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

Topics S-Factor Simulation Outlook

S-Factor

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

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

Gamow-Peak Fusion reaction below Coulomb barrier 𝑘𝑇∼15 keV @ 𝑇=2⋅ 10 8 K Transmission probability governed by tunnel efffect Gamow-Peak 𝐸 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

S-Factor Nonresonant Cross section 𝜎 𝐸 = 1 𝐸 𝑒 − 2𝜋 𝑍 1 𝑍 2 𝛼𝑐 𝑣 𝑆(𝐸) 𝜎 𝐸 = 1 𝐸 𝑒 − 2𝜋 𝑍 1 𝑍 2 𝛼𝑐 𝑣 𝑆(𝐸) 𝑒− Factor = probability to tunnel through Coulomb barrier 𝑣= velocity between the two nuclei 𝛼= fine structure constant 𝑍 1 ,𝑍 2 = Proton number of the nuclei 𝑆 𝐸 = Deviation Factor from trivial model

Gamow-Peak for 12 C 𝛼,𝛾 16 O Gamow-Peak (𝑇≈2⋅ 10 8 K) 𝑆 𝐸 =𝐸⋅ 𝑒 𝑏/ 𝐸 𝜎(𝐸) Gamow-Peak (𝑇≈2⋅ 10 8 K) 𝐸 0 = 1 2 𝑏⋅𝑘⋅𝑇 2 3 ≈300 keV 𝑘= Bolzmann constant 𝑏=𝜋𝛼 𝑍 1 𝑍 2 2𝜇 𝑐 2 𝜇= 𝑀 1 𝑀 2 𝑀 1 + 𝑀 2 reduced mass Gamow Width Δ=4 𝐸 0 𝑘𝑇/3

Cross section 𝜎( 𝐸 0 )~ 10 −17 barn Precise low-energy measurements required MAGIX@MESA Direct measurements never done @ 𝐸 cm <0.9 MeV Cp. Simulation of Ugalde 2013

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

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

Inverse Kinematik Time reversed reaction: 𝜎( 𝐸 0 )~ 10 −15 barn High Energy resolution required MAGIX 𝐸 0 Cp. Simulation of Ugalde 2013

Simulation

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

Kinematik Momentum transfer 𝑞 2 =−4𝐸 𝐸 ′ sin 2 𝜃 2 Photon Energy 𝜈= 𝑊 2 − 𝑀 2 − 𝑞 2 2𝑀 with 𝑊 2 = p 𝛾 𝜇 + p 𝑂 𝜇 2 invariant mass of photon and oxygen 𝑀= Oxygen mass Inelastic scattering cross section 𝑑 2 𝜎 𝑑Ω𝑑𝐸′ = 4𝛼 2 𝐸 ′2 𝑞 4 𝑊 2 𝑞 2 ,𝜈 ⋅ cos 2 𝜃 2 +2 𝑊 1 𝑞 2 ,𝜈 ⋅sin 2 𝜃 2

Virtual Photon flux Relation beween structural functions and the transversal / longitudinal part of the virtual photon cross section 𝜎 𝑇 , 𝜎 𝐿 𝑊 1 = 𝜅 4 𝜋 2 𝛼 𝜎 𝑇 𝑊 2 = 𝜅 4 𝜋 2 𝛼 1− 𝜈 2 𝑞 2 −1 ( 𝜎 𝐿 + 𝜎 𝑇 ) with 𝜅= 𝑊 2 − 𝑀 2 2𝑀 So we get 𝑑 3 𝜎 𝑑Ω𝑑 𝐸 ′ =Γ 𝜎 𝑇 +𝜀 𝜎 𝐿 with Γ= 𝛼𝜅 2 𝜋 2 𝑞 2 ⋅ 𝐸 ′ 𝐸 ⋅ 1 1−𝜀 𝜀= 1−2 𝜈 2 − 𝑞 2 𝑞 2 tan 2 𝜃 2 −1 For 𝑞 2 →0 : 𝜎 𝐿 vanish and 𝜎 𝑇 → 𝜎 tot 𝛾 ∗ + 16 O →𝑋 𝑑 5 𝜎 𝑑 Ω 𝑒 𝑑 𝐸 ′ 𝑑 Ω ∗ =Γ 𝑑 𝜎 𝑣 𝑑 Ω ∗ Cp. Halzen & Martin: Quarks and Leptons

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

Time reversal Factor Phase space examination under T-symmetry invariance 𝜎 𝑖→𝑓 𝜎 𝑓→𝑖 = (2 𝐼 3 +1)(2 𝐼 4 +1) (2 𝐼 1 +1)(2 𝐼 2 +1) ⋅ | 𝑝 | 𝑓 2 | 𝑝 | 𝑖 2 Spinstatistic: I=0 for even–even nuclides ( 4 He, 12 C, 16 O ) in ground state 2 𝐼 𝛾 +1 =2 for photon. So we get 𝜎( 16 O 𝛾,𝛼 12 C) = 1 2 𝑊 2 − 𝑚 He + 𝑚 C 2 𝑊 2 − 𝑚 He − 𝑚 C 2 𝑊 2 − 𝑚 O 2 𝑊 2 − 𝑚 O 2 ⋅𝜎( 12 C (𝛼,𝛾) 16 O) Cp. Mayer-Kuckuk Kernphysik: Chapter 7.3

