10 -1
15N
10 -2
10 -3
-4
10
p
10 -3
-4
10
-4
10 -6
10 -6
10
-7
10
-7
10
-7
10
-8
10
-8
10
-8
Abundance
10 -6
10 -9
10 -9
10 -1
16O
10 -2
10 -5
Zone 28
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))
10 -1
Zone 1
10 -5
Zone 8
Abundance
Abundance
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))
Zone 28
10 -2
Zone 8
10 -3
10 -4
Zone 4
10 -5
10 -9
10 -8
10 -9
10 -10
10 -11
10 -11
10 -11
10 -12
10 -12
10 -12
10 -12
10 -13
10 -13
10 -13
10 -13
10 -14
10 -14
10 -14
10 -14
10 -15
10
-15
10
-15
10 -15
10
-16
10
-16
Zone 1
10
-16
-20
-17
-14
-11
-8
-5
-2
1
4
7
10
-20
-17
-14
-11
Time (sec)
-8
-5
-2
1
4
7
-20
10
-17
-14
-11
-8
-5
-2
Time (sec)
Time (sec)
Energy production and nucleosynthesis in stars
Hydrogen burning in the stars produces energy by two processes, 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 the stars,
and produces energy to make the stars shine.
Proton-proton chain
p(p,e+,ν)d
d(p,γ)3He
86%
14%
3He(3He,2p)4He
3He(α,γ)7Be
14%
.02%
7Be(p,
γ)8B
ν)8Be
8Be(α)4He
7Be(e-,ν)7Li
7Li(p,
8B(e+,
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
abundance, compared to heavier elements.
CNO Cycles
(p,α)
(p,γ)
13
14
C
17
(p,γ)
(e+,ν)
13
I
N
15
(p,γ)
18
O
F
(e+,ν)
II
O
17
(e+,ν)
(e+,ν)
III
F
(p,γ)
(p,γ)
12
15
C
(p,α)
16
N
(p,γ)
18
O
7
10
10 -16
-20
-17
-14
-11
-8
-5
-2
1
4
7
10
Time (sec)
Experimental S(E) factor for 15N(p,γ)16O
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 Jp=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 1 leak per 880 CN cycles. S(0 keV)=64±6 keV b
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 a
reaction. 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].
20
F
rnuc
 203.235 fm
E
 R15N  1.3 A1/ 3  3.2 fm
f
Nonresonant reactions:
In nonresonant (direct-capture) reactions, a proton goes directly to the
state in the final compound nucleus, and g-radiation 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 twostep process which occurs at fixed proton energies.
,l f 1/ 2
(r ) bound state wave function
W f ,l f 1/ 2 (r ) Whitaker function
 l (r ) scattering wave function
i
blue g 1  8.7eV g 2  40eV
black g 1  8.7eV g 2  44eV
Reaction Rates for 15N(p,γ)16O
100
Reaction Rates
10
16
-
Ji = 1
O(1- )
Ji=Jf
li = 0
15
red g 1  10eV g 2  40eV
DC
100
-
N(1/2 )
15N(p,g)16O
1
1
0.1
0.01
0.001
0.0001
1E-05
1E-06
1E-07
RRate
1E-09
S ( Ec.m. ) 
( 2 J R 1)
p
2k
2
( 2 J x 1)( 2 J A
with two interfering
2 p
E
e
c .m.
1)
Jp=1-
 MRi( Ec.m. )
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
1E-05
1E-06
1E-07
NACRE low
1E-08
NACRE high
NACRE adopt
RRTotal
1E-12
0.1
0.2
0.3
0.4
0
T9
resonances at Ecm=312 keV and 960 keV.
a 1
0.0001
1E-11
0
 p2
0.01
0.001
1E-10
1E-12
 p1
0.1
1E-09
RRNarrowRes
1E-11
i 1, 2
1/a i2 ( Ec.m. )1/pi2 ( Ec.m. )
MRi ( Ec.m. ) 
 (E )
Ec.m.  ERi  i i c.m.
2
RRTotal
1E-10
2
Reaction Rates 15N(p,g)16O
10
1E-08
from 15N(p,g)16O
0.1
0.2
0.3
0.4
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.
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.
rcl >rnuc
Proton Capture
f
1
kinematical factor
(2 L  1)!!
magenta g 1  12eV g 2  40eV
VCB [MeV]>E Astro [keV]
Nuclear Potential
Coulomb parameter
C amplitude of the tail ANC
p(1/2 )
a
r0  5.0 fm  rnuc

