As pointed in Ref. adamian , the calculations of the nucleus-
nucleus potentials of spherical nuclei in the framework of DNS model have the
Coulomb barriers which are at least 5 MeV lower than the phenomenological Bass
barriers bass . The values of barriers of the nucleus-nucleus potential for
very asymmetric systems with medium or small values of $Z_{1}Z_{2}$ calculated
in our approach are close to the Bass barriers d2015 . Therefore, such
difference between values of the barrier $B^{\rm DNS,tr}_{0,{\rm CN}}$
obtained in different approaches is reasonable.

Figure 8:  A comparison of our theoretical calculation of the cross section
for reactions 208Pb(70Zn,$1n)^{277}$Cn with available experimental data. The
cross sections for this reaction are measured in Refs. h ; 112gsi  (GSI) and
112riken ; 112riken1  (RIKEN). The theoretical calculation of the cross
section for reaction 208Pb(70Zn,$2n)^{276}$Cn is also presented.

The mechanism of this reaction is taken into account the contribution of the
intermediate state. This intermediate state locates in the well between the
inner and outer fission barriers, see Fig. 2. The height of the inner fission
barrier is lower the height of the outer fission barrier on approximately 1
MeV, see Fig. 2. The height of barrier related to the decay back to the DNS is
higher than the heights of the inner fission barriers, see Fig. 2 and Tables
3,4. Therefore, the decay of the intermediate state to the compound nucleus is
the most preferable for this reaction. Due to this we put $P_{is}\approx
0.88(0.85)$ for $1n(2n)$ reactions for a simplification of the calculation.

###  III.7 Reaction 208Pb(78Ge,$xn)^{286-x}$Fl

The values of the cross sections for reaction 208Pb(78Ge,$xn)^{285}$Fl for
$x=1$ and 2 have not been measured up to now. We calculate the cross sections
for these reactions in the framework of our model and present it in Fig. 9.
The values of SHN production cross section are very small for these reactions.

Figure 9:  The results of calculations of the cross sections for reactions
208Pb(78Ge,$xn)^{286-x}$Fl for $x=1$ and 2.

The parameters of the model used in the calculation of reactions
208Pb(78Ge,$xn)^{285}$Fl are given in Tables 2–4. The values of parameters are
chosen using a similar procedure as before.

The mechanism of compound nucleus formation in this reaction is also passed
through the intermediate state, see Fig. 2. The height of the inner fission
barrier is slightly higher than the height of the outer fission barrier, see
Fig. 2. The height of barrier related to the decay back to the DNS is higher
than the ones of the inner or outer fission barriers, see Fig. 2 and Tables
3,4. Therefore, the decay of the intermediate state into the quasi-fission
fragments is more preferable than decays into the compound nucleus or the DNS.
Due to this we put $P_{is}\approx 0.4$ for a simplification. The decay of the
intermediate state into the quasi-fission fragments for reaction
208Pb(78Ge,$xn)^{286-x}$Fl is more probable in a comparison to reaction
208Pb(70Zn,$xn)^{278-x}$Cn.

###  III.8 The probability of compound nucleus formation

Let us discuss the mechanisms of compound nucleus formation in our model by
comparing the barrier heights of different DNS decay processes. The various
barrier heights are given in Tables 1 and 5.

The barrier heights related to the transfer $B^{\rm DNS,tr}_{0,{\rm CN}}$ and
fusion $b_{0,{\rm CN}}^{\rm DNS,f}=B^{\rm DNS,f}_{0,{\rm CN}}-Q_{\rm CN}$
paths of the compound nucleus formation obey the inequality $B^{\rm
DNS,tr}_{0,{\rm CN}}>b_{\rm 0,CN}^{\rm DNS,f}$. We can conclude from this
inequality that $\Gamma_{\rm CN}^{\rm DNS,tr}(E,\ell)=0$ at small collision
energies and $\Gamma_{\rm CN}^{\rm DNS,f}(E,\ell)\gg\Gamma_{\rm CN}^{\rm
DNS,tr}(E,\ell)$ at high collision energies. Therefore, the transfer mechanism
of compound nucleus formation has a small contribution at high collision
energies only.

