Differences Between Reactions in the Laboratory and Reactions in Stars

1. Differences Between Reactions in the Laboratory and Reactions in Stars Thomas Rauscher University of Basel Switzerland

2. the 3rd minute cataclysmic binaries stellar evolution Supernovae AGB stars Hot “stellar” plasmas… →Nuclear reactions →Change in elemental and isotopic abundances…

3. Reactions in (Fully) Ionized Plasmas — Outline • Effect of free electrons • Modification of half-lives • Screening (charge shielding) • Reaction rates and effective energies • Definition • Gamow peak and Gamow window • Stellar cross section and stellar rate • Definition • Stellar “enhancement” • Reciprocity, detailed balance • Problem with isomeric states • Capture rate is preferred to photodisintegration • Coulomb suppression of stellar enhancement

4. t=77 d t = 140 d Half-life Modifications • Laboratory: Atoms • Electron capture (from K-shell) (e-+p→n+ne) • b--decay with e- emission to continuum (n→p+e-+anti-ne) • (b+ decay) • Plasma: Fully or partially ionized nuclei • Capture from atomic shell suppressed or impossible • HL ↑ • Bound-state decay possible • HL ↓? • Phase-space for emission to continuum smaller (Pauli principle) • HL ↑ (at high density) • Decay of thermally populated excited states (180Ta) • HL ↕ • Temperature dependent!! • May convert stable nucleus to unstable one!

5. Electron screening The nuclei in an astrophysical plasma undergoing nuclear reactions are fully ionized. However, they are immersed in a dense electron gas, which leads to some shieldingof the Coulomb repulsion between projectile and target for charged particle reactions. Charged particle reaction rates are therefore enhanced in a stellar plasma, comparedto reaction rates for bare nuclei. The Enhancement depends on the stellar conditions Bare nucleusCoulomb Extra Screeningpotential (attractive,so <0) (Clayton Fig. 4-24)

6. For weak screening, each ion is surrounded by a sphere of ions and electronsthat are somewhat polarized by the charge of the ion (Debeye Huckel treatment) More positive ions (average changeof charge distributiondue to test charge) Ion underconsideration(test charge) More electrons RD Debye Radius Exp: Quicker drop offdue to screening Then potential around ion With Thus, complete screening for r>>RD .

7. The previous equations describe a corrected Coulomb barrier for the astrophysical environment. One can show that the impact of the correction on the barrier penetrability and therefore on the astrophysical reaction rate can be approximated through a Screening factor f: In weak screening U0 << kT and therefore Summary weak screening:

8. Other screening cases: Strong screening: Average coulomb energy larger than kT – for high densities and low temperatures Again simple formalism available. Intermediate screening: Average Coulomb energy comparable to kT – more complicated but formalismsavailable in literature Dynamic screening: Kinetic motion of electron cloud relative to nuclei can also be considered. Depends on plasma conditions and hydrodynamic details. Complicated. Usually not followed completely in nucleosynthesis models.

9. Screening in Laboratory Experiments: When measuring reaction rates in the laboratory with atomic targets (always), then atomic (or molecular) electrons screen as well. In the laboratory one measures screened reaction rates. BUT the screening is different from the screening in the stellar plasma. • In the star it depends on temperature, density and composition • In the lab it depends on the material (and temperature ?) Measured reaction rates need to be corrected to obtain bare reaction rates. Theseare employed in stellar models that then include the formalism to calculate the screening correction in the astrophysical plasma.

10. In the laboratory, screening is described with screening potential Ue: Example: d(d,p)t withd-implantedTa target Bare (theory) F. Raiola et al, Eur. Phys. J. A 13 (2002) 377

11. Reactions in (Fully) Ionized Plasmas — Outline • Effect of free electrons • Modification of half-lives • Screening (charge shielding) • Reaction rates and effective energies • Definition • Gamow peak and Gamow window • Stellar cross section and stellar rate • Definition • Stellar “enhancement” • Reciprocity, detailed balance • Problem with isomeric states • Capture rate is preferred to photodisintegration • Coulomb suppression of stellar enhancement

13. Reaction Networks I Reactions i(j,k)m lead to change in plasma composition: • NN reactions: • Ng, NL reactions, decays:

14. Network equations: (with M species in the plasma we obtain M equations) Reaction Networks II Want density independent measure, interested in changes caused by reactions, not density fluctuations  use abundances Yk(nk(t),r(t))=nk(t)/(r(t)NA):

15. The velocity distribution depends on the particle statistics and can be derived fromthermodynamics. Thermonuclear Reaction Rates Definition: Number of reactions per volume and timebetween two components of the stellar plasma:

16. Particle Statistics Occupation probabilities of states with energy E and chemical potential m: Low r + high T, -m/kT- , then MB applies (H-, He-burning).

