Effective Interaction Theory: A Historical Perspective, Slides of Astrophysics

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Effective Interaction Theory

Ch. Elster

Lecture 7

Intuitive ideas leading to effective interaction theory

Basic ideas for dealing with the many-body, stong (non-perturbative) nuclear interaction problem began with scattering.

A seminal idea was due to Leslie L. Foldy working on sonar during WWII.

Foldy described projectile scattering from a nucleus as a wave propagating through many, dense scattering sources with a complex (absorptive) index of refraction. His essential idea was to express the total scattered wave in terms of individual N+N scattered waves, rather than in terms of the very strong N+N interaction which can not be expanded in a perturbation theory.

L.L. Foldy, Phys. Rev. 67 , 107 (1945)

check out: The Journal of the Acoustical Society of America, Vol. 132, 1960 (2012) Mulltiple Scattering in the Spirit of Leslie Foldy

Intuitive ideas leading to effective interaction theory

In 1951 Melvin Lax extended these approaches to obtain an effective interaction potential, later called the “optical potential” to represent the effective p+A interaction.

First representation of such an effective potential. Introduced the socalled “tρ” form, where ρ is the nuclear density and t represents an effective N+N interaction

Global Phenomenological Optical Potential

Remark: Same importance as NN phase shift analysis

Phenomenological Optical Potential

Check out yourself: http://home.eckerd.edu/~weppnesp/optical/

Coulomb term

Phenomenological Optical Potential

For all functions go to website

Phenomenological Optical Potential

Phenomenological Optical Potential

Ideas leading to effective interaction theory

In 1953 Kenneth M. Watson gathered up emerging ideas and published the first formal scattering solution for the p+A problem

Later more explicit description

Ideas leading to effective interaction theory

In 1959 Arthur Kerman, Hugh McManus and Roy Thaler modified the Watson theory by re-organizing the expansion and paved the way an accurate (numerical) application for Chew’s impulse approximation:

Scattering Problem p → A

• Transition amplitude: T = V + V G 0 T

• Hamiltonian: H = H 0 + V

• Free Hamiltonian: H 0 = h 0 + H A

• h 0 : kinetic energy of projectile ‘0’
• HA: target hamiltonian with HA |Φ〉 = E (^) A |Φ〉

• V: interactions between projectile ‘0’ and target

nucleons ‘i’ V = ΣAi=0 v0i

• Propagator is (A+1) body operator

• G 0 (E) = (E – h 0 – HA + iε) -

A

Watson Series:

• T = ΣAi=0 T0i
  • with T (^) 0i = voi + v0i G 0 (E) T

with t 0i = v0i + v0i G 0 (E) t0i

Watson series for multiple scattering

Spectator Expansion explicitly

2 nd^ order term:

Single scattering approximation:

Scattering from pairs

Pauli Principle: Antisymmetrize in active pairs for 1 st^ order

Elastic Scattering

• In- and Out-States have the target in ground state Φ 0
• Projector on ground state P = |Φ 0 〉〈Φ 0 |
  • With 1=P+Q and [P,G 0 ]=
• For elastic scattering one needs
• P T P = P U P + P U P G 0 (E) P T P
• Or
  • T = U + U G 0 (E) P T
  • U = V + V G 0 (E) Q U ⇐ optical potential

Standard: U (1)^ ≈ Σ Ai=0 τ 0i (1st^ order)

with τ 0i = v0i + v0i G 0 (E) Q

Spectator expansion for U: Chinn, Elster, Thaler, Weppner, PRC52, 1992 (1995)