Effect of Barrier Height on Nuclear Fusion

IOSR Journal of Applied Physics (IOSR-JAP)
e-ISSN: 2278-4861.Volume 9, Issue 1 Ver. I (Jan. – Feb. 2017), PP 08-16
www.iosrjournals.org
DOI: 10.9790/4861-0901010816 www.iosrjournals.org 8 | Page
Effect of Barrier Height on Nuclear Fusion
G. S. Hassan1
, A. Abd-EL-Daiem, and A. M. Mahmoud
1
physics Department, Assiut University, Assiut , Egypt
2
Physics Department, Sohag University, Sohag , Egypt
Abstract: The enhancement of sub-barrier fusion has been interpreted due to coupling between the relative
motion and other degrees of freedom. The coupling gives rise to the distribution of fusion barriers and passage
over the lowest barrier which is responsible for fusion enhancement at energies below the barrier. There are
several orders of magnitude could be considered due to the tunneling through the barrier. The barrier height
could be deduced from the measured cross section data for different energies, as well as using many empirical
forms for incomplete and complete fusion of two massive nuclei. Firstly, we present a formula for barrier height
(ODEFF) and check, over wide ranges of interacting pairs the percentage agreement with those calculated or
measured values for all pairs within ZPZT ≤ 3000. Secondly, the more recently measured excitation functions
are studied using four models of nuclear forces, indicating that most of them can be used for wide energy
range while the others failed to do so .We refer this notice to the theory deducing the model . For this, the 14
undertaken pairs recover the range18 ≤ ZPZT ≤ 1320
Keywords: fusion barrier, ODEFF function , excitation, nuclear potential, WKB approximation
PACS: 25.60.Pj ; 25.70.Gh ; 25.70.Jj
I. Introduction
The sub-barrier fusion provides a method to test the nuclear potential on the inner side of the
interaction barrier and to gain information on the influence of nuclear structure upon the behavior of nuclear
matter and dynamics of nuclear reactions, specially for energies where penetrability effects are considered. The
sub-barrier fusion cross section presents an unexpected enhancement, as compared with conventional models of
tunneling through a one-dimensional penetration model, which successfully describes fusion above the Coulomb
barrier.
For sub-barrier fusion of two massive nuclei, it was discovered that there was several orders of
magnitude more than sub-barrier, could be accounted in terms of quantal tunneling through the fusion barrier.
The quantum mechanical barrier penetration effects play a central role in near- and sub-barrier fusion reactions,
where the fusion cross section has been vanished suddenly as the bombarding energy becomes less than the
interaction barrier.
1.1 Fusion Barrier
The probability of fusion of two heavy ions at energies below their mutual interaction barrier, is
defined by a barrier radius given due to different reasonable forms [1,2]as:
)( 3/13/1
Tpifus AArR  where ri is referring to the height and extension of the barrier, and
has a critical value [3] as rc =1.3 . The total energy VT required for a specified reaction channel [3,4,5,6] is
related to barrier height VB by:
VT ( Rfus , L ) = VB+ VL = VC + VN + VL (1)
where Vc , Vn and VL are the Coulomb, nuclear and centrifugal forces respectively. The motion of the binary
system is then described by Schrödinger equation:
2 2
2
( ) ( ) 0
2
d
V R E R
dR
 
     
 

(2)

where E is the excitation energy in the center of mass system. In order for fusion reactions to occur, the barrier
height VB created by the strong cancellation between the repulsive Coulomb force VC and the attractive nuclear
interaction Vn has to be overcome. The nuclear force may be used in different forms and also based on different
interpretations for the attraction between nuclei. Four forms of them are in high range of usage, namely:
1-The proximity potential which is based [4] upon the liquid drop model:





 


b
ccr
ccr
ccbc
rV
Tp
TP
sTP
n
)(
)(
)( 2
0
'
 (3)
2-The unified model, which is based [4] on the collective model:
aSPT
n e
r
R
a
s
FDrV /
)()( 
 (4)
3- Woods – Saxon form represents [5] that force as
)/)exp(1/(16)( 0 aRrRarV dn   (5)
Rd,R0 are the reduced and half density radius , a is the diffusivity and
 is the average value of both the projectile and target surface tension .
4- R. Bass potential[6], presented in terms of the liquid drop model as :
P T
n fus
P T
R R -S
V ( R ) 4 d exp
R R d

