We show that asymptotic normalization coefficients (ANC) can be extracted
from one nucleon breakup reactions of loosely bound nuclei at 30-300 MeV/u. In
particular, the breakup of ^{8}B is described in terms of an extended Glauber
model. The 8B ANC extracted for the ground state of this nucleus from breakup
data at several energies and on different targets, C^2 = 0.450+/-0.039} fm^-1,
leads to the astrophysical factor S_{17}(0)= 17.4+/-1.5 eVb for the key
reaction for solar neutrino production 7Be(p,gamma)8B. The procedure described
here is more general, providing an indirect method to determine reaction rates
of astrophysical interest with beams of loosely bound radioactive nuclei.Comment: 4 pages, RevTex, 3 figures revised version to appear in Phys Rev Let

A. Azhari

A. Navin

B. Blank

B. Davids

C. A. Gagliardi

C. Bertulani

E. Bauge

E. G. Adelberger

E. Sauvan

F. Carstoiu

F. Hammache

F. Negoita

G. Bertsch

G. R. Satchler

H. Esbensen

H. M. Xu

I. Pecina

J. H. Kelley

J. N. Bahcall

J. P. Jeukenne

J. S. Al-Khalili

K. Hencken

L. Trache

N. Iwasa

P. G. Hansen

R. E. Tribble

T. Motobayashi

English

arXiv

Trache, L.

Carstoiu, F.

Gagliardi, C. A.

Tribble, R. E.

