1 Nuclear Reactions – 1/2 DTP 2010, ECT*, Trento 12 th April -11 th June 2010 Jeff Tostevin, Department of Physics Faculty of Engineering and Physical Sciences University of Surrey, UK

2 Notes/Resources Please let me know if there are problems.

3 The Schrodinger equation In commonly used notation: and defining With bound states scattering states

4 Optical potentials – the role of the imaginary part

5 Recall - the phase shift and partial wave S-matrix and beyond the range of the nuclear forces, then Scattering states regular and irregular Coulomb functions

6 Phase shift and partial wave S-matrix: Recall If U(r) is real, the phase shifts are real, and […] also Ingoing waves outgoing waves survival probability in the scattering absorption probability in the scattering Having calculate the phase shifts and the partial wave S-matrix elements we can then compute all scattering observables for this energy and potential (but later).

7 Ingoing and outgoing waves amplitudes 0

8 Semi-classical models for the S-matrix - S(b) b for high energy/or large mass, semi-classical ideas are good k b , actually  +1/2 1 b 1 absorption transmission b=impact parameter

9 Eikonal approximation: point particles Approximate (semi-classical) scattering solution of assume valid when  high energy Key steps are: (1) the distorted wave function is written all effects due to U(r), modulation function (2) Substituting this product form in the Schrodinger Eq. small wavelength

10 Eikonal approximation: point neutral particles 1D integral over a straight line path through U at the impact parameter b The conditions  imply that and choosing the z-axis in the beam direction with solution b r z Slow spatial variation cf. k phase that develops with z

11 Eikonal approximation: point neutral particles So, after the interaction and as z  Eikonal approximation to the S-matrix S(b) S(b) is amplitude of the forward going outgoing waves from the scattering at impact parameter b theory generalises simply to few-body projectiles Moreover, the structure of the b r z

12 Eikonal approximation: point particles (summary) b z limit of range of finite ranged potential

13 Semi-classical models for the S-matrix - S(b) b for high energy/or large mass, semi-classical ideas are good k b , actually  +1/2 1 b 1 absorption transmission b=impact parameter

14 Point particle – the differential cross section Using the standard result from scattering theory, the elastic scattering amplitude is with is the momentum transfer. Consistent with the earlier high energy (forward scattering) approximation

15 Bessel function Point particles – the differential cross section So, the elastic scattering amplitude Performing the z- and azimuthal  integrals is approximated by

16 Point particle – the Coulomb interaction Treatment of the Coulomb interaction (as in partial wave analysis) requires a little care. Problem is, eikonal phase integral due to Coulomb potential diverges logarithmically. Must ‘screen’ the potential at some large screening radius overall unobservable screening phase usual Coulomb (Rutherford) point charge amplitude nuclear scattering in the presence of Coulomb See e.g. J.M. Brooke, J.S. Al-Khalili, and J.A. Tostevin PRC nuclear phase Due to finite charge distribution

17 Accuracy of the eikonal S(b) and cross sections J.M. Brooke, J.S. Al-Khalili, and J.A. Tostevin PRC

18 Accuracy of the eikonal S(b) and cross sections J.M. Brooke, J.S. Al-Khalili, and J.A. Tostevin PRC

19 Point particle scattering – cross sections All cross sections, etc. can be computed from the S-matrix, in either the partial wave or the eikonal (impact parameter) representation, for example (spinless case): and where (cylindrical coordinates) etc. b z

20 Total interaction energy Eikonal approximation: several particles (preview) b1b1 z b2b2 with composite objects we will get products of the S-matrices

21 Eikonal approach – generalisation to composites Total interaction energy

22 Folding models are a general procedure Pair-wise interactions integrated (averaged) over the internal motions of the two composites

23 Folding models from NN effective interactions Single folding Single folding A B B Only ground state densities appear Double folding Double folding

24 Effective interactions – Folding models Double folding Double folding Single folding Single folding A B B

25 The M3Y interaction – nucleus-nucleus systems Double folding Double folding A B M.E. Brandan and G.R. Satchler, The Interaction between Light Heavy-ions and what it tells us, Phys. Rep. 285 (1997) originating from a G-matrix calculation and the Reid NN force resulting in a REAL nucleus-nucleus potential

26 t-matrix effective interactions – higher energies Double folding Double folding A B M.E. Brandan and G.R. Satchler, The Interaction between Light Heavy-ions and what it tells us, Phys. Rep. 285 (1997) At higher energies – for nucleus-nucleus or nucleon-nucleus systems – first order term of multiple scattering expansion resulting in a COMPLEX nucleus-nucleus potential nucleon-nucleon cross section

27 Skyrme Hartree-Fock radii and densities W.A. Richter and B.A. Brown, Phys. Rev. C67 (2003)

28 Double folding models – useful identities proofs by taking Fourier transforms of each element

29 Effective NN interactions – not free interactions B nuclear matter include the effect of NN interaction in the “nuclear medium” – Pauli blocking of pair scattering into occupied states But as E  high Fermi momentum

30 JLM interaction – local density approximation nuclear matter For finite nuclei, what value of density should be used in calculation of nucleon-nucleus potential? Usually the local density at the mid-point of the two nucleon positions complex and density dependent interaction B

31 JLM interaction – fine tuning Strengths of the real and imaginary parts of the potential can be adjusted based on experience of fitting data. p + 16 O J.S. Petler et al. Phys. Rev. C 32 (1985), 673

32 JLM predictions for N+ 9 Be cross sections A. Garcıa-Camacho, et al. Phys. Rev. C 71, (2005)

33 JLM folded nucleon-nucleus optical potentials J.S. Petler et al. Phys. Rev. C 32 (1985), 673

34 Cluster folding models – the halfway house can use fragment-target interactions from phenomenological fits to experimental data or the nucleus-nucleus or nucleon- nucleus interactions just discussed to build the interaction of the composite from that of the individual components. for a two-cluster projectile (core +valence particles) as drawn

35 Cluster folding models – useful identities proofs by taking Fourier transforms of each element

36 So, for a deuteron for example