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Chapter 2. Relativistic Eikonal A(p, pN) Formalism 172.2 Relativistic Optical Model Eikonal Approximation2.2.1 Nucleon-Nucleus ScatteringFollowing the discussion of Refs. [56,70], √we consider the time-independent Dirac equation fora particle with relativistic energy E = k 2 + MN 2 and spin state ∣ 12 m s〉subject to a sphericalLorentz scalar V s (r) and vector potential V v (r)Ĥ Ψ (+)⃗ k,ms(⃗r) = [⃗α · ˆ⃗p + βM N + βV s (r) + V v (r)] Ψ (+)⃗ k,ms(⃗r) = E Ψ (+)⃗ k,ms(⃗r) , (2.41)where Ψ (+)⃗ k,ms(⃗r) is the unbound (scattered) Dirac state and ˆ⃗p represents the impulse operator.The influence of the nuclear medium on the particle is twofold: the scalar potential V s (r) shiftsthe particle mass to an effective value, while the vector potential V v (r) affects the energy term.The scattering wave function Ψ (+)⃗ k,ms(⃗r) is decomposed in an upper and a lower component,u (+)⃗ k,ms(⃗r) and w (+)⃗ k,ms(⃗r). Some straightforward manipulations lead to a Schrödinger-like equationfor the upper component][− ∇2+ V c (r) + V so (r) (⃗σ ·2M ⃗L − i⃗r · ˆ⃗p) u (+)⃗(⃗r) =Nk,mswhile the lower component is related to the upper one throughw (+)⃗ k,ms(⃗r) =k22M Nu (+)⃗ k,ms(⃗r) , (2.42)1E + M N + V s (r) − V v (r) ⃗σ · ˆ⃗p u (+)⃗ k,ms(⃗r) . (2.43)Here, the central and spin-orbit potentials V c (r) and V so (r) are defined asV c (r) = V s (r) + E V v (r) + V s 2 (r) − Vv 2 (r),M N 2M N11 dV so (r) =2M N [E + M N + V s (r) − V v (r)] r dr [V v(r) − V s (r)] . (2.44)So far, no approximations have been made. In the relativistic DWIA frameworks [24,25,29,30], the scattering wave function is expanded in partial waves√Ψ (+)E + MN ∑⃗(⃗r) = 4πi l 1〈lm l k,ms 2M N2 m s|jm〉 Ylm ∗l(Ω k ) Ψ m κ (⃗r) , (2.45)κmm lwhere Ψ m κ (⃗r) are four-spinors of the same form as the bound-state wave functions of Eq. (B.4),and Eq. (2.41) is solved numerically using optical potentials. This partial-wave procedure becomesimpractical as the energy increases. Therefore, at higher energies, the Schrödinger-typeequation (2.42) is solved in the eikonal approximation [56, 70].Following the method outlined in Ref. [70], the average momentum ⃗ K and the momentumtransfer ⃗ ∆ which occur during the nucleon-nucleus collision, are defined in terms of thenucleon’s initial ( ⃗ k i ) and final momentum ( ⃗ k f )⃗K = ⃗ k i + ⃗ k f,2⃗∆ = ⃗ k i − ⃗ k f . (2.46)