Texas A&M University

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1
Asymptotic normalization coefficient of 8B from breakup reactions and the S17
astrophysical factor
L. Trache1, F. Carstoiu2,3, C. A. Gagliardi1, R. E. Tribble1
1Cyclotron Institute, Texas A&M University, College Station, TX-77843,
USA
2Institute of Physics and Nuclear Engineering H. Hulubei, Bucharest,
Romania
3Laboratoire de Physique Corpusculaire, F-14050 Caen Cedex, France
(September 19, 2018)
We show that asymptotic normalization coefficients can
be extracted from one nucleon breakup reactions of loosely
bound nuclei at 30-300 MeV/u. In particular, the breakup
of 8B is described in terms of an extended Glauber model.
The 8B ANC extracted from breakup data at several energies
and on different targets, C2tot = 0.450 ± 0.039 fm
−1, leads
to the astrophysical factor S17(0) = 17.4 ± 1.5 eV·b for the
key reaction for solar neutrino production 7Be(p,γ)8B. The
procedure described provides an indirect method to determine
reaction rates of astrophysical interest with beams of loosely
bound radioactive nuclei.
8B, produced in the 7Be(p,γ)8B reaction, is the ma-
jor source of the high-energy neutrinos observed by solar
neutrino detectors [1]. Results of recent direct and indi-
rect determinations of the astrophysical factor S17(0) (see
[2–6] and references therein) do not fully agree, but span
the full adopted range, 19+4
−2 eVb [7]. The asymptotic
normalization coefficient (ANC) approach [8] is based on
the fact that the 7Be(p,γ)8B reaction is highly periph-
eral. Thus, what one needs most is a precise quanti-
tative description of the tail or periphery of the 7Be +
p ↔ 8B overlap function, which is given by the corre-
sponding ANC, rather than of the full many-body wave
function inside the core. The advantage of this approach
is that ANCs may be determined from other peripheral
nuclear reactions that have cross sections orders of mag-
nitude larger than the radiative capture reaction of in-
terest. This technique has been applied to determine S17
from measurements of two (7Be,8B) proton transfer reac-
tions [6], after being tested to give results to better than
9% [9] for 16O(p,γ)17F.
In the present paper we show that one can extract
asymptotic normalization coefficients for the 8B ground
state wave function from data on one-proton removal (or
breakup) reactions at energies of 30-300 MeV/u [10–13].
Improvements in the nuclear reaction model used make
a quantitative description possible for the energy range
and the ANC extracted reliable. Then, this ANC can
be used to obtain an independent determination of the
astrophysical factor S17(0).
Recent advances in experimental techniques permit
the selection of the final state of the remaining core in
breakup reactions at large energies. Typically an exotic
nucleus B = (Ap), where B is a bound state of the core
A and the proton p, is produced by fragmentation from
a primary beam, separated and used to bombard a sec-
ondary target. After the breakup occurs, the breakup
cross section and the parallel momentum distribution of
the core A are measured. By detecting the final state of
the core, e.g. using coincident gamma rays, one can de-
termine important spectroscopic information about the
ground state of the exotic projectile [14,15]. The mea-
sured momentum distribution of the core can be related
to the momentum distribution of the bound nucleon.
In all reactions where the core survives (either transfer
or one-nucleon breakup) the matrix elements include the
overlap integral IBAp(~r) for the nuclei A, p, and B, ob-
tained after the integration over the internal coordinates
of fully antisymmetric wave functions, with ~r the vector
connecting the center of mass of nucleus A with p [16,8].
The overlap integrals are not normalized to unity, but to
the spectroscopic factors Snlj . At asymptotic distances
where nuclear forces are vanishingly small, r > RN , the
overlap integrals behave as
IBAplj(r)→ S
1/2
nlj ϕnlj(r)→ C
B
Aplj
W
−η,l+1/2(2kr)
r
. (1)
Here CBAplj is the asymptotic normalization coefficient
defining the amplitude of the tail of the overlap integral,
W is the Whittaker function, k is the wave number, and
η is the Sommerfeld parameter for the bound state (Ap).
The asymptotic normalization coefficients CBAplj can be
extracted from any peripheral observables that are mea-
sured experimentally.
For the reaction model calculations we assume that
the ground state of the projectile (Jpi) can be approxi-
mated by a superposition of configurations of the form
[Ψpicc ⊗ nlj]
Jpi , where Ψpicc denote the core states and
nlj are the quantum numbers for the single particle
wave function ϕnlj(r) in a spherical mean field poten-
tial. These single particle states are normalized to unity
and have the asymptotic behavior given by Eq. 1, with
the single particle asymptotic normalization coefficients
bnlj . When more than one configuration contributes to a
selected core state, the total cross section for one-nucleon
breakup is written as an incoherent superposition of sin-
gle particle cross sections:
1
σ−1p =
∑
S(c, nlj)σsp(nlj). (2)
A similar relation holds for the momentum distribution.
Typically the nucleon is not measured, therefore the cal-
culated cross sections σsp(nlj) contain a stripping term
(the loosely bound nucleon is absorbed by the target and
the core is scattered and detected), a diffraction dissoci-
ation term (the nucleon is scattered away by the target,
the core is scattered by the target and is detected) and
a Coulomb dissociation term:
σsp =
∫
∞
0
2πbdb(Pstr(b) + Pdiff (b)) + σCoul (3)
In previous analyses of breakup reactions, a structure for
the projectile (the spectroscopic factors S(c, nlj)) was
assumed and agreement between the calculated and ex-
perimental values was considered a validation both of the
assumed nuclear structure and of the reaction model cal-
culations used [11,17,18]. In the present case we use an
extended Glauber model, in the eikonal approximation
with non-eikonal correction terms up to the second order
[10,19]. Realistic nucleon-target and core-target S-matrix
elements are used in the evaluation of the impact parame-
ter dependent probabilities. We find that the largest con-
tributions to the cross sections σsp(nlj) come from large
impact parameters and therefore the phenomena are pe-
