Properties of the first 1/2+ state in 9Be from electron scattering and astrophysical implications

Abstract

Spectra of the 9Be(e,e') reaction have been measured at the S-DALINAC at an electron energy E0 = 73 MeV and scattering angles of 93° and 141° with high energy resolution up to excitation energies Ex = 8 MeV. The astrophysically relevant 9Be(gamma,n) cross sections have been extracted from the (e,e') data. The resonance parameters of the first 1/2+ state in 9Be have been derived in a one-level R-matrix approximation resulting in a resonance energy ER = 1.748(6) MeV and a resonance width GammaR = 274(8) keV These parameters are in mutual agreement with those obtained from the latest real photon experiments but with improved uncertainties.

Full text

PHYSICAL REVIEW C 82, 015808 (2010)
Resonance parameters of the first 1/2+state in 9Be and astrophysical implications
O. Burda and P. von Neumann-Cosel*
Institut f¨
ur Kernphysik, Technische Universit¨
at Darmstadt, D-64289 Darmstadt, Germany
A. Richter
Institut f¨
ur Kernphysik, Technische Universit¨
at Darmstadt, D-64289 Darmstadt, Germany and European Centre for Theoretical Studies in
Nuclear Physics and Related Areas (ECT∗), Villa Tambosi, I-38123 Villazzano (Trento), Italy
C. Forss´
en
Fundamental Physics, Chalmers University of Technology, SE-412 96 G¨
oteborg, Sweden
B. A. Brown
Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University,
East Lansing, Michigan 48824-1321, USA
(Received 8 June 2010; published 19 July 2010)
Spectra of the 9Be(e,e) reaction have been measured at the Superconducting Darmstadt Electron Linear
Accelerator at an electron energy of E0=73 MeV and scattering angles of 93◦and 141◦with high-energy
resolution up to excitation energies of Ex=8 MeV. The astrophysically relevant resonance parameters of the
first excited 1/2+state of 9Be have been extracted in a one-level approximation of R-matrix theory, resulting in
resonance energy ER=1.748(6) MeV and width R=274(8) keV, which are in good agreement with the latest
9Be(γ,n) experiment but with considerably improved uncertainties. However, the reduced B(E1) transition
strength deduced from an extrapolation of the (e,e) data to the photon point is smaller by a factor of two.
Implications of the new results for possible production of 12C in neutron-rich astrophysical scenarios are discussed.
DOI: 10.1103/PhysRevC.82.015808 PACS number(s): 25.30.Dh, 27.20.+n, 26.30.Ef, 26.20.Kn
I. INTRODUCTION
The nucleus 9Be is a loosely bound system formed by two α
particles and a neutron in which no two constituents alone can
form a bound system. It has the lowest neutron threshold, Sn=
1.6654 MeV, of all stable nuclei. Already the first excited state
is some tens of kilo-electron-volts above the neutron threshold,
and thus all excited states are unstable with respect to neutron
decay.
The properties of the first excited state are of particular
interest because they determine the importance of 9Be pro-
duction for the synthesis of 12C seed material, triggering the
rprocess in type II supernovae [1–4]. In stellar burning, the
triple-αprocess dominates the production of 12C, where at
sufficiently high temperatures a small equilibrium amount
of the short-lived 8Be is formed, which can capture the
third αparticle and form 12C. In explosive nucleosynthe-
sis, such as a core-collapse supernova, the reaction path
8Be(n,γ )9Be(α,n)12C may provide an alternative route for
building up heavy elements.
A direct measurement of the cross sections of the
8Be(n,γ )9Be reaction is impossible because of the short
lifetime, about 10−16 s, of the 8Be ground state (g.s.), but
they can be deduced from photodisintegration cross sections
on 9Be using the principle of detailed balance. At low energies,
the photodisintegration cross section is dominated by the
properties of the 1/2+resonance just above the 8Be +n
*vnc@ikp.tu-darmstadt.de
threshold in 9Be. The description of this unbound level (viz.,
its resonance energy and width) is a long-standing problem.
