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Reaction Rates

Reaction Rates. Marialuisa Aliotta. School of Physics University of Edinburgh. principles of stellar structure and evolution general features of thermonuclear reactions experimental approach. Second European Summer School on Experimental Nuclear Astrophysics

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Reaction Rates

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  1. Reaction Rates Marialuisa Aliotta School of Physics University of Edinburgh • principles of stellar structure and evolution • general features of thermonuclear reactions • experimental approach Second European Summer School on Experimental Nuclear Astrophysics St. Tecla, Sept. 28th – Oct. 5th 2003

  2. The Macro-cosmos: some observables Luminosity vs. surface temperature Hertzsprung-Russel (HR) Diagram Surprise! SUPERGIANTS No chaos, but order! GIANTS Stefan´s law: L = 4R2T4 sun Luminosity WHITE DWARFS MAIN SEQUENCE  Temperature [K] • ~ 95% of all stars in MAIN SEQUENCE • highest probability of observing them in this stage (cf. adulthood for human beings) Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  3. The Macro-cosmos: some observables Mass-Luminosity relationship (for main sequence stars only) L ~ M4  more massive stars evolve more rapidly mass (Msun) lifetime (years) 1 ~1010 5 ~108 10 ~107 L, T, M cannot take up ANY values ORDER! Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  4. The Macro-cosmos: some observables Abundance curve of the elements Data sources: Earth, Moon, meteorites, stellar (Sun) spectra, cosmic rays... • Features: • distribution everywhere similar • 12 orders-of-magnitude span • H ~ 75% • He ~ 23% • C  U ~ 2% (“metals”) • D, Li, Be, B under-abundant • exponential decrease up to Fe • peak near Fe • almost flat distribution beyond Fe Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  5. Experimental nuclear astrophysics (EXPERIMENTAL) NUCLEAR ASTROPHYSICS • study energy generation processes in stars • study nucleosynthesis of the elements • What is the origin of the elements? • How do stars and galaxies form and evolve? • What powers the stars? • How old is the universe? • … MACRO-COSMOS intimately related to MICRO-COSMOS NUCLEAR PHYSICS KEY for understanding Courtesy: M. Arnould Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  6. Quiescent stages of stellar evolution Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  7. Principles of stellar structure and evolution: quiescent evolution • Stellar structure and evolution controlled by: • Gravity  collapse • Internal pressure  expansion Star composed of many particles (~1057 in the Sun) Total energy: a) mutual gravitational energy of particles () b) internal (kinetic) energy of particles (including photons) (U) For an ideal gas in hydrostatic equilibrium: 2U +  = 0 virial theorem Assume pressure imbalance gravitational contraction sets in amount of energy released - internal energy change to restore equilibrium U = - ½  gas temperature increases energy excess - ½  lost from star in form of radiation Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  8. Principles of stellar structure and evolution: quiescent evolution gravitational contraction of gas (mainly H) increase of central temperature if T high enough “nuclear burning”takes place HYDROGEN BURNING(1st equilibrium) 4H  4He + 2b+ + 2n + 26 MeV ash of nuclear burning energy source gravitational collapse is halted  star undergoes phase of hydrostatic equilibrium MAIN SEQUENCE STARS Here: T ~ 10 – 15 X106 K and r ~ 102 gcm-3 are required M > 0.1 M (Jupiter = failed star) Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  9. Principles of stellar structure and evolution: quiescent evolution Hydrogen burning Two main mechanisms: proton-proton chain and CNO cycle Energy production rate M < 1.5 M  T6 < 30  p-p chain M  1.5 M  T6 > 30  CNO cycle (also depends on CNO abundance) (Fiorentini’s lecture) Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  10. Principles of stellar structure and evolution: quiescent evolution H exhausted in core isothermal He core contraction sets in temperature increases H-burning shell R ~ 10-100 Ri Ts ~ 3-4x103 K Wien’s law: maxT = const. RED GIANT STARS contracting core expanding envelope when T ~ 108 K and r ~ 103 gcm-3(minimum mass ~ 0.5 M) HELIUM BURNING(2nd equilibrium) 3a 12C 12C(a,g)16O + 8 MeV energy source nuclear burning ashes Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  11. Principles of stellar structure and evolution: quiescent evolution 12C/16O BURNING … 12C ashes = Ne, Na, Mg … 16O ashes = Al, … Si major ash = 28Si SUPER RED-GIANT STARS … A = 40-65 28Si MELTING PRE-SUPERNOVA STARS major ash = 56Fe further reactions become endothermic final gravitational collapse SUPERNOVA EXPLOSION (type II) T, r M  8 M remnant: neutron star or black hole Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  12. Principles of stellar structure and evolution: summary Evolution stages of a 25 Mstar Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  