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Classical Description of Two-Cluster Transfer Reactions in a Coulomb Field.

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Annalen der Physik. 7. Folge, Band 34, Heft 2, 1977, S. 91-98
J. A. Barth, Leipzig
Classical Description of Two-Cluster Transfer Reactions
in a Coulomb Field
By M. EL-NADIand A. OSMAN
Cairo University, Physics Department, Faculty of Science, Cairo (Egypt)
A b s t r a c t . The theory of two-cluster transfer reactions is reconsidered on the basis of a simple
semiclassical approximation. The obtained expression for the differential cross section is in a closed
form so as to be easily calculated. A theoretical study with calculations for the angular distrihution
for different incoming and outgoing energies, and also for different angular momentum transfer is
introduced. Comparing our theoretical calculations with experimental data, satisfactory agreement is
obtained especially in the backward direction.
Hlassische Beschreibung yon Zwei-,,Cluster-Transfer"-Reaktionen
in eineni Coulomb-Feld
I n h a l t s u b e r s i c h t . Es wird eine Theorie der Schwerionenreaktionen mit MehrnukleonenTransfer angegeben. Dabei wird das relativ einfache Problem bei Energien unterhalb der Coulombschwelle betrachtet, bei denen die Coulombwechselwirkung den entscheidenden Beitrag liefert. Die
Reaktion wird halbklassisch beschrieben, wobei ein geschlossener Ausdruck fur die Winkelverteilung
erhalten wird.
1. Introdiiction
The theory of two-cluster transfer reactions a t energies well below the Coulomb
barrier was considered by EL-NADI
[l-31. Due t o the large Coulomb repulsion between
the projectile and the target nucleus, the motion of the incoming and outgoing particles
is highly distorted. Thus, in such a case the expression of the angular distribution for a
given nuclear reaction will largely depend on the Coulomb force. Considering the motion
of the particles in the semi-classical approximation, one has t o consider the classical
trajectories in the external Coulomb field of the nucleus. The work of ALDERet al. [4]
on Coulomb excitation and of LEMMER
[ 5 ] on (d, p ) stripping show that in many cases
the classical treatment of the Coulomb stripping may lead to simple quantitative expressions for the cross section, which explain the main features of the experimental
data. The semi-classical approximation has the advantage of leading t o a closed expression for the angular distribution. I n the quantum mechanical treatment one has t o
deal with the complicated Coulomb wave functions which are difficult to compute.
I n the present work, we shalluse a classical description for the charged particle motion
in the external Coulomb field of the nucleus. The energies of both the incident and outgoing particles are well below the Coulomb barrier; hence following Lemmer's work,
one should consider a semi-classical description for the incoming and outgoing particles.
92
M. EL-NADI
and A. OSMAN
2. Theory
The transition niatrix element for two-cluster transfer reactions is given by [l- 31 :
and
The transition matrix element for two-cluster transfer reactions is given in ref.
[l--31 where we are left withtheintegral i including the Coulomb wave functions f(f)(r)
Y
f(-)(r)and the bound state wave function G J f M f ( ras:
)
f = J dr eiQ.r GhPlff(r)f ( + ) ( r )/*(-)(r)
(1)
where
Expanding Coulomb wave functions in partial wavesfor f ( f ) ( r )and f7-)(r) and introducing a MORINIGO[6] form for the wave function
we obtain an expression
for with an integral Rn.a refered to as the radial matrix element given by
hff
R
~= J~dr r+'> e-firFni
~
~(k,r)~Flf(kfr).
(2)
I n case when the energies of both the projectile and outgoing particles are well below
the Couloinb barrier, i.e. when 7
1, the radial matrix element (2) could be treated
by the WKB approximation following the work of ALDERet al. [4] and LEMMER[ 5 ] .
The WBK method, which gives good accuracy, gives:
with
Classical Description of Two-Cluster Transfer Reactions in a Coulomb Field
93
where r,, is the classical turning point given by f(ro) = 0.
After a lengthy but straightforward calculation which has no place here, we get for
the differential cross-section the following expression :
94
M. EL-NADI
and A. OSMAN
where
q
= it - k, - k,
f 2k4
+- 2k,,
5 = Vf - Ti,
Y = PrT
and
I n the expression (6) for the differential cross-section the symbols 0 stands for the
spectroscopic factors. j a and j b are the total spins of the two clusters in the projectile
while Ja and Jb are their spins in the residual nucleus. I , L and 2 are the relative, centre
of mass and total angular momentums of the transfered clusters in the projectile, while
If, Lfand gf are the corresponding angular momentums in the residual nucleus.
3. Results and Discussion
I n some cases of nuclear reactions, the transfered clusters are found in the projectile
such that there is no transfer in angular momentum. I n these cases of nuclear reactions,
it is found that the transfered relative, I , centre of mass, L,and also the total, 2, angular
momentums, are zero. This is very interesting because we find that the very long mathematical expression (6) for the differential cross section reduces t o a very short and closed
expression.
I n csae of interaction between complex ions where no transfer of angular momentum
takes place, i.e. when 1 = L = 2 = 0, we get for the differential cross-section the expression :
This forniula is calculated for the nuclear reaction 018(C12,oc) MgZ4with I = L =
2 = 0. It is investigated in case of different C12energies (Fig. 1) and for different excited
states of MgZ4(Fig. 2). Similarly, calculations for the differential cross-section for the
same reaction were carried out for different angular momentum transfers (Fig. 3).