Because of its closeness to the neutron threshold, the resonance
has a strongly asymmetric line shape.
Several experiments have investigated the 9Be(γ,n)re-
action, either with real photons from bremsstrahlung or
from laser-induced Compton backscattering, and with virtual
photons from electron scattering (see Ref. [5] for a discussion
and references). Despite the sizable body of data, there still
exist considerable uncertainties of the resonance parameters.
Utsunomiya et al. [6] measured the photoneutron cross section
for 9Be with real photons in the whole energy range of
astrophysical relevance. The deduced resonance parameters
for the 1/2+state are shown in Table Iin comparison
with results from earlier electron-scattering experiments [7,8].
Another recent result from 9Li βdecay [ER=1.689(10) MeV,
R=224(7) keV] is quoted in Ref. [9], but it is not clear
whether these values refer to the true resonance parameters or
the peak and full width at half maximum (FWHM). Further-
more, the results have been questioned in [10]. Table Iprovides
a summary of resonance parameters deduced from the various
experiments.
Obviously, there are significant differences between the
results obtained from photonuclear and electron-scattering
experiments. The discrepancy in the B(E1) transition strength
amounts to a factor of about two when comparing with
the results of (e,e) measurements [7,12] at low momentum
transfers q, whereas it is reduced by ∼30% in an (e,e)
experiment [8] at larger q. The reason for these discrepancies
0556-2813/2010/82(1)/015808(7) 015808-1 ©2010 The American Physical Society

O. BURDA et al. PHYSICAL REVIEW C 82, 015808 (2010)
TABLE I. Summary of resonance parameters and reduced tran-
sition probability of the 1/2+state in 9Be deduced from different
experiments. Reference [11] contains a reanalysis of the data of [7].
Reaction Ref. ER(MeV) R(keV) B(E1)↑(e2fm2)
(e,e)[7]1.684(7) 217(10) 0.027(2)
(e,e)[8]1.68(15) 200(20) 0.034(3)
(γ,n)[6]1.750(10) 283(42) 0.0535(35)
(e,e)[11]1.732 270 0.0685
βdecay [9]1.689(10) 224(7) –
(e,e) Present work 1.748(6) 274(8) 0.027(2)
between the strengths deduced from the real photon and virtual
photon experiments is unknown. Furthermore, Barker [11]
reanalyzed the (e,e) data of [7] and extracted resonance
parameters that differ considerably from those quoted in the
original paper (see Table I).
To resolve these discrepancies, high-resolution measure-
ments of the 9Be(e,e) reaction were performed at the
Superconducting Darmstadt Electron Linear Accelerator
(S-DALINAC), and new resonance parameters for the 1/2+
state were extracted. Furthermore, an independent reanalysis
of the electron-scattering data of [7] was performed. Finally,
the temperature dependence of the 8Be(n,γ )9Be reaction rate
including the new results was derived and compared to that
of the European Compilation of Reaction Cross Sections for
Astrophysics (NACRE) compilation [13], which serves as
standard for most network calculations.
II. EXPERIMENT
The 9Be(e,e) experiment was carried out using the high-
resolution 169◦magnetic spectrometer of S-DALINAC. Data
were taken at an incident electron beam energy of E0=
73 MeV and scattering angles of Lab =93◦and 141◦with
typical beam currents of 2 µA. For the measurements, a self-
supporting 9Be target with an areal density of 5.55 mg/cm2
was used. The properties of the spectrometer are described
in Ref. [14]. A new focal plane detector system based on
silicon microstrip detectors was recently implemented [15].
In dispersion-matching mode, an energy resolution of E 
30 keV (FWHM) was achieved in the measurements.
Figure 1presents the spectra of the 9Be(e,e) reaction
measured up to an excitation energy of about 8 MeV. There is
only one narrow peak visible in the spectra, which corresponds
to the excitation of the Jπ=5/2−state at Ex=2.429 MeV.
