Scaling laws and higher-order effects in Coulomb excitation of neutron halo nuclei

Abstract

Essential properties of halo nuclei can be described in terms of a few low-energy constants. For neutron halo nuclei, analytical results can be found for wave functions and electromagnetic transition matrix-elements in simple but well-adapted models. These wave functions can be used to study nuclear reactions; an especially simple and instructive example is Coulomb excitation. A systematic expansion in terms of small parameters can be given. We present scaling laws for excitation amplitudes and cross sections. The results can be used to analyze experiments like ${}^{11}$Be Coulomb excitation. They also serve as benchmark tests for more involved reaction theories.

Full text

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arXiv:0808.3478v1 [nucl-th] 26 Aug 2008
EPJ manuscript No.
(will be inserted by the editor)
Scaling laws and higher-order effects in Coulomb excitation of
neutron halo nuclei
S. Typel1,2and G. Baur3,4
1Excellence Cluster Universe, Technische Universit¨ at M¨ unchen, Boltzmannstraße 2, D-85748 Garching, Germany
2Gesellschaft f¨ ur Schwerionenforschung (GSI) mbH, Planckstraße 1, D-64291 Darmstadt, Germany
3Institut f¨ ur Kernphysik, Forschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany
4J¨ ulich Centre for Hadron Physics, Forschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany
Received: date / Revised version: date
Abstract. Essential properties of halo nuclei can be described in terms of a few low-energy constants.
For neutron halo nuclei, analytical results can be found for wave functions and electromagnetic transition
matrix-elements in simple but well-adapted models. These wave functions can be used to study nuclear
reactions; an especially simple and instructive example is Coulomb excitation. A systematic expansion
in terms of small parameters can be given. We present scaling laws for excitation amplitudes and cross
sections. The results can be used to analyze experiments like11Be Coulomb excitation. They also serve as
benchmark tests for more involved reaction theories.
PACS. 25.70.De Coulomb excitation – 23.20.Js Multipole matrix elements – 27.20.+n 6 ≤ A ≤ 19
1 Introduction
Exotic nuclei are available as secondary beams at many
radioactive beam facilities around the world. These un-
stable nuclei are generally weakly bound with few, if any,
excited states. A well developed method to study halo nu-
clei is Coulomb excitation. An instructive example is the
excitation of the 1/2−bound state in11Be from the 1/2+
ground state [1,2,3,4].
Halo nuclei are a low-energy phenomenon and can be
described effectively in terms of a few low-energy param-
eters. The ratios of core size to the sizes of the halo states
(to be defined in eq. (1) below) serve as small expansion
parameters. Wave functions and matrix elements can be
given in terms of these parameters. These wave functions
can be used in reaction models and simple and realistic
formulae are obtained.
In a recent paper the B(E1;1/2+→ 1/2−) strength
for11Be has been determined from intermediate energy
Coulomb excitation measurements [1]. In order to analyse
such kind of data in terms of electromagnetic matrix ele-
ments it is certainly necessary to use rather sophisticated
codes which take higher-order electromagnetic and nuclear
effects into account. These codes can be checked and val-
idated by comparing their results to limiting cases where
analytical results can be obtained, e.g., pure Coulomb ex-
citation that shows some rather simple features. This can
provide a useful guide for more sophisticated approaches.
It is the purpose of this paper to provide such analytical
Send offprint requests to: S.Typel
results. In the theoretical analysis of [1], quite a compli-
cated XCDCC method was used, which is required for
a quantitative description of the experimental data, but
it may tend to obscure the understanding of the simple
physical mechanism of Coulomb excitation.
archetype of a halo nucleus with a10Be core and a sin-
gle halo neutron in the 2s1/2state. There is a strong E1
transition to the 1/2−bound state, itself a p-wave neutron
halo state. This dipole transition was previously studied
by Coulomb excitation at GANIL, RIKEN, and MSU [2,
3,4].
