Similarity between nuclear rainbow and meteorological rainbow

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Similarity between nuclear rainbow and meteorological
rainbow: Evidence for nuclear ripples
Ohkubo, S.; Hirabayashi, Y.
Physical Review C, 89(6): 61601
2014-06-27
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http://hdl.handle.net/2115/56861
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©2014 American Physical Society
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Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP
RAPID COMMUNICATIONS
PHYSICAL REVIEW C 89, 061601(R) (2014)
Similarity between nuclear rainbow and meteorological rainbow: Evidence for nuclear ripples
S. Ohkubo1,* and Y. Hirabayashi2
1
Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan
2
Information Initiative Center, Hokkaido University, Sapporo 060-0811, Japan
(Received 14 May 2014; revised manuscript received 10 June 2014; published 27 June 2014)
We present evidence for the nuclear ripples superimposed on the Airy structure of the nuclear rainbow,
which is similar to the meteorological rainbow. The mechanism of the nuclear ripples is also similar to that of
the meteorological rainbow, which is caused by the interference between the externally reflective waves and
refractive waves. The nuclear ripple structure was confirmed by analyzing the elastic angular distribution in
16
O + 12 C rainbow scattering at EL = 115.9 MeV using the coupled channels method by taking account of
coupling to the excited states of 12 C and 16 O with a double folding model derived from a density-dependent
effective nucleon-nucleon force with realistic wave functions for 12 C and 16 O. The coupling to the excited states
plays the role of creating the external reflection.
DOI: 10.1103/PhysRevC.89.061601
PACS number(s): 25.70.Bc, 24.10.Eq, 42.25.Gy
Descartes [1] and subsequently Newton [2] explained the
rainbow in optics by reflection and refraction in the raindrops.
Airy [3] understood the supernumerary rainbow by the wave
nature of light. The mechanism of the meteorological rainbow
was understood precisely only recently by Nussenzveig using
the electromagnetic theory of light [4]. In analogy with the
meteorological rainbow the nuclear rainbow was predicted
theoretically [5] and observed in α particle scattering [6].
The rainbow has been observed also in other systems such
as in atom-atom collisions, atom-molecule collisions [7],
electron-molecule collisions [8], and atom scattering from
crystal surfaces [9]. Although the mechanism of Newton’s
zero-order (p = 1 in Fig. 1) nuclear rainbow [10], where
only refraction is active, is very different from that of the
meteorological rainbow (p = 2 in Fig. 1), a similar Airy
structure has been observed. As shown in Fig. 2, the precise
description of the meteorological rainbow given by solving
Mie scattering shows the rapidly oscillating structure, the
high-frequency ripple structure, superimposed on the Airy
structure of the rainbow [4,11,12]. The ripple structure is not
predicted by the semiclassical theory of the nuclear rainbow of
Ref. [5] and no attention has been paid to its possible existence.
Here we report for the first time evidence for the existence of
the ripple structure in the observed nuclear rainbow and explain
its mechanism.
The Airy structure of nuclear rainbows has been studied extensively especially for heavy ion scattering such as
16
O + 16 O, 16 O + 12 C, and 12 C + 12 C [13,14]. For the most
typical 16 O + 16 O system, similar to the typical α + 16 O
and α + 40 Ca scattering [15,16], a global deep potential has
been determined uniquely from the rainbow scattering. It
reproduces the experimental data over a wide range of energies
from negative energy to the incident energy EL = 1120
MeV—that is, the rainbows [17], prerainbows [18], molecular
resonances and cluster structures with the superdeformed configuration [19]—in a unified way. Unfortunately the observed
Airy structure in the angular distributions is obscured due to
symmetrization of two identical bosons.
