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Analysis of Low-energy -4He Elastic Scattering Process in Direct Channel Regge-Pole Formalism.

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Annalen der Physik. 7. Folge, Band 41, Heft 3, 1984, S. 218-221
J. A. Barth, Leipzig
Analysis of Low-energy u - ~ Elastic
H ~
Scattering Process
in Direct Channel Regge-Pole Formalism
By D. MAJUMDAR,A. ROYCHOFVDHURY,
and T. ROY
High Energy Physics Division, Department of Physics, Jadavpur University, Calcutta/India
Abstract. The elastic scattering of low-energy a-particles by a 4He target has been analysed
within the frame-work of a modified Regge-Pole model which reproduces the partial wave phaseshifts in an elegant and simple way and describes all other relevant features of the scattering phenomena.
Analyse des elastiiehen a-4He-Streuungsprozesses bei niedrigen Energien im Direkt-
Kanel-Regge-Pol-Forma~mus
Inhaltsiibersicht. Die elastische Streuung von a-Teilchen niedriger Energie an einem 4HeTarget wurde im Rahmen eines modifizierten Regge-Pol-Modellsanalysiert ;dieses Modell stellt die
Phasenverschiebungen der Partialwellen in eleganter und einfacher Weise dar und beschreibt alle
anderen wesentlichen Aspekte des Streuvorganges.
The interaction between two a-particles is of utmost importance in nuclear structure
theory since the a-particle is believed to be an important substructure in complex nuclei.
Again, from such a study informations about the levels of even spin and panty of *Be
can be obtained. I n the complex angular momentum approach the modified pole model
devised by GRUSHIN
and NIZLITIN
[l]was applied by us in the analysis of the nuclear
scattering of a spin-1 projectile by a spinless target [2] and also that of the scattering
of a spinless projectile by a non-identical spinless target [3]. We have, therefore, here
considered the same model to study the nuclear scattering of two identical spinless
particles i.e., the resonant elastic scattering induced by 6 to 12 MeV (Lab.) a-particles
on 4He considering the contribution from a single-dominant pole only. From the experimental data [4] we note that the prominent intermediate state in the energy range
considered is the well known D-state resonance (J" = 2f) at 3.18 MeV in 8Be; we have,
therefore, taken into account the contribution from this resonance only. The differential
scattering cross-sections for the elastic scattering of two identical spinless bosons is
given by
o(E, 4 = If(&
(1)
Z,IZ
where the fictitious amplitude [5] f(E, z ) is given by
1
f ( E ,z) = - 2 (21 1)exp (id,) sin (6,) (P,(cos 0)
2=0,2.4,
...
+
+ P,(cos
7c
- @)}
(2)
where 6, are 'quasi-nuclear' phase-shifts. Applying Sommerfeld-Watsontransformation
D. MAJUMDARet al., Low-energy m4He Elastic Scattering
219
on (2) and Reggeizing according to GRUSHINand NIKITIN[l] we get
+
+
where 1 is the pole defined by L(E) = a ( E ) i p ( E ) ,a(1, E ) = R(A,E ) (21 1);
P2 as discussed in [3] and R(1, E ) is the residue at the
Cosh E = 1
=1 -
+ 2k2
+ 2k2
pole considered. In the case of resonance the leading pole is situated very close to the
real axis in the angular-momentum plane and hence p is very very small and this
smallness is corroborated by our numerical analysis as discussed later, so that in the
first approximation we can assume a(L,E ) to be a real quantity. Now projecting an
individual partial-wave out of (3) and remembering that
for
and using the well-known relation
(XI)
$1 PA(-z)P&) dz = n(L -2 Isin
) (A + I + 1) ’
-1
we obtain the following expression for the partial-wave amplitude,
a(l,E ) exp (E(L - I ) )
fdE) = (1 exp W))2 . sin dl (1- I) (21 + 1)
-
+
Now, on satisfying the condition of unitarity I f,(E)l2 = 1 we get,
2 (21 1)R1 exp (-&a - 1))
a(&E ) =
+
C
S
f
and
+ @)
t ’0°”
4
50’-
oo-.a A
n
A,
.8i
Y
n
,64
(4)
Ann. Physik Leipzig 41 (1984) 3
220
Corresponding to the contribution from a resonance state of SBe the theoretical onepole formula given by (3) can be rewritten as [l]
where T,(z) are functions as defined in [3]. Sin (6,) given by the equation (5) are evaluated by a least-square fit with the experimental data [4] for 5, d, and g partial-waves
only which are most effectively excited in the energy region considered. The angular
distributions given by the formula (1) along with the formula (2) with the experimental
phase-shifts [41 are compared with those given by the theoretical one-pole formulation
given by the equation (6).
