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IV. Evolution and Nucleosynthesis in Stars, and Cross Sections — Hydro Static Burning

IV. Evolution and Nucleosynthesis in Stars, and Cross Sections — Hydro Static Burning

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Cyclotron Institute, Texas A&M University, College Station, Texas '7'7843



Institute for Nuclear Physics, Czech Academy of Sciences, Prague-Rei, Czech



Institute for Atomic Physics, Bucharest, Romania

Asymptotic normalization coefficients (ANCs) have proven to be useful for determining reaction rates of interest in nuclear astrophysics. These coefficients, which

provide the normalization of the tail of the overlap function, determine 5' factors

for direct capture reactions at astrophysical energies. They also can be related to

resonance capture rates and are particularly useful for determining proton-capture

reaction rates that involve subthreshold resonance states. Dnring the past eight,

years, many ANCs have been measured by peripheral transfer reactions. Recent

proton transfer reaction measurements have yielded ANCs for 1 4 0 --f 13N p , 150

+ 14N p and "Na + "Ne

p . These results have been used to find S factors

for 13N(p,7)140, l4N(p,7)l50 and 20Ne(p,7)21Na. Using mirror symmetry, the

13C('Li,8Li)'2C reaction has been used to obtain a new measurement of the ANC


an introduction to ANCs, the

that defines the S factor for 7Be(p, Y ) ~ BFollowing

recent experiments are discussed along with the astrophysical implications of these





1. Introduction

Stellar evolution is a complex process involving an interplay between nuclear burning, hydrodynamics, plasma physics and gravity. Nuclear burning

provides the energy that ultimately dictates the fate of a star. The burning

process is itself complicated, involving sequences of capture and transfer



reactions and beta decays, and depends on the density, temperature and

nuclear abundances. Reaction and decay cycles, beginning with the p p

chain and extending to the CNO, Ne-Na, etc., cycles, process the nuclear

fuel, primarily through hydrogen and helium burning, yielding increasingly

massive nuclei and producing energy. Predicting the evolution of a star

requires knowing the important reaction rates and half lives.

Direct proton-capture reactions of astrophysical interest usually involve

systems where the binding energy of the captured proton is low. Hence

at stellar energies, the capture proceeds through the tail of the nuclear

overlap function. The shape of the overlap function in this tail region is

completely determined by the Coulomb interaction, so the amplitude of

the overlap function alone dictates the rate of the capture reaction. The

asymptotic normalization coefficient (ANC), C, for A p f) B specifies

the amplitude of the tail of the overlap function for the system. In previous communicationsii2, we have pointed out that astrophysical 5’ factors

for peripheral direct radiative capture reactions can be determined through

measurements of ANC’s using traditional nuclear reactions such as peripheral nucleon transfer. In many systems, resonant and nonresonant capture

compete. We have shown3 that the ANC can be used to determine the

external part of rr. Thus the ANC is connected to both the resonant and

nonresonant capture amplitudes. Also the ANC can be used to determine

astrophysical S factors when the capture occurs through a subthreshold

resonance state3.

Below we describe the ANC technique citing a test that has been carried

out to verify it. We then discuss four recent measurements, present their

results and discuss their astrophysical implications.


2. Introduction t o ANCs

It is well known that proton capture at stellar energies occurs at distances

that are large compared to the nuclear radius. Direct capture rates depend

on the normalization of the overlap function which is fhed by the appropriate ANCs. The connection between ANCs and the direct proton capture

rate at low energies is straightforward to obtain. The cross section for the

direct capture reaction A + p -+ B y can be written as


where X contains kinematical factors, I f p is the overlap function for

B -+ A p , 0 is the electromagnetic transition operator, and




the scattering wave in the incident channel. If the dominant contribution

to the matrix element comes from outside the nuclear radius, the overlap

function may be replaced by

where C defines the amplitude of the tail of the radial overlap function I:p,

W is the Whittaker function, q and I are the Coulomb parameter and orbital

angular momentum for the bound state B = A+p, and n is the bound state

wave number. The required (5”s are just the ANCs; the direct capture cross

sections are directly proportional to the squares of these ANCs. The ANC

can also be related to the external or channel part of the resonance width

for resonant capture3. The internal part of the width, however, depends

on the strength of the overlap function in the nuclear interior. If resonance

parameters are known either from measurements or calculations and ANCs

are known, the resonant and nonresonant components can be used together

in an R-matrix calculation to obtain capture cross sections.

