1. University of Colorado
Nuclear reactions in stars, and how we measure their reaction rates in the lab.
R. J. (Jerry) Peterson
University of Colorado

2. “Kitchens in the Cosmos”
Nuclear fusion reactions generate the energy that drives stars. Nuclear fusion reactions in stars generate many heavy nuclei, up to A=56 (14 times 4). Stellar explosions, initiated by gravity, create heavier nuclei, with high fluxes of neutrons, protons, alpha particles (4He). All of these reactions involve ‘reaction rates’ which increase with temperature. Source of that fusion energy?

3. Curve of binding energy

4. A better curve of binding energy
fusion
fission

5. Fission
Fission energy is from the electrostatic repulsion of two positive nuclei formed by nuclear fission.

6. The Coulomb barrier to fusion
Like charges repel, so positive particles need to have kinetic energy to get near enough for nuclear processes to happen.
This is a barrier penetration problem.

9. Maxwell Boltzmann thermal distribution
T9=temperature in 109 K

11. keV

12. Coulomb barrier penetration probabilities=cross sections
GK=109K =T9

13. S-factor
These reactions are dominated by the Coulomb barrier, so remove it—
s(E)= S(E) exp(-2ph) / E
h=Sommerfeld parameter=Z1Z2e2 / (4pe0) hbar v
v= relative velocity = SQRT(2E/A)
E= center of mass energy
A=reduced mass
Numerically s(E)=S(E) exp( Z1Z2 SQRT(A/E)/E
Units of S(E)—MeV barn

15. P-P chain
26.73 MeV

16. The CNO Cycle
26.73 MeV

17. Reaction Rates [AB] = R(T) [A] [B] [13N] = R(T) [proton] [12C]
Example—
12C (p.g) 13N (g.s.)
The start of the stellar CNO cycle to burn hydrogen to helium in stars.
[13N] = R(T) [proton] [12C]
In nuclear units—R(T) = NA <s(E) v>, in cm3 mole -1 sec -1
s(E) = radiative capture cross section
v= relative velocity
NA = 6.02 x 10 23—Avogadro’s number.
Averaged over a thermal distribution of energies/velocities at temperature T.
NA <s v> = NA (8 / p m (kT)3)1/2 ∫0∞ E exp(-E/kT) s(E) dE

18. Nucleosynthesis pathways
Neutron slow process—capture a neutron on A, to A+1, which may decay. Repeat.
Neutron fast process- capture a neutron on A, and before A+1 can decay, capture another. Repeat to the drip line.
Proton process (there are lots of protons in stars, with mechanisms to accelerate them)-capture a proton on A, making A+1
Alpha process (there are also many alphas in stars)-capture an alpha on A to make A+4. (More Coulomb repulsion, but efficient)

19. The usual way
Make a thin sample (thin is hard at low beam energies, since energy losses dE/dx are large)
Put a single energy proton or alpha beam on the sample.
Measure the cross sections ds/dW at some angles to integrate over all angles for s for that reaction.
Repeat at more energies.
Integrate <s v > for the reaction rate.

20. Thick TargetYields Beam into a thick sample, stopping the beam.
Measure number of final radioactive nuclei produced, by off-line gamma counting.
On a natural target, several reactions may be measured, with different gamma energies and halflives..
Yield= final nuclei per beam particle, at energies from the beam energy down to zero.
Y(E) = NA/A ∫0 Ebeam [-dE/d(rx)] -1 s(E) dE

21. Thermonuclear Reaction Rates at any T
NA <s v> = NA (8 / p m (kT)3)1/2 ∫0∞ E exp(-E/kT) s(E) dE
= ∫0∞ W2(E) s(E) dE
And yield:
Y(E) = NA/A ∫0 Ebeam [-dE/d(rx)] -1 s(E) dE
= ∫0Ebeam W1(E) s(E) dE
Integrate by parts:
NA <sv> = ∫0∞ Y(E) d[-W2(E)/W1(E)]/dE dE
= ∫0∞ f(E) dE
NA<sv> = ∫0Emin f(E) dE +
∫EminEmax f(E) dE +
∫Emax∞ f(E) dE.
Few protons above Emax, small cross sections below Emin for a given T9.
Thanks to Paul Ingalls

