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

β–Ά
nucleosynthesis 12 c 16 o
SMART_READER_LITE
LIVE PREVIEW

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

Sep 21, 2022 247 likes β€’553 views

Nucleosynthesis 12 C(, ) 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)

slide-1
SLIDE 1

Nucleosynthesis

12C(𝛽, 𝛿)16O

at MAGIX/MESA

Stefan Lunkenheimer MAGIX Collaboration Meeting 2017

slide-2
SLIDE 2

Topics

2

S-Factor Simulation Outlook

slide-3
SLIDE 3

S-Factor

3

slide-4
SLIDE 4
  • Hydrogen Burning (PPI-III & CNO Chain)
  • Fuel: proton
  • π‘ˆ β‰ˆ 2 β‹… 107 K
  • Main product: 4He
  • Helium Burning
  • Fuel: 4He
  • π‘ˆ β‰ˆ 2 β‹… 108 K
  • Main product: 12C, 16O

4

Stages of stellar nucleosynthesis

slide-5
SLIDE 5

Helium Burning in red giants

  • Main reactions:

3𝛽 β†’ 12C + 𝛿

12C 𝛽, 𝛿 16O

  • 12C/16O abundance ratio
  • Further burning states
  • Nucleosynthesis in massive stars

5

  • Cp. Hammache: 12C 𝛽, 𝛿 16O in massive star stellar evolution
slide-6
SLIDE 6
  • Fusion reaction below Coulomb barrier

π‘™π‘ˆ ∼ 15 keV @ π‘ˆ = 2 β‹… 108K

  • Transmission probability governed by

tunnel efffect

  • Gamow-Peak 𝐹0
  • Convolution of probability distribution
  • Maxwell-Boltxmann
  • QM Coulomb barrier transmission
  • Depends on reaction and temperature

6

Gamow-Peak

  • Cp. Marialuisa Aliotta: Exotic beam studies in Nuclear Astrophyiscs
slide-7
SLIDE 7

S-Factor

  • Nonresonant Cross section

𝜏 𝐹 = 1 𝐹 π‘“βˆ’2πœŒπ‘Ž1π‘Ž2𝛽𝑑

𝑀

𝑇(𝐹)

  • 𝑓 βˆ’ Factor = probability to tunnel through Coulomb barrier

𝑀 = velocity between the two nuclei 𝛽 = fine structure constant π‘Ž1, π‘Ž2 = Proton number of the nuclei

  • 𝑇 𝐹 = Deviation Factor from trivial model

7

slide-8
SLIDE 8

Gamow-Peak for 12C 𝛽, 𝛿 16O

  • Gamow-Peak (π‘ˆ β‰ˆ 2 β‹… 108K)

𝐹0 = 1 2 𝑐 β‹… 𝑙 β‹… π‘ˆ

2 3

β‰ˆ 300 keV

  • 𝑙 = Bolzmann constant
  • 𝑐 = πœŒπ›½π‘Ž1π‘Ž2 2πœˆπ‘‘2
  • 𝜈 =

𝑁1𝑁2 𝑁1+𝑁2 reduced mass

  • Gamow Width

Ξ” = 4 𝐹0π‘™π‘ˆ/3

8

𝑇 𝐹 = 𝐹 β‹… 𝑓𝑐/ 𝐹𝜏(𝐹)

slide-9
SLIDE 9

Cross section

  • 𝜏(𝐹0)~10βˆ’17barn
  • Precise low-energy

measurements required

  • MAGIX@MESA
  • Direct measurements never

done @𝐹cm < 0.9 MeV

9

  • Cp. Simulation of Ugalde 2013
slide-10
SLIDE 10

Approximate 𝑇(300 keV)

  • Buchmann (2005)
  • 102 βˆ’ 198 keVβ‹…b
  • Caughlan and Fowler (1988)
  • 120 βˆ’ 220 keV β‹… b
  • Hammer (2005)
  • 162 Β± 39 keVβ‹…b

10

Measurement of S-Factor

slide-11
SLIDE 11

Measurement at MAGIX@MESA

  • Time reverted reaction 16O(𝛿, 𝛽)12C
  • Cross section gain a factor of Γ— 100
  • Inelastic π‘“βˆ’ scattering on oxygen gas
  • Measurement of coincidence (π‘“βˆ’, 𝛽)
  • suppress background
  • 𝛽-Particle with low energy
  • High Luminosity

11

slide-12
SLIDE 12

Inverse Kinematik

12

  • Time reversed reaction:

𝜏(𝐹0)~10βˆ’15barn

  • High Energy resolution required
  • MAGIX
  • Cp. Simulation of Ugalde 2013

𝐹0

slide-13
SLIDE 13

Simulation

13

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

14

Introduction

slide-15
SLIDE 15

Kinematik

  • Momentum transfer

π‘Ÿ2 = βˆ’4𝐹𝐹′ sin2 πœ„

2

  • Photon Energy

πœ‰ =

𝑋2βˆ’π‘2βˆ’π‘Ÿ2 2𝑁

with

  • 𝑋2 = p𝛿

𝜈 + p𝑃 𝜈 2

invariant mass of photon and oxygen

  • 𝑁 = Oxygen mass
  • Inelastic scattering cross section

𝑒2𝜏 𝑒Ω𝑒𝐹′ = 4𝛽2𝐹′2 π‘Ÿ4 𝑋

2 π‘Ÿ2, πœ‰ β‹… cos2 πœ„

2 + 2𝑋

1 π‘Ÿ2, πœ‰ β‹… sin2 πœ„

2

15

slide-16
SLIDE 16

Relation beween structural functions and the transversal / longitudinal part of the virtual photon cross section πœπ‘ˆ, πœπ‘€

𝑋

1 = πœ† 4𝜌2𝛽 πœπ‘ˆ

𝑋

2 = πœ† 4𝜌2𝛽 1 βˆ’ πœ‰2 π‘Ÿ2 βˆ’1

(πœπ‘€ + πœπ‘ˆ) with πœ† =

𝑋2βˆ’π‘2 2𝑁 So we get

𝑒3𝜏 𝑒Ω𝑒𝐹′ = Ξ“ πœπ‘ˆ + πœπœπ‘€

with

Ξ“ =

π›½πœ† 2𝜌2 π‘Ÿ2 β‹… 𝐹′ 𝐹 β‹… 1 1βˆ’πœ

𝜁 = 1 βˆ’ 2

πœ‰2βˆ’π‘Ÿ2 π‘Ÿ2

tan2

πœ„ 2 βˆ’1

For π‘Ÿ2 β†’ 0 : πœπ‘€ vanish and πœπ‘ˆ β†’ 𝜏tot π›Ώβˆ— + 16O β†’ π‘Œ

𝑒5𝜏 π‘’Ξ©π‘“π‘’πΉβ€²π‘’Ξ©βˆ— = Ξ“ π‘’πœπ‘€ π‘’Ξ©βˆ—

16

Virtual Photon flux

  • Cp. Halzen & Martin: Quarks and Leptons
slide-17
SLIDE 17
  • Direct cross section -> Measurement
  • Compare with inverse cross section -> extract the S-Factor
  • Calculate time reversal factor

17

Time reversal Factor

slide-18
SLIDE 18

Time reversal Factor

Phase space examination under T-symmetry invariance

πœπ‘—β†’π‘” πœπ‘”β†’π‘— = (2𝐽3+1)(2𝐽4+1) (2𝐽1+1)(2𝐽2+1) β‹… | π‘ž|𝑔

2

| π‘ž|𝑗

2

Spinstatistic: I=0 for even–even nuclides ( 4He, 12C, 16O) in ground state 2𝐽𝛿 + 1 = 2 for photon. So we get 𝜏(16O 𝛿, 𝛽 12C) = 1 2 𝑋2 βˆ’ 𝑛He + 𝑛C

2

𝑋2 βˆ’ 𝑛He βˆ’ 𝑛C

2

𝑋2 βˆ’ 𝑛O

2

𝑋2 βˆ’ 𝑛O

2

β‹… 𝜏(12C(𝛽, 𝛿)16O)

18

  • Cp. Mayer-Kuckuk Kernphysik: Chapter 7.3
slide-19
SLIDE 19
  • Simulation correlate to the results of

Ugalde

  • 4𝜌 βˆ’ Simulation
  • ∼ 0.1 mHz Reaction Rate by 𝐹0 with

𝑀 ∼ 1034 π‘‘π‘›βˆ’2π‘‘βˆ’1

  • Worst case Luminosity (see later talks)
  • Now simulation with

π‘“βˆ’, 𝛽 βˆ’Acceptance needed.

