All orders proton breakup from exotic nuclei
ABSTRACT We present a semiclassical method to treat the proton breakup from a weakly bound state in an exotic nucleus. The Coulomb interactions between the proton, core and target are treated to all orders and including the full multipole expansion of the Coulomb potential. The nuclear proton-target interaction is also treated to all orders. The core-target interaction is included as an absorption. The method is semi-analytical thus allowing for a detailed understanding of the short range and long range effects of the interactions in the reaction dynamics. It explains also the origin of possible asymmetries in the core parallel momentum distributions when the full multipole expansion of the Coulomb potential is used. Calculations are compared to results of other, fully numerical, methods and to experimental data in order to establish the accuracy and reliability of the method.
arXiv:0706.1222v1 [nucl-th] 8 Jun 2007
All orders proton breakup from exotic nuclei
A. Garc´ ıa-Camacho(a), G. Blanchon(a), A. Bonaccorso(a), D. M. Brink(b)
(a)INFN, Sez. di Pisa and Dipartimento di Fisica, Universit` a di Pisa,
Largo Pontecorvo 3, 56127 Pisa, Italy.
(b)Department of Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, U. K.
We present a semiclassical method to treat the proton breakup from a weakly bound state in
an exotic nucleus. The Coulomb interactions between the proton, core and target are treated
to all orders and including the full multipole expansion of the Coulomb potential. The nuclear
proton-target interaction is also treated to all orders. The core-target interaction is included as
an absorption. The method is semi-analytical thus allowing for a detailed understanding of the
short range and long range effects of the interactions in the reaction dynamics. It explains also the
origin of possible asymmetries in the core parallel momentum distributions when the full multipole
expansion of the Coulomb potential is used. Calculations are compared to results of other, fully
numerical, methods and to experimental data in order to establish the accuracy and reliability of
Pacs 21.10.Jx, 24.10.-i, 25.60.Gc, 27.30.+t
The study of exotic nuclei through knockout reactions
has provided important results along the last decades.
Among those, one proton removal reactions have been
proven as a useful tool for the understanding of light,
proton rich, exotic nuclei. From the reaction point of
view, the inclusion of a long-range part in the interaction
between the proton and the target, i.e.
field, provides a significant complication with respect to
the neutron case as well as a challenge for reaction theory.
This work is an extension of the method presented in
Ref. . There, a formalism to treat neutron breakup
to all orders, including the whole multipole expansion
of the Coulomb interaction giving the recoil of the core,
was presented. Here that theoretical framework is com-
pleted in order to account for proton breakup as well.
For proton halo nuclei Coulomb breakup reactions in the
laboratory have been used to get indirect information
on the radiative capture, since it has been shown that
the Coulomb breakup cross section is proportional to the
radiative capture cross section . A number of experi-
mental papers - have reported experiments on pro-
ton breakup from8B.8B is partner in (p,γ) radiative
capture reactions of great astrophysical interest for the
understanding of the neutrino flux from the sun (see for
example the discussion and references of ).17F is an-
other candidate for proton halo which is still under inves-
tigation both experimentally [14, 15, 16] as well as the-
oretically [17, 18, 19, 20]. Furthermore17F plays an im-
portant role in understanding explosive nucleo-synthesis
in X-ray bursts and novae, as it enters in the sequence
14O(α,p)17F(p,γ)18Ne(α,p)21Na, where cross sections are
The existence of a proton halo has sometimes been
questioned  and results from different experiments
might seem to be contradictory [22, 23]. Some of the
apparent discrepancies in experimental results and their
FIG. 1: (Colour online) Coordinate system.
analysis have been recently discussed and in large part
solved in Ref. .
