Elastic and Total Reaction Cross Sections of Oxygen Isotopes in Glauber Theory

Abstract

We systematically calculate the total reaction cross sections of oxygen isotopes, $^{15-24}$O, on a $^{12}$C target at high energies using the Glauber theory. The oxygen isotopes are described with Slater determinants generated from a phenomenological mean-field potential. The agreement between theory and experiment is generally good, but a sharp increase of the reaction cross sections from ^{21}O to ^{23}O remains unresolved. To examine the sensitivity of the diffraction pattern of elastic scattering to the nuclear surface, we study the differential elastic-scattering cross sections of proton-^{20,21,23}O at the incident energy of 300 MeV by calculating the full Glauber amplitude. Comment: 9 pages, 8 figures

Full text

arXiv:0810.0320v1 [nucl-th] 2 Oct 2008
Elastic and total reaction cross sections of oxygen isotopes in Glauber theory
B. Abu-Ibrahim,1, 2 S. Iwasaki∗,3W. Horiuchi,3A. Kohama,1and Y. Suzuki4
1RIKEN Nishina Center, RIKEN, Wako-shi, Saitama 351-0198, Japan
2Department of Physics, Cairo University, Giza 12613, Egypt
3Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan
4Department of Physics, Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan
We systematically calculate the total reaction cross sections of oxygen isotopes, 15−24O, on a 12 C
target at high energies using the Glauber theory. The oxygen isotopes are described with Slater
determinants generated from a phenomenological mean-field potential. The agreement between
theory and experiment is generally good, but a sharp increase of the reaction cross sections from 21 O
to 23O remains unresolved. To examine the sensitivity of the diffraction pattern of elastic scattering
to the nuclear surface, we study the differential elastic-scattering cross sections of proton -20,21,23O
at the incident energy of 300 MeV by calculating the full Glauber amplitude.
PACS numbers: 25.40Cm, 21.10.Gv, 25.60.Dz, 25.60.-t, 24.10.Ht
Keywords:
I. INTRODUCTION
Studies on neutron-rich unstable nuclei have been at-
tracting much attention both experimentally and theo-
retically. These studies are motivated, for example, by
that we want to understand the nuclear structure and ex-
citation mode of the neutron-rich nuclei as well as a role
played by them in forming heavy elements in stars. Bind-
ing energy, radius and density distribution, among oth-
ers, are basic quantities to determine the nuclear prop-
erty. Reactions of unstable neutron-rich nuclei with a
proton target are, therefore, of current interest as they
are at present a major means to probe the matter densi-
ties of exotic nuclei, particularly the region of the nuclear
surface. If one appropriately selects the incident energies,
protons could be more sensitive to the neutron distribu-
tions than to the proton distributions of nuclei.
The Glauber theory [1] offers a powerful and handy
framework for the description of high energy nuclear re-
actions. This theory describes proton-nucleus reactions
very well. Usually a calculation based on the Glauber
theory is performed using a one-body density calculated
from a nuclear wave function. However, Bassel et al. [2]
calculated Glauber’s scattering amplitude using a Slater
determinant wave function, and were able to examine the
effect of Pauli blocking, which is impossible to discuss us-
ing just the one-body density.
Recently, we have analyzed the total reaction cross sec-
tions of carbon isotopes on both 12C [3] and proton [4]
targets using the Glauber theory. The densities of the
carbon isotopes are constructed from Slater determinants
generated from a phenomenological mean-field potential
or two types of dynamical models, core+nand core+n+n
models, which go beyond the mean-field model.
Data on the cross sections of the oxygen isotopes up
∗Present address: Intertrade Co., Ltd., Chuo-ku, Tokyo 104-0032,
Japan
to the dripline nucleus 24O are available at high ener-
gies [5, 6]: The cross sections show a gentle increase with
increasing neutron number from 16O to 21O and then an
abrupt increase up to 23 O. On the other hand, several
theoretical studies on the matter radii predict a mild in-
crease of the radii with increasing neutron number. The
calculations have been performed for the isotopes up to
N= 12 [7], and much further [8, 9, 10, 11]. As the
spin-parity systematics of the ground states of the oxy-
gen isotopes indicates that the d5/2single-particle orbit
seems fairly stable, we first analyze the cross section data
using a model similar to that of the previous study, that
is, assuming the mean-field potentials that reproduce the
nucleon separation energies. The interest of this analysis
is to examine the extent to which the theory can explain
the characteristic behavior of the cross sections.
