Nucleosynthesis 12 C(, ) 16 O at MAGIX/MESA Stefan Lunkenheimer - PowerPoint PPT Presentation

Nucleosynthesis 12 C(𝛽, 𝛿) 16 O at MAGIX/MESA Stefan Lunkenheimer MAGIX Collaboration Meeting 2017

Topics S-Factor Simulation Outlook 2

S-Factor 3

Stages of stellar nucleosynthesis • Hydrogen Burning (PPI-III & CNO Chain) • Fuel: proton • 𝑈 ≈ 2 ⋅ 10 7 K • Main product: 4 He • Helium Burning • Fuel: 4 He • 𝑈 ≈ 2 ⋅ 10 8 K • Main product: 12 C , 16 O 4

Helium Burning in red giants • Main reactions: 3𝛽 → 12 C + 𝛿 12 C 𝛽, 𝛿 16 O • 12 C/ 16 O abundance ratio • Further burning states • Nucleosynthesis in massive stars Cp. Hammache: 12 C 𝛽, 𝛿 16 O in massive star stellar evolution 5

Gamow-Peak • Fusion reaction below Coulomb barrier 𝑙𝑈 ∼ 15 keV @ 𝑈 = 2 ⋅ 10 8 K • Transmission probability governed by tunnel efffect • Gamow-Peak 𝐹 0 • Convolution of probability distribution  Maxwell-Boltxmann  QM Coulomb barrier transmission • Depends on reaction and temperature Cp. Marialuisa Aliotta: Exotic beam studies in Nuclear Astrophyiscs 6

S-Factor • Nonresonant Cross section 𝜏 𝐹 = 1 𝐹 𝑓 −2𝜌𝑎 1 𝑎 2 𝛽𝑑 𝑇(𝐹) 𝑤 • 𝑓 − Factor = probability to tunnel through Coulomb barrier 𝑤 = velocity between the two nuclei 𝛽 = fine structure constant 𝑎 1 , 𝑎 2 = Proton number of the nuclei • 𝑇 𝐹 = Deviation Factor from trivial model 7

Gamow-Peak for 12 C 𝛽, 𝛿 16 O 𝑇 𝐹 = 𝐹 ⋅ 𝑓 𝑐/ 𝐹 𝜏(𝐹) • Gamow-Peak ( 𝑈 ≈ 2 ⋅ 10 8 K ) 2 1 3 𝐹 0 = 2 𝑐 ⋅ 𝑙 ⋅ 𝑈 ≈ 300 keV • 𝑙 = Bolzmann constant • 𝑐 = 𝜌𝛽𝑎 1 𝑎 2 2𝜈𝑑 2 𝑁 1 𝑁 2 𝑁 1 +𝑁 2 reduced mass • 𝜈 = • Gamow Width Δ = 4 𝐹 0 𝑙𝑈/3 8

Cross section • 𝜏(𝐹 0 )~10 −17 barn • Precise low-energy measurements required  MAGIX@MESA • Direct measurements never done @ 𝐹cm < 0.9 MeV Cp. Simulation of Ugalde 2013 9

Measurement of S-Factor Approximate 𝑇(300 keV) • Buchmann (2005) • 102 − 198 keV⋅b • Caughlan and Fowler (1988) • 120 − 220 keV ⋅ b • Hammer (2005) • 162 ± 39 keV⋅b 10

Measurement at MAGIX@MESA • Time reverted reaction 16 O(𝛿, 𝛽) 12 C  Cross section gain a factor of × 100 • Inelastic 𝑓 − scattering on oxygen gas • Measurement of coincidence ( 𝑓 − , 𝛽 )  suppress background  𝛽 -Particle with low energy • High Luminosity 11

Inverse Kinematik • Time reversed reaction: 𝜏(𝐹 0 )~10 −15 barn • High Energy resolution required  MAGIX 𝐹 0 Cp. Simulation of Ugalde 2013 12

Simulation 13

Introduction • MXWare (see talk Caiazza) • Monte Carlo Integration • Fix Beam Energy • Target at Rest • Simulation acceptance 4𝜌 14

