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18 Chapter 3. Relativistic Bound State Wave Functions ρ [fm -3 ] 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 r [fm] Scalar Baryon Proton 12 C 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 ρ [fm -3 ] 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 r [fm] Scalar Baryon Proton 16 O 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Figure 3.1 The calculated scalar (ρ s ), baryon (ρ B ) and proton (ρ P ) density distributions in the 12 C and 16 O nuclei. The rho density (ρ 3 ) is not depicted as it is several orders of magnitudes smaller.
3.2. Results 19 have verified that these results are comparable to those produced by the TIMORA code [16], which is widely used to solve the set of Eq. (3.7). The scalar potential is entirely governed by the scalar meson field (σ meson), and is responsible for the attractive force that keeps the nucleons bound within the nucleus. The repulsive vector potential has contributions from the vector (ω and ρ) meson fields and the Maxwell field. It should be noted that the contributions from the ρ meson field and the Maxwell field are several orders of magnitude smaller than the contribution from the ω meson field. Since the potential experienced by the neutrons differs only from the one experienced by the protons in these two minor terms, the total scalar and vector potentials for protons and neutrons are almost exactly the same. That is why we have only depicted the proton potentials in Fig. 3.2. These very same potentials will be used in Sec. 5 to derive the eikonal scattering states. A point of high interest in nuclear physics are the nucleon spectral functions and momentum distributions. In calculating these contributions, we closely follow the method outlined in Ref. [38]. In free space the nucleon’s motion is governed by the free particle solution of the Dirac equation. These plane wave states with momentum ⃗ k and energy ɛ k = √ k 2 + M 2 are given by φ ⃗k,s (⃗r) = √ [ ɛ k + M 2M 1 E+M ] 1 ⃗σ · ⃗k e ı⃗k·⃗r χ 1 . (3.10) ms 2 Since the Dirac-Hartee potentials are spherically symmetric, we expand this free space solution in terms of the spin-spherical harmonics of Eq. (3.6) : φ ⃗k,s (⃗r) = × √ ɛ k + M ∑ 4π(+ı) l 1 < lm l 2M κmm l 2 m s|l 1 2 jm > Y lm ⋆ l (Ω ⃗k ) kr [ ] ıgκ (kr) Y κm , (3.11) −f κ (kr) Y −κm where the usual spherical harmonics Y lml are given in terms of the associated Legendre polynomials P m l l : √ ( ) (2l + 1) (l − ml )! Y lml (θ, φ) = (−1) m l e ımlφ P m l l (cos θ) . (3.12) 4π (l + m l )! The radial components of the wave function are written in terms of Ricatti-Bessel functions (these are the spherical Bessel functions times their argument) : g κ (kr) = ĵ l (kr) f κ (kr) = ±k ɛ k + M ĵl∓1(kr) , (3.13)
Page 1: De Eikonale Benadering in Dirac-Ge
Page 4 and 5: 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
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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: 3.2. Results 17 12 C 16 O proton ne
Page 25 and 26: 3.2. Results 21 where the positive
Page 27: 3.2. Results 23 Figure 3.4 Nucleon
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Page 35 and 36: Chapter 5 The Eikonal Final State I
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Page 41 and 42: 5.3. Optical Potentials and the Eik
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Page 45 and 46: 5.4. The Glauber Approach 41 where
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Page 51 and 52: 5.4. The Glauber Approach 47 ε pp
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Page 61 and 62: Chapter 6 Final State Interactions
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6.1. 16 O(e, e ′ p) 15 N 69 0.4 0
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6.2. 12 C(e, e ′ p) 11 B 71 0 -0.
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6.2. 12 C(e, e ′ p) 11 B 73 ρ [(
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6.2. 12 C(e, e ′ p) 11 B 75 P n P
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Chapter 7 Off-Shell Ambiguities A m
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79 momentum region. Since these exc
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81 R L [fm 3 ] R T [fm 3 ] R TT [fm
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83 R L [fm 3 ] R T [fm 3 ] 0.04 0.0
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85 manner, we will from here on onl
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87 R CC1/CC2 R CC1/CC2 R CC1/CC2 R
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89 The relation obtained in Eq. (7.
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91 A LT 1 0.5 0 OMEA (CC2) OMEA (CC
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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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