19

Result of first simulations

Nonresonant cross section 𝜏(16𝑃(𝛿, 𝛽)12𝐷)

slide-20
SLIDE 20

Outlook

20

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

21

Simulation

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

22

𝛽-Detection

slide-23
SLIDE 23

THANK YOU FOR YOUR ATTENTION!

http://magix.kph.uni-mainz.de

slide-24
SLIDE 24

BACKUP

slide-25
SLIDE 25

Production factor

25

Waver and Woosley Phys Rep 227 (1993) 65

slide-26
SLIDE 26

Two-Body Reaction

In the center of mass frame 16𝑃(π›Ώβˆ—, 𝛽)12𝐷 𝐹3 =

𝑋2+𝑛3

2βˆ’π‘›4 2

2𝑋

𝐹4 =

𝑋2+𝑛4

2βˆ’π‘›3 2

2𝑋

π‘ž = 𝐹2 βˆ’ 𝑛2 =

𝑋2βˆ’ 𝑛3+𝑛4 2 𝑋2βˆ’ 𝑛3βˆ’π‘›4 2 2𝑋

26

slide-27
SLIDE 27

Electron scattering

Cross section inelastic scattering (cp. Chapter 7.2) 𝑒2𝜏 𝑒Ω𝑒𝐹′ = π‘’πœ 𝑒Ω

βˆ—

Mott 𝑋

2 π‘Ÿ2, πœ‰ + 2𝑋 1 π‘Ÿ2, πœ‰ tan2 πœ„

2 With structural functions 𝑋

1, 𝑋 2

And Mott crossection (in this case) π‘’πœ 𝑒Ω Mott

βˆ—

= 4𝛽2𝐹′2 π‘Ÿ4 cos2 πœ„ 2 We get (cp. Halzen & Martin Chapter 8) 𝑒2𝜏 𝑒Ω𝑒𝐹′ = 4𝛽2𝐹′2 π‘Ÿ4 𝑋

2 π‘Ÿ2, πœ‰ β‹… cos2 πœ„

2 + 2𝑋

1 π‘Ÿ2, πœ‰ β‹… sin2 πœ„

2

27

slide-28
SLIDE 28

Basic of Simulation

Connection between count rate and cross section 𝑂 =

Ξ©

𝐡 Ξ© d𝜏 𝑒Ω 𝑒Ω β‹… 𝑀𝑒𝑒 + 𝑂BG With 𝑀 : Luminosity 𝑂 : Number of counts 𝐡 Ξ© ∢ Acceptance (1 full accepted, 0 not detected)

28

slide-29
SLIDE 29

Definition of mean value in volume π‘Š: 𝑔 = 1 π‘Š

π‘Š

𝑔 𝑦 π‘’π‘œπ‘¦ Estimator for mean value: 𝑔 β‰ˆ 1 𝑂

𝑗=1 𝑂

𝑔(𝑦𝑗) Monte-Carlo Integration:

π‘Š

𝑔 𝑦 π‘’π‘œπ‘¦ = 𝑔 β‰ˆ π‘Š 𝑂

𝑗=1 𝑂

𝑔 𝑦𝑗 Β± π‘Š 𝑂 𝑔2 βˆ’ 𝑔 2 Strategies for numerical improvements:

  • Improve convergence 1/ 𝑂
  • Improve variance

𝑔2 βˆ’ 𝑔 2

29

Monte Carlo Integration

slide-30
SLIDE 30

π‘’πœ π‘’Ξ©π‘“π‘’πΉπ‘“π‘’Ξ©βˆ— π‘’Ξ©π‘“π‘’πΉπ‘“π‘’Ξ©βˆ— Transform Ξ©, 𝐹 β†’ 𝑋, 1/π‘Ÿ2, 𝜚 with det 𝐾 = π‘Ÿ4

𝑋 2𝑁𝐹𝐹′

With Monte-Carlo Integration: π‘’πœ π‘’Ξ©π‘“π‘’πΉπ‘“π‘’Ξ©βˆ— π‘’Ξ©π‘“π‘’πΉπ‘“π‘’Ξ©βˆ— = V N

𝑗

det 𝐾 β‹… π‘’πœ π‘’Ξ©π‘“π‘’πΉπ‘“π‘’Ξ©βˆ— 𝑋, 1/π‘Ÿ2, 𝜚, Ξ©βˆ— Define πœ•π‘— = π‘Š β‹… det 𝐾 β‹…

π‘’πœ π‘’Ξ©π‘“π‘’πΉπ‘“π‘’Ξ©βˆ—

So we get πœ•π‘— = π‘Ÿ4 𝑋 2𝑁𝐹𝐹′ β‹… 𝑀 β‹… Ξ”πœš β‹… Δ𝑋 β‹… Ξ” cos πœ„βˆ— β‹… Ξ”πœšβˆ— β‹… Ξ“ β‹… π‘’πœπ‘€ π‘’Ξ©βˆ—

30

Cross section simulation