There are two somewhat separated questions: the influ-
ence of nuclear excitation in grazing collisions and higher-
order electromagnetic effects in distant collisions where
nuclear interaction effects can safely be neglected. We deal
here with the second question: higher-order electromag-
netic excitation. This problem was studied for the11Be
case in the mid-nineties by two groups [5,6]. In view of
recent advances in the description of the electromagnetic
properties of halo nuclei, see e.g. [7,8,9,10], it seems ap-
propriate to update this work using the recent analytical
results for halo wave functions and present scaling laws
for Coulomb excitation of halo nuclei. Effective field the-
ory methods are also applied successfully to halo nuclei,
see [11,12].
In section 2 we give the main theoretical formulae for
our model of the Coulomb excitation of neutron halo nu-
clei from s- to p-states. In section 3 scaling rules are dis-
cussed. Then we give an application to the case of11Be.
This can serve as a benchmark for more involved studies
like XCDCC or time-dependent approaches [13]. Conclu-
11Be is an

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2S. Typel, G. Baur: Scaling laws and higher-order effects in Coulomb excitation of neutron halo nuclei
sions are given in section 4. A preliminary account of part
of this work was published in the proceedings of the con-
ference ’Nuclear Physics in Astrophysics III’ in Dresden,
March 2007 [14].
2 Analytically solvable model for Coulomb
excitation of neutron halo nuclei
A few low-energy parameters are sufficient to characterize
halo nuclei. Let us consider the single-particle excitation
of a neutron from a ground state i = 0 to a bound excited
state i = 1 with neutron separation energies Ei> 0. (For
11Be we have E0= 504 keV and E1= 184 keV.) The size
of the single-particle wave functions is determined by the
bound-state constants qi =√2µEi/¯ h with the reduced
mass µ = mnmc/(mn+ mc) of the nucleon+core system.
With the radius R of the core beyond which the nuclear
interaction is assumed to vanish, it is possible to introduce
dimensionless parameters
γi= qiR (1)
that are a measure for the ratio of the core size to the
size of the neutron wave functions. These parameters are
small for halo nuclei, see, e.g., [7,8], and can be used as
convenient expansion parameters. In this spirit, we use
this single-particle model for the jπ
1/2−states in order to evaluate the matrix element for
the E1 electromagnetic excitation. It is dominated by the
exterior contributions.
i = 1/2+and jπ
f=
2.1 Halo wave functions and matrix elements
The single-particle wave functions are given by
Φi(r) =fi(r)
r
Ylis
jimi(ˆ r) (2)
for the ground state (i = 0) and the excited state (i =
1). The angular dependence is described by the spinor
spherical harmonics Yls
and notation of [8] assuming zero spin of the core. The
radial wave functions of the two states are given in the
exterior region (r > R) by
jm(s = 1/2). We use the results
f0(r) = C0q0rh(1)
0(iq0r) = −C0exp(−q0r) (3)
and
f1(r) = C1iq1rh(1)
1(iq1r)(4)
= −C1exp(−q1r)
?
1 +
1
q1r
?
with Hankel functions h(1)
states are halo states and the normalization constants are
given in the halo limit by C0=√2q0and C1=
respectively [8].
l
of imaginary argument. Both
?2q2
1R/3,
The parameters q0and q1of the shallow bound states
in a halo nucleus are closely related to the scattering length
aland the effective range parameter rl for partial waves
l ≥ 0 in the effective range expansion k2l+1cot(δl) =
−1/al+rlk2/2+.... The S matrix Sl= [cot(δl)+i]/[cot(δl)−
i] has a pole at k = ikB, i.e. cot[δl(kB)] = i for a bound
state with kB= qi. This gives the desired relation
(−1)l+1q2l+1
l
= −1
al
−1
2rlq2
l+ ... . (5)
Now there is a difference between l = 0 and l > 0. For
l = 0 and small q0 we have in lowest order the relation
q0 = 1/a0 and the effective-range term is a small cor-
rection. For l > 0 the left hand side of (5) is smaller
than the first two terms individually on the right hand
side and we have an enhanced value (as compared to “di-
mensional considerations”) of the scattering length that
is given in lowest order by al= −2/(rlq2
In [7] the scattering length in the l = 1, j = 1/2 chan-
nel of11Be was determined to be 457(67,−66) fm3. With
q1= 0.0895 fm−1in this channel we find an effective-range
parameter r1= −0.547 fm−1of “natural order” R−1.