In this respect rainbow scattering of the asymmetric
16
O + 12 C system is important and has been thoroughly investigated [20–25]. A global deep potential could describe well the
rainbows in the high-energy region, prerainbows [10], molecular resonances, and cluster structures with the 16 O + 12 C
configuration in the quasibound energy region in a unified
way [26]. However at energies around EL = 100 MeV the
global optical potential calculations [23] only reproduced the
experimental angular distributions in a qualitative way at larger
angles. Also, to reproduce the high-frequency oscillations,
imaginary potentials—with a thin-skinned volume term and
an extraordinary small diffuseness parameter around 0.1 fm
accompanying a surface term peaked at a larger radius—were
needed [23,25].
The purpose of this paper is to show that the high-frequency
oscillations superimposed on the Airy structure are nothing
but the ripple structure of the nuclear rainbow, and can
be explained by fully taking account of coupling to the
excited states of 12 C and 16 O by using the microscopic
wave functions and the extended double folding model. The
mechanism of the ripple structure and the role of coupling to
the excited states is clarified, and the similarity between the
macroscopic meteorological rainbow and the quantum nuclear
rainbow, despite the difference of the underlying interactions,
is discussed.
We study 16 O + 12 C scattering with the coupled channels
method using an extended double folding (EDF) model that
describes all the diagonal and off-diagonal coupling potentials
derived from the microscopic realistic wave functions for 12 C
and 16 O using a density-dependent nucleon-nucleon force. The
diagonal and coupling potentials for the 16 O + 12 C system
are calculated using the EDF model without introducing a
normalization factor:
Vij,kl (R) =
*
O)
12
(r1 ) ρkl( C) (r2 )
×vNN (E,ρ,r1 + R − r2 ) dr1 dr2 ,
[email protected]
0556-2813/2014/89(6)/061601(5)
16
ρij(
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(1)
©2014 American Physical Society
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S. OHKUBO AND Y. HIRABAYASHI
FIG. 1. (Color online) Illustrative figure of path of an incident
ray (i) in a spherical raindrop in geometrical optics. The rays p = 0,
p = 1, and p = 2 correspond to reflection, refraction only, and the
primary rainbow, respectively.
16
where ρij( O) (r) is the diagonal (i = j ) or transition (i = j )
nucleon density of 16 O taken from the microscopic α + 12 C
cluster model wave functions calculated in the orthogonality
condition model (OCM) in Ref. [27]. This model uses a
realistic size parameter both for the α particle and 12 C,
and is an extended version of the OCM α cluster model of
Ref. [28], which reproduces almost all the energy levels well
up to Ex ≈ 13 MeV and the electric transition probabilities in
16
O. We take into account the important transition densities
available in Ref. [27], i.e., g.s. ↔ 3− (6.13 MeV) and
2+ (6.92 MeV) in addition to all the diagonal potentials.
12
ρkl( C) (r) represents the diagonal (k = l) or transition (k = l)
nucleon density of 12 C calculated using the microscopic
three-α cluster model in the resonating group method [29].
This model reproduces the structure of 12 C well, and the
wave functions have been checked for many experimental
data [29]. In the coupled channels calculations we take
+
into account the 0+
(4.44 MeV), and 3−
1 (0.0 MeV), 2
12
(9.64 MeV) states of C. The mutual excitation channels
in which both 12 C and 16 O are excited simultaneously are
not included. For the effective interaction vNN we use the
DDM3Y-FR interaction [30], which takes into account the
FIG. 2. (Color online) Cross sections for the primary rainbow
in Fig. 1. The dashed line is by classical theory in optics and the
dark region before the critical angle (θc = 138◦ ) is displayed by a
black shade. The dotted and solid lines show calculations of Mie
scattering [11] and in the Airy approximation [11] in the wave theory
of light, respectively.
θ
FIG. 3. (Color online) The experimental cross sections (points)
in 16 O + 12 C scattering at EL = 115.9 MeV [23] are compared
with the calculations (solid line) using the EDF potential: (a) the
coupled channels calculations with aW = 0.2 (blue line), (b) the
single channel calculations with aW = 0.6, and (c) the single channel
calculations using the extremely thin-skinned volume-type imaginary
potential with aW = 0.1. For comparison, in (a) the coupled channels
calculations with aW = 0.4 are displayed by the green line. The
calculated cross sections (solid line) are decomposed into the farside
(dashed line) and nearside (dotted line) components.
finite-range nucleon exchange effect. An imaginary potential
(nondeformed) is introduced phenomenologically to take into
account the effect of absorption due to other channels.