One of the advantages of the Glauber theory is that the
same input as used in the reaction cross section calcula-
tion is readily applicable for the calculation of the differ-
ential cross section of elastic-scattering. The diffraction
pattern in the differential cross section is expected to de-
pend on the diffuseness of the nuclear surface. Therefore
we also study elastic scattering of the oxygen isotopes on
a proton target. The case of p-16O scattering serves as a
testing ground of our model. We then predict the differ-
ential cross sections of p-20,21,23O elastic scattering. The
differential elastic-scattering cross section data of p-20 O
is available only at low energy, such as ∼30 MeV per
nucleon [12].
This paper is organized as follows. The reaction mod-
els for the calculation of the total reaction and elastic-
scattering cross sections are recapitulated in Sect. II. The
construction of the densities of the oxygen isotopes is ex-
plained in Sect. III. We present the calculation of the
total reaction cross sections of the oxygen isotopes in
Sect. IV. The differential cross sections for neutron-rich
oxygen isotopes are given in Sect. V. A summary is given
in Sect. VI. Appendix A presents the densities of the oxy-
gen isotopes which are obtained using the Slater deter-
minants constructed from the harmonic-oscillator single-

2
particle orbits. Appendix B explains the calculation of
the p-nucleus elastic-scattering amplitude using a Slater
determinant.
II. PHASE SHIFT FUNCTIONS IN THE
GLAUBER MODEL
The Glauber theory provides us with an excellent
framework to describe high energy reactions [1]. The
p-nucleus elastic-scattering amplitude is given by
F(q) = iK
2πZdbeiq·b1−eiχN(b),(1)
where bis the impact parameter vector perpendicular
to the beam (z) direction, Kthe initial momentum of
the relative motion, and qthe momentum transferred
from the projectile to the target. Here we neglect the
p-nucleus Coulomb potential, because its effect appears
only in the extreme forward direction for a light target.
The nuclear part of the optical phase-shift function is
given by
eiχN(b)=hΨ0|
A
Y
i=1 h1−1 + τ3i
2Γpn(b−si)
−1−τ3i
2Γpp(b−si)i|Ψ0i,(2)
where Ψ0is the intrinsic (translation-invariant) A-
nucleon wave function of the nuclear ground state, and si
is the projection onto the xy-plane of the nucleon coor-
dinate relative to the center-of-mass of the nucleus. Here
τ3iis 1 for neutron and −1 for proton.
The profile function, ΓpN , for the pp and pn elastic
scatterings, is usually parameterized in the form
ΓpN (b) = 1−iαpN
4πβpN
σtot
pN exp −b2
2βpN ,(3)
where αpN is the ratio of the real to the imaginary part
of the pN scattering amplitude in the forward direction,
σtot
pN is the pN total cross sections, and βpN is the slope
parameter of the pN differential cross section. These
parameters for different incident energies are tabulated
in Refs. [3, 4].
The differential cross section of the p-nucleus elastic-
scattering is given by
dσ
dΩ=
F(q)

2,(4)
and the total reaction cross section of the p-nucleus col-
lision is calculated from
σR=Zdb1−
eiχN(b)

2.(5)
In the optical limit approximation (OLA), the p-
nucleus scattering phase-shift function is simply given
using the proton density ρp(r) and the neutron density
ρn(r) as follows
eiχOLA(b)= exp [iχp(b) + iχn(b)] ,(6)
where χp(χn) denotes the phase shift due to the protons
(neutrons) inside the nucleus
iχp(b) = −Zdrρp(r)Γpp(b−s),
iχn(b) = −Zdrρn(r)Γpn(b−s).(7)
We will evaluate Eq. (2) completely for a Ψ0that
is given by a Slater determinant comprising harmonic-
oscillator single-particle orbits, and use it in Sect. V.
III. DENSITIES OF OXYGEN ISOTOPES
Compared to the carbon isotopes, where the compe-
tition of the 1s1/2and 0d5/2orbits appears to play an
important role to determine their structure, the spin-
parity systematics of the oxygen isotopes indicates that
the 0d5/2orbit is lower than the 1s1/2orbit in the ground
states up to 22O. The configurations for the oxygen iso-
topes are thus assumed to be given according to the shell-
model as follows: The nucleus 16O is a doubly magic nu-
cleus occupying the 0s1/2, 0p3/2and 0p1/2orbits, and
15O has a neutron hole in the 0p1/2orbit. The nucleus
AO with 16< A ≤22 has A−16 valence neutrons in the
0d5/2orbit, and for A=23, 24 it has six neutrons in the
0d5/2orbit as well as A−22 neutrons in the 1s1/2orbit.
