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20 Chapter 3. Relativistic Bound State Wave Functions Proton potentials [MeV] 800 600 400 200 0 -200 -400 Scalar Vector Scalar+Vector -600 -800 12 C 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 r [fm] Proton potentials [MeV] 800 600 400 200 0 -200 -400 Scalar Vector Scalar+Vector -600 -800 16 O 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 r [fm] Figure 3.2 The radial dependence of the scalar and vector proton potentials as obtained from relativistic Hartree calculations for 12 C and 16 O. The neutron potentials are quasi identical to those of the proton’s and have not been plotted.
3.2. Results 21 where the positive (negative) sign corresponds to κ > 0 (κ < 0). The overlap between a free plane wave spinor characterized by a momentum ⃗ k and the meanfield eigenstates of Eq. (3.5), summed over all spin projections, is then given by Making use of the following identity : j∑ m=−j ρ ⃗k,ɛ,κ ≡ ∑ ∫ ∣ d⃗r φ † ⃗ (⃗r)ψ ɛκm (⃗r) k,s ∣ m s,m 2 . (3.14) ( ) 2j + 1 Y κmY † κ ′ m = δ κκ ′ , κ ′ = ±κ , (3.15) 4π we finally arrive at the following expression for the nucleonic momentum distribution : ( ) ( ) 1 ɛk + M ρ ⃗k,ɛ,κ = (2j + 1) 2π 2 k 2 |G ɛκ (k) + F ɛκ (k)| 2 , (3.16) 2M where G ɛκ (k) = F ɛκ (k) = ∫ ∞ 0 ∫ ∞ 0 dr G ɛκ (r)g κ (kr) , dr F ɛκ (r)f κ (kr) , (3.17) are the Fourier transforms of the radial components of the bound state wave function. Eq. (3.16) can be seen as the relativistic generalization of the nucleon momentum distribution. Fig. 3.3 shows the momentum distributions for all occupied levels in 12 C and 16 O. Since the proton and neutron results can be hardly distinguished from each other, the latter are not shown. The nucleon momentum distributions are subject to sizeable corrections when correlations beyond the mean-field Hartree approach are implemented. In an uncorrelated Fermi gas all states up to the Fermi momentum k F are occupied, and the states above k F remain empty. In a correlated system, states below the Fermi surface become partially depleted and states above the Fermi surface get a non-vanishing occupation probability. The nucleon-nucleon correlations are mainly induced by the short-range and tensor components of the nucleon-nucleon interaction. Evidence for the depletion of the single-particle states was obtained from the analysis of (e,e’p) reactions for a large number of target nuclei [1]. Another possible source for the fragmentation of the single-particle strength is the mixing between the positive and negative energy states, or, equivalently, between particles and antiparticles [38]. For small values of the momenta (k < 300 MeV/c), it has been shown that the nucleon spectral function is dominated by positive energy
Page 1: De Eikonale Benadering in Dirac-Ge
Page 4 and 5: ii CONTENTS 10 Conclusion and Outlo
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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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