ripheral. It then follows that we can express the results
in terms of the asymptotic normalization coefficients and
reverse the process: we can use the experimental results
to extract the ANCs.
Calculations were done for several 8B breakup reac-
tions for which data exist in the literature. The model
is similar to that developed by Bertsch et al. [20,21]. It
has been tested before on 23 different reactions in the
p-sd shell [10,22]. The loosely bound proton and the
core moving on an eikonal trajectory interact indepen-
dently with the target nucleus, an assumption valid at
these energies. For the proton-target interaction we used
that of Jeukenne, Lejeune and Mahaux (JLM) [23], in
the updated version of Bauge et al. [24]. For the target-
core nucleus-nucleus interaction we use the double fold-
ing procedure described in [25]: the same JLM inter-
action is folded with Hartree-Fock nuclear matter dis-
tributions of the core and of the target. Subsequently
the double folding potentials were renormalized to repro-
duce a variety of elastic scattering data for light nuclei.
We found there that the real part needed a substantial
renormalization at about 10 MeV/u (NV=0.366), but the
imaginary part did not (NW=1.00). We have checked
the procedure on a much wider set of data from litera-
ture at higher energies, and found a similar conclusion
for the imaginary part, while the renormalization of the
real part approaches unity around 50 MeV/u. Therefore,
in the present calculations we adopted the procedure of
[25], with the JLM(1) interaction, and NW=1.00. The
S-matrix calculations that enter the first two terms of
Eq. 3 are expected to depend primarily on the imagi-
nary part of the interaction. Indeed, calculations with
large variations of the renormalization of the real poten-
tial (NV =0.366 and NV=0.80) give the same cross sec-
tions. The integrated Coulomb term is treated in a per-
turbative method that retains the dipole and quadrupole
terms, equivalent with that of Ref. [26], but using radial
matrix elements calculated with realistic Woods-Saxon
radial wave functions.
The impact parameter dependence of the first two
terms in Eq. 3 is plotted in Fig. 1 for the case of the
breakup of 8B on a Si target at 38 MeV/u [10]. Clearly,
both terms are dominated by the periphery but have con-
tributions from small impact parameters. The Coulomb
term is even more peripheral, due to its long range. To
investigate the influence of the nuclear interior on the
extracted ANC, different single particle wave functions
were used for the outer proton to calculate the breakup
cross section in the same reaction model. We chose a
range of radii and diffusenesses (R = 2.20 − 2.60 fm
and a = 0.50 − 0.70 fm) for the Woods-Saxon poten-
tials and repeated the calculations. A correct spin-orbit
term was included, and in all cases we adjusted the depth
of the potential to reproduce the proton separation en-
ergy Sp = 137 keV. The radial behavior of 1p1/2 and
1p3/2 orbitals is identical at large distances and, for a
given (R, a), differs at small radii by much less than the
variation associated with the choice of Woods-Saxon po-
tential. Thus, for simplicity, only the 1p3/2 component
was included, and we rewrite Eq. 2 as:
σ−1p = (Sp3/2 + Sp1/2)σsp = (C
2
p3/2
+ C2p1/2)σsp/b
2
p, (4)
where bp is the ANC of the normalized 1pj radial single
particle wave function. The experimental value for the
breakup cross section is 222±15 mb. The calculated cross
section varies from 226 mb to 326 mb with the choice of
the single particle wave function used (i.e. (R, a) the
radius and diffuseness parameters of the Woods-Saxon
potential used), which is equivalent to 44% variation if a
spectroscopic factor Stot = Sp3/2 + Sp1/2 is extracted us-
ing the first part of Eq. 4. However if the square of the
ANC C2tot = C
2
p3/2
+C2p1/2 is extracted instead, the result
is very stable, as shown in Fig. 2 (where for convenience
the results are plotted against the single particle ANC
bp calculated for each geometry assumed). The varia-
tion in the ANC over the full range considered here is
<±3%. A flat curve for C2 in Fig. 2 would be a sig-
nature of a purely peripheral reaction; the small slope
reflects the participation of the interior of the nucleus.
Similar calculations were done for the same target and
two other energies (35 and 28 MeV/u), and for other
targets (12C, Sn and 208Pb) at other energies: 40, 142
and 285 MeV/u [12,11]. Pictures similar to Fig. 2 are
obtained in each and every case. These experiments have
not determined the yield to the 7Be first excited state. A
Coulomb breakup experiment carried out at 50 MeV/u
on a Pb target [14] found it to be small, about 5% of
the total. From this we estimated the ANC for the core
excitation part in the wave function, then calculated its
2
contribution to the one-proton removal cross section on
each target and subtracted it (e.g. 7.5% for the Si target).
The corrected ANCs extracted are presented in Table I.
With the exception of two data points on 12C, there is
good agreement among the ANCs which come from data
over a wide range of incident energies and both low and
high Z targets. In order to extract an average ANC, we
have done an unweighted average of the individual mea-
surements. Using all of the data points, this results in
C2tot = 0.450(30) fm
−1 where the uncertainty is the stan-
dard error of the mean. If the two 12C data points (4 and
5 in the table) at 40 and 142MeV/u are removed from our
average (note that they fail a simple test of the expected
energy dependence), we find C2tot = 0.456(14) fm
−1. Sev-
eral correlated uncertainties must be added: 4% for the
renormalization coefficients used for the optical model
parameters, 3% for the variation in C2 as a function of
bp and 2% for the uncertainty in the excited state con-
tribution. Including these we find C2tot = 0.450(39) fm
−1
for the full data set and C2tot = 0.456(28) fm
−1 when the
two 12C data points are removed. The values are similar
and, without additional information, we shall adopt the
first.
Using a wave function with the asymptotic normaliza-
tion as extracted above, we have calculated the distri-
bution of the parallel momentum of the core measured
in the breakup of 8B at 41 MeV/u on a 9Be target [13].
