Rainbow effects in nuclear scattering* Angular distributions. Phase rules

31 Pages·2004·0.39 MB·English

Rainbow effects in nuclear scattering*
Angular distributions. Phase rules
R. da Silveira
Division de Physique Théorique*
Institut de Physique Nucléaire
F-91406 Orsay Cedex
IPNO/TH 89-38 July 1989
"Invited talk given at the International Workshop on Rainbow Scattering. Cavtat, Yu-
goslavia, August 14-16, 1989.
, tUnité de Recherche des Universités Paris 11 et Paris 6 associée au C.N.R.S.
Contents
Introduction
1. The angular distribution patterns
1- The uniform approximation of the scattering amplitude.
2- The role of the absorption in changing the Airy-Iike shape of the
angular distributions.
2. Feynman path approach to phase rules
1- The collision model.
2- The scattering amplitude. Classical and quantal regimes.
3- Interference effects. Phase rules.
Introduction
The usual way to identify the rainbow effect is by observing the visual
resemblance between the shape of the angular distributions and that of
the square of an Airy function, i.e., an oscillatory pattern- the illuminated
region- and beyond it, a smooth exponential decrease- the shadow region.
This kind of behaviour is easily identified in the angular distributions
displayed in figures 1, 2, 3 and 4. These are examples of elastic (1,2) and
non-elastic (3,4) rainbows.
Ir I
i ]
Ar40^Sn120 i
'5 !-
ir 1760 MeV j
1
• [
'•"S J
i
A '
L / \ J
I
-"^V•' "• X /i ^ \ I'
'[ ;
' \ ]
|_
!
\ ;
\
L \ -;
i
\ J
5 -
I ,
L \ J
* (
\
1
ri -J
L '\ J
2 3 i 3r 5 3 go
Fig.l : Elastic angular distribution for Ar40 + Sn120 at E = 1760 MeV. 6 is the
r
Coulomb rainbow angle. The curve is to guide the eye. Prom ref.5.
da
da,
iff1
a+ NI
KOMeV
id'
30 70 e 90
Fig.2 : Elastic scattering of a-particles on JVt58 at E = 140 MeV, The angular
distribution corresponds to the "negative-angle" scattering. 9 is the nuclear rain-
r
bow angle. This angular distribution has been obtained from the experimental
data18, using the "envelope method". See ref.ll for details.
c_ C1SBa'38 64Mev
"S/
|10
/" \
C /,-. \
? 1 •', x \
A//: \ \
-a F 7/2 .y \/i ; \ '
p 3/2 VJ ! Vransf
X
\ «• », ' .""*•,
a/a
R elastic
10'
i i < i i i i i i i
30 50 90
6cm
Fig.3 ; Angular distributions for the elastic scattering and the one-neutron trans-
fer reactions (p 3/2 and f 7/2 in Ba139) in the collision of C14 on Ba138 at 64 MeV.
The curves are drawn to guide the eye. The data is from ref.6
0> C12 i r~> i 208 Ox1NyI
+ rb_ 96Mev
/ \ F 5/2
10
\r\ \
-Oa M \ X hransf
1 i/2
M \ p
a/a
R
i
-^««•...-" "-. —
'
\ elashc
IQ"
!O
- •-• ^-. '•
1
10"2
'inelast
10 "
1 I -3
0
i i i i i i t « i
20 40 60 80
0cm
Fig-4 : Angular distributions for the elastic scattering, the inelastic scattering
(2+ and 3~ in Po208J and the one-neutron transfer reactions ((p 1/2)"1 and (f
5/2)-1 in Pb201) in the collision of C11 on P6208. The incident energy is 96 MeV
for elastic scattering and 98 MeV for inelastic and transfer reactions. The curves
are the result of a DWBA analysis. The data is from ref.[7,8,9].
Besides the specific behaviour displayed in these figures, the rainbow ef-
fect has another less familiar signature. This signature which is embodied
in what we shall call "phase rules" [1,2,3], arises as a direct consequence of
the interferences originating from the rainbow situation. Examples of these
"phase rules" can be seen hi figures 3 and 4, by comparing the undula-
tions of the elastic cross-section with those of the non-elastic cross-section,
inelastic scattering and one neutron transfer.
From this comparison, it is apparent that,
i) The oscillations of the elastic angular distributions are :
- in phase with those of the neutron transfer [3],
- 180° out-of-phase with those of the inelastic scattering [1,2].
ii) for the non-elastic channels, the oscillations of the angular distribu-
tions corresponding to different spin states are in phase with each
other.
In the course of my talk I would like :
i) to elucidate the role of the collision mechanism in determining the
angular distribution patterns,
ii) to give a simple explanation of the "phase rules".
