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Exotic beam studies in Nuclear Astrophysics

Exotic beam studies in Nuclear Astrophysics. Marialuisa Aliotta School of Physics - University of Edinburgh. 11th Euro Summer School on Exotic Beams – Guilford, August 19-27 2004. Lectures layout. introduction the need for RIBs in Astrophysics the tools of the trade

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Exotic beam studies in Nuclear Astrophysics

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  1. Exotic beam studies in Nuclear Astrophysics Marialuisa Aliotta School of Physics - University of Edinburgh 11th Euro Summer School on Exotic Beams – Guilford, August 19-27 2004

  2. Lectures layout • introduction • the need for RIBs in Astrophysics • the tools of the trade • the synthesis of the trans-iron elements • the s-process • the r-process • explosive hydrogen burning • novae and X-ray bursts • the rp- and ap-processes • experimental investigations • techniques and detection systems • selected examples

  3. why does one kilogram of gold cost so much more than one kilogram of iron? Abundance curve of the elements Fe 7 orders of magnitude less abundant! Au WHY? “The 11 Greatest Unanswered Questions of Physics” based on National Academy of Science Report, 2002 [Committee for the Physics of the Universe (CPU)] Question 3 How were the elements from iron to uranium made?

  4. nucleosynthesis processes The Synthesis of the Elements in Stars Burbidge, Burbidge, Fowler & Hoyle (B2FH): Rev. Mod. Phys. 29 (1957) 547 from: M. Wiescher, JINA lectures on Nuclear Astrophysics

  5. nuclear processes charged-particle induced reaction mainly neutron capture reaction during quiescent stages of stellar evolution mainly during explosive stages of stellar evolution involve mainly STABLE NUCLEI involve mainly UNSTABLE NUCLEI

  6. Overview of main astrophysical processes M.S. Smith and K.E. Rehm, Ann. Rev. Nucl. Part. Sci, 51 (2001) 91-130 the vast majority of reactions encountered in these processes involve UNSTABLE species hence the need for Radioactive Ion Beams

  7. Lecture 1: The tools of the trade YEAH, LIKE WHAT MAKES ASTRONOMY DIFFERENT FROM ASTROLOGY? WELCOME TO BASIC ASTRONOMY. BEFORE WE START, ARE THERE ANY QUESTIONS? LOTS AND LOTS OF MATHS http://www.univie.ac.at/strv-astronomie/unterhaltung.html • notations and definitions • reaction mechanisms • reaction cross sections • stellar reaction rates

  8. nuclear reaction rates nuclear reactions in stars: a) produce energy b) synthesise elements stars = cooking pots of the Universe for reaction: T(p,e)R (T=target, p=projectile, e=ejectile, R=recoil) reaction Q-value: Q=[(mp+mT)-(me+mR)]c2 (energy per single reaction) if Q > 0  net production of energy • reaction rate r • (number of reactions per unit time and volume) Ni = number density of interacting species v = relative velocity f(v) = velocity distribution in plasma s(v) = reaction cross section KEY QUANTITY • energy production rate e e = rQ/r typical units: MeV g-1 s-1

  9. example of nuclear reactions in stars CNO cycle 15 O (p,g) 13 14 15 N (e+n) (p,a) C 12 13 6 7 8 12C(p,g)13N(e+)13C(p,g)14N(p,g)15O(e+)15N(p,a)12C cycle limited by  decay of 13N (t ~ 10 min) and 15O (t ~ 2 min) CNO isotopes act as catalysts net result: 4p  4He + 2e+ + 2n + Qeff Qeff= 26.73 MeV nucleosynthesis energy production changes in stellar conditions  changes in energy production and nucleosynthesis need to knowREACTION RATEat all temperatures to determineENERGY PRODUCTION

  10. 3 is produced 1 is destroyed abundance changes and lifetimes reactions are random processes with constant probability (cross section) for given conditions  abundance change is governed by same laws of radioactive decay consider reaction 1+2  3 where 1 is destroyed through capture of 2 and 3 is produced example 15 O 13 14 15 N C 12 13 6 7 8 define: lifetime of 1 against destruction by with 2: need to knowREACTION RATEat all temperatures to determineNUCLEOSYNTHESIS