z x z Ae2 
k
spin factor
g 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.
rcl 
z p z Ae 2
l (r )  CW
+
p
 2.30 MeV
(2 J R 1)
(2 J x 1)(2 J A 1)
total width
Z e
1 Z e
 ( L  1)(2 L  1) 
   L  xL  (1) L AL  (kg i r0 ) L 1/ 2 

k
mA 
L


 mx
For the reaction
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
(-) (+) (+) = (-)
To find the resonant parameters, we fit direct data for
Direct data is from A. Redder et al., Z. Phys. A305, 325 (1982).
2
reduced mass
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 )
15N(p,g)16O:
15N(p,a)12C
Radius
mx m A
mx  m A
i 1,2,3, 6
remove the Gamow factor from the cross section s
Resonances
Coulomb Barrier
rnuc
relative momentum

SDi( Ec.m. )
Interference between resonances and direct capture 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.
Fit of Direct Data
Resonant Parameters
rcl
i 1,2

Interference
Energy
Direct Capture
)  MD0( Ec.m. ) 
Fitting the Data for 15N(p,γ)16O
The leak reaction involves proton capture. An impinging proton
experiences two forces: the long-range electromagnetic force (Coulomb
potential) outside 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 Ep<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.
Ep
Resonant Capture
c . m.
1/ 2
Energy Levels 16O
Ec
 MRi( E
r0
Mechanics
z p z Ae
(2 J x 1)( 2 J A
Ne
IV
VCB 
2
MDi( E )    dr l f (r ) r li (r ) direct capture amplitude to bound state i
Rolfs, C., and Rodney, W.S., Cauldrons in the Cosmos (The University of Chicago Press, Chicago, 1988).
For 15 N+p at E=50 keV
Ee
2p
2k
2p
E
e
c.m.
1)
2

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.
rnuc
k  2 E
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 1 leak per 2200 CN cycles. S(25 keV)=32±5.76 keV b
C. Angulo, Nucl. Phys. A656 (1999) 3-183.
(p,α)
IMPORTANT LEAK REACTION
S ( Ec.m. ) 
(2 J R 1)
p
(p,γ)
19
O
(p,α)
S(E) = s (E) E e2p
S(E) Factor Expression for 15N(p,γ)16O
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 asymptotic normalization coefficients (ANCs) and resonant parameters. The ANCs were recently measured in
Prague by Nuclear Physics Institute, Czech Republic; Catania National Lab, Italy; and Cyclotron Institute, TAMU, USA. The
resonant parameters were fit for both channels 15N(p,a)12C and 15N(p,g)16O.
(p,γ)
N
4
Adele Plunkett, Middlebury College
REU 2007, Cyclotron Institute, Texas A&M University
Mentor: A.M. Mukhamedzhanov
15N
Astrophysical S(E) Factor
Why the stars shine
In very old first-generation stars,
energy is produced by
burning hydrogen via the
proton-proton (p-p) chain.
1
16O
10 -7
10 -10
10 -11
Astrophysical Factor for the
15
16
CNO Cycle Reaction N(p,γ) O
p
10 -6
Zone 1
10 -10
10 -10
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))
Reaction Rates cm3 mol-1 s-1
10
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 0
10 0
Reaction Rates cm3 mol-1 s-1
10 -3
Zone 28
10 0
Abundance
10 -2
www.nucastrodata.org
Element abundance in the Sun
10 -1
Abundance vs. Time
Abundance vs. Time
Abundance vs. Time
Abundance vs. Time
10 0
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.
•Systematic uncertainties of resonant S(E) factors in the RR analysis are not given. We estimate
15% uncertainties.
•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(0 keV)=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.