Theory of Nuclear reactions. Scattering theory: multi-channel!

Transcription

1 Theory of Nuclear reactions Scattering theory: multi-channel! 58

2 Multichannel coupled equations t! R x p! within the same partition, the Schrodinger equations becomes a coupled equation:!

3 Integrated channel cross section o channel cross section o total outgoing non-elastic cross section 60

4 Reaction cross section o flux leaving the elastic channel (depends only on elastic S-matrix elements) o the total cross section is elastic plus reaction cross sections o absorption cross section 61

5 elastic scattering: multi-channel 62 Courtesy of Antonio Moro!

6 detailed balance Consequence of hermiticity: S-matrix is unitary! Even if the S-matrix is not unitary, it may be that:! Symmetry condition is sufficient for detailed balance:! 63

7 Theory of Nuclear reactions Scattering: Integral formulation 64

8 Integral forms and T-matrix approach (E - T) ψ=vψ! G=(E-T) -1! Lippmann-Schwinger equation! φ is incoming free wave (only non zero for elastic channel)! ψ is full wavefunction! Partial wave T-matrix! Vector T-matrix! Scattering amplitude! 65

9 two potential formula: definitions Consider your potential can be split into two parts: U=U 1 +U 2! 66

10 two potential formula: derivation 1 67

11 two potential formula: derivation 2 68

12 two potential formula: result Remember the Coulomb and nuclear?! 69

13 Born series 70

14 distorted wave Born approximation (DWBA) (E T U 1 ) ψ=u 2 ψ! ψ=χ+g 1 U 2 ψ! Born series is truncated after the first term! 1 st order DWBA: U 2 appears to first order! U 2 appears to second order! 71

15 multiple orders in DWBA 72

16 QUIZ: Which of the following sentences is correct? a) DWBA treats the transition potential U 2 perturbatively b) DWBA treats the distorted waves perturbatively c) DWBA treats the full projectile-target interaction perturbatively d) DWBA is not a perturbative theory!

17 T-matrix Lippmann-Schwinger Equation Define a transition matrix (t-matrix) such that: Remember the Born series? Multiply by: and we can obtain an operator form of the equation in terms of the t-matrix often used in few-body methods 74 Wong, Introduction to Nuclear Physics, Wiley

18 Theory of Nuclear reactions Three-body methods 75

19 Three-body methods in direct reactions 3-body Hamiltonian for the problem:! 3 jacobi coordinate sets! Faddeev Equations:!

20 reduction to one jacobi set Expand wfn in eigenstates of projectile s internal Hamiltonian:! Expand in partial waves:! H proj φ = εφ p p Radial wavefunctions for projectile:!

21 continuum bins average method! non overlaping continuum intervals - continuum bins are orthogonal! square integrable analytic form if potential is zero and l=0:!

22 breakup: continuum discretized coupled channels CDCC 3-body wavefunction:! Coupled channel equations:! c! r! v R! t! Coupling potentials:! Energies:!

23 nuclear reactions and astrophysics direct measurement 14 C(n,γ) 15 C Coulomb dissociation n low relative energy 15 C γ! 208 Pb 14 C

24 Coulomb dissociation for (n,γ) 208 Pb( 15 C, 14 C+n) 208 MeV/u CDCC + set of single particle parameters! extract ANC from χ 2 minimum! error from ε=χ min2 +1! Yao, JPG33 (2006) 1 ANC =1.32 ± 0.07 fm -1/2 Summers et al., PRC78(2009) Reifarth 14 C(n,γ) 15 C Nakamura et al, NPA722(2003)301c Reifarth et al, PRC77, (2008)

25 one nucleon transfer:coordinates Initial and final bound states:!

26 one nucleon transfer: operator

27 one nucleon transfer: auxiliary potential binding potential! remnant term! U i is an auxiliary potential and therefore the solution is independent of that choice! " standard choice in DWBA U i is optical potential reproducing elastic scattering " standard choice in CDCC U i is U ct + U xt folded over the bound state c+x " other possible choise U i = U cc (R cc ) to cancel the remnant term " etc

28 Features of transfer reactions Q-value matching: similar k in incident and exit distorted waves Angular momentum dependence in zero-range approximation

29 QUIZ: Which line corresponds to larger Q-value? a) Solid! b) Dotted c) Dashed d) Dot-dashed

30 QUIZ: Which line corresponds to larger Q-value? a) Solid! b) Dotted c) Dashed d) Dot-dashed

31 Angular momentum dependence ISF whitepaper (2006)

32 Theory of Nuclear reactions Typical Approximations: Eikonal Adiabatic 89

33 (d,p) reactions: zero range adiabatic wave approx. (ADWA) DWBA! dominating term in ν f is V np (r), short range thus exact wfn only needed for small r!! ADWA! Johnson and Soper potential!

34 (d,p) reactions: finite range adiabatic wave approx Johnson and Soper potential is based on the zero-range approx! Tandy and Johnson reformulated the adiabatic model without the zero-range approx and obtain a new adiabatic potential for the deuteron including breakup effect and finite-range:! V TJ ( R) = φ d V np φ d ( U n V np + U φ d p ) φ d Effective diffuseness increases!!

35 QUIZ: Which of the following sentences is correct? a) ADWA treats deuteron breakup pertubatively b) ADWA is valid for high beam energies c) ADWA takes the np relative energies to be the beam energy d) ADWA can be used to calculate deuteron elastic scattering!

36 Example of using (d,p) to probe halos 10 Be(d,p) MeV! DWBA entrance channel! DWBA exit channel! ADWA! Schmitt et al, PRL 108, (2012), PRC 88, (2013)

37 Testing three-body methods Faddeev AGS: EXACT all three Jacobi components are included elastic, breakup and rearrangement channels are fully coupled computationally expensive Deltuva and Fonseca, Phys. Rev. C79, (2009). CDCC: only one Jacobi component elastic and breakup fully coupled (no rearrangement) computationally expensive 3 jacobi coordinate sets! ADWA: only one Jacobi component elastic and breakup fully coupled (no rearrangement) approximation for breakup one term in Weinberg expansion runs on desktop practical use!

38 Comparing transfer 10 Be(d,p) 11 Be! 12 C(d,p) 13 C! 48 Ca(d,p) 49 Ca! 132 Sn(d,p) 133 Sn! CDCC is ok for transfer at ~10MeV/u agreement deteriorates for higher energy! 5 Upadhyay, Deltuva and Nunes, PRC 85, (2012)

39 The dependence on the optical potential 12 C(p,p) 12 C! 12 C(d,p) 13 C! Constraining p-a elastic reduces uncertainties but remaining uncertainty not neglegible 208 Pb(p,p) 208 Pb! 208 Pb(d,p) 209 Pb! Important to include good optical potential information

40 theory opportunities with FRIB FRIB-CDR, 2010!

41 theory opportunities with FRIB FRIB Theory Center Steering Committee, Dec 2013!