The tiny peak at about 4.2 MeV in both spectra corresponds
to the first excited state in 12C. (The deviation between the
observed and the true excitation energy Ex=4.439 MeV
stems from the difference of the recoil correction for nuclei
with mass A=9 and 12, respectively.) The bumps around
5.1 MeV in the top spectrum and around 8.1 MeV in the
bottom spectrum are due to elastic scattering on hydrogen.
The broad bump between 6 and 7 MeV results from the
overlap of resonances at Ex=6.38 MeV (Jπ=7/2−) and
Ex=6.76 MeV (Jπ=9/2+)in9Be [8]. The asymmetric
line shape of the 1/2+state at Ex≈1.7 MeV (marked
02468
0
3
6
0
12
24
36
Excitation Ener
g
y (MeV)
Counts / mC
9Be(e,e´)
E = 73 MeV
= 93
0
Lab
Θ°
Θ
lab
= 141
°
(b)
(a)
FIG. 1. Spectra of the 9Be(e,e) reaction at E0=73 MeV and
Lab =93◦(top) and 141◦(bottom) and their decomposition. Solid
lines: Fits to experimentally known resonances. Dashed lines: Radia-
tive tail from elastic scattering. The arrows indicate the transition to
the first excited state, whose asymmetric line shape is clearly visible.
by arrows in Fig. 1) is already clearly visible in the raw
spectra.
In the decomposition of the spectra, the line shape of the
narrow state was described by the function given in Ref. [16],
whose parameters were determined by a fit to the elastic line.
This also determines the background from the radiative tail of
the elastic line in the region of interest indicated by the dashed
lines in Fig. 1. The line shapes of the broad resonances were
assumed to correspond to an energy-dependent Breit-Wigner
function. The energies and widths of the resonances (taken
from the latest compilation [5]) were kept fixed during the fit
except for the parameters of the first excited state. Finally, the
1/2+resonance was treated in a one-level R-matrix formalism
as explained in the next section.
III. ANALYSIS
Because the state of interest lies above the neutron thresh-
old, we first discuss an extraction of the relevant parameters as
if it was excited in a (γ,n) reaction. In Sec. III B, the relation
to the (e,e) data is explained. Finally, Sec. III C describes the
extraction of the resonance parameters.
A. One-level R-matrix approximation
The contribution to the (γ,n) cross section from an isolated
level of spin Jlocated near threshold in the one-level
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RESONANCE PARAMETERS OF THE FIRST 1/2+... PHYSICAL REVIEW C 82, 015808 (2010)
approximation of R-matrix theory [17] is given by
σγ,n(Eγ)=π
2k2
γ
2J+1
2I+1
γn
(Eγ−Eλ−(E))2+2
4
,(1)
where kγ=Eγ/¯hc stands for the photon wave number, Iis
g.s. spin, γis the g.s. radiative width, nis the neutron
decay width, the total decay width is =γ+n, and Eλ
corresponds to the energy eigenvalue. The level shift (E)is
given by
(E)=−γ2[S(E)−B],(2)
with the reduced width γ2, the shift factor S(E), and the
boundary condition parameter B(see Ref. [17]).
Then for a 1/2+level in 9Be excited by E1γradiation
and decaying by s-wave neutrons, and for an energy E=
Eγ−Sn>0, one has
γ=16π
9e2k3
γB(E1,k)↓,(3)
n=2(Eγ−Sn),(4)
with kγ=Ex/¯hc being the photon momentum transfer (called
photon point), B(E1,k)↓being the reduced transition strength
at the photon point for the decay, =2µa2γ4/¯h2>0, where
µand aare the reduced mass and the 8Be +nchannel radius,
respectively, and Sn(9Be) =1.6654 MeV [5] being the neutron
threshold energy. The boundary condition parameter Bis taken
to be zero and the shift factor S(E)=0fors-wave neutrons
[17], and thus (E)=0.
