Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 27–30, 2002

27

The Effects of Deformed Projectile in Threshold Anomaly and Fusion Reaction
for 19 F + 208 Pb System
C. J. Lin,∗ H. Q. Zhang, Z. H. Liu, J. C. Xu, and F. Yang
China Institute of Atomic Energy, P.O. Box 275(10), Beijing 102413, China
Received: November 13, 2001; In Final Form: November 13, 2001
Angular distributions of elastic scatterings for 19 F + 208 Pb system have been measured at six energies around the
Coulomb barrier. It is found that the real and imaginary potentials show pronounced energy dependence in termsof the phenomenological optical model analysis. The real parts of potentials extracted from the fusion data aresimilar to those from the elastic data. A comparison with the neighbouring systems of 16 O + 208 Pb and 16 O + 209 Bi
reveals that the 19 F deformations play an important role in the fusion reaction. Based on the deformed and energy
dependent barrier penetration model, the calculated fusion cross sections and mean-square spin distribution agreewith the experimental data well. Moreover, it is indicated that the enhancement of fusion cross sections attributes tothe dynamic polarization effects around the barrier, and to the static deformation effects further below the barrier.

1. Introduction
It is well known that the threshold anomaly (TA) comes fromthe coupled-channels (CC) effects and plays an important rolein heavy ion reaction at the energies around the Coulomb barrier.1, 2 The existence of the TA in the interacting potential forthe elastic scattering has been formed for many systems. Thisanomaly shows up as a localized peak in the real part of theinteracting potential around the Coulomb barrier and it is associated with the decrease of the imaginary part of the interacting
potential in the same energy range. It has been shown that thiscorrelation between the real and imaginary parts of the interacting potential is due to causality and consequently that they obey
the dispersion relation.
Due to the difficulty to calculate the dynamic polarization potential in theory, TA is still a purely experimental matter up to
now. Most of systems studied are spherical or near spherical.
Very little work has been done for well deformed system3 withattempt to understand the effects of nuclear structure, especiallyfor the deformed projectile. In this work, the 19 F + 208 Pb system was chosen because 19 F nucleus has quite large static deformations (β2 = 0.44 and β4 = 0.14).4 The CC effects in this system should be pronounced and appreciably influence on the fusion channel. Since the neighboring systems of 16 O + 208 Pb and16
O + 209 Bi are well investigated,5–7 it is interesting to make acomparison with these systems. Furthermore, the fusion data of19
F + 208 Pb system are available in the literatures.8, 9 It is meaningful to compare these data with the calculations of barrier penetration model (BPM) modified by the deformations and energy
dependence.

equal to the Rutherford cross sections at the forward angles.
The energy resolutions of the detectors allow to separate all
the Ex ≥ 1.3457 MeV states of 19 F and all the excitation states of208
Pb from ground states, while the two lower excitation states
0.1099 MeV and 0.1971 MeV of 19 F cannot be resolved fromthe measured elastic scatterings. Thus the elastic spectra include the inelastic scatterings of these two low-lying excitation
states. The differential elastic scattering cross sections normalized to the Rutherford cross sections for all energies are shownin Figure 2. The overall error is 3% for the forward angles andgradually increases to 10% for the backward angles.
3. Energy Dependent Potentials
3.1. Optical Model Analysis of Elastic Scatterings. In
this work the phenomenological analyses are utilized to get
the polarization optical potentials. Since the low-lying excitedstates are not resolved, the coupled-channels code ECIS9510 isemployed to fit the elastic scattering angular distributions in order to get the pure elastic scattering potentials. In these calculations only the couplings between the Ex = 0.1971 MeV, J π = 5/2+
state and ground state which belong to the K = 1/2+ ground state
rotational band are taken into account. The same optical potentials are assumed for these two coupling nuclear states. Another excitation state Ex = 0.1099 MeV, J π = 1/2− which belongs
to K = 1/2− band is not included, because the strength coupled
to the ground state band is very weak.
In the calculations, the Woods-Saxon shapes of the realand imaginary potentials are kept the same (r0v = r0w = r0 ,1800

2. Experimental Procedure

Elastic

1600∗Corresponding author. E-mail: cjlin@iris.ciae.ac.cn. FAX: +8610-6935-7787.