This work is the extension to the proton breakup
problem of the formalism developed in  for neutron
breakup.Similar formulae to those of the neutron
breakup can be derived, since the Coulomb potential can
be written as
V (? r,?R) =
|?R − β1? r|
|?R + β2? r|
where Vc = ZcZte2, Vv = ZvZte2and V0 = (Zv +
Zc)Zte2. β1 and β2 are the mass ratios of proton and
core, respectively, to that of the projectile. The coordi-
nate system used in this paper is shown in Fig. 1. Fol-
lowing a procedure analogous to that of , where χ was
defined as χpert=1
?dteiωtV (? r,t), the Coulomb phase
?v(Vceiβ1ωz/vK0(ωbc/v) − V0K0(ωR⊥/v)
for the proton is shown to be
with ω = (εf− ε0)/? and ε0is the neutron initial bound
state energy while εfis the final neutron-core continuum
energy. Since V0= Vc+ Vv, Eq. (2) can be written as
χp= χ(β1,Vc) + χ(−β2,Vv) (3)
χ(β,V ) =2V
eiβωz/vK0(ωb/v) − K0(ωR⊥/v)
The Coulomb phase Eq.(3) is therefore the sum of
two terms: one of them describes the recoil of the core
whereas the other accounts for the direct proton-target
Coulomb interaction. Of course, in the case of the neu-
tron the latter vanishes and the phase reduces to the one
derived in .
The expansion of Eq. (4) to first order in ? r yields the
dipole approximation to the phase:
which only differs from the neutron breakup case in the
different constant factor, which is now (Vcβ1− Vvβ2) in-
stead of V0β1of Ref. .
Our expression for the differential cross-section is
d?bc|Sct(bc)|2|grec+ gdir+ gnuc|2. (6)
where |Sct(bc)|2is the core-target elastic scattering prob-
ability discussed later. Such a form of the cross section
has been introduced for relativistic energy electromag-
netic excitations in Ref. where however a first or-
der perturbation theory amplitude was considered. It
has been derived in an eikonal approach to the halo-
projectile scattering [27, 28] and it has been widely used
also for normal projectiles in the case of peripheral re-
actions such as transfer and inelastic excitations .
Under the conditions that the core-target interaction is
mainly absorptive it has been re-obtained by Bertulani
in a relativistic approach to proton breakup . This
paper has also shown that relativistic corrections to dy-
namical approaches such as the continuum discretised
coupled channel method (CDCC) can increase the pro-
ton breakup cross section of as much as 15% at the inci-
dent energy of 250 A.MeV. Our formalism however fol-
lows closely Ref. where relativistic kinematics is easily
implemented and it has been used in the numerical cal-
culations presented in this paper. As we shall see in the
following our results agree indeed rather well with high
energy experimental data.
The probability amplitude in Eq.(6) has been written
as the sum of three pieces: the recoil term,
d? re−i?k·? rφi(? r)(ei2Vc
R⊥ − 1 − i2Vc
where, according to the discussions in [1, 25, 27, 28],
the sudden limit has been used in order to include all
orders in the interaction. Similarly, the second term in
our probability amplitude is the direct proton Coulomb
interaction. It has the same form as Eq.(7) but for the
substitution Vc→ Vv, bc→ bvand β1→ −β2.
Finally, the nuclear part is
d? re−i?k·? rφi(? r)
A number of papers have addressed the problem of
asymmetry in the core parallel momentum distribution
after proton knockout [4, 5, 6, 19, 31, 32, 33]. The fact
that this asymmetry comes from high order terms can be
directly extracted from our formalism. If the Coulomb
part of the amplitude gCou= grec+ gdiris simply ex-
panded to first order in χ, it becomes
−V0K0(ωR⊥/v) + Vve−iβ2ωz/vK0(ωbv/v)), (9)
d? re−i?k·? rφi(? r)2
which can be written, in terms of the one-dimensional
Fourier transform in z−directionˆφi, as
− V0K0(ωR⊥/v)ˆφi(? r⊥,kz)
+ VvK0(ωbv/v)ˆφi(? r⊥,kz− β2ω/v)
d? r⊥ei?k⊥·? r⊥2
VcK0(ωbc/v)ˆφi(? r⊥,kz+ β1ω/v)
Thus the Coulomb breakup probability amplitude can be
regarded as a coherent sum of three terms, each of which
contains a shifted z−Fourier transform. The shifts are in
opposite directions, −β1ω/v and β2ω/v, but they are not
visible in the calculated momentum distributions shown
in this paper as ω depends on k itself. The amount of
deformation will also depend on the explicit form of the
wave function φi(? r). Moreover, the 1/v factor indicates
that the asymmetry decreases as the beam energy in-
In dipole approximation, however, the amplitude is
d? r⊥e−i?k⊥·? r⊥2(β1Vc− β2Vv)
which does not contain any asymmetry for the momen-
tum distribution as it involves square moduli ofˆφi(? r⊥,kz)
separately. Hence we can confirm analytically that the
asymmetry in Coulomb breakup parallel momentum dis-
tributions is due to the presence of higher multipole
FIG. 2: Calculated inclusive momentum distribution of7Be
fragments after proton-removal from
A.MeV. The calculations are the sum of contributions lead-
ing to the ground and excited state of7Be weighted by the
respective spectroscopic factors and include both the diffrac-
tion and stripping parts of nuclear breakup, besides Coulomb
breakup. Data are from .