Kinematik • Momentum transfer 𝑟 2 = −4𝐹𝐹 ′ sin 2 𝜄 2 • Photon Energy 𝑋 2 −𝑁 2 −𝑟 2 𝜉 = 2𝑁 with 𝜈 + p 𝑃 𝜈 2 𝑋 2 = p 𝛿 • invariant mass of photon and oxygen 𝑁 = Oxygen mass • • Inelastic scattering cross section 𝑒Ω𝑒𝐹′ = 4𝛽 2 𝐹 ′2 𝑒 2 𝜏 2 𝑟 2 , 𝜉 ⋅ cos 2 𝜄 1 𝑟 2 , 𝜉 ⋅ sin 2 𝜄 𝑋 + 2𝑋 𝑟 4 2 2 15

Virtual Photon flux Relation beween structural functions and the transversal / longitudinal part of the virtual photon cross section 𝜏 𝑈 , 𝜏 𝑀 −1 𝜉 2 𝑋 2 −𝑁 2 𝜆 𝜆 with 𝜆 = 𝑋 1 = 4𝜌 2 𝛽 𝜏 𝑈 𝑋 2 = 4𝜌 2 𝛽 1 − (𝜏 𝑀 + 𝜏 𝑈 ) 𝑟 2 2𝑁 So we get 𝑒 3 𝜏 𝑒Ω𝑒𝐹 ′ = Γ 𝜏 𝑈 + 𝜁𝜏 𝑀 with −1 𝐹 ′ 𝜉 2 −𝑟 2 𝛽𝜆 1 𝜄 tan 2 Γ = 2𝜌 2 𝑟 2 ⋅ 𝐹 ⋅ 𝜁 = 1 − 2 𝑟 2 1−𝜁 2 For 𝑟 2 → 0 : 𝜏 𝑀 vanish and 𝜏 𝑈 → 𝜏 tot 𝛿 ∗ + 16 O → 𝑌 𝑒 5 𝜏 𝑒Ω 𝑓 𝑒𝐹 ′ 𝑒Ω ∗ = Γ 𝑒𝜏 𝑤 𝑒Ω ∗ Cp. Halzen & Martin: Quarks and Leptons 16

Time reversal Factor • Direct cross section -> Measurement • Compare with inverse cross section -> extract the S-Factor • Calculate time reversal factor 17

Time reversal Factor Phase space examination under T-symmetry invariance 2 | 𝑞| 𝑔 𝜏 𝑗→𝑔 (2𝐽 3 +1)(2𝐽 4 +1) 𝜏 𝑔→𝑗 = (2𝐽 1 +1)(2𝐽 2 +1) ⋅ 2 | 𝑞| 𝑗 Spinstatistic : I=0 for even – even nuclides ( 4 He, 12 C, 16 O ) in ground state for photon. 2𝐽 𝛿 + 1 = 2 So we get 2 2 𝑋 2 − 𝑛He + 𝑛C 𝑋 2 − 𝑛He − 𝑛C 𝜏( 16 O 𝛿, 𝛽 12 C) = 1 ⋅ 𝜏( 12 C(𝛽, 𝛿) 16 O) 𝑋 2 − 𝑛O 𝑋 2 − 𝑛O 2 2 2 Cp. Mayer-Kuckuk Kernphysik: Chapter 7.3 18

Result of first simulations Nonresonant cross section 𝜏( 16 𝑃(𝛿, 𝛽) 12 𝐷) • Simulation correlate to the results of Ugalde • 4𝜌 − Simulation • ∼ 0.1 mHz Reaction Rate by 𝐹 0 with 𝑀 ∼ 10 34 𝑑𝑛 −2 𝑡 −1  Worst case Luminosity (see later talks) • Now simulation with 𝑓 − , 𝛽 − Acceptance needed. 19

Outlook 20

Simulation • Finish simulation  electron acceptance  𝛽 -Particle acceptance • Preliminary results  Need measurement on angles smaller than Spectrometer coverage  0 degree scattering -> New Theoretic calculations 21

𝛽 -Detection • Low kinetic energy  ∼ 20 MeV • Needs specialized detector  Silicon-Strip-Detector • Choose and Test Silicon-Strip-Detectors in the Lab 22