1400

Counts

The experiment was carried out at HI-13 tandem acceleratorof China Institute of Atomic Energy, Beijing. A 208 Pb target ofthickness about 100 µg/cm2 evaporated onto a 20 µg/cm2 carbon foil backing was bombarded by a collimated 19 F beam. The
beam energies were 88, 91, 93, 96, 98, and 102 MeV, respectively. The reaction products were detected by three pairs of∆E-E telescope detectors with the total energy resolution lessthan 1.3%. A typical spectrum cut by the Z = 9 banana in the∆E-E matrix at EL = 102 MeV and θL = 86◦ is shown in Figure 1.
A Si(Au) surface barrier detector was mounted at −42.4◦ withrespect to the beam direction as a monitor to detect the elastic
scatterings. All the reaction cross sections were normalized tothe monitor counts, assuming that the elastic cross sections are

19

208

1200

F+ Pb
ELab=102MeV

1000

θ Lab=86

o

800
600
400

Inelastic

200
0
700

750

800

850

ChannelFigure 1. A typical elastic and inelastic spectrum.

c 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002

900

28

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

Lin

TABLE 1: The sensitive radii and the results of the best fits obtained at r0 = 1.24 fm and a0 = 0.53 fm for elastic scatterings.

Elab
/ MeV
102
98
96
93
91
88

RSV
/ fm
12.86
12.62
12.58
12.50
12.32
12.24

RSW
/ fm
13.16
12.75
12.89
12.20
12.12

−V0

−W0

χ2 /pt

/ MeV
73.2
77.2
77.5
85.1
88.6
78.4

/ MeV
44.5
23.4
22.5
10.2
3.44
1.57

7.67
10.5
27.1
14.2
8.87
1.25

σR
/ mb
1169
897.2
838.3
673.2
579.9
512.0

av = aw = a0 ). In order to assess the influences of the potential
parameters, two sets of fits are performed with the fixed radiusparameters r0 = 1.20 fm and r0 = 1.24 fm, respectively. For eachset, a grid search is made on the different diffuseness parameters (a0 = 0.43, 0.48, 0.53, 0.58, 0.63, 0.68 fm) to obtain the best
V0 and W0 values with the minimum χ2 . It is found that the χ2values reach their minimum when r0 = 1.24 fm and a0 = 0.53 fm
for all energies. The results of the best fits in this case are illustrated in Figure 2 as the solid lines and the parameters arelisted in Table 1. All the lines of the real or imaginary potentials
with different parameters converge at a certain distance, namely
sensitive radius (RS ). At the RS , the potentials have a minimum
uncertainty and are nearly independent of geometrical shapes.
It is realized that the RS values depend on the energies and havesome differences between the real and imaginary parts of potentials. For consistency, RS are taken as 12.5 fm for both real andimaginary parts of potentials at all energies. The real and imaginary parts of potentials at the RS (VS and WS ) are illustrated in
Figure 3 with the circle symbols. The errors are derived fromthe χ2 values. It is clear that the potentials are energy dependent
around the Coulomb barrier, as the typical TA behaviour.
3.2. Extract TA Parameters in Linear Model. For convenience, the linear schematic model of the two straight line
segments is utilized to describe the TA behaviour.1, 2 WS (E) isdefined as 0
E − EaWS (E) =W0S
Eb − EaW0S

E ≤ EaEa < E < Eb ,(1)

E ≥ Eband according to the dispersion relation, VS (E) is
VS (E) = V0S + ∆VS (E)= V0S − (W0S /π)[ εa ln |εa | − εb ln |εb | ] .

