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3.1. The IFSI Factor 36sence of initial- and final-state interactions the real part of the IFSI factor equals one, whereasthe imaginary part vanishes identically.The A(p, pN) IFSI factor is a function of three independent variables (r, θ, φ). The z axis ischosen along the direction of the incoming beam ⃗p 1 , the y axis lies along ⃗p 1 × ⃗ k 1 , and the xaxis lies in the scattering plane defined by the proton momenta ⃗p 1 and ⃗ k 1 . θ and φ denote thepolar and azimuthal angles with respect to the z axis and the x axis, respectively. The radialcoordinate r represents the distance relative to the center of the target nucleus.3.1.1 Polar-Angle DependenceTo gain a better insight into the dependence of the IFSI factor on r, θ, and φ, we calculated thecontribution of the three distorting functions Ŝp1, Ŝ k1 , and Ŝk2 to the IFSI factor. In Figs. 3.1and 3.2 results are displayed for the computed real and imaginary part of S IFSI (r, θ, φ = 0) forproton emission from the Fermi level in 12 C. The results were computed within the ROMEAframework, using the EDAD1 optical-potential parametrization of [46].The θ dependence can be interpreted as follows. For a given r, the distance that the incomingproton travels through the target nucleus before colliding hard with a target nucleondecreases with increasing angle θ. As a consequence, small values of θ induce the largest ISI.For the FSI of the scattered proton, the opposite holds true, and θ = 180 ◦ for a large r valuecorresponds to an event whereby the hard collision transpires at the outskirts of the nucleusand the scattered proton has to travel through the whole nucleus before it becomes asymptoticallyfree, thus giving rise to the smallest (largest) values for the real (imaginary) part of theIFSI factor. Unlike the scattered proton, which moves almost collinear to the z axis, the ejectednucleon leaves the nucleus under a large scattering angle θ 2 . Hence, the FSI are minimal forθ close to 0 ◦ or 180 ◦ and maximal for θ around 180 ◦ − θ 2 . Finally, the θ dependence of thecomplete IFSI factor is the result of the interplay between the three distorting effects, with thestrongest scattering and absorption observed at θ close to 0 ◦ , 180 ◦ − θ 2 , and 180 ◦ .3.1.2 Radial DependenceFig. 3.3 displays the real part of the 40 Ca IFSI factor as a function of r at various values of θ.The ROMEA calculations were performed for the same kinematics as in Figs. 3.1 and 3.2, andemployed the EDAI optical-potential fit of [46].The upper left panel suggest that the ISI effects increase with growing r for θ = 0 ◦ . Forθ = 45 ◦ and increasing r, initially, the growing distance the proton has to travel through thenucleus leads to a decrease of the real part of Ŝp1. This is followed by an increase for larger rup to Ŝp1 = 1. This reduction in ISI effects with increasing r is brought about by the incomingproton’s path through the nucleus moving away from the nuclear interior and closer to theless dense nuclear surface. The other curves of the upper left figure reveal a general trend for90 ◦ ≤ θ ≤ 180 ◦ : as r increases, the real part of the IFSI factor grows correspondingly. As canbe appreciated from Fig. 3.3 as well as from the previous figures, the global behavior of the Ŝk1
Chapter 3. IFSI and A(p, pN) Differential Cross Sections 37real part of S IFSI10.80.60.40.2real part of S IFSI10.80.60.40.2150θ (deg)10050012345r (fm)6150θ (deg)10050012345r (fm)6real part of S IFSI0.80.60.4real part of S IFSI0.80.60.40.2150θ (deg)10050012345r (fm)6150θ (deg)10050012345r (fm)6Figure 3.1 The radial and polar-angle dependence of the real part of the IFSI factor S IFSI in the scatteringplane (φ = 0 ◦ ) for proton knockout from the Fermi level in 12 C. The upper left panel is the contributionfrom the impinging proton (Ŝp1), while the upper right panel shows the effect of the FSI of the scatteredproton (Ŝk1). In the bottom left figure, the effect of the FSI of the ejected proton (Ŝk2) is presented andthe bottom right figure shows the complete IFSI factor (Ŝk1 Ŝk2 Ŝp1). The kinematics was T p1 = 1 GeV,T k1 = 870 MeV, θ 1 = 13.4 ◦ , and θ 2 = 67 ◦ .factor describing the scattered proton’s FSI can be related to that of the ISI factor Ŝp1 through thesubstitution θ → 180 ◦ − θ. This approximate symmetry can be attributed to the small scatteringangle θ 1 , i.e., the scattered proton leaves the nucleus almost parallel to the incoming proton’sdirection. In the bottom left panel, the additional curve (θ = 115 ◦ , i.e., close to 180 ◦ − θ 2 )represents the situation of maximal FSI of the ejected nucleon. For this θ value, the path of theejected nucleon passes through the center of the nucleus, and the distance traveled throughthe nucleus increases with r. Accordingly, the real part of Ŝk2 is a monotonously decreasingfunction of r. The other extreme is the θ = 0 ◦ case, where increasing r means less FSI. For theother θ values, the absorption reaches its maximum for some intermediate r value. Again, thecombination of Ŝp1, Ŝk1, and Ŝk2 determines the total IFSI factor, with the strongest attenuationpredicted in the nuclear interior.
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
Page 36 and 37: 2.4. Approximated RMSGA 2812121010
Page 38 and 39: 2.5. Second-Order Eikonal Correctio
Page 46 and 47: 3.1. The IFSI Factor 38imaginary pa
Page 48 and 49: 3.2. Radial and Polar-Angle Contrib
Page 50 and 51: 3.3. A(p, pN) Differential Cross Se
Page 62 and 63: 54Figure 4.1 The beam-momentum depe
Page 64 and 65: 4.1. High-Momentum-Transfer Wide-An
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Page 68 and 69: 4.3. Color Transparency 60of CT. Th
Page 70 and 71: 4.3. Color Transparency 62with grou
Page 72 and 73: 4.3. Color Transparency 64with p th
Page 74 and 75: 4.4. Nuclear Transparency Results 6
Page 80 and 81: 4.5. Outlook 72the 7 Li and 12 C da
Page 82 and 83: 4.5. Outlook 7410.812 C0.6Transpare
Page 84 and 85: 4.5. Outlook 76
Page 86 and 87: 5.1. The A(e, e ′ p) Matrix Eleme
Page 88 and 89: 5.2. Nuclear Transparency 80112 C0.
Page 90 and 91: 5.3. Induced Normal Polarization 82
Page 92 and 93: 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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Nederlandstalige samenvatting 1246
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