1. Methods to directly measure non-resonant stellar reaction rates
Tanja Geib

2. Outline Theoretical background:
Reaction rates
Maxwell-Boltzmann-distribution of velocity
Cross-section
Gamow-Window
Experimental application using the example of the pp2-chain reaction in the Sun
Motivation and some more theory
Historical motivation
3He(α,γ)7Be as important onset reaction
Prompt and activation method

3. Reaction Rates Nuclear Reaction Rate: particle density of type X
reaction cross section
flux of particles of type a as seen by particles X
Important: this reaction rate formula only holds when the flux of particles has a mono-energetic (delta-function) velocity distribution of just

4. Generalization to a Maxwell-Boltzmann velocity-distribution
Sun
Inside a star, the particles clearly do not move with a mono-energetic velocity distribution. Instead, they have their own velocity distributions.
Looking at the figure, one can see, that particles inside the Sun (as well inside stars) behave like an ideal gas. Therefore their velocity follows a Maxwell-Boltzmann distribution.

5. Generalization to a Maxwell-Boltzmann velocity-distribution
The reaction rate of an ideal gas velocity distribution is the sum over all reaction rates for the fractions of particles with fixed velocity:
Here the Maxwell-Boltzmann distribution enters via

6. Generalization to a Maxwell-Boltzmann velocity-distribution
After some calculation, including the change into CMS, one obtains:
is entered to avoid double-counting of particle pairs if it should happen that 1 and 2 are the same species
In terms of the relative energy (E=1/2 μv2 ) this means

7. Cross-Section
The only quantity in the reaction rate that we have not treated yet is the cross-section, which is a measure for the probabitlity that the reaction takes place if particles collide. We will now motivate its contributions.
Tunneling/ Transmission through the potential barrier
repulsive square-well potential

8. Cross-Section Radial Schrödinger equation for s-waves
is solved by the ansatz
This leads to transmission coefficient
for low-energy s-wave transmission at a square-barrier potential

9. Cross-Section
We generalize this to a Coulomb-potential by dividing the shape of the Coulomb-tail into thin slices of width
Total transmission coefficient for s-wave:
Reminder: If angular momentum not equal zero, then V(r)  V(r) + centrifugal barrier

10. Cross-Section Inserting the Coulomb potential, one obtains:
Solving the integral, and again using that the incident s-wave has small energies compared to the Coulomb barrier height, we get:

11. Cross-Section
Quantum-mechanical interaction between two particles is always proportional to a geometrical factor:
deBroglie wavelength
We account for the corrections arising from higher angular momenta by inserting the “Astrophysical S-Factor” S(E), which “absorbs” all of the fine details that our approximations have omitted.
Finally, our considerations lead to defining the cross-section at low energies as:

12. Cross-Section
12C(p,g)13N
The figure on the left shows the measured cross section as a function of the laboratory energy of protons striking a target. The observed peak corresponds to a resonance.

13. Gamow-Window
Entering the cross section into the reaction rate, we obtain:
with
Using mean value theorem for integration, we bring the equation to the form
to pull out the essential physics/ evolve the Gamow-window.

14. Gamow-Window We know that area under the curve
Log scale plot
Gamow-Window
We know that
area under the curve
This is where the action happens in thermonuclear burning!
This overlap function is approximated by a Gaussian curve: the Gamow-Window.
The Gamow-Window provides the relevant energy range for the nuclear reaction.
Linear scale plot

15. Gamow-Window D
A Gaussian curve is characterized by its expectation value and its width : 6 6
tells us where we find the Gamow-window. provides us with the relevant energy range.
Knowing the temperature of a star, we are able to determine where we have to measure in the laboratory.

16. Astro-Physical S-Factor (12C(p,g)13N)
How does look like?
A given temperature defines the Gamow-window. For stars, inside the Gamow-window, S(E) is slowly varying.
Approximate the astro-physical factor by its value at :

17. Nuclear Reactions in the Sun
core temperature: 15 Mio K
main fusion reactions to convert hydrogen into helium:
proton-proton-chain
CNO-cycle
nuclear reactions in the Sun are non-resonant

18. Proton-Proton-Chain Netto: 4p  4He + 2e+ + 2n + Qeff
p + p  d + e+ + n
p + d  3He + g
86%
14%
3He + 3He  4He + 2p
3He + 4He  7Be + g
99.7%
0.3%
PP-I
Qeff= MeV
7Be + e-  7Li + n
7Be + p  8B + g
7Li + p  2 4He
8B  8Be + e+ + n
PP-II
Qeff= MeV
2 4He
PP-III
Qeff= MeV

19. Homestake-Experiment
Basic idea: if we know which reactions produce neutrinos in the Sun and are able to calculate their reaction rates precisely, we can predict the neutrino flux.
Same idea by Raymond Davis jr and John Bahcall in the late 1960´s: Homestake Experiment
purpose: to collect and count neutrinos emitted by the nuclear fusion reactions inside the Sun
theoretical part by Bahcall: expected number of solar neutrinos had been computed based on the standard solar model which Bahcall had helped to establish and which gives a detailed account of the Sun's internal operation.