17. Relevant Energies – Gamov Window for charged particle reactions Gamov Peak Note: relevant cross sectionin tail of M.B. distribution, much larger thankT (very different from n-capture !)

18. „Gamow peak“ for neutrons Rolfs & Rodney 1985 Neutrons havetypical energykT=T9/11.605MeV. Iliadis 2006

19. Regimes: • Overlapping resonances: statistical model (Hauser-Feshbach) • Single resonances: Breit-Wigner, R-matrix • Without or in between resonances: Direct reactions Determined by nucl. level density Reaction Mechanisms

20. The stellar reaction rate of a nuclear reaction is determined by the sum of • sum of direct transitions to the various bound states • sum of all narrow resonances in the relevant energy window • tail contribution from higher lying resonances Or as equation: (Rolfs & Rodney) Caution: Interference effects are possible (constructive or destructive addition) among • Overlapping resonances with same quantum numbers • Same wave direct capture and resonances

21. Rate of reaction through a narrow resonance Narrow means: In this case, the resonance energy must be “near” the relevant energy range DE to contribute to the stellar reaction rate. Recall: and For a narrow resonance assume: M.B. distribution constant over resonance constant over resonance All widths G(E) constant over resonance

22. Then one can carry out the integration analytically and finds: For the contribution of a single narrow resonance to the stellar reaction rate: The rate is entirely determined by the “resonance strength” Which in turn depends mainly on the total and partial widths of the resonance at resonance energies. Often , then for And reaction rate is determined by the smaller width !

23. Limitation of Gamow peak concept Narrow resonances can also be important below the Gamow window when width of exit channel smaller than width of entrance channel! Iliadis 2006

24. Rate for broad resonances ornon-resonant reactions Often (for example with theoretical reaction rates) one approximates the rate calculation by assuming the S-factor is constant over the Gamow Window: S(E)=S(E0) Then one finds the useful equation: (AR reduced mass number A1A2/(A1+A2))

25. Transmission coefficients are solutions of Schrödinger equation: Width fluctuation corrections account for non-statistical correlations between entrance and exit channels; formally: Hauser-Feshbach Averaged Cross Section(Statistical Model)

26. 110Sn 110Sn S-Widths T9 Relative importance of widths • Average widths (=transmission coefficients) determine the Hauser-Feshbach cross section • g-widths not necessarily the smallest ones at astrophysical energies!

27. Applicability of the Statistical Model Neutron induced reactions Rauscher et al. 1997

28. Applicability of Statistical Model a-induced reactions Proton induced reactions Rauscher et al. 1997

29. Connection to capture rate by detailed balance: Nucleus-Photon Rate With Planck distribution of photons:

30. Reactions in (Fully) Ionized Plasmas — Outline • Effect of free electrons • Modification of half-lives • Screening (charge shielding) • Reaction rates and effective energies • Definition • Gamow peak and Gamow window • Stellar cross section and stellar rate • Definition • Stellar “enhancement” • Reciprocity, detailed balance • Problem with isomeric states • Capture rate is preferred to photodisintegration • Coulomb suppression of stellar enhancement

31. Number of reactions per time and volume Reaction Rate (MB) Stellar cross section

32. Thermally excited target nuclei Ratio of nuclei in a thermally populated excited state to nuclei in the ground state is given by the Saha Equation: Ratios of order 1 for Ex~kT In nuclear astrophysics, kT=1-100 keV, which is small compared to typicallevel spacing in nuclei at low energies (~ MeV). • -> usually only a very small correction, but can play a role in select cases if: • a low lying (~100 keV) excited state exists in the target nucleus • temperatures are high • the populated state has a very different rate (for example due to very different angular momentum or parity or if the reaction is close to threshold and the slight increase in Q-value ‘tips the scale’ to open up a new reaction channel) The correction for this effect has to be calculated.