(2)Here εi = (E − Ei )/(Eb − Ea ) with i = a, b, respectively. In theabove equations the variables with subscription S indicate thevalues at the RS . Thus, in the linear model the TA can be described by four parameters: Ea , Eb , W0S , and V0S . In general,these parameters except V0S are determined by fitting the imaginary potentials with eq 1, and the V0S is obtained at the reference
point of which the real potential is almost energy independent.
Due to the energy limit, the present work lacks of some highenergy points as the reference to extract the energy-independent
potential W0S . Moreover, no elastic scattering data in the literature are available for this system. Anyway the real and imaginary potentials can be simultaneously fit by eq 1 and eq 2, assuming the above equations are tenable (assuredly, this assumption is trustworthy). Thus the four parameters Ea = 89.2 MeV,
Eb = 101.7 MeV, W0S = −1.20 MeV, and V0S = −2.19 MeV werereliably extracted by the fits. The results are shown in Figure 3
with the solid lines.
3.3. Extract TA Parameters From Fusion Data. In fact,
the energy dependent potentials can be extracted from the fusion excitation function.11 It is interesting to compare them withthose from elastic scatterings. The BPM model with the consideration of nuclear deformation is used under the parabolic
approximation. The partial cross section at each angle is given
3.0

5

10

4

X10

91MeV
X10

4

93MeV
10

2.5

19

F+

208

Pb

2.0

1.5

3

96MeV
10

2

10

1

10

0

98MeV

X10

1.0

3

(RS=12.5fm)

X10

2

0.5

1.2

102MeV

10

-1

10

-2

-W S / MeV

dσ el/dσ Ru

5-VS / MeV

88MeV
10

X10

19

208

60

80

F+ Pb
Elastic Scatterings
40

100

0.8

elastic
fusion

0.4

120

140

160

θ c.m. / deg
Figure 2. Angular distributions of elastic scatterings.

0.0
80

100

120

140

160

180

200

ELab / MeV
Figure 3. The real and imaginary potentials extracted from elastic andfusion channels.

The Effects of Deformed Projectile in Threshold Anomaly

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 200229

by
π(2l + 1)
σlf (E, θ) =
2



k




2π
1 + exp
VB (θ) − Ecmh̄ω(θ) −1l(l + 1)h̄2+,
2µR2B (θ)

VC (r, θ) =Z1 Z2 e21
1 + 2 ∑ R20irr i=P,T(3)+

where k is the wave number, µ is the reduced mass, VB (θ), RB (θ),
and h̄ω(θ) the barrier parameters (height, radius, and curvature)for different orientation, respectively. Summing over the partial
waves, the above equation becomesσ f (E, θ) = ∑9β2i P2 (cos θi )20π
3
[ β2i P2 (cos θi ) ]2
7π
1β4i P4 (cos θi ) ,4π

1+ 4 ∑ R40i
r i=P,T(7)


VN (r, θ) = V0 1 + exp

σlf (E, θ)r − R0l=
R2B (θ)h̄ω(θ)
2πln 1 + exp(Ecm − VB (θ))
2Ecmh̄ω(θ)

.

i=P,T

(4)
Then the total cross section is given by integration over the angles
Z π/2
σ f (E, θ)dθ .
(5)
σ f (E) =0

On the basis of Wong model,12 the symmetrical deformationsof 19 F up to hexadecapole are taken into account in the calculations. The radial radius is expressed as
R(θ) = R0 [ 1 + β2Y2 (θ) + β4Y4 (θ) ] .

(6)Here R0 is the radius of the spherical nucleus with equivalent
volume, θ is the orientation angle of deformed nucleus with respect to the collision axis, and the Y (θ) is the spherical harmonic. The interaction potentials, including the Coulomb potential and the nuclear potential are taken as the deformed form,

10

1

10

0

,

(8)