In Fig. 3(a) the angular distributions of elastic 16 O + 12 C
scattering at EL = 115.9 MeV calculated using the coupled
channels method (blue solid line) are compared with the
experimental data. We found that the EDF potential works
well without introducing a normalization factor. The volume
integral per nucleon pair of the ground state diagonal part,
JV = 317.7 MeV fm3 , is consistent with those used in other
optical potential model calculations and belongs to the same
global potential family found in the EL = 62−1503 MeV
region [23,24,31]. The parameters used in the imaginary
potential with a Woods-Saxon volume-type form factor displayed in Fig. 4 are WV = 14 MeV, RW = 5.6 fm, and
aW = 0.20 fm with a volume integral per nucleon pair JW =
54.3 MeV fm3 , in the conventional notation. We see that
the refractive farside scattering dominates at the intermediate
and large angles. The calculation reproduces well the two
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FIG. 4. (Color online) The real folding and Woods-Saxon
volume-type imaginary potentials used in the coupled channels
calculations [Fig. 3(a)] in 16 O + 12 C scattering are displayed by the
red solid line and the blue solid line, respectively. The imaginary
potentials used in the single calculations in Figs. 3(b) and 3(c)
are displayed by the green dotted line and the black dashed line,
respectively.
broad Airy maxima in the angular range, θ = 60–90◦ (Airy
maximum A2) and θ = 100–140◦ (Airy maximum A1) in the
experimental angular distribution, which are brought about
by the refracted farside component. Also the high-frequency
oscillations superimposed on the two broad Airy maxima, A1
and A2, in the experimental data are reproduced well. We
see that the high-frequency oscillations are brought about by
the interference between the farside and nearside scattering
components. The investigation of the contributions of each
channel reveals that none is overwhelmingly dominant and
that the contribution of the excited states of 16 O is as much
as that of 12 C, which is quite different from the higher energy
region around EL = 300 MeV where coupling to the 2+ state
of 12 C contributed dominantly in creating the secondary bow in
the classically forbidden dark side of the primary rainbow [32].
Also neither the extremely thin-skinned volume-type imaginary potential with aW =0.1 nor the surface imaginary potential
peaked at a larger radius used in Refs. [23,25] were needed.
In Fig. 3(b) the angular distributions calculated in the
single channel calculation using the readjusted imaginary
potential (displayed in Fig. 4), WV = 11.5 MeV, RW = 5.9 fm,
and aW = 0.6 fm (JW = 52.8 MeV fm3 ), which is similar
to Ref. [31], are shown. Although the farside scattering is
dominant, similar to Fig. 3(a), and the gross behavior of
the Airy structure of the experimental angular distribution is
reproduced, the high-frequency oscillations are missing. By
comparing Fig. 3(a) (blue solid line) and Fig. 3(b), we note
that the channel coupling to the excited states of 16 O and
12
C contributes in generating the high-frequency oscillations,
although the rather small aW = 0.2 is needed. In Fig. 3(b) we
note that the nearside component is damped more than two
order of magnitude compared with the farside component,
and no interference between them occurs resulting no highfrequency oscillations. While the JW values are almost the
same for Figs. 3(a) and 3(b), the nearside scattering is retained
significantly in Fig. 3(a). This means that the channel coupling
in Fig. 3(a) plays a role of increasing the nearside scattering
component, i.e., reflection.
