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 "B^{2}FH," the "bible" of nuclear astrophysics.

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 socalled "triplealpha"
capture to form ^{12}C, followed by the subsequent capture of
alpha particles to form ^{16}O. In essence, helium burning
terminates there, because further alpha captures (to form
^{20}Ne, for example) occur too slowly at these temperatures and
densities to be significant. The termination of helium burning at ^{16}O was not realized at the time of B^{2}FH, because of uncertainty in the level structure of ^{20}Ne near the α+^{16}O 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 ^{16}O(α,γ)^{20}Ne is far below that for ^{12}C(α,γ)^{16}O for T9≤0.2. Therefore, the blocking of further alpha captures for normal heliumburning conditions allows us to concentrate on triplealpha and α+^{16}O 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. 
This reaction actually occurs in two stages: first, two alpha particles
resonate in the lowlying (but unbound) state that forms the ground state
of ^{8}Be. This state is sufficiently longlived
(τ_{1/2}=0.968x10^{16} s)
that there is a nonneglible probability that a third alpha particle will be
captured before it disintegrates, forming ^{12}C^{**}
(E_{x}=7.6542 MeV,
J^{π}=0^{+}).
A level diagram showing the relevant states and thresholds for
^{12}C is shown below.
Because of its quantum numbers, there is only a small probability that this excited state will deexcite (rather than decay back into three alpha particles), either by e^{+}  e^{} pair production, or by a γray cascade through the first excited state, leaving ^{12}C in its 0^{+} ground state. The prediction [Hoy53], and subsequent experimental verification, of the properties of ^{12}C^{**} in order to account for the observed abundance of ^{12}C remains one of the most impressive accomplishments of nuclear astrophysics. The rate per unit volume for the deexcitation process at temperature T is given by the resonance form [Rol88]
with N_{α} the alphaparticle number density, M_{α} its mass,
and Γ _{α} and Γ_{rad} the decay widths for α and (β+ γ) emission, respectively, which sum to the total width Γ. Since Γ_{rad} Γ _{α} ≅ Γ, the triplealpha rate depends only on Γ_{rad}= Γ _{γ}+ &/Gamma;_{pair}. The current values [Ajz90] of these widths are Γ _{γ}= (3.64 +/ 0.50) meV and Γ_{pair}=(60.5 +/ 3.9) μeV. The underestimation of this rate by B^{2}FH due to their value of Γ_{ γ}=1 meV was compensated in part by having at that time a smaller Qvalue (372 keV) in the exponential factor in the rate equation. 
This is probably the most important reaction in nuclear astrophysics
today. Its rate at stellar temperatures, relative to that of
"triplealpha" capture, determines how much of the ^{12}C
formed is converted to ^{16}O, and thereby the carbon/oxygen
abundance ratio in red giant stars. The relative amounts of carbon
and oxygen at the end of the heliumburning phase set the initial
conditions for the next phase, which is heavyion burning.
Depending on the mass of the star, heavy ionburning 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+^{12}C capture rate [Wea93],
so that the abundances of the mediummass 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+^{12}C capture is given by the familiar expression involving the number densities N_{α} and N_{12C}, and the Maxwellianaveraged <σv> for temperature T,
The cross section is parameterized as
in terms of Sommerfeld's Coulomb parameter η for α+^{12}C, and the Sfactor for the capture reaction,
The <σv> integral is mainly determined by the value of the cross section (or Sfactor) at the Gamow energy, which for heliumburning temperatures (T9 ≅ 0.2) is E_{0}=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+^{12}C, 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=E_{0}, both being influenced by the presence of subthreshold levels in ^{16}O having J^{π}=1^{} and 2^{+}, respectively. The levels of ^{16}O near the α+^{12}C threshold are shown below.