−1/3
with R0 = R0P + R0T + 0.29 fm, R0i = (1.233A1/3) fm,
i −0.98Ai
i = P, T , and a = 0.63 fm. P(cos θ) is the Legendre polynomial
whose orders are considered up to 4, neglecting the higher order terms and other square terms. The nuclear potential only
contains the real part for the fusion channel.
In terms of eq 3–8, V0 can be determined by tuning it when
the calculation value equals to the experimental fusion cross section.8, 9 In order to compare with the TA behaviour of elastic
scattering, the V0S values are calculated at the same sensitive
radius 12.5 fm. The results are also shown in Figure 3 withthe square symbols. In the figure, the dashed line in the realpart of potentials represents the best fit by eq 2 with TA parameters Ea = 89.5 MeV, Eb = 99.1 MeV, W0S = −1.03 MeV, and
V0S = −2.23 MeV. The dashed line in the imaginary part of potentials is the results calculated by eq 1. It should be pointed out

CCDEF
Def. + Ene. dep.
Energy dependent
Deformed
Bare

-1

19

F+

208

Pb

10

5

10

4

10

3

10

2

10

1

10

0

σ QEx10

σF

2

3

19

H.Q. Zhang et al.
B.B. Back et al.

2

10
80

90

100

110

120

ELab / MeV
Figure 4. The fusion excitation functions and mean square spin distributions.

208

F+ Pb
16
208
O+ Pb
16
209
O+ Bi

2

< > / 10/aσ Rx100
210

9
β4i P4 (cos θi )
4π

3

10

10

+

σ / mbσ F / mb

105
β2i P2 (cos θi )4π
 −1− ∑ R0i

-1

-10

-5

0

5

10

15

Ec.m.-VB / MeV
Figure 5. Comparison with the neighbouring systems.

20

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J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