We assume the spin-parity of the ground state of 21 O is
5
2
+though it is not yet confirmed experimentally.
The single-particle orbits are generated from the mean-
field potential containing central and spin-orbit poten-
tials. The Coulomb potential is added for protons. The
strength of the spin-orbit potential is set to follow the
standard value, whereas the strength of the central part
is chosen so as to reproduce the separation energy of the
last nucleon. See Ref. [3] for details. This prescription is
called an “Snmodel” hereafter. This model apparently
ignores the pairing effect, which gives larger separation
energy for the even-Nnucleus than for the odd-Nnu-
cleus. Therefore the Snmodel tends to predict a too
large size for the odd-Nisotope. To remedy this prob-
lem, we also test another one, called an “hSnimodel”,
which fits the average separation energy for the nucle-
ons in the last orbit. For example, in the case of 19O in
which three neutrons occupy the 0d5/2orbit, the average
neutron separation energy to be fitted is one third of a
sum of the neutron separation energies of 17,18,19O, and
likewise the average proton separation energy to be fitted
is one half of a sum of the proton separation energies of
19O and 18 N. The center-of-mass motion is taken into ac-
count in order to obtain the intrinsic density. See Ref. [3]
for detail.

3
2.4
2.6
2.8
3.0
3.2
3.4
15 16 17 18 19 20 21 22 23 24
mass number A
rms radius (fm)
neutron
proton
2.4
2.6
2.8
3.0
3.2
3.4
Sn model
matter
empirical
15 16 17 18 19 20 21 22 23 24
2.4
2.6
2.8
3.0
3.2
3.4
neutron
proton
2.4
2.6
2.8
3.0
3.2
3.4
<Sn> model
matter
empirical
FIG. 1: The matter, neutron and proton root-mean-square radii of the oxygen isotopes calculated with the Snand hSnimodels.
The empirical values are taken from Ref. [5].
We also use harmonic-oscillator single-particle orbits
in Sect. V to examine the differential cross section.
Figure 1 compares the matter, neutron and proton
root-mean-square (rms) radii of the oxygen isotopes (rm,
rn, and rp, respectively) calculated in the Snand hSni
models. Clearly the strong even-odd staggering seen in
the neutron radius of the Snmodel becomes mild in the
hSnimodel, and the matter radii in the latter model seem
to be in better agreement with those extracted from a
model-dependent analysis of the interaction cross section
data [5]. Noticeable discrepancies appear at 21 O and 23O.
The theory predicts a smaller radius for 21 O but a larger
radius for 23O than the empirical values. We will see
in Sect. IV that this discrepancy directly appears in the
comparison of the reaction cross sections. The charge
radii of 16,17,18O calculated in the hSnimodel using the
finite size correction of the nucleon are found to be 2.64,
2.63, 2.62 fm, which are slightly smaller than the mea-
sured values, 2.718(21), 2.662(26), 2.727(20) fm [13], re-
spectively.
IV. RESULTS FOR TOTAL REACTION CROSS
SECTIONS
Before showing results of calculation for the total reac-
tion cross sections of the oxygen isotopes, we first com-
pare in Fig. 2 the theoretical p-16 O total reaction cross
section to experiment. The OLA approximation is used.
The cross sections calculated at higher energies are in
reasonable agreement with the data. A slight underes-
timation is consistent with the fact that the density of
16O gives a slightly smaller radius compared to the ex-
perimental value. The theory predicts about 15% larger
cross sections at lower energies, which is very similar to
the one found in the analysis of the p-12C total reaction
cross section [4]. The reason for this discrepancy is due
1000
800
600
400
200
0
σ
R
(mb)
100 1000
E (MeV)
p -
16
O
FIG. 2: Comparison of the p+16O total reaction cross sections
calculated in the OLA with experiment. The data are taken
from Ref. [14, 15].

4
900
1000
1100
1200
1300
1400
15 16 17 18 19 20 21 22 23 24
σ (mb)
mass number A
Sn
<Sn>
σI(exp.)