Because nγ, the total resonance width ≈n, and
the energy dependence of the photoabsorption cross section of
Eq. (1) is reduced to
σγ,n(Eγ)=16π2
9
e2
¯hc
2J+1
2I+1B(E1,k)↓
×Eγ(Eγ−Sn)
(Eγ−ER)2+(Eγ−Sn).(5)
The resonance energy ERis calculated from
ER=Eλ+, (6)
and the resonance width using Eq. (4)is
R(ER)=2(ER−Sn).(7)
It should be noted that because of the asymmetric line shape,
the resonance energy ERdoes not coincide with the excitation
energy at the maximum of the cross section, and the resonance
width Rdiffers from the FWHM.
B. Extraction of equivalent (γ,n) cross sections and B(E1)
transition strength from the (e,e)data
Equation (5) holds also for the relation between the
(e,e) cross sections and the reduced transition strength if
B(E1,k) is replaced by the corresponding value at finite
momentum transfer B(E1,q). (Note that in the following
B(E1,q)↑= (2Jf+1)/(2Ji+1)B(E1,q)↓is given, where
Ji,f denote the spins of initial and final state, respectively.)
If interference with transitions to higher lying 1/2+res-
onances can be neglected, the equivalent σγ,n cross sec-
NCSM
(d /d )/σΩ(d /d )σΩ
Mott
10-4
10-3
0.2 0.4 0.6 0.8
q (fm )
-1
9Be(e,e )
E = 1.748 MeV
S-DALINAC
´
x
SM
FIG. 2. Ratio of the measured cross sections to the Mott cross
section of the transition to the 1/2+state in 9Be as a function of
momentum transfer. Data are from Ref. [12] (triangles), Ref. [7]
(circles), and the present work (squares). Solid and dashed lines are
theoretical predictions of the shell-model (SM) and no-core shell-
model (NCSM) calculations described in the text normalized to the
data.
tions can be determined from the electron-scattering results
by extrapolating the reduced transition strength B(E1,q)
measured at finite momentum transfer qto the photon
point k=Ex/¯hc.
Figure 2presents the momentum-transfer dependence of
the measured (e,e) cross sections normalized to the Mott cross
section for the transition to the first 1/2+state in 9Be. Besides
the data from the present work, displayed as squares, results
of previous experiments at comparable momentum transfers,
shown as triangles (Ref. [12]) and circles (Ref. [7]), are
included. In first-order perturbation theory, inclusive electron-
scattering cross sections factorize in a longitudinal (C) and a
transverse (E) part, reflecting the respective polarization of the
exchanged virtual photon. The kinematics of the data shown in
Fig. 2favor longitudinal excitation, and thus B(C1,q) rather
than B(E1,q) is determined. Both quantities can be related by
Siegert’s theorem B(E1,q)=(k/q)2B(C1,q); that is, they
should be equal at the photon point q=k.
There are two methods to perform the extrapolation from
finite momentum transfer to the photon point: (i) based
on microscopic model calculations or (ii) the plane-wave
Born approximation (PWBA) for a nearly model-independent
extraction. The latter method is valid only at small momentum
transfers (q<1fm
−1) and for small atomic numbers Z
(αZ 1).
For an application of the first method, shell-model (SM)
calculations of the electroexcitation of the 1/2+state were
performed with the interaction of [18] coupling 1pand
2s1dshells. The formalism for calculating electron-scattering
form factors from the SM one-body transition densities is
described in Ref. [19]. A similar calculation of an E1
longitudinal form factor for a transition in 12C is de-
scribed in Ref. [20]. Spurious states are removed with the
Gloeckner-Lawson method [21]. The SM one-body transition
density is dominated by the 0p1/2→1s1/2neutron transi-
tion. For these two orbitals, we used Hartree-Fock radial
wave functions obtained with the SKX Skyrme interaction
[22] with their separation energies constrained to be 1.665
and 0.2 MeV, respectively. Harmonic oscillator (HO) radial
015808-3

O. BURDA et al. PHYSICAL REVIEW C 82, 015808 (2010)
wave functions were used for all other orbitals. The result
normalized to the data is shown in Fig. 2as a dashed
line.