The result is compared in Fig. 3 with the experimental
data. A very good description is obtained. Notably, un-
like the black disk model [27] calculation which describes
the width of the parallel momentum distribution [28],
the extended Glauber model calculation also matches the
large momentum tails due to the nuclear interior. This
gives further credibility to our calculations and the entire
approach.
The value extracted here from the breakup of 8B at
30-300 MeV/u agrees very well with the one extracted
from transfer reactions at 12 MeV/u [6]. We can use the
ANC extracted to evaluate the astrophysical S factor for
the reaction 7Be(p,γ)8B at very low energies, following
the procedure of Ref. [8]. Using the value from breakup
we find S17(0) = 17.4 ± 1.5 eV·b. We can also use the
ANC extracted here to determine the rms radius for the
8B proton halo, following the procedure of [28]. We find
a rh = 4.20± 0.21 fm.
In conclusion, reliable spectroscopic information can
be extracted from one-nucleon breakup reactions of
loosely bound nuclei at energies around and above the
Fermi energy. However, we have shown that, despite a
more transparent meaning of the spectroscopic factors,
the values obtained are not unambiguous, and a bet-
ter quantitative description is achieved in terms of the
asymptotic normalization coefficients. In turn, these can
be used to calculate any observables that are dominated
by the periphery of the nucleus, notably rms radii for
halos and astrophysical S-factors. Calculations using an
extended Glauber model for the breakup data of 8B on a
wide range of targets and energies lead to an unambigu-
ous value for the ANC and an astrophysical factor S17(0)
in very good agreement with the values from recent deter-
minations from direct measurements [2] and with those
using indirect methods [5,6]. New measurements for the
elastic scattering of 8B, a more accurate determination
of the breakup cross sections (eventually separating the
stripping and diffraction dissociation components) and
a precise determination of the core excitation contribu-
tion, can increase the reliability of the ANC extracted.
The validity of the procedure is wider than for the 8B
case discussed above. In addition to peripherality, en-
sured more or less for the halo nuclei, the requirements
are good absolute values for the breakup cross sections,
with the identification of the final state of the core, and
reliable cross section calculations. The method can be
used to extract valuable information for nuclear astro-
physics. Very difficult or even impossible direct measure-
ments that would involve bombarding short lived targets
with very low energy protons can be replaced or supple-
mented by indirect methods seeking the relevant ANCs,
rather than complete knowledge of the ground state wave
function of these exotic nuclei. In addition, the indi-
rect ANC method is subject to different systematic er-
rors than the direct measurements, and therefore redun-
dance of the results is very much welcome, particularly
for critical astrophysical S-factors, such as that for the
7Be(p,γ)8B reaction.
One of us (FC) acknowledges the support of IN2P3 for
a stay at Laboratoire de Physique Corpusculaire in Caen,
during which part of this work was completed. The work
was supported by the U. S. Department of Energy un-
der Grant No. DE-FG03-93ER40773, by the Romanian
Ministry for Research and Education under contract no
555/2000, and by the Robert A. Welch Foundation.
[1] J. N. Bahcall, M. H. Pinsonneault and S. Basu, Astroph.
J. 555, 990 (2001).
[2] F. Hammache et al., Phys. Rev. Lett. 86, 3985 (2001).
[3] A. R. Junghans et al., INPC 2001, Berkeley, July 2001,
paper 4.6, and to be published.
[4] N. Iwasa et al., Phys. Rev. Lett. 83, 2910 (1999).
[5] B. Davids et al., Phys. Rev. C 63, 065806 (2001).
[6] A. Azhari et al., Phys. Rev. C 63, 055803 (2001).
[7] E. G. Adelberger et al., Rev. Mod. Phys. 70, 1265 (1998).
[8] H. M. Xu et al., Phys. Rev. Lett. 73, 2027 (1994).
[9] C. A. Gagliardi et al., Phys. Rev. C 59, 1149 (1999).
[10] F. Negoita et al., Phys. Rev. C 54,1787 (1996).
[11] B. Blank et al., Nucl. Phys. A624, 242 (1997).
[12] I. Pecina et al., Phys. Rev. C 52, 191 (1995).
[13] J. H. Kelley et al., Phys. Rev. Lett. 77, 5020 (1996).
[14] T. Motobayashi et al., Nucl. Phys. A682, 345c (2001).
[15] A. Navin et al., Phys. Rev. Lett. 81, 5089 (1998).
[16] G. R. Satchler, Direct Nuclear Reactions, Clarendon
3
Press, Oxford Univ. Press, N.Y., 1983.
[17] J. S. Al-Khalili and J. A. Tostevin, Phys. Rev. Lett. 76,
3903 (1996).
[18] H. Esbensen and K. Hencken, Phys. Rev. C 61, 054606
(2000).
[19] F. Carstoiu et al., to be published.
[20] G. Bertsch, H. Esbensen and A. Sustich, Phys. Rev. C
42, 758 (1990).
[21] K. Hencken, G. Bertsch and H. Esbensen, Phys. Rev. C
54, 3043 (1996).
[22] E. Sauvan et al., Phys. Lett. B491, 1 (2000).
[23] J. P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rev.
C 16, 80 (1977).
[24] E. Bauge, J. P. Delaroche and M. Girod, Phys. Rev. C
58, 1118 (1998).
[25] L. Trache et al., Phys. Rev. C 61, 024612 (2000).
[26] C. Bertulani and G. Baur, Nucl. Phys.A480, 615 (1988).
[27] P. G. Hansen, Phys. Rev. Lett. 77, 1016 (1995).
[28] F. Carstoiu et al., Phys. Rev. C 63, 054310 (2001).
FIG. 1. The stripping and diffraction dissociation parts of
the breakup probability as a function of the impact parame-
ter.
FIG. 2. Comparison of the spectroscopic factors Stot(dots)
and of the ANC C2tot (triangles) extracted from the
8B
breakup data on Si at 38 MeV/u [10], for different param-
eters of the single particle Woods-Saxon potentials.
FIG. 3. Calculated parallel momentum distributions for
core-like fragments in a breakup reaction of 8B on a 9Be target
at 41 MeV/u, are compared with experimental data [13]. The
curve labeled ”intr” is the result of a calculation with the
Serber model in the transparent limit, that labeled ”disk”
with the black disk model.
TABLE I. Summary of the ANC extracted from different
8B breakup reactions.
Target E/A exp. c.s. Ref. C2tot
[MeV/u] [mb] [fm−1]
28Si 28 244(15) [10] 0.435
35 225(15) [10] 0.420
38 222(15) [10] 0.423
12C 40 80(15) [12] 0.250
142 109(1) [11] 0.597
285 89(2) [11] 0.482
Sn 142 502(6) [11] 0.547
285 332(6) [11] 0.464
208Pb 142 744(9) [11] 0.421
285 542(9) [11] 0.460
aver 0.450(30)
4
8B→7Be+p 38 MeV/u
str.
diff.
b(fm)
 