FIG. 3: (Color online) Total reaction cross sections for the
oxygen isotopes incident on a 12C target. The incident energy
is ab out 1 GeV p er nucleon except for 15 O. The calculation
is based on the NTG approximation. The experimental data
are taken from Ref. [5].
to the limitation of applying the Glauber theory to the
reaction at the energy lower than 100 MeV [16].
Figure 3 displays the total reaction cross sections of the
oxygen isotopes incident on a 12 C target. The calculated
values are based on the NTG approximation [3], which
includes higher-order corrections missing in the OLA cal-
culation, and they are compared to the interaction cross
sections measured at incident energies around 1 GeV per
nucleon [5]. The energy per nucleon is, however, about
710 MeV in the case of 15O. The density used for 12C is
the same as in Ref. [3]. The hSnimodel seems to give
slightly better results than the Snmodel.
For a comparison between theory and experiment, it
is important to realize that it is the total reaction cross
section that is calculated and not the interaction cross
section. The interaction cross section does not include
the contribution from the inelastic excitation of a pro jec-
tile nucleus to a particle-bound state, whereas the total
reaction cross section includes this contribution. It is in
general not trivial to estimate the difference between the
two cross sections [17], but they are not expected to be
very different at high energies [18, 19]. As already noted
in the comparison of the matter radii, the theory can
reproduce the increase of the cross sections up to 20O
but fails to reproduce the sharp rise of the cross section
from 21O to 23 O. It was noted in Ref. [20] that the latter
discrepancy cannot be explained in any theoretical mod-
els. See Fig. 4 of Ref. [20] as well as the calculation of
Ref. [21].
The sharp rise in the interaction cross section at 23O
has led to some controversial issues for the understanding
of its origin. It was conjectured in Refs. [20, 22] that a
core nucleus 22O could be substantially modified in 23 O
if described with a 22 O+nmodel and also that the spin-
parity of the ground state of 23 O might be 5
2
+rather than
1
2
+. Motivated by this, several experiments have been
performed, which all support 1
2
+[23, 24, 25, 26, 27], as
adopted in the present study. Namely, the configuration
of the ground state of 23 O is dominated by a 1s1/2neu-
tron coupled to 22O(0+) and an excited configuration of
(1s1/2)2(0d5/2)−1appears as a resonance lying very close
to the n+22O threshold [25, 27]. The enhancement of
the cross section from 21 O to 23O thus remains an open
question to be explained.
As another possibility for getting information on nu-
clear size, particularly near its surface, we will examine
in the next section the p-nucleus elastic scattering.
V. ELASTIC-SCATTERING CROSS SECTIONS
In the previous section we calculated the total reac-
tion cross section for the oxygen isotopes using the phe-
nomenological mean-field potential. Though the hSni
model was found to give nuclear sizes that reproduce
the cross sections reasonably well, the increase of the
cross section from 21 O to 23O was not explained. The
p-20O elastic scattering has recently been measured at
the National Institute of Radiological Sciences (NIRS) in
Japan [28]. In this section we study the differential cross
section of the p-oxygen isotope elastic-scattering in order
to examine how sensitive the differential cross section is
to the nuclear size.
To this end, it is more convenient if we can adjust
the nuclear size more flexibly. Thus we use harmonic-
oscillator single-particle wave functions instead of the
orbits determined from the separation energies as was
done in Sect. III. The size parameters νpand νnof the
harmonic-oscillator wave functions are first determined
so as to reproduce the proton and neutron rms radii of
the hSnimodel and then, by changing the νnvalue to
fit the total reaction cross section, we examine the ex-
tent to which the differential cross section is altered. Ta-
ble I lists the radii and the total reaction cross sections
of the oxygen isotopes for several sets of νpand νnval-
ues adopted in this section. The rm, rn, and rpvalues
in each row containing a parenthesis correspond to the
values of the hSnimodel. The density formula obtained
from the harmonic-oscillator functions are summarized
in Appendix A.
The calculation of the Glauber amplitude (1) for the
p-nucleus elastic scattering is explained in Appendix B.
A. Test of the Glauber amplitude: 16 O
We present numerical results of the differential cross
section for the p-16O elastic-scattering, and compare it
to experiment to study how precisely we can discuss the
nuclear size within the Glauber theory.
We start with the calculation of the differential cross
section for p-16O at an incident energy of 1.0 GeV, where

5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
d
σ
/d
Ω
(mb/sr)
302520151050
θ
c.m.
