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9. Direct reactions - for example direct capture:

9. Direct reactions - for example direct capture:. a + A -> B + g. Direct transition from initial state |a+A> to final state <f| (some state in B). geometrical factor (deBroglie wave length of projectile - “size” of projectile).

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9. Direct reactions - for example direct capture:

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  1. 9. Direct reactions - for example direct capture: a + A -> B + g Direct transition from initial state |a+A> to final state <f| (some state in B) geometrical factor(deBroglie wave length of projectile - “size” of projectile) Penetrability: probabilityfor projectile to reach the target nucleus forinteraction. Depends on projectileAngular momentum land Energy E Interaction matrixelement III.25

  2. 2 effects that can strongly reduce penetrability: Penetrability: 1. Coulomb barrier V for a projectile with Z2 anda nucleus with Z1 Coulomb Barrier Vc Potential R r or Example: 12C(p,g) VC= 3 MeV Typical particle energies in astrophysics are kT=1-100 keV ! Therefore, all charged particle reaction rates in nuclear astrophysics occur way below the Coulomb barrier – fusion is only possible through tunneling

  3. 2. Angular momentum barrier Incident particles can have orbital angular momentum L Momentum pImpact parameter d Classical: p d In quantum mechanics the angular momentum of an incident particle can havediscrete values: And parity of thewave function: (-1)l With s-wave l = 0 p-wave l = 1 d-wave l = 2 … For radial motion (with respect to the center of the nucleus), angular momentumconservation (central potential !) leads to an energy barrier for non zero angularmomentum. Classically, if considering the radial component of the motion, d is decreasing, which requires an increase in momentum p and therefore in energy to conserve L.

  4. Energy E of a particle with angular momentum L (still classical) Similar here in quantum mechanics: m : reduced mass of projectile-target system Peaks again at nuclear radius (like Coulomb barrier) Or in MeV using the nuclear radius and mass numbers of projectile A1 andtarget A2:

  5. 9.1. Direct reactions – the simplest case: s-wave neutron capture No Coulomb or angular momentum barriers: Vl=0 VC=0 s-wave capture therefore always dominates at low energies But, change in potential still causes reflection – even without a barrierRecall basic quantum mechanics: Incoming wave transmitted wave Reflected wave Potential Transmission proportional to

  6. Therefore, for direct s-wave neutron capture: Penetrability Cross section (use Eq. III.19): Or Example: 7Li(n,g) thermalcross section<s>=45.4 mb(see Pg. 27) ~1/v Deviationfrom 1/vdue to resonantcontribution

  7. Why s-wave dominated ? Level scheme: 2.063 3/2- + 1/2+ E1 g 7Li + n 0.981 1+ E1 g 0 2+ 8Li Angular momentum and parity conservation: l Entrance channel 7Li + n : 3/2- + 1/2++ l(-1) = 1-, 2- ( l=0 for s-wave ) Exit channel 8Li + g : 2+ + ? (photon spin/parity) Recall: Photon angular momentum/parity depend on multiploarity: For angular momentum L (=multipolarity) electric transition EL parity (-1)Lmagnetic transition ML parity (-1)L+1

  8. Also recall: • E.M. Transition strength increases: • for lower L • for E over M • for higher energy Entrance channel 7Li + n : 3/2- + 1/2++ l(-1) = 1-, 2- ( l=0 for s-wave ) = 1-, 2-, 3- 2+ + 1- Exit channel 8Li + g : E1 photonlowest EL that allows to fulfillconservation laws match possible Same for 1+ state “At low energies 7Li(n,g) is dominated by (direct) s-wave E1 capture”.

  9. Stellar reaction rate for s-wave neutron capture: Because Thermal neutron cross section: Many neutron capture cross sections have been measured at reactors usinga “thermal” (room temperature) neutron energy distribution atT= 293.6 K (20 0C), kT=25.3 meV The measured cross section is an average over the neutron flux spectrum F(E) used: (all Lab energies) For a thermal spectrum so

  10. ? Why is a flux of thermalized particles distributed as The number density n of particles in the beam is Maxwell Boltzmann distributed BUT the flux is the number of neutrons hitting the target per second and area. Thisis a current density j = n * v therefore The cross section is averaged over the neutron flux (the number of neutronshitting the target for each energy bin) because that is what determines the event rate. This is the same situation in the center of a star. The number density of particlesis M.B. distributed, but the number of particles passing through an area per secondis distributed, and so is the stellar reaction rate !