Alternatively, a no-core shell-model (NCSM) calculation
was performed in the framework of the model described in
Ref. [23] (solid line in Fig. 2). This calculation utilized the
realistic nucleon-nucleon interaction CD-Bonn 2000 using
very large model spaces, namely 8(9) ¯hω for the 3/2−(1/2+)
state, and an HO frequency of 12 MeV. Despite the large
model spaces and improved convergence techniques [24], no
convergence was achieved for the wave function of the 9Be,
1/2+state. One should note that these calculations treat the
1/2+state in a quasibound approximation.
The two calculations predict a very similar momentum-
transfer dependence that describes the data well. However,
the absolute magnitudes are underpredicted by factors of
3.6 (SM) and 1.7 (NCSM), respectively. By normalizing the
theoretical predictions [B(E1,k)↑= 0.008 e2fm2(SM) and
0.016 e2fm2(NCSM), respectively] to the experimental data,
one finds B(E1,k)↑=0.027(2) e2fm2using the NCSM and
B(E1,k)↑=0.029(2) e2fm2using the SM form factors. Both
results agree with each other within error bars.
An alternative independent method to derive the E1 transi-
tion strength is based on a PWBA analysis (see, e.g., Ref. [25]).
At low momentum transfers, the form factor can be expanded
in a power series of q
B(E1,q)=B(E1,0) 1−R2
trq2
10 +R4
trq4
280 −···,
(8)
where higher powers of qare negligible in the momentum
transfer range studied in the present experiment. The so-called
transition radius Rtr is given by R2
tr =rλ+2tr/rλtr , where
rλtr denotes the moments of the transition density
rλtr =4πρtrrλ+2dr. (9)
An additional assumption is made that R4
tr can be parame-
terized in the form R4
tr =a(R2
tr)2, where the parameter ais
determined using theoretical transition densities.
Because relation (8) holds in the plane wave limit only,
distorted wave Born approximation (DWBA) correction fac-
tors have been calculated based on the NCSM results to
convert the measured cross sections into equivalent PWBA
cross sections. Corrections on the order of 10% are obtained.
Figure 3presents the corrected data as a function of the squared
momentum transfer. The solid line shows a fit of Eq. (8) with
parameters √B(C1,0) =0.164(12) efm and Rtr =2.9(3) fm.
Extrapolation of the transition strength to the photon point
yields B(E1,k)↑=0.027(4) e2fm2, in agreement with the
results obtained from the analysis based on microscopic form
factors.
A significant difference to the corresponding B(E1,k)↑
strength deduced from the real-photon experiment is observed,
which finds 0.0535(35) e2fm2(cf. Table I), larger than the
present result by about a factor of two. This implies a
severe violation of Siegert’s theorem. Its origin is presently
unclear, but possible explanations could lie in the quasibound
0.1 0.3 0.5
q (fm )
2-2
0.0
0.2
0.3
0.1
B(C1,q) (e fm)
9Be(e,e )
E = 1.748 MeV
S-DALINAC
´
x
FIG. 3. Ratio of the measured cross sections of the transition to
the 1/2+state in 9Be to the Mott cross sections as a function of the
squared momentum transfer. The solid line is a fit of Eq. (8) with
parameters √B(C1,0) =0.164(12) efm and Rtr =2.9(3) fm.
approximation used in the SM calculations and/or a need
to modify the E1 operator. A detailed discussion of this
interesting problem is postponed to a future publication.
C. Resonance parameters
Figure 4shows the photoneutron cross sections of the first
excited state in 9Be extracted from the present work (top and
middle) together with the previous (bottom) result of Ref. [7].
The data are summed in 15-keV bins. All three data sets are in
good agreement with each other.