Table 5: The values of barrier heights for the DNS decay branches. $B^{\rm fus}_{0}$ is the barrier for the DNS decay into the incident channel, $b_{\rm 0,CN}^{\rm DNS,f}$ is the barrier for the DNS decay into the compound nucleus by the fusion pass, $B^{\rm DNS,tr}_{0,{\rm CN}}$ is the barrier for the DNS decay into the compound nucleus by the nucleon transfer way, $B^{\rm DNS}_{0,{\rm DIC}}$ is the barrier for the DNS decay into the more symmetric nuclear system than the incident channel, and $B^{\rm DNS}_{0,{\rm def}}$ is the barrier for the DNS decay into the incident deformed nuclei. $\delta_{B}=b_{\rm 0,CN}^{\rm DNS,f}-B^{\rm DNS}_{0,{\rm def}}$. The barriers are evaluated relatively the nucleus-nucleus interaction energy on the infinite distance between them at $\ell=0$. All values are given in MeVs. Reactions | $B^{\rm fus}_{0}$ | $b_{\rm 0,CN}^{\rm DNS,f}$ | $B^{\rm DNS,tr}_{0,{\rm CN}}$ | $B^{\rm DNS}_{0,{\rm DIC}}$ | $B^{\rm DNS}_{0,{\rm def}}$ | $\delta_{B}$
---|---|---|---|---|---|---
50Ti + 208Pb | 189.8 | 182.0 | 198.7 | 187.1 | 175.8 | 6.2
52Cr + 208Pb | 207.2 | 198.4 | 212.3 | 199.1 | 191.3 | 7.1
54Cr + 208Pb | 206.0 | 198.2 | 215.7 | 202.1 | 189.1 | 9.1
58Fe + 208Pb | 222.0 | 215.6 | 232.6 | 217.5 | 204.0 | 11.6
64Ni + 208Pb | 236.6 | 231.1 | 251.9 | 233.9 | 216.2 | 14.9
70Zn + 208Pb | 251.1 | 246.9 | 269.3 | 248.0 | 228.4 | 18.5
78Ge + 208Pb | 264.4 | 270.3 | 289.9 | 265.4 | 239.9 | 30.4

We see in Table 5 that for each reaction the height of barrier $B^{\rm
DNS}_{0,{\rm def}}$ are the lowest and $B^{\rm DNS,tr}_{0,{\rm CN}}>b_{\rm
0,CN}^{\rm DNS,f}$. Therefore, the values of the widths obey the inequalities:
$\Gamma_{\rm def}^{\rm DNS}(E,\ell)\gg\Gamma_{\rm CN}^{\rm DNS,tr}(E,\ell)$,
$\Gamma_{\rm def}^{\rm DNS}(E,\ell)\gg\Gamma_{\rm DIC}^{\rm DNS}(E,\ell)$, and
$\Gamma_{\rm def}^{\rm DNS}(E,\ell)\gg\Gamma_{\rm sph}^{\rm DNS}(E,\ell)$. As
a result, $\Gamma^{\rm tot}_{\rm CN}(E,\ell)\approx\Gamma_{\rm def}^{\rm
DNS}(E,\ell)$ and the probability of compound nucleus formation may be
approximated as

$\displaystyle P(E,\ell)\approx\frac{\Gamma_{\rm CN}^{\rm
DNS,f}(E,\ell)}{\Gamma_{\rm def}^{\rm DNS}(E,\ell)}.$ (60)

At the high excitation energy $E^{*}=E+Q_{\rm CN}$ of the compound nucleus
this expression can be presented in a simple form

$\displaystyle P(E^{*},\ell)$ $\displaystyle\approx$
$\displaystyle\frac{\Gamma_{\rm CN}^{\rm DNS,f}(E^{*},\ell)}{\Gamma_{\rm
def}^{\rm DNS}(E^{*},\ell)}$ (61) $\displaystyle\propto$
$\displaystyle\frac{\exp{\left\\{2\left[a_{\rm dens}\left(E^{*}-B_{0,{\rm
CN}}^{\rm DNS,f}\right)\right]^{1/2}\right\\}}}{\exp{\left\\{2\left[a_{\rm
dens}\left(E^{*}-(B_{0,{\rm def}}^{\rm DNS}+Q_{\rm
CN})\right)\right]^{1/2}\right\\}}}$ $\displaystyle\approx$
$\displaystyle\exp{\left\\{-\left[\frac{a_{\rm
dens}}{E^{*}}\right]^{1/2}\delta_{B}\right\\}},$

where $\delta_{B}=b_{\rm 0,CN}^{\rm DNS,f}-B^{\rm DNS}_{0,{\rm def}}$.

The difference between barriers $\delta_{B}$ increases from 6.2 MeV for
reaction 50Ti + 208Pb to 30.4 MeV for reaction 78Ge + 208Pb, see Table 5. Due
to this and Eq. (57), the probability of compound nucleus formation
$P(E^{*},\ell)$ strongly decreases with the increasing of the mass of the
projectile, see also Fig. 10. The dependencies of the probabilities of
compound nucleus formation $P(E^{*},\ell)$ on the excitation energy of the
compound nucleus formed in reactions 50Ti, 52,54Cr, 58Fe, 64Ni, 70Zn, and 78Ge
+ 208Pb for $\ell=0$ are presented in Fig. 10.

According to the statistical approach, see Eq. (57), the probability of
compound nucleus formation increases with the rising of energy $E^{*}$. This
rising is very strong at the energies near the barrier of the compound nucleus
formation by fusion trajectory pass. This tendency is clearly seen in Fig. 10.
The probability of compound nucleus formation rises smoother at higher
energies.