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 (E_{x}=9.585 MeV;
all level energies and widths for
^{16}O are taken from [Til93]). The importance of the
subthreshold 1^{} state at E=45 keV
(E_{x}=7.11685 MeV) in
fitting and extrapolating the E1 cross section was first
demonstrated by [Dye74]. Most subsequent analyses have fit
Sfactors extracted from the measurements along with
the Pwave phase shift obtained from
α+^{12}C elastic scattering
measurements [Pla87], using either Rmatrix or Kmatrix
theory. More recently, several groups have acted on the
longstanding suggestion of Barker [Bar71] to measure the
βdelayed
α spectrum from the decay
of ^{16}N 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 positiveenergy resonance (originally, [Oue92] had found destructive interference), leading to extrapolated Sfactors at E=0.3 MeV that range from 10 to 200 keVb. The extrapolated Sfactors 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 Rmatrix or Kmatrix fitting, or using the "hybrid" Rmatrix 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.
βdelayed alpha spectrum from the decay of ^{16}N At least three groups [Buc93, Zha93, Zha95] have measured the delayed alpha spectrum from the βdecay of ^{16}N 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 S_{E1}(E_{0}) in the range 8095 keVb, as favored by the sensitivity study [Wea93] of the dependence of calculated elemental abundances on the extrapolated α+^{12}C capture cross section. The functional form of the spectrum can be obtained from the usual Rmatrix relation of the scattering states to the level matrix, giving
in which f_{β} is Fermi's integrated βdecay function for Z=8, P_{α} is the α+^{12}C penetrability, B_{λ} is the dimensionless βfeeding amplitude for level λ, A is the level matrix, and γ_{ λ'α} is the reducedwidth amplitude in the α+^{12}C channel for level λ'. The product of the first two terms in the expression gives the phasespace behavior of the spectrum, while the third term gives the structure. A similar expression results from the Kmatrix 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 B_{λ}. However, as was mentioned in the previous section, all the direct crosssection measurements indicate that the interference is constructive. The magnitude of the lowenergy peak in the spectrum is sensitive to the value of the reducedwidth amplitude in the α+^{12}C channel for the subthreshold level, assuming that the one in the γ+^{16}O channel, and the βfeeding amplitude for the state have been fixed by γdecay 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 lowenergy peak in the Yale/UConn measurement, the shape of the highenergy peak in the TRIUMF data compared to an earlier measurement [Neu74], and the effect of targetthickness 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 Fwave contribution to the spectrum is required in order to fill in the minimum, thus decreasing the Pwave contribution (and the extrapolated value of S_{E1}) correspondingly. However, [Hal96] has shown that allowing the βfeeding amplitude of the positiveenergy 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. 
The extrapolated E2 capture cross section is also determined by
the interaction of positiveenergy levels (and "direct capture"
contributions) with a subthreshold 2^{+} state at E=245 keV
(E_{x}=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
(E_{x}=11.52 MeV). In addition, there is a narrow resonance
(Γ=0.625 keV) at
E_{x}=9.8445 MeV that is not even visible in many of the
measurements. The approximate centroid of the E2 strength appears to be at
E_{x}=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 S_{E2}(0.3 MeV) ranging from 14 to 96 keVb, 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 nonnegligible. Most of the analyses used some form of microscopic or potential model (usually of Gaussian form) to do the extrapolation, since a pure Rmatrix or Kmatrix fit would have involved too many parameters for the number of available data points. In some cases, the potentialmodel amplitudes were combined with singlelevel parametrizations of the subthreshold level. However, the present E2capture data are simply not good enough to allow an unambiguous separation of the resonant and direct effects, even when considered simultaneously with the elasticscattering data.
It is not known with certainty, for example, whether the interference between the subthreshold 2^{+} state at E=245 keV and the distantlevel (or "direct") contribution is constructive or destructive. The recent analysis of [Tra96] is one of the few that has attempted to include the two positiveenergy 2^{+} resonances at E_{x}≤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 Sfactor 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 ^{16}O beam by a ^{208}Pb target. The data analysis to obtain the equivalent capture cross sections from the α^{12}C 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. 
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 αparticle spectrum from the
βdecay of ^{16}N.
These analyses often included theoretical refinements of the usual
Rmatrix and Kmatrix fitting procedures, such as taking
into account the channelregion contributions to the photon widths in
Rmatrix theory, and explored the parameter spaces of the
representations used. The extrapolated Sfactors obtained range
from 30 to 260 keVb for S_{E1}, and from 7 to 120 keVb
for S_{E2}. Two recent analyses considered only the available primary data in order to avoid the correlations introduced by using derived quantities (such as elasticscattering phase shifts) in determining the extrapolated E1 and E2 Sfactor values. The analysis of [Buc96] included the differential crosssection measurements for ^{12}C(α,α) scattering and ^{12}C(α,γ) capture, along with the &betaldelayed alpha spectrum of [Azu94]. Their results for the E1 part of the cross section essentially confirmed their earlier extrapolated value S_{E1}(E_{0})=(79 +/ 21) keVb, and for the E2 part, established the upper limit S_{E2}(E_{0})<140 keVb. [Hal96] recently reported an Rmatrix analysis of the E1 cross section alone that also used the elastic scattering angulardistribution data of [Pla87] rather than the Pwave scattering phase shift. This analysis indicated that a rather low value of S_{E1}(E_{0}), 20 keVb, was consistent with all the measured data, including those of the βdelayed alpha spectrum. However, after that analysis was done, the paper of [Oue96] appeared, reporting corrections to earlier data [Oue92] that significantly raised their lowenergy cross sections. Because the uncertainties were so small, their earlier data had been a determining factor in the extrapolated value of S_{E1}(E_{0}) obtained by [Hal96]. Therefore, the recent changes in those data will likely raise Hale's extrapolated E1 Sfactor to at least 60 keVb. At this point, it appears that the best values to recommend for the extrapolated Sfactors for α+^{12}C capture are the recent ones of [Oue96]. The energydependent Sfactors 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).
The curves extrapolate at E_{0}=300 keV to the values: S_{E1}(E_{0})=(79 +/ 16) keVb, S_{E2}(E_{0})=(36 +/ 6) keVb, giving in round numbers, S_{cap}(E_{0})=(120 +/ 40) keVb. 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 Sfactor. While the uncertainties of these important parameters are gradually decreasing with time, they remain well outside of the 1015% level that is desirable for astrophysical applications. Clearly, more work remains to be done on α+^{12}C capture, especially concerning the extrapolation of the E2 Sfactor. 

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