(deg)
p -
16
O
1.0 GeV
FIG. 4: (Color online) Comparison of the numerical results
with the differential elastic-scattering cross section data for p-
16O at Ep= 1.0 GeV. The solid curve represents the full calcu-
lation with the harmonic-oscillator shell-model wave function.
The dashed curve represents the calculation with OLA. The
data are taken from Ref. [5].
10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
d
σ
/d
Ω
(mb/sr)
6050403020100
θ
c.m.
(deg)
p -
16
O
317 MeV
FIG. 5: (Color online) Comparison of the numerical results
with the differential elastic-scattering cross section data for
p-16O at Ep= 317 MeV. See the caption of Fig. 4.
the parameters of the nucleon-nucleon scattering ampli-
tude are well determined. The numerical results of the
differential cross sections are shown in Fig. 4. The solid
curve represents the result of the full calculation with
the harmonic-oscillator shell-model wave function, while
the dashed curve shows the cross section calculated in
the OLA. Both results agree satisfactorily with the data.
The two calculations give no large difference up to the
second minimum. This analysis confirms that the p-16 O
elastic-scattering cross section is reproduced using the
wave function which has a correct size together with the
profile function used in the present study.
As the experimental study of p-nucleus elastic scatter-
ing at around 300 MeV is considered an interesting and
promising project at RIKEN, it is useful to assess the
suitability of the profile function at this energy as well.
For this purpose we calculate the differential cross section
of the p-16O elastic scattering in this energy region. Fig-
ure 5 displays the results of the differential cross section.
Both the exact and OLA curves reproduce the data fairly
well, similarly to the case of 1 GeV. From these compar-
isons at 300 MeV and 1 GeV, we can conclude that the
p-nucleus elastic scattering can reliably be described in
the OLA at least up to the second minimum using the
present parameters of the profile function.
B. The cross sections of 21,23O
As was shown in the previous section, the total reac-
tion cross sections of the oxygen isotopes with a 12 C tar-
get are reproduced reasonably well, except for the cases
of 21,23O. In particular, the total reaction cross section
of 23O+12 C is calculated to be about 2% smaller than
the measured interaction cross section. This may sug-
gest that the hSnimodel slightly underestimates the mat-
ter radius of 23O. As mentioned at the beginning of the
present section, we increase the radius to reproduce the
interaction cross section. Three sets of νpand νnvalues
are listed in Table I. In the case of 21 O, the total reaction
cross section calculated with the NTG approximation is
about 5% larger than the interaction cross section. As
21O has four particle-bound excited states, the difference
between the two cross sections could be reduced to some
extent. Table I lists a set which fits the interaction cross
section of 21O+12C.
Now we examine whether or not the p-nucleus elastic
scattering can give us useful information on the matter
radius of the nucleus. Figure 6 displays the differential
cross section of p-23 O elastic-scattering at Ep=300 MeV
for the three different matter radii listed in Table I. We
focus on the angle of the first peak and the magnitude
of the cross section at this angle. As the matter radius
increases, the first peak moves to a smaller angle. The
angle in degrees (the magnitude of the cross section in
mb/sr) of the first peak for the solid, dashed, and dotted
curves are 21.5 (4.7), 21.1 (4.2), and 20.5 degree (3.3
TABLE I: The rms radii, in fm, of the matter, neutron and
proton density distributions for several oxygen isotopes. The
νpand νnvalues given in fm are the size parameters of the
harmonic-oscillator functions for proton and neutron. See Ap-
pendix A. The seventh and eighth columns denote the total
reaction cross sections, in units of mb, calculated in the OLA
for proton and 12C targets at the incident energy of 1.0 GeV
per nucleon. The interaction cross sections taken from Ref. [5]
are given in parentheses.
νpνnrmrnrpσR(with p)σR(with 12C)
16O 1.71 1.71 2.51 2.51 2.51 305 1003 (982±6)
20O 1.70 1.83 2.78 2.94 2.51 369 1135 (1078±10)
21O 1.70 1.87 2.85 3.04 2.51 386 1173 (1098±11)
1.70 1.67 2.64 2.72 2.51 357 1098
23O 1.70 1.98 3.05 3.30 2.51 429 1267 (1308±16)
1.70 2.05 3.13 3.41 2.51 440 1302
1.70 2.15 3.25 3.58 2.51 457 1353

6
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
d
σ
/d
Ω
(mb/sr)
403020100
θ
c.m.
