Helium Burning in Stars

This article is also available as a section of a long paper by G. Wallerstein et al., Reviews of Modern Physics 69, October 1997, prepared to mark the 40th anniversary of the appearance of the seminal work on nuclear astrophysics in that journal in 1957. That publication [E. M. Burbidge, G. R. Burbidge, W. A. Fowler, and F. W. Hoyle, Rev. Mod. Phys, 29, 547 (1957)] became widely known as "B2FH," the "bible" of nuclear astrophysics.

Gerry Hale,

As the hydrogen burns in a star, a hot, dense core of helium is formed that fuels the nucleosynthesis of the heavier elements. The first stage of this process is the so-called "triple-alpha" capture to form 12C, followed by the subsequent capture of alpha particles to form 16O. In essence, helium burning terminates there, because further alpha captures (to form 20Ne, for example) occur too slowly at these temperatures and densities to be significant.

The termination of helium burning at 16O was not realized at the time of B2FH, because of uncertainty in the level structure of 20Ne near the α+16O threshold. However, it was later discovered that the levels closest to the threshold had the wrong parity and/or angular momentum to make a large resonant contribution to the rate. A calculation by Fowler et al. [Fow75] showed that the rate for 16O(α,γ)20Ne is far below that for 12C(α,γ)16O for T9≤0.2. Therefore, the blocking of further alpha captures for normal helium-burning conditions allows us to concentrate on triple-alpha and α+16O capture in the following discussion. Although the formulas used to describe the rates of these reactions are essentially unchanged from forty years ago, the major advance has been the experimental determination, with great precision in some cases, of the nuclear parameters upon which the expressions depend.

Triple-Alpha Capture

This reaction actually occurs in two stages: first, two alpha particles resonate in the low-lying (but unbound) state that forms the ground state of 8Be. This state is sufficiently long-lived (τ1/2=0.968x10-16 s) that there is a non-neglible probability that a third alpha particle will be captured before it disintegrates, forming 12C** (Ex=7.6542 MeV, Jπ=0+). A level diagram showing the relevant states and thresholds for 12C is shown below.

C-12 Level Diagram

Because of its quantum numbers, there is only a small probability that this excited state will de-excite (rather than decay back into three alpha particles), either by e+ - e- pair production, or by a γ-ray cascade through the first excited state, leaving 12C in its 0+ ground state. The prediction [Hoy53], and subsequent experimental verification, of the properties of 12C** in order to account for the observed abundance of 12C remains one of the most impressive accomplishments of nuclear astrophysics.

The rate per unit volume for the de-excitation process at temperature T is given by the resonance form [Rol88]

r= (0.5 Nα3 33/2(2π hbar2/ (MαkT))3α Γrad/ (hbarΓ)) exp(-Q/(kT)),

with Nα the alpha-particle number density, Mα its mass,

Q=(M12C**- 3Mα) c2=(379.5 +/- 0.3) keV,

and Γ α and Γrad the decay widths for α and (β+ γ) emission, respectively, which sum to the total width Γ. Since Γrad << Γ α ≅ Γ, the triple-alpha rate depends only on Γrad= Γ γ+ &/Gamma;pair. The current values [Ajz90] of these widths are Γ γ= (3.64 +/- 0.50) meV and Γpair=(60.5 +/- 3.9) μeV. The under-estimation of this rate by B2FH due to their value of Γ γ=1 meV was compensated in part by having at that time a smaller Q-value (372 keV) in the exponential factor in the rate equation.

alpha + 12C capture

This is probably the most important reaction in nuclear astrophysics today. Its rate at stellar temperatures, relative to that of "triple-alpha" capture, determines how much of the 12C formed is converted to 16O, and thereby the carbon/oxygen abundance ratio in red giant stars. The relative amounts of carbon and oxygen at the end of the helium-burning phase set the initial conditions for the next phase, which is heavy-ion burning. Depending on the mass of the star, heavy ion-burning results in a number of different possibilities, including white dwarfs, and supernovae. In the process, many of the heavier elements up through iron are synthesized. All of these processes have been found to be quite sensitive to the α+12C capture rate [Wea93], so that the abundances of the medium-mass elements, and even the final evolution of massive stars that explode as supernovae, depend critically on its determination.