 
  
  
1/3 1/3 fus PT
S P T
PT
R Rd
a A A exp ,
R d
 
  
 
(6)
With d = 1.35 and δ are the diffusivity and the specific surface energy and PT P TR R R  is
the sum of the half-maximum density radii. Both of these potentials are actively used for various ranges of ion
masses and excitation energies. The many degrees of freedom quantum tunneling which is often called
macroscopic quantum tunneling was firstly treated by Dasso and ,Broglia [7,8], in which the tunneling degree of
freedom ( the elastic or entrance channel ) couples to the internal degrees of freedom ( the transfer and inelastic
channels ). The concept of the distribution of barriers can be easy visualized classically when one of the
interacting nuclei is deformed; this results in a dependence of the fusion barrier height on the orientation of the
deformed nucleus and leads to a continuous distribution of potential barriers which extends below and above the
conventional coulomb barrier
1.2 Fusion Cross Section
The reaction cross section through a definite channel of an energy E has been given by WKB approximation as a
summation over all penetrating partial waves [9].




0
2
)()()12(
l
llrec EPETl (7)
where  , Tl (E) and Pl (E) are reduced De Broglie wave length of the incident ion, the transmission
coefficient and the probability of penetration respectively . For fusion we assume rec = fus and 1lP . The
upper limit in the last equation becomes maxl ,[ 1] and σfus reads





 



max
0
2
)(2
exp1
)12(l
l cm
fus
EV
l




 (8.a)
Where  is the harmonic oscillator frequency or curvature parameter. A logarithmic form is given by Wong as
:
fus(E) = ( ћ R
2
/ 2E ) lin { 1+ exp[( 2  / ћ  )( E – VB(r) )] } (8.b)
A sharp cut-off approximation assumes that relative angular momentum l smaller than a particular critical
angular momentum lcr contribute to complete fusion, while higher values of lfus are associated with direct
(peripheral) process [10]
2
2
)1(
2
 fus
cm
fus l
E



(9)
Heavy-ion-induced fusion reactions can be treated classically and the cross sections are decomposed into partial
ones corresponding to orbital angular momentum. This approximation gives the fusion cross section [10] similar
to that given by equation(8) replacing lmax by lfus as shown in equation (9 ). When applying the form on the
measured data from more recent references we can deduce the critical lcr values as :
2
2
)1(
2
 cr
cm
fus l
E



(10)
1.3 Angular Momentum Limits
The formed composite nucleus by the complete fusion will decay either by fission or by evaporation. The
evaporation residues cross section will represent :
2
2
)1(
2
 er
CM
ER l
E



(11)
Where as, ERer ll  for ERll max , or maxller  for ERll max , ERl is the specific angular
momentum at which the partial level width for fission is equal to that for evaporation . For l › maxl the real
potential no longer has a pocket, and so the cross section formula, eq. (12) in Sharp Cut-Off approximation
tends to be
 fus
= 10  R2
fus
(1-vB /Ecm) =  g (1-vB /Ecm)
 fus
/ g = - vB (1/Ecm) + 1 (12)
The linear relation (fus , 1/Ecm ) leads to extraction of fusion radius as the maximum distance at which fusion
can take place. This form has been used for a long time to predict the compound nucleus formation cross-section
and it is also commonly used for the heavy- ion fusion reaction (figs. 1.a,1.b).
II. One Dimension Empirical Formula for Fusion (ODEFF)
Many years ago, using an empirical model called elastic model given by Scalia, we tried to make check
and extension[11] on the study of fusion excitation functions for wider range of energies as well as wider range
of interacting pairs and found that it is more significant and simpler for use. Similarly, it is well known that the
barrier height could be deduced, using recently measured data, as the slope of the linear plot of eq.(12) . When
applying this method for some recently measured data we calculate [12] for Li
6
+ Sm
144
a slope = 0.99 and v
b
= 21.4 Mev and [13] for Ca
40
+ Sn
124
a slope = 0.99 and v
b
= 111.1 Mev (see figs.(1.a,1.b). In addition,
many empirical forms were introduced to calculate the barrier height of sub-barrier fusion of two massive
nuclei. A recently deduced empirical form for fusion barrier was given by Kumari and Puri [14] depending on
two dimensions ( Z and A of the interacting pair) as :
V
B
(x) = α x
2
+ β x + γ (13)
Where X= 1.44 z /( A
P
1/3
+ A
T
1/3
) , Z = ZPZT, α = 4.53 x 10
-4
, β= 0.93 , γ = -1.01
Our new form for barrier height is deduced by applying the least square method on wide range of measured data
to give a function of one parameter Z only:

