Astrophysical Factor for the
CNO Cycle Reaction 15N(p,g)16O
Adele Plunkett, Middlebury College
REU 2007, Cyclotron Institute, Texas A&M University
Mentor: A.M. Mukhamedzhanov
"For my own part, I declare I know nothing whatever about it, but looking at the
stars always makes me dream, as simply as I dream over the black dots
representing villages and towns on a map. Why, I ask myself, shouldn't the
shining dots of the sky be as accessible as the black dots on the map of France?"
Vincent Van Gogh, 1889
Why the stars shine
Energy production and nucleosynthesis in stars
Hydrogen burning in the stars produces energy by two cycles, with
the same net result 4p  4He + 2e+ + 2 n + Q (Q=26.73 Mev).
The fusion of hydrogen into helium fuels the luminosity of stars,
and produces energy to make the stars shine.
Proton-proton chain
In first-generation stars,
energy is produced by burning
hydrogen via the
proton-proton (p-p) chain.
p(p,e+,ν)d
d(p,γ)3He
86%
3He(3He,2p)4He
14%
3He(α,γ)7Be
14%
7Be(e-,ν)7Li
7Li(p, α)4He
.02%
7Be(p, γ)8B
8B(e+, ν)8Be
8Be(α)4He
In second-generation stars, energy is produced by hydrogen burning
of heavier elements in the CN cycle. Carbon and nitrogen are
heavier elements than helium with relatively small Coulomb barriers
and high abundances compared to heavier elements.
CNO Cycles
(p,γ)
13
(p,α)
14
C
(p,γ)
17
N
(p,γ)
(e+,ν)
13
N
I
15
(p,γ)
O
18
O
F
(e+,ν)
II
17
(e+,ν)
F
(e+,ν)
III
(p,γ)
(p,γ)
12
C
15
(p,α)
N
16
(p,γ)
18
O
(p,α)
IMPORTANT LEAK REACTION
(p,γ)
19
O
F
20
Ne
IV
(p,α)
Catalytic material is lost from the process via the leak reaction 15N(p,g)16O. Subsequent
reactions restore the catalytic material to the cycle, creating oxygen-16 and heavier
elements. The nucleosynthesis of heavier elements can be explained by the relative
abundance and reaction rates of CN elements.
The important leak reaction 15N(p,g)16O is the subject of this research.
Rolfs, C., and Rodney, W.S., Cauldrons in the Cosmos (The University of Chicago Press, Chicago, 1988)
Mechanics
The leak reaction involves proton radiative capture. An impinging proton experiences two forces: the long-range
electromagnetic force (Coulomb potential) outside of the nucleus, and the short-range strong force (nuclear
potential) inside the nucleus. In classical mechanics, E=T+V, and thus rcl is the classical closest approach distance.
In quantum mechanics, due to the tunnel effect, a proton can penetrate the Coulomb barrier with energy E p<Ec. At
astrophysical energies, the Coulomb barrier is the main obstacle for the reaction to occur, and the probability to
penetrate the barrier is small.
Energy
For 15 N+p at E=50 keV
Coulomb Barrier
Ec
rcl =
Resonant Capture
rnuc
=203.235 fm
E
R15N =1.3A1/3 =3.2 fm
p
Ep
Direct
Capture rcl
z p z A e2
Radius
rcl >R15N
VCB =
z p z A e2
rnuc
=2.30 MeV
r0 =5.0 fm> rnuc
VCB [MeV]>E Astro [keV]
Nuclear Potential
Proton Capture
Nonresonant reactions:
In nonresonant (direct-capture) reactions, a proton goes directly to the state in the final compound nucleus, and gradiation is emitted. This process can occur for all proton energies.
Resonant reactions:
In resonant reactions, an excited state of the compound nucleus is first formed, and then g-decays to the final
compound state. This is a two-step process which occurs at fixed proton energies.
Astrophysical S(E) Factor
S(E) = s (E) E e2
The cross section s is the effective geometrical area of a
projectile and target nucleus interaction, and it is directly
related to the probability of the reaction under consideration.
For projectile energies below the Coulomb barrier, such as
astrophysical energies, the cross section drops by many orders
of magnitude with decreasing energy. We use the
astrophysical S(E) factor in our theory to extrapolate to lower
astrophysical energies, essentially to zero energy. The
astrophysical S(E) factor has units [Energy-Area][keV-b].
Our Analysis
•In our analysis, the astrophysical S(E) factor is contributed by two resonances AND
direct captures.
•The S(E) factor is determined by resonant parameters and asymptotic normalization
coefficients (ANCs).
•The resonant parameters were fit for both channels 15N(p,a)12C and 15N(p,g)16O.
•The ANCs were recently measured in Prague
•Nuclear Physics Institute, Czech Republic
•Catania National Lab, Italy
•Cyclotron Institute, TAMU, USA.
Experimental S(E) factor for 15N(p,γ)16O
13
14
C
17
N
(p,γ)
(e+,ν)
13
N
I
15
(p,γ)
O
C
15
(p,α)
N
F
(e+,ν)
II
17
(e+,ν)
12
18
O
F
(e+,ν)
III
(p,γ)
16
(p,γ)
18
O
(p,α)
IMPORTANT LEAK REACTION
(p,γ)
(p,γ)
19
O
F
20
Ne
IV
(p,α)
C. Rolfs and W. S. Rodney, Nucl. Phys. A235 (1974) 450.
Rolfs and Rodney (RR) measured at higher energies and extrapolated data to zero
energy. From their direct measurements, the S(E) factor is contributed by two J=1resonances at Ecm=317 keV and 964 keV and nonresonant captures. Their result was
very sensitive to direct capture; the direct capture amplitude was varied arbitrarily to
fit the data. They conclude S(0keV)=64±6 keV b, thus 1 leak per 880 CN cycles.
D.F. Hebbard, Nucl. Phys. 15 (1960) 289.
Hebbard measured an S(E) factor contributed by two resonances at Ecm=317 keV and
947 keV. The experimental data were renormalized at the second resonance with 42%
error, contributing 18% uncertainty at lower energies. Hebbard concludes
S(25keV)=32±5.76 keV b, thus 1 leak per 2200 CN cycles.
C. Angulo, Nucl. Phys. A656 (1999) 3-183.
Resonances
For the reaction 15N(p,g)16O:
15N has spin ½, proton has spin ½, and relative orbital angular momentum l is 0.
i
16
The angular momentum Jf of the resonant state O is 1.
15N(1/2) + p(1/2) + l (0) = J (1)
i
f
15N has parity -, proton has parity +, and parity of l is +. Parity of 16O is -.
i
(-) (+) (+) = (-)
p(1/2+ )
16
-
J i =1
O(1- )
Ji =J f
li =0
15
N(1/2- )
Interference
Interference between resonant and direct amplitudes depends on selection rules – angular
momentum and parity conservation laws. Total angular momentum is the sum of channel
spins and relative orbital angular momentum. Resonances and direct capture amplitudes
interfere if they have the same quantum numbers in the initial state.
Energy Levels
16O
Fit of Direct Data
Resonant Parameters
To find the resonant parameters, we fit direct data for 15N(p,a)12C with two interfering J=1resonances at Ecm=312 keV and 960 keV.
S ( Ec.m. ) 
(2 J R 1)

