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Electron scattering on the Hoyle state and

carbon production in stars

M. Chernykh

∗

, H.P. Blok

†

, H. Feldmeier

∗∗

, T. Neff

∗∗

,

P. von Neumann-Cosel

∗

and A. Richter

∗

∗

Institut für Kernphysik, Technische Universität Darmstadt, Germany

†

Department of Physics and Astronomy, Vrije Universiteit, Amsterdam, The Netherlands

∗∗

Gesellschaft für Schwerionenforschung (GSI), Darmstadt, Germany

Abstract. High-resolution inelastic electron scattering experiments were performed at the

S-DALINAC for precise determination of the partial pair width Γ

π

of the second J

π

= 0

+

state,

the so-called Hoyle state, in

12

C. Results for the monopole matrix element (directly related to Γ

π

)

from a nearly model-independent analysis based on an extrapolation of low-q data to zero momen-

tum transfer are presented. Additionally, a Fourier-Bessel analysis of the transition form factor is

discussed. The combined result of both methods leads to a partial pair width Γ

π

= 62.2(10)

µ

eV.

Keywords: 3

α

process,

α

clustering, electron scattering, Hoyle state

PACS: 21.10.Tg, 23.20.Ra, 25.30.Dh, 26.20.Fj, 27.20.+n

INTRODUCTION

The production of the element carbon is a key reaction of stellar nucleosynthesis. The

creation of its most abundant isotope

12

C is the result of the triple-alpha process [1],

which passes through the J

π

= 0

+

resonance at an excitation energy E

x

= 7.65 MeV in

12

C. The state is called Hoyle state as it was postulated [2] by the astrophysicist Fred

Hoyle to explain the observed abundance of carbon in the Universe. The Hoyle state has

unusual structure considered as a pure example of an

α

cluster condensate [3, 4].

Despite itsastrophysical relevance, the carbon production rate through the triple-alpha

process is known with insufﬁcient precision only [5, 6]. For temperatures above 10

8

K

the reaction rate r

3

α

can be written as

r

3

α

∝ Γ

rad

exp

−

Q

3

α

kT

, (1)

where Γ

rad

is the radiative width of the Hoyle state, k the Boltzmann constant, T the

temperature, and Q

3

α

the energy released in the decay of the Hoyle state into three

alpha particles.

As seen from Eq. (1), for a precise determination of r

3

α

only Γ

rad

and Q

3

α

need

to be known. The released energy Q

3

α

= 379.38 keV is measured with a precision

±0.20 keV (see Ref. [7] and references therein). The uncertainty of the 3

α

process

due to this factor is very small and corresponds to only ±1.2% at 2×10

8

K, decreasing

as 1/T with increasing temperature. Consequently, the main uncertainty in r

3

α

is due to

the uncertainty in Γ

rad

which, however, cannot be measured directly. It is determined as

a product of three independently measured quantities

Γ

rad

= Γ

γ

+ Γ

π

=

Γ

γ

+ Γ

π

Γ

·

Γ

Γ

π

· Γ

π

, (2)

where Γ = Γ

α

+ Γ

γ

+ Γ

π

is the total decay width taking into account

α

,

γ

and e

±

decay

of the Hoyle state. At present these three quantities are known with an accuracy (left to

right) of ±2.7% (see Ref. [8] and references therein), ±9.2% (see Ref. [9] and references

therein), and ±6.4% (from combination of values quoted in Refs. [10, 11]), leading to

a total r

3

α

uncertainty of ±11.6%. A new experiment for a precise measurement of the

second quantity is discussed in Refs. [5, 12]. Additionally, a new result for the third

quantity, i.e. the pair width Γ

π

, with a quoted accuracy of ±2.7% is given by Crannell

et al. [13]. It would reduce the total uncertainty of r

3

α

to ±10.0%. Unfortunately, the

new value is inconsistent within error bars with the earlier values of Γ

π

[10, 11].

A main purpose of the present work is to resolve the existing inconsistency in the

partial pair width Γ

π

of the Hoyle state using high-resolution electron scattering mea-

surements at low momentum transfers.

EXPERIMENTAL DETAILS

The experiment was performed at the 169

◦

magnetic spectrometer of the S-DALINAC.

Excitation energy spectra were taken at initial electron energies between 29.3 MeV

and 78.3 MeV and scattering angles between 69

◦

and 141

◦

. This corresponds to the

momentum transfer range for the Hoyle state between 0.21 fm

−1

and 0.66 fm

−1

. A self-

supporting carbon targetwith natural isotopic content and an areal density of 6.4 mg/cm

2

was used. Typical beam currents were about 1

µ

A. In the energy-loss mode an energy

resolution ∆E ≈ 28 keV (full width at half maximum, FWHM) was achieved.