This can be confirmed in Fig. 3(c) where the angular
distributions calculated in the single channel using both the
extremely thin-skinned volume-type (Woods-Saxon squared)
imaginary potential with WV = 16 MeV, RW = 4.4 fm,
and aW = 0.1 fm and the surface imaginary potential with
WD = 7 MeV, RD = 6.1 fm, and aD = 0.46 fm (JW =
59.8 MeV fm3 ) (displayed in Fig. 4), which are similar to
those in Refs. [23] and [25], are shown. We note that the
values of JW are almost the same for the three cases (a),
(b), and (c). The high-frequency oscillations superimposed
on the Airy structure are recovered only by using this
extremely thin-skinned imaginary potential. We see that the
nearside component needed to bring about the high-frequency
oscillations is significantly increased compared with Fig. 3(b).
The sharper the diffuseness of the imaginary potential is, the
more the nearside component is increased. This can be checked
by decreasing the ratio aW /WV ; that is, if we increase the
strength of the imaginary potential in Fig. 3(c) by 50% to
WV = 24 MeV from the original 16 MeV, the magnitude of
the calculated cross section and its farside component decrease
as expected from the increase of absorption. However, the
magnitude of the nearside component is increased. This means
that for the nearside scattering the imaginary potential does
not act as absorption but acts as “divergence,” i.e., increasing
reflective waves under the small diffuseness aW = 0.1. Thus
the increased nearside component is found to be reflective in
origin. The same behavior is observed for the coupled channels
calculations in Fig. 3(a) with aW = 0.2. We see in Fig. 3(a)
that the calculated cross sections with a moderate smooth
diffuseness aW = 0.4 (green line) show no high-frequency
oscillations, which means that no reflective waves are created.
We note that the magnitudes of the S matrix in Figs. 5(a) (blue
line) and 5(a) (green line) are similar to those in Figs. 5(c)
and 5(b), respectively.
We show that the high-frequency oscillations superimposed
on the Airy structure in Fig. 3(a) are nothing but the ripple
structure of the nuclear rainbow. In Fig. 2 the ripple structure
in the meteorological rainbow is generated by the interference
between the p = 0 external direct reflection and the p = 2 refractive rainbow rays with one internal reflection [4]. However
it has been considered that in the nuclear rainbow, which is
caused by a Luneberg lens [10] of a nuclear potential (p = 1)
with a smoothly diffused surface, direct reflection scarcely
occurs. This is seen in Fig. 3(b) where no high-frequency
oscillations appear in the calculations using the real and
imaginary potentials with a smooth surface. On the other
hand, we see in Fig. 3(c) that the high-frequency oscillations
are created by the interference between the reflective nearside
component caused by the sharp-edged imaginary potential and
the farside component. Thus the high-frequency oscillations
superimposed on the Airy structure are considered to be the
nuclear ripples because they have the same physical origin
as those of the meteorological rainbow. In Fig. 3(a) it is
found that the coupling to the excited states also contributes
in creating the reflective nearside waves that are caused by
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S. OHKUBO AND Y. HIRABAYASHI
FIG. 5. (Color online) The moduli of the calculated S matrices
in Figs. 3(a), 3(b), and 3(c). The lines are explained in the caption
of Fig. 3.
the sharp-edged imaginary potential in Fig. 3(c). The nearside
scattering waves that are responsible for the generation of the
high-frequency oscillations in Figs. 3(a) and 3(c) correspond
to the externally reflected waves in Mie scattering of the
meteorological rainbow. Now the physical meaning and the
origin of the extremely thin-skinned volume-type imaginary
potential needed in Refs. [18,25] are clear. They were needed
to mimic the effect of the channel coupling to enhance the
reflective waves.
In the single channel calculations, the increase of the
nearside component that corresponds to the externally reflective waves is only attained by using the extraordinary small
diffuseness parameter aW = 0.1 for the imaginary potential.
This necessarily accompanies introducing the additional surface imaginary potential at the large radius to preserve the
net absorption, i.e., the volume integral of the imaginary
potential. In the coupled channel calculations in Fig. 3(a),
in fact, no surface imaginary potential at the large radius was
needed. It is important to treat channel coupling to the excited
states microscopically to avoid the unphysically sharp-edged
volume-type imaginary potential and the surface imaginary
potential at the large radius.