Due to the lack of experimental data a t energies well below the Coulonib barrier,
the expression (11) is calculated for the reaction C12(01e,oc) MgZ4which was carried out
experimentally a t energies 6.75, 7.393 and 7.929 MeV, all slightly less than the barrier
height (= 9.582 MeV), by GROCEand LAWRENCE
[7] (Figs. 4, 5 and 6). Cn Figs. 7, 8 and
9 , the expression (6) is applied t o the reaction Bes(Li6,a ) Bll for different excited states
of Bll, with Lie energy 1.95 MeV and a Couloinb barrier height of 2.956 MeV. The experimental observations are those of BLAIR[8].
Although the energies in Figs. 4-9 are not sufficiently below the Coulomb barrier
height, these curves show significantly a backward rise in the angular distribution which
is the characteristic features of Coulomb stripping reactions.
The principal result of this work, as may be seen from Figs. 1-3, is that the Coulomb
effects become dominating, and the angular distribution changes markedly from the
Butler usual forward distributions to a predominantly backward distribution which
becomes more peaked as the Coulomb parameter q Land qf increases. Thus, in that typical example one sees that for large q, and qf the angular distribution becomes Gaussian
Classical Description of Two-Cluster Transfer Reactions in a Coulomb Field
0 Icm I
Fig. 1
95
elcml
Fig. 2
Fig. 1 Calculated angular distribution of the ejected a-particles in the nuclear reaction Ola(C12,a ) MgZ4leaving the Mg24 nucleus in its ground state. The calculations are
performed at different C12 energies of about one third, half, and near the Coulomb
barrier height between the 0 1 % and the C12 nuclei
Fig. 2 Calculated angular distribution of the ejected a-particles in the nuclear reaction
016(Cla,a)Mgz4 leaving the Mg24 nucleus in the ground, second excited, and the fourth
excited state. The C12energy is taken to be 3 MeV
Fig. 3 Calculated angular distribution of the ejected a-particles in the nuclear
reaction OlB(C12,a ) Mg24 leaving the MgZ4nucleus in different excited states with different transfered angular momentum L = 0,2, and 4. The Clzenergy is taken to be 4.5 MeV
X. EL-NADI
and A. OSMAN
96
20.70 -
B
($0.07 p- z f
-Ti
I
0.007
3
?;,
I
I
I
0 Icm I
Fig. 4 Angular distribution of the emitted alpha particles from the reaction
C12(016,
a)MgZ4 leaving the residual nucleus in its ground state. The energy of the incident 0 l 6 ions is 15.75 MeV.
The solid curve in Fig. 4-6 reprzsent our present theoretical calculations. The
[7]
experimental data are due to GROCEand LAWRENCE
10.0
1
I
0.07 I
0
I
I
30
60
1
I
720
150
I
90
0 lcml
1
780
Fig. 6 Angular distribution of the emitted alpha particles from the reaction
leaving the residual nucleus in it,s first excited state. The energy of the
incident 0 1 6 ions is 17.00 MeV
C12(016,
a ) Mg24
I
30
I
I
60
90
8 icm)
I
I
120
750
780
Fig. 6 Angular distribution of the emitted alpha particles from the reaction
C12(016,
a)Mg24leaving the residual nucleus in its ground state. The energy of the incident 0 1 6 ions is 18.60 MeV
Classical Description of Two-Cluster Transfer Reactions in a Coulomb Field
T
0.700 I
--
97
I
I
I
I
I
I
QO
60
30
.
I
I
I
720
758
mo
8 lcml
Fig. 7 Bngular distribution of the emitted alpha particles from the reaction
Be9(Li6,a ) B**leaving the residual nucleus in its ground state. The solid curve in
Fig. 7-9 is our present theoretical calculations. The experimental data are due t o
B ~ a m[8]. ELi6= 3.26 MeV
0'
0
I
I
I
I
I
30
60
90
720
750
780
0 lcm 1
Fig. 8 Sngular distribution of the emitted alpha particles from the reaction
Be9(Li6,x) B" leaving the residual nucleus in its first excited state
0.700
d
I
30
I
60
I
go
I
720
I
150
1
8 lcml
Fig. 9 Angular distribution of the emitted alpha particles from the reaction
Be9(Li6,x) Bl' leaving the residual nucleus in the second and third excited state
7
,\iiti.
Pliysik. 7. Folge, Bd. 3'1
M. EL-NADIand A. OSMAN
98
in form about the backward direction, with a width that decreases as q increases. I n the
case of L
0, and for low values of q , Butler forward peaks are displaced slightly to
larger angles. The same holds for large values of q as is clear from Pig. 3.
This shows that the semi-classical treatment of the Coulomb stripping reactions lead
to a simple closed forni for the cross-section.
+
References
[l]31.EL-XADI,A. OSMANand T. H. RIHAK,Acta Phys. Hung. 29, 127 (1970).
M. EL-NADI,A. OSMANand T. H. RIHAX,Acta Phys. Hung. 29,143 (1970).
$1. EL-NADI,A. OSMAN
and T. H. RIHAK,
U.A.R. J. Physics 5, 45 (1971).
K. ALDER,A. BOD, T. Huns, B. MOTTELSOH
and A. W’INTHER, Rev. Mod. Phys. 28,432 (1956).
R. H. LEYMER,
Nucl. Phys. 39, 680 (1962).
[GI F. B. MORINIGO,
Phys. Rev. 134, B 1243 (1964).
[‘i] D. E. GROCE
and G. P. LAWRENCE,
Nucl. Phys. 65, 277 (1965).
[a] J. W.BLAIR,“Proceedings of the Second Conference on Reactions Between Complex Nuclei”,
[2]
[3]
[4]
[5]
1960, p. 138.
Bei der Redaktion eingegangen am 4. Dezember 1975.
Anschr. d. Tierf.: Dr. A. OSMAN
Kuwait University, Physics Department,
Faculty of Science,
P.O. Box 5969,
Kuwait
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