(deg)
p
-
23
O
300 MeV
10
2
10
1
10
0
10
-1
242220181614
FIG. 6: (Color online) Comparison of the differential elastic-
scattering cross sections of p-23O at Ep= 300 MeV for differ-
ent values of nuclear matter radii. The solid curve represents
the case with a radius of 3.05 fm, the dashed curve with a
radius of 3.15 fm, and the dotted curve with a radius of 3.25
fm. The calculations are carried out with the OLA.
mb/sr), respectively.
The difference, at each first peak angle, between the
magnitudes of the cross sections denoted by the solid and
dotted curves is about 30%. This value is not small, but,
unfortunately the presently estimated experimental un-
certainty at this angle for the p-20O elastic scattering is
comparable to this value [28]. The angle shift of the first
peak position between the two curves is about 1.0 degree,
which might be too small to distinguish experimentally.
We have carried out a similar analysis for the case of
21O. The result is displayed in Fig. 7. The angle (the
magnitude of the cross section) of the first peak for the
solid and dashed curves are 23.5 (7.3), 22.3 degree (6.4
mb/sr), respectively. Again a high precision experiment
will be needed to distinguish which of the above two is
favorable.
C. Prediction of the cross section of 20O
Recently the differential cross section of 20O elastic
scattering on a proton has been measured at the energy
of 300 MeV p er nucleon at NIRS [28]. Motivated by this
experiment, we predict the elastic and total reaction cross
sections of p-20O.
The Glauber amplitude is calculated fully using the
Slater determinant consisting of the harmonic-oscillator
single-particle orbits. The size parameter of the harmonic
oscillator wave function is chosen to reproduce the radius
of 20O given in the hSnimodel, as listed in Table I. The
result is shown in Fig. 8. The first p eak appears at 24.8 ◦
and the cross section at the peak is 2.43 mb/sr. As shown
in Fig. 3, the reaction cross section calculated in the NTG
approximation is about 30 mb larger than the interaction
cross section. It is not clear, however, whether this dif-
ference between the two cross sections indicates that the
radius used for 20 O is slightly too large or not, because
a considerable contribution to the reaction cross section
is expected from the 12 particle-bound excited states of
20O. In any case the predicted differential cross section
near the peak appears consistent with the preliminary
data [28].
VI. SUMMARY
We have systematically calculated the total reaction
cross sections of the oxygen isotopes, 15−24O, on a 12C
target at high energies using the Glauber theory. The
oxygen isotopes are described with the Slater determi-
nants generated from a phenomenological mean-field po-
tential, which is an extension of our previous work for
describing the carbon isotopes [3, 4]. We have intro-
duced two schemes, the Snmodel and the hSnimodel.
The agreement between theory and experiment is gen-
erally good, especially in the hSnimodel, but the sharp
increase of the reaction cross sections from 21 O to 23 O
remains an open question.
As a possible cross section which may depend on the
nuclear radius more sensitively than the reaction cross
section, we have examined the differential cross sections
of the p-20,21,23O elastic-scatterings at the incident en-
ergy of Ep= 300 MeV using the full and OLA Glauber
amplitudes. The differential cross sections calculated
from the two amplitudes are not very different up to the
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
d
σ
/d
Ω
(mb/sr)
403020100
θ
c.m.
(deg)
p -
21
O
300 MeV
10
2
10
1
10
0
10
-1
28262422201816
FIG. 7: (Color online) Comparison of the differential elastic-
scattering cross sections of p-21O at Ep= 300 MeV for differ-
ent values of nuclear matter radii. The solid curve represents
the case with a radius of 2.64 fm, while the dashed curve with
a radius of 2.85 fm. The calculations are carried out with the
OLA.

7
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
d
σ
/d
Ω
(mb/sr)
50403020100
θ
c.m.
(deg)
p -
20
O
300 MeV
FIG. 8: Prediction of the differential elastic-scattering cross
section for p-20O at Ep= 300 MeV. The curve represents the
full calculation with the harmonic-oscillator shell-model wave
function.
second minimum of the angular distribution. We have
calculated the total reaction cross sections by varying
the nuclear radii so as to be consistent with the observed
interaction cross sections, and then analyzed the sensitiv-
ity of the differential cross sections to the radii. We find
a considerable change in the cross section, but whether
it can really be observed or not strongly depends on the
precision of the experimental data.