Concerning the analysis of the angular distributions, the starting point
will be the quasi-classical limit of the high-energy scattering amplitude in
which the real and the imaginary parts of the optical potential are properly
included. Using simple parametrizations for both the classical deflection
function and the 5-matrix, the resulting scattering amplitude is evaluated
in the uniform approximation. This allows to obtain for the scattering
cross-section a closed analytical expression suitable for understanding the
angular distributions patterns [13]. This will be summarized in chapter 1.
Chapter 2 will be devoted to the study of the interference effects giving
rise to the "phase rules". The study of the "phase rules" involves both
elastic and non-elastic channels. Inelastic scattering and nucléon transfer
are usually analyzed in the framework of the "Distorted Wave Born Ap-
proximation" . This is done either with the help of numerical codes or by
using semi-classical approximations. However, as long as our purpose is the
understanding of some general features arising from the interference effects,
the use of the D.W.B.A. is somewhat cumbersome. Rather, we shall use
a Feynman path approach applied to a collision model [12] which through
very simple retains enough of the important physical features to reproduce
the interference effects responsible for the "phase rules". This model is
inspired by the conceptual experiment of passage of particles through a slit
as discussed by Feynman and Hibbs [4].
1 The angular distribution patterns
1.1 The uniform approximation of the high-energy
scattering amplitude. Closed form expressions
Let us assume that the incident energy is high enough that V/ E « 1 (V
is a measure of the absolute magnitude of the potential) and also that the
contributing £-values are very large, kR » I. The scattering amplitude is
given approximately by [20]
1(8) = IMJ (U) exp ~ V [(P/k* + z2)1/2]^ (1)
o 0 0p
where t/k — b is the impact parameter and V = V(r) + iW(r) is the
0p
optical potential.
Furthermore, we also assume that the regime of the scattering process
is a quasi classical one, i.e.,
(V I E) xkR»l (2)
As the angular range of the approximation is 6 ~ V /E, one has from (2) ,
OkR » 1. This allows the replacement of J (x) by (2/îrz)1/2 CQS(I - Tr/4).
0
The scattering amplitudes becomes,
= f (8)+f.(8)
+
where the f±(0) are given by
/±w (2 6] 1 il 2dee e>io± ee (3)
~ ik * J " * '
0
(4)
(5)
The square modulus of /±(0) gives the scattering cross-section through
"positive" (-) and "negative" (+) angles. The function exp(w(£)j is the
modulus of the S-matrix, \S(i)\ = exp[w(£)].
In the classical limit, the integral (3) can be evaluated with accuracy by
the stationary phase method if exp[o/(£)] is a smoothly varying function of
L However if rainbow scattering occurs , the stationary points of <p±(£,0)
can either be well separated or coalesce. To account for both situations we
use the uniform approximation [21,15,27], The integral to be considered is
of the general form,
J-OO
It is assumed that <p± has two stationary points I — £,• (» = 1,2) which can
come close together or coalesce as O varies. The case where more than two
stationay points coalesce [14,15,16] is unlikely to occur in the energy range
of interest. The lower limit in (6) has been extended to — oo because the
main contributions come from the neighbourhood of £,-.
The uniform approximation for (6), is [15]
e)i (7)
(8)
1»- _i_ \-i i - f • i-J-\-*j "/1 V /
2
Ai(— C) and A'i(— C) being the Airy function and its derivative respectively.
The uniform approximation includes as special cases the non-uniform
results : the stationary phase method (f » O) , and the Airy approxima-
tion (£ ~ O). Now, the stationary points of <p±(t,6) are those satisfying
= O, or, from (4) and (5),
_ J-A F [(t2/k2 + z2)1/2]d* =- T i/k ( dV)dr = ±6
V
(10)
In (10) one recognizes the expression for the classical deflection function in
the limit V /E « 1 [22,23]. Equation (10) is written
G(£) = ±9 (11)
The lower limit of the right-hand side integral in (10) tells us that the
distance of closest approach of a trajectory of impact parameter b = i/k is
6 itself. This means that the projectile moves on a straight-line trajectory. It
should be said that in heavy-ion scattering at intermediate energy, one often
deals with non négligeable Coulomb repulsion for trajectories of impact
parameter b < R. This slightly incurve the incident trajectory within the
range of the nuclear forces. To account for such effects one evaluates [17]
the contribution of the nuclear potential to the integral (10) for an effective
impact parameter b = b + TJ /k (»7 is the Sommerfeld parameter). This is
e
almost equivalent to considering incident particles of impact parameter 6
impinging on a target of radius R — TJ /k rather than R.
To proceed further one chooses simple parametrized forms for both €>(£)
and exp[w(£)].
i) 0(£) is parametrized by
0(£) = 0 + g(£-£)2 , (12)
r r
where 9 ,q and i have their usual meanings. It should be noted that the
T r
use of a parabola is a much better approximation for the nuclear rainbow
than for the Coulomb one (see figure 5).
ii) exp[w(£)] is parametrized by ,
(13)
where I is the strong absorption angular momentum,
0
10