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

2. 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

3. References [1] http://www.pbs.org/wgbh/nova/elegant/
[1]
[2]
[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]
[7] Phy. Rev. C, , 1993

4. Fundamental Interactions
[1]
[2]

5. 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]

6. 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

7. 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

8. 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

9. 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.

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

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

12. 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

13. 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

14. 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

15. 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.

16. 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

17. 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

18. 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.