Download INSTITUTE OF NUCLEAR PHYSICS

The Henryk Niewodnicza
nski
INSTITUTE OF NUCLEAR PHYSICS
ul. Radzikowskiego 152, 31-342 Krakow, Poland
www.ifj.edu.pl/reports/2001.html
Krakow, October 2001
Report No 1882/PL
Coulomb Amplitude Representation
and Nuclear Diraction and Refraction
H. Wojciechowski
The Henryk Niewodniczanski Institute of Nuclear Physics, 31-342 Krakow, Poland
e-mail: [email protected]
Abstract
A new, consistent with the wave optics, method of treating diraction and refraction in nuclear
elastic scattering is proposed and discussed. The method is based on the decomposition of the total
elastic scattering amplitude, written in dierent than usual form, into the amplitude describing pure
diraction and the rest i.e. refraction and reection. It is shown that nuclear diraction, being a
kind of always present "background, is due to the incompleteness of the pure Coulomb scattering
amplitude and is described by it irrespectively of the studied system.
1 Introduction
Nuclear elastic scattering, the simplest nuclear reaction, has been studied extensively from the early beginning of nuclear physics. The experimental dierential cross section for this reaction can be reproduced
quite well using, for example, the optical model, where the elastic scattering matrix elements are obtained
by solving the Schrodinger equation for complex nuclear plus Coulomb potentials. From the early studies
it is known that the elastic dierential cross sections show structure resembling diractional patterns
well known from the wave optics. Many semiclassical models, based on parameterization of the elastic
scattering amplitude or of the elastic scattering matrix elements, were used to describe these patterns
successfully.
The present paper presents how these patterns can be treated in the pure quantum way by modifying
slightly the elastic scattering amplitude and its interpretation.
2 Total elastic amplitude
According to the quantum scattering theory, the dierential cross section for charged particle elastic
scattering is calculated from the total elastic scattering amplitude of the form:
1
fel () =
n exp[;in ln(sin2 1 ) + i + 2i ] +
0
2
2k sin2 21 1 1 (2l + 1)S C (S ; 1)P (cos )
l
l l
2ik 0
X
(1)
where: is the scattering angle, n is the Sommerfeld parameter, 0 = arg ;(1+in), k is the wave number,
SlC are the Coulomb scattering matrix elements and Sl are the nuclear scattering matrix elements.
Instead of using the analytical form of the Coulomb scattering amplitude, we can use one of the series
which, as a distribution, converges to it [1, 2, 3]. This series is:
X
1 1 (2l + 1)S C P (cos );
fC () = 2ik
l l
0
(2)
In the absence of Coulomb interaction (SlC = 1, i.e. neutral particle scattering), this reduces to the
"residual Coulomb amplitude":
X
1 1 (2l + 1)P (cos );
f0 () = 2ik
l
0
(3)
which behaves as [1 ; cos()].
If we now insert (2) into (1), we obtain the following simple expression for total elastic scattering
amplitude for charged particles:
X
1 1 (2l + 1)S C S P (cos )
fel () = 2ik
l l l
0
(4)
This is the most general form of the elastic scattering amplitude. In the absence of nuclear force
(Sl = 1), we have the Coulomb amplitude (2). By including nuclear force, one simply modies the
low partial amplitudes by complex nuclear scattering functions Sl . Outside the nuclear region (above a
certain value of angular momentum), we have pure Coulomb partial amplitudes.
3 Diractive and refractive scattering amplitudes
The practical application of the above new forms (4) of scattering amplitude is a very simple, almost
trivial, treatment of nuclear diraction and refraction.
Let us rst consider the nucleus as a sharp-edged object of "radius" lgr in angular momentum space.
In this case, ReSl and ImSl have certain values for l lgr , and ReSl = 1 and ImSl = 0 for l > lgr .
The elastic scattering amplitude (6) can be then split into two parts:
X
X1 (2l + 1)SlC Pl(cos)
+ 1
1 lgr (2l + 1)S C S P (cos )
fel () = 2ik
l l l
0
2ik lgr +1
(5)
The above decomposition is unique and is a consequence of the behavior of the real parts of Sl . The
second term is the incomplete Coulomb scattering amplitude of form (2), with all partial waves up to lgr
removed.
In optics we can nd the following denition of diraction, just perfect for the case of nuclear scattering
[4]: "The rays passing along an object form a plane wave front, from which a part, in the form and size
of the object is removed by it. The incomplete wave front gives rise, by Huygens' principle, to a certain
angular distribution of intensity called diraction pattern". Exactly the same situation occurs in nuclear
scattering, where we have also a plane wave for neutral or Coulomb-distorted plane wave for charged
particles incident on atomic nucleus.
2
This second term describes then the nuclear diraction in terms of the denition presented above.
This term is nothing but the Blair model [5] amplitude for black nucleus, the rst semi-classical model
of elastic scattering.