With the radial wave functions (3) and (4) we calculate
the B(E1) value for the 1/2+→ 1/2−-transition as well
as the higher-order effects in electromagnetic excitation.
We propose this to be a model study and leave the spec-
troscopic factors equal to one. (They could be adjusted,
which would result in a quasi-realistic description of the
11Be system for our purpose.) The B(E1) value is given
by
1
4π
l) (“fine tuned”).
B(E1) =
?
Z(1)
effe
?2???R(1)
01
???
2
,(6)
where Z(1)
charge number and the radial dipole integral is given by
eff
= Zcmn/(mn+ mc) is the dipole effective
R(1)
01=
?∞
0
?γ0
dr f∗
1(r)rf0(r)(7)
= 2
3
(γ0+ 2γ1)
(γ0+ γ1)2R .
In the present paper we are only interested in the halo
limit, i.e., we keep only the lowest-order term in the ex-
pansion in γ0 and γ1. In this approximation we can use
the exterior radial wave functions, eqs. (3) and (4) in the
integral eq. (7) down to r = 0. The correction terms from
using the correct interior wave functions are of higher or-
der in the expansion parameters γ0 and γ1. For R → 0
the radial dipole integral goes to zero because the normal-
ization of the p-wave function tends to zero in this limit.
Thus R must be kept finite; we choose R = 2.78 fm as
a realistic value for11Be [7]. This value determines the
asymptotic normalization of the p-wave bound state. We
find B(E1) = 0.193 e2fm2, to be compared to the value of
B(E1) = 0.105(12) e2fm2obtained from an analysis of the
GANIL data, see [1]. This value is consistent with other
Coulomb dissociation experiments at RIKEN and MSU
and the value obtained by the Doppler shift attenuation
method [15].

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S. Typel, G. Baur: Scaling laws and higher-order effects in Coulomb excitation of neutron halo nuclei3
2.2 Coulomb excitation of neutron halo nuclei and
scaling laws
We treat electromagnetic excitation in the semiclassical
approximation. For high beam energies the classical tra-
jectory can be taken as a straight line with impact param-
eter b. In the sudden approximation one can take higher-
order effects into account in a convenient way. Like the
related Glauber approximation it is applicable for high
beam energies and low excitation energies. This is reason-
ably well fulfilled for11Be, even at GANIL energies. In [5]
higher-order effects in the electromagnetic excitation of
11Be to the 1/2−bound state were studied using the sud-
den approximation. In [6] a continuum discretized coupled
channels approach was adopted. In [16] an analytically
solvable model for higher-order effects in the electromag-
netic dissociation of neutron halo nuclei was presented. In
that work, there was only the transition from an s-wave
bound state to the continuum. Now we consider the case
where there is, in addition, a p-wave bound state, as it is
the case in11Be. In the sudden approximation the dipole
excitation amplitude from the ground state i to the final
state f is given by
asudden= ?f|exp(−iqCoul· r)|i?
where the transfered momentum is
(8)
¯ hqCoul=2ZZ(1)
effe2
vb
ez. (9)
The impact parameter b is chosen to be in the z-direction;
this is convenient for the following calculation using polar
coordinates. The target charge number is denoted by Z
and v is the beam velocity. The dipole approximation is
quite well fulfilled, since the dipole effective charge Z(1)
is much larger than the corresponding quadrupole charge.
The neutron and core masses are denoted by mnand mc
respectively, the charge of the core is given by Zc. The sud-
den approximation is applicable for ξ ≡ ωb/v ≪ 1 , where
¯ hω = E0−E1is the excitation energy. Even for the com-
paratively low GANIL energies of about 40 MeV/nucleon
this is reasonably well fulfilled. The most important inter-
mediate states are expected to be in the low-energy con-
tinuum, where the dipole strength has a peak, at around
1 MeV excitation energy. Thus the adiabaticity condition
ξ ≪ 1 will also apply to the important intermediate states.