Result of first simulations Nonresonant cross section 𝜎( 16 𝑂(𝛾,𝛼) 12 𝐶) Simulation correlate to the results of Ugalde 4𝜋− Simulation ∼0.1 mHz Reaction Rate by 𝐸 0 with 𝐿∼ 10 34 𝑐 𝑚 −2 𝑠 −1 Worst case Luminosity (see later talks) Now simulation with 𝑒 − , 𝛼−Acceptance needed.

Outlook

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

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

BACKUP

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

Two-Body Reaction In the center of mass frame 16 𝑂( 𝛾 ∗ ,𝛼) 12 𝐶 𝐸 3 = 𝑊 2 + 𝑚 3 2 − 𝑚 4 2 2𝑊 𝐸 4 = 𝑊 2 + 𝑚 4 2 − 𝑚 3 2 2𝑊 𝑝 = 𝐸 2 − 𝑚 2 = 𝑊 2 − 𝑚 3 + 𝑚 4 2 𝑊 2 − 𝑚 3 − 𝑚 4 2 2𝑊

Electron scattering Cross section inelastic scattering (cp. Chapter 7.2) 𝑑 2 𝜎 𝑑Ω𝑑𝐸′ = 𝑑𝜎 𝑑Ω ∗ Mott 𝑊 2 𝑞 2 ,𝜈 +2 𝑊 1 𝑞 2 ,𝜈 tan 2 𝜃 2 With structural functions 𝑊 1 , 𝑊 2 And Mott crossection (in this case) 𝑑𝜎 𝑑Ω Mott ∗ = 4𝛼 2 𝐸 ′2 𝑞 4 cos 2 𝜃 2 We get (cp. Halzen & Martin Chapter 8) 𝑑 2 𝜎 𝑑Ω𝑑𝐸′ = 4𝛼 2 𝐸 ′2 𝑞 4 𝑊 2 𝑞 2 ,𝜈 ⋅ cos 2 𝜃 2 +2 𝑊 1 𝑞 2 ,𝜈 ⋅sin 2 𝜃 2

Basic of Simulation Connection between count rate and cross section 𝑁= Ω 𝐴 Ω d𝜎 𝑑Ω 𝑑Ω⋅ 𝐿𝑑𝑡+ 𝑁 BG With 𝐿 : Luminosity 𝑁 : Number of counts 𝐴 Ω : Acceptance (1 full accepted, 0 not detected)

Monte Carlo Integration Definition of mean value in volume 𝑉: 𝑓 = 1 𝑉 𝑉 𝑓 𝑥 𝑑 𝑛 𝑥 Estimator for mean value: 𝑓 ≈ 1 𝑁 𝑖=1 𝑁 𝑓( 𝑥 𝑖 ) Monte-Carlo Integration: 𝑉 𝑓 𝑥 𝑑 𝑛 𝑥 = 𝑓 ≈ 𝑉 𝑁 𝑖=1 𝑁 𝑓 𝑥 𝑖 ± 𝑉 𝑁 𝑓 2 − 𝑓 2 Strategies for numerical improvements: Improve convergence 1/ 𝑁 Improve variance 𝑓 2 − 𝑓 2

Cross section simulation 𝑑𝜎 𝑑 Ω 𝑒 𝑑 𝐸 𝑒 𝑑 Ω ∗ 𝑑 Ω 𝑒 𝑑 𝐸 𝑒 𝑑 Ω ∗ Transform Ω,𝐸 → 𝑊,1/ 𝑞 2 ,𝜙 with det 𝐽= 𝑞 4 𝑊 2𝑀𝐸 𝐸 ′ With Monte-Carlo Integration: 𝑑𝜎 𝑑 Ω 𝑒 𝑑 𝐸 𝑒 𝑑 Ω ∗ 𝑑 Ω 𝑒 𝑑 𝐸 𝑒 𝑑 Ω ∗ = V N 𝑖 det 𝐽⋅ 𝑑𝜎 𝑑 Ω 𝑒 𝑑 𝐸 𝑒 𝑑 Ω ∗ 𝑊,1/ 𝑞 2 ,𝜙, Ω ∗ Define 𝜔 𝑖 =𝑉⋅ det 𝐽⋅ 𝑑𝜎 𝑑 Ω 𝑒 𝑑 𝐸 𝑒 𝑑 Ω ∗ So we get 𝜔 𝑖 = 𝑞 4 𝑊 2𝑀𝐸 𝐸 ′ ⋅𝑣⋅Δ𝜙⋅Δ𝑊⋅Δ cos 𝜃 ∗ ⋅Δ 𝜙 ∗ ⋅Γ ⋅ 𝑑 𝜎 𝑣 𝑑 Ω ∗