# Analytical parametrization of fusion barriers using proximity potentials

Ishwar Dutt  Rajeev K. Puri  rkpuri<EMAIL_ADDRESS>Department of
Physics, Panjab University, Chandigarh 160014, India

###### Abstract

Using the three versions of proximity potentials, namely proximity 1977,
proximity 1988, and proximity 2000, we present a pocket formula for fusion
barrier heights and positions. This was achieved by analyzing as many as 400
reactions with mass between 15 and 296. Our parametrized formula can
reproduced the exact barrier heights and positions within an accuracy of $\pm
1\%$. A comparison with the experimental data is also in good agreement.

###### pacs:

24.10.-i, 25.70.Jj, 25.70.-z.

##  I Introduction

In the low energy heavy-ion collisions, fusion of colliding nuclei and related
phenomena has always been of central interestrkp1 . Depending upon the
incident energy of the projectile as well as angular momentum and impact
parameter, the collision of nuclei can lead to several interesting phenomena
such as incomplete fusion rkp1 , multifragmentation rkp2 ; rkp3 , subthreshold
particle production rkp4 , nuclear flow rkp5  as well as formation of the
superheavy elements rkp6 . Since fusion is a low density phenomenon, several
mean field models rkp1 ; rkp6 ; blocki77 ; wr94 ; ms2000 ; wang06 ; deni02
have been developed in the recent past at microscopic/macroscopic level and
have been robust against the vast experimental data ms2000 ; wang06 ; expt
that range from symmetric to highly asymmetric colliding nuclei. The study of
mass dependence has always guided the validity of various models irrespective
of the energy range. The essential idea of developing a model is to understand
the physical mechanism behind a process or phenomenon. Extension of the
physics is also reported toward isospin degree of freedom. At the same time,
accumulation of huge experimental data ms2000 ; wang06 ; expt  (that include
all kinds of masses and asymmetry of colliding nuclei) puts stringent test for
any theoretical model.

As fusion process occurs at the surface of colliding nuclei, any difference
occurring in the interior part of the potential does not make any difference
toward the fusion. One always tries to parametrize the potential in terms of
some known quantities such as the masses and charges of colliding nuclei rkp1
; deni02 ; bass73 ; ngo75 . At intermediate energies, several forms of density
dependent potentials are also available rkp2 ; rkp3 ; rkp4 ; rkp5 . Generally,
the benchmark is to parameterized the outcome in proximity fashion blocki77 .
By adding the Coulomb potential to the parameterized form of the nuclear ion-
ion potential, one obtains total ion-ion potential and ultimately, the fusion
barriers and cross sections.

Alternatively, one calculates the barrier heights as well as positions of
large number of reactions and then tries to parametrize these in terms of some
known quantities like the charges and masses of the colliding nuclei rkp1 ;
skg82 . Recently, even neutron excess dependence has also been incorporated in
some attempts ng04 . Similarly, an analytical expression to determine the
barrier heights and positions are also presented in Ref. rm01 . The cost of
such attempts was in the form of more complicated parametrized form. The
utility of such direct parametrization is that one can use these pocket
formula to find out the fusion barriers instantaneously.

As is evident from the literature, several modifications over the original
proximity potential have also been suggested in the recent years wr94 ; ms2000
. We shall here attempt to present a direct parametrization of the fusion
barrier positions as well as heights using different proximity potentials.
This attempt will introduce great simplification in obtaining the fusion
barrier positions and heights. Section II describes the models in brief, Sec.
III depicts the results, and a summary is presented in Sec. IV.

##  II  The Model

All proximity potentials are based on the proximity force theorem. According
to which, _“the force between two gently curved surfaces in close proximity is
proportional to the interaction potential per unit area between the two flat
surfaces”_. The nuclear part of the interaction potential in different
proximity potentials is described as a product of geometrical factor
representing the mean curvature of the interacting surfaces and an universal
function depending on the separation distance.