Because all three measurements shown in Fig. 4were
independent, the data can be averaged. The resulting averaged
(γ,n) cross sections are presented in the upper part of Fig. 5.
The solid line is a fit with Eq. (5). To account for the detector
response, the theoretical form is folded with the experimental
resolution function. Because the experimental resolution was
much smaller than the resonance width, the influence of the
resolution function is small except for energies around the
0.0
σ
γn
(mb)
0.5
1.0
0.0
0.5
1.0
0.5
1.0
0.0
1.5 1.8 2.1 2.4
9Be(e,e )
E = 73 MeV
= 93
´
°
0
Lab
Θ
E = 73 MeV
= 141
0
Lab
Θ°
E = 49 MeV
= 117
0
Lab
Θ°
E(MeV)
γ
(a)
(b)
(c)
FIG. 4. Photoneutron cross sections extracted from the present
(top and middle) and older (bottom) (e,e)data[7].
015808-4

RESONANCE PARAMETERS OF THE FIRST 1/2+... PHYSICAL REVIEW C 82, 015808 (2010)
0.0
σγn(mb)
0.7
1.4 9Be(e,e )´
1.7 1.9 2.1
E (MeV)
γ
0.0
0.7
1.4 9Be( ,n)γ
(a)
(b)
FIG. 5. Averaged photoneutron cross sections extracted from the
(e,e) data shown in Fig. 4in comparison with the cross sections
extracted from the latest 9Be(γ,n) experiments [6]. The solid lines
are the corresponding fits with Eq. (5) with the parameters given in
the text.
maximum of the cross sections. The fit results in resonance
energy ER=1.748(6) MeV and width R=274(8) keV in
contradiction to the results of Ref. [7] but in agreement with the
reanalysis of Ref. [11]. In fact, because the data of [7] are very
close to those of the present work (cf. Fig. 4), an independent
reanalysis yields resonance parameters very similar to the ones
from the new data. The most likely explanation for the values
given in Ref. [7] is that the maximum energy and FWHM
instead of the true resonance parameters were quoted. The
final results are included in Table I.
The lower part of Fig. 5shows the measured 9Be(γ,n)
cross sections of Ref. [6]. Application of Eq. (5) leads to
comparable resonance parameters ER=1.750(10) MeV and
R=283(42) keV, but the present work provides values with
considerably improved uncertainties.
IV. ASTROPHYSICAL IMPLICATIONS
To calculate the thermonuclear reaction rate of α(αn,γ )9Be
in a wide range of temperatures, we numerically integrate the
thermal average of cross sections N2
Aσv(as defined, e.g.,
in Ref. [13]), assuming two-step formation of 9Be through a
metastable 8Be. The formation through 5He followed by an α
capture is generally neglected because of the short lifetime of
5He except for the work of Ref. [26], which indicates relevance
of this channel at TT9(see, however, the criticism in
10-3 10-2 10-1 100101
T9
0.00
0.25
0.50
0.75
1.00
Rate(i) / Rate(total)
FIG. 6. Contributions of the lowest-lying states (i) in 9Be to the
α(αn,γ)9Be reaction rate.
Ref. [27]). The same formulation to the 9Be formation is
also used in the NACRE compilation [13]. Resonant and
nonresonant contributions from the α+α→8Be reaction are
taken into account. The g.s. of 8Be is described by a resonance
energy ER=0.0918 MeV with respect to the α+αthreshold
and a width of α=5.57(25) eV taken from Ref. [5]. Elastic
cross sections of αα scattering were treated as described in
Ref. [28].
The resonance properties (energy, γ, and neutron decay
widths) of the lowest excited states in 9Be with the corre-
sponding g.s. branching ratios fincluded into the calculation
of the α(αn,γ )9Be reaction rate are summarized in Table II.
An energy dependence of the partial decay widths was
taken into account only for the 1/2+resonance. Reaction
rates calculated at representative temperatures are given in
Table III.