8B against Pb at 936
tot, nuc-Cou, exc, dipole
tot, nuc-Cou, gs, dipole
tot, nuc-Cou, excited state
tot, nuc-Cou, gs
FIG. 3: (Colour online) Proton momentum distribution after
Coulomb breakup of
dipole and full-multipole approximations, for both ground and
excited state. Spectroscopic factors are not included.
8B against Pb at 936 A.MeV in both
terms, in agreement with earlier works [4, 5, 6, 19, 31, 32].
However, the presence of the nuclear interaction intro-
duces an interference that does depend on the sign of kz
and thus an additional asymmetry to that due to higher
multipole terms in the Coulomb interaction.
This formalism has been applied to proton breakup of
8B against Pb and C targets at a beam energy of 936
A.MeV, experiment reported in . The projectile8B is
taken as a two-body object. Its ground state has com-
ponents with a proton coupled to the ground 3/2−state
and to the excited E∗= 429keV 1/2−state of7Be. The
proton separation energy is 137 keV when the7Be is left
in its ground state and 137 + 429 keV when it is left in
an excited state.
Radial wave functions have been obtained by numeri-
cal solution of the Schr¨ odinger equation in Woods-Saxon
potentials with depths adjusted to reproduce the corre-
sponding separation energies. The radius parameter of
these Woods-Saxon potentials have been taken as 1.3 fm
with a diffuseness of 0.6 fm.
The nuclear breakup is described in this formalism
through the proton-target and core-target S-matrices.
These have been calculated by taking the optical limit in
Glauber theory . The effective two-body interactions
are obtained by folding effective nucleon-nucleon interac-
tions with the densities of the nuclei involved. Pb density
was approximated by a Woods-Saxon profile fitted to re-
produce the experimental root mean square radius 5.5 fm
root mean square radius 2.31 and 2.35 fm respectively
. The stripping contribution to nuclear breakup has
been calculated in the eikonal formalism .
With these ingredients and the formulae described in
the previous section, the single particle cross-sections of
Table I were calculated. Interference between Coulomb
and elastic nuclear part are included automatically in
the amplitude, but stripping must be added incoherently.
When comparing to experimental data, spectroscopic fac-
tors must be taken into account. Following , the spec-
troscopic factor for the excited state is Sex = 0.101
in this work, while for the ground state we have used
Sgs = 0.841. The latter has been obtained by adding
together the spectroscopic factors for two states which
differ in core intrinsic spin , since our formalism can-
not distinguish between them. These numbers provide
an exclusive/inclusive ratio of Rtheo= 0.089 for the Pb
target, in good agreement with the experimental result
Rexp= 0.085±0.021. Such agreement is also good for the
C target, as the experimental result is Rexp= 0.13±0.03,
while we obtain Rtheo = 0.098. The inclusive momen-
tum distribution on lead is shown in Fig. 2 with no scale
factors. In this calculation the stripping contribution is
assumed to have the same shape as the diffractive one.