  11. With these definitions one can show that the measured averaged cross sectionand the stellar reaction rate are related simply by (most frequent velocity, corresponding to ECM=kT) with for reactor neutrons (thermal neutrons) vT=2.20e5 cm/s that’s usually tabulated as“thermal cross section” and For s-wave neutron capture (which is generally the only capture mechanism for room temperature neutrons) one can relate the thermal cross section to the actual cross section value at the energy kT which is for example useful to read cross section graphs

  12. 9.2. Direct reactions – neutron captures with higher orbital angular momentum For neutron capture, the only barrier is the angular momentum barrier The penetrability scales with and therefore the cross section (Eq III.19) for l>0 cross section decreases with decreasing energy (as there is a barrier present) Therefore, s-wave capture in general dominates at low energies, in particular at thermal energies. Higher l-capture usually plays only a role at higher energies. What “higher” energies means depends on case to case - sometimes s-wave is strongly suppressed because of angular momentum selection rules (as it wouldthen require higher gamma-ray multipolarities)

  13. Example: p-wave capture in 14C(n,g)15C (from Wiescher et al. ApJ 363 (1990) 340)

  14. Why p-wave ? 14C+n 0.74 5/2+ 1/2+ 0 15C Exit channel (15C + g) g total to 1/2+ total to 5/2+ strongest ! E1 1/2- 3/2- 3/2- 5/2- 7/2- 1- 1+ 3/2+ 5/2+ 7/2+ M1 1/2+ 3/2+ E2 1/2+ 3/2+ 5/2+ 7/2+ 9/2+ 2+ 3/2+ 5/2+ Entrance channel: strongest possible Exit multipole total into 1/2+ into 5/2+ n lp 14C M1 E2 s-wave 0+ 1/2+ 1/2+ 0+ 3/2- p-wave 1/2- E1 1- 0+ 1/2+ E1 despite of higher barrier, for relevant energies (1-100 keV) p-wave E1 dominates.At low energies, for example thermal neutrons, s-wave still dominates. But herefor example, the thermal cross section is exceptionally low (<1mb limit known)

  15. 9.2.1. What is the energy range the cross section needs to be known to determine the stellar reaction rate for n-capture ? This depends on cross section shape and temperature: s-wave n-capture: of the order of KT (somewhat lower than MB distribution) Example: kT=10 keV M.B. distribution Y(E) s(E) Y(E) E = E1/2 exp(-E/kT) relevant for stellar reaction rate s(E)

  16. p-wave n-capture: of the order of KT (close to MB distribution) Example: kT=10 keV M.B. distribution Y(E) s(E) Y(E) E = E3/2exp(-E/kT)relevant for stellar reaction rate s(E)

  17. 9.2.2 The concept of the astrophysical S-factor (for n-capture) recall: III.25 “trivial” strongenergydependence “real” nuclear physicsweak energy dependence(for direct reactions !) S-factor concept: write cross section as strong “trivial” energy dependence X weakly energy dependent S-factor • The S-factor can be • easier graphed • easier fitted and tabulated • easier extrapolated • and contains all the essential nuclear physics Note: There is no “universally defined S-factor - the S-factor definition depends on Pl(E) and therefore on the type of reaction and the dominant l-value !!!

  18. For neutron capture with strong s-wave dominance with corrections. Then define S-factor S(E) and expand S(E) around E=0 as powers of with denoting in practice, these are tabulated fitted parameters typical S(E) units with this definition: barn MeV1/2

  19. and for the astrophysical reaction rate (after integrating over the M.B. distribution) of course for pure s-wave capture for pure s-wave capture the S-factor is entirely determined by the thermal cross section measured with room temperature reactor neutrons: using one finds (see Pg. 35)

  20. For neutron capture that is dominated by p-wave, such as 14C(p,g) one can definea p-wave S-factor: or which leads to a relatively constantS-factor because of (typical unit for S(E) is then barn/MeV1/2) S-factor

  21. 9.3. Charged particle induced direct reactions 9.3.1Cross section and S-factor definition (for example proton capture - such as 12C(p,g) in CN cycle) incoming projectile Z1 A1 (for example proton or a particle) target nucleus Z2 A2 again but now incoming particle has to overcome Coulomb barrier. Therefore with (from basic quantum mechanical barrier transmission coefficient)

  22. so the main energy dependence of the cross section (for direct reactions !)is given by therefore the S-factor for charged particle reactions is defined via typical unit for S(E): keV barn So far this all assumed s-wave capture. However, the additional angular momentumbarrier leads only to a roughly constant addition to this S-factor that strongly decreaseswith l Therefore, the S-factor for charged particle reactions is defined independentlyof the orbital angular momentum

  23. Given here is the partial proton width R=nuclear radiusQ = reduced width (matrix element) (from Rolfs & Rodney)

  24. Example:12C(p,g) cross section need cross sectionhere !

  25. S-Factor: Need rateabout here From the NACRE compilation of charged particle induced reaction rates on stable nuclei from H to Si (Angulo et al. Nucl. Phys. A 656 (1999) 3

  26. 9.3.2. Relevant cross section - 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 !)

  27. The Gamov peak can be approximated with a Gaussian centered at energy E0 and with 1/e width DE Then, the Gamov window or the range of relevant cross section can be easily calculated using: with A “reduced mass number” and T9 the temperature in GK

  28. Example: Note:kT=2.5 keV !

  29. 9.3.3. Reaction rate from S-factor Often (for example with theoretical reaction rates) one approximates the rate calculation by assuming the S-factor is constant over the Gamov WIndow S(E)=S(E0) then one finds the useful equation: Equation III.53 (A reduced mass number !)

  30. better (and this is often done for experimental data) one expands S(E) around E=0as powers of E to second order: If one integrates this over the Gamov window, one finds that one can useEquation III.46 when replacing S(E0) with the effective S-factor Seff and E0 as location of the Gamov Window (see Pg. 51) with

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