Chapter 4
Treatment **of** **the** **fission** channel
4.1. Double humped **fission** barriers
**The** main concepts **of** nuclear ﬁssion theory are based essentially on **the** liquid -drop model. According to this model, **the** competition between **the** surface tension forces **of** a nuclear liquid drop and **the** Coulomb repulsion forces related to **the** nuclear charge leads to **the** formation **of** a single humped energy barrier which can determine spontaneous decay **of** **the** nucleus. **The** penetrability **of** **the** barrier determines **the** half-life for spontaneous ﬁssion. Early studies showed that **the** liquid-drop model cannot explain **the** major peculiarity **of** spontaneous and low-energy ﬁssion **of** **the** actinides, namely **the** asymmetric mass distribution **of** ﬁssion fragments, nor **the** sub-barrier **fission** **of** fertile actinides or **the** isomeric **fission**. **The** spontaneously – ﬁssioning americium isomers and **the** intermediate resonance structures observed in neutron-induced ﬁssion **cross** sections, required radical changes in **the** ﬁssion model. Calculations **of** nuclear deformation energies based on **the** shell correction method by Strutinsky ( Strutinsky, 1967 and 1968 ) played a crucial role. Applying **the** Strutinski shell corrections a double humped barrier is obtained. **The** ﬁssion barriers calculated for **the** actinides consisted **of** a double-hump curve with a rather deep potential well between **the** humps.

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(F ), C I
(F eff ), C I + ∆C I
(F)} as independent pa- rameters. In Kin-1 and Kin-2, due to **the** lower photon energy E γ (Table I), our acceptance, trigger, and readout did not record a comprehensive set **of** ep → eπ 0 X events. For those events we were able to reconstruct, we found only a few percent contribution to dΣ, but a larger con- tribution to dσ. For Kin-1,2, we only present results on dΣ. Our systematic errors in **the** **cross**-**section** measure- ments are dominated by **the** following contributions: 3% from HRS×PbF 2 acceptance and luminosity; 3% from H(e, e ′ γ)γX (π 0 ) background; 2% from radiative correc- tions; and 3% from inclusive H(e, e ′ γ)N π . . . background. **The** total, added in quadrature, is 5.6%. **The** dΣ results contain an additional 2% systematic uncertainty from **the** beam polarization. In order to compute **the** BH contri- bution in **the** dσ analysis we used Kelly’s parametrization **of** form factors [26], which reproduce elastic **cross**-**section** world data in our t range with 1% error and 90% CL.

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3.2. Culture characterization
**The** cell size distribution **of** C. vulgaris was measured using 165 microscope images captured using a digital camera (AxioCam MRc) mounted on an optical microscope (Zeiss Axio Scope A1). **The** dia- meters **of** 2873 cells were measured manually using an image pro- cessing software (Axio Vision Routine) and their volume equivalent sphere radius was reported for **the** sake **of** completeness. These results were not used in either experimental approaches investigated. **The** culture dry biomass concentration X (in g/L) was determined gravimetrically by ﬁltering 5 mL **of** culture through a pre-dried and pre-weighed 0.45 μ m pore size glass- ﬁber ﬁlter (Whatman GF/F). **The** ﬁlters were dried overnight in an oven at 105 °C and weighed, in an analytical balance (Mettler Toledo XA204, Columbus, OH) with a sensitivity **of** 0.1 mg, after being cooled in a desiccator for 10 min. **The** samples were analyzed in triplicates and **the** reported biomass con- centration corresponded to **the** mean value. In addition, **the** cell density N T (in #/m 3 ) was counted using a 200 μ m deep Malassez

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factors F i only depend on x. This behaviour goes under **the** name Bjorken scaling.
2.2 Neutrino-nucleus scattering
Theoretical modelling **of** **the** neutrino-nucleus scattering face many complications. **The** first is **the** initial state **of** **the** nucleons inside a nucleus. They are indeed constantly moving around inside **the** nuclear potential, changing their momentum and direction in relation to an incoming neutrino, which affects both **the** kinematics and **the** **cross**-**section**. Unfortunately **the** initial momentum spectra **of** nucleons is not well known, and can vary significantly even between similar mass nuclei. An additional complication is that **the** neutrino can interact not only with individual nucleons, but **the** interaction can include correlated nucleon pairs or any combination **of** nucleons in a quasi-bound state. After **the** final-state particles have been created from an interaction, they need to propagate out through **the** nucleus. During **the** propagation they can interact with **the** other nucleons inside **the** nucleus. These processes, called “final state interactions” (FSI), can alter **the** particles type and number.