The rate per unit volume for α+12C capture is given by the familiar expression involving the number densities Nα and N12C, and the Maxwellian-averaged <σv> for temperature T,

ralpha+12C = Nα N12Ccapv>T.

The cross section is parameterized as

σ cap(E) = Scap(E) (1/E) exp(-2πη)

in terms of Sommerfeld's Coulomb parameter η for α+12C, and the S-factor for the capture reaction,

Scap(E) = SE1(E) + SE2(E).

The <σv> integral is mainly determined by the value of the cross section (or S-factor) at the Gamow energy, which for helium-burning temperatures (T9 ≅ 0.2) is E0=0.3 MeV. Since present measurements cannot be extended to such low energies in the presence of Coulomb barriers as large as that for α+12C, the rate must be found by theoretical extrapolation. Due to their different energy dependence, the E1 and E2 multipole components of the cross section are extrapolated separately to E=E0, both being influenced by the presence of sub-threshold levels in 16O having Jπ=1- and 2+, respectively. The levels of 16O near the α+12C threshold are shown below.

Levels of 16O

E1 Capture

Much effort has been devoted to extracting just the E1 part of the capture cross section, which is dominated by a broad 1- resonance at E=2.4 MeV (Ex=9.585 MeV; all level energies and widths for 16O are taken from [Til93]). The importance of the subthreshold 1- state at E=-45 keV (Ex=7.11685 MeV) in fitting and extrapolating the E1 cross section was first demonstrated by [Dye74]. Most subsequent analyses have fit S-factors extracted from the measurements along with the P-wave phase shift obtained from α+12C elastic scattering measurements [Pla87], using either R-matrix or K-matrix theory. More recently, several groups have acted on the long-standing suggestion of Barker [Bar71] to measure the β-delayed α spectrum from the decay of 16N with the hope of better determining the contribution of the subthreshold state, which is evident in the spectrum as a secondary maximum at low energies.

Direct measurements

Several direct measurements [Dye74, Red87, Kre88, Oue92] of the E1 capture cross section have been made in the c.m. energy range E=1.0 - 3.0 MeV. With the recently reported corrections in the data of Ouellet et al. [Oue96], these measurements are in relatively good agreement at energies between 1.3 and 3.0 MeV. They now all imply constructive interference of the subthreshold level with the positive-energy resonance (originally, [Oue92] had found destructive interference), leading to extrapolated S-factors at E=0.3 MeV that range from 10 to 200 keV-b. The extrapolated S-factors obtained from these measurements, along with their assigned uncertainties, are given in the table below. The table also notes briefly the methods used to make and analyze the measurements. Most of the analyses involved doing standard R-matrix or K-matrix fitting, or using the "hybrid" R-matrix method [Koo74], in which a potential is used to represent the resonances at 2.4 MeV and above, in order to reduce the uncertainty in the ``background" contribution to the R matrix.

Extrapolated values of SE1(E0=0.3 MeV) obtained from direct measurements of the differential capture cross section.
Reference SE1(E0) Methods
[Dye74] measured σ(90o); corr. for E2 with direct capt. calc.

140 (+140)(-120) 3-level R-matrix

80 (+50)(-40) hybrid R matrix
[Red87] measured σ (θ); separated E1 and E2

200 (+270)(-110) 3-level R-matrix

80 (+120)(-80) hybrid R matrix
[Kre88] measured σ(90o); using gamma recoil coinc.

10 (+130)(-10) 3-level R-matrix
[Oue96] measured σ(θ); separated E1 and E2

79 +/- 16 K-matrix fits,
includes beta-delayed alpha spectra

β-delayed alpha spectrum from the decay of 16N

At least three groups [Buc93, Zha93, Zha95] have measured the delayed alpha spectrum from the β-decay of 16N down to energies low enough to see the secondary maximum attributed to the presence of the subthreshold 1- state. Some of these new spectral measurements have been included in the fitting along with direct measurements of the E1 cross section, in order to better constrain the parameters (in particlular, the reduced alpha width) of the subthreshold state. These constrained analyses give extrapolated values for SE1(E0) in the range 80-95 keV-b, as favored by the sensitivity study [Wea93] of the dependence of calculated elemental abundances on the extrapolated α+12C capture cross section.