2p
bP
(b)
 (f
m)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 2 4 6 8 10 12 14 16 18 20
single part. ANC b (fm-1/2)
Sp
ec
tr.
 fa
cto
r S
 an
d C
2  
(fm
-
1 )
0
0.2
0.4
0.6
0.8
1
0.6 0.65 0.7 0.75 0.8 0.85 0.9
kz (MeV/c)
ds
/d
k z
(co
un
ts) 9Be(8B,7Be)
Elab=41A MeV
total
stripp
diff
coul
intr
disk
0
50
100
150
200
250
300
350
400
450
500
1700 1750 1800 1850 1900 1950 2000 2050 2100


Physical Review Letters

Asymptotic Normalization Coefficient of 8 B from Breakup Reactions and the S 17 Astrophysical Factor

Crossref

Asymptotic Normalization Coefficient of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow /><mml:mrow><mml:mn>8</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mi>B</mml:mi></mml:math>from Breakup Reactions and the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="italic">S</mml:mi></mml:mrow><mml:mrow><mml:mn>17</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>Astrophysical Factor

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:nucl-ex/0108002v2  8 Nov 2001Asymptotic normalization coefficient of 8B from breakup reactions and the S17astrophysical factorL. Trache1, F. Carstoiu2,3, C. A. Gagliardi1, R. E. Tribble11Cyclotron Institute, Texas A&M University, College Station, TX-77843,USA2Institute of Physics and Nuclear Engineering H. Hulubei, Bucharest,Romania3Laboratoire de Physique Corpusculaire, F-14050 Caen Cedex, France(September 19, 2018)We show that asymptotic normalization coefficients canbe extracted from one nucleon breakup reactions of looselybound nuclei at 30-300 MeV/u. In particular, the breakupof 8B is described in terms of an extended Glauber model.The 8B ANC extracted from breakup data at several energiesand on different targets, C2tot = 0.450 ± 0.039 fm−1, leadsto the astrophysical factor S17(0) = 17.4 ± 1.5 eV·b for thekey reaction for solar neutrino production 7Be(p,γ)8B. Theprocedure described provides an indirect method to determinereaction rates of astrophysical interest with beams of looselybound radioactive nuclei.8B, produced in the 7Be(p,γ)8B reaction, is the ma-jor source of the high-energy neutrinos observed by solarneutrino detectors [1]. Results of recent direct and indi-rect determinations of the astrophysical factor S17(0) (see[2–6] and references therein) do not fully agree, but spanthe full adopted range, 19+4−2 eVb [7]. The asymptoticnormalization coefficient (ANC) approach [8] is based onthe fact that the 7Be(p,γ)8B reaction is highly periph-eral. Thus, what one needs most is a precise quanti-tative description of the tail or periphery of the 7Be +p ↔ 8B overlap function, which is given by the corre-sponding ANC, rather than of the full many-body wavefunction inside the core. The advantage of this approachis that ANCs may be determined from other peripheralnuclear reactions that have cross sections orders of mag-nitude larger than the radiative capture reaction of in-terest. This technique has been applied to determine S17from measurements of two (7Be,8B) proton transfer reac-tions [6], after being tested to give results to better than9% [9] for 16O(p,γ)17F.In the present paper we show that one can extractasymptotic normalization coefficients for the 8B groundstate wave function from data on one-proton removal (orbreakup) reactions at energies of 30-300 MeV/u [10–13].Improvements in the nuclear reaction model used makea quantitative description possible for the energy rangeand the ANC extracted reliable. Then, this ANC canbe used to obtain an independent determination of theastrophysical factor S17(0).Recent advances in experimental techniques permitthe selection of the final state of the remaining core inbreakup reactions at large energies. Typically an exoticnucleus B = (Ap), where B is a bound state of the coreA and the proton p, is produced by fragmentation froma primary beam, separated and used to bombard a sec-ondary target. After the breakup occurs, the breakupcross section and the parallel momentum distribution ofthe core A are measured. By detecting the final state ofthe core, e.g. using coincident gamma rays, one can de-termine important spectroscopic information about theground state of the exotic projectile [14,15]. The mea-sured momentum distribution of the core can be relatedto the momentum distribution of the bound nucleon.In all reactions where the core survives (either transferor one-nucleon breakup) the matrix elements include theoverlap integral IBAp(~r) for the nuclei A, p, and B, ob-tained after the integration over the internal coordinatesof fully antisymmetric wave functions, with ~r the vectorconnecting the center of mass of nucleus A with p [16,8].The overlap integrals are not normalized to unity, but tothe spectroscopic factors Snlj . At asymptotic distanceswhere nuclear forces are vanishingly small, r > RN , theoverlap integrals behave asIBAplj(r)→ S1/2nlj ϕnlj(r)→ CBApljW−η,l+1/2(2kr)r. (1)Here CBAplj is the asymptotic normalization coefficientdefining the amplitude of the tail of the overlap integral,W is the Whittaker function, k is the