Cesare Chiosi Department of Astronomy University of Padova, Italy

1. Stellar Evolution in General and in Special Effects:Core Collapse, C-Deflagration, Dredge-up Episodes Cesare Chiosi Department of Astronomy University of Padova, Italy

2. PART AGENERALITIES

3. Elementary Theory of Nuclear reactions 4 H He

4. Energy at separation Energy in bound state p+n  D The case of p+n D & 4 H He

5. Binding energy per nucleon

6. Kinetic vs Binding Energies Very low!!!!!

7. Typical dimensions of the Nuclei

8. Standard notation • A particle a impinges on a nucleus X, producing a nucleus Y and liberating a particle b • a + X  Y + b or X(a,b)Y • Q-value (liberated energy) Q=[Ma + Mx – My - Mb]c^2

9. Schematic Potential

10. How does a reaction happen? Consider two nuclei of charge Za and Zb and masses Ma and Mb at the ever changing distance r (they are in motion). The repulsive coulomb potential is for r > rN (the nuclear radius), whereas it is strongly actractive for r < rN . The minimum relative distance between the two nuclei for energy E of the relative motion at infinite distance is

11. Barrier Thickness The coulombian potential at the border of the nucleus is the particle sees a very thick potential barrier.

12. How many nuclei with E=FRN?

13. Barrier crossing probability

14. However…… This is only part of the story because we must evaluate • The probability that two nuclei may encounter; • The probability that once the encounter has occurred and the barrier is crossed the final result is Y+b; • Take into account that nuclei are moving with different relative velocities. To this aim we introduce two simple but powerful concepts A) The Bohr model of a nuclear reaction B) The Cross Section

15. Bohr Model • Formation of a composite nucleus in an excited state a + X  C* Y+b • The two steps are fully independent; the second step depends only on the energy and angular momentum of C* and not on the first step a + X (i.e. the energy of the incident particle). • A reaction is favoured if the first step brings C* in (or close to) a quasi stationary state.

16. Quasi-Stationary States Quasi-stationary states Typical of composite nucleus. Explains resonances.

17. Cross Section • Medium with number density of targets n • The probability that an incoming particle interacts is PDX=snDx • The probability of no interaction is Pno,Dx=1-snDx . . . . . . . . . . . . . . . . . . . . . . . . . . Dividing a finite distance x into x/N intervals we get Pno,x=lim[1-snx/N]^N  exp(-snx)

18. The cros section s is an effective area proportional to the probability of the reaction occurring in a collision Mean Free Path & s

19. PG: De Broglie Term

20. PBP: Barrier Penetration

21. The nuclear term is proportional to the probability that the composite • nucleus C* decays into the end product Y+b. • To describe the energy situation of C* we make use of the so-called • Shell Model in analogy to the classical description of electrons. The • nucleus C* has a set of energy levels to disposal each of which has an • intrinsic width PN: Nuclear Term

22. FR: Correction for Resonances

23. Consider two type of nuclei, A and B, with number densities nA and nB. Suppose B at rest and A moving with velocity v. The mean free path of particles B isl=1/nBsand mean lifetimetB=l/v=1/nBsv. Therefore in the unit volume, nA nuclei react with nB nuclei at the rate RAB=nA nBs v per s In reality both A and B are moving with relative velocity v which obeys the probability distribution P(v) expressed by the Mawell-Boltzmann Law, therefore the product sv has to be weighed on P(v) Nuclear Reaction Rates

24. Maxwell-Boltzmann

25. Approximate Expression

26. The PP Chains

27. Lifetimes in the pp-chain

28. CNO Cycle CN Cycle The CN-CNO cycle

29. 0.282 7.65 g 0.099 4.43 Pair emission Rest Mass Energies (Mev) 0 He-burning 3a C g

30. Reactions in He-burning…..

31. Companion reactions Heavy elements start being synthetized !

32. C-burning New, p, n and a particles are created…..very important

33. Ne-photodissociation (burning) For the first time photo-dissociation becomes important

34. Oxygen-burning Followed by ….

35. Si-burning In reality Si is fused in heavier and heavier nuclei by means of many reactions in which p, n, and a emitted by photo-dissociations are rapidly captured. A sort of equilibrium condition is established in which Si is converted to elements of the Iron-group (for which the binding energy per nucleon is maximum). The end of the nuclear exo-energetic history of a star. THE END NUCLEAR REACTIONS AS LARGE SCALE ENERGY SOURCES

36. Burnings in Tc vs rc plane

37. Electron Screening • The coulomb interaction energy has been evaluated considering bare • nuclei, i.e. neglecting the effect of the electron gas in which they are • immmersed. • At high densities the nuclei tend to (locally) attract the electrons • which form a cloud of negative charge around and shield the charge • the nuclear charge. • This will lower the effective nuclear charge and the coulomb barriers • in turn thus favouring the nuclear reactions. • Polarization of the charge induced by a nucleus of charge Ze: the • number density of electrons ne in vicinity of a nucleus is slightly higher • than the mean value <ne>. The other nuclei are pushed away and the • local number density of nuclei ni is lower than the mean value <ni>.

38. For a non degenerate gas, the number density of particles with charge q in presence of a potential f is modified according to In most cases |qf|<< KT so that the exponential can be approximated to 1-qf/KT, therefore which show the increase and decrease of electron and nuclei densities. Electron screening: continue

39. Let us consider all types of nucleus and derive the total charge density s. In absence of f s=0. In presence of f we have We may write the above relations in the following way Electron screening: continue

40. Charge density s and potential f are related by the Poisson equation (in spherical symmetry) RD is the radius of the electron cloud effectively shielding the nucleus Electron screening: continue

41. The lower barrier makes the ractions easier. This occurs through the term exp(-2ph) in calculation of PB, which in fact depends on the quantity EC-E, where EC is the coulomb interaction energy Comparison between RO and RD

42. Final Correction

43. Starting from C-burning in all major nuclear steps p, n, a particles together with light nuclei rare produced . For instance The generated photons may dissociate another Ne. We get an equilibrium state in which the abundances of are regulated by law similar to the Saha relationship between ionic species and electrons Nuclear Statistical Equilibrium

44. In general …… • In reality this is only one of the many possible reactions. The problem becomes identical to that of simultaneous ionization • and recombination of many atomic species. The processes are • are mutually correlated as they produce electrons that affect • the recombination rate. • At high T many nuclear species are present whose abundances • are each other correlated. As 56Fe is the most stable nucleus (maximum binding energy per nucleon), it plays the pivotal role • in establishing the equilibrium state among the nuclear species (Nuclear Statistical Equilibrium, NSE).

45. Let us consider the case of Iron photo-dissociation we want to find the ratio . Iron photo-dissociation

46. Together with the condition For given T , r and the ratio nn/na, the above relations constitute a system in nFe and na From these considerations we get the important result that the equilibrium conditions a low T require the formation of Fe, whereas the same at high T require the break-down of Fe in P, n, a Consequences…

47. Nuclei organized in a crystal lattice • Nuclei oscillate about an equilibrium position • Reactions can occur even at T=0 due to the very high density Ions separated by 2Ro RN radius of nuclei Nuclear Reactions at High Densities

48. Nuclear reactions at high densities

49. Nuclear reactions at high densities

50. Nuclear reactions at high densities