Clearly the agreement is not perfect, but in view of the
very simple two-body forms of our initial and final wave
functions which do not contain any three-body effects, it
can be considered reasonable. Regarding absolute cross-
sections, in the case of the Pb target, our result for the
excited state is 57.8 mb, in agreement with the exclu-
sive measurement of 56±5 mb, while the inclusive result
662 ± 60 mb is rightly described by our calculation of
652.4 mb (Table I). For the C target we predict 9.1 (92.6)
mb for the exclusive (inclusive) measurement, consistent
with the experimental result 12 ± 3 (94 ± 9) mb.
This formalism allows a direct check of the dipole ap-
proximation for the Coulomb interaction, by compar-
ing the results provided by the phase Eq.(3) to those
of Eq.(5). Fig. 3 shows the calculated momentum dis-
tributions of protons in the projectile reference frame af-
ter breakup of8B. This does not include stripping. The
dipole approximation, yields narrower momentum distri-
bution and smaller single particle cross-section. High or-
der effects, especially in the heavy target case, are not
7Be and12C densities were taken as gaussian with
TABLE I: Single particle cross-sections in mb. Beam energies
TABLE II: Barrier penetration factors w1 and w2 for8B and
17F respectively. Proton-core energies, εf, and momenta, ?k,
in MeV. n1 and n2 are dimensionless Sommerfeld parameters.
0.5 30.7 0.89 0.22 1.78 0.013
1.0 43.5 0.63 0.58 1.26 0.158
1.5 53.2 0.51 0.81 1.03 0.39
2.0 61.5 0.44 0.94 0.89 0.61
negligible, in agreement with , although in their cal-
culation the effect appears smaller due to their choice of
large impact parameters.
We have also performed preliminary calculations for
17F. They differ from those of Ref. by about an order
of magnitude for the g.s. breakup and by about a fac-
tor three for the breakup from17F∗. The reason for this
discrepancy is in large part due to the use of a the plane
wave as final state which neglects the Coulomb proton-
core interaction. An estimate of the possible correction
can be obtained by calculating a barrier penetration fac-
tor w at the core radius Rcaccording to Ref..
In Table II we give the values of the penetration fac-
tors w1 and w2 for8B and17F projectiles respectively.
We give also the values of the corresponding Sommerfeld
parameters n1and n2, with n=ZpZte2/?v. These values
show that for a peak in the proton-core energy spectrum
around 1.5-2MeV there is a possible error in the cross
sections of 10-20% in the8B projectile case while the
error is of 40-60% for17F. Such estimates are indepen-
dent from the initial binding energy and from the beam
energy. Our method is an high energy approximation
and therefore further corrections should be expected if
the method is applied at the low energies considered in
In this paper we have introduced a semiclassical and
analytical method to treat proton breakup to all orders
and all multipoles in the Coulomb potential. As previ-
ously done by some of us [1, 27, 28], the present method is
based on the use of a final nucleon wave function which
is an extention of the traditional eikonal-type.
Coulomb potential is neglected, our breakup amplitude
reduces to the well known eikonal form for the nuclear
breakup. On the other hand if only the first order term
of the Coulomb phase is considered, our Coulomb am-
plitude reduces to the multipole expansion in which the
dipole term is treated by perturbation theory, while high
order and higher multipole terms are treated in the sud-
den approximation according to . To test the method
we have applied it to the study of one proton breakup
from both the ground and first excited states of8B. Our
results are in very good agreement with experimental
data at relativistic energies. Preliminary calculations for
17F at lower energies (65 A.MeV) have shown discrepan-
cies of about one order of magnitude with respect to the
results of Ref.. We believe this is mainly due to our
choice of a plane-wave for the final state wave-function.
However, the use of the plane-wave has allowed analyti-
cal calculations leading to Eq.(10) which is perhaps the
most interesting result of this work. It shows explicitly
the origin of a possible asymmetry in core parallel mo-
mentum distributions as contained in the Fourier trans-
form of the initial state wave function. In agreement with
other numerical methods  we have demonstrated that
the asymmetry is due to the higher order multipole terms
in the expansion of the Coulomb potential, while it dis-
appears in the dipole approximation.
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