22. The experiment
Measure thick target radioactive yields Y(E) from Emin to Emax, by counting protons into a thick (stopping) target and counting the radioactivity of the final nucleus in a well-calibrated gamma spectrometer.
From tables of dE/dx in that material, and for some assumed Maxwell-Boltzman temperature, compute R(T).
Vary that temperature until the uncertainties due to unmeasured energies gets too large.

23. Realities
We used a variable energy (28 MeV protons) cyclotron on odd harmonics to reach proton energies down to a very few hundred keV.
A cyclotron is a splendid momentum analyzer.
Correct for radioactive half-lives during the bombardment and counting.
Count gammas several times, to be sure of the right half-life.

24. 12C(p,g)13N First ingredient of the CNO cycle
13N decay with t1/2=9.97 minutes by positron emission (no nuclear gamma rays)
That positron annihilates on a regular electron to make two 511 keV photons
Cross sections also well determined by thin target methods

25. First Result 12C(p,g)13N
10-9
Yield
13N per proton
1 MeV

26. What are the jumps? The yield must be a monotonically rising curve.
At resonant energies exactly, cross sections rise.
A test of the method, and a calibration.

27. Thin target (usual) data.
1 MeV
Rolfs and Azuma :Nucl. Phys. A 227, 3 (1974).

28. Phys. Rev. C 64,

29. Typical data

30. Danger! 56Fe(p,g)57Co (271.1 days) (91.7%) Or 57Fe(p,n)57Co (2.1%)
Opens at lab. energy of MeV.
Nuclear Structure—
Isobaric analog state at Coulomb energy difference from target ground state.
At a resonant beam energy of MeV for the p-wave ground state analog.

31. 95Mo(p,g)96Tc vs 96Mo(p,n)96Tc
Pure 95Mo

32. Thermonuclear Reaction Rates
Recap
NA <s v> = NA (8 / p m (kT)3)1/2 ∫0∞ E exp(-E/kT) s(E) dE
= ∫0∞ W2(E,T) s(E) dE
And yield:
Y(E) = NA/A ∫0 Ebeam [-dE/d(rx)] -1 s(E) dE
= ∫0Ebeam W1(E) s(E) dE
Integrate by parts:
NA <sv> = ∫0∞ Y(E) d[-W2(E,T)/W1(E)]/dE dE
= ∫0∞ f(E) dE
NA<sv> = ∫0Emin f(E) dE +
∫EminEmax f(E) dE +
∫Emax∞ f(E) dE.

33. C. Angulo et al., Nucl.Phys. A 656, 3 (1999)
12C(p,g)13N
C. Angulo et al., Nucl.Phys. A 656, 3 (1999)

34. Test vs. newest thin target data N. Burtebaev et al. Phys. Rev
Test vs. newest thin target data N. Burtebaev et al. Phys. Rev. C 78, (2008)
12C(p,g)13N

35. Ditto-- 4He-induced reactions
2H,3He-induced reactions for plasmas (UCB and CSM)
(backwards, to determine temperatures)
Proton yields, reaction rates, N A Roughton et al. Atomic Data and Nuclear Data Tables 23, 177 (1980)
4He yields and rates N A Roughton et al., At. Data Nucl. Data Tables 28,341 (1983)

38. Conclusions-
The thick target method has been used to mass-produce nuclear reaction rates with protons and 4He, measuring the induced radioactivity, with half-lives as short as 0.6 sec (41Sc). Results agree with other methods.

39. acknowledgements
Norbert Roughton, Clyde Zaidins, Carl Hansen, Martin Fritts Tom Intrator/plasma diagnostics. F. E. Cecil (CSMines)