We note that for the virtual excitation of the high-lying
giant dipole state the opposite limit ξ ≫ 1 is realized, this
would lead to a real polarization potential, see e.g. [17]. It
influences the classical trajectory of the projectile and it
is not important in our context. It is also neglected in our
model space. The sudden approximation has the advan-
tage that intermediate states are treated by closure, thus
one only needs a model for the initial and final states,
and not for all the intermediate states. In lowest order
in qCoul the first-order dipole approximation is obtained
corresponding to a single-photon exchange between target
and projectile. It is shown in [5] that third-order E1 ex-
citation is more important than second-order E1-E2 ex-
citation. The role of higher multipole excitations in the
eff
electromagnetic excitation of one-neutron halo nuclei is
also studied in [18].
In the appendix we show that
asudden= (−1)ji+miC0C1
2i
?
1
qCoulI1(z) +1
q1I2(z)
?
(10)
where
I1(z) = 2
?
1 −arctan(z)
z
?
(11)
and
I2(z) =
?
1 +1
z2
?
arctan(z) −1
z
(12)
depending on the dimensionless parameter
z =
qCoul
q0+ q1
=qCoulR
γ0+ γ1
. (13)
This quantity is proportional to the usual strength param-
eter
χ(1)=Ze?f||M(E1)||i?
¯ hvb
(14)
that characterizes the importance of higher-order effects
in Coulomb excitation processes. We note that this param-
eter becomes large in the halo limit (γ → 0). In contrast
to χ(1), the adiabaticity parameter ξ = ¯ hω/(vb) with the
excitation energy ¯ hω = ¯ h2(q2
the halo limit and the application of the sudden approxi-
mation (8) is very well justified.
We can expand the excitation amplitude (10) in powers
of z with the help of the power series of the arctan func-
tion. This power series expansion converges for |z| ≤ 1
and one obtains
0−q2
1)/(2µ) becomes small in
asudden= a1+ a3+ ...(15)
with the lowest-order term
a1= (−1)ji+mi2
3i
?γ0
3
γ0+ 2γ1
γ0+ γ1
z (16)
and the next-to-leading order term
a3= (−1)ji+mi2i
15
?γ0
3
γ0+ 4γ1
γ0+ γ1
z3. (17)
The even terms vanish due to the parity selection rule.
Note that in the sudden approximation (8) there is no
change in the magnetic quantum number, since the vector
qCoulis in the z-direction.
Following Ref. [5] we can introduce a reduction factor
r(z) =|asudden|2
|a1|2
≈ 1 −2
5
γ0+ 4γ1
γ0+ 2γ1z2+ ... . (18)
In figure 1 the dependence of the reduction factor r on z is
shown for the present case with γ1/γ0=?E1/E0≈ 0.604
by the solid line. We find r(z) ≈ 1 − 0.619 z2in low-
est order in z. It seems of interest to compare the re-
sults of [5] with the present one. In [5] two different mod-
els (denoted by I and II) were used. Expanding in the

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4S. Typel, G. Baur: Scaling laws and higher-order effects in Coulomb excitation of neutron halo nuclei
0.0
0.5
1.0
z
1.5
2.0
0.0
0.2
0.4
0.6
0.8
1.0
r(z)
Fig. 1. Reduction factor r(z) = |asudden|2/|a1|2as a function
of the scaling parameter z as defined in eq. (13) for values of 0
(dashed line), 0.604 (solid line) and 1 (dotted line) of the ratio
γ1/γ0 =?
E1/E0.
strength parameter z we get rI(z) = 1 − 6z2/5 + ..., and
rII(z) = 1 − 12z2/5 + .... The present higher-order cor-
rections are smaller than the one found in [5]. In that ref-
erence rather crude models were used for the first excited
1/2−-state; those wave functions extended much further
out than the present wave function (4), which takes the
angular momentum barrier into account properly. In the
higher-order terms the importance of the outer region is
enhanced due to the weighting with a higher power of r. It
should be recalled that the main aim of [5] was to provide
an upper limit for higher-order effects in order to under-
stand the results of [2]. In the meantime, other Coulomb
excitation experiments were performed [1,3,4] that clari-
fied the situation.
In the general case of a Coulomb excitation from a
bound s-wave to a bound p-wave neutron halo state, the
energy E1 of the excited state is limited from below by
zero (state at the breakup threshold) and from above by
E0(zero-energy excitation). This corresponds to the inter-
vall [0,1] for the ratio γ1/γ0. The two limiting cases are
depicted in figure 1 by dashed and dotted lines.