###  II.1 $\rm Proximity~{}1977~{}(Prox~{}77)$

According to the original version of proximity blocki77 , the interaction
potential $V_{N}(r)$ between two surfaces can be written as

$V_{N}^{Prox~{}77}(r)=4\pi\gamma
b\overline{R}\Phi\left(\frac{{r}-C_{1}-C_{2}}{b}\right){~{}\rm MeV},$ (1)

where the surface energy coefficient $\gamma$ taken from the Lysekil mass
formula $(~{}\rm in~{}MeV/fm^{2})$ is written as

$\gamma=\gamma_{0}\left[1-k_{s}I^{2}\right],$ (2)

with $I=\left(\frac{N-Z}{A}\right)$; $N$, $Z$, and $A$ refer to the neutron,
proton and total mass of two interacting nuclei. Though the proximity
potential Prox 77, in principle, is for zero-neutron excess, the factor
$\gamma$ takes care of some neutron excess content. In the above formula,
$\gamma_{0}$ is the surface energy constant and $k_{s}$ is the surface-
asymmetry constant. Both constants were first parametrized by Myers and
Świa̧tecki ms66  by fitting the experimental binding energies. The first set
of these constants yielded values $\gamma_{0}$ and
$k_{s}=1.01734~{}\rm~{}MeV/fm^{2}$ and 1.79, respectively. Later on, these
values were revised to ${~{}\rm\gamma_{0}}$ = 0.9517 $~{}\rm MeV/fm^{2}$ and
$k_{s}=1.7826$ ms67 . Interestingly, most of the modified proximity type
potentials use different values of the parameter $\gamma$ wr94 ; ms2000 . The
mean curvature radius, $\overline{R}$ in Eq. (1) has the form

$\overline{R}=\frac{C_{1}C_{2}}{C_{1}+C_{2}},$ (3)

quite similar to the one used for reduced mass. Here

$C_{i}=R_{i}\left[1-\left(\frac{b}{R_{i}}\right)^{2}+\cdots\cdots\right],$ (4)

${\rm R_{i}}$, the effective sharp radius, reads as

$R_{i}=1.28A^{1/3}_{i}-0.76+0.8A^{-1/3}_{i}{~{}\rm fm}~{}~{}(i=1,2).$ (5)

The universal function $\Phi\left(\xi\right)$ was parametrized with the
following form:

$\Phi\left(\xi\right)=\left\\{\begin{array}[]{l}-\frac{1}{2}\left(\xi-2.54\right)^{2}-0.0852\left(\xi-2.54\right)^{3},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\xi\leq 1.2511$ },\\\ -3.437\exp\left(-\frac{\xi}{0.75}\right),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\xi\geq 1.2511$ },\end{array}\right.$ (6)

with $\xi$ = $(r-C_{1}-C_{2}$)/$b$. The width $b$ has been evaluated close to
unity. Using the above form, one can calculate the nuclear part of the
interaction potential ${V_{N}(r)}$. This model is referred as Prox 77 and
corresponding potential as $V_{N}^{Prox~{}77}(r)$.

###  II.2 $\rm Proximity~{}1988~{}(Prox~{}88)$

Later on, using the more refined mass formula of Möller and Nix mn81 , the
value of coefficients $\gamma_{0}$ and $k_{s}$ were modified yielding their
values =1.2496 $\rm MeV/fm^{2}$ and 2.3, respectively. Reisdorf wr94  labeled
this modified version as ‘Proximity 1988’. Note that this set of coefficients
give stronger attraction compared to the above sets. Even a more recent
compilation by Möller and Nix mn95  yields similar values. We marked this
potential as Prox 88.

###  II.3 $\rm Proximity~{}2000~{}(Prox~{}00)$

Recently, Myers and Świa̧tecki ms2000  modified Eq. (1) by using up-to-date
knowledge of nuclear radii and surface tension coefficients using their
droplet model concept. The prime aim behind this attempt was to remove
discrepancy of the order of $4\%$ reported between the results of Prox 77 and
experimental data ms2000 . Using the droplet model ms80 , matter radius
$C_{i}$ was calculated as

$C_{i}=c_{i}+\frac{N_{i}}{A_{i}}t_{i}~{}~{}~{}~{}(i=1,2),$ (7)

where $c_{i}$ denotes the half-density radii of the charge distribution and
$t_{i}$ is the neutron skin of the nucleus. To calculate $c_{i}$, these
authors ms2000  used two-parameter Fermi function values given in Ref. dv87
and remaining cases were handled with the help of parametrization of charge
distribution described below. The nuclear charge radius (denoted as $R_{00}$
in Ref. bn94 ), is given by the relation:

$R_{00i}=\sqrt{\frac{5}{3}}\left<r^{2}\right>^{1/2}$
$\displaystyle=1.240A_{i}^{1/3}\left\\{1+\frac{1.646}{A_{i}}-0.191\left(\frac{A_{i}-2Z_{i}}{A_{i}}\right)\right\\}{~{}\rm
fm}~{}~{}$ $\displaystyle(i=1,2),$ (8)

where $<r^{2}>$ represents the mean square nuclear charge radius. According to
Ref. bn94 , Eq. (8) was valid for the even-even nuclei with $8\leq Z<38$ only.
For nuclei with $Z\geq 38$, the above equation was modified by Pomorski _et
al_. bn94  as

$R_{00i}=1.256A_{i}^{1/3}\left\\{1-0.202\left(\frac{A_{i}-2Z_{i}}{A_{i}}\right)\right\\}{~{}\rm
fm}.$ (9)

These expressions give good estimate of the measured mean square nuclear
charge radius $<r^{2}>$. In the present model, authors used only Eq. (8). The
half-density radius, $c_{i}$ was obtained from the relation:

$c_{i}=R_{00i}\left(1-\frac{7}{2}\frac{b^{2}}{R_{00i}^{2}}-\frac{49}{8}\frac{b^{4}}{R_{00i}^{4}}+\cdots\right)~{}~{}~{}~{}~{}~{}~{}(i=1,2).$
(10)

Using the droplet model ms80 , neutron skin $t_{i}$ reads as

$t_{i}=\frac{3}{2}r_{0}\left[\frac{JI_{i}-\frac{1}{12}c_{1}Z_{i}A^{-1/3}_{i}}{Q+\frac{9}{4}JA^{-1/3}_{i}}\right](i=1,2).$
(11)

Here $r_{0}$ is $1.14$ fm, the value of nuclear symmetric energy coefficient
$J=32.65$ MeV and $c_{1}=3e^{2}/5r_{0}=0.757895$ MeV. The neutron skin
stiffness coefficient $Q$ was taken to be 35.4 MeV. The nuclear surface energy
coefficient $\gamma$ in terms of neutron skin was given as;

$\gamma=\frac{1}{4\pi
r^{2}_{0}}\left[18.63{\rm(MeV)}-Q\frac{\left(t^{2}_{1}+t^{2}_{2}\right)}{2r^{2}_{0}}\right],$
(12)

where $t_{1}$ and $t_{2}$ were calculated using Eq. (11). The universal
function $\Phi(\xi)$ is reported as

$\Phi\left(\xi\right)=\left\\{\begin{array}[]{ll}-0.1353+\sum\limits_{n=0}^{5}\left[c_{n}/\left(n+1\right)\right]\left(2.5-\xi\right)^{n+1},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for \quad$0<\xi\leq 2.5$},\\\
-0.09551\exp\left[\left(2.75-\xi\right)/0.7176\right],\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{
for $\quad\xi\geq 2.5$}.\end{array}\right.$ (13)

The values of different constants $c_{n}$ were: $c_{0}=-0.1886$,
$c_{1}=-0.2628$, $c_{2}=-0.15216$, $c_{3}=-0.04562$, $c_{4}=0.069136$, and
$c_{5}=-0.011454$. For $\xi>2.74$, the above exponential expression is the
exact representation of the Thomas-Fermi extension of the proximity potential.
This potential is marked as Prox 00.

##  III Results and Discussion

As a first step, we calculated the nuclear part of the ion-ion potential using
Prox 77, Prox 88, and Prox 00 potentials and then by adding the Coulomb
potential (= $\frac{Z_{1}Z_{2}e^{2}}{r}$), total ion-ion potential $V_{T}(r)$
for spherical colliding pair is obtained. The fusion barrier is then extracted
using conditions

$\frac{dV_{T}(r)}{dr}|_{r=R_{B}}=0,~{}~{}{\rm{and}}~{}~{}\frac{d^{2}V_{T}(r)}{dr^{2}}|_{r=R_{B}}\leq
0.$ (14)