Peripheral transfer reactions provide an excellent way to determine

ANCs. Consider the proton transfer reaction a + A -+ c + B, where

a = c p , B = A + p . As was previously shown4 we can write the DWBA

cross section in the form




is the reduced DWBA cross section and j,,1%are the total

and orbital angular momenta of the transferred proton in nucleus i. The

factors bcplaja and bApisjB are the ANC’s of the bound state proton wave

functions in nuclei a and B. which are related to the corresponding ANC

of the overlap function by

where S:pi,jais the spectroscopic factor. If the reaction under consideration

is peripheral, the ratio

is independent of the single particle ANC’s b A p l B j B and b c p l , j , . Thus for

surface reactions the DWBA cross section is best parametrized in terms

of the product of the square of the ANCs of the initial and final nuclei


( C B ) 2 ( C a ) 2The

. ANCs are just those needed in Eq. 1 to determine the

capture reaction cross section.

We have measured ANCs in 160(3He,d)17Fand compared them to radiative capture measurements t o test our technique. The 160(3He,d)17F

reaction was measured previously at a beam energy of 25 MeV5. We repeated the measurement at 29.75 MeV in order t o obtain better angular

coverage and to have a measurement at a second energy, both of which

were necessary for extracting reliable -4NC's. Data at laboratory scattering angles between 6.5" and 25" were obtained using Si solid state detectors

and a 3He beam, incident on a 134 pg/cm2 Mylar target, from the U-120M

isochronous cyclotron of the Nuclear Physics Institute (NPI) of the Czech

Academy of Sciences. Additional data at laboratory angles between 1" and

11" were obtained using the MDM magnetic spectrometer and a molecular (3He-d)+ beam, incident on a 540 pg/cm2 Mylar target, from the

Texas .4&M University K500 superconducting cyclotron. Absolute cross

sections were determined at the NPI using their detection system which

has been well calibrated for (3He,d) reaction studies. The data obtained

at Tz4MU were normalized t o the data from the NPI measurement in the

region where the two data sets overlapped. ANC's were extracted using

finite-range DWBA calculations. Details of the experiment and the DWBA

analysis can be found elsewhere6.

Good agreement between predictions for S factors based on ANCs and

experimental results were obtained. From these comparison we have verified

that the ANC technique works to better than 10%. The largest uncertainty

came from optical model parameters used to calculate the DWBA cross


3. Recent ANC Measurements


We have measured ANCs for 150+ 14N + p and 21Na + "Ne

p using

the 14N(3He,d)150and 2oNe(3He,d)21Nareactions. The experiments were

carried out with 3He beams supplied by the U-120M isochronous cyclotron

of the NPI. A 26.3 MeV beam was used with a 260 pg/cm2 melamine

target (C3H6N6) to measure the 14N(3He,d)150.Reaction products were

measured in a AE-E Si detector telescope. A second Si detector telescope

fixed at 19" was used to monitor the target for degradation during the

experiment. ANCs were obtained for the ground and five excited states in

150.The state that dominates the reaction rate at stellar energies, through

an s-wave capture, is a subthreshold state at E, = 6.79 MeV. The angular


Figure 1. Angular distribution for the excited state at E, = 6.79 MeV in

14N(3He,d)150reaction. The solid line is a DWBA fit to the data.



distribution for the important subthreshold state is shown in Fig. 1 along

with a DWBA prediction. More details are given elsewhere7.

A gas target cell filled with isotopic 20Negas (99.99%) was bombarded

by a 25.83 MeV 3He beam in order to study the 20Ne(3He,d)21Nareaction.

The gas cell had entrance and exit windows of 3.05 pm havar and was operated at a pressure of 195 mbar. Both the temperature and pressure of

the gas cell were continuously monitored during the experiment to verify

that the target thickness remained constant. Also a Si monitor detector

was set at 19" to act as an additional check on the system. A cooled AE-E

Si detector telescope was used to observe the outgoing reaction products.

A double collimating system was used to carefully define the detector solid

angle. The stellar capture rate below T9 of 0.2 is dominated by a subthreshold state at E, = 2.425 MeV which is only 7 keV below the proton

threshold. The ANC for the subthreshold state has been extracted from

the angular distribution following the same procedure described above for

the 14N(3He,d)150reaction.