V
B
(Z) = a Z
3
+ b Z
2
+ c Z + d (14.a)
With Z = ZPZT and a = 2.926 x 10
-8
,b = 2.479 x 10
-8
, C = 0.0641 , d = 9.706 .
Firstly, fig.(2.a,2.b), indicates that the two curves calculated by eq.(14.a) are similar to those calculated
using b = 0, which means that the second term could be neglected without any variation on the curve smoothing
and equation (14.a) can be reduced to the form
V
B
(Z) = a Z
3
+ c Z + d (14.b)
for the range Z ≤ 800, while using (a =b=0. ), the form
V
B
(Z) = c Z + d (14.c)
is the more applicable for the range 800 ‹ Z ≤ 3000 as we will check in the next section. We make the
required comparisons using either the calculated or measured data. Three sets of data are taken in consideration
for comparisons, the recently calculated barriers using both of unified or proximity nuclear forms, those
calculated using Kumari and Puri empirical[14] form eq.(13 ),and those given by Ishiwara Dutt and R. K. Puri
[15].
III. Results and Discussion
In this work, we deduce a new empirical function ( ODEFF function ) and make checks and normalization to
be:
V
B
(Z) = a Z
3
+ b Z
2
+ c Z + d
where ( Z = ZPZT ) , a = 2.926 x 10
-8
,b = 2.479 x 10
-8
, C = 0.0641 , d = 9.706 . For Z ≤ 800 and a = b
= 0 , C = 0.0641 , d = 9.706 for 800 ‹ Z ≤ 3000
In the table(1), we defined 14 studied pairs in addition to the corresponding lmax values, while in
figs.(1.a and 1.b) we make use for equation ( 12 ) to deduce the barrier height from excitation functions of the
pairs taken from the corresponding references. The relations appear some agreement for the straight shape. The
check of our deduced form with three sets of data are given in figs.(3.a,3.b) when comparing with those given
by authors using both unified and R. Bass nuclear form but in figs.(4.a,4.b) it is clear that a higher agreement is
found with calculated data using the two dimensional empirical form eq.(13). The final agreement and the best
is that shown on figs.( 5.a,5.b ) with those calculated by R. Bass model [15].
At energies near and above the Coulomb barriers, using four nucleus-nucleus interaction potentials, and
three different forms for fus are employed eqs.(8.a,8.b,9) in order to fit available measured fusion cross
sections as shown in Table(1). It is found enhanced values in comparison with the 1D BPM predictions. It is
found also that the degree of enhancement strongly depends on the type of the target nucleus; spherical or
deformed. We found also, that the unified and R. Bass nuclear formula are the joker for fitting measured
excitation functions near the barrier or even when exceeding up to twice its value. The unified potential points
out successful predictions of data for the intermediate ion reactions through both formulas. On the other hand,
the success of the proximity potential comes next to that of R. Bass and unified models under the same
circumstances and poor fits with data are obtained by it. Also we found that when the charge product ZPZT has a
large value (heavy nuclei reactions), the smooth cut-off approximation (8.a,8.b) is the best to successfully
predict the experimental data. For light nuclei systems (relatively low charge product ZPZT), both of the smooth
cut-off approximation and Wong formulas produce reasonable predictions using either unified or R. Bass. On
the other hand, the proximity potential through the smooth cut-off approximation successfully reproduces the
experimental data. Predictions of the Bass potential reveal reasonable agreement with data. This result may
resemble the significant difference in depth between this potential and the others and the low level agreement
for some pairs may be interpreted as results of nature and strength of the couplings. Thus lies in the distribution
of fusion barrier and the experimental determination of this distribution, which are major steps for
understanding heavy-ion fusion. The results of fitting are shown on figs.(6.1-6.14). The last note will be the
main point of interest for research in the preceding work to interpret the effect of barrier distribution on fusion
excitation functions.
IV. Conclusion
The ODEFF function has been checked and normalized to give the final form (14.a and 14.c ), is a
simple and easier form. Also it is a very high accurate form predicting the barrier height of fusion for any X + Y
interacting pair. The cross sections at low energies then result from passage over the lower barriers rather than
penetration through the single barrier. Information on the nature and strength of the couplings thus lies in the