  11. stellar reaction rate need: a) velocity distribution b) cross section a) velocity distribution • interacting nuclei in plasma are in thermal equilibrium at temperature T • also assume non-degenerate and non-relativistic plasma • Maxwell-Boltzmann velocity distribution reduced mass with v = relative velocity kT ~ 8.6 x 10-8 T[K] keV example: Sun T ~ 15x106 K  kT ~ 1 keV

  12. reaction cross sections b) cross section no nuclear theory available to determine reaction cross section a priori • depends sensitively on: • the properties of the nuclei involved • the reaction mechanism and can vary by orders of magnitude, depending on the interaction examples: 1 barn = 10-24 cm2 = 100 fm2 in practice, need experiments AND theory to determine stellar reaction rates

  13. nuclear properties relevant to reaction rates recall: nucleons in nuclei arranged in quantised shells of given energy  nucleus’s configuration as a whole corresponds to discrete energy levels excitation energy [MeV] spin parity Jp example 4.97 2- 3rd excited state 4.25 4+ 2nd excited state excitation energy 1.63 2+ 1st excited state 0 0+ ground state 0 20Ne any nucleus in an excited state will eventually decay either by g, p, n ora emission with a characteristic lifetime twhich corresponds to a width G in the excitation energy of the state Heisenberg’s relationship the lifetime for each individual exit channel is usually given in terms of partial widths with Gg, Gp, Gn and Ga

  14. reaction mechanisms: I. direct reactions II. resonant reactions

  15. reaction mechanisms I. direct process one-step process direct transition into a bound state example: radiative capture A(x,g)B Ecm g A+x Q B Hg = electromagnetic operator describing the transition • reaction cross section proportional to single matrix element • can occur at all projectile energies • smooth energy dependence of cross section other direct processes: stripping, pickup, charge exchange, Coulomb excitation

  16. reaction mechanisms II. resonant process two-step process example: resonant radiative capture A(x,g)B 1. Compound nucleus formation (in an unbound state) 2. Compound nucleus decay (to lower excited states) Ecm Er G Sx g A+x Q B B compound formation probability  Gx compound decay probability  Gg • reaction cross section proportional to two matrix elements • only occurs at energies Ecm ~ Er - Q • strong energy dependence of cross section N. B. energy in entrance channel (Q+Ecm) has to match excitation energy Er of resonant state, however all excited states have a width  there is always some cross section through tails

  17. reaction mechanisms example: resonant reaction A(x,a)B 1. Compound nucleus formation (in an unbound state) 2. Compound nucleus decay (by particle emission) a Ecm G Sx A+x Sa B+a B C C compound formation probability  Gx compound decay probability  Ga N. B. energy in entrance channel (Sx+Ecm) has to match excitation energy Er of resonant state, however all excited states have a width  there is always some cross section through tails

  18. cross section • cross section expressions • fordirect reactions • with neutrons • with charged particles

  19. cross sections for direct reactions example: direct capture A + x  B + g penetrability/transmission probability for projectile to reach target for interaction depends on projectile’s angular momentum l and energy E “geometrical factor” de Broglie wavelength of projectile matrix element contains nuclear properties of interaction s = (strong energy dependence) x (weak energy dependence) S(E) = astrophysical factor contains nuclear physics of reaction + can be easily: graphed, fitted, extrapolated (if needed) need expression for Pl(E) • factors affecting transmission probability: • Coulomb barrier (for charged particles only) • centrifugal barrier (both for neutrons and charged particles)

  20. Coulomb barrier for projectile and target charges Z1 and Z2 V Coulomb potential Ekin ~ kT (keV) height of Coulomb barrier tunnel effect in numerical units: R r nuclear well example: 12C(p,g) VC= 3 MeV • average kinetic energies in stellar plasmas: kT ~ 1-100 keV! • fusion reactions between charged particles take place well below Coulomb barrier • transmission probability governed by tunnel effect for E<<VC and zero angular momentum, tunnelling probability given by: with Gamow factor

  21. angular momentum barrier classical treatment: p p = projectile linear momentum d = impact parameter incident particle d z-axis target nucleus L=pd incident particle can have orbital angular momentum angular momentum is conserved in central potential  linear momentum p (and hence energy) must increase as distance d decreases quantum-mechanical treatment: (discrete values only) • = 0 s-wave • = 1 p-wave • = 2 d-wave … with parity of wave-function: p = (-1)l angular momentum is conserved in central potential  non-zero angular momentum implies “angular momentum energy barrier” Vl • = reduced mass of projectile-target system r= radial distance from centre of target nucleus

