The Strong Force: NN Interaction Febdian Rusydi Student Seminar 2 Nov 05 KVI - RUG

Outline Yukawa theory of nuclear force NN Interaction Partial Wave Analysis: phase shift, scattering length, and cross-section. Lippman-Schwinger equation: T-matrix Nijmegen analysis

References [1] http://www.pbs.org/wgbh/nova/elegant/ [1] http://www.pbs.org/wgbh/nova/elegant/ [2] http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html [3] Pohv et. al., Particles and Nuclei, Springer, 2002 [4] Frauenfelder, Henley, Subatomic Physics, Prentice Hall, 1991 [5] Krane, Introductory Nuclear Physics, John Wiley & Sons, 1988 [6] http://fermi.la.asu.edu/PHY577/ [7] Phy. Rev. C, 48 792, 1993

Fundamental Interactions [1] [2]

Yukawa theory of nuclear forces Early ’30: existence of a very strong force known in nuclei at distance ~ 2 [fm] ‘34: Yukawa suggested a new sort of quantum, a meson, with 100 [MeV/c2] mass. ‘38: found a new particle suspected to be the meson.  the muon () ’78: The real meson found, the pion or -meson. [3]

Yukawa Potential [4] Classical electrodynamics Yukawa Interaction Strength (dimensionless) Hamiltonian: Substitution: Poisson Eq.: Hfree Hint Add. term Solution: x’  0 Maxwell Eq.: What is k? Poisson Eq.: Solution: k  inverse of Compton wavelength

NN Interactions What we do know: 2 nuclei interact  meson exchange, with minimum energy ~ mc2. Distance ~ 1.4 [fm] Hold nucleus together At some senses analogue to EM interaction. How do we know: Scattering method Wave mechanics m  reduced mass E  kinetic energy = - binding energy

NN Interactions Many experiments have been done to study NN interaction. Deuteron system pp scattering np scattering What we do not know: Interaction potential Some methods to analyze the potential: Phase shift method Lippman – Schwinger equation

Phase shift method NN Scattering at low energy: [5] NN Scattering at low energy: Incident particle with speed ~ v Angular momentum ~ If: Corresponding kinetic energy: Partial wave analysis Features: Incident wave is plane wave  I(r-2), A(r-1) Scattered wave is diffracted  I(, ) Detector records both incident and scattered waves.

Partial wave analysis [5] Radial part of Schrödinger equation: Solution, assume the potential well and l = 0: Boundary condition at r = R: inside outside

Phase shift & scattering length [5] outgoing phase shift incoming Scattering length Phase shift Amplitude

Cross section Cross-section Scattering gives information on the interaction between incoming and target particles, such as: Reaction rates, Energy spectrum Angular distribution Cross-section Current of particles per unit area: Cross section Scattering length

pp/np scattering (E  20 MeV) [5] pp scattering: Isovector Charge  easy to detect np/pn scattering: Isovector and isoscalar Neutral  difficult to detect V V(r), determined by measuring E-dependent of NN parameters (such as phase shift) Spin-dependent  only 1S0 Charge symmetry  pp & nn interactions are identical, nearly charge-independent

E density of outgoing particle Lippman – Schwinger [6] Schrödinger eq. No interaction Interaction T-matrix Lippman-Schwinger eq. E density of outgoing particle Cross section Scattering length

Nijmegen Analysis [7] Fact: pp scattering analysis (multi-energy, m.e.) is easier than np scattering. pp scattering analysis is established. np scattering analysis: parameterize isoscalar lower partial wave. substitute isovector lower partial wave from pp scattering result General method: Long-range part of NN interaction is well known: EM & Yukawa interaction  higher J are understood. Short-range is sufficiently short for higher partial wave to be screened by central barrier  small number of lower partial waves need to be parameterized.

Long-range interaction [7] The long-range potential consists of an EM part VEM and a nuclear part VN EM part Nuclear part Coupling constants: OPE = One Pion Exchange HBE = Heavier Boson Exchange

Short-range interaction [7] Boundary-condition parameterization at r = R E-dependent of pp 3P0 phase shift Quantum number (l, s, J) Radial wave function no param. 1 param. Treatment isovector and isoscalar np phase parameter Parameterize isoscalar lower partial waves. Calculate pp phase shift by solving Schrödinger eq. (VN & VC) OPE (pp) replaced by OPE (np) Nijmegen. solution What to be fitted? an, according to the data Neutral-to-charged m difference

What we have so far… Strong force in nuclei is studied by scattering and partial-wave analysis. Potential of NN interaction is not well understood; there are many approaches (scattering length, T-matrix). Nijmegen potential: OPE and HBE Nijmegen analysis: long- and short-range. Not covered yet: Bonn and Argonne Potentials.