The height of the barrier and position is marked, respectively, as $V_{B}$ and
$R_{B}$. For the present analysis, all kind of the reactions involving
symmetric $(N=Z,~{}A_{1}=A_{2})$ as well as asymmetric $(N\neq Z,~{}A_{1}\neq
A_{2})$ nuclei are considered. In all, 400 reactions covering almost whole of
the periodic table are taken into account. All nuclei considered here are
assumed to be spherical in nature, however, deformation as well as orientation
of the nuclei also affect the fusion barriers deni07 . The lightest reaction
considered here is ${}^{6}Li+^{9}Be$ whereas the heaviest one is
${}^{48}Ca+^{248}Cm$. As reported in Ref. ms2000 , proximity Prox 77
overestimate experimental data by $4\%$. It was reported to be better for
newer versions.

Once fusion barrier heights and positions were calculated, a search was made
for their parametrization. Since it is evident that barrier positions depend
on the size of the colliding systems, the best way is to parametrize them in
terms of the radius dependence i.e. in terms of $A^{1/3}$. In the literature,
several attempts exist that parametrize $R_{B}$ directly either as
$A^{{}^{\prime}}+B^{{}^{\prime}}(A_{1}^{1/3}+A_{2}^{1/3})$ anjos02 ; broglia81
; kovar79 ; cw76  or as $r_{B}~{}(=\frac{R_{B}}{A_{1}^{1/3}+A_{2}^{1/3}}$)
cngo75 ; ngo80 . We have also tried similar fits. Unfortunately, the
scattering around the mean curve was quite significant in both the cases,
therefore, we discard this kind of parametrizations. Alternatively, we plotted
the reduced fusion barrier positions $s_{B}=R_{B}-C_{1}-C_{2}$, as a function
of $\frac{Z_{1}Z_{2}}{A_{1}^{1/3}+A_{2}^{1/3}}$ for all three versions of
proximity potentials (see Fig. 1). Very encouragingly, the reduced barrier
positions $s_{B}$ of all the reactions fall on the mean curve that can be
parametrized in terms of exponential function. We noted that the scattering
around the mean positions is very small. Due to the weak Coulomb force in
lighter colliding nuclei, lesser attractive potential is needed to
counterbalance it. As a result, separation distance increases in lighter
colliding nuclei. As we go to heavier nuclei, stronger Coulomb contribution
demands more and more penetration, therefore, decreasing the value of $s_{B}$.
In other words, the fusion in lighter nuclei occurs at the outer region
compared to the heavier nuclei where $s_{B}$ is much smaller.

If we compare (a) and (b) parts of the Fig. 1, we notice that $s_{B}$, the
separation distance between nuclei is slightly more in Prox 88 compared to
Prox 77. This is due to the fact that Prox 88 has stronger surface energy
coefficient $\gamma$ [see Eq. (2) with $\gamma_{0}$ = $1.2496~{}\rm
MeV/fm^{2}$ and $k_{s}=2.3$ respectively]. This results in more attractive
nuclear potential compared to Prox 77 and therefore, counterbalancing happens
at larger distances. From the figure, it is also evident that latest proximity
potential has shallow nuclear potential compared to the other two versions.
All three proximity potentials follow similar mass/ charge dependence and can
be parametrized in terms of following function:

${s_{B}^{par}}=\alpha\exp\left[-\beta\left(x-2\right)^{1/4}\right].$ (15)

Here, $x$ = $\frac{Z_{1}Z_{2}}{A_{1}^{1/3}+A_{2}^{1/3}}$ and $\alpha$, $\beta$
are the constants whose values depend on the model one is using. The values of
$\alpha$, are 5.184 19, 5.374 57, and 5.087 58, whereas the values of $\beta$
are 0.339 79, 0.313 26, and 0.295 18 for Prox 77, Prox 88, and Prox 00,
potentials, respectively. The analytical parametrized fusion barrier positions
therefore, read as

${R_{B}^{par}}={s_{B}^{par}}+C_{1}+C_{2}.$ (16)

The quality of our parametrized fusion positions can be judged by analyzing
the percentage deviation defined as

$\Delta R_{B}~{}(\%)=\frac{R_{B}^{par}-R_{B}^{exact}}{R_{B}^{exact}}\times
100.$ (17)

