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2.3. Relativistic Multiple-Scattering Glauber Approximation 24small, the spectator nucleons can be approximated by fixed scattering centers (the so-calledfrozen approximation). As mentioned before, we adopt a spinless Glauber theory, i.e., the fastparticle only interacts with the scatterers by means of two-body spin-independent interactions.Exchange effects between the fast particle and the spectator nucleons are neglected as well.For a multiple-scattering event leading from an initial state |i〉 to a final state |f〉, the Glauberscattering amplitude readsF multi ( ∆) ⃗ = iK ∫2πwhere the total Glauber phase-shift functionχ tot (K, ⃗ b, ⃗ b 2 , . . . , ⃗ b A ) =d ⃗ b e i⃗ ∆·⃗b 〈f| 1 − e iχtot(K,⃗ b, ⃗ b 2 ,..., ⃗ b A ) |i〉 , (2.65)A∑χ j (K, ⃗ b − ⃗ b j ) , (2.66)j=2is the sum of the phase shifts χ j contributed by each of the spectator scatterers. Here, ⃗ b denotesthe impact parameter of the fast nucleon and ( ⃗ b 2 , ⃗ b 3 , . . . , ⃗ b A ) those of the frozen spectator nucleons.The phase-shift additivity property of Eq. (2.66) is a direct result of the following assumptions:the one-dimensional nature of the relative motion and the neglect of three- and morebodyforces, scatterer motion, and longitudinal momentum transfer. Moreover, as Eq. (2.65) isbased on the eikonal approximation, it is only valid when the energy transfer is small comparedto the incident particle energy, i.e., for elastic and mildly inelastic collisions. Expression (2.65)does not apply to deeply inelastic collisions in which the nature of the particles is modified orthe number of particles is altered during the collision.Under the assumption of phase-shift additivity, the eikonal wave function of Eq. (2.51) canbe generalized to many-body scattering. The scattering wave function for a fast nucleon withmomentum k and spin state ∣ 12 m s〉which scatters from A − 1 residual nucleons reads√ []Ψ (+) E + MN 1⃗(⃗r) =k,ms 2M N1E+M N⃗σ · ˆ⃗pŜ e i⃗k·⃗r χ 1 , (2.67)ms2where the operator Ŝ implements the subsequent collisions of the fast nucleon with the frozenspectator nucleonsŜ (⃗r, ⃗r 2 , ⃗r 3 , . . . , ⃗r A ) ≡A∏j=2e −i M NKR z−∞ dz ′ V c( ⃗ b− ⃗ b j ,z ′ −z j ) . (2.68)As in the ROMEA framework, only the central spin-independent contribution V c ( ⃗ b, z) is retained,the impulse operator is replaced by the nucleon momentum, and the dynamical enhancementof the lower component has been neglected since E + M N ≫ |V s (r) − V v (r)| for thehigh energies at which Glauber theory is applied. In this general form, the multiple-scatteringwave function can not be directly related to the individual profile function for NN scattering(2.64).
