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52 Chapter 5. The Eikonal Final State where ρ i denotes the density distribution of the nucleon with quantumnumbers i. One thus arrives at an expression where the profile functions are folded with the nucleonic densities. In deriving Eq. (5.52) we have supposed an independent-particle behaviour. A more realistic description needs the introduction of nucleon-nucleon correlations. This can now simply be adopted by the substitution ρ i (⃗r i ) → ρ i (⃗r i )g(⃗r − ⃗r i ) , (5.53) where g(⃗r − ⃗r i ) is the (central) correlation function. The important role played by these short-range correlations has been extensively discussed in literature [8, 74, 75, 76, 77]. We will address these correlations further in Sec. 9, when discussing nuclear transparencies, as they appear to play a significant role there. We use the correlation function proposed by Gearheart and Dickhoff [78, 79] as plotted in Fig. 5.12. This choice is based on the fact that this correlation function was shown to produce a favorable agreement with 12 C(e, e ′ p) data [80]. Also, in comparison with other model predictions for the central correlation function, the one obtained by Gearhart and Dickhoff can be classified between the categories of “hard” (with a core at short internucleonic distances) and “soft” (characterized by a finite probability to observe nucleon pairs at very short internucleonic distances) correlations. 5.4.4 Relativistic Extension of the Glauber Approximation In this section a relativistic generalization of the non-relativistic Glauber framework will be outlined. We start from the scattering amplitude [57] F ss ′( ⃗ k i , ⃗ k f , E) = − M 2π < Φ ⃗ ki ,s |(βV s + V v )|ψ (+) ⃗ kf ,s ′ > , (5.54) with a relativistic scattered wave of the type √ [ ψ (+) ⃗ = E + M 1 kf ,s ′ 1 2M E+M+V s−V v ⃗σ · ⃗p and the free Dirac solution √ [ Φ ⃗ki ,s = E + M 2M ] ] 1 1 E+M ⃗σ · ⃗k i After some straightforward algebraic manipulations one obtains where ∫ F ( ⃗ k i , ⃗ k f , E) = −iK e ı⃗ k f ·⃗r e ıS(⃗r) χ 1 2 m s ′ , (5.55) e ı⃗ k i·⃗r χ 1 . (5.56) ms 2 d ⃗ b 2π ei⃗q·⃗b (e iχ(⃗ b) − 1) , (5.57) F ss ′( ⃗ k i , ⃗ k f , E) =< χ s ′|F ( ⃗ k i , ⃗ k f , E)|χ s > . (5.58)
5.4. The Glauber Approach 53 1.2 1 0.8 g(r) 0.6 0.4 0.2 0 -0.2 0 0.5 1 1.5 2 2.5 3 r [fm] Figure 5.12 The Gearheart-Dickoff correlation function as a function of the relative internucleonic range r.
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De Eikonale Benadering in Dirac-Ge
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ii CONTENTS 10 Conclusion and Outlo
Page 6 and 7: 2 Chapter 1. Introduction the stron
Page 8 and 9: 4 Chapter 1. Introduction a wide Q
Page 10 and 11: 6 Chapter 2. Reaction Observables a
Page 12 and 13: 8 Chapter 2. Reaction Observables a
Page 14 and 15: 10 Chapter 2. Reaction Observables
Page 17 and 18: Chapter 3 Relativistic Bound State
Page 19 and 20: 3.1. Formalism 15 content to the me
Page 21 and 22: 3.2. Results 17 12 C 16 O proton ne
Page 23 and 24: 3.2. Results 19 have verified that
Page 25 and 26: 3.2. Results 21 where the positive
Page 27: 3.2. Results 23 Figure 3.4 Nucleon
Page 30 and 31: 26 Chapter 4. Off-Shell Electron-Pr
Page 32 and 33: 28 Chapter 4. Off-Shell Electron-Pr
Page 35 and 36: Chapter 5 The Eikonal Final State I
Page 37 and 38: 5.2. A Consistent Approach to Final
Page 39 and 40: 5.2. A Consistent Approach to Final
Page 41 and 42: 5.3. Optical Potentials and the Eik
Page 43 and 44: 5.3. Optical Potentials and the Eik
Page 45 and 46: 5.4. The Glauber Approach 41 where
Page 47 and 48: 5.4. The Glauber Approach 43 Figure
Page 49 and 50: 5.4. The Glauber Approach 45 shape
Page 51 and 52: 5.4. The Glauber Approach 47 ε pp
Page 53 and 54: 5.4. The Glauber Approach 49 1250 1
Page 55: 5.4. The Glauber Approach 51 so tha
Page 59 and 60: 5.5. Higher Order Eikonal Correctio
Page 61 and 62: Chapter 6 Final State Interactions
Page 63 and 64: 6.1. 16 O(e, e ′ p) 15 N 59 d 5
Page 65 and 66: 6.1. 16 O(e, e ′ p) 15 N 61 ρ [(
Page 67 and 68: 6.1. 16 O(e, e ′ p) 15 N 63 12 C(
Page 69 and 70: 6.1. 16 O(e, e ′ p) 15 N 65 CEA w
Page 71 and 72: 6.1. 16 O(e, e ′ p) 15 N 67 Figur
Page 73 and 74: 6.1. 16 O(e, e ′ p) 15 N 69 0.4 0
Page 75 and 76: 6.2. 12 C(e, e ′ p) 11 B 71 0 -0.
Page 77 and 78: 6.2. 12 C(e, e ′ p) 11 B 73 ρ [(
Page 79 and 80: 6.2. 12 C(e, e ′ p) 11 B 75 P n P
Page 81 and 82: Chapter 7 Off-Shell Ambiguities A m
Page 83 and 84: 79 momentum region. Since these exc
Page 85 and 86: 81 R L [fm 3 ] R T [fm 3 ] R TT [fm
Page 87 and 88: 83 R L [fm 3 ] R T [fm 3 ] 0.04 0.0
Page 89 and 90: 85 manner, we will from here on onl
Page 91 and 92: 87 R CC1/CC2 R CC1/CC2 R CC1/CC2 R
Page 93 and 94: 89 The relation obtained in Eq. (7.
Page 95: 91 A LT 1 0.5 0 OMEA (CC2) OMEA (CC
Page 98 and 99: 94 Chapter 8. Relativistic Effects
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Chapter 9 Nuclear Transparency The
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9.1. Theoretical Background 105 Req
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9.1. Theoretical Background 107 few
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9.2. Review of Experimental Results
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9.2. Review of Experimental Results
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9.3. Nuclear Transparency Calculati
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Chapter 10 Conclusion and Outlook I
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123 with increasing Q 2 the uncerta
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Bibliography [1] V. Pandharipande,
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Bibliography 127 [42] J. Kelly, Phy
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Bibliography 129 [85] A. Saha, W. B
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