How the ripple structure emerges on the Airy structure
by the reflective waves is shown in Fig. 6 where the angular
distributions—calculated by replacing the SL matrices with
the orbital angular momentum L = 0 ∼ Lc generated by the
potential of Fig. 3(b) by those generated by the potential of
Fig. 3(c)—are displayed. With around Lc = 8 (corresponding
impact parameter b = 2.1 fm) the ripple structure starts to
emerge on the A1 Airy peak in the backward angles beyond
θ ≈ 100◦ and with around Lc = 14 (b = 3.6 fm) they also
θ
FIG. 6. (Color online) The angular distributions in 16 O + 12 C
scattering at EL =115.9 MeV calculated by replacing the SL with
L = 0 ∼ Lc among the SL -matrices generated by the potential used
for Fig. 3(b) by those generated by the potential used for Fig. 3(c)
(solid line) and its farside (dashed line) and nearside (dotted line)
components are compared with the experimental data (points) [23].
appear on the Airy peak A2 in the intermediate angles before
θ ≈ 90◦ . For Lc = 14–17 (b = 3.6–4.3 fm), which corresponds to the radius of the sharp edge of the imaginary potential
in Fig. 4 (black dashed line), the high-frequency oscillations in
phase with the experimental data are reproduced. The impact
parameters of these partial waves are significantly smaller than
b = 5.9 fm of the grazing partial wave L = 24 for which
|SL | = 0.5 (see Fig. 5). The channel coupling in Fig. 3(a)
plays two roles; that is, enhancing the reflective waves and
enhancing absorption at the surface, which makes it possible
to use the relatively larger diffuseness parameter aW = 0.2 fm
and no surface imaginary potential needed in Fig. 3(c).
Finally we mention that the high-frequency oscillations are
not due to the elastic transfer of the α particle [31]. In fact,
we see in the detailed coupled reaction channels calculations
of 16 O + 12 C scattering at EL = 115.9 MeV in Ref. [33]
that the contribution of the elastic transfer is three orders of
magnitude smaller than the experimental data. Also the present
calculations take into account the one-nucleon exchange effect,
which is suggested to prevail over other transfer reactions [33],
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by using the effective interaction DDM3Y, in which the
knock-on exchange effect is incorporated [14,30].
To summarize, we have calculated 16 O + 12 C scattering
with the Airy structure at EL = 115.9 MeV using a coupled
channels method with an extended double folding (EDF)
potential that is derived by using the microscopic realistic
wave functions for 12 C and 16 O by taking account of excited
states of the 2+ (4.44 MeV) and 3− (9.64 MeV) states of 12 C
and the 3− (6.13 MeV) and 2+ (6.92 MeV) states of 16 O.
Our calculations reproduce the high-frequency oscillations
superimposed on the Airy structure. It is found that the
high-frequency oscillations are nothing but the nuclear ripples
similar to those superimposed on the Airy structure in Mie
scattering of the meteorological rainbow. The nuclear ripples
are generated by the interference between the refractive waves
and the externally reflected waves. The coupling to the excited
states of 16 O and 12 C plays the role of creating external
reflection. Although the active interactions in the nuclear and
the meteorological rainbows are very different, we see the similarity in that both have the ripple structure on the Airy structure
due to the same origin of the interference between refractive
waves and the externally reflected waves. It is startling that
a classical concept of a ripple in the meteorological rainbow
persists in the quantum nuclear rainbow.
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One of the authors (S.O.) thanks the Yukawa Institute
for Theoretical Physics for the hospitality extended during
a stay in Spring 2014. Part of this work was supported
by the Grant-in-Aid for the Global COE Program “The
Next Generation of Physics, Spun from Universality and
Emergence” from the Ministry of Education, Culture, Sports,
Science and Technology (MEXT) of Japan.
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