Methods to directly measure non-resonant stellar reaction rates

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

2. Outline • Theoreticalbackground: • Reactionrates • Maxwell-Boltzmann-distribution ofvelocity • Cross-section • Gamow-Window • Experimental applicationusingtheexampleofthe pp2-chain reaction in the Sun • Motivation andsomemoretheory • Historical motivation • 3He(α,γ)7Be asimportantonsetreaction • Prompt andactivationmethod

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-mechanicalinteractionbetweentwoparticlesisalways proportional to a geometricalfactor: 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 Enteringthecrosssectionintothereaction rate, weobtain: 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. 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= 26.20 MeV 7Be + e- 7Li +n 7Be + p  8B + g 7Li + p  24He 8B 8Be + e+ +n PP-II Qeff= 25.66 MeV 24He PP-III Qeff= 19.17 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 partby Davis: • in Homestake Gold Mine, 1 478 m underground (toprotectfromcosmicrays) • 380 m3ofperchloroethylene (bigtargettoaccountforsmallprobabiltiyofsuccessfulcapture) • determinationofcapturedneutrinos via countingofradioactive isotope ofargon, whichisproducewhenneutrinosandchlorinecollide • result: only 1/3 ofthepredictednumberofelectronneutrinosweredetected Solar neutrino puzzle: discrepancies in the measurements of actual solar neutrino types and what the Sun's interior models predict.

21. Homestake-Experiment • Possibleexplanations: • The experiment was wrong. • The standard solar model was wrong. • Reactionratesare not accurateenough. • The standardpictureofneutrinos was wrong. Electronneutrinoscouldoscillatetobecomemuonneutrinos, whichdon'tinteractwithchlorine (neutrinooscillations). 3He + 4He 7Be + g 99.7% 0.3% Necessarytomeasurereactionrates at high accuracy. Here: withthehelpof3He(α,γ)7Beastheonsetofneutrino-producingreactions 7Be + e- 7Li +n 7Be + p  8B + g 7Li + p  24He 8B 8Be + e+ +n

22. Motivation We will take a look at the3He(α,γ)7Bereactionas: • The nuclearphysicsinputfromitscrosssectionis a majoruncertainty in thefluxesof7Be and8B neutrinosfromthe Sun predictedby Solar models • As well: majoruncertainty in 7Li abundanceobtained in big-bang nucleosynthesiscalculations Critical link: importanttoknowwith high accuracy

23. Measuring the reaction rate of 3He(α,γ)7Be 429 keV Q= 1,586 MeV Therearetwowaystomeasurethatthe3He(α,γ)7Be reactionoccured: • prompt γmethod: measuringtheγ´s emittedasthe7Be* γ-decaysintothe 1st excitedorthegroundstate • 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: • theyieldisthenumberofγeventscounted • NBeamisthenumberof beam particlescounted • ρisthenumberoftargetparticles per unitarea

25. Background reduction surface • underground, at theenergyrangeweareinterested in: about10 h toseeonebackgroundevent • usingtheequationmentionedbefore, wecanapproximatethatour3He(α,γ)7Be reactionprovidesabout70 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 Schematicviewofthetargetchamber

28. Prompt-γ-Method 1st GS 1st background GS signal GS 1st Measuredγ-rayspectrum at Gran Sasso LUNA acceleratorfacility

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

30. Activation Method Experimental Set-Up at Gran Sasso LUNA2 Schematicviewofthetargetchamberusedfortheirradiations

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 thetemperatureof e.g. the Sun, wecanspecifythe relevant energyrangefor a nuclearreaction • An importantreactiontoresearchtheinteriorofthe Sun aswellasbig-bang nucleosynthesisis3He(α,γ)7Be • Energiesrelatedto Sun temperaturesaretechnically not feasible: extrapolationdemands high accuracymeasurements • Necessarytoreducebackground • The weightedaverageoverresultsofbothmethods (prompt andactivation) provides an extrapolated S-factorof

34. References • Donald D. Clayton, Principlesof Stellar Evolution andNucleosynthesis(University of Chicago Press, Chicago, 1983) • Christian Iliadis, NuclearPhyicsof Stars (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2007) • F. Confortolaet al., arXiv: 0705.2151v1 (2007) • F. Confortolaet al., Phys. Rev. C 75, 065803 (2007) • Gy. Gyürkyet al., Phys. Rev. C 75, 035805 (2007) • C. Arpesella, Appl. Radiat. Isot. Vol. 47, No. 9/10, pp. 991-996 (1996) • D. Bemmereret al., arXiv: 0609013v1 (2006)

35. Zusatz-Folie Example: using a α-Beam at an energyof 300 keV, whichcorrespondsto an relative energyof 129 keVaccordsto a temperatureof 207 MK (whichismorethantentimeshigherthan in the Sun: needforextrapolation)