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tions:
D(e, e 0 π 0 )X = d(e, e 0 π 0 )d+n(e, e 0 π 0 )n+p(e, e 0 π 0 )p. (5) We subtract **the** p(e, e 0 π 0 )p yield from **the** deuterium data by normalizing our H(e, e 0 π 0 )X data to **the** lumi- nosity **of** **the** LD2 data. **The** Fermi-momentum ~ p F **of** bound protons inside **the** deuteron is statistically added to **the** LH2 data following **the** distribution given in [22] since this effect is intrinsically present in **the** M X 02 spec- trum **of** **the** LD2 data. **The** result **of** **the** subtraction **of** **the** H(e, e 0 π 0 )X data from **the** D(e, e 0 π 0 )X yield is shown in Fig. 2. **The** d(e, e 0 π 0 )d and n(e, e 0 π 0 )n chan- nels are in-principle kinematically separated by ∆M X 02 = t(1 − M/M d ) ≈ t/2 where M d is **the** deuteron mass. This kinematic shift is exploited in **the** procedure described below to separate **the** contributions **of** **the** quasi-free neu- tron and coherent deuteron channels in **the** total π 0 elec- troproduction **cross** **section**.

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It should be noted that both **the** total and elastic **cross** **section** datasets include discordant data from different experiments. This is quantified by a simple consistency check that fits generic quadratic polynomials in log s to each dataset and computes **the** resulting χ 2 . Table I shows **the** results with both **the** elastic and total **cross** sections running up χ 2 =d:o:f noticeably greater than 1. Thus, one obtains a minimum combined χ 2 **of** 47.1. This is a well- known problem with these data, first addressed in [30] and later in [31] . At present, however, **the** number **of** data points is simply too small to identify individual outliers, and hence there is little one can do for lack **of** better experimental results. We shall thus neither filter nor sieve **the** data, but remember that **the** best possible χ 2 is rather high.

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Following this line **of** research, some quantities have still been left aside from this global correlation effort, namely **fission** yields. In **the** following, **the** term “**fission** yield” will designate without distinction independent or cumu- lative **fission** yields (called FY). It is a known deficiency that current evaluated libraries do not provide correlation matrices for **fission** yields, neither for a specific **fission**- ing isotope (such as all **fission** yields from 235 U(n,f)), or between fissioning isotopes. This represents a practical problem for nuclear data users, as uncertainties on quanti- ties dependent **of** FY (e.g. long-lived **fission** products from spent fuel) are believed to be too large, due to **the** lack **of** FY correlations [ 31 – 34 ]. As mentioned before, **the** current evaluation format is not defined to propose such correla- tions for FY; there are nevertheless efforts to update such format.

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5 Sensitivity / uncertainty analysis
5.1. Calculation **of** sensitivity coefficients
**The** correlated sampling [7] feature **of** **the** TRIPOLI4 code was used to calculate sensitivity coefficients, defined as **the** relative change in **the** ITR i due to a relative change **of** 1% in **the** **cross** **section** σ i **of** 238 U - with σ i = capture, **fission**, elastic, inelastic or (n,xn) reactions (for **the** time being, it is not possible to get **the** sensitivity coefficient due to prompt neutron multiplicity neither due to double differential data like for instance **the** energy or angular distribution **of** inelastic scattering). **The** 1-group sensitivity coefficients for 238 U and 235 U reactions are presented in Table 5 while **the** sensitivity profile as a function **of** neutron incident energy is plotted in Fig. 5 for **the** ITR obtained from **the** activation **of** 54 Fe. **The** sensitivity coefficient breakdown shows a