that the potentials extracted from the fusion data correspond to
the inner radius and it has a large degree of uncertainty to extrapolate to an outer radius, for example, to the sensitive radius.
Anyhow, it can be seen from the figure that these two kinds ofTA behaviours are similar. This reflects that they have the samecause, namely CC effects.
4. Discussion
4.1. Fusion Excitation Function and Spin Distribution.
In general, the energy dependent BPM model can well reproduce the enhancement of fusion cross sections near and below
the barrier, but fails to describe the broadening of spin distribution. In addition to the energy dependence, the deformation
effects are considered in the present work. The TA parameters extracted from elastic scatterings are employed in the calculations. The results well reproduce both cross sections andmean-square spins, as shown in Figure 4. The CC results calculated by CCDEF code are also illustrated in the figure forcomparison. In the CC calculation, six inelastic channels areincluded, as done by Zhang et al.,9 the 0.19714, 1.3457, 1.5540,
and 2.7798 MeV excited states of 19 F with β2 = 0.55, β3 = 0.33,
β2 = 0.58, β4 = 0.22, as well as the 2.6146 and 3.1977 MeV excited states of 208 Pb with β3 = 0.12 and β5 = 0.05, respectively.
It is clear that the increases of fusion cross sections and meansquare spins around the Coulomb barrier attribute to the energy
dependent potential, in other words, to the dynamic polarization
effects. While in the energy region further below the barrier,the static deformations of the nucleus should be considered toplay an important role. The small bump around the barrier inthe mean-square spin distribution results from the deformations.
4.2. Comparison with 16 O + 208 Pb and 16 O + 209 Bi Systems.
The reaction, quasi-elastic and fusion cross sections as a function of Ecm − VB for these three systems are plotted in Figure 5. The data of 16 O + 208 Pb and 16 O + 209 Bi systems are takenfrom References 5–7. Here the Bass barriers13 of 85.4 MeV,
77.1 MeV, and 78.0 MeV are used for 19 F + 208 Pb, 16 O + 208 Pb,
and 16 O + 209 Bi systems, respectively. As shown in Figure 5,
both reaction and quasi-elastic cross sections are comparable forthe 16 O + 208 Pb and 16 O + 209 Bi systems, while obviously larger
for the 19 F + 208 Pb system. Maybe, the larger probability of thequasi-elastic reactions results from the structure of projectile 19 F.
The fusion cross sections are comparable for these three systems
at near and well above barrier energies, but at energies lower
than a certain value (i.e. 3 MeV below the barrier) the fusioncross sections of 19 F + 208 Pb system are obviously larger thanthose of the other two systems which are still comparable. It isdifficult to understand why 16 O + 208 Pb and 16 O + 209 Bi systems
have the same tendencies in fusion excitation function while19
F + 208 Pb system does not. As shown in References 2,7, the fusion excitation functions can be reproduced by consideration of
the energy dependent potentials for 16 O + 208 Pb and 16 O + 209 Bi
systems. But for 19 F + 208 Pb system, the above analysis showsthat it is not enough to explain the larger fusion cross sections
at low energies. When the effects of 19 F deformations are also
taken into account in the calculations, the fusion excitation function can be reproduced quite well. Hence, we may conclude thatLin
the nuclear deformations give rise to some effects in the processof fusion below the Coulomb barrier.
5. Conclusion
The angular distributions of elastic scatterings are presented at six energies near and below the Coulomb barrier for19
F + 208 Pb system. The phenomenological optical model analysis gives the depths of the real and imaginary potentials, showingthat both real and imaginary potentials have pronounced energy
dependence. The real parts of potentials derived from the fusion
channel also show the energy dependence and are quite similar
to those from the elastic channel. In the framework of the linear
model, the TA parameters are extracted from both elastic andfusion channels. Comparison with the neighbouring systems
shows that the projectile deformations have effects on the fusion
reaction at low energy range. Using the TA parameters extracted
from the elastic channels, the fusion excitation function and themean-square spin distribution are reproduced by the deformed
and energy dependent BPM model. In addition, the calculations
indicate that the dynamic polarization effects play roles aroundthe barrier, while the static deformations are at work at energies
further below the barrier. From the above conclusion, we draw adeduction that the deformed configuration is of great benefit tosynthesis of heavy or super heavy nuclide at low energy.
Acknowledgement. This work was supported by the Nuclear Industry Science Foundation of China under Contract No.
92A01010 and partly by the Major State Basic Research Development Program under Grand No. G200077400.
References(1) C. Mahaux, H. Ngô, and G. R. Satchler, Nucl. Phys. A 449,
354 (1986).
(2) G. R. Satchler, Phys. Rep. 199, 147 (1991).
(3) H. Leucker, K. Becker, K. Blatt, W. Korsch, W. Luck, H.
G. Völk, D. Ficḱ, R. Butsch, H. J. Jänsch, H. Reich, and Z.
Moroz, Phys. Lett. B 233, 277 (1989).
(4) D. R. Tilley, H. R. Weller, C. M. Cheves, and R. M.
Chasteler, Nucl. Phys. A 595, 1 (1995).
(5) F. Videbæk, R. B. Goldstein, L. Grodzins, S. G. Steadman,
T. A. Belote, and J. D. Garrett, Phys. Rev. C 15, 954 (1977).
(6) E. Vulgaris, L. Grodzins, S. G. Steadman, and R. Ledoux,
Phys. Rev. C 33, 2017 (1986).
(7) P. Singh, S. Kailas, A. Chatterjee, S. S. Kerekatte, A. Navin,
A. Nijasure, and B. John, Nucl. Phys. A 555, 606 (1993).
(8) B. B. Back, R. R. Betts, J. E. Gindler, B. D. Wilkins, S.
Saini, M. B. Tsang, C. K. Gelbke, W. G. Lynch, M. A.
McMahan, and P. A. Baisden, Phys. Rev. C 32, 195 (1985).
(9) H. Q. Zhang, Z. H. Liu, J. C. Xu, K. Xu, J. Lu, and M.
Ruan, Nucl. Phys. A 512, 531 (1990).
(10) J. Raynal, Computing as a Language of Physics (IAEA, Vienna, 1972), p. 281.
(11) C. J. Lin, J. C. Xu, H. Q. Zhang, Z. H. Liu, F. Yang, and L.
X. Lu, Phys. Rev. C 63, 064606 (2001).
(12) C. Y. Wong, Phys. Rev. Lett. 31, 766 (1973).
(13) R. Bass, Phys. Rev. Lett. 39, 265 (1977).