We plot in Fig. 2, the percentage deviation $\Delta R_{B}~{}(\%)$ as a
function of the product of charges $Z_{1}Z_{2}$. Very encouragingly, we see
that in all three cases, our analytical parametrized form gives very good
results within $\pm 1\%$ of the actual exact barriers positions. The average
deviations calculated over 400 reactions are -0.01%, -0.02%, and 0% for Prox
77, Prox 88, and Prox 00, respectively. This is very encouraging since it is
for the first time that such accurate parametrization has been obtained. Note
that our parametrizations depend on the charges and masses of the colliding
nuclei only. This definitely introduces great simplification in the
calculation of fusion barrier positions within proximity concept.

In Fig. 3, we parametrize the fusion barrier heights $V_{B}$ as a function of
$\frac{1.44Z_{1}Z_{2}}{R_{B}^{par}}(1-\frac{0.75}{R_{B}^{par}})$, similar to
the one reported in Refs. ng04 ; broglia81 . The first part is the Coulomb
contribution whereas the second part is the reduction due to the nuclear
potential. We see that the fusion barrier heights in all three proximity
potentials can be parametrized using the following relation:

$V^{par}_{B}=\delta[\frac{1.44Z_{1}Z_{2}}{R_{B}^{par}}(1-\frac{0.75}{R_{B}^{par}})].$
(18)

Where $\delta$ is a constant having values 0.99903, 0.99868, and 1.002 for
Prox 77, Prox 88, and Prox 00, respectively. Here second term in the above
relation is introduced to take care of the deviations that happen in the lower
tail of the fusion barrier heights. We see that one can parametrize the
barrier heights very closely. The quality of our analytical parametrization is
tested in the Fig. 4, where again percentage difference between parametrized
and exact values are shown. Mathematically,

$\Delta V_{B}~{}(\%)=\frac{V_{B}^{par}-V_{B}^{exact}}{V_{B}^{exact}}\times
100.$ (19)

Very encouragingly, we see that our fits are within $\pm 1\%$ of the actual
values. Some slight deviations can be seen for lighter masses. This may also
be due to the limitations of proximity potentials in handling the lighter
masses where surface is of the order of nuclear radius. It is very encouraging
to note that our parametrized form give barrier heights and positions within
$\pm 1\%$ of the actual values. The average deviations are -0.10%, -0.12%, and
0.07% for Prox 77, Prox 88, and Prox 00, respectively. In Table 1, we display
the actual and analytical parametrized values of some selected collisions for
all three versions of proximity potentials. We note that our results are in
very close agreement with the actual value and therefore, introduces great
simplification in the calculation of fusion barriers. Finally, we compare our
outcome with experimental data in Fig. 5. Here we display our analytically
parametrized calculated fusion barrier heights $V_{B}^{par}$ [Eq. (18)] with
experimentally extracted fusion barrier heights $V_{B}^{expt}$. The
experimentally extracted fusion barrier heights displayed in this figure are
obtained in the approach, when shapes of both colliding nuclei are spherical.
The experimental data are taken from Refs ms2000 ; wang06 ; expt . It is clear
from the figure that our results are in good agreement with experimental data.
In a recent attempt id , we presented comparison of 16 different proximity
based potentials and found that potentials due to Bass wr94 , Aage Winther
aw95 , and Denisov id  (marked as Bass 80, AW 95, and Denisov DP in Ref. id )
were performing better than other proximity based potentials. The analytical
parametrizations of such potentials will be presented elsewhere idd .

##  IV Summary

Using three versions of proximity potentials, we obtained analytical relations
for the fusion barrier heights and positions. Our analysis is based on the
calculations of 400 reactions. Our analytical parametrized values are in very
close agreement with actual as well as experimental values. Therefore,
introducing great simplifications in the calculation of fusion barrier heights
and positions. These results can be used as a guide line for estimating the
fusion barriers in those cases where measurements do not exist and also for
the study of new nuclei yet unexplored.

##  V Acknowledgments

This work was supported by a research grant from the Department of Atomic
Energy, Government of India.