2k
2
(2 J x 1)(2 J A 1)
Ec.m.e2
 MRi( Ec.m. )
2
i 1,2
1/a i2 ( Ec.m. )1/pi2 ( Ec.m. )
MRi ( Ec.m. ) 
 (E )
Ec.m.  ERi  i i c.m.
2
Direct data from
A. Redder et al., Z. Phys. A305, 325 (1982)
 p1
α
 p2
a 1
a 2
g 1
g 2
keV
keV
keV
keV
eV
eV
Rolfs
1.1
100
98
45
12±2
32±5*
Hebbard
1.1
100
93
40
12.8
88
Theory
1.1
95
90
45
8.7
40
from 15N(p,a)12C
from 15N(p,g)16O
*Inelastic electron scattering g2=31±8 eV
R-Matrix Approach
In the R-Matrix approach, the configuration space is divided into an
internal and an external region, divided at a radius r0. The nuclear
parameters inside combine resonant and nonresonant capture. The
collision matrix outside includes only nonresonant capture from r0∞.
The advantage of the R-Matrix approach is the essential independence
of astrophysical results on the radius r0 division of configuration space.
S(E) Factor Expression for 15N(p,g)16O
S ( Ec.m. )  2k
k  2 E
Ee
2
(2 J R 1)
2
(2 J x 1)( 2 J A
2
E
e
1) c.m.
relative momentum
 MRi(E
c . m.
2
)  MD0( Ec.m. ) 
i 1,2