wave number, andη is the Sommerfeld parameter for the bound state (Ap).The asymptotic normalization coefficients CBAplj can beextracted from any peripheral observables that are mea-sured experimentally.For the reaction model calculations we assume thatthe ground state of the projectile (Jpi) can be approxi-mated by a superposition of configurations of the form[Ψpicc ⊗ nlj]Jpi , where Ψpicc denote the core states andnlj are the quantum numbers for the single particlewave function ϕnlj(r) in a spherical mean field poten-tial. These single particle states are normalized to unityand have the asymptotic behavior given by Eq. 1, withthe single particle asymptotic normalization coefficientsbnlj . When more than one configuration contributes to aselected core state, the total cross section for one-nucleonbreakup is written as an incoherent superposition of sin-gle particle cross sections:1σ−1p =∑S(c, nlj)σsp(nlj). (2)A similar relation holds for the momentum distribution.Typically the nucleon is not measured, therefore the cal-culated cross sections σsp(nlj) contain a stripping term(the loosely bound nucleon is absorbed by the target andthe core is scattered and detected), a diffraction dissoci-ation term (the nucleon is scattered away by the target,the core is scattered by the target and is detected) anda Coulomb dissociation term:σsp =∫∞02πbdb(Pstr(b) + Pdiff (b)) + σCoul (3)In previous analyses of breakup reactions, a structure forthe projectile (the spectroscopic factors S(c, nlj)) wasassumed and agreement between the calculated and ex-perimental values was considered a validation both of theassumed nuclear structure and of the reaction model cal-culations used [11,17,18]. In the present case we use anextended Glauber model, in the eikonal approximationwith non-eikonal correction terms up to the second order[10,19]. Realistic nucleon-target and core-target S-matrixelements are used in the evaluation of the impact parame-ter dependent probabilities. We find that the largest con-tributions to the cross sections σsp(nlj) come from largeimpact parameters and therefore the phenomena are pe-ripheral. It then follows that we can express the resultsin terms of the asymptotic normalization coefficients andreverse the process: we can use the experimental resultsto extract the ANCs.Calculations were done for several 8B breakup reac-tions for which data exist in the literature. The modelis similar to that developed by Bertsch et al. [20,21]. Ithas been tested before on 23 different reactions in thep-sd shell [10,22]. The loosely bound proton and thecore moving on an eikonal trajectory interact indepen-dently with the target nucleus, an assumption valid atthese energies. For the proton-target interaction we usedthat of Jeukenne, Lejeune and Mahaux (JLM) [23], inthe updated version of Bauge et al. [24]. For the target-core nucleus-nucleus interaction we use the double fold-ing procedure described in [25]: the same JLM inter-action is folded with Hartree-Fock nuclear matter dis-tributions of the core and of the target. Subsequentlythe double folding potentials were renormalized to repro-duce a variety of elastic scattering data for light nuclei.We found there that the real part needed a substantialrenormalization at about 10 MeV/u (NV=0.366), but theimaginary part did not (NW=1.00). We have checkedthe procedure on a much wider set of data from litera-ture at higher energies, and found a similar conclusionfor the imaginary part, while the renormalization of thereal part approaches unity around 50 MeV/u. Therefore,in the present calculations we adopted the procedure of[25], with the JLM(1) interaction, and NW=1.00. TheS-matrix calculations that enter the first two terms ofEq. 3 are expected to depend primarily on the imagi-nary part of the interaction. Indeed, calculations withlarge variations of the renormalization of the real poten-tial (NV =0.366 and NV=0.80) give the same cross sec-tions. The integrated Coulomb term is treated in a per-turbative method that retains the dipole and quadrupoleterms, equivalent with that of Ref. [26], but using radialmatrix elements calculated with realistic Woods-Saxonradial wave functions.The impact parameter dependence of the first twoterms in Eq. 3 is plotted in Fig. 1 for the case of thebreakup of 8B on a Si target at 38 MeV/u [10]. Clearly,both terms are dominated by the periphery but have con-tributions from small impact parameters. The Coulombterm is even more peripheral, due to its long range. Toinvestigate the influence of the nuclear interior on theextracted ANC, different single particle wave functionswere used for the outer proton to calculate the breakupcross section in the same reaction model. We chose arange of radii and diffusenesses (R = 2.20 − 2.60 fmand a = 0.50 − 0.70 fm) for the Woods-Saxon poten-tials and repeated the calculations. A correct spin-orbitterm was included, and in all cases we adjusted the depthof the potential to reproduce the proton separation en-ergy Sp = 137 keV. The radial behavior of 1p1/2 