By William Hornyak (Auth.)
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These potentials are derived from a multiple-scattering expansion summed over the nucleons in the nucleus. For example, one rather satisfactory fit to 12C (π, π) scattering at E = 120-280 MeV is given by Kisslinger and Tabakin (1974). Rather simple considerations based on conservation principles and the assignment of quantum numbers lead to a ready understanding of the Δ (1236) resonance. Since at this energy C^LAB ~ 200 MeV) elastic scattering with or without charge exchange is the only energetically allowed process, we restrict ourselves to considering π + + p -» π + + p (l-69a) π" + p -► π" + p (l-69b) π" + p -► π° + n.
We relegate a discussion of this and other mesons to later. B. ν 1 )(σ 2 ·ν 2 ) μ a P2(r-r2)eXP(|r^ r l) ' p 1 ( r ' - r 1 ) d*r d3r>, (1-39) and vi2 = Vw Σ k= + 1 , 0 , - 1 This last sum involves the factor Σ*= + ι,ο, - I fk%\S3) τ ^(1)· Taking/ 2 x =/ 2 1 results in a charge symmetric theory and incidentally ensures that Vi2 is Hermitian. We further note that τ+ι(2)τ*+1(1) + τ - I ( 2 ) T U 1 ) = -2[ί + 1 (2)ί_ 1 (1) + /- 1 (2)/ + 1 (1)] (1-40) and using (1-7), -2[ί + 1 (2)/_ 1 (1) + ί. 1 (2)/ + 1 (1)] = ^ ( 1 ) ^ ( 2 ) + τ 2 (1)τ 2 (2).
44 / 900 1100 1300 NUCLEON-NUCLEON 1500 1700 1900 2100 Mass of 7Γ - p system (MeV) 2300 FORCES 2500 FIG. 5. Total cross sections for π + -ρ and π"-ρ scattering [taken from Gasiorowicz (1966)]. This state is, in fact, observed as a strong pion-nucleon elastic scattering resonance and might be considered to constitute the "first excited" state of the nucleon, denoted by Δ(1236). 5 shows the observed total cross sections in π*-ρ scattering. , see Particle Data Group (1973). An energy level diagram showing some of the more prominent pion-proton resonances with hypercharge Y = 1 is given in Fig.