We have predicted the differential cross section for p-
20O elastic-scattering, which has recently been measured.
Our prediction appears consistent with the preliminary
data. This implies that it is possible to calculate to a
good approximation the total reaction and elastic scat-
tering cross sections at high energies.
We are grateful to B. V. Carlson for his careful reading
of the manuscript. We acknowledge T. Motobayashi for
his encouragement during the course of this work. B. A-
I. was supported in part by Japan International Cultural
Exchange Foundation (JICEF). W. H. is supported by
the Japan Society for the Promotion of Science for Young
Scientists. This work was in part supported by a Grant
for Promotion of Niigata University Research Projects
(2005–2007), and a Grant-in Aid for Scientific Research
for Young Scientists (No. 19·3978). One of the authors
(Y. S.) thanks the JSPS core-to-core program, Exotic
Femto Systems.
APPENDIX A: HARMONIC-OSCILLATOR
DENSITY
We briefly explain the method of calculating the neu-
tron or proton intrinsic density which is employed in our
paper [3]. We assume a Slater determinant constructed
from harmonic-oscillator wave functions. Denoting the
Slater determinant by Ψ, we first calculate the neutron
or proton density which contains the effect of the center-
of-mass motion
˜ρ(r) = hΨ|
A
X
i=1
δ(ri−r)Pi|Ψi,(A1)
where riis the nucleon single-particle coordinate and Pi
is a projector to either neutron or proton. The single-
particle wave function is given by the harmonic-oscillator
function
ψnljm =Rnlj (r)[Yl(ˆ
r)χ1/2]jm,(A2)
where χ1/2denotes the spin function. If the orbit spec-
ified by nlj is not completely filled, we take an average
over the zcomponent m, obtaining a spherical density.
After some angular momentum algebra we have
˜ρ(r) = 1
4πX
nlj
Nnlj R2
nlj (r),(A3)
where Nnlj is the occupation number of the nlj orbit.
To obtain the intrinsic density which has no contribu-
tion from the center-of-mass motion, we make use of the
fact that the center-of-mass motion Ψcm(X) contained
in Ψ is factored out as
Ψ = Ψ0Ψcm(X),(A4)
where Xis the center-of-mass coordinate and
Ψcm(X) = A
πν23/4
exp −A
2ν2X2.(A5)
Then the intrinsic density ρ(r) defined by
ρ(r) = hΨ0|
A
X
i=1
δ(ri−X−r)Pi|Ψ0i(A6)
is obtained from the following relation [3]
Zdreik·rρ(r) = [hΨcm|eik·X|Ψcmi]−1Zdreik·r˜ρ(r).
(A7)
The radial functions which we need for the oxygen iso-
topes are
R00(r) = 4
√πν31/2exp−r2
2ν2,
R01(r) = 8
3√πν51/2rexp−r2
2ν2,
R02(r) = 16
15√πν71/2r2exp−r2
2ν2,
R10(r) = 8
3√πν31/2r2
ν2−3
2exp−r2
2ν2.(A8)
We give the examples of 21,23O below. The neutron or
proton density of 21O is
˜ρ(r) = 1
4πh2R2
00 + 6R2
01 + (M−8)R2
02i
=2
π√πν3exp−r2
ν2h1 + 2 r2
ν2+2(M−8)
15
r4
ν4i,
(A9)

8
were Mis the number of protons Z=8 or the number of
neutrons N=13. The neutron density for 23O is
˜ρ(r) = 1
4πh2R2
00 + 6R2
01 + 6R2
02 +R2
10i
=2
π√πν3exp−r2
ν2h7
4+r2
ν2+17
15
r4
ν4i.(A10)
The corresponding intrinsic density is obtained from
the relation (A7). The results are
ρ(r) = 2u
πν23/2exp−ur2
ν2
×h1 + 3(1 −u) + 15
2α(1 −u)2
+2u2(1+ 5α(1−u)) r2
ν2+ 2αu4r4
ν4i,(A11)
for 21O and
ρ(r) = 2u
πν23/2exp−ur2
ν2h15
2−10u+17u2
4
+20u2
3−17u3
3r2
ν2+17u4
15
r4
ν4i,(A12)
for 23O. Here u=A/(A−1) and α= (M−8)/15.