The excitation probability is given by
P(b) =
1
2ji+ 1
?
mimf
δmimf|asudden|2
(19)
= PLO(b) + PNLO(b) + ... .
The lowest-order term
PLO(b) =4γ0(γ0+ 2γ1)2
27(γ0+ γ1)2z2≡A2
b2
(20)
in the excitation probability is proportional to z2. The
most important higher-order contribution comes from a3.
Its interference with the lowest-order term a1leads to the
next term
PNLO(b) = −8γ0(γ0+ 2γ1)(γ0+ 4γ1)
135(γ0+ γ1)2
z4≡ −A4
b4
(21)
in the expansion of P(b) in z, of the order of z4. The
constants A2and A4can be written as
A2=4γ0(γ0+ 2γ1)2
27(γ0+ γ1)4
?
2ηmn
mc+ mn
?2
R2
(22)
and
A4=8γ0(γ0+ 2γ1)(γ0+ 4γ1)
135(γ0+ γ1)6
?
2ηmn
mc+ mn
?4
R4
(23)
where the Coulomb parameter is given by η = ZZce2/(¯ hv).
Total cross sections are obtained by integration over the
impact parameter, starting from a minimum impact pa-
rameter bmin. The sudden approximation fails for large im-
pact parameters, and an adiabatic cut-off bmax= γbeamv/ω
with the Lorentz-factor by γbeam= 1/?1 − (v/c)2has to
be introduced for the lowest-order result. For the higher-
order terms this is not necessary, the convergence in b is
fast enough. We get
σLO= 2π
?bmax
bmin
PLO(b) b db = 2πA2lnbmax
bmin
(24)
and
σNLO= 2π
?∞
bmin
PNLO(b) b db = −πA4
b2
min
.(25)
(A somewhat different method was used in eqs. (16) and
(17) of [5], where the ξ-dependence of the first-order am-
plitude a1is taken into account. Then there is no need for
the adiabatic cutoff bmax.)
We note that the strength parameter z is proportional
to 1/v, i.e. the leading-order term decreases like 1/E, the
next-to-leading-order term like 1/E2, where E is the beam
energy. In the halo limit (γ ≡ γ0,γ1→ 0) the probabilities
scale as PLO∝ 1/γ and PNLO∝ 1/γ3(z scales as 1/γ) and
the excitation probability tends to infinity. However, the
NLO contribution tends to infinity even faster and higher-
order corrections become more important. Still, for realis-
tic values of γ the higher order effects are quite small (see
below). It seems intuitively understandable that higher-
order effects tend to increase with decreasing γ. In fact,
this increase with decreasing γ is faster than the increase
in lowest order, as our analysis shows.
For the total cross section there is the additional well
known enhancement lnγ in LO, which is absent in NLO,
due to the fast convergence in b.
In [16] and [19] the scaling properties of the Coulomb
dissociation of halo nuclei were investigated in a related
approach. The strength parameter defined there corre-
sponds to the one defined now. Using the present nota-
tion we found in that case that PLO∝ 1/γ3and PNLO∝
1/γ5. The difference to the present case arises because the
bound-continuum and bound-bound state dipole matrix
elements show different scaling properties with γi.
3 Application to the11Be Coulomb excitation
experiments
In the case of11Be we have neutron separation energies
of E0 = 504 keV and E1 = 184 keV for the ground

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S. Typel, G. Baur: Scaling laws and higher-order effects in Coulomb excitation of neutron halo nuclei5
Table 1. Projectile energy per nucleon E/A, velocity v, max-
imum impact parameter bmax, maximum parameter zmax and
cross section reduction factor R(v) for the11Be experiments.
E/A
[MeV]
v
[c]
bmax
[fm]
zmax
R(v)
GANIL [1]
GANIL [2]
MSU [4]
RIKEN [3]
38.6
43.0
60.0
64.0
0.2861
0.3009
0.3507
0.3611
184.1
194.6
230.9
238.8
0.6549
0.6227
0.5343
0.5189
0.8902
0.9025
0.9321
0.9366
and excited bound state. This leads to an excitation en-
ergy of ¯ hω = 320 keV for the 1/2−state. With a radius
R = 2.78 fm one obtains the numerical values γ0= 0.4116
and γ1= 0.2487, respectively, for the dimensional scaling
parameters that appear in the excitation amplitude. De-
pending on the projectile energy, the projectile velocities
v and the maximum impact parameters bmax= γbeamv/ω
are found for the conditions of the experiments at GANIL
[1,2], RIKEN [3] and MSU [4]. They are given in ta-
ble 1. These adiabatic cutoff radii are impressively large.