The ANCs for 1 4 0 -+ 13N + p have been extracted from the

14N(13N,140)13Creaction. The experiment was carried out at Texas A&M

University. A beam of 13C at 15 MeV/A from the K500 superconducting

cyclotron was used to bombard a 9 cm long gas target filled with 2 atmospheres of H2 gas cooled to LN2 temperature. Recoil ions were collected

by the MARS recoil spectrometer to produce a pure (>99%) secondary


e,, W 9 )

Figure 2. Angular distribution for 14N('3N,140)13Cg.s.. The dashed curve is the

DWBA prediction. The solid curve is the smeared prediction taking into account the

finite size of the secondary beam spot and the angular spread in the beam. The upper

and lower dashed-dotted lines represent the 14N(p1/2)+ I4O(p1/2) and 14N(p3/2) --t

l 4O (pl 1 2 ) proton transfer reactions, respectively.

beam of 13N at 11.8 MeV/A. A 1.5 mg/cm2 melamine target was placed at

the focal plane of the recoil spectrometer and reaction products from the

14N(13N,140)13Creaction were observed in 5 cm x 5 cm AE-E Si detector

telescopes. The AE counters were 60 pm thick strip detectors and the E

counters were 500 pm thick solid Si. A plastic scintillator detector was used

to measure the beam particles. The angular distribution for 140reaction

products populating the ground state of 13C is shown in Fig. 2. The ANCs

for 1 4 0 + 13N p have been extracted and used to predict the direct capture contribution to the 13N(p,~)140

reaction as described below. More

details about the experiment can be found elsewhere'.

In addition to the three proton transfer reactions discussed above, we

have measured the ANCs for 'Li -+ 7Li + n from the neutron transfer reaction 13C(7Li,8Li)12C.By mirror symmetry, these ANCs can be related to

those for the 8B + 7Be + p systemg. Thus the neutron transfer reaction can

be used to determine the Sfactor for 7Be(p,y)aB.The 13C(7Li,'Li)12Creaction was carried out at Texas A&M University with a 7Li beam at 9 MeV/A

from the K500 superconducting cyclotron bombarding a 300 pg/cm2 13C

target. After passing through the beam analysis system, the primary beam

was focused at the target chamber of the MDM magnetic spectrometer.

Reaction products from the 13C target were observed in the focal plane of



ec.m,( deg )

Figure 3. The angular distribution for the '3C(7Li,8Li)'2C reaction. The data are

shown as points and the solid line is the best fit. The p 1 / 2 + p l l Z component is shown

as the dotted line, and the p l p --t p 3 p component is the dashed line.

the spectrometer. The high-quality beam from the analysis system allowed

us to measure the angular distribution for the reaction to 0". This made

it possible to separate the l p , p and l p 3 p components in the transfer. In

previous measurements of the mirror (7Be,8B)reactionlo, we were not able

to separate the two components and had to rely on microscopic model calculations to fix the ratio. The angular distribution for the reaction and

the DWBA fit are shown in Fig. 3. Optical model parameters were taken

from an analysis of the 7Li elastic scattering on 13C. The resulting ANCs

for the neutron transfer reaction are Ci,/2(8Li)= 0.384 f 0.038 fm-' and

C ~ l / 2 ( 8 L=i )0.048 0.006 fm-'. The ratio of the two ANCs is 0.13(2)

which is in excellent agreement with the model calculations". More details

can be found elsewhere".


4. S factors from the ANCs and their astrophysical


The l4N(p,y)l5O reaction is one of the most important reactions in the

CNO cycle. As the slowest reaction in the cycle, it defines the rate of energy p r o d ~ c t i o n ' and,


hence, the lifetime of stars that are governed by

hydrogen burning via CNO processing. The most recent measurement of

the 14N(p,y)150 reaction was carried out14 in 1987 and a total astrophys-


3/2', 6.79 MeV

Figure 4. The i 4 N ( p , 7 ) 1 5 0Sfactor for the important subthreshold state. The squares

are the data points and the solid line is the best fit R-matrix solution.

ical factor S(0) = 3.20 f 0.54 keV b was deduced. This measurement led

to a new understanding of the reaction, however, since it was found that


capture at low energies is dominated by capture through the

~ 259.5 keV (the resonance energy in the c.m.) and

first resonance at E R =

a subthreshold state at E, = -504 keV. At very low energies appropriate

for stellar burning, the reaction was found to be dominated by a combination of direct and resonant capture and interference from the tail of the

subthreshold resonance and the first resonance. Recently the first measurement of the radiative width of the subthreshold state in I5O to the ground

state was reported15. The new result for the width, 0.41?:::4,

eV, is about

15 times smaller than the value used by Schroder et al. in the analysis of

their data. Consequently only direct capture to the subthreshold state in

150is important for the S factor. The ANC to the subthreshold state thus

determines the reaction rate at stellar energies.

The ANCs that we determined from the transfer reaction have been

used in an R-matrix analysis to determine the S factor for 1 4 N ( p , ~ ) 1 5 0 .

The result for the transition to the subthreshold state is shown in Fig. 4.