Figure 10:  The dependencies of the probability of compound nucleus formation
on the excitation energy of the compound nucleus formed in reactions 50Ti,
52,54Cr, 58Fe, 64Ni, 70Zn, and 78Ge + 208Pb for $\ell=0$.

The value of probability $P(E^{*},\ell)$ for reaction 78Ge + 208Pb is strongly
smaller than the one for reaction 70Zn + 208Pb, see Fig. 10. This is related
to the rise of the difference between barriers $\delta_{B}$ for reaction with
78Ge projectile, see Table 5. Due to this the cross sections for reaction 78Ge
+ 208Pb is strongly smaller than the one for reaction 70Zn + 208Pb, compare
results presented in Figs. 9 and 10.

We remind that the values of the barrier height between deformed incident
nuclei $B^{\rm DNS,f}_{0,{\rm def}}$ as well as the width $\Gamma_{\rm
def}^{\rm DNS}(E,\ell)$ depend strongly on the surface stiffness coefficient.
Therefore, the value of the surface stiffness coefficient affects considerably
on the probability of the compound nucleus formation. Due to this, the choice
of incident nuclei is very important for the values of compound nucleus
production cross section.

The crucial role of the transfer mechanism of the compound nucleus formation
in the framework of the DNS model is related to the small height of the
barrier $B^{\rm DNS,tr}_{0,{\rm CN}}$ in this model, see discussion of this
barrier height in Sec. III.F. The DNS path of compound nucleus formation is
main, when $b^{\rm DNS,f}_{0,{\rm CN}}>B^{\rm DNS,tr}_{0,{\rm CN}}$. In our
model $b^{\rm DNS,f}_{0,{\rm CN}}<B^{\rm DNS,tr}_{0,{\rm CN}}$, therefore the
fusion path of the compound nucleus production is basic. So, the role of
different mechanisms of SHN production depends the potential landscape, which
is defined by the model(s) for the calculation of the potential energy for
one- and two-body nuclear shapes.

##  IV Conclusion

We have presented the new model of the SHN production in the cold-fusion
reactions. This model takes into account the competition between the DNS
multi-nucleon transfer and fusion trajectories of the compound nucleus
formation. The available experimental data are well described in our model.

The compound nucleus is mainly formed by the fusion path, because the barrier
related to this fusion way is lower than the barrier related to the multi-
nucleon transfer from the light nucleus to heavy one (the DNS trajectory).

We show the correlation between the surface stiffnesses of nuclei involved in
the SHN production and the reaction cross sections. The using of more stiff
nuclei leads to higher cross section value due to reduction of the DNS decay
width to deformed nuclei.

The competition between the compound nucleus formation and the true quasi-
fission occurred along the fusion trajectory is taken into account for heavy
nucleus-nucleus systems leading to the SHN. This competition is related to the
existence of the intermediate state and connected to the landscape of the
potential energy surface. The intermediate state is important for reactions
with heavy projectiles. The quasi-fission is linked to the decay of the
intermediate state into fragments bypassing the formation of the compound
nucleus.

The yields of the various reaction processes in the model depend on the
relative values of the corresponding barrier heights. The values of the
barrier heights $B_{\rm 0,CN}^{\rm DNS,f}$ and $B_{\rm f}$ depend on the
choice of the nuclear structure model for calculation of these barriers and
the one-body shape parametrization. The uncertainty of the fission barrier
height $B_{\rm f}$ obtained in the framework of different models is several
MeVs, see Tabl. 3. So, we may expect similar uncertainty for the barrier
$B_{\rm 0,CN}^{\rm DNS,f}$. The values of the barriers $B^{\rm fus}_{0}$,
$B^{\rm DNS,tr}_{0,{\rm CN}}$, $B^{\rm DNS}_{0,{\rm DIC}}$ and $B^{\rm
DNS}_{0,{\rm def}}$ are determined by the choice of the nuclear part of the
nucleus-nucleus potential and the stiffness parameter of the nuclei. The
difference between the interaction barrier heights of spherical nuclei leading
to SHNs calculated in various approaches to the nuclear interaction part may
reach 20 MeV dn . We emphasize the barrier values calculated by using the
potential (6) agree well with the ones, extracted from the quasi-elastic
scattering, see Table 1, and with the empirical barriers for light and medium
heavy-ion systems d2015 . The values of stiffness parameter are only known for
some nuclei. Thus, the obtained results are model dependent, and the using
more accurate approaches are encouraged for further studies.

## Acknowledge

V. Yu. Denisov is grateful to Academician Yu. Ts. Oganessian and Professor S.
Hofmann for the stimulated discussions. V. Yu. Denisov is grateful to
Professor S. Hofmann for a digital form of the experimental data for reactions
208Pb(58Fe,$1n)^{265}$Hs and 208Pb(64Ni,$1n)^{271}$Ds 108gsi , which were
presented in Ref. dh  too.

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