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BACKUP

Production factor Waver and Woosley Phys Rep 227 (1993) 65 25

Two-Body Reaction In the center of mass frame 16 𝑃(𝛿 ∗ , 𝛽) 12 𝐷 𝑋 2 +𝑛 3 2 −𝑛 4 2 𝑋 2 +𝑛 4 2 −𝑛 3 2 𝐹 3 = 𝐹 4 = 2𝑋 2𝑋 𝑋 2 − 𝑛 3 +𝑛 4 2 𝑋 2 − 𝑛 3 −𝑛 4 2 𝐹 2 − 𝑛 2 = 𝑞 = 2𝑋 26

Electron scattering Cross section inelastic scattering (cp. Chapter 7.2) ∗ 𝑒 2 𝜏 𝑒𝜏 1 𝑟 2 , 𝜉 tan 2 𝜄 2 𝑟 2 , 𝜉 + 2𝑋 𝑒Ω𝑒𝐹′ = 𝑋 𝑒Ω 2 Mott With structural functions 𝑋 1 , 𝑋 2 And Mott crossection (in this case) ∗ = 4𝛽 2 𝐹 ′2 𝑒𝜏 cos 2 𝜄 𝑟 4 𝑒Ω Mott 2 We get (cp. Halzen & Martin Chapter 8) 𝑒Ω𝑒𝐹′ = 4𝛽 2 𝐹 ′2 𝑒 2 𝜏 2 𝑟 2 , 𝜉 ⋅ cos 2 𝜄 1 𝑟 2 , 𝜉 ⋅ sin 2 𝜄 𝑋 + 2𝑋 𝑟 4 2 2 27

Basic of Simulation Connection between count rate and cross section 𝐵 Ω d𝜏 𝑂 = 𝑒Ω 𝑒Ω ⋅ 𝑀𝑒𝑢 + 𝑂BG Ω With 𝑀 : Luminosity 𝑂 : Number of counts 𝐵 Ω ∶ Acceptance (1 full accepted, 0 not detected) 28

Monte Carlo Integration Definition of mean value in volume 𝑊 : 𝑔 = 1 𝑔 𝑦 𝑒 𝑜 𝑦 𝑊 𝑊 Estimator for mean value: 𝑂 𝑔 ≈ 1 𝑂 𝑔(𝑦 𝑗 ) 𝑗=1 Monte-Carlo Integration: 𝑂 𝑔 𝑦 𝑒 𝑜 𝑦 = 𝑔 ≈ 𝑊 ± 𝑊 𝑔 2 − 𝑔 2 𝑂 𝑔 𝑦 𝑗 𝑂 𝑊 𝑗=1 Strategies for numerical improvements: • Improve convergence 1/ 𝑂 𝑔 2 − 𝑔 2 • Improve variance 29

Cross section simulation 𝑒𝜏 𝑒Ω 𝑓 𝑒𝐹 𝑓 𝑒Ω ∗ 𝑒Ω 𝑓 𝑒𝐹 𝑓 𝑒Ω ∗ 𝑋 Transform Ω, 𝐹 with → 𝑋, 1/𝑟 2 , 𝜚 det 𝐾 = 𝑟 4 2𝑁𝐹𝐹 ′ With Monte-Carlo Integration: 𝑒Ω 𝑓 𝑒𝐹 𝑓 𝑒Ω ∗ 𝑒Ω 𝑓 𝑒𝐹 𝑓 𝑒Ω ∗ = V 𝑒𝜏 𝑒𝜏 𝑋, 1/𝑟 2 , 𝜚, Ω ∗ N det 𝐾 ⋅ 𝑒Ω 𝑓 𝑒𝐹 𝑓 𝑒Ω ∗ 𝑗 𝑒𝜏 Define 𝜕 𝑗 = 𝑊 ⋅ det 𝐾 ⋅ 𝑒Ω 𝑓 𝑒𝐹 𝑓 𝑒Ω ∗ So we get 2𝑁𝐹𝐹 ′ ⋅ 𝑤 ⋅ Δ𝜚 ⋅ Δ𝑋 ⋅ Δ cos 𝜄 ∗ ⋅ Δ𝜚 ∗ ⋅ Γ ⋅ 𝑒𝜏 𝑤 𝑋 𝜕 𝑗 = 𝑟 4 𝑒Ω ∗ 30