Chapter 2. Relativistic Eikonal A(p, pN) Formalism 25The zero-range approximation along the scattering direction offers the possibility to expressthe multiple-scattering wave function in terms of the experimentally determined profilefunction, thereby avoiding the technical complications with respect to potential scattering. Thezero-range approximationV c ( ⃗ b − ⃗ b j , z ′ − z j ) ≃ V ⊥c ( ⃗ b − ⃗ b j ) δ(z ′ − z j ) , (2.69)amounts to neglecting the finite longitudinal dimension of the NN interaction region. Afterexpanding expression (2.68) for Ŝ and adopting the zero-range approximation, one obtainsŜ ≃A∏j=2(1 − i M )NK V c ⊥ ( ⃗ b − ⃗ b j ) θ(z ′ − z j ). (2.70)Using the relation (2.58) between the profile function and the phase-shift function, a similarreasoning leads toΓ NN (K, ⃗ b) ≃ i M NK V ⊥c ( ⃗ b) . (2.71)Finally, the Glauber operator becomesŜ (⃗r, ⃗r 2 , ⃗r 3 , . . . , ⃗r A ) ≡A∏ [1 − Γ NN(K, ⃗ b − ⃗ )b jj=2]θ (z − z j ), (2.72)where the step function θ (z − z j ) ensures that the nucleon only interacts with other nucleonsif they are localized in its forward propagation path.2.3.3 RMSGA for A(p, pN) ReactionsThe Glauber operators in Eq. (2.36) take the formsŜ p1 (⃗r, ⃗r 2 , ⃗r 3 , . . . , ⃗r A ) =Ŝ k1 (⃗r, ⃗r 2 , ⃗r 3 , . . . , ⃗r A ) =Ŝ k2 (⃗r, ⃗r 2 , ⃗r 3 , . . . , ⃗r A ) =A∏ [1 − Γ pN(p 1 , ⃗ b − ⃗ )b jj=2A∏j=2A∏j=2]θ (z − z j )( [1 − Γ pN k 1 , ⃗ b ′ − ⃗ )b ′ j θ ( z ′ j − z ′)] ,[ (1 − Γ Nk2 N k 2 , ⃗ b ′′ − ⃗ )b ′′j, (2.73a)θ ( z ′′j − z ′′)] ,(2.73b)(2.73c)where N k2 = p (n) for A(p, 2p) (A(p, pn)) reactions. Further, ⃗r denotes the collision pointand (⃗r 2 , ⃗r 3 , . . . , ⃗r A ) are the positions of the frozen spectator protons and neutrons in the target.The ( ⃗ b, z), ( ⃗ b ′ , z ′ ), and ( ⃗ b ′′ , z ′′ ) coordinate systems are defined as in Section 2.2.2. The stepfunctions guarantee that the incoming proton can only interact with those spectator nucleonsit encounters before the hard collision and that the outgoing protons can only interact with thespectator nucleons they find in their forward propagation paths.
Page 1: FACULTEIT WETENSCHAPPENA RELATIVIST
Page 8 and 9: Contentsiv
Page 10 and 11: 2mechanism is obtained than in incl
Page 12 and 13: 4rely on partial-wave expansions of
Page 15 and 16: Chapter 2Relativistic Eikonal Descr
Page 18 and 19: 2.1. Observables and Kinematics 10w
Page 20 and 21: 2.1. Observables and Kinematics 12B
Page 22 and 23: 2.1. Observables and Kinematics 14M
Page 24 and 25: 2.1. Observables and Kinematics 16d
Page 26 and 27: 2.2. Relativistic Optical Model Eik
Page 28 and 29: 2.2. Relativistic Optical Model Eik
Page 30 and 31: 2.3. Relativistic Multiple-Scatteri
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Page 36 and 37: 2.4. Approximated RMSGA 2812121010
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Page 44 and 45: 3.1. The IFSI Factor 36sence of ini
Page 46 and 47: 3.1. The IFSI Factor 38imaginary pa
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Page 62 and 63: 54Figure 4.1 The beam-momentum depe
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Page 80 and 81: 4.5. Outlook 72the 7 Li and 12 C da
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4.5. Outlook 7410.812 C0.6Transpare
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4.5. Outlook 76
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5.1. The A(e, e ′ p) Matrix Eleme
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5.2. Nuclear Transparency 80112 C0.
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5.3. Induced Normal Polarization 82
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5.5. Differential Cross Section 84w
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5.5. Differential Cross Section 860
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5.5. Differential Cross Section 88d
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90framework, which is a multiple-sc
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92be transformed into an amplitude
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A.2. Pauli Matrices 94A.2 Pauli Mat
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96pseudo-scalar π mesons, as well
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980.11ρ chg(e/fm 3 )0.080.060.0440
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100with∫D r,θ (r, θ) ≡dφ sin
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Bibliography 102[12] R. A. Arndt, I
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Bibliography 104[43] E. D. Cooper,
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Bibliography 106[71] R. J. Glauber,
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Bibliography 108[104] S. J. Wallace
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Bibliography 110[132] C. Baglin et
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Bibliography 112[163] K. Wijesooriy
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Bibliography 114[195] L. L. Frankfu
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Bibliography 116[227] J. Botts, J.-
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Bibliography 118
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Nederlandstalige samenvatting 120
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Nederlandstalige samenvatting 122we
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