33. 100.0 keV 7/2- 5/2- 75.0 keV 3/2- 74.3 keV 3/2- 9.75 keV 1/2- 187Os 187Os stellar rate Thermal population at 30 keV: P(gs) = 33% P(1st) = 47% P(all others) = 20% The inelastic scattering cross section for the first excited level allows to define the competition by the scattering channels under stellar conditions

34. Calculated SEF

35. Reciprocity relation Lab cross section; no reciprocity with Reciprocity of stellar rates

36. Reciprocity again!! or Reciprocity relation For fun, let’s postulate “effective” cross section: But: unmeasureable!

37. From Saha equation; G(T) is partition function Stellar cross section Effective c.s. PAm Reciprocity relation Step 2: Let’s add thermal population of excited states → Detailed Balance

38. PBn Effective c.s. PAm Reciprocity relation Step 3: Insert in stellar rate Stellar cross section Stellar rates obey reciprocity! This implies thermal equilibrium in BOTH nuclei A, B!

39. PBn Reciprocity relation for stellar rates: (Similar for photodisintegration) Stellar cross section PAm Prerequisite: Fast thermal equilibration in all channels! Fulfilled in most cases, unless there are isomeric states. Always determine rate in direction of positive QAa , to minimize SEF and numerical errors. For numerical stability in reaction networks, forward and backward rates have to be computed from ONE source!

40. Reciprocity in Stellar Rates Some considerations: • Detailed balance: thermalization required • Problematic for nuclei with isomeric states • e.g., 26Al, 180Ta • Use “internal” network to follow all particle and photon transitions between states in a nucleus • ONE source for forward and reverse reaction in network for numerical stability and proper equilibria • Usually direction of positive Q value • Photodisintegration in lab tests only few transitions, better use capture and compute reverse rate

41. Connection to capture rate by detailed balance: Nucleus-Photon Rate With Planck distribution of photons:

42. Simulating Photodisintegration • Bremsstrahlung spectra or mono-energetic • Simulate Photon-Bath by superposition • Can only probe ground-state transition: unrealistic rate • Tests only few transitions! Mohr et al. 2001, Vogt et al. 2003, Sonnabend et al. 2003

43. Stellar enhancement in photon-induced rates (g,n) Utsonomiya et al. (2006)(MST and NS from Goriely and Rauscher, respectively)

44. Stellar enhancement factor: MB population: transition probability: Coulomb suppression of stellar enhancement It is usually assumed that fforw<frev and therefore a measurement of the forward reaction will be closer to stellar cross section. However, low energy transitions of charged particles will be suppressed even when they are favored by spin selection. Thus, for reactions with different Coulomb barriers in the channels, an inversion is possible! forward reaction Q>0 reverse reaction Q<0

45. Coulomb suppression of stellar enhancement II Prerequisite: Q value is sufficiently small with respect to Coulomb barrier in order to have only few transitions from excited states. Example: Plot Q values of (p,n) and (a,n) reactions with fforw>frev and frev≈1: When considering all kinds of reactions between proton and neutron drip from Ne to Bi, >1200 reactions of this type are found! Interesting with respect to experiments, for example: 85Rb(p,n)85Sr (Q<0) Phys. Rev. Lett. 101 (2008) 191101

46. Reactions in (Fully) Ionized Plasmas — Outline • Effect of free electrons • Modification of half-lives • Screening (charge shielding) • Reaction rates and effective energies • Definition • Gamow peak and Gamow window • Stellar cross section and stellar rate • Definition • Stellar “enhancement” • Reciprocity, detailed balance • Problem with isomeric states • Capture rate is preferred to photodisintegration • Coulomb suppression of stellar enhancement

48. (De-) Population by photon absorption/emission Population by particle reactions Population of 180Ta Levels • States can be populated and depopulated through particle or photon channels! • When detailed balance is not fulfilled, need to explicitly follow all transitions in “internal reaction network”