and1p3/2 orbitals is identical at large distances and, for agiven (R, a), differs at small radii by much less than thevariation associated with the choice of Woods-Saxon po-tential. Thus, for simplicity, only the 1p3/2 componentwas included, and we rewrite Eq. 2 as:σ−1p = (Sp3/2 + Sp1/2)σsp = (C2p3/2+ C2p1/2)σsp/b2p, (4)where bp is the ANC of the normalized 1pj radial singleparticle wave function. The experimental value for thebreakup cross section is 222±15 mb. The calculated crosssection varies from 226 mb to 326 mb with the choice ofthe single particle wave function used (i.e. (R, a) theradius and diffuseness parameters of the Woods-Saxonpotential used), which is equivalent to 44% variation if aspectroscopic factor Stot = Sp3/2 + Sp1/2 is extracted us-ing the first part of Eq. 4. However if the square of theANC C2tot = C2p3/2+C2p1/2 is extracted instead, the resultis very stable, as shown in Fig. 2 (where for conveniencethe results are plotted against the single particle ANCbp calculated for each geometry assumed). The varia-tion in the ANC over the full range considered here is<±3%. A flat curve for C2 in Fig. 2 would be a sig-nature of a purely peripheral reaction; the small slopereflects the participation of the interior of the nucleus.Similar calculations were done for the same target andtwo other energies (35 and 28 MeV/u), and for othertargets (12C, Sn and 208Pb) at other energies: 40, 142and 285 MeV/u [12,11]. Pictures similar to Fig. 2 areobtained in each and every case. These experiments havenot determined the yield to the 7Be first excited state. ACoulomb breakup experiment carried out at 50 MeV/uon a Pb target [14] found it to be small, about 5% ofthe total. From this we estimated the ANC for the coreexcitation part in the wave function, then calculated its2contribution to the one-proton removal cross section oneach target and subtracted it (e.g. 7.5% for the Si target).The corrected ANCs extracted are presented in Table I.With the exception of two data points on 12C, there isgood agreement among the ANCs which come from dataover a wide range of incident energies and both low andhigh Z targets. In order to extract an average ANC, wehave done an unweighted average of the individual mea-surements. Using all of the data points, this results inC2tot = 0.450(30) fm−1 where the uncertainty is the stan-dard error of the mean. If the two 12C data points (4 and5 in the table) at 40 and 142MeV/u are removed from ouraverage (note that they fail a simple test of the expectedenergy dependence), we find C2tot = 0.456(14) fm−1. Sev-eral correlated uncertainties must be added: 4% for therenormalization coefficients used for the optical modelparameters, 3% for the variation in C2 as a function ofbp and 2% for the uncertainty in the excited state con-tribution. Including these we find C2tot = 0.450(39) fm−1for the full data set and C2tot = 0.456(28) fm−1 when thetwo 12C data points are removed. The values are similarand, without additional information, we shall adopt thefirst.Using a wave function with the asymptotic normaliza-tion as extracted above, we have calculated the distri-bution of the parallel momentum of the core measuredin the breakup of 8B at 41 MeV/u on a 9Be target [13].The result is compared in Fig. 3 with the experimentaldata. A very good description is obtained. Notably, un-like the black disk model [27] calculation which describesthe width of the parallel momentum distribution [28],the extended Glauber model calculation also matches thelarge momentum tails due to the nuclear interior. Thisgives further credibility to our calculations and the entireapproach.The value extracted here from the breakup of 8B at30-300 MeV/u agrees very well with the one extractedfrom transfer reactions at 12 MeV/u [6]. We can use theANC extracted to evaluate the astrophysical S factor forthe reaction 7Be(p,γ)8B at very low energies, followingthe procedure of Ref. [8]. Using the value from breakupwe find S17(0) = 17.4 ± 1.5 eV·b. We can also use theANC extracted here to determine the rms radius for the8B proton halo, following the procedure of [28]. We finda rh = 4.20± 0.21 fm.In conclusion, reliable spectroscopic information canbe extracted from one-nucleon breakup reactions ofloosely bound nuclei at energies around and above theFermi energy. However, we have shown that, despite amore transparent meaning of the spectroscopic factors,the values obtained are not unambiguous, and a bet-ter quantitative description is achieved in terms of theasymptotic normalization coefficients. In turn, these canbe used to calculate any observables that are dominatedby the periphery of the nucleus, notably rms radii forhalos and astrophysical S-factors. Calculations using anextended Glauber model for the breakup data of 8B on awide range of targets and energies lead to an unambigu-ous value for the ANC and an astrophysical factor S17(0)in very good agreement with the values from recent deter-minations from direct measurements [2] and with thoseusing indirect methods [5,6]. New