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Figure 1: Reduced fusion barrier positions $s_{B}~{}\rm(fm)$ (defined as
$s_{B}=R_{B}-C_{1}-C_{2}$) as a function of the
$\frac{Z_{1}Z_{2}}{A_{1}^{1/3}+A_{2}^{1/3}}$. Parts (a), (b), and (c) show the
results with Prox 77, Prox 88, and Prox 00 versions of the proximity
potential. Our parametrized fits are shown as solid curves. The values of
constants $\alpha$ and $\beta$ are given in the text. Figure 2: The percentage
difference $\Delta R_{B}~{}(\%)$ [defined in Eq. (17)] as a function of the
product of charges of colliding pair $Z_{1}Z_{1}$. Parts (a), (b), and (c)
show the results with Prox 77, Prox 88, and Prox 00 versions of the proximity
potential. Figure 3: The fusion barrier heights $V_{B}$ (MeV), as a function
of $\frac{1.44Z_{1}Z_{2}}{R_{B}^{par}}(1-\frac{0.75}{R_{B}^{par}})$. Parts
(a), (b), and (c) show the results with Prox 77, Prox 88, and Prox 00 versions
of the proximity potential. Our parametrized fits are shown as solid curves.
The value of the constant $\delta$ is given in the text. Figure 4: Same as
Fig. 2, but for $\Delta V_{B}~{}(\%$). Figure 5: The variation of the
parametrized fusion barrier heights $V_{B}^{par}~{}\rm(MeV)$ as a function of
experimental fusion barrier heights $V_{B}^{expt}~{}\rm(MeV)$. Parts (a), (b),
and (c) show the results with Prox 77, Prox 88, and Prox 00 versions of the
proximity potential. The experimental values are taken from Refs. ms2000 ;
wang06 ; expt .