  22. reactions with neutrons

  23. neutron capture simplest case: s-wave neutrons Vl = 0 and also VC = 0 discontinuity in potential gives rise to partial reflection of incident wave incident wave transmitted wave reflected wave V=0 attractive nuclear potential -V0 for l = 0 and hence: transmission probability: for l 0 and hence: consequences: s-waveneutron capture usually dominates at low energies (except if hindered by selection rules) higher l neutron capture only plays role at higher energies (or if l=0 capture suppressed)

  24. l dependence of penetrability through centrifugal barrier lower l values dominate reaction rate note: arbitrary scale between different l values neutron capture l dependence of neutron capture cross section cross section decreases strongly with decreasing energy (effect of barrier)

  25. stellar reaction rate stellar reaction rates for neutron capture energy range of interest for astrophysics depends on: temperature and cross section shape s-wave neutron capture energy range of interest E ~ kT sth = measured cross section for thermal neutrons most probable velocity, corresponding to Ecm = kT

  26. case: l = 0 s-wave neutron capture thermal cross sections>= 45.4 mb example: 7Li(n,g)8Li deviation from 1/v trend due to resonant contribution (see later)

  27. reactions with charged particles

  28. charged-particle capture probability of tunnelling through Coulomb barrier for charged particle reactions at energies E << Vcoul Sommerfeld parameter • assumes: • full ion charges • zero orbital angular momentum units of S(E): keV barn, MeV barn … additional angular momentum barrier leads to a roughly constant addition to the S-factor that strongly decreases with l  S-factor definition for charged particle reactions is independent of orbital angular momentum (unlike neutron capture processes!)

  29. Maxwell-Boltzmann distribution  exp(-E/kT) tunnelling through Coulomb barrier  exp(- ) MeV Gamow peak MeV relative probability E0 energy kT E0 stellar reaction rates for charged particle capture and substituting for s: maximum reaction rate at E0: Gamow peak E0 = relevant energy for astrophysics >> kT N.B. Gamow energy depends on reactionand temperature

  30. cross section cross section expressions forresonant reactions (neutrons and charged particles)

  31. cross section for resonant reactions for a single isolated resonance: resonant cross section given by Breit-Wigner expression for reaction: 1 + T  C  F + 2 • strongly energy-dependent term • G1 = partial width for decay via emission of particle 1 • = probability of compound formation via entrance channel • G2 = partial width for decay via emission of particle 2 • = probability of compound decay via exit channel • = total width of compound’s excited state = G1 + G2 + Gg+ … Er = resonance energy spin factor w J = spin of CN’s state J1 = spin of projectile JT = spin of target geometrical factor  1/E what about penetrability considerations?  look for energy dependence in partial widths! partial widths are NOT constant but energy dependent!

  32. energy dependence of partial widths ql = “reduced width” (contains nuclear physics info) particle widths Pl gives strong energy dependence example: 16O(p,g)17F energy dependence of proton partial width Gp as function of l particle partial widths have approximately same energy dependence as penetrability function seen in direct reaction processes

  33. reaction rate for resonant processes here Breit-Wigner cross section integrate over appropriate energy region E ~ kT for neutron induced reactions E ~ Gamow window for charged particle reactions if compound nucleus has an exited state (or its wing) in this energy range  RESONANT contribution to reaction rate (if allowed by selection rules) • typically: • resonant contribution dominates reaction rate • reaction rate critically depends on resonant state properties • two simplifying cases: • narrow (isolated) resonances • broad resonances

  34. reaction rate for: • narrow resonances • broad resonances

  35. narrow resonance case  << ER • resonance must be near relevant energy range DE0 to contribute to stellar rate • MB distribution assumed constant over resonance region • partial widths also constant, i.e. Gi(E)  Gi(ER) reaction rate for a single narrow resonance • NOTE • exponential dependence on energy means: • rate strongly dominated by low-energy resonances (ER kT) if any • small uncertainties in ER (even a few keV) imply large uncertainties in reaction rate

  36. some considerations… rate entirely determined by “resonance strength” wgand energy of the resonance ER resonance strength (Gi values at resonant energies) (= integrated cross section over resonant region) often reaction rate is determined by the smaller width ! • experimental info needed: • partial widths Gi • spin J • energy ER note: for many unstable nuclei most of these parameters are UNKNOWN!