20. Homestake-Experiment
experimental part by Davis:
in Homestake Gold Mine, m underground (to protect from cosmic rays)
380 m3 of perchloroethylene (big target to account for small probabiltiy of successful capture)
determination of captured neutrinos via counting of radioactive isotope of argon, which is produce when neutrinos and chlorine collide
result: only 1/3 of the predicted number of electron neutrinos were detected
Solar neutrino puzzle: discrepancies in the measurements of actual solar neutrino types and what the Sun's interior models predict.

21. Homestake-Experiment
Possible explanations:
The experiment was wrong.
The standard solar model was wrong.
Reaction rates are not accurate enough.
The standard picture of neutrinos was wrong. Electron neutrinos could oscillate to become muon neutrinos, which don't interact with chlorine (neutrino oscillations).
3He + 4He  7Be + g
99.7%
0.3%
Necessary to measure reaction rates at high accuracy. Here: with the help of 3He(α,γ)7Be as the onset of neutrino-producing reactions
7Be + e-  7Li + n
7Be + p  8B + g
7Li + p  2 4He
8B  8Be + e+ + n

22. Motivation Critical link: important to know with high accuracy
We will take a look at the 3He(α,γ)7Be reaction as:
The nuclear physics input from its cross section is a major uncertainty in the fluxes of 7Be and 8B neutrinos from the Sun predicted by Solar models
As well: major uncertainty in 7Li abundance obtained in big-bang nucleosynthesis calculations
Critical link: important to know with high accuracy

23. Measuring the reaction rate of 3He(α,γ)7Be
Q= 1,586 MeV
429 keV
There are two ways to measure that the 3He(α,γ)7Be reaction occured:
prompt γ method: measuring the γ´s emitted as the 7Be* γ-decays into the 1st excited or the ground state
activation method: measuring the γ´s that are emitted when the radioactive 7Be decays

24. Basic Measuring Idea Experimentally we get the cross section over:
where:
the yield is the number of γ events counted
NBeam is the number of beam particles counted
ρ is the number of target particles per unit area

25. Background reduction surface
underground, at the energy range we are interested in: about 10 h to see one background event
using the equation mentioned before, we can approximate that our 3He(α,γ)7Be reaction provides about 70 events an hour.
thanks to the shielding: the yield is significantly higher than the background and can therefore be clearly seperated from it

26. Laboratory for Underground Nuclear Astrophysics at Laborati Nazionali del Gran Sasso (LNGS)
Luna
target
accelerator
detector
Credits to Matthias Junker at LNGS-INFN for making the LNGS picture available

27. Prompt-γ-Method Experimental Set-Up
Schematic view of the target chamber

28. Prompt-γ-Method 1st GS 1st background GS signal 1st GS
Measured γ-ray spectrum at Gran Sasso LUNA accelerator facility

29. Prompt-γ-Method
Overview on available S-factor values and extrapolation

30. Activation Method Experimental Set-Up at Gran Sasso LUNA2
Schematic view of the target chamber used for the irradiations

31. Activation Method
Offline γ-counting spectra from detector LNGS1

32. Activation Method
Astrophysical S-factor at lower panel, uncertainties at upper panel

33. Summary
Knowing the temperature of e.g. the Sun, we can specify the relevant energy range for a nuclear reaction
An important reaction to research the interior of the Sun as well as big-bang nucleosynthesis is 3He(α,γ)7Be
Energies related to Sun temperatures are technically not feasible: extrapolation demands high accuracy measurements
Necessary to reduce background
The weighted average over results of both methods (prompt and activation) provides an extrapolated S-factor of 𝑆 0 =0.560±0.017 𝑘𝑒𝑉

34. References
Donald D. Clayton, Principles of Stellar Evolution and Nucleosynthesis (University of Chicago Press, Chicago, 1983)
Christian Iliadis, Nuclear Phyics of Stars (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2007)
F. Confortola et al., arXiv: v1 (2007)
F. Confortola et al., Phys. Rev. C 75, (2007)
Gy. Gyürky et al., Phys. Rev. C 75, (2007)
C. Arpesella, Appl. Radiat. Isot. Vol. 47, No. 9/10, pp (1996)
D. Bemmerer et al., arXiv: v1 (2006)

35. Zusatz-Folie
Example: using a α-Beam at an energy of 300 keV, which corresponds to an relative energy of 129 keV accords to a temperature of 207 MK (which is more than ten times higher than in the Sun: need for extrapolation)