Table 1: Fusion barrier heights VB (in MeV) and positions RB (in fm), calculated using different proximity potentials along with their corresponding parametrized values are displayed for few cases. Reaction | Prox 77 | Prox 88 | Prox 00 | Prox 77 | Prox 88 | Prox 00
---|---|---|---|---|---|---
| R${}_{B}^{exact}$ | R${}_{B}^{par}$ | R${}_{B}^{exact}$ | R${}_{B}^{par}$ | R${}_{B}^{exact}$ | R${}_{B}^{par}$ | V${}_{B}^{exact}$ | V${}_{B}^{par}$ | V${}_{B}^{exact}$ | V${}_{B}^{par}$ | V${}_{B}^{exact}$ | V${}_{B}^{par}$
6Li + 9Be | 7.01 | 7.03 | 7.26 | 7.27 | 6.74 | 6.81 | 2.21 | 2.20 | 2.14 | 2.13 | 2.29 | 2.26
10B + 12C | 7.22 | 7.21 | 7.47 | 7.45 | 6.99 | 7.03 | 5.36 | 5.36 | 5.19 | 5.20 | 5.54 | 5.50
16O + 16O | 7.65 | 7.65 | 7.90 | 7.90 | 7.51 | 7.54 | 10.86 | 10.86 | 10.55 | 10.55 | 11.10 | 11.03
20Ne + 20Ne | 7.95 | 7.97 | 8.20 | 8.21 | 8.42 | 8.28 | 16.39 | 16.35 | 15.94 | 15.92 | 15.68 | 15.85
24Mg + 26Mg | 8.40 | 8.37 | 8.65 | 8.61 | 8.86 | 8.73 | 22.54 | 22.53 | 21.95 | 21.96 | 21.47 | 21.75
24Mg + 34S | 8.61 | 8.61 | 8.86 | 8.85 | 8.89 | 8.80 | 29.34 | 29.28 | 28.60 | 28.55 | 28.64 | 28.80
16O + 64Ni | 9.01 | 9.03 | 9.26 | 9.27 | 9.05 | 9.08 | 35.17 | 35.06 | 34.33 | 34.22 | 35.08 | 34.99
6Li + 238U | 10.87 | 10.97 | 11.07 | 11.21 | 10.81 | 10.93 | 34.07 | 33.72 | 33.46 | 33.04 | 34.28 | 33.94
12C + 124Sn | 9.88 | 9.94 | 10.13 | 10.18 | 9.97 | 10.00 | 40.31 | 40.14 | 39.49 | 39.26 | 40.20 | 40.04
16O + 110Pd | 9.88 | 9.90 | 10.08 | 10.13 | 10.02 | 10.01 | 49.60 | 49.42 | 48.56 | 48.38 | 49.12 | 49.07
30Si + 64Ni | 9.63 | 9.60 | 9.83 | 9.84 | 9.71 | 9.65 | 54.13 | 54.16 | 52.94 | 52.92 | 53.93 | 54.06
48Ca + 48Ca | 9.89 | 9.81 | 10.09 | 10.05 | 9.89 | 9.83 | 53.96 | 54.18 | 52.84 | 52.97 | 53.93 | 54.24
32S + 58Ni | 9.40 | 9.45 | 9.65 | 9.68 | 9.50 | 9.53 | 63.04 | 62.79 | 61.60 | 61.40 | 62.64 | 62.49
40Ar + 60Ni | 9.82 | 9.78 | 10.02 | 10.02 | 10.00 | 9.94 | 68.40 | 68.45 | 66.91 | 66.92 | 67.37 | 67.64
16O + 166Er | 10.64 | 10.66 | 10.84 | 10.89 | 10.77 | 10.76 | 68.56 | 68.25 | 67.25 | 66.89 | 67.93 | 67.87
16O + 186W | 10.86 | 10.90 | 11.06 | 11.13 | 11.18 | 11.15 | 73.09 | 72.76 | 71.74 | 71.34 | 71.39 | 71.45
36S + 90Zr | 10.30 | 10.28 | 10.55 | 10.50 | 10.41 | 10.36 | 82.99 | 83.03 | 81.30 | 81.39 | 82.35 | 82.69
35Cl + 92Zr | 10.25 | 10.25 | 10.50 | 10.47 | 10.39 | 10.36 | 88.58 | 88.45 | 86.75 | 86.71 | 87.64 | 87.85
32S + 110Pd | 10.43 | 10.45 | 10.68 | 10.68 | 10.65 | 10.65 | 94.21 | 94.05 | 92.33 | 92.15 | 92.43 | 92.70
64Ni + 64Ni | 10.48 | 10.47 | 10.73 | 10.70 | 10.60 | 10.57 | 99.84 | 100.00 | 97.86 | 97.98 | 98.90 | 99.43
40Ar + 110Pd | 10.75 | 10.73 | 10.95 | 10.95 | 11.07 | 10.98 | 103.19 | 103.25 | 101.21 | 101.30 | 100.61 | 101.37
32S + 138Ba | 10.87 | 10.87 | 11.07 | 11.09 | 10.93 | 10.96 | 110.71 | 110.40 | 108.62 | 108.33 | 109.73 | 109.89
40Ar + 130Te | 11.05 | 11.03 | 11.25 | 11.26 | 11.22 | 11.18 | 113.63 | 113.78 | 111.56 | 111.58 | 111.96 | 112.69
24Mg + 208Pb | 11.41 | 11.44 | 11.61 | 11.66 | 11.73 | 11.69 | 116.04 | 115.63 | 114.02 | 113.56 | 113.09 | 113.66
