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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

015808-2

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

γn

(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 nγ, 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+2tr/rλtr , where

rλtr denotes the moments of the transition density

rλtr =4πρtrrλ+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

Aσv(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 TT9(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]

015808-5

O. BURDA et al. PHYSICAL REVIEW C 82, 015808 (2010)

TABLE III. The thermonuclear reaction rate N2

Aσvof

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

0.2T95. 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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