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CHAPTER 7. CONCLUSION & PERSPECTIVES 7.1. OVERVIEW **OF** **THE** WORK
felt by **the** nucleons. This parameterized potential is chosen to be consistent with **the** former macroscopic description. This model is **the** outcome **of** many successive improve- ments that have refined **the** original liquid-drop model. Implementing this model in **the** CONRAD code has been a colossal work because **of** **the** model complexity and **the** related 50 year **of** legacy. An operational FRLDM has been added to **the** CONRAD toolbox and verified by various means. Despite this pragmatical choice, deepest care has been brought in **the** implementation in order to meet **the** speed requirements **of** evaluation. Advanced numerical and computational (hardware) solutions have been found and put into effect. **The** PES obtained using **the** macroscopic-microscopic FRLDM describes **the** deformation energy **of** a nucleus allowed to deform according to a related geometrical shape parame- terization. At least three parameters are necessary to satisfactorily describe **the** shape **of** **the** **fission** barrier: elongation, neck size and mass-asymmetry. **The** corresponding mul- tidimensional PES must be reduced into a one-dimensional deformation potential. Two algorithms have been implemented in **the** CONRAD code to obtain such a one-dimensional path: **the** least-energy and least-action paths searching methods. To study **the** barrier penetrability, **the** deformation inertia “along” **the** **fission** path must be defined. In many studies this parameter is explicitly or implicitly chosen to be deformation-independent. **The** natural **fission** abscissa describing **the** position along **the** path provided by these meth- ods has been discussed. **The** present conclusions advocate for **the** least-action method as both **the** **fission** abscissa and **the** inertia parameter are naturally provided by **the** model. Yet **the** least-energy path provides, for **the** time being, results closer to **the** experimental data. It is this pragmatical approach that has been selected in **the** last **section** **of** this chapter to investigate **the** impact **of** some parameters on **the** eventual **fission** **cross** sec- tion. **The** one-dimensional deformation potential and **the** related inertia parameter are obtained with both **of** **the** reduction methods and their effect on barrier penetrability has been compared with special focus set on **the** impact **of** **the** deformation dependency **of** **the** inertia parameter.

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We also determine **the** number **of** events with an iso- lated muon arising from jets in **the** eµ and µµ channels. This number is estimated as n µ f = N loose f µ , where N loose is **the** number **of** events in **the** same sign sample with loose isolation criteria on **the** muon: E T µ,iso /p µ T < 0.5 and p µ,iso T /p µ T < 0.5, and f µ is **the** misidentification rate for isolated muons. In **the** µµ final state, we apply these loose isolation criteria only to one randomly cho- sen muon. In **the** eµ channel, **the** number **of** events with jets misidentified as electrons in **the** same sign sample is subtracted from N loose . **The** misidentification rate, f µ , is determined in a dimuon sample with at least one jet. In this sample we require one muon to be close to **the** jet (R(µ, jet) < 0.5) with reversed isolation criteria E T µ,iso /p µ T > 0.15 and p µ,iso T /p µ T > 0.15. **The** other muon defined as **the** probe, should pass **the** loose isolation crite- ria E T µ,iso /p µ T < 0.5 and p µ,iso T /p µ T < 0.5. We compute f µ as **the** ratio **of** **the** number **of** events in which **the** probe muon passes **the** tight isolation criteria to **the** total num- ber **of** events in this same sign sample. **The** systematic uncertainty on **the** f µ determination is about 10% and results mainly from **the** statistical uncertainty due to **the** limited size **of** **the** sample used for **the** muon misidenti- fication rate calculation and **the** potential dependence **of** **the** misidentification rate on p T and / p T .

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5. Numerical study
**The** fact that **the** set **of** equations to be solved in order to obtain one value **of** **the** **cross** **section** is finite when **the** scatterers are pointlike makes this type **of** scatterer especially suitable for numerical studies in which a large number **of** such values is needed. In this **section** we present and discuss **the** results **of** a numerical study about **the** scattering by a disordered target consisting **of** pointlike scatterers. **The** target has been modelled by a set **of** N scatterers having each an equal probability **of** being at any position inside a sphere whose radius R may be modified; samples fitting in **the** same volume have been assigned **the** same value **of** **the** density ρ = 3N/4πR 3 . We have studied **the** variation **of**

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is approximately 85% from quark-antiquark annihilation (q ¯q → t¯t) and 15% from gluon-gluon fusion (gg → t¯t).
SM predictions for inclusive t¯t production at **the** Tevatron, calculated to different orders in perturbative quantum chromodynamics (QCD), are available in Refs. [16 –24] . **The** first calculations at full next-to-leading-order (NLO) QCD were performed before **the** discovery **of** **the** top quark [16] , and have been updated using **the** more recent CTEQ6.6 parton distribution functions (PDF) [17] . These full NLO calculations were further improved by adding resummations **of** logarithmic corrections to **the** **cross** **section** from higher-order soft-gluon radiation, in particular by including next-to-leading logarithmic (NLL) soft-gluon resummation [18] and **the** more recent PDF [19] . Also available are NLO calculations with soft-gluon resum- mation to next-to-next-to-leading logarithmic (NNLL) accuracy, and approximations at next-to-next-to-leading- order (NNLO) obtained by reexpanding **the** result from NLO þ NNLL in a fixed-order series in **the** strong coupling constant α s (NNLO approx ) [20–23,25,26] . Differences

