1 Nuclear Reactions 1 2 DTP 2010 ECT

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

Notes/Resources http: //www. nucleartheory. net/DTP_material/ Please let me know if there are problems. 2

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

Optical potentials – the role of the imaginary part 4

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

Phase shift and partial wave S-matrix: Recall If U(r) is real, the phase shifts are real, and […] also Ingoing 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). 6

Ingoing and outgoing waves amplitudes 0 7

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

Eikonal approximation: point particles Approximate (semi-classical) scattering solution of assume valid when small wavelength 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. 9

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

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

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

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

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 14

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

Point particle – the Coulomb interaction 16 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 usual Coulomb nuclear scattering in the presence (Rutherford) point screening phase of Coulomb charge amplitude nuclear phase Due to finite charge distribution See e. g. J. M. Brooke, J. S. Al-Khalili, and J. A. Tostevin PRC 59 1560

Accuracy of the eikonal S(b) and cross sections J. M. Brooke, J. S. Al-Khalili, and J. A. Tostevin PRC 59 1560 17

Accuracy of the eikonal S(b) and cross sections J. M. Brooke, J. S. Al-Khalili, and J. A. Tostevin PRC 59 1560 18

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): etc. and where (cylindrical coordinates) b z

Eikonal approximation: several particles (preview) b 1 Total interaction energy b 2 with composite objects we will get products of the S-matrices z 20

Eikonal approach – generalisation to composites Total interaction energy 21

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

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

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

The M 3 Y interaction – nucleus-nucleus systems 25 Double folding A B originating from a G-matrix calculation and the Reid NN force resulting in a REAL nucleus-nucleus potential M. E. Brandan and G. R. Satchler, The Interaction between Light Heavy-ions and what it tells us, Phys. Rep. 285 (1997) 143 -243.

t-matrix effective interactions – higher energies Double folding A B At higher energies – for nucleus-nucleus or nucleon-nucleus systems – first order term of multiple scattering expansion nucleon-nucleon cross section resulting in a COMPLEX nucleus-nucleus potential M. E. Brandan and G. R. Satchler, The Interaction between Light Heavy-ions and what it tells us, Phys. Rep. 285 (1997) 143 -243. 26

Skyrme Hartree-Fock radii and densities W. A. Richter and B. A. Brown, Phys. Rev. C 67 (2003) 034317 27

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

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

30 JLM interaction – local density approximation complex and density dependent interaction 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 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

JLM predictions for N+9 Be cross sections A. Garcıa-Camacho, et al. Phys. Rev. C 71, 044606(2005) 32

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

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

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

So, for a deuteron for example 36