distribution of fusion barrier and the experimental determination of this distribution is a major step towards
understanding heavy-ion fusion. The last note will be the main point of interest for research in the preceding
work to interpret the effect of barrier distribution on the excitation functions for fusion.
References
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[2]. L C Vaz and J. M. Alexander, Phys. Rev. ,C 18 ,2152, 1978
[3]. R Bass , Nucl.Phys. ,A231 ,45, 1974
[4]. H J Krappe, et al , Phys .Rev. , C 20, 992, 1979
[5]. H Esbensen et al., Phys. Rev. , C 40 ,2046, 1989
[6]. R Bass , Phys. Rev. Lett. , 30, 265, 1977
[7]. C H Dasso et al , Nucl. Phys. , A405 381 , 1983 and C H Dasso et al Nucl. Phys. , A407, 221, 1983
[8]. R A Broglia et al , Phys. Rev. C27, 233 , 1983
[9]. M Hugi, et al, Nucl. Phys. , A368, 173, 1981
[10]. A M Stefanini , et al, Nucl. Phys. , A548 453,1992
[11]. G S Hassan , et al , Acta Phys. Pol. , B31(8),1799,2000
[12]. P K Rath et al , phys. Rev. , C79 , 51610,2009
[13]. J J Kolata, et al , phys. Rev. , C85 , 054603,2012
[14]. R Kumari and R. Puri , Nucl. Phys. , A993,135, 2015
[15]. Ishiwara Dutt and R K Puri, ,(http://arxiv.org/abs/1005.5213),2010
[16]. P A De Young,et al, Phys. Rev. , C28,692,1983
[17]. L C Dennis, et al, Phys. Rev. , C26,981,1982
[18]. D G Kovar,et al, Phys. Rev. , C20,1305,1979
[19]. Y D Chan,et al, Nucl. Phys. , A303,500,1978
[20]. R M Anjos,et al, Phys. Rev. , C42,354 ,1990
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[22]. V K C Cheng , et al, Nucl. Phys. , A322 168,1979
[23]. S Gary, et al, Phys. Rev. , C25,1877,1982
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[25]. M Hugi, et al, Nucl. Phys. , A368, 173,1981
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[27]. N Knyazheva et al, Phys. Rev. , C75 064602,2007
[28]. J Khuyagbaatar et al. Phys.Rev. C86 (2012) 064602
Table(I) Barrier radius Rfus (fm) and heights VB(Mev) and maximum angular momentum of the concerned
interacting pairs. In the three columns it is given the nuclear model used to fit, fig. number and reference for
measured data.
Pair ZPZT Rfus
(fm)
VB
Proximity
VB W-S VB
Unified
VB
R. Bass
l
max
Model
used
Fig. Ref
10
B+14
N 35 7.60 6.15 5.69 5.78 6.55 12 R.Bass 6.1 16
6
Li+12
C 18 7.12 3.29 3.02 3.03 3.59 9 R.Bass 6.2 17
12
C+12
C 36 7.61 6.31 5.84 5.99 6.73 12 R.Bass 6.3 18
12
C+16
O 48 7.84 8.23 7.65 7.80 8.71 13 Unified 6.4 19
12
C+19
F 54 8.03 9.08 8.46 8.63 9.58 14 R.Bass 6.5 20
14
N+14
N 49 7.85 8.40 7.80 7.66 8.88 13 R.Bass 6.6 21
16
O+16
O 64 8.07 10.74 10.01 10.24 11.30 15 Unified 6.7 22
24
Mg+24
Mg 144 8.75 22.53 21.14 21.76 32.47 20 Unified 6.8 23
27
Al+35
Cl 221 9.23 32.86 35.07 31.89 34.14 25 Unified 6.9 24
28
Si+9
Be 56 8.26 9.18 8.61 8.76 9.66 14 R.Bass 6.10 25
16
O+144
Sm 496 10.76 63.54 60.33 62.31 65.81 29 R.Bass 6.11 26
64
Ni+96
Zr 1120 11.17 135.34 349.27 133.67 142.33 44 R.Bass 6.12 10
48
Ca+154
Sm 1240 11.61 143.70 664.90 142.61 151.50 43 Unified 6.13 27
36
S+206
Pb 1312 11.83 149.27 621.23 147.86 157.41 39 Unified 6.14 28