It means that Coulomb excitation extends really far out
and there is an amply large region where the interac-
tion is purely electromagnetic, and our approach works
very transparently. Of course, there is a Coulomb and nu-
clear interference zone, quite moderate in extension, close
to the minimum impact parameter bmin = 1.2(111/3+
2081/3) fm = 9.78 fm which is not considered in our ap-
proach.
The maximum value of the dimensionless strength pa-
rameter (13) is given by zmax= 2ZZ(1)
γ1)] with the target charge number Z = 82 and the effec-
tive charge number Z(1)
eff= 4/11. The numerical values for
zmaxare given again in table 1. They are smaller than one
and decrease with the projectile energy. The range where
the reduction factor is actually needed is well approxi-
mated by the NLO correction, which corresponds to the
approximation of r(z) in figure 1 by an inverted parabola.
In the present approach we are more certain about the
nuclear structure input that determines the opening pa-
rameter of this parabola.
Similar as in [5] we define a reduction factor R(v) =
σ(∞)/σLOof the total excitation cross section σ(∞)caused
by higher-order effects. This factor R(v) can be used to
take higher order effects into account in an analysis of the
experimental data.
Since the parameter z is always smaller than one in
the experiments, the full cross section in sudden approxi-
mation σ(∞)= σLO+σNLO+... is well approximated by
the NLO correction and the reduction factor becomes
effe2R/[¯ hvbmin(γ0+
R(v) ≈ 1 +σNLO
σLO
= 1 −
A4
minlnbmax
2A2b2
bmin
. (26)
The numerical values as given in table 1 show that the
reduction of the first-order cross section is in the order
of 6 to 11 % for the various experiments. In figure 2 the
dependence of the reduction on the projectile velocity v
0.00.2 0.4
0.6
0.8
1.0
v/c
0.6
0.7
0.8
0.9
1.0
R(v)
Fig. 2. Reduction factor R(v) = 1 + σNLO/σLO as a function
of the velocity v for values of 0 (dashed line), 0.604 (solid line)
and 1 (dotted line) of the ratio γ1/γ0 =?
E1/E0.
is depicted for the actual value of the ratio γ1/γ0and the
two limiting cases as in figure 1. The curves are shown
only for velocities where the parameter z does not exceed
the value one, the radius of convergence for the expansion
of the amplitude (15). Note that an increase of γ1 from
zero (dashed line) to one (dotted line) leads to a reduction
of the parameter z and correspondingly to a reduction of
the higher-order corrections in the cross section for the
same velocity v.
4 Conclusion
We have presented a realistic model for higher-order ef-
fects in the Coulomb excitation of neutron halo nuclei.
With quite simple methods a reliable value for the re-
duction of the cross section could be given. In addition,
this should be very useful as a benchmark, for valuable
and necessary tests of more sophisticated approaches like
CDCC or time-dependent calculations. These kinds of cal-
culations require a lot of computation, yet they should
reproduce the present results in certain well defined lim-
its. Using an effective-range point of view we updated our
previous work [5]. Other angular momentum combinations
can be treated in a similar way. However, for higher an-
gular momenta the halo nature is less pronounced and
the importance of the core size parameter R increases.
Higher-order terms in γ will become more important and
the method less useful. It is quite remarkable that our
results for11Be Coulomb excitation depend only on the
binding energies (they determine the parameters q0 and
q1) and a parameter R which characterizes the core size.
This stresses the fact that halo nuclei depend only on a
few low-energy constants, and not on details of the shape
of the potential. This is in accord with low-energy scat-
tering, which is also characterized by a few low-energy
parameters in the effective-range expansion.
In view of the discovery of the neutron-rich isotopes
40Mg and42Al [20,21] one may expect that more neutron-