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96 Chapter 8. Relativistic Effects effects, in terms of our definition, introduces more absorption, resulting in higher spectroscopic factors. In contrast with other studies [45, 53, 90, 91] that have estimated the effect of relativity, we find an overall enhancement of the differential cross section due to coupling of the lower components in the bound and scattering wave function. This should be attributed to the fact that these other studies explicitly made a comparison of a non-relativistic Schrödinger approach with their fully relativistic models, while we estimate the effect of relativity by omitting the coupling term of the lower components in the fully relativistic expression of the transition matrixelement. The genuine relativistic effect stemming from this coupling is larger in the longitudinal than in the transverse channel. It is noteworthy that in the cross section the impact of the “relativistic dynamical effects” never exceeds the 10 % level. This also applies to the longitudinal structure function R L . In the transverse response R T the magnitude of the effects is at the percent level. However, in this channel, the net effect of introducing relativity into the calculation is a reduction of the strength. If we turn our attention to the interference structure functions R T L and R T T , the relativistic effects grow in importance. Especially for the R T L structure function the effects are large and extend to the largest values of Q 2 considered here. This enhanced sensitivity of the R T L response to relativistic effects, even when relatively low values of Q 2 are probed, complies with the conclusions drawn in other studies [45, 90, 104, 105, 106]. The enhancement of the R T L structure function after including the dynamical coupling between the lower components in the nucleon spinors, gives rise to an enhanced left-right asymmetry in quasi-perpendicular kinematics. This enhancement of the R T L structure function makes that the relativistic predictions for the differential cross section are less symmetric than the nonrelativistic predictions. We conclude the discussion of Fig. 8.2 with remarking that the enhanced sensitivity to relativistic effects when using the CC1 version of the off-shell electron-nucleon coupling (Eq. (4.2)), can be attributed to the momentum-dependent part of that operator. The contribution of this term in the matrixelement < k f s f |J µ cc1 |k is i > experiences severe reductions when neglecting the coupling between the lower components. A quantity that can be relatively easy accessed experimentally, and, at the same time, depends heavily upon the R T L structure function is the left-right asymmetry A LT that was defined earlier in Eq. (6.12). The role played by the lower components in this dynamical enhancement of the left-right asymmetry can be further clearified by looking at the results of Fig. 8.3. In this figure, we plot the left-right asymmetry for 1p 3/2 knockout from 12 C, for different Q 2 and quasi-elastic conditions. Looking at the fully relativistic curves, we observe a gradual decrease of the asymmetry with increasing Q 2 . At the same time, the relative contribution of the “non-relativistic” contribution to A LT diminishes. This indicates that the asymmetry is almost exclusively generated by the coupling between the lower components as Q 2 increases.
97 A quantity similar to the left-right asymmetry A LT is the A LT ′ observable A LT ′ = 2v T L ′R T L ′ v L R L + v T R T + v T L R T L + v T T R T T , (8.1) and can be measured by taking the electron’s helicity into account. At high energies the reaction process becomes independent of the electron’s helicity and therefore it is expected that A LT ′ will fall to zero if higher and higher energies are probed. In Fig. 8.4 we display A LT ′ for a number of Q 2 for scattering in a plane that is perpendicular to the electron scattering plane. This choice maximizes the contribution from the R T L ′ structure function which has a sin φ dependence. Several observations emerge from Fig. 8.4. First, the asymmetry A LT ′ decreases as Q 2 increases. Second, we observe relativistic effects only at higher momenta, a feature that was already established in literature [45, 90, 91, 92, 103]. Third, in line with the left-right asymmetry A LT , the nonrelativistic calculations predict much smaller asymmetries than the fully relativistic ones, indicating that the asymmetry is again mainly generated by this dynamical coupling of the lower components. If one assumes equivalent central and spin-orbit potentials in a Schrödinger and a Dirac approach, another relativistic effect stems from the so-called Darwin term. Recalling the eikonal phase of Eq. (5.16) ıS( ⃗ b, z) = −ı M K ∫ z −∞ dz ′ [V c ( ⃗ b, z ′ ) + V so ( ⃗ b, z ′ )[⃗σ · ( ⃗ b × ⃗ K) − ıKz ′ ]], (8.2) it is the impact of the last term in this expression on the cross section and structure functions that we want to measure. For that reason, we have performed calculations for the 16 O(e, e ′ p) experiment of Ref. [15]. In Fig. 8.5 we present results for a fully relativistic eikonal model with Cooper potentials, once with and once without the Darwin term included. As a comparison, we have also plotted results where both the spin-orbit and Darwin term were set to zero. These results are also interesting to compare with Glauber calculations, which typically only incorporate a central scattering part. As expected, the effects of the entire spin-dependent term (i.e. including the Darwin term) on the relatively spin-insensitive response functions R L and R T is minimal. A slight reduction of the response functions is observed when spin-orbit and Darwin terms are taken into account. The effects on the interference structure functions R T L and R T T , on the other hand, are significant as far as the spin-orbit term is concerned. The net effect of the Darwin term is rather modest, and more pronounced in the 1p 1/2 knockout reaction than in the 1p 3/2 knockout reaction. In the 1p 1/2 knockout reaction, the exclusion of the Darwin term causes a modest enhancement of the interference structure functions, while omitting the spinorbit part results in a rather large decrease of strength in the R T L and R T T response
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De Eikonale Benadering in Dirac-Ge
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ii CONTENTS 10 Conclusion and Outlo
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2 Chapter 1. Introduction the stron
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4 Chapter 1. Introduction a wide Q
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6 Chapter 2. Reaction Observables a
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8 Chapter 2. Reaction Observables a
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10 Chapter 2. Reaction Observables
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Chapter 3 Relativistic Bound State
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3.1. Formalism 15 content to the me
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3.2. Results 17 12 C 16 O proton ne
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3.2. Results 19 have verified that
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3.2. Results 21 where the positive
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3.2. Results 23 Figure 3.4 Nucleon
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26 Chapter 4. Off-Shell Electron-Pr
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28 Chapter 4. Off-Shell Electron-Pr
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Chapter 5 The Eikonal Final State I
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5.2. A Consistent Approach to Final
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5.2. A Consistent Approach to Final
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5.3. Optical Potentials and the Eik
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5.3. Optical Potentials and the Eik
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5.4. The Glauber Approach 41 where
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5.4. The Glauber Approach 43 Figure
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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
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Page 69 and 70: 6.1. 16 O(e, e ′ p) 15 N 65 CEA w
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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
Page 107 and 108: Chapter 9 Nuclear Transparency The
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Page 125 and 126: Chapter 10 Conclusion and Outlook I
Page 127 and 128: 123 with increasing Q 2 the uncerta
Page 129 and 130: Bibliography [1] V. Pandharipande,
Page 131 and 132: Bibliography 127 [42] J. Kelly, Phy
Page 133 and 134: Bibliography 129 [85] A. Saha, W. B
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