mx m A
mx  m A
reduced mass
remove the Gamow factor from the cross section s

SDi( Ec.m. )
i 1,2,3, 6
(2 J R 1)
(2 J x 1)(2 J A 1)

z x z Ae2 
k
spin factor
Coulomb parameter
1/g i 2 ( E )1/pi2 ( E ) capture amplitude
 pi ( E ) proton partial resonant width
MRi ( E ) 
i ( E ) (through resonance i) g i ( E ) radiative width
E  ERi  i
2
i ( E )   pi ( E )  a i ( E )  g i ( E )

MDi( E )    dr l f (r ) r li (r )
total width
direct capture amplitude to bound state i
r0
1/ 2
1 L  Z xe
1
L Z Ae 
L 1/ 2  ( L  1)(2 L  1) 
    L  (1)
(
k
r
)

kinematical factor
gi 0


L
k
m
m
L
(2
L

1)!!


A 
 x
l (r )  CW
f
f
,l f 1/ 2
(r ) bound state wave function
C amplitude of the tail ANC
W f ,l f 1/ 2 (r ) Whitaker function
 l (r ) scattering wave function
i
Fitting the Data for
15N(p,γ)16O
magenta g 1  12eV g 2  40eV
blue g 1  8.7eV g 2  40eV
DC
red g 1  10eV g 2  40eV
black g 1  8.7eV g 2  44eV
g for the 1-st and 2-nd resonances are fixed by the resonant peaks
- - - Nonresonant part
■ Rolfs, C., and Rodney, W.S., Nucl. Phys. A235 (1974) 450.
▲ D.F. Hebbard, Nucl. Phys. 15 (1960) 289.
Reaction Rates for
15N(p,γ)16O
100
100
Reaction Rates 15N(p,g)16O
Reaction Rates 15N(p,g)16O
10
1
1
Reaction Rates cm3 mol-1 s-1
Reaction Rates cm3 mol-1 s-1
10
0.1
0.01
0.001
0.0001
1E-05
1E-06
1E-07
RRate
1E-08
1E-09
0.001
0.0001
1E-05
1E-06
1E-07
NACRE low
1E-08
NACRE high
NACRE adopt
1E-10
RRNarrowRes
1E-11
0.01
1E-09
RRTotal
1E-10
0.1
RRTotal
1E-11
1E-12
1E-12
0
0.1
0.2
0.3
T9
0.4
0
0.1
0.2
0.3
T9
Reaction rates are used to determine relative abundance of elements in the CNO cycle.
Because the theoretical S(E) factor is lower than experimental data at lower energies,
theoretical reaction rates (RRTotal) are also lower than experimental (NACRE). At 0.3 T9,
reaction rates increase due to a narrow resonance. More resonances contribute to the hot
CNO cycle (above 0.1 T9). New measurements at low energies and new calculations of
reaction rates should consider all resonances.
(NACRE) C. Angulo et al., Nucl. Phys. A 656 (1999) 3-183.
0.4
Abundance vs. Time
10 0
10 -1
10
-2
10 -3
10 -4
10 -5
O16 (ORNL Canonical (zone_01))
N15 (ORNL Canonical (zone_01))
p (ORNL Canonical (zone_01))
O16 (ORNL Canonical (zone_02))
N15 (ORNL Canonical (zone_02))
p (ORNL Canonical (zone_02))
O16 (ORNL Canonical (zone_08))
N15 (ORNL Canonical (zone_08))
p (ORNL Canonical (zone_08))
O16 (ORNL Canonical (zone_28))
N15 (ORNL Canonical (zone_28))
p (ORNL Canonical (zone_28))
p
16O
10 -6
Abundance
Abundance of
elements in the sun
10 -7
10 -8
10 -9
10 -10
15N
10 -11
10 -12
10 -13
10 -14
10 -15
10 -16
-20
-17
-14
-11
-8
-5
-2
1
4
7
10
Time (sec)
Abundance vs. Time
10 -1
10 -2
10 -3
10
Zone 28
10 0
N15 (ORNL Canonical (zone_01))
