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

2. Topics
S-Factor
Simulation
Outlook

3. S-Factor

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

5. 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

6. Gamow-Peak
Fusion reaction below Coulomb barrier 𝑘𝑇∼15 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

7. 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

8. 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 = 𝑏⋅𝑘⋅𝑇 ≈300 keV
𝑘= Bolzmann constant
𝑏=𝜋𝛼 𝑍 1 𝑍 2 2𝜇 𝑐 2
𝜇= 𝑀 1 𝑀 2 𝑀 1 + 𝑀 reduced mass
Gamow Width
Δ=4 𝐸 0 𝑘𝑇/3

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

10. 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

11. 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

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

13. Simulation

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

15. 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 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 𝑊 1 𝑞 2 ,𝜈 ⋅sin 2 𝜃 2

16. 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 −1
For 𝑞 2 →0 : 𝜎 𝐿 vanish and 𝜎 𝑇 → 𝜎 tot 𝛾 ∗ O →𝑋
𝑑 5 𝜎 𝑑 Ω 𝑒 𝑑 𝐸 ′ 𝑑 Ω ∗ =Γ 𝑑 𝜎 𝑣 𝑑 Ω ∗
Cp. Halzen & Martin: Quarks and Leptons

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

18. 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) = 𝑊 2 − 𝑚 He + 𝑚 C 𝑊 2 − 𝑚 He − 𝑚 C 𝑊 2 − 𝑚 O 2 𝑊 2 − 𝑚 O 2 ⋅𝜎( 12 C (𝛼,𝛾) 16 O)
Cp. Mayer-Kuckuk Kernphysik: Chapter 7.3

19. 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∼ 𝑐 𝑚 −2 𝑠 −1
Worst case Luminosity (see later talks)
Now simulation with 𝑒 − , 𝛼−Acceptance needed.

20. Outlook

21. 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

22. 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

24. BACKUP

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

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

27. 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

28. 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)

29. 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

30. 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 ∗