The functional form of the spectrum can be obtained from the usual R-matrix relation of the scattering states to the level matrix, giving

dNα/dE = fβ(E) Pα(E) Abs ( Sumλ Bλ Sumλ' Aλ λ'(E) γ λ' α )2,

in which fβ is Fermi's integrated β-decay function for Z=8, Pα is the α+12C penetrability, Bλ is the dimensionless β-feeding amplitude for level λ, A is the level matrix, and γ λ'α is the reduced-width amplitude in the α+12C channel for level λ'. The product of the first two terms in the expression gives the phase-space behavior of the spectrum, while the third term gives the structure. A similar expression results from the K-matrix formalism [Hum91].

The interference minimum between the two maxima in the spectrum cannot fix the interference of the levels in the cross section, due to the undetermined relative phases of the Bλ. However, as was mentioned in the previous section, all the direct cross-section measurements indicate that the interference is constructive. The magnitude of the low-energy peak in the spectrum is sensitive to the value of the reduced-width amplitude in the α+12C channel for the subthreshold level, assuming that the one in the γ+16O channel, and the β-feeding amplitude for the state have been fixed by γ-decay data.

The spectra measured by the TRIUMF [Buc93] and Yale/Connecticut (UConn) [Zha93] groups are generally in good qualitative agreement. However, questions have been raised on both sides about important experimental details, such as the shape of the low-energy peak in the Yale/UConn measurement, the shape of the high-energy peak in the TRIUMF data compared to an earlier measurement [Neu74], and the effect of target-thickness corrections on comparisons of the two measurements.

Two new measurements of the spectrum have recently been done at the University of Washington in Seattle [Zha95], and at Yale/UConn [Fra96]. These new measurements are said to agree well with each other, being as much as a factor of two higher than the TRIUMF data in the minimum between the two peaks. This difference could imply that a larger F-wave contribution to the spectrum is required in order to fill in the minimum, thus decreasing the P-wave contribution (and the extrapolated value of SE1) correspondingly. However, [Hal96] has shown that allowing the β-feeding amplitude of the positive-energy level to have a small imaginary part also has the effect of filling in the minumum between the peaks. Therefore, the impact of these new measurements will depend on how they are analyzed.

E2 capture

The extrapolated E2 capture cross section is also determined by the interaction of positive-energy levels (and "direct capture" contributions) with a subthreshold 2+ state at E=-245 keV (Ex=6.9171 MeV). In this case, the only broad 2+ level in the region is quite far above the threshold, at E=4.36 MeV (Ex=11.52 MeV). In addition, there is a narrow resonance (Γ=0.625 keV) at Ex=9.8445 MeV that is not even visible in many of the measurements. The approximate centroid of the E2 strength appears to be at Ex=15 MeV, with a spread of about 15 MeV. Therefore, one expects the dominant E2 contribution above threshold to come from distant levels and direct capture terms.

The direct measurements lead to extrapolated values of SE2(0.3 MeV) ranging from 14 to 96 keV-b, as are shown in table below. The extrapolated value of [Red87] includes also contributions from the cascade transitions to the ground state, which were found to be non-negligible. Most of the analyses used some form of microscopic or potential model (usually of Gaussian form) to do the extrapolation, since a pure R-matrix or K-matrix fit would have involved too many parameters for the number of available data points. In some cases, the potential-model amplitudes were combined with single-level parametrizations of the subthreshold level. However, the present E2-capture data are simply not good enough to allow an unambiguous separation of the resonant and direct effects, even when considered simultaneously with the elastic-scattering data.