N15 (ORNL Canonical (zone_02))
N15 (ORNL Canonical (zone_04))
N15 (ORNL Canonical (zone_08))
N15 (ORNL Canonical (zone_16))
N15 (ORNL Canonical (zone_28))
10 -1
15N
10 -2
10 -3
-4
10
10 -5
p (ORNL Canonical (zone_01))
p (ORNL Canonical (zone_02))
p (ORNL Canonical (zone_04))
p (ORNL Canonical (zone_08))
p (ORNL Canonical (zone_16))
p (ORNL Canonical (zone_28))
10 0
10 -1
Zone 1
p
10 -2
10 -3
-4
10
10 -6
10 -6
10 -6
-7
-7
10 -7
Zone 28
10 -8
10 -9
10 -10
10
Abundance
10 -5
10
10 -8
10 -9
10 -11
10 -11
10 -12
10 -12
10 -12
10 -13
10 -13
10 -13
-14
-14
10 -14
10 -15
10 -15
10 -15
10 -16
-20
10 -16
-20
10
10
-17
-14
-11
-8
-5
Time (sec)
-2
1
4
7
10
-17
-14
-11
-8
-5
Time (sec)
-2
1
4
7
www.nucastrodata.org
10
Zone 28
Zone 8
Zone 4
Zone 1
10 -9
10 -10
Zone 1
16O
10 -8
10 -10
10 -11
O16 (ORNL Canonical (zone_01))
O16 (ORNL Canonical (zone_02))
O16 (ORNL Canonical (zone_04))
O16 (ORNL Canonical (zone_08))
O16 (ORNL Canonical (zone_16))
O16 (ORNL Canonical (zone_28))
-4
10 -5
Zone 8
Abundance
Abundance
Abundance vs. Time
Abundance vs. Time
10 0
10 -16
-20
-17
-14
-11
-8
-5
Time (sec)
-2
1
4
7
10
Considerations
We consider factors which may contribute to discrepancies with experimental and
theoretical data.
•Stopping power and straggling has little effect on proton energy when a thin target
is used.
•The Rolfs & Rodney (RR) analysis contains inconsistent use of lab and center of
mass systems.
•Systematic uncertainties of resonant S(E) factors are not given. We estimated
uncertainties of 15%.
•The RR analysis theory is not specified, and a significant overestimation of the
nonresonant capture may be due to incorrect division of configuration space;
channel radius is not specified.
Conclusions
The calculated astrophysical factor S(0keV)=38 keV b. The result does not depend
on channel radius. The ratio of S(E) factors for 15N(p,a)12C and 15N(p,g)16O is
1436:1. Our analysis calls for new measurements of 15N(p,g)16O.
Paper
Results of this analysis included in the paper to be submitted to Phys Rev C
Asymptotic Normalization Coefficients From the 15 N( 3 He,d) 16 O Reaction
and Astrophysical S Factor for15 N(p, γ) 16 O
A. M. Mukhamedzhanov, C. A. Gagliardi, A. Plunkett, L. Trache, R. E. Tribble,
Cyclotron Institute, Texas A&M University, College Station, TX 77843
P. Bem, V. Burjan Z. Hons, V. Kroha, J. Novak, S. Piskor, E. Simeckova,
F. Vesely, J.Vincour,
Nuclear Physics Institute, Czech Academy of Sciences, 250 68 Rez near Prague, Czech Republic
M. La Cognata, R. G. Pizzone, S. Romano, C. Spitaleri,
Universitá di Catania and INFN Laboratori Nazionali del Sud, Catania, Italy
THANKS!
• Dr. Zhanov – physics and soccer advisor.
• Changbo Fu, Dr. Goldberg, Dr. Tribble –
insightful experimentalists.