0.0260416 0.0325520 0.0434027
0
0.1
0.2
0.3
0.4
0.5
0.6
0.0074184 0.0081967 0.0087108
0
0.05
0.1
0.15
0.2
(1.a) (1.b)
0 400 800 1200 1600 2000 2400 2800
0
200
400
600
800
1000
1200
0 400 800 1200 1600 2000 2400 2800
0
200
400
600
800
1000
1200
(2.a) (2.b)
0 80 160 240 320 400 480 560 640 720 800
0
20
40
60
80
100
120
800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000
0
50
100
150
200
250
300
350
(3.a) (3.b)
0 80 160 240 320 400 480 560 640 720 800
0
20
40
60
80
100
120
800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000
0
50
100
150
200
250
300
350
(4.a) (4.b)

0 80 160 240 320 400 480 560 640 720 800
0
20
40
60
80
100
120
800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000
0
50
100
150
200
250
300
350
(5.a) (5.b)
5.166 9.166 13.166 17.166 21.166 25.166
0
200
400
600
800
1000
2.675 6.675 10.675 14.675 18.675 22.675
0
200
400
600
800
1000
(6.1) (6.2)
6.852 10.352 13.852 17.352 20.852 24.352 27.852 31.352
0
200
400
600
800
1000
1200
7.223 11.223 15.223 19.223
0
100
200
300
400
500
600
700
(6.5) (6.6)

9.576 11.176 12.776 14.376 15.976 17.576 19.176
0
100
200
300
400
500
600
700
20.994 22.594 24.194 25.794 27.394 28.994 30.594
0
100
200
300
400
500
600
700
(6.7) (6.8)
58.175 59.775 61.375 62.975 64.575 66.175 67.775
0
100
200
300
400
127.386 128.986 130.586 132.186 133.786 135.386 136.986
0
20
40
60
80
100
120
140
(6.11) (6.12)
136.1 142.1 148.1 154.1 160.1 166.1 172.1 178.1 184.1 190.1
0
200
400
600
800
1000
1200
140.8 151.8 162.8 173.8 184.8
0
200
400
600
800
1000
1200
(6.13) (6.14)
Figure Caption
Fig(1) linear representation (1/E
CM
, σ
Fus
/σ
G
) : (a) for Li
6
+ Sm
144
with slope = 0.99 and v
b
= 21.4 Mev , the
data are taken from [12]
(b) for Ca
40
+ Sn
124
With slope = 0.99 and v
b
= 111.1 Mev , the data are taken from [13]

Fig (2) The one dimensional empirical formula for fusion ODEFF function for ( 0 ‹ Z ‹ 3000 ) : (a) is the form
(14.a) in comparison with those calculated by eq.(14.c) (b) is the form (14.b) in comparison with those
calculated by eq.(14.c)
Fig (3) Barrier height by ODEFF function, in comparison with those calculated using unified nuclear potential
or proximity nuclear form (a) eq.(14.a) for Z ≤ 800 and (b) eq.(14.c) for 800 ‹ Z ≤ 3000
Fig (4) Barrier height by ODEFF function in comparison with those calculated using the empirical [14] form,
eq.(13) given by R. Kumari and R. Puri, (a) eq.(14.a) for Z ≤ 800 and (b) eq.(14.c)for 800 ‹ Z ≤ 3000
Fig (5) Barrier height by ODEFF function in comparison with those calculated by Dutt and Puri using Bass80
[15] form eq.( 1, 3 ) (a) eq.(14.a) for Z ≤ 800 and (b) eq.(14.c)for 800 ‹ Z ≤ 3000
Figs (6.1-6.14) Calculated excitation functions for undertaken pairs in comparison with measured data (Table
1). The unified and R. Bass nuclear formula are the joker for fit near and above up to twice the barrier. When
ZPZT has a large value, the smooth cut-off approximation is the best for successful predictions. For low charge
product ZPZT, both of the smooth cut-off approximation and Wong formulas produce reasonable predictions. On
the other hand, the proximity potential and smooth cut-off approximation, fits successfully the experimental
data. The low level agreement for some pairs may be interpreted as results of nature and strength of the
couplings.

Effect of Barrier Height on Nuclear Fusion

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