Extrapolated values of SE2(E0=0.3 MeV) obtained from direct measurements of the differential capture cross section.
Reference SE2(E0) Methods
[Red87] measured σ(θ); separated E1 and E2

96 (+24)(-36) 3-level R-matrix

80 +/- 25 hybrid R matrix
[Oue96] measured σ(θ); separated E1 and E2

36 +/- 6 microscopic cluster mode
[Tra96] measured σ(θ); separated E1 and E2

14.5 (+96)(-14) R matrix, including 2+ levels
below Ex=11.52 MeV

It is not known with certainty, for example, whether the interference between the subthreshold 2+ state at E=-245 keV and the distant-level (or "direct") contribution is constructive or destructive. The recent analysis of [Tra96] is one of the few that has attempted to include the two positive-energy 2+ resonances at Ex≤11.52 MeV. The uncertain nature of the interferences of these levels was a major contributing factor to the rather large uncertainty on the extrapolated S-factor listed in table. Thus, the uncertain role of resonances in the predominantly direct reaction mechanism, and the increased reliance on theoretical models, introduce even more uncertainty into the E2 extrapolation than that for the E1 contribution. An attempt has also been made recently [Kie96] to extract the E2 cross section from the inverse process, Coulomb dissociation of an 16O beam by a 208Pb target. The data analysis to obtain the equivalent capture cross sections from the α-12C coincidence spectra is heavily dependent on theoretical calculations, but the qualitative agreement with the capture measurements achieved at this stage shows the promise this method may hold for obtaining new information about the E2 cross section.

Other analyses and recommended values

A number of analyses not connected with any particular measurement have been done in recent years [Bar87, Fil89, Bar91, Hum91] that consider several data sets simultaneously, including the earlier measurement [Neu74] of the α-particle spectrum from the β-decay of 16N. These analyses often included theoretical refinements of the usual R-matrix and K-matrix fitting procedures, such as taking into account the channel-region contributions to the photon widths in R-matrix theory, and explored the parameter spaces of the representations used. The extrapolated S-factors obtained range from 30 to 260 keV-b for SE1, and from 7 to 120 keV-b for SE2.

Two recent analyses considered only the available primary data in order to avoid the correlations introduced by using derived quantities (such as elastic-scattering phase shifts) in determining the extrapolated E1 and E2 S-factor values. The analysis of [Buc96] included the differential cross-section measurements for 12C(α,α) scattering and 12C(α,γ) capture, along with the &betal-delayed alpha spectrum of [Azu94]. Their results for the E1 part of the cross section essentially confirmed their earlier extrapolated value SE1(E0)=(79 +/- 21) keV-b, and for the E2 part, established the upper limit SE2(E0)<140 keV-b.

[Hal96] recently reported an R-matrix analysis of the E1 cross section alone that also used the elastic scattering angular-distribution data of [Pla87] rather than the P-wave scattering phase shift. This analysis indicated that a rather low value of SE1(E0), 20 keV-b, was consistent with all the measured data, including those of the β-delayed alpha spectrum. However, after that analysis was done, the paper of [Oue96] appeared, reporting corrections to earlier data [Oue92] that significantly raised their low-energy cross sections. Because the uncertainties were so small, their earlier data had been a determining factor in the extrapolated value of SE1(E0) obtained by [Hal96]. Therefore, the recent changes in those data will likely raise Hale's extrapolated E1 S-factor to at least 60 keV-b.

At this point, it appears that the best values to recommend for the extrapolated S-factors for α+12C capture are the recent ones of [Oue96]. The energy-dependent S-factors from their analysis are shown in the following two figures compared with various direct measurements for the E1 component (in the first figure) for the E2 contribution (in the second).

plot se1

Figure above: S-factor plot for α+12C E1 capture from the R-matrix fit of [Oue96]. The data shown are those of [Oue96, Red87, Dye74, Kre88].

plot se2

Figure above: S-factor plot for α+12C E2 capture from the cluster-model calculation of [Oue96]. The data shown are those of [Oue96, Red87].

The curves extrapolate at E0=300 keV to the values: SE1(E0)=(79 +/- 16) keV-b, SE2(E0)=(36 +/- 6) keV-b, giving in round numbers, Scap(E0)=(120 +/- 40) keV-b. These values are quite consistent with the best estimates obtained by the TRIUMF group, with many of the earlier direct measurements, and with the results of the sensitivity study [Wea93]. However, it should be remembered that some of the recent data and analyses continue to indicate lower values of the E1 S-factor. While the uncertainties of these important parameters are gradually decreasing with time, they remain well outside of the 10-15% level that is desirable for astrophysical applications. Clearly, more work remains to be done on α+12C capture, especially concerning the extrapolation of the E2 S-factor.


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