Outline 1. Theoretical background: – Reaction rates – Maxwell-Boltzmann-distribution of velocity – Cross-section – Gamow-Window 2. Experimental application using the example of the pp2-chain reaction in the Sun – Motivation and some more theory – Historical motivation – 3 He(α,γ) 7 Be as important onset reaction – Prompt and activation method
Reaction Rates Nuclear Reaction Rate: reaction cross section particle density of type X 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
Generalization to a Maxwell-Boltzmann velocity-distribution Inside a star, the particles clearly do not move with a mono-energetic velocity distribution. Instead, they have their own velocity distributions. Sun 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.
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
is entered to avoid double-counting of particle pairs if it should happen that 1 and 2 are the same species Generalization to a Maxwell-Boltzmann velocity-distribution After some calculation, including the change into CMS, one obtains: In terms of the relative energy (E=1/2 μv 2 ) this means
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
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
We generalize this to a Coulomb-potential by dividing the shape of the Coulomb-tail into thin slices of width. Cross-Section Reminder: If angular momentum not equal zero, then V(r) V(r) + centrifugal barrier Total transmission coefficient for s-wave:
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:
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. Cross-Section Quantum-mechanical interaction between two particles is always proportional to a geometrical factor: deBroglie wavelength Finally, our considerations lead to defining the cross-section at low energies as:
Cross-Section 12 C(p, ) 13 N 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.
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.
Gamow-Window This is where the action happens in thermonuclear burning! Log scale plot Linear scale plot We know that area under the curve This overlap function is approximated by a Gaussian curve: the Gamow-Window. The Gamow-Window provides the relevant energy range for the nuclear reaction.
Gamow-Window A Gaussian curve is characterized by its expectation value and its width : 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. 6 6
A given temperature defines the Gamow-window. For stars, inside the Gamow-window, S(E) is slowly varying. Astro-Physical S-Factor ( 12 C(p,g) 13 N) How does look like? Approximate the astro- physical factor by its value at :
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
PP-I Q eff = MeV p + p d + e + + p + d 3 He + 3 He + 3 He 4 He + 2p 86%14% 3 He + 4 He 7 Be He 7 Be + e - 7 Li + 7 Li + p 2 4 He 7 Be + p 8 B + 8 B 8 Be + e %0.3% PP-II Q eff = MeV PP-III Q eff = MeV Netto: 4p 4 He + 2e Q eff Proton-Proton-Chain
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.standard solar model
Homestake-Experiment experimental part by Davis: in Homestake Gold Mine, m underground (to protect from cosmic rays) 380 m 3 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.
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). 3 He + 4 He 7 Be + 7 Be + e - 7 Li + 7 Li + p 2 4 He 7 Be + p 8 B + 8 B 8 Be + e %0.3% Necessary to measure reaction rates at high accuracy. Here: with the help of 3 He(α,γ) 7 Be as the onset of neutrino- producing reactions
Motivation We will take a look at the 3 He(α,γ) 7 Be reaction as: The nuclear physics input from its cross section is a major uncertainty in the fluxes of 7 Be and 8 B neutrinos from the Sun predicted by Solar models As well: major uncertainty in 7 Li abundance obtained in big-bang nucleosynthesis calculations Critical link: important to know with high accuracy
Measuring the reaction rate of 3 He(α,γ) 7 Be Q= 1,586 MeV 429 keV There are two ways to measure that the 3 He(α,γ) 7 Be reaction occured: prompt γ method: measuring the γ´s emitted as the 7 Be* γ-decays into the 1st excited or the ground state activation method: measuring the γ´s that are emitted when the radioactive 7 Be decays
Basic Measuring Idea Experimentally we get the cross section over: where: the yield is the number of γ events counted N Beam is the number of beam particles counted ρ is the number of target particles per unit area
Background reduction 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 3 He(α,γ) 7 Be 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 surface
Luna accelerator detector target Credits to Matthias Junker at LNGS-INFN for making the LNGS picture available Laboratory for Underground Nuclear Astrophysics at Laborati Nazionali del Gran Sasso (LNGS)
Prompt-γ-Method Schematic view of the target chamber Experimental Set-Up
Prompt-γ-Method Measured γ-ray spectrum at Gran Sasso LUNA accelerator facility GS background signal 1st
Prompt-γ-Method Overview on available S-factor values and extrapolation
Activation Method Experimental Set-Up at Gran Sasso LUNA2 Schematic view of the target chamber used for the irradiations
Activation Method Offline γ-counting spectra from detector LNGS1
Activation Method Astrophysical S-factor at lower panel, uncertainties at upper panel
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)
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)