Typical spectra measured at a beam energy 73 MeV are presented in Fig. 1. The

prominent peaks correspond to the elastic line, the 2

+

1

state at E

x

= 4.44 MeV and the

FIGURE 1. Spectra of the

12

C(e,e

′

) reaction measured at a beam energy E

0

= 73 MeV and scattering

angles

θ

= 93

◦

(top) and

θ

= 141

◦

(bottom).

Hoyle state at E

x

= 7.65 MeV in

12

C. For an energy calibration of the measured spectra

electron scattering on

9

Be,

28

Si and

94

Mo was performed under the same kinematics.

PWBA ANALYSIS

In the plane wave Born approximation (PWBA) the monopole matrix element ME for

the transition to the Hoyle state can be extracted in a nearly model-independent way [14]

by a least squares ﬁt of the reduced transition probabilities using the equation

p

4

π

B(C0,q) =

q

2

6

(ME)

h

1−

q

2

20

R

2

tr

+ · · ·

i

, (3)

where B(C0,q) is the reduced transition probability, ME the monopole matrix element,

R

tr

the transition radius, and q the momentum transfer. The outcome of the ﬁt procedure

FIGURE 2. Extraction of the monopole matrix element by extrapolation of the reduced transition

probability to zero momentum transfer. Data are from Ref. [11] (circles) and the present work (squares).

is presented in Fig. 2, where the experimental values

p

4

π

B(C0,q)/q

2

and the ﬁt

function are plotted as a function of q

2

. Data measured in the present work (squares)

are shown together with older data (circles) from Ref. [11]. The solid curve represents

the ﬁt to the data resulting in ME = 5.37(7) fm

2

and a transition radius R

tr

= 4.30(12) fm.

In the analysis the data points were weighted by their error bars and terms up to order

q

6

were considered. The resulting partial pair width Γ

π

= 59.6(16)

µ

eV is in a good

agreement with the values extracted by Crannell et al. [10] and Strehl [11], but with a

signiﬁcantly reduced uncertainty.

FOURIER-BESSEL ANALYSIS

An alternative method to determine Γ

π

is a reconstruction of the 0

+

1

→ 0

+

2

transition

density by means of a Fourier-Bessel analysis, which then allows to extract the transition

charge density and the matrix element ME. For an accurate extraction of the transition

density, data should be available over a broad momentum transfer range. This is the case

for the Hoyle state in a region q = 0.2 − 3.1 fm

−1

(Refs. [11, 15] and the present work),

as shown in Fig. 3.

A detailed description of the ﬁt procedure can be found in Ref. [16]. In the present

analysis a bias tail starting from the radius R

t

= 5 fm with a weight P = 20 was used.

Due to program limitations only 100 data points could be used in the analysis. These

were chosen to be 22 Darmstadt low-q data points, 13 high-q points from HEPL and the

rest was taken randomly from the Bates and NIKHEF data. Comparison of different data

sets in regions where they overlap has shown that the HEPL data were systematically

15% higher than the data from Bates and NIKHEF, the latter being in good agreement

with each other. The used HEPL data points were reduced by a factor of 1.15.

FIGURE 3. Transition form factor for the transition from the ground state in

12

C to the Hoyle state

extracted by Fourier-Bessel analysis in comparison with the experimental data.

Figure 3 shows the transition form factor to the Hoyle state obtained by the ﬁt pro-

cedure in comparison with the experimental data. The corresponding transition density

yields a monopole matrix element of the transition from the ground state in

12

C to the

Hoyle state ME = 5.55(5) fm

2

. This leads to a partial pair width Γ

π

= 63.7(12)

µ

eV in

disagreement with Ref. [13].

Additional details concerning both analyses are contained in Ref. [17].

EXTRACTION OF THE PAIR WIDTH

As the low-q extrapolation and the Fourier-Bessel analysis can be considered indepen-

dent of each other, the obtained results are averaged. This leads to a weighted mean of

the partial pair width for the transition from the ground state to the Hoyle state in

12

C

Γ

π

= 62.2(10)

µ

eV, (4)

the most precise result to date.

FIGURE 4. Summary of partial pair widths derived over the last 40 years for the Hoyle state in

12

C.

Figure 4 shows a comparison of the results for the partial pair width of the Hoyle state

obtained in the present work with the data available in the literature. The newly obtained

partial pair width agrees well with older data – except the one from Ref. [13] – but has

a signiﬁcantly reduced uncertainty.

ACKNOWLEDGMENTS

We acknowledge H.-D. Gräf, R. Eichorn and the S-DALINAC team for preparing

excellent beams. This work has been supported by the DFG under contract SFB 634.

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