measurements for theelastic scattering of 8B, a more accurate determinationof the breakup cross sections (eventually separating thestripping and diffraction dissociation components) anda precise determination of the core excitation contribu-tion, can increase the reliability of the ANC extracted.The validity of the procedure is wider than for the 8Bcase discussed above. In addition to peripherality, en-sured more or less for the halo nuclei, the requirementsare good absolute values for the breakup cross sections,with the identification of the final state of the core, andreliable cross section calculations. The method can beused to extract valuable information for nuclear astro-physics. Very difficult or even impossible direct measure-ments that would involve bombarding short lived targetswith very low energy protons can be replaced or supple-mented by indirect methods seeking the relevant ANCs,rather than complete knowledge of the ground state wavefunction of these exotic nuclei. In addition, the indi-rect ANC method is subject to different systematic er-rors than the direct measurements, and therefore redun-dance of the results is very much welcome, particularlyfor critical astrophysical S-factors, such as that for the7Be(p,γ)8B reaction.One of us (FC) acknowledges the support of IN2P3 fora stay at Laboratoire de Physique Corpusculaire in Caen,during which part of this work was completed. The workwas supported by the U. S. Department of Energy un-der Grant No. DE-FG03-93ER40773, by the RomanianMinistry for Research and Education under contract no555/2000, and by the Robert A. Welch Foundation.[1] J. N. Bahcall, M. H. Pinsonneault and S. Basu, Astroph.J. 555, 990 (2001).[2] F. Hammache et al., Phys. Rev. Lett. 86, 3985 (2001).[3] A. R. Junghans et al., INPC 2001, Berkeley, July 2001,paper 4.6, and to be published.[4] N. Iwasa et al., Phys. Rev. Lett. 83, 2910 (1999).[5] B. Davids et al., Phys. Rev. C 63, 065806 (2001).[6] A. Azhari et al., Phys. Rev. C 63, 055803 (2001).[7] E. G. Adelberger et al., Rev. Mod. Phys. 70, 1265 (1998).[8] H. M. Xu et al., Phys. Rev. Lett. 73, 2027 (1994).[9] C. A. Gagliardi et al., Phys. Rev. C 59, 1149 (1999).[10] F. Negoita et al., Phys. Rev. C 54,1787 (1996).[11] B. Blank et al., Nucl. Phys. A624, 242 (1997).[12] I. Pecina et al., Phys. Rev. C 52, 191 (1995).[13] J. H. Kelley et al., Phys. Rev. Lett. 77, 5020 (1996).[14] T. Motobayashi et al., Nucl. Phys. A682, 345c (2001).[15] A. Navin et al., Phys. Rev. Lett. 81, 5089 (1998).[16] G. R. Satchler, Direct Nuclear Reactions, Clarendon3Press, Oxford Univ. Press, N.Y., 1983.[17] J. S. Al-Khalili and J. A. Tostevin, Phys. Rev. Lett. 76,3903 (1996).[18] H. Esbensen and K. Hencken, Phys. Rev. C 61, 054606(2000).[19] F. Carstoiu et al., to be published.[20] G. Bertsch, H. Esbensen and A. Sustich, Phys. Rev. C42, 758 (1990).[21] K. Hencken, G. Bertsch and H. Esbensen, Phys. Rev. C54, 3043 (1996).[22] E. Sauvan et al., Phys. Lett. B491, 1 (2000).[23] J. P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rev.C 16, 80 (1977).[24] E. Bauge, J. P. Delaroche and M. Girod, Phys. Rev. C58, 1118 (1998).[25] L. Trache et al., Phys. Rev. C 61, 024612 (2000).[26] C. Bertulani and G. Baur, Nucl. Phys.A480, 615 (1988).[27] P. G. Hansen, Phys. Rev. Lett. 77, 1016 (1995).[28] F. Carstoiu et al., Phys. Rev. C 63, 054310 (2001).FIG. 1. The stripping and diffraction dissociation parts ofthe breakup probability as a function of the impact parame-ter.FIG. 2. Comparison of the spectroscopic factors Stot(dots)and of the ANC C2tot (triangles) extracted from the8Bbreakup data on Si at 38 MeV/u [10], for different param-eters of the single particle Woods-Saxon potentials.FIG. 3. Calculated parallel momentum distributions forcore-like fragments in a breakup reaction of 8B on a 9Be targetat 41 MeV/u, are compared with experimental data [13]. Thecurve labeled ”intr” is the result of a calculation with theSerber model in the transparent limit, that labeled ”disk”with the black disk model.TABLE I. Summary of the ANC extracted from different8B breakup reactions.Target E/A exp. c.s. Ref. C2tot[MeV/u] [mb] [fm−1]28Si 28 244(15) [10] 0.43535 225(15) [10] 0.42038 222(15) [10] 0.42312C 40 80(15) [12] 0.250142 109(1) [11] 0.597285 89(2) [11] 0.482Sn 142 502(6) [11] 0.547285 332(6) [11] 0.464208Pb 142 744(9) [11] 0.421285 542(9) [11] 0.460aver 0.450(30)48B→7Be+p 38 MeV/ustr.diff.b(fm) 2pbP(b) (fm)00.20.40.60.811.21.41.61.820 2 4 6 8 10 12 14 16 18 20single part. ANC b (fm-1/2)Spectr. factor S and C2  (fm-1 )00.20.40.60.810.6 0.65 0.7 0.75 0.8 0.85 0.9kz (MeV/c)ds/dk z(counts) 9Be(8B,7Be)Elab=41A MeVtotalstrippdiffcoulintrdisk0501001502002503003504004505001700 1750 1800 1850 1900 1950 2000 2050 2100

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Asymptotic normalization coefficient of ^{8}B from breakup reactions and
  the S_{17} astrophysical factor

Asymptotic normalization coefficient of ^{8}B from breakup reactions and the S_{17} astrophysical factor

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