2 Outline Theoretical background: Reaction ratesMaxwell-Boltzmann-distribution of velocityCross-sectionGamow-WindowExperimental application using the example of the pp2-chain reaction in the SunMotivation and some more theoryHistorical motivation3He(α,γ)7Be as important onset reactionPrompt and activation method
3 Reaction Rates Nuclear Reaction Rate: particle density of type X reaction cross sectionflux of particles of type a as seen by particles XImportant: 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 SunInside 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 speciesIn terms of the relative energy (E=1/2 μv2 ) this means
7 Cross-SectionThe 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 barrierrepulsive square-well potential
8 Cross-Section Radial Schrödinger equation for s-waves is solved by the ansatzThis leads to transmission coefficientfor low-energy s-wave transmission at a square-barrier potential
9 Cross-SectionWe generalize this to a Coulomb-potential by dividing the shape of the Coulomb-tail into thin slices of widthTotal 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-SectionQuantum-mechanical interaction between two particles is always proportional to a geometrical factor:deBroglie wavelengthWe 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-Section12C(p,g)13NThe 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-WindowEntering the cross section into the reaction rate, we obtain:withUsing mean value theorem for integration, we bring the equation to the formto pull out the essential physics/ evolve the Gamow-window.
14 Gamow-Window We know that area under the curve Log scale plotGamow-WindowWe know thatarea under the curveThis 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-WindowDA Gaussian curve is characterized by its expectation value and its width :66tells 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 Kmain fusion reactions to convert hydrogen into helium:proton-proton-chainCNO-cyclenuclear reactions in the Sun are non-resonant
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 Experimentpurpose: to collect and count neutrinos emitted by the nuclear fusion reactions inside the Suntheoretical 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 collideresult: only 1/3 of the predicted number of electron neutrinos were detectedSolar 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 + g99.7%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 reactions7Be + e- 7Li + n7Be + p 8B + g7Li + p 2 4He8B 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 modelsAs well: major uncertainty in 7Li abundance obtained in big-bang nucleosynthesis calculationsCritical link: important to know with high accuracy
23 Measuring the reaction rate of 3He(α,γ)7Be Q= 1,586 MeV429 keVThere 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 stateactivation 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 countedNBeam is the number of beam particles countedρ 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 eventusing 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) LunatargetacceleratordetectorCredits 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-MethodOverview 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 MethodOffline γ-counting spectra from detector LNGS1
32 Activation MethodAstrophysical S-factor at lower panel, uncertainties at upper panel
33 SummaryKnowing the temperature of e.g. the Sun, we can specify the relevant energy range for a nuclear reactionAn important reaction to research the interior of the Sun as well as big-bang nucleosynthesis is 3He(α,γ)7BeEnergies related to Sun temperatures are technically not feasible: extrapolation demands high accuracy measurementsNecessary to reduce backgroundThe weighted average over results of both methods (prompt and activation) provides an extrapolated S-factor of 𝑆 0 =0.560±0.017 𝑘𝑒𝑉
34 ReferencesDonald 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-FolieExample: 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)