Extrapolating to stellar energies, we find S ( 0 ) = 1.40 h 0.20 keVb for the

contribution from the subthreshold state. Including all contributions, we

find the total calculated astrophysical factor at zero energy to be S ( 0 ) =

1.70f0.22 keV b7. This rate is about a factor of 2 smaller than that obtained

from the analysis of Schroder et al. and makes the energy production in

the CNO cycle smaller than previously estimated.


With increasing temperature, the rates of the proton capture reactions

in the cold CNO cycle exponentially increase. Eventually the reaction rate

for 13N(p,y)140exceeds the rate of 13N decay (tlp=9.965 min) and the

hot (Plimited) CNO-cycle takes over. But 14N(p,y)150is still the slowest

reaction and it controls the energy generation rate. As the temperature continues to increase, all of the proton capture reaction rates in the hot CNO

cycle exceed the B-decay rates of 140and 150(with half lifes tIp=70.6 s

and 122s, respectively). The energy generation rate of the hot CNO cycle

will remain constant until some break out processes begin to occur. Indeed the thermonuclear runaway in novae is driven by the energy release

of the hot CNO cycle. Because the peak temperature in the thermonuclear

runaway is typically below 3.5 x 10' K, break out processes are inhibited

by their limited reaction rates. Observation of the abundance distribution

in nova ejecta indicate large over abundances of nitrogen produced by the

P-decay of the bottleneck nuclei, 140and 150,in the hot CNO cycle16.

For Tg=0.2, the Gamow window for the 13N(p,y)140reaction is located

at 148 keV with a width of 117 keV. At this energy the reaction is dominated

by the low-energy tail of the s-wave capture on the broad 1- resonance at

E,=0.529 MeV7. The direct capture contribution is significantly smaller

than that due to the tail of the resonance within the Gamow window.

But since both resonant and nonresonant capture proceed via s waves and

then decay by E l transitions, there is an interference between the two

components. Thus the resonant tail can be enhanced through constructive

interference or reduced through destructive interference.

The cross section for direct capture and resonant capture through the

broad first excited state was calculated from the measured ANC and the

experimental resonance parameters using the R-matrix approach'. In the

calculation, the same parameters (E;iim.= 527.9 f 1.7 keV, rtotaz= 37.3 f

0.9 keV and rr = 3.36 f0.72 eV) as Magnus et al." adopted were used for

the first resonance. A theoretical calculation by D e s c o u ~ e m o n tsuggested


that constructive interference occurs between the resonant and nonresonant

amplitudes for the low energy tail, but there is no experimental confirmation

of this so far. Thus both constructive and destructive interference were

calculated and are shown, together with the result from Decrock et a1.17,

in Fig. 5. The relatively flat lower solid line, which is our result for direct

capture alone, is about 30% larger than the result obtained by Decrock et

al. which is shown as the lowest dash-dotted line in Fig. 5. The two results

just agree within the quoted uncertainties. At Ec.m. = 140 keV where the

Gamow peak is located for Tg=0.2, our updated result with constructive



Figure 5. S factor for 13N(p,7)i40. The relatively flat solid line is the direct c a p

ture contribution determined from the ANC. This result is about 30% higher than that

obtained by Decrock et al. which is shown as the lowest dash-dotted line.

interference, shown as the upper solid line, is about 38% higher than the

previous result, which is shown as the upper dash-dotted line in Fig. 5. This

is due to the larger direct capture contribution from the ANC measurement.

In Decrock's calculation, the resonance energy was taken as 526 keV which

is smaller than Magnus's recommendation'* (Eirn' = 527.9 + 1.7 keV).

Therefore the peak of the resonance is shifted upward slightly with a higher

resonance energy. The estimated destructive interference, shown as the

lower solid line in Fig. 5 is smaller than the constructive interference result

by a factor of 3 at Ec.m.= 140 keV. Verifying by a direct measurement

that constructive interference is indeed correct would be quite useful.

As material leaks out of the CNO cycle, the Ne-Na cycle can begin.

The 20Ne(p,y)"Na reaction is one of the reactions in the Ne-Na cycle. Its

reaction rate is extremely hard to measure since, like the 14N(p,7)150 reaction, it is dominated by a subthreshold state. The important state in

21Naat E, = 2.425 MeV is only 7 keV below threshold. Thus even a small

r r will result in a substantial resonant contribution to the reaction rate.

The reaction rate at stellar energies was estimated previously to be about

3500 keV b20. The ANC that we have determined for this state fixes the direct capture through the subthreshold state. Figure 6 shows a preliminary

R-matrix fit to the data from direct measurements20. At proton energies

below about 150 keV, the resonant capture through the subthreshold state

dominates the S factor. Based on the present information available about

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