29Si + 178Hf | 11.27 | 11.28 | 11.47 | 11.50 | 11.55 | 11.49 | 120.24 | 120.00 | 118.08 | 117.83 | 117.75 | 118.32
34S + 168Er | 11.35 | 11.32 | 11.55 | 11.55 | 11.39 | 11.40 | 129.16 | 129.10 | 126.86 | 126.67 | 128.04 | 128.65
64Ni + 96Zr | 11.13 | 11.08 | 11.33 | 11.30 | 11.21 | 11.19 | 135.37 | 135.58 | 132.87 | 133.07 | 134.04 | 134.74
38S + 181Ta | 11.69 | 11.64 | 11.89 | 11.87 | 11.79 | 11.78 | 134.80 | 135.05 | 132.51 | 132.56 | 133.21 | 133.96
48Ca + 154Sm | 11.61 | 11.59 | 11.86 | 11.80 | 11.72 | 11.68 | 143.72 | 143.95 | 141.26 | 141.51 | 142.55 | 143.35
40Ar + 180Hf | 11.65 | 11.66 | 11.90 | 11.88 | 11.81 | 11.80 | 149.63 | 149.61 | 147.07 | 146.98 | 147.58 | 148.40
38S + 208Pb | 11.98 | 11.94 | 12.18 | 12.16 | 12.00 | 12.00 | 147.89 | 148.15 | 145.47 | 145.60 | 147.31 | 147.90
64Ni + 124Sn | 11.55 | 11.52 | 11.75 | 11.73 | 11.68 | 11.68 | 163.23 | 163.45 | 160.37 | 160.67 | 160.85 | 161.84
40Ar + 206Pb | 11.93 | 11.94 | 12.18 | 12.16 | 12.11 | 12.10 | 166.66 | 166.67 | 163.89 | 163.79 | 164.19 | 165.10
86Kr + 100Mo | 11.59 | 11.57 | 11.84 | 11.79 | 11.68 | 11.70 | 175.40 | 175.81 | 172.33 | 172.69 | 173.67 | 174.51
90Zr + 90Zr | 11.42 | 11.42 | 11.67 | 11.64 | 11.56 | 11.59 | 188.23 | 188.32 | 184.79 | 184.94 | 185.53 | 186.30
40Ar + 238U | 12.31 | 12.28 | 12.51 | 12.49 | 12.30 | 12.35 | 182.29 | 182.15 | 179.41 | 179.22 | 181.07 | 181.72
96Mo + 100Mo | 11.75 | 11.72 | 11.95 | 11.93 | 11.81 | 11.86 | 202.39 | 202.67 | 198.85 | 199.28 | 200.05 | 201.03
54Cr + 196Os | 12.22 | 12.19 | 12.42 | 12.40 | 12.34 | 12.34 | 201.86 | 202.01 | 198.62 | 198.75 | 199.21 | 200.31
51V + 208Pb | 12.23 | 12.24 | 12.48 | 12.45 | 12.36 | 12.40 | 208.11 | 208.09 | 204.75 | 204.73 | 205.18 | 206.18
54Cr + 209Bi | 12.33 | 12.32 | 12.53 | 12.53 | 12.59 | 12.61 | 218.37 | 218.45 | 214.85 | 214.95 | 212.95 | 214.38
96Zr + 124Sn | 12.15 | 12.13 | 12.40 | 12.34 | 12.28 | 12.29 | 222.18 | 222.53 | 218.53 | 218.91 | 219.15 | 220.48
55Mn + 208Pb | 12.35 | 12.32 | 12.55 | 12.53 | 12.24 | 12.35 | 224.74 | 224.80 | 221.13 | 221.20 | 224.89 | 224.96
70Zn + 176Yb | 12.35 | 12.31 | 12.55 | 12.52 | 12.36 | 12.41 | 230.12 | 230.47 | 226.42 | 226.76 | 228.67 | 229.41
58Fe + 208Pb | 12.39 | 12.40 | 12.64 | 12.61 | 12.38 | 12.47 | 232.38 | 232.38 | 228.67 | 228.68 | 231.26 | 231.85
59Co + 208Pb | 12.42 | 12.41 | 12.62 | 12.62 | 12.50 | 12.57 | 241.20 | 241.15 | 237.34 | 237.30 | 237.99 | 238.98
59Co + 209Bi | 12.43 | 12.42 | 12.63 | 12.63 | 12.62 | 12.69 | 244.02 | 243.90 | 240.10 | 240.01 | 238.47 | 239.75
63Cu + 197Au | 12.39 | 12.37 | 12.59 | 12.57 | 12.20 | 12.36 | 250.40 | 250.29 | 246.33 | 246.46 | 251.22 | 251.22
64Ni + 208Pb | 12.56 | 12.54 | 12.76 | 12.75 | 12.53 | 12.64 | 247.56 | 247.65 | 243.66 | 243.74 | 245.68 | 246.54
70Zn + 208Pb | 12.71 | 12.67 | 12.91 | 12.87 | 12.76 | 12.85 | 262.60 | 262.78 | 258.53 | 258.86 | 259.01 | 260.10
86Kr + 208Pb | 12.99 | 12.98 | 13.24 | 13.18 | 12.92 | 13.09 | 308.05 | 308.27 | 303.40 | 303.77 | 306.16 | 306.75