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(sinφ θ /ω) ¯ M θ , (2.31)
As can be seen in equation (2.31), **the** stiffness term is now incorporated in **the** flow-induced dam- ping term. From a physical point **of** view, equation (2.31) shows that **the** phase-lag-related damping is associated to **the** time needed for **the** viscous flow to adjust to position changes **of** **the** structure. **The** second term **of** equation (2.31) demonstrates that **the** unsteady effects can not be neglected. In order to use **the** quasi-steady theory to analyze torsional galloping properly, **the** phase delay needs to be considered. However, there is no reliable tool to predict **the** phase delay. Nakamura and Mizota [20] experimentally measured **the** M θ ˙ for different aspect ratios. Their results showed that

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couple to a single quark. **The** fixed-θ cm scaling power is
considerably larger than that predicted by perturbative QCD.
We thank P. Kroll, J. M. Laget, and G. Miller for pro- ductive discussions, and acknowledge **the** Jefferson Lab staff for their outstanding contributions. This work was supported **the** US Department **of** Energy under contract DE-AC05-84ER40150, Modification No. M175, under which **the** Southeastern Universities Research Associa- tion (SURA) operates **the** Thomas Jefferson National Accelerator Facility. We acknowledge additional grants from **the** U.S. National Science Foundation, **the** UK Engi- neering and Physical Science Research Council, **the** Ital- ian INFN, **the** French CNRS and CEA, and **the** Israel Science Foundation.

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This raises a question: how ubiquitous is **the** flow-induced vibration behaviour **of** a purely circular cylinder? Navrose et al. ( 2014 ) studied **the** vortex-induced vibration **of** an elliptical **cross** **section** using two-dimensional direct numerical simulations for ellipses **of** varying aspect ratio. They found that **the** response regimes present, characterised by amplitude **of** oscillation, forces, and vortex shedding modes, were a strong function **of** **the** aspect ratio. To understand **the** response **of** circular cylinders yawed with respect to **the** incoming flow, Franzini et al. ( 2009 ) conducted an experimental study on **the** vortex-induced vibration **of** an elliptical cylinder with an aspect ratio **of** 1.41, **the** same aspect ratio as **the** **cross** **section** **of** a cylinder yawed at an angle **of** π / 4. With **the** long axis aligned with **the** flow, this study showed that **the** response **of** **the** body was reduced compared to **the** circular **cross** **section**. It was hypothesised that this reduction is due to **the** increased added mass **of** **the** flatter transversely oscillating body. Both **of** these studies investigated situations where **the** ellipse was mounted symmetrically with respect to **the** incoming flow, i.e, with **the** major or minor axis aligned with **the** flow.

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This raises a question: how ubiquitous is **the** flow-induced vibration behaviour **of** a purely circular cylinder? Navrose et al. ( 2014 ) studied **the** vortex-induced vibration **of** an elliptical **cross** **section** using two-dimensional direct numerical simulations for ellipses **of** varying aspect ratio. They found that **the** response regimes present, characterised by amplitude **of** oscillation, forces, and vortex shedding modes, were a strong function **of** **the** aspect ratio. To understand **the** response **of** circular cylinders yawed with respect to **the** incoming flow, Franzini et al. ( 2009 ) conducted an experimental study on **the** vortex-induced vibration **of** an elliptical cylinder with an aspect ratio **of** 1.41, **the** same aspect ratio as **the** **cross** **section** **of** a cylinder yawed at an angle **of** π / 4. With **the** long axis aligned with **the** flow, this study showed that **the** response **of** **the** body was reduced compared to **the** circular **cross** **section**. It was hypothesised that this reduction is due to **the** increased added mass **of** **the** flatter transversely oscillating body. Both **of** these studies investigated situations where **the** ellipse was mounted symmetrically with respect to **the** incoming flow, i.e, with **the** major or minor axis aligned with **the** flow.

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