The above separation of the scattering amplitude is equivalent to the decomposition of the ReSl in
the following way:
ReSl = ReSlref + Sldiff
(6)
where Sldiff (real) in the sharp cut-o case are zero up to lgr and one above lgr , while Slref = ReSlref +
ImSl up to lgr and vanish (as ImSl do) above lgr . In reality, Sldiff must vary gradually from zero to one
around lgr , and must be associated with the function describing the "shape" of the nucleus in angular
momentum
space. As long as we do not know the exact shape of the nucleus in angular momentum space,
Sldiff can be calculated by tting the tail of ReSl using, for instance, the following function widely used
in semiclassical models:
Sldiff = [1 + exp lgr; l ];1
l = 1; 2; 3; :::
(7)
where lgr and are parameters obtained from the t.
General separation of the elastic amplitude is then:
X
X1
+ 1 (2l + 1)SlC Sldiff Pl (cos)
1 1 (2l + 1)S C S ref P (cos )
fel () = f ref () + f diff () = 2ik
l l l
0
(8)
2ik 0
Again, the second term in (8) is an incomplete Coulomb scattering amplitude (Sldiff are real and
play the role of dumping coecients) describing diraction, and is just the smooth cut-o Blair model
amplitude for black nucleus.
The rst part in (8), where Slref = ReSlref + ImSl , is the refractive nuclear amplitude describing
refraction and reection.
Exactly the same separation can be done for the scattering of neutral particles. In that case, the nal
result is again equation (8) where all SlC are equal to one.
The main point here is that the diraction term is always present, always the same, whether the
nucleus is black, opaque or transparent. This is then a kind of "background" around the beam axis. For
opaque or transparent nucleus, the rst term describes refraction (and reection). Experimentally we
observe the superposition of diraction and refraction.
To illustrate the eect of the above separation, the optical model elastic scattering amplitudes, giving
the best ts to some typical elastic scattering angular distributions (Sommerfeld parameter n varies from 0
to 10:13) found in literature [6, 7, 8, 9], have been chosen and decomposed. The optical model parameters
used for obtaining these amplitudes are presented in Table 1. The decompositions of the elastic matrix
elements Sl were performed by tting the tail of ReSl (for l where ReSl > 0:7) with function (7) to
determine the lgr and parameters. The example of such a decomposition for 208Pb + 4 He is presented
in Fig 1.
Table 1: Optical model (Wood-Saxon) parameters, and Fl function parameters lgr and , used for
decomposition of some typical dierential elastic cross sections, into diractive and refractive parts.
Reaction
Elab
n
U
ru
au
W
rw aw
lgr
(MeV)
(MeV) (fm) (fm) (MeV) (fm) (fm)
28Si + n
14.0
0
53.55 1.12 0.80 11.49 1.31 0.46 3.93 0.37
27Al + 4 He
145.0 0.68 107.10 1.23 0.79 19.70 1.63 0.62 25.29 2.40
28Si + 4 He
26.5 1.71 216.00 1.42 0.49 19.77 1.42 0.49 10.99 0.52
208Pb + 4He
26.0 10.13 19.75 1:22b 0.57 7.41 1.22 0.57 8.74 1.29
3
The refractive and diractive cross-sections were calculated using the f ref and f diff of relation (8)
respectively. The results are shown in Fig 2. As can be seen, the "diractional background" is always
very large, much larger than the refraction.
It has to be pointed out that the general pictures presented in Fig 2 change very little when tting
the tail of ReSl from the point where ReSl > 0:5 or ReSl > 0:8.
4 Conclusions
From the mathematical point of view, amplitudes (1) and (4) are identical. Amplitude (1) is perfect
for numerical evaluation of the dierential elastic cross section, while its general form (4) is perfect
for interpretation of the elastic scattering phenomenon. Amplitude (4) converges very slowly, and is
inconvenient for numerical evaluation of the cross section. We can, however, apply to it a common
method of speeding up convergence of the series [10], leading exactly to relation (1).
The above general form of the elastic scattering amplitude has been also successfully used to derive
the Generalized Sum-of-Dierences formula [2] for the total reaction cross section r :
GSOD =
Z (jf ()j ; jf (j )d(
) = ;
C
2
el
2
r
(9)
The presented here formalism is than complete.
The presented above method of the decomposition of elastic scattering amplitude is similar to the
existing in literature two other semiclassical methods. The rst one, proposed by Fuller [11], decomposes
scattering amplitude into nearside and farside parts. Presented above diractive and refractive amplitudes
are conceptually similar to the nearside (responsible for classical trajectories in Coulomb forces, outside
the nucleus) and farside (responsible for trajectories penetrating nuclear forces) amplitudes respectively.
The second method introduced by Brink and Takigawa [12] decomposes the scattering matrix elements
Sl into "barrier" and "internal" parts. This method looks almost identical to presented here and gives
very similar results [13, 14]. There is however, a substantial dierence between the presented method
and that of Brink. In presented here method, the Sldiff (which correspond to Brink's SB ) must be pure
real, since diraction is due to the incompleteness of the Coulomb (or residual Coulomb) amplitude.