As is evident, the above derivation assumes that the
harmonic-oscillator constant νis the same for protons
and neutrons. When we use different values for these, as
in Table I, the intrinsic densities of Eqs. (A11) and (A12)
will no longer be exact. However, we have still used these
equations to calculate the intrinsic densities in that case.
APPENDIX B: ELASTIC SCATTERING
AMPLITUDE
In this appendix, following the method of Refs. [2] we
explain our calculation scheme of the proton-nucleus elas-
tic scattering amplitude for a Slater determinant wave
function constructed from harmonic-oscillator single-
particle orbits.
Let us start with the p-nucleus elastic-scattering am-
plitude already given by Eq. (1),
F(q) = iK
2πZdbeiq·b1−eiχN(b),(B1)
where bis the impact parameter vector perpendicular to
the beam (z) direction, Kthe initial momentum of the
relative motion, and qthe momentum transferred from
the projectile to the target.
As we have shown in Eq. (2), χN(b) is the nuclear part
of the optical phase-shift function given by
eiχN(b)=hΨ0|
A
Y
i=1 h1−1 + τ3i
2Γpn(b−si)
−1−τ3i
2Γpp(b−si)i|Ψ0i
=hΨ0|O(b−s)|Ψ0i,(B2)
where Ψ0is the intrinsic (translation-invariant) A-
nucleon wave function of the nuclear ground state. The
coordinate siis the projection onto the xy-plane of the
nucleon coordinate relative to the center-of-mass of the
nucleus, i.e.,si=s′
i−X, where s′
iis the coordinate of a
nucleon in the nucleus projected onto the xy-plane, and
Xis the center-of-mass coordinate of the nucleus. τ3iis
1 for neutron and −1 for proton. For later convenience,
here we introduce an operator O(b−s).
The main idea is that, for the harmonic-oscillator
Slater determinant wave function, the elastic scattering
amplitude F′(q) referring to the coordinate origin is fac-
torized into a center-of-mass part and the intrinsic am-
plitude F(q) [2], where F′(q) is defined by
F′(q) = iK
2πZdbeiq·bhΨ|δ(3)(X−(1/A)X
j
rj)
×(1 −O(b−s′)) |Ψi,(B3)
For the harmonic oscillator, |Ψiis exactly factorized as
|Ψi=|Ψcmi|Ψ0i.
Then, the intrinsic amplitude which is free from the
center-of-mass motion is easily calculated from F′(q) as
F′(q) = hΨcm|e−iq·X|ΨcmiF(q),(B4)
where
hΨcm|e−iq·X|Ψcmi= exp[−ν2q2/4A],(B5)
and νis the size parameter of the harmonic-oscillator
potential. The separability of the elastic scattering am-
plitude, Eq. (B3), is valid when the size parameter of the
harmonic well for protons, νp, is equal to that for the neu-
trons, νn, which is the case for 16 O. On the other hand,
when νp6=νn, which is the case for 20,21,22O, we only
assume this separability and adopt the following relation
[3]:
ν2=Z
Aν2
p+N
Aν2
n.(B6)
Now, we focus on the calculation of the elastic scat-
tering amplitude F′(q) using the nucleon coordinates re-
ferring to the coordinate origin. Since we represent the
ground-state wave function as a Slater determinant,
Ψ = (A!)−1/2det[ψm(r′
n)],(B7)
the antisymmetrization is only required for the bra, such
as
hΨ|= (A!)−1/2det[δmnψ†
m(r′
n)].(B8)
As one can see from Eq. (B2), the operator O(b−s′)
is factorizable into the operators that act on one particle
subspaces, so that
hΨ|O(b−s′)|Ψi= det[Onm]ndet[Onm ]p,(B9)

9
where det[Onm]pis the Z×Zdeterminant for Zprotons
(in this work, Z= 8) and det[Onm ]nis the N×Nde-
terminant for Nneutrons. Onm is the matrix element
defined by
Onm =δnm −Zψ∗
m(r′)Γ(b+s′)ψn(r′)dr′.(B10)
We have calculated the matrix element, Onm , analyti-
cally and obtained the determinants [2]. Concerning the
neutron part, det[Onm]n, we take the average of the de-
terminants of different configurations. For example, for
20O, which has a 0+ground state, we have 12 neutrons,
four of which are in the 0d5/2state. Therefore, we have
three different determinants that give a 0+ground state
of 20O.
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