13. Quiescent burning Pre-Supernova Super Giant ~ 3x104 y O, Ne … Red Giant ~ 3x108 y Main Sequence ~ 1010 y C He H Gravitational Contraction Principles of stellar structure and evolution: quiescent evolution Main parameters: 1) initial mass (  central temperature) 2) initial chemical composition (  nuclear processes) Energy generation rate  ~ Tn n ~ 4 (H-burning) n ~ 30 (C-burning) innermost regions only contribute to nuclear burning e.g. 1/10 M for H-burning less for subsequent stages H-burning  MAIN SEQUENCE longest stage of star’s lifetime Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  14. Explosive stages of stellar evolution Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  15. Principles of stellar structure and evolution: explosive evolution NOVAE = sudden increase in star’s luminosity (L ~ 104 – 106 Li and t ~ 1 h – 1 d) semi-detached binary system: White Dwarf + less evolved star (e.g. Red Giant) H-rich mass transfer from RG to WD degenerate matter  P and T uncoupled thermonuclear runaway  cataclysmic explosion temperature and density increase on WD’s surface (p,) and (,p) reactions on proton-rich nuclei T > 108 K  > 103 g cm-3 determine nature of nova phenomenon nucleosynthesis up to A ~ 60 mass region Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  16. Principles of stellar structure and evolution: explosive evolution X-RAY BURSTERS & X-RAY PULSARS semi-detached binary system: Neutron star + less evolved star T ~ 109 K  ~ 106 g cm-3 (,p) and (p,) reactions on proton-rich nuclei nucleosynthesis up to A ~ 80-100 mass region intense X-ray fluxes CORE-COLLAPSE SUPERNOVAE end stage of M ~ 8-30 Mstars core collapse & rebound shock wave outer layers blown off Neutron Star or Black Hole remnants T > 109 K n > 1020 g cm-3 “seed” nuclei in Fe region (n,) reactions on neutron-rich nuclei followed by  decays nucleosynthesis of n-rich elements through r-process Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  17. Stellar life cycle BIRTH gravitational contraction Interstellar gas Stars mixing of interstellar gas thermonuclear reactions • energy production • stability against collapse • synthesis of “metals” abundance distribution explosion DEATH Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  18. Thermonuclear reactions in stars Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  19. Q > 0 Q < 0 Thermonuclear reactions in stars: properties of nuclei Aston: measurements of atomic masses E = Mnc2 Mnucl < mp + mn enormous energy stored in nuclei! Rutherford (1919): discovery of nuclear reactions • liberate nuclear energy source • complex nuclides formed through reactions amount of energy liberated in nuclear reaction: Q =[(m1+m2)-(m3+m4)]c2 > 0 Binding energy curve spontaneous nuclear processes: Q > 0 fusion up to Fe region fission of heavy nuclei H most abundant element in the Universe FUSION reactions most effective in stars Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  20. Thermonuclear reactions in stars: general features & definitions Consider reaction: 1 + 2  3 + 4 Q12 > 0 (  known from atomic mass tables) Reaction cross section   probability for a reaction to occur Dimension: area Unit: barn (b) = 10-24 cm2 In general: not possible to determine reaction cross section from first principles However: • cross sections depend on nature of force involved • cross sections are energy (i.e. velocity) dependent Reaction rate: v(v) r= N1N2 Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  21. <v>12 = (E) exp E dE v(v)(v)dv <v>12 = Reaction rate per particle pair: (v) velocity distribution Thermonuclear reactions in stars: general features & definitions In stellar plasma: velocity of particles varies over wide range Quiescent stellar burning: non-relativistic, non-degenerate gas in thermodynamic equilibrium at temperature T Maxwell-Boltzmann distribution (v)  exp = exp Probability (E) (E)  exp(-E/kT) • = reduced mass v = relative velocity (E)  E kT Energy Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  22. = Total reaction rate: R12 = (1+12)-1 N1N2<v>12reactions cm-3 s-1 Ni = number density Thermonuclear reactions in stars: general features & definitions Energy production rate: 12 = R12 Q12 Mean lifetime of nuclei X against destruction by nuclei a energy production as star evolves change in abundance of nuclei X <v> = KEY quantity to be determined from experiments and/or theoretical considerations as star evolves, T changes  evaluate <sv> for each temperature NEED ANAYLITICAL EXPRESSION FOR ! Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  23. V Er+1 incident nucleus Er E 0  E2 r0 r E1 Thermonuclear reactions in stars: reaction mechanisms Consider reaction: a + X  b + Y (b = particle or photon) Non-resonant process One-step process leading to final nucleus Y   |<b+Y lHl a+X>|2 single matrix element • occurs at all interaction energies • cross section has WEAK energy dependence Resonant process Two-step process: 1) compound nucleus formation a + X  C* 2) decay of compound nucleus C* b + Y   |<b+Y lH’l C*>|2 |<C* lHl a+X>|2 two matrix elements • occurs at specific energies • cross section has STRONG energy dependence Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  24. nucleosynthesis up to Fe typically quiescent stages Reactions between charged particles Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  25. in numerical units: 2ph = 31.29 Z1Z2(m/E)½ m in amu and Ecm in keV Thermonuclear reactions in stars: charged particles charged particles Coulomb barrier energy available: from thermal motion V Coulomb potential Ekin ~ kT (keV) Ecoul ~ Z1Z2 (MeV) kT ~ 8.6 x 10-8 T[K] keV tunnel effect r0 r T ~ 15x106 K (e.g. our Sun)  kT ~ 1 keV T ~ 1010 K (Big Bang)  kT ~ 2 MeV nuclear well reactions occur through TUNNEL EFFECT tunneling probability P  exp(-2) during quiescent burnings: kT << Ec 2ph = GAMOW factor determines exponential drop in abundance curve! Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  26. Thermonuclear reactions in stars: non-resonant reactions Non-resonant reactions geometrical factor (particle’s de Broglie wavelength) interaction matrix element penetrability probabilitydepends on projectile’sangular momentum  and energy E (E) = exp(-2) S(E) (for s-waves only!) non-nuclear origin STRONG energy dependence nuclear origin WEAK energy dependence Above relation defines ASTROPHYSICAL S(E)-FACTOR N.B. If angular momentum is non zero  centrifugal barrier must also be taken into account Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  27. Maxwell-Boltzmann distribution  exp(-E/kT) tunnelling through Coulomb barrier  exp(- ) Gamow peak relative probability E0 energy kT E0 Thermonuclear reactions in stars: Gamow peak With above definition of cross section: <v>12 = S(E) exp dE f(E) varies smoothly with energy governs energy dependence MAXIMUM reaction rate: E0 < E0 only small energy range contributes to reaction rate  OK to set S(E) ~ S(E0) = const. Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  28. E0 ± E0/2 Thermonuclear reactions in stars: Gamow peak Gamow peak: most effective energy region for thermonuclear reactions energy window of astrophysical interest E0 = f(Z1, Z2, T) varies depending on reaction and/or temperature Examples: T ~ 15x106 K (T6 = 15) area of Gamow peak (height x width) ~ <v> STRONG sensitivity to Coulomb barrier separate stages: H-burning He-burning C/O-burning … Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  29.  << ER 1. Narrow resonances a b (E-Er)2 + (/2)2 (E)BW = π2(1+12) Experiment: determine and ER Thermonuclear reactions in stars: resonant reactions Resonant reactions Breit-Wigner formula insert in expression for reaction rate, integrate and get: <v>12 = exp (for single resonance) resonance strength (integrated cross section over resonant region) low-energy resonances (ER kT) dominate reaction rate Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  30. 2. Broad resonances  ~ ER Thermonuclear reactions in stars: resonant reactions Breit-Wigner formula + energy dependence of partial a(E), b(E) and total (E) widths N.B. Overlapping broad resonances of same Jπ interference effects Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  31. Thermonuclear reactions in stars: resonant reactions 3. Sub-threshold resonances any exited state has a finite width  ~ h/ high energy wing can extend above particle threshold cross section can be entirely dominated by contribution of sub-threshold state(s) Example: 12C(,)16O (Gialanella’s lecture & Schürmann’s talk) TOTAL REACTION RATE vtot = vr + vnr Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  32. nucleosynthesis beyond Fe typically explosive stages Reactions with neutrons Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  33. Thermonuclear reactions in stars: neutron captures NO Coulomb barrier neutrons produced in stars are quickly thermalised E0 ~ kT = relevant energy (e.g. T ~ 1-6x108 K  E0 ~ 30 keV) accounts for almost flatabundance distribution beyond iron peak <sv> ~ const = <sTvT> Typically: neutron-capture cross sections can be measured DIRECTLY at relevant energies Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  34. Experimental approach & laboratory requirements Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  35. Experimental approach: general features Quiescent burning stages of stellar evolution T ~ 106 - 108 KE0 ~ 100 keV << Ecoul  tunnel effect 10-18 barn <  < 10-9 barn  average interaction time  ~ <v>-1 ~ 109 y unstable species DO NOT play significant role FEATURES 10-18 b <  < 10-9 b poor signal-to-noise ratio  major experimental challenge  extrapolation procedure required PROBLEMS poor signal-to-noise ratio long measurements  ultra pure targets  high beam intensities  high detection efficiency REQUIREMENTS Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  36. (E) resonance LOG SCALE non-resonant many orders of magnitude direct measurements  E0 Ecoul Coulomb barrier extrapolation needed ! Experimental approach: extrapolation Experimental procedure measure (E) over as wide a range as possible, then EXTRAPOLATE down to Gamow energy region around E0! CROSS SECTION Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  37. Thermonuclear reactions in stars: non resonant reactions Example: cross section S-factor Data EXTRAPOLATION down to astrophysical energies REQUIRED! Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  38. low-energy tail of broad resonance Er Experimental approach: extrapolation S(E)-FACTOR S(E) extrapolation direct measurement LINEAR SCALE non resonant process sub-threshold resonance -Er 0 interaction energy E DANGER OF EXTRAPOLATION ! Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  39. Experimental approach: extrapolation ALTERNATIVE SOLUTIONS • Go UNDERGROUND  reduce (cosmic) background example: LUNA facility  (Junker’s lecture) • Use INDIRECT methods  (Figuera’s lecture) INTRINSIC LIMITATION At lower and lower energies ELECTRON SCREENING EFFECT sets in Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  40. The electron screening Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  41. (E) = S(E) exp(-2) Experimental approach: electron screening penetration through Coulomb barrier between BAREnuclei • in stellar plasmas:ions in sea of free electrons Ec bare Coulomb potential Debye-Hückel radius RD ~ (kT/)½ E + Ue screened E RD 0 Rn Rt Ue = electron screening potential Similarly: • in terrestrial laboratories: interaction between ions (projectiles) and atoms or molecules (target) Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  42. S(E) screened S(E) (E) fit to measured low-energy data  Ue screened high-energy data extrapolation bare S(E) bare 0 E E plasma(E) fplasma(E) = exp(Ue/E)  1 bare(E) Experimental approach: electron screening cross-section enhancement factor: BUT: electron screening in lab DIFFERENT fromelectron screening in plasma need to understand flab(E) improve calculation of fplasma(E) PROBLEM:experimental Ue>>theoretical Ue Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  43. Experimental approach: general features Explosive burning stages of stellar evolution T > 108 KE0 ~ 1 MeV ~ Ecoul 10-6 barn <  < 10-3 barn  NO extrapolation needed  average interaction time  ~ <v>-1 ~ seconds unstable species DO GOVERN nuclear processes FEATURES  ~ <v>-1 ~ seconds  unknown nuclear properties  low beam intensities (5-10 o.d.m. lower thanfor stable beams)  beam-induced background PROBLEMS unstable species RIBs production and acceleration  large area detectors  high detection efficiency REQUIREMENTS Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  44. Experimental approach: data needs NUCLEAR DATA NEEDS knowledge required: reactions involving: A < 30 A > 30 cross-section dependence: individual resonances nuclear properties statistical properties Hauser-Feshbach calculations excitation energies spin-parity & widths decay modes masses level densities part. separation energy experimental constraints wherever possible Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  45. Experimental approach: explosive nucleosynthesis EXAMPLES rp-process r-process rapid proton captures X(p,)Y rapid neutron captures X(n,)Y synthesis of neutron-rich nuclei A > 60 synthesis of proton-rich nuclei A ~ 100 proton capture neutron capture - decay + decay Z stable N unstable (Shotter’s lecture) Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  46. Overview of main astrophysical processes M.S. Smith and K.E. Rehm, Ann. Rev. Nucl. Part. Sci, 51 (2001) 91-130 Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  47. Maxwell-Boltzmann distribution  exp(-E/kT) tunnelling through Coulomb barrier  exp(- ) Gamow peak relative probability E0 energy E0 kT Thermonuclear reactions in stars: overview Hydrostatic equilibrium T ~ 106 - 108 K average interaction time  ~ <v>-1 ~ 109 y unstable species do not play significant role stellar reactions take place through TUNNEL effect kT << E0 << Ecoul 10-18 barn <  < 10-9 barn Extrapolation NEEDED Solutions: underground measurements indirect approaches BUT! Electron screening problem Explosive phenomena T > 108 K average interaction time  ~ <v>-1 ~ seconds unstable nuclei govern nuclear reaction processes • Sophisticated techniques for RIBs production and acceleration • Ad-hoc detection systems required E0 ~ Ecoul 10-6 barn <  < 10-3 barn Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

  48. Further reading W.D. Arnett and J.W. Truran Nucleosynthesis The University of Chicago Press, 1968 J. Audouze and S. Vauclair An introduction to Nuclear Astrophysics D. Reidel Publishing Company, Dordrecth, 1980 E. Böhm-Vitense Introduction to Stellar Astrophysics, vol. 3 Cambridge University Press, 1992 D.D. Clayton Principles of stellar evolution and nucleosynthesis The University of Chicago Press, 1983 H. Reeves Stellar evolution and Nucleosynthesis Gordon and Breach Sci. Publ., New York, 1968 C.E. Rolfs and W.S. Rodney Cauldrons in the Cosmos The University of Chicago Press, 1988 (…the “Bible”) Copies of this lecture at: www.ph.ed.ac.uk/~maliotta/teaching Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

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