ХИМИЧЕСКАЯ ФИЗИКА, 2ÚÚ4, том 23, № 2, с. б-12
ЭЛЕМЕНТАРНЫЕ ФИЗИКО-ХИМИЧЕСКИЕ ПРОЦЕССЫ
yflK 539.188
IMPROVED NEARSIDE-FARSIDE DECOMPOSITION OF ELASTIC SCATTERING AMPLITUDES
© 2004 r. R. Anni, J. N. L. Connor*, C. Noli*
Dipartimento di Fisica dell' Universita and Istituto Nazionale di Fisica Nucleare, 173100 Lecce, Italy * Department of Chemistry, University of Manchester, Manchester M13 9PL, United Kingdom Received 16.11.2002
A recently proposed technique is described that provides improved nearsie-farside (NF) decompositions of elastic scattering amplitudes. The technique, involving a new resummation formula for Legendre partial wave series, reduces the importance of unphysical contributions to NF subamplitudes, which can appear in more conventional NF decompositions. Applications are made to a strong absorption model that arises in chemical and nuclear physics, as well as to a O + 12C optical potential at Elab = 132 MeV.
1. INTRODUCTION
In nuclear, atomic and molecular collisions, an elastic differential cross section a(0), where 0 is the scattering angle, is often characterized by a complicated interference pattern. In some cases, semiclassical methods [1] explain the scattering pattern as the interference between simpler, and slowly varying, subamplitudes. Ignoring the complication that, in some angular regions, uniform asymptotic techniques are often necessary, then the semiclassical subamplitudes arise mathematically from saddle points or poles which account physically for contributions from reflected, refracted or generalized diffracted semiclassical trajectories [2]. These subamplitudes can be conveniently grouped into two types: those arising from semiclassical trajectories which initially move in the same half plane as the detector (N or nearside trajectories) and those from the opposite half plane (F or farside trajectories).
Semiclassical methods are not always simple to apply and sometimes they have a limited range of applicability. In order to overcome these difficulties it is possible to apply a simple NF decomposition to the elastic scattering amplitude f(0), that was proposed by Fuller [3] more than 25 years ago. Fuller's NF method uses only the quantum mechanical scattering matrix elements Sl, therefore bypassing the difficulties in using semiclassical methods. The aim of a NF method is to split the full scattering amplitude f(0) into the sum of two subamplitudes, f(-)(0) and f(+)(0), describing the contributions from the N and F scattered particles, respectively.
The Fuller NF method was introduced to treat nucleus-nucleus scattering, where a long range Coulombic
term is present in the interaction potential. Its starting point is the usual partial wave series (PWS) for /(0)
/(б) = 2kE aPl( cos б),
(1)
l = 0
where k is the wavenumber, P(cos 0) is the Legendre polynomial of degree l and al is given in terms of the scattering matrix element Sl by:
al = ( 2l +1 )( Sl -l ).
(2)
We recall, in fact, that the PWS in (1), considered as a distribution, is convergent also if Sl is asymptotically Coulombic [4].
In Section 2, we outline the Fuller NF method and discuss some of its limitations. Section 3 describes an improved NF method using resumation techniques that reduces the importance of unphysical contributions to the NF subamplitudes. Results from our improved NF method for the elastic scattering of two collision systems are presented in Section 4. Our conclusions are in Section 5.
2. FULLER NF METHOD AND ITS LIMITATIONS
The Fuller NF decomposition is realized by splitting P;(cos 0), considered as a standing angular wave, into traveling angular wave components
P¡( cos б) = QÍ-)( cos б) + Ql+( cos б), where (for б Ф 0, n)
Q(T)( cos б) = --
Electronic address: J.N.L. Connor@Manchester.ac.uk
Pl ( cos б)± ^Q, ( cos б)
(3)
(4)
*
б
with Q^cos 0) the Legendre function of the second kind of degree l.
Inserting (3) into (1), splits /(0) into the sum of two subamplitudes /(T) (0). For lsin0 > 1, the Qf] (cos0) behave as
Q(,)( cos e)~J
Xe -
n
(5)
example is the angular distribution for a strong absorption model (SAM) with a two parameter (A and A) symmetric S-matrix element and Fermi-like form factors [7]
S = S(X) =
■ lA - X
i + exp i -a
+
l A + X 1 + exp i -A
(6)
with X = l + 1/2. This asymptotic behavior suggests that
the f(-) (e) should correspond to NF trajectories respectively appearing in the complete semiclassical decomposition off(e) (Ref. [1], p. 121).
The NF decomposition is in general less satisfactory than the full semiclassical one. For example, if two or more N or F semiclassical trajectories contribute to the same e, interference effects may appear in the N or F cross sections. The Fuller NF decomposition has, however, the merit of being simple and, although inspired by the semiclassical theories, it uses only quantities calculated within the exact quantum mechanical treatment. The NF method therefore bypasses problems associated with the applicability and validity of semiclassical theories.
