Download Thesis in Pdf Format - Theoretical Nuclear Physics and ...

Chapter 2. Relativistic Eikonal A(p, pN) Formalism 23with βNN 2 the slope parameter. This Gaussian parametrization is based on the diffractive pattern,with broad maxima and diffractive dips, observed in elastic NN collisions at GeV energies.Similar to Fraunhofer diffraction in optics, the diffraction phenomenon occurs when thewavelength of the projectile is short compared to the size of the interaction region. The differentialcross section dσ/dt, with t ≡ (k µ f − kµ i )2 the Mandelstam variable, is extremely forwardpeaked and drops exponentially over many orders. The slope parameter βNN 2 describes the tdependence of the elastic NN differential cross section at forward angles:dσ elNNdt≈ dσel NNdt∣ exp ( −βNN 2 |t| ) . (2.62)t=0Using the optical theorem Im F (θ = 0, φ = 0) = k σtot4π, one findsA( ∆) ⃗ = k )σtot NN4π(ɛ NN + i) exp(− β2 NN ∆2, (2.63)2with ɛ NN the ratio of the real to the imaginary part of the scattering amplitude.The inverse Fourier transform of Eq. (2.57) brings about the following expression for theprofile function:Γ NN(k, ⃗ )b = σtot NN (k) (1 − iɛ NN (k))4πβNN 2 (k)( )⃗ b2exp −2βNN 2 (k) . (2.64)The profile function for central elastic NN scattering depends on the momentum k throughthree parameters: the total NN cross sections σNN tot (k), the slope parameters β2 NN(k), and theratios of the real to the imaginary part of the scattering amplitude ɛ NN (k). These parameterscan be determined directly from the elementary proton-proton and proton-neutron scatteringdata, and will be discussed in Section 2.3.4.At lower energies, that part of the profile function proportional to ɛ NN (k) is non-Gaussianand makes significant contributions to nuclear scattering. Rather than Eq. (2.64), a parametrizationin terms of the Arndt NN phases [12,13] is appropriate at lower energies. For the Glaubercalculations presented here, which address higher energies, the Gaussian-like real part ofΓ NN(k, ⃗ )b is the dominant contributor, and the use of Eq. (2.64) is justified.2.3.2 Glauber Multiple-Scattering Extension of the EAAssuming that the sequential scatterings are independent, the eikonal method can easily beextended to multiple scattering. This constitutes the basis of Glauber theory [62–64]. The scatteringoff a composite system (the IFSI in A(p, pN) reactions) is modeled as the scattering ofa fast particle (the incident, scattered, or ejected nucleon) with the scattering centers of thecomposite system (the spectator nucleons in the residual nucleus). We assume that the fastparticle’s momentum is much larger than that of the spectator nucleons. Then, the particle undergoesa negligible deflection and its trajectory can be approximated by a straight line (thisis the EA). In addition, because the time it takes the particle to traverse the nucleus is very