The energy behaviour of the three resonant phase-shifts evaluated from the least
square fit to (5) are compared with the experimental data [4] and are depicted in Fig. 1.
The agreement is found to be very satisfactory over the entire energy region considered,
a
b
7'88
uuu
I
d
7'47 MeV
6'96
6'47
9 c.m
C
8' 87
9'88
1088
11'88 MeV
50 100150' 50 100 150' 50 100 150" 50 100150" 50 100150"
e
f
9
h
0c.m
Fig. 2. Centre of maw differential scattering cross-sections at different laboratory energies. The
circles are experimental cross-sections and the line represents the theoretical results
D. MAJUMDAR et al., Low-energy ,+He Elastic Scattering
221
on the other hand the phase-shift analysis of the experimental data reveals that only
8, d and g waves s c a t t e h g occurs in the said energy region. The values of the real part
of the angular momentum (corresponding to D-state in 8Be) are found to vary from
2.0104 to 2.0140, while those of the imaginary part vary from 0.0010 to 0.0030. The
theoretical angular distributions are compared with the experimental ones at different
bombarding energies from 6 to 12 MeV laboratory energies and are plotted in Fig. 2a
to h. The experimental scattering in the centre of mass system is found to be symmetric
7r
about 0 = -, as is evident from equations (1)and (2) when the scattering is indepen2
dent of the azimuthal angle. The experimental values a t 6.47, 6.96 and 7.47 MeV are
multiplied by 4 for the sake of convenience of drawing. A careful scrutiny of all the angular distribution curves reveals that these are decisively in favour of the experimental
ones with the distinct diffraction pattern satisfactorily reproduced. But the theoretical
minima are found to be too deep. This may be due to the fact that the experimental
arrangements cannot detect such very sharp minima. Again, the agreement would,
perhaps, have been better with the inclusion of the contributions from more than one
pole, but in that caae the mathematical analysis would have become much more
complicated and hence the simplicity of the method would have been lost. However,
considering a simplified pole model which furnishes proper asymptotic behaviour, satisfies unitarity condition etc. as are required by the exact partial-wave amplitude, we
have succeeded in proving that the inclusion of the contribution from one dominant
resonance only leads a t least to a qualitative best fit to the experimental angular
distribution data over the whole of the angular range and to the reproduction of other
relevant features of the nuclear scattering phenomena of two spinless identical particles
at low energy.
References
[l] GRUSEIN, V. V.; NIELITIN,
Yu. P.: Sov. J. Nucl. Phya. 15 (1971) 89; (Yad. Rz. 18 (1971) 166).
A.; ROY,T.: J. Phya. G 1 (1975) 646.
[2] GUPTA,K.; ROY~HOWDWRY,
[3] MAJUMDAR,
D.; ROYCHOWDHURY,A.; ROY,T.: Z. Naturforach. 86 A (1981) 443.
T. A.; SENHOUSE,
L. S.: Phya. Rev. 129 (1963) 2262.
[4] TOMBBELLO,
M.; CARRASSI,M.; PASSATORE,
G.: Nuovo Cimento 86 (1965) 954.
[5] BERTERO,
Bei der Redaktion eingegangen am 26. September 1980, revidiertes Manuskript 5. Oktober 1983.
Anschr. d. Verf.: D. MAJUBWAR,
A. ROYCEOWDEURYand T. ROY
High Energy Phyaioa Division
Department of Physica
Jadavpur University
Calcutta-700032/India
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channel, process, elastica, low, scattering, 4he, direct, energy, formalism, analysis, pole, regge
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