The conclusions derived from the above methods are also dierent from that presented in this paper.
There is, however, an open eld for studies how the presented method would explain such a well known
phenomena, present in elastic scattering, like rainbow and Airy structures.
The diraction phenomena in nuclear scattering have also been extensively studied by Frahn [15]
using semiclassical approximation, but he uses dierent formula for Fresnel and Fraunhofer types of
diraction. Here, as one can see, the incomplete Coulomb scattering amplitude describes diraction
completely without any semiclassical approximations.
The proposed above method of separation diractive and refractive phenomena in nuclear scattering
process can be regarded as a another interpretation of quantum scattering process, interpretation from
the point of view of the classical wave optics.
All of the above considerations show that diraction is always present in the scattering experiment,
and the only information one can derive from it is the shape of the nucleus. Here, one can learn how to
separate this "large diractional background" theoretically at the elastic scattering amplitude level. The
question is, can we achieve such a separation experimentally? In order to try it, we must set up a new
class of scattering experiments. So far, we have been using two classes of scattering experiments. The
rst one are experiments where xed targets are bombarded by particle beams. The other one, commonly
used in high-energy particle physics, are head-on collisions of two particle beams. A third possible class
of experiments are crossed-beam experiments, where two beams of particles, one regarded as a "target
beam" and the other one as a "particle beam", cross at a certain angle . Here the centre-of-mass axis is
separated perfectly from both beam axes. The diraction pattern, produced by the incompleteness of the
plane waves (or Coulomb-distorted plane waves) should be centred into the directions of both beams, while
all reactions going through the compound nucleus (compound elastic, complete and incomplete fusions,
etc.) should be symmetrical around the centre-of-mass axis. The question is, can we separate in this
way the nuclear refraction from the "diractional background"? Particles can penetrate the nucleus for a
very short time, without forming a compound nucleus, and leaving the system they can still "remember"
their initial direction. This separation might then depend strongly on the time of interaction, and can
be obtained successfully for the compound elastic cross sections.
4
The only disadvantage of that class of experiments will be low intensity at the output. This disadvantage, however, is only a purely technical matter, i.e. a technical problem in producing colliding beams of
high enough density.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
L.I. Shi, Quantum Mechanics, (Third Edition) McGraw-Hill Kogakusha, Tokyo 1968, p. 143;
H. Wojciechowski, Europhysics Letters, 51 (2000) 275;
J.R. Taylor, Nuovo Cimento 25B, 313 (1974);
H.C. van de Hulst, Light Scattering by Small Particles, Wiley, New York 1957, p. 104;
J.S. Blair, Phys.Rev. 95, 1218 (1954);
S. Kliczewski and Z. Lewandowski, Nucl.Phys. A304, 269 (1978);
S. Wiktor, C. Mayer-Boricke, A. Kiss, M. Rogge, P. Turek and H. Dabrowski, Acta Phys. Polonica
B12, 491 (1981);
K. Chyla, H. Niewodniczanski, J. Szymakowski, U. Tomza and H. Wojciechowski, INP Report No
669/PL (1970);
H. Wojciechowski, D.E. Gustafson, L.R. Medsker, and R.H. Davis, Physics Letters, 63B, 413 (1976);
A. Bjork and G. Dahlquist, Numerical Methods, Prentice-Hall, 1974 (Polish edition p. 74);
R.C. Fuller, Phys. Rev. C 12, 1561 (1975);
D.M. Brink and N. Takigawa, Nucl. Phys. A279, 159 (1977);
J. Albinski and F. Michel, Phys. Rev. C 25, 213 (1982);
F. Michel, G. Reidemeister, and S. Ohkubo, Phys. Rev. C 63, 34620 (2001);
W.E. Frahn, Diractive processes in nuclear physics, Clarendon Press, Oxford 1985;
ReSl
1.0
0.8
0.6
- Re Sl
- ReSref
0.4
-S
diff
0.2
0.0
0
5
10
15
20
l
Fig. 1: The decomposition of the ReSl (circles) into ReSlref (triangles) and Sldiff (crosses) for 208Pb +
4 He system.
5
( )
1000
[mb=sr]
100
...................
..
......
.
.....
10
1
0:1
=c
0
0
0:01
0
30
60
90
120
20
40
150
180
27 Al + E = 145:0MeV
n = 0:68
60
Si + E = 26:5MeV
n = 1:71
80
120
180
28
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(c)
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208
Pb + E = 26:5MeV
n = 10:13
........................................................................................................ ...............................
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30
60
90
120
150
180
c:m:[deg]
Fig. 2: The decomposition of the optical model (solid line) dierential cross sections into diractive
parts (dotted line, calculated using diractive scattering amplitude), and refractive parts (dashed line,
calculated using refractive scattering amplitude), for some typical (Sommerfeld parameter varies from
zero to ten) elastic angular distributions.
6