  37. example: 24Mg(p,g)25Al the cross section

  38. … and the corresponding S-factor non-constant S-factor resonant contribution almost constant S-factordirect capture contribution Note varying widths of resonant states !

  39. reaction rate through: • narrow resonances • broad resonances

  40. broad resonance case  ~ ER broader than the relevant energy window for the given temperature resonances outside the energy range can also contribute through their wings Breit-Wigner formula + energy dependence of partial and total widths assume: G2=const, G = const and use simplified same energy dependenceas in direct process for E << ER very weak energy dependence N.B. overlapping broad resonances of same Jpinterference effects

  41. summary • stellar reaction rate of nuclear reaction determined by the sum of contributions due to • direct transitions to the various bound states • all narrow resonances in the relevant energy window • broad resonances (tails) e.g. from higher lying resonances • any interference term total rate Rolfs & Rodney Cauldrons in the Cosmos, 1988

  42. some considerations example resonant and non resonant contributions to stellar reaction rates typically • lower temperature rates by direct reactions • higher temperature rates dominated by resonances note: level density in nuclei increases with excitation energy 130-220 keV 330-670 keV the Gamow window moves to higher energies with increasing temperature  different resonances play a role at different temperatures 500-1100 keV Gamow region

  43. Lecture 2: The synthesis of the trans-iron elements • the s-process • the r-process • their astrophysical sites • nuclear data needs http://www.univie.ac.at/strv-astronomie/unterhaltung.html

  44. nuclear processes charged-particle induced reaction mainly neutron capture reaction during quiescent stages of stellar evolution mainly during explosive stages of stellar evolution involve mainly STABLE NUCLEI involve mainly UNSTABLE NUCLEI

  45. Q < 0 Q > 0 why neutron capture processes for the synthesis of heavy elements? • exponential abundance decrease up to Fe  exponential decrease in tunnelling probability for charged-particle reactions • almost constant abundances beyond Fe  non-charged-particle reactions • binding energy curve  fusion reactions beyond iron are endothermic • characteristic abundance peaks at magic neutron numbers • neutron capture cross sections for heavy elements increasingly larger • large neutron fluxes can be made available during certain stellar stages

  46. nucleosynthesis beyond iron start with Fe seeds for neutron capture • whenever an unstable species is produced one of the following can happen: • the unstable nucleus decays (before reacting) • the unstable nucleus reacts (before decaying) • the two above processes have comparable probabilities if tn >> tb unstable nucleus decays if tn<<tb unstable reacts if tn ~ tb branchings occur with: mean lifetime of nucleus X against destruction by neutron capture mean lifetime of nucleus X against b decay NOTE:tn varies depending upon stellar conditions (T, r)  different processes dominate in different environments ALSO:tb can be affected too by physical conditions of stellar plasma!

  47. aside factors influencing the b-decay lifetime of an unstable nucleus • both b- and b+ decay are hampered in the presence of electron or positron degeneracy • b- and b+ decays may occur from excited isomeric states maintained in equilibrium with ground state by radiative transitions • electron-capture rates are affected by temperature and density through population of • the K electronic shell example: 7Be 7Be nucleus can only decay by electron capture with a lifetime: tEC ~ 77 d in the Sun, T ~ 15x106 K kT ~ 1.3 keV  low-Z nuclei almost completely ionized e.g. binding energy of innermost K-shell electrons in 7Be: Eb = 0.22 keV  if no electrons available7Be becomes essentially STABLE! in fact free electrons present in the plasma can be captured for solar conditions: tEC ~ 120 d factor 1.6 larger thanin terrestrial laboratory

  48. 80Br, t1/2=17 min, 92% (b-),8% (b+) p-process proton number 85Kr, t1/2=11 y 79Se, t1/2=65 ky r-process 64Cu, t1/2=12 h, 40% (b-),60% (b+) 63Ni, t1/2=100 y neutron number (from Rene Reifarth) s-only the s-process Zr Y Sr (n,g) Rb p-only Kr Br (b-) Se As (b+) Ge Ga Zn r-only Cu Ni Co Fe

  49. s-only, r-only and p-only isotopes help to disentangle the individual contributions A=85 A=140 A=208 abundance peaks at A = 85,140, and208 how to explain abundance curve with the s process? s-process abundances

  50. the s-process • the process • its astrophysical site(s) • nuclear data needs • (experimental equipment and techniques)

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