Figure 6shows the individual contributions of the excited
states considered in Table II to the total reaction rate as a
function of temperature. The 1/2+state (solid line) dominates
in the temperature range T9=0.04 −3. The role of the 5/2−
state (dotted line) is negligibly small at all temperatures.
At values of T9<0.04, the low-energy tails of the broad
1/2−(dashed line) and 5/2+(dashed-dotted line) resonances
become increasingly important. Temperatures in supernova II
scenarios reach values well above T9. Under these conditions,
the maximum of the photon spectrum is shifted to energies
above the 1/2+state, and the 5/2+state starts to dominate
when approaching T9=10.
The ratio of the present reaction rates to the latest NACRE
compilation [13] is shown in Fig. 7. Deviations ranges from
TABLE II. Low-lying states in 9Be considered in the calculations of the α(αn,γ)9Be reaction rate.
The quantity fdenotes the branching ratio of the corresponding state into the n+8Be decay channel.
JπER(MeV) γ(eV) n(MeV) f(%) Ref.
1/2+1.748(6) 0.302(45) 0.274(8) 100 Present work
5/2−2.4294(13) 0.089(10) 0.78(13) 7(1) [5]
1/2−2.78(12) 0.45(36) 1.08(11) 100 [5,13]
5/2+3.049(9) 0.90(45) 0.282(110) 87(13) [5,13]
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O. BURDA et al. PHYSICAL REVIEW C 82, 015808 (2010)
TABLE III. The thermonuclear reaction rate N2
Aσvof
α(αn,γ)9Be at representative temperatures.
T9Rate T9Rate T9Rate
0.001 4.67 ×10−59 0.04 7.53 ×10−16 0.5 3.93 ×10−07
0.002 2.82 ×10−47 0.05 1.07 ×10−13 0.6 3.91 ×10−07
0.003 1.45 ×10−41 0.06 2.74 ×10−12 0.7 3.70 ×10−07
0.004 5.77 ×10−38 0.07 2.68 ×10−11 0.8 3.41 ×10−07
0.005 2.11 ×10−35 0.08 1.43 ×10−10 0.9 3.11 ×10−07
0.006 1.90 ×10−33 0.09 5.17 ×10−10 1 2.81 ×10−07
0.007 6.97 ×10−32 0.1 1.41 ×10−09 1.25 2.18 ×10−07
0.008 1.36 ×10−30 0.11 3.17 ×10−09 1.5 1.71 ×10−07
0.009 1.69 ×10−29 0.12 6.12 ×10−09 1.75 1.37 ×10−07
0.01 1.49 ×10−28 0.13 1.06 ×10−08 2 1.12 ×10−07
0.011 9.96 ×10−28 0.14 1.67 ×10−08 2.5 7.90 ×10−08
0.012 5.38 ×10−27 0.15 2.46 ×10−08 3 6.00 ×10−08
0.013 2.44 ×10−26 0.16 3.43 ×10−08 3.5 4.81 ×10−08
0.014 9.60 ×10−26 0.18 5.86 ×10−08 4 4.00 ×10−08
0.015 3.34 ×10−25 0.2 8.79 ×10−08 5 2.97 ×10−08
0.016 1.05 ×10−24 0.25 1.71 ×10−07 6 2.32 ×10−08
0.018 7.98 ×10−24 0.3 2.50 ×10−07 7 1.87 ×10−08
0.02 4.65 ×10−23 0.35 3.11 ×10−07 8 1.53 ×10−08
0.025 1.86 ×10−21 0.4 3.54 ×10−07 9 1.27 ×10−08
0.03 1.97 ×10−19 0.45 3.80 ×10−07 10 1.06 ×10−08
+20% to −60% depending on the temperature. Besides using
the improved resonance parameters of the 1/2+state, there
are some differences between the present calculation and the
one described in Ref. [13]. The 5/2−state is neglected in the
latter case. However, as can be seen in Fig. 6, its contributions
are very small. Also the 8Be g.s. parameters taken from [5]
differ from those used in Ref. [13]. The pronounced kink at
T9=0.03 in Fig. 7marks the onset of resonant contributions in
the α+α→8Be cross sections. Rates from a semimicroscopic
three-body model [29] are also available for temperatures
0.2T95. These are typically about 20% larger than the
NACRE results.