The physical meaning attributed to the f(e) is implicitly based on the (unproven) hypothesis that it is possible to perform on the PWS, written in terms of the
(cos e), the same manipulations that are used in deriving the complete semiclassical decomposition of f(e). These manipulations are path deformations in X of the integrals into which (1) can be transformed, using either the Poisson summation formula or the Watson transformation. The consequences of these path deformations depend on the properties of the terms in the PWS when they are continued to real or complex values of X from the initial half integer X values. The splitting of P^cos e) into Q(+) (cos e) modifies these properties and can cause the appearance of unphysical contributions in the f(+) (e) which cancel out inf(e).
In spite of these possible limitations, the Fuller NF decomposition is widely used, as demonstrated by the fact that the ISI Web of Science reports about 140 citations since 1981 to the original Fuller work. In many nucleus-nucleus scattering cases [5, 6], the Fuller method effectively decomposes f(e) into simpler subamplitudes, which are free from the unphysical contributions that can arise from the above mathematical difficulties. However for a few examples, the NF subamplitudes can be directly compared with the corresponding semiclassi-cal results and it is found that the Fuller and semiclassical decompositions predict different results. One classic example is pure Coulomb scattering. For repulsive Coulomb potentials only a N contribution is expected semi-classically ([1], p. 56), whereas the Fuller NF decomposition yields also a F contribution [3]. Another important
with X = l + 1/2. For a fixed value of the cut-off parameter A and for a sufficiently large value of the diffuse-ness parameter A, the Fuller NF cross sections agree with the semiclassical results only up to a certain value of 0, which decreases with increasing A.
Fortunately, the Fuller NF subamplitudes contain information that allows one to recognize the unphysical nature of the undesired contributions. Suppose /(+)(0), or /-)(0), contains a single contribution from a stationary phase point at X(0). Then the derivative with respect to 0 of the phase of/(+)(0), or /-)(0), is equal to X(0), or —X(0) respectively. Following Fuller we will call this derivative the Local Angular Momentum (LAM) for the N (or F) subamplitude; it depends on 0 ([1], p. 57). Only for certain generalized diffracted trajectories is the LAM expected to be constant, equal to the angular momentum of the incoming particle responsible for the diffraction. In the semiclassical regime, this constant value is expected to be large. Because of this, if we observe that in a certain 0 range LAM = 0, this can be considered the signature of the unphysical nature of the N or F subamplitudes in that range of 0. This occurs for the LAM of the Fuller Coulomb F subamplitude, and for the NF subamplitudes of the SAM in the angular region where the Fuller NF cross sections differ from the semiclassical results. In both cases this decoupling of 0 from LAM suggests the unphysical nature of the subamplitudes. Thus an analysis of the LAM can avoid misleading interpretations of cross sections obtained from the Fuller Nf decomposition. However the problem of obtaining more satisfactory NF decompositions remains open.
3. IMPROVED NF METHOD USING RESUMMATION THEORY
A possible solution to the problem was proposed by Hatchell [7], who used a modified NF decomposition. The modifications consisted of, first, writing /(0) in the resumed form (0 ^ 0)
f (e) =
1
1
2 ik( i-cos e)
a( r) Pt ( cos e),
(7)
l = 0
r = 1, 2, ..., and, second, using a different splitting for the Legendre polynomials into travelling waves.
The use of the resummed form (7) for/(0) was originally proposed [8] by Yennie, Ravenhall, and Wilson (YRW) to speed up the convergence of the PWS for high-energy electron-nucleus scattering. Equation (7)
is an exact resummation formula, of order r, which is derived from the recurrence relation for Legendre polynomials. The YRW resummation formula can be derived by iterating r times, starting from a(0) = a, the re-summation identity
Ia('-1)Pi(cos= ^^Ia(°P(cos6), (8) 2a('"Pl(cos6) = ex ! + (b' co s 92a(°P
One possible solution to this puzzle was recently found by introducing an improved NF method [15, 16], based, first, in dropping the 1 from the term (Sl _ 1) in the PWS for /(0), and, second, in considering (8) only as a particular case of a modified resummation identity [17]
a(0Pl(cos 0), (10)
i = o
i = 1, 2, ..., where
i = o
(i) l (i -1) (i -1) l +1 (i -1) ,n\
a( = "27^1 a-1 + a "27+3 a( +1' (9)
with a
(i -i) -1
= 0.
with a + P' cos 6 ^ 0 and
( i) n l ( i -1)
al = Pi Sl-! al -i + aal
('-1) + PiJr+1 a(+i1). (11)
i2l + 3
Note that f(0) is independent of r, unlike the Fuller NF subamplitudes which do depend on the value of r used. This
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