The difference observed for the γdecay width of the
1/2+resonance between the measurements of [6] and the
present work have a non-negligible impact on the reaction
rates. In general, taking a larger γthe contribution of the
Present / NACRE
0.4
0.8
1.2
10-3 10-2 10-1 100101
T9
FIG. 7. The ratio of the present rate to the latest NACRE
compilation [13]. Deviations ranges from +25% to −60% depending
on the temperature.
1/2+resonance will increase reducing the deviations from the
NACRE result at high temperatures. It should also be noted
that nonresonant contributions to the 8Be(n,γ )9Be neglected
in both approaches discussed previously may be relevant [30].
The calculations described in Refs. [30,31] suggest sizable
effects while Ref. [26] finds it to be of minor importance.
Finally, there is a recent claim [32] that the picture of a
sequential formation is incorrect for the near-threshold 1/2+
state and that it should be described as a genuine three-body
process [33]. This would modify the resonance parameters
considerably.
V. CONCLUDING REMARKS
The astrophysically relevant 9Be(γ,n) cross sections have
been extracted from 9Be(e,e) data. The resonance parameters
of the first excited 1/2+state in 9Be are derived in a
one-level R-matrix approximation. The resonance parameters
averaged over all available (e,e) data are ER=1.748(6) keV
and R=274(8) keV, which are in agreement with the
latest direct (γ,n) experiment [6] but with much improved
uncertainties. However, the deduced γdecay width is smaller
by about a factor of two. Rates for the temperature-dependent
formation of 9Be under stellar conditions are given. They
differ significantly from the values adopted in the NACRE
compilation [13]. Further improvements of the reaction rate
require the inclusion of direct capture reactions.
The difference in the B(E1) transition strength obtained
from electron- and photon-induced reaction presents an in-
triguing problem. Because the present result is extracted from
the longitudinal form factor, it might indicate a violation of
Siegert’s theorem at the photon point. A similar problem
was observed in the electroexcitation of 1−levels in 12C
[20], 16O[34,35], and 40Ca [36]. There, isospin mixing was
offered as an explanation leading to modified form factors
of longitudinal and transverse electron scattering at small
momentum transfers. Another explanation could be the need
for a modification of the E1 operator due to meson-exchange
currents. A detailed study of the weak transverse form factor
of the transition to the 1/2+resonance in 9Be would be highly
desirable to clarify the origin of the discrepancy.
The SM calculations seem to describe the momentum
transfer dependence of the electron-scattering data for the
measured qrange but fall short of the experimental transition
strength. One possible explanation may be the quasibound
approximation applied in the description of the 1/2+state.
Near-threshold α-cluster states are expected to have an
increased size, which amplifies the dependence on tails of
the wave function like, for example, that observed for the case
of the Hoyle state in 12C[37,38]. Calculations with improved
radial wave function would be important. Also, the role of
direct three-body decay needs to be further explored.
ACKNOWLEDGMENTS
We thank H.-D. Gr¨
af and the S-DALINAC team for
preparing excellent beams and M. Chernykh for help in
collecting data. The experiment originated from a discussion
of AR with the late Fred Barker on an inconsistency of the
analysis of the data in Ref. [7], and we are grateful for his
015808-6

RESONANCE PARAMETERS OF THE FIRST 1/2+... PHYSICAL REVIEW C 82, 015808 (2010)
advice. Discussions with A. S. Jensen, G. Mart´
ınez-Pinedo,
A. Mengoni, and S. Typel are gratefully acknowledged. This
work has been supported by the DFG under Contract SFB 634
and by the NSF under Grant PHY-0758099. CF acknowledges
financial support from the Swedish Research Council and the
European Research Council under the FP7.
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